[    {        "title": "1311.4925v1.Asymptotic_Improvement_of_the_Gilbert_Varshamov_Bound_on_the_Size_of_Permutation_Codes.pdf",        "content": "arXiv:1311.4925v1  [math.CO]  20 Nov 20131\nAsymptoticImprovement oftheGilbert-Varshamov\nBoundontheSize ofPermutation Codes\nMichael Tait,AlexanderVardy,and JacquesVerstra¨ ete\nAbstract—Given positive integers nand d, let M(n,d)denote\nthemaximumsizeofapermutationcodeoflength nandminimum\nHamming distance d. The Gilbert-Varshamov bound asserts that\nM(n,d)/greaterorequalslantn!/V(n,d−1)where V(n,d)is the volume of a Ham-\nmingsphereofradius dinSn.Recently,Gao,Yang,andGeshowed\nthat this bound can be improved by a factor Ω(logn), when dis\nfixedand n→∞.Herein,weconsiderthesituationwheretheratio\nd/nis fixed and improve the Gilbert-Varshamov bound by a fac-\ntor thatis linearin n.Thatis, weshow that if d/n<0.5,then\nM(n,d)/greaterorequalslantcnn!\nV(n,d−1)\nwhere cis a positive constant that depends only on d/n. To estab-\nlish this result, we follow the method of Jiang and Vardy. Nam ely,\nwe recast the problemof bounding M(n,d)intoagraph-theoretic\nframework andprove that theresultinggraphis locallyspar se.\nIndexTerms —Ajtai-Koml ´os-Szemer ´edibound,Gilbert-Varsha-\nmov bound,locallysparse graphs, permutationcodes\nI. INTRODUCTION\nLETSnbe the symmetric group of permutationson nele-\nments. A permutation code Cis a subset of Sn. For σ,τ\ninSn,theHammingdistance d(σ,τ)betweenthemisthenum-\nberofpositionswheretheydiffer,namely:\nd(σ,τ)def=/braceleftbig\ni∈[n]:σ(i)/n⌉}ationslash=τ(i)/bracerightbig\n(1)\nWe saythatapermutationcode C⊆ S nhasminimumdistance\ndif each pair of permutations in Cis at Hamming distance at\nleast d. The maximum number of codewords in a permutation\ncodeofminimumdistance dwill bedenoted M(n,d).\nPermutationcodeswere first investigatedin [4,5,12,13,26 ].\nInrecentyears,theyhavereceivedconsiderablerenewedat ten-\ntion, due to their application in data transmission over pow er\nlines[9,10,16,24,28].Otherapplicationsofpermutation codes\nincludedesignofblockciphers[11]andcodingforflashmemo -\nries[3,7,8](althoughthelatterapplicationinvolvesthe Kendall\nτ-metricin lieuoftheHammingdistance).\nAkeyprobleminthetheoryofpermutationcodesistodeter-\nmine M(n,d).Variousboundson M(n,d)havebeenproposed\nin[10,13–15,17,18,23].Forsmallvaluesof d,computersearch\nhasbeenusedtofindexactvaluesof M(n,d)in[23].However,\nin general, the problem is very difficult, and little progres shas\nbeenmadefor d>4.\nMichael Tait and Jacques Verstra¨ ete are with the Departmen t of Mathemat-\nics, University of California San Diego, La Jolla, CA 92093, U.S.A. (e-mail:\nmtait@math.ucsd.edu ,jacques@ucsd.edu ).\nAlexander VardyiswiththeDepartmentofElectrical andCom puter Engine-\nering, the Department of Computer Science and Engineering, and the Depart-\nmentofMathematics, University of California San Diego, La Jolla, CA 92093,\nU.S.A.(e-mail: avardy@ucsd.edu ).Forx∈R,let[x]denotetheoperationofroundingtothene-\narest integer. Then the number of derangements of kelements\nisgivenby Dk= [n!/e]. UndertheHammingdistancedefined\nin(1),a sphereofradius rhasvolume\nV(n,r)def=r\n∑\nk=0/parenleftbiggn\nk/parenrightbigg\nDk\nThe Gibert-Varshamov bound [19,27] and the sphere-packing\nbound[22]nowgivethefollowing.\nTheorem1.\nn!\nV(n,d−1)/lessorequalslantM(n,d)/lessorequalslantn!\nV/parenleftbig\nn,⌊d−1\n2⌋/parenrightbig(2)\nThe Gilbert-Varshamovbound is used extensively in coding\ntheory[22,25].Asiswellknown,improvingthisboundasymp -\ntoticallyisadifficulttask [25].\nRecently, Gao, Yang, and Ge [18] showed that the Gilbert-\nVarshamovboundin(2)canbeimprovedbyafactor Ω(logn),\nwhen dis fixed and n→∞. In this paper, we complementthis\nwork, focusing on the more natural case where d/nis a fixed\nratioand ntendstoinfinity.Ourmaintheoremisthefollowing.\nTheorem2. Letd/nbeafixedratiowith 0<d/n<0.5.Then\nasn→∞,we have\nM(n,d)/greaterorequalslantΩ/parenleftbiggn!\nV(n,d−1)n/parenrightbigg\n(3)\nToprovethistheorem,wefollowthemethodof[20].Namely,\nwe will construct a graph in which every independent set cor-\nrespondsto a permutationcode of minimum distance d. Using\ntechniquesfromgraphtheory,wewillthenobtainalowerbou nd\nonthesize ofthelargestindependentset inthisgraph.\nII. GRAPHTHEORY\nOurgraphnotationisstandard;thereaderisreferredto[30 ]for\nany undefined terms. A graph Gconsists of a set of vertices\nV(G)anda set of edges E(G),which arepairsofvertices.We\nassumethroughoutthatthesets V(G)andE(G)arefinite.Two\nvertices uandvareadjacentif{u,v} ∈ E(G). For a subset S\nofV(G),thesubgraphinducedby Sisthegraphthathasvertex\nsetSandedgeset/braceleftbig{u,v} ∈E(G):u,v∈S/bracerightbig\n.Theneighbor-\nhoodof a vertex vis the set of all vertices adjacent to v, and\ntheneighborhood graph of vis the subgraph induced by the\nneighborhood of v. Thedegreeof a vertex vis the size of the\nneighborhood of v. We let ∆(G)denote the maximum vertex\ndegreeinagraph G.Aset K⊆V(G)isacliqueifthesubgraph\ninduced by Khas every possible edge, and a set I⊆V(G)is2\nanindependent set if the subgraph induced by Ihas no edges.\nAtriangleis a clique of size 3. The maximum number of ver-\nticesinanindependentsetof Giscalledthe independencenum-\nberanddenotedby α(G).\nConsiderthegraph Gwhosevertexsetis Snwithtwopermu-\ntations σ,τbeingadjacentifandonlyif 1/lessorequalslantd(σ,τ)<d.Then\nit is clear that any permutation code of length nand minimum\ndistance disanindependentsetin G. Conversely,anyindepen-\ndent set in Gis a permutation code of length nand minimum\ndistance d.Thisbijectionprovesthefollowing.\nLemma3. M(n,d) =α(G).\nObservethateveryvertexin Ghasthesamedegree,givenby\n∑d−1\nk=1(n\nk)Dk. Itisgraphtheoryfolklorethatforanygraph G,\nα(G)/greaterorequalslant|V(G)|\n∆(G)+1(4)\nIt is evident that (4) along with Lemma3 immediately recov-\ners the Gilbert-Varshamovbound in (2). We note that we have\nrecoveredthisboundusingverylittle informationabout G.\nOurstrategytoproveTheorem2istousetherelativesparse-\nness of the neighborhoodgraph of every vertex in V(G)in or-\ndertostrengthen(4).Thistechniquehasbeenintroducedin [20]\nto improve the Gilbert-Varshamovbound on the size of binary\ncodes.It waslater appliedto improvelowerboundsonthe siz e\nofq-arycodes[29] andspherepackings[21].\nWewillneedsomeresultsaboutlocallysparsegraphs.Ajtai ,\nKoml´ os,andSzemer´ edi[1]showedthat onecanimprove(4)i f\nGhas no triangles. The following lemma is from [1] (but see\nalso[2,p.272]foramuchshorterproofofthe sameresult).\nLemma4. LetGbeagraphwithmaximumdegree ∆,andsup-\nposethat Ghasnotriangles.Then\nα(G)/greaterorequalslant|V(G)|\n8∆log2∆ (5)\nThisresultwasextendedin[6,Lemma15,p.296]fromgraphs\nwithnotrianglestographswith relativelyfewtriangles.\nLemma5. LetGbe a graphwith maximumdegree ∆andsup-\nposethat Ghasat most Ttriangles.Then\nα(G)/greaterorequalslant|V(G)|\n10∆/parenleftbigg\nln∆−1\n2ln/parenleftbiggT\n|V(G)|/parenrightbigg/parenrightbigg\n(6)\nA graph has no triangles if and only if the neighborhoodof\nevery vertex is an independent set, and a graph has relativel y\nfew triangles if the neighborhoodsof its vertices are relat ively\nsparse.We makethispreciseinthefollowingcorollary.\nCorollary6. LetGbeagraphwithmaximumdegree ∆andsup-\npose that forall v∈V(G), the subgraphinducedby the neigh-\nborhoodof vhasatmost Eedges.Then\nα(G)/greaterorequalslant|V(G)|\n10∆/parenleftbigg\nln∆−1\n2ln/parenleftbiggE\n3/parenrightbigg/parenrightbigg\n(7)\nProof:Thenumberoftrianglesincidentwithavertex viseq-\nualtothenumberofedgesinthesubgraphinducedbytheneigh -\nborhoodof v. Therefore,forevery v∈V(G), thereare at most\nEtriangles incident with v. Summing over the vertex set of G\nand noting that each triangle is incident with exactly 3vertices\ngivestheresult. /squaresolidIII. PROOF OF THE MAINTHEOREM\nInordertouseCorollary6,wemustcountthenumberofedges\nintheneighborhoodofeveryvertexin V(G).Todoso,firstnote\nthatGisvertextransitivebecauseforall σ,τ∈ S n,we have\nd(σ,τ) = d(id,στ−1)\nwhere idis the identity element of Sn. This implies that the\nnumberofedgesintheneighborhoodofavertex v∈V(G)does\nnotdependon v.Therefore,tosimplifythecalculation,wewill\nconsideronlytheneighborhoodoftheidentitypermutation .\nBy Corollary6, in order to prove our main result in (3), it\nwouldsufficeto showthat\nlog/parenleftbigg∆2\nE/parenrightbigg\n=Ω(n) (8)\nwhere Eisthenumberofedgesin thesubgraphinducedbythe\nneighborhoodoftheidentity,and ∆isgivenby\n∆def=d\n∑\nk=1/parenleftbiggn\nk/parenrightbigg\nDk (9)\nNotethat,fornotationalconvenience,thesumin(9)extend sup\ntodrather d−1.Since d/nisfixed,thisdoesnotmatter.\nItfollowsfrom(8)thattheproofofTheorem2reducestoes-\ntimating Eand considering the asymptotics of the ratio ∆2/E.\nWe count the number of edges in the subgraph induced by the\nneighborhoodoftheidentityasfollows.\n1) Verticesintheneighborhoodgraphoftheidentityhavedi s-\ntancebetween 1andd−1fromtheidentity.Wewillcount\ntheedgesincidentwithpermutations σthatareatdistance\nsfrom id,andthensumover s=1, 2, . . . , d−1.Notethat,\ngiven a fixed distance s, there are exactly (n\ns)Dspermuta-\ntionsat distance sfromtheidentity.\n2) Let us fix a permutation σat distance sfrom the identity\nand count how many edges in the neighborhoodgraph of\nthe identity are incident with σ. To do this, we will sum\nover permutations τthat are at distance tfrom the iden-\ntity, for t=1, 2, . . . , d−1. Note that for τto be incident\nwith σ,wemust alsohave d(τ,σ)<d.\n3) Tocounthowmanypermutations τsatisfytheaboverequ-\nirements,welet mbethenumberofindiceswhere σandτ\narederangingthesameposition.Thatis,given σandτ,let\nmdef=/vextendsingle/vextendsingle{i∈[n]:σ(i)/n⌉}ationslash=i,τ(i)/n⌉}ationslash=i}/vextendsingle/vextendsingle(10)\nBy assumption, σis deranging spositions while τis de-\nranging tpositions.Thus m/lessorequalslantmin(s,t).Also noticethat\nin the s−mpositions where σis deranging but τis not,\nthetwopermutationsnecessarilydiffer.Samegoesforthe\nt−mpositionswhere τisderangingbut σisnot.Hence\n(s−m) + ( t−m)/lessorequalslantd(σ,τ)<d(11)\nIt follows that the parameter mdefined in (10) is in the\nrange⌈(s+t−d)/2⌉+<m/lessorequalslantmin(s,t),where⌈x⌉+de-\nnotesthesmallest nonnegativeinteger kwith k/greaterorequalslantx.\n4) Now let us suppose that s,t,mare fixed, and count those\npositions where σandτare both deranging,but still map\ntothesame value.Thatis, let\nrdef=/vextendsingle/vextendsingle{i∈[n]:σ(i)/n⌉}ationslash=i,τ(i)/n⌉}ationslash=i,σ(i) =τ(i)}/vextendsingle/vextendsingle(12)3\nNoticethat d(σ,τ) =s+t−m−r.Itfollowsthatthepa-\nrameter rdefinedin(12)satisfies r>⌈s+t−m−d⌉+.\nOurstrategyforcounting Eistofix s,t,m,randcountthenum-\nberofpairsofpermutations σandτthathavetheseparameters.\nWe will thensumover s,t,m,rintheappropriateranges.\nAs already noted above, there are exactly (n\ns)Dspermuta-\ntions σat a given distance sfrom the identity. Given σ, there\nare(s\nm)ways to select the mpositions where both σandτare\nderanging.Giventhese mpositions,thereare (m\nr)waysto pick\ntherpositionswherebothpermutationsarederangingbuthave\nthe same image, as in (12). We have now chosen our permu-\ntation σ, and part of our permutation τ. In particular, we have\nchosen mpositionswhere τisderanging.Tospecifytherestof\nτ, we first choosethe other t−mpositionswhere τis derang-\ning.Thiscanbedoneinexactly (n−s\nt−m)ways.Nowthatwehave\nchosen the tpositions where τis deranging, we must pick the\nimage of t−rout of the tpositions (the image of τis already\nspecifiedasequaltotheimageof σonrpositions).Thiscanbe\ndone in less than (t−r)!ways. Note that we are overcounting\nhere because we must derange the t−rpositions and because\nthere are some restrictions on the indices where σandτover-\nlap.However,thisovercountingwill nothurtourfinalbound .\nTosummarizethe foregoingdiscussion,if wedefine\ng(s,t,m,r)def=/parenleftbiggn\ns/parenrightbigg\nDs/parenleftbiggs\nm/parenrightbigg/parenleftbiggm\nr/parenrightbigg/parenleftbiggn−s\nt−m/parenrightbigg\n(t−r)!(13)\nthenwehavethat\nE/lessorequalslantd\n∑\ns=1d\n∑\nt=1min{s,t}\n∑\nm=⌈s+t−d\n2⌉+m\n∑\nr=⌈s+t−m−d⌉+g(s,t,m,r)(14)\nOurnexttaskistoobtainanupperboundon g(s,t,m,r)in(14).\nIndeed,ifwecanshowthat g(s,t,m,r)/lessorequalslantGwhenever s,t,m,r\nsatisfy theconstraints\n1/lessorequalslants/lessorequalslantd (15)\n1/lessorequalslantt/lessorequalslantd (16)\n/ceilingleftbiggs+t−d\n2/ceilingrightbigg+\n/lessorequalslantm/lessorequalslantmin{s,t} (17)\n⌈s+t−m−d⌉+/lessorequalslantr/lessorequalslantm (18)\nthenwecanconcludefrom(14)that E/lessorequalslantd4G.Infact,sincewe\nareinterestedonlyintheasymptoticsof Easn→∞,anasymp-\ntoticupperboundon g(s,t,m,r)wouldsuffice.Thenextlemma\nshows that the value of g(s,t,m,r)ats=t=m=d,r=0\ncanserveassuchabound.\nLemma7. Supposethat s,t,m,rsatisfytheconstraintsinequa-\ntions(15)–(18) . Then for all real εin the range 0<ε<1/6\nandforall sufficientlylarge n, we have\nlog2/parenleftbiggg(d,d,d, 0)\ng(s,t,m,r)/parenrightbigg\n/greaterorequalslant−3nh2(3ε) (19)\nProof:Let a positive ε<1/6be fixed, and recall that\nd=δnfor a positive constant δ<0.5. We first show that\n(19)holdsvac-uouslyunless t/greaterorequalslant(1−ε)d.Indeed,\ng(d,d,d, 0)\ng(s,t,m,r)/greaterorequalslant(n\nd)Ddd!\n(n\ns)Ds(n\nn/2)3t!/greaterorequalslantd!\n23nt!(20)Itiseasytoseethatif t/lessorequalslant(1−ε)d,thentheRHSof(20)grows\nwithoutboundas n→∞.Byasimilarargument,(19)holdsun-\nlesss/greaterorequalslant(1−ε)d.We have\ng(d,d,d, 0)\ng(s,t,m,r)/greaterorequalslant(n\nd)Ddd!\n(n\ns)Ds(n\nn/2)3t!/greaterorequalslantDd\n23nDs(21)\nwhichgrowswithoutboundas n→∞ifs/lessorequalslant(1−ε)d.Wenext\nshow that (19) again holds vacuously unless r/lessorequalslantεt. Indeed, if\nr>εt,then\ng(d,d,d, 0)\ng(s,t,m,r)/greaterorequalslant(n\nd)Ddd!\n(n\ns)Ds(n\nn/2)3((1−ε)t)!/greaterorequalslantd!\n23n((1−ε)d)!\nwhich grows without bound as n→∞. It remains to consider\nthecase where\ns/greaterorequalslant(1−ε)d,t/greaterorequalslant(1−ε)dand r/lessorequalslantεd(22)\nObservethat in this case, we must also have m/greaterorequalslant(1−3ε)din\nviewof(18). Furtherobservethat\ng(s,t,m,r)/lessorequalslant/parenleftbiggn\nd/parenrightbigg\nDd/parenleftbiggd\nm/parenrightbigg/parenleftbiggm\nr/parenrightbigg/parenleftbiggn\nd−m/parenrightbigg\nd!(23)\nforall s,t,m,rsatisfyingtheconstraints(15)–(18).Combining\nthiswith (22)and m/greaterorequalslant(1−3ε)d,weobtain\ng(s,t,m,r)/lessorequalslantg(d,d,d, 0)/parenleftbiggd\n3εd/parenrightbigg/parenleftbiggd\nεd/parenrightbigg/parenleftbiggn\n3εd/parenrightbigg\n(24)\n/lessorequalslantg(d,d,d, 0)/parenleftbiggn\n3εn/parenrightbigg3\n(25)\n/lessorequalslantg(d,d,d, 0)23nh2(3ε)(26)\nwhere h2(x) =−xlog2x−(1−x)log2(1−x)isthebinary\nentropyfunction.Thelemmanowfollowsfrom(26). /squaresolid\nWearenowinapositiontocompletetheproofofTheorem2.\nRecall thatit wouldsufficeto showthat\nlog2/parenleftbigg∆2\nE/parenrightbigg\n=Ω(n)\nasin(8),where ∆isgivenby(9)and Eisupper-boundedby(14).\nInordertosimplifytheboundin(14),let usdefine\nG(n,d)def=max\ns,t,m,rg(s,t,m,r)\nwhere the maximum is over s,t,m,rsatisfying the constraints\nin(15)–(18).Then E/lessorequalslantd4G(n,d).Asalowerboundon ∆,we\nusethelargestterminthesum of(9).Thus\n∆/greaterorequalslant/parenleftbiggn\nd/parenrightbigg\nDd\nCombiningthiswithLemma7,weconcludethatforall εinthe\nrange 0<ε<1/6andforall sufficientlylarge n, we have\nlog2/parenleftbigg∆2\nE/parenrightbigg\n/greaterorequalslantlog2/parenleftBigg\n(n\nd)2D2\nd\nd4G(n,d)/parenrightBigg\n=log2/parenleftBigg\n(n\nd)2D2\nd\nd4g(d,d,d, 0)/parenrightBigg\n+log2/parenleftbiggg(d,d,d, 0)\nG(n,d)/parenrightbigg\n/greaterorequalslantlog2/parenleftBigg\n(n\nd)2D2\nd\nd4g(d,d,d, 0)/parenrightBigg\n−3nh2(3ε)(27)4\nSince lim ε→03h2(3ε) =0,we canconcludefrom(8) and(27)\nthatit remainstoshow\nlog2/parenleftBigg\n(n\nd)2D2\nd\nd4g(d,d,d, 0)/parenrightBigg\n=log2/parenleftBigg\n(n\nd)2D2\nd\nd4(n\nd)Ddd!/parenrightBigg\n=Ω(n)\nThisiseasily accomplishedasfollows:\nlog2/parenleftBigg\n(n\nd)2D2\nd\nd4(n\nd)Ddd!/parenrightBigg\n/greaterorequalslantlog2/parenleftbigg(n\nd)\n3d4/parenrightbigg\n/greaterorequalslantdlog2/parenleftBign\nd/parenrightBig\n−4 log2(d)−log23\n=δnlog2/parenleftbigg1\nδ/parenrightbigg\n−4 log2(δn)−log23\nSince δ=d/nis a positive constant strictly less than 0.5, we\nseethat δlog2(1/δ)ispositiveand,hence,theexpressionabove\nisΩ(n).ThiscompletestheproofofTheorem2.\nREFERENCES\n[1] M.Ajtai, J.Koml´ os, and E.Szemer´ edi, “A note on Ramsey numbers,” J.\nCombinatorial Theory Ser.A ,vol.29, pp.354–260, 1980.\n[2] N.AlonandJ.Spencer, TheProbabilisticMethod ,2ndedition,NewYork:\nJohn Wiley and Sons,2000.\n[3] A.Barg and A.Mazumdar,“Codes in permutations and error correction\nfor rank modulation,” IEEE Trans. Inform. Theory , vol.56, pp.3158–\n3165, July 2010.\n[4] I.F.Blake, “Permutation codes for discrete channels,” IEEE Trans. In-\nform. Theory , vol.20, pp.138–140, 1974.\n[5] I.F.Blake, G.Cohen,andM.Deza,“Coding withpermutati ons,”Inf.Con-\ntrol, vol.43,no.1, pp.1–19, 1979.\n[6] B.Bollob´ as, RandomGraphs ,1stedition,London:AcademicPress,1985.\n[7] J.Bruck, A.Jiang, R.Mateescu, and M.Schwartz, “Rank mo duation for\nflash memories,” Proc. IEEE Int. Symp. Information Theory , pp.1731–\n1735, Toronto, Canada, 2008.\n[8] J.Bruck, A.Jiang, and M.Schwartz, “Error-correcting c odes for rank\nmodulation,” Proc. IEEE Int. Symp. Information Theory , pp.1736–1740,\nToronto, Canada, 2008.\n[9] W.Chu,C.J.Colbourn,andP.Dukes,“Constructions forp ermutationcod-\nes in powerline communications,” Des. Codes Cryptogr. , vol.32, pp.51-\n64, 2004.\n[10] C.J.Colbourn, T.Kløve, and A.C.H.Ling, “Permutation arrays for pow-\nerline communications and mutually orthogonal Latin squar es,”IEEE\nTrans. Inform. Theory , vol.50, no.6,pp.1289–1291, June2004.\n[11] C.J.Colbourn, A.C.H.Ling, D.R. de la Torre, “An applic ation of permu-\ntation arrays to block ciphers,” Proc. Southeastern International Confer-\nence on Combinatorics, Graph theory and Computing , vol.145, pp.5–7,\nBoca Raton, FL,2000.\n[12] M.Deza and P.Frankl, “On the maximum number of permutat ions with\ngiven maximal or minimal distance,” J. Combinatorial Theory Ser. A ,\nvol.22, no.3, pp.352–360, 1977.\n[13] M.Deza and S.A.Vanstone, “Bounds for permutation arra ys,”J. Statist.\nPlann. Inference , vol.2,no.2, pp.197–209, 1978.\n[14] C.Ding, F.-W.Fu, T.Kløve, and V.K.-W.Wei, “Construct ion of permu-\ntation arrays,” IEEE Trans. Inform. Theory , vol.48, no.4, pp. 977–980,\n2002.\n[15] P.Dukes and N.Sawchuck, “Bounds on permutation codes o f distance\nfour,”J. Algebraic Combin. , vol.31,no.1, pp.143–158, 2010.\n[16] H.C.Ferreira and A.J.H.Vinck, “Inference cancellati on with permutation\ntrellis arrays,” Proc. IEEE Vehicular Technology Conf. , pp.2401-2407,\nBoston, MA,September 2000.\n[17] F.-W.Fuand T.Kløve, “Twoconstructions ofpermutatio ns arrays,” IEEE\nTrans. Inform. Theory , vol.50, no.5,pp.881–883, 2004.\n[18] F.Gao, Y.Yang, and G.Ge, “An improvement to the Gilbert -Varshamov\nbound for permutation codes,” IEEE Trans. Inform. Theory , vol.59,\npp.3059–3063, May 2013.\n[19] E.N.Gilbert, “A comparison of signalling alphabets,” Bell Syst. Tech. J. ,\nvol.31, pp.504–522, 1952.\n[20] T.Jiang and A.Vardy, “Asymptotic improvement of the Gi lbert-Varsha-\nmovboundonthesizeofbinarycodes,” IEEETrans.Inform.Theory ,vol.\n50, pp.1655–1664, August2004.[21] M.Krivelevich, S.Litsyn,andA.Vardy,“Alower boundo nthedensity of\nspherepackings viagraph theory,” International Math. ResearchNotices ,\nvol.43, pp.2271–2279, June2004.\n[22] F.J.MacWilliams and N.J.A.Sloane, The Theory of Error Correcting\nCodes, Amsterdam: North-Holland/Elsevier, 1977.\n[23] R.MontemanniandD.H.Smith,“Anewtableofpermutatio ncodes,” Des.\nCodes Cryptogr. , vol.63, no.2,pp.241–253, 2012.\n[24] N.Pavlidou, A.J.H.Vinck, J.Yazdani, and B.Honary, “P ower line com-\nmunications: State of the art and future trends,” IEEE Commun. Mag. ,\nvol.41, pp.34–40, 2003.\n[25] V.S.Pless and W.C.Huffman (Editors), Handbook of Coding Theory ,\nAmsterman: North-Holland/Elsevier, 1998.\n[26] D. Slepian, “Permutation modulation,” Proc. IEEE ,vol.53, pp.228–236,\nMarch 1965.\n[27] R.R.Varshamov, “Estimate of the number of signals in er ror correcting\ncodes,”Dokl. Acad.Nauk , vol.117, pp.739–741, 1957 (in Russian).\n[28] A.J.H.Vinck,“Codedmodulationforpowerlinecommuni cations,”A.E.¨U.\nInt. J. Electron. Commun. , vol.54,no.9,pp.3200–3208, 2005.\n[29] V.H.Vu and L.Wu, “Improving the Gilbert-Varshamov bou nd for q-ary\ncodes,”IEEETrans.Inform. Theory , vol.51,no.9pp.3200–3208, 2005.\n[30] D.B.West, Introduction to Graph Theory , Prentice Hall, 1996."    },    {        "title": "1311.6305v1.Spin_wave_excitation_and_propagation_in_microstructured_waveguides_of_yttrium_iron_garnet__YIG__Pt_bilayers.pdf",        "content": "arXiv:1311.6305v1  [cond-mat.mes-hall]  25 Nov 2013Spin-wave excitation and propagation in microstructured w aveguides of yttrium\niron garnet (YIG) /Pt bilayers\nP. Pirro,1T. Brächer,1, 2A. Chumak,1B. Lägel,1C. Dubs,3O. Surzhenko,3P. Görnet,3\nB. Leven,1and B. Hillebrands1\n1)Fachbereich Physik and Landesforschungszentrum OPTIMAS,\nTechnische Universität Kaiserslautern, D-67663 Kaisersl autern,\nGermany\n2)Graduate School Materials Science in Mainz, Gottlieb-Daim ler-Strasse 47,\nD-67663 Kaiserslautern, Germany\n3)Innovent e.V., Prüssingstraße 27B, 07745 Jena, Germany\n(Dated: 7 May 2018)\nWe present an experimental study of spin-wave excitation an d propagation in\nmicrostructured waveguides patterned from a 100 nm thick yt trium iron garnet\n(YIG)/platinum (Pt) bilayer. The life time of the spin waves is found to be more than\nan order of magnitude higher than in comparably sized metall ic structures despite\nthe fact that the Pt capping enhances the Gilbert damping. Ut ilizing microfocus\nBrillouin light scattering spectroscopy, we reveal the spi n-wave mode structure for\ndifferent excitation frequencies. An exponential spin-wav e amplitude decay length of\n31µm is observed which is a significant step towards low damping , insulator based\nmicro-magnonics.\n1The concept of magnon spintronics, i.e., the transport and m anipulation of pure spin\ncurrents in the form of spin-wave quanta, called magnons, ha s attracted growing interest\nin the recent years1–12. One of the key advantages of magnon spin currents is their la rge\ndecay length which can be several orders of magnitude higher than the spin diffusion length\nin conventional spintronic devices based on spin-polarize d electron currents. Considering\npossible applications, the miniaturization of magnonic ci rcuits is of paramount importance.\nUp to now, downscaling has been achieved using metallic ferr omagnets like NiFe or Heusler\ncompounds2–7. But even the best metallic ferromagnets exhibit a damping w hich is two or-\nders of magnitude larger than for Yttrium Iron Garnet (YIG), a ferrimagnetic insulator1,13,14.\nHowever, to the best of our knowledge, as high quality YIG film s could only be grown with\nthicknesses in the range of microns, no microstructured YIG devices have been fabricated so\nfar. A big step forward has been taken with the recent introdu ction of methods to produce\nhigh quality, low damping YIG films with thicknesses down to s everal nanometers9,15–17.\nIn this Letter, we show that microscaled waveguides (see Fig . 1) can be fabricated from\nliquid phase epitaxy (LPE) grown YIG films of 100nm thickness whose high quality has\nbeen confirmed by ferromagnetic resonance spectroscopy (FM R). Studying the excitation\nand propagation of spin-waves in these waveguides by microf ocus Brillouin light scattering,\nwe demonstrate that the damping of the unstructured film can b e preserved during the\nstructuring process.\nAnother key feature of magnon spintronics is its close relat ionship to a multitude of phys-\nical phenomena like spin-pumping, spin-transfer torque, s pin Seebeck effect, and (inverse)\nspin Hall effect, which allow for the amplification, generati on and transformation between\ncharge currents and magnonic currents7–12,15–24. Hetero-structures of YIG covered with a\nthin layer of platinum (Pt) have proven to show these effects w hich opens a way to a new class\nof insulator based spintronics. Therefore, we directly stu dy bilayers of YIG/Pt, providing a\nbasis for further studies utilizing the described effects.\nThe used YIG film is prepared by liquid phase epitaxy from a PbO -B2O3-FeO3flux\nmelt using a standard isothermal dipping technique with a gr owth rate of 20nm/min. The\nincorporation of Pb and Pt ions into the garnet lattice allow s for a low relative lattice\nmismatch of 3·10−4.\nWe determine the magnetic properties of the film using FMR and compare the results\nto measurements performed after the deposition of a 9nmPt film onto YIG using plasma\n2FIG. 1. Sample schematic: In a 5µm wide waveguide patterned from a bilayer of YIG/Pt\n(100nm /9nm), spin waves are excited using the dynamic Oersted fields of a microwave current\nflowing in a copper antenna. An external bias field Hextis applied along the short axis of the\nwaveguide. The spin-wave intensity is detected using micro focus Brillouin light scattering spec-\ntroscopy.\ncleaning and RF sputtering. From the resonance curve HFMR(fFMR), a saturation magne-\ntization of Ms= 144±2kA/mhas been determined for the pure YIG film. We find that\nthe deposition of Pt slightly reduces the resonance field µ0HFMR(for example by 1mT for\nfFMR= 7.0GHz ) compared to the pure YIG film. This shift agrees with the rece nt findings\nof Ref. 15, where a proximity induced ferromagnetic orderin g of Pt combined with a static\nexchange coupling to YIG has been proposed as possible expla nation.\nFigure 2 shows the ferromagnetic resonance linewidth (FWHM )µ0∆Hwith and with-\nout Pt and the corresponding fits to evaluate the effective Gil bert damping parameter α\naccording to22\nµ0∆H=µ0∆H0+2αfFMR\nγ(1)\nwith the gyromagnetic ratio γ= 28GHz /T. The Gilbert damping αincreases by almost a\nfactor of 5 due to the deposition of Pt: from (2.8±0.3)×10−4to(13.0±1.0)×10−4. The\ninhomogeneous linewidth µ0∆H0is unchanged within the accuracy of the fit ( 0.16±0.02mT\nand0.14±0.04mT , respectively). Please note that the increase of the dampin g cannot be\nexplained exclusively by spin pumping from YIG into Pt. Othe r interface effects, like the\nalready mentioned induced ferromagnetic ordering of Pt in c ombination with a dynamic\nexchange coupling may play a role15,23. Using the spin mixing conductance for YIG/Pt\n(g↑↓≈1.2×1018m−2from22,24), we find that the expected increase in Gilbert damping due\nto spin pumping11,20–22isαsp= 1.25×10−4,i.e., it is by a factor of 8 smaller than the\n3/s48 /s50 /s52 /s54 /s56 /s49/s48 /s49/s50 /s49/s52/s48/s46/s50/s48/s46/s52/s48/s46/s54/s48/s46/s56/s49/s46/s48\n/s89/s73/s71/s61/s50/s46/s56/s32/s120/s49/s48/s45/s52/s32/s89/s73/s71/s47/s80/s116/s32/s98/s105/s108/s97/s121/s101/s114\n/s32/s112/s117/s114/s101/s32/s89/s73/s71/s181\n/s48/s32 /s72 /s32/s40/s109/s84/s41\n/s32/s102\n/s70/s77/s82/s40/s71/s72/s122/s41/s89/s73/s71/s47/s80/s116/s61/s49/s51/s46/s48/s32/s120/s49/s48/s45/s52\nFIG. 2. Linewidth µ0∆Has a function of the ferromagnetic resonance frequency fFMRfor the\npure YIG film (blue circles) and the YIG/Pt bilayer (red squar es). The deviations from the linear\nincrease of µ0∆HwithfFMR(fit according to Eqn. 1) are mainly due to parasitic modes cau sing a\nsmall systematical error in the measurement of the linewidt h.\nmeasured increase. This clearly demonstrates the importan ce of additional effects15,23. As\nshown recently in Ref.15, this additional damping can be strongly reduced by the intr oduction\nof a thin copper (Cu) layer in between YIG and Pt, which does no t significant influence the\nspin-pumping efficiency.\nThe micro structuring of the YIG/Pt waveguide is achieved us ing a negative protective\nresist mask pattered by electron beam lithography and physi cal argon ion beam etching. As\nlast production step, a microwave antenna (width 3.5µm,510nm thickness) made of copper\nis deposited on top of the waveguide (see Fig. 1).\nTo experimentally detect the spin waves in the microstructu red waveguide, we employ\nmicrofocus Brillouin light scattering spectroscopy (BLS)3–8. This method allows us to study\nthe spin-wave intensity as a function of magnetic field and sp in-wave frequency. In addition,\nit provides a spatial resolution of 250nm , which is not available in experiments using spin\npumping and inverse spin Hall effect8–10as these methods integrate over the detection area\n(and also over the complete spin-wave spectrum12).\nTo achieve an efficient spin-wave excitation, we apply a stati c magnetic field of 70mT\nperpendicular to the long axis of the waveguide. The dynamic Oersted field of a microwave\ncurrent passing through the antenna exerts a torque on the st atic magnetization. This config-\nuration results in an efficient excitation of Damon-Eshbach l ike spin waves which propagate\n4/s51/s46/s52 /s51/s46/s53 /s51/s46/s54 /s51/s46/s55 /s51/s46/s56/s48/s46/s49/s49/s32/s99/s101/s110/s116/s101/s114/s32/s114/s101/s103/s105/s111/s110\n/s32/s101/s100/s103/s101/s32/s114/s101/s103/s105/s111/s110/s78/s111/s114/s109/s97/s108/s105/s122/s101/s100/s32/s66/s76/s83/s32/s105/s110/s116/s101/s110/s115/s105/s116/s121\n/s102\n/s77/s87/s32/s40/s71/s72/s122/s41\n/s49 /s49 /s32/s181 /s109\n/s97/s110/s116/s101/s110/s110/s97/s89/s73/s71/s47/s80/s116 \n/s72 \n/s101/s120/s116 \nFIG. 3. Normalized BLS intensity (log scale) as a function of t he applied microwave frequency fMW\n(external field µ0Hext= 70mT ). The blue line (circular dots) shows the spectrum measured at\nthe edges of the waveguide (see inset). The red line (rectang ular dots) is an average of the spectra\nrecorded in the center of the waveguide.\nperpendicular to the static magnetization. A microwave pow er of0dBm (pulsed, duration\n3µs, repetition 5µs) in the quasi-linear regime, where nonlinearities are not s ignificantly\ninfluencing the spin-wave propagation, has been chosen. To o btain a first characterization\nof the excitation spectrum, BLS spectra as a function of the a pplied microwave frequency\n(fMW) have been taken at different positions across the width of th e waveguide at a distance\nof11µm from the antenna. Figure 3 shows the spectrum of the edge re gions (blue circles)\nand of the center of the waveguide (red squares, see sketch in the inset). The main excitation\nin the center of the waveguide takes place at frequencies bet weenfMW= 3.49−3.66GHz\nand we will refer to these spin-wave modes as the waveguide modes . At the borders of the\nwaveguide, edge modes have their resonance around fMW= 3.44GHz . The reason for the\nappearance of these edge modes is the pronounced reduction o f the effective magnetic field\nHeffat the edges by the demagnetization field and the accompanyin g inhomogeneity of the\nz-component of the static magnetization. This situation has been analyzed experimentally\nand theoretically in detail for metallic systems25,26.\nTo get a better understanding of the nature of the involved sp in-wave modes, mode profiles\nat different excitation frequency measured at a distance of 6µm from the antenna are shown\nin Fig. 4. The evolution of the modes can be seen clearly: for f requencies below fMW=\n3.45GHz , the spin-wave intensity is completely confined to the edges of the waveguide. In\n5/s48 /s49 /s50 /s51 /s52 /s53/s51/s46/s52/s48/s51/s46/s52/s53/s51/s46/s53/s48/s51/s46/s53/s53/s51/s46/s54/s48/s51/s46/s54/s53\n/s80/s111/s115/s105/s116/s105/s111/s110/s32 /s122 /s32/s97/s108/s111/s110/s103/s32/s119/s105/s100/s116/s104/s32/s111/s102/s32/s119/s97/s118/s101/s103/s117/s105/s100/s101/s32/s40/s181/s109/s41/s102\n/s77/s87/s40/s71/s72/s122/s41\n/s48/s48/s46/s51/s48/s46/s53/s48/s46/s56/s49/s32\n/s66/s76/s83/s32/s105/s110/s116/s101/s110/s115/s105/s116/s121/s32/s40/s97/s114/s98/s46/s32/s117/s110/s105/s116/s41\nFIG. 4. BLS intensity (linear scale) as a function of the posit ion along the width of the waveguide for\ndifferent excitation frequencies fMW(µHext= 70mT ). Frequencies below 3.45GHz show strongly\nlocalized edge modes which start to extend into the center of the waveguide for frequencies between\n3.45−3.50GHz . For higher fMW, waveguide modes appear which have their local intensity ma xima\nin the center of the waveguide. The dashed lines indicate the calculated minimal frequencies of the\nwaveguide modes shown in Fig. 5.\nthe range fMW= 3.45−3.50GHz , the maximum of the intensity is also located near the\nedges, but two additional local maxima closer to the center o f the waveguide appear. For\nfrequencies in the range of 3.50−3.57GHz , three spin-wave intensity maxima symmetrically\ncentered around the center of the waveguide are observed. Th is mode is commonly labeled\nas the third waveguide mode n= 3(ndenotes the number of maxima across the width of\nthe waveguide). For higher fMW, only one intensity maximum is found in the center of the\nwaveguide (first waveguide mode, n= 1).\nFor the waveguide modes, we can compare the experimental res ults to theoretical consid-\nerations. The theory for spin waves in thin films27with the appropriate effective field from\nmicromagnetic simulations and a wave-vector quantization over the waveguide’s short axis\nprovides an accurate description of the spin-wave mode disp ersions4–6. Figure 5 shows the\ndispersion relations and the excitation efficiencies of the w aveguide modes n= 1,3,5. Only\nodd waveguide modes can be efficiently excited4,5(even modes have no net dynamic mag-\nnetic moment averaged over the width of the waveguide) using direct antenna excitation.\nThe minimal frequencies of these three modes are indicated a s dashed lines for comparison\nin Fig. 4. Comparing Fig. 4 and Fig. 5, we find a reasonable agre ement between theory and\nexperiment for the first and the third waveguide mode. The n= 5and higher waveguide\n6/s51/s46/s52/s51/s46/s53/s51/s46/s54/s51/s46/s55/s51/s46/s56/s51/s46/s57\n/s48/s46/s48 /s48/s46/s53 /s49/s46/s48 /s49/s46/s53 /s50/s46/s48/s48/s46/s49/s49/s102/s32/s40/s71/s72/s122/s41/s32/s49\n/s32/s51\n/s32/s53/s119/s97/s118/s101/s103/s117/s105/s100/s101/s32/s109/s111/s100/s101/s69/s120/s99/s46/s32/s101/s102/s102/s46/s32/s40/s97/s114/s98/s46/s117/s110/s105/s116/s41\n/s32/s87/s97/s118/s101/s32/s118/s101/s99/s116/s111/s114 /s32/s107\n/s120/s32/s40/s114/s97/s100/s47/s181/s109/s41/s48/s46/s48 /s48/s46/s53 /s49/s46/s48 /s49/s46/s53 /s50/s46/s48/s32\nFIG. 5. (color online) Dispersion relations and amplitude e xcitation efficiencies for the first three\nodd waveguide modes of a transversally magnetized YIG waveg uide and an antenna width of 3.5µm\n(external field µHext= 70mT , width of waveguide 5µm, further parameters see28).\nmodes are not visible in the experiment. Due to the fact that t he excitation efficiency and\nthe group velocity of the spin-wave modes decreases with inc reasingn(see Ref. 4 and 5 for\ndetails), this can be attributed to a small amplitude of thes e modes.\nTo visualize the influence of the spatial decay of the spin wav es on the mode composition,\nFig. 6 (a) shows 2D spin-wave intensity maps for two exemplar y excitation frequencies. For\nfMW= 3.45GHz , edge modes can be detected for distances larger than 20µm. Different\nhigher order waveguide modes are also excited, but they can o nly be detected within 5µm\nfrom the antenna. From this findings, we can conclude that the edge modes are dominating\nthe propagation in this frequency range because of their hig h group velocities (proportional\nto the decay length) compared to the available waveguide mod es (n≥5).\nThe situation is completely different for fMW= 3.60GHz . Here, the preferably excited\nn= 1waveguide mode is interfering with the weaker n= 3waveguide mode causing a\nperiodic beating effect3–5of the measured spin-wave intensity. In this frequency rang e, no\nsignificant contribution of modes confined to the edges is vis ible.\nAn important parameter for magnonic circuits and applicati ons is the exponential de-\ncay length δampof the spin-wave amplitude. To determine δexper\nampforfMW= 3.60GHz , we\nintegrate the spin-wave intensity over the width of the wave guide (Fig. 6 (b)) and obtain\nδexper\namp= 31µm which is substantially larger than the reported decay len gths in metallic mi-\ncrostructures made of Permalloy or Heusler compounds3,4. This value can be compared\n7FIG. 6. (a) BLS intensity maps (linear scale) for two different excitation frequencies ( µ0Hext=\n70mT ). (b) Integrated BLS intensity (logarithmic scale) over the width of the waveguide for fMW=\n3.60GHz including a fit to determine the exponential amplitude decay length (δamp= 31µm).\nto the expected theoretical value δtheo\namp=vgτwherevgis the group velocity and τis\nthe life time of the spin wave. The Gilbert damping of the unpa tterned YIG/Pt bilayer\nα= 1.3·10−3measured by FMR corresponds to a life time τ≈28ns for our experi-\nmental parameters. The group velocity vgcan be deduced from the dispersion relations\nin Fig. 5 or from dynamic micromagnetic simulations yieldin gvg≈1.0−1.1µm/ns, thus\nδtheo\namp= 28−31µm. The agreement with our experimental findings δexper\namp= 31µm is excel-\nlent, especially if one considers that the plain film values o fαandMs, which might have\nbeen changed during the patterning process, have been used f or the calculation. This indi-\ncates that possible changes of the material properties due t o the patterning have only an\nnegligible influence on the decay length of the waveguide mod es and that the damping of the\nspin waves due to the Pt capping is well described by the measu red increase of the Gilbert\ndamping.\nTo conclude, we presented the fabrication of micro-magnoni c waveguides based on high\nquality YIG thin films. Spin-wave excitation and propagatio n of different modes in a mi-\ncrostructured YIG/Pt waveguide was demonstrated. As expec ted, the enhancement of the\nGilbert damping due to the Pt deposition leads to a reduced li fe time of the spin waves com-\npared to the pure YIG case. However, the life time of the spin w aves in the YIG/Pt bilayer is\nstill more than an order of magnitude larger than in the usual ly used microstructured metallic\n8systems. This leads to a high decay length reaching δexper\namp= 31µm for the waveguide modes.\nOne can estimate that the achievable decay length for a simil ar microstructured YIG/Pt\nwaveguide is δamp= 100 µm if a Cu interlayer is introduced to suppress the damping eff ects\nwhich are not related to spin pumping15(α=αYIG+αsp). Going further, from YIG thin\nfilms having the same damping than high quality, micron thick LPE films ( α≈4×10−5,\nµ0∆H≈0.03mT , Ref. 13 and 14), the macroscopic decay length of δAmp= 1mm for\nmicro-magnonic waveguides of pure YIG might be achieved.\nOur studies show that downscaling of YIG preserving its high quality is possible. Thus,\nthe multitude of physical phenomena reported for macroscop ic YIG can be transferred to\nmicrostructures which is the initial step to insulator base d, microscaled spintronic circuits.\nREFERENCES\n1A.A. Serga, A.V. Chumak, and B. Hillebrands, J. Phys. D: Appl . Phys. 43, 264002 (2010).\n2B. Lenk, H. Ulrichs, F. Garbs, and M. Münzenberg, Physics Rep orts,507, 107-136 (2011).\n3T. Sebastian, Y. Ohdaira, T. Kubota, P. Pirro, T. Brächer, K. Vogt, A.A. Serga, H.\nNaganuma, M. Oogane, Y. Ando, and B. Hillebrands, Appl. Phys . Lett. 100, 112402\n(2012).\n4P. Pirro, T. Brächer, K. Vogt, B. Obry, H. Schultheiss, B. Lev en, and B. Hillebrands, Phys.\nStatus Solidi B, 248No.10, 2404-2408, (2011).\n5V.E. Demidov, M.P. Kostylev, K. Rott, P. Krzysteczko, G. Rei ss, and S.O. Demokritov,\nAppl. Phys. Lett. 95, 112509 (2009).\n6T. Brächer, P. Pirro, B. Obry, B. Leven, A.A. Serga, and B. Hil lebrands, Appl. Phys. Lett.\n99, 162501 (2011).\n7H. Ulrichs, V.E. Demidov, S.O. Demokritov, W.L. Lim, J. Mela nder, N. Ebrahim-Zadeh,\nand S. Urazhdin, Appl. Phys. Lett. 102, 132402 (2013).\n8M.B. Jungfleisch, A.V.Chumak, V.I. Vasyuchka, A.A. Serga, B . Obry, H. Schultheiss,\nP.A. Beck, A.D. Karenowska, E. Saitoh, and B. Hillebrands, A ppl. Phys. Lett. 99, 182512\n(2011).\n9O. d’Allivy Kelly, A. Anane, R. Bernard, J. Ben Youssef, C. Ha hn, A.H. Molpeceres,\nC. Carrétéro, E. Jacquet, C. Deranlot, P. Bortolotti, R. Leb ourgeois, J.C. Mage, G. de\nLoubens, O. Klein, V. Cros, and A. Fert, Appl. Phys. Lett. 103, 082408 (2013).\n910A.V. Chumak, A.A. Serga, M.B. Jungfleisch, R. Neb, D.A. Bozhk o, V.S. Tiberkevich, and\nB. Hillebrands, Appl. Phys. Lett. 100, 082405 (2012).\n11M.B. Jungfleisch, V. Lauer, R. Neb, A.V .Chumak, and B. Hilleb rands, Appl. Phys. Lett.\n103, 022411 (2013).\n12C.W. Sandweg, Y. Kajiwara, A.V. Chumak, A.A. Serga, V.I. Vas yuchka, M.B. Jungfleisch,\nE. Saitoh, and B. Hillebrands, Phys. Rev. Lett. 106(21), 216601 (2011).\n13V. Cherepanov, I. Kolokolov, and V. L’Vov, Physics Reports 229, 81-144 (1993).\n14H.L. Glass and M.T. Elliot, Journal of Crystal Growth, 34, 285-288 (1976).\n15Y. Sun, H. Chang, M. Kabatek, Y.-Y. Song, Z. Wang, M. Jantz, W. Schneider, M. Wu,\nE. Montoya, B. Kardasz, B. Heinrich, S.G.E. te Velthuis, H. S chultheiss, and A. Hoffmann,\nPhys. Rev. Lett. 111, 106601 (2013).\n16C. Hahn, G. de Loubens, O. Klein, , M. Viret, V. V. Naletov, and J. Ben Youssef, Phys.\nRev. B 87, 174417 (2013).\n17V. Castel, N. Vlietstra, J. Ben Youssef, and B.J. van Wees, ar Xiv: cond-mat.mtrl-sci,\n1304.2190 (2013)\n18Y. Tserkovnyak, A. Brataas, and G.E.W. Bauer, Phys. Rev. Let t.88, 117601 (2002).\n19M.V. Costache, M. Sladkov, S.M. Watts, C.H. van der Wal, and B .J. van Wees, Phys. Rev.\nLett.97, 216603 (2006).\n20H. Nakayama, K. Ando, K. Harii, T. Yoshino, R. Takahashi, Y. K ajiwara, K. Uchida, and\nY. Fujikawa, and E. Saitoh, Phys. Rev. B 85, 144408 (2012).\n21O. Mosendz, J.E. Pearson, F.Y. Fradin, G.E.W. Bauer, S.D. Ba der, and A. Hoffmann,\nPhys. Rev. Lett. 104, 046601 (2010).\n22B. Heinrich, C. Burrowes, E. Montoya, B. Kardasz, E. Girt, Y. -Y. Song, Y. Sun, and M.\nWu, Phys. Rev. Lett. 107, 066604 (2011).\n23S.M.Rezende, R.L. Rodríguez-Suárez, M.M. Soares, L.H. Vil ela-Leão, D. Ley Domínguez,\nand A. Azevedo, Appl. Phys. Lett. 102, 012402 (2013).\n24Z. Qiu, K. Ando, K. Uchida, Y. Kajiwara, R. Takahashi, H. Naka yama, T. An, Y. Fujikawa,\nand E. Saitoh, Appl. Phys. Lett. 103, 092404 (2013).\n25G. Gubbiotti, M. Conti, G. Carlotti, P. Candeloro, E. D. Fabr izio, K.Y. Guslienko, A.\nAndré, C.Bayer, and A.N. Slavin, J. Phys.: Condens. Matter 16, 7709 (2004).\n26C. Bayer, J.P. Park, H. Wang, M. Yan, C.E. Campbell, and P.A. C rowell, Phys. Ref. B\n69, 134401 (2004).\n1027B.A. Kalinikos and A.N. Slavin, Journal of Physics C: Solid S tate Physics, 19pp. 7013,\n(1986).\n28Dispersion relations calculated according to Ref. 27 with p arameters Beff= 68mT and\neffective width = 4µm from a micromagnetic simulation29,Ms= 144kA /m,100nm thick-\nness, exchange constant A= 3.5pA/m.\n29M.J. Donahue and D.G. Porter, Interagency Report NISTIR 6376, National Institute of\nStandards and Technology, Gaithersburg, MD (Sept 1999).\n11"    },    {        "title": "1312.4781v2.Control_of_the_in_plane_anisotropy_in_off_stoichiometric_NiMnSb.pdf",        "content": "arXiv:1312.4781v2  [cond-mat.mes-hall]  11 Mar 2014Control of the magnetic in-plane anisotropy in off-stoichio metric NiMnSb\nF. Gerhard, C. Schumacher, C. Gould, L.W. Molenkamp1\nPhysikalisches Institut (EP3), Universit¨ at W¨ urzburg,\nAm Hubland, D-97074 W¨ urzburg, Germany\n(Dated: 10 April 2018)\nNiMnSb is a ferromagnetic half-metal which, because of its rich aniso tropy and very\nlow Gilbert damping, is a promising candidate for applications in informat ion tech-\nnologies. Wehaveinvestigatedthein-planeanisotropypropertieso fthin, MBE-grown\nNiMnSbfilms asafunctionoftheir Mnconcentration. Using ferromag neticresonance\n(FMR)todetermine theuniaxialandfour-foldanisotropyfields,2KU\nMsand2K1\nMs, wefind\nthat a variationin composition can change the strength of the four -fold anisotropy by\nmore than an order of magnitude and cause a complete 90◦rotation of the uniaxial\nanisotropy. This provides valuable flexibility in designing new device geo metries.\n1INTRODUCTION\nNiMnSb is a half-metallic ferromagnetic material offering 100% spin pola rization in its\nbulk1, and was therefore long considered a very promising material for s pintronic appli-\ncations such as spin injection. Experience has shown however that preserving sufficiently\nhigh translation symmetry to maintain this perfect polarization at su rfaces and interface\nis a major practical challenge, reducing its atractiveness for spin in jection. The material\nnevertheless continues to be very promising for use in other spintr onic applications; in par-\nticular in spin torque devices such as spin-transfer-torque (STT) controlled spin valves and\nspin torque oscillators (STO). This promise is based on its very low Gilbe rt damping, of\norder 10−3or lower2which should enhance device efficiency, as well as on its rich and stron g\nmagnetic anisotropy which allows for great flexibility in device engineer ing.\nFor example, it has been shown that STO oscillators formed from two layers of orthogo-\nnal anisotropy can yield significantly higher signal than those with co -linear magnetic easy\naxis3–6. Being able to tune the magnetic anisotropy of individual layers is clea rly useful for\nthe production of such devices.\nPrevious results have shown a dependence of the anisotropy of NiM nSb on film thickness7,\nwhich offers some control possibilities when device geometries allow fo r appropriate layer\nthicknesses, but that is not always possible due to other design or lit hography limitations.\nHere we show how the anisotropy of layers of a given range of thickn ess can effectively be\ntuned by slight changes in layer composition, achieved by adjusting t he Mn flux.\nEXPERIMENTAL\nThe NiMnSb layers are grown epitaxial by molecular beam epitaxy (MBE ) on top of\na 200 nm thick (In,Ga)As buffer on InP (001) substrates. All sample s have a protective\nnon-magnetic metal cap (Ru or Cu) deposited by magnetron sputt ering before the sample\nis taken out of the UHV environment, in order to avoid oxidation and/ or relaxation of the\nNiMnSb8. The flux ratio Mn/Ni, and thus the composition, is varied between sa mples by\nadjusting the Mn cell temperature while the flux ratio Ni/Sb is kept co nstant. The thickness\nof most of the studied NiMnSb layers is 38 ±2 nm. Two samples have a slightly larger film\nthickness (45 nm, marked with ( ) in Fig. 3a), caused by the change in growth rate due\n2/s49/s52/s46/s53 /s49/s53/s46/s48 /s49/s53/s46/s53/s49/s48/s48/s49/s48/s50/s49/s48/s52/s49/s48/s54/s49/s48/s56/s49/s48/s49/s48/s49/s48/s49/s50/s49/s48/s49/s52/s49/s48/s49/s54\n/s49/s52/s46/s56/s52 /s49/s52/s46/s56/s56 /s49/s52/s46/s57/s50/s49/s48/s49/s49/s48/s50/s49/s48/s51\n/s66\n/s67\n/s32 /s32/s40/s176/s41/s32/s108/s111/s119/s101/s115/s116/s32/s77/s110/s32/s99/s111/s110/s99/s101/s110/s116/s114/s97/s116/s105/s111/s110/s32/s40/s115/s97/s109/s112/s108/s101/s32/s65/s41\n/s32/s109/s101/s100/s105/s117/s109/s32/s77/s110/s32/s99/s111/s110/s99/s101/s110/s116/s114/s97/s116/s105/s111/s110/s32/s40/s115/s97/s109/s112/s108/s101/s32/s66/s41\n/s32/s104/s105/s103/s104/s101/s115/s116/s32/s77/s110/s32/s99/s111/s110/s99/s101/s110/s116/s114/s97/s116/s105/s111/s110/s32/s40/s115/s97/s109/s112/s108/s101/s32/s67/s41\n/s32/s32/s32/s105/s110/s116/s101/s110/s115/s105/s116/s121/s32/s40/s99/s111/s117/s110/s116/s115/s41/s65/s32/s32/s32\n/s32 /s32/s40/s176/s41\nFIG. 1. HRXRD ω-2θ-scans of 3 NiMnSb samples with various Mn concentrations. T he curves are\nvertically offset for clarity. Inlet: ω-scans showing high crystal quality.\nto the change in Mn flux. We verified that there is no correlation betw een anisotropy and\nsample thickness in this range.\nHigh Resolution X-Ray Diffraction (HRXRD) measurements of the (00 2) Bragg reflection\nare used to determine the vertical lattice constant of each sample . Fig.1 shows standard\nω-2θ-scans of the (002) Bragg reflection on layers with the lowest and h ighest Mn concen-\ntrations used in the study, as well as a scan for a sample with medium M n concentration.\nThe sample with the lowest Mn content has a vertical lattice constan t of 5.939 ˚A (sample A)\nand that with the highest Mn content (sample C) has a vertical lattic e constant of 6.092 ˚A.\nTo get an estimate of the vertical lattice constant of stoichiometr ic NiMnSb in our layer\nstacks, we used an XRD measurement of a stoichiometric, relaxed s ample9. We determine\na relaxed lattice constant of arel= (5.926 ±0.007)˚A. Together with the lattice constant\nof our InP/(In,Ga)As substrate, 5.8688 ˚A, and an estimated Poisson ratio of 0.3 ±0.03,\nwe get the minimal and maximal values for the vertical lattice consta nt of stoichiometric\nNiMnSb: a⊥,max= 5.999˚A,a⊥,min= 5.957˚A. The vertical lattice constant of the sample\nwith medium Mn concentration (sample B, 5.968 ˚A) lies in this range. We conclude that\n3the composition of sample B is approximately stoichiometric.\nIn Ref. 10 and 11, the effects of off-stoichiomteric defects in NiMnS b are discussed. Among\nthe possible defects related to Mn, Mn Ni(Mn substituting Ni) is most likely (it has lowest\nformation energy) and the predicted decrease of the saturation magnetization is consistent\nwith our observation (see Fig. 3b). Furthermore, an increase of t he lattice constant with\nincreasing concentration of this kind of defect is predicted theore tically and observed ex-\nperimentally. Thus, we can use the (vertical) lattice constant as a m easure for the Mn\nconcentration in our samples.\nThe crystal quality is also assessed by the HRXRD measurements. T he inset in Fig. 1 shows\ntheω-scans of the same three NiMnSb layers. The ω-scans of both the low and medium Mn\nconcentration sample are extremely narrow with a full width half-ma ximum (FWHM) of 15\nand 14 arcsec, respectively. A broadening for the sample with highe st Mn concentration\ncan be seen (FWHM of 35 arcsec). Reasons for the broadening can be partial relaxation of\nthe layer due to the increased lattice mismatch with the (In,Ga)As Bu ffer, and/or defects\nrelated to the surplus of Mn.\nUsing the experimental data of the lattice constant in Ref. 11, we c an estimate a difference\nin Mn concentration between sample A and C (extreme samples) of ab out 40%. For sample\nC (extreme high Mn concentration), we determine a saturation mag netization of 3.4 µBohr\n(see Fig. 3b). According to Ref. 11, this corresponds to a crysta l where about 20% of Ni is\nreplaced by Mn. It should be noted that we investigated the effect o f extreme surplus/deficit\nof Mn within the limits of acceptable crystal quality. As can be seen in F ig. 3a, already a\nmuch smaller change incomposition canchange the strength andorie ntationof the magnetic\nanisotropy significantly.\nTo map out the in-plane anisotropy of our samples, we use frequenc y-domain ferromagnetic\nresonance (FMR) measurements at a frequency of 12.5 GHz. The r esonance fields are de-\ntermined as a function of an external magnetic field applied at fixed a ngles ranging from 0◦\n(defined as the [100] crystal direction) to 180◦. Fig. 2 shows results of these measurements\nfor four different samples with four distinct types of anisotropy: S ample A and D both ex-\nhibit large uniaxial anisotropies with anadditional four-foldcompone nt, however of opposite\nsign. The hard axis of sample A is along the [1 ¯10] crystal direction, where for sample D the\nhard axis is along the [110] crystal direction. Sample B and C both sho w mainly uniaxial\nanisotropies, again with opposite signs.\n4/s50/s46/s48/s50/s46/s50/s50/s46/s52/s50/s46/s54\n/s50/s46/s48/s50/s46/s50/s50/s46/s52/s50/s46/s54\n/s48 /s52/s53 /s57/s48 /s49/s51/s53 /s49/s56/s48/s49/s46/s56/s56/s49/s46/s57/s50/s49/s46/s57/s54/s50/s46/s48/s48/s50/s46/s48/s52\n/s48 /s52/s53 /s57/s48 /s49/s51/s53 /s49/s56/s48/s50/s46/s48/s50/s46/s49/s50/s46/s50/s50/s46/s51/s32/s32\n/s32/s114/s101/s115/s111/s110/s97/s110/s99/s101/s32/s102/s105/s101/s108/s100/s32/s40/s107/s79/s101/s41/s32/s65\n/s32/s32\n/s32\n/s32/s32/s67\n/s32/s32 /s32/s32\n/s32/s66\n/s32/s32\n/s32/s91/s49/s48/s48/s93/s32/s32/s32/s91/s49/s49/s48/s93/s32/s32/s91/s48/s49/s48/s93/s32/s32/s32/s91/s49/s49 /s48/s93/s32/s32/s91/s49/s48/s48/s93/s32/s32/s32/s91/s49/s49/s48/s93/s32/s32/s91/s48/s49/s48/s93/s32/s32/s32/s91/s49/s49 /s48/s93/s32/s32\n/s32/s68\n/s97/s110/s103/s108/s101/s32/s98/s101/s116/s119/s101/s101/s110/s32/s101/s120/s116/s101/s114/s110/s97/s108/s32/s109/s97/s103/s110/s101/s116/s105/s99/s32/s102/s105/s101/s108/s100/s32\n/s97/s110/s100/s32/s99/s114/s121/s115/s116/s97/s108/s32/s100/s105/s114/s101/s99/s116/s105/s111/s110/s32/s91/s49/s48/s48/s93/s32/s40/s176/s41\nFIG. 2. FMR measurements and simulation for four different sam ples. The symbols are measure-\nments of the resonance frequency for magnetic fields along sp ecific crystal directions, where 0◦lies\nalong [100]. The lines are simulations (see below) and also s erve as a guide to the eye. Sample A,\nB and C correspond to the samples with lowest, medium and high est Mn concentration shown in\nFig. 1. Sample D completes the various kinds of anisotropy ob served in NiMnSb.\nTheFMRdatacanbesimulated withasimple phenomenological magneto staticmodel toex-\ntract the anisotropy components (derivation taken from Ref. 12 ). The free energy equation\nfor thin films of cubic materials is given by:\nǫc=−K/bardbl\n1\n2(α4\nx+α4\ny)−K⊥\n1\n2α4\nz−Kuα2\nz (1)\nwhereαx,αyandαzdescribe the magnetization with respect to the crystal directions [100],\n[010] and [001]. K/bardbl\n1is the four-fold in-plane anisotropy constant, KuandK⊥\n1represent\nthe perpendicular uniaxial anisotropy (second and fourth order r espectively). In our in-\nplane FMR geometry, the fourth order perpendicular anisotropy t ermK⊥\n1can be neglected.\nInstead, an additional uniaxial in-plane anisotropy term is added:\nǫu=−K/bardbl\nu(ˆn·ˆM)2\nM2s\nwith the unit vector ˆ nalong the uniaxial anisotropy and the saturation magnetization Ms,\nˆM. The Zeeman term coupling to the external field H0and a demagnetization term origi-\nnating from the thinness of the sample, are defined as\nǫZ=−M·H0, ǫdemag=−4πDM2\n⊥\n2(2)\n5and added as well to the free energy. The effective magnetic field\nHeff=−∂ǫtotal\n∂M(3)\nwith\nǫtotal=ǫc+ǫu+ǫZ+ǫdemag (4)\nis used to solve the Landau-Lifshitz-Gilbert-Equation (LLG):\n−1\nγ∂M\n∂t= [M×Heff]−G\nγ2M2s[M×∂M\n∂t] (5)\nwith the gyromagnetic ratio γ=gµB\n/planckover2pi1and the Gilbert damping constant G. The resonance\ncondition can be found by calculating the susceptibility13,χ=∂M\n∂H:\n(ω\nγ)2=BeffH∗\neff (6)\nIn the following, we neglect the damping contribution sinceG\nγMsin our samples is of the\norder of 10−3or lower. Thus, BeffandH∗\neffin our case can be found to be:\nH∗\neff=H0cos[φM−φH]+2K/bardbl\n1\nMscos[4(φM−φF)]\n+2K/bardbl\nU\nMscos[2(φM−φU)] (7)\nBeff=H0cos[φM−φH]+K/bardbl\n1\n2Ms(3+cos[4(φM−φF)])\n+4πDMs−2K⊥\nU\nMs+K/bardbl\nU\nMs(1+cos[2(φM−φU])) (8)\nHere,φM,φHandφUdefine the angles of the magnetization, external magnetic field and in-\nplane easy axis of the uniaxial anisotropy, respectively, with respe ct to the crystal direction\n[100].φFaccounts for the angle of the four-fold anisotropy. At the magne tic fields used in\nthese studies, it is safe to assume φM=φH14. In equation (8), 4 πDMs−2K⊥\nU\nMScan be defined\nas an effective magnetization 4 πMeff, containing the out-of-plane anisotropy. It is used as\na constant in our simulation.\nFor each sample, we extract2K1\nMsand2KU\nMs, the four-fold and uniaxial in-plane anisotropy\nfield, from the simulation and plot them versus the vertical lattice co nstant (Fig. 3a). The\nvertical, dotted lines mark the range where stoichiometric NiMnSb is e xpected. For vertical\nlattice constants in the range from 5.96 to 6.00 ˚A, both anisotropy fields are relatively small.\nThe four-fold contribution increases for samples with decreasing v ertical lattice constant\n6/s45/s52/s48/s48/s45/s51/s48/s48/s45/s50/s48/s48/s45/s49/s48/s48/s48/s49/s48/s48/s50/s48/s48/s51/s48/s48/s52/s48/s48\n/s53/s46/s57/s50 /s53/s46/s57/s54 /s54/s46/s48/s48 /s54/s46/s48/s52 /s54/s46/s48/s56 /s54/s46/s49/s50/s51/s46/s48/s51/s46/s53/s52/s46/s48/s52/s46/s53\n/s32/s32\n/s32\n/s67/s66/s65\n/s65/s66\n/s68/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32 /s75\n/s85\n/s32\n/s40/s98/s41/s97/s110/s105/s115/s111/s116/s114/s111/s112/s121/s32/s102/s105/s101/s108/s100/s115/s32\n/s50/s75\n/s49/s47/s77\n/s83/s32/s44/s32 /s50/s75\n/s117/s47/s77\n/s83/s32/s40/s79/s101/s41/s32/s32/s32/s32/s32/s32/s32/s32/s32 /s32/s32/s75\n/s85\n/s32\n/s40/s97/s41/s50/s75\n/s49/s47/s77\n/s115\n/s50/s75\n/s85/s47/s77\n/s115\n/s67/s40/s32/s32/s41\n/s32/s32/s40/s32/s32/s41/s115/s97/s116/s46/s32/s109/s97/s103/s110/s101/s116/s105/s122/s97/s116/s105/s111/s110\n/s77\n/s115/s32/s40/s105/s110/s32/s181/s66/s111/s104/s114/s41\n/s118/s101/s114/s116/s105/s99/s97/s108/s32/s108/s97/s116/s116/s105/s99/s101/s32/s99/s111/s110/s115/s116/s97/s110/s116/s32/s40 /s197 /s41\nFIG. 3. (a) Uniaxial anisotropy field2KU\nMsand four-fold anisotropy field2K1\nMsfor NiMnSb layers with\nvarious Mn concentrations. The vertical lattice constant i s used as a gauge of the Mn content.\nSamples with a rotated RHEED pattern (see last section) are i ndicated by open symbols. The\ndotted lines mark the range where stoichiometric NiMnSb is e xpected. The samples of lowest,\nmedium and highest Mn concentration (A, B and C) together wit h sample D are marked. The\ntwo samples marked with ( ) exhibit slightly higher film thick ness than the other samples. (b)\nSaturation magnetization Msdepending on the vertical lattice constant.\n(lower Mn concentration) but remains small for larger vertical latt ice constant (increasing\nMn concentration). The uniaxial anisotropy gets more strongly ne gative with increasing\nvertical lattice constant, whereas in samples with lower vertical lat tice constants, the uni-\naxial field can be either positive or negative while its absolute value gro ws significantly with\ndecreasing vertical lattice constant. The change in sign of the unia xial anisotropy field at a\nvertical lattice constant of about 5.99 ˚A corresponds to a rotation of the easy axis from the\n[110] direction (positive anisotropy fields) to the [1 ¯10] direction. One can see that already\na small change of the vertical lattice constant (small change in com position) is sufficient to\nrotate the uniaxial anisotropy as well as to induce a significant four -fold anisotropy.\nThe fitting accuracy of the extracted anisotropy fields is ∼5%, giving error bars smaller than\n7the symbols in Fig. 3a. It should be noted that in order to exactly ext ract the anisotropy\nconstants K1andKUfrom the anisotropy fields, the saturation magnetization Msof each\nsample is needed. This can be determined by SQUID measurements. W e have performed\nsuch measurements on a representative fraction of the samples ( Fig. 3b). Samples with\nmedium Mn concentration show saturation magnetizations which, to experimental accuracy\nof about 8% are consistent with the theoretically expected 4.0 µBohrper unit formula for\nstoichiometric NiMnSb15. The estimated measurement accuracy of 8% accounts for un-\ncertainty in the sample thickness extracted from the HRXRD data o f about 5%, as well\nas errors in determining the exact sample area, SQUID calibration an d SQUID response\ndue to finite sample size. Our samples with highest and lowest magnetiz ation show a slight\ndecrease in saturation magnetization, of order 12%. This change is sufficiently small to be\nneglected in the overall assesment of the anisotropy vs. vertical lattice constant of Fig. 3a.\nIn an attempt to understand the effect of higher or lower Mn conce ntration on the crystal\nstructure in our samples, we consider the possible non-stoichiomet ric defects which can exist\nin NiMnSb, asdiscussed in Ref. 10. Formationenergies, magnetic mom ent change and effect\non the half-metallic character are presented there for each type of defect. Mn-related defects\nare a) Mn substituting Ni or Sb (Mn Ni, MnSb), b) Mn on a vacancy position (Mn I), c) Ni or\nSb substituting Mn (Ni Mn, SbMn) or d) a vacancy position at the Mn site (vac Mn). With a\nsurplus of Mn, both Mn substituting Ni or Sb and Mn incorporated on the vacancy position\nseem plausible. However, the formation energy of Mn Sbis more than three times larger than\nfor the other defects, suggesting it should be very rare. On the o ther hand, in the case of a\nMn deficiency, either Ni or Sb could substitute Mn or vacancies can b e built into the crystal.\nAll those three defects have similar formation energies, making the m equally possible.\nExcept for Mn Iand Mn Sb, all of these possible defects reduce the magnetic moment per\nformula unit. Our observations of a lower magnetic moment for samp les with either high or\nlow Mn flux, are thus consistent with the defects Mn Ni, NiMn, SbMnand vac Mn. The posi-\ntive contribution of Mn Ito the magnetic moment is however some 5 times smaller than the\ndecrease induced by the other defects, so some fraction of defe cts of the Mn Ivariety could\nalso be present in the samples. A detailed discussion on the transition from stoichiometric\nNiMnSb towards off-stoichiometric Ni 1-xMn1+xSb is given in Ref. 11. It is shown that the\nlattice constant of off-stoichiometric NiMnSb increases for increas ing substitution of Ni by\nMn. This behavior is clearly seen in our samples for increasing Mn conce ntration and we\n8FIG. 4. Typical RHEED reconstruction of the NiMnSb surface i llustrating the two reconstructions\ndiscussed in the text.\nconclude that this kind of defect is most prominent in our samples. An explanation for a\ndecreasing lattice constant for decreasing Mn concentration is ye t to be found.\nAfurtherobservationwhichmayprovideinsightintotheobserveda nisotropybehaviorcomes\nfrom Reflective High Energy Electron Diffraction (RHEED), which is us ed to monitor the\nsurface of the sample in-situ during the growth. RHEED provides inf ormation about the\nsurface reconstruction, which turns out to be sensitive to the Mn content. In all samples,\nat the beginning of the growth (after approximately one minute), t he surface reconstruction\nexhibits a clear 2 ×1 pattern, meaning a d/2 reconstruction in the [110] crystal direction and\nad/1 reconstruction along [1 ¯10] direction (see Fig.4). How this pattern then evolves during\ngrowth depends on the Mn flux. For ideal Mn flux, the pattern is sta ble throughout the\nentire 2 hour growth time corresponding to a 40 nm layer. A reduced Mn flux results in a\nmore blurry RHEED pattern, but does not lead to any change in the s urface reconstruction.\nA higher Mn flux, on the other hand, causes a change of the recons truction such that the\nd/2 pattern also becomes visible along the [1 ¯10] direction and fades over time in the [110]\ndirection until a 90◦rotation of the original pattern has been completed. The length of\ntime (and thus the thickness) required for this rotation depends s trongly on the Mn flux.\nA slightly enhanced Mn flux causes a very slow rotation of the recons truction that can last\nthe entire growth time, whereas a significant increase of the Mn flux (sample with vertical\nlattice constants above 6.05 ˚A) will cause a rotation of the reconstruction within a few min-\nutes of growth start, corresponding to a thickness of only very f ew monolayers. Based on\nthese observations, our samples can be split into two categories: s amples with a stable 2 ×1\n9reconstruction and those with a 2 ×1 reconstruction that rotates during growth. In Fig.\n3a, samples with a stable RHEED pattern are indicated with filled symbo ls while empty\nsymbols show samples with a rotated RHEED reconstruction. It is int eresting to note that\nall samples with a rotated reconstruction exhibit a very low four-fo ld anisotropy field. In\naddition, the sooner the rotation of the RHEED pattern occurs, t he stronger the uniaxial\nanisotropy is.\nSUMMARY\nWe have shown that the anisotropy of NiMnSb strongly depends on t he composition\nof the material. A variation of the Mn flux results in different (vertica l) lattice constants\n(measured by HRXRD) that can be used for a measure of the Mn con centration. RHEED\nobservations (in-situ) during thegrowthalreadygive anindication o fhighorlowMn concen-\ntration. The anisotropy shows a clear trend for increasing Mn cont ent. Using this together\nwith the RHEED observations, NiMnSb layers with high crystal quality and anisotropies\nas-requested can be grown. The microscopic origin of this behavior remains to be under-\nstood, and it is hoped that this paper will stimulate further efforts in this direction. The\nphenomenology itself is nevertheless of practical significance in tha t it provides interesting\ndesign opportunitiesfordevices such asspin-valves thatcouldbem adeoftwo NiMnSb layers\nwith mutually parallel or orthogonal magnetic easy axes as desired.\nACKNOWLEDGEMENTS\nWe thank T. Naydenova for assistance with the SQUID measuremen ts. This work was\nsupported by the European Commission FP7 contract ICT-257159 “MACALO”.\nREFERENCES\n1R. A. de Groot, F. M. Mueller, P. G. v. Engen, and K. H. J. Buschow, Phys. Rev. Lett.\n50, 2024 (1983).\n2A.Riegler, Ferromagneticresonancestudyof the Half-Heusler alloyNi MnSb: Thebenefitof\nusing NiMnSb as a ferromagnetic layer in pseudo spin-valve b ased spin-torque oscillators ,\nPh.D. thesis, Universitaet Wuerzburg (2011).\n103T. Devolder, A. Meftah, K. Ito, J. A. Katine, P. Crozat, and C. Ch appert, Journal of\nApplied Physics 101, 063916 (2007).\n4D. Houssameddine, U. Ebels, B. Dela¨ et, B. Rodmacq, I. Firastrau , F. Ponthenier,\nM. Brunet, C. Thirion, J.-P. Michel, L. Prejbeanu-Buda, M.-C. Cyrille , O. Redon, and\nB. Dieny, Nature Materials 6, 447 (2007).\n5G. Consolo, L. Lopez-Diaz, L. Torres, G. Finocchio, A. Romeo, and B. Azzerboni, Applied\nPhysics Letters 91, 162506 (2007).\n6S. M. Mohseni, S. R. Sani, J. Persson, T. N. Anh Nguyen, S. Chung, Y. Pogoryelov,\nand J.˚Akerman, physica status solidi (RRL) Rapid Research Letters 5, 432 (2011) and\nreferences therein\n7A. Koveshnikov, G. Woltersdorf, J. Q. Liu, B. Kardasz, O. Mosend z, B. Heinrich, K. L.\nKavanagh, P. Bach, A. S. Bader, C. Schumacher, C. R¨ uster, C. Gould, G. Schmidt, L. W.\nMolenkamp, and C. Kumpf, Journal of Applied Physics 97, 073906 (2005).\n8C. Kumpf, A. Stahl, I. Gierz, C. Schumacher, S. Mahapatra, F. Lo chner, K. Brunner,\nG. Schmidt, L. W. Molenkamp, and E. Umbach, physica status solidi ( c)4, 3150 (2007).\n9W. Van Roy and M. W´ ojcik, inHalf-metallic Alloys , Lecture Notes in Physics, Vol. 676,\nedited by I. Galanakis and P. Dederichs (Springer Berlin Heidelberg, 2 005) pp. 153–185.\n10B. Alling, S. Shallcross, and I. A. Abrikosov, Phys. Rev. B 73, 064418 (2006).\n11M. Ekholm, P. Larsson, B. Alling, U. Helmersson, and I. A. Abrikosov , Journal of Applied\nPhysics108, 093712 (2010).\n12B. Heinrich and J. A. C. Bland, eds., Ultrathin Magnetic Structures II (Springer Berlin\nHeidelberg, 1994).\n13C. Kittel, Phys. Rev. 73, 155 (1948).\n14We have confirmed from frequency dependent measurements tha t this assumption leads\nto errors smaller than the size of the symbols in Fig. 3a.\n15T. Graf, C. Felser, and S. S. Parkin, Progress in Solid State Chemist ry39, 1 (2011).\n11"    },    {        "title": "1401.1672v1.Dynamic_exchange_via_spin_currents_in_acoustic_and_optical_modes_of_ferromagnetic_resonance_in_spin_valve_structures.pdf",        "content": "1 \n Dynamic exchange via spin currents in acoustic and optical modes of \nferromagnetic resonance in spin -valve structures  \n \nA.A. Timopheev1, Yu.G. Pogorelov2, S. Cardoso3, P.P. Freitas3, G.N. Kakazei2,4, N.A . Sobolev1 \n1Departamento de Física and  I3N, Universidade de Aveiro, 3810 -193 Aveiro, Portugal  \n2IFIMUP and IN -Institute of Nanoscience and Nanotechnology, Departamento de Física e Astronomia, \nUniversidade do Porto, 4169 -007 Porto, Portugal  \n3INESC -MN and IN -Institute of Nanoscience and Nanotechn ology, 1000 -029 Lisbon, Portugal  \n4Institute of Magnetism, NAS of Ukraine, 03142 Kiev, Ukraine  \n     \ne-mail:  andreyt@ua.pt  \nTwo ferromagnetic layer s magnetically decoupled by a thick normal metal spacer layer can be, \nnevertheless, dynamically coupled via spin currents emitted by the spin -pump and absorbed through the \nspin-torque effects at the neighboring interfaces. A decrease of damping in both layers due to a partial \ncompensation of the angular momentum leakage in e ach layer was previously observed at the coincidence \nof the two ferromagnetic resonances. In case of non -zero magnetic coupling, such a dynamic exchange \nwill depend on the mutual precession of the magnetic moments in the layers. A difference in the linewid th \nof the resonance peaks is expe cted for the acoustic and optical regimes of precession. However, the \ninterlayer coupling hybridizes the resonance responses of the layers and therefore can also change their \nlinewidths. The interplay between the two mechan isms has never been considered before. In the present \nwork, the joint influence of the hybridization and non -local damping on the linewidth has been studied in \nweakly coupled NiFe/CoFe/Cu/CoFe/MnIr spin -valve multilayers. It has been found that the dynamic  \nexchange by spin currents is different in the optical and acoustic modes, and this difference is dependent \non the interlayer coupling strength. In contrast to the acoustic precession mode, the dynamic exchange in \nthe optical mode works as an additional da mping source. A simulation in the framework of the Landau -\nLifshitz -Gilbert formalism for two ferromagnetic layers coupled magnetically and by spin currents has \nbeen done to separate the effects of the non -local damping from the resonance modes hybridizatio n. In \nour samples both mechanisms bring about linewidth changes of the same order of magnitude, but lead to \na distinctly different angular behavior. The obtained results are relevant  for a broad class of coupled \nmagnetic multilayers  with ballistic regime o f the spin transport . \n    \n1. Introduction  \nSpin current, a flow of angular momentum , is a basic concept in spintronics  and spin caloritronics [1, \n2]. Spin current generation is experimentally accessible via spin pumping [3-5], spin Seebek effect [6], spin \nHall effect [7, 8] and acoustic wave propagation in the case of magnetic insulators [9]. The spin -orbit \ninteraction plays a fundamental role in these effects. The presence of a spin current in a normal metal 2 \n (NM) or semiconductor can be detected by the inverse spin Hall effect [10-12] or as a change of the \neffective damping in an adjacent ferromagnetic (FM) layer [3-5]. The latter effect allows one to alter the \nswitching field of the FM layer and even sustain  a stable precession in it [13-15]. It is hard to overestimate \nthe fundamental and practical importance of the issues emerging from the investigation  of the spin \ncurrents.  \nA precessing magnetic moment in a FM layer acts as a spin battery [16] injecting a pu re spin current in \na neighboring NM layer through the FM/NM interface. This spin current can then  return to  the NM/FM \ninterface bringing the carried angular momentum  back to the precessing spins of the FM layer. Depending \non the spin -orbit interaction stre ngth and the layer thickness, the normal metal will absorb a certain part \nof the angular momentum flow via the spin -flip relaxation processes. Thus, the backflow through the \nNM/FM interface will be always weaker than the direct flow, which results in an enhanced  precession  \ndamping  [3-5]. The spin diffusion length of the normal metal and the spin mixing  interface conductance \ncan be evaluated in this way [3-5, 17]. \nAn interesting result has been obtained for a FM/NM/FM trilayer [18] having non -identical FM layers. \nThe asymmetry provided different angular dependences of the ferromagnetic resonance (FMR) fields of \nthe FM layers. When the external magnetic field was directed at an angle for which the FMR peak \npositions coincide, a narrowing of both resonances w as observed. The explanation of this effect is that, \nfor the case of separately precessing FM layers, the spin current generated in  a precessing FM layer is \nabsorbed in the other, non -resonating FM layer, which causes , in a full analogy to the written  abov e, a \ndamping enhancement,  while for the case of a mutual resonant precession this spin current leakage is \npartially compensated by the spin current from the other FM layer. In this experiment , the NM spacer was \nthin enough for the spin current to be consid erable at the second NM/FM interface, but thick enough to \nexclude any possible magnetic coupling between the FM layers.  \nIndeed, the magnetic coupling between two FM layers complicates the analysis of the spin -current -\ninduced non -local damping. If the coupling is strong enough, the resonance response of the system is \nrepresented by the collective acoustic and optical modes w hich are the in -phase and out -of-phase mutual \nprecession modes in the  FM layers. There is no separate precession in such a regime – the precession in \none layer drags the magnetic moment in the other one. Moreover, the linewidths of the resonance peaks \nare dependent o n the field separation betwe en them, and usually these parameters are angular dependent. \nAnd finally, the interaction fundamentally forbids the peaks to have a crossing point, i.e. the anticrossing \nis a characteristic feature here. The stronger the interlayer coupling, the larger is the anticrossing \nseparation between  the modes. From this point of view, the difference of damping  for the acoustic and \noptical modes in  a FM/NM/FM trilayer as a result of a dynamic spin currents exchange, theoreticall y \npredicted by Kim and Chappert [19], seems to be experimentally unachievable. Nevertheless , in several \nrecent papers [20-22] experimental observations of this effect  have been already claimed.  There  is, 3 \n however, a full ignorance of the fact that the FMR p eaks hybridization will also influence the linewidth \neven if a separate measurement of the precession in each layer can be done.  \nMotivated by this, we have performed  a comprehensive  study of weakly coupled spin -valve (SV) \nmultilayers, where the hybridizati on is weak and the layers behave almost independently, conserving at \nthe same time the main features of the acoustic and optical modes of the collective magnetic response. \nOne important objective is to separate the hybridization -induced change of the FMR linewidth from the \nspin-current -induced one and to check in this way the difference between the spin -current -induced \ndamping in the optical and acoustic regime s of precession. W e present an  experimental study of the FMR \nin NiFe/CoFe/Cu/CoFe/MnIr SV  multilayers  conducted using a standard X -band EPR spectrometer. Our \nstudy is acc ompanied by a simulation of the microwave absorption in such a magnetically coupled system \nin the presence of dynamical exchange by spin currents  in the framework of the Landau -Lifshitz -Gilbert \nformalism.  \n \n2. Experimental details  \nFMR was measured  at room temp erature using a Bruker ESP 300E E SR spectrometer at a microwave \nfrequency  of 9.67 GHz. The f irst derivative of the microwave absorp tion by the magnetic field  was \nregistered. For each sample, a series of in-plane FMR spectra were collected for different ang les of  the \nmagnetic field in the film plane with respect to the internal exchange bias  field. Each FMR spectrum , \nexperimentally measured or simulated,  was fitted by Lorentzian functions to obtain  angular dependences \nof the resonance field and linewidth.  The least -squares method was employed.  \nThe SV multilayers were grown by the ion -beam deposition in a Nordiko  3000 system.  The cobalt -\niron fixed layer is exchange coupled to the MnIr  antiferromagnet (AF), the free layer is a bilayer \ncomposed of a permalloy  and a cobalt -iron sublayers, and the copper spacer separates the free and fixed \nlayers. Two series of samples were used in the study:  \n1) Glass / Ta(30 Å) / Ni 80Fe20(30 Å) / Co 80Fe20(25 Å) / Cu ( dCu) / Co 80Fe20(25 Å) / Mn 82Ir18(80 Å) / \nTa(30 Å) – the average thickness of the copper spacer , dCu, varies from 17 to 28 Å in 1  Å steps.  \n2) Glass / Ta(30 Å) / Ni80Fe20(56 ‒ dF) / Co 80Fe20(dF) / Cu(22 Å) / Co 80Fe20(25 Å) / Mn 82Ir18(80 Å) / \nTa(50 Å)  – the relative thicknesses of the permalloy and cobal -iron sublayers var y within the \n56 Å thick free layer by setting the parameter dF to 8, 16, 24, 32 and 40  Å. \nAdditionally, separate free layers ( Glass / Ta(30 Å) / Ni80Fe20(56 ‒ dF) / Co 80Fe20(dF) / Cu(22 Å) / \nTa(50 Å)) of the first and second series were grown to serve as reference samples.  \nThe first series was already studied in Refs. [23, 24]. It has been shown that the samples with tCu > 16 Å \nare in the weak coupling regime, and the main interlayer coupling mechanism here is  Néel’s “orange -\npeel” magnetostatic interaction  [25]. When the copper spacer thickness grows from 17 to 28 Å, the 4 \n interlayer coupling energy is reduced from 1.1×102 erg/cm2 to 4×103 erg/cm2, which corresponds to a \nvariation of the effective interaction field on the free layer from  17 to 6 Oe.   \nThe second series has a fixed metallic spacer thickness, tCu = 22 Å, while the free layer effective \nmagnetization, 4π Meff, determined by the Kittel formula, gradually varies from 15 kG to 8.5 kG.  In this \nway the angular dependence of the free layer resonance field can be  vertically  shifted  with respect to that \nof the fixed layer . \n \n3. Simulation of the microwave absorption spectrum  \nA SV is considered as a system of two cou pled FM layers  consisting of a free and a fixed layer with \nthe thicknesses d1, d2, volume saturation magnetization s Ms1, Ms2, and in -plane uniaxial magnetic \nanisotropy constants K1, K2, respectively. The e xchange coupling of the fixe d layer to the AF  layer with \nthe interface coupling energy Eex is defined by a unidirectional anisotropy with the effective field \nex 2 s2E d M\n. The e asy axes of all three anisotropies lay in  the sample plane and have the same direction \nalong the magnetic field applied at annealing. The m agnetizations in  both layers are assumed  to be  \nuniform, thus the bilayer magnetic state is completel y described by the unit vectors \nˆˆ,12mm  of their \ninstantaneous directions. The layers are coupled by the Heisenberg exchange interaction , Eic.  \nThen the magnetic energy density per unit area of the considered system can be written as:  \n \n  \n\n22\ntot 1 s1 1 ext s1 mw s1\n22\ns2 2 ext s2\n2 icex\nmw s2\n2 s2ˆˆ ˆ ˆ ˆ ˆ ˆ 2\nˆ ˆ ˆ ˆ ˆ 2\nˆˆ .ˆˆˆ  U d M K H M h M\nM K H M\ndEEhMdM\n    \n   \n1 1 1 0 1 1\n2 2 2 0\n12\n2 2 2m ·n m ·û m ·h m ·h\nm ·n m ·û m ·h\nm ·m\nm ·h m ·û  (1) \nThere are also included four unit vectors determining the spatial orientation of the effective fields:  the \neasy axis \nˆûn  of the uniaxial and unidirectional anisotropies  (here \nˆn  is the normal to the multilayer \nplane) , the direction \nˆ\n0h  of the external magnetic field Hext, and the direction \n1ˆh  of the microwave \nmagnetic  field hmw.  \nThe spin -pump / spin -sink mechanism  in our SVs is considered as follows. The CoFe/Cu and \nCu/CoFe interfaces are assumed to be identical and t o give rise to an effective spin mixing conductance in \nthe FM1/NM/FM2 structure characterized  by the parameter  AFNF [26] which is in  a gener ic case \ndependent on the relative magnetization orienta tions in the layers, \nˆˆ,12mm . Since the copper spacer is \nmuch thinner than the  the spin -diffusion length ( λsd ~ 0.4 µm at T = 300  K), the transfer of the angular \nmomentum from one FM layer to the other occurs in a purely ballistic regime, i.e. the spin current emitted 5 \n at the first CoFe /Cu interface is fully absorbed at the second Cu/CoFe interface . The spin current \nbackflow is not considered separately: it just renormalizes the parameter  AFNF. The spin -pump  / spin-\ntorque induced damping \nsp  for each layer is influenced by its thickness, saturation magnetization and g-\nfactor.  The dynamics of such a structure can be described by a system of coupled Landau -Lifshitz -Gilbert \nequations with additional spin -pump / spin -torque induced Gilbert -like damping  terms [5]: \n \n1 eff sp\ntot\neff\nFNF\nsp B\nsˆ ˆ ˆ ˆˆ ˆ ˆ ˆ ,\n1,ˆ\n,4\n, 1,2,\n.i\niis\ni\nit t t t\nU\ndM\nAgdM\nij\nij  \n                   \n\n\n\nm m m mm H m m m\nHmii\ni\nij i i i\ni i i i j\ni  (2) \nThe microwave field, \nmwˆjthe1h , is linear ly polarized and directed along the multilayer \nnormal, \n1ˆˆ||hn , while the external static magnetic field lies in the film plane,  \n0ˆˆhn , making an angle h \nwith the system’s easy axis \nû . A linear response of the system relates to small angle deviations from the \nequilibrium, \n2ˆ ˆ ˆ ˆ ,    1 1 2 1 1 2 2mδm , m δm δm m , δm m . The  complex vectors \njte12δm , δm  \ncan be found from a linear 4 ×4 system by Eqs. (2) linearized near the equilibrium . This system is too \ncomplicated for an analytical treatment but easily solved numerically using  a standard  desktop computer. \nA certain simplification can be achieved using spherical coordinates. The microwave absorption is \nproportional to the imaginary part of the microwave susceptibility in the direction of the  microwave field : \n \n 1 s1 2 s2 212\n1 2 Cu mw 1 2ˆˆ\nIm()d M d M dd\nd d d h d d        1 1 1δm h δm h . (3) \nTo treat the volume microwa ve susceptibility of a SV, the metallic  spacer width, dCu, was added in Eq. \n(3). Then a full cycle of calculations in each simulation consists of : i) finding the equilibrium orientation \nof the magnetic moments by the minimization of Eq. (1) ; ii) numerical solution of Eq. (2) linearized near \nthe equilibrium; iii) combining the obtained precession amplitudes in the volume susceptibility by Eq. (3) . \nThe separate susceptibility of each layer can be obtained if the thickness of the other layer is set to zero at \nthe last step of calculations. This can be useful in the analysis of experimental data obtained by the \nelement -specific X -ray magnetic circular dichroism, time -resolved Kerr microscopy and other techniques \nallowing to separately measure the microwave responses  of the layers [22, 27, 28]. 6 \n The m agnetic parameters in our simulations were set in accordance  to the experiment. In the studied \nsamples , the in -plane effective fields of the free and fixed layers are several times lower than the \nresonance field of the free layer ( Hres > 600 Oe), whose FMR linewidth will be the main discussion  issue  \nin the present paper. This implies that at the free layer’s resonance conditions the magnetic fi eld almost \naligns both magnetic moments. Thus, the dynamic exchange via spin currents will be considered in the \ncollinear regime , and the parameter AFNF is assumed to be independent of the in -plane magnetic field \norientation.  \n \n4. General features of the FM R in both SV series  \nThe d ynamic s of two coupled FM layers can be described in terms of acoustic and optical modes, a \nhybridized response of the system to the exciting microwave field. These modes are the in-phase and out -\nof-phase mutual precession of the magnetic moments in the FM layers.  The acoustic mode bears averaged \nmagnetic parameters of the system, while the optical one gives information about the system’s \nasymmetry. The interlayer coupling shifts the optical mode away from the acoustic one, therefor e, the \ncoupling strength can be determined if the other effective fields in the system are known. However, this is \na strong coupling regime which has few similarities with the FMR of standard SV multilayers, including \nthe samples used in this study, where the effective inte rlayer coupling does not exceed  several tens of \nOersted.   \n \n-1.5 -1.0 -0.5 0.0 0.5 1.0 1.5 2.0fixed layerfree layerspin valve\n  '', arb. units\nHext, kOeh= 0o,\nEic=10-2erg/cm2,\nhmw= 1 Oe,\n/2 = 9.67 GHz.\n \nFig. 1.  FMR spectrum calculated for a SV in the weak coupling regime (top curve). The m iddle and \nbottom curves show separated responses from the free and fixed layers in the SV. The layer  parameters \ncorrespond to the  first series of SVs: d1 = 5.5nm, α1 = 0.012, Ms1 = 1155 emu/cm3, K1 = 5.7103 \nerg/cm3; d2 = 2.5 nm, α2 = 0.055, Ms2 = 1175 emu/cm3, K2 = 1.7104 erg/cm3, Eex = 0.094 erg/cm2 and \nEic = 0.01 erg/cm2. \n 7 \n The samples  under study are in a weak coupling regime provided by Néel’s “orange -peel” \nmagnetostatic interaction [25]. The determined effective interlayer coupling field, acting from one layer to \nanother, is in the 10 to 30 Oe range for both layers [24] in all sample s of the two series. The main \ninteraction effect is a constant decrease of the resonance field in each laye r. This and other  related effects \nare thoroughly discussed in Ref.  [24].  \nTo support the ideology of the weak coupling regime, a simulation of the microwave response has \nbeen done using a parameter  set for the first series and the interlayer coupling stren gth Eic = 0.01 erg/cm2. \nThe spin -pump  / spin-sink mechanism was switched off: αsp1 = αsp2 = 0. Fig. 1 shows a typical microwave \nabsorption spectrum of a SV multilayer and respective separated responses  of each layer in it. The \nmagnetic moments are precessing almost independently , and therefore each peak can be associated with \nthe precession of the magnetization in a specific layer . The a symmetry of the thicknesses, damping  \nparameter s and magnetizations is clearly manifested  in these spectra. A fixed layer  with half the thickness \nof the free one  is much easier dragged by the precessing free layer. However , the inverse effect, a drag of \nthe free layer by the reson ance precession in the fixe d layer, is not so pronounced: only a small \nasymm etry on the wings of the free layer peak is observed. A four times strong er damping, mainly that \ndue to the contact with an antiferromagnet [24], produces a much lower precession amplitude of the fixed \nlayer. The situation gets even worse because the free layer is twice  as thick as the fixed one, t hus, the \neffective interlayer coupling field, acting on the free layer from the preces sing fixed layer, is about two \ntimes  lower. Leaping ahead, it is evid ent that the spin -pump  / spin-torque effect will be more pronounced \nin the free layer . \nA very important feature is that, despite the almost independent precession of the layers, an optical -\nlike and acoustic -like behavior is still present in the dynamics. A precessing layer drags the magnetization \nof the other layer either in the “in -phase” or in the “out -of-phase” regime . For the case of \nferromagnetically coupled layers, the optical mode (an out -of-phase mutual precession) has , in a given \nmagnetic field,  a higher precession frequen cy than the acoustic mode (an in -phase mutual precession). \nTherefore , the optical mode will be observe d, at a given microwave frequency,  in lower r esonance fields. \nA specifics of the first sample series is that, for the parallel and antiparallel orientatio ns of Hext (φh = 0º \nand 180 º), the resonance field of the fixed layer is respectively lower (~  300 Oe) or higher (~  1000 Oe) \nthan th at of the free layer (~  700 Oe in both cases). As seen from Fig. 1 , this brings  about  an interesting \nbehavior: the precession of the free layer in the parallel Hext (Hext > 0, φh = 0º) drags the fixed layer “in -\nphase”, while in the antiparallel orientation ( Hext < 0, φh = 0º) it drags the fixed layer “out -of-phase”, i.e. \nin the optical mode.   \nIt is evident that the switching between the acoustic and optical “drag” regimes would disappear with \nthe fixed layer resonance peak being  below that of the free layer. This justifies our choice of the sample \nseries: a variation of the interlayer coupling in the first series sho uld influence the intensity of the dragged 8 \n precession, while varying the effective magnetization of the free layer in the second series will tune the \nresonance field of the free layer with respect to that of the fixed one.  \nFig. 2  shows the evolution of the angular dependences of the resonance field in both series . The \ngeneral properties of the samples are as follow s. The effective field of unidirectional anisotropy for the \nfixed layer is about 300 Oe , and it is the main in -plane anisotropic contribution he re. The free layer has a \nweak in -plane unidirectional anisotropy of 5 to 20 Oe, var ying with the NiFe/CoFe composition. The \nmagnetic parameters of the free layer are less fluctuating than those of the fixed one since the former  is \nthicker and always deposi ted on the same surface. The i ncreased roughness of the fixed layer also \nstrongly influe nces the AF /FM interface, giving rise to fluctuations not only of the fixed layer ’s effective \nmagnetization but also of the exchange bias coupling. It is hard  as well  to prepare reference samples for \nthe fixed layer . Our previous investigation ha s shown that a separately deposited fixed layer has \nconsiderably different magnetic parameters [24]. The s trong angular variation of the resonance field and \nthe direct contact wi th the AF has also a strong  influence  on the angular dependence of the linewidth even \nin a separately deposited fixed layer. Moreover, as the linewidth is extracted using the least -squares \nmethod, the accuracy of the fitti ng for the low -intensity peak stemming from the fixed layer will be m uch \nlower than for the free layer . Due to  these reasons and the asymmetry discussed above, the following  \ndiscussion of the experimental results is mostly focused on the linewidth, ∆Hfr, of the free -layer -related  \npeak and on its angular dependence, ∆Hfr(φh). \n \n0 60 120 1800,40,60,81,0\nfixed layer's peaks\n  Hres, kOe\nh, deg. dCu= 28 A\n dCu= 24 A\n dCu= 21 A\n dCu= 17 Afree layer's peaks\n   \n0 60 120 1800,20,40,60,81,0\nfree layer's \npeaks\nfixed layer's\n peaks\n  Hres, kOe\nh, deg.Ni80Fe20/Co80Fe20:\n 48 A /   8A\n 40 A / 16A\n 32 A / 24A\n 16 A / 40A  \nFig. 2.  Angular dependences of the FMR peaks for the free and fixed layer: the first series where the \ninterlayer coupling strength is varied by gradual ly changing the metal  spacer thickness dCu (left panel); \nthe second series where the mean FMR  field of the free layer  is varied by gradual ly changing the free \nlayer effective magnetization, Ms1 (right panel).  \n \n5. Analysis of angular dependences  9 \n Additional reference samples  which completely duplicate the free layer and the next nearest \nnonmagnetic layers in each SV sample  have been grown and used as a reference in the analysis of the \nangular dependences of the free layer FMR linewidth, ∆Hfr(φh). It has been found that < ∆Hfr> (averaged \nover the whole φh range) of each reference sample is at least 20% lower than < ∆Hfr> in the corresponding \nSV sample. However, the increased damping in the presence of a second FM layer (i.e. fixed layer) \ncannot be uniquely associ ated with the spin -pump  / spin -sink mechanism [5, 26], because a non -zero \ninterlayer coupling causes a hybridization of the resonance modes. Though the layers are weakly  coupled, \neach layer’s resonance mode bears a small portion of the magnetic behavior of the layer coupled to it. As \nthe free layer’s damping parameter is several times lower than the fixed -layer -related  one, the observed \nFMR line broadening in the SV can have both origins , and it demands a quantitative analysis. At the s ame \ntime, the shape of the ∆Hfr(φh) dependence in the SV samples deserves  additional attention .  \n \n0 60 120 18064728088\n  \nHfr, Oe\nh, deg. dCu=28A,\n dCu=24A,\n dCu=21A,\n dCu=17A.  \nreference sample\n   \n16 18 20 22 24 26 28481216Relative step height ,  %\ndCu, Å0.011 0.0088 0.0076 0.0064 0.0052 0.01 Eic, erg/cm2\n0.004\n72747678808284\n<Hfr >, Oe  \nFig. 3.  Linewidth of the free layer in the first sample series.  Left panel: The angular dependence for \ndifferent copper spacer thicknesses. The reference sample  curve  does not show step s. Right panel: The \nrelative step height  and mean linewidth versus the interlayer coupling strength . \n \nFig. 3  shows experimental results obtained on the first series of s amples, where the interlayer \ncoupling has been gradually tuned by changing the copper spacer thickness. The reference layer does not \nshow any noticeable ∆Hfr(φh) dependence . In contrast, a step -like shape of  the ∆Hfr(φh) dependence has \nbeen observed in all  SVs. A noticeable growth of ∆Hfr is observed for the antiparal lel orientation of the \nmagnetic field (90 º < φh < 270 º). The transition from the weaker  damped to the stronger  damped regime is \nquite smooth and occurs within the angular range where the fixed layer peak crosses the free layer’s one \n(see Fig. 2 ). The relative step height in the ∆Hfr(φh) dependence has been found to decrease w ith \nincreasing  copper spacer  thickness , dCu. In other words, with decreasing interlayer coupling, assumed to \nbe the only parameter influenc ing the free layer in this series, the observed step height  also decreases. As \nseen from Fig. 3 , the relative step height monotonously decreases from 12% to  4% with decreasing \ninterlayer coupling. It should be noted that, among the other extracted SV parameters analyzed as a 10 \n function of dCu, this one has the smoothest  dependence. As an example, we show the thickness \ndependence of < ∆Hfr> averaged over the whole [0,  360º] range of angles ( Fig. 3 ). Though the scattering \nof experimental points is several times higher, this parameter also shows a tendency to decrease, whose \nnature is hard to identify at present. A degree of resonance modes hybridization, weaken ing with \ndecreasing interlayer coupling, seems to be the most probable source of this effect.  The free layer \nresonance precession drags the magnetic moment of the fixed layer , and this could be itself an additional \nsource of increased linewidth. A more det ailed discussion o f a simultaneous influence of hybridization \nand spin -pump  / spin-sink effects on the linewidth will be given in the next Section.  \n0 60 120 180556065707580859095\nNi80Fe20/Co80Fe20:  48 A /   8A,  40 A / 16A,\n 32 A / 24A,  24 A / 32A,  16 A / 40A.\n  \nH, Oe\nh, Deg.reference samples\n \nFig. 4.  Angular dependences of the linewidth for the free layer in the second sa mple series and in the \nrespective reference samples.  \nIn the second SV series , an increase of the < ∆Hfr> parameter in comparison with the reference layers \nis also clearly seen (see Fig. 4 ). At the same time, the observed step -like ∆Hfr(φh) dependence has \nrevealed  additional features. The step from the weaker  damped to strong er damped regime is shifted to \nhigher angles as the mean resonance field of the free layer get s higher. The observed shift completely \nmatches that of the crossing angle, i.e., the angle  where the resonances of the free and fixed layers \ncoincide (see Fig. 2 ). The most important feature is the absence of step -like behavior in  the ∆Hfr(φh) \ndependence for the sample with  the Ni80Fe20(48 Å) / Co 80Fe20(8 Å) free layer. Fig. 2 shows that the \nresonances are not crossing there at all: the free layer’s resonance field is always higher than the  fixed \nlayer’s one.  \nAs compared to the first series, there are also additional peculiarities in the ∆Hfr(φh) dependences, \ndistorting the step -like shape. Th ey, however, are linked to the intrinsic  angular dependence of ∆Hfr of a \nconcrete free layer. An analysis of the reference samples shows that the increase of the Co 80Fe20 / Ni80Fe20 \nthickness ratio causes a noticeable increase in the angular variation of ∆Hfr. Also a considerable variation \nof the damping parameter is observed in the reference samples , however , of a nonsystematic character . \nThese intrinsic features, as seen from Fig. 4 , are conserved also in the SV samples.  \n 11 \n  \nThus, the observed experimental results can be res umed as follows. When the fixed layer resonance \nfield is higher than the free layer’s one, the linewidth of the free layer  peak, ∆H fr, get s larger. The \nrespective angular dependence,  ∆H fr(φh), shows a step -like shape with the threshold an gular position \ncorresponding to the crossing region of the free and fixed layer resonances. The step height decreas es \nwith decreas ing interlayer coupling strength. This effect is absent in the reference samples containing \nonly the free layer , as well as it  disappears in the SVs where the resonances of the free and fixed layers do \nnot cross.  \n \n6. Hybridization versus non -local damping  \nTo clarify the interpretation of the experiment, a series of in -plane FMR spectra w ere simulated as a \nfunction of the in -plane magnetic field direction φh employing the formalism described in Sec. 3 . The \nsimulated spectr a display the resonance peaks by the free and fixed layer (as shown, e.g., in Fig. 1 ). By \nfitting a set of overlapping Lor entzians to the simulated spectrum, the resonance peaks’ parameters were \ndeduced. Then the angular dependence of the linewidth of the free layer, ∆Hfr(φh), was analyzed. For the \nfirst sample series, the layer  parameters and coupling were determined in our previous work  [24] on \nexactly the same samples. For the second series, these parameters were chosen to reproduce the \nexperiment as close as possible , and the interlayer coupling was fixed to Eic = 0.01 erg/cm2 in all SV s. \nFluctuating parameters of the fixe d layer and a slight variation of the internal damping of the free layer \n0 60 120 18066697281848790\n00 - SP\nIC - SP\nIC - 00\n  Hfr, Oe\nh, deg.00 - 00 \nFig. 5.  Simulated angular dependences of the free layer’s FMR linewidth. Four different regimes are \nshown: “00 -00”: Eic=  0 and \nFNFA  = 0; “IC -00”: Eic = 0.01 erg/cm2 and \nFNFA  = 10; “00 -SP”: Eic = 0 \nand \nFNFA  = 1.11015 cm‒2; “IC -SP”: Eic = 0.01 erg/cm2 and \nFNFA  = 1.11015 cm‒2. The layer  \nparameters refer to the first series of SVs, as they are already listed in the caption to Fig. 1 . 12 \n noted in the experiment were ignored in the simulation. In both series, the effective spin -mixing \nconductance for the whole FM/NM/FM structure is assumed  to be \nFNFA  = 1.11015 cm2 (which is \nslightly lower than in case of a single Co/Cu interface ~ 1.41015 cm2 [26]), in units of e2/h. \nRelative contributions of the hybridization and spin -pump  / spin-sink effects to the linewidth of a \nweakly coupled SV system are the central  object of th e present  investigation. Referring to a SV from the \nfirst series, we have done four different simulations (see Fig. 5 ) of the ∆Hfr(φh) dependences. First , both \nthe interlayer coupling (IC) and the spin mixing conductivity (SP) were set to zero (the “00 -00” curve). \nThis has demonstrated that the fitting procedure  correctly extracts the linewidth , and the free layer’s ∆Hfr \ndoes not depend on the peaks separa tion between the free and fixed layers (when fully uncoupled). It has \nbeen found that a small increase of ∆Hfr is observed in the crossing region. This increase, however, is \nlower than 0.3%, thus being at least one order of magnitude lower than the other f actors relevant for the \n∆Hfr(φh) dependence , both in the experiment and simulation. Therefore , this factor was ignored in the \nabove experimental data and will be omitted  in the further  consideration s.  \nThe next simulation has been made with Eic = 0.01 erg/ cm2 and \nFNFA  = 0 (the “IC -00” curve). In this \ncase, a noticeable increase (~  7%) in ∆Hfr is observed in the crossing region. This effect can be only \nattributed to an enhanced hybridization of the resonance peaks in th is region. When increasing the \nlinewidth of the free layer peak, the hybridizat ion also makes the fixed layer  peak narrower. The \ndependence of the hybridization degree on the distance between  the resonance peaks is also responsible \nfor the fact that  the ∆Hfr value for the antiparallel orientation ( φh = 180 º) is slightly higher ( by ~ 1.3%) \nthan that for the parallel orientation ( φh = 0º). As seen from Fig. 2 , the resonance peaks are indeed closer \nto each other in the antiparallel orientation . It is worth noti ng that the shape of the ∆Hfr(φh) dependence is \nquite different from the exp erimentally observed step -like profil e.  \nA pure spin -pump / spin -sink regime has been set in the next simulation, i.e. with Eic = 0 and \nFNFA  = \n1.11015 cm‒2. The corresponding ∆Hfr(φh) dependence is labeled “00 -SP”. In comparison with the \npreviously discussed regime, ∆Hfr is depressed (by ~  2%) in the crossing region. This effect was observed \nexperimentally in a FM/NM/FM system and has been interpreted as a p artial compensation of the spin \ncurrent leakage which occurs when  both FM layers are in resonance precession [5] and thus emit the spin \ncurrents. Without discussing t his in details, we note only two points: i) due to the considerably  thicker \nFM layers in our SVs , the observed effect is much weaker than in the above mentioned paper  [5]. Since \nthe spin torque effect is of interfacial origin, its influence scales with the  inverse layer thickness; ii) the \nspin-pump  / spin-sink and hybridi zation effect s work in the opposite  senses  in the crossing region.  \n \n 13 \n \n0.000 0.013 0.0266080100120\n0FNFA\n    h=00\n   h=1800\n  \nHfr, Oe\nEic, erg/cm2\n15 21.1 10 cmFNFA  \nFig. 6.  Linewidth of the free layer in the parallel and antiparallel orientation versus the interlayer \ncoupling strength simulated through spin conductiv ity (and without it). The layer  parameters are set \nfor the first series of SVs, as they are already listed in the caption to Fig. 1 . \nThe last simulation, labeled “IC -SP”, shows a simultaneous action of the interlayer coupling and sp in-\npump / spin -sink effect,  i.e. Eic = 0.01 erg/cm2 and \nFNFA  = 1.11015 cm‒2. As seen from Fig. 5 , there is a \ngood agreement with the experiment. The step size in the ∆Hfr(φh) dependence is ~  8%, also very close to \nthe experimental values. In the parallel orientation ( φh = 0º), the ∆Hfr value  is almost the same as in the \ncrossing region for the case of the pure spin -pump  / spin-sink effect. This means that a partial \ncompensation of the spin current leakage takes place in the whole range of angles for the acoustic regime \nof precession ( ‒90º < φh < 90º). On the contrary , in t he optical regime ( ‒110º > φh > 110 º) the free layer \nsuffers additional damping, absent in the previously discussed “00 -SP” simulation. The explanation is as \nfollow s. The p recession can be  geometrically separated in a transversal and a longitudinal component of \nmagnetization with respect to its equilibrium orientation. The c onservation of angular momentum allows \nthe same separation for the generated spin current. For a small -angle precession, the transversal \ncomponent of magnetization (  sin(θprec)) is larger than the longitudinal one ( sin2(θprec/2)). The \ntransversal part varies in time, while the longitudinal does not (at least in the linear response \napproximation , neglecting, e. g., a possible nutation). The i mportance of the time -dependent  transversal \npart of the spin current has been recently sho wn in Ref.  [29]. Both components are transferred by the spin \ncurrent from one FM layer to the other. In the acoustic precession mode (a s well as in the crossing point \nfor the “00 -SP” case), the transversal component of the spin current from the second layer is in -phase \nwith the transversal part of that from the first layer. Therefore , the spin current absorbed at the interface \nshould act in an “anti -damping” manne r. On the contrary, in the optical pr ecession regi me the transversal \ncomponent of the absorbed spin current is out -of-phase with the magnetic moment precession , and \ntherefore an extra damping occurs. An increase of the non-local damping in the optical precession regime \nin a magnetically  coupled FM/NM/FM trilayer has been predicted by Kim in Ref.  [19]. Probably this \neffect was observed in several  papers [20-22]. However , its interpretation in these papers fully ignores the \nhybridization of resonance modes , and therefore it is hard to draw  some clear  conclusions.  14 \n The weak interlayer coupling and an almost symmetri cal position of the free layer  peak with respect \nto the fixed one in the first SV series play an important role  in the non -local damping effect. Fig. 6  shows \nthe calculated ∆Hfr parameter versus the interlayer coupling strength for φh = 0º and φh = 180 º, with and \nwithout spin -pump  / spin-sink effect. It is seen that, for Eic < 0.013 erg/cm2, the increase of ∆Hfr occurs  \nmerely due to the non-local damping effect, while for a stronger coupling  the hybridization takes a \ncomparable role , and the se two contributions are hardly separable in a real experiment . From this \nsimulation  it is also seen that the dynamic exchang e via spin currents is quite different in the optical and \nacoustic precession modes . The i ncrease of ∆Hfr due to increasing hybridization is suppressed in the \nacoustic mode ( φh = 0º) by “anti -damping”, i.e., in-phase interaction between the transversal components \nof magnetization and the absorbed spin current. On the  contrary, in the  optical  precession  mode  (φh = \n180º) the effect of non -local damping is considerably enhanced, as the transversal components of the \nprecessing magnetization and of the absorbed spin current are out -of-phase.  \n \n0 60 120 1807580859095\n2\n15 2\nFNF0.01erg/cm ,\n1.1 10 cm .icE\nA \n\nMs1 = 1600 emu/cm3Ms1 = 1200 emu/cm3Ms1 = 1000 emu/cm3Ms1 = 800 emu/cm3 \n \nHfr, Oe\nh, deg.Ms1 = 700 emu/cm3\n   \n0 60 120 180666870727476\n  \nHfr, Oe\nh, deg. Ms1 = 700 emu/cm3\n Ms1 = 800 emu/cm3\n Ms1 = 1000 emu/cm3\n Ms1 = 1200 emu/cm3\n Ms1 = 1600 emu/cm3\n2\nFNF0.01erg/cm ,\n0.icE\nA\n  \nFig. 7.  Angular behavior of the linewidth in the second series of SVs, with a gradual variation of the \neffective magnetization of the free layer , simulated considering the  spin conductivity and without it. For \nthe red and black curves, the fixed layer resonance does  not cross that of the free layer  anymore . The \nparameters set is the same as for the first se ries and with Ms2 = 1525 emu/cm3 and Eex = 0.12 erg/cm2. \n \nA simulation of the ∆Hfr(φh) dependence in the second series, where the effective magnetization of \nthe free layer,  Ms1, is gradually changed, complete s the discussion. A comparison of the simulation ( Fig. \n7) with the experiment ( Fig. 2 , right panel) allows one to conclude that the effects of non -local  damping \nare also clearly seen here. First, when the free layer ’s saturation magnetization is such low that the fixed \nlayer pe ak does not cros s the free layer resonance, and therefor e, the precessing free layer drags the fixed \nlayer always in -phase (ac oustic mode), a characteristic step -like feature in the ∆Hfr(φh) dependence \ndisappears. I n these regime, the calculated ∆Hfr(φh) dependences are fundament ally different , \nirrespectively of whether the spi n conductivity exist s in the system or not. For the case of \nFNFA  = 0, the 15 \n fixed layer  peak approaching the free layer  one at φh = 180 º induces an enhanced hybridization , and ∆Hfr \ngrow s, while for \nFNFA  = 1.11015 cm‒2 the enhanced hybridization is fully suppressed by the described \nabove “anti -damping” feature of the acoustical mode of precession in the presence of spin conductivity. A \ndecrease of ∆Hfr is observed when the fixed layer peak is approaching. T he closer i s the fixed layer  \nresonance  to the free  layer  one, the higher is the precession amplitude in the fixed layer , and thus the \nhigher is the generated spin current. Therefore , a decrease of ∆Hfr is observed. Another distinct feature of \nthe non -local damping is  a continuous growth of the low -angle part of the ∆Hfr(φh) dependence (which \ncorresponds to the acoustical  precession mode) with decreas ing Ms1. As Ms1 decreases, all effective fields \narising from the interface , as well as the spin torque emerging from the absorbed spin current , will \nincrease. For the case of zero spin conductivity, the low -angle part of the ∆Hfr(φh) dependence remains \nalways the same. Both these features are clearly seen in  the experiment ( Fig. 2 , right panel ). \n \n7. Conclusions  \nIn-plane angular dependences of the free layer’s FMR linewidth have been studied in two series of \nspin-valve  multilayers , where the free and fixed layers are weakly coupled by N éel’s “orange peel” \nmagneto static interaction. In the first series, the interlayer coupling strength was  varied by changing the \nmetal  spacer thickness, while in the second series the in -plane resonance field of the free layer was tuned \nby changing the Ni 80Fe20/Co 80Fe20 thickness rat io.  \nThe main experimental results are as follow s. The a ngular dependence of the linewidth of the free \nlayer displays a characteristic step -like feature. When the resonance field of the fixed layer is higher than \nthat of the free layer , the damping increase s. The transition from the weakly  damped to strongly  damped \nregime occurs in the angular region of the peaks crossing. The reference  samples, containing only a free \nlayer and an adjacent nonmagnetic layer, do not show such a behavior. Similarly, no step is observed in \nthe samples from the second series , where the fixed layer  peak does not cross that of the free layer at all. \nThe step size decreases with decreasing interlayer coupling strength .  \nA comparison with simulation s has shown th at the observed effect is due to the non -local damping \neffect. In the weakly coupled regime, the hybridization of the resonance peaks is low , and each peak can \nbe attributed to the resonance precession of a particular layer. At the same time, due to a non -zero \nmagnetic coupling, the resonant precession in one layer induces a small correlated precession (“drag”) in \nthe other one. Depending on the relative fi eld position of the free layer  resonance peak with respect to the \nfixed one, the fixed layer  magnetic moment is “dragged” either in the acoustic -like (“in -phase” precession \nin both layers) or optical -like (“out -of-phase” mutual precession) regime. Therefore, varying the in -plane \nangle between the external magnetic field and the exchange bias field  and chan ging in this way the \nrelative peaks field position , one can switch between these two regimes. In case of ballistic regime of spin \ntransport, a dditionally to the time -independent longitudinal component, the spin current generated by the 16 \n dragged fixed layer has a time -varying transversal component, being “in-phase” or “out -of-phase” with \nthe time -varying transversal component of the free layer’s precessing magnetization. The resulting spin -\ntorque effect on the free layer will be either of “anti -damping” or “e xtra-damping” type, experimentally \nobservable as an additional increase/decrease of the linewidth in the antiparallel/parallel orientation. It is \nworth noting  that the acoustic regime is in a full analogy to the case of a magnetically  uncoupled \nFM1/NM/FM2 system [5] when the resonances coincide.  Another important point is  that diffusive regime \nof the spin transport will suppress the above described effects  due to averaging of transversal  components \nof the spin currents . \nOur study has also shown that the hybridization effect on the linewidth is of the same magnitude as \nthe non -local damping effect in the case of weak interlayer coupling , and that the hybridization fully \ndominates in the case of strongly coupled magnetic layer s. A separation of these two contributions, \nhowever, is possible due to  their different angula r behavior. In general case, contribution of the \nhybridization to the linewidth parameter will be dependent on degree of asymmetry of layers.  Thus, one \ncan expec t that, if the free and fixed layers would have the same damping , the influence of the \nhybridization would be considerably suppressed.  \n \nAcknowledgements  \nThis work was partially supported by the FCT of Portugal through the projects PEst/CTM/LA0025/2011, \nRECI/FIS -NAN/0183/2012, PTDC /CTM -NAN/112672/2009,  PTDC/FIS/120055/2010 , and grants \nSFRH/BPD/74086/2010 (A.A.T.)  and IF/00981/2013 (G.N.K.) as well as by the Euro pean FP7 project \n“Mold -Nanonet” .17 \n References  \n1 S. Maekawa, S. O. Valenzuela, E. Saitoh, and T. Kimura, Spin Current  (OUP Oxford, 2012).  \n2 E. Y. Tsymbal and I. Zutic, Handbook of Spin Transport and Magnetism  (Taylor & Francis, \n2011).  \n3 S. Mizukami, Y. Ando, and T. Miyazaki, Japanese Journal of Applied P hysics 40, 580 (2001).  \n4 R. Urban, G. Woltersdorf, and B. Heinrich, Physical Review Letters 87, 217204 (2001).  \n5 B. Heinrich, Y. Tserkovnyak, G. Woltersdorf, A. Brataas, R. Urban, and G. Bauer, Physical \nReview Letters 90, 187601 (2003).  \n6 K. Uchida, S. Takahashi, K. Harii, J. Ieda, W. Koshibae, K. Ando, S. Maekawa, and E. Saitoh, \nNature 455, 778 (2008).  \n7 Y. K. Kato, R. C. Myers, A. C. Gossard, and D. D. Awschalom, Science 306, 1910 (2004).  \n8 J. Wunderlich, B. Kaestner, J. Sinova, and T. Ju ngwirth, Physical Review Letters 94, 047204 \n(2005).  \n9 K. Uchida, H. Adachi, T. An, T. Ota, M. Toda, B. Hillebrands, S. Maekawa, and E. Saitoh, \nNature materials 10, 737 (2011).  \n10 A. A. Bakun, B. P. Zakharchenya, A. A. Rogachev, M. N. Tkachuk, and V. G. Fle ǐsher, JETP \nLetters 40, 464 (1984).  \n11 E. Saitoh, M. Ueda, H. Miyajima, and G. Tatara, Applied Physics Letters 88, 182509 (2006).  \n12 S. O. Valenzuela and M. Tinkham, Nature 442, 176 (2006).  \n13 L. Liu, O. J. Lee, T. J. Gudmundsen, D. C. Ralph, and R. A. Buh rman, Physical Review Letters \n109, 096602 (2012).  \n14 M. Buhl, A. Erbe, J. Grebing, S. Wintz, J. Raabe, and J. Fassbender, Scientific reports 3, 2945 \n(2013).  \n15 V. E. Demidov, S. Urazhdin, H. Ulrichs, V. Tiberkevich, A. Slavin, D. Baither, G. Schmitz, and \nS. O. Demokritov, Nature materials 11, 1028 (2012).  \n16 A. Brataas, Y. Tserkovnyak, G. Bauer, and B. Halperin, Physical Review B 66 (2002).  \n17 M. Weiler, et al., Physical Review Letters 111, 176601 (2013).  \n18 G. Woltersdorf, O. Mosendz, B. Heinrich, and C. Back, Physical Review Letters 99, 246603 \n(2007).  \n19 J.-V. Kim and C. Chappert, Journal of Magnetism and Magnetic Materials 286, 56 (2005).  \n20 X. Joyeux, T. Devolder, J. V. Kim, Y. G. de la Torre, S. Eimer, an d C. Chappert, J Appl Phys \n110, 063915 (2011).  \n21  . Salikhov,  .  brudan,  .  r ssing, S.  uschhorn, M. Ewerlin,  . Mishra,  .  adu, I.  . \nGarifullin, and H. Zabel, Applied Physics Letters 99, 092509 (2011).  \n22 R. Salikhov, R. Abrudan, F. Brüssing, K. Gr oss, C. Luo, K. Westerholt, H. Zabel, F. Radu, and I. \nA. Garifullin, Physical Review B 86, 144422 (2012).  \n23 A. A. Timopheev, N. A. Sobolev, Y. G. Pogorelov, S. A. Bunyaev, J. M. Teixeira, S. Cardoso, P. \nP. Freitas, and G. N. Kakazei, J Appl Phys 113, 17D7 13 (2013).  \n24 A. A. Timopheev, N. A. Sobolev, Y. G. Pogorelov, A. V. Talalaevskij, J. M. Teixeira, S. \nCardoso, P. P. Freitas, and G. N. Kakazei, J Appl Phys 114, 023906 (2013).  \n25 L. Nèel, Compt. Rend. 255, 1676 (1962).  \n26 M. Zwierzycki, Y. Tserkovnyak, P. Kelly, A. Brataas, and G. Bauer, Physical Review B 71, \n064420 (2005).  \n27 M. K. Marcham, et al., Physical Review B 87, 180403 (2013).  \n28 O. Mosendz, G. Woltersdorf, B. Kardasz, B. Heinrich, and C. Back, Physical Review B 79, \n224412 (2009).  \n29 H. Jiao and G. E. W. Bauer, Physical Review Letters 110, 217602 (2013).  \n \n \n "    },    {        "title": "1401.3396v1.Damping_of_Terahertz_Plasmons_in_Graphene_Coupled_with_Surface_Plasmons_in_Heavily_Doped_Substrate.pdf",        "content": "arXiv:1401.3396v1  [cond-mat.mes-hall]  15 Jan 2014Damping of Terahertz Plasmons in Graphene\nCoupled with Surface Plasmons in Heavily-Doped Substrate\nA. Satou1,2,∗Y. Koseki1, V. Ryzhii1,2, V. Vyurkov3, and T. Otsuji1,2\n1Research Institute of Electrical Communication, Tohoku Un iversity, Sendai 980-8577, Japan\n2CREST, Japan Science and Technology Agency, Tokyo 107-0075 , Japan\n3Institute of Physics and Technology, Russian Academy of Sci ences, Moscow 117218, Russia\nCoupling of plasmons in graphene at terahertz (THz) frequen cies with surface plasmons in a\nheavily-doped substrate is studied theoretically. We reve al that a huge scattering rate may com-\npletely damp out the plasmons, so that proper choices of mate rial and geometrical parameters are\nessential to suppress the coupling effect and to obtain the mi nimum damping rate in graphene.\nEven with the doping concentration 1019−1020cm−3and the thickness of the dielectric layer be-\ntween graphene and the substrate 100 nm, which are typical va lues in real graphene samples with\na heavily-doped substrate, the increase in the damping rate is not negligible in comparison with\nthe acoustic-phonon-limited damping rate. Dependence of t he damping rate on wavenumber, thick-\nnesses of graphene-to-substrate and gate-to-graphene sep aration, substrate doping concentration,\nand dielectric constants of surrounding materials are inve stigated. It is shown that the damping\nrate can be much reduced by the gate screening, which suppres ses the field spread of the graphene\nplasmons into the substrate.\nI. INTRODUCTION\nPlasmons in two-dimensional electron gases (2DEGs)\ncan be utilized for terahertz (THz) devices. THz\nsources and detectors based on compound semicon-\nductor heterostructures have been extensively investi-\ngated both experimentally and theoretically.1–8The two-\ndimensionality, which gives rise to the wavenumber-\ndependent frequency dispersion, and the high electron\nconcentration on the order of 1012cm−2allow us to have\ntheir frequency in the THz rangewith submicron channel\nlength. Most recently, a very high detector responsivity\noftheso-calledasymmetricdouble-grating-gatestructure\nbased on an InP-based high-electron-mobility transistor\nwas demonstrated.9However, resonant detection as well\nas single-frequency coherent emission have not been ac-\ncomplished so far at room temperature, mainly owning\nto the damping rate more than 1012s−1in compound\nsemiconductors.\nPlasmons in graphene have potential to surpass those\nin the heterostructures with 2DEGs based on the stan-\ndard semiconductors, due to its exceptional electronic\nproperties.10Massiveexperimental and theoretical works\nhave been done very recently on graphene plasmons in\nthe THz and infrared regions (see review papers Refs. 11\nand 12 and references therein). One of the most im-\nportant advantages of plasmons in graphene over those\nin heterostructure 2DEGs is much weaker damping rate\nclose to 1011s−1at room temperature in disorder-free\ngraphene suffered only from acoustic-phonon scatter-\ning.13That is very promising for the realization of the\nresonant THz detection14and also of plasma instabili-\nties, which can be utilized for the emission. In addition,\ninterbandpopulation inversioninthe THz rangewaspre-\ndicted,15,16andithasbeeninvestigatedfortheutilization\nnot only in THz lasers in the usual sense but also in THz\nactive plasmonic devices17,18and metamaterials.19Many experimental demonstrations of graphene-based\ndevices have been performed on graphene samples with\nheavily-doped substrates, in order to tune the carrier\nconcentration in graphene by the substrate as a back\ngate. Typically, either peeling or CVD graphene trans-\nferred onto a heavily-doped p+-Si substrate, with a SiO 2\ndielectric layer in between, is used (some experiments\non graphene plasmons have adapted undoped Si/SiO 2\nsubstrates20,21). Graphene-on-silicon, which is epitax-\nial graphene on doped Si substrates,22is also used.\nFor realization of THz plasmonic devices, properties of\nplasmons in such structures must be fully understood.\nAlthough the coupling of graphene plasmons to sur-\nface plasmons in perfectly conducting metallic substrates\nwith/without dielectric layers in between have been the-\noretically studied,23,24the influence of the carrier scat-\ntering in a heavily-doped semiconductor substrate (with\nfinite complex conductivity) has not been taken into ac-\ncount sofar. Since the scatteringratein the substratein-\ncreases as the doping concentration increases, it is antic-\nipated that the coupling of graphene plasmons to surface\nplasmonsin the heavily-dopedsubstrate causesundisired\nincrease in the damping rate.\nThe purpose of this paper is to study theoretically the\ncoupling between graphene plasmons and substrate sur-\nface plasmons in a structure with a heavily-doped sub-\nstrate and with/without a metallic top gate. The paper\nis organized as follows. In the Sec. II, we derive a dis-\npersion equation of the coupled modes of graphene plas-\nmons and substrate surface plasmons. In Sec. III, we\nstudy coupling effect in the ungated structure, especially\nthe increase in the plasmon damping rate due to the cou-\npling and its dependences on the doping concentration,\nthe thickness of graphene-to-substrate separation, and\nthe plasmon wavenumber. In Sec. IV, we show that the\ncoupling in the gated structures can be less effective due\nto the gate screening. We also compare the effect in\nstructures having different dielectric layers between the2\nFIG. 1. Schematic views of (a) an ungated graphene structure\nwith a heavily-doped Si substrate where the top surface is\nexposed on the air and (b) a gated graphene structure with a\nheavily-doped Si substrate and a metallic top gate.\ntop gate, graphene layer, and substrate, and reveal the\nimpact of values of their dielectric constants. In Sec. V,\nwe discuss and summarize the main results of this paper.\nII. EQUATIONS OF THE MODEL\nWe investigateplasmonsinanungatedgraphenestruc-\nture with a heavily-doped p+-Si substrate, where the\ngraphene layer is exposed on the air, as well as a gated\ngraphene structure with the substrate and a metallic top\ngate, whichareschematicallyshowninFigs.1(a)and(b),\nrespectively. The thickness of the substrate is assumed\nto be sufficiently larger than the skin depth of the sub-\nstrate surface plasmons. The top gate can be considered\nasperfectlyconducting metal, whereasthe heavily-doped\nSisubstrateischaracterizedbyitscomplexdielectriccon-\nstant.\nHere, we use the hydrodynamic equations to describe\nthe electronmotion in graphene,26while using the simple\nDrude model for the hole motion in the substrate (due\nto virtual independence of the effective mass in the sub-\nstrate on the electron density, in contrast to graphene).\nIn addition, these are accompanied by the self-consistent\n2D Poisson equation (The formulation used here almost\nfollows that for compound semiconductor high-electron-\nmobility transistors, see Ref. 25). Differences are the\nhydrodynamic equations accounting for the linear dis-\npersion of graphene and material parameters of the sub-\nstrate and dielectric layers. In general, the existence of\nboth electrons and holes in graphene results in various\nmodes such as electrically passive electron-hole sound\nwaves in intrinsic graphene as well as in huge damping\nof electrically active modes due to the electron-hole fric-\ntion, as discussed in Ref. 26. Here, we focus on the case\nwhere the electron concentration is much higher than thehole concentration and therefore the damping associated\nwith the friction can be negligibly small. Besides, for the\ngeneralization purpose, we formulate the plasmon dis-\npersion equation for the gated structure; that for the un-\ngated structure can be readily found by taking the limit\nWt→ ∞(see Fig. 1).\nThen, assuming the solutions of the form exp( ikx−\niωt), where k= 2π/λandωare the plasmon wavenum-\nber and frequency ( λdenotes the wavelength), the 2D\nPoisson equation coupled with the linearized hydrody-\nnamic equations can be expressed as follows:\n∂2ϕω\n∂z2−k2ϕω=−8πe2Σe\n3meǫk2\nω2+iνeω−1\n2(vFk)2ϕωδ(z),\n(1)\nwhereϕωis the ac (signal) component of the potential,\nΣe,me, andνeare the steady-state electron concentra-\ntion, the hydrodynamic “fictitious mass”, and the col-\nlision frequency in graphene, respectively, and ǫis the\ndielectric constant which is different in different layers.\nThe electron concentration and fictitious mass are re-\nlated to each other through the electron Fermi level, µe,\nand electron temperature, Te:\nΣe=/integraldisplay∞\n02ε\nπ/planckover2pi12v2\nF/bracketleftbigg\n1+exp/parenleftbiggε−µe\nkBTe/parenrightbigg/bracketrightbigg−1\ndε,(2)\nme=1\nv2\nFΣe/integraldisplay∞\n02ε2\nπ/planckover2pi12v2\nF/bracketleftbigg\n1+exp(ε−µe\nkBTe)/bracketrightbigg−1\n.(3)\nIn the following we fix Teand treat the fictitious mass as\na function of Σ e. The dielectric constant can be repre-\nsented as\nǫ=\n\nǫt, 0< z < W t,\nǫb, −Wb< z <0,\nǫs[1−Ω2\ns/ω(ω+iνs)], z <−Wb,(4)\nwhereǫt,ǫb, andǫsare thestaticdielectric constants of\nthe top and bottom dielectric layers and the substrate,\nrespectively, Ω s=/radicalbig\n4πe2Ns/mhǫsis the bulk plasma\nfrequency in the substrate with Nsandmhbeing the\ndoping concentration and hole effective mass, and νsis\nthe collision frequency in the substrate, which depends\non the doping concentration. The dielectric constant in\nthe substrate is a sum of the static dielectric constant of\nSi,ǫs= 11.7 and the contribution from the Drude con-\nductivity. The dependence of the collision frequency, νs,\non the doping concentration, Ns, is calculated from the\nexperimental data for the hole mobility at room temper-\nature in Ref. 27.\nWe use the following boundary conditions: vanish-\ning potential at the gate and far below the substrate,\nϕω|z=Wt= 0 and ϕω|z=−∞= 0; continuity conditions\nof the potential at interfaces between different layers,\nϕω|z=+0=ϕω|z=−0andϕω|z=−Wb+0=ϕω|z=−Wb−0;\na continuity condition of the electric flux density at\nthe interface between the bottom dielectric layer and3\nthe substrate in the z-direction, ǫb∂ϕω/∂z|z=−Wb+0=\nǫs∂ϕω/∂z|z=−Wb−0; and a jump of the electric flux den-\nsity at the graphene layer, which can be derived from\nEq. (1). Equation (1) together with these boundary con-\nditions yield the following dispersion equation\nFgr(ω)Fsub(ω) =Ac, (5)\nwhere\nFgr(ω) =ω2+iνeω−1\n2(vFk)2−Ω2\ngr,(6)\nFsub(ω) =ω(ω+iνs)−Ω2\nsub, (7)\nAc=ǫ2\nb(H2\nb−1)\n(ǫbHb+ǫtHt)(ǫs+ǫbHb)Ω2\ngrΩ2\nsub,(8)\nΩgr=/radicalBigg\n8πe2Σek\n3meǫgr(k), ǫgr(k) =ǫtHt+ǫbǫb+ǫsHb\nǫs+ǫbHb,(9)\nΩsub=/radicalBigg\n4πe2Ns\nmhǫsub(k), ǫsub(k) =ǫs+ǫbǫb+ǫtHtHb\nǫbHb+ǫtHt,\n(10)\nandHb,t= cothkWb,t. In Eq. (5), the term Acon the\nright-handsiderepresentsthecouplingbetweengraphene\nplasmons and substrate surface plasmons. If Acwere\nzero, the equations Fgr(ω) = 0 and Fsub(ω) = 0 would\ngive independent dispersion relations for the former and\nlatter, respectively. Qualitatively, Eq. (8) indicates that\nthe coupling occurs unless kWb≫1 orkWt≪1,\ni.e., unless the separation of the graphene channel and\nthe substrate is sufficiently large or the gate screening\nof graphene plasmons is effective. Note that the non-\nconstant frequency dispersion of the substrate surface\nplasmon in Eq. (10) is due to the gate screening, which\nis similar to that in the structure with two parallel metal\nelectrodes.28Equation (5) yields two modes which have\ndominantpotentialdistributionsnearthegraphenechan-\nnel and inside the substrate, respectively. Hereafter, we\nfocus on the oscillating mode primarily in the graphene\nchannel; we call it “channel mode”, whereas we call the\nother mode “substrate mode”.\nIII. UNGATED PLASMONS\nFirst, we study plasmons in the ungated structure.\nHere, the temperature, electron concentration, and col-\nlision frequency in graphene are fixed to Te= 300 K,\nΣe= 1012cm−2, andνe= 3×1011s−1. With these val-\nues of the temperature and concentration the fictitious\nmass is equal to 0 .0427m0, wherem0is the electron rest\nmass. The value of the collision frequency is typical to\nthe acoustic-phonon scattering at room temperature.13\nAs for the structural parameters, we set ǫt= 1 and\nWt→ ∞, and we assume an SiO 2bottom dielectric layer\nwithǫb= 4.5. Then Eq. (5) is solved numerically.(a)\n(b) 0 1 2 3 4 5 6 7\n1017101810191020Damping rate, 1012 s-1\nDoping concentration, cm-3 0 1 2 3 4 5 6 7\n1017101810191020Damping rate, 1012 s-1\nDoping concentration, cm-3 0 1 2 3 4 5 6 7\n1017101810191020Damping rate, 1012 s-1\nDoping concentration, cm-3 0 1 2 3 4 5 6 7\n1017101810191020Damping rate, 1012 s-1\nDoping concentration, cm-3 0 1 2 3 4 5 6 7\n1017101810191020Damping rate, 1012 s-1\nDoping concentration, cm-3 0 1 2 3 4 5 6 7\n1017101810191020Damping rate, 1012 s-1\nDoping concentration, cm-30.150.200.250.30\n10191020\n 0 0.2 0.4 0.6 0.8 1 1.2\n1017101810191020Frequency, THz\nDoping concentration, cm-3Wb = 50, 100, 200, \n \t \t \t \t 300, 400 nm\nFIG. 2. Dependencesof(a) theplasmon dampingrate and(b)\nfrequencyonthesubstrate dopingconcentration, Ns, withthe\nplasmon wavenumber k= 14×103cm−1(the wavelength λ=\n4.5µm)and with differentthicknesses of thebottom dielectric\nlayer,Wb, in the ungated graphene structure. The inset in (a)\nshows the damping rate in the range Ns= 1019−1020cm−3\n(in linear scale).\nFigures 2(a) and (b) show the dependences of the plas-\nmondamping rateandfrequencyon the substratedoping\nconcentrationwiththeplasmonwavenumber k= 14×103\ncm−1(i.e., the wavelength λ= 4.5µm) and with differ-\nent thicknesses of the bottom dielectric layer, Wb. The\nvalue of the plasmon wavelength is chosen so that it gives\nthe frequency around 1 THz in the limit Ns→0. They\nclearlydemonstratethatthereisahugeresonantincrease\nin the damping rate at around Ns= 3×1017cm−3as\nwell as a drop of the frequency. This is the manifestation\nofthe resonantcoupling ofthe grapheneplasmonand the\nsubstratesurfaceplasmon. The resonancecorrespondsto\nthe situation where the frequencies of graphene plasmons\nand substrate surface plasmons coincide, in other words,\nwhere the exponentially decaying tail of electric field of\ngraphene plasmons resonantly excite the substrate sur-\nface plasmons.\nAt the resonance, the damping rate becomes larger\nthan 1012s−1, over 10 times larger than the contribu-\ntion from the acoustic-phonon scattering in graphene,\nνe/2 = 1.5×1011s−1. For structures with Wb= 504\nand 100 nm, even the damping rate is so large that the\nfrequency is dropped down to zero; this corresponds to\nan overdamped mode. It is seen in Figs. 2(a) and (b)\nthat the coupling effect becomes weak as the thickness\nof the bottom dielectric layer increases. The coupling\nstrength at the resonance is determined by the ratio of\nthe electric fields at the graphene layer and at the inter-\nface between the bottom dielectric layer and substrate.\nIn the case of the ungated structure with a relatively low\ndoping concentration, it is roughly equal to exp( −kWb).\nSinceλ= 4.5µm is much larger than the thicknesses of\nthebottomdielectriclayerinthestructuresunderconsid-\neration, i.e., kWb≪1, the damping rate and frequency\nin Figs. 2(a) and (b) exhibit the rather slow dependences\non the thickness.\nAway from the resonance, we have several nontrivial\nfeatures in the concentration dependence of the damp-\ning rate. On the lower side of the doping concentra-\ntion, the damping rate increase does not vanish until\nNs= 1014−1015cm−3. This comes from the wider field\nspread of the channel mode into the substrate due to\nthe ineffective screening by the low-concentration holes.\nOn the higher side, one can also see a rather broad\nlinewidthoftheresonancewithrespecttothedopingcon-\ncentration, owning to the large, concentration-dependent\ndamping rate of the substrate surface plasmons, and a\ncontribution to the damping rate is not negligible even\nwhen the doping concentration is increased two-orders-\nof-magnitude higher. In fact, with Ns= 1019cm−3,\nthe damping rate is still twice larger than the contribu-\ntion from the acoustic-phonon scattering. The inset in\nFig. 2(a) indicates that the doping concentration must\nbe at least larger than Ns= 1020cm−3for the cou-\npling effect to be smaller than the contribution from\nthe acoustic-phonon scattering, although the latter is\nstill nonnegligible. It is also seen from the inset that,\nwith very high doping concentration, the damping rate\nis almost insensitive to Wb. This originates from the\nscreeningbythe substratethat stronglyexpandsthe field\nspread into the bottom dielectric layer.\nAs for the dependence of the frequency, it tends to a\nlower value in the limit Ns→ ∞than that in the limit\nNs→0, as seen in Figure 2(b), along with the larger de-\npendence on the thickness Wb. This corresponds to the\ntransition of the channel mode from an ungated plasmon\nmode to a gated plasmon mode, where the substrate ef-\nfectively acts as a back gate.\nTo illustrate the coupling effect with various frequen-\nciesinthe THzrange, dependencesoftheplasmondamp-\ning rate and frequency on the substrate doping concen-\ntration and plasmon wavenumber with Wb= 300 nm\nare plotted in Figs. 3(a) and (b). In Fig. 3(a), the peak\nof the damping rate shifts to the higher doping concen-\ntration as the wavenumber increases, whereas its value\ndecreases. The first feature can be understood from\nthe matching condition of the wavenumber-dependent\nfrequency of the ungated graphene plasmons and the\ndoping-concentration-dependent frequency of the sub-\n(a)\n(b) 0.2 0.4 0.6 0.8 1 1.2 1.4\n 5 10 15 20 25 30 35Ns = 2 x 1018 cm-3\n5 x 1018 cm-3\n1 x 1019 cm-3\nFIG. 3. Dependencesof(a) theplasmon dampingrate and(b)\nfrequency on the substrate doping concentration, Ns, and the\nplasmon wavenumber, k, with different the thickness of the\nbottom dielectric layer Wb= 300 nm in the ungated graphene\nstructure. The inset of(a) shows thewavenumberdependence\nof the damping rate with certain doping concentrations. The\nregion with the damping rate below 0 .2×1012s−1is filled\nwith white in (a).\nstrate surface plasmons, i.e., Ω gr∝k1/2, roughlly speak-\ning, and Ω sub∝N1/2\ns. The second feature originates\nfrom the exponential decay factor, exp( −kWb), of the\nelectricfield ofthe channelmode at the interfacebetween\nthe bottom dielectric layer and the substrate; since the\ndoping concentrationis /lessorsimilar1018cm−3at the resonancefor\nany wavevector in Fig. 3, the exponential decay is valid.\nAlso, with a fixed doping concentration, say Na>1019\ncm−3, the damping rate has a maximum at a certain\nwavenumber, resulting from the first feature (see the in-\nset in Fig. 3(a)).\nIV. GATED PLASMONS\nNext, we study plasmons in the gated structures.\nWe consider the same electron concentration, ficticious\nmass, and collision frequency, Σ e= 1012cm−2,me=\n0.0427m0, andνe= 3×1011s−1, as the previous sec-\ntion. As examples of materials for top/bottom dielectric5\n(a)\n(b)0.150.250.350.450.550.65\n1017101810191020Damping rate, 1012 s-1\nDoping concentration, cm-30.150.250.350.450.550.65\n1017101810191020Damping rate, 1012 s-1\nDoping concentration, cm-30.150.250.350.450.550.65\n1017101810191020Damping rate, 1012 s-1\nDoping concentration, cm-30.150.250.350.450.550.65\n1017101810191020Damping rate, 1012 s-1\nDoping concentration, cm-30.150.250.350.450.550.65\n1017101810191020Damping rate, 1012 s-1\nDoping concentration, cm-30.150.200.250.30\n10191020\n0.951.001.051.10\n1017101810191020Frequency, THz\nDoping concentration, cm-3Wb = 50, 100, 200, \n \t \t \t \t 300, 400 nm0.150.200.250.300.35\n1017101810191020Damping rate, 1012 s-1\nDoping concentration, cm-30.150.200.250.300.35\n1017101810191020Damping rate, 1012 s-1\nDoping concentration, cm-30.150.200.250.300.35\n1017101810191020Damping rate, 1012 s-1\nDoping concentration, cm-30.150.200.250.300.35\n1017101810191020Damping rate, 1012 s-1\nDoping concentration, cm-30.150.200.250.300.35\n1017101810191020Damping rate, 1012 s-1\nDoping concentration, cm-30.150.20\n10191020\n0.780.790.800.810.820.830.840.85\n1017101810191020Frequency, THz\nDoping concentration, cm-3Wb = 50, 100, 200, \n \t \t \t \t 300, 400 nm\nFIG. 4. Dependences of (a) the plasmon damping rate and\n(b)frequency on thesubstrate doping concentration, Ns, with\nthe plasmon wavelength λ= 1.7µm (the wavenumber k=\n37×103cm−1), with thicknesses of the Al 2O3top dielectric\nlayerWt= 20 and 40 nm (left and right panels, respectively),\nand with different thicknesses of the SiO 2bottom dielectric\nlayer,Wb, in the gated graphene structure. The insets in (a)\nshow the damping rate in the range Ns= 1019−1020cm−3\n(in linear scale).\nFIG. 5. Dependence of the plasmon damping rate on the sub-\nstrate doping concentration, Ns, and the plasmon wavenum-\nber,k, with different the thicknesses of the Al 2O3top dielec-\ntric layer Wt= 20 nm and the SiO 2bottom dielectric layer\nWb= 50 nm in thegated graphene structure. The region with\nthe damping rate below 0 .2×1012s−1is filled with white.\nlayers, we examine Al 2O3/SiO2and diamond-like carbon\n(DLC)/3C-SiC. These materials choices not only reflect\nthe realistic combination of dielectric materials available\ntoday, but also demonstrate two distinct situations for\nthe coupling effect under consideration, where ǫt> ǫbfor\nthe former and ǫt< ǫbfor the latter.Figures 4(a) and (b) show the dependences of the plas-\nmondamping rateandfrequencyon the substratedoping\nconcentration with the wavenumber k= 37×103cm−1\n(the plasmon wavelength λ= 1.7µm), with thicknesses\nof the Al 2O3top dielectric layer Wt= 20 and 40 nm, and\nwith different thicknesses of the SiO 2bottom dielectric\nlayer,Wb. As seen, the resonant peaks in the damping\nrate as well as the frequency drop due to the coupling\neffect appear, although the peak values are substantially\nsmaller than those in the ungated structure (cf. Fig. 2).\nThe peak value decreases rapidly as the thickness of the\nbottom dielectric layerincreases; it almostvanishes when\nWb≥300 nm. These reflect the fact that in the gated\nstructuretheelectricfieldofthechannelmodeisconfined\ndominantly in the top dielectric layer due to the gate\nscreening effect. The field only weakly spreads into the\nbottom dielectric layer, where its characteristic length is\nroughtly proportional to Wt, rather than the wavelength\nλas in the ungated structure. Thus, the coupling effect\non the damping rate together with on the frequency van-\nishesquicklyas Wbincreases, evenwhen the wavenumber\nis small and kWb≪1. More quantitatively, the effect is\nnegligible when the first factor of Acgiven in Eq. (8) in\nthe limit kWb≪1 andkWt≪1,\nǫ2\nb(H2\nb−1)\n(ǫbHb+ǫtHt)(ǫs+ǫbHb)≃1\n1+(Wb/ǫb)/(Wt/ǫt)(11)\nis small, i.e., when the factor ( Wb/ǫb)/(Wt/ǫt) is much\nlarger than unity. A rather strong dependence of the\ndamping rate on Wbcan be also seen with high doping\nconcentration, in the insets of Fig. (4)(a).\nFigure 5 shows the dependence of the plasmon damp-\ning rate on the substrate doping concentration and plas-\nmon wavenumber, with dielectric layer thicknesses Wt=\n20 andWb= 50 nm. As compared with the case of\nthe ungated structure (Fig. 3(a)), the peak of the damp-\ning rate exhibits a different wavenumber dependence;\nit shows a broad maximum at a certain wavenumber\n(around 150 ×103cm−1in Fig. 5) unlike the case of\nthe ungatedstructure, where the resonantpeakdecreases\nmonotonically as increasing the wavenumber. This can\nbe explained by the screening effect of the substrate\nagainst that of the top gate. When the wavenumber is\nsmall and the doping concentration corresponding to the\nresonance is low, the field created by the channel mode\nis mainly screened by the gate and the field is weakly\nspread into the bottom direction. As the doping concen-\ntration increases (with increase in the wavevector which\ngivesthe resonance),thesubstratebeginstoactasaback\ngateand the field spreadsmoreinto the bottom dielectric\nlayer,sothatthecouplingeffectbecomesstronger. When\nthe wavenumberbecomessolargethat kWb≪1doesnot\nhold, the field spread is no longer governed dominantly\nby the substrate or gate screening, i.e., the channel mode\nbegins to be “ungated” by the substrate. Eventually,\nthe coupling effect on the damping rate again becomes\nweak, with the decay of the field being proportional to\nexp(−kWb).6\n0.00.51.01.52.02.5\n1017101810191020Damping rate, 1012 s-1\nDoping concentration, cm-30.00.51.01.52.02.5\n1017101810191020Damping rate, 1012 s-1\nDoping concentration, cm-30.00.51.01.52.02.5\n1017101810191020Damping rate, 1012 s-1\nDoping concentration, cm-30.00.51.01.52.02.5\n1017101810191020Damping rate, 1012 s-1\nDoping concentration, cm-30.00.51.01.52.02.5\n1017101810191020Damping rate, 1012 s-1\nDoping concentration, cm-30.150.200.250.30\n10191020\n0.600.700.800.901.001.10\n1017101810191020Frequency, THz\nDoping concentration, cm-3Wb = 50, 100, 200, \n \t \t \t \t 300, 400 nm0.00.20.40.60.81.01.2\n1017101810191020Damping rate, 1012 s-1\nDoping concentration, cm-30.00.20.40.60.81.01.2\n1017101810191020Damping rate, 1012 s-1\nDoping concentration, cm-30.00.20.40.60.81.01.2\n1017101810191020Damping rate, 1012 s-1\nDoping concentration, cm-30.00.20.40.60.81.01.2\n1017101810191020Damping rate, 1012 s-1\nDoping concentration, cm-30.00.20.40.60.81.01.2\n1017101810191020Damping rate, 1012 s-1\nDoping concentration, cm-30.150.20\n10191020\n0.500.550.600.650.700.750.80\n1017101810191020Frequency, THz\nDoping concentration, cm-3Wb = 50, 100, 200, \n \t \t \t \t 300, 400 nm(a)\n(b)\nFIG. 6. The same as Figs. 4(a) and (b) but with the DLC\ntop and 3C-SiC bottom dielectric layer.\nAs illustrated in Eq. (11), the coupling effect in\nthe gated strcture is characterized by the factor\n(Wb/ǫb)/(Wt/ǫt) when the conditions kWb≪1 and\nkWt≪1 are met. This means that not only the thick-\nnesses ofthe dielectric layersbut alsotheir dielectric con-\nstants are very important parameters to determine the\ncoupling strength. For example, if we adapt the high-k\nmaterial,e.g.,HfO 2inthetopdielectriclayer,itresultsin\nthe more effective gate screening than in the gated struc-\nture with the Al 2O3top dielectric layer, so that the cou-\npling effect can be suppressed even with the same layer\nthicknesses. The structure with the DLC top and 3C-SiC\nbottom dielectric layers (with ǫt= 3.129andǫb= 9.7)\ncorresponds to the quite opposite situation, where the\ngate screening becomes weak and the substrate screening\nbecomes more effective, so that stronger coupling effect\nis anticipated. Figures 6(a), (b), and 7 show the same\ndependences as in Figs 4(a), (b), and 5, respectively, for\nthe structure with the DLC top and 3C-SiC bottom di-\nelectric layers. Comparing with those for the structure\nwith the Al 2O3top and SiO 2bottom dielectric layers,\nthe damping rate as well as the frequency are more in-\nfluenced by the coupling effect in the entire ranges of the\ndoping concentration and wavevector. In particular, the\nincrease in the damping rate with high doping concen-\ntrationNs= 1019−1020cm−3and the thickness of the\nbottom layer Wb= 50−100 nm, which are typical val-\nues in real graphene samples, is much larger. However,\nthis increase can be avoided by adapting thicker bottom\nlayer, say, Wb/greaterorsimilar200 nm or by increasing the doping\nconcentration.\nFIG. 7. The same as Fig. 5 but with the DLC top and 3C-SiC\nbottom dielectric layer.\nV. CONCLUSIONS\nIn summary, we studied theoretically the coupling of\nplasmons in graphene at THz frequencies with surface\nplasmonsinaheavily-dopedsubstrate. Wedemonstrated\nthat in the ungated graphene structure there is a huge\nresonant increase in the damping rate of the “channel\nmode” at a certain doping concentration of the substrate\n(∼1017cm−2) and the increase can be more than 1012\ns−1, due to the resonant coupling of the graphene plas-\nmon and the substrate surface plasmon. The depen-\ndences of the damping rate on the doping concentration,\nthe thickness of the bottom dielectric layer, and the plas-\nmon wavenumber are associated with the field spread of\nthe channel mode into the bottom dielectric layer and\ninto the substrate. We revealed that even with very high\ndoping concentration (1019−1020cm−2), away from the\nresonance, the coupling effect causes nonnegligible in-\ncrease in the damping rate compared with the acoustic-\nphonon-limited damping rate. In the gated graphene\nstructure, the coupling effect can be much reduced com-\npared with that in the ungated structure, reflecting the\nfact that the field is confined dominantly in the top di-\nelectric layer due to the gate screening. However, with\nvery high doping concentration, it was shown that the\nscreening by the substrate effectively spreads the field\ninto the bottom dielectric layer and the increase in the\ndamping rate can be nonnegligible. These results suggest\nthat the structural parameters such as the thicknesses\nand dielectric constants of the top and bottom dielectric\nlayers must be properly chosen for the THz plasmonic\ndevices in order to reduce the coupling effect.\nACKNOWLEDGMENTS\nAuthorsthankM.SuemitsuandS.Sanbonsugeforpro-\nviding information about the graphene-on-silicon struc-\nture and Y. Takakuwa, M. Yang, H. Hayashi, and T. Eto7\nfor providing information about the diamond-like-carbon\ndielectric layer. This work was supported by JSPSGrant-in-Aid for Young Scientists (B) (#23760300), by\nJSPS Grant-in-Aid for Specially Promoted Research\n(#23000008), and by JST-CREST.\n∗a-satou@riec.tohoku.ac.jp\n1M. Dyakonov and M. Shur, Phys. Rev. Lett. 71, 2465\n(1993).\n2M. Dyakonov and M. Shur, IEEE Trans. Electron Devices\n43, 380 (1996).\n3M. S. Shur and J.-Q. L¨ u, IEEE Trans. Micro. Th. Tech.\n48, 750 (2000).\n4M. S. Shur and V. Ryzhii, Int. J. High Speed Electron.\nSys.13, 575 (2003).\n5F. Teppe, D. Veksler, A. P. Dmitriev, X. Xie, S. Rumyant-\nsev, W. Knap, and M. S. Shur, Appl. Phys. Lett. 87,\n022102 (2005).\n6T. Otsuji, Y. M. Meziani, T. Nishimura, T. Suemitsu, W.\nKnap, E. Sano, T. Asano, V. V. and Popov, J. Phys.:\nCondens. Matter 20, 384206 (2008).\n7V. Ryzhii, A. Satou, M. Ryzhii, T. Otsuji, and M. S. Shur,\nJ. Phys.: Condens. Matter 20, 384207 (2008).\n8V. V. Popov, D. V. Fateev, T. Otsuji, Y. M. Meziani, D.\nCoquillat, and W. Knap, Appl. Phys. Lett. 99, 243504\n(2011).\n9T. Watanabe, S. Boubanga Tombet, Y. Tanimoto, Y.\nWang, H. Minamide, H. Ito, D. Fateev, V. Popov, D. Co-\nquillat, W. Knap, Y. Meziani, and T. Otsuji, Solid-State\nElectron. 78, 109 (2012).\n10V. Ryzhii, Jpn. J. Appl. Phys. 45, L923 (2006); V. Ryzhii,\nA. Satou, and T. Otsuji, J. Appl. Phys. 101, 024509\n(2007).\n11T.Otsuji, S.A.BoubangaTombet, A.Satou, H.Fukidome,\nM. Suemitsu, E. Sano, V.Popov, M. Ryzhii, andV.Ryzhii,\nJ. Phys. D: Appl. Phys. 45, 303001 (2012).\n12A. N. Grigorenko, M. Polini, and K. S. Novoselov, Nat.\nPhoton. 6, 749 (2012).\n13E. Hwang and S. Das Sarma, Phys. Rev. B 77, 115449\n(2008).\n14V. Ryzhii, T. Otsuji, M. Ryzhii, and M. S. Shur, J. Phys.\nD: Appl. Phys. 45, 302001 (2012).15V. Ryzhii, M. Ryzhii, and T. Otsuji, J. Appl. Phys. 101,\n083114 (2007).\n16M. Ryzhii and V. Ryzhii, Jpn. J. Appl. Phys. 46, L151\n(2007).\n17A. A. Dubinov, V. Ya Aleshkin, V. Mitin, T. Otsuji, and\nV. Ryzhii, J. Phys.: Condens. Matter 23, 145302 (2011).\n18V. Popov, O. Polischuk, A. Davoyan, V. Ryzhii, T. Otsuji,\nand M. Shur, Phys. Rev. B 86, 195437 (2012).\n19Y. Takatsuka, K. Takahagi, E. Sano, V. Ryzhii, and T.\nOtsuji, J. Appl. Phys. 112, 033103 (2012).\n20L. Ju, B. Geng, J. Horng, C. Girit, M. Martin, Z. Hao, H.\nBechtel, X. Liang, A. Zettl, Y. R. Shen, and F. Wang, Nat.\nNanotechnol. 6, 630 (2011).\n21J. H. Strait, P. Nene, W.-M. Chan, C. Manolatou, S. Ti-\nwari, F. Rana, J. W. Kevek, and P. L. Mceuen, Phys. Rev.\nB87, 241410 (2013).\n22M. Suemitsu, Y. Miyamoto, H. Handa, and A. Konno, e-J.\nSurf. Sci. Nanotech. 7, 311 (2009).\n23N. Horing, Phys. Rev. B 80, 193401 (2009).\n24J. Yan, K. Thygesen, and K. Jacobsen, Phys. Rev. Lett.\n106, 146803 (2011).\n25A. Satou, V. Vyurkov, and I. Khmyrova, Jpn. J. Appl.\nPhys.43, 566 (2004).\n26D. Svintsov, V. Vyurkov, S. Yurchenko, T.Otsuji, and V.\nRyzhii, J. Appl. Phys. 111, 083715 (2012).\n27C. Bulucea, Solid-State Electron. 36, 489 (1993).\n28S. A. Maier, Plasmonics: Fundamentals and Applications\n(Springer Science, NY, 2007).\n29The dielectric constant of DLC varies in the range between\n3.1 and 7.8, depending on its growth condition.30Here, we\nchoose thelowest valuefor demonstration ofthecase where\nǫt≪ǫb.\n30H. Hayashi, S. Takabayashi, M. Yang, R. Jeˇ sko, S. Ogawa,\nT. Otsuji, and Y. Takakuwa, “Tuning of the dielec-\ntric constant of diamond-like carbon films synthesized by\nphotoemission-assisted plasma-enhanced CVD,” in 2013\nInternational Workshop on Dielectric Thin Films for Fu-\nture Electron Devices -Science and Technology- (2013\nIWDTF), Tokyo, Japan, November 7-9, 2013."    },    {        "title": "1401.6467v2.Wavenumber_dependent_Gilbert_damping_in_metallic_ferromagnets.pdf",        "content": "arXiv:1401.6467v2  [cond-mat.mtrl-sci]  24 Jan 2016Wavenumber-dependent Gilbert damping in metallic ferroma gnets\nY. Li and W. E. Bailey\nDept. of Applied Physics & Applied Mathematics,\nColumbia University, New York NY 10027, USA\n(Dated: November 21, 2021)\nA wavenumber-dependentdissipative term to magnetization dynamics, mirroring the conservative\nterm associated with exchange, has been proposed recently f or ferromagnetic metals. We present\nmeasurements ofwavenumber-( k-)dependentGilbert dampinginthree metallic ferromagnet s, NiFe,\nCo, and CoFeB, using perpendicular spin wave resonance up to 26 GHz. In the thinnest films\naccessible, where classical eddy-current damping is negli gible, size effects of Gilbert damping for the\nlowest and first excited modes support the existence of a k2term. The new term is clearly separable\nfrom interfacial damping typically attributed to spin pump ing. Higher-order modes in thicker films\ndo not show evidence of enhanced damping, attributed to a com plicating role of conductivity and\ninhomogeneous broadening. Our extracted magnitude of the k2term, ∆α∗\nkE= ∆α∗\n0+A∗\nkk2where\nA∗\nk=0.08-0.1 nm2in the three materials, is an order of magnitude lower than th at identified in prior\nexperiments on patterned elements.\nThe dynamical behavior of magnetization for ferro-\nmagnets (FMs) can be described by the Landau-Lifshitz-\nGilbert (LLG) equation[1]:\n˙ m=−µ0|γ|m×Heff+αm×˙ m (1)\nwhereµ0is the vacuum permeability, m=M/Msis the\nreduced magnetization unit vector, Heffis the effective\nmagnetic field, γis the gyromagnetic ratio, and αis the\nGilbert damping parameter. The LLG equation can be\nequivalently formulated, for small-angle motion, in terms\nof a single complex effective field along the equilibrium\ndirection, as ˜Heff=Heff-iαω/|γ|; damping torque is in-\ncluded in the imaginary part of ˜Heff.\nFor all novel spin-transport related terms to the LLG\nidentified so far[2–7], each real (conservative) effective\nfield term is mirrored by an imaginary (dissipative)\ncounterpart. In spin-transfer torque, there exist both\nconventional[2, 3] and field-like[8] terms in the dynamics.\nIn spin-orbit torques (spin Hall[4] and Rashba[6] effect)\ndampinglike and fieldlike components have been theoret-\nically predicted[9] and most terms have been experimen-\ntally identified[5, 6]. For pumped spin current[7], theory\npredictsrealandimaginaryspinmixingconductances[10]\ng↑↓\nrandg↑↓\niwhich introduce imaginary and real effective\nfields, respectively.\nIt is well known that the exchange interaction, respon-\nsible for ferromagnetism, contributes a real effective field\n(fieldlike torque) quadratic in wavenumber kfor spin\nwaves[11]. It isthennaturaltoaskwhetheracorrespond-\ning imaginary effective field might exist, contributing a\ndampinglike torque to spin waves. Theoretically such an\ninteraction has been predicted due to the intralayer spin-\ncurrent transport in a spin wave[12–15], reflected as an\nadditional term in Eq. (1):\n˙ m=···−(|γ|σ⊥/Ms)m×∇2˙ m (2)\nwhereσ⊥is the transverse spin conductivity. This term\nrepresents a continuum analog of the well-established in-terlayer spin pumping effect[7, 16, 17]. For spin wave\nresonance (SWR) with well-defined wavenumber k, Eq.\n(2) generates an additional Gilbert damping ∆ α(k) =\n(|γ|σ⊥/Ms)k2. In this context, Gilbert damping refers\nto an intrinsic relaxation mechanism in which the field-\nswept resonance linewidth is proportional to frequency.\nRemarkably, the possible existence of such a term has\nnot been addressed in prior SWR measurements. Previ-\nous studies of ferromagneticresonance (FMR) linewidths\nof spin waves[18–21] were typically operated at fixed fre-\nquency, not allowing separation of intrinsic (Gilbert) and\nextrinsic linewidths. Experiments have been carried out\non thick FM films, susceptible to a large eddy current\ndamping contribution[22]. Any wavenumber-dependent\nlinewidth broadening in these systems has been at-\ntributed to eddy currents or inhomogeneous broadening,\nnot intrinsic torques which appear in the LLG equation.\nIn this Manuscript, wepresent a study of wavenumber-\ndependent Gilbert damping in the commonly applied\nferromagnetic films Ni 79Fe21(Py), Co, and CoFeB. A\nbroad range of film thicknesses (25-200 nm) has been\nstudied in order to exclude eddy-current effects. We\nobserve a thickness-dependent difference in the Gilbert\ndamping for uniform and first excited spin wave modes\nwhich is explained well by the intralayer spin pump-\ning model[14]. Corrections for interfacial damping, or\nconventional spin pumping, have been applied and are\nfound to be small. The measurements show that the\nwavenumber-dependent damping, as identified in contin-\nuousfilms, isinreasonableagreementwith thetransverse\nspin relaxation lengths measured in Ref. [23], but an or-\nder of magnitude smaller than identified in experiments\non sub-micron patterned Py elements[24].\nTwo different types of thin-film heterostructures were\ninvestigated in this study. Films were deposited by\nUHV sputtering with conditions given in Ref. [23, 25].\nMultilayers with the structure Si/SiO 2(substrate)/Ta(5\nnm)/Cu(5 nm)/ FM(tFM)/Cu(5 nm)/Ta(5 nm), where2\nFM= Py, Co and CoFeB and tFM= 25-200 nm, were\ndesigned to separate the effects of eddy-current damp-\ning and the intralayer damping mechanism proposed in\nEq. (2). The minimum thickness investigated here is\nour detection threshold for the first SWR mode, 25 nm.\nA second type of heterostructure focused on much thin-\nner Py films, with the structure Si/SiO 2(substrate)/Ta(5\nnm)/Cu(5 nm)/Py( tPy)/Cu(5 nm)/ X(5 nm),tPy= 3-30\nnm. Here the cap layer X= Ta or SiO 2was changed,\nfor two series of this type, in order to isolate the effect\nof interfacial damping (spin pumping) from Cu/Ta inter-\nfaces.\nTo study the Gilbert damping behavior of finite-\nwavenumber spin waves in the samples, we have\nexcited perpendicular standing spin wave resonance\n(PSSWR)[26] using a coplanar waveguide from 3 to 26\nGHz. The spin-wave mode dispersion is given by the\nKittel equation ω(k)/|γ|=µ0(Hres−Ms+Hex(k)); the\neffective field from exchange, µ0Hex(k) = (2Aex/Ms)k2\nwithAexas the exchange stiffness, gives a precise mea-\nsurement of the wavenumber excited ((Fig. 1 inset)).\nPSSWR modes are indexed by the number of nodes p,\nwithk=pπ/tFMin the limit of unpinned surface spins.\nThe full-width half-maximum linewidth, ∆ H1/2, is fit-\nted using µ0∆H1/2(ω) =µ0∆H0+ 2αω/|γ|to extract\nthe Gilbert damping α. Forp= 1 modes we fix µ0∆H0\nas the values extracted from the corresponding p= 0\nmodes for ( tFM≤40 nm), because frequency ranges are\nreduced due to large exchange fields. In unconstrained\nfits for films of this thickness, the inhomogeneous broad-\neningµ0∆H0of thep= 1 modes does not exhibit a\ndiscernible trend with 1 /t2\nFM(ork2)[19–21], justifying\nthis approximation[27].\nTo fit our data, we have solved Maxwell’s equations\nand the LLG equation (Eq. 1), including novel torques\nsuch as those given in Eq. (2), according to the method\nof Rado[28]. The model (designated ’EM+LLG’) is de-\nscribed in the Supplemental Information. Values calcu-\nlated using the EM+LLG model are shown with curves\nin Fig. 1 and dashed lines in Fig. 4. Comparison with\nsuch a model has been necessary since in our first type of\nsample series, tFM= 25-200 nm, eddy-current damping\nis negligible for thinner films (25 nm), the Akk2contri-\nbution is negligible for thicker films (200 nm), but the\ntwo effects coexist for the intermediate region.\nIn Fig. 1(a-c)we comparethe measuredGilbert damp-\ning for the uniform ( p= 0,αu) and first excited ( p= 1,\nαs)spinwavemodes. Thedominantthickness-dependent\ncontribution to Gilbert damping of the uniform modes of\nPy, Co, and CoFeB is clearly due to eddy currents which\narequadraticinthickness. Notethateddy-currentdamp-\ning is negligible for the thinnest films investigated (25\nnm), but quite significant for the thickest films (200 nm).\nThis term sums with the bulk Gilbert damping α0[29].\nThe simulation of αu, shown by black curves in Fig. 1,\nmatches closely with the analytical expression for bulkand eddy-current damping only[30] of αu=αu0+αE0,\nwhereαE0=µ2\n0γMst2\nFM/12ρcdenotes the eddy-current\ndamping for uniform modes. Fittings of αuyield resis-\ntivitiesρc= 16.7, 26.4 and 36.4 µΩ·cm for Py, Co and\nCoFeB, respectively.\nUnlike the uniform-mode damping, the 1st SWR\n(×10 -3 )\nuu\ns\nu\ns(a) (b) \n(c) Co \nCoFeB Py μ0HB (T). . .p=0 p=1\nPy 75nm  μ 0Hex \nx50\ns\n(×10 -3 )\nkus-=\nFIG. 1. Thickness dependence of αuandαsfor (a) Py, (b)\nCo and (c) CoFeB thin films. Curves are calculated from\na combined solution of Maxwell’s equations and the LLG\n(EM+LLG). For αuthe values of µ0Ms,α(Table I), effective\nspin mixing conductance (Supplemental Information Sectio n\nC) g-factor (2.12 for Py and CoFeB and 2.15 for Co) and ρc\n(from analytical fitting) are used. For αsthe values of ∆ α∗\nkE\nand ∆α∗\nk0(Table I) are also included in the simulation. Inset:\n10 GHz FMR spectra of p= 0 and p= 1 modes in Py 75 nm\nfilm.\nmode damping αsis found to exhibit a minimum as a\nfunction of thickness. For decreasing thicknesses below\n75 nm,αsis increased. This behavior indicates an addi-\ntional source of Gilbert damping for the 1st SWR modes.\nIn CoFeB the increased αsis less visible in Fig. 1(c) due\nto fluctuations in damping for samples of different thick-\nness, but is evident in the difference, αs−αu, plotted in\nFig. 2.\nIn order to isolate this new damping mechanism, we\nplot in Fig. 2 the increased damping for the 1st SWR\nmode, ∆ αk=αs−αu, side-by-side with exchange field\nµ0Hexas a function of ( π/tFM)2taken as the wavenum-\nberk2. When π/tFMis large, a linear k2dependence\nof ∆αkin all three ferromagnets mirrors the linear de-\npendence of µ0Hexonk2. This parallel behavior reflects\nthe wavenumber-dependent imaginary and real effective\nfields acting on magnetization, respectively. To quantify\nthe quadratic wavenumber term in ∆ αk, we also show\nthe eddy-current-corrected values ∆ αkE=∆αk−∆αEin\nFig. 2(a). Here ∆ αE=αE1−αE0denotes the differ-\nence in eddy current damping between p= 1 and p= 0\nmodes according to the theory of Ref. [30], for weak sur-\nface pinning, where αE1≈0.23αE0(See Supplemental\nInformation for more details). We then fit this eddy-3(×10 -3 )\nPy \nCo \nCoFeB \nPy 150 nm (a)\n(b)\n(π/t FM )2 (×10 16  m -2 )Py \nCo \nCoFeB kk\nFIG. 2. Imaginary (damping, a) and real (exchange, b) ef-\nfective fields as a function of k2for Py, Co and CoFeB. (a)\nAdditional SWR damping ∆ αk(circle) and eddy-current cor-\nrected value ∆ αkE(cross) as a function of ( π/tFM)2. Solid\nlines are guides to eye and dashed lines are fits to Eq. (3). (b)\nExchange field µ0Hexas a function of ( π/tFM)2((pπ/tFM)2,\np=0-6, for Py 150 nm). Lines are fits to µ0Hex= (2A/Ms)k2.\ncurrent-corrected value to a linearization of Eq. (2), as:\n∆αkE= ∆αk0+Akk2(3)\nwithAk=|γ|σ⊥/Msand ∆αk0a constant offset. The\nvalues of Akestimated this way are 0 .128±0.022 nm2,\n0.100±0.011 nm2and 0.100±0.018 nm2for Py, Co and\nCoFeB.\nRecently, Kapelrud et al.[31] have predicted that\ninterface-localized (e.g. spin-pumping) damping terms\nwill also be increased in SWR, with interfacial terms\nforp≥1 modes a factor of two greater than those for\nthep= 0 mode. Using the second series of thinner\nPy films, we have applied corrections for the interfacial\nterm to our data, and find that these effects introduce\nonly a minor ( ∼20%) correction to the estimate of\nAk. Thep= 0 mode damping associated with the\nCu/Ta interface has been measured from the increase\nin damping upon replacement of SiO 2with Ta at the\ntop surface (Fig. 3, inset). Here Cu/SiO 2is taken as\na reference with zero interfacial damping; insulating\nlayers have been shown to have no spin pumping\ncontribution[32]. We find the damping enhancement to\nbe inversely proportional to tFM, indicating an interfa-\ncial damping term quantified as spin pumping into Ta[7]\nwith ∆αsp=γ¯h(g↑↓/S)/4πMstFM. Using the values\nin Table I yields the effective spin mixing conductance\nasg↑↓\nPy/Cu/Ta/S=2.5 nm−2, roughly a factor of three\nsmaller than that contributed by Cu/Pt interfaces[17].\nUsing the fitted g↑↓\nFM/Cu/Ta/S, we calculate andcorrect for the additional spin pumping contribution to\ndamping of the p= 1 mode, 2∆ αsp(from top and bot-\ntom interfaces). The corrected values for the 1st SWR\ndamping enhancement, ∆ α∗\nkE= ∆αkE−2∆αsp, are\nplotted for Py(25-200nm)in Fig. 3. These correctionsdo\nnot change the result significantly. We fit the k2depen-\ndence of ∆ α∗\nkEto Eq. (3) to extract the corrected values\nA∗\nkand ∆α∗\nk0. The fitted value, A∗\nk= 0.105±0.021 nm2\nfor Py, is slightly smaller than the uncorrected value\nAk. Other extracted interfacial-corrected values A∗\nkare\nlisted in Table I. Note that the correction of wavenumber\nby finite surface anisotropy will only introduce a small\ncorrection of AkandA∗\nkwithin errorbars. We also\nshow the EM+LLG numerical simulation results for\nthe uniform modes and the first SWR modes in Fig. 1\n(solid curves). Those curves coincide with the analytical\nexpressions of eddy-current damping plus k2damping\n(not shown) and fit the experimental data points nicely.\nThe negative offsets ∆ α∗\nk0between uniform modes\n(π/t FM )2 (×10 16  m -2 )(×10 -3 )*(= )\n**u0 u0 \ntFM  (nm)\nFIG. 3. Interfacial damping correction for Py. Main panel:\n∆αkEand ∆α∗\nkEas a function of ( π/tFM)2. Dashed lines\nare fits to k2-dependent equation as Eq. (3); ∆ α∗\nk0are ex-\ntracted from ∆ α∗\nkEfits.Inset:size effect of uniform-modes\nGilbert damping in Py/Cu/Ta and Py/Cu/SiO 2samples (cir-\ncles). The dashed curve is the theoretical reproduction of\nPy/Cu/SiO 2usingαu0+ ∆αsp(tFM). The shadow is the\nsame reproduction using αu0+ ∆αsp(tFM) +A∗\nkk2where\nthe error of shadow is from A∗\nk. Here kis determined by\nAexk2= 2Ks/tFM.\nand spin wave modes for Py and CoFeB are attributed\nto resistivitylike intrinsic damping[33]: because ˙ mis\naveraged through the whole film for uniform modes\nand maximized at the interfaces for unpinned boundary\ncondition, the SWR mode experiences a lower resistivity\nnear low-resistivity Cu and thus a reduced value of\ndamping. For Co a transition state between resistivity-\nlike and conductivitylike mechanisms[34] corresponds to\nnegligible ∆ α∗\nk0as observed in this work.\nIn addition to the thickness-dependent comparison of\np= 0 and p= 1 modes, we have also measured Gilbert4\ndamping for a series of higher-order modes in a thick\nPy (150 nm) film. Eddy-current damping ( αE∼0.003)\nis the dominant mode-dependent contribution in this\nfilm. The wavenumber kfor the mode p= 6 is roughly\nequal to that for the first SWR, p= 1, in the 25 nm\nfilm. Resonance positions are plotted with the dashed\nlines in Fig. 2(b), as a function of k, and are in good\nagreement with those found from the p= 1 data. In Fig.\n4 we plot the mode-related Gilbert damping αpup to\np= 6, which gradually decreases as pincreases. We have\nagain conducted full numerical simulations using the\nEM+LLG method with ( A∗\nk= 0.105 nm2) or without\n(A∗\nk= 0) the intralayer spin pumping term, shown in\nred and black crosses, respectively. Neither scenario fits\nthe data closely; an increase at p= 3 is closer to the\nmodel including the k2mechanism, but experimental α\natp= 6 falls well below either calculation.\nWe believe there are two possibilities why the α∝p2\ndamping term is not evident in this configuration. First,\nthe effective exchange field increases with p, resulting\nin a weaker (perpendicular) resonance field at the same\nfrequency. When the perpendicular biasing field at\nresonance is close to the saturation field, the spins\nnear the boundary are not fully saturated, which might\nproduce an inhomogeneous linewidth broadening at\nlower frequencies and mask small Gilbert contributions\nfrom wavenumber effect. From the data in Fig. 4 inset\nthe high- pSWR modes is more affected by this inhomo-\ngeneous broadening and complicate the extraction of k2\ndamping. Second, high- pmodes in thick films are close\nto the anomalous conductivity regime, kλM∼1, where\nλMis the electronic mean free path. The Rado-type\nmodel such as that applied in Fig. 4 is no longer valid\nin this limit[35], beyond which Gilbert damping has\nbeen shown to decrease significantly in Ni and Co[36].\nBased on published ρλMproducts for Py[37] and our\nexperimental value of ρc= 16.7µΩ·cm, we find λM∼8\nnm andkλM∼1 for the p= 6 mode in Py 150 nm. For\nthe 1st SWR mode in Py 25 nm, on the other hand,\neddy currents are negligible and the anomalous behavior\nis likely suppressed due to surface scattering, which\nreducesλM.\nAn important conclusion of our work is that the\nintralayer spin pumping, as measured classically through\nPSSWR, is indeed present but more than 10 times\nsmaller than estimated in single nanoscale ellipses[24].\nThe advantages of the PSSWR measurements presented\nin this manuscript are that the one-dimensional mode\nprofile is well-defined, two-magnon effects are reduced,\nif not absent[39], and there are no lithographic edges to\ncomplicate the analysis. The lower estimates of A∗\nkfrom\nPSSWR aresensible, basedonphysicalparametersofPy,\nCo, and CoFeB. The polarization of continuum-pumped\nspins in a nearly uniformly magnetized film, like that\nof pumped spin current in a parallel-magnetized F/N/F\nstructure, is transverse to the magnetization[14]. Fromthe measured transverse spin conductance σ⊥we extract\nthat the relaxation lengths of pumping intralayer spin\ncurrent are 0.8-1.9 nm for the three ferromagnets[27], in\ngood agreement with the small transverse spin coherence\nlengths found in these same ferromagneticmetals[23, 40].\nFinally, we show that the magnitude of the intralayer\nmT p=1, tFM =25-200 nm \np=0-6, t FM =150 nm \n(pπ/t FM )2 (×1016  m -2 )\np*\n*\nFIG.4. Mode-dependentdamping αpfor Py(150nm), 0 ≤p≤\n6. Crosses are EM+LLG calculated values with and without\nthe wavenumber-dependent damping term. Inset: Inhomoge-\nneous broadening ∆ H0vs 0≤p≤6, 150nm film. Larger,\nk-dependent values are evident, compared with those in the\nthickness series ( tFM=25-200 nm).\nspin pumping identified here is consistent with the\ndamping size effect notattributable to interlayer spin\npumping, in layers without obvious spin sinks. For the\np= 0 mode, a small but finite wavenumber is set by the\nsurface anisotropy through[30, 41] Aexk2= 2Ks/tFM.\nThe damping enhancement due to intralayer spin\npumping will, like the interlayer spin pumping, be\ninverse in thickness, leading to an ’interfacial’ term as\nα= 2Ks(A∗\nk/Aex)t−1\nFM. This contribution is indicated\nby the grey shadow in Fig. 3 insetand provides a\ngood account of the additional size effect in the SiO 2-\ncapped film. Here we use Ks=0.11 mJ/m2extracted\nby fitting the thickness-dependent magnetization to\nµ0Meff=µ0Ms−4Ks/MstFM. While alternate\ncontributions to the observed damping size effect for the\nSiO2-capped film cannot be ruled out, the data in Fig.\n3insetplace an upper bound on A∗\nk.\nIn summary, we have identified a wavenumber-\ndependent, Gilbert-type damping contribution to spin\nwaves in nearly uniformly magnetized, continuous\nfilms of the metallic ferromagnets Py, Co and CoFeB\nusing classical spin wave resonance. The term varies\nquadratically with wavenumber, ∆ α∼A∗\nkk2, with the\nmagnitude, A∗\nk∼0.08-0.10 nm2, amounting to ∼20% of\nthe bulk damping in the first excited mode of a 25 nm\nfilm of Py or Co, roughly an order of magnitude smaller\nthan previously identified in patterned elements. The\nmeasurements quantify this texture-related contribution5\nto magnetization dynamics in the limit of nearly homo-\ngeneous magnetization.\nµ0Ms(T)α0Aex(J/m) A∗\nk(nm2)∆α∗\n0\nPy 1.00 0.0073 1.2×10−110.11±0.02 -0.0008\nCo 1.47 0.0070 3.1×10−110.08±0.01 -0.0002\nCoFeB 1.53 0.0051 1.8×10−110.09±0.02 -0.0011\nTABLE I. Fit parameters extracted from resonance fields and\nlinewidths of uniform and 1st SWR modes. Values of A∗\nk\nand ∆α∗\n0for Co and CoFeB are calculated using the spin\nmixing conductances measured in FM/Cu/Pt[17]. See the\nSupplemental Material for details.\n[1] T. L. Gilbert, IEEE Trans. Magn. 40, 3443 (2004).\n[2] J. C. Slonczewski, J. Magn. Magn. Mater. 159, L1\n(1996).\n[3] L. Berger, Phys. Rev. B 54, 9353 (1996).\n[4] J. E. Hirsch, Phys. Rev. Lett. 83, 1834 (1999).\n[5] S. O. Valenzuela and M. Tinkham, Nature442, 176\n(2006).\n[6] I. M. Miron, G. Gaudin, S. Auffret, B. Rodmacq, A.\nSchuhl, S. Pizzini, J. Vogel and P. Gambardella, Nature\nMater.9, 230 (2010).\n[7] Y. Tserkovnyak, A. Brataas and G. E. W. Bauer, Phys.\nRev. Lett. 88, 117601 (2002).\n[8] S. Zhang, P. M. Levy and A. Fert, Phys. Rev. Lett. 88,\n236601 (2002).\n[9] P. M. Haney, H. W. Lee, K. J. Lee, A. Manchon and M.\nD. Stiles, Phys. Rev. B 87, 174411 (2013).\n[10] M. Zwierzycki, Y. Tserkovnyak, P. J. Kelly, A. Brataas\nand G. E. W. Bauer, Phys. Rev. B 71, 064420 (2005).\n[11] C. Kittel, Phys. Rev. 81, 869 (1951).\n[12] E. M. Hankiewicz, G. Vignale and Y. Tserkovnyak, Phys.\nRev. B78, 020404(R) (2008).\n[13] J. Foros, A. Brataas, Y. Tserkovnyak and G. E. W.\nBauer,Phys. Rev. B 78, 140402(R) (2008).\n[14] Y. Tserkovnyak, E. M. Hankiewicz and G. Vignale, Phys.\nRev. B79, 094415 (2009).[15] S. Zhang and Steven S.-L. Zhang, Phys. Rev. Lett. 102,\n086601 (2009).\n[16] S. Mizukami, Y. Ando, and T. Miyazaki, Phys. Rev. B\n66, 104413 (2002).\n[17] A. Ghosh, J. F. Sierra, S. Auffret, U. Ebels and W. E.\nBailey,Appl. Phys. Lett. 98, 052508 (2011).\n[18] P. E. Wigen, Phys. Rev. 133, A1557 (1964).\n[19] T. G. Phillips and H. M. Rosenberg, Phys. Lett. 8, 298\n(1964).\n[20] G. C. Bailey, J. Appl. Phys. 41, 5232 (1970).\n[21] F. Schreiber and Z. Frait, Phys. Rev. B 54, 6473 (1996).\n[22] P. Pincus, Phys. Rev. 118, 658 (1960).\n[23] A. Ghosh, S. Auffret, U. Ebels and W. E. Bailey, Phys.\nRev. Lett. 109, 127202 (2012).\n[24] H. T. Nembach, J. M. Shaw, C. T. Boone and T. J. Silva,\nPhys. Rev. Lett. 110, 117201 (2013).\n[25] Y. Li, Y. Lu and W. E. Bailey, J. Appl. Phys. 113,\n17B506 (2013).\n[26] M. H. Seavey and P. E. Tannenwald, Phys. Rev. Lett. 1,\n168 (1958).\n[27] See the Supplemental Information for details.\n[28] W. S. Ament and G. T. Rado, Phys. Rev. 97, 1558\n(1955).\n[29] C. Scheck, L. Cheng and W. E. Bailey, Appl. Phys. Lett.\n88, 252510 (2006).\n[30] M. Jirsa, phys. stat. sol. (b) 113, 679 (1982).\n[31] A. Kapelrud and A. Brataas, Phys. Rev. Lett. 111,\n097602 (2013).\n[32] O. Mosendz, J. E. Pearson, F. Y. Fradin, S. D. Bader\nand A. Hoffmann, Appl. Phys. Lett. 96, 022502 (2009).\n[33] K. Gilmore, Y. U. Idzerda and M. D. Stiles, Phys. Rev.\nLett.99, 027204 (2007).\n[34] S. M. Bhagat andP.Lubitz, Phys. Rev. B 10, 179(1974).\n[35] G. T. Rado, J. Appl. Phys. 29, 330 (1958).\n[36] V. Korenman and R. E. Prange, Phys. Rev. B 6, 2769\n(1972).\n[37] B. A. Gurney, V. S. Speriosu, J.-P. Nozieres, H. Lefakis ,\nD. R. Wilhoit and O. U. Need, Phys. Rev. Lett. 71, 4023\n(1993).\n[38] J. N. Lloyd and S. M. Bhagat, Solid State Commun. 8,\n2009 (1970).\n[39] R. D. McMichael, M. D. Stiles, P. J. Chen and W. F.\nEgelhoff Jr., J. Appl. Phys. 83, 7037 (1998).\n[40] J. Zhang, P. M. Levy, S. F. Zhang and V. Antropov,\nPhys. Rev. Lett. 93, 256602 (2004).\n[41] R. F. Soohoo, Phys. Rev. 131, 594 (1963)."    },    {        "title": "1402.6899v1.On_the_longitudinal_spin_current_induced_by_a_temperature_gradient_in_a_ferromagnetic_insulator.pdf",        "content": "arXiv:1402.6899v1  [cond-mat.mtrl-sci]  27 Feb 2014On the longitudinal spin current induced by a temperature gr adient in a\nferromagnetic insulator\nS. R. Etesami,1,2L. Chotorlishvili,2A. Sukhov,2and J. Berakdar2\n1Max-Planck-Institut f¨ ur Mikrostrukturphysik, 06120 Hal le, Germany\n2Institut f¨ ur Physik, Martin-Luther-Universit¨ at Halle- Wittenberg, 06120 Halle, Germany\n(Dated: March 3, 2022)\nBased on the solution of the stochastic Landau-Lifshitz-Gi lbert equation discretized for a ferro-\nmagnetic chain subject to a uniform temperature gradient, w e present a detailed numerical study\nof the spin dynamics with a focus particularly on finite-size effects. We calculate and analyze the\nnet longitudinal spin current for various temperature grad ients, chain lengths, and external static\nmagnetic fields. In addition, we model an interface formed by a nonuniformly magnetized finite-size\nferromagnetic insulator and a normal metal and inspect the e ffects of enhanced Gilbert damping on\nthe formation of the space-dependent spin current within th e chain. A particular aim of this study\nis the inspection of the spin Seebeck effect beyond the linear response regime. We find that within\nour model the microscopic mechanism of the spin Seebeck curr ent is the magnon accumulation effect\nquantified in terms of the exchange spin torque. According to our results, this effect drives the spin\nSeebeck current even in the absence of a deviation between th e magnon and phonon temperature\nprofiles. Our theoretical findings are in line with the recent ly observed experimental results by M.\nAgrawal et al., Phys. Rev. Lett. 111, 107204 (2013).\nPACS numbers: 85.75.-d, 73.50.Lw, 72.25.Pn, 71.36.+c\nI. INTRODUCTION\nThermal magneto- and electric effects have a long his-\ntory and are the basis for a wide range of contemporary\ndevices. Research activities revived substantially upon\nthe experimental demonstration of the correlation be-\ntween an applied temperature gradient and the observed\nspindynamics, includingaspincurrentalongthetemper-\naturegradientinanopen-circuitmagneticsample,theso-\ncalled spin Seebeck effect (SSE)1. Meanwhile an impres-\nsive body work has accumulated on thermally induced\nspin- and spin-dependent currents1–11(for a dedicated\ndiscussion we referto the topical review12). The SSE was\nobserved not only in metallic ferromagnets (FMs) like\nCo2MnSi or semiconducting FMs, e.g. GaMnAs, (Ref.4),\nbut also in magnetic insulators LaY 2Fe5O12(Ref.5) and\n(Mn, Ze)Fe 2O4(Ref.7). The Seebeck effect is usually\nquantified by the Seebeck coefficient Swhich is defined,\nin a linear response manner, as the ratio of the gener-\nated electric voltage ∆ Vto the temperature difference\n∆T:S=−∆V\n∆T. The magnitude of the Seebeck coeffi-\ncientSdepends on the scattering rate and the density\nof electron states at the Fermi level, and thus it is ma-\nterial dependent variable. In the case of SSE, the spin\nvoltage is formally determined by µ↑−µ↓, whereµ↓(↑)\nare the electrochemical potentials for spin-up and spin-\ndown electrons, respectively. The density of states and\nthe scattering rate for spin-up and spin-down electrons\nare commonly different, which results in various Seebeck\nconstants for the two spin channels. In a metallic magnet\nsubjected to a temperature gradient, one may think of\nthe electrons in different spin channels to generate differ-\nent driving forces, leading to a spin voltage that induces\na nonzero spin current. When a magnetic insulator is\nin contact with a normal metal (NM) and the system issubjected to a thermal gradient, the total spin current\nflowing through the interface is a sum of two oppositely\ndirected currents. The current emitted from the FM into\nthe NM, is commonly identified as a spin pumping cur-\nrentIspand originatesfromthe thermally activatedmag-\nnetization dynamics in the FM, while the other current\nIflis associated with the thermal fluctuations in the NM\nand is known as spin torque13. The competition between\nthe spin pump and the spin torque currents defines the\ndirectionofthetotalspincurrentwhichisproportionalto\nthe thermal gradient applied to the system. The theory\nofthe magnon-drivenSSE5presupposesthat the magnon\ntemperature follows the phonon temperature profile and\ninalinearresponseapproximationprovidesagoodagree-\nment with experiments.\nIn a recent study14the theory of the magnon-driven\nSSE was extended beyond the linear response approxi-\nmation. In particular, it was shown that the nonlinearity\nleads to a saturation of the total spin current and nonlin-\near effects become dominant when the following inequal-\nity holds H0/Tm\nF< kB/(MsV), where H0is the constant\nmagnetic field applied to the system, Tm\nFis the magnon\ntemperature, Msis the saturation magnetization and V\nis the volume of the sample. The macrospin formulation\nofthestochasticLandau-Lifshitz-Gilbert(LLG)equation\nand the Fokker-Planck approach utilized in Ref.14is in-\nappropriate for non-uniformly magnetized samples with\ncharacteristic lengths exceeding several 10 nm. Beyond\nthe macrospin formulation the SSE effect for nonuni-\nformly magnetized samples can be described by intro-\nducing a local magnetization vector15/vector m(/vector r,t). In this\ncase, however, the correspondingFokker-Planckequation\nturns into an integro-differential equation and can only\nbe solved after a linearization16. Recently17, the lon-\ngitudinal SSE was studied in a NM-FM-NM sandwich2\nstructure in the case of a nonuniform magnetization pro-\nfile. The linear regime, however, can not totaly embrace\nnontrivial and affluent physics of the SSE.\nIn the present study we inspect the SSE for a nonuni-\nformlymagnetized finite-size FM-NM interfacesubjected\nto an arbitrary temperature gradient. Our purpose is to\ngo beyond linear response regime which is relevant for\nthe nonlinear magnetization dynamics. It is shown that\nin analogy with the macrospin case14the spin current in\nthe nonlinear regime depends not only on the tempera-\nture gradient, but on the absolute values of the magnon\ntemperature as well. In finite-size non-uniformly magne-\ntized samples, however, the site-dependent temperature\nprofile may lead to new physical important phenomena.\nFor instance, we show that the key issue for the spin cur-\nrent flowing through a nonuniformly magnetized mag-\nnetic insulator is the local exchange spin torque and the\nlocal site-dependent magnon temperature profile, result-\ning in a generic spatial distribution of the steady state\nspin current in a finite chain subject to a uniform tem-\nperaturegradient. The maximalspincurrentispredicted\nto be located at the middle of the chain.\nII. THEORETICAL FRAMEWORK\nFor the description of the transversal magnetiza-\ntion dynamics we consider propagation of the normal-\nized magnetization direction /vector m(/vector r,t) as governed by the\nLaundau-Lifshitz-Gilbert (LLG) equation18,36\n∂/vector m\n∂t=−γ/bracketleftBig\n/vector m×/vectorHeff/bracketrightBig\n+α/bracketleftbigg\n/vector m×∂/vector m\n∂t/bracketrightbigg\n−γ/bracketleftBig\n/vector m×/vectorh(/vector r,t)/bracketrightBig\n,(1)\nwhere the deterministic effective field /vectorHeff=−1\nMSδF\nδ/vector mde-\nrives from the free energy density Fand is augmented by\na Gaussian white-noise random field h(/vector r,t) with a space-\ndependent local intensity and autocorrelation function.\nαis the Gilbert damping, γ= 1.76·1011[1/(Ts)] is the\ngyromagnetic ratio and MSis the saturation magnetiza-\ntion.Freads\nF=1\nV/integraldisplay/bracketleftbiggA\n2|/vector∇m|2+Ea(/vector m)−µ0MS/vectorH0·/vector m/bracketrightbigg\ndV,(2)\nwhere/vectorH0is the external constant magnetic field, Ea(/vector m)\nis the anisotropy energy density and Ais the exchange\nstiffness. Vis the system volume. We employ a dis-\ncretized version of the integro-differential equation (1)\nby defining Ncells with a characteristic length a= /radicalbig\n2A/µ0M2sof the exchange interaction between the\nmagnetic moments. a3= Ω0is the volume of the re-\nspective cell. Assuming negligible variations of /vector m(/vector r,t)\nover a small a, one introduces a magnetization vector\n/vectorMnaveraged over the nth cell/vectorMn=MS\nV/integraltext\nΩ0/vector m(/vector r,t)dVand the total energy density becomes\nε=−/vectorH0·/summationdisplay\nn/vectorMn+K1\nM2\nS/summationdisplay\nn/parenleftbig\nM2\nS−(Mz\nn)2/parenrightbig\n−2A\na2M2\nS/summationdisplay\nn/vectorMn·/vectorMn+1.(3)\n/vectorH0is the external magnetic field and K1is the uniaxial\nanisotropy density (with the easy axis: /vector ez). The effective\nmagnetic field actingon the n-th magnetic moment reads\n/vectorHeff\nn=−∂ε\n∂/vectorMn=/vectorH0+2K1\nM2\nSMz\nn/vector ez\n+2A\na2M2\nS/parenleftBig\n/vectorMn+1+/vectorMn−1/parenrightBig\n.(4)\nThermal activation is introduced by adding to the total\neffectivefieldastochasticfluctuatingmagneticfield /vectorhn(t)\nso that\n/vectorHeff\nn(t) =/vectorH0+/vectorHanis\nn+/vectorHexch\nn+/vectorhn(t).(5)\nHere/vectorHanis\nnis the magnetic anisotropy field, /vectorHexch\nnis the\nexchange field. The random field /vectorhn(t) has a thermal\norigin and simulates the interaction of the magnetization\nwith a thermal heat bath (cf. the review Ref. [19] and\nreferences therein). The site dependence of /vectorhn(t) reflects\ntheexistenceofthelocalnonuniformtemperatureprofile.\nOn the scale ofthe volumeΩ 0the heat bath is considered\nuniform at a constant temperature. The random field is\ncharacterized via the standard statistical properties of\nthe correlation function\n/angb∇acketlefthik(t)/angb∇acket∇ight= 0,\n/angb∇acketlefthik(t)hjl(t+∆t)/angb∇acket∇ight=2kBTiαi\nγMSa3δijδklδ(∆t).(6)\niandjdefine the corresponding sites of the FM-chain\nandk,lcorrespond to the cartesian components of the\nrandom magnetic field, Tiandαiare the site-dependent\nlocal temperature and the dimensionless Gilbert damp-\ning constant, respectively, kB= 1.38·10−23[J/K] is the\nBoltzmann constant.\nIn what follows we employ for the numerical calcu-\nlations the material parameters related to YIG, e.g. as\ntabulated in Ref.6(Table I). Explicitlythe exchangestiff-\nness isA≈10 [pJ/m], the saturation magnetization has\na value of 4 πMS≈106[A/m]. The anisotropy strength\nK1can be derived from the estimate for the frequency\nω0=γ2K1/MS≈10·109[1/s]6. The size of the FM\ncell is estimated from a=/radicalbig\n2A/µ0M2syielding about 20\n[nm]. Fordampingparameterwetakethevalue α= 0.01,\nwhich exceeds the actual YIG value5,6. This is done to\noptimize the numerical procedure in order to obtain rea-\nsonable calculation times. We note that although the\nquasi-equilibrium is assured when tracking the magne-\ntization trajectories on the time scale longer than the\nrelaxation time, the increased αquantitatively alters the3\nFIG. 1: a) Schematics of the FM chain considered in the\ncalculations. b) Suggested alignment for measurements.\nstrength in the correlation function (eq. (6)) and there-\nfore indirectly has an impact on the values of the spin\ncurrent.\nWe focus on a system representing a junction of a FM\ninsulatorand a NM which is schematically shownin FIG.\n1. This illustration mimics the experimental setup for\nmeasuring the longitudinal SSE20, even though the anal-\nysis performed here does not include all the aspects of\nthe experimental setting. The direction of the magnetic\nmoments in the equilibrium is parallel to the FM-NM in-\nterface. Experimentally it was suggested to pick up the\nlongitudinal spin current by means of the inverse spin\nHall effect20. If it is so possible then, the electric field\ngenerated via the inverse spin Hall effect (ISHE) reads− →E=D− →Is×− →σ. Here− →Edenotes electric field related\nto the inverse spin Hall effect,− →Isdefines the spatial di-\nrection of the spin current, and− →σis spin polarization of\nthe electrons in the NM, and Dis the constant. We note,\nhowever, that our study is focused on the spin dynamics\nonly and makes no statements on ISHE.\nIII. DEFINITION OF THE SPIN CURRENT\nFor convenience we rewrite the Gilbert equation with\nthe total energy density (3) in the form suggested in\nRef.17\n∂/vectorSn\n∂t+γ/bracketleftBig\n/vectorSn×/parenleftBig\n/vectorHeff\nn(t)−/vectorHex/parenrightBig/bracketrightBig\n+αγ\nMS/bracketleftBigg\n/vectorSn×∂/vectorSn\n∂t/bracketrightBigg\n+∇·/vectorJ/vector s\nn= 0,\n(7)where/vectorSn=−/vectorMn/γand the expression for the spin cur-\nrent density tensor reads\n∇·/vectorJ/vector s\nn=γ/bracketleftBig\n/vectorSn×/vectorHex\nn/bracketrightBig\n. (8)\nHere\n/vectorQn=−γ[/vectorSn×/vectorHex\nn] (9)\nis the local exchange spin torque.\nFor the particular geometry (FIG. 1) the only nonzero\ncomponents of the spin current tensor are Isxn,Isy\nn,Iszn.\nTaking into account eqs. (4) and (7), we consider a dis-\ncrete version of the gradient operator and for the com-\nponents of the spin current tensor Is\nn=a2Js\nnwe deduce:\nIα\nn=Iα\n0−2Aa\nM2\nSn/summationdisplay\nm=1Mβ\nm(Mγ\nm−1+Mγ\nm+1)εαβγ,(10)\nwhereεαβγis the Levi-Civita antisymmetric tensor,\nGreek indexes define the current components and the\nLatin ones denote sites of the FM-chain. In what fol-\nlows we will utilize eq. (10) for quantifying the spin cur-\nrent in the spin chain. We consider different temperature\ngradients applied to the system taking into account the\ndependence of the magnon temperature on the phonon\ntemperature profile5. Since the temperature in the chain\nis not uniform, we expect a rich dynamics of different\nmagnetic moments /vectorMn. In this case only nonuniform\nsite-dependent spin current Incan fulfil the equation (7).\nIn order to prove this we will consider different configu-\nrations of magnetic fields for systems of different lengths.\nModeling the interface effects between the FM insulator\nand the NM proceeds by invoking the concept of the en-\nhanced Gilbert damping proposed in a recent study21.\nThe increased damping constant in the LLG equation of\nthe last magnetic moment describes losses of the spin\ncurrent due to the interface effect. In order to evaluate\nthe spin current flowing from the NM to the FM insula-\ntor we assume that the dynamics of the last spin in the\ninsulator chain is influenced by the spin torque flowing\nfrom the NM to the magnetic insulator. The magnetic\nanisotropy is considered to have an easy axis5.\nIV. NUMERICAL RESULTS ON ISOLATED\nFERROMAGNETIC INSULATOR CHAIN\nFor the study of thermally activated magnetization\ndynamics we generate from 1000 to 10000 random tra-\njectories for each magnetic moment of the FM-chain.\nAll obtained observables are averaged over the statis-\ntical ensemble of stochastic trajectories. The number\nof realizations depends on the thermal gradient applied\nto the system. For long spin chains (up to 500 mag-\nnetic moments) the calculations are computationally in-\ntensive even for the optimized advanced numerical Heun-\nmethod22, which converges in quadratic mean to the so-\nlution of the LLG equation when interpreted in the sense4\nof Stratonovich23. For the unit cell of the size 20 [nm],\nthe FM-chain of 500 spins is equivalent to the magnetic\ninsulator sample of the width around 10 [ µm]. We make\nsure in our calculations that the magnetization dynam-\nics is calculated on the large time scale exceeding the\nsystem’s relaxation time which can be approximated via\nτrel≈MS/(γ2K1α)≈10 [ns]24.\nA. Role of the local temperature and local spin\nexchange torque\nPrior to studying a realistic finite-size system we con-\nsider a toy model of three coupled magnetic moments.\nOur aim is to better understand the role of local tem-\nperature and local exchange spin torque Qn(eq. (9)) in\nthe formation of the spin current In. Considering eqs.\n(8, 9), we can utilize a recursive relation for the site-\ndependent spin current Inand the local exchange spin\ntorqueIn=In−1+a3\nγQnfor different temperatures of\nthe site in the middle of the chain above T2> Tavand\nbelowT2< Tav. The mean temperature in the system is\nTav=/parenleftbig\nT1+T2+T3/parenrightbig\n/3. The calculations are performed\nfor different values of the site temperatures. We find that\nthe exchange spin torques Qnrelated to magnetic mo-\nmentsMnwith a temperature above the mean temper-\natureTn> Tavhave a positive contribution to the spin\ncurrent in contrast to the exchange spin torques Qmof\nthe on-average-”cold”magnetic momentswith Tm< Tav.\nThis finding hints on the existence of a maximum spin\ncurrent in a finite chain of magnetic moments and/or\nstrong temperature gradient. This means that the site-\ndependent spin current Inincreases if Qn>0 until the\nlocal site temperature drops below the mean tempera-\ntureTn< Tav, in which case the exchange spin torque\nbecomes negative Qn<0 and the spin current decreases.\nIn order to prove that the negative contribution in the\nspin current of the on-average-cold magnetic moments\nis not an artefact of the three magnetic moments only,\nwe studied long spin chains which mimic non-uniformly\nmagnetized magnetic insulators. In the thermodynamic\nlimit for a large number of magnetic moments N≫1\nwe expect to observe a formation of the equilibrium pat-\nterns in the spin current profile correspondingto the zero\nexchange spin torque Qn= 0 between nearest adjacent\nmoments.\nB. Longitudinal spin current\nIn FIG. 2 a dependence of distinct components of the\nspin current on the site is plotted. As inferred from the\nfigure the current is not uniformly distributed along the\nchain. Evidently, the spin current has a maximum in\nthe middle of the chain. The site-dependent spin cur-\nrent is an aftermath of the nonuniform magnon temper-/SolidSquare/SolidSquare\n/SolidSquare\n/SolidSquare/SolidSquare\n/SolidSquare/SolidSquare/SolidSquare/SolidSquare\n/SolidSquare/SolidSquare/SolidSquare\n/SolidSquare/SolidSquare/SolidSquare/SolidSquare/SolidSquare\n/SolidSquare/SolidSquare\n/SolidSquare/SolidSquare/SolidSquare/SolidSquare/SolidSquare/SolidSquare\n/SolidSquare/SolidSquare/SolidSquare/SolidSquare/SolidSquare/SolidSquare/SolidSquare/SolidSquare\n/SolidSquare/SolidSquare\n/SolidSquare/SolidSquare/SolidSquare/SolidSquare/SolidSquare/SolidSquare/SolidSquare/SolidSquare/SolidSquare/SolidSquare/SolidSquare/SolidSquare/SolidSquare\n/SolidSquare/SolidSquare/Square/Square/Square/Square/Square/Square\n/Square/Square/Square\n/Square/Square\n/Square/Square/Square/Square\n/Square/Square/Square/Square/Square/Square/Square/Square/Square/Square\n/Square/Square/Square/Square/Square/Square\n/Square\n/Square/Square/Square\n/Square/Square/Square/Square\n/Square/Square/Square/Square/Square/Square/Square/Square/Square/Square/Square/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle\n/SolidCircle\n/SolidCircle\n/SolidCircle\n/SolidCircle\n/SolidCircle\n/SolidCircle\n/SolidCircle\n/SolidCircle\n/SolidCircle\n/SolidCircle/SolidSquareInSx\n/SquareInSy\n/SolidCircleInSz\n0 10 20 30 40 5001234\nSitenumber, nSpincurrent/Multiply1011/LBracket1/HBars/Minus1/RBracket1\nFIG. 2: Different cartesian components of the statistically\naveraged longitudinal spin current as a function of the site\nnumber. Numerical parameters are ∆ T= 50 [K], α= 0.01\nandH0= 0 [T]. The temperature gradient is defined ∆ T=\nT1−T50, whereT1= 50 [K]. The only nonzero component of\nthe spin current is ISzn. Other two components ISxn, ISy\nnare\nzero because of the uniaxial magnetic anisotropy field which\npreserves XOYsymmetry of the magnetization dynamics.\nature profile applied to the system. This effect was not\nobserved in the single macro spin approximation and is\nonlyrelevantforthe non-uniformlymagnetizedfinite-size\nmagnetic insulator sample. In addition one observesthat\nthe amplitude of the spin current increases with increas-\ningthe thermalgradient. Thisispredictablynatural; less\nsohowever, is the presenceof amaximum ofthe spin cur-\nrent observed in the middle of the chain. We interpret\nthis observation in terms of a collective cumulative av-\neraged influence of the surrounding magnetic moments\non particular magnetic moment. For a linear tempera-\nture gradient, as in FIG. 2, we have ∆ T=T1−TN\naNwhich\nmeans that half of the spins with i < N/2 possess tem-\nperatures abovethe mean temperature of the chain T1/2,\nwhile the other half have temperatures below the mean\ntemperature. Further, the main contributors in the total\nspin current are the hot magnetic moments with temper-\natures above the mean temperature Tn> Tavand with\na positive exchange spin torque Qn>0. While magnetic\nmoments with a temperature below the mean tempera-\ntureTn< Tav,Qn<0absorbthe spincurrentandhavea\nnegativecontribution in the total spin current. This non-\nequivalence of magnetic moments results in a maximum\nof the total spin current in the center of the chain. In\nwhat follows the magnetic moments with temperatures\nhigher than the mean temperature in the chain are re-\nferred to as hot magnetic moments, while the magnetic\nmoments with temperatures lower than Tavwe refer to\nas cold magnetic moments (i.e., our reference tempera-\nture isTav). The idea we are following is that the hot\nmagnetic moments form the total spin current which is\npartly utilized for the activation of the cold magnetic\nmoments. FIG. 3 illustrates the motivation of this state-\nment. The maximum of the spin current (solid circles) is5\n/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle\n/SolidCircle\n/SolidCircle\n/SolidCircle\n/SolidCircle\n/SolidCircle\n/SolidCircle\n/SolidCircle\n/SolidCircle\n/SolidCircle\n/SolidCircle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidCircleInSz\n/SolidUpTriangle/ScriptA3\nΓQnz\n0 10 20 30 40 5001234\nSitenumber, n1011/LBracket1/HBars/Minus1/RBracket1\nFIG. 3: Z-componentofthestatistically averaged spincurr ent\nISzn(blue solid circles) and the distribution of the exchange\nspin torquea3\nγQz\nn(red solid triangles), both site-dependent.\nDirect correlation between the behavior of the spin current\nand the exchange spin torque can be observed: the change\nof the sign of the exchange spin torque exactly matches the\nmaximum of the spin current.\nobserved in the vicinity of the sites where the exchange\nspin torque term Qnchanges its sign from positive to\nnegative (solid triangles), highlighting the role of the hot\nand cold magnetic moments in finite-size systems. To\nfurther affirm we consider two different temperature pro-\nfiles - linear and exponential - with slightly shifted values\nof the mean temperature (FIG. 4). The dependence of\nthe maximum spin current on the mean temperature is\na quite robust effect and a slight shifts of the mean tem-\nperature to the left lead to a certain shifting of the spin\ncurrent’s maximum. The effect of the nonuniform spin\ncurrent passing through the finite-size magnetic insula-\ntormightbetestedexperimentallyusingtheSSEsetupin\nwhich the spin current’s direction is parallel to the tem-\nperature gradient. One may employ the inverse spin Hall\neffect using FM insulator covered by a stripe of param-\nagnetic metal, e.g. Ptat different sites (cf Ref.20), albeit\nthe chain must be small ( <∼1µm).\nFurthermore, from FIG. 2 we infer that the only\nnonzero component of the spin current is Iz\nn. Due to\nthe uniaxial magnetic anisotropy all orientations of the\nmagnetic moments in the XOYplane are equivalent and\nIx\nn,Iy\nncomponents of the spin current vanish.\nC. Role of boundary conditions\nTo elaborate on the origin of the observed maximum\nof the spin current we inspect the role of boundary con-\nditions. In fact, in spite of employing different bound-\nary conditions for the chain we observe the same effect\n(FIG. 5), from which we can conclude that the effect of/SolidSquare/SolidSquare/SolidSquare/SolidSquare/SolidSquare/SolidSquare/SolidSquare/SolidSquare/SolidSquare/SolidSquare/SolidSquare/SolidSquare/SolidSquare/SolidSquare/SolidSquare/SolidSquare/SolidSquare/SolidSquare/SolidSquare/SolidSquare/SolidSquare/SolidSquare/SolidSquare/SolidSquare/SolidSquare/SolidSquare/SolidSquare/SolidSquare/SolidSquare/SolidSquare/SolidSquare/SolidSquare/SolidSquare/SolidSquare/SolidSquare/SolidSquare/SolidSquare/SolidSquare\n/SolidSquare/SolidSquare\n/SolidSquare\n/SolidSquare\n/SolidSquare\n/SolidSquare\n/SolidSquare\n/SolidSquare\n/SolidSquare\n/SolidSquare\n/SolidSquare\n/SolidSquare/Circle/Circle/Circle/Circle/Circle/Circle/Circle/Circle/Circle/Circle/Circle/Circle/Circle/Circle/Circle/Circle/Circle/Circle/Circle/Circle/Circle/Circle/Circle/Circle/Circle/Circle/Circle/Circle/Circle/Circle/Circle/Circle/Circle/Circle/Circle/Circle/Circle/Circle/Circle/Circle\n/Circle\n/Circle\n/Circle\n/Circle\n/Circle\n/Circle\n/Circle\n/Circle\n/Circle\n/Circle/SolidSquareTn/EquΑlLinear\n/CircleTn/EquΑlExponential\n0 10 20 30 40 500.00.51.01.52.02.5\nSitenumber, nSpincurrent/Multiply1011/LBracket1/HBars/Minus1/RBracket1\n1 25 502031.234.450\nnTn/LBracket1K/RBracket1Temperature\nprofile\nFIG. 4: Z-component of the statistically averaged spin cur-\nrent for the linear ∆ T=T1−T50and exponential ∆ T(n) =\n50[K]e−(n−1)/50temperature gradients. The slight shift of the\nmean temperature to the left leads to a certain shifting of th e\nmaximum spin current to the left.\nthe cold and hot magnetic moments is inherent to the\nspin dynamics within the chain, which is independent\nfrom the particular choice of the boundary conditions.\nFurthermore, we model the situation with the extended\nregion at the ends of the FM chain (FIG. 6), in which\nthe end temperatures are constant (i.e., one might imag-\nine the heat reservoirs to have finite spatial extensions).\nModeling the ends of the FM-chain with zero temper-\nature gradient by means of the LLG equations is cer-\ntainly an approximation, which can be improved by em-\nploying the Landau-Lifshitz-Bloch equations reported in\nRef.12. It captures, however, the main effects at rela-\ntively low temperatures: the flow of the spin current for\nthe decaying spin density away from the T= const-∆ T-\ninterfaceand anon-zerointegralspin currentfor the sites\n0< n <50 and 150 < n <200. As we see even in the\nfragments of the chain with a zero temperature gradient\nthespincurrentisnotzero. Thereasonisthattheforma-\ntion of the spin current profile is a collective many body\neffect of the interacting magnetic moments. Therefore,\nthe fragment of the chain with nonzero temperature gra-\ndient (sites 50 < n <150) has a significant influence on\nthe formation of the spin current profiles in the left and\nrightregionsofthe chainwhere the temperaturegradient\nvanishes.\nD. Temperature dependence of the longitudinal\nspin current\nInFIG.7thedependenceofthe z-componentoftheav-\neraged longitudinal spin current on the temperature gra-\ndient is shown. The dependence In(∆T) (inset ofFIG. 7)\nis linear and the amplitude of the spin current increases6\n/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle\n/SolidCircle\n/SolidCircle\n/SolidCircle\n/SolidCircle\n/SolidCircle\n/SolidCircle\n/SolidCircle\n/SolidCircle\n/SolidCircle\n/SolidCircle/UpTriangle/UpTriangle/UpTriangle/UpTriangle/UpTriangle/UpTriangle/UpTriangle/UpTriangle/UpTriangle/UpTriangle/UpTriangle/UpTriangle/UpTriangle/UpTriangle/UpTriangle/UpTriangle/UpTriangle/UpTriangle/UpTriangle/UpTriangle/UpTriangle/UpTriangle/UpTriangle/UpTriangle/UpTriangle/UpTriangle/UpTriangle/UpTriangle/UpTriangle/UpTriangle/UpTriangle/UpTriangle/UpTriangle/UpTriangle/UpTriangle/UpTriangle/UpTriangle/UpTriangle/UpTriangle/UpTriangle/UpTriangle\n/UpTriangle\n/UpTriangle\n/UpTriangle\n/UpTriangle\n/UpTriangle\n/UpTriangle\n/UpTriangle\n/UpTriangle\n/UpTriangle/DownTriangle/DownTriangle/DownTriangle/DownTriangle/DownTriangle/DownTriangle/DownTriangle/DownTriangle/DownTriangle/DownTriangle/DownTriangle/DownTriangle/DownTriangle/DownTriangle/DownTriangle/DownTriangle/DownTriangle/DownTriangle/DownTriangle/DownTriangle/DownTriangle/DownTriangle/DownTriangle/DownTriangle/DownTriangle/DownTriangle/DownTriangle/DownTriangle/DownTriangle/DownTriangle/DownTriangle/DownTriangle/DownTriangle/DownTriangle/DownTriangle/DownTriangle/DownTriangle/DownTriangle/DownTriangle/DownTriangle/DownTriangle\n/DownTriangle\n/DownTriangle\n/DownTriangle\n/DownTriangle\n/DownTriangle\n/DownTriangle\n/DownTriangle\n/DownTriangle\n/DownTriangle/Square/Square/Square/Square/Square/Square/Square/Square/Square/Square/Square/Square/Square/Square/Square/Square/Square/Square/Square/Square/Square/Square/Square/Square/Square/Square/Square/Square/Square/Square/Square/Square/Square/Square/Square/Square/Square/Square/Square/Square/Square\n/Square\n/Square\n/Square\n/Square\n/Square\n/Square\n/Square\n/Square\n/Square/SolidCircleM0/EquΑlMN/Plus1/EquΑl/LParen10,0,0/RParen1\n/UpTriangleM0/EquΑl/LParen10,0,0/RParen1,MN/Plus1/EquΑl/LParen10,0,Ms/RParen1\n/DownTriangleM0/EquΑl/LParen10,0,Ms/RParen1,MN/Plus1/EquΑl/LParen10,0,0/RParen1\n/SquareM0/EquΑlMN/Plus1/EquΑl/LParen10,0,Ms/RParen1\n0 10 20 30 40 5001234\nSitenumber, nSpinnurrent/Multiply1011/LBracket1/HBars/Minus1/RBracket1\nFIG. 5: Effect of different boundary conditions on the aver-\naged spin current. Numerical parameters are ∆ T= 50 [K],\nα= 0.01 andH0= 0 [T]. The temperature gradient is defined\n∆T=T1−T50, whereT1= 50 [K]. Inspite of different bound-\nary conditions we observe the same maximal spin current for\nthe site number corresponding to the mean temperature of\nthe system. Thus, the effect of the cold and hot magnetic\nmoments is independent of the particular choice of boundary\nconditions.\n/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet\n/Bullet\n0 50 100 150 200/Minus1012345\nSitenumber, nSpinnurrent/Multiply1011/LBracket1/HBars/Minus1/RBracket1\n1501001502002060100\nnTn/LBracket1K/RBracket1Temperature\nprofile\nFIG. 6: Effect of boundary conditions in the case of different\ntemperature profiles at the boundaries: linear temperature\ngradient (thick curve), constant temperature for 0 < n <50\nand 150 < n <200 (thin curve). Even in the fragments of\nthe chain with zero temperature gradient the spin current is\nnot zero, which results from the formation of the spin cur-\nrent profile as a collective many body effect of the interactin g\nmagnetic moments. Therefore, the fragment of the chain with\nnonzero temperature gradient (sites 50 < n <150) has a sig-\nnificant influence on the formation of the spin current profile s\nin the left and right zero temperature gradient parts of the\nchain.\nwith the temperature gradient. This result is consistent\nwith the experimental facts (Refs.4,5) and our previous\nanalytical estimations obtained via the single macrospin\nmodel14./Circle/Circle/Circle/Circle/Circle/Circle/Circle/Circle/Circle/Circle/Circle/Circle/Circle/Circle/Circle/Circle/Circle/Circle/Circle/Circle/Circle/Circle/Circle/Circle/Circle/Circle/Circle/Circle/Circle/Circle/Circle/Circle/Circle/Circle/Circle/Circle/Circle/Circle/Circle/Circle/Circle/Circle/Circle/Circle/Circle/Circle/Circle/Circle/Circle/Circle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/Square/Square/Square/Square/Square/Square/Square/Square/Square/Square/Square/Square/Square/Square/Square/Square/Square/Square/Square/Square/Square/Square/Square/Square/Square/Square/Square/Square/Square/Square/Square/Square/Square/Square/Square/Square/Square/Square/Square/Square/Square/Square/Square/Square/Square/Square/Square/Square/Square/Square/SolidSquare/SolidSquare/SolidSquare/SolidSquare/SolidSquare/SolidSquare/SolidSquare/SolidSquare/SolidSquare/SolidSquare/SolidSquare/SolidSquare/SolidSquare/SolidSquare/SolidSquare/SolidSquare/SolidSquare/SolidSquare/SolidSquare/SolidSquare/SolidSquare/SolidSquare/SolidSquare/SolidSquare/SolidSquare/SolidSquare/SolidSquare/SolidSquare/SolidSquare/SolidSquare/SolidSquare/SolidSquare/SolidSquare/SolidSquare/SolidSquare/SolidSquare/SolidSquare/SolidSquare/SolidSquare/SolidSquare/SolidSquare/SolidSquare/SolidSquare/SolidSquare/SolidSquare\n/SolidSquare\n/SolidSquare\n/SolidSquare\n/SolidSquare\n/SolidSquare/UpTriangle/UpTriangle/UpTriangle/UpTriangle/UpTriangle/UpTriangle/UpTriangle/UpTriangle/UpTriangle/UpTriangle/UpTriangle/UpTriangle/UpTriangle/UpTriangle/UpTriangle/UpTriangle/UpTriangle/UpTriangle/UpTriangle/UpTriangle/UpTriangle/UpTriangle/UpTriangle/UpTriangle/UpTriangle/UpTriangle/UpTriangle/UpTriangle/UpTriangle/UpTriangle/UpTriangle/UpTriangle/UpTriangle/UpTriangle/UpTriangle/UpTriangle/UpTriangle/UpTriangle/UpTriangle/UpTriangle/UpTriangle\n/UpTriangle\n/UpTriangle\n/UpTriangle\n/UpTriangle\n/UpTriangle\n/UpTriangle\n/UpTriangle\n/UpTriangle\n/UpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle\n/SolidUpTriangle\n/SolidUpTriangle\n/SolidUpTriangle\n/SolidUpTriangle\n/SolidUpTriangle\n/SolidUpTriangle\n/SolidUpTriangle\n/SolidUpTriangle\n/SolidUpTriangle\n/SolidUpTriangle\n/SolidUpTriangle\n/SolidUpTriangle\n0 10 20 30 40 5001234\nSitenumber, nSpincurrent/Multiply1011/LBracket1/HBars/Minus1/RBracket1\n/Circle /Circle/SolidCircle /SolidCircle/Square /Square/SolidSquare /SolidSquare/UpTriangle /UpTriangle/SolidUpTriangle /SolidUpTriangle\n0 5003\n/DifferenceDeltaT/LBracket1K/RBracket1I26/Multiply1011/LBracket1/HBars/Minus1/RBracket1\n/DifferenceDeltaT=50 [K]\n/DifferenceDeltaT=40 [K]\n/DifferenceDeltaT=0 [K]/DifferenceDeltaT=10 [K]/DifferenceDeltaT=20 [K]/DifferenceDeltaT=30 [K]\nFIG. 7: Dependence of the averaged spin current on the\nstrength of the temperature gradient. Numerical parameter s\nareα= 0.01 andH0= 0 [T]. The temperature gradient is\ndefined as ∆ T=T1−T50, whereT1= 50 [K]. The inset shows\nthe averaged spin current for the 26-th site. The maximum\ncurrent increases with elevating the temperature gradient .\nE. Finite-size effects\nFinite-size effects are considered relevant for the ex-\nperimental observations (e.g. Ref.4). In the thermody-\nnamic limit N≫1 we expect the formation of equilib-\nrium patterns in the spin currentprofile correspondingto\nthe zero exchange spin torque Qn= 0 between nearest\nadjacent moments. To address this issue, the spin cur-\nrent for chains of different lengths is shown in FIG. 8. As\nwe see in the case of N= 500 magnetic moments large\npattern of the uniform spin current corresponding to the\nsites 50< n <450 is observed. In order to understand\nsuch a behavior of the spin current for a large system\nsize, we plotted the dependence on the site number of\nthe exchange spin torque Qn(FIG. 9). As we see, the\nexchange spin torque corresponding to the spin current\nplateau is characterizedby largefluctuations aroundzero\nvalue, while nonzero positive (negative) values of the ex-\nchange spin torque Qnobserved at the left (right) edges\ncorrespond to the nonmonotonic left and right wings of\nthe spin torque profile. One may try to interpret the ob-\nservedresults in terms ofthe so called magnon relaxation\nlength (MRL) λm≈2/radicalBig\n(DkBT/¯h2)τmmτmp(Refs.5,6),\nwhereDis the spin-wave stiffness constant and τmm,mp\nare the magnon-magnon, and the magnon-phonon relax-\nation times, respectively. The MRL is a characteristic\nlength which results from the solution of the heat-rate\nequation for the coupled magnon-phonon system5. The\nphysical meaning of λmis an exponential drop of the\nspace distribution of the local magnon temperature for\nthe given external temperature gradient ∆ T. In other\nwords, although the externally applied temperature bias\nis kept constant, the thermal distribution for magnons\nis not necessarily linear. In general, one may suggest a\nsinh(x)-like spatial dependence5and a temperature de-7\npendence λm(T). Estimates of the MRL for the material\nparameters related to YIG (suppl. mater. of Ref.5) and\nTN= 0.2 [K] yield the following λm≈10 [µm]26. As\nseen from FIG. 8 the length starting from which the sat-\nuration of the spin current comes into play as long as\nthe FM-chain exceeds the length 20 [nm] ×100≈2 [µm].\nHowever,werecallthat MRLisawitnessofthe deviation\nbetween the magnon and phonon temperature profiles.\nTherefore, for interpretingthe nonmonotonicpartsofthe\nspin current profile (FIG. 8) in terms of the MRL one has\nto prove the pronounced deviation between magnon and\nphonontemperaturesattheboundaries. Forfurtherclar-\nification we calculate the magnon temperature profile.\nThis can be done self-consistently via the Langevin func-\ntion< Mz\nn>=L/parenleftbig\n< Mz\nn> Hn/kBTm\nn/parenrightbig\n. HereHz\nnis the\nzcomponent of the local magnetic field which depends\non the external magnetic field and the mean values of the\nadjacent magnetic moments < Mz\nn−1>, < Mz\nn+1>(\nsee eq. (4)). As inferred from the FIG. 10 the magnon\ntemperature profile follows the phonon temperature pro-\nfile. Prominent deviation between the phonon and the\nmagnon temperatures is observed only at the beginning\nof the chain and gradually decreases and becomes small\non the MRL scale. Close to the end of the chain the\ntemperature difference becomes almost zero. This means\nthat left nonmonotonic parts of the spin current profile\nFIG. 8 can be interpreted in terms of none-equilibrium\nprocesses. Comparing this result with the exchange spin\ntorque profile (FIG. 9) we see that in this part of the\nspin chain the exchange spin torque is positive. This\nis the reason why the spin current Inis increasing with\nthe site number n. The saturated plateau of the spin\ncurrent shown in FIG. 8 corresponds to the zero ex-\nchange spin torque Qn= 0 (cf. FIG. 9) and the decay\nof the spin Seebeck current Inat the right edge corre-\nsponds to the negative spin exchange torque Qn<0.\nThus, for the formation of the convex spin current profile\nthe key issue is not the difference between magnon and\nphonontemperatures,whichasweseeisprettysmall,but\nthe magnon temperature profile itself. The existence of\nthe hot(cold) magnetic moments with the local magnon\ntemperature up (below) the mean magnon temperature\ngenerates the spin current. This difference in the local\nmagnon temperature of the different magnetic moments\ndrives the spin current in the chain. On the other hand\nany measurement of the spin current done in the vicin-\nity of the right edge of the current profile will demon-\nstrate a non-vanishing spin current in the absence of the\ndeviation between the magnon and phonon temperature\nprofiles. This may serve as an explanation of the re-\ncent experiment25, where a non-vanishing spin current\nwas observed in the absence of the deviation between the\nmagnon and the phonon temperature profiles. We note\nthat zero values of the spin current shown in FIG. 8 is\nthe artefact of isolated magnetic insulator chain. Real\nmeasurement of the spin currents usually involve FM-\ninsulator/NM-interfaces. As will be shown below the in-\nterface effect described by an enhanced Gilbert dampingN=50N=500\nN=200\nN=150\nN=100\n1 100 200 300 400 500012\nSitenumber, nSpincurrent/Multiply1011/LBracket1/HBars/Minus1/RBracket1\nFIG. 8: The dependence of the averaged spin current on the\nlength of the FM-chain. Numerical parameters are α= 0.01\nandH0= 0 [K]. The temperature gradient is linear and the\nmaximum temperature is on the left-hand-side of the chain\n(T1= 100 [K]). In all cases the per-site temperature gradient\nis ∆T/N= 0.2 [K].\n0100200300400500/Minus0.10/Minus0.050.000.050.100.150.20\nSitenumber, n/ScriptA3\nΓQnz/Multiply1011/LBracket1/HBars/Minus1/RBracket1\nFIG. 9: The dependence of the exchange spin torque on\nthe site number. Numerical parameters are α= 0.01 and\nH0= 0 [K]. The temperature gradient is linear and the\nmaximum temperature is on the left-hand-side of the chain\n(T1= 100 [K]). The per-site temperature gradient is ∆ T/N=\n0.2 [K]. The exchange spin torque profile consists of three\nparts, the positive part corresponds to the high temperatur e\ndomain and low temperature domain corresponds to the neg-\native exchange spin torque. In the middle of the chain where\nthe spin current is constant, the exchange spin torque fluctu -\nates in the vicinity of the zero value.\nand the spin torque lead to a nonzero spin current at the\ninterfaces which is actually measured in the experiment.\nF. Role of the external magnetic filed ( H0/negationslash= 0)\nIt follows from our calculations that the dependence of\nthe longitudinal spin current on the magnetic field is not8\n/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bu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200 300 400 500020406080100\nSitenumber, nTemperature/LBracket1K/RBracket1\nFIG. 10: The magnon temperature profile (line) formed in the\nsystem. Numerical parameters are α= 0.01 andH0= 0 [K].\nBlue line corresponds to the applied linear phonon tempera-\nture profile. The maximum temperature on the left-hand-side\nof the chain is( T1= 100 [K]). The per-site temperature gradi-\nent is ∆T/N= 0.2 [K]. The maximal deviation between the\nphonon and magnon temperatures is observed only at the left\nedge of chain. The difference between temperatures graduall y\ndecreases and becomes almost zero for the sites with n >400.\ntrivial. Once the external static magnetic field is applied\nperpendicularly to the FM-chain and along the easy axis\nat the same time, we can suppress the spin current at ele-\nvated magnetic fields (FIG. 11). The threshold magnetic\nfield is - asexpected - the strength ofthe anisotropyfield,\ni.e. 2K1/MS∼0.056 [T]. By applying magnetic fields\nmuch higher than 0 .056 [T], the magnetic moments are\nfully aligned along the field direction and hence the X-,\nY-components of the magnetization required to form the\nZ-component of the longitudinal averaged spin current\nvanish.\nIn the case of the magnetic field being applied perpen-\ndicularly to the easy axis, the behavior becomes more\nrich (FIG. 12). In analogy with the situation observed\nin FIG. 11 there are no sizeable changes for the In(∆T)-\ndependence at low static fields. This is the regime where\nthe anisotropy field is dominant. In contrast to the Hz\n0\napplied field, the spin current does not linearly depend\non the strength of the field (inset of FIG. 11), which is\nexplained by the presence of different competing contri-\nbutions in the total energy density and not a simple cor-\nrection of the Z-component of the anisotropy field illus-\ntrated in the previous figure. Surprisingly, the magnetic\nfield oriented along the FM-chain can also suppress the\nappearance of the spin current’s profile. Also in this case\nthe strong magnetic field destroys the formation of the\nmagnetization gradient resulting from the applied tem-\nperature bias./SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle\n/Circle/Circle/Circle/Circle/Circle/Circle\n/SolidSquare/SolidSquare/SolidSquare/SolidSquare/SolidSquare/SolidSquare\n/Square/Square/Square/Square/Square/Square/SolidCircleH0z/EquΑl0/LBracket1T/RBracket1\n/CircleH0z/EquΑl10/Minus2/LBracket1T/RBracket1\n/SolidSquareH0z/EquΑl10/Minus1/LBracket1T/RBracket1\n/SquareH0z/EquΑl1/LBracket1T/RBracket1\n0 10 20 30 40 500123\n/DifferenceDeltaT/LBracket1K/RBracket1Spincurrent I26/Multiply1011/LBracket1/HBars/Minus1/RBracket1\nFIG. 11: Effect of the external magnetic field applied paralle l\nto the easy axis on the averaged spin current. Numerical\nparameters are ∆ T= 50 [K], α= 0.01 andN= 50. The\ntemperature gradient is linear and the maximum temperature\nis on the left-hand-side of the chain.\n/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle\n/Circle/Circle/Circle/Circle/Circle/Circle\n/SolidSquare/SolidSquare/SolidSquare/SolidSquare\n/SolidSquare/SolidSquare /Square/Square/Square/Square/Square/Square/SolidCircleH0x/EquΑl0/LBracket1T/RBracket1\n/CircleH0x/EquΑl10/Minus2/LBracket1T/RBracket1\n/SolidSquareH0x/EquΑl10/Minus1/LBracket1T/RBracket1\n/SquareH0x/EquΑl1/LBracket1T/RBracket1\n0 10 20 30 40 5001234\n/DifferenceDeltaT/LBracket1K/RBracket1Spincurrent I26/Multiply1011/LBracket1/HBars/Minus1/RBracket1\n/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle\n/SolidUpTriangle/SolidUpTriangle\n/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/DifferenceDeltaT/EquΑl50/LBracket1K/RBracket1\n0.020.040.060.080.1001234\nH0x/LBracket1T/RBracket1I26/Multiply1011/LBracket1/HBars/Minus1/RBracket1\nFIG. 12: Effect of the external magnetic field applied perpen-\ndicularly to the easy axis on the averaged spin current. Nu-\nmerical parameters are ∆ T= 50 [K], α= 0.01 andN= 50.\nThe temperature gradient is linear and the maximum tem-\nperature is on the left-hand-side of the chain.\nV. INTERFACE EFFECTS\nThe experimental setup to detect the spin current\nmight involve a NM adjacent to the spin-current gener-\nating substance, e.g. a FM insulator. This NM converts\nthe injected spin current from the FM to an electric cur-\nrent via ISHE1,5,27. So it is of interest to see the effect\nof the adjacent NM on the generated spin current in the\nconsidered chain. Obviously, the main effects appear in\nthe FM-NM interface. The interface effect can be di-\nvided into two parts which is described in the following\nsubsections.9\nA. Spin pumping and enhanced Gilbert damping\nInmagnetic insulators , chargedynamicsislessrelevant\n(in our model, anyway), and in some cases the dissipa-\ntive losses associated with the magnetization dynamics\nare exceptionally low (e.g. in YIG28α= 6.7×10−5).\nWhen a magnetic insulator is brought in contact with\nanormal metal , magnetization dynamics results in spin\npumping, which in turn causes angular momentum being\npumped to the NM. Because of this nonlocal interaction,\nthe magnetization losses become enhanced21.\nIf we consider the normal metal as a perfect spin\nsinkwhich remains in equilibrium even though spins are\npumped into it (which means there is a rapid spin relax-\nation and no back flow of spin currents to the magnetic\ninsulator), the magnetization dynamics is described by\nthe LLG equation with an additional torque originating\nfrom the FM-insulator/NM interfacial spin pumping21\n∂/vectorM\n∂t=−γ/bracketleftBig\n/vectorM×/vectorHeff/bracketrightBig\n+α\nMS/bracketleftBigg\n/vectorM×∂/vectorM\n∂t/bracketrightBigg\n+/vector τsp,(11)\nwhere\n/vector τsp=γ¯h\n4πM2\nSgeffδ(x−L)/bracketleftBigg\n/vectorM×∂/vectorM\n∂t/bracketrightBigg\n,(12)\nwhereLis the position of the interface, eis the elec-\ntron charge and geffis the real part of the effective\nspin-mixing conductance. In the YIG-Pt bilayer the\nmaximum measured effective spin-mixing conductance is\ngeff= 4.8×1020[m−2] Ref.21. In fact if the spin pumping\ntorque should be completely described, one should add\nanother torque containing the imaginary part of geff29.\nHowever, we omit this imaginary part here because it\nhas been found to be too small at FM-NM interfaces30.\nThe aforementioned spin pumping torque concerns the\ncases that we characterized with /vectorM. In our discrete\nmodel which includes a chain of Nferromagnetic cells,\nwe describe the above phenomena as follows\n∂/vectorMn\n∂t=−γ/bracketleftBig\n/vectorMn×/vectorHeff\nn/bracketrightBig\n+α\nMS/bracketleftBigg\n/vectorMn×∂/vectorMn\n∂t/bracketrightBigg\n+/vector τsp\nn,\n(13)\nwhere\n/vector τsp\nn=γ¯h2\n2ae2M2\nSg⊥δnN/bracketleftbigg\n/vectorMn×∂Mn\n∂t/bracketrightbigg\n,(14)\nwhich means the spin pumping leads to an enhanced\nGilbert damping in the last site\n∆α=γ¯h\n4πaMsgeff. (15)\nAs mentioned, the above enhanced Gilbert damping\ncould solely describe the interfacial effects as long as\nwe treat the adjacent normal metal as a perfect spinsink without any back flow of the spin current from the\nNM17,21. The latter is driven by the accumulated spins\nin the normal metal. If we model the normal metal as\na perfect spin sink for the spin current, spin accumu-\nlation does not build up. This approximation is valid\nwhen the spin-flip relaxation time is very small and so\nit prevents any spin-accumulation build-up. So the spins\ninjected by pumping decayand/orleavethe interfacesuf-\nficiently fast and there won’t be any backscattering into\nthe ferromagnet13,31. We note by passing that in a re-\ncentstudy concerningthis phenomena, it hasbeen shown\nthat spin pumping (and so enhanced Gilbert damping)\ndepends on the transverse mode number and in-plane\nwave vector21.\nB. Spin transfer torque\nIt was independently proposed by Slonczewski32and\nBerger33that the damping torque in the LLG equation\ncould have a negative sign as well, corresponding to a\nnegative sign of α. This means that the magnetization\nvector could move into a final position antiparallel to the\neffective field. In order to achieve this, energy has to be\nsupplied to the FM system to make the angle between\nthe magnetization and the effective field larger. This en-\nergy is thought to be provided by the injection of a spin\ncurrent/vectorIincidentto the FM13,29,34\n/vector τs=−γ\nM2\nSV/bracketleftBig\n/vectorM×/bracketleftBig\n/vectorM×/vectorIinjected/bracketrightBig/bracketrightBig\n,(16)\nwhich describes the dynamics of a monodomain ferro-\nmagnet of volume Vthat is subject to the spin current\n/vectorIincidentand modifies the right-hand side of the LLG\nequation as a source term. In general, a torque-term\nadditional to the Slonczewskis torque (eq. (16)) is also\nallowed29,35\n/vector τsβ=−γ\nMSVβ/bracketleftBig\n/vectorM×/vectorIincident/bracketrightBig\n, (17)\nwhereβgives the relative strength with respect to the\nSlonczewski’s torque (eq. (16)).\nFor the case of a FM-chain, again we assume that the\nabovespin-transfertorques act solelyon the last FM cell.\nC. Numerical results for interface effects\nInordertosimulatetheenhancedGilbertdampingand\nthe spin–transfer torque we assume that they act only on\nthechainend(motivatedbytheiraforementionedorigin).\nSo the dynamics of our FM-chain is described by the\nfollowing LLG18,36equations\n∂/vectorMn\n∂t=−γ\n1+α2/bracketleftBig\n/vectorMn×/vectorHeff\nn/bracketrightBig\n−γα\n(1+α2)MS/bracketleftBig\n/vectorMn×/bracketleftBig\n/vectorMn×/vectorHeff\nn/bracketrightBig/bracketrightBig\n,\nn= 1,...,(N−1),(18)10\nand\n∂/vectorMN\n∂t=−γ\n1+α2\nN/bracketleftBig\n/vectorMN×/vectorHeff\nN/bracketrightBig\n−γαN\n(1+α2\nN)MS/bracketleftBig\n/vectorMN×/bracketleftBig\n/vectorMN×/vectorHeff\nN/bracketrightBig/bracketrightBig\n−γ\nM2\nSa3/bracketleftBig\n/vectorMN×/bracketleftBig\n/vectorMN×/vectorIinjected/bracketrightBig/bracketrightBig\n−γ\nMSa3β/bracketleftBig\n/vectorMN×/vectorIincident/bracketrightBig\n,(19)\nwhereαN=α+γ¯hgeff/(4πaMs).\nEq.(18) and (19) describe the magnetizationdynamics\nin the presence of the interface effects and include both\nspinpump and spintorqueeffects. Results inthe absence\nof the spin torque are presented at the FIG. 13. The en-\nhanced Gilbert damping captures losses of the spin cur-\nrent associated with the interface effect. A nonzero spin\ncurrent corresponding to the last n= 500 spin quantifies\nthe amount of the spin current pumped into the normal\nmetal from the magnetic insulator. However, the con-\nvex profile of the spin current is observed as well in the\npresence ofthe interface effects. The influence ofthe spin\ntorqueonthespincurrentprofileisshowninFIG.14. We\nsee from these results, the large spin torque reduces the\ntotalspincurrentfollowingthroughtheFM-insulato/NM\ninterfaces. The spin torque current is directed opposite\nto the spin pump current and therefore compensates it.\n/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSqu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ircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle\n/SmallCircle\n0 100 200 300 400 500012\nSitenumber, nSpincurrent/Multiply1011/LBracket1/HBars/Minus1/RBracket1\n/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle\n/SmallCircle\n450 475 500012\nnWith enhanced \ndamping, /DifferenceDeltaΑ=0.5\nWithout enhanced \ndamping, /DifferenceDeltaΑ=0.5\nFIG. 13: Statistically averaged spin current in the chain of\nN= 500-sites. Numerical parameters are ∆ T= 100 [K],\nα= 0.01 andH0= 0 [T]. The temperature gradient is lin-\near and the maximum temperature is on the left-hand-side\nof the chain ( T1). The blue curve shows the averaged spin\ncurrent when no enhanced Gilbert damping and no spin–\ntransfer torque is present. The red curve shows the aver-\naged spin current when the enhanced Gilbert damping with\ngeff= 1.14×1022[m−2] is present. The inset shows the aver-\naged spin current of the last fifty sites only./SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare\n/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidS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SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle\n/SmallCircle\n0 100 200 300 400 500012\nSitenumber, nSpincurrent/Multiply1011/LBracket1/HBars/Minus1/RBracket1\n/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare\n/SmallSolidSquare/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle\n/SmallCircle\n450 475 500012\nnIn both cases /DifferenceDeltaΑ=0.5\nincidentIincident/EquΑl\n1.03/Multiply1015/LParen1/Minus1, 0, 0/RParen1/LBracket2/HBars/Minus1/RBracket2\nIincident/EquΑl\n5.15/Multiply1015/LParen1/Minus1, 0, 0/RParen1/LBracket2/HBars/Minus1/RBracket2\nFIG. 14: Statistically averaged spin current in the chain of\nN= 500 when there are both the enhanced Gilbert damp-\ning and the spin–transfer torque. Numerical parameters are\n∆T= 100 [K], α= 0.01,H0= 0 [T], geff= 1.14×1022[m−2]\nandβ= 0.01. The temperature gradient is linear and the\nmaximum temperature is on the left-hand-side of the chain\n(T1). The blue curve has /vectorIincident= 1×1015(−1,0,0) [¯hs−1]\nand the red curve is with /vectorIincident= 5×1015(−1,0,0) [¯hs−1].\nVI. MECHANISMS OF THE FORMATION OF\nSPIN EXCHANGE TORQUE AND SPIN\nSEEBECK CURRENT\nIn the previous sections we demonstrated the direct\nconnection between the spin Seebeck current profile and\nthe exchange spin torque. Here we consider the mecha-\nnisms of the formation of the exchange spin torque. For\nthis purpose we investigate changes in the magnetization\nprofile associatedwith the changeof the magnontemper-\nature<∆Mz\nn>=< Mz\nn>−< Mz\n0n>, where< Mz\nn>\nis the mean component of the magnetization moment\nfor the case of the applied linear thermal gradient, while\n< Mz\n0n>correspondstothemeanmagnetizationcompo-\nnent in the absence of thermal gradient ∆ T= 0. Quan-\ntity<∆Mz\nn>defines the magnon accumulation as the\ndifferencebetweentherelativeequilibriummagnetization\nprofile and excited one Ref.37and is depicted in FIG. 15.\nWe observe a direct connection between the magnon ac-\ncumulation effect and the exchange spin torque. A pos-\nitive magnon accumulation, meaning an excess of the\nmagnonscomparedtotheequilibriumstateisobservedin\nthe high temperature part of the chain. While in the low\ntemperature part the magnon accumulation is negative\nindicatingalackofmagnonscomparedtotheequilibrium\nstate. The exchange spin torque is positive in the case of\nthe positive magnon accumulation and is negative in the\ncase of the negative magnon accumulation (the exchange\nspin torque vanishes in the equilibrium state). From the\nphysical point of view, the result is comprehensible: the\nspin Seebeck current is generated by the magnon accu-\nmulation, transmitted through the equilibrium part of\nthe chain and partially absorbed in the part of the chain\nwith a negative magnon accumulation.11\n/SolidCircle/SolidCircle/SolidCircle\n/SolidCircle/SolidCircle\n/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle\n/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle\n/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle\n/SolidCircle/SolidCircle/SolidCircle\n/SolidCircle\n/SolidCircle\n50 100 150 200/Minus4/Minus202\nSitenumber, nMagnon accumulation10/Minus5/Multiply/LParen1m/Minusm0/RParen1\n15010015020020\nnTn/LBracket1K/RBracket1\n050100150200/Minus0.10/Minus0.050.000.050.100.15\nn/ScriptA3\nΓQnz/Multiply1011/LBracket1/HBars/Minus1/RBracket1\nFIG. 15: Site dependence of the exchange spin torque and\nthe magnon accumulation effect. We observe a direct connec-\ntion between magnon accumulation effect and the exchange\nspin torque. A positive magnon accumulation, i.e. excess of\nthe magnons, is observed in the high temperature part of the\nchain. While in the low temperature part magnon accumu-\nlation is negative (lack of magnons compare to the equilib-\nrium state). The exchange spin torque is positive for posi-\ntive magnon accumulation, and negative for negative magnon\naccumulation. The spin Seebeck current is generated by ex-\ncess magnons, transmitted through the equilibrium part of\nthe chain and partially absorbed in the region with magnon\ndrain.\nVII. CONCLUSIONS\nBased on the solution of the stochastic Landau-\nLifshitz-Gilbert equation discretized for a ferromagnetic\nchain in the presence of a temperature gradient formed\nalong the chain, we studied the longitudinal spin See-\nbeck effect with a focus on the space-dependent effects.\nIn particular, we calculated a longitudinal averaged spin\ncurrent as a function of different temperature gradients,\ntemperature gradient strengths, distinct chain lengths\nand differently oriented external static magnetic fields.\nOur particular interest was to explain the mechanisms\nof the formation of the spin Seebeck current beyond the\nlinear response regime. The merit was in pointing out a\nmicroscopicmechanismfortheemergenceofthespinSee-beck current in a finite-size system. We have shown that,\nwithin our model, the microscopic mechanism of the spin\nSeebeck current is the magnon accumulation effect quan-\ntified in terms of the exchange spin torque. We proved\nthat the magnon accumulation effect drives the spin See-\nbeck current even in the absence of significant deviation\nbetween magnon and phonon temperature profiles. Our\ntheoretical findings are in line with recently observed ex-\nperimental results25where non-vanishing spin Seebeck\ncurrent was observed in the absence of a temperature\ndifference between phonon and magnon baths.\nConcerningthe influence ofthe external constant mag-\nnetic fields on the spin Seebeck current we found that\ntheir role is nontrivial: An external static magnetic field\napplied perpendicularly to the FM-chain and along the\neasy axis may suppress the spin current at elevated mag-\nnetic fields (FIG. 11). The threshold magnetic field has a\nstrengthofthe anisotropyfield, i.e. 2 K1/MS∼0.056[T].\nIn the case of the magnetic field applied perpendicu-\nlarly to the easy axis, we observe a more complex be-\nhavior (FIG. 12). In analogy with the situation seen in\nFIG. 11 there are no sizeable changes for the In(∆T)-\ndependence at low static fields. This is the regime where\nthe anisotropy field is dominant. In contrast to the Hz\n0\napplied field, it does not linearly depend on the strength\nof the field (inset of FIG. 11), which is explained by the\npresence of different competing contributions in the total\nenergyand not a simple correctionof the Z-componentof\nthe anisotropy field. Notably, the magnetic field oriented\nalong the FM-chain can also suppress the emergence of\nthe spin current’s profile. Also in this case a strong mag-\nnetic field destroys the formation of the magnetization\ngradient resulting from the applied temperature bias.\nIn addition, we modeled an interface formed by a\nnonuniformly magnetized finite size ferromagnetic insu-\nlator and a normal metal (e.g., YIG-Platinum junction)\nto inspect the effects of the enhanced Gilbert damping\non the formation of space-dependent spin current within\nthe chain.\nVIII. ACKNOWLEDGEMENTS\nThe financial support by the Deutsche Forschungsge-\nmeinschaft (DFG) is gratefully acknowledged.\n1K. Uchida, S. Takahashi, K. Harii, J. Ieda, W. Koshibae,\nK. Ando, S. Maekawa, and E. Saitoh, Nature 455, 778\n(2008).\n2M. Hatami, G. E. W. Bauer, Q. Zhang, and P. J. Kelly,\nPhys. Rev.B 79, 174426 (2009); A. D.Avery, M. R. Pufall,\nand B. L. Zink, Phys. Rev. Lett. 109, 196602 (2012); C.\nH. Wong, H. T. C. Stoof, and R. A. Duine, Phys. Rev. A\n85, 063613 (2012).\n3S. Bosu, Y. Sakuraba, K. Uchida, K. Saito, T. Ota, E.Saitoh, and K.Takanashi, Phys. Rev. B 83, 224401 (2011);\nC. M. Jaworski, R. C. Myers, E. Johnston-Halperin, J. P.\nHeremans, Nature 487, 210 (2012); D. G. Rothe, E. M.\nHankiewicz, B. Trauzettel, and M. Guigou, Phys. Rev. B\n86, 165434 (2012).\n4C. M. Jaworski, J. Yang, S. Mack, D. D. Awschalom, J. P.\nHeremans and R. C. Myers, Nat. Mater. 9, 898 (2010); A.\nSlachter, F. L. Bakker, and B. J. van Wees, Phys. Rev. B\n84, 020412(R) (2011); F. K. Dejene, J. Flipse, and B. J.12\nvan Wees, Phys. Rev. B 86, 024436 (2012).\n5K. Uchida, J. Xiao, H. Adachi, J. Ohe, S. Takahashi, J.\nIeda, T. Ota, Y. Kajiwara, H. Umezawa, H. Kawai, G.\nBauer, S. Maekawa, and E. Saitoh, Nat. Mater. 9, 894\n(2010).\n6J. Xiao, G. E. W. Bauer, K. Uchida, E. Saitoh, and S.\nMaekawa, Phys. Rev. B 81, 214418 (2010).\n7K. Uchida, T. Nonaka, T. Ota, H. Nakayama, and E.\nSaitoh, Appl. Phys. Lett. 97, 262504(2010); D. Qu, S.Y.\nHuang, J. Hu, R. Wu, and C. L. Chie, Phys. Rev. Lett.\n110, 067206 (2013); M. Weiler, H. Huebl, F. S. Goerg, F.\nD. Czeschka, R. Gross, and S. T. B. Goennenwein, Phys.\nRev. Lett. 108, 176601 (2012); M. R. Sears and W. M.\nSaslow, Phys. Rev. B 85, 035446 (2012).\n8M. R. Sears and W. M. Saslow Phys. Rev. B 85, 035446\n(2012); R. Jansen, A. M. Deac, H. Saito, and S. Yuasa,\nPhys. Rev. B 85, 094401 (2012).\n9J. Torrejon, G. Malinowski, M. Pelloux, R. Weil, A. Thi-\naville, J. Curiale, D. Lacour, F. Montaigne, and M. Hehn,\nPhys. Rev. Lett. 109, 106601 (2012).\n10N. Li, J. Ren, L. Wang, G. Zhang, P. H¨ anggi, B. Li, Rev.\nMod. Phys. 84, 1045 (2012); J. Borge, C. Gorini, and R.\nRaimondi, Phys. Rev. B 87, 085309 (2013); S.Y. Huang,\nX. Fan, D. Qu, Y. P. Chen, W. G. Wang, J. Wu, T.Y.\nChen, J. Q. Xiao, and C. L. Chien, Phys. Rev. Lett. 109,\n107204 (2012).\n11C.-L. Jia and J. Berakdar, Phys. Rev. B 83, 180401R\n(2011).\n12H. Adachi, K. Uchida, E. Saitoh, and S. Maekawa, Rep.\nProg. Phys. 76, 036501 (2013).\n13Y. Tserkovnyak, A. Brataas, and G. E. W. Bauer, Phys.\nRev. Lett. 88, 117601 (2002).\n14L. Chotorlishvili, Z. Toklikishvili, V. K. Dugaev, J. Bar-\nnas, S. Trimper, and J. Berakdar Phys. Rev. B 88, 144429\n(2013).\n15Nonlinear Magnetization Dynamics in Nanosystems , G.\nBertotti, C. Serpico, I. D. Mayergoyz, Elsevier, Amster-\ndam (2009).\n16T. Bose and S. Trimper, Phys. Rev. B 81, 104413 (2010).\n17S. Hoffman, K. Sato, and Y. Tserkovnyak, Phys. Rev. B\n88, 064408 (2013).\n18T. L. Gilbert, Phys. Rev. 100, 1243 (1955) (abstract only);\nIEEE Trans. Magn. 40, 3443 (2004).\n19W. T. Coffey, Y. P. Kalmykov, J. Appl. Phys. 112, 121301\n(2012).\n20T. Kikkawa, K. Uchida, Y. Shiomi, Z. Qiu, D. Hou, D.Tian, H. Nakayama, X.-F. Jin, and E. Saitoh, Phys. Rev.\nLett.110, 067207 (2013).\n21A. Kapelrud and A. Brataas, Phys. Rev. Lett. 111, 097602\n(2013).\n22Numerical Solution of SDE through Computer Experi-\nments, P. E. Kloeden, E. Platen, H. Schurz, Springer,\nBerlin (1991).\n23Stochastic Processes in Physics and Chemistry , N. G. van\nKampen, Elsevier, Amsterdam (2007).\n24A. Sukhov, and J. Berakdar, J. Phys.: Condens. Matter\n20, 125226 (2008).\n25M. Agrawal, V. I. Vasyuchka, A. A. Serga, A. D.\nKarenowska, G. A. Melkov, and B. Hillebrands, Phys. Rev.\nLett.111, 107204 (2013).\n26We note that one should properly rescale the magnon-\nmagnon and the magnon-phonon relaxation times given\nin Ref.5, since we use in the numerical calculations the op-\ntimized values for damping. It is shown in Ref.6, that the\nrelaxation times are inversely proportional to the magne-\ntization damping.\n27E. Saitoh, M. Ueda, H. Miyajima, and G. Tatara, Appl.\nPhys. Lett. 88, 182509 (2006).\n28Y. Kajiwara, K. Harii, S. Takahashi, J. Ohe, K. Uchida,\nM. Mizuguchi, H. Umezawa, H. Kawai, K. Ando, K.\nTakanashi, S. Maekawa, and E. Saitoh, Nature 464, 262\n(2010).\n29A. Brataas, A. D. Kent, and H. Ohno, Nat. Mater. 11, 372\n(2012).\n30X. Jia, K. Liu, K. Xia, and G. E. W. Bauer, Europhys.\nLett.96, 17005 (2011).\n31Y. Tserkovnyak, A. Brataas, G. E. Bauer, and B. I.\nHalperin, Rev. Mod. Phys. 77, 1375 (2005).\n32J. C. Slonczewski, J. Magn. Magn. Mater. 159, L1 (1996).\n33L. Berger, Phys. Rev. B 54, 9353 (1996).\n34L. Chotorlishvili, Z. Toklikishvili, A. Sukhov, P. P. Horle y,\nV. K. Dugaev, V. R. Vieira, S. Trimper, and J. Berakdar,\nJ. Appl. Phys. 114, 123906 (2013).\n35Magnetism: From Fundamentals to Nanoscale Dynamics ,\nJ. St¨ ohr and H. C. Siegmann, Springer, Berlin, (2006).\n36L. D. Landau, E. M. Lifshitz, Phys. Z. Sowjetunion 8, 153\n(1935).\n37D. Hinzke, and U. Nowak, Phys. Rev. Lett. 107, 027205\n(2011); U. Ritzmann, D. Hinzke, and U. Nowak, Phys.\nRev. B.89, 024409 (2014)."    },    {        "title": "1402.7237v1.A_new_way_to_evaluate_x_ray_Brillouin_scattering_data.pdf",        "content": "arXiv:1402.7237v1  [cond-mat.dis-nn]  28 Feb 2014A new way to evaluate x-ray Brillouin scattering data\nU. Buchenau∗\nJ¨ ulich Center for Neutron Science, Forschungszentrum J¨ u lich\nPostfach 1913, D–52425 J¨ ulich, Federal Republic of German y\n(Dated: February 28, 2014)\nMaking use of the classical second moment sum rule, it is poss ible to convert a series of constant- Q\nx-ray Brillouin scattering scans ( Qmomentum transfer) into a series of constant frequency scan s\nover the measured Qrange. The method is applied to literature results for the ph onon dispersion in\nliquid vitreous silica and in glassy polybutadiene. It turn s out that the constant frequency scans are\nagain well fitted by the damped harmonic oscillator function , but now in terms of a Q-independent\nphonon damping depending exclusively on the frequency. At l ow frequency, the sound velocity and\nthe damping of both evaluations agree, but at higher frequen cies one gets significant differences.\nThe results in silica suggest a new interpretation of x-ray B rillouin data in terms of a strong mixing\nof longitudinal and transverse phonons toward higher frequ encies. The results in polybutadiene\nenlighten the crossover from Brillouin to Umklapp scatteri ng.\nPACS numbers: 78.35.+c, 63.50.Lm\nOur knowledge of the sound waves at and above the\nboson peak in glasses is to a large part due to x-ray Bril-\nlouin scattering measurements [1–5], which allow to see\nthe longitudinal partofthe soundwavemotionin the fre-\nquency range between 2 and 20 meV. The experimental\narrangement makes scans of S(Q,ω) at constant momen-\ntum transfer Qmuch easier than constant- ωscans [3]. It\nis usual to fit such a constant- Qscan in terms of the\ndamped harmonic oscillator function, the so-called DHO\nS(Q,ω)\nS(Q)=fQδ(ω)+1−fQ\nπΩ2\nQΓQ\n(ω2−Ω2\nQ)2+ω2Γ2\nQ.(1)\nHeresymbolswiththeindex Qdependonthemomentum\ntransferQ, butnoton the frequency ω. ΩQis the sound\nwave frequency, which defines the sound velocity cQ=\nΩQ/Qat thisQ; ΓQis the damping of the sound wave,\nandfQis the elastic (in liquids quasielastic) fraction of\nthe scattering at this Q.\nThe weak point of this evaluation is the following: The\nstrong damping which one fits to the sound waves above\nthe boson peak is not a real physical damping of the vi-\nbrations at the sound wave frequency. Instead, it reflects\na deviation of the eigenvectors from a perfect sine func-\ntioninspace. Thus, it isnotadampingforallfrequencies\nat fixedQ, as supposed by eq. (1), but rather a distribu-\ntion of sound wave vectors around an average one at the\ngiven frequency. It is a property of the frequency win-\ndow rather than a property of the momentum transfer\nwindow. In fact, this weak point can be directly seen at\nlargerQ, where the DHO fit has too much intensity close\nto the elastic line [1].\nOn the other hand, at most points in the relevant\n(Q,ω)-space, the DHO manages to fit the data very well.\nThus it certainly supplies a good parameter set for the\n∗Electronic address: buchenau-juelich@t-online.dedescription of S(Q,ω). The question is only whether the\nparameters are indeed meaningful.\nFortunately, it is easy to translate a set of DHO\nmeasurements at a series of different Qinto the set of\nconstant- ωscanswhichonewouldliketohave. Onenotes\nfirst that for a DHO (1 −fQ)S(Q) is fixed to the value\n(1−fQ)S(Q) =kBT\nMc2\nQ(2)\nby the classical second moment sum rule [6]\n/integraldisplay∞\n−∞ω2S(Q,ωdω=kBTQ2\nM, (3)\nwhereMis the average atomic mass.\nWith this equation, one can calculate a constant- ω\nscan ofS(Q,ω) for any ωin absolute units, each DHO-\nscan supplying a point at its Q-value. The result is best\nplottedintermsofthedimensionlessdynamicalstructure\nfactorFω(Q) defined by\nFω(Q) =Mω3S(Q,ω)\nkBTQ2, (4)\nwhich in terms of the DHO parameters is given by\nFω(Q) =1\nπΓQω3\n(ω2−Ω2\nQ)2+ω2Γ2\nQ. (5)\nFig. 1 shows scans of Fω(Q) for the beautiful Brillouin\nx-ray data of Baldi, Giordano, Monaco and Ruta [5] in\nvitreous silica at 1620 K. In such scans, the broadening\nof the phonons in Qcan be directly seen in their own\nfrequency window.\nIn order to fit these data, one uses the dynamic struc-\nture factor of a damped longitudinal phonon [6, 7]\nFω(Q) =fω\nπ(Γω/ω)Q2Q2\nB\n(Q2−Q2\nB)2+(Γω/ω)2Q4(6)2\nwhich is again the DHO, but now with parameters which\nno longer depend on Q. Instead, they depend on ωas\nthey should (remember that the phonon broadening is a\nproperty of the phonon in a given frequency window, be-\ncause it corresponds to a broadening in wavevectorspace\nandnotin frequency). The Brillouin wavevector QBde-\nfines the frequency-dependent longitudinal sound veloc-\nitycω=ω/QB.\nAsseenfromFig. 1, onegetsexcellentfitswitheq. (6).\nHowever, one can no longer reckon with the normaliza-\ntion property of the second moment sum rule. Therefore,\none needs not only the two parameters cωand Γω, but an\nadditional normalization factor fωas well. fωis found\nto decrease from 1 at low frequency to 0.75 at 25 meV,\nshowing a slow disappearance of the acoustic correlation\ntoward higher frequencies.\nThe fitting in terms of the three parameters fω,cω\nand Γωworks so well that one even gets reliable numbers\nwhen the phonon peak begins to leave the measured Q-\nrange, as in Fig. 1 (e). The fitted values for cωand Γω\nare plotted in Fig. 2. Below 10 meV, the exchange of\nscattering laws does not bring anything new: The new\nparameters cωand Γ ωagree within experimental error\nwith the old cQand Γ Qdetermined directly from the\nDHO scans. Nevertheless, even at these low frequencies\nthe exercise is useful: It is not enough to know, one must\nalso know that one can trust what one knows.\n/s48/s49/s50\n/s48/s46/s48/s48/s46/s53\n/s48/s46/s48/s48/s46/s50/s48/s46/s52\n/s48 /s49 /s50 /s51 /s52/s48/s46/s48/s48/s46/s50/s48/s46/s52/s40/s97/s41/s32/s53/s32/s109/s101/s86\n/s40/s98/s41/s32/s49/s48/s32/s109/s101/s86\n/s70\n/s105/s110/s99/s40/s99/s41/s32/s49/s53/s32/s109/s101/s86/s70 /s40/s81/s41\n/s70\n/s105/s110/s99\n/s109/s111/s109/s101/s110/s116/s117/s109/s32/s116/s114/s97/s110/s115/s102/s101/s114/s32/s81/s32/s40/s110/s109/s45/s49\n/s41/s115/s105/s108/s105/s99/s97/s32/s49/s54/s50/s48/s32/s75\n/s40/s100/s41/s32/s50/s48/s32/s109/s101/s86\nFIG. 1: Constant- ωscans of the dynamic structure factor\nFω(Q) calculated from the x-ray Brillouin scattering data of\nBaldi et al [5] in vitreous silica at 1620 K. The lines are fits\nin terms of eq. (6).Between10and25meV,thenewevaluationrevealsnot\nonlyasmallerdampingofthephonons,butalsoamarked\nincrease of the sound velocity with increasing frequency.\nA similar result has been obtained in the conventional\nway in glycerol [4], though there the initial decrease of\nthe sound velocity is more pronounced than in silica and\nthe subsequent hardening is less pronounced. This will\nalso be seen in our second example, polybutadiene.\nThe initial decrease and the subsequent hardening of\nthe sound velocity can be understood by considering a\nmechanism which so far has not been taken into account,\nnamely the possible interaction between longitudinal and\ntransversesoundwaves. Thereasonforthe dampingmay\nbe controversial, but whatever it is, the damping mech-\nanism affects both longitudinal and transverse phonons.\nThus one must expect a coupled longitudinal-transverse\nmode as soon as Γ Qis of the order of the difference of the\nsound wavefrequencies at the given Q(or, what amounts\nto the same, the wavevector broadening becomes of the\norder of the wavevector difference of longitudinal and\ntransverse phonons at the given frequency ω). This cou-\npled mode will be expected to consist to one third of\nlongitudinal and two thirds of transverse modes, because\nthere are two transverse modes.\nIn fact, if one extrapolates the measured values for the\nhigh frequency sound velocity in Fig. 2 (a) back to the\n/s48/s46/s53/s49/s46/s48/s49/s46/s53\n/s48 /s53 /s49/s48 /s49/s53 /s50/s48 /s50/s53/s48/s46/s48/s48/s46/s53/s49/s46/s48/s49/s46/s53/s40/s97/s41/s32/s115/s111/s117/s110/s100\n/s118/s101/s108/s111/s99/s105/s116/s105/s101/s115\n/s99\n/s116/s47/s99\n/s108/s115/s105/s108/s105/s99/s97/s32/s49/s54/s50/s48/s32/s75/s99/s47/s99\n/s108\n/s40/s98/s41/s32/s100/s97/s109/s112/s105/s110/s103/s47\n/s104 /s32/s40/s109/s101/s86/s41\nFIG. 2: Comparison of the DHO results of Baldi et al [5]\n(open symbols) with those fitted here (full symbols) for (a)\nthe sound velocity in units of the longitudinal light scatte ring\nBrillouin sound velocity cl(b) for the damping ratio Γ /ω.3\nfrequency zero, one arrives at the point corresponding to\na sound velocity cl/3+ 2ct/3, the sound velocity which\none expects for a mixed mode which is one third longi-\ntudinal and two thirds transverse.\nThisfindingsuggeststhefollowinginterpretationofthe\ndata: There is a marked increase of the sound velocities\nwith increasing frequency at all frequencies. However,\nat small frequency this increase is masked by the grad-\nual transformation of the longitudinal sound waves into\nmixed ones, which leads to a concomitant increase of the\nwavevectorofthe mixed phonon in a Brillouinscattering\nexperiment. In glycerol [4], this wavevectorincrease even\novercompensates the hardening, leading to an apparent\nsoftening of the phonons with increasingfrequency at the\nbeginning of the dispersion curve.\nThe fit of the silica data in Fig. 1 leaves an important\nopen question: Why does one not see any indication of\nthe Umklapp scattering, which dominates the dynamic\nstructure factor at higher momentum transfer? Natu-\nrally, x-ray scattering is coherent scattering, but if we\nhad incoherent scattering, then in the one-phonon ap-\nproximation\nSinc(Q,ω) =kBTQ2\n2Mg(ω)\nω2, (7)\nwhereg(ω) is the vibrationaldensity ofstates. This leads\n/s48/s46/s48/s48/s46/s53/s49/s46/s48/s49/s46/s53\n/s48/s46/s48/s48/s46/s53\n/s48/s46/s48/s48/s46/s53\n/s48 /s50 /s52 /s54 /s56 /s49/s48/s48/s46/s48/s48/s46/s53/s40/s97/s41/s32/s51/s32/s109/s101/s86\n/s70\n/s105/s110/s99/s40/s98/s41/s32/s53/s32/s109/s101/s86\n/s70\n/s105/s110/s99\n/s40/s99/s41/s32/s55/s32/s109/s101/s86/s70 /s40/s81/s41\n/s70\n/s105/s110/s99\n/s109/s111/s109/s101/s110/s116/s117/s109/s32/s116/s114/s97/s110/s115/s102/s101/s114/s32/s81/s32/s40/s110/s109/s45/s49\n/s41/s112/s111/s108/s121/s98/s117/s116/s97/s100/s105/s101/s110/s101/s32/s49/s52/s48/s32/s75\n/s40/s100/s41/s32/s57/s32/s109/s101/s86\nFIG. 3: Constant- ωscans of the dynamic structure factor\nFω(Q) calculated from the x-ray Brillouin scattering data of\nFioretto et al [9] in polybutadiene at 140 K. The continuous\nlines are fits in terms of eq. (6) with an added Umklapp term\n(the dashed lines in (c) and (d)) as explained in the text.to the constant dynamic structure factor Finc\nFinc=1\n2ωg(ω) (8)\naround which the coherent dynamic structure factor os-\ncillates at higher Q.\nFor vitreous silica, g(ω) has been measured [8] at 1673\nK, close enough to the temperature of the x-ray Brillouin\nexperiment [5] of 1620 K to calculate a reliable Finc. The\nvalues are shown by arrows in Fig. 1 (c) and (d), where\nthey begin to be of the order of the measured ones. Why\nis it then possible to evaluate the data without any Umk-\nlapp contribution?\nThe question is answeredby the evaluation ofBrillouin\nx-ray data for polybutadiene [9] in Fig. 3. In this case,\nthe data stretch to higher Q-values and the Umklapp\nscattering becomes indeed visible at higher momentum\ntransfer in Fig. 3 (c) and (d). But even though one mea-\nsures at more than twice the momentum transfer of Fig.\n1, the Umklapp scattering is still a factor of six to seven\nweakerthan the Finccalculated from a measurement [10]\nofg(ω) at 140 K.\nThe reason for this is the fact that the coherent scat-\ntering from any vibrational eigenmode must begin with\nQ4, because the sum over the atomic displacements in\nthe mode equals zero due to momentum conservation.\nThis implies that the coherent dynamic structure factor\n/s48/s46/s54/s48/s46/s56/s49/s46/s48/s49/s46/s50\n/s48 /s53 /s49/s48/s48/s49/s50/s40/s97/s41/s32/s115/s111/s117/s110/s100\n/s118/s101/s108/s111/s99/s105/s116/s105/s101/s115/s112/s111/s108/s121/s98/s117/s116/s97/s100/s105/s101/s110/s101/s32/s49/s52/s48/s32/s75/s99/s47/s99\n/s108\n/s40/s98/s41/s32/s100/s97/s109/s112/s105/s110/s103/s47\n/s104 /s32/s40/s109/s101/s86/s41\nFIG. 4: Comparison of the DHO results of Fioretto et al [9]\n(open symbols) with those fitted here (full symbols) for (a)\nthe sound velocity in units of the longitudinal light scatte ring\nBrillouin sound velocity cl(b) for the damping ratio Γ /ω.4\nFω(Q) begins with a Q2-term. Obviously, in the momen-\ntum transfer rangeofFig. 1 and Fig. 3, one is still in this\ninitial region and can fit the Umklapp scattering with an\nadditional fUQ2. This has been done for the data of Fig.\n3. The dashed curves in Fig. 3 (c) and (d) show the\nUmklapp contribution alone.\nFig. 4 compares the resulting sound velocities and\ndampings with those obtained by the conventional evalu-\nation. Again, the new evaluation provides more accurate\ndata for the sound velocity, showing the initial softening\nand even a subsequent small hardening. The hardening,\nhowever, is barely seen, similar to the findings in glycerol\n[4].\nFrom the two examples shown, it is obvious that one\ngets more (and more accurate) information from the new\nevaluation method proposed here, not only because it\nis better adapted to the physics, but also because it al-\nlows fora quantitative combination ofseveralconstant- Q\nscans to calculate the dynamic structure factor on an ab-\nsolute scale. For future experiments, it is naturally not\nnecessaryto fit with the DHO, because one can apply thesecond moment sum rule directly to the measured data.\nThis should allow to extend the method beyond the Bril-\nlouin scattering range, where the data are no longer well\nfitted by the DHO.\nTo conclude, the classical second moment sum rule al-\nlows to calculate constant energy Brillouin scans from\ndamped harmonicoscillatorfits ofconstant- Qscans. Ap-\nplying the method to measurements in liquid vitreous\nsilica, one finds a pronounced hardening of the Brillouin\nphonons at higher frequency, which suggests a radical\nchange of previous interpretations. Instead of an es-\nsentially frequency-independent or even softening sound\nvelocity, it now appears that one has to reckon with\na marked hardening towards higher frequencies in the\nwhole frequency range, a hardening which is only masked\nat the beginning by the mixing process of longitudinal\nand transverse phonons. The Umklapp scattering, not\nyet visible at the low momentum transfer of the vit-\nreous silica scans, could be identified in measurements\nof polybutadiene at higher momentum transfer and was\nfound to be surprisingly low in the Brillouin range.\n[1] B. Ruffl´ e, M. Foret, E. Courtens, R. Vacher, and G.\nMonaco, Phys. Rev. Lett. 90, 095502 (2003)\n[2] B. Ruffl´ e, G. Guimbretiere, E. Courtens, R. Vacher, and\nG. Monaco, Phys. Rev. Lett. 96, 045502 (2006)\n[3] C. Masciovecchio, A. Mermet, G. Ruocco and F. Sette,\nPhys. Rev. Lett. 85, 1266 (2000)\n[4] G. Monaco and V. M. Giordano, Proc. New York Ac. Sci.\n106, 3659 (2009)\n[5] G. Baldi, V. M. Giordano, G. Monaco, and B. Ruta,\nPhys. Rev. Lett. 104, 195501 (2010)\n[6] J.-P. Hansen and I. R. McDonald, Theory of simple liq-\nuids, 2nd ed. (Academic Press, New York 1986). Second\nmoment sum rule: ch. 7.4, eq. (7.4.40), p. 220; Rayleigh-Brillouin scattering: ch. 8.5, p. 275 ff.\n[7] J. P. Boon and S. Yip, Molecular Hydrodynamics , (Mc-\nGraw-Hill, New York 1980), ch. 6\n[8] A. Wischnewski, U. Buchenau, A. J. Dianoux, W. A.\nKamitakahara, and J. L. Zarestky, Phys. Rev.B 57, 2663\n(1998)\n[9] D. Fioretto, U. Buchenau, L. Comez, A.Sokolov, C. Mas-\nciovecchio, A. Mermet, G. Ruocco, F. Sette, L. Willner,\nB. Frick, D. Richter, and L. Verdini, Phys. Rev. E 59,\n4470 (1999)\n[10] U. Buchenau, Prog. Theor. Phys. Suppl. 126, 151 (1997)"    },    {        "title": "1403.3914v2.Interpolating_local_constants_in_families.pdf",        "content": "arXiv:1403.3914v2  [math.NT]  20 Aug 2015INTERPOLATING LOCAL CONSTANTS IN FAMILIES\nGILBERT MOSS\n1.Introduction\nLetG=GLn(F), letkbe an algebraically closed field of characteristic ℓ, and\nW(k) its ring of Witt vectors. By an ℓ-adic family of representations we mean an\nA[G]-moduleVwhereAis a commutative W(k)-algebra with unit; then each point\npofAgives aκ(p)[G]-moduleV⊗Aκ(p) whereκ(p) denotes the residue field at p.\nIn [EH12], Emerton and Helm conjecture a local Langlands correspo ndence for ℓ-\nadicfamiliesofadmissiblerepresentations. ToanycontinuousGaloisr epresentation\nρ:GF→GLn(A), they conjecturally associate an admissible smooth A[G]-module\nπ(ρ), which interpolates the local Langlands correspondence for poin tsA→κwith\nκcharacteristic zero. They prove that any A[G]-module which is subject to this\ninterpolation property and a short list of representation-theore tic conditions (see\n[EH12, Thm 6.2.1]) must be unique.\nIn [Hel12b], Helm further investigates the structure of π(ρ) by taking the list of\nrepresentation-theoretic conditions in [EH12, Thm 6.2.1] as a start ing point for the\ntheory of “co-Whittaker” A[G]-modules (see Section 2.5 below for the definitions).\nUsing this theory, he is able to reformulate the conjecture in terms of the existence\nof a certain homomorphism between the integral Bernstein center and a universal\ndeformation ring ([Hel12b, Thm 7.8]).\nRoughly speaking, representations of GLn(F) overCare completely determined\nby data involving only local constants ([Hen93]), and in particular th e bijections\nof the classical local Langlands correspondence are uniquely dete rmined using L-\nand epsilon-factors (see, for example, [Jia13]). However, L- and epsilon-factors are\nabsent from the the local Langlands correspondence in families. Th us it is natural\nto ask whether it is possible to attach L- and epsilon-factors to an ℓ-adic family\nsuch asπ(ρ) as in [EH12], or more generally any co-Whittaker A[G]-module, in a\nway that interpolates the L- and epsilon factors at each point.\nOverC,L-factorsL(π,X) arise as the greatest common denominator of the zeta\nintegralsZ(W,X;j) of a representation πasWvaries over the space W(π,ψ) of\nWhittakerfunctions (seeSections2.2, 3.1fordefinitions). Epsilon- factorsǫ(π,X,ψ)\nare the constant of proportionality (i.e. not depending on W) in a functional\nequation relating the modified zeta integralZ(W,X)\nL(π,X)to its pre-composition with\na Fourier transform. Here, the formal variable Xreplaces the complex variable\nq−(s+n−1\n2)appearingin [JPSS79] and otherliterature, and weconsiderthese objects\nas formal series.\nIt appears difficult to construct L-factors in a way compatible with arbitrary\nchange of coefficients. To see this, consider the following simple exam ple: letq≡\n1 modℓ, and letχ1,χ2:F×→W(k)×be smooth characters such that χ1is\nDate: August 17, 2021.\n12 GILBERT MOSS\nunramifiedbut χ2isramified, andsuchthat χ1≡χ2modℓ. Followingtheclassical\nprocedure (see for example [BH06, 23.2]) for finding a generator of the fractional\nideal of zeta integrals, we get L(χi,X)∈W(k)(X) and find that L(χ1,X) =\n1\n1−χ1(̟F)X, andL(χ2,X) = 1. Now let Abe the Noetherian local ring {(a,b)∈\nW(k)×W(k) :a≡bmodℓ}, which has two characteristic zero points p1,p2and\na maximal ideal ℓA. Letπbe theA[F×]-moduleA, with the action of F×given\nbyx·(a,b) = (χ1(x)a,χ2(x)b). Interpolating L(χ1,X) andL(χ2,X) would mean\nfindinganelement L(π,X)inA[[X]][X−1] suchthatL(π,X)≡L(χi,X) modℓfor\ni= 1,2, but such a task is impossible because L(χ1,X) andL(χ2,X) are different\nmodℓ.\nOn the otherhand, zeta integralsthemselvesseem to be much more well-behaved\nwith respect to specialization. Classically, zeta integrals form elemen ts of the quo-\ntient field C(X) ofC[X,X−1]. Our first result is identifying, for more arbitrary\ncoefficient rings A, the correct fraction ring in which our naive generalization of\nzeta factors will live:\nTheorem 1.1. SupposeAis aNoetherian W(k)-algebra. Let Sbe the multiplicative\nsubset ofA[X,X−1]consisting of polynomials whose first and last coefficients ar e\nunits. Then if Vis a co-Whittaker A[G]-module,Z(W,X;j)lies in the fraction\nringS−1(A[X,X−1])for allW∈ W(V,ψ)and for0≤j≤n−2.\nThe proof of rationality in the setting of representations over a fie ld relies on\na useful decomposition of a Whittaker function into “finite” functio ns ([JPSS79,\nProp 2.2]). In the setting of rings, such a structure theorem is lack ing, but certain\nelements of its proof can be translated into a question about the fin iteness of the\n(n−1)st Bernstein-Zelevinsky derivative. This finiteness property, c ombined with\na simple translation property of the zeta integrals, yields Theorem 1 .1 (see§3.2).\nClassically, zeta integrals satisfy a functional equation which does n ot involve\ndividingbythe L-factor. Theconstantofproportionalityin thisfunctionalequat ion\nis called the gamma-factorand equals ǫ(π,X,ψ)L(πι,1\nqnX)\nL(π,X), when theL-factormakes\nsense. Our second main result is that gamma-factors interpolate in ℓ-adic families\n(see§4.1 for details on the notation):\nTheorem 1.2. SupposeAis a Noetherian W(k)-algebra and suppose Vis a prim-\nitive co-Whittaker A[G]-module. Then there exists a unique element γ(V,X,ψ)of\nS−1(A[X,X−1])such that\nZ(W,X;j)γ(V,X,ψ) =Z(/tildewidew′W,1\nqnX;n−2−j)\nfor anyW∈ W(V,ψ)and for any 0≤j≤n−2.\nToproveTheorem1.2weusethe theoryoftheintegralBernsteinc entertoreduce\nto the characteristic zero case of [JPSS79].\nThe question of interpolating local constants in ℓ-adic families has been inves-\ntigated in a simple case by Vigneras in [Vig00]. For supercuspidal repre sentations\nofGL2(F) overQℓ, Vigneras notes in [Vig00] that it is known that epsilon factors\ndefine elements of Zℓ, and proves that for two supercuspidal integral representa-\ntions to be congruent modulo ℓit is necessary and sufficient that they have epsilon\nfactors which are congruent modulo ℓ(we call a representation with coefficients\nin a local field Eintegral if it stabilizes an OE-lattice). The classical epsilon and\ngamma factors are equal in the supercuspidal case, so when the s pecialization of anINTERPOLATING LOCAL CONSTANTS IN FAMILIES 3\nℓ-adic family at a characteristic zero point is supercuspidal, the gamm a factor we\nconstruct in this paper specializes to the epsilon factor of [JPSS79, Vig00]. Since\ntwo representations V1,V2overOEwhich are congruent mod mEdefine a family\nV1×VV2over the connected W(k)-algebra OE×kEOE, Theorems 1.1 and 1.2 give\nthe following corollary (implying the “necessary” part of [Vig00]):\nCorollary 1.3. LetKdenote the fraction field of W(k). Ifπandπ′are absolutely\nirreducible integral representations of GLn(F)over a coefficient field Ewhich is a\nfinite extension of K, then:\n(1)γ(π,X,ψ)andγ(π′,X,ψ)have coefficients in the fraction ring\nS−1(OE[X,X−1]).\n(2) IfmEis the maximal ideal of OE, andπ≡π′modmE, thenγ(π,X,ψ)≡\nγ(π′,X,ψ) modmE.\nThe question of extending the theory of zeta integrals to the ℓ-modular setting\nhas been investigated in [M ´12], and very recently in [KM14] for the Rankin-Selberg\nintegrals. The question of deforming local constants over polynom ial rings over\nChas been investigated by Cogdell and Piatetski-Shapiro in [CPS10], an d the\ntechniques of this paper owe much to those in [CPS10].\nAnalogous to the results of Bernstein and Deligne in [BD84] for RepC(G), Helm\nshows in [Hel12a, Thm 10.8] that the category RepW(k)(G) has a decomposition\ninto full subcategories known as blocks. Our third main result is cons tructing for\neach block a gamma factor which is universal in the sense that it gives rise via\nspecialization to the gamma factor for any co-Whittaker module in th at block. We\nwill now state this result more precisely.\nEach block of the category RepW(k)(G) corresponds to a primitive idempotent\nin the Bernstein center Z, which is defined as the ring of endomorphisms of the\nidentity functor. It is a commutative ring whose elements consist of collections of\ncompatible endomorphisms of every object, each such endomorph ism commuting\nwith all morphisms. Choosing a primitive idempotent eofZ, the ringeZis the\ncenter of the subcategory e·RepW(k)(G) of representations satisfying eV=V. The\nringeZhas an interpretation as the ring of regular functions on an affine alg ebraic\nvariety over W(k), whosek-points are in bijection with the set of unramified twists\nof a fixed conjugacy class of supercuspidal supports in Repk(G). See [Hel12a]\nfor details. In [Hel12b], Helm determines a “universal co-Whittaker m odule” with\ncoefficientsin eZ, denoted hereby eW, whichgivesrisetoanyco-Whittakermodule\nvia specialization (see Proposition 2.31 below). By applying our theory of zeta\nintegrals to eWwe get a gamma factor which is universal in the same sense:\nTheorem 1.4. SupposeAis any Noetherian W(k)-algebra, and suppose Vis a\nprimitive co-Whittaker A[G]-module. Then there is a primitive idempotent e, a\nhomomorphism fV:eZ →A, and an element Γ(eW,X,ψ)∈S−1(eZ[X,X−1])\nsuch thatγ(V,X,ψ) =fV(Γ(eW,X,ψ)).\nInterpolating gamma factors of pairs may be the next step in obtain ing a local\nconversetheorem for ℓ-adic families. By capturing the interpolation property, fami-\nlies ofgammafactorsmight givean alternativecharacterizationoft he co-Whittaker\nmoduleπ(ρ) appearing in the local Langlands correspondence in families.\nThe author would like to thank his advisor David Helm for suggesting th is prob-\nlem and for his invaluable guidance, Keenan Kidwell for his helpful conv ersations,\nand Peter Scholze for his helpful questions and comments at the MS RI summer4 GILBERT MOSS\nschool on New Geometric Techniques in Number Theory in 2013. He wo uld also\nlike to thank the referee for her/his very helpful comments and su ggestions.\n1.1.Notation and Conventions. LetFbe a finite extension of Qp, letqbe the\norder of its residue field, and let kbe an algebraically closed field of characteristic\nℓ, whereℓ/ne}ationslash=pis an odd prime. Denote by W(k) the ring of Witt vectors over\nk. The assumption that ℓis odd is made so that W(k) contains a square root\nofq. Whenℓ= 2 all the arguments presented will remain valid, after possibly\nadjoining a square root of qtoW(k). The letter GorGnwill always denote the\ngroupGLn(F). Throughout the paper Awill be a Noetherian commutative ring\nwhich is aW(k)-algebra, with additional properties in various sections, and κ(p)\nwill denote the residue field of a prime ideal pofA. For any locally profinite group\nH, RepA(H) denotes the category of smooth representations of Hover the ring A,\ni.e.A[H]-modules for which every element is stabilized by an open subgroup of H.\nEven when this category is not mentioned, all representations are presumed to be\nsmooth. When Hisa closedsubgroupof G, wedefine the non-normalizedinduction\nfunctor IndG\nH(resp. c-IndG\nH) : RepA(H)→RepA(G) sendingτto the smooth part\noftheA[G]-module, underrighttranslation, offunctions(resp. functionsc ompactly\nsupported modulo H)f:G→τsuch thatf(hg) =τ(h)f(g),h∈H,g∈G.\nThe integral Bernstein center of [Hel12a] (see the discussion prec eding Theorem\n1.4) will alwaysbe denoted by Z. IfVis in RepA(G), then it is also in RepW(k)(G),\nand we frequently use the Bernstein decomposition of RepW(k)(G) to interpret\nproperties of V.\nIfAhas a nontrivial ideal I, thenI·Vis anA[H]-submodule of V, which shows\nthat most content would be missing if we developed the representat ion theory of\nRepA(H) around the notion of irreducible objects, or simple A[H]-modules. Thus\nconditions appear throughout the paper which in the traditional se tting are implied\nby irreducibility:\nDefinition 1.5. VinRepA(H)will be called\n(1)Schurif the natural map A→EndA[G](V)is an isomorphism;\n(2)G-finiteif it is finitely generated as an A[G]-module.\n(3)primitive if there exists a primitive idempotent ein the Bernstein center Z\nsuch thateV=V.\nWe say a ring is connected if it has connected spectrum or, equivalen tly, no\nnontrivial idempotents, for example any local ring or integral doma in. Note that if\nAis connected, Corollary2.32implies all co-Whittaker A[G]-modules are primitive.\nDenote by Nnthe subgroup of Gnconsisting of all unipotent upper-triangular\nmatrices. Let ψ:F→W(k)×be an additive character of Fwith kerψ=p. Then\nψdefines a character on any subgroup of Nn(F) by\n(u)i,j/mapsto→ψ(u1,2+···+un−1,n);\nwe abusively denote this character by ψas well.\nIfHis asubgroupnormalizedby anothersubgroupgroup K, andθis a character\nof the group H, denote by θkthe character given by θk(h) =θ(khk−1) forh∈H,\nk∈K. ForVin RepA(H), denote by VH,θthe quotient V/V(H,θ) whereV(H,θ)\nis the sub-A-module generated by elements of the form hv−θ(h)vforh∈Hand\nv∈V; it isK-stable ifθk=θ,k∈K. Given a standard Levi subgroup M⊂GnINTERPOLATING LOCAL CONSTANTS IN FAMILIES 5\nwith unipotent radical U, and1the trivial character, we denote by JMthe non-\nnormalized Jacquet functor RepA(G)→RepA(M) :V/mapsto→VU,1.\nFor eachm≤n, letGmdenoteGLm(F) and embed it in Gvia (Gm0\n0In−m).\nWe let{1}=P1⊂ ··· ⊂Pndenote the mirabolic subgroups of G1⊂ ··· ⊂Gn,\nwherePmis given by/braceleftbig\n(gm−1x\n0 1) :gm−1∈Gm−1, x∈Fm−1/bracerightbig\n. We also have the\nunipotent upper triangular subgroup UmofPmgiven by/braceleftbig\n(Im−1x\n0 1) :x∈Fm−1/bracerightbig\n.\nIn particular, Um≃Fm−1andPm=UmGm−1. Note that this is different from\nthe groups N(r) defined in Proposition 2.3.\nSinceGncontains a compact open subgroup whose pro-order is invertible in\nW(k), there exists a unique (for that choice of subgroup) normalized H aar mea-\nsure, defining integration on the space C∞\nc(G,A) of smooth compactly supported\nfunctionsG→A([Vig96, I.2.3]).\n2.Representation Theoretic Background\n2.1.Co-invariants and Derivatives. Asin[EH12,BZ77], wedefinethefollowing\nfunctors with respect to the character ψ.\nΦ+:RepA(Pn−1)→RepA(Pn) Ψ+:Rep(Gn−1)→Rep(Pn)\nV/mapsto→c-IndPn\nPn−1UnV(Unacts viaψ)V/mapsto→V(Unacts trivially)\nˆΦ+:Rep(Pn−1)→Rep(Pn) Ψ−:Rep(Pn)→Rep(Gn−1)\nV/mapsto→IndPn\nPn−1UnV V /mapsto→V/V(Un,1)\nΦ−:Rep(Pn)→Rep(Pn−1)\nV/mapsto→V/V(Un,ψ)\nNote that we give these functors the same names as the ones origin ally defined\nin [BZ76], but we use the non-normalized induction functors, as in [BZ7 7, EH12],\nbecause they are simpler for our purposes. As observed in [EH12], t hese functors\nretain the basic adjointness properties proved in [BZ77, §3.2]. This is because the\nmethods of proof in [BZ76, BZ77] use properties of l-sheaves which carry over to\nthe setting of smooth A[G]-modules where Ais a Noetherian W(k)-algebra.\nProposition 2.1 ([EH12],3.1.3) .(1) The functors Ψ−,Ψ+,Φ−,Φ+,ˆΦ+are ex-\nact.\n(2)Φ+is left adjoint to Φ−,Ψ−is left adjoint to Ψ+, andΦ−is left adjoint to\nˆΦ+.\n(3)Ψ−Φ+= Φ−Ψ+= 0\n(4)Ψ−Ψ+,Φ−ˆΦ+, andΦ−Φ+are naturally isomorphic to the identity functor.\n(5) For each VinRep(Pn)we have an exact sequence\n0→Φ+Φ−(V)→V→Ψ+Ψ−(V)→0.\n(6) (Commutativity with Tensor Product) If Mis anA-module and FisΨ−,Ψ+,\nΦ−,Φ+, orˆΦ+, we haveF(V⊗AM)∼=F(V)⊗AM.\nFor 1≤m≤nwe define the mth derivative functor\n(−)(m):= Ψ−(Φ−)m−1: Rep(Pn)→Rep(Gn−m).6 GILBERT MOSS\nThis gives a functor Rep( Gn)→Rep(Gn−m) by first restricting representations to\nPnand then applying ( −)(m); this functor is also denoted ( −)(m). The zero’th de-\nrivativefunctor ( −)(0)is the identity. We candescribe the derivativefunctor ( −)(m)\nmore explicitly by using the following lemma on the transitivity of coinvar iants:\nLemma 2.2 ([BZ76]§2.32).LetHbe a locally profinite group, θa character of\nH, andVa representation of H. SupposeH1,H2are subgroups of Hsuch that\nH1H2=HandH1normalizes H2. Then/parenleftig\nVH2,θ|H2/parenrightig\nH1,θ|H1=VH,θ.\nDefineN(r) to be the group of matrices whose first rcolumns are those of the\nidentity matrix, and whose last n−rcolumns are those of elements of Nn(recall\nthatNnis the group of unipotent upper triangular matrices). For 2 ≤r≤nwe\nhaveUrN(r) =N(r−1) andUrnormalizes N(r). AsN(r) is contained in Nn, we\ndefineψonN(r) via its superdiagonal entries. We can also define a character /tildewideψon\nN(r) slightly differently from the usual definition: /tildewideψwill be given as usual via ψ\non the last n−r−1 superdiagonal entries, but trivially on the ( r,r+1) entry, i.e.\n/tildewideψ(x) :=ψ(0+xr+1,r+2+···+xn−1,n) forx∈N(r).\nThe functors (Φ−)mand (−)(m), defined above, can be described more explicitly.\nLetm=n−r. By applying Lemma 2.2 repeatedly with H1=Ur, andH2=\nN(r−1), we get\nProposition 2.3 ([Vig96] III.1.8) .(1)(Φ−)mVequals the module of coinvariants\nV/V(N(n−m),ψ).\n(2)V(m)equals the module of coinvariants V/V(N(n−m),/tildewideψ).\nIn particular, if m=n, this gives V(n)=V/V(Nn,ψ). Note that V(n)is simply\nanA-module.\n2.2.Whittaker and Kirillov Functions. The character ψ:Nn→A×defines a\nrepresentationof NnintheA-moduleA, whichwealsodenoteby ψ. ByProposition\n2.3 we have Hom A(V(n),A) = Hom Nn(V,ψ).\nDefinition 2.4. ForVinRepA(Gn), we say that Vis of Whittaker type if V(n)\nis free of rank one as an A-module. As in [EH12, Def 3.1.8] , ifAis a field we refer\nto representations of Whittaker type as generic.\nIfVis of Whittaker type, Hom Nn(V,ψ) is free of rank one, so we may choose\na generator λ. The image of λunder the Frobenius reciprocity isomorphism\nHomNn(V,ψ)∼→HomGn(V,IndGn\nNnψ) is the map v/mapsto→WvwhereWv(g) =λ(gv).\nTheA[G]-module formed by the image of the map v/mapsto→Wvis independent of the\nchoice ofλ.\nDefinition 2.5. The image of the homomorphism V→IndGn\nNnψis called the space\nof Whittaker functions of Vand is denoted W(V,ψ)or justW.\nChoosing a generator of V(n)and allowing Nnto act via ψ, we get an iso-\nmorphismV(n)∼→ψ. Composing this with the natural quotient map V→V(n)\ngives anNn-equivariant map V→ψ, which is a generator λ. Note that the map\nV→ W(V,ψ) is surjective but not necessarily an isomorphism, unlike the setting\nofirreducible generic representations with coefficients in a field. Different A[G]-\nmodules of Whittaker type can have the same space of Whittaker fu nctions:INTERPOLATING LOCAL CONSTANTS IN FAMILIES 7\nLemma 2.6. SupposeV′,VinRepA(G)are of Whittaker type, and suppose there\nis aG-equivariant homomorphism α:V′→Vsuch thatα(n): (V′)(n)→V(n)\nis an isomorphism. Then W(V′,ψ)is the subrepresentation of W(V,ψ)given by\nW(α(V′),ψ).\nProof.Letq′:V′→V′/V′(Nn,ψ) andq:V→V/V(Nn,ψ) be the quotient maps.\nChoosing a generator for V(n)gives isomorphisms η,η′such that the following\ndiagram commutes.\nVq>V(n)η>A\nV′α∨\nq′\n>(V′)(n)α(n)\n∨ η′>\nGivenv′∈V′we get\nWα(v′)(g) =η(q(gαv′)) =η((q◦α)(gv′)) =η′(q′(gv′)) =Wv′(g), g∈G.\nThis shows W(V′,ψ) =W(α(V′),ψ)⊂ W(V,ψ). /square\nIfVin RepA(Gn) is Whittaker type and v∈V, we will denote by Wv|Pnthe\nrestriction of the function Wvto the subgroup Pn⊂Gn.\nDefinition 2.7. The image of the Pn-equivariant homomorphism V→IndPn\nNnψ:\nv/mapsto→Wv|Pnis called the Kirillov functions of Vand is denoted K(V,ψ)or justK.\nThe following properties of the Kirillov functions are well known for Re pC(G),\nbut we will need them for RepA(G):\nProposition 2.8. LetVbe of Whittaker type in RepA(Pn), and choose a generator\nofV(n)in order to identify V(n)withA. Then the following hold:\n(1)(Φ+)n−1V(n)= c-IndPn\nNnψand(ˆΦ+)n−1V(n)= IndPn\nNnψ.\n(2) The composition (Φ+)n−1V(n)→V→IndPn\nNnψdiffers from the inclusion\nc-IndPn\nNnψ֒→IndPn\nNnψby multiplication with an element of A×.\n(3) The Kirillov functions K(V,ψ)contains c-IndPn\nNnψas a sub-A[Pn]-module.\nProof.The proof in [BZ76] Proposition 5.12 (g) works to prove (1) in this con text.\nLetS= (Φ+)n−1V(n). There is an embedding S→Vby Proposition 2.1\n(5); denote by tthe composition S→V→Indψ. Thent(n):S(n)→Indψ(n)\nis a nonzero homomorphism between free rank one A-modules, hence given by\nmultiplication with an element aofA. By Proposition 2.1 (6), For any maximal\nidealmofA,t(n)⊗κ(m) must be an isomorphism because it is a nonzero element\nof\nHomκ(m)((S(V)⊗κ(m))(n),(Indψ⊗κ(m))(n)) =κ(m).\nThusais nonzero in κ(m) for allm, hence a unit, so t(n)is an isomorphism. On the\nother hand there is the natural embedding c-Ind ψ→Indψ, which we will denote\ns. Sinces(n)is an isomorphism by [BZ77, Prop 3.2 (f)], we have s(n)=ut(n)\nfor someu∈A×. Thus, if K:= ker(s−ut) thenK(n)=S(V)(n)=V(n),\nwhence Hom P(S(V)/K,Indψ)∼=HomA((S(V)/K)(n),A) = Hom A({0},A) = 0,\nwhich implies s−ut≡0.\nTo prove (3), note that since K(V,ψ) is defined to be the image of the map\nV→IndPn\nNnψ, this follows from (2). /square8 GILBERT MOSS\nDefinition 2.9 ([EH12],§3.1 ).IfVis inRep(Pn), the image of the natural inclu-\nsion(Φ+)n−1V(n)→Vis called the Schwartz functions of Vand is denoted S(V).\nForVinRep(Gn)we also denote by S(V)the Schwartz functions of Vrestricted\ntoPn.\nWe can ask how the functor Φ−is reflected in the Kirillov space of a represen-\ntation. First we observe that Φ−commutes with the functor K:\nLemma 2.10. For0≤m≤n, theA[Pm]-modules K((Φ−)n−mV,ψ)and\n(Φ−)n−mK(V,ψ)are identical.\nProof.The image of the Pn−m-submodule V(N(m),ψ) in the map V→ Kequals\nthe submodule K(N(m),ψ). The lemma then follows from Proposition 2.3 /square\nFollowing [CPS10], we can explicitly describe the effect of the functor Φ−on the\nKirillov functions K. Recall that K(Un,ψ) denotes the A-submodule generated by\n{uW−ψ(u)W:u∈Un, W∈ K}and Φ−K:=K/K(Un,ψ).\nProposition 2.11 ([CPS10] Prop 1.1) .\nK(Un,ψ) ={W∈ K:W≡0on the subgroup Pn−1⊂Pn}.\nProof.The proof of [CPS10, Prop 1.1] carries over in this setting. It utilizes the\nJacquet-Langlandscriterion for an element vof a representation Vto be in the sub-\nspaceV(Uni,ψ), which remains valid over more general coefficient rings Abecause\nall integrals are finite sums. /square\nThus Φ−has the same effect as restriction of functions to the subgroup Pn−1\ninsidePn:\nΦ−K∼=/braceleftbig\nW/parenleftbigp0\n0 1/parenrightbig\n:W∈ W(V,ψ), p∈Pn−1/bracerightbig\n.\nBy applying for each m= 1,...,n−2 the argument of [CPS10, Prop 1.1] to the\nPn−m+1representation\n/braceleftig\nW/parenleftig\np0\n0Im−1/parenrightig\n:W∈ W(V,ψ), p∈Pn−m+1/bracerightig\ninstead of to K, we can describe (Φ−)mK:\nCorollary 2.12. Form= 1,...,n−1,\n(Φ−)mK∼=/braceleftbig\nW/parenleftbigp0\n0Im/parenrightbig\n:W∈ W(V,ψ), p∈Pn−m/bracerightbig\n.\n2.3.Partial Derivatives. Given a product H1×H2of subgroups of G, and a\ncharacterψon the unipotent upper triangular elements of H2, we can define “par-\ntial” versionsof the functors Φ±, Ψ±as follows: given Vin RepA(H1×H2), restrict\nit to a representation of H1={1}×H2, then apply the functor Φ±or Ψ±, and\nobservethat H1×{1}acts naturally on the result, since it commutes with {1}×H2.INTERPOLATING LOCAL CONSTANTS IN FAMILIES 9\nMore precisely:\nΦ+,2:RepA(Gn−m×Pm−1)→RepA(Gn−m×Pm)\nV/mapsto→c-IndGn−m×Pm\nGn−m×Pm−1Um(V), with {1}×Umacting viaψ\nˆΦ+,2:Rep(Gn−m×Pm−1)→Rep(Gn−m×Pm)\nV/mapsto→c-IndGn−m×Pm\nGn−m×Pm−1Um(V)\nΦ−,2:Rep(Gn−m×Pm)→Rep(Gn−m×Pm−1)\nV/mapsto→V/V({1}×Um,ψ)\nΨ+,2:Rep(Gn−m×Gm−1)→Rep(Gn−m×Pm)\nV/mapsto→V({1}×Umacts trivially)\nΨ−,2:Rep(Gn−m×Pm)→Rep(Gn−m×Gm−1)\nV/mapsto→V/V({1}×Um,1)\nBecauseH1×{1}commutes with {1}×H2, we immediately get\nLemma 2.13. The analogue of Proposition 2.1 (1)-(6) holds for Φ+,2,ˆΦ+,2,Φ−,2,\nΨ+,2, andΨ−,2.\nDefinition 2.14. We define the functor (−)(0,m): RepA(Gn−m×Gm)→\nRepA(Gn−m)to be the composition Ψ−,2(Φ−,2)m−1.\nThe proof of the following Proposition holds for W(k)-algebrasA:\nProposition 2.15 ([Zel80] Prop 6.7, [Vig96] III.1.8) .LetM=Gn−m×Gm.\nFor0≤m≤nthem’th derivative functor (−)(m)is the composition of the\nJacquet functor JM: Rep(Gn)→Rep(Gn−m×Gm)with the functor (−)(0,m):\nRepA(Gn−m×Gm)→RepA(Gn−m).\nLemma 2.16. LetVbe inRepA(Gn−m×Gm). ThenVcontains an A-submodule\nisomorphic to V(0,m).\nProof.The image of the natural embedding (Φ+,2)m−1Ψ+,2(V(0,m))→V, which\nis given by Proposition 2.13 (5), will be denoted S0,2(V). By Proposition 2.13 (4),\nthe natural surjection V→V(0,m)restricts to a surjection S0,2(V)→V(0,m). By\nProposition 2.13 (6), the map of A-modules S0,2(V)→V(0,m)arises from the map\n(Φ+,2)m−1Ψ+,2(A)→Aby tensoring over AwithV(0,m). Take the A-submodule\ngenerated by any element of (Φ+,2)m−1Ψ+,2(A) that maps to the identity in A;\nthen tensor with V(0,m). /square\n2.4.Finiteness Results. In this subsection we gathercertain finiteness results in-\nvolving derivatives, most of which are well-known when Ais a field of characteristic\nzero.\nLetHbeanytopologicalgroupcontainingadecreasingsequence {Hi}i≥0ofopen\nsubgroups whose pro-order is invertible in A, and which forms a neighborhood base\nof the identity in H. IfVis a smooth A[H]-module we may define a projection\nπi:V→VHi:v/mapsto→/integraltext\nHihvfor a Haar measure on HiwhereHihas total measure\n1. TheA-submodules Vi:= ker(πi)∩VHi+1then satisfy/circleplustext\niVi=V.\nLemma 2.17 ([EH12] Lemma 2.1.5,2.1.6) .A smoothA[H]-moduleVis admissible\nif and only if each A-moduleViis finitely generated. In particular, quotients of\nadmissibleA[H]-modules by A[H]-submodules are admissible.10 GILBERT MOSS\nThus the following version of the Nakayama lemma applies to admissible A[H]-\nmodules:\nLemma2.18 ([EH12]Lemma3.1.9) .LetAbe a Noetherian local ring with maximal\nidealm, and suppose that Mis a submodule of a direct sum of finitely generated\nA-modules. If M/mMis finite dimensional then Mis finitely generated over A.\nIfVis admissible, then it is G-finite if and only if V/mVisG-finite. To see this,\ntakeS⊂V/mVan (A/m)[H]-generating set, let Wbe theA[H]-span of a lift to\nV. SinceV/Wis admissible, we can apply Nakayama to each factor ( V/W)ito\nconcludeV/W= 0.\nProposition 2.19 ([EH12] 3.1.7) .Letκbe aW(k)-algebra which is a field, and\nVan absolutely irreducible admissible representation of Gn. ThenV(n)is zero or\none-dimensional over κ, and is one-dimensional if and only if Vis cuspidal.\nProposition 2.20 ([Vig96] II.5.10(b)) .Letκbe aW(k)-algebra which is a field.\nIfVis aκ[G]-module, then Vis admissible and G-finite if and only if Vis finite\nlength over κ[G].\nProof.SupposeVis admissible and G-finite. Ifκwere algebraically closed of char-\nacteristic zero (resp. characteristic ℓ), this is [BZ77, 4.1] (resp. [Vig96, II.5.10(b)]).\nOtherwise, let κbe an algebraic closure, then V⊗κis finite length, so Vis finite\nlength.\nIfVis finite length, so is V⊗κκ. Overan algebraicallyclosed field ofcharacteris-\ntic different from p, irreducible representations are admissible ([BZ77, 3.25],[Vig96,\nII.2.8]). Since admissibility is preserved under taking extensions V⊗κbeing finite\nlength implies it is admissible, hence Vis admissible. Thus we can reduce prov-\ningG-finiteness to proving that, given any exact sequence of admissible objects,\n0→W0→V→W1→0 whereW0andW1areG-finite, then VisG-finite. But\nthere is a compact open subgroup Usuch thatW0andW1are generated by WU\n0\nandWU\n1, respectively. It follows that that Vis generated by VU. /square\nLemma 2.21. Letκbe aW(k)-algebra which is a field. If Vis an absolutely\nirreducible κ[Gn]-module, then for m≥0,V(m)is finite length as a κ[Gn−m]-\nmodule.\nProof.We follow [Vig96, III.1.10]. Given j,kpositive integers, let M=Gj×Gk\nand letP=MNbe the associated standard parabolic subgroup. Given τin\nRepκ(Gj) andσin Repκ(Gk), we define τ×σto be the normalized induction\nc-IndP(δ1/2\nN(σ⊗τ)) in Repκ(Gj+k), whereδNdenotes the modulus character of\nN(for the definition of δNsee [BZ77, 1.7]). There exists a multiset {π1,...,π r}\nof irreducible cuspidals such that V⊂π1× ··· ×πr. The Liebniz formula for\nderivatives says that ( π1×π2)(t)has a filtration whose successive quotients are\nπ(t−i)\n1×π(i)\n2. Its proof, given in [BZ77, §7], carries over in this generality. Then\nV(m)⊂(π1× ··· ×πr)(m), which is finite length by induction, using Proposition\n2.19 combined with the Liebniz formula. /square\nProposition 2.22 ([Hel12b] Prop 9.15) .LetMbe a standard Levi subgroup of G.\nIfVinRepA(G)is admissible and primitive, then JMVinRepA(M)is admissible.\nCorollary 2.23. IfAis a local Noetherian W(k)-algebra and Vis admissible and\nG-finite, then V(m)is admissible and G-finite for 0≤m≤n.INTERPOLATING LOCAL CONSTANTS IN FAMILIES 11\nProof.LetM=Gn−m×Gm. By Proposition 2.15, V(m)= (JMV)(0,m), so by\nLemma 2.16, there is an embedding V(m)→JMV. Admissibility and G-finiteness\nmeanVis generated over A[G] by vectors in VKfor some compact open subgroup\nK. SinceVKis finite over A,eVKis nonzero for only a finite set of primitive\nidempotents eofthe Bernstein center, so eV/ne}ationslash= 0for at most finitely many primitive\nidempotents eof the integral Bernstein center. Therefore, Proposition 2.22 ap plies,\nshowingV(m)embeds in an admissible module. Thus by Lemma 2.18, we are\nreduced to proving the result for V:=V/mV. SinceVis admissible and G-finite,\nandA/mischaracteristic ℓ, Lemma2.20shows Visfinite length, thereforeitfollows\nfrom Lemma 2.21 that V(m)is finite length. Applying Lemma 2.20 once more, we\nhave the result. /square\nLoosely speaking, the ( n−1)st derivative describes the restriction of a Gn-\nrepresentation to a G1-representation (see Corollary 2.12). The next result shows\nthat this restriction intertwines a finite set of characters:\nTheorem 2.24. IfAis a localW(k)-algebra and VinRepA(G)is admissible and\nG-finite, then V(n−1)is finitely generated as an A-module.\nProof.ByLemma2.18andCorollary2.23itissufficienttoshowthat V(n−1)isfinite\nover the residue field κ. We know V(n−1)isG-finite and admissible by Corollary\n2.23, hence finite length asa κ[G1]-module by Proposition2.20. Since G1is abelian,\nall composition factors are 1-dimensional, so V(n−1)being finite length implies it\nis finite dimensional over κ. /square\nSince the hypotheses of being admissible and G-finite are preserved under local-\nization by Proposition 2.1 (6), we can go beyond the local situation:\nCorollary 2.25. LetAbe a Noetherian W(k)-algebra and suppose that Vis ad-\nmissible and G-finite. Then for every pinSpecA,V(n−1)\npis finitely generated as\nanAp-module.\n2.5.Co-Whittaker A[G]-Modules. In this subsection we define co-Whittaker\nrepresentations and show that every admissible A[G]-moduleVof Whittaker type\ncontains a canonical co-Whittaker subrepresentation.\nDefinition 2.26 ([Hel12b] 3.3) .Letκbe a field of characteristic different from p.\nAn admissible smooth object UinRepκ(G)is said to have essentially AIG dual if it\nis finite length as a κ[G]-module, its cosocle cos(U)is absolutely irreducible generic,\nandcos(U)(n)=U(n)(the cosocle of a module is its largest semisimple quotient) .\nThis condition is equivalent to U(n)being one-dimensional over κand having\nthe property that W(n)/ne}ationslash= 0 for any nonzero quotient κ[G]-moduleW(see [EH12,\nLemma 6.3.5] for details).\nDefinition 2.27 ([Hel12b] 6.1) .An objectVinRepA(G)is said to be co-Whittaker\nif it is admissible, of Whittaker type, and V⊗Aκ(p)has essentially AIG dual for\neachp.\nProposition 2.28 ([Hel12b] Prop 6.2) .LetVbe a co-Whittaker A[G]-module.\nThen the natural map A→EndA[G](V)is an isomorphism.12 GILBERT MOSS\nLemma 2.29. SupposeVis admissible of Whittaker type and, for all primes p,\nany non-generic quotient of V⊗κ(p)equals zero. Then Vis generated over A[G]\nby a single element.\nProof.Letxbe a generator of V(n), and let ˜x∈Vbe a lift of x. IfV′is the\nA[G]-submodule of Vgenerated by ˜ x, then (V/V′)(n)= 0. Since any non-generic\nquotient of V⊗κ(p) equals zero, ( V/V′)⊗κ(p) = 0 for all p. SinceV/V′is\nadmissible, we can apply Lemma 2.18 over the local rings Apto conclude V/V′is\nfinitely generated, then apply ordinary Nakayama to conclude it is ze ro. /square\nThus every co-Whittaker module is admissible, Whittaker type, G-finite (in fact\nG-cyclic), and Schur, so satisfies the hypotheses of Theorem 3.5, b elow. Moreover,\nevery admissible Whittaker type representation contains a canonic al co-Whittaker\nsubmodule:\nProposition 2.30. LetVinRepA(G)be admissible of Whittaker type. Then the\nsub-A[G]-module\nT:= ker(V→/productdisplay\n{U⊂V: (V/U)(n)=0}V/U)\nis co-Whittaker.\nProof.(V/T)(n)= 0 soTis Whittaker type. Since Vis admissible so is T. Let\npbe a prime ideal and let T:=T⊗κ(p). We show that cos( T) is absolutely\nirreducible and generic. By its definition, cos( T) =/circleplustext\njWjwithWjan irre-\nducibleκ(p)[G]-module. Since the map T→/circleplustext\njWjis a surjection and ( −)(n)\nis exact and additive, the map ( T)(n)→/circleplustext\njW(n)\njis also a surjection. Hence\ndimκ(p)(/circleplustext\njW(n)\nj)≤dimκ(p)(T(n)). SinceTis Whittaker type and T(n)=T(n)is\nnonzero, there can only be one jsuch thatW(n)\njis potentially nonzero. On the\nother hand, suppose some W(n)\njwere zero, then Wjis a quotient appearing in the\ntarget of the map\nV→/productdisplay\n{U⊂V: (V /U)(n)=0}V/U,\nhence as a quotient of Tit would have to be zero, a contradiction. Hence precisely\noneWjis nonzero. Now applying [EH12, 6.3.4] with Abeingκ(p) andVbeing\ncos(T), we have that End G(cos(T))∼=κ(p) hence absolutely irreducible. It also\nshows that cos( T)(n)=W(n)\nj/ne}ationslash= 0. Hence T(n)= cos(T)(n). By Lemma 2.29, Tis\nκ(p)[G]-cyclic; since it is admissible, it is finite length by Lemma 2.20. /square\n2.6.The Integral Bernstein Center. IfAis a Noetherian W(k)-algebra and\nVis anA[G]-module, then in particular Vis aW(k)[G]-module, so we use the\nBernstein decomposition of RepW(k)(G) to studyV.\nLetWbe theW(k)[G]-module c-IndGn\nNnψ. Ifeis a primitive idempotent of Z,\nthe representation eWlies in the block eRepW(k)(G), and we may view it as an\nobject in the category RepeZ(G). With respect to extending scalars from eZtoA,\nthe module eWis “universal” in the following sense:\nProposition 2.31 ([Hel12b] Thm 6.3) .LetAbe a Noetherian eZ-algebra. Then\neW⊗eZAis a co-Whittaker A[G]-module. Conversely, if Vis a primitive co-\nWhittakerA[G]module in the block eRepW(k)(G), andAis aneZ-algebra viaINTERPOLATING LOCAL CONSTANTS IN FAMILIES 13\nfV:eZ →A, then there is a surjection α:W⊗A,fVA→Vsuch thatα(n):\n(W⊗A,fVA)(n)→V(n)is an isomorphism.\nIf we assume Ahas connected spectrum (i.e. no nontrivial idempotents), then\nthe mapfV:Z →Awould factor through a map eZ →Afor some primitive\nidempotent e, hence:\nCorollary 2.32. IfAis a connected Noetherian W(k)-algebra and Vis co-\nWhittaker, then Vmust be primitive for some primitive idempotent e.\nRemark 2.33. Theorems 1.1, 1.2, and 1.4 remain true if the hypothesis that Vis\nprimitive is replaced with the hypothesis that Ais connected.\n3.Zeta Integrals\nIn this section we use the representation theory of Section 2 to de fine zeta inte-\ngrals and investigate their properties.\n3.1.Definition of the Zeta Integrals. We first propose a definition of the zeta\nintegral which is the analog of that in [JPSS79], and then check that t he definition\nmakes sense.\nDefinition 3.1. ForW∈ W(V,ψ)and0≤j≤n−2, letXbe an indeterminate\nand define\nZ(W,X;j) =/summationdisplay\nm∈Z(qn−1X)m/integraldisplay\nx∈Fj/integraldisplay\na∈UFW/bracketleftbigg/parenleftbigg\n̟ma0 0\nx Ij0\n0 0In−j−1/parenrightbigg/bracketrightbigg\nd×adx,\nandZ(W,X) =Z(W,X;0)\nWe first show that Z(W,X;0) defines an element of A[[X]][X−1].\nLemma 3.2. LetWbe any element of IndG\nNnψ. Then there exists an integer\nN <0such thatW(a0\n0In−1)is zero for vF(a)< N. Moreover if Wis compactly\nsupported modulo Nn, then there exists an integer L >0such thatW(a0\n0In−1)is\nzero forvF(a)>L\nProof.There is an integer jsuch that/parenleftbigg\n1pj0\n0 1 0\n0 0In−2/parenrightbigg\nstabilizesW. Forxinpj, we\nhave\nW/parenleftiga0 0\n0 1 0\n0 0In−2/parenrightig\n=W/parenleftig/parenleftiga0 0\n0 1 0\n0 0In−2/parenrightig/parenleftig1x0\n0 1 0\n0 0In−2/parenrightig/parenrightig\n=ψ/parenleftig1ax0\n0 1 0\n0 0In−2/parenrightig\nW/parenleftiga0 0\n0 1 0\n0 0In−2/parenrightig\nWhenevervF(a) is negative enough that axlands outside of ker ψ=p, we get that\nψ/parenleftig1ax0\n0 1 0\n0 0In−2/parenrightig\nis a nontrivial p-power root of unity ζinW(k), hence 1 −ζis the\nlift of something nonzero in the residue field k, and defines an element of W(k)×.\nThis shows that W(a0\n0In−1) = 0. /square\nJust as in [JPSS79], the next two lemmas show that Z(W,X;j) defines an ele-\nment ofA[[X]][X−1] when 0<j <n−2 by reducing it to the case of Z(W,X;0).\nLemma 3.3 ([JPSS79] Lemma4.1.5) .LetHbe a function on G, locally fixed under\nright translation by G, and satisfying H(ng) =ψ(n)H(g)forg∈G,n∈Nn. Then\nthe support of the function on Fjgiven by\nx/mapsto→H/bracketleftbigg/parenleftbigga0 0\nx Ij0\n0 0In−j−1/parenrightbigg/bracketrightbigg14 GILBERT MOSS\nis contained in a compact set independent of a∈F×.\nCorollary 3.4. Ifρdenotes right translation (ρ(g)φ)(x) =φ(xg), andUis the\nunipotent radical of the standard parabolic subgroup of typ e(1,n−1), then there is\na finite set of elements u1,...,u rofUsuch that\nZ(W,X;j) =r/summationdisplay\ni=1Z(ρ(tui)W,X;0)\nfor anyW∈IndG\nNnψ.\nIn [JPSS79], the zeta integrals form elements of the field C((X)) and it is proved\nthat in fact they are elements of the subfield C(X) of rational functions. Whereas\nC((X)) (resp. C(X)) is the fraction field of the domain C[[X]] (resp. C[X,X−1]),\nour ringsA[[X]][X−1] andA[X,X−1] are not in general domains. The first main\nresult of this paper is determining the appropriate fraction ring of A[X,X−1] in\nwhich the zeta integrals Z(W,X;j) live:\nTheorem 3.5. SupposeAis aNoetherian W(k)-algebra. Let Sbe the multiplicative\nsubset ofA[X,X−1]consisting of polynomials whose first and last coefficients ar e\nunits. Then if Vis admissible, Whittaker type, and G-finite, then Z(W,X;j)lies\ninS−1A[X,X−1]for allWinW(V,ψ)for0≤j≤n−2.\nIn particular, the result holds if Vis co-Whittaker, as in Theorem 1.1. The proof\nof Theorem 3.5 will occupy the remainder of this section. The key idea is that the\nzeta integrals Z(W,X) are completely determined by the values W/parenleftbiga0\n0In−1/parenrightbig\nfor\na∈F×, and asWranges over W(V,ψ), the set of these values is equivalent to the\ndata of the P2-representation (Φ−)n−2K. Determining the rationality of Z(W,X)\nwill then reduce to a finiteness result for the quotient K(n−1), or more generally for\nV(n−1).\n3.2.Proof of Rationality. Denote byτthe righttranslationrepresentationof G1\nonK(n−1). LetBbe the commutative A-subalgebra of End A(K(n−1)) generated\nbyτ(̟) andτ(̟−1), where̟is a uniformizer of F. It follows from Corollary\n2.25 that K(n−1)\npis finitely generated over Ap. For every pof SpecA, the inclusion\nBp⊂End(K(n−1))p֒→End(K(n−1)\np), showsBpis finitely generated as an Ap-\nmodule.\nLemma 3.6. Bis finitely generated as an A-module.\nProof.Bis the image of the map A[X,X−1]→EndA(K(n−1)) sendingXtoτ(̟).\nBpis the image of the localized map Ap[X,X−1]→/parenleftbig\nEndA(K(n−1))/parenrightbig\np, which is\nfinitely generated. Thus for every p,τ(̟) andτ(̟−1) satisfy monic polynomials\nsp(X),tp(X) with coefficients in Ap. Sincespandtphave finitely many coefficients\nthere exists a global section fp/∈psuch thatsp(X),tp(X) lie inAfp[X]. The open\nsubsetsD(fp) cover Spec Aand we can take a finite subset {f1,...,f n} ⊂ {fp}\nsuch that (fi) = 1. Since τ(̟) andτ(̟−1) satisfy monic polynomials over Afi, we\nhave thatBfiis finitely generated over Afifor eachi. It follows that Bis finitely\ngenerated over A. /square\nSinceBis finitely generated over A,τ(̟) andτ(̟−1) satisfy monic polynomials\nc0+c1X+...cr−1Xr−1+Xrandb0+b1X+···+bs−1Xs−1+Xsrespectively.INTERPOLATING LOCAL CONSTANTS IN FAMILIES 15\nThe degrees randsare nonzero because τ(̟) andτ(̟−1) are units in B. Adding\nthese together we have\n0 =τ(̟)−s+bs−1τ(̟)−s+1+···+b0+c0+...cr−1τ(̟)r−1+τ(̟)r,\nhenceτ(̟) satisfies a Laurent polynomial whose first and last coefficients are units.\nThe final ingredient in proving rationality is the following transformat ion prop-\nerty.\nLemma 3.7. Z(̟nW,X) =X−nZ(W,X)for anyW∈ W(V,ψ), and any integer\nn.\nProof of Lemma. Denotebybmthecoefficient/integraltext\nUFW(̟mu)d×u. ThenZ(̟nW,X)\nis/summationtext\nm∈ZXmbm+n, which can be rewritten X−nZ(W,X). /square\nDeducing Theorem 3.5. The representation K(n−1)is formed by restricting the\nright translation representation on (Φ−)n−2KfromP2toG1, then taking the quo-\ntient by the G1-stable submodule (Φ−)n−2K(U2,1). By Corollary 2.12, the right\ntranslation representation on (Φ−)n−2Kis given by translations of the restricted\nKirillov functions W|(x0\n0I), denotedW(x) for short. As an endomorphism of the\nquotientmodule K(n−1),τ(̟) satisfiesapolynomial Xn−an−1Xn−1−···−a1X−a0\n(in fact we can take a0to be−1). Hence for any restricted Kirillov function W(x)\nwe have\n̟nW(x) =n−1/summationdisplay\ni=0ai̟iW(x)+W1(x),\nfor some element W1of ((Φ−)n−2K)(U2,1). Therefore we get a relation\nZ(̟nW,X) =n−1/summationdisplay\ni=0aiZ(̟iW,X)+Z1(X)\nwithZ1(X) being a Laurent polynomial by Lemma 3.2. Using Lemma 3.7, then\nmultiplying through by Xnand rearranging we get Z(W,X)(1−/summationtextn−1\ni=0aiXn−i) =\nXnZ1(X) which completes the proof since a0is a unit. /square\n4.Functional Equation and Gamma Factor\n4.1.Contragredient Whittaker Functions. There isan analogueofthe contra-\ngredient which is reflected on the level of Whittaker functions by a t ransform/tildewidest(−);\nthe functional equation will relate the zeta integral of Wto that of its transform.\nWe will need the following two matrices:\nw=\n0···0 1\n0··· −1 0\n...\n(−1)n−1···0 0\n, w′=\n(−1)n0···0\n0 0 ···(−1)n−2\n......\n0 (−1)0···0\n\nFor any element Wof IndG\nNnψ, define the transform /tildewiderWofWas/tildewiderW(g) :=W(wgι),\nwheregι:=tg−1.\nObservation 4.1. IfVis of Whittaker type, then for v∈V,/tildewiderWvis an element of\nIndG\nNψbecause\n/tildewiderW(ng) =W(w(ng)ι) =W(wnιw−1wgι) =ψ(wnιw−1)W(wgι) =ψ(n)/tildewiderW(g).16 GILBERT MOSS\nThusZ(/tildewidew′W,X;j) lands inA[[X]][X−1] by Lemma 3.2. In this section we state\nthe second main result and recover the rationality properties of Se ction 2.2 for\nZ(/tildewidew′W,X;j). The second main result is as follows:\nTheorem 4.2. SupposeAis a Noetherian W(k)-algebra, and suppose Vin\nRepA(G)is co-Whittaker and primitive. Let Sdenote the multiplicative subset of\nTheorem 3.5. Then there exists a unique element γ(V,X,ψ)ofS−1A[X,X−1]such\nthat for any W∈ W(V,ψ),\nZ(W,X;j)γ(V,X,ψ) =Z(/tildewidew′W,1\nqnX;n−2−j)\nfor0≤j≤n−2.\nThe proof of Theorem 4.2 is in Section 5. We now verify that Z(/tildewidew′W,1\nqnX;j)\nalways lives in S−1A[X,X−1].\nProposition 4.3. SupposeVinRepA(G)is admissible, Whittaker type, G-finite,\nSchur, and primitive. Let Vιdenote the smooth A[G]-module whose underlying A-\nmodule isVand whoseG-action is given by g·v=gιv. ThenVιis also admissible,\nWhittaker type, G-finite, Schur, and primitive.\nProof.Consider the map Hom Nn(V,ψ)→HomNn(Vι,ψ) given by λ/mapsto→/tildewideλ, where\n/tildewideλ:x/mapsto→λ(wx). We have /tildewideλ(n·v) =λ(wnιw−1wv) =ψ(n)/tildewideλ(v), which shows\n/tildewideλindeed defines an element of Hom Nn(Vι,ψ). Sincew2= (−1)n−1In, it is an\nisomorphism of A-modules. Admissibility, G-finiteness, and Schur-ness all hold for\nVιsinceg/mapsto→gιis a topological automorphism of the group G. SinceVis Schur,\nAmust be connected, hence Vmust be primitive since it is Schur. /square\nIn particular, ( Vι)(n)=Vι/Vι(Nn,ψ) is free of rank one and we may define\n(/tildewiderW)v(g) =/tildewideλ(gιv) and take W(Vι,ψ) to be the A-module {(/tildewiderW)v:v∈Vι}as\nbefore. Note that this is precisely the same as {/tildewide(Wv) :v∈V}. We record this\nsimple observation as a Lemma:\nLemma 4.4. Ifλis a generator of HomNn(V,ψ)then/tildewideλ:x/mapsto→λ(wx)is a generator\nofHomNn(Vι,ψ)and defines W(Vι,ψ). There is an isomorphism of G-modules\nW(V,ψ)→ W(Vι,ψ)sendingWto/tildewiderW.\nThus all the hypotheses for the results of the previous sections, in particular\nTheorem 3.5, apply to Vιwhenever they apply to V, so we get Z(/tildewidew′W,X;j)\nis inS−1A[X,X−1]. Now we can make the substition1\nqnXforXin the ratio of\npolynomials Z(/tildewidew′W,X;j)to getZ(/tildewidew′W,1\nqnX;j). It will againbe in S−1A[X,X−1]\nbecause this process swaps the first and last coefficients in the den ominator (and q\nis a unit in Asinceqis relatively prime to ℓ).\n4.2.Zeta Integrals and Tensor Products. The goal of this subsection is to\ncheck that the formation of zeta integrals commutes with change o f base ring A.\nFor anyf:A→B, denote by ψA⊗Bthe free rank one B-module with action\ngiven by the character f◦ψ. The group action on V⊗ABis given by acting in\nthe first factor. Let idenote the map V→V⊗AB. Proposition 2.1 (6), gives the\nfollowing lemma.\nLemma 4.5. (1) IfVis of Whittaker type, so is V⊗AB.INTERPOLATING LOCAL CONSTANTS IN FAMILIES 17\n(2) Letλgenerate HomA[N](V,ψ)as anA-module. Then λ⊗idis a generator of\nHomB[N](V⊗B,ψ⊗B).\n(3) LetWv⊗b(g) := (f◦λ)(gv)⊗bdefine elements of W(V⊗B,ψ⊗B). Then\nf◦Wv=Wi(v)for anyv∈V.\nFrom the definition of integration given in §1.1, it follows that if Φ kis the\ncharacteristic function of some Hk, then/integraltext\n(f◦Φk)d(f◦µ×) = (f◦µ×)(Hk) =\nf/parenleftbig/integraltext\nΦkd(µ×)/parenrightbig\n. It follows from the definitions that ( f◦/tildewiderW)(x) =/tildewiderf◦W(x).\nCorollary 4.6. LetFdenote the map of formal Laurentseries rings A[[X]][X−1]→\nB[[X]][X−1]induced by f, then we have\nF(Z(Wv,X;j)) =Z(f◦W,X;j) =Z(Wi(v),X;j) (1)\nF/parenleftig\nZ(/tildewidew′W,X;j)/parenrightig\n=Z(f◦/tildewidew′W,X;j) =Z(/tildewiderw′(f◦W),X;j) (2)\nfor anyWinW(V,ψ), and for 0≤j≤n−2.\nThe next proposition follows from the linearity of the zeta integrals a nd the\ntransform/tildewidest(−).\nProposition 4.7. Suppose there is an element γ(V,X,ψ)inA[[X]][X−1]satisfying\na functional equation as in Theorem 4.2 for all Wv∈ W(V,ψ). Then the element\nF(γ(V,X,ψ))∈B[[X]][X−1]satisfies the functional equation for all W∈ W(V⊗\nB,ψ⊗B).\n4.3.Construction of the Gamma Factor. We define the gamma factor to be\nwhat it must in order to satisfy the functional equation of Theorem 4.2 for a single,\nparticularly simple Whittaker function W0. We seek a W0such thatZ(W,X;0) is\na unit inS−1A[X,X−1].\nBy Proposition2.8 and Lemma 2.10, we have that c-IndP2\nU2ψ⊂(Φ−)n−2K. Since\nc-IndP2\nU2ψis isomorphic to C∞\nc(F×,A) via restriction to G1(recall that C∞\nc(F×,A)\ndenotes the locally constant compactly supported functions F×→A), we find the\nfollowing:\nProposition 4.8. SupposeVinRepA(G)is of Whittaker type. Then the charac-\nteristic function of U1\nFis realized as a restricted Whittaker function W0(g0\n0In−1)for\nsomeW0inW(V,ψ).\nFrom now on, the symbol W0will denote a choice of element in W(V,ψ) whose\nrestriction to (g0\n0In−1) is the characteristic function of U1\nF. ThenZ(W0,X) is/integraltext\nUFW1(a0\n0 1)d×a=µ×(U1\nF) = 1. Since we want our gamma factor to satisfy the\nfunctional equation for W0, we are left with no choice:\nDefinition 4.9 (The Gamma Factor) .LetAbe any Noetherian W(k)-algebra\nand suppose VinRepA(G)is of Whittaker type. We define the gamma factor\nofVwith respect to ψto be the element of A[[X]][X−1]given byγ(V,X,ψ) :=\nZ(/tildewiderw′W0,1\nqnX;n−2).\nWhenVis co-Whittaker and primitive the uniqueness of this gamma factor will\nfollow from the functional equation: if γandγ′both satisfy the functional equation\nfor all Whittaker functions, then γ=Z(/tildewiderw′W0,1\nqnX,n−2) =γ′. In particular for\nsuch representations our construction of the gamma factor doe s not depend on the\nchoice ofW0.18 GILBERT MOSS\n4.4.Functional Equation for Characteristic Zero Points. If the residue field\nκ(p) ofphas characteristic zero, the reduction modulo pofZ(W,X;j) forms an\nelement of κ(p)(X). Asκ(p) is an uncountable algebraically closed field of char-\nacteristic zero, we may fix an embedding C֒→κ(p). The proof of [JPSS83, Thm\n2.7(iii)(2)] (which occurs in [JPSS83, §2.11]) carries over verbatim to the setting\nwhereπandπ′are admissible, Whittaker type, G-finite representations over any\nfield containing C, hence for representations over κ(p). Thus the reduction modulo\npof Ψ(W,X;j) is precisely the integral Ψ( s,W;j) of [JPSS79, §4.1], after replac-\ning the complex variable q−(s+n−1\n2)with the indeterminate X, and there exists a\nunique element, which we will call γp(s,V⊗κ(p),ψ), inκ(p)(q−s) such that for all\nW∈ W(Vp⊗κ(p),ψp) and for all j≥0,\nΨ(1−s,/tildewidew′W;n−2−j) =γp(s,V⊗κ(p),ψp)Ψ(s,W,j).\nThe changeofvariable s/mapsto→1−scan be re-written as −(s+n−1\n2)/mapsto→(s+n−1\n2)−n, so\nwriting the functional equation in terms of Xwe have shown the following Lemma:\nLemma 4.10. SupposeVis admissible of Whittaker type, and G-finite. For each\nprimepofAwith residue characteristic zero, there exists a unique ele mentγp(V⊗\nκ(p),X,ψp)inκ(p)(X)such that for all WinW(V⊗κ(p),ψp)and for0≤j≤n−2\nwe have\nZ(/tildewidew′W,1\nqnX;n−2−j) =γp(X,V⊗κ(p),ψp)Z(W,X;j).\nMoreover,γp(V⊗κ(p),X,ψp) =γ(V,X,ψ) modpby uniqueness in [JPSS79] .\n4.5.Proof of Functional Equation When Ais Reduced and ℓ-torsion Free.\nIn the case that Ais reduced and ℓ-torsion free as a W(k)-algebra, we get a slightly\nstronger result than that of Theorem 4.2.\nTheorem 4.11. IfAis a Noetherian W(k)-algebra and Ais reduced and ℓ-torsion\nfree, then the conclusion of Theorem 4.2 holds for any VinRepA(G)which is\nG-finite, and admissible of Whittaker type.\nProof.Letpbe any characteristic zero prime, and let fp:A→κ(p) be reduction\nmodulop. Corollary 4.6 and Lemma 4.10 tell us that\nfp/parenleftbigg\nγ(V,X,ψ)Z(W,X)−Z(/tildewidew′W,1\nqnX;n−2)/parenrightbigg\n= 0\nforanyWinW(V,ψ), not justW0. This shows that the difference\nγ(V,X,ψ)Z(W,X)−Z(/tildewidew′W,1\nqnX;n−2)\nis in the intersection of all characteristic zero primes of A. WhenAis reduced its\nzero divisors are the union of its minimal primes, so it is ℓ-torsion free if and only\nif all minimal primes have residue characteristic zero. Thus when Ais reduced and\nℓ-torsion free, the intersection of all characteristic zero primes o fAequals zero, so\nthe functional equation holds for any WinW(V,ψ).\nWe now prove uniqueness. If there were another element γ′satisfying the same\nproperty, it would satisfy the functional equation in κ(p) for allWi(v)by reduction,\nso it satisfies the functional equation for all WinW(V⊗κ(p),ψp). But uniqueness\nin Lemma 4.10 then shows fp(γ(V,X,ψ)−γ′) = 0 for all characteristic zero primes\npofA. Again, this means γ′=γ(V,X,ψ).INTERPOLATING LOCAL CONSTANTS IN FAMILIES 19\nWe get rationality by observing that whenever Vis admissible of Whittaker\ntype, it has a canonical co-Whittaker submodule Tby Proposition 2.30, which is\nprimitive if Vis primitive. Since γ(T,X,ψ) satisfies the functional equation for all\nWinW(T,ψ), we must have γ(T,X,ψ) =γ(V,X,ψ) by the construction of the\ngamma factor. But γ(T,X,ψ) is inS−1A[X,X−1] by Theorem 3.5, which holds for\nprimitive co-Whittaker modules. /square\n5.Universal Gamma Factors\nWhenVis primitive and co-Whittaker, we can remove the hypothesis that A\nis reduced and ℓ-torsion free by specializing the gamma factor for the universal\nco-Whittaker module eW.\nTheorem 5.1 ([Hel12a] Thm 12.1) .Any block eZof the Bernstein center of\nRepW(k)(G)is a finitely generated (hence Noetherian), reduced, ℓ-torsion free\nW(k)-algebra.\nBy Proposition 2.31, eWis co-Whittaker, and since it is clearly primitive, all\nthe hypotheses of Proposition 4.11 are satisfied. Hence (Thm 4.11) there exists\na unique gamma factor in S−1(eZ[X,X−1]), which we will denote Γ( eW,X,ψ),\nsatisfying the functional equation for all WinW(eW,ψ).\nProof of Theorem 4.2. SinceVis primitive and co-Whittaker, there is a (unique)\nprimitive idempotent eofZand a ring homomorphism fV:eZ →EndG(V)∼→A,\nand a surjection of A[G]-moduleseW⊗fVA→Vpreserving the top derivative,\nso thatfV(Γ(eW,X,ψ)) =γ(eW⊗fVA,X,ψ). Since Γ( eW,X,ψ) satisfies the\nfunctional equation for all WinW(eW,ψ), we can apply Proposition 4.7 again to\nconcludethat γ(eW⊗A,X,ψ)satisfiesthe functionalequationforall WinW(eW⊗\nA, ψ). SinceeW⊗Ahas a surjection onto Vpreserving the top derivative, Lemma\n2.6 tells us that W(V,ψ) =W(eW⊗A,ψ). The functional equation shows that\nDefinition 4.9 gives a unique gamma factor, hence γ(V,X,ψ) =γ(eW⊗A,X,ψ); it\nsatisfies the functional equation for all WinW(V,ψ). Note that since Γ( eW,X,ψ)\nis inS−1(eZ[X,X−1]), its image in fVis in the corresponding fraction ring of\nA[X,X−1]. This proves Theorem 4.2. /square\nWe can extend the uniqueness and rationality result to a larger class of rep-\nresentations, though with a weaker functional equation coming on ly from the co-\nWhittaker case:\nCorollary 5.2. LetVbe admissible, primitive, of Whittaker type and let Tbe\nits canonical co-Whittaker submodule. Then there exists a u nique gamma factor\nγ(V,X,ψ)inS−1(A[X,X−1])which equals γ(T,X,ψ), and satisfies the functional\nequation for all WinW(T,ψ).\nProof.WhenVis admissible of Whittaker type it has a canonical co-Whittaker\nsub by Proposition 2.30. We have just shown that its gamma factor γ(T,X,ψ)\nsatisfies the functional equation for all WinW(T,ψ). Applying Proposition 2.6\nwithα:T→Vbeing the inclusion map, we conclude that W(T,ψ)⊂ W(V,ψ).\nThe coefficients of the series Z(/tildewiderw′W0,1\nqnX;n−2) in Definition 4.9 are determined\nbyG-translates of the Whittaker function W0, so occurs already in W(T,ψ), so by\ndefinitionγ(T,X,ψ) =γ(V,X,ψ). In particular γ(V,X,ψ) lies inS−1A[X,X−1]\nand satisfies the functional equation for all WinW(T,ψ). /square20 GILBERT MOSS\nWe can make precise the sense in which we have created a universal g amma\nfactor:\nCorollary 5.3. SupposeAis a Noetherian W(k)-algebra, and suppose Vis a\nco-Whittaker A[G]-module in the subcategory eRepW(k)(G)ofRepW(k)(G). Then\nthere is a homomorphism fV:eZ →AandfV(Γ(eW,X,ψ))equals the unique\nγ(V,X,ψ)satisfying a functional equation for all WinW(V,ψ).\nAgain,wecanbroadentheclassofrepresentationsatthecostof amorerestrictive\nfunctional equation:\nTheorem 5.4. SupposeAis any Noetherian W(k)-algebra, and suppose Vis an\nadmissibleA[G]-module of Whittaker type in the subcategory eRepW(k)(G). Then\nthere is a homomorphism fV:eZ →Aand the gamma factor of Corollary 5.2\nequalsfV(Γ(eW,X,ψ)).\nProof.We define fVto be the homomorphism eZ →EndG(T)∼→AwhereT\nis the canonical co-Whittaker submodule of Proposition 2.30. Since Tlies in\neRepW(k)(G),eW⊗fVAsurjectsonto T, andwehave fV(Γ(eW,X,ψ) =γ(T,X,ψ)\n(Prop2.6), andsince Tinjects into V(with topderivativepreserved),againbyProp\n2.6, we have\nfV(Γ(eW,X,ψ)) =γ(eW⊗eZ,fVA,X,ψ) =γ(T,X,ψ) =γ(V,X,ψ).\n/square\nReferences\n[BD84] J. Bernstein and P. Deligne. Le “centre” de Bernstein .Representations des groups re-\nductifs sur un corps local, Traveux en cours , 1984.\n[BH06] Colin Bushnell and Guy Henniart. The Local Langlands Conjecture for GL(2). Berlin\nHeidelberg: Springer-Verlag, 2006.\n[BZ76] I. N. Bernshtein and A. V. Zelevinskii. Representati ons of the group GL(n,F) where F\nis a nonarchimedean local field. Russian Math. Surveys 31:3 (1976), 1-68 , 1976.\n[BZ77] I. N. Bernshtein and A. V. Zelevinskii. Induced repre sentations of reductive p-adic\ngroups. I. Annales scientifiques de l’ ´E.N.S., 1977.\n[CPS10] James Cogdell and I.I. Piatetski-Shapiro. Derivat ives and L-functions for GL(n).The\nHeritage of B. Moishezon, IMCP , 2010.\n[EH12] Matthew Emerton and David Helm. The local Langlands c orrespondence for GL(n) in\nfamilies. Ann. Sci. E.N.S. , 2012.\n[Hel12a] David Helm. The Bernstein center of the category of smooth W(k)[GLn(F)]-modules.\narxiv:1201.1874 , 2012.\n[Hel12b] David Helm. Whittaker models and the integral Bern stein center for GL(n).\narXiv:1210.1789 , 2012.\n[Hen93] Guy Henniart. Caracterisation de la correspondanc e de langlands locale par les facteurs\nǫde paires. Inventiones Mathematicae , 1993.\n[Jia13] Dihua Jiang. On the local Langlands conjecture and r elated problems over p-adic local\nfields.Proceedings of the 6th International Congress of Chinese Ma thematicians , 2013.\n[JPSS79] Herve Jacquet, Ilja Iosifovitch Piatetski-Shapi ro, and Joseph Shalika. Automorphic\nforms on GL(3) I.The Annals of Mathematics , 1979.\n[JPSS83] Herve Jacquet, Ilja Iosifovitch Piatetski-Shapi ro, and Joseph Shalika. Rankin-Selberg\nconvolutions. American Journal of Mathematics , 1983.\n[KM14] Robert Kurinczuk and Nadir Matringe. Rankin-selber g local factors modulo ℓ.\narXiv:1408.5252v2 , 2014.\n[M´12] Alberto M´ ınguez. Fonctions Zˆ eta ℓ-modulaires. Nagoya Math. J. , 208, 2012.\n[Vig96] Marie-France Vigneras. Representations ℓ-modulaires d’un groupe reductif p-adique avec\nℓdifferent de p. Boston: Birkhauser, 1996.INTERPOLATING LOCAL CONSTANTS IN FAMILIES 21\n[Vig00] Marie-France Vigneras. Congruences modulo ℓbetween ǫfactors for cuspidal represen-\ntations of GL(2).Journal de theorie des nombres de Bordeaux , 2000.\n[Zel80] A. V. Zelevinsky. Induced representations of reduc tivep-adic groups. II. on irreducible\nrepresentations of GL(n).Annales scientifiques de l’ ´E.N.S., 1980."    },    {        "title": "1403.4996v1.The_effects_of_time_dependent_dissipation_on_the_basins_of_attraction_for_the_pendulum_with_oscillating_support.pdf",        "content": "The e\u000bects of time-dependent dissipation on the basins of\nattraction for the pendulum with oscillating support\nJames A. Wright1, Michele Bartuccelli1, Guido Gentile2\n1Department of Mathematics, University of Surrey, Guildford, GU2 7XH, UK\n2Dipartimento di Matematica e Fisica, Universit\u0012 a di Roma Tre, 00146 Roma, Italy\nAbstract\nWe consider a pendulum with vertically oscillating support and time-dependent damping\ncoe\u000ecient which varies until reaching a \fnite \fnal value. Although it is the \fnal value which\ndetermines which attractors eventually exist, however the sizes of the corresponding basins of\nattraction are found to depend strongly on the full evolution of the dissipation. In particular\nwe investigate numerically how dissipation monotonically varying in time changes the sizes of\nthe basins of attraction. It turns out that, in order to predict the behaviour of the system,\nit is essential to understand how the sizes of the basins of attraction for constant dissipation\ndepend on the damping coe\u000ecient. For values of the parameters where the systems can be\nconsidered as a perturbation of the simple pendulum, which is integrable, we characterise\nanalytically the conditions under which the attractors exist and study numerically how the sizes\nof their basins of attraction depend on the damping coe\u000ecient. Away from the perturbation\nregime, a numerical study of the attractors and the corresponding basins of attraction for\ndi\u000berent constant values of the damping coe\u000ecient produces a much more involved scenario:\nchanging the magnitude of the dissipation causes some attractors to disappear either leaving\nno trace or producing new attractors by bifurcation, such as period doubling and saddle-node\nbifurcation. Finally we pass to the case of an initially non-constant damping coe\u000ecient, both\nincreasing and decreasing to some \fnite \fnal value, and we numerically observe the resulting\ne\u000bects on the sizes of the basins of attraction: when the damping coe\u000ecient varies slowly\nfrom a \fnite initial value to a di\u000berent \fnal value, without changing the set of attractors, the\nslower the variation the closer the sizes of the basins of attraction are to those they have for\nconstant damping coe\u000ecient \fxed at the initial value. Furthermore, if during the variation\nof the damping coe\u000ecient attractors appear or disappear, remarkable additional phenomena\nmay occur. For instance it can happen that, in the limit of very large variation time, a \fxed\npoint asymptotically attracts the entire phase space, up to a zero measure set, even though\nno attractor with such a property exists for any value of the damping coe\u000ecient between the\nextreme values.\nKeywords: action-angle variables, attractors, basins of attraction, dissipative systems,\nnon-constant dissipation, periodic motions, simple pendulum.\nMathematical Subject Classi\fcation (2000) 34C60, 34C25, 37C60, 58F12, 70K40, 70K50.\n1 Introduction\nConsider the ordinary di\u000berential equations\nx+G(x;t) +\r_x= 0; \u0012+F(\u0012;t) +\r_\u0012= 0; (1.1)\n1arXiv:1403.4996v1  [math.DS]  19 Mar 2014where (x;_x)2R2and (\u0012;_\u0012)2T\u0002R, with T=R=2\u0019Z. The functions FandGare smooth and\n2\u0019-periodic in time t(Fis also 2\u0019-periodic in \u0012); the dots denote derivatives with respect to time.\nEquations to describe the motion of one-dimensional physical systems are often of this form, in\nwhich case the functions G(x;t) andF(\u0012;t) can be considered as an external driving force and the\nparameter\rrepresents the damping coe\u000ecient, which we shall assume positive.\nFirst, for convenience, let us summarise some of the already known ideas regarding systems of\nthe form (1.1), which can be found in the literature [1, 3, 5]. When \ris \fxed at zero, the system is\nHamiltonian and no attractors are present. For \r >0 numerical experiments show that a \fnite set\nof attractors exist: this is consistent with Palis' conjecture [31, 16, 33]. The number of attractors\npresent and the percentage of phase space covered by their basins of attraction depend upon the\nchosen values of the parameters (perturbation parameter \"and damping coe\u000ecient \r), but, for all\nvalues of the parameters, the union of the corresponding basins of attraction completely \fll the phase\nspace, up to a set of zero measure. Moreover if the system is a perturbation of an integrable system\n(perturbation regime), all attractors found numerically turn out to be either \fxed points or periodic\nsolutions with periods that are rational multiples of the forcing period (subharmonic solutions); we\ncannot exclude the presence of chaotic attractors [20, 15], but apparently they either do not arise or\nseem to be irrelevant.\nGenerally in the literature the damping coe\u000ecient is taken as constant, but in many physical\nsystems it changes non-periodically over time. This can be due to several factors, such as the\nheating or cooling of a mechanical system and the wear out or rust on mechanical parts. Despite\nthis, usually models and numerical simulations of such systems only take the \fnal value of dissipation\ninto account when calculating basins of attraction. The recent paper [3] puts forward the idea that,\nalthough the \fnal value of dissipation determines which attractors exist, the relative sizes of their\nbasins of attraction depend on the evolution of the dissipation. In particular the e\u000bect of dissipation\nincreasing to some constant value over a given time span induces a signi\fcant change to the sizes of\nthe basins of attraction in comparison to those when dissipation is constant.\nLet us illustrate in more detail the phenomenology. Suppose that for two values \r0and\r1of\nthe damping coe\u000ecient, with \r06=\r1, the same set of attractors exists. Provided the di\u000berence\nbetween the two values is su\u000eciently large, the relative sizes of the basins of attraction under the\ntwo coe\u000ecients will in general be appreciably di\u000berent. If we allow the damping coe\u000ecient \rto\ndepend on time, \r=\r(t), and vary from \r0to\r1over an initial period of time T0, after which it\nremains constant at the value \r1, then the sizes of the basins of attraction will be di\u000berent from those\nwhere the system has constant coe\u000ecient \r1throughout. Moreover if T0is taken larger, the sizes of\nthe basins of attraction tend towards those for the system under constant \r=\r0: this re\rects the\nfact that the damping coe\u000ecient remains close to \r0for longer periods of time.\nNow consider two values \r0and\r1of the damping coe\u000ecient for which the corresponding sets of\nattractorsA0andA1are not the same. As a system evolving under dissipation is expected to have\nonly \fnitely many attractors, there can only be a \fnite number of attractors which exist for one\nof the two values and not for the other one. What happens is that, by varying \r(t) from\r0to\r1,\nan attractor can either appear or disappear, and in the latter case it can disappear either without\nleaving any trace or being replaced by a new attractor by bifurcation. Suppose, for instance, that\nthe only di\u000berence between A0andA1is that the attractor a02A 0simply disappears, that is\nA0nA1=fa0g; then, if the time T0over which \r(t) varies is large, each remaining attractor tends\nto have a basin of attraction not smaller than that it has for \r\fxed at\r0: the reason being, again,\nthat the damping coe\u000ecient remains close to \r0for a long time and, moreover, the trajectories which\nwould be attracted by a0at\r=\r0will move towards some other attractor when a0disappears. If,\ninstead, the only di\u000berence between the sets of attractors A0andA1is that the attractor a02A 0\nis replaced by an attractor a1, say by period doubling bifurcation, then, letting \r(t) vary from \r0to\n\r1over a su\u000eciently large time T0causes the size of the basin of attraction of a1to tend towards\nthat of the basin of attraction that a0has for\r=\r0.\nWe summarise our results by the following statements.\n1. IfA0, the set of attractors at \r=\r0, is a subset ofA1, the set of attractors which exist at\n\r=\r1, that isA0\u0012A 1, then, as the time T0over which \r(t) is varied from \r0to\r1is taken\nlarger, the basins of attraction tend towards those when \ris kept constant at \r=\r0. In\n2particular, if an attractor belongs to A1nA0, then the larger T0the more negligible is the\ncorresponding basin of attraction.\n2. If the set of attractors at \fxed \r=\r1is a proper subset of those which exist at \r=\r0, that\nisA1\u001aA 0, then, asT0is taken larger, the basins of attraction for the attractors which exist\nat both\r0and\r1change so that for \r(t) varying from \r0to\r1they tend to become greater\nthan or equal to those for constant \r=\r0.\n3. If an attractor a0exists for\r=\r0but is destroyed as \r(t) tends towards \r1, and a new\nattractora1is created from it by bifurcation (we will explicitly investigate the case of saddle-\nnode or period doubling bifurcations), then the size of the basin of attraction of a1, asT0is\ntaken larger, tends towards that of a0at constant \r=\r0.\n4. IfA01is the set of attractors which exist at both \r=\r0and\r=\r1, that isA01=A0\\A 1,\nand none of the elements in A0nA01are linked by bifurcation to elements in A1nA01, then, as\nT0is taken larger, the phase space covered by the basins of attraction of the attractors which\nbelong toA01tends towards 100%. Moreover, all such attractors have a basin of attraction\nlarger than or equal to that they have when the coe\u000ecient of dissipation is \fxed at \r=\r0.\nThe main model used in [3] to convey some of the ideas above is a version of the forced cubic\noscillator, which is of the form of the \frst equation in (1.1), with G(x;t) = (1 +\"cost)x3. This\nsystem, considered in the perturbation regime (both \"and\rsmall), apart from the \fxed point and\nas far as the numerics fortells, exhibits only oscillatory attractors with di\u000berent periods depending\non the parameter values. Also discussed in [3] is the relevance to the spin-orbit problem, describing\nan asymmetric ellipsoidal satellite moving in a Keplerian elliptic orbit around a planet [28]: the\ncorresponding equations of motion are of the form of the second equation in (1.1), with the tidal\nfriction term \r(t) (_\u0012\u00001) instead of \r_\u0012, with\r(t) slowly increasing in time because of the the cooling\nof the satellite.\nIn the present paper we wish to extend the discussion to the pendulum with periodically oscil-\nlating support [25, 32]. The latter is a system which has been already extensively studied in the\nliterature (we refer to [5] for a list of references): it o\u000bers a wide variety of dynamics and, because\nof the separatrix of the unperturbed system, in the perturbation regime, unlike the cubic oscillator,\nalso includes rotatory attractors in addition to the oscillatory attractors. An important di\u000berence\nwith respect to the results in [1] is the following. In [1], if an attractor exists for some value of \r, it\nis found to exist for smaller values of \rtoo. This is not always true for the pendulum considered in\nthe present paper, where we will see that, at least for some values of the parameters, both increas-\ning and decreasing \rcan destroy attractors as well as create new ones. However, this occurs away\nfrom the perturbation regime, where the system can no longer be considered as a perturbation of\nan integrable one: the appearance and disappearance of attractors would occur also in the case of\nthe cubic oscillator for larger values of the forcing. In addition to the case of increasing dissipation\nstudied in [3], here we also include the case where the damping coe\u000ecient decreases to a constant\nvalue, which is appropriate for physical systems where joints are initially tight and require time to\nloosen. In this case similar phenomena are expected. For instance, as the value of variation time\nT0is taken larger, the amount of phase space covered by each of the basins of attraction should\ntend towards that corresponding to original value \r0of\r, providing the set of attractors remains\nthe same.\nThe non-linear pendulum with vertically oscillating support is described by\n\u0012+f(t) sin\u0012+\r_\u0012= 0; f (t) =\u0012g\n`\u0000b!2\n`f0(!t)\u0013\n; (1.2)\nwheref0is a smooth 2 \u0019-periodic function and the parameters `,b,!andgrepresent the length,\namplitude and frequency of the oscillations of the support and the gravitational acceleration, re-\nspectively, all of which remain constant; for the sake of simplicity we shall take f0(!t) = cos(!t)\nin (1.2), as in [5, 6, 7]. As mentioned above, the parameter \rrepresents the damping coe\u000ecient,\n3which, for analysis where it remains constant, we shall model as \r=Cn\"n, where\"is small and n\nis an integer. We shall consider (1.2) as a pair of coupled \frst order non-autonomous di\u000berential\nequations by letting x=\u0012andy= _x, such that the phase space is T\u0002Rand the system can be\nwritten as\n_x=y; _y=\u0000\u0012g\n`\u0000b!2\n`cos (!t)\u0013\nsinx\u0000\ry:\nThe system described by (1.2) can be non-dimensionalised by taking\n\u000b=g\n`!2; \f =b\n`; \u001c =!t;\nso that it becomes\n\u001200+f(\u001c) sin\u0012+\r\u00120= 0; f (\u001c) = (\u000b\u0000\fcos\u001c); (1.3)\nor, written as a system of \frst order di\u000berential equations,\nx0=y; y0=\u0000f(\u001c) sinx\u0000\ry; (1.4)\nwhere the dashes represent di\u000berentiation with respect to the new time \u001cand\rhas been normalised\nso as not to contain the frequency. Linearisation of the system about either \fxed point results in a\nsystem of the form of Mathieu's equation, see for instance [27]. When the downwards \fxed point is\nlinearly stable, it is possible, for certain parameter values, to prove analytically the conditions for\nwhich the \fxed point attracts a full measure set of initial conditions; see Appendix A.\nIn the Sections that follow we shall use the non-dimensionalised version of the system (1.3),\npreferable for numerical implementation as it reduces the number of parameters in the system. In\nSection 2 we detail the calculations of the threshold values for the attractors, that is the values of\nconstant\rbelow which periodic attractors exist in the perturbation regime (small \f). As we shall\nsee, because of the presence of the separatrix for the unperturbed pendulum, this will be of limited\navail for practical purposes: the persisting periodic solutions found to \frst order are in general too\nclose to the separatrix for the perturbation theory to converge. In Section 3 we present numerical\nresults, in the case of both constant and non-constant (either increasing or decreasing) dissipation,\nfor values of the parameters in the perturbation regime. Since for such values the downwards position\nturns out to be stable, we shall refer to this case as the downwards pendulum. Next, in Section 4 we\nperform the numerical analysis for values of the parameters for which the upwards position is stable\n(hence such a case will be referred to as the inverted pendulum). Of course such parameter values\nare far away from the perturbation regime: as a consequence additional phenomena occur, including\nperiod doubling and saddle-node bifurcations. In Section 5 we include a discussion of numerical\nmethods used. Finally in Section 6 we draw our conclusions and brie\ry discuss some open problems\nand possible directions for future investigation.\n2 Thresholds values for the attractors\nThe method used below to calculate the threshold values of \rbelow which given attractors exist\nfollows that described in [1, 3], where it was applied to the damped quartic oscillator and the spin-\norbit model. We consider the system (1.4), with \f=\"and\r=C1\", where\";C1>0. This approach\nis well suited to compute the leading order of the threshold values. In general, it would be preferable\nto write\ras a function of \"of the form \r=C1\"+C2\"2+:::(bifurcation curve), and \fx the constants\nCkby imposing formal solubility of the equations to any perturbation order, see [18]; however this\nonly produces higher order corrections to the leading order value.\nFor\"= 0 the system reduces to the simple pendulum \u001200+\u000bsin\u0012= 0, which admits periodic\nsolutions inside the separatrix (librations or oscillations) and outside the separatrix (rotations). In\nterms of the variables ( x;y) the equations (1.4) become x0=y,y0=\u0000sinx: the librations are\ndescribed by(\nxosc(\u001c) = 2 arcsin [ k1sn (p\u000b(\u001c\u0000\u001c0);k1)];\nyosc(\u001c) = 2k1p\u000bcn (p\u000b(\u001c\u0000\u001c0);k1);k1<1; (2.1)\n4while the rotations are described by\n(\nxrot(\u001c) = 2 arcsin [sn (p\u000b(\u001c\u0000\u001c0)=k2;k2)];\nyrot(\u001c) = 2k\u00001\n2p\u000bdn (p\u000b(\u001c\u0000\u001c0)=k2:k2);k2<1; (2.2)\nwhere cn (\u0001;k), sn(\u0001;k) and dn(\u0001;k) are the Jacobi elliptic functions with elliptic modulus k[8, 11,\n26, 35], and k1andk2are such that k2\n1= (E+\u000b)=2\u000bandk2\n2= 1=k2\n1, withEbeing the energy of\nthe pendulum. From (2.1) and (2.2) it can be seen that the solutions are functions of ( \u001c\u0000\u001c0), so\nthat the phase of a solution depends on the initial conditions. We can \fx the phase of the solution\nto zero without loss of generality by instead writing f(\u001c) in equation (1.4) as f(\u001c\u0000\u001c0). This moves\nthe freedom of choice in the initial condition to the phase of the forcing.\nThe dynamics of the simple pendulum can be conveniently written in terms of action-angle\nvariables (I;'), for which we obtain two sets of variables: for the librations inside the separatrix\none expresses the action as\nI=8\n\u0019p\u000bh\n(k2\n1\u00001)K(k1) +E(k1)i\n; (2.3)\nwhere K(k) and E(k) are the complete elliptic integrals of the \frst and second kind, respectively,\nand writes\nx= 2 arcsin\u0014\nk1sn\u00122K(k1)\n\u0019';k1\u0013\u0015\n; y = 2k1p\u000bcn\u00122K(k1)\n\u0019';k1\u0013\n; (2.4)\nwithk1obtained by inverting (2.3), while for the rotations outside the separatrix one expresses the\nactions as\nI=4\nk2\u0019p\u000bE(k2); (2.5)\nand writes\nx= 2 arcsin\u0014\nsn\u0012K(k2)\n\u0019';k2\u0013\u0015\n; y =2\nk2p\u000bdn\u0012K(k2)\n\u0019';k2\u0013\n; (2.6)\nwithk2obtained by inverting (2.5); further details can be found in Appendix B.\nFor\"small, in order to compute the thresholds values, we \frst write the equations of motion for\nthe perturbed system in terms of the action-angle coordinates ( I;') of the simple pendulum, then\nwe look for solutions in the form of power series expansions in \",\nI(\u001c) =1X\nn=0\"nI(n)(\u001c); ' (\u001c) =1X\nn=0\"n'(n)(\u001c); (2.7)\nwhereI(0)(\u001c) and'(0)(\u001c) are the solutions to the unperturbed system, that is, see Appendix B,\n(I(0)(\u001c);'(0)(\u001c)) = (Iosc;'osc(\u001c)) and (I(0)(\u001c);'(0)(\u001c)) = (Irot;'rot(\u001c)), in the case of oscillations\nand rotations, respectively, with\nIosc=8\n\u0019p\u000bh\n(k2\n1\u00001)K(k1) +E(k1)i\n; ' osc(\u001c) =\u0019\n2K(k1)p\u000b\u001c;\nIrot=4\nk2\u0019p\u000bE(k2); ' rot(\u001c) =\u0019\nK(k2)p\u000b\u001c\nk2;(2.8)\nand with given k1=k(0)\n1andk2=k(0)\n2.\nAs the solution (2.7) is found using perturbation theory, its validity is restricted to the system\nwhere\"is comparatively small. In particular this limitation has the result that the calculations of\nthe threshold values are not valid for the inverted pendulum, where large \"is required to stabilise\nthe system. On the other hand the regime of small \"has the advantage that we can characterise\nanalytically the attractors and hence allows a better understanding of the dynamics with respect to\nthe case of large \", where only numerical results are available.\n52.1 Librations\nWe \frst write the equations of motion (1.4) in action-angle variables, see Appendix D, as\n'0=\u0019p\u000b\n2K(k1)\u0000\"\u0019\n2K(k1)p\u000b\u0014\nsn2(\u0001) +k2\n1sn2(\u0001) cn2(\u0001)\n1\u0000k2\n1\u0000Z(\u0001) sn(\u0001) cn(\u0001) dn(\u0001)\n1\u0000k2\n1\u0015\ncos(\u001c\u0000\u001c0)\n+C1\"\u0019cn(\u0001)\n2K(k1)\u0014sn(\u0001)\ndn(\u0001)+k2\n1sn(\u0001) cn2(\u0001)\n(1\u0000k2\n1) dn(\u0001)\u0000Z(\u0001) cn(\u0001)\n1\u0000k2\n1\u0015\n;\nI0=8\"k2\n1K(k1)\n\u0019cos(\u001c\u0000\u001c0) sn(\u0001) cn(\u0001) dn(\u0001)\u00008C1\"k2\n1p\u000bK(k1)\n\u0019cn2(\u0001);(2.9)\nwhere Z(\u0001) is the Jacobi zeta function, see [26]. Here and throughout Section 2.1 to save clutter we\nde\fne (\u0001) =\u0010\n2K(k1)\n\u0019';k1\u0011\n. Note that in (2.9), the dependence on Iof the vector \feld is through\nthe variable k1, according to (2.3).\nThe coordinates for the unperturbed system ( \"= 0) satisfy\n'0=dE\ndI:= \n(I) =\u0019p\u000b\n2K(k1); I0= 0: (2.10)\nLinearising around ( '(0)(\u001c);I(0)(\u001c)) = (\n(I(0))\u001c;I(0)), we have\n\u000e'0=@\n@I(I(0))\u000eI; \u000eI0= 0; (2.11)\nwhere, see Appendix B,\n\u0010(I) :=@\n@I(I) =\u0000\u00192\n16k2\n1K3(k1)\u0014E(k1)\n1\u0000k2\n1\u0000K(k1)\u0015\n: (2.12)\nSinceI=I(k1), that is the action is a function of k1, settingI=I(0)\fxesk1=k(0)\n1, yielding\n\u0010(I(0)) =\u0010(0), with\u0010(0)given by (2.12) with k1=k(0)\n1.\nThe linearised system (2.11) can by written in compact form as\n\u0012\n\u000e'0\n\u000eI0\u0013\n=\u0012\n0\u0010(0)\n0 0\u0013\u0012\n\u000e'\n\u000eI\u0013\n: (2.13)\nThe Wronskian matrix W(\u001c) is de\fned as the solution of the unperturbed linear system\nW0(\u001c) =\u0012\n0\u0010(0)\n0 0\u0013\nW(\u001c); W (0) =I;\nwhere Iis the 2\u00022 identity matrix. Hence\nW(\u001c) =\u0012\n1\u0010(0)\u001c\n0 1\u0013\n; (2.14)\nwith (1;0) and (\u0010(0)\u001c;1) two linearly independent solutions to (2.13).\nWe now look for periodic solutions ( '(\u001c);I(\u001c)) to (2.9) with period T= 2\u0019q= 4K(k1)p=p\u000b,\nwith p=q2Q, of the form (2.7); see also [18, 19] for a more general discussion. A solution of this\nkind will be referred to as a p:qresonance.\nThe functions ( '(n)(\u001c);I(n)(\u001c)) are formally obtained by introducing the expansions (2.7) into\nthe equations (2.9) and equating the coe\u000ecients of order n. This leads to the equations\n\u0012('(n))0\n(I(n))0\u0013\n=\u0012\n\u0010(0)I(n)\n0\u0013\n+ \nF(n)\n1(\u001c)\nF(n)\n2(\u001c)!\n(2.15)\n6withF(n)\n1(\u001c) andF(n)\n2(\u001c) given by\nF(n)\n1(\u001c) =\u0014\u0019p\u000b\n2K(k1)\u0000\u0010(0)I\u0015(n)\n+\"\n\u0000\u0019\n2K(k1)p\u000b\u0014\nsn2(\u0001) +k2\n1\n1\u0000k2\n1sn2(\u0001) cn2(\u0001)\n\u0000Z(\u0001)\n1\u0000k2\n1sn(\u0001) cn(\u0001) dn(\u0001)\u0015\ncos(\u001c\u0000\u001c0)\n+C1\u0019cn(\u0001)\n2K(k1)\u0014sn(\u0001)\ndn(\u0001)+k2\n1sn(\u0001) cn2(\u0001)\n(1\u0000k2\n1) dn(\u0001)\u0000Z(\u0001) cn(\u0001)\n1\u0000k2\n1\u0015#(n\u00001)\n;\nF(n)\n2(\u001c) =\"\n8k2\n1K(k1)\n\u0019cos(\u001c\u0000\u001c0) sn(\u0001) cn(\u0001) dn(\u0001)\u00008C1k2\n1p\u000bK(k1)\n\u0019cn2(\u0001)#(n\u00001)\n:\nThe notation [ :::](n)means that one has to take all terms of order nin\"of the function inside\n[:::]. By construction, F(n)\n1(\u001c) andF(n)\n2(\u001c) depend only on the coe\u000ecients '(p)(\u001c) andI(p)(\u001c), with\np<n , so that (2.15) can be solved recursively.\nThen, by using the Wronskian matrix (2.14), see [30], one can integrate (2.15) so as to obtain\n\u0012'(n)(\u001c)\nI(n)(\u001c)\u0013\n=W(\u001c)\u0012\u0016'(n)\n\u0016I(n)\u0013\n+W(\u001c)Z\u001c\n0d\u001bW\u00001(\u001b) \nF(n)\n1(\u001b)\nF(n)\n2(\u001b)!\n(2.16)\nwhere \u0016'(n)and \u0016I(n)are thenthorder in the \"-expansion of the initial conditions for 'andI,\nrespectively. In the last term of (2.16) we have\nW(\u001c)Z\u001c\n0d\u001bW\u00001(\u001b) \nF(n)\n1(\u001b)\nF(n)\n2(\u001b)!\n=Z\u001c\n0d\u001bW(\u001c\u0000\u001b) \nF(n)\n1(\u001b)\nF(n)\n2(\u001b)!\n:\nThis yields\n'(n)(\u001c) = \u0016'(n)+\u0010(0)\u001c\u0016I(n)+Z\u001c\n0d\u001bF(n)\n1(\u001b) +\u0010(0)Z\u001c\n0d\u001bZ\u001b\n0d\u001b0F(n)\n2(\u001b0);\nI(n)(\u001c) =\u0016I(n)(\u001c) +Z\u001c\n0d\u001bF(n)\n2(\u001b):(2.17)\nFor a periodic function g, let us denote the average of gwithhgiand the zero-average function\ng\u0000hgiwith \u0015g. Suppose that\nhF(n)\n2i:=1\nTZT\n0d\u001cF(n)\n2(\u001c) = 0; (2.18)\nwhereT= 4K(k1)p; we will check later on the validity of (2.18). Then we may write\nF(n)\n1(\u001c) =Z\u001c\n0d\u001bF(n)\n1(\u001b) =hF(n)\n1i\u001c+Z\u001c\n0d\u001b\u0015F(n)\n1(\u001b);\nF(n)\n2(\u001c) =Z\u001c\n0F(n)\n2(\u001b) d\u001b=Z\u001c\n0d\u001b\u0015F(n)\n2(\u001b);\nand subsequently rewrite (2.17) as\n'(n)(\u001c) = \u0016'(n)+\u0010(0)\u001c\u0016I(n)+hF(n)\n1i\u001c+Z\u001c\n0d\u001b\u0015F(n)\n1(\u001b) +\u0010(0)hF(n)\n2i\u001c+\u0010(0)Z\u001c\n0d\u001bF(n)\n2(\u001b);\nI(n)(\u001c) =\u0016I(n)+Z\u001c\n0d\u001b\u0015F(n)\n2(\u001b);\nin which all the terms which are not linear in \u001care periodic. If we choose our initial conditions \u0016I(n)\nsuch that they satisfy\n\u0016I(n)=\u00001\n\u0010(0)hF(n)\n1i\u0000hF(n)\n2i;\n7the above reduces to\n'(n)(\u001c) = \u0016'(n)(\u001c) +Z\u001c\n0d\u001b\u0015F(n)\n1(\u001b) +\u0010(0)Z\u001c\n0d\u001bF(n)\n2(\u001b);\nI(n)(\u001c) =\u0016I(n)+Z\u001c\n0d\u001b\u0015F(n)\n2(~\u001c);\nso that both '(n)(\u001c) andI(n)(\u001c) are periodic functions with period T, provided (2.18) holds.\nLemma 1 Consider the series (2.7) . If p=q= 1=2m,m2NandC1is small enough, then it is\npossible to \fx the initial conditions ( \u0016'(n);\u0016I(n))in such a way that (2.18) holds for all n\u00151. If\np=q6= 1=2mfor allm2N, then (2.18) can be satis\fed only for C1= 0.\nProof Forn= 1 we have\nF(1)\n2(\u001c) =8k2\n1K(k1)\n\u0019cos(\u001c\u0000\u001c0) sn(p\u000b\u001c;k 1) cn(p\u000b\u001c;k 1) dn(p\u000b\u001c;k 1)\n\u00008C1k2\n1p\u000bK(k1)\n\u0019cn2(p\u000b\u001c;k 1);\nwithk1=k(0)\n1here and henceforth. Moreover set, see Appendix E,\n\u0001 :=p\u000b\n4K(k1)Z4K(k1)=p\u000b\n0d\u001ccn2(p\u000b\u001c;k 1)\n=1\n2K(k1)Z2K(k1)\n0d\u001ccn2(\u001c;k1) =1\nk2\n1\u00141\n2K(k1)E(2K(k1);k1)\u0000(1\u0000k2\n1)\u0015\n;(2.19)\nwhere E(u;k) is the incomplete elliptic integral of the second kind, and \u0000 1(\u001c0;p;q) := sin(\u001c0)G1(p;q),\nwith\nG1(p;q) =1\nTZT\n0sn(p\u000b\u001c;k 1) cn(p\u000b\u001c;k 1) dn(p\u000b\u001c;k 1) sin(\u001c)\n=1\n4K(k1)pZ4K(k1)p\n0d\u001csn(\u001c;k1) cn(\u001c;k1) dn(\u001c;k1) sin(\u001c=p\u000b):(2.20)\nUnder the resonance condition \u0019\u000b=2K(k1) = p=q, one has\nsin(\u001c=p\u000b) = sin\u0012\u0019\u001c\n2K(k1)q\np\u0013\n;\nwhere pand qare relatively prime. By expanding the Jacobi elliptic functions in Fourier series, see\nAppendix E, we \fnd that p;qmust also satisfy the condition\np\u0010\n\u0006(2m1\u00001)\u0006(2m2\u00001)\u00062m3\u0011\n\u0006q= 0\nforG1(p;q) to be non-zero. Thus q= 2mp,m2N, that is p= 1 and q22N, andhF(1)\n2i= 0\nprovidedC1and\u001c0satisfy\nC1=sin(\u001c0)p\u000b\u0001G1(p;q): (2.21)\nNote that the existence of a value of \u001c0satisfying (2.21) is possible only if\njC1j\u0014C1(p=q) :=1p\u000b\u0001G1(p;q):\nSome values of the constants C1(p=q) for\u000b= 0:5 are listed in Table 1.\n8qk1G1(1=q) \u0001 C1(1=q)\n2 0.885201568846 0.172135 0.407121 0.597944\n4 0.998888384493 0.077675 0.224342 0.489649\n6 0.999986981343 0.051734 0.150043 0.487616\n8 0.999999846887 0.038800 0.112539 0.487578\n10 0.999999998199 0.031040 0.090032 0.487577\n12 0.999999999979 0.025867 0.075026 0.487577\nTable 1: Constants for the oscillating attractors with \u000b= 0:5.\nFor alln\u00152 we can write F(n)\n2(\u001c) as\nF(n)\n2(\u001c) =8k2\n1K(k1)\n\u0019@\n@'\u0000\ncos(\u001c\u0000\u001c0) sn(\u0001) cn(\u0001) dn(\u0001)\u0000p\u000bC1cn2(\u0001)\u0001\f\f\f\f\n'='(0)\u0016'(n\u00001)+R(n)(\u001c);\nwhereR(n)(\u001c) is a suitable function which does not depend on \u0016 '(n\u00001). It can be seen that hF(n)\n2i= 0\nif and only if\nhR(n)i=\u00008k2\n1K(k1)\n\u0019 \n1\nTZT\n0d\u001c2K(k1)p\u000b\u0019@\n@\u001c\u0010\nsn(p\u000b\u001c) cn(p\u000b\u001c) dn(p\u000b\u001c)\u0011\ncos(\u001c\u0000\u001c0)\n\u0000p\u000bC1\nTZT\n0d\u001c2K(k1)p\u000b\u0019@\n@\u001c\u0010\ncn2(p\u000b\u001c)\u0011!\n\u0016'(n\u00001):\nThis can be rewritten as\nhR(n)i=\u000016k2\n1K2(k1)p\u000b\u00192cos(\u001c0)G1(p;q) \u0016'(n\u00001):\nWe refer the reader to Appendix E for more details on the evaluation of the integrals. The coe\u000ecient\nof \u0016'(n\u00001)is non-vanishing for \u001c0chosen such that (2.21) is satis\fed. Therefore it is possible to \fx\nthe initial conditions \u0016 '(n\u00001)in such a way that one has hF(n)\n2i= 0 at all orders, thus completing\nthe proof of the lemma. 2\nLemma 1 implies that the threshold values of the p:qresonances are \r(p=q) =C1(p=q)\"forp= 1\nand qeven, with the constants C1(p=q) in Table 1, while the threshold values of the other resonances\nare at least O(\"2).\n2.2 Rotations\nSimilarly for the rotating scenario, again further details can be found in Appendix D, the perturbed\nsystem can be written as\n'0=\u0019p\u000b\nk2K(k2)+\"\u0019k2p\u000bK(k2)\u0014k2\n2sn2(\u0001) cn2(\u0001)\n1\u0000k2\n2\u0000Z(\u0001) sn(\u0001) cn(\u0001) dn(\u0001)\n1\u0000k2\n2\u0015\ncos(\u001c\u0000\u001c0)\n\u0000C1\"\u0019\nK(k2)\u0014k2\n2sn(\u0001) cn(\u0001) dn(\u0001)\n1\u0000k2\n2\u0000Z(\u0001) dn2(\u0001)\n1\u0000k2\n2\u0015\n;\nI0=4\"K(k2)\n\u0019cos(\u001c\u0000\u001c0) sn(\u0001) cn(\u0001) dn(\u0001)\u00004C1\"p\u000bK(k2)\n\u0019k2dn2(\u0001):(2.22)\nHere and throughout this subsection we set ( \u0001) =\u0010\nK(k2)\n\u0019';k2\u0011\n. In this scenario, the Wronskian\nmatrixW(\u001c) can be written as in (2.14), with \u0010(0)given by, see Appendix B,\n\u0010(I) =\u0000\u00192p\u000b\n4K3(k2)\u0014E(k2)\n1\u0000k2\n2\u0000K(k2)\u0015\n; (2.23)\n9fork2=k(0)\n2. We again look for solutions ( '(\u001c);I(\u001c)) with period T= 2\u0019q= 4k2K(k1)p=p\u000b\ncorresponding to a resonance p:q, of the form (2.7), the functions '(n)(\u001c) andI(n)(\u001c) being de\fned\nas in (2.16), with F(n)\n1(\u001c) andF(n)\n2(\u001c) de\fned as\nF(n)\n1(\u001c) =\u0014\u0019p\u000b\nk2K(k2)\u0000\u0010(0)I\u0015(n)\n+\"\n\u0019k2p\u000bK(k2)\u0014k2\n2sn2(\u0001) cn2(\u0001)\n1\u0000k2\n2\n\u0000Z(\u0001) sn(\u0001) cn(\u0001) dn(\u0001)\n1\u0000k2\n2\u0015\ncos(\u001c\u0000\u001c0)\n\u0000C1\u0019\nK(k2)\u0014k2\n2sn(\u0001) cn(\u0001)\n1\u0000k2\n2\u0000Z(\u0001) dn(\u0001)\n1\u0000k2\n2\u0015#(n\u00001)\n;\nF(n)\n2(\u001c) =\u00144K(k2)\n\u0019cos(\u001c\u0000\u001c0) sn(\u0001) cn(\u0001) dn(\u0001)\u00004C1p\u000bK(k2)\n\u0019k2dn2(\u0001)\u0015(n\u00001)\n:\nThe theory goes through exactly as previously shown for the case of libration and we must show\nthathF(n)\n2i= 0.\nLemma 2 Consider the series (2.7) . If p=q= 1=2m,m2N, andC1is small enough, then it is\npossible to \fx the initial conditions ( \u0016'(n);\u0016I(n))in such a way that hF(n)\n2i= 0 for alln\u00151. If\np=q6= 1=2mfor allm2NthenhF(n)\n2i= 0only whenC1= 0.\nProof One has\nF(1)\n2(\u001c) =4K(k2)\n\u0019cos (\u001c\u0000\u001c0) sn\u0012p\u000b\nk2\u001c;k2\u0013\ncn\u0012p\u000b\nk2\u001c;k2\u0013\ndn\u0012p\u000b\nk2\u001c;k2\u0013\n\u00004C1p\u000bK(k2)\n\u0019k2dn2\u0012p\u000b\nk2\u001c;k2\u0013\n;\nwithk2=k(0)\n2here and henceforth. De\fne, see Appendix E,\n\u0001 :=p\u000b\n2k2K(k2)Z2k2K(k2)=p\u000b\n0d\u001cdn2\u0012p\u000b\nk2\u001c;k2\u0013\n=1\n2K(k2)E\u0010\n2K(k2);k2\u0011\n:\nand \u0000 1(\u001c0;p;q) := sin(\u001c0)G1(p;q), where\nG1(p;q) =1\nTZT\n0d\u001csn\u0012p\u000b\nk2\u001c;k2\u0013\ncn\u0012p\u000b\nk2\u001c;k2\u0013\ndn\u0012p\u000b\nk2\u001c;k2\u0013\nsin(\u001c)\n=1\n4K(k2)pZ4K(k2)p\n0d\u001csn(\u001c;k2) cn(\u001c;k2) dn(\u001c;k2) sin(k2\u001c=p\u000b);\nthen use the resonance condition to set\nsin\u0012k2\u001cp\u000b\u0013\n= sin\u0012\u0019\u001c\n2K(k2)q\np\u0013\n:\nOn inspection of the above we see that \u0000 1(\u001c0;p;q) can be calculated similarly to the case inside the\nseparatrix. It follows that the same applies and p=q= 1=2mform2N. ThenhF(1)\n2i= 0 if\nC1=k2sin(\u001c0)p\u000b\u0001G1(p;q); (2.24)\nwhich requires\njC1j\u0014C1(p=q) :=k2p\u000b\u0001G1(p;q):\n10Some values of the constants C1(p=q) for\u000b= 0:5 are listed in Table 2.\nqk2G1(1=q) \u0001 C1(1=q)\n2 0.924397052341 0.156774 0.474414 0.432005\n4 0.998899257272 0.077612 0.225808 0.485542\n6 0.999986983601 0.051734 0.150063 0.487439\n8 0.999999846887 0.038800 0.112540 0.487577\n10 0.999999998199 0.031040 0.090032 0.487577\n12 0.999999999978 0.025867 0.075026 0.487577\nTable 2: Constants for the rotating attractors with \u000b= 0:5.\nForn\u00152 one has\nF(n)\n2(\u001c) =4K(k2)\n\u0019@\n@'\u0000\ncos(\u001c\u0000\u001c0) sn(\u0001) cn(\u0001) dn(\u0001)\u0000p\u000bC1dn2(\u0001)\u0001\f\f\f\f\n'='(0)\u0016'(n\u00001)+R(n)(\u001c);\nwhere again R(n)(\u001c) will be a suitable function which does not depend on \u0016 '(n\u00001). Similarly to the\ncase of libration, hF(n)\n2i= 0 if and only if\nhR(n)i=\u00004K(k2)\n\u0019 \n1\nTZT\n0k2K(k1)p\u000b\u0019@\n@\u001c\u0010\nsn(p\u000b\u001c) cn(p\u000b\u001c) dn(p\u000b\u001c)\u0011\ncos(\u001c\u0000\u001c0) d\u001c\n\u0000p\u000bC1\nTZT\n0k2K(k1)p\u000b\u0019@\n@\u001c\u0010\ndn2(p\u000b\u001c)\u0011\nd\u001c!\n\u0016'(n\u00001)=\u00004k2K2(k2)p\u000b\u00192cos(\u001c0)G1(p;q) \u0016'(n\u00001);\nso that the coe\u000ecient of \u0016 '(n\u00001)turns out to be non-vanishing for \u001c0chosen such that equation (2.24)\nis satis\fed. Therefore it is possible to \fx the initial conditions \u0016 '(n\u00001)in such a way that one has\nhF(n)\n2(\u001c)i= 0 at all orders, thus completing the proof. 2\nLemma 2 implies that the threshold values of the p:qresonances are \r(p=q) =C1(p=q)\"forp= 1\nand qeven, with the constants C1(p=q) in Table 2, while the threshold values of the other resonances\nare at least O(\"2).\nNote, in Tables 1 and 2 it is apparent that, for \u000b= 0:5, increasing qcauses the value of C1(1=q)\nto converge to 0.487577 in both cases. However this does not mean that for \r <0:487577\"there\nare in\fnitely many attracting solutions with increasing period: this would be a counter-example to\nPalis' conjecture! The explanation for this seeming paradox is as follows: As qincreases the solutions\nmove closer and closer to the separatrix (this can be seen by the corresponding values of k1andk2),\nwhere the power series expansions (2.7) for the solutions I(\u001c) and'(\u001c) which were constructed with\nperturbation theory converge only for very small values of \": the larger q, the smaller must be \". In\nparticular, for any \fxed \"there is only a \fnite number of periodic solutions which can be studied by\nperturbation theory. In particular, for the chosen parameters the only periodic solution corresponds\nto the resonance 1:2 inside the separatrix. We also note that the above analysis applies only to\nperiodic attractors with p= 1 and qeven. However we shall see that the numerical simulations\nprovide also rotating attractors with period 2 \u0019, that is the same period as the forcing: we expect\nthat continuing the analysis to second order and writing \r=C2\"2, see [1], would give the threshold\nvalues for these periodic attractors.\n3 Numerics for the downwards pendulum\nWe shall investigate the system (1.3) in the same region of phase space used in [5], namely \u00122[\u0000\u0019;\u0019],\n\u001202[\u00004;4] and calculate the relative areas of the basins of attraction, that is the percentage of phase\nspace they cover relative to this region.\n11Throughout this Section we \fx the parameters \u000b= 0:5 and\f= 0:1. These parameter values,\nalso investigated in [5], correspond to a stable region of the stability tongues for the system linearised\naround\u0012= 0, see [22], so that the downwards con\fguration is stable. The chosen values for the\ndamping coe\u000ecient span values between \r= 0:002 and 0:06, of which only \r= 0:03 was previously\ninvestigated in [5]. For some values of \r, the system exhibits three non-\fxed-point attractors,\nexamples of which are shown in Figure 1, as well as the downwards \fxed point attractor. Here\nand henceforth, for brevity, we shall say that a solution has period nif it comes back to its initial\nvalue after nperiods of the forcing. Of course the upwards \fxed point also exists as a solution to\nthe system, however it is unstable and thus does not attract any non-zero measure subset of phase\nspace. It can also be seen from Figure 1 that the attractive solutions are near the separatrix of the\nunperturbed system: this is evident as the curves described by the two rotating attractors are close\nto that of the oscillating attractor and the separatrix lies between them. This observation con\frms\nthe reasoning as to why the computation of the threshold values can only produce valid results for\nthe period 2 oscillating attractor (see the conclusive remarks in Section 2).\nFigure 1: Attracting solutions for the system (1.3) with \u000b= 0:5,\f= 0:1 and\r= 0:02, namely two period\n1 rotations and one period 2 solution which oscillates about the downwards \fxed point. Periods can be\ndeduced by circles corresponding to the Poincar\u0013 e map.\nFor\r < \u0016\r0, for a suitable \u0016 \r02(0:4;0:5), the system has three periodic attractors, in addition\nto the downwards \fxed point: one oscillating and two rotating attractors. For \r\u0015\u0016\r0the two\nrotating attractors no longer exist, leaving just the oscillatory attractor and the \fxed point. The\nbasins of attraction for \r= 0:02, 0:03, 0:04 and 0:05 are shown in Figure 2, from which we can\nsee that the entire phase space is covered: this suggests that no other attractors exist, at least\nfor the values of the parameters considered. The corresponding relative areas, as estimated by the\nnumerical simulations, are given in Table 3 and plotted in Figure 3. The relative areas of the basins\nof attraction for positive and negative rotations have been listed in the same column: numerical\nsimulations found a di\u000berence in size no greater than 10\u00002% and, due to the symmetries of the\nsystem, it is expected that this di\u000berence is numerically induced by the selection of initial conditions\nThe basins of attraction were estimated using numerical simulations with 600 000 random initial\nconditions in phase space. More notes on the numerics used can be found in Section 5.\nIt can be seen that the results in Table 3 are in agreement with the calculations for the threshold\nvalue for the period 2 oscillatory attractor. The calculations in Appendix A predict that, for the\nchosen values \u000b= 0:5 and\f= 0:1, taking\r > \u0016\r1\u00190:1021 ensures for the origin to capture a\nfull measure set of initial conditions. From Table 3 a stronger result emerges numerically: the \fxed\npoint attracts the full phase space, up to a zero-measure set, for \r\u0015\u0016\r2\u00190:06. Upon comparing\nresults in Table 3, we see that, essentially, the basin of attraction for the \fxed point becomes smaller\nwith an increase in \r, up to approximately 0 :035, after which it grows again. Similarly, by increasing\nthe value of \r, the basins of attraction of the oscillating and rotating solutions attractors increase\ninitially, up to some value (about 0 :025 and 0:035, respectively), after which they become smaller.\n12Furthermore, the variations of the relative areas of the basins of attraction are never monotonic, as\none observes slight oscillations for small variations of \r. These features seem contrary to systems\nsuch as the cubic oscillator, where decreasing dissipation seems to cause the relative area of the\nbasin of attraction of the \fxed point to decrease monotonically, while the basins of attraction of\nthe periodic attractors reach a maximum value, after which their relative areas slightly decrease,\nsee for instance Table III in [3]. We note, however, that a more detailed investigation shows that\noscillations occur also in the case of the cubic oscillator. This was already observed for some values\nof the parameters (see Table IX in [3]), but the phenomenon can also be observed for the parameter\nvalues of Table III, simply by considering smaller changes of the value of \rwith respect to the values\nin [3]. For instance, by varying slightly \raround 0:0005 (see Table III in [3] for notations), one \fnds\nfor the main attractors the relative areas in Table 4.\n(a)\n (b)\n(c)\n (d)\nFigure 2: Basins of attraction for the system (1.3) with \u000b= 0:5,\f= 0:1 and (a)\r= 0:02, (b)\r= 0:03,\n(c)\r= 0:04 and (d) \r= 0:05. The \fxed point (FP) is shown in blue, the positive and negative rotating\nsolutions (PR and NR) are shown in red and yellow, respectively. and the oscillating solution (OSC) in\ngreen.\nIn conclusion, for the pendulum, apart from small oscillations, by decreasing the value of \rfrom\n0:06 to 0:002, the basins of attraction of the periodic attractors, after reaching a maximum value,\nbecomes smaller. A similar phenomenon occurs also in the cubic oscillator (albeit less pronounced).\nHowever, a new feature of the pendulum, with respect to the cubic oscillator, is that the basin\nof attraction of the origin after reaching a minimum value increases again: the increase seems to\nbe too large to be ascribed simply to an oscillation, even though this cannot be excluded. In the\ncase of the cubic oscillator the slight decrease of the sizes of the basins of attraction of the periodic\nattractors was due essentially to the appearance of new attractors and their corresponding basins\n13of attraction. It would be interesting to investigate further, in the case of the pendulum, how the\nbasins of attractions, in particular that of the \fxed point, change by taking smaller and smaller\nvalues of\r. We intend to come back to this in the future [36].\nBasin of attraction %\nFP PR/NR OSC\n\r0.0020 84.57 3.35 8.73\n0.0050 79.91 3.88 12.32\n0.0100 72.24 4.60 18.57\n0.0200 71.95 4.57 18.90\n0.0230 70.73 5.18 18.90\n0.0250 69.28 5.19 20.35\n0.0300 69.94 4.42 21.23\n0.0330 68.92 3.75 23.59\n0.0350 68.77 3.16 24.90\n0.0400 73.84 1.42 23.32\n0.0500 85.61 0.00 14.39\n0.0590 96.96 0.00 3.04\n0.0597 98.59 0.00 1.41\n0.0600 100:00 0.00 0.00\nTable 3: Relative areas of the basins of\nattraction with \u000b= 0:5,\f= 0:1 and con-\nstant\r.\nFigure 3: Plot of the relative areas of the basins of\nattraction with constant \ras per Table 3.\nBasin of attraction %\n0 1/2 1/4 1a 1b 1/6 3a 3b\n\r0.00052 39.03 41.73 14.72 1.22 1.22 1.59 0.25 0.25\n0.00051 39.73 41.70 13.88 1.24 1.24 1.66 0.27 0.27\n0.00050 38.72 41.85 14.65 1.28 1.28 1.65 0.29 0.29\n0.00049 39.26 41.96 13.81 1.29 1.29 1.77 0.27 0.32\n0.00048 38.48 41.87 14.60 1.30 1.30 1.75 0.34 0.34\nTable 4: Relative areas of the basins of attraction of the main attractors for\nthe system  x+ (1 +\"cost)x3+\r_x= 0, with\"= 0:1 and\raround 0:0005. The\nbasins of attraction were estimated using numerical simulations with 1000000\nrandom initial conditions in the square [ \u00001;1]\u0002[\u00001;1] in phase space.\n3.1 Increasing dissipation\nIn this section we shall investigate the case where dissipation increases with time, up to a time T0,\nafter which it remains constant. We will consider a linear increase in dissipation from a value \r0at\ntimet= 0 up to\r1at timeT0, that is (see Figure 4)\n\r=\r(t) =(\n\r0+ (\r1\u0000\r0)\u001c\nT0;0\u0014\u001c <T 0;\n\r1; T 0\u0014\u001c:(3.1)\nAlthough this is a greatly simpli\fed model of what might take place in reality, it serves the purpose\nof demonstrating the signi\fcant e\u000bects of initially non-constant dissipation on the \fnal basins of\nattraction. Below we will consider explicitly the cases \r0= 0:2 and\r1= 0:3, 0:4 and 0:5.\nAs previously mentioned, we expect that increasing the value of T0results in the relative areas\nof the basins of attraction moving along the curves plotted for constant \r. The movement along\nthese curves starts at \r1and goes towards \r0. In particular, any values of the relative area of a\nbasin of attraction for constant values of the damping coe\u000ecient between \r1and\r0are traced as the\nvalue ofT0varies for time-dependent dissipation. This movement along the curve is not linear with\n14the value of T0but asymptotic, with the relative area of the basin of attraction tending towards\nthe value at \r=\r0asT0!1 , providing the attractors existing at \r=\r0also persist at \r1.\nWhen the attractors which persist at \r=\r1are a proper subset of those which exist at \r0, we\nexpect the persisting attractors to absorb the remaining phase space left by the attractors which\nhave disappeared: thus their basins of attraction should be greater than or equal to those at \r=\r0.\nFor the values of the parameters in the chosen range, only these two cases may occur as the solutions\nwhich exist for \r=\r1also exist at \r=\r0, see Table 3.\nFigure 4: Plot of equation (3.1) with \r0= 0:1,\r1= 0:2 and varying T0.\nThe results in Tables 5, 6 and 7 are in agreement with the expectations above. It can be seen\nfrom Tables 5 and 6 that the relative areas of the basins of attraction trace those of constant \r. In\nparticular, the relative area of the basins of attraction of the rotating attractors tends towards that\nat\r=\r0from above, despite having a smaller basin of attraction for the chosen values of \r1. More\nprecisely, the longer T0, the closer is the relative area of the basin of attraction to the value it has\nat\r=\r0. However, the convergence to the asymptotic value is rather slow: for instance in Table\n5, evenT0= 2000 is not enough to reach the values corresponding to \r= 0:02. The simulations for\ntime varying dissipation have in general taken 300 000 or 400 000 initial conditions in phase space.\nIn some cases more points were used for additional accuracy.\nBasin of Attraction %\nFP PR/NR OSC\nT00 69.94 4.42 21.23\n25 69.80 4.42 21.36\n50 69.57 4.45 21.52\n75 69.40 4.64 21.33\n100 68.84 4.85 21.47\n200 68.82 5.10 20.99\n500 69.86 5.17 19.80\n1000 70.65 5.18 18.99\n2000 71.17 5.11 18.61\nTable 5: Relative areas of the basins of\nattraction with \r0= 0:02,\r1= 0:03 and\nT0varying.\nFigure 5: Plot of the relative areas of the basins of\nattraction as per Table 5.\nIn Table 7, we see that for \r0= 0:02 and\r1= 0:05 only two attractors are present: indeed\nthe oscillating attractors no longer exist for \r= 0:05. Hence, when \r(t) increases and crosses the\nvalue at which those attractors disappear, all the trajectories that up to this time were converging to\nthem, will fall into the basins of attraction of the persisting attractors, that is the \fxed point and the\noscillating solution. In particular the corresponding basins of attraction will acquire relative areas\nlarger than those they have at constant \r=\r0, because of the absorption of all these trajectories.\nIt is di\u000ecult to predict how such trajectories are distributed among the persisting attractors. In\n15the case of Table 7 they seem to be attracted slightly more by the \fxed point, even though the\npercentage increase is larger for the oscillating solution.\nBasin of Attraction %\nFP PR/NR OSC\nT00 73.84 1.42 23.32\n25 73.66 1.44 23.45\n50 73.37 1.50 23.63\n75 72.15 2.22 23.41\n100 68.69 3.50 24.31\n200 67.46 4.76 23.03\n500 69.05 5.02 20.92\n1000 69.85 5.17 19.81\n2000 70.63 5.18 19.02\nTable 6: Relative areas of the basins of\nattraction with \r0= 0:02,\r1= 0:04 and\nT0varying.\nFigure 6: Plot of the relative areas of the basins of\nattraction as per Table 6.\nBasin of Attraction %\nFP OSC\nT00 85.61 14.39\n25 86.01 13.99\n50 86.18 13.82\n75 84.19 15.87\n100 80.42 19.58\n200 75.47 24.53\n500 77.95 22.06\n1000 77.55 22.45\n1500 78.03 21.97\nTable 7: Relative areas of the basins\nof attraction with \r0= 0:02,\r1= 0:05\nandT0varying.\nFigure 7: Plot of the relative areas of the basins of\nattraction as per Table 7.\n3.2 Decreasing dissipation\nIn this section we conversely look at the damping coe\u000ecient decreasing from some value \r0> \r 1,\nwith di\u000berent rates of decrease, see Figure 8. We will consider the cases \r0= 0:04 and\r1= 0:02,\n\r0= 0:04 and\r1= 0:03,\r0= 0:05 and\r1= 0:02.\nFigure 8: Plot of equation (3.1) with \r0= 0:23,\r1= 0:2 and varying T0.\n16In this situation it is possible that more attractors exist at \r1than at\r0, see Table 3. We again\nexpect that increasing T0causes the relative areas of the basins of attraction to tend towards those\nat\r0. The result of this is that solutions which do not exist at \r0will attract less and less of the\nphase space as T0increases, and for T0large enough their basins of attraction will tend to zero.\nBasin of Attraction %\nFP PR/NR OSC\nT00 71.95 4.57 18.90\n25 71.85 4.60 18.94\n50 72.36 4.48 18.69\n75 73.64 4.43 17.51\n100 74.10 4.32 17.27\n200 72.31 2.99 21.71\n500 71.51 2.09 24.31\n1000 72.61 1.79 23.81\n2000 73.11 1.63 23.64\nTable 8: Relative areas of the basins of\nattraction with \r0= 0:04,\r1= 0:02 and\nT0varying.\nFigure 9: Plot of the relative areas of the basins of\nattraction as per Table 8.\nBasin of Attraction %\nFP PR/NR OSC\nT00 69.94 4.42 21.23\n25 69.73 4.50 21.28\n50 70.78 4.23 20.77\n75 72.03 3.45 21.07\n100 71.77 2.95 22.33\n500 72.61 1.79 23.81\n1000 73.11 1.63 23.64\nTable 9: Relative areas of the basins of\nattraction with \r0= 0:04,\r1= 0:03 and\nT0varying.\nFigure 10: Plot of the relative areas of the basins of\nattraction as per Table 9.\nBasin of Attraction %\nFP PR/NR OSC\nT00 71.95 4.57 18.90\n25 72.13 4.58 18.72\n50 72.97 4.24 18.55\n75 76.14 3.15 17.56\n100 77.02 2.18 18.62\n200 77.71 0.31 21.67\n500 81.94 0.00 18.06\nTable 10: Relative areas of the basins of\nattraction with \r0= 0:05,\r1= 0:02 andT0\nvarying.\nFigure 11: Plot of the relative areas of the basins of\nattraction as per Table 10.\n17Tables 8 and 9 illustrate cases in which the system admits the same set of attractors for both\nvalues\r0and\r1of the damping coe\u000ecient. An example of what happens when an attractor exists\nat\r1but not at \r0can be seen in the results of Table 10, where \r(t) varies from 0 :05 to 0:02. As\nthe damping coe\u000ecient starts o\u000b at a larger value, then decreases to some smaller value, we also\nexpect the change in the basins of attraction to happen over shorter values of T0. The reasoning for\nthis is simply that larger values of dissipation cause trajectories to move onto attractors in less time.\nIncreasingT0results in the system remaining at higher values of dissipation for more time and thus\ntrajectories land on the attractors in less time.\n4 Numerics for the inverted pendulum\nThe upwards \fxed point of the inverted pendulum can be made stable for large values of \f, i.e when\nthe amplitude of the oscillations is large relative to the length of the pendulum. In this section we\nnumerically investigate the system (1.3) for parameter values for which this happens. For simplicity,\nas mentioned in the introduction, we refer to this case as the inverted pendulum. It can be more\nconvenient to set x=\u0019+\u0018, so as to centre the origin at the upwards position of the pendulum.\nThen the equations of motion become\n\u001800+f(\u001c) sin\u0018+\r\u00180= 0; f (\u001c) =\u0000(\u000b+\fcos\u001c); (4.1)\n\u000b=g\n`!2; \f =b\n`; \u001c =!t:\nThe di\u000berence between equations (1.3) and (4.1) is that here the parameter \u000bhas changed sign.\n(a)\n (b)\n(c)\n (d)\nFigure 12: Attracting solutions for the system (4.1) with \u000b= 0:1 and\f= 0:545. Figure (a) shows an\nexample of the positive and negative rotating attractors with period 2, taken for \r= 0:05. Figure (b) shows\nthe positive and negative attractors with period 1 when \r= 0:23. Figures (c) and (d) show the oscillatory\nattractors with periods 2 and 4 respectively when \ris taken equal to 0.2725. These solutions oscillate about\nthe downwards \fxed point \u0018=\u0019and the axis has been shifted to show a connected curve in phase space.\nThe period of each solution can be deduced from the circles corresponding to the Poincar\u0013 e map.\n18The stability of the upwards \fxed point creates interesting dynamics to study numerically, how-\never it means that the system is no longer a perturbation of the simple pendulum system. This\nin turn has the result that the analysis in Section 2 to compute the thresholds of friction cannot\nbe applied. However, we shall see that the very idea that attractors have a threshold value below\nwhich they always exist does not apply to the inverted pendulum: both increasing and decreasing\nthe damping coe\u000ecient can create and destroy solutions.\nFor numerical simulations of the inverted pendulum throughout we shall take parameters \u000b= 0:1\nand\f= 0:545, which are within the stable regime for the upwards position. For these parameter\nvalues the function f(\u001c) changes sign. As such, the analysis in Appendix A cannot be applied.\nAgain these particular parameter values were also investigated in [5], but with a small value for\nthe damping coe\u000ecient, that is \r= 0:08, where only three attractors appeared in the system: the\nupwards \fxed point and the left and right rotating solutions. We have opted to focus on larger\ndissipation because the range of values considered for \rallows us to incorporate already a a wide\nvariety of dynamics, in which remarkable phenomena occur, and, at the same time, larger values of\n\rare better suited to numerical simulation because of the shorter integration times. We note that,\nfor the values of the parameters chosen, no strange attractors arise: numerically, besides the \fxed\npoints, only periodic attractors are found.\nFor constant dissipation we provide results for \r2[0:05;0:6]. These values of \rare considered\nto correspond to large dissipation, however non-\fxed-point solutions still persist due to the large\ncoe\u000ecient\fof the forcing term. Some examples of the persisting non-\fxed-point solutions can be\nseen in Figure 12; of course, the exact form of the curves depends on the particular choices of \r. For\n\rvarying in the range considered the following attractors arise (we follow the same convention as in\nSection 3 when saying that a solution has period n): the upwards \fxed point (FP), the downwards\n\fxed point (DFP), a positively rotating period 1 solution (PR), a negatively rotating period 1\nsolution (NR), a positively rotating period 2 solution (PR2), a negatively rotating period 2 solution\n(NR2), an oscillating period 2 solution (DO2) and an oscillating period 4 solution (DO4). However,\nas we will see, the solution DO2 deserves a separate, more detailed discussion.\n(a)\n (b)\n (c)\nFigure 13: Basins of attraction for constant dissipation with \r= 0:2, 0:23 and 0:2725 from left to right\nrespectively. The \fxed point (FP) is shown in blue, the positively rotating solution (PR) in red, the\nnegatively rotating solution (NR) in yellow, the downwards oscillation with period 2 (DO2) in green and\n\fnally the downwards oscillation with period 4 (DO4) in orange.\nThe basins of attraction corresponding to the values \r= 0:2, 0:23 and 0:2725 are shown in Figure\n13. The relative areas of the basins of attraction for \r2[0:05;0:6] are listed in Table 11. Again the\npositive and negative rotations have been listed together as any di\u000berence in the size of their basins\nof attraction is expected to be due to numerical inaccuracies.\nIn Figure 14, there is a large jump in the relative area of the basin of attraction for the upwards\n\fxed point (FP) between the values of \r= 0:22 and 0:225, approximately: this is due to the\nappearance of the oscillatory solution which oscillates about the downwards pointing \fxed point\n(\u0018=\u0019). For values of \rslightly larger than 0 :22 large amounts of phase space move close to the\nsolution DO2, where they remain for long periods of time; however they do not land on the solution\n19and are eventually attracted to FP. The percentage of phase space which does this is marked in\nFigure 14 by a dotted line, which becomes solid when the trajectories remain on the solution for all\ntime (however, see comments below).\nBasin of Attraction %\nFP DFP PR/NR PR2/NR2 DO2 DO4\n\r0.0500 4.30 0.00 0.00 47.85 0.00 0.00\n0.0750 5.08 0.00 0.00 47.46 0.00 0.00\n0.0900 7.41 0.00 0.00 46.30 0.00 0.00\n0.1000 8.51 0.00 45.74 0.00 0.00 0.00\n0.1700 49.65 0.00 25.17 0.00 0.00 0.00\n0.2000 64.31 0.00 17.84 0.00 0.00 0.00\n0.2230 72.09 0.00 13.95 0.00 0.00 0.00\n0.2250 27.60 0.00 13.59 0.00 45.22 0.00\n0.2300 25.00 0.00 12.68 0.00 49.61 0.00\n0.2500 15.87 0.00 8.49 0.00 67.16 0.00\n0.2690 16.50 0.00 2.13 0.00 79.25 0.00\n0.2694 17.26 0.00 0.00 0.00 82.74 0.00\n0.2700 17.28 0.00 0.00 0.00 82.73 0.00\n0.2725 17.21 0.00 0.00 0.00 79.44 3.35\n0.2800 17.30 0.00 0.00 0.00 82.70 0.00\n0.2900 17.30 0.00 0.00 0.00 82.70 0.00\n0.3000 16.97 0.00 0.00 0.00 83.03 0.00\n0.4600 9.61 0.00 0.00 0.00 90.39 0.00\n0.4700 9.80 90.20 0.00 0.00 0.00 0.00\n0.5000 10.06 89.94 0.00 0.00 0.00 0.00\n0.5500 10.32 89.68 0.00 0.00 0.00 0.00\n0.6000 8.79 91.21 0.00 0.00 0.00 0.00\nTable 11: Relative areas of the basins of attraction with constant damping coe\u000ecient \r. The solutions are\nnamed as per Figures 12 and 13 with the addition of the downwards \fxed point (DFP) and the rotating\nperiod 2 solutions (PR2/NR2). For details on the DO2 solution we refer to the text.\nFigure 14: Relative sizes of basins of attraction with constant \ras per Table 11. The lines are labeled as in\nTable 11 and regions in which a bifurcation takes place are marked with a dot. The basin of attraction for\nthe oscillatory solution with period 4 (DO4), has not been included due to its small size and the solutions\nlow range of persistence with respect to \r. The broken lines for FP and DO2 represent areas of transition\njust before DO2 (and the solutions created by the period doubling bifurcation) becomes stable, see text.\n20The solution DO4 listed in Table 11 is found to persist only in the interval [0 :272;0:27422], where\nit only attracts a small amount of the phase space. As such it has not been included in Figure 14.\n(a)\n (b)\n (c)\n (d)\nFigure 15: The transition from the period 2 rotating solutions to the period 1 rotating solutions. \r= 0:09,\n0.094, 0.096 and 0.098 from (a) to (d) respectively.\nNumerical simulations can be used to \fnd estimates for the values of \rat which solutions\nappear/disappear. This is done by starting with initial conditions on a solution, then allowing the\nparameter\rto be varied to see for which value that solution vanishes. In Figure 15 the transition\nfrom the period 2 rotating solution to the period 1 rotating solution can be observed: by moving\ntowards smaller values of \r, this corresponds to a period doubling bifurcation [21, 13] (period halving,\nif we think of \ras increasing). Similarly, starting on the period 1 rotating attractor and increasing\nfurther\r, the solution disappears at \r\u00190:2694.\nThe same analysis can be done for the oscillatory solutions. We \fnd that the downwards oscilla-\ntory solution labeled DO2 persists for \rin the interval [0 :224;0:46], approximately. However such a\nsolution is really a period 2 solution only for \rgreater than \r\u00190:24. In the interval [0 :224;0:24] the\ntrajectory is \\thick\", see Figure 16: only due to its similarity to the solution DO2 and to prevent\nTable 11 having yet more columns, the basin of attraction of these solutions in that range has also\nbeen listed under that of DO2. Nevertheless, by moving \rbackwards starting from 0 :24 we have\na sequence of period doubling bifurcations, corresponding to values of \rcloser and closer to each\nother. A period doubling cascade is expected to lead to a chaotic attractor, which, however, may\nsurvive only for a tiny window of values of \r(at\r= 0:223 it has already de\fnitely disappeared)\nand has a very small basin of attraction (for \rgetting closer to the value 0.223 its relative area\ngoes to zero). The appearance of chaotic attractors for small sets of parameters and with small\nbasins of attraction has been observed in similar contexts of multistable dissipative systems close\nto the conservative limit [15]. For the value \r= 0:223 numerical simulations \fnd that trajectories\nremain in the region of phase space occupied by DO2 for a long time, before eventually moving onto\nthe \fxed point. As \rincreases further towards \r\u00190:46, the amplitude of the period 2 oscillatory\nsolution gradually decreases and taking \rlarger causes a slow spiral into the downwards \fxed point,\nwhich now becomes stable.\n(a)\n (b)\n (c)\n (d)\nFigure 16: The solution DO2 for di\u000berent values of time independent \r. As the damping coe\u000ecient is\nincreased, the solution becomes more clearly de\fned: this is due to a period halving bifurcation which stops\nwhen the period becomes 2. The damping coe\u000ecient is \r= 0:23, 0.235, 0.239 and 0.24 from (a) to (d),\nrespectively. The position of the trajectory at every 2 \u0019, i.e the period of the forcing, is shown by circles.\n214.1 Increasing dissipation\nAs mentioned in Section 3.1, as \rincreases from \r0to\r1it is expected that taking T0larger causes\nthe sizes of the basins of attraction to tend towards the sizes of the corresponding basins when\n\r=\r0, when the set of attractors remains the same for all values in between. If an attractor is\nreplaced by a new attractor (by bifurcation), then the new attractor inherits the basin of attraction\nof the old one.\nWe shall begin by \fxing \r1= 0:2 and\r02[0:05;0:2], as for\r=\r1the basins of attraction are\nnot so sensitive to initial conditions, see Figure 13(a), and for \rin that range the set of attractors\nconsists only of the the upwards \fxed point and two rotating solutions; moreover the pro\fles of the\ncorresponding relative areas plotted in Figure 14 are rather smooth and do not present any sharp\njumps.\nBasin of Attraction %\nFP PR/NR\nT00 64.31 17.84\n25 51.41 24.30\n50 32.09 33.95\n75 23.48 38.26\n100 17.09 41.45\n200 6.85 46.58\n500 4.35 47.82\n1000 4.15 47.92\n1500 4.09 47.96\nTable 12: Relative areas of the basins of\nattraction with \r0= 0:05,\r1= 0:2 andT0\nvarying.\nFigure 17: Plot of the relative areas of the basins of\nattraction as per Table 12.\nBasin of Attraction %\nFP PR/NR\nT00 64.31 17.84\n25 49.06 25.47\n50 38.54 30.73\n75 31.25 34.37\n100 25.67 37.16\n150 18.12 40.94\n200 14.14 42.93\n500 9.81 45.09\n1000 9.45 45.27\nTable 13: Relative areas of the basins of\nattraction with \r0= 0:1,\r1= 0:2 andT0\nvarying.\nFigure 18: Plot of the relative areas of the basins of\nattraction as per Table 13.\nTables 12, 13 and 14 show the relative area of each basin of attraction as \rincreases from 0.05,\n0.1 and 0.17, respectively, to 0.2 with varying T0. It can be seen from the results in Tables 13 to 14\nthat the numerical simulations are in agreement with the above expectation. With the exception of\nTable 12 the relative areas of the basins of attraction tend towards those when \ris kept constant at\n\r0. The exception of the case of Table 12 is due to the fact that the set of attractors has changed as\n\rpasses from 0 :05 to 0:2: the period 2 rotating solutions have been destroyed and replaced by the\nperiod 1 rotating solutions. However, when the transition occurs, the new attractors are located in\n22phase space very close to the previous ones and we \fnd that the initial conditions which were heading\ntowards or had indeed landed on the period 2 rotating solutions move onto the now present period\n1 rotating solutions. On the other hand, when the damping coe\u000ecient crosses the value \r\u00190:1,\nthe attractor undergoes topological changes, but, apart from that, the transition is rather smooth:\nthe location in phase space and the basin of attraction change continuously. In conclusion, we \fnd\nthat the relative areas of the basins of attraction for the two period 1 rotating attractors (PR/NR)\ntend towards those the now destroyed period 2 rotating attractors (PR2/NR2) had at \r=\r0. As in\nSection 3 we expect that the sizes of the basin of attraction at \r=\r0are recovered asymptotically\nasT0!1 . Nevertheless, once more, the larger T0the smaller is the variation in the relative area:\nfor instance in Table 12 for T0= 100 the relative area of the basin of attraction of the \fxed point\nhas become nearly 1 =4 of the value for \r= 0:2 constant, while in order to have a further reduction\nby a factor of 4 one has to take T0= 1000.\nBasin of Attraction %\nFP PR/NR\nT00 64.31 17.84\n25 58.54 20.73\n50 56.78 21.62\n100 55.41 22.29\n200 53.69 23.15\n500 51.93 24.04\n1000 50.80 24.60\n1500 50.62 24.69\nTable 14: Relative areas of the basins of\nattraction with \r0= 0:17,\r1= 0:2 andT0\nvarying.\nFigure 19: Plot of the relative areas of the basins of\nattraction as per Table 14.\nBasin of Attraction %\nFP PR/NR DO2\nT00 25.00 12.68 49.61\n10 24.86 13.81 47.52\n15 24.34 14.84 45.98\n20 24.43 15.57 44.43\n25 24.77 16.04 43.15\n50 27.60 16.98 38.45\n75 30.12 17.28 35.32\n100 33.08 17.42 32.08\n150 38.29 17.56 26.58\n200 42.60 17.66 22.08\n300 49.36 17.73 15.18\n400 54.20 17.75 10.30\n500 57.37 17.77 7.08\n1000 63.39 17.78 1.06\n1500 64.26 17.79 0.16\n2000 64.38 17.80 0.03\nTable 15: Relative areas of the basins of\nattraction with \r0= 0:2,\r1= 0:23 and\nT0varying.\nFigure 20: Plot of the relative areas of the basins of\nattraction as per Table 15.\n23We now consider the case where either \r0= 0:2 and\r1= 0:23 or 0:2725 or\r0= 0:23 and\n\r1= 0:2725. Such values for \ro\u000ber more complexities as not only are there more attractors to\nconsider, but one may have attractors (PR and NR) that are destroyed without leaving any trace.\nWhen this happens it is not obvious which persisting attractor will inherit their basins of attraction.\nThe result of this could cause the \fnal basins of attraction to be drastically di\u000berent from those for\nconstant\rand even not monotonically increasing or decreasing as the value of T0is increased.\nIn Table 15 we see that initially, for values of T0not too large, the basin of attraction of FP\nslightly reduces in size, while those of the rotating solutions PR/NR increase substantially. Instead,\nfor larger values of T0, the basin of attraction of FP increases appreciably, while those of PR/NR\nincrease very slowly. Apparently, the rotating solutions react more quickly as \ris varied, attracting\nphase space faster, so that the relative areas of their basins of attraction tend towards the values at\n\r=\r0for shorter initial times T0. It would be interesting to study further this phenomenon.\nWhen\r(t) varies from \r= 0:2 to\r= 0:2725 and from \r= 0:23 to\r= 0:2725, the rotating\nsolutions PR/NR disappear, so that their basins of attractions are absorbed by the persisting at-\ntractors. In Table 16 one sees a very slow movement towards global attraction of the upwards \fxed\npoint, which is the only attractor persisting for both \r0and\r1. However even taking T0= 5000\nis not enough for the asymptotic behaviour to be approached. The results in Table 17 show that,\nby takingT0larger and larger, the relative areas of the basins of attraction of FP and DO4 both\ntend to the values corresponding to \r0= 0:23 (in particular the basin of attraction of DO4 becomes\nnegligible). Nearly all trajectories which were converging towards the rotating solutions before the\nlatter disappeared are attracted by the period two oscillations. This could be due to the fact that\nDO2 is the closest attractor in phase space which persists at both \r0and\r1.\nBasin of Attraction %\nFP DO2 DO4\nT00 17.21 79.44 3.35\n25 18.63 78.33 3.04\n50 20.32 79.29 0.39\n75 23.38 76.52 0.12\n100 25.71 74.27 0.02\n150 28.45 71.55 0.01\n200 30.92 69.08 0.00\n300 35.70 64.30 0.00\n400 39.82 60.18 0.00\n500 42.76 57.24 0.00\n980 54.19 45.81 0.00\n990 54.39 45.81 0.00\n995 54.30 45.70 0.00\n1000 90.11 9.89 0.00\n1005 54.58 45.43 0.00\n1010 54.52 45.48 0.00\n1020 90.34 9.66 0.00\n1030 54.81 45.19 0.00\n1050 55.13 44.87 0.00\n1500 59.78 40.22 0.00\n2000 62.12 37.88 0.00\n3000 63.89 36.11 0.00\n5000 64.29 35.71 0.00\nTable 16: Relative areas of the basins of\nattraction with \r0= 0:2,\r1= 0:2725 and\nT0varying.\nFigure 21: Plot of the relative areas of the basins of\nattraction as per Table 16.\nHowever, the more striking feature of Figures 21 and 22 are the jumps corresponding T0= 1000\n24in the prior and T0= 100 and T0= 500 in the latter. Moreover such jumps are very localised: for\ninstance in Figure 22 for T0= 99 andT0= 100 the basins of attraction of FP and DO2 are found\nto be about 44% and 56%, respectively, whereas by slightly increasing or decreasing T0they settle\naround 20% and 80%. The quantity of phase space exchanged in these instances is roughly equal\nto that attracted to the rotating solutions for \r0. For particular values of T0when the rotating\nattractors disappear their trajectories move to the upwards \fxed point rather than the period 2\noscillations. The reason for this to happen is not clear. Moreover, note that in principle there could\nbe other jumps, corresponding to values of T0which have not been investigated: however, it seems\nhard to make any prediction as far as it remains unclear how the disappearing basins of attractions\nare absorbed by the persisting ones.\nBasin of Attraction %\nFP DO2 DO4\nT00 17.21 79.44 3.35\n25 17.96 79.45 2.60\n50 16.04 83.36 0.60\n75 18.39 80.27 1.34\n90 19.56 80.38 0.05\n95 20.05 79.91 0.04\n97 20.08 79.89 0.04\n98 20.01 79.96 0.03\n99 44.14 55.83 0.03\n100 43.96 56.01 0.03\n101 20.54 79.42 0.03\n105 20.40 79.57 0.03\n110 20.79 79.19 0.02\n125 21.55 78.45 0.01\n150 22.46 77.54 0.00\n200 23.33 76.67 0.00\n300 24.06 75.94 0.00\n400 24.36 75.64 0.00\n490 24.54 75.46 0.00\n500 49.45 50.55 0.00\n510 24.42 75.58 0.00\n1000 24.81 75.19 0.00\nTable 17: Relative areas of the basins of\nattraction with \r0= 0:23,\r1= 0:2725 and\nT0varying.\nFigure 22: Plot of the relative areas of the basins of\nattraction as per Table 17.\n4.2 Decreasing dissipation\nTables 18, 19, 20 and 21 and the corresponding Figures 23, 24, 25 and 26 illustrate the cases when\ndissipation decreases over an initial period of time T0. We have considered the cases with \r0= 0:23,\n0:02725 and\r1= 0:2, with\r0= 0:2725 and\r1= 0:23 and with \r0= 0:3 and\r1= 0:2725.\nIn particular they show that if the set of attractors at \r=\r1is a proper subset of the set of\nattractors which exist at \r=\r0, then, asT0!1 , the basin of attraction of each attractor which\nexists at\r1turns out to have a relative area which tend to be greater than or equal to that found\nfor\r=\r0. In Table 18 we consider the situation in which the attractor DO2, which has a large\nbasin of attraction for \r0= 0:23, is no longer present when \r(t) has reached the \fnal value \r1= 0:2:\nas a consequence the trajectories which would be attracted by DO2 at \r=\r0end up onto the other\nattractors: in fact most of them are attracted by the \fxed point.\n25Basin of Attraction %\nFP PR/NR\nT00 64.31 17.84\n25 70.69 14.65\n50 72.57 13.72\n75 73.24 13.38\n100 73.57 13.22\n200 74.10 12.95\n500 74.42 12.79\nTable 18: Relative areas of the basins of\nattraction with \r0= 0:23,\r1= 0:2 andT0\nvarying.\nFigure 23: Plot of the relative areas of the basins of\nattraction as per Table 18.\nBasin of Attraction %\nFP PR/NR\nT00 64.31 17.84\n5 65.80 17.10\n10 69.52 15.24\n15 74.40 12.80\n20 77.90 11.05\n25 80.38 9.81\n50 86.22 6.89\n75 88.64 5.68\n100 90.07 4.97\n200 92.79 3.61\n500 95.92 2.04\n1000 99.34 0.33\n1500 99.35 0.32\nTable 19: Relative areas of the basins of\nattraction with \r0= 0:2725,\r1= 0:2 and\nT0varying.\nFigure 24: Plot of the relative areas of the basins of\nattraction as per Table 19.\nBasin of Attraction %\nFP PR/NR DO2\nT00 25.00 12.68 49.61\n25 24.73 7.44 60.40\n50 19.67 5.35 69.63\n75 17.13 4.44 73.99\n100 16.42 3.88 75.83\n200 16.29 2.72 78.28\n500 17.03 0.76 81.45\nTable 20: Relative areas of the basins of\nattraction with \r0= 0:2725,\r1= 0:23 and\nT0varying.\nFigure 25: Plot of the relative areas of the basins of\nattraction as per Table 20.\nWe also notice the interesting features in Table 19: as the \fxed point is the only attractor which\n26exists for both \r0and\r1, we \fnd that as T0increases its basin of attraction tends towards 100%,\nwhich corresponds to attraction of the entire phase space, up to a zero-measure set. This happens\ndespite the fact that \r(t) does not pass through any value for which global attraction to the \fxed\npoint is satis\fed. It also suggests that it is possible to provide conditions on the intersection of the\ntwo setsA0andA1of the attractors corresponding to \r0and\r1, respectively, in order to obtain\nthat all trajectories move towards the same attractor when the time T0over which \r(t) is varied is\nsu\u000eciently large. In particular, it is remarkable that it is possible to create an attractor for almost\nall trajectories by suitably tuning the damping coe\u000ecient as a function of time.\nIn Table 20, the relative areas of the rotating solutions, which are absent at \r=\r0, tend to\nbecome negligible when T0is large. Similarly, in Table 21, the basin of attraction of the period\n4 oscillating attractor, which exists only for the \fnal value \r1of the damping coe\u000ecient, tends\nto disappear when T0is taken large enough. This con\frms the general expectation: the basin of\nattraction of the disappearing attractor is absorbed by the closer attractor, that is the solution DO2\nin this case.\nBasin of Attraction %\nFP DO2 DO4\nT00 17.21 79.44 3.35\n25 17.15 80.99 1.86\n50 16.83 83.03 0.14\n75 16.75 83.24 0.01\n100 16.79 83.21 0.00\n200 16.81 83.19 0.00\n500 16.80 83.20 0.00\nTable 21: Relative areas of the basins of\nattraction with \r0= 0:3,\r1= 0:2725 and\nT0varying.\nFigure 26: Plot of the relative areas of the basins of\nattraction as per Table 21.\n5 Numerical Methods\nThe two main numerical methods implemented for the simulations throughout were a variable order\nAdams-Bashforth-Moulton method and the method of analytic continuation [10, 23, 12, 17, 34, 36];\nthe latter consists in a numerical implementation of the Frobenius method. Also used to check\nthe results was a Runge-Kutta method. The Adams-Bashforth-Moulton integration scheme used is\nthe built in integrator found in matlab , ODE113, whereas the programs based on the method of\nanalytic continuation and Runge-Kutta scheme were written in C. Of the three methods, the slowest\nwas the Adams-Bashforth-Moulton method, however it was found that the method worked well for\nthe system with the chosen parameter values and the results produced were reliable.\nBoth the Adams-Bashforth-Moulton and Runge-Kutta methods are standard methods for solving\nODEs of this type. When implementing these two integrators to calculate the basins of attraction,\ntwo di\u000berent methods for both choosing initial conditions in phase space and classifying attractors\nwere used. The \frst method for picking initial conditions was to take a mesh of equally spaced points\nin the phase space: this method ensures uniform coverage of the phase space. When taking this\napproach a mesh of either 321 141 or 503 289 points was used depending on the desired accuracy. The\nsecond method was to take random initial conditions: this can be done by choosing initial conditions\nfrom a stream of random points, which allows the user to use the same random initial conditions\nin each simulation if required. This method is often preferred as the accuracy of the estimates\nof the relative areas of the basins of attraction compared to the number of initial conditions used\ncan be calculated [29]; also it is easier to run additional simulations for extra random points to\nimprove estimates later on, when needed. When using random points, the number of points used\nto calculate the basins of attraction was 300 000 or 400 000 depending on the expected complexity\n27of the system under given parameters. In some cases where extra accuracy was required due to\nsome attractors having particularly small basins of attraction, additional 200 000 or 300 000 random\npoints were used. This allowed us to obtain an error less than 0.20 on the relative areas of the basins\nof attraction. In the most delicate cases, where more precise estimates were needed to distinguish\nbetween values very close to each other (for instance in Table 4), the error was made smaller by\nincreasing the number of points. We decided to express in all cases the relative areas up to the\nsecond decimal digit because often further increasing the number of points did not alter appreciably\nthat digit.\nTo detect and classify solutions, two methods can be used. The \frst method consists in \fnding\nall attractors as a \frst step, before computing the corresponding basins of attraction: this required\na complete characterisation of both the period and the location in phase space of the attractors. In\nprinciple, this works very well, but has the downside of having to \fnd initially all the attractors,\nand di\u000berentiate between those that occupy the same region in phase space. The second method\nfor classifying the solutions was to create a library of solutions. This was created as the program\nran and built up as new solutions were found. The solution of each integration was then checked\nagainst the library and, if not already known, was added. In this way, the program \fnds solutions\nas it goes, so has the advantage of the user not having to know the existing solutions in the system\nprior to calculating the basins of attraction.\nThe method of analytic continuation was implemented and produced results very similar to those\nof the Adams-Bashforth-Moulton method, but in general was much quicker to run. When imple-\nmenting this method of integration, we only used random initial conditions in the phase space and\nthe library method for classifying solutions. The reason for using the Adams-Bashforth-Moulton\nmethod, despite being the slowest of the three integrators, was for comparison with the method of\nanalytic continuation. Analytic continuation has not previously been used for numerically integrat-\ning an ODE of the form (1.2), that is an equation with an in\fnite polynomial nonlinearity that\nsatis\fes an addition formula, thus it was good to have a reliable method to check results with. The\nmethod for using analytic continuation for integrating ODE's that have in\fnite polynomial nonlin-\nearity that satisfy an addition formula will be described more extensively in [36]. The similarity in\nresults of these completely di\u000berent methods for integration, initial condition selection and solution\nclassi\fcation provides reassurance and con\fdence that the results produced are accurate.\n6 Concluding remarks\nIn this work we have numerically shown the importance of not only the \fnal value of dissipation but\nits entire time evolution, for understanding the long time behaviour of the pendulum with oscillating\nsupport. This extends the work done in [3] to a system which, even for values of the parameters\nin the perturbation regime, exhibits richer and more varied dynamics, due to the presence of the\nseparatrix in the phase space of the unperturbed system. In addition we have considered also values\nof the parameters beyond the perturbation regime (the inverted pendulum), where the system cannot\nbe considered a perturbation of an integrable one. In particular this results in a more complicated\nscenario, with bifurcation phenomena and the appearance of attractors which exist only for values\nof the damping coe\u000ecient \rin \fnite intervals away from zero, say \r2[\r1;\r2], with\r1>0.\nWe have preliminarily studied the behaviour of the system in the case of constant dissipation.\nFirstly, in the perturbation regime, we have analytically computed to \frst order the threshold values\nbelow which the periodic attractors exist. We have also discussed why this approach fails due to\nintrinsic perturbation theory limitations, in particular why the method cannot be applied to stable\ncases of the upwards con\fguration or to solutions too close to the unperturbed separatrix. Next,\nwe have studied numerically the dependence of the sizes of the basins of attraction on the damping\ncoe\u000ecient.\nThen we have explicitly considered the case of damping coe\u000ecient varying monotonically between\ntwo values and outlined a few expectations for the way in which the basins of attraction accordingly\nchange with respect to the case of constant dissipation. These expectations were later illustrated\nand backed up with numerical simulations: in particular the relevance of the study of the dynamics\n28at constant dissipation was argued at length. While the expectations account for many features\nobserved numerically, there are still some facts which are di\u000ecult to explain, even at an heuristic\nlevel, and which would deserve further investigation, such as the relationship between the \fxed point\nsolution and the oscillating attractors, to better understand why in some cases they exchange large\nareas of their basins of attraction when the damping coe\u000ecient varies in time. More generally, an\nin-depth numerical study of the system with constant dissipation, also for other parameter values,\nwould be worthwhile. In particular it would be interesting to perform a more detailed bifurcation\nanalysis with respect to the parameter \rand to study the system for very small values of the damping\ncoe\u000ecient\r(which relates to the spin-orbit model in celestial mechanics), both in the perturbation\nregime and for large values of the forcing amplitude. We think that, in order to study cases with\nvery small dissipation, the method of analytical continuation brie\ry described in Section 5 could be\nparticularly fruitful.\nSome interesting features appeared in our analysis which would deserve further consideration\nare:\n\u000fthe increase in the basin of attraction of the \fxed point observed in Figure 3 when the damping\ncoe\u000ecient becomes small enough;\n\u000fthe appearance of the period 4 solution for a thin interval of values of the damping coe\u000ecient,\nas emerges in Table 11;\n\u000fthe rate at which the values of the relative areas of the basins attraction corresponding to the\ninitial value \r0of\r(t) are approached as the variation time T0increases;\n\u000fthe way in which the basins of attraction of the disappearing attractors distribute among the\npersisting ones;\n\u000fthe oscillations through which the relative areas of the basins of attraction approach the\nasymptotic value when taking larger and larger values of the variation time T0, as observed\nfor instance in Tables 16 and 17;\n\u000fthe jumps corresponding to T0= 1000 in Figure 21 and T0= 100,T0= 500 in Figure 22;\n\u000fthe computation of the threshold values to second order, so as to include the rotating solutions\nfound numerically in the perturbation regime investigated in Section 3.\nFinally, investigating analogous systems such as the pendulum with periodically varying length\nto see if similar dynamics occur would also be fascinating in its own right. Another interesting model\nto investigate further, especially in the case of very small dissipation, is the spin-orbit model already\nconsidered in [3], which is expected to be of relevance to understand the locking into the resonance\n3 : 2 of the system Mercury-Sun.\nAcknowledgements\nThe Adams-Bashforth-Moulton method used was MATLAB 's ODE113. We thank Jonathan Deane\nfor helpful conversations on analytic continuation and support with coding in C. This research was\ncompleted as part of an EPSRC funded PhD.\nA Global attraction to the two \fxed points\nTo compute the conditions for attraction to the origin we use the method outlined in [4]; see also\n[2]. We de\fne f(\u001c) as in (1.3) and require f(\u001c)>0: the consequences of this restriction are that\nthe method can only be applied to the downwards pointing pendulum when \u000b>\f . Then we apply\nthe Liouville transformation\n~\u001c=Z\u001c\n0p\nf(s)ds (A.1)\n29and write our equation (1.3) in terms of the new time ~ \u001cas\n\u0012~\u001c~\u001c+0\n@~f(~\u001c)~\u001c\n2~f(~\u001c)+\rq\n~f(~\u001c)1\nA\u0012~\u001c+ sin\u0012= 0; (A.2)\nwhere the subscript ~ \u001crepresents derivative with respect to the new time ~ \u001cand ~f(~\u001c) :=f(\u001c). This\ncan be represented as the two-dimensional system on T\u0002R, by setting x(~\u001c) =\u0012(~\u001c) and writing\nx~\u001c=y; y ~\u001c=\u0000yq\n~f0\n@~f~\u001c\n2q\n~f+\r1\nA\u0000sinx; (A.3)\nfor which we have the energy E(x;y) = 1\u0000cosx+y2=2. By setting H(~\u001c) =E(x(~\u001c);y(~\u001c)), one \fnds\nH~\u001c=\u0000y2\nq\n~f0\n@~f~\u001c\n2q\n~f+\r1\nA; (A.4)\nthusH~\u001c\u00140, i.ex,yare bounded given that \rsatis\fes\n\r >\u0000min\n~\u001c\u00150~f~\u001c\n2q\n~f=\u0000min\n\u001c\u00150f0\n2f: (A.5)\nMoreover we have that for all ~ \u001c >0\nH(~\u001c) +Z~\u001c\n0y2\nq\n~f0\n@~f0\n2q\n~f+\r1\nAds=H(0); (A.6)\nso that, as ~ \u001c!1 , using the properties above we can arrive at\nmin\ns\u001502\n41q\n~f0\n@~f~\u001c\n2q\n~f+\r1\nA3\n5Z1\n0y2(s)ds<1: (A.7)\nHencey!0 as time tends to in\fnity. There are two regions of phase space to consider. Any level\ncurve ofHstrictly inside the separatrix of the unperturbed pendulum is the boundary of a positively\ninvariant set Dcontaining the origin: since S=f(x(~\u001c);y(~\u001c)) :H~\u001c= 0g[Dconsists purely of the\norigin, we can apply the local Barbashin-Krasovsky-La Salle theorem [24] to conclude that every\ntrajectory that begins strictly inside the separatrix will converge to the origin as ~ \u001c!+1.\nOutside of the separatrix we may use equation (A.4), which shows the energy to be strictly\ndecreasing while y6= 0, provided \ris chosen large enough, coupled with y!0 as time tends to\nin\fnity. The result is that all trajectories tend to the invariant points on the x-axis as time tends\nto in\fnity. One of two cases must occur: either the trajectory moves inside the separatrix or it does\nnot. In the \frst instance we have already shown that the limiting solution is the origin. In the latter\nthere is only one possibility. As all points on the x-axis are contained within the separatrix other\nthan the unstable \fxed point, the trajectory must move onto such a \fxed point and hence belongs\nto its stable manifold, which is a zero-measure set. Therefore we conclude that a full measure set\nof initial conditions are attracted by the origin. Reverting back to the original system with time \u001c,\nwe conclude that for that system too the basin of attraction of the origin has full measure, provided\n\f <\u000b and\rsatis\fes (A.5).\n30B Action-angle variables\nIn this section we detail the calculation of the action-angle variables for the simple pendulum in\ntime\u001c. More details on calculating action-angle variables can be found in [14, 32, 9]. The simple\npendulum has equation of motion given by\n\u001200+\u000bsin\u0012= 0; (B.1)\nwhere the dashes represent derivative with respect to the scaled time \u001c. The Hamiltonian for the\nsimple pendulum in this notation is\nE=H(\u0012;\u00120) =1\n2(\u00120)2\u0000\u000bcos\u0012 (B.2)\nor, in terms of the usual notation for Hamiltonian dynamics,\nE=H(p;q) =1\n2p2\u0000\u000bcosq; (B.3)\nwhereq=\u0012andp=q0=\u00120. Rearranging this for pwe obtainp=\u0006p(E;q), with\np(E;q) =p\n2(E+\u000bcosq) =p\n2\u000b(E0+ cosq); (B.4)\nwhereE0=E=\u000b. It is clear that there are two types of dynamics, oscillatory dynamics when\nE0<1 and rotational dynamics when E0>1, separated at a separatrix when E0= 1, for which no\naction-angle variables exist.\nB.1 Librations\nWe \frst consider the case E0<1. The action variable is\nI=1\n2\u0019I\npdq=2\n\u0019p\n2\u000bZq1\n0p\nE0+ cosqdq=8\n\u0019p\u000bh\n(k2\n1\u00001)K(k1) +E(k1)i\n; (B.5)\nwherek2\n1= (E0+1)=2 andq1= arccos(\u0000E0). The functions K(k) and E(k) are the complete elliptic\nintegrals of the \frst and second kinds respectively.\nThe angle variable 'can be found as follows\n'0=@H\n@I=dE\ndI=\u0012dI\ndE\u0013\u00001\n; (B.6)\nso that\ndI\ndE=d\ndE2\n\u0019Zq1\n0p\nE+\u000bcosqdq=2\n\u0019p\u000bK(k1): (B.7)\nHence we have\n'(\u001c) =\u0019\n2K(k1)p\u000b(\u001c\u0000\u001c0): (B.8)\nTakes= sin (q=2); then using equation (B.2) it is easy to show that\n(s0)2=g\nl(1\u0000s2)\u0000\nk2\n1\u0000s2\u0001\n: (B.9)\nIntegrating using the Jacobi elliptic functions\ns(\u001c) =k1sn\u0012rg\nl(\u001c\u0000\u001c0);k1\u0013\n; (B.10)\nthe expresssion can then be rearranged to achieve the following result:\nq= 2 arcsin\u0014\nk1sn\u00122K(k1)\n\u0019';k1\u0013\u0015\n; p = 2k1p\u000bcn\u00122K(k1)\n\u0019';k1\u0013\n; (B.11)\nwhich coincide with equations (2.4). By using (E.3) in Appendix E, one obtains from (B.5)\n@I\n@k1=8\n\u0019k1K(k1)p\u000b; (B.12)\na relation which has been used to derive (2.12).\n31B.2 Rotations\nIn the case of rotational dynamics we have\nI=1\n2\u0019Z2\u0019\n0pdq=1\n2\u0019p\u000bZ2\u0019\n0p\nE0+ cosqdq=4\nk2\u0019p\u000bE(k2); (B.13)\nwhere this time we let k2\n2= 2=(E0+ 1) = 1=k2\n1. The angle variable 'can similarly be found using\n(B.6), where d I=dEcan be similarly calculated as\ndI\ndE=d\ndE1\n2\u0019Z2\u0019\n0p\nE+\u000bcosqdq=k2\n\u0019p\u000bK(k2); (B.14)\nwhich hence gives\n'(\u001c) =\u0019\nK(k2)p\u000b(\u001c\u0000\u001c0)\nk2: (B.15)\nUsing (B.9) and the de\fnition of k2, for the rotating solutions we \fnd that\ns(\u001c) = sn\u0012p\u000b(\u001c\u0000\u001c0)\nk2;k2\u0013\n; (B.16)\nand similarly, by simple rearrangement, we \fnd that\nq= 2 arcsin\u0014\nsn\u0012K(k2)\n\u0019';k2\u0013\u0015\n; p =2\nk2p\u000bdn\u0012K(k2)\n\u0019';k2\u0013\n; (B.17)\nwhich again yields equations (2.6) By using (E.3) in Appendix E, one obtains from (B.13)\n@I\n@k2=\u00004\n\u0019k2\n2K(k2)p\u000b: (B.18)\nwhich has been used to derive (2.23).\nC Jacobian determinant\nHere we compute the entries of the Jacobian matrix Jof the transformation to action-angle variables,\nwhich will be used in the next Appendix. As a by-product we check that Jdeterminant equal to 1,\nthat is@q\n@'@p\n@I\u0000@q\n@I@p\n@'= 1: (C.1)\nFor further details on the proof of (C.1) we refer the reader to [9], where the calculations are given\nin great detail. The derivative with respect to 'is straightforward in both the libration and rotation\ncase, however the dependence of pandqon the action Iis less obvious. That said, the dependence of\npandqonk1in the oscillating case and k2in the rotating case is clear and we know the relationship\nbetweenIandkin both cases, hence the derivative of the Jacobi elliptic functions can be calculated\nby using that\n@\n@I=@k\n@I@\n@k+@u\n@I@\n@u=@k\n@I\u0012@\n@k+@u\n@k@\n@u\u0013\n; (C.2)\nwhereuis the \frst argument of the functions, i.e sn( u;k), etc. Then for the oscillations we have\n@q\n@I=\u0019\n4k1K(k1)p\u000b\u0014sn(\u0001)\ndn(\u0001)+2E(k1)'cn(\u0001)\n\u0019k02\n1+k2\n1sn(\u0001) cn2(\u0001)\nk02\n1dn(\u0001)\u0000E(\u0001) cn(\u0001)\nk02\n1\u0015\n;\n@p\n@I=\u0019\n4k1K(k1)\u0014\ncn(\u0001)\u00002E(k1)'sn(\u0001) dn(\u0001)\n\u0019k02\n1\u0000k2\n1sn2(\u0001) cn(\u0001)\nk02\n1+E(\u0001) sn(\u0001) dn(\u0001)\nk02\n1\u0015\n;\n@q\n@'=4k1K(k1) cn(\u0001)\n\u0019;\n@p\n@'=\u0000p\u000b4k1K(k1) sn(\u0001) dn(\u0001)\n\u0019;(C.3)\n32where (\u0001) =\u0010\n2K(k1)'\n\u0019;k1\u0011\nandk0\n1=p\n1\u0000k2\n1. From the above it is easy to check that equation (C.1)\nis satis\fed. Similarly, for the rotations we have\n@q\n@I=\u0000\u0019k2\n2\n2K(k2)p\u000b\u0014'E(k2) dn(\u0001)\n\u0019k2k02\n2+k2sn(\u0001) cn(\u0001)\nk02\n2\u0000E(\u0001) dn(\u0001)\nk2k02\n2\u0015\n;\n@p\n@I=\u0019k2\n2\n2K(k2)\u0014dn(\u0001)\nk2\n2+'E(k2) sn(\u0001) cn(\u0001)\n\u0019k02\n2+sn2(\u0001) dn(\u0001)\nk02\n2\u0000E(\u0001) sn(\u0001) cn(\u0001)\nk02\n2\u0015\n;\n@q\n@'=2K(k2) dn(\u0001)\n\u0019;\n@p\n@'=\u0000p\u000b2k2K(k2) sn(\u0001) cn(\u0001)\n\u0019;(C.4)\nwhere (\u0001) =\u0010\nK(k2)\n\u0019';k2\u0011\nandk0\n2=p\n1\u0000k2\n2. It is once again easily checked from the above that\n(C.1) is satis\fed.\nD Equations of motion for the perturbed system\nBy (C.1) one has\u0012@'=@q @'=@p\n@I=@q @I=@p\u0013\n=\u0012@p=@I\u0000@q=@I\n\u0000@p=@' @q=@':\u0013\n(D.1)\nWe rewrite the equation (1.3) in the action-angle coordinates introduced in Appendix B as follows.\nD.1 Librations\nIn this section we want to write (1.3) in terms of the action-angle introduced in Appendix B. By\ntaking into account the forcing term \u0000\fcos\u001ccos\u0012in (1.3) one \fnds\nI0=@I\n@qq0+@I\n@pp0=\u0000@p\n@'q0+@q\n@'p0\n=8\fk2\n1K(k1)\n\u0019cos(\u001c\u0000\u001c0) sn(\u0001) cn(\u0001) dn(\u0001);\n'0=@'\n@qq0+@'\n@pp0=@p\n@Iq0\u0000@q\n@Ip0\n=\u0019p\u000b\n2K(k1)\u0000\u0019\f\n2K(k1)p\u000b\u0014\nsn2(\u0001) +2E(k1)'sn(\u0001) cn(\u0001) dn(\u0001)\n\u0019k02\n1\n+k2\n1sn2(\u0001) cn2(\u0001)\n1\u0000k2\n1\u0000E(\u0001) sn(\u0001) cn(\u0001) dn(\u0001)\n1\u0000k2\n1\u0015\ncos(\u001c\u0000\u001c0);(D.2)\nwhere we have used the properties of the Jacobi elliptic functions in Appendix E. As in Appendix\nC, we are shortening ( \u0001) =\u0010\n2K(k1)'\n\u0019;k1\u0011\n.\nWe then wish to add the dissipative term \r\u00120. This results in the following equations:\nI0=8\fk2\n1K(k1)\n\u0019cos(\u001c\u0000\u001c0) sn(\u0001) cn(\u0001) dn(\u0001)\u00008\rk2\n1p\u000bK(k1)\n\u0019cn2(\u0001);\n'0=\u0019p\u000b\n2K(k1)\u0000\u0019\f\n2K(k1)p\u000b\u0014\nsn2(\u0001) +2E(k1)'sn(\u0001) cn(\u0001) dn(\u0001)\n\u0019(1\u0000k2\n1)\n+k2\n1sn2(\u0001) cn2(\u0001)\n1\u0000k2\n1\u0000E(\u0001) sn(\u0001) cn(\u0001) dn(\u0001)\n1\u0000k2\n1\u0015\ncos(\u001c\u0000\u001c0)\n+\r\u0019cn(\u0001)\n2K(k1)\u0014sn(\u0001)\ndn(\u0001)+2E(k1)'cn(\u0001)\n\u0019(1\u0000k2\n1)+k2\n1sn(\u0001) cn2(\u0001)\n(1\u0000k2\n1) dn(\u0001)\u0000E(\u0001) cn(\u0001)\n1\u0000k2\n1\u0015\n:(D.3)\nUsing the property that, see [26], E(u;k) =E(k)u=K(k) +Z(u;k) we arrive at equations (2.9). The\nfunction Z(u;k) is the Jacobi zeta function, which is periodic with period 2 K(k) inu.\n33D.2 Rotations\nThe presence of the forcing term leads to te equations\nI0=\u0000@p\n@'q0+@q\n@'p0=4\fK(k2)\n\u0019cos(\u001c\u0000\u001c0) sn(\u0001) cn(\u0001) dn(\u0001);\n'0=@p\n@I_q\u0000@q\n@I_p=\u0019p\u000b\nk2K(k2)+\u0019k2\fp\u000bK(k2)\u0014E(k2)'sn(\u0001) cn(\u0001) dn(\u0001)\n\u0019(1\u0000k2\n2)\n+k2\n2sn2(\u0001) cn2(\u0001)\n1\u0000k2\n2\u0000E(\u0001) sn(\u0001) cn(\u0001) dn(\u0001)\n1\u0000k2\n2\u0015\ncos(\u001c\u0000\u001c0):(D.4)\nAgain, if we wish to add a dissipative term, we arrive at the equations\nI0=4\fK(k2)\n\u0019cos(\u001c\u0000\u001c0) sn(\u0001) cn(\u0001) dn(\u0001)\u00004\rp\u000bK(k2)\n\u0019k2dn2(\u0001);\n'0=\u0019p\u000b\nk2K(k2)+\u0019k2\fp\u000bK(k2)\u0014E(k2)'sn(\u0001) cn(\u0001) dn(\u0001)\n\u0019(1\u0000k2\n2)\n+k2\n2sn2(\u0001) cn2(\u0001)\n1\u0000k2\n2\u0000E(\u0001) sn(\u0001) cn(\u0001) dn(\u0001)\n1\u0000k2\n2\u0015\ncos(\u001c\u0000\u001c0)\n\u0000\r\u0019\nK(k2)\u0014E(k2)'dn2(\u0001)\n\u0019(1\u0000k2\n2)+k2\n2sn(\u0001) cn(\u0001) dn(\u0001)\n1\u0000k2\n2\u0000E(\u0001) dn2(\u0001)\n1\u0000k2\n2\u0015\n:(D.5)\nAgain using that E(u;k) =E(k)u=K(k) +Z(u;k) we arrive at the equations (2.22).\nE Useful properties of the elliptic functions\nThe complete integrals of the \frst and second kind are, respectively,\nK(k) =Z\u0019=2\n0d p\n1\u0000k2sin2 ; E(k) =Z\u0019=2\n0d q\n1\u0000k2sin2 ; (E.1)\nwhereas the incomplete elliptic integral of the second kind is\nE(u;k) =Zsn(u;k)\n0dxp\n1\u0000k2x2\np\n1\u0000x2: (E.2)\nOne has\n@K(k)\n@k=1\nk\u0012E(k)\n1\u0000k2\u0000K(k)\u0013\n;@E(k)\n@k=1\nk(E(k)\u0000K(k)): (E.3)\nThe following properties of the Jacobi elliptic functions have been used in the previous sections.\nThe derivatives with respect to the \frst arguments are\n@\n@usn(u;k) = cn(u;k) dn(u;k);\n@\n@ucn(u;k) =\u0000sn(u;k) dn(u;k);\n@\n@udn(u;k) =\u0000k2sn(u;k) cn(u;k);(E.4)\nwhile the derivatives with respect to the elliptic modulus are\n@\n@ksn(u;k) =u\nkcn(u;k) dn(u;k) +k\nk02sn(u;k) cn2(u;k)\u00001\nkk02E(u;k) cn(u;k) dn(u;k);\n@\n@kcn(u;k) =\u0000u\nksn(u;k) dn(u;k)\u0000k\nk02sn2(u;k) cn(u;k) +1\nkk02E(u;k) sn(u;k) dn(u;k);\n@\n@kdn(u;k) =\u0000kusn(u;k) cn(u;k)\u0000k\nk02sn2(u;k) dn(u;k) +k\nk02E(u;k) sn(u;k) cn(u;k);(E.5)\n34wherek02= 1\u0000k2.\nFinding the value of \u0001 for rotations in Section 2 requires use of\nZx1\n0dn2(x;k) dx=Zsn(x1;k)\n0p\n1\u0000k2^x2\np\n1\u0000^x2d^x=E(x1;k): (E.6)\nIn the case of librations we also require the relation k2cn2(\u0001) + (1\u0000k2) = dn2(\u0001).\nThe integral for \u0000 1(\u001c0;p;q) in equation (2.20) is found by\n\u00001(\u001c0;p;q) =1\nTZT\n0sn(p\u000b\u001c) cn(p\u000b\u001c) dn(p\u000b\u001c) cos(\u001c\u0000\u001c0) d\u001c\n=cos(\u001c0)\nTZT\n0sn(p\u000b\u001c) cn(p\u000b\u001c) dn(p\u000b\u001c) cos(\u001c) d\u001c\n+sin(\u001c0)\nTZT\n0sn(p\u000b\u001c) cn(p\u000b\u001c) dn(p\u000b\u001c) sin(\u001c) d\u001c\n=sin(\u001c0)\nTZT\n0sn(p\u000b\u001c) cn(p\u000b\u001c) dn(p\u000b\u001c) sin(\u001c) d\u001c;(E.7)\nwhereT= 2\u0019q= 4K(k1)p.\nThe Jacobi elliptic functions can be expanded in a Fourier series as\nsn(u;k) =2\u0019\nkK(k)1X\nn=1qn\u00001=2\n1\u0000q2n\u00001sin\u0012(2n\u00001)\u0019u\n2K(k)\u0013\n;\ncn(u;k) =2\u0019\nkK(k)1X\nn=1qn\u00001=2\n1 +q2n\u00001cos\u0012(2n\u00001)\u0019u\n2K(k)\u0013\n;\ndn(u;k) =\u0019\n2K(k)+2\u0019\nK(k)1X\nn=1qn\n1\u0000q2ncos\u00122n\u0019u\n2K(k)\u0013\n;(E.8)\nwhere qis the nome, de\fned as\nq= exp\u0012\n\u0000\u0019K(k0)\nK(k)\u0013\n;\nwithk0=p\n1\u0000k2.\nIn the calculation of hR(n)iforn\u00152, when the pendulum is in libration, we require the evaluation\nof the integrals\n1\nTZT\n02K(k1)p\u000b\u0019@\n@\u001c\u0010\ncn2(p\u000b\u001c)\u0011\nd\u001c= 0; (E.9)\nand\n1\nTZT\n02K(k1)p\u000b\u0019@\n@\u001c\u0010\nsn(p\u000b\u001c) cn(p\u000b\u001c) dn(p\u000b\u001c)\u0011\ncos(\u001c\u0000\u001c0) d\u001c\n=1\nTZT\n02K(k1)p\u000b\u0019sn(p\u000b\u001c) cn(p\u000b\u001c) dn(p\u000b\u001c) sin(\u001c\u0000\u001c0) d\u001c\n=cos(\u001c0)\nTZT\n02K(k1)p\u000b\u0019sn(p\u000b\u001c) cn(p\u000b\u001c) dn(p\u000b\u001c) sin(\u001c) d\u001c\n\u0000sin(\u001c0)\nTZT\n02K(k1)p\u000b\u0019sn(p\u000b\u001c) cn(p\u000b\u001c) dn(p\u000b\u001c) cos(\u001c) d\u001c;(E.10)\nwhereT= 2\u0019q= 4K(k1)p. The integral multiplying sin( \u001c0) vanishes due to parity and hence\n2K(k1)p\u000b\u0019TZT\n0@\n@\u001c\u0010\nsn(p\u000b\u001c) cn(p\u000b\u001c) dn(p\u000b\u001c)\u0011\ncos(\u001c\u0000\u001c0) d\u001c=2K(k1)p\u000b\u0019cos(\u001c0)G1(p;q):(E.11)\n35References\n[1] M.V. Bartuccelli, A. Berretti, J.H.B. Deane, G. Gentile, S.A. Gourley, Selection rules for periodic orbits\nand scaling laws for a driven damped quartic oscillator , Nonlinear Anal. Real World Appl. 9(2008),\nno. 5, 1966{1988.\n[2] M.V. Bartuccelli, J.H.B. Deane, G. Gentile, Globally and locally attractive solutions for quasi-\nperiodically forced systems , J. Math. Anal. Appl. 328(2007), no. 1, 699{714.\n[3] M.V. Bartuccelli, J.H.B. Deane, G. Gentile, Attractiveness of periodic orbits in parametrically forced\nsystems with time-increasing friction , J. Math. Phys. 53(2012), no.10, 102703, 27 pp.\n[4] M.V. Bartuccelli, J.H.B. Deane, G. Gentile, S.A. Gourley, Global attraction to the origin in a\nparametrically-driven nonlinear oscillator , Appl. Math. Comput. 153(2004), no. 1, 1{11.\n[5] M.V. Bartuccelli, G. Gentile, K.V. Georgiou, On the dynamics of a vertically driven damped planar\npendulum , Proc. R. Soc. Lond. A 457(2001), no. 2016, 3007{3022.\n[6] M.V. Bartuccelli, G. Gentile, K.V. Georgiou, On the stability of the upside-down pendulum with damp-\ning, Proc. R. Soc. Lond. A 458(2002), no. 2018, 255{269.\n[7] M.V. Bartuccelli, G. Gentile, K.V. Georgiou, KAM theory, Lindstedt series and the stability of the\nupside-down pendulum , Discrete Contin. Dyn. Syst. 9(2003), no. 2, 413{426.\n[8] F. Bowman, Introduction to elliptic functions with applications , English Universities Press, London,\n1953.\n[9] A.J. Brizard, Jacobi zeta function and action-angle coordinates for the pendulum , Commun. Nonlinear\nSci. Numer. Simul. 18(2013), 511{518.\n[10] K. Brown, Topics in calculus and probability , Self-published on Lulu (2012), 64{77. Also available at:\nwww.mathpages.com.\n[11] P.F. Byrd, M.D. Friedman, Handbook of elliptic integrals for engineers and scientists , Second edition -\nrevised, Springer, New York-Heidelberg, 1971.\n[12] J.R. Cannon, K. Miller, Some problems in numerical analytic continuation , J. Soc. Indust. Appl. Math.\nSer. B Numer. Anal. 2(1965), no. 1, 87{98.\n[13] Sh.N. Chow, J.K. Hale, Methods of bifurcation theory , Grundlehren der Mathematischen Wissenschaften\n251, Springer, New York-Berlin, 1982.\n[14] A. Fasano, S. Marmi, Analytical mechanics , Oxford University Press, Oxford, 2006.\n[15] U. Feudel, C. Grebogi, Why are chaotic attractors rare in multistable systems , Phys. Rev. E 91(2003),\n134102, 4pp.\n[16] U. Feudel, C. Grebogi, B.R. Hunt, J.A. Yorke, Map with more than 100 coexisting low-period periodic\nattractors , Phys. Rev. E 54(1996), 71{81.\n[17] E. Gallicchio, S.A. Egorov, B.J. Berne, On the application of numerical analytic continuation methods\nto the study of quantum mechanical vibrational relaxation processes , J. Chem. Phys. 109 (1998), no.\n18, 7745{7755.\n[18] G. Gentile, M. Bartuccelli, J. Deane, Bifurcation curves of subharmonic solutions and Melnikov theory\nunder degeneracies , Rev. Math. Phys. 19(2007), no. 3, 307{348.\n[19] G. Gentile, G. Gallavotti, Degenerate elliptic resonances , Comm. Math. Phys. 257(2005), no. 2, 319{\n362\n[20] M. Ghil, G. Wolansky, Non-Hamiltonian perturbations of integrable systems and resonance trapping ,\nSIAM J. Appl. Math. 52(1992), no. 4, 1148{1171.\n[21] J. Guckenheimer, Ph. Holmes, Nonlinear oscillations, dynamical systems, and bifurcations of vector\n\felds , Revised and corrected reprint of the 1983 original, Applied Mathematical Sciences 42, Springer,\nNew York, 1990.\n36[22] D.W. Jordan, P. Smith, Nonlinear ordinary di\u000berential equations. An introduction for scientists and\nengineers , Fourth edition, Oxford University Press, Oxford, 2007.\n[23] K. Kodaira, Complex analysis , Cambridge University Press, Cambridge, 2007.\n[24] N.N. Krasovsky, Problems of the Theory of Stability of Motion , Stanford University Press, Stanford,\n1963.\n[25] L. D. Landau, E. M. Lifshitz, Mechanics , Third Editon, Pergamon Press, Oxford, 1976.\n[26] D.F. Lawden, Elliptic functions and applications , Springer, New York, 1989.\n[27] W. Magnus, S. Winkler, Hill's equation , Second edition - reprint, Dover Publications, New York, 2004.\n[28] C. D. Murray, S. F. Dermott, Solar System Dynamics , Cambridge University Press, Cambridge, 2001.\n[29] R.H. Myers, S.L. Myers, R.E. Walpole, Probability and statistics for engineers and scientists , Sixth\nedition, Prentice Hall, London, 1998.\n[30] R.K. Nagle, E.B. Sa\u000b, A.D. Snider, Fundamentals of di\u000berential equations , Eight edition, Addison-\nWesley, Boston, 2012.\n[31] J. Palis, A global view of dynamics and a conjecture on the denseness of \fnitude of attractors , Ast\u0013 erisque\n261, (2000), 339{351.\n[32] I. Percival, D. Richards, Introduction to dynamics , First edition - reprint, Cambridge University Press,\nCambridge, 1994.\n[33] Ch.S. Rodrigues, A.P.S. de Moura, C. Grebogi, Emerging attractors and the transition from dissipative\nto conservative dynamics , Phys. Rev. E 80(2009), no. 2, 026205, 4pp.\n[34] I. Stewart, D. Tall, Complex analysis , Cambridge University Press, Cambridge, 1983.\n[35] G.N. Watson, E. T. Whittaker, A course of modern analysis , Fourth edition - reprint, Cambridge\nUniversity Press, Cambridge, 1947.\n[36] J.A. Wright, J.H.B. Deane, M. Bartuccelli, G. Gentile, Analytic continuation applied to the problem of\nthe pendulum with vertically oscillating support , in preparation.\n37"    },    {        "title": "1404.1488v2.Gilbert_damping_in_noncollinear_ferromagnets.pdf",        "content": "arXiv:1404.1488v2  [cond-mat.mtrl-sci]  27 Nov 2014Gilbert damping in noncollinear ferromagnets\nZhe Yuan,1,∗Kjetil M. D. Hals,2,3Yi Liu,1Anton A. Starikov,1Arne Brataas,2and Paul J. Kelly1\n1Faculty of Science and Technology and MESA+Institute for Nanotechnology,\nUniversity of Twente, P.O. Box 217, 7500 AE Enschede, The Net herlands\n2Department of Physics, Norwegian University of Science and Technology, NO-7491 Trondheim, Norway\n3Niels Bohr International Academy and the Center for Quantum Devices,\nNiels Bohr Institute, University of Copenhagen, 2100 Copen hagen, Denmark\nThe precession and damping of a collinear magnetization dis placed from its equilibrium are well\ndescribed by the Landau-Lifshitz-Gilbert equation. The th eoretical and experimental complexity\nof noncollinear magnetizations is such that it is not known h ow the damping is modified by the\nnoncollinearity. We use first-principles scattering theor y to investigate transverse domain walls\n(DWs) of the important ferromagnetic alloy Ni 80Fe20and show that the damping depends not only\non the magnetization texture but also on the specific dynamic modes of Bloch and N´ eel DWs in ways\nthat were not theoretically predicted. Even in the highly di sordered Ni 80Fe20alloy, the damping is\nfound to be remarkably nonlocal.\nPACS numbers: 72.25.Rb, 75.60.Ch, 75.78.-n, 75.60.Jk\nIntroduction. —The key common ingredient in various\nproposed nanoscale spintronics devices involving mag-\nnetic droplet solitons [ 1], skyrmions [ 2,3], or magnetic\ndomain walls (DWs) [ 4,5], is a noncollinear magneti-\nzation that can be manipulated using current-induced\ntorques (CITs) [ 6]. Different microscopic mechanisms\nhave been proposed for the CIT including spin trans-\nfer [7,8], spin-orbit interaction with broken inversion\nsymmetry in the bulk or at interfaces [ 9–11], the spin-\nHalleffect[ 12]orproximity-inducedanisotropicmagnetic\nproperties in adjacent normal metals [ 13]. Their contri-\nbutions are hotly debated but can only be disentangled\nif the Gilbert damping torque is accurately known. This\nis not the case [ 14]. Theoretical work [ 15–19] suggest-\ning that noncollinearity can modify the Gilbert damping\ndue to the absorption of the pumped spin current by the\nadjacent precessing magnetization has stimulated exper-\nimental efforts to confirm this quantitatively [ 14,20]. In\nthis Letter, we use first-principles scattering calculations\nto show that the Gilbert damping in a noncollinear alloy\ncan be significantly enhanced depending on the partic-\nular precession modes and surprisingly, that even in a\nhighly disordered alloy like Ni 80Fe20, the nonlocal char-\nacterofthe dampingis verysubstantial. Ourfindingsare\nimportant for understanding field- and/or current-driven\nnoncollinear magnetization dynamics and for designing\nnew spintronics devices.\nGilbert damping in Ni 80Fe20DWs.—Gilbert damping\nis in general described by a symmetric 3 ×3 tensor.\nFor a substitutional, cubic binary alloy like Permalloy,\nNi80Fe20, this tensor is essentially diagonal and isotropic\nand reduces to scalar form when the magnetization is\ncollinear. A value of this dimensionless scalar calculated\nfrom first-principles, αcoll= 0.0046, is in good agree-\nment with values extracted from room temperature ex-\nperiments that range between 0.004 and 0.009 [ 21]. In a\none-dimensional (1D) transverse DW, the Gilbert damp-ing tensor is still diagonal but, as a consequence of the\nlowered symmetry [ 22], it contains two unequal compo-\nnents. The magnetization in static N´ eel or Bloch DWs\n(a) \n(b) \n(c) \nφ\nθ\n x\n y z\nφ\nθ\n0 0.1 0.2 0.3 0.4 \n1/( /h w) (nm -1 )00.01 0.02 0.03 _eff Néel \nBloch \njSO =0 50 20 10 5 3 /h w (nm) \n_oeff _ieff \nFIG. 1. (color online). Sketch of N´ eel (a) and Bloch (b)\nDWs. (c) Calculated effective Gilbert damping parameters\nfor Permalloy DWs (N´ eel, black lines; Bloch, red lines) as a\nfunction of the inverse of the DW width λw. Without spin-\norbit coupling, calculations for the two DW types yield the\nsame results (blue lines). The green dot represents the valu e\nof Gilbert damping calculated for collinear Permalloy. For\neach value of λw, we typically consider 8 different disorder\nconfigurations and the error bars are a measure of the spread\nof the results.2\nliesinsidewelldefinedplanesthatareillustratedinFig. 1.\nAn angle θrepresents the in-plane rotation with respect\nto the magnetizationin the left domainand it variesfrom\n0 toπthrough a 180◦DW. If the plane changes in time,\nas it does when the magnetization precesses, an angle φ\ncan be used to describe its rotation. We define an out-\nof-plane damping component αocorresponding to varia-\ntion inφ, and an in-plane component αicorresponding\nto time-dependent θ. Rigid translation of the DW, i.e.\nmaking the DW center rwvary in time, is a specific ex-\nample of the latter.\nFor Walker-profile DWs [ 23], an effective (dimension-\nless) in-plane ( αeff\ni) and out-of-plane damping ( αeff\no) can\nbe calculated in terms of the scattering matrix Sof the\nsystem using the scattering theory of magnetization dis-\nsipation [ 24,25]. Both calculated values are plotted in\nFig.1(c) as a function of the inverse DW width 1 /λwfor\nN´ eel and Bloch DWs. Results with the spin-orbit cou-\npling (SOC) artificially switched off are shown for com-\nparison; because spin space is then decoupled from real\nspace, the results for the two DW profiles are identical\nand both αeff\niandαeff\novanish in the large λwlimit con-\nfirming that SOC is the origin of intrinsic Gilbert damp-\ning for collinear magnetization. With SOC switched on,\nN´ eel and Bloch DWs have identical values within the\nnumerical accuracy, reflecting the negligibly small mag-\nnetocrystalline anisotropy in Permalloy. Both αeff\niand\nαeff\noapproach the collinear value αcoll[21], shown as a\ngreen dot in the figure, in the wide DW limit. For finite\nwidths, theyexhibit aquadraticandapredominantlylin-\near dependence on 1 /(πλw), respectively, both with and\nwithoutSOC;forlargevaluesof λw, thereisahintofnon-\nlinearity in αeff\no(λw). However, phenomenological theo-\nries [15–17] predict that αeff\nishould be independent of λw\nand equal to αcollwhileαeff\noshould be a quadratic func-\ntion of the magnetization gradient. Neither of these pre-\ndicted behaviours is observed in Fig. 1(c) indicating that\nexisting theoretical models of texture-enhanced Gilbert\ndamping need to be reexamined.\nTheαeffshown in Fig. 1(c) is an effective damping\nconstant because the magnetization gradient dθ/dzof a\nWalker profile DW is inhomogeneous. Our aim in the\nfollowing is to understand the physical mechanisms of\ntexture-enhanced Gilbert damping with a view to deter-\nmining how the local damping depends on the magneti-\nzation gradient, as well as the corresponding parameters\nfor Permalloy, and finally expressing these in a form suit-\nable for use in micromagnetic simulations.\nIn-plane damping αi.—To get a clearer picture of how\nthe in-plane damping depends on the gradient, we calcu-\nlate the energy pumping Er≡Tr/parenleftBig\n∂S\n∂rs∂S†\n∂rs/parenrightBig\nfor a finite\nlengthLof a Bloch-DW-type spin spiral (SS) centered\natrs. In this SS segment (SSS), dθ/dzis constant ex-\ncept at the ends. Figure 2(b) showsthe resultscalculated\nwithout SOC for a single PermalloySSS with dθ/dz= 6◦0 10 20 30 40 \nL (nm) 020 40 Er (nm -2 )Without smearing \nWith smearing 0 4 2 6Winding angle ( /)\n0 1 2 3 4 \nNumber of SSSs z0n//L de/dz L L \n(c) (a) \n(b) \nFIG. 2. (color online). (a) Sketch of the magnetization gra-\ndient for two SSSs separated by collinear magnetization wit h\n(green, dashed) and without (red, solid) a broadening of the\nmagnetization gradient at the ends of the SSSs. The length\nof each segment is L. (b) Calculated energy pumping Eras a\nfunction of Lfor asingle Permalloy Bloch-DW-typeSSSwith-\nout SOC. The upper horizontal axis shows the total winding\nangle of the SSS. (c) Calculated energy pumping Erwithout\nSOC as a function of the number of SSSs that are separated\nby a stretch of collinear magnetization.\nper atomic layer; Fig. 1(c) shows that SOC does not in-\nfluence the quadratic behaviour essentially. Eris seen\nto be independent of Lindicating there is no dissipation\nwhendθ/dzis constant in the absence of SOC. In this\ncase, the only contribution arises from the ends of the\nSSS where dθ/dzchanges abruptly; see Fig. 2(a). If we\nreplace the step function of dθ/dzby a Fermi-like func-\ntion with a smearing width equal to one atomic layer, Er\ndecreasessignificantly(greensquares). Formultiple SSSs\nseparated by collinear magnetization, we find that Eris\nproportional to the number of segments; see Fig. 2(c).\nWhat remains is to understand the physical origin of\nthe damping at the ends of the SSSs. Rigid translation\nof a SSS or of a DW allows for a dissipative spin cur-\nrentj′′\ns∼ −m×∂z∂tmthat breaks time-reversal sym-\nmetry [19]. The divergence of j′′\nsgives rise to a local\ndissipative torque, whose transverse component is the\nenhancement of the in-plane Gilbert damping from the\nmagnetizationtexture. After straightforwardalgebra, we\nobtain the texture-enhanced in-plane damping torque\nα′′/bracketleftbig\n(m·∂z∂tm)m×∂zm−m×∂2\nz∂tm/bracketrightbig\n,(1)\nwhereα′′is a material parameter with dimensions\nof length squared. In 1D SSs or DWs, Eq. ( 1)\nleads to the local energy dissipation rate ˙E(r) =\n(α′′Ms/γ)∂tθ∂t(d2θ/dz2) [25], where Msis the satura-\ntion magnetization and γ=gµB//planckover2pi1is the gyromagnetic\nratio expressed in terms of the Land´ e g-factor and the\nBohr magneton µB. This results shows explicitly that\nthe in-plane damping enhancement is related to finite\nd2θ/dz2. Using the calculated data in Fig. 1(c), we ex-3\ntract a value for the coefficient α′′= 0.016 nm2that is\nindependent of specific textures m(r) [25].\nOut-of-plane damping αo.—We begin our analysis of\nthe out-of-plane damping with a simple two-band free-\nelectron DW model [ 25]. Because the linearity of the\ndamping enhancement does not depend on SOC, we ex-\namine the SOC free case for which there is no differ-\nence between N´ eel and Bloch DW profiles and we use\nN´ eel DWs in the following. Without disorder, we can\nuse the known φ-dependence of the scattering matrix for\nthis model [ 31] to obtain αeff\noanalytically,\nαeff\no=gµB\n4πAMsλw/summationdisplay\nk/bardbl/parenleftbigg/vextendsingle/vextendsingle/vextendsinglerk/bardbl\n↑↓/vextendsingle/vextendsingle/vextendsingle2\n+/vextendsingle/vextendsingle/vextendsinglerk/bardbl\n↓↑/vextendsingle/vextendsingle/vextendsingle2\n+/vextendsingle/vextendsingle/vextendsingletk/bardbl\n↑↓/vextendsingle/vextendsingle/vextendsingle2\n+/vextendsingle/vextendsingle/vextendsingletk/bardbl\n↓↑/vextendsingle/vextendsingle/vextendsingle2/parenrightbigg\n≈gµB\n4πAMsλwh\ne2GSh,. (2)\nwhereAis the cross sectional area and the convention\nused for the reflection ( r) and transmission ( t) probabil-\nity amplitudes is shown in Fig. 3(a). Note that |tk/bardbl\n↑↓|2and\n|tk/bardbl\n↓↑|2are of the order of unity and much larger than the\nothertwotermsbetweenthebracketsunlesstheexchange\nsplitting is very large and the DW width very small. It\nis then a good approximation to replace the quantities in\nbracketsbythenumberofpropagatingmodesat k/bardbltoob-\ntain the second line of Eq. ( 2), where GShis the Sharvin\nconductance that only depends on the free-electron den-\nsity. Equation ( 2) shows analytically that αeff\nois pro-\nportional to 1 /λwin the ballistic regime. This is repro-\nduced by the results of numerical calculations for ideal\nfree-electron DWs shown as black circles in Fig. 3(b).\nIntroducing site disorder [ 32] into the free-electron\nmodel results in a finite resistivity. The out-of-plane\ndamping calculated for disordered free-electron DWs ex-\nhibits a transition as a function of its width. For narrow\nDWs (ballistic limit), αeff\nois inversely proportional to λw\nand the green, red and blue circles in Fig. 3(b) tend to\nbecomeparalleltothevioletlineforsmallvaluesof λw. If\nλwis sufficiently large, αeff\nobecomes proportional to λ−2\nw\nin agreement with phenomenological predictions [ 15–17]\nwhere the diffusive limit is assumed. This demonstrates\nthe different behaviour of αeff\noin these two regimes.\nWe can construct an expression that describes both\nthe ballistic and diffusive regimes by introducing an ex-\nplicit spatial correlation in the nonlocal form of the out-\nof-plane Gilbert damping tensor that was derived using\nthe fluctuation-dissipation theorem [ 15]\n[αo]ij(r,r′) =αcollδijδ(r−r′)+α′D(r,r′;l0)\n×[m(r)×∂zm(r)]i[m(r′)×∂z′m(r′)]j.(3)\nHereα′isamaterialparameterwithdimensionsoflength\nsquared and Dis a correlation function with an effective\ncorrelation length l0. In practice, we use D(r,r′;l0) =\n1√πAl0e−(z−z′)2/l2\n0, which reduces to δ(r−r′) in the dif-\nfusive limit ( l0≪λw) and reproduces earlier results [ 15–\n17]. In the ballistic limit, both α′andl0are infinite,0.01 0.05 0.1 0.5 \n1/( /h w) (nm -1 )10 -4 10 -3 10 -2 10 -1 _oeff \nBallistic \nl=2.7 !1 cm \nl=25 !1 cm \nl=94 !1 cm 100 50 30 20 10 5 2/h w (nm) \n~1/ hw\n~1/ hw2(a) \n(b) \nand . By definition, for weak splitting 1, but for all commonplace \ns the Fermi wavelength 2 is orders of magnitude smaller than . This \nimplies a wall resistance that is vanishingly small, because of the exponential depen- \ndence. For the example of iron, 2 is only 1 or 2 A , depending on which band \nis in question, whilst the wall thickness is some thousands of A . This leads to a \n10 . The physical reason for this is that waves are only scattered very much \nby potential steps that are abrupt on the scale of the wavelength of that wave, as \nsketched in figure 13. \nFor strong splitting ( it was found to be necessary to restrict the \nculation to a very narrow wall, viz. me 1. In practice this means \nmic abruptness. In this case a variable ¼ ð ÞÞ , trivially \nconnected to the definitions of in equations (2) and (3), determines the DW \nce. The obvious relationship with the Stearns definition of polarisation, \nequation (3), emphasises that the theory is essentially one of tunnelling between \none domain and the next. The DW resistance vanishes as 1, as might be \nd. As !1 uivalent to unity), the material becomes half-metallic \nand the wall resistance also !1 . A multi-domain half-metal, with no opportunity \nfor spin relaxation, is an insulator, no matter how high is. \nCabrera and Falicov satisfied themselves that, once the diamagnetic Lorentz \nforce e that give rise to additional resistance at the wall were properly treated \n[178], their theory could account for the results found in the Fe whiskers. However, \nit does not describe most cases encountered experimentally because the condition Abrupt \nFigure 13. Spin-resolved potential profiles and resulting wavefunctions at abrupt \nand wide (adiabatic) domain walls. The wavefunctions are travelling from left to right. In the \nadiabatic case, the wavelengths of the two wavefunctions are exchanged, but the change in \npotential energy is slow enough that there is no change in the amplitude of the transmitted \nwave. When the wall is abrupt the wavelength change is accompanied by substantial reflection, \nlting in a much lower transmitted amplitude (the reflected part of the wavefunction is not \nshown). This gives rise to domain wall resistance. C. H. Marrows Downloaded By: [University of California, Berkeley] At: 14:36 9 June 2010  V↑ V↓\n↓\n↑  e±ik↑z e±ik↓z\n e±ik↓z e±ik↑z\n. By definition, for weak splitting 1, but for all commonplace \nmi wavelength 2 is orders of magnitude smaller than . This \na wall resistance that is vanishingly small, because of the exponential depen- \ne of iron, 2 is only 1 or 2 A , depending on which band \nis in question, whilst the wall thickness is some thousands of A . This leads to a \n10 . The physical reason for this is that waves are only scattered very much \nby potential steps that are abrupt on the scale of the wavelength of that wave, as \nd in figure 13. \nit was found to be necessary to restrict the \nto a very narrow wall, 1. In practice this means \nabruptness. In this case a variable ¼ ð ÞÞ , trivially \nto the definitions of in equations (2) and (3), determines the DW \n. The obvious relationship with the Stearns definition of polarisation, \non (3), emphasises that the theory is essentially one of tunnelling between \nDW resistance vanishes as 1, as might be \nd. As !1 to , the material becomes half-metallic \n!1 . A multi-domain half-metal, with no opportunity \nis an insulator, no matter how high \nto additional resistance at the wall were properly treated \nld account for the results found in the Fe whiskers. However, \nit does not describe most cases encountered experimentally because the condition at abrupt \nto right. In the \nof the two wavefunctions are exchanged, but the change in \nis slow enough that there is no change in the amplitude of the transmitted \nis abrupt the wavelength change is accompanied by substantial reflection, \nin a much lower transmitted amplitude (the reflected part of the wavefunction is not \nto domain wall resistance. C. H. Marrows Downloaded By: [University of California, Berkeley] At: 14:36 9 June 2010 \n t↑↑ t↓↓\n↓↓\n t↑↓  t↓↑\nFIG. 3. (color online). (a) Cartoon of electronic transport\nin a two-band, free-electron DW. The global quantization\naxis of the system is defined by the majority and minority\nspin states in the left domain. (b) Calculated αeff\nofor two-\nband free-electron DWs as a function of 1 /(πλw) on a log-log\nscale. The black circles show the calculated results for the\nclean DWs, whichare in perfect agreement with theanalytica l\nmodel Eq. ( 2), shown as a dashed violet line. When disorder\n(characterized by the resistivity ρcalculated for the corre-\nsponding collinear magnetization) is introduced, αeff\noshows a\ntransition from a linear dependence on 1 /λwfor narrow DWs\ntoaquadraticbehaviourfor wideDWs. The solid lines arefits\nusing Eq. ( S24). The dashed orange lines illustrate quadratic\nbehaviour.\nbut the product α′D(r,r′;l0) =α′/(√πAl0) is finite and\nrelated to the Sharvin conductance of the system [ 33],\nconsistent with Eq. ( 2). We then fit the calculated val-\nues ofαeff\noshown in Fig. 3(b) using Eq. ( S24) [25]. With\nthe parameters α′andl0listed in Table I, the fit is seen\nto be excellent over the whole range of λw. The out-\nof-plane damping enhancement arises from the pumped\nspin current j′\ns∼∂tm×∂zmin a magnetization tex-\nture [15,17], where the magnitude of j′\nsis related to the\nTABLEI. Fitparameters usedtodescribe thedampingshown\nin Fig.1for Permalloy DWs and in Fig. 3for free-electron\nDWs with Eq. ( S24). The resistivity is determined for the\ncorresponding collinear magnetization.\nSystem ρ(µΩ cm) α′(nm2)l0(nm)\nFree electron 2 .69 45 .0 13 .8\nFree electron 24 .8 1 .96 4 .50\nFree electron 94 .3 0 .324 2 .78\nPy (ξSO= 0) 0 .504 23 .1 28 .3\nPy (ξSO/negationslash= 0) 3 .45 5 .91 13 .14\nconductivity [ 15]. This is the reason why α′is larger in\na system with a lower resistivity in Table I.l0is a mea-\nsure of how far the pumped transverse spin current can\npropagate before being absorbed by the local magnetiza-\ntion. It is worth distinguishing the relevant characteris-\ntic lengths in microscopic spin transport that define the\ndiffusive regimes for different transport processes. The\nmean free path lmis the length scale for diffusive charge\ntransport. The spin-flip diffusion length lsfcharacterizes\nthe length scale for diffusive transport of a longitudinal\nspin current, and l0is the corresponding length scale for\ntransverse spin currents. Only when the system size is\nlarger than the corresponding characteristic length can\ntransport be described in a local approximation.\nWe can use Eq. ( S24) to fit the calculated αeff\noshown\nin Fig.1for Permalloy DWs. The results are shown in\nFig.S4. Since the values of αeff\nowe calculate for N´ eel\nand Bloch DWs are nearly identical, we take their aver-\nage for the SOC case. Intuitively, we would expect the\nout-of-plane damping for a highly disordered alloy like\nPermalloy to be in the diffusive regime corresponding to\na shortl0. But the fitted values of l0are remarkably\nlarge, as long as 28.3 nm without SOC. With SOC, l0\nis reduced to 13.1 nm implying that nonlocal damping\ncan play an important role in nanoscale magnetization\ntextures in Permalloy, whose length scale in experiment\nis usually about 100 nm and can be reduced to be even\nsmaller than l0by manipulating the shape anisotropy of\nexperimental samples [ 34,35].\nAs shown in Table I,l0is positively correlated with\nthe conductivity. The large value of l0and the low re-\nsistivity of Permalloy can be qualitatively understood in\nterms of its electronic structure and spin-dependent scat-\ntering. The Ni and Fe potentials seen by majority-spin\nelectrons around the Fermi level in Permalloy are almost\nidentical [ 25] so that they are only very weakly scattered.\nThe Ni and Fe potentials seen by minority-spin electrons\nare howeverquite different leading to strongscattering in\ntransport. The strong asymmetric spin-dependent scat-\ntering can also be seen in the resistivity of Permalloy\ncalculated without SOC, where ρ↓/ρ↑>200 [21,36]. As\na result, conduction in Permalloy is dominated by the\nweakly scattered majority-spin electrons resulting in a\nlow total resistivity and a large value of l0. This short-\ncircuit effect is only slightly reduced by SOC-induced\nspin-flipscatteringbecausetheSOCin3 dtransitionmet-\nals is in energy terms small compared to the bandwidth\nand exchange splitting. Indeed, αeff\no−αcollcalculated\nwith SOC (the red curve in Fig. S4) shows a greater cur-\nvature at large widths than without SOC, but is still\nquite different from the quadratic function characteristic\nofdiffusive behaviourforthe widest DWs wecould study.\nBothαeff\niandαeff\nooriginate from locally pumped spin\ncurrents proportional to m×∂tm. Because of the spa-\ntially varying magnetization, the spin currents pumped\ntotheleftandrightdonotcancelexactlyandthenetspin0.02 0.05 0.1 0.2 0.5\n1/(πλw) (nm-1)0.0010.010.05αoeff-αcoll\nξSO≠0\nξSO=040 30 20 15 10 5 3 2πλw (nm)\n~1/λw\n~1/λw2\nFIG. 4. (color online). Calculated out-of-plane damping\nαeff\no−αcollfrom Fig. 1plotted as a function of 1 /(πλw) on a\nlog-log scale. The solid lines are fitted using Eq. ( S24). The\ndashed violet and orange lines illustrate linear and quadra tic\nbehaviour, respectively.\ncurrent contains two components, j′′\ns∼ −m×∂z∂tm[19]\nandj′\ns∼∂tm×∂zm[15,17]. For out-of-plane damping,\n∂zmis perpendicular to ∂tmso there is large enhance-\nment due to the lowest order derivative. For the rigid\nmotion of a 1D DW, ∂zmis parallel to ∂tmso thatj′\ns\nvanishes. The enhancement of in-plane damping arising\nfromj′′\nsdue to the higher-orderspatial derivative of mag-\nnetization is then smaller.\nConclusions.— We have discovered an anisotropic\ntexture-enhanced Gilbert damping in Permalloy DWs\nusing first-principles calculations. The findings are ex-\npressed in a form [Eqs. ( 1) and (S24)] suitable for ap-\nplication to micromagnetic simulations of the dynamics\nof magnetization textures. The nonlocal character of the\nmagnetization dissipation suggests that field and/or cur-\nrentdrivenDW motion, whichis alwaysassumedto be in\nthe diffusive limit, needs to be reexamined. The more ac-\ncurate form of the damping that we propose can be used\nto deduce the CITs in magnetization textures where the\nusual way to study them quantitatively is by comparing\nexperimental observations with simulations.\nCurrent-drivenDWs movewith velocities that arepro-\nportional to β/αwhereβis the nonadiabatic spin trans-\nfer torque parameter. The order of magnitude spread in\nvalues of βdeduced for Permalloy from measurements of\nthe velocities of vortex DWs [ 37–40] may be a result of\nassumingthat αis a scalarconstant. Ourpredictions can\nbe tested by reexamining these studies using the expres-\nsions for αgiven in this paper as input to micromagnetic\ncalculations.\nWe would like to thank Geert Brocks and Taher Am-\nlaki for useful discussions. This work was financially\nsupported by the “Nederlandse Organisatie voor Weten-\nschappelijk Onderzoek” (NWO) through the research\nprogramme of “Stichting voor Fundamenteel Onderzoek\nder Materie” (FOM) and the supercomputer facilities5\nof NWO “Exacte Wetenschappen (Physical Sciences)”.\nIt was also partly supported by the Royal Netherlands\nAcademy of Arts and Sciences (KNAW). A.B. acknowl-\nedges the Research Council of Norway, grant no. 216700.\n∗Present address: Institut f¨ ur Physik, Johannes\nGutenberg–Universit¨ at Mainz, Staudingerweg 7, 55128\nMainz, Germany; zyuan@uni-mainz.de\n[1] S. M. Mohseni, S. R. Sani, J. Persson, T. N. A. Nguyen,\nS. Chung, Y. Pogoryelov, P. K. Muduli, E. Iacocca,\nA. Eklund, R. K. Dumas, S. Bonetti, A. Deac, M. A.\nHoefer, and J. ˚Akerman, Science339, 1295 (2013) .\n[2] X. Z. Yu, Y. Onose, N. Kanazawa, J. H. Park,\nJ. H. Han, Y. Matsui, N. Nagaosa, and Y. Tokura,\nNature465, 901 (2010) .\n[3] A. Fert, V. Cros, and J. Sampaio,\nNature Nanotechnology 8, 152 (2013) .\n[4] L. Thomas, R. Moriya, C. Rettner, and S. S. P. Parkin,\nScience330, 1810 (2010) .\n[5] J. H. Franken, H. J. M. Swagten, and B. Koopmans,\nNature Nanotechnology 7, 499 (2012) .\n[6] A. Brataas, A. D. Kent, and H. Ohno,\nNature Materials 11, 372 (2012) .\n[7] J.C.Slonczewski, J. Magn. & Magn. Mater. 159, L1 (1996) .\n[8] L. Berger, Phys. Rev. B 54, 9353 (1996) .\n[9] I. M. Miron, G. Gaudin, S. Auffret, B. Rodmacq,\nA. Schuhl, S. Pizzini, J. Vogel, and P. Gambardella,\nNature Materials 9, 230 (2010) .\n[10] S. Emori, U. Bauer, S.-M. Ahn, E. Martinez, and\nG. S. D. Beach, Nature Materials 12, 611 (2013) .\n[11] O.Boulle, S.Rohart, L.D.Buda-Prejbeanu, E.Ju´ e, I. M .\nMiron, S. Pizzini, J. Vogel, G. Gaudin, and A. Thiaville,\nPhys. Rev. Lett. 111, 217203 (2013) .\n[12] L. Liu, C.-F. Pai, Y. Li, H. W. Tseng, D. C. Ralph, and\nR. A. Buhrman, Science336, 555 (2012) .\n[13] K.-S. Ryu, L. Thomas, S.-H. Yang, and S. Parkin,\nNature Nanotechnology 8, 527 (2013) .\n[14] H. T. Nembach, J. M. Shaw, C. T. Boone, and T. J.\nSilva,Phys. Rev. Lett. 110, 117201 (2013) .\n[15] J. Foros, A. Brataas, Y. Tserkovnyak, and G. E. W.\nBauer,Phys. Rev. B 78, 140402 (2008) .\n[16] Y. Tserkovnyak and C. H. Wong,\nPhys. Rev. B 79, 014402 (2009) .\n[17] S. Zhang and Steven S.-L. Zhang,\nPhys. Rev. Lett. 102, 086601 (2009) .\n[18] E. M. Hankiewicz, G. Vignale, and Y. Tserkovnyak,\nPhys. Rev. B 78, 020404(R) (2008) .\n[19] Y. Tserkovnyak, E. M. Hankiewicz, and G. Vignale,\nPhys. Rev. B 79, 094415 (2009) .\n[20] Y. Li and W. E. Bailey, arXiv:1401.6467 (2014).\n[21] A. A. Starikov, P. J. Kelly, A. Brataas,\nY. Tserkovnyak, and G. E. W. Bauer,\nPhys. Rev. Lett. 105, 236601 (2010) .\n[22] K. M. D. Hals and A. Brataas,\nPhys. Rev. B 89, 064426 (2014) .\n[23] Z. Yuan, Y. Liu, A. A. Starikov, P. J. Kelly, and\nA. Brataas, Phys. Rev. Lett. 109, 267201 (2012) .\n[24] A. Brataas, Y. Tserkovnyak, and G. E. W.\nBauer, Phys. Rev. Lett. 101, 037207 (2008) ;\nPhys. Rev. B 84, 054416 (2011) ; K. M. D.Hals, A. K. Nguyen, and A. Brataas,\nPhys. Rev. Lett. 102, 256601 (2009) .\n[25] See Supplemental Material at [URL] for computational\ndetails, extracting α′′, the free-electron model using\nmuffin-tin orbitals, fitting α′andl0and band structures\nof Ni and Fe in Permalloy, which includes Refs. [ 26–30].\n[26] K. Xia, M. Zwierzycki, M. Talanana, P. J. Kelly, and\nG. E. W. Bauer, Phys. Rev. B 73, 064420 (2006) .\n[27] O. K. Andersen, Z. Pawlowska, and O. Jepsen,\nPhys. Rev. B 34, 5253 (1986) .\n[28] I. Turek, V. Drchal, J. Kudrnovsk´ y, M. ˇSob, and\nP. Weinberger, Electronic Structure of Disordered Al-\nloys, Surfaces and Interfaces (Kluwer, Boston-London-\nDordrecht, 1997).\n[29] P. Soven, Phys. Rev. 156, 809 (1967) .\n[30] S. Wang, Y. Xu, and K. Xia,\nPhys. Rev. B 77, 184430 (2008) .\n[31] R. A. Duine, Phys. Rev. B 79, 014407 (2009) .\n[32] A. K. Nguyen and A. Brataas,\nPhys. Rev. Lett. 101, 016801 (2008) .\n[33] It is analogous to the relation between the Drude con-\nductivity σand the mean free path lm,σ=2e2\nhk2\nF\n3πlm.\nIn ballistic systems, we have σ→ ∞andlm→ ∞,\nσ/lm=2e2\nhk2\nF\n3π=4GSh\n3A. Here we have used the Sharvin\nconductance of free electrons GSh=2e2\nhAk2\nF\n4π.\n[34] O. Boulle, G. Malinowski, and M. Kl¨ aui,\nMat. Science and Eng. R 72, 159 (2011) .\n[35] A. Ben Hamida, O. Rousseau, S. Petit-Watelot, and\nM. Viret, Europhys. Lett. 94, 27002 (2011) .\n[36] J. Banhart, H. Ebert, and A. Vernes,\nPhys. Rev. B 56, 10165 (1997) .\n[37] M. Hayashi, L. Thomas, C. Rettner, R. Moriya, and\nS. S. P. Parkin, Appl. Phys. Lett. 92, 162503 (2008) .\n[38] S. Lepadatu, A. Vanhaverbeke, D. Atkin-\nson, R. Allenspach, and C. H. Marrows,\nPhys. Rev. Lett. 102, 127203 (2009) .\n[39] M. Eltschka, M. W¨ otzel, J. Rhensius, S. Krzyk,\nU.Nowak, M. Kl¨ aui, T. Kasama, R.E.Dunin-Borkowski,\nL. J. Heyderman, H. J. van Driel, and R. A. Duine,\nPhys. Rev. Lett. 105, 056601 (2010) .\n[40] J.-Y. Chauleau, H. G. Bauer, H. S. K¨ orner, J. Stiglo-\nher, M. H¨ artinger, G. Woltersdorf, and C. H. Back,\nPhys. Rev. B 89, 020403(R) (2014) .1\nSupplementary Material for “Gilbert damping in noncolline ar ferromagnets”\nZhe Yuan,1,∗Kjetil M. D. Hals,2,3Yi Liu,1Anton A. Starikov,1Arne Brataas,2and Paul J. Kelly1\n1Faculty of Science and Technology and MESA+Institute for Nanotechnology,\nUniversity of Twente, P.O. Box 217, 7500 AE Enschede, The Net herlands\n2Department of Physics, Norwegian University of Science and Technology, NO-7491 Trondheim, Norway\n3Niels Bohr International Academy and the Center for Quantum Devices,\nNiels Bohr Institute, University of Copenhagen, 2100 Copen hagen, Denmark\nI. COMPUTATIONAL DETAILS.\nTaking the concrete example of Walker profile domain\nwalls (DWs), the effective (dimensionless) in-plane and\nout-of-plane damping parameters can be expressed in\nterms of the scattering matrix Sof the system as, re-\nspectively,\nαeff\ni=gµBλw\n8πAMsTr/parenleftbigg∂S\n∂rw∂S†\n∂rw/parenrightbigg\n, (S1)\nαeff\no=gµB\n8πAMsλwTr/parenleftbigg∂S\n∂φ∂S†\n∂φ/parenrightbigg\n,(S2)\nusing the scattering theory of magnetization dissipation\n[S1,S2]. Heregis the Land´ e g-factor,µBis the Bohr\nmagneton, λwdenotes the DW width, Ais the cross sec-\ntional area, and Msis the saturation magnetization.\nIt is interesting to compare the scheme for calculat-\ning the Gilbert damping of DWs using Eqs. ( S1) and\n(S2) [S1,S2] with that used for collinear magnetiza-\ntion [S3,S4]. Both of them are based upon the energy\npumping theory [ S2,S3]. To calculate the damping αcoll\nfor the collinear case, the magnetization is made to pre-\ncessuniformlyandthelocalenergydissipationishomoge-\nneous throughout the ferromagnet. The total energy loss\ndue to Gilbert damping is then proportional to the vol-\nume of the ferromagnetic material and the homogenous\nlocal damping αcollcan be determined from the damp-\ning per unit volume. When the magnetization of a DW is\nmade to change either by moving its center rwor varying\nits orientation φ, this results in a relatively large preces-\nsion at the center of the DW; the further from the center,\nthe less the magnetization changes. The local contribu-\ntion to the total energydissipationofthe DWis weighted\nby the magnitude of the magnetization precession when\nrworφvaries. For a fixed DW width, the total damping\nis not proportional to the volume of the scattering region\nbut converges to a constant once the scattering region is\nlarge compared to the DW. In practice, αeff\niandαeff\nocal-\nculated using Eqs. ( S1) and (S2) are well converged for\na scattering region 10 times longer than λw. Effectively,\nαeffcan be regardedas a weighted averageof the (dimen-\nsionless) damping constant in the region of a DW. In the\nwide DW limit, αeff\niandαeff\noboth approach αcollwith\nspin-orbit coupling (SOC) and vanish in its absence.\nTo evaluate the effective Gilbert damping of a DWusing Eqs. ( S1) and (S2), we attached semiinfinite (cop-\nper) leads to a finite length of Ni 80Fe20alloy (Permal-\nloy, Py) and rotated the local magnetization to make\na 180◦DW using the Walker profile. Specifically, we\nusedm= (sechz−rw\nλw,0,tanhz−rw\nλw) for N´ eel DWs and\nm= (−tanhz−rw\nλw,−sechz−rw\nλw,0) for Bloch DWs. The\nscatteringpropertiesofthedisorderedregionwereprobed\nby studying how Bloch waves in the Cu leads incident\nfrom the left or right sides weretransmitted and reflected\n[S4,S5]. Thescatteringmatrixwasobtainedusingafirst-\nprinciples “wave-function matching” scheme [ S6] imple-\nmented with tight-binding linearized muffin-tin orbitals\n(TB-LMTOs) [ S7]. SOC was included using a Pauli\nHamiltonian. The calculations were rendered tractable\nby imposing periodic boundary conditions transverse to\nthe transport direction. It turned out that good results\ncould be achieved even when these so-called “lateral su-\npercells” were quite modest in size. In practice, we used\n5×5 lateral supercells and the longest DW we consid-\nered was more than 500 atomic monolayers thick. After\nembedding the DW between collinear Py and Cu leads,\nthe largest scattering region contained 13300 atoms. For\nevery DW width, we averaged over about 8 random dis-\norder configurations.\nA potential profile for the scattering region was con-\nstructed within the framework of the local spin den-\nsity approximation of density functional theory as fol-\nlows. For a slab of collinear Py binary alloy sandwiched\nbetween Cu leads, atomic-sphere-approximation (ASA)\npotentials [ S7] were calculated self-consistently without\nSOC using a surface Green’s function (SGF) method im-\nplemented [ S8] with TB-LMTOs. Chargeand spin densi-\nties for binary alloy AandBsites were calculated using\nthe coherent potential approximation [ S9] generalized to\nlayer structures [ S8]. For the scattering matrix calcu-\nlation, the resulting ASA potentials were assigned ran-\ndomly to sites in the lateral supercells subject to mainte-\nnance of the appropriate concentration of the alloy [ S6]\nand SOC was included. The exchange potentials are ro-\ntated in spin space [ S10] so that the local quantization\naxis for each atomic sphere follows the DW profile. The\nDW width is determined in reality by a competition be-\ntween interatomic exchange interactions and magnetic\nanisotropy. For a nanowire composed of a soft mag-\nnetic material like Py, the latter is dominated by the2\nshape anisotropy that arises from long range magnetic\ndipole-dipole interactions and depends on the nanowire\nprofile. Experimentallyitcanbetailoredbychangingthe\nnanowire dimensions leading to the considerable spread\nof reported DW widths [ S11]. In electronic structure cal-\nculations, that do not contain magnetic dipole-dipole in-\nteractions, we simulate a change of demagnetization en-\nergy by varying the DW width. In this way we can study\nthe dependence of Gilbert damping on the magnetization\ngradient by performing a series of calculations for DWs\nwith different widths.\nFor the self-consistent SGF calculations (without\nSOC), the two-dimensional(2D) Brillouin zone (BZ) cor-\nresponding to the 1 ×1 interface unit cell was sampled\nwith a 120 ×120 grid. The transport calculations includ-\ning SOC were performed with a 32 ×32 2D BZ grid for a\n5×5 lateral supercell, which is equivalent to a 160 ×160\ngrid in the 1 ×1 2D BZ.\nII. EXTRACTING α′′\nWe first briefly derive the form of the in-plane damp-\ning. It has been argued phenomenologically [ S12] that\nfor a noncollinear magnetization texture varying slowly\nin time the lowest order term in an expansion of the\ntransverse component of the spin current in spatial and\ntime derivatives that breaks time-reversal symmetry and\nis therefore dissipative is\nj′′\ns=−ηm×∂z∂tm, (S3)\nwhereηis a coefficient depending on the material and\nmis a unit vector in the direction of the magnetization.\nThe divergence of the spin current,\n∂zj′′\ns=−η/parenleftbig\n∂zm×∂z∂tm+m×∂2\nz∂tm/parenrightbig\n,(S4)\ngives the corresponding dissipative torque exerted on the\nlocal magnetization. While the second term in brackets\nin Eq. (S4) is perpendicular to m, the first term contains\nboth perpendicular and parallel components. Since we\nare only interested in the transverse component of the\ntorque, we subtract the parallel component to find the\ndamping torque\nτ′′=−η/braceleftbig\n(1−mm)·(∂zm×∂z∂tm)+m×∂2\nz∂tm/bracerightbig\n=−η/braceleftbig\n[m×(∂zm×∂z∂tm)]×m+m×∂2\nz∂tm/bracerightbig\n=η/bracketleftbig\n(m·∂z∂tm)m×∂zm−m×∂2\nz∂tm/bracketrightbig\n.(S5)\nThe Landau-Lifshitz-Gilbert equation including the\ndamping torque τ′′reads\n∂tm=−γm×Heff+αcollm×∂tm+γτ′′\nMs\n=−γm×Heff+αcollm×∂tm\n+α′′/bracketleftbig\n(m·∂z∂tm)m×∂zm−m×∂2\nz∂tm/bracketrightbig\n,(S6)where the in-plane damping parameter α′′≡γη/Mshas\nthe dimension of length squared.\nIn the following, we explain how α′′can be extracted\nfrom calculations on Walker DWs and show that it is ap-\nplicable to other profiles. The formulation is essentially\nindependent of the DW type (Bloch or N´ eel) and we use\na Bloch DW in the following derivation for which\nm(z) = [cosθ(z),sinθ(z),0], (S7)\nwhereθ(z) represents the in-plane rotation (see Fig. 1 in\nthe paper). The local energy dissipation associated with\na time-dependent θis given by [ S2]\nγ\nMs˙E(z) =αcoll∂tm·∂tm\n+α′′/bracketleftbig\n(m·∂z∂tm)∂tm·∂zm−∂tm·∂2\nz∂tm/bracketrightbig\n.(S8)\nFor the one-dimensional profile Eq. ( S7), this can be sim-\nplified as\nγ\nMs˙E(z) =αcoll/parenleftbiggdθ\ndt/parenrightbigg2\n−α′′dθ\ndtd\ndt/parenleftbiggd2θ\ndz2/parenrightbigg\n.(S9)\nSubstituting into Eq. ( S9) the Walker profile\nθ(z) =−π\n2−arcsin/parenleftbigg\ntanhz−rw\nλw/parenrightbigg\n,(S10)\nthat we used in the calculations, we obtain for the total\nenergy dissipation associated with the motion of a rigid\nDW for which ˙θ= ˙rwdθ/drw,\n˙E=/integraldisplay\nd3r˙E(z) =2MsA\nγλw/parenleftbigg\nαcoll+α′′\n3λ2w/parenrightbigg\n˙r2\nw.(S11)\nComparing this to the energy dissipation expressed in\nterms of the effective in-plane damping αeff\ni[S2]\n˙E=2MsA\nγλwαeff\ni˙r2\nw, (S12)\nwe arrive at\nαeff\ni(λw) =αcoll+α′′\n3λ2w. (S13)\nUsing Eq. ( S13), we perform a least squares linear fitting\nofαeff\nias a function of λ−2\nwto obtain αcollandα′′. The\nfitting is shown in Fig. S1and the parameters are listed\nin Table SI. Note that αcollis in perfect agreement with\nindependent calculations for collinear Py [ S4].\nTo confirm that α′′is independent of texture, we con-\nsider another analytical DW profile in which the in-plane\nrotation is described by a Fermi-like function,\nθ(z) =−π+π\n1+ez−rF\nλF. (S14)\nHererFandλFdenote the DW center and width, re-\nspectively. Substituting Eq. ( S14) into Eq. ( S9), we find\nthe energy dissipation for “Fermi” DWs to be\n˙E=π2MsA\n6γλF/parenleftbigg\nαcoll+α′′\n5λ2\nF/parenrightbigg\n˙r2\nF,(S15)3\n0 0.5 1.0 1.5\n1/λw2 (nm-2)00.0050.0100.015αieff\nBloch\nNéel\nξSO=0Walker\nFIG. S1. Calculated αeff\nifor Walker-profile Permalloy DWs.\nN´ eel DWs: black circles, Bloch DWs: red circles. Without\nSOC, calculations for the twoDWtypesyield thesame results\n(blue circles). The dashed lines are linear fits using Eq. ( S13).\nwhich suggests the effective in-plane damping\nαeff\ni(λF) =αcoll+α′′\n5λ2\nF. (S16)\nEq. (S16) is plotted as solid lines in Fig. S2with the\nvalues of αcollandα′′taken from Table SI.\nSince the energy pumping can be expressed in terms\nof the scattering matrix Sas\n˙E=/planckover2pi1\n4πTr/parenleftbigg∂S\n∂t∂S†\n∂t/parenrightbigg\n=/planckover2pi1\n4πTr/parenleftbigg∂S\n∂rF∂S†\n∂rF/parenrightbigg\n˙r2\nF,(S17)\nwe can calculate the effective in-plane damping for a\nFermi DW from the Smatrix to be\nαeff\ni=3/planckover2pi1γλF\n2π3MsATr/parenleftbigg∂S\n∂rF∂S†\n∂rF/parenrightbigg\n.(S18)\nWe plot the values of αeff\nicalculated using the derivative\nof the scattering matrix Eq. ( S18) as circles in Fig. S2.\nThe good agreement between the circles and the solid\nlines demonstratesthe validity ofthe form ofthe in-plane\ndamping torque in Eq. ( S6) and that the parameter α′′\ndoes not depend on a specific magnetization texture.\nTABLE SI. Fit parameters to describe the in-plane Gilbert\ndamping in Permalloy DWs.\nDW type αcoll α′′(nm2)\nBloch (4.6 ±0.1)×10−30.016±0.001\nN´ eel (4.5 ±0.1)×10−30.016±0.001\nξSO=0 (2.0 ±1.0)×10−60.017±0.0010 0.5 1.0 1.5 2.0 2.5 3.0\n1/λF2 (nm-2)00.0050.0100.015αieff\nBloch\nξSO=0Fermi\nFIG. S2. Calculated αeff\nifor Permalloy Bloch DWs (red cir-\ncles) with the Fermi profile Eq. ( S14). The blue circles are\nresults calculated without SOC. The solid lines are the an-\nalytical expression Eq. ( S16) using the parameters listed in\nTableSI.\nIII. THE FREE-ELECTRON MODEL USING\nMUFFIN-TIN ORBITALS\nWe take constant potentials, V↑=−0.2 Ry,V↓=\n−0.1Ry inside atomic sphereswith an exchangesplitting\n∆V= 0.1 Ry between majority and minority spins and a\nFermi level EF= 0. The atomic spheres are placed on a\nface-centered cubic (fcc) lattice with the lattice constant\nof nickel, 3.52 ˚A. The magnetic moment on each atom is\nthen 0.072µB. The transport direction is along the fcc\n[111]. In the scattering calculation, we use a 300 ×300\n01020 30 4050 60\nL (nm)306090102030AR (fΩ m2)456(a)\n(b)\n(c)ρ=2.69±0.06 µΩ cm\nρ=24.8±0.5 µΩ cm\nρ=94.3±4.4 µΩ cm\nFIG.S3. Resistancecalculatedforthedisorderedfree-ele ctron\nmodel as a function of the length of the scattering region for\nthree values of V0, the disorder strength: 0.05 Ry (a), 0.15\nRy (b) and 0.25 Ry (c). The lines are the linear fitting used\nto determine the resistivity.4\nk-point mesh in the 2D BZ. The calculated Sharvin con-\nductances for majority and minority channels are 0.306\nand 0.153 e2/hper unit cell, respectively, compared with\nanalytical values of 0.305 and 0.153.\nTo mimic disordered free-electron systems, we intro-\nduce a 5 ×5 lateral supercell and distribute constant\npotentials uniformly in the energy range [ −V0/2,V0/2]\nwhereV0is some given strength [ S13] and spatially at\nrandom on every atomic sphere in the scattering re-gion. The calculated total resistance as a function of the\nlengthLof the (disordered) scattering region is shown in\nFig.S3withV0= 0.05 Ry (a), 0.15 Ry (b) and 0.25 Ry\n(c). The resistivity increases with the impurity strength\nas expected and can be extracted with a linear fitting\nAR(L) =AR0+ρL. For each system, we calculate about\n10randomconfigurationsand takethe averageofthe cal-\nculatedresults. Wellconvergedresultsareobtainedusing\na 32×32k-point mesh for the 5 ×5 supercell.\nIV. FITTING α′ANDl0\nWith a nonlocal Gilbert damping, α(r,r′), the energy dissipation rate is given by [ S2]\n˙E=Ms\nγ/integraldisplay\nd3r˙m(r)·/integraldisplay\nd3r′α(r,r′)·˙m(r′). (S19)\nIf we consider the out-of-plane damping of a N´ eel DW, i.e. for which the angle φvaries in time (see Fig. 1 in the\npaper), we have\n˙m(r) =˙φsechz−rw\nλwˆy. (S20)\nConsidering again a Walker profile, we find the explicit form of the out- of-plane damping matrix element\nαo(z,z′) =αcollδ(z−z′)+α′\nλ2wsechz−rw\nλwsechz′−rw\nλw1√πAl0e−(z−z′\nl0)2. (S21)\nSubstituting Eq. ( S21) and Eq. ( S20) into Eq. ( S19), we obtain explicitly the energy dissipation rate\n˙E=2MsAλw\nγαcoll˙φ2+MsAα′˙φ2\n√πγl0λ2w/integraldisplay\ndzsech2z−rw\nλw/integraldisplay\ndz′sech2z′−rw\nλwe−(z−z′\nl0)2\n. (S22)\nThe calculated effective out-of-plane Gilbert damping for a DW with th e Walker profile is related to the energy\ndissipation rate as [ S2]\n˙E=2MsAλw\nγαeff\no˙φ2. (S23)\nComparing Eqs. ( S22) and (S23), we arrive at\nαeff\no=αcoll+α′\n2√πλ3wl0/integraldisplay\ndzsech2z−rw\nλw/integraldisplay\ndz′sech2z′−rw\nλwe−(z−z′\nl0)2\n. (S24)\nThe last equation is used to fit α′andl0toαeff\nocalculated for different λw. For Bloch DWs, it is straightforward to\nrepeat the above derivation and find the same result, Eq. ( S24).\nV. BAND STRUCTURES OF NI AND FE IN\nPERMALLOY\nIn the coherent potential approximation (CPA) [ S8,\nS9], the single-site approximation involves calculating\nauxiliary (spin-dependent) potentials for Ni and Fe self-\nconsistently. In our transport calculations, these auxil-\niary potentials are distributed randomly in the scattering\nregion. It is instructive to place the Ni potentials (for\nmajority- and minority-spin electrons) on an fcc latticeand to calculate the band structure non-self-consistently.\nThen we do the same using the Fe potentials. The cor-\nresponding band structures are plotted in Fig. S4. At\nthe Fermi level, where electron transport takes place,\nthe majority-spin bands for Ni and Fe are almost identi-\ncal, including their angular momentum character. This\nmeans that majority-spin electrons in a disordered al-\nloy see essentially the same potentials on all lattice sites\nand are only very weakly scattered in transport by the\nrandomly distributed Ni and Fe potentials. In contrast,5\n-9-6-303E-EF (eV)Majority Spin Minority Spin\nX Γ L-9-6-303E-EF (eV)\nX Γ LNi Ni\nFe Fe\nFIG. S4. Band structures calculated with the auxiliary Ni\nand Fe atomic sphere potentials and Fermi energy that were\ncalculated self-consistently forNi 80Fe20usingthecoherentpo-\ntential approximation. The red bars indicates the amount of\nscharacter in each band.\nthe minority-spin bands are quite different for Ni and Fe.\nThiscanbeunderstoodintermsofthe differentexchange\nsplitting between majority- and minority-spin bands; the\ncalculated magnetic moments of Ni and Fe in Permalloy\nin the CPA are 0.63 and 2.61 µB, respectively. The ran-\ndom distribution of Ni and Fe potentials in Permalloy\nthen leads to strong scattering of minority-spin electrons\nin transport.∗Present address: Institut f¨ ur Physik, Johannes\nGutenberg–Universit¨ at Mainz, Staudingerweg 7, 55128\nMainz, Germany; zyuan@uni-mainz.de\n[S1] K. M. D. Hals, A. K. Nguyen, and A. Brataas,\nPhys. Rev. Lett. 102, 256601 (2009) .\n[S2] A. Brataas, Y. Tserkovnyak, and G. E. W. Bauer,\nPhys. Rev. B 84, 054416 (2011) .\n[S3] A. Brataas, Y. Tserkovnyak, and G. E. W. Bauer,\nPhys. Rev. Lett. 101, 037207 (2008) .\n[S4] A. A. Starikov, P. J. Kelly, A. Brataas,\nY. Tserkovnyak, and G. E. W. Bauer,\nPhys. Rev. Lett. 105, 236601 (2010) .\n[S5] Z. Yuan, Y. Liu, A. A. Starikov, P. J. Kelly, and\nA. Brataas, Phys. Rev. Lett. 109, 267201 (2012) .\n[S6] K. Xia, M. Zwierzycki, M. Talanana, P. J. Kelly, and\nG. E. W. Bauer, Phys. Rev. B 73, 064420 (2006) .\n[S7] O. K. Andersen, Z. Pawlowska, and O. Jepsen,\nPhys. Rev. B 34, 5253 (1986) .\n[S8] I. Turek, V. Drchal, J. Kudrnovsk´ y, M. ˇSob, and\nP. Weinberger, Electronic Structure of Disordered Al-\nloys, Surfaces and Interfaces (Kluwer, Boston-London-\nDordrecht, 1997).\n[S9] P. Soven, Phys. Rev. 156, 809 (1967) .\n[S10] S. Wang, Y. Xu, and K. Xia,\nPhys. Rev. B 77, 184430 (2008) .\n[S11] O. Boulle, G. Malinowski, and M. Kl¨ aui,\nMat. Science and Eng. R 72, 159 (2011) .\n[S12] Y. Tserkovnyak, E. M. Hankiewicz, and G. Vignale,\nPhys. Rev. B 79, 094415 (2009) .\n[S13] A. K. Nguyen and A. Brataas,\nPhys. Rev. Lett. 101, 016801 (2008) ."    },    {        "title": "1405.2267v1.Current_induced_magnetization_dynamics_in_two_magnetic_insulators_separated_by_a_normal_metal.pdf",        "content": "Current-induced magnetization dynamics in two magnetic insulators separated by a\nnormal metal\nHans Skarsv\u0017 ag1, Gerrit E. W. Bauer2;3and Arne Brataas1\n1Department of Physics, Norwegian University of Science and Technology, NO-7491 Trondheim, Norway\n2Institute for Materials Research, Tohoku University, Sendai 980-8577, Japan\n3Kavli Institute of NanoScience, Delft University of Technology, 2628 CJ Delft, The Netherlands\n(Dated: March 14, 2022)\nWe study the dynamics of spin valves consisting of two layers of magnetic insulators separated\nby a normal metal in the macrospin model. A current through the spacer generates a spin Hall\ncurrent that can actuate the magnetization via the spin-transfer torque. We derive expressions\nfor the e\u000bective Gilbert damping and the critical currents for the onset of magnetization dynamics\nincluding the e\u000bects of spin pumping that can be tested by ferromagnetic resonance experiments.\nThe current generates an amplitude asymmetry between the in-phase and out-of-phase modes. We\nbrie\ry discuss superlattices of metals and magnetic insulators.\nPACS numbers: 76.50.+g,75.30.Ds,75.70.-i,75.76.+j\nI. INTRODUCTION\nElectric currents induce spin-transfer torques in het-\nerogeneous or textured magnetic systems.1In this con-\ntext, magnetic insulators such as yttrium iron garnet\n(YIG) combined with normal metal contacts exhibit-\ning spin-orbit interactions, such as Pt, have recently\nattracted considerable interest, both experimentally2{8\nand theoretically.9{15Since the discovery of non-local\nexchange coupling and giant magnetoresistance in spin\nvalves, i.e., a normal metal sandwiched between two fer-\nromagnetic metals, these systems have been known to\ndisplay rich physics. Some of these e\u000bects, such as the dy-\nnamic exchange interaction,19should also arise when the\nmagnetic layers are insulators. The spin Hall magnetore-\nsistance (SMR) is predicted to be enhanced in such spin\nvalves,10although experimental realizations have not yet\nbeen reported. Here, we consider multilayer structures\nwith ferromagnetic but electrically insulating (FI) layers\nand normal metal (N) spacers. In-plane electric currents\napplied to N generate perpendicular spin currents via the\nspin Hall e\u000bect (SHE). When these spin currents are ab-\nsorbed at the NjFI interfaces, the ensuing spin-transfer\ntorques can induce magnetization dynamics and switch-\ning. We consider ground state con\fgurations in which\nthe magnetizations are parallel or antiparallel to each\nother. For thin magnetic layers, even small torques can\ne\u000bectively modify the (Gilbert) damping, which can be\nobserved as changes in the line width of the ferromagnetic\nresonance (FMR) spectra. We employ the macrospin\nmodel for the magnetization vectors that is applicable\nfor su\u000eciently strong and homogeneous magnetic \felds,\nwhile extensions are possible.13{15Our results include\nthe observation of e\u000bective (anti)damping resulting from\nin-plane charge currents in FI jNjFI trilayers, magnetic\nstability analysis in the current-magnetic \feld parame-\nter space and a brief analysis of the dynamics for cur-\nrents above the critical value. We also consider current-\ninduced e\u000bects in superlattices. Our paper is organized as\nfollows. In Section II, we present our model for a FI jNjFIspin valve including the SHE spin current generation\nand spin pumping, modeled as additional torques in the\nLandau-Lifshitz-Gilbert equation. We proceed to formu-\nlate the linearized magnetization dynamics and the spin\naccumulation in N in Section III. In Section IV, we calcu-\nlate the eigenmodes and the current-controlled e\u000bective\nGilbert damping and determine the critical currents at\nwhich the magnetic precession becomes unstable. We dis-\ncuss the current-induced dynamics of \u0001\u0001\u0001jFIjNjFIjNj\u0001\u0001\u0001\nsuperlattices in Section V. Finally, we summarize our\nconclusions and provide an outlook in Section VI.\nII. MODEL\nFI1jNjFI2 denotes the heterostructure composed of a\nnormal metal (N) layer sandwiched between two layers of\nferromagnetic insulators (FIs) (see Fig. 1). We denote the\nthicknesses of FI1, N and FI2 by d1,dNandd2, respec-\ntively. We adopt a macrospin model of spatially constant\nmagnetization Miin each layer. The magnetization dy-\nnamics of the two layers are described by the coupled\nLandau-Lifshitz-Gilbert-Slonczewski (LLGS) equations:\n_Mi=\u0000\rMi\u0002\u0012\nHe\u000b;i+J\ndiMS;iMj\u0013\n+\u000biMi\u0002_Mi\n+\u001cDSP\ni+\u001cISP\ni+\u001cSH\ni; (1)\nwhere Miis the unit vector in the direction of the magne-\ntization in the left/right layer with indices i= 1;2;MS;i\nis the saturation magnetization; \ris the gyromagnetic\nratio;\u000biis the Gilbert damping constant; Jis the in-\nterlayer dipolar and exchange energy areal density, with\nj= 1(2) when i= 2(1); and He\u000b;iis an e\u000bective mag-\nnetic \feld:\nHe\u000b;i=Hext+Han;i(Mi) (2)\nconsisting of the external magnetic \feld Hextas well as\nthe anisotropy \felds Han;ifor the left/right layer. We dis-\ntinguish direct (DSP) and indirect spin pumping (ISP).arXiv:1405.2267v1  [cond-mat.mes-hall]  9 May 20142\nDSP generates the spin angular momentum current jDSP\n1(2)\nthrough the interfaces of FI1(2). A positive spin current\ncorresponds to a spin \row toward the FI from which it\noriginates. The DSP spin current is expressed as\njDSP\ni=~\neg?;iMi\u0002_Mi; (3)\nwhereg?;iis the real part of the spin-mixing conduc-\ntance of the NjFI1(2) interface per unit area for i= 1(2),\nrespectively, and \u0000eis the electron charge. This angu-\nlar momentum loss causes a damping torque (here and\nbelow in CGS units):\n\u001cDSP\ni=\r~2g?;i\n2e2MS;idiMi\u0002_Mi: (4)\nIn ballistic systems, the spin current emitted by the\nneighboring layer is directly absorbed and generates an\nindirect spin torque on the opposing layer:19\n\u001cISP\ni;ball=\u0000\r~2g?;i\n2e2MS;idiMi\u0002_Mi: (5)\nIn the presence of an interface or bulk disorder, the trans-\nport is di\u000buse, and the ISP is\n\u001cISP\ni=\u0000\r~\n2e2MS;idig?;iMi\u0002(Mi\u0002\u0016SP(zi));(6)\nwhere \u0016SP(zi) is the spin pumping contribution to the\nspin accumulation (di\u000berence in chemical potentials) at\nthe interface in units of energy, with zi\u0011 \u0007dN=2 for\ni= 1;2.\u0016SPis the solution of the spin di\u000busion equation\nin N as discussed below.\nDue to the SHE, an in-plane DC charge current pro-\nduces a transverse spin current that interacts with the\nFIjN interfaces. Focusing on the di\u000busive regime, the\nareal density of charge current jcas well as the spin jSH\nk\ncurrent in the k-direction, where jSH\nk=\f\fjSH\nk\f\fis the spin\npolarization unit vector, can be written in terms of a\nsymmetric linear response matrix:10\n0\nBB@jc\njSH\nx\njSH\ny\njSH\nz1\nCCA=\u001b0\nB@1 \u0002 SH^x\u0002\u0002SH^y\u0002\u0002SH^z\u0002\n\u0002SH^x\u0002 1 0 0\n\u0002SH^y\u0002 0 1 0\n\u0002SH^z\u0002 0 0 11\nCA\n0\nBB@\u0000r\u0016c=e\n\u0000r\u0016SH\nx=(2e)\n\u0000r\u0016SH\ny=(2e)\n\u0000r\u0016SH\nz=(2e)1\nCCA; (7)\nwhere \u0002SHis the spin Hall angle, \u001bis the electrical\nconductivity and \u0016cis the charge chemical potential.\n\u0016SH= (\u0016SH\nx;\u0016SH\ny;\u0016SH\nz) is the spin accumulation induced\nby re\rection of the spin currents at the interfaces. The\nspin transfer torques \u001cSH\niat the FI interfaces ( i= 1;2)\nare then expressed as\n\u001cSH\ni=\u0000\r~\n2e2MS;idig?;iMi\u0002\u0000\nMi\u0002\u0016SH(zi)\u0001\n:(8)The polarization of \u0016SHand thereby \u001cSH\nican be con-\ntrolled by the charge current direction. In the following\nsections, we assume that the shape anisotropy and ex-\nchange coupling favor parallel or antiparallel equilibrium\norientations of M1andM2. For small current levels, the\ntorques normal to the magnetization induce tilts from\ntheir equilibrium directions and, at su\u000eciently large cur-\nrents, trigger complicated dynamics, while torques di-\nrected along the equilibrium magnetization modify the\ne\u000bective damping and induce magnetization reversal.\nHere, we focus on the latter con\fguration, in which the\nspin accumulation in N is collinear to the equilibrium\nmagnetizations.\nIn the following equations, we take the thickness, satu-\nration magnetization, Gilbert damping and spin-mixing\nconductance to be equal in the two layers FI1 and FI2,\nwith an out-of-plane hard axis and an in-plane internal\n\feld:\nHe\u000b;1=!H\n\r^x\u0000!M\n\r(M1)z^z; (9a)\nHe\u000b;2=s!H\n\r^x\u0000!M\n\r(M2)z^z; (9b)\nwith!H=\r(Hext+ (Han;i)x) and!M= 4\u0019\rMS. Pure\ndipolar interlayer coupling with J <0 favors an antipar-\nallel ground state con\fguration, while the exchange cou-\npling oscillates as a function of dN.\nm\n1m1\nM\n1m\n2m2M2\n FI1                    NM              FI2\nx\n                      z\nyz=-d1-dN /2 z=-dN/2 z=dN/2z=dN/2+d2\njc jSH(SHE spin current direction)\nj1SP ~ M1 x m1j2SP ~ M2 x m2\nFIG. 1: (Color online) Spin valve of ferromagnetic insulators\n(FIs) sandwiching a normal metal (N). The equilibrium mag-\nnetizations M1andM2are collinear, i.e., parallel or antipar-\nallel. A spin-Hall-induced spin current \rows in the z-direction\nand is polarized along x.\nIII. SPIN-TRANSFER TORQUES\nThe spin-pumping and spin-transfer torques \u001cDSP\niand\n\u001cISP\ni(Eqs. (4) and (6)) cause dynamic coupling between\nthe two magnetizations. To leading order, these torques\ncan be treated separately. We now derive expressions3\nfor disordered systems that support spin accumulations\n\u0016X(z) (X = SH;SP) governed by the spin-di\u000busion equa-\ntion:\n_\u0016X=D@2\nz\u0016X\u0000\u0016X\n\u001csf: (10)\nHere,Dis the di\u000busion constant, and \u001csfis the spin-\n\rip relaxation time. The di\u000buse spin current in the\nz-direction related to this spin accumulation follows\nEq. (7):\njX=\u0000\u001b\n2e@\u0016X\n@z; (11)\nwhere\u001bis the conductivity of N.\nA. Spin-pumping-induced torques\nThe total spin current into an FI is the sum of the\nspin-transfer and spin-pumping currents. Disregarding\ninterface spin-\rip scattering, the boundary conditions for\nthe left/right layer are\n\u00001\neg?Mi\u0002\u0000\nMi\u0002\u0016SP(zi)\u0001\n+jDSP\ni=\u0007jSP(zi):(12)\nThe -(+) sign on the right-hand side is due to the oppo-\nsite \row direction of the spin currents at the left (right)\ninterface. We expand the magnetization direction around\nthe equilibrium con\fguration as\nM1=^ x+m1; (13a)\nM2=s^ x+m2; (13b)\nas long asjmij\u001cjMijormi\u0001Mi=O\u0010\njmij2\u0011\n. The\nparameters= 1 when the equilibrium con\fguration is\nparallel;s=\u00001 when it is antiparallel. The FMR fre-\nquency is usually much smaller than the di\u000buse electron\ntraversal rate D=d2\nNand spin-\rip relaxation 1 =\u001csfrate;\nthus, retardation of the spin \row may be disregarded. In\nthe steady state, the left-hand side of Eq. (10) vanishes.\nWe solve Eq. (10) for the adiabatic magnetization dy-\nnamics with boundary conditions Eq. (12) to obtain the\nspin accumulation:\n\u0016SP=\u0000~\n2^ x\u0002[(_ m1+s_ m2)\u00001(z)\n\u0000(_ m1\u0000s_ m2)\u00002(z)]; (14)\nwherelsf=pD\u001csfis the spin-di\u000busion length and\n\u00001(z)\u0011cosh (z=lsf)\ncosh (z=lsf) +\u001bsinh (z=lsf)=2g?lsf;(15a)\n\u00002(z)\u0011sinh (z=lsf)\nsinh (z=lsf) +\u001bcosh (z=lsf)=2g?lsf:(15b)\nThe torques are\n\u001cISP\ni=\r~\n2e2MSdg?\u0016SP(zi): (16)Because the spin accumulation is generated by the dy-\nnamics of both ferromagnets, we obtain spin-pumping-\ninduced dynamic coupling that is quenched when dN\u001d\nlsf. In the limit of vanishing spin-\rip scattering, the spin\naccumulation is spatially constant and is expressed as\n\u0016SPdN\u001clsf! \u0000~\n2^ x\u0002(_ m1+s_ m2): (17)\nThe corresponding di\u000busive torque is then a simple av-\nerage of the contributions from the two spin-pumping\ncurrents, in contrast to the ballistic torque that depends\nonly on the magnetization on the opposite side.\nB. Current-induced torques\nA charge current in the y-direction causes a spin Hall\ncurrent in the z-direction that is polarized along the x-\ndirection (see Fig. 1). At the interfaces, the current in-\nduces a spin-accumulation \u0016SHthat satis\fes the di\u000busion\nEq. (10) and drives a spin current (dropping the index z\nfrom now on):\njSH=\u0000\u001b\n2e@\u0016SH\n@z\u0000jSH\n0^x; (18)\nwherejSH\n0= \u0002 SHjc:Angular momentum conservation at\nthe left/right boundaries leads to\n\u00001\neg?Mi\u0002\u0000\nMi\u0002\u0016SH(zi)\u0001\n=\u0007jSH(zi): (19)\nWhen Mik\u0016SH, the spin Hall current is completely\nre\rected and the spin current at the interface van-\nishes, while the absorption and torque are maximal when\nMi?\u0016SH. Spin currents and torques at the interface scale\nfavor mifor small magnetization amplitudes. Let us de-\n\fne a time-independent \u0016SH\n0for collinear magnetizations\nand spin current polarization. For small dynamic mag-\nnetizations, then\n\u0016SH=\u0016SH\n0+\u000e\u0016SH; (20)\nwhere\u000e\u0016SH\u0018mi. We will show that the spin-Hall in-\nduced spin accumulation leads to a (anti)damping torque\nin the trilayer, while it gives a contribution to the real\npart of the frequency for superlattices (see Sec. V).\nSolving the di\u000busion Eq. (10) with boundary condi-\ntions, Eq. (19) yields\n\u0016SH\n0=\u00002elsf\n\u001bjSH\n0sinh(z=lsf)\ncosh(dN=2lsf)^ x: (21)\nThe dynamic correction\n\u000e\u0016SH=\u00001\n22elsf\n\u001bjSH\n0tanh(dN=2lsf)\n[(m1+sm2)\u00002(z)\u0000(m1\u0000sm2)\u00001(z)] (22)4\nleads to SHE torques [Eq. (8)]:\n\u001cSH\ni=\u0000\r~\n2e2MSdg?\u0002\nmi(\u0016SH\n0\u0001^x)\u0000\u000e\u0016SH(zi)\u0003\n:(23)\nEq. (1) then reduces to four coupled linear \frst-order\npartial di\u000berential equations for mi.\nIV. EIGENMODES AND CRITICAL\nCURRENTS\nAfter linearizing Eq. (1) and Fourier transforming to\nthe frequency domain _Mi!i!^mi, Eq. (1) becomes\nMv= 0; (24)\nwhere vT= ( ^m1;y;^m1;z;^m2;y;^m2;z) andMis a 4\u00024\nfrequency-dependent matrix that can be decomposed as\nM=M0+JMJ+(\u000b+\u000b0)Md+\u000b0MSP+jSH\n0MSH;(25)\nwith\nM0=0\nB@\u0000i!\u0000~!H\u0000!M 0 0\n~!H\u0000i! 0 0\n0 0\u0000i!\u0000s~!H\u0000s!M\n0 0 s~!H\u0000i!1\nCA;(26a)\nMd=0\nB@0\u0000i! 0 0\ni! 0 0 0\n0 0 0\u0000is!\n0 0is! 01\nCA; (26b)\nMJ=0\nB@0 0 0 !x\n0 0\u0000!x0\n0s!x0 0\n\u0000s!x0 0 01\nCA; (26c)\nMISP=0\nB@0i!F00is!G0\n\u0000i!F00\u0000is!G00\n0i!G00is!F0\n\u0000i!G00\u0000is!F001\nCA; (26d)\nMSH=0\nB@\u0000F0\u0000sG 0\n0\u0000F 0\u0000sG\nG 0sF 0\n0G 0sF1\nCA: (26e)\nHere,M0describes dissipationless precession in the e\u000bec-\ntive magnetic \felds, and Mdarises from Gilbert damping\nand the direct e\u000bect of spin pumping with a renormalized\ndamping coe\u000ecient ~ \u000b=\u000b+\u000b0and\n\u000b0=\r~2\n2e2Msdg?; (27)\nMJrepresents interlayer exchange coupling, MISPrep-\nresents spin-pumping-induced spin transfer, and MSH\nrepresents the spin transfer caused by the spin Hall cur-\nrent. The external and possible in-plane anisotropy \felds\nare modi\fed by the interlayer coupling, !H!~!H=\n!H+!x, where!x=\rJ=(Msd). The matrix elementsF0,G0,FandGare generalized susceptibilities extracted\nfrom Eqs. (16) and (23):\nF0=1\n\u000b0@(\u001cST\n1)y\n@_m1;z; (28a)\nG0=1\n\u000b0@(\u001cST\n1)y\n@(s_m2;z); (28b)\nF=\u00001\njSH\n0@(\u001cST\n1)y\n@m1;y; (28c)\nG=1\njSH\n0@(\u001cST\n1)y\n@(sm2;y): (28d)\nThe explicit expressions given in Appendix A are simpli-\n\fed for very thick and thin N spacers.\nThin N layer: WhendN\u001clsf, the interlayer coupling\nG0due to spin pumping approaches F0;, the intralayer\ncoupling:\nG0!F0!1\n2; (29)\nwhich implies that the incoming and outgoing spin cur-\nrents are the same. This outcome represents the limit of\nstrong dynamic coupling in which the additional Gilbert\ndamping due to spin pumping vanishes when the mag-\nnetization motion is synchronized.16In this regime, the\nSHE becomes ine\u000bective because FandGscale asdN=lsf.\nF=G!2 becauseFcontains a contribution from both\nthe static as well as the dynamic spin accumulation.\nThick N layer: In the thick \flm limit, dN\u001dlsf, the\ninterlayer coupling vanishes as G!0 andG0!0, while\nF0!1\n1 +\u001b\n2g?lsf; (30a)\nF!\r~\n2eMSd1\n1 +\u001b\n2g?lsf: (30b)\nIntroducing the spin conductance Gsf\u0011A\u001b=2lsfG?=\nAg?andRtot= (G?+Gsf)\u00001, the total resistance of the\ninterface and the spin active region of N, F0!RtotG?,\nrepresents the back\row of pumped spins. The same holds\nfor the part of Fthat originates from the dynamic part\nof\u0016SH, while the static part approaches a constant value\nwhendNbecomes large (see Appendix A). In this limit,\nthe system reduces to two decoupled FI jN bilayers.\nThe eigenmodes of the coupled system are the solutions\nof det [M(!n)] = 0 with complex eigenfrequencies !n.\nThe SHE spin current induces spin accumulations with\nopposite polarizations at the two interfaces. In the paral-\nlel case, the torques acting on the two FIs are exerted in\nopposite directions. The torques then stabilize one mag-\nnetization, but destabilize the other. When the eigenfre-\nquencies acquire a negative imaginary part, their ampli-\ntude grows exponentially in time. We de\fne the thresh-\nold current jSH\n0;thrby the value at which Im[ !n\u0010\njSH\n0;thr\u0011\n] =\n0:Because the total damping has to be overcome at the5\nthreshold,jSH\n0;thr\u0018~\u000b. We treat the damping and ex-\nchange coupling perturbatively, thereby assuming ~ \u000b\u001c1\nand!x\u001c!0;where!0=p\n~!H(~!H+!M) is the FMR\nfrequency. The spin Hall angle is usually much smaller\nthan unity; thus, jSH\n0is treated as a perturbation for cur-\nrents up to the order of the threshold current, implying\nthat\f\fIm[!n\u0000\njSH\n0\u0001\n]\f\f\u001c\f\fRe[!n\u0000\njSH\n0\u0001\n]\f\f.\nThe exchange coupling !x=\rJ=(Msd) for YIGjPtj\nYIG should be weaker than that of the well-studied\nmetallic magnetic monolayers, where it is known to be-\ncome very small for d&3 nm.17In the following sections,\nwe assume that !x\u001c!Mmay be treated as a perturba-\ntion.\nTo treat the damping, spin pumping, spin-Hall-\ninduced torques and static exchange perturbatively, we\nintroduce the smallness parameter \u000fand let\u000b!\u000f\u000b,\n\u000b0!\u000f\u000b0,jSH\n0!\u000fjSH\n0,!x!\u000f!x. In the following sec-\ntions, a \frst-order perturbation is applied by linearizing\nin\u000fand subsequently setting \u000f= 1.\nWe transformMby the matrixUthat diagonalizes\nM0with eigenvalues ( !0;!0;\u0000!0;\u0000!0). We then ex-\ntract the part corresponding to the real eigenfrequencies,\nwhich yields the following equation:\n\f\f\f\f(D)11(D)12\n(D)21(D)22\f\f\f\f= 0; (31)\nwhereD=U\u00001MU. We thus reduce the fourth-order\nsecular equation in !to a second-order expression. To\nthe \frst order, we \fnd for the parallel ( s= 1) case,\n!P= ~!0+i\u000bP\ne\u000b\n2(2~!H+!M); (32)\nwhere we introduced a current-controlled e\u000bective\nGilbert damping:\n\u000bP\ne\u000b=\u000b+\u000b0(1\u0000F0)\n\u0006s\u0012\n\u000b0G0\u0000i!x\n!0\u00132\n+4(F2\u0000G2)\u0000\njSH\n0\u00012\n(2~!H+!M)2:(33)\nThe imaginary part of the square root in Eq. (33) causes a\n\frst-order real frequency shift that we may disregard, i.e.,\nRe\u0002\n!P\u0003\n\u0019~!0\u0019!0. We thus \fnd two modes with nearly\nthe same frequencies but di\u000berent e\u000bective broadenings.\nThe critical current jSH;P\n0;thris now determined by requir-\ning that\u000bP\ne\u000bvanish, leading to\njSH;P\n0;thr=\u0006q\n(\u000b+\u000b0(1\u0000F0))2\u0000(\u000b0G0)2\n2p\nF2\u0000G2\ns\n1 +\u0012!x=!0\n\u000b+\u000b0(1\u0000F0)\u00132\n(2~!H+!M);(34)\nwhile the critical charge current is jP\nc;thr=jS;P\n0;thr=\u0002SH.\nSpin pumping and spin \rip dissipate energy, leading to a\nhigher threshold current, which is re\rected by 1 \u0000F0\u0015\nG0. The reactive part of the SHE-induced torque ( G)suppresses the e\u000bect of the applied current and thereby\nincreases the critical current as well. The static exchange\ncouples M1andM2, hence increasing jSH;P\n0;thr. The criti-\ncal spin current decreases monotonically with increasing\ndN=lsf, implying that the spin valve (with parallel mag-\nnetization) has a larger threshold current than the FI jN\nbilayer (with thick dN).\nAnalogous to the parallel case, we \fnd two eigenmodes\nfor the antiparallel case ( s=\u00001), with eigenfrequencies\n!AP=!0+\u0012\n\u0006\u0000!x\n2!0+i\u000bAP\ne\u000b\n2\u0013\n(2~!H+!M) (35)\nand corresponding e\u000bective Gilbert damping parameters\n\u000bAP\ne\u000b=\u000b+\u000b0(1\u0000F0)\n\u0006\u000b0G0!M\n2~!H+!M+2\n2~!H+!MFjSH\n0;(36)\nwhich depend on the magnetic con\fguration because the\ndynamic exchange coupling di\u000bers, while the resonance\nfrequency is a\u000bected by the static coupling. In the AP\ncon\fgurations, the spin Hall current acts with the same\nsign on both layers due to the increase/decrease in damp-\ning on both sides depending on the applied current direc-\ntion. The corresponding threshold current is expressed\nas\njSH;AP\n0;thr=\u0000(\u000b+\u000b0(1\u0000F0)) (2~!H+!M)\u0000\u000b0G0!M\n2F;\n(37)\nwithjSH;AP\nc;thr=jSH;AP\n0;thr=\u0002SH. Again, the threshold for\ncurrent-induced excitation is increased by the spin pump-\ning.\nm2M0\nIncreasing current FI1 FI2 Nm2\nm1M0m1\nFIG. 2: (Color online) The acoustic mode for the parallel case\nfor di\u000berent applied currents, ranging from zero to just below\nthe critical current. For large currents the oscillations of the\ntwo FIs become out of phase.\nTo zeroth order in the smallness parameter \u000f, we \fnd\nthat the eigenvectors for the parallel con\fguration take6\nthe form vP= (u;\fu)T, where uis the 2-component\nvector\nu=\u0012\nip\n1 +!M=~!H\n1\u0013\n: (38)\nThe imbalance in the amplitudes of both layers is param-\neterized by\n\f=2jSH\n0F\u0007r\n4(F2\u0000G2)(jSH\n0)2+\u0010\n\u000b0G0\u0000i!x\n!0\u00112\n(2~!H+!M)2\n\u00002jSH\n0G+\u0010\n\u000b0G0\u0000i!x\n!0\u0011\n(2~!H+!M);\n(39)\nwhere\u0007corresponds to the \u0006in Eq. (32). For the sym-\nmetric case, the applied current favors out-of-phase os-\ncillations. It can be demonstrated that in the limit of\nlarge currents and low spin-memory loss, the correspond-\ning amplitude di\u000berence is \f=\u00001, withjSH\n0= 0, and\nan interlayer coupling dominated by either dynamic or\nstatic exchange \f=\u00071, which correspond to an optical\nand an acoustic mode, respectively. We use the labels\n\\acoustic\" and \\optic\"even though the phase di\u000berence\nis not precisely 0 or \u0019due to the static exchange inter-\naction. Note that \f(\u0000jSH\n0) = 1=\f(jSH\n0) is required by\nsymmetry; inverting the current direction is equivalent\nto interchanging FI1 and FI2. For !x= 0,\f(jSH\n0) is a\npole or node depending on the current direction for the\nacoustic mode in which the magnetization in one layer\nvanishes. Above this current, \fchange signs, and both\nmodes have a phase di\u000berence of \u0019. The critical cur-\nrent lies above the current corresponding to the node at\nwhich the acoustic mode becomes unstable. The ballistic\nmodel also supports acoustic and optical modes,19with\nthe optical mode being more e\u000eciently damped.\nIn the antiparallel case, acoustic and optical modes can\nare characterized by amplitudes\nvAP\nA=0\nBB@i!0\n~!H\n1\ni!0\n~!H\n\u000011\nCCA;vAP\nO=0\nBB@i!0\n~!H\n1\n\u0000i!0\n~!H\n11\nCCA; (40)\nwhere the optical (acoustic) mode corresponds to the +(-\n) sign in Eq. (35). The labels optical and acoustic are\nkept because of the di\u000berence in e\u000bective damping; a 180\u000e\nrotation about the yaxis of FI2 map these modes to the\ncorresponding modes for the parallel case.\nWhen the composition of the spin valve is slightly\nasymmetric, the dynamics of the two layers can still be\nsynchronized by the static and dynamic coupling. How-\never, at some critical detuning \u0001 !=!2\u0000!1, this tech-\nnique no longer works, as illustrated by the eigenfrequen-\ncies for the asymmetric spin valve in Fig. 4. Here, we\nemploy YIGjPtjYIG parameters but tune the FMR fre-\nquency of the right YIG layer. In practice, the tuning\ncan be achieved by varying the direction of the applied\nmagnetic \feld.16When the FMR frequencies of the two\nlayers are su\u000eciently close, the precessional motions in\nFI1 FI2 NM0\nFI1 FI2 N(a)\n(b)m1M0m1\nm2m2\nm1M0m1\nM0\nm2\nm2FIG. 3: (Color online) The eigenmodes of the antiparallel con-\n\fguration. (a)/(b) corresponds to the acoustic/optical mode\nof Eq. (40). For the acoustic/optical mode the in-plane/out-\nof-plane component is equal in the two layers, and opposite\nfor the out-of-plane/in-plane component.\nTABLE I: Physical parameters used in the numerical calcu-\nlations\nConstant Value Units\ng?a3:4\u00011015cm\u00002e2=h\n\u001bb5:4\u00011017s\u00001\n4\u0019MSc1750 G\nHint 0:2\u00014\u0019MS G\n\u000bc3\u000110\u00004\nlsf 10 nm\nd1;dN;d210, 5, 10 nm\na) Ref. [20], b) Ref. [21], c) Ref. [22]\nthe two layers lock to each other. The asymmetry in-\ntroduced by higher currents is observed to suppress the\nsynchronization.\nThe non-linear large-angle precession that occurs for\ncurrents above the threshold is not amenable to analyti-\ncal treatments; however, numerical calculations can pro-\nvide some insights. Because the dissipation of YIG is\nvery low the number of oscillations required to achieve\na noticable change in the precession angle is very large.\nTo speed up the calculations and make the results more\nreadable we rescale both g?and\u000bby a factor 0 :005=\u000b, in\nthis way the e\u000bective damping is rescaled. Fig. IV shows\nthe components of the magnetization in the two layers as\na function of time when a large current is switched on for\nan initially parallel magnetization along xwith a slight\ncanting ofMi;y= 0:01 fori= 1;2. We apply a current\njSH\n0=jP;SH\n0;thr= 110% at t= 0. For 5 T.t <40T, the\nprecession is out of phase, and the amplitude gradually\nincreases. At t= 40T, the applied current is ramped up\ntojSH\n0=jP;SH\n0;thr= 130%. At t\u001860T, the precession angle is\nno longer small, and our previous perturbative treatment\nbreaks down. However, we can understand that the right\nlayer precesses with a large angle, while the left layer7\nFIG. 4: (Color online) The lowest resonance frequencies of a\nparallel FI1jNjFI2 spin valve as a function of the detuning of\nthe FMR frequencies of the individual layers and for di\u000berent\ncurrentsjSH\n0=jSH\n0;thr= 0;10%;50% for the solid, black dashed\nand grey dashed lines, respectively. At zero applied current\nthe two layers lock when detuning is small. The current sup-\npresses synchronization almost completely when reaching the\nthreshold value.The inset shows the corresponding broaden-\nings.\nstays close to the initial equilibrium from the opposite\ndirection of the interface spin accumulations \u0016SH\n0.\nV. SUPERLATTICES\nA periodic stack of FIs coupled through Ns supports\nspin wave excitations propagating in the perpendicular\ndirection. The coupling between layers is described by\nEq. (24); however, each FI is coupled through the N lay-\ners to two neighboring layers. The primitive unit cell\nof the superlattice with collinear magnetization is the\nFIjN bilayer for the parallel con\fguration (two bilayers\nin the antiparallel con\fguration). For equivalent satu-\nration magnetizations in all FI layers, we can write for\ni2Z\nMi=si^x+mi; (41)\nwheresi= 1 for the parallel and si= (\u00001)ifor the\nantiparallel ground state. We can then linearize the ex-\npression with respect to the small parameters mi. An\nin-plane charge current causes accumulations of opposite\nsign in each N layer. The long-wavelength excitations of\nthe superlattice magnetization can be treated in the con-\ntinuum limit. Denoting the total thickness of a unit cell\nb=dN+dFI, we \fnd for the parallel case ( si= 1)\n@tm=^x\u0002[!Hm+!Mmz^z+ (\u000b+ 2\u000b0(1\u0000F0\u0000G0))@tm\n\u0000\u000b0G02@t;zzm\u0000!xb2@zzm+ 2jSH\n0Gb@z^x\u0002m\u0003\n:(42)\nFIG. 5: (Color online) Magnetization dynamics for the par-\nallel con\fguration and currents above the threshold. (a)/(b)\nthe magnetization in the left/right layer as a function of time\nin units of T= 2\u0019=! 0. The e\u000bective damping is rescaled by\nlettingg?!g?0:005=\u000band\u000b!\u000b0:005=\u000b. The numeri-\ncal calculation was carried out by a 4th order Runge-Kutta\nmethod with a step size \u0001 t=T=50.\nForm=m0ei(!t\u0000kzz), the linearized dispersion relation\nis\n!=p\n!H(!H+!M) +1\n2(2!H+!M)!x\n!0b2k2\nz\u00002jSH\n0Gbkz\n+i1\n2(2!H+!M)(\u000b+ 2\u000b0(1\u0000F0\u0000G0) +\u000b0G0k2\nzb2):(43)\nThe applied current thus adds a term that is linear in\nkzto the real part of the frequency. The direct e\u000bect\nof the SHE now vanishes because the torques on both\nsides of any FI cancel. However, when m06=0, a net\nspin current \rows normal to the stack, which a\u000bects the\ndispersion. In the ferromagnetic layers, this phenomenon\nis equivalent to a pure strain \feld on the magnetization\nand is therefore non-dissipative. While generating jSH\n0\ncauses Ohmic losses, the magnetization dynamics in this\nlimit do not add to the energy dissipation, explaining\nthe contribution to Re[ !]:In this regime, there are no\nexternal current-induced contributions or instabilities.\nAntiferromagnetic superlattices appear to be di\u000ecult\nto realize experimentally because a staggered external\nmagnetic \feld would be required. The unit cell is dou-\nbled as is the number of variables in the equation of mo-\ntion. Determining the coupling coe\u000ecients from Eq. (26)8\nis straightforward but cumbersome and is not presented\nhere. Naively, one could expect that the SHE-induced\ntorque would act very di\u000berently in the antiferromag-\nnetic case. The SHE acts in a symmetric manner on the\nFI(\")jNjFI(#) system, stabilizing or destabilizing both\nlayers simultaneously. However, similarly to the ferro-\nmagnetic superlattice, the direct SHE vanishes also in\nthe antiferromagnetic superlattice. Each FI is in contact\nwith an N, with spin accumulations of opposite sign on\nthe left and right side of the interfaces, which leads to\nthe same cancellation of the direct SHE-induced torque\npresented for the ferromagnetic superlattice.\nWe can also envision a multilayer in which individual\nmetallic layers can be contacted separately and indepen-\ndently. NjFIjN structures have been predicted to display\na magnon drag e\u000bect through the ferromagnetic \flm,23\ni.e. a current in one layer induces an emf in the other one.\nA drag e\u000bect does also exists in our macrospin model: if\nwe induce dynamics by a current in one layer by the spin\nHall e\u000bect, the spin pumping and inverse spin Hall e\u000bect\ngenerates a current in the other layer, but only above a\ncurrent threshhold.\nWith separate contacts to the layers one may drive op-\nposite currents through neighboring \flms. In that case,\nthe spin currents absorbed by a ferromagnetic layer is\nrelatively twice as large as in the FI jN bilayer, thereby\nreducing the critical currents for the parallel con\fgura-\ntion, but of opposite sign for neighboring magnetic lay-\ners. A staggered current distribution in the superlattice\ndestabilizes the ferromagnetic con\fguration, but it can\nstabilize an antiferromagnetic one even in the absence of\nstatic exchange coupling. This leads to intricate dynam-\nics when competing with an applied magnetic \feld.VI. CONCLUSIONS\nWe study current-induced magnetization dynamics in\nspin valves and superlattices consisting of insulating mag-\nnets separated by metallic spacers with spin Hall ef-\nfect. The current-induced torques experienced by the two\nmagnetic layers in an FI( \")jNjFI(\") spin valve caused by\nthe spin Hall e\u000bect are opposite in sign. A charge current\nin N normal to the magnetization this leads to a damp-\ning and an antidamping, stabilizing one and destabilizing\nthe other magnetization. We calculate the magnetiza-\ntion dynamics when the two layers are exchange coupled\nand in the presence of the dynamic exchange coupling in-\nduced by spin pumping. In an antiparallel con\fguration\nFI(\")jNjFI(#) the interlayer couplings play a minor role\nin the current-induced e\u000bects. The threshold currents at\nwhich self-oscillation occur are higher for parallel than\nantiparallel spin valves. We predict interesting current-\ninduced e\u000bects for superlattices and multilayers in which\nthe metallic spacer layers can be individually contacted.\nAcknowledgments\nH.S. and A. B. acknowledge support from the Re-\nsearch Council of Norway, project number 216700. This\nwork was supported by KAKENHI (Grants-in-Aid for\nScienti\fc Research) Nos. 25247056 and 25220910, FOM\n(Stichting voor Fundamenteel Onderzoek der Materie),\nthe ICC-IMR, the EU-RTN Spinicur, EU-FET grant\nInSpin 612759 and DFG Priority Program 1538 \"Spin-\nCaloric Transport\" (GO 944/4).\n1A. Brataas, A. D. Kent, and H. Ohno, Nature Mat., 373\n(2012).\n2H. Nakayama, M. Althammer, Y.-T. Chen, K. Uchida, Y.\nKajiwara, D. Kikuchi, T. Ohtani, S. Gepr ags, M. Opel,\nS.Takahashi, R. Gross, G.E. W. Bauer, S. T. B. Goennen-\nwein and E. Saitoh, Phys. Rev. Lett., 110206601 (2013).\n3Y. Kajiwara, K. Harii, S. Takahashi, J. Ohe, K. Uchida,\nM. Mizuguchi, H. Umezawa, H. Kawai, K. Ando, K.\nTakanashi, S. Maekawa, and E. Saito, Nature 464, 7269\n(2010).\n4C. W. Sandweg, Y. Kajiwara, K. Ando, E. Saitoh, and B.\nHillebrands, Appl. Phys. Lett. 97, 252504 (2010).\n5C. W. Sandweg, Y. Kajiwara, A. V. Chumak, A. A. Serga,\nV. I. Vasyuchka, M. B. Jung\reisch, E. Saitoh, and B. Hille-\nbrands, Phys. Rev. Lett. 106, 216601 (2011).\n6L. H. Vilela-Leao, C. Salvador, A. Azevedo, S. M. Rezende,\nAppl. Phys. Lett. 99, 102505 (2011).\n7C. Burrowes, B. Heinrich, B. Kardasz, E. A. Montoya, E.\nGirt, Yiyan Sun, Young-Yeal Song, and Mingzhong Wu,\nAppl. Phys. Lett. 100, 092403 (2012).\n8S. M. Rezende, R. L. Rodriguez-Suarez, M. M. Soares, L.\nH. Vilela-Leao, D. Ley Dominguez, and A. Azevedo, Appl.\nPhys. Lett. 102, 012402 (2012).9J. Xiao, G. E. W. Bauer, K.-C. Uchida, E. Saitoh, and S.\nMaekawa, Phys. Rev. B 81, 214418 (2010).\n10Y.-T. Chen, S. Takahashi, H. Nakayama, M. Althammer,\nS. T. B. Goennenwein, E. Saitoh, and G. E. W. Bauer,\nPhys. Rev. B 87, 144411 (2013).\n11J. C. Slonczewski, Phys. Rev. B 82, 054403 (2010).\n12J. Xingtao, L. Kai, K. Xia, and G. E. W. Bauer, EPL 96,\n17005 (2011).\n13A. Kapelrud and A. Brataas, Phys. Rev. Lett. 111, 097602\n(2013).\n14Y. Zhou, H. J. Jiao, Y. T. Chen, G. E. W. Bauer, and J.\nXiao, Phys. Rev. B 88, 184403 (2013).\n15J. Xiao and G. E. W. Bauer, Phys. Rev. Lett. 108, 217204\n(2012).\n16B. Heinrich, Y. Tserkovnyak, G. Woltersdorf, A. Brataas,\nR. Urban and G. E. W. Bauer, Phys. Rev. Lett. 90, 187601\n(2003).\n17Z. Q. Qiu, J. Pearson and S. D. Bader, Phys. Rev. B 46,\n8659 (1992).\n18J.-M.L. Beaujour, W. Chen, A. D. Kent and J. Z. Sun, J.\nAppl. Phys. 99, 08N503 (2006).\n19Y. Tserkovnyak, A. Brataas, G. E. W. Bauer and B. I.\nHalperin, Rev. Mod. Phys. 771375 (2005).9\n20M. B. Jung\reisch, V. Lauer, R. Neb, A. V. Chumak and\nB. Hillebrands, Appl. Phys. Lett. 103, 022411 (2013).\n21D. Giancoli, \"25. Electric Currents and Resistance\". In Jo-\ncelyn Phillips. Physics for Scientists and Engineers with\nModern Physics (4th ed.), (2009) [1984]\n22A. A. Serga, A. V. Chumak and B. Hillebrands, J. Phys.\nD: App. Phys. 43, 264002 (2010)\n23Steven S.-L. Zhang and S. Zhang, Phys. Rev. Lett. 109,\n096603 (2012).\nAppendix A: Matrix Elements\nHere, we derive the response coe\u000ecients F,G,F0and\nG0that determine the torques, depending on the prop-\nerties of the normal metal. Let us \frst discuss the coef-\n\fcients related to the torques induced by the SHE. The\nfunctionsFandGare extracted from the derivatives of\nEq. (23) with respect to the transverse components of the\ndynamic magnetizations mi.Fgoverns the SHE-induced\ntorque in one layer due to displaced magnetization in the\nsame layer and can be computed as\n@(\u001cSH\n1)y\n@m1;y=@(\u001cSH\n1)z\n@m1;z=\u0000@(\u001cSH\n2)y\n@(sm2;y)=\u0000@(\u001cSH\n2)z\n@(sm2;z)=\u0000FjSH\n0:\n(A1)\nThus,\nF=\r~\n2e2MSdg?2elsf\n\u001btanh(dN=2lsf)\n\u0014\n1\u00001\n2\u00001(dN=2)\u00001\n2\u00002(dN=2)\u0015\n:(A2)\nSimilarly, we can identify G, which governs the cross-\ncorrelation of the SHE-induced torque in one layer arising\nfrom a displaced magnetization in the other layer from\n@(\u001cSH\n1)y\n@(sm2;y)=@(\u001cSH\n1)z\n@(sm2;z)=\u0000@(\u001cSH\n2)y\n@m1;y=\u0000@(\u001cSH\n2)z\n@m1;z=GjSH\n0;\n(A3)Thus\nG=\r~\n2e2MSdg?2elsf\n\u001btanh(dN=2lsf)\n1\n2[\u00001(dN=2)\u0000\u00002(dN=2)]: (A4)\nTorques generated by spin pumping contain terms of the\nform^ x\u0002miand couple the y- andz-components of the\nmagnetization dynamics. We \fnd\n@(\u001cST\n1)y\n@_m1;z=\u0000@(\u001cST\n1)z\n@_m1;y=@(\u001cST\n2)y\n@(s_m2;z)=\u0000@(\u001cST\n2)z\n@(s_m2;y)=F0\u000b0;\n(A5)\nwhere\n2F0= \u00001(dN=2) + \u0000 2(dN=2): (A6)\nSimilarly,\n@(\u001cST\n1)y\n@(s_m2;z)=\u0000@(\u001cST\n1)z\n@(s_m2;y)=@(\u001cST\n2)y\n@_m1;z=\u0000@(\u001cST\n2)z\n@_m1;y=G0\u000b0;\n(A7)\nwhere\n2G0= \u00001(dN=2)\u0000\u00002(dN=2): (A8)\nWe \fnally note that some of the coe\u000ecients are related:\nG\nG0\u000b0=1\n~2elsf\n\u001btanh(dN=2lsf): (A9)"    },    {        "title": "1405.2347v1.Magnetization_dynamics_and_damping_due_to_electron_phonon_scattering_in_a_ferrimagnetic_exchange_model.pdf",        "content": "Magnetization Dynamics and Damping due to Electron-Phonon Scattering in a\nFerrimagnetic Exchange Model\nAlexander Baral,\u0003Svenja Vollmar, and Hans Christian Schneidery\nPhysics Department and Research Center OPTIMAS,\nKaiserslautern University, P. O. Box 3049, 67663 Kaiserslautern, Germany\n(Dated: June 4, 2018)\nWe present a microscopic calculation of magnetization damping for a magnetic \\toy model.\"\nThe magnetic system consists of itinerant carriers coupled antiferromagnetically to a dispersionless\nband of localized spins, and the magnetization damping is due to coupling of the itinerant carriers\nto a phonon bath in the presence of spin-orbit coupling. Using a mean-\feld approximation for\nthe kinetic exchange model and assuming the spin-orbit coupling to be of the Rashba form, we\nderive Boltzmann scattering integrals for the distributions and spin coherences in the case of an\nantiferromagnetic exchange splitting, including a careful analysis of the connection between lifetime\nbroadening and the magnetic gap. For the Elliott-Yafet type itinerant spin dynamics we extract\ndephasing and magnetization times T1andT2from initial conditions corresponding to a tilt of the\nmagnetization vector, and draw a comparison to phenomenological equations such as the Landau-\nLifshitz (LL) or the Gilbert damping. We also analyze magnetization precession and damping for\nthis system including an anisotropy \feld and \fnd a carrier mediated dephasing of the localized spin\nvia the mean-\feld coupling.\nPACS numbers: 75.78.-n, 72.25.Rb, 76.20.+q\nI. INTRODUCTION\nThere are two widely-known phenomenological ap-\nproaches to describe the damping of a precessing mag-\nnetization in an excited ferromagnet: one introduced\noriginally by Landau and Lifshitz1and one introduced\nby Gilbert,2which are applied to a variety of prob-\nlems3involving the damping of precessing magnetic mo-\nments. Magnetization damping contributions and its in-\nverse processes, i.e., spin torques, in particular in thin\n\flms and nanostructures, are an extremely active \feld,\nwhere currently the focus is on the determination of novel\nphysical processes/mechanisms. Apart from these ques-\ntions there is still a debate whether the Landau-Lifshitz\nor the Gilbert damping is the correct one for \\intrin-\nsic\" damping, i.e., neglecting interlayer coupling, inter-\nface contributions, domain structures and/or eddy cur-\nrents. This intrinsic damping is believed to be caused\nby a combination of spin-orbit coupling and scattering\nmechanisms such as exchange scattering between s and d\nelectrons and/or electron-phonon scattering.4{6Without\nreference to the microscopic mechanism, di\u000berent macro-\nscopic analyses, based, for example, on irreversible ther-\nmodynamics or near equilibrium Langevin theory, prefer\none or the other description.7,8However, material param-\neters of typical ferromagnetic heterostructures are such\nthat one is usually \frmly in the small damping regime so\nthat several ferromagnetic resonance (FMR) experiments\nwere not able to detect a noticeable di\u000berence between\nLandau Lifshitz and Gilbert magnetization damping. A\nrecent analysis that related the Gilbert term directly to\nthe spin-orbit interaction arising from the Dirac equa-\ntion does not seem to have conclusively solved this dis-\ncussion.9\nThe dephasing term in the Landau-Lifshitz form isalso used in models based on classical spins coupled\nto a bath, which have been successfully applied to\nout-of-equilibrium magnetization dynamics and magnetic\nswitching scenarios.10The most fundamental of these\nare the stochastic Landau-Lifshitz equations,10{13from\nwhich the Landau-Lifshitz Bloch equations,14,15can be\nderived via a Fokker-Planck equation.\nQuantum-mechanical treatments of the equilibrium\nmagnetization in bulk ferromagnets at \fnite temper-\natures are extremely involved. The calculation of\nnon-equilibrium magnetization phenomena and damp-\ning for quantum spin systems in more than one dimen-\nsion, which include both magnetism and carrier-phonon\nand/or carrier-impurity interactions, at present have to\nemploy simpli\fed models. For instance, there have been\nmicroscopic calculations of Gilbert damping parameters\nbased on Kohn-Sham wave functions for metallic ferro-\nmagnets16,17and Kohn-Luttinger p-dHamiltonians for\nmagnetic semiconductors.18While the former approach\nuses spin density-functional theory, the latter approach\ntreats the anti-ferromagnetic kinetic-exchange coupling\nbetween itinerant p-like holes and localized magnetic\nmoments originating from impurity d-electrons within a\nmean-\feld theory. In both cases, a constant spin and\nband-independent lifetime for the itinerant carriers is\nused as an input, and a Gilbert damping constant is ex-\ntracted by comparing the quantum mechanical result for\n!!0 with the classical formulation. There have also\nbeen investigations, which extract the Gilbert damping\nfor magnetic semiconductors from a microscopic calcula-\ntion of carrier dynamics including Boltzmann-type scat-\ntering integrals.19,20Such a kinetic approach, which is of\na similar type as the one we present in this paper, avoids\nthe introduction of electronic lifetimes because the scat-\ntering is calculated dynamically.arXiv:1405.2347v1  [cond-mat.mtrl-sci]  9 May 20142\nThe present paper takes up the question how the spin\ndynamics in the framework of the macroscopic Gilbert\nor Landau-Lifshitz damping compare to a microscopic\nmodel of relaxation processes in the framework of a rel-\natively simple model. We analyze a mean-\feld kinetic\nexchange model including spin-orbit coupling for the itin-\nerant carriers. Thus the magnetic mean-\feld dynamics is\ncombined with a microscopic description of damping pro-\nvided by the electron-phonon coupling. This interaction\ntransfers energy and angular momentum from the itin-\nerant carriers to the lattice. The electron-phonon scat-\ntering is responsible both for the lifetimes of the itiner-\nant carriers and the magnetization dephasing. The lat-\nter occurs because of spin-orbit coupling in the states\nthat are connected by electron-phonon scattering. To be\nmore speci\fc, we choose an anti-ferromagnetic coupling\nat the mean-\feld level between itinerant electrons and\na dispersion-less band of localized spins for the magnetic\nsystem. To keep the analysis simple we use as a model for\nthe spin-orbit coupled itinerant carrier states a two-band\nRashba model. As such it is a single-band version of the\nmulti-band Hamiltonians used for III-Mn-V ferromag-\nnetic semiconductors.18,21{24The model analyzed here\nalso captures some properties of two-sublattice ferrimag-\nnets, which are nowadays investigated because of their\nmagnetic switching dynamics.25,26The present paper is\nset apart from studies of spin dynamics in similar mod-\nels with more complicated itinerant band structures19,20\nby a detailed comparison of the phenomenological damp-\ning expressions with a microscopic calculation as well as\na careful analysis of the restrictions placed by the size\nof the magnetic gap on the single-particle broadening in\nBoltzmann scattering.\nThis paper is organized as follows. As an extended\nintroduction, we review in Sec. II some basic facts con-\ncerning the Landau-Lifshitz and Gilbert damping terms\non the one hand and the Bloch equations on the other.\nIn Sec. III we point out how these di\u000berent descriptions\nare related in special cases. We then introduce a micro-\nscopic model for the dephasing due to electron-phonon\ninteraction in Sec. IV, and present numerical solutions\nfor two di\u000berent scenarios in Secs. V and VI. The \frst\nscenario is the dephasing between two spin subsystems\n(Sec. V), and the second scenario is a relaxation process\nof the magnetization toward an easy-axis (Sec. VI). A\nbrief conclusion is given at the end.\nII. PHENOMENOLOGIC DESCRIPTIONS OF\nDEPHASING AND RELAXATION\nWe summarize here some results pertaining to a single-\ndomain ferromagnet, and set up our notation. In equilib-\nrium we assume the magnetization to be oriented along\nits easy axis or a magnetic \feld ~H, which we take to\nbe thezaxis in the following. If the magnetization\nis tilted out of equilibrium, it starts to precess. As\nillustrated in Fig. 1 one distinguishes the longitudinal\nFIG. 1. Illustration of non-equilibrium spin-dynamics in pres-\nence of a magnetic \feld without relaxation (a) and within\nrelaxation (b).\ncomponent Mk, inzdirection, and the transverse part\nM?\u0011q\nM2\u0000M2\nk, precessing in the x-yplane with the\nLarmor frequency !L.\nIn connection with the interaction processes that re-\nturn the system to equilibrium, the decay of the trans-\nverse component is called dephasing. There are three\nphenomenological equations used to describe spin de-\nphasing processes:\n1. The Bloch(-Bloembergen) equations27,28\n@\n@tMk(t) =\u0000Mk(t)\u0000Meq\nT1(1)\n@\n@tM?(t) =\u0000M?(t)\nT2(2)\ndescribe an exponential decay towards the equilib-\nrium magnetization Meqinzdirection. The trans-\nverse component decays with a time constant T2,\nwhereas the longitudinal component approaches its\nequilibrium amplitude with T1. These time con-\nstants may be \ft independently to experimental\nresults or microscopic calculations.\n2. Landau-Lifshitz damping1with parameter \u0015\n@\n@t~M(t) =\u0000\r~M\u0002~H\u0000\u0015~M\nM\u0002\u0000~M\u0002~H\u0001\n(3)\nwhere\ris the gyromagnetic ratio. The \frst term\nmodels the precession with a frequency !L=\rj~Hj,\nwhereas the second term is solely responsible for\ndamping.\n3. Gilbert damping2with the dimensionless Gilbert\ndamping parameter \u000b\n@\n@t~M(t) =\u0000\rG~M\u0002~H+\u000b\u0010~M\nM\u0002@t~M\u0011\n(4)\nIt is generally accepted that \u000bis independent of\nthe static magnetic \felds ~Hsuch as anisotropy\n\felds,18,29and thus depends only on the material\nand the microscopic interaction processes.3\nThe Landau-Lifshitz and Gilbert forms of damping are\nmathematically equivalent2,7,30with\n\u000b=\u0015\n\r(5)\n\rG=\r(1 +\u000b2) (6)\nbut there are important di\u000berences. In particular, an in-\ncrease of\u000blowers the precession frequency in the dynam-\nics with Gilbert damping, while the damping parameter\n\u0015in the Landau-Lifshitz equation has no impact on the\nprecession. In contrast to the Bloch equations, Landau-\nLifshitz and Gilbert spin-dynamics always conserve the\nlengthj~Mjof the magnetization vector.\nAn argument by Pines and Slichter,31shows that there\nare two di\u000berent regimes for Bloch-type spin dynamics\ndepending on the relation between the Larmor period and\nthe correlation time. As long as the correlation time is\nmuch longer than the Larmor period, the system \\knows\"\nthe direction of the \feld during the scattering process.\nStated di\u000berently, the scattering process \\sees\" the mag-\nnetic gap in the bandstructure. Thus, transverse and\nlongitudinal spin components are distinguishable and the\nBloch decay times T1andT2can di\u000ber. If the correlation\ntime is considerably shorter than the Larmor period, this\ndistinction is not possible, with the consequence that T1\nmust be equal to T2. Within the microscopic approach,\npresented in Sec. IV D, this consideration shows up again,\nalbeit for the energy conserving \u000efunctions resulting from\na Markov approximation.\nThe regime of short correlation times has already been\ninvestigated in the framework of a microscopic calcula-\ntion by Wu and coworkers.32They analyze the case of\na moderate external magnetic \feld applied to a non-\nmagnetic n-type GaAs quantum well and include di\u000ber-\nent scattering mechanisms (electron-electron Coulomb,\nelectron-phonon, electron-impurity). They argue that\nthe momentum relaxation rate is the crucial time scale\nin this scenario, which turns out to be much larger than\nthe Larmor frequency. Their numerical results con\frm\nthe identity T1=T2expected from the Pines-Slichter\nargument.\nIII. RELATION BETWEEN\nLANDAU-LIFSHITZ, GILBERT AND BLOCH\nWe highlight here a connection between the Bloch\nequations (1, 2) and the Landau-Lifshitz equation (3).\nTo this end we assume a small initial tilt of the mag-\nnetization and describe the subsequent dynamics of the\nmagnetization in the form\n~M(t) =0\n@\u000eM?(t) cos(!Lt)\n\u000eM?(t) sin(!Lt)\nMeq\u0000\u000eMk(t)1\nA (7)\nwhere\u000eM?and\u000eMjjdescribe deviations from equilib-\nrium. Putting this into eq. (3) one gets a coupled set ofequations.\n@\n@t\u000eM?(t) =\u0000\u0015HMeq\u0000\u000eMk(t)\nj~M(t)j\u000eM?(t) (8)\n@\n@t\u000eMk(t) =\u0000\u0015H1\nj~M(t)j\u000eM2\n?(t) (9)\nEq. (8) is simpli\fed for a small deviation from equilib-\nrium, i.e.,\u000eM(t)\u001cMeqandj~M(t)j\u0019Meq:\n\u000eM?(t) =Cexp(\u0000\u0015Ht) (10)\n\u000eMk(t) =C2\n2Meqexp(\u00002\u0015Ht) (11)\nwhereCis an integration constant. For small excitations\nthe deviations decay exponentially and Bloch decay times\nT1andT2result, which are related by\n2T1=T2=1\n\u0015H: (12)\nOnly this ratio of the Bloch times is compatible with a\nconstant length of the magnetization vector at low exci-\ntations. By combining Eqs. (12) and (5) one can connect\nthe Gilbert parameter \u000band the dephasing time T2\n\u000b=1\nT2!L: (13)\nIf the conditions for the above approximations apply, the\nGilbert damping parameter \u000bcan be determined by \ft-\nting the dephasing time T2and the Larmor frequency !L\nto computed or measured spin dynamics. This dimen-\nsionless quantity is well suited to compare the dephasing\nthat results from di\u000berent relaxation processes.\nFigure 2 shows the typical magnetization dynamics\nthat results from (3), i.e., Landau-Lifshitz damping. As\nan illustration of a small excitation we choose in Fig. 2(a)\nan angle of 10\u000efor the initial tilt of the magnetization,\nwhich results in an exponential decay with 2 T1=T2.\nFrom the form of Eq. (3) it is clear that this behavior\npersists even for large !Land\u0015. Obviously the Landau-\nLifshitz and Gilbert damping terms describe a scenario\nwith relatively long correlation times (i.e., small scat-\ntering rates), because only in this regime both decay\ntimes can di\u000ber. The microscopic formalism in Sec. IV\nworks in the same regime and will be compared with\nthe phenomenological results. For an excitation angle\nof 90\u000e, the Landau-Lifshitz dynamics shown in Fig. 2(b)\nbecome non-exponential, so that no well-de\fned Bloch\ndecay times T1,T2exist.\nIV. MICROSCOPIC MODEL\nIn this section we describe a microscopic model that in-\ncludes magnetism at the mean-\feld level, spin-orbit cou-\npling as well as the microscopic coupling to a phonon\nbath treated at the level of Boltzmann scattering inte-\ngrals. We then compare the microscopic dynamics to4\n0 5000.51δM⊥/Meq\ntime (ps)0 5000.51\ntime (ps)δM/bardbl/Meq0 5000.010.02δM/bardbl/Meq\ntime (ps)0 5000.10.2\ntime (ps)δM⊥/Meq\n  \nT1= 5.02 ps(a)\n(b)T2= 10.04 ps\nFIG. 2. Dynamics of \u000eM?and\u000eMkcomputed using to\nLandau-Lifshitz damping ( !L= 1 ps\u00001,H= 106A\nm\u0019\n1:26\u0001104Oe,\u0015= 10\u00007m\nA ps). (a) An angle of 10\u000eleads to\nexponential an exponential decay with well de\fned T1andT2\ntimes. (b). For an angle of 90\u000e, the decay (solid line) is not\nexponential as comparison with the exponential \ft (dashed\nline) clearly shows.\nthe Bloch equations (1), (2), as well as the Landau-\nLifshitz (3) and Gilbert damping terms (4). The mag-\nnetic properties of the model are de\fned by an anti-\nferromagnetic coupling between localized magnetic im-\npurities and itinerant carriers. As a prototypical spin-\norbit coupling we consider an e\u000bectively two-dimensional\nmodel with a Rashba spin-orbit coupling. The reason\nfor the choice of a model with a two-dimensional wave\nvector space is not an investigation of magnetization dy-\nnamics with reduced dimensionality, but rather a reduc-\ntion in the dimension of the integrals that have to be\nsolved numerically in the Boltzmann scattering terms.\nSince we treat the exchange between the localized and\nitinerant states in a mean-\feld approximation, our two-\ndimensional model still has a \\magnetic ground state\"\nand presents a framework, for which qualitatively dif-\nferent approaches can be compared. We do not aim at\nquantitative predictions for, say, magnetic semiconduc-\ntors or ferrimagnets with two sublattices. Finally, we\ninclude a standard interaction hamiltonian between the\nitinerant carriers and acoustic phonons. The correspond-\ning hamiltonian reads\n^H=^Hmf+^Hso+^He\u0000ph+^Haniso: (14)\nOnly in Sec. VI an additional \feld ^Haniso is included,\nwhich is intended to model a small anisotropy.A. Exchange interaction between itinerant carriers\nand localized spins\nThe \\magnetic part\" of the model is described by the\nHamiltonian\n^Hmf=X\n~k\u0016~2k2\n2m\u0003^cy\n~k\u0016^c~k\u0016+J^~ s\u0001^~S: (15)\nwhich we consider in the mean-\feld limit. The \frst term\nrepresents itinerant carriers with a k-dependent disper-\nsion relation. In the following we assume s-like wave\nfunctions and parabolic energy dispersions. The e\u000bective\nmass is chosen to be m\u0003= 0:5me, wheremeis the free\nelectron mass, and the ^ c(y)\n~k\u0016operators create and annihi-\nlate carriers in the state j~k;\u0016iwhere\u0016labels the itinerant\nbands, as shown in Fig. 3(a).\nThe second term describes the coupling between itiner-\nant spins~ sand localized spins ~Svia an antiferromagnetic\nexchange interaction\n^~ s=1\n2X\n~kX\n\u0016\u00160h~k;\u00160j^~ \u001bj~k;\u0016i^cy\n~k\u0016^c~k\u00160 (16)\n^~S=1\n2X\n\u0017\u00170h\u00170j^~ \u001bj\u0017iX\n~K^Cy\n~K\u0017^C~K\u00170 (17)\nHere, we have assumed that the wave functions of the lo-\ncalized spins form dispersionless bands, i.e., we have im-\nplicitly introduced a virtual-crystal approximation. Due\nto the assumption of strong localization there is no or-\nbital overlap between these electrons, which are therefore\nconsidered to have momentum independent eigenstates\nj\u0017iand a \rat dispersion, as illustrated in Fig. 3(a). The\ncomponents of the vector ^~ \u001bare the Pauli matrices ^ \u001biwith\ni=x;y;z , and ^C(y)\n~K\u0017are the creation and annihilation op-\nerators for a localized spin state.\nWe do notinclude interactions among localized or itin-\nerant spins, such as exchange scattering. For simplicity,\nwe assume both itinerant and localized electrons to have\na spin 1=2 and therefore \u0016and\u0017to run over two spin-\nprojection quantum numbers \u00061=2. In the following we\nchosse an antiferromagnetic ( J > 0) exchange constant\nJ= 500 meV, which leads to the schematic band struc-\nture shown in Fig. 3(b).\nIn the mean \feld approximation used here, the itiner-\nant carriers feel an e\u000bective magnetic \feld ^Hloc\n~Hloc=\u0000J\u0016B\u0016\ng~S (18)\ncaused by localized moments and vice versa. Here \u0016B\nis the Bohr magneton and g= 2 is the g-factor of the\nelectron. The permeability \u0016is assumed to be the vac-\nuum permeability \u00160. This time-dependent magnetic\n\feld~Hloc(t) de\fnes the preferred direction in the itiner-\nant sub-system and therefore determines the longitudinal\nand transverse component of the itinerant spin at each\ntime.5\nr#k (a) (b) E(k) \nk \n\u0010\nk \n\u000e\u0010,k\n\u000e,kE(k) \nEF \nFIG. 3. Sketch of the band-structure with localized (\rat\ndispersions) and itinerant (parabolic dispersions) electrons.\nAbove the Curie-Temperature TCthe spin-eigenstates are de-\ngenerate (a), whereas below TCa gap between the spin states\nexists.\nB. Rashba spin-orbit interaction\nThe Rashba spin-orbit coupling is given by the Hamil-\ntonian\n^Hso=\u000bR(^\u001bxky\u0000^\u001bykx) (19)\nA Rashba coe\u000ecient of \u000bR= 10 meV nm typical for semi-\nconductors is chosen in the following calculations. This\nvalue, which is close to the experimental one for the\nInSb/InAlSb material system,33is small compared to the\nexchange interactions, but it allows the exchange of an-\ngular momentum with the lattice.\nC. Coherent dynamics\nFrom the above contributions (15) and (19) to the\nHamiltonian we derive the equations of motion contain-\ning the coherent dynamics due to the exchange interac-\ntion and Rashba spin-orbit coupling as well as the inco-\nherent electron-phonon scattering. We \frst focus on the\ncoherent contributions. In principle, one has the choice\nto work in a basis with a \fxed spin-quantization axis or\nto use single-particle states that diagonalize the mean-\n\feld (plus Rashba) Hamiltonian. Since we intend to use\na Boltzmann scattering integral in Sec. IV D we need to\napply a Markov approximation, which only works if one\ndeals with diagonalized eigenenergies. In our case this is\nthe single-particle basis that diagonalizes the entire one-\nparticle contribution of the Hamiltonian ^Hmf+^Hso. In\nmatrix representation this one-particle contribution for\nthe itinerant carriers reads:\n^Hmf+^Hso= \n~2k2\n2m\u0003+ \u0001loc\nz(\u0001loc\n++R~k)\u0003\n\u0001loc\n++R~k~2k2\n2m\u0003\u0000\u0001loc\nz!\n(20)\nwhere we have de\fned \u0001loc\ni=J1\n2h^SiiandR~k=\n\u0000i\u000bRkexp(i'k) with'k= arctan(ky=kx). The eigenen-\nergies are\n\u000f\u0006\n~k=~2k2\n2m\u0003\u0007q\nj\u0001loczj2+jR~k+ \u0001loc\n+j2: (21)and the eigenstates\nj~k;+i=\u0012\n1\n\u0018~k\u0013\n;j~k;\u0000i=\u0012\u0000\u0018\u0003\n~k\n1\u0013\n(22)\nwhere\n\u0018~k=\u0001loc\n++R~k\n\u0001locz+q\nj~\u0001locj2+jR~kj2(23)\nIn this basis the coherent part of the equation of mo-\ntion for the itinerant density matrix \u001a\u0016\u00160\n~k\u0011 h^cy\n~k\u0016^c~k\u00160i\nreads\n@\n@t\u001a\u0016\u00160\n~k\f\f\f\ncoh=i\n~\u0000\n\u000f\u0016\n~k\u0000\u000f\u00160\n~k\u0001\n\u001a\u0016\u00160\n~k: (24)\nNo mean-\feld or Rashba terms appear explicitly in these\nequations of motion since their contributions are now hid-\nden in the time-dependent eigenstates and eigenenergies.\nSince we are interested in dephasing and precessional\ndynamics, we assume a comparatively small spin-orbit\ncoupling, that can dissipate angular momentum into the\nlattice, but does not have a decisive e\u000bect on the band-\nstructure. Therefore we use the spin-mixing only in the\ntransition matrix elements of the electron-phonon scat-\nteringM~k0\u00160\n~k\u0016(31). For all other purposes we set R~k= 0.\nIn particular, the energy-dispersion \u000f\u0006\n~kis assumed to be\nuna\u000bected by the spin-orbit interaction and therefore it\nis spherically symmetric.\nWith this approximation the itinerant eigenstates are\nalways exactly aligned with the e\u000bective \feld of the local-\nized moments ~Hloc(t). Since this e\u000bective \feld changes\nwith time, the diagonalization and a transformation of\nthe spin-density matrix in \\spin space\" has to be re-\npeated at each time-step. This e\u000bort makes it easier\nto identify the longitudinal and transverse spin compo-\nnents with the elements of the single-particle density\nmatrix: The o\u000b-diagonal entries of the density matrix\n\u001a\u0006\u0007\n~k, which precess with the k-independent Larmor fre-\nquency!L= 2\u0001loc=~, always describe the dynamics of\nthe transverse spin-component. The longitudinal compo-\nnent, which does not precess, is represented by the diag-\nonal entries \u001a\u0006\u0006\n~k. Since both components change their\nspatial orientation continuously, we call this the rotating\nframe. The components of the spin vector in the rotating\nframe are\nh^ski=1\n2X\n~k\u0000\n\u001a++\n~k\u0000\u001a\u0000\u0000\n~k\u0001\n(25)\nh^s?i=X\n~k\f\f\u001a+\u0000\n~k\f\f (26)\nThe components in the \fxed frame are obtained from\nEq. (16)\nh^~ si=1\n2X\n~kX\n\u0016\u00160h~k;\u00160j^~ \u001bj~k;\u0016i\u001a\u0016\u00160\n~k(27)6\nIn this form, the time-dependent states carry the infor-\nmation how the spatial components are described by the\ndensity matrix at each time step. No time-independent\n\\longitudinal\" and \\transverse\" directions can be identi-\n\fed in the \fxed frame.\nIn a similar fashion, the diagonalized single-particle\nstates of the localized spin system are obtained. The\neigenenergies are\nE\u0006=\u0007\f\f~\u0001itin\f\f (28)\nwhere \u0001itin\ni=J1\n2h^siiis the localized energy shift caused\nby the itinerant spin component si. The eigenstates are\nagain always aligned with the itinerant magnetic mo-\nment. In this basis the equation of motion of the localized\nspin-density matrix \u001a\u0017\u00170\nloc\u0011P\n~Kh^Cy\n~K\u0017^C~K\u00170iis simply\n@\n@t\u001a\u0017\u00170\nloc=i\n~(E\u0017\u0000E\u00170)\u001a\u0017\u00170\nloc (29)\nand does not contain explicit exchange contributions.\nEqs. (25), (26), and (27) apply in turn to the components\nhSkiandhS?iof the localized spin and its spin-density\nmatrix\u001a\u0017\u00170\nloc.\nD. Electron-phonon Boltzmann scattering with\nspin splitting\nRelaxation is introduced into the model by the interac-\ntion of the itinerant carriers with a phonon bath, which\nplays the role of an energy and angular momentum sink\nfor these carriers. Our goal here is to present a derivation\nof the Boltzmann scattering contributions using stan-\ndard methods, see, e.g., Refs. 34 and 36. However, we\nemphasize that describing interaction as a Boltzmann-\nlike instantaneous, energy conserving scattering process\nis limited by the existence of the magnetic gap. Since we\nkeep the spin mixing due to Rashba spin-orbit coupling\nonly in the Boltzmann scattering integrals, the resulting\ndynamical equations describe an Elliott-Yafet type spin\nrelaxation.\nThe electron-phonon interaction Hamiltonian reads34\n^He\u0000ph=X\n~ q~!ph\nq^by\n~ q^b~ q\n+X\n~k~k0X\n\u0016\u00160\u0000\nM~k0\u00160\n~k\u0016^cy\n~k\u0016^b~k\u0000~k0^c~k0\u00160+ h.c.\u0001(30)\nwhere ^b(y)\n~ qare the bosonic operators, that create or an-\nnihilate acoustic phonons with momentum ~ qand linear\ndispersion!ph(q) =cphj~ qj. The sound velocity is taken\nto becph= 40 nm/ps and we use an e\u000bectively two-\ndimensional transition matrix element35\nM~k0\u00160\n~k\u0016=Dq\nj~k\u0000~k0jh~k;\u0016j~k0;\u00160i (31)\nwhere the deformation potential is chosen to be D=\n60 meVnm1=2. The scalar-product between the initialstatej~k0;\u00160iand the \fnal state j~k;\u0016iof an electronic\ntransition takes the spin-mixing due to Rashba spin-orbit\ncoupling into account.\nThe derivation of Boltzmann scattering integrals for\nthe itinerant spin-density matrix (24) leads to a memory\nintegral of the following shape\n@\n@t\u001aj(t)\f\f\f\ninc=1\n~X\nj0Zt\n\u00001ei(\u0001Ejj0+i\r)(t\u0000t0)Fjj0[\u001a(t0)]dt0;\n(32)\nregardless whether one uses Green's function36or\nequation-of-motion techniques.34Since we go through a\nstandard derivation here, we highlight only the impor-\ntant parts for the present case and do not write the equa-\ntions out completely. In particular, for scattering process\nj0=j\u00160;~k0i!j=j\u0016;~ki, we useFjj0[\u001a(t0)] as an abbre-\nviation for a product of dynamical electronic spin-density\nmatrix elements \u001a, evaluated at time t0<t, and equilib-\nrium phononic distributions. The corresponding energy\ndi\u000berence is denoted by \u0001 Ejj0=Ej\u0000Ej0\u0006~!ph(j~k\u0000~k0j),\nand\rdescribes the decay of the exponential function due\nto dissipation and/or higher order correlation functions.\nIn general, the integral has to be evaluated numerically\nand contains memory e\u000bects. To apply the Markov ap-\nproximation one needs to compare two time scales: the\n\\memory depth\" 1 =\r, i.e., the time scale on which the\nexp[\u0000\r(t\u0000t0)] factor essentially cuts o\u000b the integral, and\nthe typical time scale on which the Fterm changes. In\nthis paper we deal with relaxation processes not too far\naway from equilibrium, so that the typical time scale of\nthe components of the spin-density matrix contained in\nFis set by the Bloch times T1andT2. We can thus\napproximate Fjj0[\u001a(t0)] byFjj0[\u001a(t)], for all transitions\nlabeled byjandj0, if the memory depth is shorter than\nthe Bloch time(s), or\n\r\u001d1\nT1: (33)\nProvided condition (33) holds, the integral (32) can\nbe done using the Markov approximation Fjj0[\u001a(t0)]'\nFjj0[\u001a(t)]\n@\n@t\u001aj(t)\f\f\f\nincoh=i\n~X\nj0Fjj0[\u001a(t)]1\n\u0001Ejj0+i~\r:(34)\nAs it is customary, we neglect in the following the real\npart of the complex energy denominator, which results in\nshifts of the single-particle energies. While these shifts\nmay play an important role in non-Markovian problems\nwith discrete energy levels37, the imaginary parts yield\nthe relaxation contributions that are important for the\npresent paper\n@\n@t\u001aj(t)\f\f\f\nincoh=X\nj0Fjj0[\u001a(t)]~\r\n(\u0001Ejj0)2+ (~\r)2(35)\nAll transitions are thus weighted by a Lorentzian peaked\nat resonant transitions (\u0001 Ejj0= 0) with a broadening7\nof~\rthat may be interpreted as an energy uncertainty.\nFor relaxation processes in a system with a spin-splitting\n(due to internal \felds and/or spin-orbit coupling), this\nbroadening must not be so large as to blur the distinc-\ntion between the split bands. Consequently, only if the\nbroadening \ris smaller than the magnetic splitting, i.e,,\nif\r\u001c!L, it is possible to distinguish between longi-\ntudinal and transverse components of the spin-density\nmatrix. With Eq. (33) the inequality \r\u001c!Lyields the\ncondition\n!L\u001d1\nT1(36)\nfor the Larmor frequencies and Bloch times, for which\nit is permissible to replace the Lorentzian by an energy\nconserving \u000efunction\n~\r\n(\u0001Ejj0)2+ (~\r)2\r!0\u0000!\u0019\u000e(\u0001Ejj0): (37)This reduces the numerical e\u000bort very considerably, be-\ncause it allows one to eliminate an integration from the\nscattering term, and the energy conserving \u000efunction is\ntherefore often used without explicitly checking its valid-\nity.\nThe considerations leading to the connection between\nEqs. (36) and (37) are a microscopic version of an argu-\nment due to Pines and Slichter,31according to which T1\nandT2can di\u000ber only for correlation times that long in\ncomparison to a Larmor period. The microscopic Boltz-\nmann scattering terms, which contain the energy con-\nserving\u000efunctions and will be used in the following,\ndo not apply in a regime outside of condition (36). If\n!L'1=T1, a \fnite broadening \rhas to be taken into\naccount. Together the full equation of motion in the\nregime (36) for the itinerant-carrier spin density matrix\nthus reads\n@\n@t\u001a\u0016\u00160\n~k=i\n~\u0000\n\u000f\u0016\nk\u0000\u000f\u00160\nk\u0001\n\u001a\u0016\u00160\n~k\n+\u0019\n~X\nk0X\n\u00161\u00162\u00163M~k\u0016\n~k0\u00161M~k0\u00162\n~k\u00163\u000e\u0000\n\u0001E~k0\u00162~k\u00163\u0001h\u0000\n1 +Nph\nj~k0\u0000~kj\u0001\n\u001a\u00161\u00162\n~k0\u0000\n\u000e\u00163\u00160\u0000\u001a\u00163\u00160\n~k\u0001\n\u0000Nph\nj~k0\u0000~kj\u001a\u00163\u00160\n~k\u0000\n\u000e\u00161\u00162\u0000\u001a\u00161\u00162\n~k0\u0001i\n\u0000\u0019\n~X\n~k0X\n\u00161\u00162\u00163M~k\u0016\n~k0\u00161M~k0\u00162\n~k\u00163\u000e\u0000\n\u0001E~k\u00163~k0\u00162\u0001h\u0000\n1 +Nph\nj~k\u0000~k0j\u0001\n\u001a\u00163\u00160\n~k\u0000\n\u000e\u00161\u00162\u0000\u001a\u00161\u00162\n~k0\u0001\n\u0000Nph\nj~k\u0000~k0j\u001a\u00161\u00162\n~k0\u0000\n\u000e\u00163\u00160\u0000\u001a\u00163\u00160\n~k\u0001i\n+\u0019\n~X\n~k0X\n\u00161\u00162\u00163M~k0\u00161\n~k\u00160M~k\u00163\n~k0\u00162\u000e\u0000\n\u0001E~k0\u00162~k\u00163\u0001h\u0000\n1 +Nph\nj~k0\u0000~kj\u0001\n\u001a\u00162\u00161\n~k0\u0000\n\u000e\u0016\u00163\u0000\u001a\u0016\u00163\n~k\u0001\n\u0000Nph\nj~k0\u0000~kj\u001a\u0016\u00163\n~k\u0000\n\u000e\u00162\u00161\u0000\u001a\u00162\u00161\n~k0\u0001i\n\u0000\u0019\n~X\n~k0X\n\u00161\u00162\u00163M~k0\u00161\n~k\u00160Mk\u00163\n~k0\u00162\u000e\u0000\n\u0001E~k\u00163~k0\u00162\u0001h\u0000\n1 +Nph\nj~k\u0000~k0j\u0001\n\u001a\u0016\u00163\n~k\u0000\n\u000e\u00162\u00161\u0000\u001a\u00162\u00161\n~k0\u0001\n\u0000Nph\nj~k\u0000~k0j\u001a\u00162\u00161\n~k0\u0000\n\u000e\u0016\u00163\u0000\u001a\u0016\u00163\n~k\u0001i\n(38)\nHere, \u0001E~k\u0016~k0\u00160=\u000f\u0016\nk\u0000\u000f\u00160\nk0\u0000~!ph\nj~k\u0000~k0j, andNph\nqis the\noccupation function of a thermalized phonon bath, given\nby a Bose-Einstein distribution\nNph\nq=1\ne\f~!ph\u00001(39)\nwhere\f= 1=(kBTph). The numerical results for the mi-\ncroscopic dynamics in the following sections are obtained\nby numerically solving the equations of motion (29) and\n(38). In the numerical calculations the spin-density ma-\ntrix is transformed to the single-particle basis of the in-\nstantaneous, diagonalized eigenstates (22).\nV. DEPHASING BETWEEN LOCALIZED AND\nITINERANT SPINS: NUMERICAL RESULTS\nSince we are interested in this paper in a comparison\nof the model described above with Landau-Lifshitz and\nGilbert damping, we investigate magnetization dynam-\nics with an initial spin-density matrix that correspondsto a tilting of the spins out of their equilibrium posi-\ntion without changing the kinetic energy of the carriers,\nbecause we need initial conditions that lead to generic\nmagnetization dephasing without carrier heating and the\ncorresponding demagnetization dynamics.\nA. Initial state and spin dynamics\nThus we take as the equilibrium initial state the\nsteady-state that is reached for the coupled spins inter-\nacting with the phonon bath at low temperature ( Tph=\n1 K), as shown in Fig. 4(a). In this equilibrium state\nthe spin density matrix is characterized by shifted Fermi\nfunctions for the distributions \u001a\u0016\u0016\n~k=f(\u000f\u0016\nk\u0000EF;Tph)\nand vanishing coherences \u001a+\u0000\n~k= 0. In particular, the\nsteady-state calculation determines the equilibrium mag-\nnetic gap.\nWe then change the itinerant density matrix to that\ncorresponding to an itinerant spin tilted by \f= 10\u000e\nout of equilibrium, see Fig. 4(b). This initial condition8\nFIG. 4. (a) Localized spin h~Siand itinerant spin h~ siin ther-\nmal equilibrium. (b) Itinerant spins tilted out of equilibrium\nby an angle \f.\n0 1 2 300.51\nk(nm−1)Occupations  \n0 1 2 3−0.500.5\nk(nm−1)Coherences\n  ρ|k|− −\nρ|k|+ +\nℜ(ρ|k|− +)\nℑ(ρ|k|− +)\nFIG. 5. Initial spin density-matrix (occupations and coher-\nences) for itinerant electrons corresponding to a tilt \f= 50\u000e.\nThe deviation from equilibrium occurs only between the Fermi\nwave-vectors of the \\+\" and \\ \u0000\" bands.\nachieves a tilting of the spins without heating and avoids\ngeneric de- and remagnetization dynamics. Microscop-\nically the tilted spin corresponds to the spin density-\nmatrix shown in Fig. 5. The perturbation for distribu-\ntions and coherences exists only between the two Fermi\nwave vectors for the \u0016= + and\u0016=\u0000bands. For smaller\ntilt angles the deviation is much less pronounced.\nFrom this initial condition, both spins start to precess\naround the instantaneous direction, along which the ex-\nchange interaction tries to align them. This direction\nis determined for the itinerant carriers by the localized\nspins, and vice versa. A return back into equilibrium re-\nquires the scattering of itinerant electrons with phonons.\nIf we switch o\u000b spin mixing, the dynamics shown in\nFig. 6(a) result: No angular momentum is exchanged\nwith the phonon bath, the excited system cannot relax\ninto equilibrium and the precession goes on inde\fnitely.\nFig. 6(b) shows the same result including spin-mixed itin-\nerant states. Now angular momentum can be transferred\nfrom the itinerant sub-system into the lattice and the to-\ntal spinh~Si+h~ sichanges. In the presence of spin-orbit\ncoupling, electron-phonon scattering, which is by itself\nspin-diagonal, can return the spin system into equilib-\nrium, characterized by aligned spins and vanishing trans-\nverse components. Since we consider here a small Rashba\n(a) (b) \n5&:P; 5&:P; \nO&:P; O&:P; FIG. 6. Non-equilibrium dynamics of localized (red) and itin-\nerant (blue) spins including electron-phonon scattering with-\nout spin-mixing (a) and within spin-mixing (b).\n0 1 2 30.120.130.14\ntime (ps)|s|0 1 2 3−0.0400.04\ntime (ps)sx\n0 1 2 3−0.0400.04\ntime (ps)sy\n0 1 2 3−0.14−0.13−0.12\ntime (ps)sz\nFIG. 7. Relaxation dynamics of itinerant spins in the \fxed\nframe.\ncoupling, the interaction with the phonon bath removes\nenergy much faster than angular momentum. The car-\nrier temperature therefore stays practically equal to the\nphonon temperature Tphduring the entire relaxation pro-\ncess, and no heat-induced demagnetization processes oc-\ncur. The \fnal magnetization is, however, not necessarily\noriented in the zdirection of the \fxed frame, because\nin the results discussed in the this section there is no\nexternal \feld or anisotropy to induce such an alignment.\nWe plot the resulting dynamics of the itinerant spins\nduring the dephasing process in Fig. 7, which shows that9\n00.511.522.53−0.133−0.132−0.131\ntime(ps)s/bardbl\n00.511.522.5300.010.020.03\ntime(ps)s⊥\nFIG. 8. Relaxation dynamics of itinerant spins in the rotating\nframe. An exponential \ft determines the Bloch decay times\ntoT1= 0:5 ps andT2= 1:0 ps.\nall itinerant spatial components precess, and no spatially\n\fxed component can be considered to be longitudinal.\nFirst, the localized spin turns away from the zdirec-\ntion due to the tilted itinerant spin and subsequently\nboth spin-systems precess around each other because the\nquantization axis of each system changes continuously\ndue to the mutual interaction. During the entire relax-\nation process the absolute value of the itinerant spin is\nconserved to better than 1%. Fig. 8 shows the same\ndynamics in the rotating frame, where the longitudinal\nand transverse dynamics can be seen clearly. Both itin-\nerant components show exponential dynamics, which are\ntherefore well described by decay times T1andT2. If well-\nde\fned decay times exist, one expects a ratio T2=T1= 2\nas long as the length of the spin is conserved. The \ft for\nour numerical results indeed gives 2 T1'T2.\nFigure 9 plots the T1andT2values extracted from the\ndynamics as a function of the strength of the electron-\nphonon coupling, or deformation potential, D. The de-\npendence of the decay times on Dcan be \ft extremely\nwell by a 1=D2relation, which demonstrates the propor-\ntionality of the Bloch decay times T1;2/1=D2. Further,\nthe ratioT2=T1stays equal to 2 for all coupling strengths.\nAs discussed in Sec. IV D about the Markov approxima-\ntion, our microscopic description cannot reach regimes\nwhereT1andT2are indistinguishable and therefore\nequal. However, these results show that for small tilting\nangles, not even a pronounced electron-phonon coupling\nleads to a noticeable deviation from the T2= 2T1be-\nhavior. Because this relation between T1andT2holds,\nthe dynamics in Fig. 8 can be equally well described by\nan Landau-Lifshitz or Gilbert damping term. By \ftting\nthe dephasing time T2\u00191 ps and the Larmor-frequency\n!L\u0019281 ps\u00001, equation (13) yields the corresponding\nGilbert damping parameter \u000biso\u00193:6\u000110\u00003.\n204060801001201401.822.2\nD(meV√nm)T2/T1\n  2040608010012014010−2100102\nD(meV√nm)T1,2(ps)\n  FIG. 9. Top: Longitudinal (black squares) and the transverse\n(blue circles) Bloch decay times vs. electron-phonon coupling\nstrengthD. Two \ft curves /1=D2show thatT1;2/1=D2.\nBottom:The ratio of both decay times remains almost con-\nstantT2=T1\u00192 over the entire range of coupling.\n369  \n0 0.5 1279279.5280280.5281\n1/T2(ps−1)ω(ps−1)\n  \nLL\nGilbert\nMicroscopic\nFIG. 10. Precession frequency with respect to the damping\nstrength in terms of the decay rate 1 =T2.\nB. Precession-frequency shift due to dephasing\nIn the Landau-Lifshitz equation the contributions de-\nscribing, respectively, the precession and the damping are\ncompletely independent, so that the damping constant \u0015\nhas no impact on the precession. By contrast, an increase\nof\u000bin the Gilbert equation does not only increase the\ndephasing rate, it lowers the precession frequency as well.\nIn this section we investigate the change of the preces-\nsion frequency in the microscopic calculation and com-\npare it with the macroscopic descriptions. To this end,\nwe use the o\u000b-diagonal components of the itinerant den-\nsity matrix \u001a+\u0000(t) =P\n~k\u001a+\u0000\n~k(t), which describe the dy-\nnamics of the transverse spin components in the rotating\nframe. The modulus of its Fourier transform j\u001a+\u0000(!)j\nshows a distinct peak, which is exactly at the precession\nfrequency.\nTo compare the precession frequency for di\u000berent\ndamping parameters, we use the dependence on T2, be-\ncause all dephasing parameters can be related to T2for10\nsmall excitations. Fig. 10 plots the Larmor frequency vs.\nthe transverse relaxation rate 1 =T2. The precession pa-\nrameters of the Landau-Lifshitz and the Gilbert equation\nare chosen such that the Larmor frequency in the un-\ndamped limiting case is equal to that of the microscopic\nsimulation. In order to stay within the bounds set by con-\ndition (36), we do not extend the plot in Fig. 10 to higher\ndephasing rates. Fig. 10 shows that the microscopic cal-\nculation yields a reduction of the precession frequency\nwith the damping rate. Although the Gilbert dynamics\nalso show such a reduction, it occurs only at shorter T2.\nAs mentioned above, for the Landau-Lifshitz damping,\nthe frequency is independent of the damping parame-\nter. Even though the change of precession frequency in\nthe microscopic calculation is small, the Landau-Lifshitz\ndamping completely fails to include this e\u000bect. While\nGilbert damping does show a reduction of precession fre-\nquency, it is not at all close to the microscopic calcula-\ntion on the frequency scale considered here. Both phe-\nnomenological damping expressions thus do not repro-\nduce the dependence of the precession frequency on T2.\nEven though the numerical di\u000berences are small, these\ndi\u000berences already occur in the small-excitation regime,\nand may perhaps be detectable.\nC. Dephasing at larger excitation angles\nThe phenomenological Landau-Lifshitz and Gilbert\ndamping contributions describe an exponential decay\nonly for small excitation angles, as studied in the pre-\nvious section. In this section we investigate the e\u000bect of\nlarger excitation angles ( >10\u000e) on the spin dynamics in\nthe microscopic calculation. Apart from this the initial\ncondition of the dynamics is the same as before, in par-\nticular, the itinerant spin is tilted such that the absolute\nvalue of the spin is unchanged.\nFigure 11 shows the time development of the skand\ns?components of the itinerant spin in the rotating frame\nfor an initial tilt angle \f= 140\u000e. While the transverse\ncomponent s?in the rotating frame can be well described\nby an exponential decay, the longitudinal component sk\nshows a di\u000berent behavior. It initially decreases with a\ntime constant of less than 1 ps, but does not reach its\nequilibrium value. Instead, the eventual return to equi-\nlibrium takes place on a much longer timescale, during\nwhich the s?component is already vanishingly small.\nThe long-time dynamics are therefore purely collinear.\nFor the short-time dynamics, the transverse component\ncan be \ft well by an exponential decay, even for large ex-\ncitation angles. This behavior is di\u000berent from Landau-\nLifshitz and Gilbert dynamics, cf. Fig. 2, which both ex-\nhibit non-exponential decay of the transverse spin com-\nponent.\nIn Fig. 12 the dependence of T2on the excitation an-\ngle is shown. From small \fup to almost 180\u000e, the decay\ntime decreases by more than 50%. This dependence is\nexclusively due to the \\excitation condition,\" which in-\n0 1 2 3 4−0.100.1\ntime (ps)s/bardbl\n0 1 2 3 400.050.1\ntime (ps)s⊥FIG. 11. Dynamics of the longitudinal and transverse itiner-\nant spin components in the rotating frame (solid lines) for a\ntilt angle of \f= 140\u000e, together with exponential \fts toward\nequilibrium (dashed lines). The longitudinal equilibrium po-\nlarization is shown as a dotted line.\nvolves only spin degrees of freedom (\\tilt angle\"), but no\nchange of temperature. Although one can \ft such a T2\ntime to the transverse decay, the overall behavior with\nits two stages is, in our view, qualitatively di\u000berent from\nthe typical Bloch relaxation/dephasing picture.\nTo highlight the similarities and di\u000berences from the\nBloch relaxation/dephasing we plot in Fig. 13 the mod-\nulus of the itinerant spin vector j~ sjin the rotating\nframe, whose transverse and longitudinal components\nwere shown in Fig. 11. Over the 2 ps, during which the\ntransverse spin in the rotating frame essentially decays,\nthe modulus of the spin vector undergoes a fast initial\ndecrease and a partial recovery. The initial length of ~ s\nis recovered only over a much larger time scale of several\nhundred picoseconds (not shown). Thus the dynamics\ncan be seen to di\u000ber from a Landau-Lifshitz or Gilbert-\nlike scenario because the spin does not precess toward\nequilibrium with a constant length. Additionally they\ndi\u000ber from Bloch-like dynamics because there is a com-\nbination of the fast and slow dynamics that cannot be\ndescribed by a single set of T1andT2times. We stress\nthat the microscopic dynamics at larger excitation angles\nshow a precessional motion of the magnetization with-\nout heating and a slow remagnetization. This scenario is\nsomewhat in between typical small angle-relaxation, for\nwhich the modulus of the magnetization is constant and\nwhich is well described by Gilbert and Landau-Lifshitz\ndamping, and collinear de/remagnetization dynamics.\nVI. EFFECT OF ANISOTROPY\nSo far we have been concerned with the question\nhow phenomenological equations describe dephasing pro-\ncesses between itinerant and localized spins, where the11\n0 50 100 1500.40.60.81\nβ(◦)T2(ps)\nFIG. 12.T2time extracted from exponential \ft to s?dynam-\nics in rotating frame for di\u000berent initial tilting angles \f.\n0 0.5 1 1.5 20.040.060.080.10.120.14\ntime (ps)|s|\n  \n10°\n50°\n90°\n140°\nFIG. 13. Dynamics of the modulus j~ sjof the itinerant spin\nfor di\u000berent initial tilt angles \f. Note the slightly di\u000berent\ntime scale compared to Fig. 11.\nmagnetic properties of the system were determined by a\nmean-\feld exchange interaction only. Oftentimes, phe-\nnomenological models of spin dynamics are used to de-\nscribe dephasing processes toward an \\easy axis\" deter-\nmined by anisotropy \felds.29\nIn order to capture in a simple fashion the e\u000bects of\nanisotropy on the spin dynamics in our model, we sim-\nply assume the existence of an e\u000bective anisotropy \feld\n~Haniso, which enters the Hamiltonian via\n^Haniso =\u0000g\u0016B\u0016^~ s\u0001~Haniso (40)\nand only acts on the itinerant carriers. Its strength is\nassumed to be small in comparison to the \feld of the\nlocalized moments ~Hloc. This additional \feld ~Haniso has\nto be taken into account in the diagonalization of the\ncoherent dynamics as well, see section IV C.\nFor the investigation of the dynamics with anisotropy,\nwe choose a slightly di\u000berent initial condition, which is\nshown in Fig. 14. In thermal equilibrium, both spins\nare now aligned, with opposite directions, along the\nanisotropy \feld ~Haniso, which is assumed to point in the\nzdirection. At t= 0 they are both rigidly tilted by an\n5&:P; \nO&:P; U T \nV E *_lgqm FIG. 14. Dynamics of the localized spin ~Sand itinerant spin\n~ s. Att= 0, the equilibrium con\fguration of both spins is\ntilted (\f= 10\u000e) with respect to an anisotropy \feld ~Haniso.\nThe anisotropy \feld is only experienced by the itinerant sub-\nsystem.\n01002003004005006000.490.4950.5\ntime(ps)Sz(t)\n010020030040050060000.050.1\ntime(ps)/radicalBig\nS2x(t)+S2y(t)\n  \nFIG. 15. Relaxation dynamics of the localized spin toward the\nanisotropy direction for longitudinal component Szand the\ntransverse componentp\nS2x+S2y. An exponential \ft yields\nBloch decay times of Taniso\n1 = 67:8 ps andTaniso\n2 = 134:0 ps.\nangle\f= 10\u000ewith respect to the anisotropy \feld.\nFigure 14 shows the time evolution of both spins in the\n\fxed frame, with zaxis in the direction of the anisotropy\n\feld for the same material parameters as in the previous\nsections and an anisotropy \feld ~Haniso =\u0000108A\nm\u0001~ ez.\nThe dynamics of the entire spin-system are somewhat\ndi\u000berent now, as the itinerant spin precesses around the\ncombined \feld of the anisotropy and the localized mo-\nments. The localized spin precesses around the itinerant\nspin, whose direction keeps changing as well.\nFigure 15 contains the dynamics of the components\nof the localized spin in the rotating frame. Both com-\nponents show an exponential behavior that allows us to\nextract well de\fned Bloch-times Taniso\n1 andTaniso\n2. Again\nwe \fnd the ratio of 2 Taniso\n1\u0019Taniso\n2, because the abso-\nlute value of the localized spin does not change, as it is\nnot coupled to the phonon bath.\nIn Fig. 16 the Larmor-frequency !aniso\nL, which is the\nprecession frequency due to the anisotropy \feld, and the\nBloch decay times Taniso\n2 are plotted vs. the strength of\nthe anisotropy \feld ~Haniso. The Gilbert damping pa-12\n0 5 10 150510\nHaniso(107A/m)ωaniso\nL (ps−1)\n0 5 10 1505001000\nHaniso(107A/m)Taniso\n2 (ps)\n0 5 10 1501020\nHaniso(107A/m)αaniso (10−4)\nFIG. 16. Larmor frequency !aniso\nL and Bloch decay time Taniso\n2\nextracted from the spin dynamics vs. anisotropy \feld Haniso,\nas well as the corresponding damping parameter \u000baniso.\nrameter\u000baniso for the dephasing dynamics computed via\nEq. (13) is also presented in this \fgure.\nThe plot reveals a decrease of the dephasing time Taniso\n2\nand a almost linear increase of the Larmor frequency\n!aniso\nL with the strength of the anisotropy \feld Haniso.\nThe Gilbert damping parameter \u000baniso shows only a neg-\nligible dependence on the anisotropy \feld Haniso. This\ncon\frms the statement that, in contrast to the dephas-\ning rates, the Gilbert damping parameter is independent\nof the applied magnetic \feld. In the investigated range\nwe \fnd an almost constant value of \u000baniso'9\u000210\u00004.\nThe Gilbert damping parameter \u000baniso for the de-\nphasing toward the anisotropy \feld is about 4 times\nsmaller than \u000biso, which describes the dephasing between\nboth spins. This disparity in the damping e\u000eciency\n(\u000baniso< \u000b iso) is obviously due to a fundamental di\u000ber-\nence in the dephasing mechanism. In the anisotropy case\nthe localized spin dephases toward the zdirection with-\nout being involved in scattering processes with itinerant\ncarriers or phonons. The dynamics of the localized spins\nis purely precessional due to the time-dependent mag-\nnetic moment of the itinerant carriers ~Hitin(t). Thus,\nonly this varying magnetic \feld, that turns out to be\nslightly tilted against the localized spins during the en-\ntire relaxation causes the dephasing, in presence of the\ncoupling between itinerant carriers and a phonon bath,\nwhich acts as a sink for energy and angular momentum.\nThe relaxation of the localized moments thus occurs only\nindirectly as a carrier-meditated relaxation via their cou-\npling to the time dependent mean-\feld of the itinerant\nspin.\nNext, we investigate the dependence of the Gilbert pa-\nrameter\u000baniso on the bath coupling. Fig. 17 shows that\n0 50 100 150 20000.0040.0080.012\nD(meV√nm)αaniso\n  FIG. 17. Damping parameter \u000baniso vs. coupling constant D\n(black diamonds). The red line is a quadratic \ft, indicative\nof\u000baniso/D2.\n\u000baniso increases quadratically with the electron-phonon\ncoupling strength D.\nSince Fig. 9 establishes that the spin-dephasing rate\n1=T2for the fast dynamics discussed in the previous sec-\ntions, is proportional to D2, we \fnd\u000baniso/1=T2. We\nbrie\ry compare these trends to two earlier calculations\nof Gilbert damping that employ p-dmodels and assume\nphenomenological Bloch-type rates 1 =T2for the dephas-\ning of the itinerant hole spins toward the \feld of the\nlocalized moments. In contrast to the present paper, the\nlocalized spins experience the anisotropy \felds. Chovan\nand Perakis38derive a Gilbert equation for the dephasing\nof the localized spins toward the anisotropy axis, assum-\ning that the hole spin follows the \feld ~Hlocof the localized\nspins almost adiabatically. Tserkovnyak et al.39extract\na Gilbert parameter from spin susceptibilities. The re-\nsulting dependence of the Gilbert parameter \u000baniso on\n1=T2in both approaches is in qualitative accordance and\nexhibits two di\u000berent regimes. In the the low spin-\rip\nregime, where 1 =T2is small in comparison to the p-dex-\nchange interaction a linear increase of \u000baniso with 1=T2\nis found, as is the case in our calculations with micro-\nscopic dephasing terms. If the relaxation rate is larger\nthan thep-ddynamics,\u000baniso decreases again. Due to\nthe restriction (36) of the Boltzmann scattering integral\nto low spin-\rip rates, the present Markovian calculations\ncannot be pushed into this regime.\nEven though the anisotropy \feld ~Haniso is not cou-\npled to the localized spin ~Sdirectly, both spins precess\naround the zdirection with frequency !aniso\nL. In analogy\nto Sec. V B we study now the in\ruence of the damping\nprocess on the precession of the localized spin around\nthe anisotropy axis and compare it to the behavior of\nLandau-Lifshitz and Gilbert dynamics. Fig. 18 reveals a\nsimilar behavior of the precession frequency as a function\nof the damping rate 1 =Taniso\n2 as in the isotropic case. The\nmicroscopic calculation predicts a distinct drop of the\nLarmor frequency !aniso\nL for a range of dephasing rates\nwhere the precession frequency is unchanged according\nto the Gilbert and Landau-Lifshitz damping models. Al-\nthough Gilbert damping eventually leads to a change in\nprecession frequency for larger damping, this result shows\na qualitative di\u000berence between the microscopic and the13\n0 0.02 0.04 0.06 0.087.127.167.27.24\n1/Taniso\n2(ps−1)ωaniso(ps−1)\n  \nGilbert\nLL\nMicroscopic\nFIG. 18. Precession frequency of the localized spin around\nthe anisotropy \feld vs. Bloch decay time 1 =Taniso\n2.\nphenomenological calculations.\nVII. CONCLUSION AND OUTLOOK\nIn this paper, we investigated a microscopic descrip-\ntion of dephasing processes due to spin-orbit coupling\nand electron-phonon scattering in a mean-\feld kinetic\nexchange model. We \frst analyzed how spin-dependent\ncarrier dynamics can be described by Boltzmann scat-\ntering integrals, which leads to Elliott-Yafet type relax-\nation processes. This is only possible for dephasing rates\nsmall compared to the Larmor frequency, see Eq. (36).\nThe microscopic calculation always yielded Bloch times\n2T1=T2for low excitation angles as it should be due\nto the conservation of the absolute value of the mag-\nnetization. A small decrease of the e\u000bective precession\nfrequency occurs with increasing damping rate, which is\na fundamental di\u000berence to the Landau-Lifshitz descrip-\ntion and exceeds the change predicted by the Gilbert\nequation in this regime.We modeled two dephasing scenarios. First, a relax-\nation process between both spin sub systems was studied.\nHere, the di\u000berent spins precess around the mean-\feld of\nthe other system. In particular, for large excitation an-\ngles we found a decrease of the magnetization during the\nprecessional motion without heating and a slow remag-\nnetization. This scenario is somewhat in between typi-\ncal small angle-relaxation, for which the modulus of the\nmagnetization is constant and which is well described\nby Gilbert and Landau-Lifshitz damping, and collinear\nde/remagnetization dynamics. Also, we \fnd important\ndeviations from a pure Bloch-like behavior.\nThe second scenario deals with the relaxation of the\nmagnetization toward a magnetic anisotropy \feld expe-\nrienced by the itinerant carrier spins for small excitation\nangles. The resulting Gilbert parameter \u000baniso is inde-\npendent of the static anisotropy \feld. The relaxation of\nthe localized moments occurs only indirectly as a carrier-\nmeditated relaxation via their coupling to the time de-\npendent mean-\feld of the itinerant spin.\nTo draw a meaningful comparison with Landau-\nLifshitz and Gilbert dynamics we restricted ourselves\nthroughout the entire paper to a regime where the elec-\ntronic temperature is equal to the lattice temperature Tph\nat all times. In general our microscopic theory is also ca-\npable of modeling heat induced de- and remagnetization\nprocesses. We intend to compare microscopic simulations\nof hot electron dynamics in this model, including scat-\ntering processes between both types of spin, with phe-\nnomenological approaches such as the Landau-Lifshitz-\nBloch (LLB) equation or the self-consistent Bloch equa-\ntion (SCB)40.\nWe \fnally mention that we derived relation (13) con-\nnecting the Bloch dephasing time T2and the Gilbert\ndamping parameter \u000b. Despite its simplicity and obvious\nusefulness, we were not able to \fnd a published account\nof this relation.\n\u0003baral@rhrk.uni-kl.de\nyhcsch@physik.uni-kl.de\n1L. Landau and E. Lifshitz, Phys. Z. Sowj. 8, 153 (1935).\n2T. L. Gilbert, IEEE Trans. Magnetics, 40, 6 (2004).\n3D. L. Mills and R. Arias The damping of spin motions\nin ultrathin \flms: Is the Landau-Lifschitz-Gilbert phe-\nnomenology applicable? Invited paper presented at the\nVII Latin American Workshop on Magnetism, Magnetic\nMaterials and their Applications, Renaca, Chile, 2005.\n4D. A. Garanin, Physica A 172, 470 (1991).\n5V. V. Andreev and V. I. Gerasimenko, Sov. Phys. JETP\n35, 846 (1959).\n6A. Brataas, A. Tserkovnyak, and G. E. W. Bauer, Phys.\nRev. Lett. 101, 037207 (2008).\n7W. M. Saslow, J. Appl. Phys. 105, 07D315 (2009).\n8N. Smith, Phys. Rev. B 78, 216401 (2008).\n9S. Zhang and S.-L. Zhang, Phys. Rev. Lett. 102, 086601(2009).\n10S. Wienholdt, D. Hinzke, K. Carva. P. M. Oppeneer and\nU. Nowak, Phys. Rev. B 88, 020406(R) (2013).\n11U. Atxitia, O. Chubykalo-Fesenko, R. W. Chantrell,\nU. Nowak and A. Rebei, Phys. Rev. Lett. 102, 057203\n(2009).\n12Z. Li and S. Zhang, Phys. Rev. B 69, 134416 (2004).\n13O. Chubykalo-Fesenko, Appl. Phys. Lett. 91, 232507\n(2007).\n14D. A. Garanin, Phys. Rev. B 55, 3050-3057 (1997).\n15U. Atxitia, and O. Chubykalo-Fesenko, Phys. Rev. B 84,\n144414 (2011).\n16I. Garate and A. MacDonald, Phys. Rev. B 79, 064403\n(2009)\n17K. Gilmore, I. Garate, A. H. MacDonald, and M. D. Stiles,\nPhys. Rev. B 84, 224412 (2011)\n18J. Sinova, T. Jungwirth, X. Liu, Y. Sasaki, J. K. Furdyna,14\nW. A. Atkinson, and A. H. MacDonald, Phys. Rev. B 69,\n085209 (2004).\n19K. Shen, G. Tatara, and M. W. Wu, Phys. Rev. B 81,\n193201 (2010)\n20K. Shen and M. W. Wu, Phys. Rev. B 85, 075206 (2012)\n21T. Jungwirth, J. Sinova, J. Masek, J. Kucera and\nA. H. MacDonald, Rev. Mod. Phys. 78, 809 (2006).\n22T. Dietl, Nature Mat. 9, 2898 (2010).\n23J. Koenig, H.-H. Lin and A. H. MacDonald, Phys. Rev.\nLett. 84, 5628 (2000).\n24J. K onig, T. Jungwirth and A. H. MacDonald, Phys. Rev.\nB64, 184423 (2001).\n25A. Kirilyuk, A. V. Kimel, and T. Rasing, Rev. Mod.\nPhys. 82, 2731 (2010).\n26S. Mangin, M. Gottwald, C-H. Lambert, D. Steil, V. Uhlr,\nL. Pang, M. Hehn, S. Alebrand, M. Cinchetti, G. Mali-\nnowski, Y. Fainman, M. Aeschlimann and E. E. Fullerton,\nNature Materials 13, 286 (2014).\n27F. Bloch, Phys. Rev. 70, 460 (1946)\n28N. Bloembergen, Phys. Rev. 78, 572 (1950)\n29S. V. Vonsovskii, Ferromagnetic Resonance (Pergamon,\nOxford, 1966).\n30D. D. Stancil and A. Prabhakar, Spin Waves , (Springer,New York, 2009).\n31D. Pines and C. Slichter, Phys. Rev. 100, 1014 (1955).\n32C. L u, J. L. Cheng, M. W. Wu and I. C. da Cunha Lima,\nPhys. Lett. A 365, 501 (2007).\n33M. Leontiadou, K. L. Litvinenko, A. M. Gilbertson,\nC. R. Pidgeon, W. R. Branford, L. F. Cohen, M. Fearn,\nT. Ashley, M. T. Emeny, B. N. Murdin and S. K. Clowes,\nJ. Phys.: Condens. Matter 23, 035801 (2011).\n34H. Haug and S. W. Koch, Quantum Theory of the Optical\nand Electronic Properties of Semiconductors , 4. ed (World\nScienti\fc, Singapore, 2003).\n35K. Kaasbjerg, K. S. Thygesen and A.-P. Jauho, Phys. Rev.\nB87, 235312 (2013).\n36F. T. Vasko and O. E. Raichev, Quantum Kinetic Theory\nand Applications , (Springer, New York, 2005).\n37H. C. Schneider, W. W. Chow, and S. W. Koch, Phys.\nRev. B 70, 235308 (2004).\n38J. Chovan and I. E. Perakis, Phys. Rev. B 77, 085321\n(2008).\n39Y. Tserkovnyak, G. A. Fiete, and B. I. Halperin, Appl.\nPhys. Lett. 84, 5234 (2004).\n40L. Xu and S. Zhang, J. Appl. Phys. 113, 163911 (2013)."    },    {        "title": "1405.2842v3.Global_Existence_and_Nonlinear_Diffusion_of_Classical_Solutions_to_Non_Isentropic_Euler_Equations_with_Damping_in_Bounded_Domain.pdf",        "content": "arXiv:1405.2842v3  [math.AP]  9 Jul 2014Global Existence and Nonlinear Diffusion of Classical\nSolutions to Non-Isentropic Euler Equations with\nDamping in Bounded Domain\nFuzhou Wu∗\nMathematical Sciences Center, Tsinghua University\nBeijing 100084, China\nAbstract\nWe considered classical solutions to the initial boundary value\nproblem for non-isentropiccompressible Euler equations with damp-\ning in multi-dimensions. We obtained global a priori estimates and\nglobal existence results of classical solutions to both non-isentro pic\nEuler equations with damping and their nonlinear diffusion equa-\ntions under small data assumption. We proved the pressure and\nvelocity decay exponentially to constants, while the entropy and\ndensity can not approach constants. Finally, we proved the press ure\nand velocity of the non-isentropic Euler equations with damping\nconverge exponentially to those of their nonlinear diffusion equa-\ntions when the time goes to infinity.\nKeywords : non-isentropic Euler equation with damping, global exis-\ntence, equilibrium states, Darcy’s law, nonlinear diffusio n\nContents\n1 Introduction 2\n2 Preliminaries and Precise Statements of Main Results 6\n3 Global A Priori Estimates for Non-Isentropic Euler Equations with\nDamping 12\n4 GlobalExistenceandEquilibriumStatesofNon-IsentropicEulerEq ua-\ntions with Damping 29\n5 Global A Priori Estimates for Diffusion Equations 37\n6 Darcy’s Law and Nonlinear Diffusion of Non-Isentropic Euler Equa-\ntions with Damping 53\nReferences 56\n∗E-mail: michael8723@gmail.com; fuzhou.wu@yahoo.com\n11 Introduction\nIn this paper, we consider classical solutions to IBVP for non-isent ropic\ncompressible Euler equations with damping in three dimensions:\n\n\n̺t+u·∇̺+̺∇·u= 0,\n̺ut+̺u·∇u+∇p+a̺u= 0,\nSt+u·∇S= 0,\n(̺,u,S)(x,0) = (̺0(x),u0(x),S0(x)),\nu·n|∂Ω= 0,∀t≥0,(1.1)\nwhere̺,u,S,p denotes the density, velocity, entropy and pressure of ideal gas es,\nrespectively. The friction coefficient a >0, Ω⊂R3is a bounded domain with\nsmooth boundary ∂Ω. The physical model of the equations (1 .1) is the non-\nisentropic flow of the ideal gases in porous media, for which the pres sure law\nreads\np=A̺γeS, (1.2)\nwhereA >0,γ=Cp\nCV>1 are constants.\nAs long as ( ̺,p,v,S) in (1.1) remain classical, IBVP (1 .1) are equivalent\nto the following IBVP, where the first two equations can be symmetr ized.\n\n\npt+u·∇p+γp∇·u= 0,\nut+u·∇u+1\n̺∇p+au= 0,\nSt+u·∇S= 0,\n(p,u,S)(x,0) = (p0(x),u0(x),S0(x)),\nu·n|∂Ω= 0,∀t≥0,(1.3)\nwhere̺=̺(p,S) :=1\nγ√\nAp1\nγexp{−S\nγ}.\nThere is a huge literature about the compressible Euler equations wit h\ndamping, we introduce these results as follows:\nAs to the isothermal compressible Euler equations with damping:\n/braceleftBigg̺t+∇·(̺u) = 0,\n̺ut+̺u·∇u+ ¯σ2∇̺+a̺u= 0,(1.4)\nwhere ¯σ2=Rθ∗is constant. The equations (1 .4) describe the isothermal flow\nof ideal gases in porous media. Zhao (see [19]) proved the global ex istence of\nclassical solutions to IBVP for (1 .4) with small data. For BV solutions, see\n[3, 9]. For entropy weak solutions, see [8, 19].\nAs to the isentropic compressible Euler equations with damping:\n/braceleftBigg̺t+∇·(̺u) = 0,\n̺ut+̺u·∇u+∇p+a̺u= 0,(1.5)\n2withp(ρ) =Aργ, Sideris, Thomases and Wang (see [16]) proved the global\nexistence of classical solutions to 3D Cauchy problem for (1 .5) under small data\nassumption. They also proved the singularity formation of classical solutions\nfor a class of large data. Pan and Zhao (see [13]) proved the global e xistence\nand exponential decay of classical solutions to 3D IBVP for (1 .5) under small\ndata assumption, verified the Darcy law when the total mass of the diffusion\nequations equals the total mass of IBVP (1 .5). Due to the boundary conditions\n∂ℓ\ntu·n|∂Ω= 0 butDαu·n|∂Ωmay not be zero, the a priori estimates for IBVP\n(see [13]) are more complicated than those for Cauchy problem (see [16]).\nAll the variables in the isothermal case (1 .4) and the isentropic case (1 .5)\nhave diffusion property, which approach constants when the time g oes to in-\nfinity. While the entropy and density of the non-isentropic Euler equ ations\nwith damping (1 .1) are transported in Eulerian coordinates, which bring main\ndifficulties for the non-isentropic Euler equations with damping.\nIn ([4],[5],[6],[7], [11],[12],[14],[20]), the authors applied characteristicsanal-\nysis together with energy estimate method to study the 1D non-ise ntropic p-\nsystem with damping in Lagrangian coordinates {(y,t)}:\n\n\nVt−uy= 0,\nut+p(V,S)y=−au,\nSt= 0,(1.6)\nwhereV=1\n̺, p(V,S) =AV−γeS. While in Lagrangiancoordinates, the entropy\nS(y,t)≡S0(y), whose transportationis implicit in this coordinates. All vertical\nlines in Lagrangiancoordinates are particle paths of 1D non-isentro pic p-system\nwith dampingand its diffusion system, so that the phenomenain 1DLag rangian\ncoordinates are much simpler.\nAstothenon-isentropicEulerequationswithdampinginmulti-dimensio nal\nEulerian coordinates, the only results in the present are the global existence\nand decay properties of classical solutions to (1 .3) inR3(see [17]) and periodic\ndomainT⊂R3(see [18]). The spectral method and Duhamel’s principle are\napplied in [17] to prove p−¯p,u,Stalgebraically decay and S−¯Sis uniformly\nbounded. Due tothe convenienceofperiodicboundarycondition, s imilarenergy\nestimate method was applied in [18], where p−¯p,u,Stdecay exponentially and\nS−¯Sis uniformly bounded. While the initial boundary value problem is more\ndifficult, due to the boundary conditions ∂ℓ\ntu·n|∂Ω= 0 butDαu·n|∂Ωmay not\nbe zero.\nThe aims of this paper are as follows: (1) to study the long time behav ior\nof classical solutions to the non-isentropic Euler equations with dam ping, such\nas global existence, exponential decay, equilibrium states, singula rity formation.\n(2) to study the long time behavior of classical solutions to the nonlin ear diffu-\nsion equations. (3) to study the relationship between the solutions of the above\ntwo systems when the time is large.\nIn this paper, we assume no vacuum initially, i.e., inf\nx∈Ω̺0>0 or inf\nx∈Ωp0>0,\notherwise the degeneracy aroused by the vacuum brings about ne w difficulties,\nsuch as local existence and behavior of vacuum boundary. Then inf\nΩ×[0,T]̺(x,t)>\n30 and inf\nΩ×[0,T]p(x,t)>0 as long as the solution remains classical in the time\ninterval [0 ,T].\nWe introduce the following constants:\n¯p=/parenleftbigg\n1\n|Ω|/integraltext\nΩp1\nγ\n0dx/parenrightbiggγ\n,¯S=1\nΩ/integraltext\nΩS0dx,¯̺=1\nΩ/integraltext\nΩ̺0dx, (1.7)\nwhere̺0=1\nγ√\nAp1\nγ\n0exp{−S0\nγ}. Thus, we can express the concept of small data\nfor IBVP (1 .3), i.e., the smallness of /ba∇dbl(p0−¯p,u0−0,S0−¯S,̺0−¯̺)/ba∇dblH3(Ω).\nWe proved that if the initial data ( p0,u0,S0)∈H3(Ω) are sufficiently small\nperturbationsoftheirmeanvalues(1\n|Ω|/integraltext\nΩp0dx,0,¯S)or(¯p,0,¯S), thenIBVP(1 .3)\nadmits a unique global classical solution ( p,u,S)∈ ∩\n0≤ℓ≤3Cℓ([0,+∞),H3−ℓ(Ω)),\nmoreover, ̺=̺(p,S)∈ ∩\n0≤ℓ≤3Cℓ([0,+∞),H3−ℓ(Ω)). (p,u) converge exponen-\ntially to (¯ p,0) rather than (1\n|Ω|/integraltext\nΩp0dx,0) ast→+∞, (̺,S) are uniformly\nbounded all the time. Moreover,/summationtext\n0≤ℓ≤3(/ba∇dbl∂ℓ\nt(p−¯p)/ba∇dblH3−ℓ(Ω)+/ba∇dbl∂ℓ\ntu/ba∇dblH3−ℓ(Ω)),\n/summationtext\n1≤ℓ≤3/ba∇dbl∂ℓ\nt̺/ba∇dblH3−ℓ(Ω)and/summationtext\n1≤ℓ≤3/ba∇dbl∂ℓ\ntS/ba∇dblH3−ℓ(Ω)decay exponentially, /ba∇dbl̺−¯̺/ba∇dblH3(Ω)\nand/ba∇dblS−¯S/ba∇dblH3(Ω)are uniformly bounded.\nSince ¯p≤1\n|Ω|/integraltext\nΩp0dx,/ba∇dblp0−¯p/ba∇dblHℓ(Ω)≥ /ba∇dblp0−1\n|Ω|/integraltext\nΩp0ds/ba∇dblHℓ(Ω),ℓ≥0. Thus\nforp0, the smallness of p0−¯pimplies the smallnessof p0−1\n|Ω|/integraltext\nΩp0dx. Therefore,\neven (p0,u0,S0) are small perturbations of (1\n|Ω|/integraltext\nΩp0dx,0,¯S), the pressure pstill\nconverges to ¯ past→+∞.\nIn order to describe the equilibrium states of the global classical so lutions,\nwe introduce the following notations:\n(p∞(x),u∞(x),S∞(x),̺∞(x)) = lim\nt→∞(p(x,t),u(x,t),S(x,t),̺(x,t)).(1.8)\nWedefine S+:= max\nx∈Ω{S0(x)}, S−:= min\nx∈Ω{S0(x)}. Duetothecharacteristic\nboundary u·n|∂Ω= 0, each particle path in Ω ×{t≥0}extends to Ω ×{t=\n+∞}rather than terminating on ∂Ω×{t≥0}, andSis invariant along every\nparticle path, so max\nx∈Ω{S∞(x)}=S+,min\nx∈Ω{S∞(x)}=S−. This is a physical\nexplanation of the transportation of the entropy, but we proved mathematically\nthatS∞/ne}ationslash=constand̺∞/ne}ationslash=const, ifS+/ne}ationslash=S−. Moreover, ( p,u,S,̺) converge\nexponentially to their equilibrium states (¯ p,0,S∞(x),̺∞(x)) in|·|∞norm.\nHowever, the damping effect on the velocity makes the equations (1 .1) or\n(1.3) weakly dissipative, such that it can not prevent the formation of singular-\nities without small data assumption. We proved that for a class of lar ge initial\ndata whose support Supp(p0−¯p,u0,S0−¯S) is away from the boundary ∂Ω,\nthe singularities must form in the interior of ideal gases. These singu larities will\nhave formed before Supp(p−¯p,u,S−¯S) reaches the boundary. Our argument is\nbased on the analysis of the moment M̺(t) =/integraltext\nΩ̺u·xdxand finite propagation\n4speed of the classical solutions, this method can be extended easily to Cauchy\nproblem. However, the finite size of bounded domain Ω can not replac e the\nfinite propagation speed of the solutions in our proof.\nToward a better understanding of the large time behavior and nonlin ear\ndiffusion property of classical solutions to non-isentropic Euler equ ations with\ndamping (1 .3), we study the following nonlinear diffusion equations which are\nobtained by applying Darcy’s law to (1 .3)2,\n\n\npt+u·∇p+γp∇·u= 0,\n1\n̺∇p+au= 0,\nSt+u·∇S= 0,\n(p,S)(x,0) = (ˆp0(x),ˆS0(x)),\nu·n|∂Ω= 0,∀t≥0,(1.9)\nwhere̺=̺(p,S) :=1\nγ√\nAp1\nγexp{−S\nγ}, (ˆp0(x),ˆS0(x)) may be different from\n(p0(x),S0(x)). Here, (1 .9)2is not an evolution equation of u, thusuitself does\nnot need the initial data.\nThe physical model of the equations (1 .9) is the sufficiently slow motion\nof the ideal gases in porous media, Darcy’s law gives the relationship b etween\nthe momentum of ideal gases and the gradient of their pressure. T he system\n(1.9) is essentially a parabolic-hyperbolic system with respect to pandSafter\neliminating u:\n\npt=γp\na̺△p−γp\na̺2∇̺·∇p+1\na̺|∇p|2,\nSt−1\na̺∇p·∇S= 0,\n(p,S)(x,0) = (ˆp0(x),ˆS0(x)),\n∂p\n∂n|∂Ω= 0,∀t≥0,(1.10)\nwhere̺=̺(p,S).\nWe introduce the following constants:\nˆ¯p=/parenleftbigg\n1\n|Ω|/integraltext\nΩˆp1\nγ\n0dx/parenrightbiggγ\n,ˆ¯S=1\nΩ/integraltext\nΩˆS0dx,ˆ¯̺=1\nΩ/integraltext\nΩˆ̺0dx, (1.11)\nwhere ˆ̺0=1\nγ√\nAˆp1\nγ\n0exp{−ˆS0\nγ}. Thus, we can express the concept of small data\nfor IBVP (1 .9), i.e., the smallness of /ba∇dblˆp0−ˆ¯p/ba∇dblH4(Ω)+/ba∇dbl(ˆS0−ˆ¯S,ˆ̺0−ˆ¯̺)/ba∇dblH3(Ω).\nWe proved that if the initial data (ˆ p0,ˆS0)∈H4(Ω)×H3(Ω) are sufficiently\nsmall perturbations of their mean values (1\n|Ω|/integraltext\nΩˆp0dx,ˆ¯S) or (ˆ¯p,ˆ¯S), then IBVP\n(1.9) and (1 .10) admit a unique global classical solution (ˆ p,ˆS) satisfying\n(ˆp,ˆS)∈ ∩\n0≤ℓ≤3Cℓ([0,+∞),H4−ℓ(Ω)×H3−ℓ(Ω)),△ˆp∈C(Ω×[0,+∞)),\nmoreover,\n\n\nˆ̺=̺(ˆp,ˆS)∈ ∩\n0≤ℓ≤3Cℓ([0,+∞),H3−ℓ(Ω)),\nˆu=−1\naˆ̺∇ˆp∈ ∩\n0≤ℓ≤3Cℓ([0,+∞),H3−ℓ(Ω)),∇·ˆu∈C(Ω×[0,+∞)).\n5Then (ˆp,ˆu) converge exponentially to ( ˆ¯p,0) rather than (1\n|Ω|/integraltext\nΩˆp0dx,0) ast→\n+∞, (ˆ̺,ˆS) are uniformly bounded all the time. Moreover,/summationtext\n0≤ℓ≤3(/ba∇dbl∂ℓ\nt(ˆp−\nˆ¯p)/ba∇dblH4−ℓ(Ω)+/ba∇dbl∂ℓ\ntˆu/ba∇dblH3−ℓ(Ω)),/summationtext\n1≤ℓ≤3/ba∇dbl∂ℓ\ntˆ̺/ba∇dblH3−ℓ(Ω)and/summationtext\n1≤ℓ≤3/ba∇dbl∂ℓ\ntˆS/ba∇dblH3−ℓ(Ω)decay\nexponentially, /ba∇dblˆ̺−ˆ¯̺/ba∇dblH3(Ω)and/ba∇dblˆS−ˆ¯S/ba∇dblH3(Ω)are uniformly bounded.\nWe define ˆS+:= sup\nx∈ΩˆS0(x),ˆS−:= inf\nx∈ΩˆS0(x) and denote ( ˆS∞,ˆ̺∞) =\nlim\nt→∞(ˆS,ˆ̺). Along the particle paths determined by ˆ u, the entropy ˆSremains\ninvariant. We also proved mathematically that ˆS∞/ne}ationslash=const,ˆ̺∞/ne}ationslash=const, if\nˆS+/ne}ationslash=ˆS−. Moreover, (ˆ p,ˆu,ˆS,ˆ̺) converge exponentially to their equilibrium\nstates (ˆ¯p,0,ˆS∞(x),ˆ̺∞(x)) in|·|∞norm.\nFurthermore, we proved that if/integraltext\nΩpγ\n0dx=/integraltext\nΩˆpγ\n0dx, then ¯p=ˆ¯pand (p,u) of\nIBVP (1.3) converge exponentially to (ˆ p,ˆu) of IBVP (1 .9), namely, as t→+∞,\n/ba∇dblp−ˆp/ba∇dblH3(Ω)+/ba∇dblu−ˆu/ba∇dblH3(Ω)≤C1exp{−C2t}.\nInLagrangiancoordinates {(y,t)},ifS0(y) =ˆS0(y), thenS∞(y)≡S(y,t)≡\nS0(y) =ˆS0(y)≡ˆS(y,t)≡ˆS∞(y). While in Eulerian coordinates, ˆS∞(x)/ne}ationslash=\nS∞(x),ˆ̺∞(x)/ne}ationslash=̺∞(x) in general, due to the transportation of ˆS,ˆ̺,S,̺. For a\ngivenS0(x), whether there exists ˆS0(x) such that ˆS∞(x) =S∞(x) is still open.\nIf such a ˆS0(x) exists, ( p,u,S,̺) of IBVP (1 .3) converge to (ˆ p,ˆu,ˆS,ˆ̺) of IBVP\n(1.9) in Eulerian coordinates, as t→+∞.\nThe rest of this paper is organized as follows: In Section 2, we refor mulate\nthe equations (1 .3),(1.9) into appropriate forms and state the main results.\nIn Section 3, we prove global a priori estimates for the non-isentr opic Euler\nequations with damping (1 .3). In Section 4, we prove the global existence of\nclassical solutions to (1 .3) and singularity formation for large data. In Section\n5, we prove global a priori estimates for the diffusion equations (1 .9). In Section\n6, we prove the global existence of classical solutions to (1 .9) and the nonlinear\ndiffusion property of (1 .3).\n2 Preliminaries and Precise Statements of Main\nResults\nIn this section, we will reformulate the equations (1 .3),(1.9) into appropri-\nate forms, define some energy quantities and state precisely the m ain results of\nthis paper.\nThe following lemma mainly gives the relationship between p∞(x) and the\ninitial data ( p0(x),S0(x),̺0(x)).\nLemma 2.1.\np∞= ¯p=/parenleftbigg\n1\n|Ω|/integraltext\nΩp1\nγ\n0dx/parenrightbiggγ\n,¯p∈[inf\nx∈Ωp(t),sup\nx∈Ωp(t)]. (2.1)\n6Proof.By (1.1)1and (1.1)3, we have\n(̺exp{S\nγ})t+u·∇(̺exp{S\nγ})+̺exp{S\nγ}∇·u= 0,\nd\ndt/integraltext\nΩ̺exp{S\nγ}dx=−/integraltext\nΩ∇·(̺uexp{S\nγ})dx\n=−/integraltext\n∂Ω̺exp{S\nγ}u·ndSx= 0,\n/integraltext\nΩp1\nγdx=/integraltext\nΩp1\nγ\n0dx.(2.2)\nIn the equilibrium state, ∂t̺∞=u∞=∂tS∞= 0, plug which into the\nequations (1 .1), we have ∇p∞= 0, namely, p∞is a constant. Then\np1\nγ∞|Ω|=/integraltext\nΩp1\nγ\n0dx,\np∞= ¯p=/parenleftbigg\n1\n|Ω|/integraltext\nΩp1\nγ\n0dx/parenrightbiggγ\n.(2.3)\nIf ¯p >sup\nx∈Ωpor ¯p <inf\nx∈Ωp, it contradicts with/integraltext\nΩp1\nγdx=/integraltext\nΩ¯p1\nγdx. Thus,\n¯p∈[inf\nx∈Ωp,sup\nx∈Ωp].\nRemark 2.2. [18]pointed out the pressure pin the periodic domain Tconverges\nto¯p=1\n|T3|/integraltext\nT3p0dx. However, ¯p=/parenleftBigg\n1\n|T3|/integraltext\nT3p1\nγ\n0dx/parenrightBiggγ\nis correct.\nFor the non-isentropic Euler equations with damping (1 .3) together with\ntheirinitialdata( p0,u0,S0,̺0)andconstants ¯ p,¯S,¯̺,weintroducetheconstants:\nk1=/radicalBig\n1\nγ¯̺¯p, k2=/radicalBig\nγ¯p\n¯̺,\ndefine the variables:\nξ=p−¯p, φ=S−¯S, v=1\nk1u, ω=∇×v,(v∞,ω∞) = lim\nt→∞(v,ω).\nIn order to establish the global existence of IBVP (1 .3), we reformulate the\nequations (1 .3) into the following form:\n\n\nξt+k2∇·v=−γk1ξ∇·v−k1v·∇ξ,\nvt+k2∇ξ+av=−k1v·∇v+1\nk1(1\n¯̺−1\n̺)∇ξ,\nφt=−k1v·∇φ,\n(ξ,v,φ)(x,0) = (p0(x)−¯p,1\nk1u0(x),S0(x)−¯S),\nv·n|∂Ω= 0,(2.4)\nwhere̺=̺(ξ,φ) :=1\nγ√\nA(ξ+ ¯p)1\nγexp{−φ+¯S\nγ}.\nIn order to prove the global existence of classical solutions to IBV P (2.4)\nvia the energy method, we define the following energy quantities:\n7Definition 2.3. Define\nE[ξ](t) :=3/summationtext\nℓ=0/ba∇dbl∂ℓ\ntξ(t)/ba∇dbl2\nL2(Ω), E1[ξ](t) :=3/summationtext\nℓ=0/ba∇dbl∂ℓ\ntξ/ba∇dbl2\nL2(Ω)−/integraltext\nΩξ\npξ2\ntttdx,\nE[v](t) :=3/summationtext\nℓ=0/ba∇dbl∂ℓ\ntv(t)/ba∇dbl2\nL2(Ω), E1[v](t) :=3/summationtext\nℓ=0|∂ℓ\ntv|2+/integraltext\nΩ(̺\n¯̺−1)|vttt|2dx,\nE[ξ](t) :=/summationtext\n0≤ℓ+|α|≤3/ba∇dbl∂ℓ\ntDαξ(t)/ba∇dbl2\nL2(Ω),E[v](t) :=/summationtext\n0≤ℓ+|α|≤3/ba∇dbl∂ℓ\ntDαv(t)/ba∇dbl2\nL2(Ω),\nE[φ](t) :=3/summationtext\nℓ=0/ba∇dbl∂ℓ\ntφ(t)/ba∇dbl2\nL2(Ω),E[φ](t) :=/summationtext\n0≤ℓ+|α|≤3/ba∇dbl∂ℓ\ntDαφ(t)/ba∇dbl2\nL2(Ω),\nE1[ω](t) :=/summationtext\n0≤ℓ+|α|≤2/ba∇dbl∂ℓ\ntDαω(t)/ba∇dbl2\nL2(Ω),\nE[ξ,v](t) :=E[ξ](t)+E[v](t),E[ξ,v,φ](t) :=E[ξ,v](t)+E[φ](t).\n(2.5)\nIn order to have classical solutions, even locally in time, the initial dat a\nare required to be compatible with the boundary condition, namely, ∂ℓ\ntu(x,0)·\nn|∂Ω= 0,0≤ℓ≤3, where ∂ℓ\ntu(x,0) can be solved by the equations (1 .3) in\nterms of initial data ( p0,u0,S0).\nThe following theorem states the global existence and large time beh avior\nof classical solutions to IBVP (2 .4) and (1 .3):\nTheorem 2.4. Assume(p0,u0,S0)∈H3(Ω),inf\nx∈Ωp0(x)>0,∂ℓ\ntu(x,0)·n|∂Ω=\n0,0≤ℓ≤3. There exists a sufficiently small number δ1>0, such that if\n/ba∇dbl(p0−¯p,1\nk1u0,S0−¯S)/ba∇dblH3(Ω)≤δ1, then IBVP (2.4)admits a unique global clas-\nsical solution (ξ,v,φ)∈ ∩\n0≤ℓ≤3Cℓ([0,+∞),H3−ℓ(Ω)),moreover, ̺=̺(ξ,φ)∈\n∩\n0≤ℓ≤3Cℓ([0,+∞),H3−ℓ(Ω)). Thus, IBVP (1.3)admits a unique global classi-\ncal solution (p= ¯p+ξ,u=k1v,S=¯S+φ)./summationtext\n0≤ℓ≤3(/ba∇dbl∂ℓ\nt(p−¯p)/ba∇dblH3−ℓ(Ω)+\n/ba∇dbl∂ℓ\ntu/ba∇dblH3−ℓ(Ω)),/summationtext\n0≤ℓ≤2/ba∇dbl∂ℓ\ntω/ba∇dblH2−ℓ(Ω),/summationtext\n1≤ℓ≤3/ba∇dbl∂ℓ\nt̺/ba∇dblH3−ℓ(Ω)and/summationtext\n1≤ℓ≤3/ba∇dbl∂ℓ\ntS/ba∇dblH3−ℓ(Ω)\ndecay exponentially, /ba∇dbl̺−¯̺/ba∇dblH3(Ω)and/ba∇dblS−¯S/ba∇dblH3(Ω)are uniformly bounded.\nFurthermore, S∞(x)∈[S−,S+]exists and is unique, p∞= ¯p,u∞=v∞=\nω∞= 0,̺∞(x) =1\nγ√\nA¯p1\nγexp{−S∞(x)\nγ}. IfS+/ne}ationslash=S−, thenS∞/ne}ationslash=¯S,̺∞/ne}ationslash= ¯̺.\nAst→+∞,(p,u,S,̺)converge to (¯p,0,S∞,̺∞)exponentially in |·|∞norm.\nHowever, the damping effect on the velocity is weakly dissipative, whic h\ncan not prevent the singularity formation without small data assum ption. The\nfollowing theorem states that for a class of large initial data whose s upport\nSupp(p0−¯p,u0,S0−¯S) is away from the boundary ∂Ω, the singularities form\nin the interior of ideal gases.\nTheorem 2.5. Assume 0∈Ω,(p0,u0,S0)∈H3(Ω),inf\nx∈Ωp0(x)>0,h=\ndist{∂Ω,Supp(p0−¯p,u0,S0−¯S)}>0,(p,u,S)∈C1(Ω×[0,τ))is the classical\nsolution to IBVP (1.3)whereτ >0is the lifespan of (p,u,S). Denote\nM̺(t) =/integraltext\nΩ̺u·xdx, B 0=|Diam(Ω)|2/integraltext\nΩ̺0dx,\nB1=3AeS−\n|Ω|γ−1/parenleftbigg/integraltext\nΩ̺0dx/parenrightbiggγ\n−3/integraltext\nΩ¯pdx, r=/radicalBig\n|B1−a2B0\n4|.(2.6)\n8For any fixed Tsatisfying 0< T <min{h\nk2,π\n2B0\nr}, if\nM̺(0)>max{aB0\n1−exp{−aT},aB0\n2+rcot(rT\nB0),\naB0\n2−r+2r\n1−exp{−2rT\nB0},aB0\n2+r},(2.7)\nthenτ≤T.\nFor the diffusion equations (1 .9) with their initial data (ˆ p0,ˆu0,ˆS0,ˆ̺0) and\nconstants ( ˆ¯p,ˆ¯S,ˆ¯̺), we introduce the following constants and variables:\nˆk1=/radicalBig\n1\nγˆ¯̺ˆ¯p,ˆk2=/radicalBig\nγˆ¯p\nˆ¯̺,ˆξ= ˆp−ˆ¯p,ˆφ=ˆS−ˆ¯S,\nˆv=1\nk1ˆu,ˆω=∇׈v,(ˆv∞,ˆω∞) = lim\nt→∞(ˆv,ˆω).\nFor simplicity, we omit the symbol ˆ over all variables and constants in the\nequations, initial data and global a priori estimates, if there is no am biguity,\notherwise we will add the symbol ˆ.\nIn order to establish the global existence of IBVP (1 .9), we reformulate the\nequations (1 .9) into the following form:\n\n\nξt+k2∇·v=−γk1ξ∇·v−k1v·∇ξ,\nk2∇ξ+av=1\nk1(1\n¯̺−1\n̺)∇ξ,\nφt=−k1v·∇φ,\n(ξ,φ)(x,0) = (p0(x)−¯p,S0(x)−¯S),\nv·n|∂Ω= 0,(2.8)\nwhere̺=̺(ξ,φ) :=1\nγ√\nA(ξ+ ¯p)1\nγexp{−φ+¯S\nγ}.\nThe system (2 .8) is still a parabolic-hyperbolic system with respect to ξ\nandφ, which has the following form after eliminating v:\n\n\nξt=γp\na̺△ξ+p\na̺∇ξ·∇φ,\nφt=1\na̺∇ξ·∇φ,\n(ξ,φ)(x,0) = (p0(x)−¯p,S0(x)−¯S),\n∂ξ\n∂n|∂Ω= 0,∀t≥0,(2.9)\nwhere̺=̺(ξ,φ).\nNext, we derive the evolution equations of vfrom (2.8), which is useful for\nproving a priori estimate for E1[ω](t). Apply ∂ito (2.8)1, we get\n(∂iξ)t+k13/summationtext\nν=1vν∂ν(∂iξ)+k13/summationtext\nν=1(∂ivν)∂νξ+k1γ(∂iξ)3/summationtext\nν=1∂νvν\n+k1γp3/summationtext\nν=1∂i∂νvν= 0.(2.10)\n9Plug∂iξ=−ak1̺viinto (2.10), we have\n(̺vi)t+k13/summationtext\nν=1vν·∂ν(̺vi)+k13/summationtext\nν=1(∂ivν)(̺vν)+k1γ̺vi3/summationtext\nν=1∂νvν\n−γp\na3/summationtext\nν=1∂i∂νvν= 0.(2.11)\nPlug̺t=−k13/summationtext\nν=1vν∂ν̺−k1̺3/summationtext\nν=1∂νvνinto (2.11), we obtained the devel-\nopment equation of v:\nvt=k1(1−γ)v(∇·v)−k1v·∇v−k1\n2∇(|v|2)+γp\na̺∇(∇·v).(2.12)\nAdd (2.12) to (2 .8)2, we obtained the following equations where vis com-\npatible with ξandφ, but (2.13)2is not independent of (2 .13)1and (2.13)3.\n\n\nξt+k2∇·v=−γk1ξ∇·v−k1v·∇ξ,\nvt+k2∇ξ+av=1\nk1(1\n¯̺−1\n̺)∇ξ+k1(1−γ)v(∇·v)−k1v·∇v\n−k1\n2∇(|v|2)+γp\na̺∇(∇·v),\nφt=−k1v·∇φ,(2.13)\nwherev=−1\nak1̺∇ξ, ̺=̺(ξ,φ).\nInordertoprovetheglobalexistenceofclassicalsolutionstoIBV P(2.8)and\n(1.9) via the energy method, we define the following energy quantities b esides\nthe energy quantities which have been defined in (2 .5):\nDefinition 2.6. Define\nF[ξ](t) :=/summationtext\n0≤ℓ≤3,ℓ+|α|≤4/ba∇dbl∂ℓ\ntDαξ(t)/ba∇dbl2\nL2(Ω),\nF[v](t) :=E[v](t)+/summationtext\n0≤ℓ≤2,ℓ+|α|≤3/ba∇dbl∂ℓ\ntDα(∇·v(t))/ba∇dbl2\nL2(Ω),\nF[ξ,v](t) :=F[ξ](t)+F[v](t),F[ξ,v,φ](t) :=F[ξ,v](t)+E[φ](t).(2.14)\nIn addition, F[v](t) contains more information about ξthanF[ξ](t) itself.\nAll the definitions of energy quantities in (2 .5), (2.14) are independent of the\nequations and initial data, thus the definitions (2 .5) can be used for IBVP (2 .8).\nIn order to have classical solutions to IBVP (2 .8), we need to improve the\nregularity of the initial data, namely ( p0,S0)∈H4(Ω)×H3(Ω). Also, the\ninitial data are required to be compatible with the boundary condition , namely,\n∂ℓ\nt∇p(x,0)·n|∂Ω= 0,0≤ℓ≤3, where ∂ℓ\nt∇p(x,0) are solved by the equations\n(1.9) in terms of initial data ( p0,S0).\nThe following theorem states that the global existence and large tim e be-\nhavior of classical solutions to IBVP (2 .8) and (1 .9):\nTheorem 2.7. Assume(p0,S0)∈H4(Ω)×H3(Ω),inf\nx∈Ωp0(x)>0and∂ℓ\nt∇p(x,0)·\nn|∂Ω= 0,0≤ℓ≤3. There exists a sufficiently small number δ2>0, such that\n10if/ba∇dblp0−¯p/ba∇dblH4(Ω)+/ba∇dblS0−¯S/ba∇dblH3(Ω)≤δ2, then IBVP (2.8)admits a unique global\nclassical solution (ξ,φ)satisfying\n(ξ,φ)∈ ∩\n0≤ℓ≤3Cℓ([0,+∞),H4−ℓ(Ω)×H3−ℓ(Ω)),△ξ∈C(Ω×[0,+∞)),\nmoreover,\n\n\n̺=̺(ξ,φ)∈ ∩\n0≤ℓ≤3Cℓ([0,+∞),H3−ℓ(Ω)),\nv=−1\nak1̺∇ξ∈ ∩\n0≤ℓ≤3Cℓ([0,+∞),H3−ℓ(Ω)),∇·v∈C(Ω×[0,+∞)).\nThus, IBVP (1.9)admits a unique global classical solution (p= ¯p+ξ,S=\n¯S+φ). Moreover,/summationtext\n0≤ℓ≤3(/ba∇dbl∂ℓ\nt(p−¯p)/ba∇dblH4−ℓ(Ω)+/ba∇dbl∂ℓ\ntu/ba∇dblH3−ℓ(Ω)),/summationtext\n0≤ℓ≤2/ba∇dbl∂ℓ\ntω/ba∇dblH2−ℓ(Ω),\n/summationtext\n1≤ℓ≤3/ba∇dbl∂ℓ\nt̺/ba∇dblH3−ℓ(Ω)and/summationtext\n1≤ℓ≤3/ba∇dbl∂ℓ\ntS/ba∇dblH3−ℓ(Ω)decay exponentially, /ba∇dbl̺−¯̺/ba∇dblH3(Ω)\nand/ba∇dblS−¯S/ba∇dblH3(Ω)are uniformly bounded.\nFurthermore, S∞(x)∈[S−,S+]exists and is unique, p∞= ¯p,u∞=v∞=\nω∞= 0,̺∞(x) =1\nγ√\nA¯p1\nγexp{−S∞(x)\nγ}. IfS+/ne}ationslash=S−, thenS∞/ne}ationslash=¯S,̺∞/ne}ationslash= ¯̺.\nAst→+∞,(p,u,S,̺)converge to (¯p,0,S∞,̺∞)exponentially in |·|∞norm.\nThefollowingtheoremstatesthatthepressureandvelocityofnon -isentropic\nEuler equations with damping have nonlinear diffusion property, they converges\nto the pressure and velocity of the diffusion equations.\nTheorem 2.8. Assume(ˆp,ˆu,ˆS,ˆ̺)are variables of the diffusion equations (1.9)\nand(p,u,S,̺)are variables of non-isentropic Euler equations with dampi ng\n(1.3), the initial data (p0,u0,S0)satisfy the conditions in Theorem 2.4,(ˆp0,ˆS0)\nsatisfy the conditions in Theorem 2.7. If\n/integraltext\nΩp1\nγ\n0dx=/integraltext\nΩˆp1\nγ\n0dx, (2.15)\nthen\n/ba∇dblp−ˆp/ba∇dblH3(Ω)+/ba∇dblu−ˆu/ba∇dblH3(Ω)≤C1exp{−C2t}, (2.16)\nfor some positive C1,C2.\nRemark 2.9. In Eulerian coordinates, starting from (x0,0), the particle path\nχ(t;x0)of non-isentropic Euler equations with damping do not coinc ide with\nˆχ(t;x0)of the diffusion equations, then S∞(x)/ne}ationslash=ˆS∞(x)in general, /ba∇dblS−\nˆS/ba∇dblL2(Ω)+/ba∇dbl̺−ˆ̺/ba∇dblL2(Ω)or|S−ˆS|∞+|̺−ˆ̺|∞may not decay. While this does not\ncontradict with the results in 1D Lagrangian coordinates (s ee [5]). Both χ(t;x0)\nandˆχ(t;x0)in Eulerian coordinates correspond to the same line {(y0,t)|t≥0}\nin Lagrangian coordinates, where y0=x0/integraltext\n0̺0(x)dxifΩ = [0,1]. The entropy\nSandˆSremains constant along vertical lines, so S(y,t)≡ˆS(y,t)≡S0(y)in\nLagrangian coordinates, then (p,u,S,̺)converge to (ˆp,ˆu,ˆS,ˆ̺)in Lagrangian\ncoordinates.\nIn the sequent sections, we will use the following notations: X/lessorsimilarYdenotes\nthe estimate X≤CYfor some implied constant C >0 which may different line\nby line. ( ·)kdenotes a vector in R3, for instance, ωk=δijk∂ivj, whereδijkis\ntotally anti-symmetric tensor such that δ123=δ231=δ312= 1,δ213=δ321=\nδ132=−1, others are 0. ’R.H.S.’ is the abbreviation for ’right hand side’.\n113 Global A Priori Estimates for Non-Isentropic\nEuler Equations with Damping\nIn this section, we derive global a priori estimates for the non-isen tropic\nEuler equations with damping (2 .4).\nThe following lemma indicates ̺−¯̺,̺t,∇̺can be estimated by E[ξ,φ](t).\nLemma 3.1. For any given T∈(0,+∞], if\nsup\n0≤t≤TE[ξ,v,φ](t)≤ǫ,\nwhere0< ǫ≪1, then\nsup\n0≤t≤TE[̺−¯̺](t)/lessorsimilarǫ,sup\n0≤t≤T|̺−¯̺|∞/lessorsimilar√ǫ,\nsup\n0≤t≤T|̺t|∞/lessorsimilar√ǫ,sup\n0≤t≤T|∇̺|∞/lessorsimilar√ǫ.(3.1)\nProof.Since̺=1\nγ√\nAp1\nγexp{−S\nγ}, we have\nsup\n0≤t≤TE[̺−¯̺](t)/lessorsimilarsup\n0≤t≤T(E[ξ](t)+E[φ](t))/lessorsimilarǫ,\nsup\n0≤t≤T|̺−¯̺|∞/lessorsimilarsup\n0≤t≤TE[̺−¯̺](t)1\n2/lessorsimilar√ǫ,\nsup\n0≤t≤T|̺t|∞/lessorsimilarsup\n0≤t≤T(|ξt|∞+|φt|∞)/lessorsimilar√ǫ,\nsup\n0≤t≤T|∇̺|∞/lessorsimilarsup\n0≤t≤T(|∇ξ|∞+|∇φ|∞)/lessorsimilar√ǫ.\nThefollowinglemmainvolvestheHelmholtz-Hodgedecompositionofvec tor\nfileds, which states that ∇vis estimated by ωand∇·v. The proofof this lemma\nis standard (see [2, 13]).\nLemma 3.2. Letv∈Hs(Ω)be a vector satisfying v·n|∂Ω= 0, wherenis the\nunit outer norm of ∂Ω, then/ba∇dblv/ba∇dbls/lessorsimilar/ba∇dblω/ba∇dbls−1+/ba∇dbl∇·v/ba∇dbls−1+/ba∇dblv/ba∇dbls−1.\nLemma 3 .2 and the standard Sobolev’s inequality /ba∇dbl·/ba∇dblL4(Ω)/lessorsimilar/ba∇dbl·/ba∇dblH1(Ω)are\nwidely used to prove a priori estimates in Section 3 and Section 5.\nThe following lemma is an application of Lemma 3 .2, which states that the\nspatial derivatives are bounded by the temporal derivatives and t he vorticity,\nthen the total energy E[ξ,v](t) can be bounded by E[ξ,v](t) andE1[ω](t).\nLemma 3.3. For any given T∈(0,+∞], there exists ǫ0>0which is indepen-\ndent of(ξ0,v0,φ0), such that if\nsup\n0≤t≤TE[ξ,v,φ](t)≤ǫ,\nwhereǫ≪min{1,ǫ0}, then for ∀t∈[0,T],\nE[ξ,v](t)≤c0(E[ξ,v](t)+E1[ω](t)), (3.2)\nfor some c0>0.\n12Proof.By (2.4)2, we get\n∇ξ=−k1̺vt−ak1̺v−k2\n1̺v·∇v,\n/ba∇dbl∇ξ/ba∇dbl2\nL2(Ω)/lessorsimilar/ba∇dblvt/ba∇dbl2\nL2(Ω)+/ba∇dblv/ba∇dbl2\nL2(Ω)+/ba∇dblv·∇v/ba∇dbl2\nL2(Ω).(3.3)\nBy (2.4)1, we get\n∇·v=−1\nk1γp(ξt+k1v·∇ξ),\n/ba∇dbl∇·v/ba∇dbl2\nL2(Ω)/lessorsimilar/ba∇dblξt/ba∇dbl2\nL(Ω)+√ǫE[ξ,v](t),\n/ba∇dbl∇v/ba∇dbl2\nL2(Ω)/lessorsimilar/ba∇dblv/ba∇dbl2\nL2(Ω)+/ba∇dblω/ba∇dbl2\nL2(Ω)+/ba∇dbl∇·v/ba∇dbl2\nL2(Ω)\n/lessorsimilar/ba∇dblξt/ba∇dbl2\nL(Ω)+/ba∇dblω/ba∇dbl2\nL2(Ω)+/ba∇dblv/ba∇dbl2\nL2(Ω)+√ǫE[ξ,v](t).(3.4)\nApply∂tto (3.3)1, we get\n∇ξt=−k1ξtvt−k1̺vtt−ak1ξtv−ak1̺vt−k2\n1∂t(̺v·∇v),\n/ba∇dbl∇ξt/ba∇dbl2\nL2(Ω)/lessorsimilar/ba∇dblvtt/ba∇dbl2\nL2(Ω)+/ba∇dblvt/ba∇dbl2\nL2(Ω)+√ǫE[ξ,v](t).(3.5)\nApply∂tto (3.4)1, we get\n∇·vt=−1\nk1γp2[p(ξtt+k1v·∇ξt+k1vt·∇ξ)−ξ2\nt−k1ξtv·∇ξ],\n/ba∇dbl∇·vt/ba∇dbl2\nL2(Ω)/lessorsimilar/ba∇dblξtt/ba∇dbl2\nL2(Ω)+√ǫE[ξ,v](t),\n/ba∇dbl∇vt/ba∇dbl2\nL2(Ω)/lessorsimilar/ba∇dbl∇·vt/ba∇dbl2\nL2(Ω)+/ba∇dblωt/ba∇dbl2\nL2(Ω)+/ba∇dblvt/ba∇dbl2\nL2(Ω)\n/lessorsimilar/ba∇dblξtt/ba∇dbl2\nL2(Ω)+/ba∇dblωt/ba∇dbl2\nL2(Ω)+/ba∇dblvt/ba∇dbl2\nL2(Ω)+√ǫE[ξ,v](t).(3.6)\nApply∂ttto (3.3)1, we get\n∇ξtt=−k1ξttvt−2k1ξtvtt−k1̺vttt−k2\n1(̺v·∇v)tt\n−ak1ξttv−2ak1ξtvt−ak1̺vtt,\n/ba∇dbl∇ξtt/ba∇dbl2\nL2(Ω)/lessorsimilar/ba∇dblvttt/ba∇dbl2\nL2(Ω)+/ba∇dblvtt/ba∇dbl2\nL2(Ω)+√ǫE[ξ,v](t).(3.7)\nApply∂ttto (3.4)1, we get\n∇·vtt=−1\nk1γpξttt+1\nk1γp2ξtξtt−1\nk1γ∂t[1\np(k1v·∇ξt+k1vt·∇ξ)\n−1\np2(ξ2\nt+k1ξtv·∇ξ)],\n/ba∇dbl∇·vtt/ba∇dbl2\nL2(Ω)/lessorsimilar/ba∇dblξttt/ba∇dbl2\nL2(Ω)+√ǫE[ξ,v](t),\n/ba∇dbl∇vtt/ba∇dbl2\nL2(Ω)/lessorsimilar/ba∇dbl∇·vtt/ba∇dbl2\nL2(Ω)+/ba∇dblωtt/ba∇dbl2\nL2(Ω)+/ba∇dblvtt/ba∇dbl2\nL2(Ω)\n/lessorsimilar/ba∇dblξttt/ba∇dbl2\nL2(Ω)+/ba∇dblωtt/ba∇dbl2\nL2(Ω)+/ba∇dblvtt/ba∇dbl2\nL2(Ω)+√ǫE[ξ,v](t).(3.8)\nApplyDαto (3.3)1, where|α|= 1, we get\nDα∇ξ=−k1(Dαξ)vt−k1̺Dαvt−ak1(Dαξ)v−ak1̺Dαv−k2\n1Dα(̺v·∇v),\n/ba∇dblDα∇ξ/ba∇dbl2\nL2(Ω)/lessorsimilar/ba∇dblDαvt/ba∇dbl2\nL2(Ω)+/ba∇dblDαv/ba∇dbl2\nL2(Ω)+√ǫE[ξ,v](t)\n/lessorsimilar/ba∇dblξtt/ba∇dbl2\nL2(Ω)+/ba∇dblωt/ba∇dbl2\nL2(Ω)+/ba∇dblvt/ba∇dbl2\nL2(Ω)+/ba∇dblξt/ba∇dbl2\nL2(Ω)+/ba∇dblω/ba∇dbl2\nL2(Ω)\n+/ba∇dblv/ba∇dbl2\nL2(Ω)+√ǫE[ξ,v](t).\n(3.9)\n13ApplyDαto (3.4)1, where|α|= 1, we get\nDα∇·v=1\nk1γp2(Dαξ)(ξt+k1v·∇ξ)−1\nk1γpDα(ξt+k1v·∇ξ),\n/ba∇dblDα∇·v/ba∇dbl2\nL2(Ω)/lessorsimilar/ba∇dblDαξt/ba∇dbl2\nL2(Ω)+√ǫE[ξ,v](t)\n/lessorsimilar/ba∇dblvt/ba∇dbl2\nL2(Ω)+/ba∇dblvtt/ba∇dbl2\nL2(Ω)+√ǫE[ξ,v](t),\n/ba∇dblDα∇v/ba∇dbl2\nL2(Ω)/lessorsimilar/ba∇dbl∇·v/ba∇dbl2\nH1(Ω)+/ba∇dblω/ba∇dbl2\nH1(Ω)+/ba∇dblv/ba∇dbl2\nH1(Ω)/lessorsimilar/ba∇dblvt/ba∇dbl2\nL2(Ω)+/ba∇dblvtt/ba∇dbl2\nL2(Ω)\n+/ba∇dblω/ba∇dbl2\nH1(Ω)+/ba∇dblv/ba∇dbl2\nL2(Ω)+/ba∇dblξt/ba∇dbl2\nL2(Ω)+√ǫE[ξ,v](t).\n(3.10)\nApplyDαto (3.5)1, where|α|= 1, we get\nDα∇ξt=−k1Dα(ξtvt)−k1(Dαξ)vtt−k1̺Dαvtt−k2\n1Dα(ξtv·∇v)\n−k2\n1Dα(̺vt·∇v)−k2\n1Dα(̺v·∇vt)−ak1Dα(ξtv)\n−ak1(Dαξ)vt−ak1̺Dαvt,\n/ba∇dblDα∇ξt/ba∇dbl2\nL2(Ω)/lessorsimilar/ba∇dblDαvtt/ba∇dbl2\nL2(Ω)+/ba∇dblDαvt/ba∇dbl2\nL2(Ω)+√ǫE[ξ,v](t)\n/lessorsimilar/ba∇dblξttt/ba∇dbl2\nL2(Ω)+/ba∇dblωtt/ba∇dbl2\nL2(Ω)+/ba∇dblvtt/ba∇dbl2\nL2(Ω)+/ba∇dblξtt/ba∇dbl2\nL2(Ω)\n+/ba∇dblωt/ba∇dbl2\nL2(Ω)+/ba∇dblvt/ba∇dbl2\nL2(Ω)+√ǫE[ξ,v](t).(3.11)\nApplyDαto (3.6)1, where|α|= 1, we get\nDα∇·vt=−1\nk1γpDαξtt+1\nk1γp2(Dαξ)ξtt−1\nk1γDα[1\np(k1v·∇ξt+k1vt·∇ξ)\n−1\np2(ξ2\nt+k1ξtv·∇ξ)],\n/ba∇dblDα∇·vt/ba∇dbl2\nL2(Ω)/lessorsimilar/ba∇dblDαξtt/ba∇dbl2\nL2(Ω)+√ǫE[ξ,v](t)\n/lessorsimilar/ba∇dblvttt/ba∇dbl2\nL2(Ω)+/ba∇dblvtt/ba∇dbl2\nL2(Ω)+√ǫE[ξ,v](t),\n/ba∇dblDα∇vt/ba∇dbl2\nL2(Ω)/lessorsimilar/ba∇dbl∇·vt/ba∇dbl2\nH1(Ω)+/ba∇dblωt/ba∇dbl2\nH1(Ω)+/ba∇dblvt/ba∇dbl2\nH1(Ω)+√ǫE[ξ,v](t)\n/lessorsimilar/ba∇dblξtt/ba∇dbl2\nL2(Ω)+/ba∇dblvttt/ba∇dbl2\nL2(Ω)+/ba∇dblvtt/ba∇dbl2\nL2(Ω)+/ba∇dblωt/ba∇dbl2\nH1(Ω)\n+/ba∇dblvt/ba∇dbl2\nL2(Ω)+√ǫE[ξ,v](t).\n(3.12)\nApplyDαto (3.3)1, where|α|= 2,α=α1+α2, we get\nDα∇ξ=−k1̺Dαvt−k1/summationtext\nα1>0(Dα1ξ)Dα2vt−ak1̺Dαv\n−ak1/summationtext\nα1>0(Dα1ξ)Dα2v−k2\n1Dα(̺v·∇v),\n/ba∇dblDα∇ξ/ba∇dbl2\nL2(Ω)/lessorsimilar/ba∇dblDαvt/ba∇dbl2\nL2(Ω)+/ba∇dblDαv/ba∇dbl2\nL2(Ω)+√ǫE[ξ,v](t)\n/lessorsimilar/ba∇dblvttt/ba∇dbl2\nL2(Ω)+/ba∇dblvtt/ba∇dbl2\nL2(Ω)+/ba∇dblωt/ba∇dbl2\nH1(Ω)+/ba∇dblξtt/ba∇dbl2\nL2(Ω)+/ba∇dblvt/ba∇dbl2\nL2(Ω)\n+/ba∇dblω/ba∇dbl2\nH1(Ω)+/ba∇dblξt/ba∇dbl2\nL2(Ω)+/ba∇dblv/ba∇dbl2\nL2(Ω)+√ǫE[ξ,v](t).\n(3.13)\nApplyDαto (3.4)1, where|α|= 2,α=α1+α2, we get\nDα∇·v=−1\nk1γDα1(1\np)Dα2(ξt+k1v·∇ξ),\n14/ba∇dblDα∇·v/ba∇dbl2\nL2(Ω)/lessorsimilar/ba∇dblDαξt/ba∇dbl2\nL2(Ω)+√ǫE[ξ,v](t)\n/lessorsimilar/ba∇dblξttt/ba∇dbl2\nL2(Ω)+/ba∇dblωtt/ba∇dbl2\nL2(Ω)+/ba∇dblvtt/ba∇dbl2\nL2(Ω)+/ba∇dblξtt/ba∇dbl2\nH1(Ω)\n+/ba∇dblωt/ba∇dbl2\nL2(Ω)+/ba∇dblvt/ba∇dbl2\nL2Ω)+√ǫE[ξ,v](t),\n/ba∇dblDα∇v/ba∇dbl2\nL2(Ω)/lessorsimilar/ba∇dbl∇·v/ba∇dbl2\nH2(Ω)+/ba∇dblω/ba∇dbl2\nH2(Ω)+/ba∇dblv/ba∇dbl2\nH2(Ω)\n/lessorsimilar/ba∇dblξttt/ba∇dbl2\nL2(Ω)+/ba∇dblωtt/ba∇dbl2\nL2(Ω)+/ba∇dblvtt/ba∇dbl2\nL2(Ω)+/ba∇dblξtt/ba∇dbl2\nH1(Ω)\n+/ba∇dblωt/ba∇dbl2\nL2(Ω)+/ba∇dblω/ba∇dbl2\nH2(Ω)+/ba∇dblvt/ba∇dbl2\nL2Ω)+/ba∇dblv/ba∇dbl2\nL2Ω)\n+/ba∇dblξt/ba∇dbl2\nL2Ω)+√ǫE[ξ,v](t).(3.14)\nThus,E[ξ,v](t)≤C3E[ξ,v](t)+C3E1[ω](t)+C3√ǫE[ξ,v](t), where C3>0.\nLetǫ0=1\n4C2\n3, whenǫ≪min{1,ǫ0}, we have\nE[ξ,v](t)≤2C3{E[ξ,v](t)+E1[ω](t)}. (3.15)\nLetc0= 2C3. Thus, Lemma 3 .3 is proved.\nNext, in order to prove the exponential decay of E[ξ,v](t) andE1[ω](t), we\nneed to prove a priori estimates for E1[ω](t),E[ξ,v](t) and/integraltext\nΩ3/summationtext\nℓ=1∂ℓ\ntξ∂ℓ−1\ntξdx\nseparatively.\nThe following lemma concerns a priori estimate for E1[ω](t).\nLemma 3.4. For any given T∈(0,+∞], if\nsup\n0≤t≤TE[ξ,v,φ](t)≤ǫ,\nwhere0< ǫ≪1, then for ∀t∈[0,T],\nd\ndtE1[ω](t)+2aE1[ω](t)≤C√ǫE[ξ,v](t). (3.16)\nProof.By∇×(2.4)2, we get\nωt+aω=−k1∇×(v·∇v)+1\nk1∇×[(1\n¯̺−1\n̺)∇ξ]\n=−k1(v·∇ω−ω·∇v+ω∇·v)+1\nk1∇×[(1\n¯̺−1\n̺)∇ξ]\n=−k1(v·∇ω−ω·∇v+ω∇·v)+1\nk1(∂i̺\n̺2∂jξ−∂j̺\n̺2∂iξ)k.(3.17)\nApply∂ℓ\ntDαto (3.17), where ℓ+|α| ≤2, we get\n(∂ℓ\ntDαω)t+a∂ℓ\ntDαω=−k1∂ℓ\ntDα(v·∇ω−ω·∇v+ω∇·v)\n+1\nk1∂ℓ\ntDα(∂i̺\n̺2∂jξ−∂j̺\n̺2∂iξ)k,\n∂t(|∂ℓ\ntDαω|2)+2a|∂ℓ\ntDαω|2=−2k1∂ℓ\ntDα(v·∇ω−ω·∇v+ω∇·v)·∂ℓ\ntDαω\n+2\nk13/summationtext\nk=1∂ℓ\ntDα(∂i̺\n̺2∂jξ−∂j̺\n̺2∂iξ)k∂ℓ\ntDαωk.\n(3.18)\n15After integrating in Ω, we get\nd\ndt/integraltext\nΩ|∂ℓ\ntDαω|2dx+2a/integraltext\nΩ|∂ℓ\ntDαω|2dx=I1+I2,(3.19)\nwhere\nI1:=−2k1/integraltext\nΩ∂ℓ\ntDα(v·∇ω−ω·∇v+ω∇·v)·∂ℓ\ntDαωdx,\nI2:=2\nk1/integraltext\nΩ3/summationtext\nk=1∂ℓ\ntDα(∂i̺\n̺2∂jξ−∂j̺\n̺2∂iξ)k·∂ℓ\ntDαωkdx.(3.20)\nWhenℓ+|α|<2, it is easy to check that I1+I2/lessorsimilar√ǫE[ξ,v](t), since they\nare lower order terms.\nWhenℓ= 0,|α|= 2,\nI1=−2k1/integraltext\nΩ(Dαv·∇ω+Dα1v·∇Dα2ω+v·∇Dαω)·Dαωdx\n+2k1/integraltext\nΩ(Dαω·∇v+Dα1ω·∇Dα2v+ω·∇Dαv)·Dαωdx\n−2k1/integraltext\nΩ(Dαω∇·v+Dα1ω∇·Dα2v+ω∇·Dαv)·Dαωdx\n/lessorsimilar/ba∇dblDαv/ba∇dblL4(Ω)/ba∇dbl∇ω/ba∇dblL4(Ω)/ba∇dblDαω/ba∇dblL2(Ω)+|Dα1v|∞/ba∇dbl∇Dα2ω/ba∇dblL2(Ω)/ba∇dblDαω/ba∇dblL2(Ω)\n−2k1/integraltext\n∂Ωn·v|Dαω|2dSx+2k1/integraltext\nΩ∇·v|Dαω|2dx+|∇v|∞/ba∇dblDαω/ba∇dbl2\nL2(Ω)\n+/ba∇dblDα1ω/ba∇dblL4(Ω)/ba∇dbl∇Dα2v/ba∇dblL4(Ω)/ba∇dblDαω/ba∇dblL2(Ω)+|ω|∞/ba∇dbl∇Dαv/ba∇dblL2(Ω)/ba∇dblDαω/ba∇dblL2(Ω)\n+|∇·v|∞/ba∇dblDαω/ba∇dbl2\nL2(Ω)+/ba∇dblDα1ω/ba∇dblL4(Ω)/ba∇dbl∇·Dα2v/ba∇dblL4(Ω)/ba∇dblDαω/ba∇dblL2(Ω)\n+|ω|∞/ba∇dbl∇·Dαv/ba∇dblL2(Ω)/ba∇dblDαω/ba∇dblL2(Ω)\n/lessorsimilar√ǫE[v](t),\n(3.21)\nwhere|α1|=|α2|= 1.\nWhenℓ= 1,|α|= 1,\nI1=−2k1/integraltext\nΩ(Dαvt·∇ω+vt·∇Dαω+Dαv·∇ωt+v·∇Dαωt)·Dαωtdx\n+2k1/integraltext\nΩ(Dαωt·∇v+ωt·∇Dαv+Dαω·∇vt+ω·∇Dαvt)·Dαωtdx\n−2k1/integraltext\nΩ(Dαωt∇·v+ωt∇·Dαv+Dαω∇·vt+ω∇·Dαvt)·Dαωtdx\n/lessorsimilar/ba∇dblDαvt/ba∇dblL4(Ω)/ba∇dbl∇ω/ba∇dblL4(Ω)/ba∇dblDαωt/ba∇dblL2(Ω)+|vt|∞/ba∇dbl∇Dαω/ba∇dblL2(Ω)/ba∇dblDαωt/ba∇dblL2(Ω)\n+|Dαv|∞/ba∇dbl∇ωt/ba∇dblL2(Ω)/ba∇dblDαωt/ba∇dblL2(Ω)+/integraltext\n∂Ωn·v|Dαωt|2dSx−/integraltext\nΩ∇·v|Dαωt|2dx\n+|∇v|∞/ba∇dblDαωt/ba∇dbl2\nL2(Ω)+|ωt|L4(Ω)/ba∇dbl∇Dαv/ba∇dblL4(Ω)/ba∇dblDαωt/ba∇dblL2(Ω)\n+/ba∇dblDαω/ba∇dblL4(Ω)/ba∇dbl∇vt/ba∇dblL4(Ω)/ba∇dblDαωt/ba∇dblL2(Ω)+|ω|∞/ba∇dbl∇Dαvt/ba∇dblL2(Ω)/ba∇dblDαωt/ba∇dblL2(Ω)\n+|∇·v|∞/ba∇dblDαωt/ba∇dbl2\nL2(Ω)+/ba∇dblωt/ba∇dblL4(Ω)/ba∇dbl∇·Dαv/ba∇dblL4(Ω)/ba∇dblDαωt/ba∇dblL2(Ω)\n+/ba∇dblDαω/ba∇dblL4(Ω)/ba∇dbl∇·vt/ba∇dblL4(Ω)/ba∇dblDαωt/ba∇dblL2(Ω)+|ω|∞/ba∇dbl∇·Dαvt/ba∇dblL2(Ω)/ba∇dblDαωt/ba∇dblL2(Ω)\n/lessorsimilar√ǫE[v](t).\n(3.22)\n16Whenℓ= 2,|α|= 0,\nI1=−2k1/integraltext\nΩ(vtt·∇ω+2vt·∇ωt+v·∇ωtt)·ωttdx\n+2k1/integraltext\nΩ(ωtt·∇v+2ωt·∇vt+ω·∇vtt)·ωttdx\n−2k1/integraltext\nΩ(ωtt∇·v+2ωt∇·vt+ω∇·vtt)·ωttdx\n/lessorsimilar/ba∇dblvtt/ba∇dblL4(Ω)/ba∇dbl∇ω/ba∇dblL4(Ω)/ba∇dblωtt/ba∇dblL2(Ω)+|vt|∞/ba∇dbl∇ωt/ba∇dblL2(Ω)/ba∇dblωtt/ba∇dblL2(Ω)\n+/integraltext\n∂Ωn·v|ωtt|2dSx−/integraltext\nΩ∇·v|ωtt|2dx+|∇v|∞/ba∇dblωtt/ba∇dbl2\nL2(Ω)\n+|ωt|L4(Ω)/ba∇dbl∇vt/ba∇dblL4(Ω)/ba∇dblωtt/ba∇dblL2(Ω)+|ω|∞/ba∇dbl∇vtt/ba∇dblL2(Ω)/ba∇dblωtt/ba∇dblL2(Ω)\n+|∇·v|∞/ba∇dblωtt/ba∇dbl2\nL2(Ω)+/ba∇dblωt/ba∇dblL4(Ω)/ba∇dbl∇·vt/ba∇dblL4(Ω)/ba∇dblωtt/ba∇dblL2(Ω)\n+|ω|∞/ba∇dbl∇·vtt/ba∇dblL2(Ω)/ba∇dblωtt/ba∇dblL2(Ω)\n/lessorsimilar√ǫE[v](t).(3.23)\nWhenℓ= 0,|α|= 2,\nI2=2\nk1/integraltext\nΩ3/summationtext\nk=1[(Dα∂i̺\n̺2)∂jξ−(Dα∂j̺\n̺2)∂iξ]kDαωkdx\n+2\nk1/integraltext\nΩ3/summationtext\nk=1[(Dα1∂i̺\n̺2)(Dα2∂jξ)−(Dα1∂j̺\n̺2)(Dα2∂iξ)]kDαωkdx\n+2\nk1/integraltext\nΩ3/summationtext\nk=1[∂i̺\n̺2(Dα∂jξ)−∂j̺\n̺2(Dα∂iξ)]kDαωkdx\n/lessorsimilar|∇ξ|∞/ba∇dblDα∇̺/ba∇dblL2(Ω)/ba∇dblDαω/ba∇dblL2(Ω)+|∇̺|∞/ba∇dblDα∇ξ/ba∇dblL2(Ω)/ba∇dblDαω/ba∇dblL2(Ω)\n+/ba∇dblDα1∇̺/ba∇dblL4(Ω)/ba∇dblDα2∇ξ/ba∇dblL4(Ω)/ba∇dblDαω/ba∇dblL2(Ω)\n/lessorsimilar√ǫE[ξ,v](t),\n(3.24)\nwhere/ba∇dblDα̺i\n̺2/ba∇dblL2(Ω)/lessorsimilar/ba∇dblDα̺i/ba∇dblL2(Ω),/ba∇dblDα1̺i\n̺2/ba∇dblL2(Ω)/lessorsimilar/ba∇dblDα1̺i/ba∇dblL2(Ω), sinceǫ≪1,\n|α1|=|α2|= 1.\nWhenℓ= 1,|α|= 1,\nI2=2\nk1/integraltext\nΩ3/summationtext\nk=1[∂tDα(∂i̺\n̺2)∂jξ−∂tDα(∂j̺\n̺2)∂iξ+∂t(∂i̺\n̺2)Dα(∂jξ)\n−∂t(∂j̺\n̺2)Dα(∂iξ)+Dα(∂i̺\n̺2)∂t(∂jξ)−Dα(∂j̺\n̺2)∂t(∂iξ)\n+∂i̺\n̺2∂tDα(∂jξ)−∂j̺\n̺2∂tDα(∂iξ)]k∂tDαωkdx\n/lessorsimilar|∇ξ|∞/ba∇dbl∂tDα∇̺\n̺2/ba∇dblL2(Ω)/ba∇dblDαωt/ba∇dblL2(Ω)\n+|∇̺\n̺2|∞/ba∇dbl∂tDα∇ξ/ba∇dblL2(Ω)/ba∇dblDαωt/ba∇dblL2(Ω)\n+/ba∇dbl∂t(∇̺\n̺2)/ba∇dblL4(Ω)/ba∇dblDα∇ξ/ba∇dblL4(Ω)/ba∇dblDαωt/ba∇dblL2(Ω)\n+/ba∇dblDα(∇̺\n̺2)/ba∇dblL4(Ω)/ba∇dbl∇ξt/ba∇dblL4(Ω)/ba∇dblDαωt/ba∇dblL2(Ω)\n/lessorsimilar√ǫE[ξ,v](t).(3.25)\n17Whenℓ= 2,|α|= 0,\nI2=2\nk1/integraltext\nΩ3/summationtext\nk=1[(∂i̺\n̺2)tt∂jξ−(∂j̺\n̺2)tt∂iξ+2(∂i̺\n̺2)t(∂jξ)t−2(∂j̺\n̺2)t(∂iξ)t\n+∂i̺\n̺2(∂jξ)tt−∂j̺\n̺2(∂iξ)tt]k∂ttωkdx\n/lessorsimilar|∇ξ|∞/ba∇dbl(∇̺\n̺2)tt/ba∇dblL2(Ω)/ba∇dblωtt/ba∇dblL2(Ω)+/ba∇dbl(∇̺\n̺2)t/ba∇dblL4(Ω)/ba∇dbl∇ξt/ba∇dblL4(Ω)/ba∇dblωtt/ba∇dblL2(Ω)\n+|∇̺\n̺2|∞/ba∇dbl∇ξtt/ba∇dblL2(Ω)/ba∇dblωtt/ba∇dblL2(Ω)\n/lessorsimilar√ǫE[ξ,v](t).\n(3.26)\nSumming the above estimates for 0 ≤ℓ+|α| ≤2, we have\nd\ndtE1[ω](t)+2aE1[ω](t)/lessorsimilar√ǫE[ξ,v](t). (3.27)\nThus, Lemma 3 .4 is proved.\nThe following lemma states that E[ξ](t) andE1[ξ](t) are equivalent, E[v](t)\nandE1[v](t) are equivalent.\nLemma 3.5. For any given T∈(0,+∞], there exists ǫ1>0which is in-\ndependent of (ξ0,v0,φ0), such that if sup\n0≤t≤TE[ξ,v,φ](t)≤ǫ1, then there exist\nc1>0,c2>0such that\nc1E[ξ](t)≤E1[ξ](t)≤c2E[ξ](t),\nc1E[v](t)≤E1[v](t)≤c2E[v](t).(3.28)\nProof.Since|ξ|∞/lessorsimilar/ba∇dblξ/ba∇dblH2(Ω)≤C4√ǫ1, letC4√ǫ1≤¯p\n3, i.e.ǫ1≤¯p2\n9C2\n4, then\nξ\np=ξ/¯p\n1+ξ/¯p∈[−1\n2,1\n4],E1[ξ](t)∼=E[ξ](t).\nSince|̺−¯̺|∞/lessorsimilar/ba∇dbl̺−¯̺/ba∇dbl1\n2\nH2(Ω)/lessorsimilarE[ξ,φ](t)1\n2≤C5√ǫ1, letC5√ǫ1≤¯̺\n2, i.e.\nǫ1≤¯̺2\n4C2\n5, then̺\n¯̺−1∈[−1\n2,1\n2],E1[v](t)∼=E[v](t).\nThus, we can take ǫ1= min{¯p2\n9C2\n4,¯̺2\n4C2\n5}.\nSinceE1[ξ,v](t)∼=E[ξ,v](t), the following lemma gives an equivalent a\npriori estimate for E[ξ,v](t).\nLemma 3.6. For any given T∈(0,+∞], if\nsup\n0≤t≤TE[ξ,v,φ](t)≤ǫ,\nwhere0< ǫ≪1, then for ∀t∈[0,T],\nd\ndtE1[ξ,v](t)+2aE1[v](t)≤C√ǫE[ξ,v](t). (3.29)\n18Proof.Suppose 0 ≤ℓ≤3, apply ∂ℓ\ntto (2.4), we get\n\n\n(∂ℓ\ntξ)t+k2∇·(∂ℓ\ntv) =−γk1∂ℓ\nt(ξ∇·v)−k1∂ℓ\nt(v·∇ξ),\n(∂ℓ\ntv)t+k2∇(∂ℓ\ntξ)+a∂ℓ\ntv=−k1∂ℓ\nt(v·∇v)+1\nk1∂ℓ\nt[(1\n¯̺−1\n̺)∇ξ].(3.30)\nLet (3.30)·(∂ℓ\ntξ,∂ℓ\ntv), we get\n\n\n(|∂ℓ\ntξ|2)t+2k2∂ℓ\ntξ∇·(∂ℓ\ntv) =−2γk1∂ℓ\ntξ∂ℓ\nt(ξ∇·v)−2k1∂ℓ\ntξ∂ℓ\nt(v·∇ξ),\n(|∂ℓ\ntv|2)t+2k2∂ℓ\ntv·∇(∂ℓ\ntξ)+2a|∂ℓ\ntv|2=−2k1∂ℓ\ntv·∂ℓ\nt(v·∇v)\n+2\nk1∂ℓ\ntv·∂ℓ\nt[(1\n¯̺−1\n̺)∇ξ].\n(3.31)\nBy (3.31)1+(3.31)2, we get\n(|∂ℓ\ntξ|2+|∂ℓ\ntv|2)t+2k2∂ℓ\ntξ∇·(∂ℓ\ntv)+2k2∂ℓ\ntv·∇(∂ℓ\ntξ)+2a|∂ℓ\ntv|2\n=−2γk1∂ℓ\ntξ∂ℓ\nt(ξ∇·v)−2k1∂ℓ\ntξ∂ℓ\nt(v·∇ξ)−2k1∂ℓ\ntv·∂ℓ\nt(v·∇v)\n+2\nk1∂ℓ\ntv·∂ℓ\nt[(1\n¯̺−1\n̺)∇ξ].(3.32)\nAfter integrating (3 .32) in Ω, we get\nd\ndt/integraltext\nΩ|∂ℓ\ntξ|2+|∂ℓ\ntv|2dx+2k2/integraltext\nΩ∂ℓ\ntξ∇·(∂ℓ\ntv)+∂ℓ\ntv·∇(∂ℓ\ntξ)dx+2a/integraltext\nΩ|∂ℓ\ntv|2dx\n=/integraltext\nΩ−2γk1∂ℓ\ntξ∂ℓ\nt(ξ∇·v)−2k1∂ℓ\ntξ∂ℓ\nt(v·∇ξ)−2k1∂ℓ\ntv·∂ℓ\nt(v·∇v)\n+2\nk1∂ℓ\ntv·∂ℓ\nt[(1\n¯̺−1\n̺)∇ξ]dx.\n(3.33)\nSince∂ℓ\ntv·n|∂Ω= 0,/integraltext\nΩ∂ℓ\ntξ∇·(∂ℓ\ntv)+∂ℓ\ntv·∇(∂ℓ\ntξ)dx=/integraltext\n∂Ω∂ℓ\ntξ∂ℓ\ntv·ndSx= 0,\nd\ndt/integraltext\nΩ|∂ℓ\ntξ|2+|∂ℓ\ntv|2dx+2a/integraltext\nΩ|∂ℓ\ntv|2dx\n=/integraltext\nΩ−2γk1∂ℓ\ntξ∂ℓ\nt(ξ∇·v)−2k1∂ℓ\ntξ∂ℓ\nt(v·∇ξ)−2k1∂ℓ\ntv·∂ℓ\nt(v·∇v)\n+2\nk1∂ℓ\ntv·∂ℓ\nt[(1\n¯̺−1\n̺)∇ξ]dx:=I3.(3.34)\nWhen 0 ≤ℓ≤2, it is easy to check that I3/lessorsimilar√ǫE[ξ,v](t), sinceI3is a lower\norder term.\nWhenℓ= 3,\nI3=/integraltext\nΩ−2γk1(ξttt)2∇·v−6γk1ξtttξtt∇·vt−6γk1ξtttξt∇·vtt\n−2γk1ξtttξ∇·vttt−2k1ξtttvttt·∇ξ−6k1ξtttvtt·∇ξt\n−6k1ξtttvt·∇ξtt−2k1ξtttv·∇ξttt−2k1vttt·∇v·vttt−6k1vtt·∇vt·vttt\n−6k1vt·∇vtt·vttt−2k1v·∇vttt·vttt+2\nk1(1\n¯̺−1\n̺)vttt·∇ξttt\n+6\nk1(̺t\n̺2vttt·∇ξtt+̺̺tt−2̺2\nt\n̺3vttt·∇ξt)+2\nk1̺2̺ttt−6̺̺t̺tt+6̺3\nt\n̺4vttt·∇ξdx.\n(3.35)\n19Now we estimate I3−d\ndt/integraltext\nΩξ\npξ2\ntttdx+d\ndt/integraltext\nΩ(̺\n¯̺−1)|vttt|2dx,\nI3−d\ndt/integraltext\nΩξ\npξ2\ntttdx+d\ndt/integraltext\nΩ(̺\n¯̺−1)|vttt|2dx\n/lessorsimilar√ǫE[ξ,v](t)+/ba∇dblξtt/ba∇dblL4(Ω)/ba∇dbl∇·vt/ba∇dblL4(Ω)/ba∇dblξttt/ba∇dblL2(Ω)−2γk1/integraltext\nΩξtttξ∇·vtttdx\n+/ba∇dblvtt/ba∇dblL4(Ω)/ba∇dbl∇ξt/ba∇dbl/ba∇dblL4(Ω)/ba∇dblξttt/ba∇dbl/ba∇dblL2(Ω)−2k1/integraltext\nΩξtttv·∇ξtttdx\n+/ba∇dblvtt/ba∇dblL4(Ω)/ba∇dbl∇vt/ba∇dblL4(Ω)/ba∇dblvttt/ba∇dblL2(Ω)−2k1/integraltext\nΩv·∇vttt·vtttdx\n+2\nk1/integraltext\nΩ(1\n¯̺−1\n̺)vttt·∇ξtttdx+/ba∇dbl̺tt/ba∇dblL4(Ω)/ba∇dbl∇ξt/ba∇dblL4(Ω)/ba∇dblvttt/ba∇dblL2(Ω)\n−2/integraltext\nΩξ\npξtttξttttdx−/integraltext\nΩ∂t(ξ\np)ξ2\ntttdx+2/integraltext\nΩ(̺\n¯̺−1)vttt·vttttdx+/integraltext\nΩ̺t\n¯̺|vttt|2dx\n/lessorsimilar√ǫE[ξ,v](t)−k1/integraltext\n∂Ω(|ξttt|2+|vttt|2)v·ndSx+k1/integraltext\nΩ(|ξttt|2+|vttt|2)∇·vdx\n−2γk1/integraltext\nΩξξttt∇·vtttdx+2\nk1/integraltext\nΩ(1\n¯̺−1\n̺)vttt·∇ξtttdx\n−2/integraltext\nΩξ\npξtttξttttdx+2/integraltext\nΩ(̺\n¯̺−1)vttt·vttttdx\n/lessorsimilar−2/integraltext\nΩξ\npξttt(ξtttt+k1γp∇·vttt)dx+2\nk1/integraltext\nΩ(1\n¯̺−1\n̺)vttt·(∇ξttt+k1̺vtttt)dx\n+√ǫE[ξ,v](t).\n(3.36)\nApply∂tttto (2.4)1, we get\nξtttt+k1γp∇·vttt=−k1v·∇ξttt−3k1vt·∇ξtt−3k1vtt·∇ξt−k1vttt·∇ξ\n−3k1γ̺t∇·vtt−3k1γξtt∇·vt−k1γξttt∇·v.\n(3.37)\nPlug (3.37) into the following integral, we get\n/integraltext\nΩξ\npξttt(ξtttt+k1γp∇·vttt)dx=/integraltext\nΩξ\npξttt[R.H.S. of (3.37)]dx\n/lessorsimilar√ǫE[ξ,v](t)−k1/integraltext\n∂Ωξ\n2p|ξttt|2v·ndSx+k1\n2/integraltext\nΩ|ξttt|2∇·(ξ\npv)dx/lessorsimilar√ǫE[ξ,v](t).\n(3.38)\nApply∂tttto (2.4)2, we get\n∇ξttt+k1̺vtttt=−3k1ξtvttt−3k1ξttvtt−k1ξtttvt−k2\n1ξtttv·∇v\n−3k2\n1ξttvt·∇v−3k2\n1ξttv·∇vt−3k2\n1ξtvtt·∇v\n−3k2\n1ξtv·∇vtt−6k2\n1ξtvt·∇vt−k2\n1̺vttt·∇v\n−3k2\n1̺vtt·∇vt−3k2\n1̺vt·∇vtt−k2\n1̺v·∇vttt\n−ak1ξtttv−3k1aξttvt−3k1aξtvtt−ak1̺vttt.(3.39)\nPlug (3.39) into the following integral, we get\n/integraltext\nΩ(1\n¯̺−1\n̺)vttt·(∇ξttt+k1̺vtttt)dx=/integraltext\nΩ(1\n¯̺−1\n̺)vttt·[R.H.S. of (3.39)]dx\n/lessorsimilar−k2\n1\n2/integraltext\n∂Ω(̺\n¯̺−1)|vttt|2v·ndSx+k2\n1\n2/integraltext\nΩ|ξttt|2∇·[(̺\n¯̺−1)v]dx+√ǫE[ξ,v](t)\n/lessorsimilar√ǫE[ξ,v](t).\n(3.40)\n20Plug (3.38) and (3 .40) into (3 .36), we get\nI3−d\ndt/integraltext\nΩξ\npξ2\ntttdx+d\ndt/integraltext\nΩ(̺\n¯̺−1)|vttt|2dx/lessorsimilar√ǫE[ξ,v](t).(3.41)\nFinally, we have\nd\ndt/parenleftbigg3/summationtext\nℓ=0/integraltext\nΩ|∂ℓ\ntξ|2+|∂ℓ\ntv|2dx−/integraltext\nΩξ\npξ2\ntttdx+/integraltext\nΩ(̺\n¯̺−1)|vttt|2dx/parenrightbigg\n+2a3/summationtext\nℓ=0/integraltext\nΩ|∂ℓ\ntv|2dx/lessorsimilar√ǫE[ξ,v](t).(3.42)\nThen\nd\ndt/parenleftbigg3/summationtext\nℓ=0/integraltext\nΩ|∂ℓ\ntξ|2+|∂ℓ\ntv|2dx−/integraltext\nΩξ\npξ2\ntttdx+/integraltext\nΩ(̺\n¯̺−1)|vttt|2dx/parenrightbigg\n+2a/parenleftbigg3/summationtext\nℓ=0/integraltext\nΩ|∂ℓ\ntv|2dx+/integraltext\nΩ(̺\n¯̺−1)|vttt|2dx/parenrightbigg\n/lessorsimilar√ǫE[ξ,v](t)+2a/integraltext\nΩ|̺−¯̺|∞\n¯̺|vttt|2dx/lessorsimilar√ǫE[ξ,v](t).(3.43)\nSo we get\nd\ndtE1[ξ,v](t)+2aE1[v](t)/lessorsimilar√ǫE[ξ,v](t). (3.44)\nThus, Lemma 3 .6 is proved.\nThefollowinglemmaconcernsaprioriestimatefor/integraltext\nΩ3/summationtext\nℓ=1∂ℓ\ntξ∂ℓ−1\ntξdx, which\nintroduces E[ξ](t) to the inequality (3 .45).\nLemma 3.7. For any given T∈(0,+∞], if\nsup\n0≤t≤TE[ξ,v,φ](t)≤ǫ,\nwhere0< ǫ≪1, then there exists c3>0such that for ∀t∈[0,T],\n−d\ndt/integraldisplay\nΩ3/summationdisplay\nℓ=1∂ℓ\ntξ∂ℓ−1\ntξdx+E[ξ](t)≤C√ǫE[ξ,v](t)+c3E[v](t).(3.45)\nProof.Apply∂tto (2.4)1, then we get\nξtt=−k1v·∇ξt−k1vt·∇ξ−k1γξt∇·v−k1γp∇·vt,\n−(ξtξ)t+ξ2\nt=k1ξv·∇ξt+k1ξvt·∇ξ+k1γξξt∇·v+k1γpξ∇·vt.(3.46)\nAfter integrating (3 .46)2in Ω, we get\n−d\ndt/integraltext\nΩξtξdx+/integraltext\nΩ(ξt)2dx\n/lessorsimilar√ǫE[ξ,v](t)+k1γ/integraltext\n∂Ωpξvt·ndSx−k1γ/integraltext\nΩξvt·∇ξdx−k1γ/integraltext\nΩpvt·∇ξdx\n21/lessorsimilar√ǫE[ξ,v](t)+/ba∇dblvt/ba∇dbl2\nL2(Ω)+/ba∇dbl∇ξ/ba∇dbl2\nL2(Ω)\n/lessorsimilar√ǫE[ξ,v](t)+/ba∇dblvt/ba∇dbl2\nL2(Ω)+/ba∇dbl−k1̺vt−k2\n1̺v·∇v−ak1̺v/ba∇dbl2\nL2(Ω)\n/lessorsimilar√ǫE[ξ,v](t)+/ba∇dblv/ba∇dbl2\nL2(Ω)+/ba∇dblvt/ba∇dbl2\nL2(Ω).(3.47)\nApply∂ttto (2.4)1, then we get\nξttt=−k1v·∇ξtt−2k1vt·∇ξt−k1vtt·∇ξ−k1γξtt∇·v\n−2k1γξt∇·vt−k1γp∇·vtt,\n−(ξttξt)t+ξ2\ntt=k1ξtv·∇ξtt+2k1ξtvt·∇ξt+k1ξtvtt·∇ξ+k1γξtξtt∇·v\n+2k1γξ2\nt∇·vt+k1γpξt∇·vtt.\n(3.48)\nAfter integrating (3 .48)2in Ω, we get\n−d\ndt/integraltext\nΩξttξtdx+/integraltext\nΩ(ξtt)2dx\n/lessorsimilar√ǫE[ξ,v](t)+k1γ/integraltext\n∂Ωpξtvtt·ndSx−k1γ/integraltext\nΩξtvtt·∇ξdx−k1γ/integraltext\nΩpvtt·∇ξtdx\n/lessorsimilar√ǫE[ξ,v](t)+/ba∇dblvtt/ba∇dbl2\nL2(Ω)+/ba∇dbl∇ξt/ba∇dbl2\nL2(Ω)\n/lessorsimilar√ǫE[ξ,v](t)+/ba∇dblvtt/ba∇dbl2\nL2(Ω)+/ba∇dbl−k1̺vtt−k1ξtvt−k2\n1ξtv·∇v\n−k2\n1̺vt·∇v−k2\n1̺v·∇vt−ak1ξtv−ak1̺vt/ba∇dbl2\nL2(Ω)\n/lessorsimilar√ǫE[ξ,v](t)+/ba∇dblvt/ba∇dbl2\nL2(Ω)+/ba∇dblvtt/ba∇dbl2\nL2(Ω).\n(3.49)\nApply∂tttto (2.4)1, then we get\nξtttt=−k1v·∇ξttt−3k1vt·∇ξtt−3k1vtt·∇ξt−k1vttt·∇ξ\n−k1γξttt∇·v−3k1γξtt∇·vt−3k1γξt∇·vtt−k1γp∇·vttt,\n−(ξtttξtt)t+ξ2\nttt=k1ξttv·∇ξttt+3k1ξttvt·∇ξtt+3k1ξttvtt·∇ξt\n+k1ξttvttt·∇ξ+k1γξttξttt∇·v+3k1γξ2\ntt∇·vt\n+3k1γξtξtt∇·vtt+k1γpξtt∇·vttt.\n(3.50)\nAfter integrating (3 .50)2in Ω, we get\n−d\ndt/integraltext\nΩξtttξttdx+/integraltext\nΩ(ξttt)2dx\n/lessorsimilar√ǫE[ξ,v](t)+/integraltext\nΩk1ξttv·∇ξttt+k1γpξtt∇·vtttdx\n/lessorsimilar√ǫE[ξ,v](t)+k1/integraltext\n∂Ωξtttξttv·ndSx−k1/integraltext\nΩξtttξtt∇·vdx−k1/integraltext\nΩξtttv·∇ξttdx\n+k1γ/integraltext\n∂Ωpξttvttt·ndSx−k1γ/integraltext\nΩξttvttt·∇ξdx−k1γ/integraltext\nΩpvttt·∇ξttdx\n/lessorsimilar√ǫE[ξ,v](t)+/ba∇dblvttt/ba∇dbl2\nL2(Ω)+/ba∇dbl∇ξtt/ba∇dbl2\nL2(Ω)\n22/lessorsimilar√ǫE[ξ,v](t)+/ba∇dblvttt/ba∇dbl2\nL2(Ω)+/ba∇dbl−k1̺vttt−2k1ξtvtt−k1ξttvt\n−k2\n1ξttv·∇v−k2\n1̺vtt·∇v−k2\n1̺v·∇vtt−2k2\n1ξtvt·∇v−2k2\n1̺vt·∇vt\n−2k2\n1ξtv·∇vt−ak1ξttv−2ak1ξtvt−ak1̺vtt/ba∇dbl2\nL2(Ω)\n/lessorsimilar√ǫE[ξ,v](t)+/ba∇dblvtt/ba∇dbl2\nL2(Ω)+/ba∇dblvttt/ba∇dbl2\nL2(Ω).\n(3.51)\nBy (3.47)+(3.49)+(3.51), we get\n−d\ndt/integraldisplay\nΩ3/summationdisplay\nℓ=1∂ℓ\ntξ∂ℓ−1\ntξdx+/integraldisplay\nΩ3/summationdisplay\nℓ=1(∂ℓ\ntξ)2dx/lessorsimilar√ǫE[ξ,v](t)+3/summationdisplay\nℓ=0/ba∇dblvℓ\nt/ba∇dbl2\nL2(Ω).(3.52)\nBy Lemma 2 .1, ¯p∈[inf\nx∈Ωp,sup\nx∈Ωp], then for any t≥0, there exists xt∈Ω\nsuch that ξ(xt,t) = 0. Assume ℓ(s) is a curve with finite length parameter s\nsuch that ℓ(0) =xt, ℓ(sx) =x, then\n/ba∇dblξ(x,t)/ba∇dbl2\nL2(Ω)=/ba∇dblξ(xt,t)+sx/integraltext\n0∇ξ[ℓ(s)]·ℓ(s)ds/ba∇dbl2\nL2(Ω)\n≤C|Diam(Ω)|2/ba∇dbl∇ξ/ba∇dbl2\nL2(Ω)/lessorsimilar/ba∇dbl∇ξ/ba∇dbl2\nL2(Ω)\n/lessorsimilar√ǫE[ξ,v](t)+/ba∇dblv/ba∇dbl2\nL2(Ω)+/ba∇dblvt/ba∇dbl2\nL2(Ω).(3.53)\nSumming (3 .52) and (3 .53), we get\n−d\ndt/integraltext\nΩ3/summationtext\nℓ=1∂ℓ\ntξ∂ℓ−1\ntξdx+/integraltext\nΩ3/summationtext\nℓ=0(∂ℓ\ntξ)2dx/lessorsimilar√ǫE[ξ,v](t)+3/summationtext\nℓ=0/ba∇dblvℓ\nt/ba∇dbl2\nL2(Ω).\n(3.54)\nThen there exist two constants C >0,c3>0 such that\n−d\ndt/integraltext\nΩ3/summationtext\nℓ=1∂ℓ\ntξ∂ℓ−1\ntξdx+/integraltext\nΩ3/summationtext\nℓ=0(∂ℓ\ntξ)2dx≤C√ǫE[ξ,v](t)+c33/summationtext\nℓ=0/ba∇dblvℓ\nt/ba∇dbl2\nL2(Ω).\n(3.55)\nThus, Lemma 3 .7 is proved.\nBased on the above a priori estimates, we prove the exponential d ecay of\nE[ξ,v](t) andE1[ω](t) in the following lemma.\nLemma 3.8. For any given T∈(0,+∞], if\nsup\n0≤t≤TE[ξ,v,φ](t)≤ǫ,\nthere exists ǫ2>0, which is independent of (ξ0,v0,φ0), such that if 0< ǫ≪\nmin{1,ǫ0,ǫ1,ǫ2}, then for ∀t∈[0,T],\nE[ξ,v](t)≤β1/ba∇dbl(ξ0,v0)/ba∇dbl2\nH3(Ω)exp{−β2t},\nE1[ω](t)≤β3/ba∇dbl(ξ0,v0)/ba∇dbl2\nH3(Ω)exp{−β2t},(3.56)\nwhereβ1,β2,β3are three positive numbers.\n23Proof.In view of Lemmas 3 .4, 3.6 and 3.7, we have obtained global a priori\nestimates as follows:\n\n\nd\ndtE1[ω](t)+2aE1[ω](t)≤C√ǫE[ξ,v](t),\nd\ndtE1[ξ,v](t)+2aE1[v](t)≤C√ǫE[ξ,v](t),\n−d\ndt/integraltext\nΩ3/summationtext\nℓ=1∂ℓ\ntξ∂ℓ−1\ntξdx+E[ξ](t)≤C√ǫE[ξ,v](t)+c3E[v](t).(3.57)\nLetλ1= max{4\n3,c3\n2a}+1, we define\nE2[ξ,v](t) :=λ1E1[ξ,v](t)−3/summationtext\nℓ=1/integraltext\nΩ∂ℓ−1\ntξ∂ℓ\ntξdx. (3.58)\nwhereE2>0 by Cauchy-Schwarz inequality. Since λ1>4\n3,E2∼=E, i.e., there\nexistc4>0,c5>0 such that\nc4E[ξ,v](t)≤E2[ξ,v](t)≤c5E[ξ,v](t). (3.59)\nBy (3.57)2×λ1+(3.57)3, we get\nd\ndtE2[ξ,v](t)+(2aλ1−c3)E1[v](t)+E[ξ](t)/lessorsimilar√ǫE[ξ,v](t).(3.60)\nSinceE1[ξ](t)≤c2E[ξ](t), we have\nd\ndtE2[ξ,v](t)+(2aλ1−c3)E1[v](t)+1\nc2E1[ξ](t)/lessorsimilar√ǫE[ξ,v](t).(3.61)\nLetc6= min(2 aλ1−c3,1\nc2)>0, it follows from (3 .61) that\nd\ndtE2[ξ,v](t)+c6E1[ξ,v](t)/lessorsimilar√ǫE[ξ,v](t). (3.62)\nBy (3.57)1+(3.62), we get\nd\ndt[E2[ξ,v](t)+E1[ω](t)]+[c6E1[ξ,v](t)+2aE1[ω](t)]/lessorsimilar√ǫE[ξ,v]\n/lessorsimilarc0√ǫ(E[ξ,v](t)+E1[ω](t))≤C6c0√ǫ(1\nc1E1[ξ,v](t)+E1[ω](t)),(3.63)\nfor some C6>0.\nLetǫ2= min{c2\n1c2\n6\n4C2\n6c2\n0,a2\nC2\n6c2\n0}. Whenǫ <min{1,ǫ0,ǫ1,ǫ2}, we have\nd\ndt[E2[ξ,v](t)+E1[ω](t)]+[c6\n2E1[ξ,v](t)+aE1[ω](t)]≤0,\nd\ndt[E2[ξ,v](t)+E1[ω](t)]+[c6c1\n2c5E2[ξ,v](t)+aE1[ω](t)]≤0,\nd\ndt[E2[ξ,v](t)+E1[ω](t)]+c7[E2[ξ,v](t)+E1[ω](t)]≤0,(3.64)\nwherec7= min{c6c1\n2c5,a}.\n24After integrating (3 .64)3, we get\nE2[ξ,v](t)+E1[ω](t)≤(E2[ξ,v](0)+E1[ω](0))exp{−c7t},\nE1[ω](t)≤(E2[ξ,v](0)+E1[ω](0))exp{−c7t}\n≤(c5E[ξ,v](0)+E[v](0))exp{−c7t}\n≤(c5+1)E[ξ,v](0)exp{−c7t}.\n≤C7(c5+1)/ba∇dbl(ξ0,v0)/ba∇dbl2\nH3(Ω)exp{−c7t}.\nc4E[ξ,v](t)≤E2[ξ,v](t)≤(E2[ξ,v](0)+E1[ω](0))exp{−c7t},\nE[ξ,v](t)≤c0(E[ξ,v](t)+E1[ω](t))\n≤(c0\nc4+c0)(E2[ξ,v](0)+E1[ω](0))exp{−c7t}\n≤(c0\nc4+c0)(c5+1)E[ξ,v](0)exp{−c7t}\n≤C7(c0\nc4+c0)(c5+1)/ba∇dbl(ξ0,v0)/ba∇dbl2\nH3(Ω)exp{−c7t}(3.65)\nTakeβ1=C7(c0\nc4+c0)(c5+1),β2=c7,β3=C7(c5+1), the exponential\ndecay in (3 .56) is obtained. Thus, Lemma 3 .8 is proved.\nFinally, we prove the uniform bound of S−¯Sand its derivatives Dα∂ℓ\ntS\nand the exponential decay of ∂ℓ\ntSand|∂tS|∞under the condition that vdecays\nexponentially. S−¯SandDαSmay not decay due to the transportation of S,\nbut they are uniformly bounded.\nThefollowinglemmaconcernstheuniformboundof E[φ](t)onthecondition\nthatvdecays exponentially.\nLemma 3.9. For any given T∈(0,+∞], if\nsup\n0≤t≤TE[ξ,v,φ](t)≤ǫ,\nwhere0< ǫ≪min{1,ǫ0,ǫ1,ǫ2}, then for ∀t∈[0,T],\nd\ndtE[φ](t)≤β4E[v](t)1\n2E[φ](t). (3.66)\nIfE[v](t)≤β1/ba∇dbl(ξ0,v0)/ba∇dbl2\nH3(Ω)exp{−β2t}, thenE[φ](t)has uniform bound:\nE[φ](t)≤β5/ba∇dblφ0/ba∇dbl2\nH3(Ω)/parenleftbig\nexp{/ba∇dbl(ξ0,v0)/ba∇dblH3(Ω)}/parenrightbigc8. (3.67)\nfor some c8>0.\nProof.Apply∂ℓ\ntDαto (2.4)3,ℓ+|α| ≤3. We have\n(∂ℓ\ntDαφ)t=−k1∂ℓ\ntDα(v·∇φ). (3.68)\nWhenℓ+|α|<3, it is easy to check (3 .66), since they are lower order\nterms.\n25Whenℓ= 0,|α|= 3, assume α=α1+α2,|α1|= 1,|α2|= 2.\n(|Dαφ|2)t=−2k1[Dαv·∇φ+(Dα1v)·∇(Dα2φ)+(Dα2v)·∇(Dα1φ)\n+v·∇(Dαφ)]Dαφ.(3.69)\nIntegrating (3 .69) in Ω, we get\nd\ndt/integraltext\nΩ|Dαφ|2dx=/integraltext\nΩ[R.H.S. of (3.69)]dx\n/lessorsimilar/ba∇dblDαφ/ba∇dblL2(Ω)(|∇φ|∞/ba∇dblDαv/ba∇dblL2(Ω)+|Dα1v|∞/ba∇dbl∇(Dα2φ)/ba∇dblL2(Ω)\n+/ba∇dblDα2v/ba∇dblL4(Ω)/ba∇dbl∇(Dα1φ)/ba∇dblL4(Ω))−k1/integraltext\n∂Ω|Dαφ|2v·ndSx\n+k1/integraltext\nΩ|Dαφ|2∇·vdx/lessorsimilarE[v](t)1\n2E[φ](t).(3.70)\nWhenℓ= 1,|α|= 2, assume α=α1+α2,|α1|= 1,|α2|= 1.\n(|Dαφt|2)t=−2k1[Dαvt·∇φ+/summationtext\nα1+α2=α(Dα1vt)·∇(Dα2φ)+vt·∇(Dαφ)\n+Dαv·∇φt+/summationtext\nα1+α2=α(Dα1v)·∇(Dα2φt)+v·∇(Dαφt)]Dαφt.\n(3.71)\nIntegrating (3 .71) in Ω, we get\nd\ndt/integraltext\nΩ|Dαφt|2dx=/integraltext\nΩ[R.H.S. of (3.71)]dx\n/lessorsimilar(|∇φ|∞/ba∇dblDαvt/ba∇dblL2(Ω)+|vt|∞/ba∇dbl∇Dαφ/ba∇dblL2(Ω)+/ba∇dblDα1vt/ba∇dblL4(Ω)/ba∇dbl∇(Dα2φ)/ba∇dblL4(Ω)\n+/ba∇dblDαv/ba∇dblL4(Ω)/ba∇dbl∇φt/ba∇dblL4(Ω)+|Dα1v|∞/ba∇dbl∇(Dα2φt)/ba∇dblL2(Ω))/ba∇dblDαφt/ba∇dblL2(Ω)\n−k1/integraltext\n∂Ω|Dαφt|2v·ndSx+k1/integraltext\nΩ|Dαφt|2∇·vdx/lessorsimilarE[v](t)1\n2E[φ](t).\n(3.72)\nWhenℓ= 2,|α| ≤1,\n(|Dαφtt|2)t=−2k1[(Dαvtt)·∇φ+vtt·∇(Dαφ)+2(Dαvt)·∇φt\n+2vt·∇(Dαφt)+(Dαv)·∇φtt+v·∇(Dαφtt)]Dαφtt.(3.73)\nIntegrating (3 .73) in Ω, we get\nd\ndt/integraltext\nΩ|Dαφtt|2dx=/integraltext\nΩ[R.H.S. of (3.73)]dx\n/lessorsimilar(|∇φ|∞/ba∇dblDαvtt/ba∇dblL2(Ω)+|vtt|L4(Ω)/ba∇dbl∇Dαφ/ba∇dblL4(Ω)+/ba∇dblDαvt/ba∇dblL4(Ω)/ba∇dbl∇φt/ba∇dblL4(Ω)\n+|vt|∞/ba∇dbl∇(Dαφt)/ba∇dblL2(Ω)+|Dαv|∞/ba∇dbl∇φtt/ba∇dblL2(Ω))/ba∇dblDαφtt/ba∇dblL2(Ω)\n−k1/integraltext\n∂Ω|Dαφtt|2v·ndSx+k1/integraltext\nΩ|Dαφtt|2∇·vdx/lessorsimilarE[v](t)1\n2E[φ](t).\n(3.74)\nWhenℓ= 3,|α|= 0,\n(|φttt|2)t=−2k1[vttt·∇φ+3vtt·∇φt+3vt·∇φtt+v·∇φttt]φttt.(3.75)\n26Integrating in Ω, we get\nd\ndt/integraltext\nΩ|φttt|2dx=/integraltext\nΩ[R.H.S. of (3.75)]dx\n/lessorsimilar(|∇φ|∞/ba∇dblvttt/ba∇dblL2(Ω)+|vtt|L4(Ω)/ba∇dbl∇φt/ba∇dblL4(Ω)+|vt|∞/ba∇dbl∇φtt/ba∇dblL2(Ω))/ba∇dblφttt/ba∇dblL2(Ω)\n−k1/integraltext\n∂Ω|φttt|2v·ndSx+k1/integraltext\nΩ|φttt|2∇·vdx/lessorsimilarE[v](t)1\n2E[φ](t).\n(3.76)\nIn views of (3 .70),(3.72),(3.74),(3.76), we have, for some constant β4>0,\nd\ndtE[φ](t)≤β4E[v](t)1\n2E[φ](t),\nE[φ](t)≤ E[φ](0)exp{t/integraltext\n0β4E[v](τ)1\n2dτ}.(3.77)\nIfE[v](t)≤β1/ba∇dbl(ξ0,v0)/ba∇dbl2\nH3(Ω)exp{−β2t}, then\nE[φ](t)≤ E[φ](0)exp{t/integraltext\n0β4E[v](s)1\n2ds}\n≤ E[φ](0)exp{t/integraltext\n0β4√β1/ba∇dbl(ξ0,v0)/ba∇dblH3(Ω)exp{−β2s}1\n2ds}\n≤β5/ba∇dblφ0/ba∇dbl2\nH3(Ω)exp{2β4√β1/ba∇dbl(ξ0,v0)/ba∇dblH3(Ω)\nβ2(1−exp{−β2\n2t})}\n≤β5/ba∇dblφ0/ba∇dbl2\nH3(Ω)exp{2β4√β1\nβ2/ba∇dbl(ξ0,v0)/ba∇dblH3(Ω)}\n=β5/ba∇dblφ0/ba∇dbl2\nH3(Ω)/parenleftbig\nexp{/ba∇dbl(ξ0,v0)/ba∇dblH3(Ω)}/parenrightbigc8,(3.78)\nwherec8=2β4√β1\nβ2,β5>0.\nTherefore E[φ](t) is uniformly bounded when E[v](t) decays exponentially.\nThus, Lemma 3 .9 is proved.\nThe following lemma concerns the exponential decay of3/summationtext\nℓ=1/ba∇dbl∂ℓ\ntS/ba∇dbl2\nH3−ℓ(Ω)\non the condition that vdecays exponentially.\nLemma 3.10. For any given T∈(0,+∞], if\nsup\n0≤t≤TE[ξ,v,φ](t)≤ǫ,\nwhere0< ǫ≪min{1,ǫ0,ǫ1,ǫ2}, then for ∀t∈[0,T],\n3/summationtext\nℓ=1/ba∇dbl∂ℓ\ntφ/ba∇dbl2\nH3−ℓ(Ω)/lessorsimilarE[v](t)E[φ](t)\n≤c9/ba∇dbl(ξ0,v0)/ba∇dbl2\nH3(Ω)/ba∇dblφ0/ba∇dbl2\nH3(Ω)/parenleftbig\nexp{/ba∇dbl(ξ0,v0)/ba∇dblH3(Ω)}/parenrightbigc8exp{−β2t},(3.79)\nfor some c9>0.\nProof.It follows from Lemma 3 .8 thatE[v](t)≤β1/ba∇dbl(ξ0,v0)/ba∇dbl2\nH3(Ω)exp{−β2t}.\nIt follows from Lemma 3 .9 thatE[φ](t)≤β5/ba∇dblφ0/ba∇dbl2\nH3(Ω)/parenleftbig\nexp{/ba∇dbl(ξ0,v0)/ba∇dblH3(Ω)}/parenrightbigc8.\n27Byφt=−k1v·∇φ, we get\n/ba∇dblφt/ba∇dbl2\nH2(Ω)=k2\n1/summationtext\n|α|≤2/integraltext\nΩ|Dαv|2|∇φ|2dx+k2\n1/summationtext\n|α|≤2/integraltext\nΩ|v|2|Dα∇φ|2dx\n+k2\n1/summationtext\n|α1|≤1,|α2|≤1/integraltext\nΩ|Dα1v|2|Dα2∇φ|2dx\n/lessorsimilar/summationtext\n|α|≤2|∇φ|2\n∞/ba∇dblDαv/ba∇dbl2\nL2(Ω)+/summationtext\n|α|≤2|v|2\n∞/ba∇dblDα∇φ/ba∇dbl2\nL2(Ω)\n+/summationtext\n|α1|≤1,|α2|≤1|Dα1v|2\n∞/ba∇dblDα2∇φ/ba∇dbl2\nL2(Ω)\n/lessorsimilarE[v](t)E[φ](t).(3.80)\nwhere 0≤ |α1| ≤1,0≤ |α2| ≤1.\nByφtt=−k1vt·∇φ−k1v·∇φt, we get\n/ba∇dblφtt/ba∇dbl2\nH1(Ω)≤k2\n1/summationtext\n0≤|α|≤1/integraltext\nΩ(Dαvt·∇φ)2+(vt·∇Dαφ)2+(Dαv·∇φt)2\n+(v·∇Dαφt)2dx,\n/lessorsimilar/summationtext\n0≤|α|≤1(|∇φ|2\n∞/ba∇dblDαvt/ba∇dbl2\nL2(Ω)+|vt|2\n∞/ba∇dbl∇Dαφ/ba∇dbl2\nL2(Ω)\n+|Dαv|2\n∞/ba∇dbl∇φt/ba∇dbl2\nL2(Ω)+|v|2\n∞/ba∇dbl∇Dαφt/ba∇dbl2\nL2(Ω))\n/lessorsimilarE[v](t)E[φ](t).\n(3.81)\nByφttt=−k1vtt·∇φ−2k1vt·∇φt−k1v·∇φtt, we get\n/ba∇dblφttt/ba∇dbl2\nL2(Ω)≤k2\n1/integraltext\nΩ(vtt·∇φ)2dx+4k2\n1/integraltext\nΩ(vt·∇φt)2dx+k2\n1/integraltext\nΩ(v·∇φtt)2dx\n/lessorsimilar|∇φ|2\n∞/ba∇dblvtt/ba∇dbl2\nL2(Ω)+|vt|2\n∞/ba∇dbl∇φt/ba∇dbl2\nL2(Ω)+|v|2\n∞/ba∇dbl∇φtt/ba∇dbl2\nL2(Ω)\n/lessorsimilarE[v](t)E[φ](t).\n(3.82)\nSumming (3 .80),(3.81),(3.82), we get\n3/summationtext\nℓ=1/ba∇dbl∂ℓ\ntφ/ba∇dbl2\nH3−ℓ(Ω)/lessorsimilarE[v](t)E[φ](t)\n≤c9/ba∇dbl(ξ0,v0)/ba∇dbl2\nH3(Ω)/ba∇dblφ0/ba∇dbl2\nH3(Ω)/parenleftbig\nexp{/ba∇dbl(ξ0,v0)/ba∇dblH3(Ω)}/parenrightbigc8exp{−β2t},(3.83)\nwherec9=β1β5. Thus, Lemma 3 .10 is proved.\nRemark 3.11. WhenE[p−¯p](t)andE[S−¯S](t)are uniformly bounded, E[̺−\n¯̺](t)is also uniformly bounded due to ̺=1\nγ√\nAp1\nγexp{−S\nγ}.\nAfter differentiating this formula with respect to t, we have\n/ba∇dbl∂t̺/ba∇dbl2\nH2(Ω)≤β1/ba∇dbl(ξ0,v0)/ba∇dbl2\nH3(Ω)exp{−β2t}\n+c9/ba∇dbl(ξ0,v0)/ba∇dbl2\nH3(Ω)/ba∇dblφ0/ba∇dbl2\nH3(Ω)/parenleftbig\nexp{/ba∇dbl(ξ0,v0)/ba∇dblH3(Ω)}/parenrightbigc8exp{−β2t},\n(3.84)\nSimilarly,\n/ba∇dbl∂tt̺/ba∇dbl2\nH1(Ω)≤Cexp{−β2t}+Cexp{−2β2t},\n/ba∇dbl∂ttt̺/ba∇dbl2\nL2(Ω)≤Cexp{−β2t}+Cexp{−2β2t}+Cexp{−3β2t}.(3.85)\n28Thus, for any given T∈(0,+∞], ifsup\n0≤t≤TE[ξ,v,φ](t)≤ǫ, where0< ǫ≪\nmin{1,ǫ0,ǫ1,ǫ2}, then3/summationtext\nℓ=1/ba∇dbl∂ℓ\nt̺/ba∇dbl2\nH3−ℓ(Ω)also decays at an exponential rate of\nCexp{−β2t}.\n4 Global Existence and Equilibrium States of\nNon-Isentropic Euler Equations with Damp-\ning\nIn this section, we prove the global existence of classical solutions to the\nnon-isentropicEuler equations with damping (2 .4) under small data assumption\nand the singularity formation for a class of large data.\nThe proof of local existence of classical solutions to IBVP (2 .4) is standard\n(see [10],[15]), so we give a lemma on the local existence without proof h ere.\nLemma 4.1. (Local Existence )\nIf(ξ0,v0,φ0)∈H3(Ω),inf\nx∈Ωp0(x)>0and∂ℓ\ntv(x,0)·n|∂Ω= 0,0≤ℓ≤3, then\nthere exists a finite time T∗>0, such that IBVP (2.4) admits a unique local\nclassical solution (ξ,v,φ)∈ ∩\n0≤ℓ≤3Cℓ([0,T∗),H3−ℓ(Ω)).\nBased on the global a priori estimates for ( ξ,v,φ), we obtained the global\nexistence of classical solutions to IBVP (2 .4).\nTheorem 4.2. (Global Existence )\nAssume (ξ0,v0,φ0)∈H3(Ω),inf\nx∈Ωp0(x)>0,∂ℓ\ntv(x,0)·n|∂Ω= 0,0≤ℓ≤3.\nThere exists a sufficiently small number δ1>0, such that if /ba∇dblξ0,v0,φ0/ba∇dblH3(Ω)≤\nδ1, then IBVP (2.4)admits a unique global classical solution\n(ξ,v,φ)∈ ∩\n0≤ℓ≤3Cℓ([0,+∞),H3−ℓ(Ω)),\nmoreover, ̺=̺(ξ,φ)∈ ∩\n0≤ℓ≤3Cℓ([0,+∞),H3−ℓ(Ω)).∀t≥0,E[ξ,v](t),E1[ω](t)\nand3/summationtext\nℓ=1/ba∇dbl∂ℓ\ntφ/ba∇dbl2\nH3−ℓ(Ω)decays exponentially, E[φ](t)is uniformly bounded.\nProof.In view of Lemmas 3 .8 and 3.9, we have the following global a priori\nestimates: for any given T∈(0,+∞], if\nsup\n0≤t≤TE[ξ,v,φ](t)≤ǫ, (4.1)\nwhere 0< ǫ≪min{1,ǫ0,ǫ1,ǫ2}, then\nE[ξ,v](t)≤β1/ba∇dbl(ξ0,v0)/ba∇dbl2\nH3(Ω)exp{−β2t},\nE[φ](t)≤β5/ba∇dblφ0/ba∇dbl2\nH3(Ω)/parenleftbig\nexp{/ba∇dbl(ξ0,v0)/ba∇dblH3(Ω)}/parenrightbigc8.(4.2)\n29The constants ǫ0,ǫ1,ǫ2are independent of ( ξ0,v0,φ0), so we can choose ǫ\nwhich is independent of ( ξ0,v0,φ0).\nTakeδ1= min{√ǫ,/radicalBig\nǫ\n2β1,/radicalBig\nǫ\n2β5/parenleftBig\nexp{/radicalBig\nǫ\n2β1}/parenrightBig−c8\n2}, then if E[ξ,v,φ](0)≤\nδ1, we have\n\n/ba∇dbl(ξ0,v0)/ba∇dblH3(Ω)≤/radicalBig\nǫ\n2β1,\n/ba∇dblφ0/ba∇dblH3(Ω)≤/radicalBig\nǫ\n2β5/parenleftBig\nexp{/radicalBig\nǫ\n2β1}/parenrightBig−c8\n2.(4.3)\nDue to the estimates in (4 .2), the solutions ( ξ,v,φ) satisfy\nE[ξ,v](t)≤ǫ\n2,E[φ](t)≤ǫ\n2,∀t∈[0,T]. (4.4)\nThis implies the a priori assumption (4 .1) is satisfied, the validity of the former\na priori estimates is verified.\nDue to the global a priori estimates for ( ξ,v,φ) and Lemma 4 .1 on the\nlocal existence result, the classical solution ( ξ,v,φ) can be extended to [0 ,+∞).\nThus, Theorem 4 .2 on the global existence of classical solutions to IBVP (2.4)\nis proved.\nRemark 4.3. Our proof requires a≥C√ǫwhereC >0is large enough. If\na→0,(p0,u0)→(¯p,0)is required.\nSince (ξ,v,φ)∈C1(Ω×[0,+∞)) is the global classical solution to IBVP\n(2.4), then ( p= ¯p+ξ,u=k1v,S=¯S+φ) isthe globalclassicalsolutionto IBVP\nfor non-isentropic Euler equations with damping (1 .3). The following theorem\ndescribes the asymptotical behavior of ( p,v,S,̺) relating to their equilibrium\nstates (p∞,v∞,S∞,̺∞).\nTheorem 4.4. Assume the conditions in Theorem 4.2hold. Let (p,u,S)∈\nC1(Ω×[0,+∞))be the global classical solution to IBVP (1.3).p∞= ¯p,\nu∞=v∞=ω∞= 0. IfS0/ne}ationslash=const, thenS∞/ne}ationslash=const.̺∞(x)/ne}ationslash=const, the\ntemperature θ∞(x)/ne}ationslash=const, the internal energy e∞(x)/ne}ationslash=const. Ast→+∞,\n(p,u,S,̺)converge to (¯p,0,S∞,̺∞)exponentially in |·|∞norm.\nProof.|∇p|∞+|∇v|∞/lessorsimilarE[ξ,v](t)1\n2/lessorsimilar/ba∇dbl(ξ0,v0)/ba∇dblH3(Ω)exp{−β2\n2t} →0, ast→\n+∞. Thusp∞= ¯p, v∞=u∞= 0,ω∞= 0.\nBy Lemma 3 .10, we have\n|St|∞/lessorsimilar/parenleftBigg\n/summationtext\n1≤ℓ≤3/ba∇dbl∂ℓ\ntS/ba∇dbl2\nH3−ℓ/parenrightBigg1\n2\n/lessorsimilar/ba∇dbl(ξ0,v0)/ba∇dblH3(Ω)/ba∇dblφ0/ba∇dblH3(Ω)/parenleftbig\nexp{/ba∇dbl(ξ0,v0)/ba∇dblH3(Ω)}/parenrightbigc8\n2exp{−β2\n2t}.\n(4.5)\nSo∞/integraltext\n0Ss(x,s)dsconverges, then S∞(x) =S0(x)+∞/integraltext\n0Ss(x,s)dsis bounded.\nThus,S∞(x) exists uniquely.\nIn order to prove that S∞/ne}ationslash=constifS0/ne}ationslash=const, we assume S∞=const.\n30For anyα∈R, we have\n(̺exp{S\nγ+αS})t+∇·(̺exp{S\nγ+αS}u),\nd\ndt/integraltext\nΩ̺exp{S\nγ+αS}dx= 0,\n/integraltext\nΩ̺∞exp{S∞\nγ+αS∞}dx=/integraltext\nΩ̺0exp{S0\nγ+αS0}dx,\n/integraltext\nΩp1\nγ∞exp{αS∞}dx=/integraltext\nΩp1\nγ\n0exp{αS0}dx.(4.6)\nBy assumption S∞=const, then we have\n/integraltext\nΩp1\nγ\n0(exp{S0−S∞})αdx=/integraltext\nΩp1\nγ∞dx=/integraltext\nΩp1\nγ\n0dx,\n/parenleftbigg/integraltext\nΩ(p1\nαγ\n0exp{S0−S∞})αdx/parenrightbigg1\nα\n=/parenleftbigg/integraltext\nΩp1\nγ\n0dx/parenrightbigg1\nα(4.7)\nWhenα >0, for any 0 < δ≪1, there exists α≥max{logp0\nγlog(1+δ),logp0\nγlog(1−δ)}\nsuch that 1 −δ≤p1\nαγ\n0≤1+δ, then\n(1−δ)/parenleftbigg/integraltext\nΩ(exp{S0−S∞})αdx/parenrightbigg1\nα\n≤/parenleftbigg/integraltext\nΩp1\nγ\n0dx/parenrightbigg1\nα\n≤(1+δ)/parenleftbigg/integraltext\nΩ(exp{S0−S∞})αdx/parenrightbigg1\nα\n(4.8)\nLetδ→0,α→+∞, we have /ba∇dblexp{S0−S∞}/ba∇dbl∞= 1.\nWhenα <0,forany0 < δ≪1,thereexists α≤min{−logp0\nγlog(1+δ),−logp0\nγlog(1−δ)}\nsuch that 1 −δ≤p1\n−αγ\n0≤1+δ, then\n(1−δ)/parenleftbigg/integraltext\nΩ(exp{S∞−S0})−αdx/parenrightbigg1\n−α\n≤/parenleftbigg/integraltext\nΩp1\nγ\n0dx/parenrightbigg1\n−α\n≤(1+δ)/parenleftbigg/integraltext\nΩ(exp{S∞−S0})−αdx/parenrightbigg1\n−α\n(4.9)\nLetδ→0,α→ −∞, we have /ba∇dblexp{S∞−S0}/ba∇dbl∞= 1.\nSo,S0≡S∞=const, it contradicts with the assumption S0/ne}ationslash=const.\nThus, we proved that S∞/ne}ationslash=constifS0/ne}ationslash=const.\nMoreover, ̺∞(x) =1\nγ√\nA¯p1\nγexp{−S∞(x)\nγ} /ne}ationslash=const, due to the pressure law\n(1.2).θ∞(x) =¯p\nR̺∞(x)/ne}ationslash=const, whereRis universal gas constant. e∞(x) =\nCVθ∞(x)/ne}ationslash=const, wereCV>0 is constant.\nThe exponential decay rates of ( ξ,v,φt) provides exponential convergence\n31rates of ( p,u,S,̺) to their equilibrium states as follows:\n\n\n|p−p∞|∞=|p−¯p|∞/lessorsimilar/ba∇dbl(ξ0,v0)/ba∇dblH3(Ω)exp{−β2\n2t},\n|u−0|∞=k1|v|∞/lessorsimilar/ba∇dbl(ξ0,v0)/ba∇dblH3(Ω)exp{−β2\n2t},\n|S(x,t)−S∞(x)|∞=|−∞/integraltext\ntSs(x,s)ds|∞≤∞/integraltext\nt|φs(x,s)|∞ds\n/lessorsimilar∞/integraltext\nt/ba∇dbl(ξ0,v0)/ba∇dblH3(Ω)/ba∇dblφ0/ba∇dblH3(Ω)/parenleftbig\nexp{/ba∇dbl(ξ0,v0)/ba∇dblH3(Ω)}/parenrightbigc8\n2exp{−β2\n2s}ds\n/lessorsimilar/ba∇dbl(ξ0,v0)/ba∇dblH3(Ω)/ba∇dblφ0/ba∇dblH3(Ω)/parenleftbig\nexp{/ba∇dbl(ξ0,v0)/ba∇dblH3(Ω)}/parenrightbigc8\n2exp{−β2\n2t},\n|̺(x,t)−̺∞(x)|∞/lessorsimilarexp{−β2\n2t}.\n(4.10)\nSo (p,u,S,̺)→(¯p,0,S∞,̺∞) exponentially in |·|∞norm as t→+∞.\nRemark 4.5. For Cauchy problem, it is easier to understand that the equi-\nlibrium states are not constant states. The linear equation s of the nonlinear\nequations in (2.4)are\n\nξt+k2∇·v= 0,\nvt+k2∇ξ+av= 0,\nφt= 0.(4.11)\nLetη∈R3is a vector in Fourier space while xis a vector in physical space.\nAfter Fourier transformation of (4.11), we get\n∂t\nˆξ(k,t)\nˆv(k,t)\nˆφ(k,t)\n=\n0−ik2η⊤0\n−ik2η−aI30\n0 0 0\n\nˆξ(k,t)\nˆv(k,t)\nˆφ(k,t)\n,(4.12)\nwhere the coefficient matrix is denoted by M(η).\nThen the eigenvalues of M(η)satisfy the following equation\n|λI5−M(η)|=/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingleλ ik 2η⊤0\nik2η(λ+a)I30\n0 0 λ/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle=λ/vextendsingle/vextendsingle/vextendsingle/vextendsingleλ ik 2η⊤\nik2η(λ+a)I3/vextendsingle/vextendsingle/vextendsingle/vextendsingle= 0.(4.13)\nSo the coefficient matrix M(η)has one eigenvalue λ5= 0, other eigenvalues\nhave negative reals. λ5= 0corresponds to its eigenvector (0,···,0,1)⊤, thusφ\nmay not decay to zero.\nHowever, the damping effect on the velocity is weakly dissipative, i.e., t here\nare a class of initial data such that the classical solutions blows up. O ur proof\nis based on the analysis of the moment M̺(t) and the finite propagation speed\nof (ξ,v,φ). So, we need to prove the following lemma which states the classical\nsolutions possess finite propagation speed, especially near the bou ndary.\nLemma 4.6. Assume inf\nx∈Ωp0(x)>0,(ξ,v,φ)∈C1(Ω×[0,τ))is the classical\nsolutions to IBVP (2.4),(p,u,S)∈C1(Ω×[0,τ))is the classical solutions to\n32IBVP(1.3). For any x∈Ω∪∂Ω,0≤t1< t2< τ, if(ξ,v,φ)(y,t1) = 0in\n{|y−x| ≤k2(t2−t1)}∩Ω, then(ξ,v,φ)(x,t2) = 0. Equivalently, if (p−¯p,u,S−\n¯S)(y,t1) = 0in{|y−x| ≤k2(t2−t1)}∩Ω, then(p−¯p,u,S−¯S)(x,t2) = 0.\nProof.For any fixed ( x,t2)∈(Ω∪∂Ω)× {t2}, we define the intersection of a\ntruncated cone and Ω as\nO(s) :={(y,ι)| |y−x| ≤k2(t2−ι),t1≤ι≤s≤t2}∩Ω. (4.14)\nand the energy at the time s\ne(s) =/integraltext\n{|y−x|≤k2(t2−s)}∩Ωξ2+|v|2+φ2dy.(4.15)\nBy (2.4)·(ξ,v,φ), we get\n(ξ2+|v|2+φ2)t+2k2ξ∇·v+2k2v·∇ξ+2a|v|2\n=−2γk1ξ2∇·v−2k1ξv·∇ξ−2k1v·∇v·v+2\nk1(1\n¯̺−1\n̺)v·∇ξ−2k1φv·∇φ.\n(4.16)\nAfter integrating (4 .16) inO(s), we get\n/integraltext\n{|y−x|≤k2(t2−s)}∩Ωξ2+|v|2+φ2dy−/integraltext\n{|y−x|≤k2(t2−t1)}∩Ωξ2+|v|2+φ2dy\n+1√\n1+k2\n2s/integraltext\nt1/integraltext\n{|y−x|=k2(t−ι)}∩Ωk2(ξ2+|v|2+φ2)+2k2ξv·y−x\n|y−x|dydι\n+s/integraltext\nt1/integraltext\n∂Ω∩{|y−x|≤k2(t−ι)}2k2ξv·ndSydι+2a/integraltext\nO(s)|v|2dydι\n=−2/integraltext/integraltext\nO(s)γk1ξ2∇·v+k1ξv·∇ξ+k1v·∇v·v−1\nk1(1\n¯̺−1\n̺)v·∇ξ\n+k1φv·∇φdydι≤Cmax\nO(t2){∇ξ,∇v,∇φ}/integraltext/integraltext\nO(s)ξ2+|v|2+φ2dydι.\n(4.17)\nWhile we have the following three estimates:\ns/integraltext\nt1/integraltext\n{|y−x|=k2(t−ι)}∩Ωk2(ξ2+|v|2+φ2)+2k2ξv·y−x\n|y−x|dydι≥0,\ns/integraltext\nt1/integraltext\n∂Ω∩{|y−x|≤k2(t−ι)}2k2ξv·ndSydι= 0,\n/integraltext/integraltext\nO(s)|v|2dydι≥0.(4.18)\nBy (4.17) and (4 .18), we get\ne(s)−e(t1)≤Cmax\nO(t2){∇ξ,∇v,∇φ}s/integraltext\nt1e(ι)dι. (4.19)\nBy Gronwall’s inequality, for ∀s∈[t1,t2],\ne(s)≤e(t1)exp{Cmax\nO(t2){∇ξ,∇v,∇φ}(s−t1)}.(4.20)\n33Thus, if( ξ,v,φ)(y,t1) = 0in{|y−x| ≤k2(t2−t1)}∩Ω, then( ξ,v,φ)(x,t2) =\n0. Equivalently, if ( p−¯p,u,S−¯S)(y,t1) = 0 in{|y−x| ≤k2(t2−t1)}∩Ω, then\n(p−¯p,u,S−¯S)(x,t2) = 0. Thus, Lemma 4 .6 is proved.\nTo verify the weak dissipativity of non-isentropic Euler equations wit h\ndamping, we proved the following theorem which states that for a cla ss of\nlarge data, the singularities form in the interior of ideal gases. In ou r proof,\nSupp(p0−¯p,u0,S0−¯S) is required to be away from ∂Ω. Thus, ∂ℓ\ntu(x,0)·n|∂Ω=\n0,0≤ℓ≤3 are satisfied automatically.\nTheorem 4.7. Assume 0∈Ω,(p0,u0,S0)∈H3(Ω),inf\nx∈Ωp0(x)>0,h=\ndist{∂Ω,Supp(p0−¯p,u0,S0−¯S)}>0,(p,u,S)∈C1(Ω×[0,τ))is the classical\nsolution to IBVP (1.3)whereτ >0is the lifespan of (p,u,S). For any fixed T\nsatisfying 0< T <min{h\nk2,π\n2B0\nr}, if the condition (2.7)holds, then τ≤T.\nProof.The quantities M̺(t),B0,B1,rare defined in (2 .6). The set of initial\ndata satisfying the inequality (2 .7) and the conditions in Theorem 4 .7 is not\nempty set if u0is large enough, since the right hand side of (2 .7) does not\ncontainu0.\nIt follows from IBVP (1 .1) that for ∀t∈[0,T),\nd\ndtM̺(t)+aM̺(t)\n=/integraltext\nΩ̺tu·xdx+/integraltext\nΩ̺ut·xdx+a/integraltext\nΩ̺u·xdx\n=−/integraltext\nΩ∇·(̺u)(u·x)dx−/integraltext\nΩ̺u·∇u·x+x·∇pdx\n=−/integraltext\nΩ∇·(̺u)(u·x)dx−/integraltext\nΩ̺u·∇(u·x)−̺u·∇x·u+x·∇pdx\n=−/integraltext\n∂Ω(u·x)̺u·ndSx+/integraltext\nΩ̺|u|2dx−/integraltext\nΩx·∇(p−¯p)dx\n=/integraltext\nΩ̺|u|2dx−/integraltext\n∂Ω(p−¯p)(x·n)dSx+3/integraltext\nΩpdx−3/integraltext\nΩ¯pdx.(4.21)\nBy Lemma 4 .6 on the finite propagation speed of classical solutions near\nthe boundary, for any ( x,t)∈∂Ω×[0,T),p−¯p= 0 due to h > k2T. So in the\ntime interval [0 ,T), we have\n/integraltext\n∂Ω(p−¯p)(x·n)dSx= 0.(4.22)\nSinceSis invariant along the particle paths, S(x,t)≥S−, then we get\n/integraltext\nΩpdx≥AeS−/integraltext\nΩ̺γdx.(4.23)\nBy H¨older’s inequality, we have/integraltext\nΩ̺dx≤/parenleftbigg/integraltext\nΩ̺γdx/parenrightbigg1\nγ/parenleftbigg/integraltext\nΩ1dx/parenrightbigg1−1\nγ\n, then\nwe get\n/integraltext\nΩ̺γdx≥1\n|Ω|γ−1/parenleftbigg/integraltext\nΩ̺dx/parenrightbiggγ\n=1\n|Ω|γ−1/parenleftbigg/integraltext\nΩ̺0dx/parenrightbiggγ\n. (4.24)\n34Therefore, it follows from (4 .21) that\nd\ndtM̺(t)+aM̺(t)≥/integraltext\nΩ̺|u|2dx+3AeS−\n|Ω|γ−1/parenleftbigg/integraltext\nΩ̺0dx/parenrightbiggγ\n−3/integraltext\nΩ¯pdx\n=/integraltext\nΩ̺|u|2dx+B1.(4.25)\nWhile, by Cauchy-Schwarz inequality, we get\nM̺(t)2=/parenleftbigg/integraltext\nΩ̺u·xdx/parenrightbigg2\n≤/integraltext\nΩ̺|u|2dx/integraltext\nΩ̺|x|2dx\n≤/integraltext\nΩ̺|u|2dx/parenleftbigg\n|Diam(Ω)|2/integraltext\nΩ̺dx/parenrightbigg\n=B0/integraltext\nΩ̺|u|2dx.(4.26)\nIfB1= 0, it follows from (4 .21) that\nd\ndtM̺(t)+aM̺(t)≥/integraltext\nΩ̺|u|2dx≥M̺(t)2\nB0,\nd\ndt[M̺(t)exp(at)]≥(M̺(t)exp(at))2\nB0exp(at),\n1\nM̺(t)exp(at)−1\nM̺(0)≤t/integraltext\n0−1\nB0exp(as)ds=1\naB0(exp{−at}−1),\n1\nM̺(t)≤exp{−at}[1\nM̺(0)+1\naB0(exp{−at}−1)].(4.27)\nSinceM̺(0)>aB0\n1−exp{−aT}>0, whentissmall,M̺(t)>0byitscontinuity.\nM̺(t)≥exp{at}\n1\nM̺(0)+1\naB0(exp{−at}−1)>0. (4.28)\nWhent→T−,thedenominator1\nM̺(0)+1\naB0(exp{−at}−1)→0+,M̺(t)→\n+∞. Thusτ≤T.\nIfB1/ne}ationslash= 0, it follows from (4 .21) that\nd\ndtM̺(t)≥M̺(t)2\nB0−aM̺(t)+B1=1\nB0(M̺(t)−aB0\n2)2+B1−a2B0\n4.(4.29)\nDenote\nN̺(t) =M̺(t)−aB0\n2, r2=/vextendsingle/vextendsingle/vextendsingleB1−a2B0\n4/vextendsingle/vextendsingle/vextendsingle, r >0. (4.30)\nd\ndtN̺(t)≥\n\n1\nB0(N̺(t)2+r2), if B 1>a2B0\n4,\n1\nB0(N̺(t)2−r2), if B 1<a2B0\n4.(4.31)\nIntegrating from 0 to τ, we obtain,\nt\nB0≤\n\n1\nrarctan(N̺(t)\nr)−1\nrarctan(N̺(0)\nr), if B 1>a2B0\n4,\n1\n2rln|N̺(t)−r\nN̺(t)+r|−1\n2rln|N̺(0)−r\nN̺(0)+r|, if B 1<a2B0\n4.(4.32)\n35As to (4.32)1,\nN̺(t)≥rtan(rt\nB0)+N̺(0)\n1−N̺(0)\nrtan(rt\nB0),\nM̺(t)≥aB0\n2+rtan(rt\nB0)+M̺(0)−aB0\n2\n1−M̺(0)−aB0\n2\nrtan(rt\nB0).(4.33)\nSinceM̺(0)>aB0\n2+rcot(rT\nB0), whentis small, M̺(t)>0by its continuity.\nWhent→T−, the denominator 1 −M̺(0)−aB0\n2\nrtan(rt\nB0)→0+,M̺(t)→+∞.\nSinceT <π\n2B0\nrand [0,τ) as the lifespan of the solution is simply connected,\nτ≤T.\nAs to (4.32)2, sinceM̺(0)>aB0\n2+r,N̺(0)> r,N̺(t) is increasing due\ntod\ndtN̺(t)≥1\nB0(N̺(t)2−r2)>0. thenN̺(t)−r\nN̺(t)+r>0.\nN̺(t)≥ −r+2r\n1−exp{2rt\nB0+ln|N̺(0)−r\nN̺(0)+r|},\nM̺(t)≥aB0\n2−r+2r\n1−exp{2rt\nB0+ln|M̺(0)−aB0\n2−r\nM̺(0)−aB0\n2+r|}.(4.34)\nSinceM̺(0)>max{aB0\n2−r+2r\n1−exp{−2rT\nB0},aB0\n2+r}, whentis small,\nM̺(t)>0 by its continuity. When t→T−, the denominator 1 −exp{2rt\nB0+\nln|M̺(0)−aB0\n2−r\nM̺(0)−aB0\n2+r|} →0+,M̺(t) =M̺(τ)→+∞. Thusτ≤T.\nThus, Theorem 4 .7 on the singularity formation of classical solutions for a\nclass of large data is proved.\nRemark 4.8. We cannot remove the condition dist(∂Ω,Supp(p0−¯p,u0,S0−\n¯S))>0, otherwise the term −/integraltext\n∂Ω(p−¯p)(x·n)dSxin(4.21)cannot be bounded.\nDenote the area of ∂Ωby|∂Ω|, we have\n−/integraltext\n∂Ω(p−¯p)(x·n)dSx\n≥ −¯p|Diam(Ω)||∂Ω|−A|Diam(Ω)|eS+/integraltext\n∂Ω̺γdSx\n≥ −¯p|Diam(Ω)||∂Ω|−CA|Diam(Ω)|eS+/ba∇dbl̺/ba∇dblγ−1\nLγ(Ω)/ba∇dbl∇̺/ba∇dblLγ(Ω)\n≥ −¯p|Diam(Ω)||∂Ω|−CA|Diam(Ω)|eS+\n|Ω|(γ−1)2\nγ/parenleftbigg/integraltext\nΩ̺0dx/parenrightbiggγ−1\n/ba∇dbl∇̺/ba∇dblLγ(Ω).(4.35)\nwhere/ba∇dbl̺/ba∇dblγ\nLγ(∂Ω)≤C/ba∇dbl̺/ba∇dblγ−1\nLγ(Ω)/ba∇dbl∇̺/ba∇dblLγ(Ω)(see [1]) is applied. While /ba∇dbl∇̺/ba∇dblLγ(Ω)\nis not conserved and may not be bounded by the initial data.\n365 Global A Priori Estimates for Diffusion Equa-\ntions\nIn this section, wederiveglobalaprioriestimatesforthe nonlinear diffusion\nequations (2 .8). For simplicity, we omit the symbol ˆ over the variables and\nconstants in this section if there is no ambiguity.\nThe following lemma is also an application of Lemma 3 .2, which states that\nthespatialderivativesareboundedbythetemporalderivativesa ndthevorticity,\nthus the total energy F[ξ,v](t) can be bounded by E[ξ,v](t) andE1[ω](t).\nLemma 5.1. For any given T∈(0,+∞], there exists ε0>0which is indepen-\ndent of(ξ0,v0,φ0), such that if\nsup\n0≤t≤TF[ξ,v,φ](t)≤ε,\nwhereε≪min{1,ε0}, then for ∀t∈[0,T],\nF[ξ,v](t)≤c10(E[ξ,v](t)+E1[ω](t)), (5.1)\nfor some c10>0.\nProof.Since the equation (2 .8)1is the same with the equation (2 .4)1, similar to\nthe estimates in Lemma 3 .3, we have\n/ba∇dbl∇v/ba∇dbl2\nL2(Ω)/lessorsimilar/ba∇dblξt/ba∇dbl2\nL(Ω)+/ba∇dblω/ba∇dbl2\nL2(Ω)+/ba∇dblv/ba∇dbl2\nL2(Ω)+√εF[ξ,v](t),\n/ba∇dbl∇vt/ba∇dbl2\nL2(Ω)/lessorsimilar/ba∇dblξtt/ba∇dbl2\nL2(Ω)+/ba∇dblωt/ba∇dbl2\nL2(Ω)+/ba∇dblvt/ba∇dbl2\nL2(Ω)+√εF[ξ,v](t),\n/ba∇dbl∇vtt/ba∇dbl2\nL2(Ω)/lessorsimilar/ba∇dblξttt/ba∇dbl2\nL2(Ω)+/ba∇dblωtt/ba∇dbl2\nL2(Ω)+/ba∇dblvtt/ba∇dbl2\nL2(Ω)+√εF[ξ,v](t).(5.2)\nMoreover, when |α|= 1,\n/ba∇dblDα∇·v/ba∇dbl2\nL2(Ω)/lessorsimilar/ba∇dblDαξt/ba∇dbl2\nL2(Ω)+√εF[ξ,v](t)/lessorsimilar/ba∇dblvt/ba∇dbl2\nL2(Ω)+√εF[ξ,v](t),\n/ba∇dblDα∇v/ba∇dbl2\nL2(Ω)/lessorsimilar/ba∇dblvt/ba∇dbl2\nL2(Ω)+/ba∇dblω/ba∇dbl2\nH1(Ω)+/ba∇dblv/ba∇dbl2\nL2(Ω)+/ba∇dblξt/ba∇dbl2\nL2(Ω)+√εF[ξ,v](t).\n(5.3)\n/ba∇dblDα∇·vt/ba∇dbl2\nL2(Ω)/lessorsimilar/ba∇dblDαξtt/ba∇dbl2\nL2(Ω)+√εF[ξ,v](t)/lessorsimilar/ba∇dblvtt/ba∇dbl2\nL2(Ω)+√εF[ξ,v](t),\n/ba∇dblDα∇vt/ba∇dbl2\nL2(Ω)/lessorsimilar/ba∇dblξtt/ba∇dbl2\nL2(Ω)+/ba∇dblvtt/ba∇dbl2\nL2(Ω)+/ba∇dblωt/ba∇dbl2\nH1(Ω)+/ba∇dblvt/ba∇dbl2\nL2(Ω)\n+√εF[ξ,v](t).\n(5.4)\nWhen|α|= 2,\n/ba∇dblDα∇·v/ba∇dbl2\nL2(Ω)/lessorsimilar/ba∇dblDαξt/ba∇dbl2\nL2(Ω)+√εF[ξ,v](t)\n/lessorsimilar/ba∇dblξtt/ba∇dbl2\nH1(Ω)+/ba∇dblωt/ba∇dbl2\nL2(Ω)+/ba∇dblvt/ba∇dbl2\nL2Ω)+√εF[ξ,v](t),\n/ba∇dblDα∇v/ba∇dbl2\nL2(Ω)/lessorsimilar/ba∇dblξtt/ba∇dbl2\nH1(Ω)+/ba∇dblωt/ba∇dbl2\nH1(Ω)+/ba∇dblω/ba∇dbl2\nH2(Ω)+/ba∇dblvt/ba∇dbl2\nL2Ω)\n+/ba∇dblv/ba∇dbl2\nL2Ω)+/ba∇dblξt/ba∇dbl2\nL2Ω)+√εF[ξ,v](t).(5.5)\n37By (2.13)2, we get\n∇ξ=−ak1̺v,\n/ba∇dbl∇ξ/ba∇dbl2\nL2(Ω)/lessorsimilar/ba∇dblv/ba∇dbl2\nL2(Ω).(5.6)\nApply∂tto (5.6)1, we get\n∇ξt=−ak1ξtv−ak1̺vt,\n/ba∇dbl∇ξt/ba∇dbl2\nL2(Ω)/lessorsimilar/ba∇dblvt/ba∇dbl2\nL2(Ω)+√εF[ξ,v](t).(5.7)\nApply∂ttto (5.6)1, we get\n∇ξtt=−ak1ξttv−2ak1ξtvt−ak1̺vtt,\n/ba∇dbl∇ξtt/ba∇dbl2\nL2(Ω)/lessorsimilar/ba∇dblvtt/ba∇dbl2\nL2(Ω)+√εF[ξ,v](t).(5.8)\nApply∂tttto (5.6)1, we get\n∇ξttt=−ak1ξtttv−3ak1ξttvt−3ak1ξtvtt−ak1̺vttt,\n/ba∇dbl∇ξttt/ba∇dbl2\nL2(Ω)/lessorsimilar/ba∇dblvttt/ba∇dbl2\nL2(Ω)+√εF[ξ,v](t).(5.9)\nApplyDαto (5.6)1, where|α|= 1, we get\nDα∇ξ=−ak1(Dαξ)v−ak1̺Dαv,\n/ba∇dblDα∇ξ/ba∇dbl2\nL2(Ω)/lessorsimilar/ba∇dblDαv/ba∇dbl2\nL2(Ω)+√εF[ξ,v](t)\n/lessorsimilar/ba∇dblξt/ba∇dbl2\nL2(Ω)+/ba∇dblω/ba∇dbl2\nL2(Ω)+/ba∇dblv/ba∇dbl2\nL2(Ω)+√εF[ξ,v](t).(5.10)\nApplyDαto (5.7)1, where|α|= 1, we get\nDα∇ξt=−ak1Dα(ξtv)−ak1(Dαξ)vt−ak1̺Dαvt,\n/ba∇dblDα∇ξt/ba∇dbl2\nL2(Ω)/lessorsimilar/ba∇dblDαvt/ba∇dbl2\nL2(Ω)+√εF[ξ,v](t)\n/lessorsimilar/ba∇dblξtt/ba∇dbl2\nL2(Ω)+/ba∇dblωt/ba∇dbl2\nL2(Ω)+/ba∇dblvt/ba∇dbl2\nL2(Ω)+√εF[ξ,v](t).(5.11)\nApplyDαto (5.8)1, where|α|= 1, we get\nDα∇ξtt=−ak1Dα(ξttv)−2ak1Dα(ξtvt)−ak1(Dα̺)vtt−ak1̺Dαvtt,\n/ba∇dblDα∇ξtt/ba∇dbl2\nL2(Ω)/lessorsimilar/ba∇dblDαvtt/ba∇dbl2\nL2(Ω)+√εF[ξ,v](t)\n/lessorsimilar/ba∇dblξttt/ba∇dbl2\nL2(Ω)+/ba∇dblωtt/ba∇dbl2\nL2(Ω)+/ba∇dblvtt/ba∇dbl2\nL2(Ω)+√εF[ξ,v](t).\n(5.12)\nApplyDαto (5.6)1, where|α|= 2,α=α1+α2, we get\nDα∇ξ=−ak1̺Dαv−ak1/summationtext\nα1>0(Dα1ξ)Dα2v,\n/ba∇dblDα∇ξ/ba∇dbl2\nL2(Ω)/lessorsimilar/ba∇dblDαv/ba∇dbl2\nL2(Ω)+√εF[ξ,v](t)\n/lessorsimilar/ba∇dblvt/ba∇dbl2\nL2(Ω)+/ba∇dblω/ba∇dbl2\nH1(Ω)+/ba∇dblξt/ba∇dbl2\nL2(Ω)+/ba∇dblv/ba∇dbl2\nL2(Ω)+√εF[ξ,v](t).\n(5.13)\n38ApplyDαto (5.7)1, where|α|= 2,α=α1+α2, we get\nDα∇ξt=−ak1Dα(ξtv)−ak1/summationtext\nα1>0(Dα1̺Dα2vt)−ak1̺Dαvt,\n/ba∇dblDα∇ξt/ba∇dbl2\nL2(Ω)/lessorsimilar/ba∇dblDαvt/ba∇dbl2\nL2(Ω)+√εF[ξ,v](t)\n/lessorsimilar/ba∇dblξtt/ba∇dbl2\nL2(Ω)+/ba∇dblvtt/ba∇dbl2\nL2(Ω)+/ba∇dblωt/ba∇dbl2\nH1(Ω)+/ba∇dblvt/ba∇dbl2\nL2(Ω)\n+√εF[ξ,v](t).(5.14)\nApplyDαto (5.6)1, where|α|= 3,α=α1+α2, we get\nDα∇ξ=−ak1̺Dαv−ak1/summationtext\nα1>0(Dα1ξ)Dα2v,\n/ba∇dblDα∇ξ/ba∇dbl2\nL2(Ω)/lessorsimilar/ba∇dblDαv/ba∇dbl2\nL2(Ω)+√εF[ξ,v](t)\n/lessorsimilar/ba∇dblvtt/ba∇dbl2\nL2(Ω)+/ba∇dblξtt/ba∇dbl2\nH1(Ω)+/ba∇dblωt/ba∇dbl2\nH1(Ω)+/ba∇dblω/ba∇dbl2\nH2(Ω)\n+/ba∇dblvt/ba∇dbl2\nL2Ω)+/ba∇dblv/ba∇dbl2\nL2(Ω)+/ba∇dblξt/ba∇dbl2\nL2(Ω)+√εF[ξ,v](t).(5.15)\nBy (2.12), we have\n∇(∇·v) =a̺\nγpvt+k1(γ−1)a̺\nγpv(∇·v)+k1a̺\nγpv·∇v+k1a̺\n2γp∇(|v|2),\n/ba∇dbl∇·v/ba∇dbl2\nH3(Ω)/lessorsimilar/ba∇dbl∇(∇·v)/ba∇dbl2\nH2(Ω)/lessorsimilar/ba∇dbl̺\npvt/ba∇dbl2\nH2(Ω)+/ba∇dbl̺\npv(∇·v)/ba∇dbl2\nH2(Ω)\n+/ba∇dbl̺\npv·∇v/ba∇dbl2\nH2(Ω)+/ba∇dbl̺\np∇(|v|2)/ba∇dbl2\nH2(Ω)\n/lessorsimilar/ba∇dblvt/ba∇dbl2\nH2(Ω)+√εF[ξ,v](t)\n/lessorsimilar/ba∇dblξtt/ba∇dbl2\nL2(Ω)+/ba∇dblvtt/ba∇dbl2\nL2(Ω)+/ba∇dblωt/ba∇dbl2\nH1(Ω)+/ba∇dblvt/ba∇dbl2\nL2(Ω)+√εF[ξ,v](t).\n(5.16)\nApply∂tto (5.16)1, we get\n∇(∇·vt) =a̺\nγpvtt+a\nγ(̺\np)tvt+k1(γ−1)a\nγ∂t[̺\npv(∇·v)]+k1a\nγ∂t[̺\npv·∇v]\n+k1a\n2γ∂t[̺\np∇(|v|2)],\n/ba∇dbl∇·vt/ba∇dbl2\nH2(Ω)/lessorsimilar/ba∇dbl∇(∇·vt)/ba∇dbl2\nH1(Ω)/lessorsimilar/ba∇dblvtt/ba∇dbl2\nH1(Ω)+√εF[ξ,v](t)\n/lessorsimilar/ba∇dblξttt/ba∇dbl2\nL2(Ω)+/ba∇dblωtt/ba∇dbl2\nL2(Ω)+/ba∇dblvtt/ba∇dbl2\nL2(Ω)+√εF[ξ,v](t).\n(5.17)\nApply∂ttto (5.16)1, we get\n∇(∇·vtt) =a̺\nγpvttt+2a\nγ(̺\np)tvtt+a\nγ(̺\np)ttvt+k1(γ−1)a\nγ∂tt[̺\npv(∇·v)]\n+k1a\nγ∂tt[̺\npv·∇v]+k1a\n2γ∂tt[̺\np∇(|v|2)],\n/ba∇dbl∇·vtt/ba∇dbl2\nH1(Ω)/lessorsimilar/ba∇dbl∇(∇·vtt)/ba∇dbl2\nL2(Ω)/lessorsimilar/ba∇dblvttt/ba∇dbl2\nL2(Ω)+√εF[ξ,v](t).\n(5.18)\nThus,F[ξ,v](t)≤C8E[ξ,v](t)+C8E1[ω](t)+C8√εF[ξ,v](t), whereC8>0.\nLetε0=1\n4C2\n8, whenε≪min{1,ε0}, we have\nF[ξ,v](t)≤2C8{E[ξ,v](t)+E1[ω](t)}. (5.19)\nLetc10= 2C8. Thus, Lemma 5 .1 is proved.\n39Next, in order to prove the exponential decay of F[ξ,v](t) andE1[ω](t),\nwe need to prove a priori estimates for E1[ω](t),E[ξ](t),/integraltext\nΩ3/summationtext\nℓ=1∂ℓ\ntξ∂ℓ−1\ntξdxand\nE[v](t) separatively.\nThe following lemma concerns a priori estimate for E1[ω](t).\nLemma 5.2. For any given T∈(0,+∞], if\nsup\n0≤t≤TF[ξ,v,φ](t)≤ε,\nwhere0< ε≪1, then for ∀t∈[0,T],\nd\ndt/integraltext\nΩE1[ω](t)dx+2aE1[ω](t)≤C√εF[ξ,v](t).(5.20)\nProof.By∇×(2.13)2, we get\n∂tω+aω=k1(1−γ)[ω(∇·v)−v×∇(∇·v)]−k1v·∇ω−k1ω(∇·v)\n+k1ω·∇v+γ\na∇(p\n̺)×∇(∇·v)+1\nk1(∂i̺\n̺2∂jξ−∂j̺\n̺2∂iξ)k.\n(5.21)\nApply∂ℓ\ntDαto (5.21), where 0 ≤ℓ+|α| ≤2, we get\n∂t(∂ℓ\ntDαω)+a∂ℓ\ntDαω\n=k1(1−γ)∂ℓ\ntDα[ω(∇·v)−v×∇(∇·v)]−k1∂ℓ\ntDα[v·∇ω+ω(∇·v)\n−ω·∇v]+γ\na∂ℓ\ntDα[∇(p\n̺)×∇(∇·v)]+1\nk1∂ℓ\ntDα(∂i̺\n̺2∂jξ−∂j̺\n̺2∂iξ)k.\n(5.22)\nLet∂ℓ\ntDαω·(5.22) and integrate in Ω, we get\nd\ndt/integraltext\nΩ|∂ℓ\ntDαω|2dx+2a/integraltext\nΩ|∂ℓ\ntDαω|2dx=I4+I5,(5.23)\nwhere\nI4:= 2/integraltext\nΩk1(1−γ)∂ℓ\ntDαω·∂ℓ\ntDα[ω(∇·v)−v×∇(∇·v)]\n−k1∂ℓ\ntDαω·∂ℓ\ntDα[v·∇ω+ω(∇·v)−ω·∇v]\n+γ\na∂ℓ\ntDαω·∂ℓ\ntDα[∇(p\n̺)×∇(∇·v)]dx,\nI5:=2\nk1/integraltext\nΩ3/summationtext\nk=1∂ℓ\ntDα(∂i̺\n̺2∂jξ−∂j̺\n̺2∂iξ)k·∂ℓ\ntDαωkdx.(5.24)\nWhenℓ+|α|<2, it is easy to check that I4+I5/lessorsimilar√εF[ξ,v](t), since they\nare lower order terms.\nSinceI5has the same form with I2in the proof of Lemma 3 .4, repeat those\nestimates (3 .24),(3.25),(3.26), we have\nI5/lessorsimilar√εE[ξ,v](t)/lessorsimilar√εF[ξ,v](t). (5.25)\n40Thus, we just need to estimate I4, whenℓ= 0,|α|= 2,\nI4/lessorsimilar/ba∇dblDαω/ba∇dblL2(Ω)(|∇·v|∞/ba∇dblDαω/ba∇dblL2(Ω)+/ba∇dblDα1ω/ba∇dblL4(Ω)/ba∇dblDα2∇·v/ba∇dblL4(Ω)\n+|ω|∞/ba∇dblDα∇·v/ba∇dblL2(Ω)+|∇(∇·v)|∞/ba∇dblDαv/ba∇dblL2(Ω)\n+|Dα1v|∞/ba∇dblDα2∇(∇·v)/ba∇dblL2(Ω)+|v|∞/ba∇dblDα∇(∇·v)/ba∇dblL2(Ω)\n+/ba∇dblDαv/ba∇dblL4(Ω)/ba∇dbl∇ω/ba∇dblL4(Ω)+|Dα1v|∞/ba∇dblDα2∇ω/ba∇dblL2(Ω)\n+|∇·v|∞/ba∇dblDαω/ba∇dblL2(Ω)+/ba∇dblDα1ω/ba∇dblL4(Ω)/ba∇dblDα2∇·v/ba∇dblL4(Ω)\n+|ω|∞/ba∇dblDα∇·v/ba∇dblL2(Ω)+|ω|∞/ba∇dblDα∇v/ba∇dblL2(Ω)+/ba∇dblDα1ω/ba∇dblL4(Ω)/ba∇dblDα2∇v/ba∇dblL4(Ω)\n+|∇v|∞/ba∇dblDαω/ba∇dblL2(Ω)+|∇(p\n̺)|∞/ba∇dblDα∇(∇·v)/ba∇dblL2(Ω)\n+/ba∇dblDα1∇(p\n̺)/ba∇dblL4(Ω)/ba∇dblDα2∇(∇·v)/ba∇dblL4(Ω)+|∇(∇·v)|∞/ba∇dblDα∇(p\n̺)/ba∇dblL2(Ω))\n−2k1(/integraltext\n∂Ωv·n|Dαω|2dSx−/integraltext\nΩ∇·v|Dαω|2dx)/lessorsimilar√εF[ξ,v](t),\n(5.26)\nwhere|α1|=|α2|= 1.\nWhenℓ= 1,|α|= 1,\nI4/lessorsimilar/ba∇dblDαωt/ba∇dblL2(Ω)(|∇·v|∞/ba∇dblDαωt/ba∇dblL2(Ω)+/ba∇dblDαω/ba∇dblL4(Ω)/ba∇dbl∇·vt/ba∇dblL4(Ω)\n+/ba∇dblωt/ba∇dblL4(Ω)/ba∇dblDα∇·v/ba∇dblL4(Ω)+|ω|∞/ba∇dblDα∇·vt/ba∇dblL2(Ω)\n+|∇(∇·v)|∞/ba∇dblDαvt/ba∇dblL2(Ω)+|Dαv|∞/ba∇dbl∇(∇·vt)/ba∇dblL2(Ω)\n+|vt|∞/ba∇dblDα∇(∇·v)/ba∇dblL2(Ω)+|v|∞/ba∇dblDα∇(∇·vt)/ba∇dblL2(Ω)\n+/ba∇dblDαvt/ba∇dblL4(Ω)/ba∇dbl∇ω/ba∇dblL4(Ω)+|vt|∞/ba∇dblDα∇ω/ba∇dblL2(Ω)+|Dαv|∞/ba∇dbl∇ωt/ba∇dblL2(Ω)\n+|∇·v|∞/ba∇dblDαωt/ba∇dblL2(Ω)+/ba∇dblωt/ba∇dblL4(Ω)/ba∇dblDα∇·v/ba∇dblL4(Ω)\n+/ba∇dblDαω/ba∇dblL4(Ω)/ba∇dbl∇·vt/ba∇dblL4(Ω)+|ω|∞/ba∇dblDα∇·vt/ba∇dblL2(Ω)\n+|ω|∞/ba∇dblDα∇vt/ba∇dblL2(Ω)+/ba∇dblωt/ba∇dblL4(Ω)/ba∇dblDα∇v/ba∇dblL4(Ω)+/ba∇dblDαω/ba∇dblL4(Ω)/ba∇dbl∇vt/ba∇dblL4(Ω)\n+|∇v|∞/ba∇dblDαωt/ba∇dblL2(Ω)+|∇(p\n̺)|∞/ba∇dblDα∇(∇·vt)/ba∇dblL2(Ω)\n+/ba∇dblDα∇(p\n̺)/ba∇dblL4(Ω)/ba∇dbl∇(∇·vt)/ba∇dblL4(Ω)+/ba∇dbl∂t∇(p\n̺)/ba∇dblL4(Ω)/ba∇dblDα∇(∇·v)/ba∇dblL4(Ω)\n+|∇(∇·v)|∞/ba∇dbl∂tDα∇(p\n̺)/ba∇dblL2(Ω))−2k1(/integraltext\n∂Ωv·n|Dαωt|2dSx\n−/integraltext\nΩ∇·v|Dαωt|2dx)/lessorsimilar√εF[ξ,v](t).\n(5.27)\nWhenℓ= 2,|α|= 0,\nI4/lessorsimilar/ba∇dblωtt/ba∇dblL2(Ω)(|∇·v|∞/ba∇dblωtt/ba∇dblL2(Ω)+/ba∇dblωt/ba∇dblL4(Ω)/ba∇dbl∇·vt/ba∇dblL4(Ω)\n+|ω|∞/ba∇dbl∇·vtt/ba∇dblL2(Ω)+|∇(∇·v)|∞/ba∇dblvtt/ba∇dblL2(Ω)+|vt|∞/ba∇dbl∇(∇·vt)/ba∇dblL2(Ω)\n+|v|∞/ba∇dbl∇(∇·vtt)/ba∇dblL2(Ω)+/ba∇dblvtt/ba∇dblL4(Ω)/ba∇dbl∇ω/ba∇dblL4(Ω)+|vt|∞/ba∇dbl∇ωt/ba∇dblL2(Ω)\n+|∇·v|∞/ba∇dblωtt/ba∇dblL2(Ω)+/ba∇dblωt/ba∇dblL4(Ω)/ba∇dbl∇·vt/ba∇dblL4(Ω)+|ω|∞/ba∇dbl∇·vtt/ba∇dblL2(Ω)\n+|ω|∞/ba∇dbl∇vtt/ba∇dblL2(Ω)+/ba∇dblωt/ba∇dblL4(Ω)/ba∇dbl∇vt/ba∇dblL4(Ω)+|∇v|∞/ba∇dblωtt/ba∇dblL2(Ω)\n41+|∇(p\n̺)|∞/ba∇dbl∇(∇·vtt)/ba∇dblL2(Ω)+/ba∇dbl∂t∇(p\n̺)/ba∇dblL4(Ω)/ba∇dbl∇(∇·vt)/ba∇dblL4(Ω)\n+|∇(∇·v)|∞/ba∇dbl∂tt∇(p\n̺)/ba∇dblL2(Ω))−2k1(/integraltext\n∂Ωv·n|ωtt|2dSx−/integraltext\nΩ∇·v|ωtt|2dx)\n/lessorsimilar√εF[ξ,v](t).\n(5.28)\nAfter summing the above estimates, we have\nd\ndt/integraltext\nΩE1[ω](t)dx+2aE1[ω](t)/lessorsimilar√εF[ξ,v](t).(5.29)\nThus, Lemma 5 .2 is proved.\nSimilar to Lemma 3 .5, the following lemma states that E[ξ](t) andE1[ξ](t)\nare equivalent, E[v](t) andE1[v](t) are equivalent.\nLemma 5.3. For any given T∈(0,+∞], there exists ε1>0which is in-\ndependent of (ˆξ0,ˆv0,ˆφ0), such that if sup\n0≤t≤TF[ξ,v,φ](t)≤ε1, then there exist\nc11>0,c12>0such that\nc11E[ξ](t)≤E1[ξ](t)≤c12E[ξ](t),\nc11E[v](t)≤E1[v](t)≤c12E[v](t).(5.30)\nTo make calculations simpler, we calculated\ndtE1[ξ](t) andd\ndtE[v](t) sepa-\nrately.\nSinceE1[ξ](t)∼=E[ξ](t), the following lemma gives an equivalent a priori\nestimate for E[ξ](t).\nLemma 5.4. For any given T∈(0,+∞], if\nsup\n0≤t≤TF[ξ,v,φ](t)≤ε,\nwhere0< ε≪1, then for ∀t∈[0,T],\nd\ndtE1[ξ](t)+2aE[v](t)≤C√εF[ξ,v](t). (5.31)\nProof.Suppose 0 ≤ℓ≤3, apply ∂ℓ\ntto (2.4), we get\n/braceleftBigg(∂ℓ\ntξ)t+k2∇·(∂ℓ\ntv) =−γk1∂ℓ\nt(ξ∇·v)−k1∂ℓ\nt(v·∇ξ),\nk2∇(∂ℓ\ntξ)+a∂ℓ\ntv=1\nk1∂ℓ\nt[(1\n¯̺−1\n̺)∇ξ].(5.32)\nLet (5.32)·(∂ℓ\ntξ,∂ℓ\ntv), we get\n/braceleftBigg(|∂ℓ\ntξ|2)t+2k2∂ℓ\ntξ∇·(∂ℓ\ntv) =−2γk1∂ℓ\ntξ∂ℓ\nt(ξ∇·v)−2k1∂ℓ\ntξ∂ℓ\nt(v·∇ξ),\n2k2∂ℓ\ntv·∇(∂ℓ\ntξ)+2a|∂ℓ\ntv|2=2\nk1∂ℓ\ntv·∂ℓ\nt[(1\n¯̺−1\n̺)∇ξ].\n(5.33)\n42By (5.33)1+(5.33)2, we get\n(|∂ℓ\ntξ|2)t+2k2∂ℓ\ntξ∇·(∂ℓ\ntv)+2k2∂ℓ\ntv·∇(∂ℓ\ntξ)+2a|∂ℓ\ntv|2\n=−2γk1∂ℓ\ntξ∂ℓ\nt(ξ∇·v)−2k1∂ℓ\ntξ∂ℓ\nt(v·∇ξ)+2\nk1∂ℓ\ntv·∂ℓ\nt[(1\n¯̺−1\n̺)∇ξ].(5.34)\nAfter integrating (5 .34) in Ω, we get\nd\ndt/integraltext\nΩ|∂ℓ\ntξ|2dx+2k2/integraltext\nΩ∂ℓ\ntξ∇·(∂ℓ\ntv)+∂ℓ\ntv·∇(∂ℓ\ntξ)dx+2a/integraltext\nΩ|∂ℓ\ntv|2dx\n=/integraltext\nΩ−2γk1∂ℓ\ntξ∂ℓ\nt(ξ∇·v)−2k1∂ℓ\ntξ∂ℓ\nt(v·∇ξ)+2\nk1∂ℓ\ntv·∂ℓ\nt[(1\n¯̺−1\n̺)∇ξ]dx.\n(5.35)\nSince∂ℓ\ntv·n|∂Ω= 0,/integraltext\nΩ∂ℓ\ntξ∇·(∂ℓ\ntv)+∂ℓ\ntv·∇(∂ℓ\ntξ)dx=/integraltext\n∂Ω∂ℓ\ntξ∂ℓ\ntv·ndSx= 0,\nd\ndt/integraltext\nΩ|∂ℓ\ntξ|2dx+2a/integraltext\nΩ|∂ℓ\ntv|2dx\n=/integraltext\nΩ−2γk1∂ℓ\ntξ∂ℓ\nt(ξ∇·v)−2k1∂ℓ\ntξ∂ℓ\nt(v·∇ξ)+2\nk1∂ℓ\ntv·∂ℓ\nt[(1\n¯̺−1\n̺)∇ξ]dx:=I6.\n(5.36)\nWhen 0 ≤ℓ≤2, it is easy to check that I6/lessorsimilar√εF[ξ,v](t), sinceI6is a lower\norder term.\nWhenℓ= 3,\nI6=/integraltext\nΩ−2γk1ξttt∂ttt(ξ∇·v)−2k1ξttt∂ttt(v·∇ξ)+2\nk1vttt·∂ttt[(1\n¯̺−1\n̺)∇ξ]dx\n=/integraltext\nΩ−2γk1(ξttt)2∇·v−6γk1ξtttξtt∇·vt−6γk1ξtttξt∇·vtt\n−2γk1ξtttξ∇·vttt−2k1ξtttvttt·∇ξ−6k1ξtttvtt·∇ξt\n−6k1ξtttvt·∇ξtt−2k1ξtttv·∇ξttt+2\nk1(1\n¯̺−1\n̺)vttt·∇ξttt\n+6\nk1(̺t\n̺2vttt·∇ξtt+̺̺tt−2̺2\nt\n̺3vttt·∇ξt)+2\nk1̺2̺ttt−6̺̺t̺tt+6̺3\nt\n̺4vttt·∇ξdx.\n(5.37)\nNow we estimate I6−d\ndt/integraltext\nΩξ\npξ2\ntttdx,\nI6−d\ndt/integraltext\nΩξ\npξ2\ntttdx\n/lessorsimilar√εF[ξ,v](t)+/ba∇dblξtt/ba∇dblL4(Ω)/ba∇dbl∇·vt/ba∇dblL4(Ω)/ba∇dblξttt/ba∇dblL2(Ω)−2γk1/integraltext\nΩξtttξ∇·vtttdx\n+/ba∇dblvtt/ba∇dblL4(Ω)/ba∇dbl∇ξt/ba∇dbl/ba∇dblL4(Ω)/ba∇dblξttt/ba∇dbl/ba∇dblL2(Ω)−2k1/integraltext\nΩξtttv·∇ξtttdx\n+2\nk1/integraltext\nΩ(1\n¯̺−1\n̺)vttt·∇ξtttdx+/ba∇dbl̺tt/ba∇dblL4(Ω)/ba∇dbl∇ξt/ba∇dblL4(Ω)/ba∇dblvttt/ba∇dblL2(Ω)\n−2/integraltext\nΩξ\npξtttξttttdx−/integraltext\nΩ∂t(ξ\np)ξ2\ntttdx\n/lessorsimilar√εF[ξ,v](t)−k1/integraltext\n∂Ω|ξttt|2v·ndSx+k1/integraltext\nΩ|ξttt|2∇·vdx\n−2γk1/integraltext\nΩξξttt∇·vtttdx+2\nk1/integraltext\nΩ(1\n¯̺−1\n̺)vttt·∇ξtttdx−2/integraltext\nΩξ\npξtttξttttdx\n/lessorsimilar−2/integraltext\nΩξ\npξttt(ξtttt+k1γp∇·vttt)dx+2\nk1/integraltext\nΩ(1\n¯̺−1\n̺)vttt·∇ξtttdx+√εF[ξ,v](t).\n(5.38)\n43Apply∂tttto (2.8)1, we get\nξtttt+k1γp∇·vttt=−k1v·∇ξttt−3k1vt·∇ξtt−3k1vtt·∇ξt−k1vttt·∇ξ\n−3k1γ̺t∇·vtt−3k1γξtt∇·vt−k1γξttt∇·v.\n(5.39)\nPlug (5.39) into the following integral, we get\n/integraltext\nΩξ\npξttt(ξtttt+k1γp∇·vttt)dx=/integraltext\nΩξ\npξttt[R.H.S. of (5.39)]dx\n/lessorsimilar√εF[ξ,v](t)−k1/integraltext\n∂Ωξ\n2p|ξttt|2v·ndSx+k1\n2/integraltext\nΩ|ξttt|2∇·(ξ\npv)dx/lessorsimilar√εF[ξ,v](t).\n(5.40)\nApply∂tttto (2.8)2, we get\n∇ξttt=−ak1̺vttt−3ak1̺tvtt−3ak1̺ttvt−ak1̺ttt.(5.41)\nPlug (5.41) into the following integral, we get\n/integraltext\nΩ(1\n¯̺−1\n̺)vttt·∇ξtttdx=/integraltext\nΩ(1\n¯̺−1\n̺)vttt·[R.H.S. of (5.41)]dx/lessorsimilar√εF[ξ,v](t).\n(5.42)\nPlug (5.40),(5.40) into (5 .38), we get\nI6−d\ndt/integraltext\nΩξ\npξ2\ntttdx/lessorsimilar√εF[ξ,v](t).(5.43)\nFinally, we have\nd\ndt/parenleftbigg3/summationtext\nℓ=0/integraltext\nΩ|∂ℓ\ntξ|2dx−/integraltext\nΩξ\npξ2\ntttdx/parenrightbigg\n+2aE[v](t)/lessorsimilar√εF[ξ,v](t).(5.44)\nThus, Lemma 5 .4 is proved.\nThe following lemma concerns a priori estimate for E[v](t).\nLemma 5.5. For any given T∈(0,+∞], if\nsup\n0≤t≤TF[ξ,v,φ](t)≤ε,\nwhere0< ε≪1, then for ∀t∈[0,T],\nd\ndtE[v](t)+γ\na/integraltext\nΩp\n̺3/summationtext\nℓ=0|∇·∂ℓ\ntv|2dx/lessorsimilar√εF[ξ,v](t). (5.45)\nProof.Apply∂ℓ\ntto the equation (2 .12), where 1 ≤ℓ≤3, we get\n∂ℓ+1\ntv=∂ℓ\nt[k1(1−γ)v(∇·v)−k1v·∇v−k1\n2∇(|v|2)]\n+γ\na/summationtext\n0≤µ<ℓ[∂ℓ−µ\nt(p\n̺)∂µ\nt∇(∇·v)]+γp\na̺∂ℓ\nt∇(∇·v).(5.46)\n44Let (5.46)·∂ℓ\ntv, we get\n∂t|∂ℓ\ntv|2= 2∂ℓ\ntv·∂ℓ\nt[k1(1−γ)v(∇·v)−k1v·∇v−k1\n2∇(|v|2)]\n+2γ\na/summationtext\n0≤µ<ℓ[∂ℓ−µ\nt(p\n̺)∂µ\nt∇(∇·v)]·∂ℓ\ntv+2γp\na̺∂ℓ\nt∇(∇·v)·∂ℓ\ntv.(5.47)\nIntegrate (5 .47) in Ω, we get\nd\ndt/integraltext\nΩ|∂ℓ\ntv|2dx= 2/integraltext\nΩ∂ℓ\ntv·∂ℓ\nt[k1(1−γ)v(∇·v)−k1v·∇v−k1\n2∇(|v|2)]dx\n+2γ\na/integraltext\nΩ/summationtext\n0≤µ<ℓ[∂ℓ−µ\nt(p\n̺)∂µ\nt∇(∇·v)]·∂ℓ\ntvdx\n+2γ\na/integraltext\nΩp\n̺∂ℓ\nt∇(∇·v)·∂ℓ\ntvdx:=I7.\n(5.48)\nWhenℓ= 0,\n/integraltext\nΩp\n̺v·∇(∇·v)dx=/integraltext\n∂Ωp\n̺v·n∇·vdSx−/integraltext\nΩp\n̺|∇·v|2dx−/integraltext\nΩ∇·v̺v·∇ξ−pv·∇̺\n̺2dx\n/lessorsimilar√εF[ξ,v]−/integraltext\nΩp\n̺|∇·v|2dx\n(5.49)\nThen\nd\ndt/integraltext\nΩ|v|2dx+2γ\na/integraltext\nΩp\n̺|∇·v|2dx/lessorsimilar√εF[ξ,v](5.50)\nWhenℓ= 1,\n/integraltext\nΩp\n̺vt·∇(∇·vt)dx=/integraltext\n∂Ωp\n̺vt·n∇·vtdSx−/integraltext\nΩp\n̺|∇·vt|2dx\n−/integraltext\nΩ(∇·vt)vt·∇(p\n̺)dx\n/lessorsimilar√εF[ξ,v]−/integraltext\nΩp\n̺|∇·vt|2dx(5.51)\nThen\nd\ndt/integraltext\nΩ|vt|2dx+2γ\na/integraltext\nΩp\n̺|∇·vt|2dx/lessorsimilar√εF[ξ,v](5.52)\nWhenℓ= 2,\n/integraltext\nΩp\n̺vtt·∇(∇·vtt)dx=/integraltext\n∂Ωp\n̺vtt·n∇·vttdSx−/integraltext\nΩp\n̺|∇·vtt|2dx\n−/integraltext\nΩ(∇·vtt)vtt·∇(p\n̺)dx\n/lessorsimilar√εF[ξ,v]−/integraltext\nΩp\n̺|∇·vtt|2dx(5.53)\nThen\nd\ndt/integraltext\nΩ|vtt|2dx+2γ\na/integraltext\nΩp\n̺|∇·vtt|2dx/lessorsimilar√εF[ξ,v](5.54)\nWhenℓ= 3, we estimate each term of I7separately.\n45The first term of I7:\n2k1(1−γ)/integraltext\nΩvttt·∂ttt[v(∇·v)]dx\n/lessorsimilar/ba∇dblvttt/ba∇dblL2(Ω)(/ba∇dblvttt/ba∇dblL2(Ω)|∇·v|∞+/ba∇dblvtt/ba∇dblL4(Ω)/ba∇dbl∇·vt/ba∇dblL4(Ω)\n+|vt|∞/ba∇dbl∇·vtt/ba∇dblL2(Ω))+2k1(1−γ)/integraltext\nΩvttt·v(∇·vttt)dx\n≤C√εF[ξ,v](t)+γ\n3a/integraltext\nΩp\n̺|∇·vttt|2dx+C/integraltext\nΩ̺\np(v·vttt)2dx\n≤C√εF[ξ,v](t)+γ\n3a/integraltext\nΩp\n̺|∇·vttt|2dx.(5.55)\nThe second term of I7:\n−2k1/integraltext\nΩvttt·∂ttt[v·∇v]dx\n/lessorsimilar/ba∇dblvttt/ba∇dblL2(Ω)(/ba∇dblvttt/ba∇dblL2(Ω)|∇v|∞+/ba∇dblvtt/ba∇dblL4(Ω)/ba∇dbl∇vt/ba∇dblL4(Ω)\n+|vt|∞/ba∇dbl∇vtt/ba∇dblL2(Ω))−k1/integraltext\nΩv·∇|vttt|2dx\n/lessorsimilar√εF[ξ,v](t)−k1/integraltext\n∂Ωv·n|vttt|2dSx+k1/integraltext\nΩ∇·v|vttt|2dx/lessorsimilar√εF[ξ,v](t).\n(5.56)\nThe third term of I7:\n−k1/integraltext\nΩvttt·∂ttt[∇(|v|2)]dx\n=−k1/integraltext\n∂Ωvttt·n∂ttt(|v|2)dSx+k1/integraltext\nΩ∇·vttt∂ttt(|v|2)dx\n=k1/integraltext\nΩ∇·vttt∂ttt(|v|2)dx≤γ\n3a/integraltext\nΩp\n̺|∇·vttt|2dx+C/integraltext\nΩ̺\np|∂ttt(|v|2)|2dx\n≤C√εF[ξ,v](t)+γ\n3a/integraltext\nΩp\n̺|∇·vttt|2dx.\n(5.57)\nThe fourth term of I7:\n2γ\na/integraltext\nΩ/summationtext\n0≤µ≤2[∂3−µ\nt(p\n̺)∂µ\nt∇(∇·v)]·vtttdx/lessorsimilar|∇(∇·v)|∞/ba∇dbl(p\n̺)ttt/ba∇dblL2(Ω)/ba∇dblvttt/ba∇dblL2(Ω)\n+/ba∇dblvttt/ba∇dblL2(Ω)/ba∇dbl(p\n̺)tt/ba∇dblL4(Ω)/ba∇dbl∇(∇·vt)/ba∇dblL4(Ω)\n+|(p\n̺)t|∞/ba∇dbl∇(∇·vtt)/ba∇dblL2(Ω)/ba∇dblvttt/ba∇dblL2(Ω)/lessorsimilar√εF[ξ,v](t).\n(5.58)\nThe fifth term of I7:\n2γ\na/integraltext\nΩp\n̺vttt·∇(∇·vttt)dx=2γ\na/integraltext\n∂Ωp\n̺vttt·n∇·vtttdSx−2γ\na/integraltext\nΩp\n̺|∇·vttt|2dx\n−2γ\na/integraltext\nΩ(∇·vttt)vttt·∇(p\n̺)dx\n≤C√εF[ξ,v]−(2γ\na−γ\n3a)/integraltext\nΩp\n̺|∇·vttt|2dx+C/integraltext\nΩ̺\np|∇(p\n̺)|∞|vttt|2dx\n≤C√εF[ξ,v]−(2γ\na−γ\n3a)/integraltext\nΩp\n̺|∇·vttt|2dx.\n(5.59)\nAfter summing the fiveterms of I7, namely (5 .55)+(5.56)+(5.57)+(5.58)+\n(5.59), we get\nd\ndt/integraltext\nΩ|vttt|2dx+γ\na/integraltext\nΩp\n̺|∇·vttt|2dx/lessorsimilar√εF[ξ,v](5.60)\n46By (5.50)+(5.52)+(5.54)+(5.60), we get\nd\ndtE[v](t)+γ\na/integraltext\nΩp\n̺3/summationtext\nℓ=0|∇·∂ℓ\ntv|2dx/lessorsimilar√εF[ξ,v]. (5.61)\nThus, Lemma 5 .5 is proved.\nThefollowinglemmaconcernsaprioriestimatefor/integraltext\nΩ3/summationtext\nℓ=1∂ℓ\ntξ∂ℓ−1\ntξdx, which\nintroduces E[ξ](t) to the inequality (5 .62).\nLemma 5.6. For any given T∈(0,+∞], if\nsup\n0≤t≤TF[ξ,v,φ](t)≤ε,\nwhere0< ε≪1, then there exists c13>0such that for ∀t∈[0,T],\n−d\ndt/integraldisplay\nΩ3/summationdisplay\nℓ=1∂ℓ\ntξ∂ℓ−1\ntξdx+E[ξ](t)≤C√εF[ξ,v](t)+c13E[v](t).(5.62)\nProof.Similar to (3 .47), we have\n−d\ndt/integraltext\nΩξtξdx+/integraltext\nΩ(ξt)2dx/lessorsimilar√εF[ξ,v](t)+/ba∇dblvt/ba∇dbl2\nL2(Ω)+/ba∇dbl∇ξ/ba∇dbl2\nL2(Ω)\n/lessorsimilar√εF[ξ,v](t)+/ba∇dblv/ba∇dbl2\nL2(Ω)+/ba∇dblvt/ba∇dbl2\nL2(Ω).(5.63)\nSimilar to (3 .49), we have\n−d\ndt/integraltext\nΩξttξtdx+/integraltext\nΩ(ξtt)2dx/lessorsimilar√εF[ξ,v](t)+/ba∇dblvtt/ba∇dbl2\nL2(Ω)+/ba∇dbl∇ξt/ba∇dbl2\nL2(Ω)\n/lessorsimilar√εF[ξ,v](t)+/ba∇dblvtt/ba∇dbl2\nL2(Ω)+/ba∇dbl−ak1ξtv−ak1̺vt/ba∇dbl2\nL2(Ω)\n/lessorsimilar√εF[ξ,v](t)+/ba∇dblvt/ba∇dbl2\nL2(Ω)+/ba∇dblvtt/ba∇dbl2\nL2(Ω).(5.64)\nSimilar to (3 .51), we have\n−d\ndt/integraltext\nΩξtttξttdx+/integraltext\nΩ(ξttt)2dx/lessorsimilar√εF[ξ,v](t)+/ba∇dblvttt/ba∇dbl2\nL2(Ω)+/ba∇dbl∇ξtt/ba∇dbl2\nL2(Ω)\n/lessorsimilar√εF[ξ,v](t)+/ba∇dblvttt/ba∇dbl2\nL2(Ω)+/ba∇dbl−ak1ξttv−2ak1ξtvt−ak1̺vtt/ba∇dbl2\nL2(Ω)\n/lessorsimilar√εF[ξ,v](t)+/ba∇dblvtt/ba∇dbl2\nL2(Ω)+/ba∇dblvttt/ba∇dbl2\nL2(Ω).\n(5.65)\nBy (5.63)+(5.64)+(5.65), we get\n−d\ndt/integraldisplay\nΩ3/summationdisplay\nℓ=1∂ℓ\ntξ∂ℓ−1\ntξdx+/integraldisplay\nΩ3/summationdisplay\nℓ=1(∂ℓ\ntξ)2dx/lessorsimilar√εF[ξ,v](t)+3/summationdisplay\nℓ=0/ba∇dblvℓ\nt/ba∇dbl2\nL2(Ω).(5.66)\nBy Lemma 2 .1, ¯p∈[inf\nx∈Ωp,sup\nx∈Ωp], then for any t≥0, there exists xt∈Ω\nsuch that ξ(xt,t) = 0. Assume ℓ(s) is a curve with finite length parameter s\n47such that ℓ(0) =xt, ℓ(sx) =x, then\n/ba∇dblξ(x,t)/ba∇dbl2\nL2(Ω)=/ba∇dblξ(xt,t)+sx/integraltext\n0∇ξ[ℓ(s)]·ℓ(s)ds/ba∇dbl2\nL2(Ω)\n≤C|Diam(Ω)|2/ba∇dbl∇ξ/ba∇dbl2\nL2(Ω)/lessorsimilar/ba∇dbl∇ξ/ba∇dbl2\nL2(Ω)\n=/ba∇dbl−ak1̺v/ba∇dbl2\nL2(Ω)/lessorsimilar/ba∇dblv/ba∇dbl2\nL2(Ω).(5.67)\nSumming (5 .66) and (5 .67), we get\n−d\ndt/integraltext\nΩ3/summationtext\nℓ=1∂ℓ\ntξ∂ℓ−1\ntξdx+/integraltext\nΩ3/summationtext\nℓ=0(∂ℓ\ntξ)2dx/lessorsimilar√εF[ξ,v](t)+3/summationtext\nℓ=0/ba∇dblvℓ\nt/ba∇dbl2\nL2(Ω).\n(5.68)\nThen there exist two constants C >0,c13>0 such that\n−d\ndt/integraltext\nΩ3/summationtext\nℓ=1∂ℓ\ntξ∂ℓ−1\ntξdx+/integraltext\nΩ3/summationtext\nℓ=0(∂ℓ\ntξ)2dx≤C√εF[ξ,v](t)+c133/summationtext\nℓ=0/ba∇dblvℓ\nt/ba∇dbl2\nL2(Ω).\n(5.69)\nThus, Lemma 5 .6 is proved.\nInordertoprovetheuniformboundof∞/integraltext\n0/summationtext\n0≤ℓ≤2,ℓ+|α|≤3/ba∇dbl∂ℓ\ntDα∇·v(s)/ba∇dbl2\nL2(Ω)ds,\nwe must have the following lemma.\nLemma 5.7. For any given T∈(0,+∞], if\nsup\n0≤t≤TF[ξ,v,φ](t)≤ε,\nwhere0< ε≪1, then for ∀t∈[0,T],\n/summationtext\n0≤ℓ≤2,ℓ+|α|≤3/ba∇dbl∂ℓ\ntDα∇·v/ba∇dbl2\nL2(Ω)≤C√εF[ξ,v](t)+c14(E[ξ,v](t)+E1[ω](t)).\n(5.70)\nProof.By (5.16)2+(5.17)2+(5.18)2, we get\n/ba∇dbl∇·v/ba∇dbl2\nH3(Ω)+/ba∇dbl∇·vt/ba∇dbl2\nH2(Ω)+/ba∇dbl∇·vtt/ba∇dbl2\nH1(Ω)\n/lessorsimilar√εF[ξ,v](t)+/ba∇dblvt/ba∇dbl2\nH2(Ω)+/ba∇dblvtt/ba∇dbl2\nH1(Ω)+/ba∇dblvttt/ba∇dbl2\nL2(Ω)\n≤C√εF[ξ,v](t)+C9c10(E[ξ,v](t)+E1[ω](t)).(5.71)\nTakec14=C9c10. Thus, Lemma 5 .7 is proved.\nThe following lemma proves not only the exponential decay of F[ξ,v](t)\nandE[ω](t), but also the uniform bound ofT/integraltext\n0E[∇·v](s)ds.\nLemma 5.8. For any given T∈(0,+∞], there exists ε2>0which is indepen-\ndent of(ξ0,v0,φ0), such that if\nsup\n0≤t≤TF[ξ,v,φ](t)≤ε,\n48whereε≪min{1,ε0,ε2}, then for ∀t∈[0,T],\nF[ξ,v](t)≤β6/ba∇dblξ0/ba∇dbl2\nH4(Ω)exp{−β7t},\nE1[ω](t)≤β8/ba∇dblξ0/ba∇dbl2\nH4(Ω)exp{−β7t},\nT/integraltext\n0E[∇·v](s)ds≤β9/ba∇dblξ0/ba∇dbl2\nH4(Ω),(5.72)\nwhereβ6,β7,β8,β9are four positive numbers.\nProof.In view of Lemmas 5 .2, 5.4, 5.5, 5.6 and 5.7, we have obtained global a\npriori estimates as follows:\n\n\nd\ndt/integraltext\nΩE1[ω](t)dx+2aE1[ω](t)/lessorsimilarC√εF[ξ,v](t),\nd\ndtE1[ξ](t)+2aE[v](t)≤C√εF[ξ,v](t),\nd\ndtE[v](t)+γ\na/integraltext\nΩp\n̺3/summationtext\nℓ=0|∇·∂ℓ\ntv|2dx/lessorsimilar√εF[ξ,v](t),\n−d\ndt/integraltext\nΩ3/summationtext\nℓ=1∂ℓ\ntξ∂ℓ−1\ntξdx+E[ξ](t)≤C√εF[ξ,v](t)+c13E[v](t),\n/summationtext\n0≤ℓ≤2,ℓ+|α|≤3/ba∇dbl∂ℓ\ntDα∇·v/ba∇dbl2\nL2(Ω)≤C√εF[ξ,v](t)+c14(E[ξ,v](t)+E1[ω](t)).\n(5.73)\nLetλ2= max{4\n3,c13\na}+1, λ3= min{1\n2c14,a\nc14}. Define\nE2[ξ](t) :=λ2E1[ξ](t)−3/summationtext\nℓ=1/integraltext\nΩ∂ℓ−1\ntξ∂ℓ\ntξdx. (5.74)\nwhereE3>0 by Cauchy-Schwarz inequality. Since λ2>4\n3,E2[ξ](t)∼=E[ξ](t),\ni.e., there exists c15>0,c16>0 such that\nc15E[ξ](t)≤E2[ξ](t)≤c16E[ξ](t). (5.75)\nBy (5.73)1+(5.73)2×λ2+(5.73)3+(5.73)4+(5.73)5×λ3, we get\nd\ndt(E2[ξ](t)+E[v](t)+E1[ω](t))+(1−λ3c14)E[ξ](t)+(2a−λ3c14)E1[ω](t)\n+(2aλ2−c13−λ3c14)E[v](t)+γ\na/integraltext\nΩp\n̺3/summationtext\nℓ=0|∇·∂ℓ\ntv|2dx\n+λ3/summationtext\n0≤ℓ≤2,ℓ+|α|≤3/ba∇dbl∂ℓ\ntDα∇·v/ba∇dbl2\nL2(Ω)≤C10√εF[ξ,v](t),\n(5.76)\nfor some C10>0.\nBy Lemma 5 .1, we have\n√εF[ξ,v](t)≤c10√ε(E[ξ,v](t)+E1[ω](t)). (5.77)\nLetε2= min{(1−λ3c14)2\n4C2\n10c2\n0,(2aλ2−c13−λ3c13)2\n4C2\n10c2\n0,(2a−λ3c14)2\n4C2\n10c2\n0}, plug (5 .77) into\n49(5.76), we get\nd\ndt(E2[ξ](t)+E[v](t)+E1[ω](t))+1−λ3c14\n2E[ξ](t)+2a−λ3c14\n2E1[ω](t)\n+2aλ2−c13−λ3c14\n2E[v](t)+γ\na/integraltext\nΩp\n̺3/summationtext\nℓ=0|∇·∂ℓ\ntv|2dx\n+λ3/summationtext\n0≤ℓ≤2,ℓ+|α|≤3/ba∇dbl∂ℓ\ntDα∇·v/ba∇dbl2\nL2(Ω)≤0.(5.78)\nSince the last two terms in (5 .78) are positive, we have\nd\ndt(E2[ξ](t)+E[v](t)+E1[ω](t))+1−λ3c14\n2c16E2[ξ](t)+2a−λ3c14\n2E1[ω](t)\n+2aλ2−c13−λ3c14\n2E[v](t)≤0.\n(5.79)\nLetc17= min{1−λ3c14\n2,2a−λ3c14\n2,2aλ2−c13−λ3c14\n2}, itfollowsfrom(3 .61)that\nd\ndt(E2[ξ](t)+E[v](t)+E1[ω](t))+c17(E2[ξ](t)+E1[ω](t)+E[v](t))≤0.\n(5.80)\nAfter integrating (5 .80), we get\nE2[ξ](t)+E[v](t)+E1[ω](t)≤(E2[ξ](0)+E[v](0)+E1[ω](0))exp{−c17t},\nE1[ω](t)≤(E2[ξ](0)+E[v](0)+E1[ω](0))exp{−c17t}\n≤(c16E[ξ](0)+E[v](0)+E[v](0))exp{−c17t}\n≤(c16+1)F[ξ,v](0)exp{−c17t}\n≤C11(c16+1)/ba∇dblξ0/ba∇dbl2\nH4(Ω)exp{−c17t}.\nc15E[ξ](t)≤E2[ξ](t)≤(E2[ξ](0)+E[v](0)+E1[ω](0))exp{−c17t},\nF[ξ,v](t)≤c10(E[ξ,v](t)+E1[ω](t))\n≤(c10\nc15+2c10)(E2[ξ](0)+E[v](0)+E1[ω](0))exp{−c17t}\n≤(c10\nc15+2c10)(c16+1)F[ξ,v](0)exp{−c17t}\n≤C11(c10\nc15+2c10)(c16+1)/ba∇dblξ0/ba∇dbl2\nH4(Ω)exp{−c17t}\n(5.81)\nTakeβ6=C11(c10\nc15+ 2c10)(c16+ 1),β7=c17,β8=C11(c16+ 1), the\nexponential decay in (5 .72) is obtained.\nIt follows from (5 .78) that\nd\ndt(E2[ξ](t)+E[v](t)+E1[ω](t))+γ\na/integraltext\nΩp\n̺3/summationtext\nℓ=0|∇·∂ℓ\ntv|2dx\n+λ3/summationtext\n0≤ℓ≤2,ℓ+|α|≤3/ba∇dbl∂ℓ\ntDα∇·v/ba∇dbl2\nL2(Ω)≤0.(5.82)\n50Integrate (5 .82) from t= 0 tot=T, we get\nE2[ξ](T)+E[v](T)+E1[ω](T)+λ3T/integraltext\n0/summationtext\n0≤ℓ≤2,ℓ+|α|≤3/ba∇dbl∂ℓ\nτDα∇·v(s)/ba∇dbl2\nL2(Ω)ds\n+γ\naT/integraltext\n0/integraltext\nΩp\n̺3/summationtext\nℓ=0|∇·∂ℓ\nτv(s)|2dxds≤E2[ξ](0)+E[v](0)+E1[ω](0).\n(5.83)\nThen\nT/integraltext\n0E[∇·v](s)ds≤C12λ3T/integraltext\n0/summationtext\n0≤ℓ≤2,ℓ+|α|≤3/ba∇dbl∂ℓ\nτDα∇·v(s)/ba∇dbl2\nL2(Ω)ds\n+C12γ\naT/integraltext\n0/integraltext\nΩp\n̺3/summationtext\nℓ=0|∇·∂ℓ\nτv(s)|2dxds≤C12(E2[ξ](0)+E[v](0)+E1[ω](0))\n≤C12(c16+1)F[ξ,v](0)≤C13C12(c16+1)/ba∇dblξ0/ba∇dbl2\nH4(Ω),\n(5.84)\nwhereC12= min{1\nλ3,9aˆ¯̺\n4γˆ¯p}.\nTakeβ9=C13C12(c16+1). Thus, Lemma 5 .8 is proved.\nThefollowinglemmaconcernstheuniformboundof E[φ](t)onthecondition\nthatvdecays exponentially.\nLemma 5.9. For any given T∈(0,+∞], if\nsup\n0≤t≤TF[ξ,v,φ](t)≤ε,\nwhere0< ε≪min{1,ε0,ε1,ε2}, then for ∀t∈[0,T],\nd\ndtE[φ](t)≤β4E[v](t)1\n2E[φ](t). (5.85)\nIfE[v](t)≤β6/ba∇dblξ0/ba∇dbl2\nH4(Ω)exp{−β7t}, thenE[φ](t)has uniform bound:\nE[φ](t)≤β10/ba∇dblφ0/ba∇dbl2\nH3(Ω)/parenleftbig\nexp{/ba∇dblξ0/ba∇dblH4(Ω)}/parenrightbigc18, (5.86)\nfor some c18>0.\nProof.Similar to Lemma 3 .9, we have the following a priori estimate:\nd\ndtE[φ](t)≤β4E[v](t)1\n2E[φ](t),\nE[φ](t)≤ E[φ](0)exp{t/integraltext\n0β4E[v](τ)1\n2dτ}.(5.87)\nIfE[v](t)≤β6/ba∇dblξ0/ba∇dbl2\nH4(Ω)exp{−β7t}, then\nE[φ](t)≤ E[φ](0)exp{t/integraltext\n0β4E[v](τ)1\n2dτ}\n≤β10/ba∇dblφ0/ba∇dbl2\nH3(Ω)exp{t/integraltext\n0β4√β6/ba∇dblξ0/ba∇dblH4(Ω)exp{−β7τ}1\n2dτ}\n51≤β10/ba∇dblφ0/ba∇dbl2\nH3(Ω)exp{2β4√β6\nβ7/ba∇dblξ0/ba∇dblH4(Ω)(1−exp{−β7\n2t})}\n≤β10/ba∇dblφ0/ba∇dbl2\nH3(Ω)exp{2β4√β6\nβ7/ba∇dblξ0/ba∇dblH4(Ω)}\n=β10/ba∇dblφ0/ba∇dbl2\nH3(Ω)/parenleftbig\nexp{/ba∇dblξ0/ba∇dblH4(Ω)}/parenrightbigc18,(5.88)\nwherec18=2β4√β6\nβ7,β10>0.\nTherefore E[φ](t) is uniformly bounded when E[v](t) decays exponentially.\nThus, Lemma 5 .9 is proved.\nThe following lemma concerns the exponential decay of3/summationtext\nℓ=1/ba∇dbl∂ℓ\ntS/ba∇dbl2\nH3−ℓ(Ω)\non the condition that vdecays exponentially.\nLemma 5.10. For any given T∈(0,+∞], if\nsup\n0≤t≤TE[ξ,v,φ](t)≤ε,\nwhere0< ε≪min{1,ε0,ε1,ε2}, then for ∀t∈[0,T],\n3/summationtext\nℓ=1/ba∇dbl∂ℓ\ntφ/ba∇dbl2\nH3−ℓ(Ω)≤c19/ba∇dblξ0/ba∇dbl2\nH4(Ω)/ba∇dblφ0/ba∇dbl2\nH3(Ω)/parenleftbig\nexp{/ba∇dblξ0/ba∇dblH4(Ω)}/parenrightbigc18exp{−β7t},\n(5.89)\nfor some c19>0.\nProof.It follows from Lemma 5 .8 thatE[v](t)≤β6/ba∇dblξ0/ba∇dbl2\nH4(Ω)exp{−β7t}. It\nfollows from Lemma 5 .9 thatE[φ](t)≤β10/ba∇dblφ0/ba∇dbl2\nH3(Ω)/parenleftbig\nexp{/ba∇dblξ0/ba∇dblH4(Ω)}/parenrightbigc18.\nSimilar to Lemma 3 .10, we have the following a priori estimate:\n3/summationtext\nℓ=1/ba∇dbl∂ℓ\ntS/ba∇dbl2\nH3−ℓ(Ω)=3/summationtext\nℓ=1/ba∇dbl∂ℓ\ntφ/ba∇dbl2\nH3−ℓ(Ω)/lessorsimilarE[v](t)E[φ](t)\n≤β6β10/ba∇dblξ0/ba∇dbl2\nH4(Ω)/ba∇dblφ0/ba∇dbl2\nH3(Ω)/parenleftbig\nexp{/ba∇dblξ0/ba∇dblH4(Ω)}/parenrightbigc18exp{−β7t}.\n(5.90)\nTakec19=β6β10. Thus, Lemma 5 .10 is proved.\nRemark 5.11. Similar to the results in Lemma 3.11, we have a priori estimates\nforE[̺−¯̺](t)and3/summationtext\nℓ=1/ba∇dbl∂ℓ\nt̺/ba∇dbl2\nH3−ℓ(Ω):\nWhenF[p−¯p](t)andE[S−¯S](t)are uniformly bounded, E[̺−¯̺](t)is also\nuniformly bounded due to ̺=1\nγ√\nAp1\nγexp{−S\nγ}. For any given T∈(0,+∞], if\nsup\n0≤t≤TF[ξ,v,φ](t)≤ε, where0< ε≪min{1,ε0,ε1,ε2}, then3/summationtext\nℓ=1/ba∇dbl∂ℓ\nt̺/ba∇dbl2\nH3−ℓ(Ω)\nalso decays at an exponential rate of Cexp{−β7t}.\n526 Darcy’s Law and Nonlinear Diffusion of Non-\nIsentropic Euler Equations with Damping\nIn this section, we prove the global existence of classical solutions to the\ndiffusion equations (2 .8) under small data assumption and the nonlinear diffu-\nsion property of the non-isentropic Euler equations with damping (2 .4) when\nthe time is large. For simplicity, we omit the symbol ˆ over the variables and\nconstants in this section if there is no ambiguity, otherwise we will add the\nsymbol ˆ.\nThe proof of the local existence of classical solutions to IBVP for t he\nparabolic-hyperbolicequations(2 .9) is standard, such as using the linearization-\niteration-convergencescheme, so we give a lemma on the local exist ence without\nproof here.\nLemma 6.1. (Local Existence )\nIf(ξ0,φ0)∈H4(Ω)×H3(Ω),inf\nx∈Ωp0(x)>0and∂ℓ\nt∇ξ(x,0)·n|∂Ω= 0,0≤ℓ≤3,\nthen there exists a finite time T∗>0, such that IBVP (2.9)admits a unique\nlocal classical solution (ξ,φ)satisfying\n/braceleftBigg\n(ξ,φ)∈ ∩\n0≤ℓ≤3Cℓ([0,T∗),H4−ℓ(Ω)×H3−ℓ(Ω)),\n△ξ∈C(Ω×[0,T∗)).(6.1)\nThe above lemma implies the local existence of classical solutions to IB VP\n(2.8) as long as ( ξ,φ) remain classical, namely, ( ξ,φ)∈C1([0,T∗),C2(Ω)×\nC1(Ω)). Based on the global a priori estimates for ( ξ,v,φ), we obtained the\nglobal existence of classical solutions to IBVP (2 .8).\nTheorem 6.2. (Global Existence )\nAssume (ξ0,φ0)∈H4(Ω)×H3(Ω),inf\nx∈Ωp0(x)>0and∂ℓ\nt∇ξ(x,0)·n|∂Ω=\n0,0≤ℓ≤3. There exists a sufficiently small number δ2>0, such that if\n/ba∇dblξ0/ba∇dblH4(Ω)+/ba∇dblφ0/ba∇dblH3(Ω)≤δ2, then IBVP (2.8)admits a unique global classical\nsolution (ξ,φ)satisfying\n(ξ,φ)∈ ∩\n0≤ℓ≤3Cℓ([0,+∞),H4−ℓ(Ω)×H3−ℓ(Ω)),△ξ∈C(Ω×[0,+∞)),\n(6.2)\nmoreover,\n\n\n̺=̺(ξ,φ)∈ ∩\n0≤ℓ≤3Cℓ([0,+∞),H3−ℓ(Ω)),\nv=−1\nak1̺∇ξ∈ ∩\n0≤ℓ≤3Cℓ([0,+∞),H3−ℓ(Ω)),∇·v∈C(Ω×[0,+∞)).\n(6.3)\n∀t≥0,F[ξ,v](t),E1[ω](t)and3/summationtext\nℓ=1/ba∇dbl∂ℓ\ntφ/ba∇dbl2\nH3−ℓ(Ω)decays exponentially, F[φ](t)\nis uniformly bounded.\nProof.In view of Lemmas 5 .8 and 5.9, we have the following global a priori\nestimates: for any given T∈(0,+∞], if\nsup\n0≤t≤TF[ξ,v,φ](t)≤ε, (6.4)\n53where 0< ǫ≪min{1,ε0,ε1,ε2}, then\nF[ξ,v](t)≤β6/ba∇dblξ0/ba∇dbl2\nH4(Ω)exp{−β7t},\nE[φ](t)≤β10/ba∇dblφ0/ba∇dbl2\nH3(Ω)/parenleftbig\nexp{/ba∇dblξ0/ba∇dblH4(Ω)}/parenrightbigc18.(6.5)\nThe constants ε0,ε1,ε2are independent of ( ξ0,φ0), so we can choose ε\nwhich is independent of ( ξ0,φ0).\nTakeδ2= min{√ε,/radicalBig\nε\n2β6,/radicalBig\nε\n2β10/parenleftBig\nexp{/radicalBig\nε\n2β6}/parenrightBig−c18\n2}, then if /ba∇dblξ0/ba∇dblH4(Ω)+\n/ba∇dblφ0/ba∇dblH3(Ω)≤δ2, we have\n\n\n/ba∇dblξ0/ba∇dblH4(Ω)≤/radicalBig\nε\n2β6,\n/ba∇dblφ0/ba∇dblH3(Ω)≤/radicalBig\nε\n2β10/parenleftBig\nexp{/radicalBig\nε\n2β6}/parenrightBig−c18\n2.(6.6)\nDue to the estimates in (6 .5), (ξ,v,φ) satisfy\nF[ξ,v](t)≤ε\n2,E[φ](t)≤ε\n2,∀t∈[0,T]. (6.7)\nThis implies the a priori assumption (6 .4) is satisfied, the validity of the former\na priori estimates is verified.\nBy Lemma 5 .8, we have\nT/integraltext\n0E[∇·v](s)ds≤β9/ba∇dblξ0/ba∇dbl2\nH4(Ω), (6.8)\nwhich implies that for any given time T∈(0,+∞],\n\n\n/ba∇dbl∇·v/ba∇dbl2\nL2([0,T],H3(Ω))/lessorsimilarT/integraltext\n0E[∇·v](s)ds/lessorsimilar/ba∇dblξ0/ba∇dbl2\nH4(Ω),\n/ba∇dbl∇·vt/ba∇dbl2\nL2([0,T],H1(Ω))/lessorsimilarT/integraltext\n0E[∇·v](s)ds/lessorsimilar/ba∇dblξ0/ba∇dbl2\nH4(Ω).(6.9)\nBy Aubin-Lions’ Lemma, we obtain\n/ba∇dbl∇·v/ba∇dbl2\nC([0,T],H2(Ω))/lessorsimilar/ba∇dblξ0/ba∇dbl2\nH4(Ω), (6.10)\nwhich implies that ∇·v∈C(Ω×[0,T]) for any T >0. Then\n△ξ=−ak1̺∇·v−ak1v·∇̺∈C(Ω×[0,T]). (6.11)\nDue to the global a priori estimates for ( ξ,v,φ) and Lemma 6 .1 on the local\nexistence result, the classical solution ( ξ,φ) can be extended to [0 ,+∞). (6.11)\nholds for any given T∈(0,+∞]. Thus, Theorem 6 .2 on the global existence of\nclassical solutions to IBVP (2 .8) is proved.\nRemark 6.3. Our proof requires a≥C√εwhereC >0is large enough. If\na→0,(p0,∇p0)→(¯p,0)is required.\n54Remark 6.4. Theorem 6.2implies the global well-posedness of the diffusion\nequations (1.9)under small data assumption, thus Darcy’s law is verified whe n\nthe ideal gases are sufficiently mild and slow. While the verifi cation of Darcy’s\nlaw for 1D non-isentropic p-system with damping see [5], for isentropic Euler\nequations with damping see [13], for isothermal Euler equat ions with damping\nsee [19].\nSince (ξ,φ)∈C1(Ω×[0,+∞)) is the global classical solution to IBVP\n(2.8), then ( p= ¯p+ξ,S=¯S+φ) is the global classical solution to IBVP for\nthe diffusion equations (1 .9). The following theorem describes the asymptotical\nbehavior of ( p,v,S,̺) relating to their equilibrium states ( p∞,v∞,S∞,̺∞).\nTheorem 6.5. Assume the conditions in Theorem 6.2hold. Let (p,S)be the\nglobal classical solution to IBVP (1.9).p∞= ¯p,u∞=v∞=ω∞= 0. If\nS0/ne}ationslash=const, thenS∞/ne}ationslash=const, ̺ ∞(x)/ne}ationslash=const, θ ∞/ne}ationslash=const, e ∞/ne}ationslash=const. As\nt→+∞,(p,u,S,̺)converge to (¯p,0,S∞,̺∞)exponentially in |·|∞norm.\nProof.By Lemma 5 .10, we have\n|St|∞/lessorsimilar/parenleftbigg3/summationtext\nℓ=1/ba∇dbl∂ℓ\ntφ/ba∇dbl2\nH3−ℓ(Ω)/parenrightbigg1\n2\n/lessorsimilar/ba∇dblξ0/ba∇dblH4(Ω)/ba∇dblφ0/ba∇dblH3(Ω)/parenleftbig\nexp{/ba∇dblξ0/ba∇dblH4(Ω)}/parenrightbigc18\n2exp{−β7\n2t},(6.12)\nSo∞/integraltext\n0Sτ(x,τ)dτconverges,then S∞(x) =S0(x)+∞/integraltext\n0Sτ(x,τ)dτis bounded.\nSimilar to the proof of Theorem 4 .4, we can show p∞= ¯p,u∞=v∞=\nω∞= 0. IfS0/ne}ationslash=const, thenS∞/ne}ationslash=const, ̺ ∞(x)/ne}ationslash=const, θ ∞/ne}ationslash=const, e ∞/ne}ationslash=\nconst.\nThe exponential decay rates of ( ξ,v,φt) provides exponential convergence\nrates of ( p,u,S,̺) to their equilibrium states as follows:\n\n\n|p−p∞|∞=|p−¯p|∞/lessorsimilar/ba∇dblξ0/ba∇dblH4(Ω)exp{−β7\n2t},\n|u−0|∞=k1|v|∞/lessorsimilar/ba∇dblξ0/ba∇dblH4(Ω)exp{−β7\n2t},\n|S(x,t)−S∞(x)|∞=|−∞/integraltext\ntSs(x,s)ds|∞≤∞/integraltext\nt|φs(x,s)|∞ds\n/lessorsimilar2\nβ7/ba∇dblξ0/ba∇dblH4(Ω)/ba∇dblφ0/ba∇dblH3(Ω)/parenleftbig\nexp{/ba∇dblξ0/ba∇dblH4(Ω)}/parenrightbigc18\n2exp{−β7\n2t},\n|̺(x,t)−̺∞(x)|∞/lessorsimilarexp{−β7\n2t}.(6.13)\nSo (p,u,S,̺)→(¯p,0,S∞,̺∞) exponentially in |·|∞norm as t→+∞.\nThefollowingtheoremstatesthatthepressureandvelocityofnon -isentropic\nEuler equations with damping converge to those of the diffusion equa tions re-\nspectively, thus the pressure and velocity have nonlinear diffusion p roperty.\nTheorem 6.6. Assume(ˆp,ˆu,ˆS,ˆ̺)are variables of the diffusion equations (1.9)\nand(p,u,S,̺)are variables of non-isentropic Euler equations with dampi ng\n55(1.3), the initial data (p0,u0,S0)satisfy the conditions in Theorem 2.4,(ˆp0,ˆS0)\nsatisfy the conditions in Theorem 2.7. If\n/integraltext\nΩp1\nγ\n0dx=/integraltext\nΩˆp1\nγ\n0dx, (6.14)\nthen\n/ba∇dblp−ˆp/ba∇dblH3(Ω)+/ba∇dblu−ˆu/ba∇dblH3(Ω)≤C1exp{−C2t}, (6.15)\nfor some positive C1,C2.\nProof.The condition (6 .14) implies ¯ p=ˆ¯p.\nTakeC1= 4(1+k2\n1)max{β1/ba∇dbl(ξ0,v0)/ba∇dbl2\nH3(Ω),β6/ba∇dblξ0/ba∇dbl2\nH4(Ω)},C2= min{β2,β7}.\nBy Lemmas 3 .8 and 5.8, we have\n/ba∇dblp−ˆp/ba∇dbl2\nH3(Ω)≤2/ba∇dblp−¯p/ba∇dbl2\nH3(Ω)+2/ba∇dblˆp−ˆ¯p/ba∇dbl2\nH3(Ω)\n≤2β1/ba∇dbl(ξ0,v0)/ba∇dbl2\nH3(Ω)exp{−β2t}+2β6/ba∇dblξ0/ba∇dbl2\nH4(Ω)exp{−β7t},\n≤C1\n2exp{−C2t},\n/ba∇dblu−ˆu/ba∇dbl2\nH3(Ω)≤2k2\n1/ba∇dblv/ba∇dbl2\nH3(Ω)+2k2\n1/ba∇dblˆv/ba∇dbl2\nH3(Ω)\n≤2k2\n1β1/ba∇dbl(ξ0,v0)/ba∇dbl2\nH3(Ω)exp{−β2t}+2k2\n1β6/ba∇dblξ0/ba∇dbl2\nH4(Ω)exp{−β7t},\n≤C1\n2exp{−C2t}.\n(6.16)\nThus, Theorem 6 .6 is proved.\nReferences\n[1]R. A. Adams and J. J. F. Fournier ,Sobolev Space , vol. 140, Pure and\nApplied Mathematics series, 2 ed., 2009.\n[2]J. P. Bourguignon and H. Brezis ,Remarks on the Euler equation , J.\nFunct. Anal., 15 (1974), pp. 341–363.\n[3]C. M. Dafermos ,A system of hyperbolic conservation laws with frictional\ndamping, Z. Angew. Math. Phys., 46 (1995), pp. 294–307.\n[4]L. Hsiao and T. Luo ,Nonlinear diffusive phenomena of solutions for the\nsystem of compressible adiabatic flow through porous media , J. Differential\nEquations, 125 (1996), pp. 329–365.\n[5]L. Hsiao and R. H. Pan ,Initial boundary value problem for the system of\ncompressible adiabatic flow through porous media , J. Differential Equations,\n159 (1999), pp. 280–305.\n[6] ,The damped p-system with boundary effect , Contemp. Math., 255\n(2000), pp. 109–123.\n[7]L. Hsiao and D. Serre ,Global existence of solutions for the system of\ncompressible adiabatic flow through porous media , SIAM J. Math. Anal., 27\n(1996), pp. 70–77.\n56[8]F. M. Huang and R. H. Pan ,Asymptotic behavior of the solutions to\nthe damped compressible Euler equations with vacuum , J. Differential Equa-\ntions, 220 (2006), pp. 207–233.\n[9]M. Luskin and B. Temple ,The existence of a global weak solution to\nthe nonlinear water-hammar problem , Comm. PureAppl. Math., 35 (1982),\npp. 697–735.\n[10]A. Majda ,Compressible fluid flow and systems of conservation laws in\nseveral space variables , Applied MathematicalSciences 53, Springer-Verlag:\nNew York, 1984.\n[11]P. Marcati and R. Pan ,On the diffusive profiles for the system of com-\npressible adiabatic flow through porous media , SIAM J. MATH. ANAL., 33\n(2001), pp. 790–826.\n[12]R. Pan,Boundary effects and large time behavior for the system of com -\npressible adiabatic flow through porous media , MichiganMath.J., 49(2001),\npp. 519–539.\n[13]R. Pan and K. Zhao ,The 3D compressible Euler equations with damping\nin a bounded domain , J. Differential Equations, 246 (2009), pp. 581–596.\n[14]R. H. Pan ,Darcy’s law as long time limit of adiabatic porous media flows ,\nJ. Differential Equations, 220 (2006), pp. 121–146.\n[15]S. Schochet ,The compressible Euler equations in a bounded domain:\nexistence of solutions and the incompressible limit , Comm. Math. Phys.,\n104 (1986), pp. 49–75.\n[16]T. C. Sideris, B. Thomases, and D. H. Wang ,Long time behavior of\nsolutions to the 3D compressible Euler equations with dampi ng, Communi-\ncations in Partial Differential Equations, 28 (2003), pp. 795–816.\n[17]G. Wu, Z. Tan, and J. Huang ,Global existence and large time behavior\nfor the system of compressible adiabatic flow through porous media in R3,\nJ. Differential Equations, 255 (2013), pp. 865–880.\n[18]Y. Zhang and G. Wu ,Global existence and asymptotic behavior for the\n3D compressible non-isentropic Euler equations with dampi ng, Acta Math-\nematica Scientia, 34 (2014), pp. 424–434.\n[19]K. Zhao ,On the isothermal compressible Euler equations with fricti onal\ndamping, Communications in Mathematical Analysis, 9 (2010), pp. 77–97.\n[20]Y. Zheng ,Global smooth solutions to the adiabatic gas dynamics syste m\nwith dissipation terms , Chinese Ann. Math., 17A (1996), pp. 155–162.\n57"    },    {        "title": "1405.4677v1.Comparison_of_micromagnetic_parameters_of_ferromagnetic_semiconductors__Ga_Mn__As_P__and__Ga_Mn_As.pdf",        "content": "1 \n Comparison of micromagnetic parameters of ferromagnetic \nsemiconductors (Ga,Mn)(As,P) and (Ga,Mn)As  \n \n \nN. Tesařová1, D. Butkovi čová1, R. P. Campion2, A.W. Rushforth2, K. W. Edmonds, \nP. Wadley2, B. L. Gallagher2, E. Schmoranzerová,1 F. Trojánek1, P. Malý1, P. Motloch4, \nV. Novák3, T. Jungwirth3, 2, and P. N ěmec1,* \n \n1 Faculty of Mathematics and Physics, Charles University in Prague, Ke Karlovu 3, 121 16 Prague 2,  \nCzech Republic \n2School of Physics and Astronomy, University of Nottingham, Nottingham NG72RD, United Kingdom \n3 Institute of Physics ASCR, v.v.i., Cukrovar nická 10, 16253 Prague  6, Czech Republic \n4 University of Chicago, Chicago, IL 60637, USA \n \nWe report on the determination of microm agnetic parameters of epilayers of the \nferromagnetic semiconductor (Ga,Mn)As, which has easy axis in the sample plane, and \n(Ga,Mn)(As,P) which has easy axis perpendicula r to the sample plane. We use an optical \nanalog of ferromagnetic resonance where the laser-pulse-induced precession of \nmagnetization is measured directly in the time domain. By the analysis of a single set of pump-and-probe magneto-optical data we determined the magnetic anisotropy fields, the \nspin stiffness and the Gilbert damping consta nt in these two materials. We show that \nincorporation of 10% of phosphorus in (Ga, Mn)As with 6% of manganese leads not only \nto the expected sign change of the perpendicu lar-to-plane anisotropy field but also to an \nincrease of the Gilbert damping and to a reduction of the spin stiffness. The observed changes in the micromagnetic parameters  upon incorporating P in (Ga,Mn)As are \nconsistent with the reduced hole density, conductivity, and Curie temperature of the (Ga,Mn)(As,P) material. We report that th e magnetization precession damping is stronger \nfor the n = 1 spin wave resonance mode than for the n = 0 uniform magnetization \nprecession mode.  \n \n PACS numbers: 75.50.Pp, 75.30.Gw, 75.70.-i, 78.20.Ls, 78.47.D- \n \n \nI. INTRODUCTION \n   \n(Ga,Mn)As is the most widely studied dilute d magnetic semiconductor (DMS) with a carrier-\nmediated ferromagnetism.1 Investigation of this material system can provide fundamental \ninsight into new physical phenomen a that are present also in ot her types of magnetic materials \n– like ferromagnetic metals – where they can  be exploited in spintronic applications.2-5 \nMoreover, the carrier concentration in DMSs is  several orders of ma gnitude lower than in \nconventional FM metals which enables manipul ation of magnetization by external stimuli – \ne.g. by electric6,7 and optical8,9 fields. Another remarkable propert y of this material is a strong \nsensitivity of the magnetic anisotropy to the ep itaxial strain. (Ga,Mn)A s epilayers are usually \nprepared on a GaAs substrate where the growth-i nduced compressive strain leads to in-plane \norientation of the easy axis (EA) for Mn concentrations ≥2%.10 However, for certain \nexperiments – e.g., for a visualization of magn etization orientation by the magneto-optical \npolar Kerr effect11-17 or the anomalous Hall effect12,18 – the EA orientation in the direction \nperpendicular to the sample plane is more suitable. To achieve this, (Ga,Mn)As layers have been grown on relaxed (In,Ga)As buffer laye rs that introduce a tensile strain in \n(Ga,Mn)As.\n11,12,14,16-18 However, the growth on (In,Ga)As la yers can result in a high density \nof line defects that can lead to high coerci vities and a strong pinning of domain walls 2 \n (DW).16,17 Alternatively, tensile strain and perpendicular-to-plane or ientation of the EA can be \nachieved by incorporation of  small amount s of phosphorus in (Ga,Mn)(As,P) layers.19,20 In \nthese epilayers, the EA can be in the sample plane for the as-grown material and perpendicular to the plane for fully annealed (Ga,Mn)(As,P).\n21 The possibility of magnetic \nanisotropy fine tuning by the thermal annealing tu rns out to be a very favorable property of \n(Ga,Mn)(As,P) because it enables the preparation of materials with extr emely low barriers for \nmagnetization switching.22,23 Compared to tensile-stained (Ga,Mn)As/(In,Ga)As films,  \n(Ga,Mn)(As,P)/GaAs epilayers show weaker DW pinning, which allows observation of the \nintrinsic flow regimes of DW propagation.13,15,24 \n Preparation of uniform (Ga,Mn)As epilayers  with minimized dens ity of unintentional \nextrinsic defects is a rather challenging task  which requires  optimized growth and post-\ngrowth annealing conditions.25 Moreover, the subsequent determination of material \nmicromagnetic parameters by the standard char acterization techniques, such as ferromagnetic \nresonance (FMR), is complicated by the fact th at these techniques require rather thick films, \nwhich may be magnetically inhomogeneous.25,26 Recently, we have reported the preparation \nof high-quality (Ga,Mn)As epila yers where the individually optimized synthesis protocols \nyielded systematic doping trends, whic h are microscopically well understood.25 \nSimultaneously with the optimization of the ma terial synthesis, we developed an optical \nanalog of FMR (optical-FMR)25, where all micromagnetic pa rameters of the in-plane \n(Ga,Mn)As were deduced from a single magneto -optical (MO) pump-and-probe experiment \nwhere a laser pulse induces precession of magnetization.27,28 In this method the anisotropy \nfields are determined from the dependence of the precession frequency on the magnitude and \nthe orientation of the external magnetic field, the Gilbert damping cons tant is deduced from \nthe damping of the precession signal, and the sp in stiffness is obtained from the mutual \nspacing of the spin wave resonance modes observe d in the measured MO signal. In this paper \nwe apply this all optical-FMR to (Ga,Mn)(As,P) . We demonstrate the applicability of this \nmethod also for the determination of microma gnetic parameters in DMS materials with a \nperpendicular-to-plane orientation of the EA. By  this method we show that the incorporation \nof P in (Ga,Mn)As leads not only to the expect ed sign change of the perpendicular-to-plane \nanisotropy field but also to a considerable  in crease of the Gilbert damping and to a reduction \nof the spin stiffness. Moreover, we illustrate that the all optical-FMR can be very effectively \nused not only for an investig ation of the uniform magnetizati on precession but also for a study \nof spin wave resonances.   \nII. EXPERIMENTAL \n  \nIn our previous work we reported in de tail on the preparation and micromagnetic \ncharacterization of (Ga,Mn)A s epilayers prepared in MBE laboratory in Prague.\n25 We also \npointed out that the preparati on of (Ga,Mn)As by this highly non-equilibrium synthesis in two \ndistinct MBE laboratories in Prague and in No ttingham led to a growth of epilayers with \nmicromagnetic parameters that showed the same doping trends.25 Nevertheless, the \npreparation of epilayers with identical paramete rs (e.g., thickness, nominal Mn content, etc.) \nin two distinct MBE machines is still a nontrivial task. Therefore, in this study of the role of \nthe phosphorus incorporation to (Ga,Mn)As we opted for a dire ct comparison of materials \nprepared in one MBE mach ine. The investigated Ga 1-xMn xAs and Ga 1-xMn xAs1-yPy epilayers \nwere prepared in Nottingham20 with the same nominal amount of Mn (x = 6%) and the same \ngrowth time on a GaAs substrate (with 50 nm  thick GaAsP buffer layer in the case of \n(Ga,Mn)(As,P)]. They differ only in the incorpor ation of P (y = 10%) in the latter epilayer. 3 \n The inferred epilayer thicknesses are (24.5 േ\t1.0) nm for both (Ga,Mn)As and \n(Ga,Mn)(As,P).29 The as-grown layers, wh ich both had the EA in th e epilayer plane, were \nthermally annealed (for 48 hours at 180°C). This led to an increase in Curie temperature and \nto a rotation of the EA to the perpendicular-to-plane orientation for (Ga,Mn)(As,P).20,21 \n The magnetic anisotropy of the samples was studied using a superconducting quantum \ninterference device (SQUID) magneto meter and by the all-optical FMR.25 The hole \nconcentration was determined by fitting to Hall  effect measurements at low temperatures \n(1.8 K) for external magnetic fields from 2 T to 6 T. In this range the magnetization is \nsaturated and one can obtain th e normal Hall coefficient af ter correction for the field \ndependence of the anomalous Hall du e to the weak magnetoresistance.30 The time-resolved \npump-and-probe MO experiments were performe d using a titanium sapphire pulsed laser \n(pulse width  200 fs) with a repetition rate of 82 MHz, which was tuned ( hυ = 1.64 eV) \nabove the GaAs band gap. The energy fl uence of the pump pulses was around 30 μJcm-2 and \nthe probe pulses were at least ten times weak er. The pump pulses were circularly polarized \n(with a helicity controlled by a quarter wave plate) and the probe pulses were linearly \npolarized (in a direction perpendicular to the external magnetic field). The time-resolved MO data reported here correspond to the polariz ation-independent part of the pump-induced \nrotation of probe polarization plane, which was computed from the measured data by \naveraging the signals obtained for the opposite  helicities of circularly polarized pump \npulses.\n27, 28 The experiment was performed close to the normal-incidence geometry, where the \nangles of incidence were 9° and 3° (measured from the sample normal) for the probe and the pump pulses, respectively.   The rotation of the probe polarization plane is caused by two MO effects – the polar \nKerr effect and the magnetic linear  dichroism, which are sensitiv e to perpendicular-to-plane \nand in-plane components of  magnetization, respectively.\n31-33 For all MO experiments, samples \nwere mounted in a cryostat and cooled down to ≈ 15 K. The cryostat was placed between the \npoles of an electromagnet and the external magnetic field Hext ranging from ≈ 0 to 585 mT \nwas applied in the sample plane,  either in the [010] or [110] crystallographic di rection of the \nsample (see inset in Fig. 1 for a definition of the coordinate system). Prior to all \nmeasurements, we always prepared the magnetiza tion in a well-defined state by first applying \na strong saturating magnetic field and then  reducing it to the desired magnitude of Hext. \n  \nIII. RESULTS AND DISCUSSION \n \nA. Sample characterization \n \n The hysteresis loops measured by SQUID magnetometry for external magnetic field \napplied along the in-plane [-110] and perpendicu lar-to-plane [001] crystallographic directions \nin (Ga,Mn)As and (Ga,Mn)(As,P) samples are s hown in Fig. 1(a) and Fig. 1(c), respectively. \nThese data confirm the expected in-plane and perp endicular-to-plane orient ations of the EA in \n(Ga,Mn)As and (Ga,Mn)(As,P), respectively. Moreover, they reveal that for the \n(Ga,Mn)(As,P) sample, an external magnetic field of  250 mT is needed to rotate the \nmagnetization into the sample plane. In Fig. 1(b) and Fig. 1(d) we show the temperature \ndependences of the remanent magnetization of the samples from which the Curie temperature \nT\nc of  130 K and  110 K can be deduced. The measur ed saturation ma gnetization also \nindicates very similar density of Mn moments contributing to the ferromagnetic state in the \ntwo samples. 4 \n  \nFig. 1 (Color online): Magnetic characterization of samples:  (a), (b) (Ga,Mn)As  and (c), (d) (Ga,Mn)(As,P). (a), \n(c) Hysteresis loops measured in at 2 K for the external magnetic field applied in the sample plane (along the \ncrystallographic direction [-110]) and perpendicular to sample plane (along the crystallographic direction [001]). (b), (d) Temperature dependence of the remanent magnetization. Inset: Definition of the coordinate system. \n \n The electrical characterization of the samp les is shown in Fig. 2. The measured data \nshow a sharp Curie point singula rity in the temperature derivative of the resistivity which \nconfirms the high quality of the samples.25 The hole densities inferred from Hall \nmeasurements are (1.3  0.2)  1021 cm-3 and (0.8  0.2)  1021 cm-3 for (Ga,Mn)As and \n(Ga,Mn)(As,P), respectively. The hole density obtained for (Ga, Mn)As is in agreement with \nour previous measurements for simila r films in magnetic  fields up 14 T.30 The reduction of \nthe density of itinerant holes quantitatively correlates with the observed increase of the resistivity of the (Ga,Mn)(As,P) film as compared to the (Ga,Mn)As sample. \n \n5 \n  \nFig. 2 (Color online): Electrical char acterization of samples. Temperature dependence of the resistivity (a) and \nits temperature derivative (b). \n \n  \nB. Time-resolved magnet o-optical experiment \n \n In Fig. 3(a) and 3(b) we show the measur ed MO signals that reflect the magnetization \ndynamics in (Ga,Mn)As and (Ga,Mn)(As,P) sa mples, respectively. Th ese signals can be \ndecomposed into the oscillatory parts [Figs. 3(c) and 3(d)] and the non-oscillatory pulse-like \nbackground [Fig. 3(e) and 3(f)].27, 28 The oscillatory part arises  from the precessional motion \nof magnetization around the quasi-equilibrium EA and the pulse-like function reflects the \nlaser-induced tilt of the EA and the laser-induced demagnetization.25,31 The pump \npolarization-independent MO data  reported here, which were measured at a relatively low \nexcitation intensity of 30 μJcm-2, can be attributed to the ma gnetization precession induced by \na transient heating of the sample due to the absorption of the laser pulse.8,9 Before absorption \nof the pump pulse the magnetization is along th e EA direction. Absorptio n of the laser pulse \nleads to a photo-injection of electron-hole pa irs. The subsequent fast non-radiative \nrecombination of photo-injected  electrons induces a transi ent increase of  the lattice \ntemperature (within tens of picoseconds afte r the impact of the pu mp pulse). The laser-\ninduced change of the lattice temperature then leads to a change of the EA position.34 As a \nresult, magnetization starts to follow th e EA shift by the precessional motion. Finally, \ndissipation of the heat leads to a return of  the EA to the equilibrium position and the \nprecession of magnetization is stopped by a Gilbert damping.25  It is apparent from Fig. 3 \nthat the measured MO signals are strongly dependent on a magnit ude of the external magnetic \nfield, which was applied in the epilayer plan e along the [010] crystall ographic direction in \nboth samples. In particular, absorption of the laser pulse does not induce precession of \nmagnetization in (Ga,Mn)(As,P) unless magnetic field stronger than 20 mT is applied [see \nFig. 3(d)].  \n  \n6 \n  \n \nFig. 3 (Color online): Time-resolved magneto-optical (MO) signals measured in (Ga,Mn)As (a) and \n(Ga,Mn)(As,P) (b) for two magnitudes of the external magnetic field applied along the [010] crystallographic \ndirection. The measured MO signals were decomposed in to oscillatory parts [(c) and (d]), which correspond to \nthe magnetization precession, and to non-oscillatory part s [(e) and (f)], which are connected with the quasi-\nequilibrium tilt of the easy axis and with the demagnetization. Note different x-scales in the left and in the right \ncolumns. \n \n The magnetization dynamics is describe d by the Landau-Lifshitz-Gilbert (LLG) \nequation that is usually expressed in the form35,36: \n \n ௗࡹሺ௧ሻ\nௗ௧ൌെ ߛൣ ࡹሺݐሻൈࢌࢌࢋࡴሺݐሻ൧ఈ\nெೞቂࡹሺݐሻൈௗࡹሺ௧ሻ\nௗ௧ቃ,      ( 1 )  \n  \nwhere   = (gμB)/ћ is the gyromagnetic ratio,  g is the Landé g-factor, μB is the Bohr magneton, \nħ is the reduced Planck constant,  is the Gilbert damping constant, and Heff is the effective \nmagnetic field. Nevertheless, it is more conve nient to express this equation in spherical \ncoordinates where the directi on of the magnetization vector M is given by the polar angle θ \nand azimuthal angle φ and where Heff can be directly connected w ith angular derivatives of the \nfree energy density functional F (see the Appendix).37 For small deviations δ and δ of \nmagnetization from its equilibrium position (given by 0 and 0), the solution of LLG \nequation can be written in the form (t) = 0 + δ(t) and (t) = 0 + δ(t) as \n \n ߠሺݐሻൌߠܣఏ݁ି௧ݏܿሺ2ݐ݂ߨΦ ఏሻ,        ( 2 )  \n ߮ሺݐሻൌ߮ܣఝ݁ି௧ݏܿ൫2ݐ݂ߨΦ ఝ൯,       ( 3 )  \n \nwhere the constants A (A) and  () represent the initial amplitude and phase of  (), \nrespectively, f is the magnetization precession frequency, and kd is the precession damping \nrate (see the Appendix). The pr ecession frequency reflects the in ternal magnetic anisotropy of \nthe sample that can be characterized by the cubic ( KC), in-plane uniaxial ( Ku) and out-of-plane \nuniaxial ( Kout) anisotropy fields (see Eq. (A4) in the Appendix).10 Moreover, f depends also on \nthe magnitude and on the orientation of Hext (see the Appendix) and, therefore, the magnetic \n7 \n field dependence of f can be used to evalua te the magnetic anisotropy fields in the sample. If \nthe applied in-plane magnetic field is strong e nough to align the magnetiz ation parallel with \nHext (i.e., for Hext exceeding the saturation field in the sa mple for a particular orientation of \nHext),   = H = π/2 and  = H and if the precession damping is relatively slow , i.e. α2 ≈ 0 f  \ncan be expressed as  \n \n ݂ൌఓಳ\nඩ൬ܪ௫௧െ2ܭ௨௧ሺଷା௦ସఝ ሻ\nଶ2ܭ௨݊݅ݏଶቀ߮ுെగ\nସቁ൰\nൈሺܪ௫௧2ܭݏܿ4߮ ுെ2ܭ௨݊݅ݏ2߮ுሻ,   (4) \n \n \n \nFig. 4 (Color online): Fourier spectrum of the oscillatory  part of the MO signal measured in (Ga,Mn)As for \nexternal magnetic fields applied alon g the [010] crystallo graphic direction. f0 and f1 indicate the frequencies of \nthe uniform magnetization precession and the fi rst spin wave resona nce, respectively. \n \n In Fig. 4 we show the fast Fourier transfor m (FFT) spectra of the oscillatory parts of \nthe MO signals measured in the (Ga,Mn )As sample for different values of Hext. This figure \nclearly reveals that for all external magnetic  fields there are two distinct oscillatory \nfrequencies present in the measured data . These precession modes are the spin wave \nresonances (SWRs) – i.e., spin waves (or magno ns) that are selectively amplified by fulfilling \nthe boundary conditions: In a homogeneous thin magnetic film  with a thickness L, only the \nperpendicular standing waves with a wave vector k fulfilling the resonant condition kL = n \n(where n is the mode number) are amplified.25,38-41 In our case – using the ferromagnetic films \nwith a thickness around 25 nm – we detect only42 the uniform magnetiza tion precession with \nzero k vector (i.e. the precession where at any instant of time all magnetic moments are \nparallel over the entire sample; n = 0 at frequency f0) and the first SWR (i.e.  n = 1 at \nfrequency f1). See the inset in Fig. 8 for a schematic de piction of the modes. In Fig. 5 we plot \nthe amplitudes of the uniform magnetization precession ( A0) and of the first SWR ( A1) as a \nfunction of the exte rnal magnetic field Hext. In the (Ga,Mn)As sample, the oscillations are \npresent even when no magnetic field is applied and the precession amplitude increases \nslightly with an increasing Hext (up to  20 mT for A0 and up to  60 mT for A1). Above this \nvalue, a further increase of Hext leads to a suppression of the oscillations, but the suppression \nof the first SWR is slower than that of the uniform magnetization precession [see Fig. 5(c)]. In \n8 \n (Ga,Mn)(As,P), the oscillatory signal starts to  appear at  50 mT, reaches its maximum for \nHext  175 mT, and a further increase of Hext leads to its monotonic d ecrease, like in the case \nof (Ga,Mn)As. The observed field dependence of the precession amplitude, which expresses \nthe sensitivity of the EA position on the laser- induced sample temperature change, can be \nqualitatively understood as follows. In (Ga,Mn)As,  the position of the EA in the sample plane \nis given by a competition between the cubic and the in-plane uniaxial magnetic \nanisotropies.10,25 The laser-induced heating of the sa mple leads to a reduction of the \nmagnetization magnitude M and, consequently, it enhances th e uniaxial anisotropy relative to \nthe cubic anisotropy.9 This is because the uniaxial anisotropy component scales with \nmagnetization as ~ M2 while the cubic component scales as ~ M4. The application of Hext \nalong the [010] crystallographic di rection deepens the minimum in  the [010] direction in the \nfree energy density functional F (due to the Zeeman term in F, see Eq. (A4) in the Appendix). \nMeasured data shown in Fig. 5 reve al that in the (Ga,Mn)As sample, Hext initially (for Hext up \nto  20 mT) destabilizes the posit ion of EA but stabilizes it for large values of Hext (where the \nposition of the energy minimum in F is dominated by the Zeeman term, which is not \ntemperature dependent). In the case of (Ga,Mn)( As,P), the position of the EA is determined \nby the strong perpendicular-to-p lane anisotropy. Therefore, w ithout an external magnetic \nfield, the laser-induced heating of the sample doe s not change significantl y the position of EA \nand, consequently, does not initiate the pr ecession of magnetization [see Fig. 5(b)]. The \napplication of an in-plane fi eld moves the energy minimum in F towards the sample plane \n[see Fig. 1(c)] which makes the EA position more  sensitive to the laser-induced temperature \nchange. Finally, for a sufficiently strong Hext, the sample magnetic anisotropy is dominated by \nthe temperature-independent Zeeman term, wh ich again suppresses the precession amplitude. \nThe markedly different ratio A1/A0 in the (Ga,Mn)As and (Ga,Mn )(As,P) samples is probably \nconnected with a different surface magnetic anis otropy and/or a slight difference in magnetic \nhomogeneity in these two samples.43,44 \n  \n \n \nFig. 5 (Color online): Dependence of the amplitude of the uniform magnetization precession ( A0) and the first \nspin wave resonance ( A1) on the magnitude of the external magnetic field ( Hext) applied along the [010] \ncrystallographic direction in (Ga,Mn )As (a) and (Ga,Mn)(As,P) (b). (c) and (d) Dependence of the ratio  A1 / A0 \non Hext.  \n9 \n C. Determination of magnetic anisotropy \n \n In Fig. 6 we plot the ma gnetic field dependences of f0 and f1 for two different \norientations of Hext. The frequency f0 of the spatially uniform precession of magnetization is \ngiven by Eq. (4). For the SWRs, where the local moments are no longer para llel (see the inset \nin Fig. 8), restoring torques due to exchange interaction and internal magnetic dipolar \ninteraction have to be included in the analysis.39-41,45 For Hext along the [010] crystallographic \ndirection (i.e., for φH = /2) Eq. (4) can be written as \n \n ݂ൌఓಳ\nඥሺܪ௫௧െ2ܭ௨௧ܭ∆ܪሻሺܪ௫௧െ2ܭെ2ܭ௨∆ܪሻ ,  (5) \n \nwhere Hn is the shift of the resonant field for the nth spin-wave mode with respect to the \nn = 0 uniform precession mode. Analogically, for Hext applied in the [110] crystallographic \ndirection (i.e., for φH = /4)  \n \n ݂ൌఓಳ\nඥሺܪ௫௧െ2ܭ௨௧2ܭܭ௨∆ܪሻሺܪ௫௧2ܭ∆ܪሻ.   (6) \n \nThe lines in Fig. 6 represent the fits  of all four measured dependencies fn = fn (Hext, H) [where \nn = 0; 1 and H = /4; /2] with a single set of anisotropy constants for each of the samples, \nwhich confirms the credibility of the fitting pr ocedure. The obtained an isotropy constants at \n≈ 15 K are: KC = (17 ± 3) mT, Ku = (11 ± 5) mT, Kout = (-200 ± 20) mT for (Ga,Mn)As and KC \n= (14 ± 3) mT, Ku = (11 ± 5) mT, Kout = (90 ± 10) mT for (Ga,Mn)(As,P), respectively (in \nboth cases we considered the Mn g-factor of 2). For (Ga,Mn)As, we can now compare these \nanisotropy constants with those obtained by the same fitting procedure for samples prepared \nin a different MBE laboratory (in Prague) – see Fig. 4 in Ref. 25. We see that the previously \nreported25 doping trends of KC and Kout predict for a sample with nominal Mn doping x = 6% \nthe anisotropy fields which are the same as thos e reported in this pape r for the sample grown \nin Nottingham. This observation is in accord with the current microscopic understanding of \ntheir origin – KC reflects the zinc-blende crystal st ructure of the host semiconductor and Kout \n  \n \n \nFig. 6 (Color online): Magnetic field dependence of the precession frequencies f0 and f1 for two different \norientations of the external magnetic field (points) measured in (Ga,Mn)As (a) and (Ga,Mn)(As,P) (b). Lines are the fits by Eqs. (5) and (6). ΔH\n1 indicates the shift of the resonant field for the first spin-wave mode with respect \nto the uniform precession mode. \n10 \n  \n is a sum of the anisotropy due to  the growth-induced lattice-ma tching strain and of the thin-\nfilm shape anisotropy, which should be the sa me for equally doped and optimally synthesized \nsamples, independent of the growth chamber. On  the other hand, the micr oscopic origin of in-\nplane uniaxial anisotropy field K\nu is still not established10,25 and our data reveal that it is \nconsiderably smaller in the sample grown in Nottingham. Th e incorporation of phosphorus \ndoes not change significa ntly the values of KC and Ku but it strongly modi fies the magnitude \nand changes the sign of Kout, which is in agreement with the previous results obtained by \nFMR experiment.22  \n  \nD. Determination of spin stiffness \n \n The observation of a higher-o rder SWR enables us to also determine the exchange \nspin stiffness constant D, which is a parameter that is rather difficult to extract from other \nexperiments in (Ga,Mn)As.25,46 In homogeneous thin films, Hn is given by the Kittel \nformula43 \n \n Δܪ\tܪെܪൌ݊ଶ\nఓಳగమ\nమ,          ( 7 )  \n \nwhere L is the thickness of the magnetic film. By fitting the data in Fig. 6, we obtained H1 = \n(363 ± 2) mT for (Ga,Mn)As and (271 ± 2) mT for (Ga,Mn)(As,P) which correspond to D = \n(2.5 ± 0.2) meVnm2 and (1.9 ± 0.2) meVnm2 for (Ga,Mn)As and (Ga,Mn)(As,P), respectively \n(note that the relatively large experimental error in D is given mainly by the uncertainty of the \nepilayer thickness).29 The value obtained for (Ga,Mn)As is again in agreement with that \nreported previously for samples grown in Prague,25 which also confirms the consistent \ndetermination of the epilayer thicknesses in both MBE laboratories.29 The incorporation of \nphosphorus leads to a  reduction of D which correlates with the decrease of the hole density,47 \nand the reduced Tc in (Ga,Mn)(As,P), as compared to its (Ga,Mn)As counterpart. \n   \nE. Determination of Gilbert damping \n \n The Gilbert damping constant α can be determined by fitting the measured dynamical \nMO signals by the LLG equation.\n35,36,48 For a relatively slow precession damping and a \nsufficiently strong external magnetic field, the analytical solution of the LLG equation gives \n(see the Appendix)  \n ݇\nௗൌߙఓಳ\nଶ൬2ܪ௫௧െ2ܭ௨௧ሺଷାହ௦ସఝ ಹሻ\nଶܭ௨ሺ1െ3݊݅ݏ2߮ ுሻ൰.  (8) \n \nEq. (8) shows not only that kd is proportional to  but also that for obtaining a correct value of \n from the measured MO precession signal damp ing it is necessary to take into account a \nrealistic magnetic anisotropy of the investigated sample. Nevert heless, the correct dependence \nof kd on magnetic anisotropy was not cons idered in the previous studies35,36,48 where only one \neffective magnetic field was used, which is probably one of the reasons why mutually \ninconsistent results were obtained for Ga 1-xMn xAs with a different Mn content x. An increase \nof  from  0.02 to  0.08 for an increase of x from 3.6% to 7.5% was reported in Ref. 36. On 11 \n the contrary, in Ref. 48 values of  from 0.06 to 0.19 – without any apparent doping trend – \nwere observed for x from 2% to 11%.  \n For numerical modeling of the measured MO data, we first computed from the LLG \nequation (Eqs. (A1) and (A2) in  the Appendix with th e measured magnetic anisotropy fields) \nthe time-dependent deviations of the spherical angles [ (t) and (t)] from the corresponding \nequilibrium values ( 0, 0). Then we calculated how such changes of  and  modify the \nstatic magneto-optical response of the samp le, which is the signal that we detect \nexperimentally31 \n \n         0\n00 2sin2 2cos2 ,MLD s MLD PKEPMt MPt Pt t MO .    (9) \n \nThe first two terms in Eq. (9) are connected wi th the out-of-plane and in-plane movement of \nmagnetization, and the last term describes a change of the sta tic magneto-optical response of \nthe sample due to the laser-induced demagnetization.31 PPKE and PMLD are MO coefficients \nthat describe the MO response of the sample which we measured independently in a static \nMO experiment,32,33 and β is the probe polarization orientation with respect to the \ncrystallographic direction [100].31 To further simplify the fitting procedure, we can extract the \noscillatory parts from the measured MO data (cf. Fig. 3), which effectively removes the MO \nsignals due to the laser-induced demagnetization [i .e., the last term in Eq. (9)] and due to the \nin-plane movement of the easy axis [i.e., a part  of the MO signal desc ribed by the second term \nin Eq. (9)].31 Examples of the fitting of the precessional MO optical data are shown in Fig. \n7(a) and (b) for (Ga,Mn)As and (Ga,Mn)(As,P), respectively. We stress that in our case the \nonly fitting parameters in the modeling are the damping coefficient  and the initial \ndeviations of the spherical angles from the corresponding equilibrium values. By this \nnumerical modeling we deduced a de pendence of the damping factor  on the external \nmagnetic field for two different orientations of Hext. At smaller fields, the dependences \nobtained show a strong anisotropy w ith respect to the field angle th at can be fully ascribed to \nthe field-angle dependence of  the precession frequency.25 However, when plotted as a \nfunction of the precession frequency, the de pendence on the field-an gle disappears – see \nFig. 7(c) and (d) for (Ga,Mn)As and (Ga,Mn )(As,P), respectively. For both materials,  \ninitially decreases monotonously with f and finally it saturates at a certain value for f ≥ \n10 GHz. A frequency-dependent (or magnetic field-dependent) damping parameter was \nreported in various magnetic materials and a va riety of underlying mechanisms responsible \nfor it were suggested  as an explanation.49-51 In our case, the most probable explanation seems \nto be the one that was used by Walowski et al.  to explain the experimental results obtained in \nthin films of nickel.49 They argued that in the low field range small magnetization \ninhomogeneities can be formed – the magnetizati on does not align parallel in an externally \napplied field, but forms ripples.49 Consequently, the measured MO signal which detects \nsample properties averaged over the laser spot size, which is in our case about 30 m wide \n(FWHM), experiences an apparent oscillation damping  because the magnetic properties \n(i.e., the precession frequencies) are slightly differing within the spot size (see Fig. 6 and 7 in \nRef. 49). On the other hand, for stronger external  fields the sample is fully homogeneous and, \ntherefore, the precession damping is not de pendent on the applied field (the precession \nfrequency), as expected for the in trinsic Gilbert damping coefficient.52,53 We note that the \nobserved monotonous frequency decrease of α is in fact a signature of a magnetic \nhomogeneity of the studied epilayers.25 The obtained frequency-independent values of α are \n(0.9 ± 0.2)  10-2 for (Ga,Mn)As and (1.9 ± 0.5)  10-2 for (Ga,Mn)(As,P), respectively. The 12 \n observed enhancement of the magnetization pre cession damping due to the incorporation of \nphosphorus is also clearly apparent  directly from Figs. 7(a) and 7(b) where the MO data with \nsimilar precession frequencies are shown for (G a,Mn)As and (Ga,Mn)(As,P), respectively. In \n(Ga,Mn)As the value of α obtained is again fully in accord with the reported Mn doping trend \nin α in this material.25  In (Ga,Mn)(As,P), the determined α is similar to the value 1.2  10-2 \nwhich was reported by Cubukcu et al. for (Ga,Mn)(As,P) with a si milar concentration of Mn \nand P.22 Comparing to the doping trends in the se ries of optimized (Ga,Mn)As materials,25 the \nvalue of α i n  o u r  ( G a , M n ) ( A s , P )  s a m p l e  i s  c o n s i s tent with the measured Gilbert damping \nconstant in  lower Mn-doped (Ga,Mn)As epilayers with similar hole densities and resistivities \nto those of the  (Ga,Mn)(As,P) film.  \n \nFig. 7 (Color online): Determination of the Gilbert da mping. (a) and (b) Oscillatory part of the MO signal \n(points) measured in (Ga,Mn)As for the external magnetic field 100 mT (a) and in (Ga,Mn)(As,P) for 350 mT \n(b); magnetic field applied along the [010] crystallographic direction leads to a similar frequency ( f0  7.5 GHz) \nin both cases. Lines are fits by the Landau-Lifshitz-G ilbert equation. (c) and (d) Dependence of the damping \nfactor () on the precession frequency for two different orienta tions of the external magnetic field in (Ga,Mn)As \n(c) and (Ga,Mn)(As,P) (d). \n \n The high quality of our MO data enables us  to evaluate not only the damping of the \nuniform magnetization precession, which is addresse d above, but also the damping of the first \nSWR. To illustrate this procedure, we show  in Fig. 8(a) the MO data measured for Hext = \n13 \n 250 mT applied along the [010] crystallographic direction in (Ga,Mn)As. The experimental \ndata (points) obtained can be fitted by a sum of two expone ntially damped cosine functions \n(line) which enables us to separate,  directly in a time domain, the contributions of the individual precession modes to the measured MO signal. In this particular case, the uniform \nmagnetization precession occurs at a frequency f\n0 = 12.2 GHz and this precession mode is \ndamped with a rate constant kd0 = 0.79 ns-1. Remarkably, the first SWR, which has a \nfrequency f1 = 23.0 GHz, has a considerably la rger damping rate constant kd1 = 1.7 ns-1 – see \nFig. 8(b) where the contribution of individual modes ar e directly compared and also Fig. 8(c) \nwhere Fourier spectra computed from the measured  MO data for two diffe rent ranges of time \ndelays are shown. To convert the damping rate constant kdn obtained to the damping constant \n \n \nFig. 8 (Color online): Comparison of the Gilbert damping of the uniform magnetization precession and of the \nfirst spin wave resonance. (a) Oscillatory part of the MO  signal (points) measured in (Ga,Mn)As for the external \nmagnetic field 250 mT applied along the [010] crystallographic direction. The solid line is a fit by a sum of two exponentially damped cosine functions that are shown in (b). Inset: Schematic illustration\n39 of the spin wave \nresonances with n = 0 (uniform magnetization precession with zero k vector) and n = 1 (perpendicular standing \nwave with a wave vector k fulfilling the resonant condition kL = ) in a magnetic film with a thickness L. (c) \nNormalized Fourier spectra computed fo r the depicted ranges of time delays from the measured MO data, which \nare shown in (a). (d) Dependence of the damping factor ( n) on the precession frequency for the uniform \nmagnetization precession ( n = 0) and the first spin wave resonance ( n = 1). \n14 \n  \nn for the n-th mode, we can use the ge neralized analytic al solution of the LLG equation. For \na sufficiently strong Hext along the [010] crystallographic direction (i.e., when φ  φH = /2), \nEq. (8) can be written as \n \n ݇ௗൌߙఓಳ\nଶሺ2ܪ௫௧2∆ܪ െ2ܭ௨௧2ܭܭ௨ሻ.    (10) \n \nFor the case of MO data measured at Hext = 250 mT, the damping constants obtained for \nmodes with n = 0 and 1 are 0 = 0.009 and 1 = 0.011, respectively. [We note that the value \nof 0 obtained from the analytical solution of LLG equation is identical to that determined by \nthe numerical fitting and shown in Fig. 7(c), which confirms the consistency of this \nprocedure.] In Fig. 8(d) we show the dependence of 0 and 1 on the precession frequency. \nThese data clearly show that  even if the modes with n = 0 and 1 were oscillating with the \nsame frequency, the SWR mode with n = 1 would have a larger  damping coefficient. \nHowever, for sufficiently high fr equencies (i.e., external magnetic  fields) the damping of the \ntwo modes is nearly equal [see Fig. 8(d)].  This feature can be ascribed to the presence of an \nextrinsic contribution to the damping coeffici ent for the SWR modes. The extrinsic damping \nprobably originates from small variations of the sample thickness (< 1 nm) within the laser \nspot size54 and/or from the presence of a weak bulk inhomogeneity,43 which is apparent as \nsmall variations of ΔHn. The frequency spacing and the (Ki ttel) character of the SWR modes \nis insensitive to such small variations of ΔHn but the resulting frequency variations (see Eq. 5) \ncan still strongly affect the observed damping of the oscillations. For high enough external \nmagnetic fields, the variations of ΔHn have a negligible role a nd the damping of the SWR \nmodes is governed solely by the intrinsic Gilbert damping parameter.   \nIV. CONCLUSIONS \n \n We used the optical analog of FMR, wh ich is based on a pump-and-probe magneto-\noptical technique, for the determination of micromagnetic parameters of (Ga,Mn)As and \n(Ga,Mn)(As,P) DMS materials. The main advantage of this technique is that it enables us to \ndetermine the anisotropy constants, the spin s tiffness and the Gilbert damping parameter from \na single set of the experimental magneto-optical  data measured in films with a thickness of \nonly several tens of nanometers. To addres s the role of phosphorus incorporation in \n(Ga,Mn)As, we measured simultaneously proper ties of (Ga,Mn)As and (Ga,Mn)(As,P) with \n6% Mn-doping which were grown under identical  conditions in the sa me MBE laboratory. \nWe have shown that the laser-i nduced precession of magnetization is closely connected with a \nmagnetic anisotropy of the samples. In partic ular, in (Ga,Mn)As with in-plane magnetic \nanisotropy the laser-pulse-induced precession of  magnetization was observed even when no \nexternal magnetic field was applied. On the cont rary, in (Ga,Mn)(As,P) with perpendicular-to-\nplane magnetic EA the precession of magnetizat ion was observed only when the EA  position \nwas destabilized by an external in-plane ma gnetic field. From the measured magneto-optical \ndata we deduced the anisotropy constants, spin  stiffness, and Gilber t damping parameter in \nboth materials. We have shown that the incorp oration of 10% of P in  (Ga,Mn)As leads not \nonly to the expected sign change of the perpendi cular-to-plane anisotropy field but also to a \nconsiderable increase of the G ilbert damping which correlates with the increased resistivity \nand reduced itinerant hole density in the (Ga,Mn)(As,P) material. We also observed a reduction of the spin stiffness consistent with the suppression of T\nc upon incorporating P in 15 \n (Ga,Mn)As. Finally, we found that in small exte rnal magnetic fields the damping of the first \nspin wave resonance is sizably stronger than  that of the uniform magnetization precession.  \n  \nACKNOWLEDGEMENTS \n \n This work was supported by the Grant Agency of the Czech Republic grant no. \nP204/12/0853 and 202/09/H041, by the Grant Agency of Charles University in Prague grant \nno. 1360313 and SVV-2013-267306, by EU grant ERC Advanced Grant 268066 - 0MSPIN, \nand by Praemium Academiae of the Academy of Sciences of the Czech Republic, from the \nMinistry of Education of the Czech Re public Grant No. LM2011026, and from the Czech \nScience Foundation Grant No. 14-37427G.   \nAPPENDIX \n \n Due to symmetry reasons, it is conveni ent to rewrite the LLG equation given by \nEq. (1) in spherical coordinates where M\nS describes the magnetiza tion magnitude and polar θ \nand azimuthal φ angles characterize its orientation. We define the perpendicular-to-plane \nangle θ (in-plane angle φ) in such a way that it is counted from the [001] ([100]) \ncrystallographic direction and it is positive wh en magnetization is tilted towards the [100] \n([010]) direction (see inset of Fig. 1 for the co ordinate system definition). The time evolution \nof magnetization is given by37 \n \n ௗெೞ\nௗ௧ൌ0,           ( A 1 )  \n ௗఏ\nௗ௧ൌെఊ\nሺଵାఈమሻெೞቀߙ ∙ܣ\n௦ఏቁൌΓఏሺߠ,߮ሻ ,       ( A 2 )  \n ௗఝ\nௗ௧ൌఊ\nሺଵାఈమሻெೞ௦ఏቀܣെఈ∙\n௦ఏቁൌΓఝሺߠ,߮ሻ ,       ( A 3 )  \n \nwhere A = dF/d and B = dF/d are the derivatives of the free energy density functional F \nwith respect to  and , respectively. We express F in a form10 \n \nܨൌܯ ௌܭ݊݅ݏଶߠቀଵ\nସ݊݅ݏଶ2݊݅ݏ߮ଶߠ ݏܿଶߠቁെܭ ௨௧ݏܿଶߠെೠ\nଶ݊݅ݏଶߠሺ1െ݊݅ݏ2߮ ሻെ\nെܪ௫௧൫ߠݏܿߠݏܿ ுߠ݊݅ݏߠ݊݅ݏ ுݏܿሺ߮െ߮ுሻ൯൩, (A4) \n \nwhere KC, Ku and Kout are the constants that characterize the cubic, uniaxial and out-of-plane \nmagnetic anisotropy fields in (Ga,Mn)As, respectively. Hext is the magnitude of the external \nmagnetic field whose orientati on is described by the angles θH and φH, which are again \ncounted from the [001] and [100] crystallographic direc tions, respectively. For small \ndeviations δθ and δφ from the equilibrium values θ0 and φ0, the Eqs. (A2) and (A3) can be \nwritten in a linear form as  \n ௗఏ\nௗ௧ൌܦଵሺߠെߠሻܦଶሺ߮െ߮ሻ,        ( A 5 )  \n ௗఝ\nௗ௧ൌܦଷሺߠെߠሻܦସሺ߮െ߮ሻ,        ( A 6 )  \n \nwhere    16 \n   ܦଵൌௗഇ\nௗୀబ,ୀబ ,          ( A 7 a )  \n  ܦଶൌௗഇ\nௗୀబ,ୀబ ,          ( A 7 b )  \nand analogically for D3, D4. The solution of Eqs. (A5) and (A6) is expressed by Eqs. (2) and \n(3) where the magnetizati on precession frequency f and the damping rate kd are given by \n \n ݂ൌඥସሺభరିమయሻିሺభାరሻమ\nସగ,          ( A 8 )  \n ݇ௗൌെభାర\nଶ.           ( A 9 )  \n \nEqs. (A8) and (A9) for F in the form (A4) can be simplified when the geometry of our \nexperiment – i.e., the in-plane orientation of the external magnetic field ( θH = π/2) – is taken \ninto account.  The equilibrium orientation of magnetization is in the sample plane for \n(Ga,Mn)As ( θ0 = π/2) and the same applies for (Ga,Mn)(As, P) if sufficiently strong external \nmagnetic field (see Fig. 1) is applied ( θ0 ≈ θH = π/2). In such conditi ons, the precession \nfrequency f and the damping rate kd are given by the following equations \n \n ݂ൌఓಳ\nଶగሺଵାఈమሻ\nۣളളളളളളളളളളളളളളളለ\n൬ܪ௫௧ݏܿሺ߮െ߮ுሻെ2ܭ௨௧ሺଷା௦ସఝ ሻ\nଶ2ܭ௨݊݅ݏଶቀ߮െగ\nସቁ൰ൈ\nൈሺܪ௫௧ݏܿሺ߮െ߮ுሻ2ܭݏܿ4߮െ2ܭ ௨݊݅ݏ2߮ሻ\nߙଶ\nەۖ۔ۖۓ൬ܪ௫௧ݏܿሺ߮െ߮ுሻെ2ܭ௨௧ሺଷା௦ସఝ ሻ\nଶ2ܭ௨݊݅ݏଶቀ߮െగ\nସቁ൰ൈ\nൈሺܪ௫௧ݏܿሺ߮െ߮ுሻ2ܭݏܿ4߮െ2ܭ ௨݊݅ݏ2߮ሻെ\nെቀܪ௫௧ݏܿሺ߮െ߮ுሻെܭ௨௧ሺଷାହ௦ସఝ ሻ\nସೠሺଵିଷ௦ଶఝ ሻ\nଶቁଶ\nۙۖۘۖۗ( A10)  \n݇ௗൌߙఓಳ\nଶሺଵାఈమሻ൬2ܪ௫௧ݏܿሺ߮െ߮ுሻെ2ܭ௨௧ሺଷାହ௦ସఝ ሻ\nଶܭ௨ሺ1െ3݊݅ݏ2߮ ሻ൰. (A11) \n  17 \n  \nREFERENCES \n \n* Corresponding author; nemec@karlov.mff.cuni.cz \n1 T. Jungwirth, J. Sinova, J. Mašek, J. Ku čera, and A. H. MacDonald, Rev. Mod. Phys. 78, \n809 (2006). \n2 Editorial , Nature Materials 9, 951 (2010). \n3 H. Ohno, Nature Materials 9, 952 (2010). \n4 T. Dietl, Nature Materials 9, 965 (2010). \n5 T. Jungwirth, J. Wunderlich, V. Novak, K. Ol ejnik, B. L. Gallagher, R. P. Campion, K. W. \nEdmonds, A. W. Rushforth, A. J. Ferguson, P. Nemec, Rev. Mod. Phys. in press (2014). \n6 H. Ohno, D. Chiba, F. Matsukura, T. Omiya, E. Abe, T. Dietl, Y. Ohno, and K. Ohtani, \nNature 408, 944 (2000). \n7 D. Chiba, M. Yamanouchi, F. Matsukura, H. Ohno, Science 301, 943 (2003). \n8 P. Němec, E. Rozkotová, N. Tesa řová, F. Trojánek, E. De Rani eri, K. Olejník, J. Zemen, \nV. Novák, M. Cukr, P. Malý, and T. Jungwirth, Nature Physics 8, 411 (2012). \n9 N. Tesařová, P. Němec, E. Rozkotová, J. Zemen, T. Janda, D. Butkovi čová, F. Trojánek, \nK. Olejník, V. Novák, P. Malý, and T. J ungwirth. Nature Photonics 7, 492-498 (2013). \n10 J. Zemen, J. Ku čera, K. Olejník, and T. Jungwirth, Phys. Rev. B 80, 155203 (2009). \n11 A. Dourlat, V. Jeudy, A. Lemaître, and C. Gourdon, Phys. Rev. B 78, 161303(R) (2008). \n12 D. Chiba, M. Yamanouchi, F. Matsukura, T. Dietl, and H. Ohno, Phys. Rev. Lett. 96, \n096602 (2006).  \n13 S. Haghgoo, M. Cubukcu, H. J. von Bardelebe n, L. Thevenard, A. Lemaître, and C. \nGourdon, Phys. Rev. B 82, 041301(R) (2010). \n14\n M. Yamanouchi, D. Chiba, F. Matsukura, T.  Dietl, and H. Ohno, Phys. Rev. Lett. 96, \n096601 (2006). \n15 E. De Ranieri, P. E. Roy, D. Fang, E. K. Vehs thedt, A. C. Irvine, D. Heiss, A. Casiraghi, \nR. P. Campion, B. L. Gallagher, T. Jungw irth, and J. Wunderlich, Nature Mater. 12, 808 \n(2013). \n16 K. Y. Wang, A. W. Rushforth, V. A. Grant,  R. P. Campion, K. W. Edmonds, C. R. \nStaddon, C. T. Foxon, and B. L. Gallagher, J. W underlich, and D. A. Willi ams, J. Appl. Phys. \n101, 106101 (2007). \n17 A. Dourlat, V. Jeudy, C. Testelin, F. Bernardot, K. Khazen, C. Gourdon, L. Thevenard, \nL. Largeau, O. Mauguin, and A. Lemaître, J. Appl. Phys. 102, 023913 (2007). \n18 D. Chiba, F. Matsukura, and H. Ohno, Appl. Phys. Lett. 89, 162505 (2006). \n19A. Lemaître, A. Miard, L. Travers, O. Mauguin, L. Largeau, C. Gourdon, V. Jeudy, M. \nTran, and J.-M. George, Appl. Phys. Lett. 93, 021123 (2008). \n20A. W. Rushforth, M. Wang, N. R. S. Farle y, R. P. Campion, K. W. Edmonds, C. R. \nStaddon, C. T. Foxon, and B. L. Gallagher, J. Appl. Phys. 104, 073908 (2008). \n21 A. Casiraghi, A. W. Rushforth, M. Wang, N.  R. S. Farley, P. Wadley, J. L. Hall, \nC. R. Staddon, K. W. Edmonds, R. P. Campion, C. T. Foxon, and B. L. Gallagher, Appl. \nPhys. Lett. 97, 122504 (2010). \n22 M. Cubukcu, H. J. von Bardel eben, Kh. Khazen, J. L. Can tin, O. Mauguin, L. Largeau, and \nA. Lemaître, Phys. Rev. B 81, 041202 (2010). \n23 A. Casiraghi, P. Walker, A. V. Akimov, K. W.  Edmonds, A. W. Rushfo rth, E. De Ranieri, \nR. P. Campion, B. L. Gallagher, and A. J. Kent, Appl. Phys. Lett. 99, 262503 (2011). \n24 L. Thevenard, S. A. Hussain, H. J. von Bard eleben, M. Bernard, A. Lemaître, and C. \nGourdon, Phys. Rev. B 85, 064419 (2012). \n25 P. Němec, V. Novák, N. Tesa řová, E. Rozkotová, H. Reichlová, D. Butkovi čová, F. 18 \n Trojánek, K. Olejník, P. Malý, R. P. Campion, B. I. Gallagher, J. Sinova, and T. Jungwirth, \nNature Commun. 4, 1422 (2013). \n26 H. Son, S. Chung, S.-Y. Yea, S. Kim, T. Yoo, S. Lee, X. Liu and J. K. Furdyna, Appl. \nPhys. Lett. 96, 092105 (2010). \n27 E. Rozkotová, P. N ěmec, P. Horodyská, D. Sprinzl, F.  Trojánek, P. Malý, V. Novák, K. \nOlejník, M. Cukr, T. Jungwirth, Appl. Phys. Lett 92, 122507 (2008).  \n28 E. Rozkotová, P. N ěmec, N. Tesa řová, P. Malý, V. Novák, K. Olejník, M. Cukr, T. \nJungwirth, Appl. Phys. Lett. 93, 232505 (2008).  \n29 A accurate determination of layer thicknesses is  a nontrivial task in case of thin (Ga,Mn)As \nlayers.25 Some standard techniques (e.g., X-ray re flectivity or optica l ellipsometry) are \ninapplicable due to the weak contrast between  the (Ga,Mn)As layer and the GaAs substrate, \nor unknown optical parameters. The relative accu racy of other common techniques (e.g., of \nX-ray diffraction) does not exceed 10% because of  the small thickness of the measured layer. \nTherefore, we used a thickness estimation ba sed on the following quantities: (i) the growth \ntime and the growth rate of the GaAs buffer layer measured by the RHEED oscillations (typical accuracy of ±3%); (ii) increase in  the growth rate by adding the known Mn-flux \nmeasured by the beam-flux monitor relatively to the Ga flux (typical accuracy of ±5% of the \nMn vs. Ga flux ratio); iii) reduction of thic kness by the native oxidation (-1.5 nm ± 0.5 nm); \n(iv) reduction of thickness by thermal oxidation (-1.0 nm ± 0.5 nm). Relative accuracy of \nsteps (i) and (ii) was verified on separate calibration growths of (G a,Mn)As on AlAs, where \nan accurate X-ray reflectivity method to m easure the (Ga,Mn)As layer thickness could be \nused. Typical thicknesses of the native and the thermal oxides in steps (iii) and (iv) were \ndetermined by XPS. The resulting total accuracy of the epilayer thickness determination is \nthus 3% (relative random error) and 1 nm (systematic error).  \n30 M. Wang, K.W. Edmonds, B.L. Gallagher, A. W. Rushforth, O. Makarovsky, A. Patane, \nR.P. Campion, C.T. Foxon, V. Novak, and T. Jungwirth, Phys. Rev. B 87, 121301 (2013). \n31 N. Tesařová, P. Němec, E. Rozkotová, J. Šubrt, H. Reichlová, D. Butkovi čová, F. Trojánek, \nP. Malý, V. Novák, and T. Jungwirth, Appl. Phys. Lett. 100, 102403 (2012). \n32 N. Tesařová, J. Šubrt, P. Malý, P. N ěmec, C. T. Ellis, A. Mukherjee, and J. Cerne, Rev. Sci. \nIntrum. 83, 123108 (2012). \n33 N. Tesarova, T. Ostatnicky, V. Novak, K. Olej nik, J. Subrt, H. Reichlova, C. T. Ellis, A. \nMukherjee, J. Lee, G. M. Sipahi, J. Sinova, J. Ha mrle, T. Jungwirth, P. Nemec,  J. Cerne, and \nK. Vyborny, Phys. Rev. B 89, 085203 (2014). \n34 N. Tesařová, E. Rozkotová, H. Reichlová, P. Mal ý, V. Novák, M. Cukr,T. Jungwirth, and P. \nNěmec, J. Nanosc. Nanotechnol. 12, 7477 (2012). \n35 Y. Hashimoto, S. Kobayashi, an d H. Munekata, Phys. Rev. Lett. 100, 067202 (2008). \n36 J. Qi, Y. Xu, A. Steigerwald, X. Liu, J. K. Furdyna, I. E. Perakis, and N. H. Tolk, Phys. \nRev. B 79, 085304 (2009). \n37 J. Miltat, G. Albuquerque, and A. Thiaville, in Spin Dynamics in Confined Magnetic \nStructures  I, edited by B. Hillebrands and K. Ounadjela (Springer, Berlin, 2002), Chap. 1. \n38 M. van Kampen, C. Jozsa, J. T. Kohlhepp, P. LeClair, L. Lagae, W.J.M. de Jonge, and B. \nKoopmans, Phys. Rev. Lett. 88, 227201 (2002). \n39 B. Lenk, G. Eilers, J. Hamrle, and M. Münzberg, Phys. Rev. B 82, 134443 (2010). \n40 X. Liu and J. K. Furdyna, J. Phys.:Cond. Matt. 18, R245-R279 (2006). \n41 D. M. Wang, Y. H. Ren, X. Liu, J. K. Fur dyna, M. Grimsditch, and R. Merlin, Superlatt. \nMicrostruct. 41, 372-375 (2007).  \n42 See Fig. 6 in Ref. 25 for the observation of hi gher SWRs in thicker films of (Ga,Mn)As and \nfor the justification of the assignment of the lowest precession mode observed in the time-\nresolved MO experiment to the unifo rm magnetization precession mode with n = 0.  19 \n 43 X. Liu, Y. Y. Zhou, and J. K. Furdyna, Phys. Rev. B 75, 195220 (2007). \n44 D. M. Wang, Y. H. Ren, X. Liu, J. K. Furdyna, M. Grimsditch, and R. Merlin, Phys, Rev. \nB 75, 233308 (2007). \n45 A. Kalinikos and A. N. Slavin, J. Phys. C 19, 7013 (1986). \n46\n A. Werpachovska, T. Dietl, Phys. Rev. B 82, 085204 (2010). \n47 J. König, T. Jungwirth, and A. H.  MacDonald, Phys. Rev. B 64, 184423 (2001). \n48 S. Kobayashi, Y. Hashimoto, and H. Munekata, J. Appl. Phys. 105, 07C519 (2009). \n49 J. Walowski, M. Djordjevic Kaufmann, B. Lenk, C. Hamann, J. McCord, and M. \nMunzenberg, J. Phys. D: Appl. Phys. 41, 164016 (2008). \n50 Y. Liu, L. R. Shelford, V. V. Kruglyak, R. J. Hicken, Y. Sakuraba, M. Oogane, and Y. \nAndo, Phys. Rev. B 81, 094402 (2010). \n51 A. A. Rzhevsky, B. B. Krichevtsov, D. E. Bürgler and C. M. Schneider, Phys. Rev. B 75, \n224434 (2007). \n52 J. Walowski, G. Müller, M. Djordjevic, M. Mün zenberg, M. Kläui, C. A.  F. Vaz, and J. A. \nC. Bland, Phys. Rev. Lett. 101, 237401 (2008). \n53 G. Woltersdorf, M. Buess, B. Heinri ch, and C. H. Back, Phys. Rev. Lett. 95, 037401 \n(2005). \n54 G. C Bailey and C. Vittoria, Phys. Rev. Lett. 28, 100 (1972). "    },    {        "title": "1405.6354v2.Spin_Hall_phenomenology_of_magnetic_dynamics.pdf",        "content": "Spin Hall phenomenology of magnetic dynamics\nYaroslav Tserkovnyak and Scott A. Bender\nDepartment of Physics and Astronomy, University of California, Los Angeles, California 90095, USA\n(Dated: July 2, 2021)\nWe study the role of spin-orbit interactions in the coupled magnetoelectric dynamics of a fer-\nromagnetic \flm coated with an electrical conductor. While the main thrust of this work is phe-\nnomenological, several popular simple models are considered microscopically in some detail, includ-\ning Rashba and Dirac two-dimensional electron gases coupled to a magnetic insulator, as well as a\ndi\u000busive spin Hall system. We focus on the long-wavelength magnetic dynamics that experiences\ncurrent-induced torques and produces \fctitious electromotive forces. Our phenomenology provides\na suitable framework for analyzing experiments on current-induced magnetic dynamics and recip-\nrocal charge pumping, including the e\u000bects of magnetoresistance and Gilbert-damping anisotropies,\nwithout a need to resort to any microscopic considerations or modeling. Finally, some remarks are\nmade regarding the interplay of spin-orbit interactions and magnetic textures.\nPACS numbers: 85.75.-d\nI. INTRODUCTION\nSeveral new directions of spintronic research have\nopened and progressed rapidly in recent years. Much\nenthusiasm is bolstered by the opportunities to initiate\nand detect spin-transfer torques in magnetic metals1and\ninsulators,2which could be accomplished by variants of\nthe spin Hall e\u000bect,3along with the reciprocal electromo-\ntive forces induced by magnetic dynamics. The spin Hall\ne\u000bect stands for a spin current generated by a transverse\napplied charge current, in the presence of spin-orbit in-\nteraction. From the perspective of angular momentum\nconservation, the spin Hall e\u000bect allows angular momen-\ntum to be leveraged from the stationary crystal lattice\nto the magnetic dynamics. A range of nonmagnetic ma-\nterials from metals to topological insulators have been\ndemonstrated to exhibit strong spin-orbit coupling, thus\nallowing for e\u000ecient current-induced torques.\nFocusing on quasi-two-dimensional (2D) geometries,\nwe can generally think of the underlying spin Hall phe-\nnomena as an out-of-equilibrium magnetoelectric e\u000bect\nthat couples planar charge currents with collective mag-\nnetization dynamics. In typical practical cases, the rel-\nevant system is a bilayer heterostructure, which consists\nof a conducting layer with strong spin-orbit coupling and\nferromagnetic layer with well-formed magnetic order. In\nthis case, the current-induced spin torque re\rects a spin\nangular momentum \rux normal to the plane, which ex-\nplains the spin Hall terminology.\nThe microscopic interplay of spin-orbit interaction and\nmagnetism at the interface translates into a macroscopic\ncoupling between charge currents and magnetic dynam-\nics. A general phenomenology applicable to a variety of\ndisparate heterostructures can be inferred by considering\na course-grained 2D system, which both conducts and\nhas magnetic order as well as lacks inversion symmetry\n(or else the pseudovectorial magnetization would not cou-\nple linearly to the vectorial current density). In a bilayer\nheterostructure, the latter is naturally provided by the\nbroken re\rection symmetry with respect to its plane.II. GENERAL PHENOMENOLOGY\nLet us speci\fcally consider a bilayer heterostructure\nwith one layer magnetic and one conducting, as sketched\nin Fig. 1. The nonmagnetic layer can be tailored to\nenhance spin-orbit coupling e\u000bects in and out of equi-\nlibrium. Phenomenologically, we have a quasi-2D sys-\ntem along the xyplane, which will for simplicity be\ntaken to be isotropic and mirror-symmetric in plane while\nbreaking re\rection symmetry along the zaxis. In other\nwords, the structural symmetry is assumed to be that\nof a Rashba 2D electron gas (although microscopic de-\ntails could be more complex), subject to a spontaneous\ntime-reversal symmetry breaking due to the magnetic\norder. Common examples of such heterostructures in-\nclude a thin transition-metal1or magnetic-insulator2\flm\ncapped by a heavy metal, or a layer of 3D topological in-\ny\nz\nj˙njsHaNaFxFN\nFIG. 1. Heterostructure consisting of a magnetic top layer\nand conducting underlayer. The charge current jinduces a\ntorque \u001cacting on the magnetic dynamics, which quanti\fes\nthe spin angular-momentum transfer in the zdirection. This\ncan be thought of as a spin current jsentering the ferro-\nmagnet at the interface. Reciprocally, magnetic dynamics _n\ninduces a motive force \u000facting on the itinerant electrons in\nthe conductor.arXiv:1405.6354v2  [cond-mat.mes-hall]  25 Jul 20142\nsulator doped on one side with magnetic impurities.4\nThe course-grained hydrodynamic variables used to de-\nscribe our system are the three-component collective spin\ndensity (per unit area) s(r;t) =sn(r;t)\u0011(snx;sny;snz)\nand the two-component 2D current density (per unit\nlength) j(r;t)\u0011(jx;jy) in thexyplane. Considering\nfully saturated magnetic state well below the Curie tem-\nperature, we treat the spin density as a directional vari-\nable, such that its magnitude sis constant and orienta-\ntional unit vector nparametrizes a smooth and slowly-\nvarying magnetic texture. We will be interested in slow\nand long-wavelength agitations of the ferromagnet cou-\npled to the electron liquid along with reciprocal motive\nforces. Perturbed out of equilibrium, the temporal evolu-\ntion of the heterostructure is governed by the forces that\ncouple to the charge \row and magnetic dynamics: the\n(planar) electric \feld and magnetic \feld, respectively.\nA. Decoupled dynamics\nA uniform electric-current carrying state in the isolated\nconducting \flm, subject to a constant external vector\npotential A, has the free-energy density\nF(p;A) =F0(p)\u0000p\u0001A\nc+O(A2); (1)\nwhereF0=Lp2=2 is the free-energy density in terms of\ntheparamagnetic current p(i.e., the current de\fned in\nthe absence of the vector potential A), andLis the local\nself-inductance of the \flm (including inertial and elec-\ntromagnetic contributions). According to time-reversal\nsymmetry, in equilibrium p= 0 when A= 0. The gauge\ninvariance (which requires that the minimum of F, as a\nfunction of p, is independent of A), furthermore, dictates\nthe following form of the free energy:\nF=L\n2\u0012\np\u0000A\ncL\u00132\n: (2)\nTherefore, the phenomenological expression for the full\ncurrent density is\nj\u0011\u0000c\u000eAF=p\u0000A\ncL; (3)\nwith\u000estanding for the 2D functional derivative of the\ntotal electronic free energy F[p] =R\nd2rF(p). We con-\nclude, based on Eqs. (2) and (3), that j=L\u00001\u000epF, which\nis thus the force thermodynamically conjugate to Lp.\nGeneral quasistatic equilibration5of a perturbed electron\nsystem can now be written as\nL_p=\u0000^%j; (4)\nor, in terms of the physical current:\nL_j+ ^%j=E; (5)where E\u0011\u0000@tA=cis the electric \feld, and ^ %is identi\fed\nas the resistivity tensor. This is the familiar Ohm's law,\nwhich, in steady state, reduces to\nj= ^gE; (6)\nin terms of the conductivity tensor ^ g\u0011^%\u00001. Based on\nthe axial symmetry around z, we can generally write ^ g=\ng+gHz\u0002, wheregis the longitudinal (i.e., dissipative)\nandgHHall conductivities.\nThe isolated magnetic-\flm dynamics, on the other\nhand, are described by the Landau-Lifshitz-Gilbert\nequation:6\ns(1 +\u000bn\u0002)_n=H\u0003\u0002n; (7)\nwhere H\u0003\u0011\u000enF[n] is the e\u000bective magnetic \feld gov-\nerned by the magnetic free-energy functional F[n] =R\nd2rF(n). The (dimensionless) Gilbert damping \u000bcap-\ntures the (time-reversal breaking) dissipative processes in\nthe spin sector.\nThe total dissipation power in our combined, but still\ndecoupled, system is given by\n\u0000_F=\u0000Z\nd2r(L_p\u0001j+_n\u0001H\u0003) =Z\nd2r\u0000\n%j2+\u000bs_n2\u0001\n;\n(8)\nwhere%=g=(g2+g2\nH) is the longitudinal resistiv-\nity. According to the \ructuation-dissipation theorem,5\n\fnite-temperature \ructuations are thus determined by\nhji(r;t)ji0(r0;t0)i= 2gkBT\u000eii0\u000e(r\u0000r0)\u000e(t\u0000t0) and\nhhi(r;t)hi0(r0;t0)i= 2\u000bskBT\u000eii0\u000e(r\u0000r0)\u000e(t\u0000t0). Hav-\ning mentioned this for completeness, we will not pursue\nthermal properties any further.\nB. Coupled dynamics\nHaving recognized ( Lp;j) and ( n;H\u0003) as two pairs\nof thermodynamically conjugate variables, their coupled\ndynamics must obey Onsager reciprocity.5Charge cur-\nrent \rowing through our heterostructure in general in-\nduces a torque \u001con the magnetic moment and, vice\nversa, magnetic dynamics produce a motive force \u000facting\non the current, de\fned as follows:\ns(_n+n\u0002^\u000b_n) =H\u0003\u0002n+\u001c; (9)\nL_j+ ^%j=E+\u000f; (10)\nwhereL_j=L_p+E, according to Eq. (3). In gen-\neral, due to the spin-orbit interaction at the interface,\nGilbert damping7^\u000band resistivity tensor8^%can acquire\nanisotropic n-dependent contributions. Let us start by\nexpanding the motive force, according to the assumed\nstructural symmetries, in the Cartesian components of\nn:\n\u000f= [(\u0011+#n\u0002)_n]\u0002z; (11)3\nwhere\u0011is the reactive and#thedissipative coe\u000ecients\ncharacterizing spin-orbit interactions in our coupled sys-\ntem. While \u0011and#can generally depend on n2\nz, we will\nfor simplicity be focusing our attention on the limit when\nthey are mere constants. The dimensionless parameter\n\f\u0011#=\u0011describes their relative strengths. The Onsager\nreciprocity then immediately dictates the following form\nof the torque:\n\u001c= (\u0011+#n\u0002)(z\u0002j)\u0002n: (12)\nIn line with the existing nomenclature,1,2we can write\nthe dissipative coe\u000ecient as\n#\u0011~\n2eaNtan\u0012; (13)\nin terms of a length scale aN, which we take to corre-\nspond to the normal-metal thickness,9and dimensionless\nparameter\u0012identi\fed as the e\u000bective spin Hall angle at\nthe interface. The coe\u000ecient \u0011in Eq. (12) parametrizes\nthe so-called \feld-like torque, which could arise, for ex-\nample, as a manifestation of the interfacial Edelstein ef-\nfect.10\nAnother important e\u000bect of the nonmagnetic layer on\nthe ferromagnet is the enhanced damping of the magne-\ntization dynamics by spin pumping,11such that\n\u000b=\u000b0+a\"#\naF: (14)\n\u000b0is the bulk damping, which is thickness aFindepen-\ndent, anda\"#parametrizes the strength of angular mo-\nmentum [as well as energy, according to Eq. (8)] loss at\nthe interface. Spin pumping into a perfect spin reser-\nvoir corresponds to11a\"#=~g\"#\nr=4\u0019S, whereg\"#\nris the\n(real part of the dimensionless) interfacial spin-mixing\nconductance per unit area and S\u0011s=aFis the 3D spin\ndensity in the ferromagnet. In reality, a\"#depends on\nthe spin-relaxation e\u000eciency in the normal metal as well\nas the spin-orbit interaction at the interface, and may\ndepend on aNin a nontrivial manner (see Ref. 12 for\na di\u000busive model), so long as aN.\u0015N, where\u0015Nis\nthe spin-relaxation length in the normal metal.13With\nthese conventions in mind and focusing on the limit of\naN\u001d\u0015Nand, in the case of a metallic ferromagnet,\naF\u001d\u0015F, we will suppose that the coe\u000ecients \u0012,\f, and\na\"#de\fned above are thickness independent.14\nUnless otherwise stated, we will disregard anisotropies\nin\u000b, which may in general depend on the directions of n\nand_n, subject to the reduced crystalline symmetries and\nthe lack of re\rection asymmetry at the interface.15In the\nsame spirit, with the exception of Sec. III C, we will not\nconcern ourselves much with the n-dependent interfacial\nmagnetoresistance/proximity e\u000bects,8which would enter\nthrough the resistivity tensor ^ %(n) in Eq. (10).\nWe remark that while we considered a nonequilibrium\nmagnetoelectric coupling in terms of torque \u001cand force\n\u000fin Eqs. (9) and (10), we had retained the decoupled\nform of the free-energy density, F(p)+F(n). We excludethe possibility of a linear coupling of pto the magnetic\norder, since it would suggest a nonzero electric current in\nequilibrium.\nC. Current-induced instability\nEquations (9) and (10) encapsulate rich nonlinear dy-\nnamics. Of particular interest are the current-induced\nmagnetic instabilities and switching. For a \fxed current\nbiasj, it is convenient to multiply Eq. (9) by (1 \u0000\u000bn\u0002)\non the left to obtain\ns(1 +\u000b2)_n=h\u0002n\u0000\u000bn\u0002h0\u0002n: (15)\nHere,\nh\u0011H\u0003+ (\u0011+#\u000b)z\u0002j;h0\u0011H\u0003+ (\u0011\u0000#=\u000b)z\u0002j(16)\nare the e\u000bective Larmor and damping \felds, respectively.\nA magnetic instability (bifurcation) at an equilibrium\n\fxed point may occur, for example, when either the ef-\nfective \feld or e\u000bective damping change sign.\nTo illustrate this, consider a simple case, where a con-\nstant current is applied in the xdirection: j=jx, while\nan external magnetic \feld parametrized by His applied\nalong theyaxis: H\u0003=Hy+Knzz, where we also in-\nclude an easy-plane magnetic anisotropy K. Equations\n(16) then become\nh= [H+ (\u0011+#\u000b)j]y+Knzz; (17)\nh0= [H+ (\u0011\u0000#=\u000b)j]y+Knzz: (18)\nIn equilibrium, when j= 0:n=\u0000y. Whenjis ramped\nup, however, this \fxed point may become unstable. Let\nus consider two extreme limits: First, suppose the mag-\nnetoelectric coupling (12) is purely reactive, i.e., #= 0.\nThe e\u000bect of the torque can thus be fully captured by\na rede\fnition of the applied \feld as H!H+\u0011j. We\nthus see that when \u0000jexceedsH=\u0011, the e\u000bective \feld\nswitches sign, and the stable magnetic orientation \rips\nfrom\u0000ytoy.\nIf, on the other hand, the magnetoelectric coupling\n(12) is purely dissipative, i.e., \u0011= 0, thenH!H+\n#\u000bj according to Eq. (17), whereas H!H\u0000(#=\u000b)j\naccording to Eq. (18). Supposing, furthermore, that \u000b\u001c\n1, as is nearly always the case, the e\u000bect of #onhis\nnegligible in comparison to its e\u000bect on h0. We thus\nrewrite Eqs. (17) and (18) as\nh\u0019Hy+Knzz;h0= [H\u0000(#=\u000b)j]y+Knzz:(19)\nA simple stability analysis gives for the critical current\nat which n=\u0000ybecomes unstable:\njc=\u000b\n#\u0012\nH+K\n2\u0013\n: (20)\nIn the presence of comparable reactive and dissipative\ntorques, i.e., \f\u00181 so that\u0011\u0018#, while still \u000b\u001c1,4\nhremains essentially una\u000bected by currents of order jc\n(unlessK&H=\u000b\u001dH), so that the above dissipative\nmagnetic instability at jcis maintained. We could thus\nexpect Eq. (20) to rather generally describe the leading\nspin-torque instability threshold16for the monodomain\ndynamics.\nIt is instructive to obtain from Eq. (20) the intrinsic in-\nstability threshold for thin magnetic \flms, aF\u001ca\"#=\u000b0,\nfor which the bulk contribution, \u000b0, to the damping (14)\ncan be neglected:\nj(0)\nc=2e\n~a\"#\ntan\u0012aN\naF\u0012\nH+K\n2\u0013\n: (21)\nWriting, furthermore, j(0)\nc=J(0)\ncaN, in terms of the 3D\ncurrent density J(0)\nc;a\"#=~g\"#\nr=4\u0019S, in terms of the\ne\u000bective spin-mixing conductance g\"#\nr(including the ef-\nfects of spin back\row from the normal layer,12in case\nof an imperfect spin sink); and converting e\u000bective \feld\nto physical units: H=!BaFSandK=!KaFS, where\n!B=\rBin terms of the gyromagnetic ratio \rand ap-\nplied \feldB,!K= 4\u0019\rMswithMs=\rS, in case of\nonly the shape anisotropy, we obtain\nJ(0)\nc=e\n2\u0019g\"#\nr\ntan\u0012\u0010\n!B+!K\n2\u0011\n: (22)\nWe recall that the Kittel formula for the ferromagnetic-\nresonance frequency is !=p\n!B(!B+!K). Using\nquantities characteristic of the Pt jYIG compound:1,2\u0012\u0018\n0:1,g\"#\nr\u00185 nm\u00002,!K\u00184\u00021010s\u00001, we would get for\nthe intrinsic instability threshold (in the absence of an\napplied \feld B):J(0)\nc\u00183\u00021010A\u0001m\u00002. (Threshold\ncurrents at this order were also evaluated in Ref. 17.) In\nthe opposite limit of thick magnetic \flms, aF\u001da\"#=\u000b0\n(\u00181=2\u0016m for YIG, using \u000b0\u001810\u00004), the bulk Gilbert\ndamping dominates magnetic dissipation, and\nJc\u0019\u000b0aF\na\"#J(0)\nc=2e\n~\u000b0aFS\ntan\u0012\u0010\n!B+!K\n2\u0011\n(23)\nincreases linearly with aFbeyond the intrinsic threshold.\nIII. SIMPLE MODELS\nEquations (9)-(14) provide a general phenomenological\nframework for exploring the coupled magnetoelectric dy-\nnamics in thin-\flm magnetic heterostructures, which we\nverify by considering several simple microscopic models\nin the following.\nA. Rashba Hamiltonian\nOne of the simplest models engendering the phe-\nnomenology of interest is based on a 2D electron gas ata re\rection-asymmetric interface, which, at low energies,\nis described by the (single-particle) Rashba Hamiltonian:\n^HR=p2\n2m+vp\u0001z\u0002^\u001b: (24)\nVelocityvhere parametrizes the spin-orbit interaction\nstrength due to structural asymmetry; ^\u001bis a vector of\nPauli matrices. When the \frst (nonrelativistic) term in\nHamiltonian (24) dominates over the second (relativistic)\nterm (i.e.,v\u001cvF, the Fermi velocity), we can treat v\nperturbatively.\nTo zeroth order in v, the velocity operator is @p^HR=\np=m, such that the current density is j=\u0000enhpi=m,\nin terms of the particle-number density n=k2\nF=2\u0019=\nm2v2\nF=2\u0019~2and the positron charge e>0. On the other\nhand, to \frst order in v, Eq. (24) results in the steady-\nstate spin density\n\u001a=mv\n2\u0019~z\u0002hpi=\u0000m2v\n2\u0019~enz\u0002j; (25)\nrecalling that the 2D density of states (which de\fnes the\nspin susceptibility) is given by m=2\u0019~2. Equation (25)\nre\rects the Edelstein e\u000bect.10\nExchange coupling this Rashba 2DEG to an adjacent\nferromagnet according to the local Hamiltonian\nH0=\u0000Z\nd2r[J(nx\u001ax+ny\u001ay) +J?nz\u001az]; (26)\nwhereJandJ?are respectively the in-plane and out-of-\nplane exchange constants, we get for the torque:\n\u001c=\u000enH0\u0002n=\u0000[J(\u001axx+\u001ayy) +J?\u001azz]\u0002n:(27)\nEvaluating this torque to leading (i.e., \frst) order in the\nexchange, we need to \fnd \u001ato zeroth order, which is\ngiven by Eq. (25). We thus have:\n\u001c=\u0011(z\u0002j)\u0002n; (28)\nwhere\n\u0011=m2vJ\n2\u0019~en=~\nevJ\nv2\nF: (29)\nThe dissipative (i.e., spin Hall) coe\u000ecient #vanishes in\nthis model at this level of approximation. We should,\nhowever, expect #to arise at quadratic order in J\n[whereas at \frst order in J, it must vanish for arbitrarily\nlargev, since, in the absence of magnetism, Eq. (25) here\ndescribes the general form of spin response to dc current].\nB. Dirac Hamiltonian\nIn the opposite extreme, the spin-orbit interaction in\nEq. (24) dominates over the nonrelativistic piece, which\nformally corresponds to sending m! 1 . The corre-\nsponding 2D Dirac Hamiltonian\n^HD=vp\u0001z\u0002^\u001b (30)5\narises physically on the surfaces of strong 3D topological\ninsulators.18\nExchange coupling electrons to a magnetic order n, ac-\ncording to Eq. (26), gives the single-particle Hamiltonian\n^H0=\u0000~\n2[J(nx^\u001bx+ny^\u001by) +J?nz^\u001bz]; (31)\nwhich can be combined with Eq. (30) as follows:\n^HD+^H0=v(p\u0000A\u0003)\u0001z\u0002^\u001b\u0000m\u0003^\u001bz: (32)\nHere,\nA\u0003\u0011~J\n2vz\u0002nandm\u0003\u0011~J?\n2nz (33)\nare \fctitious vector potential and mass. The correspond-\ning electromotive force (recalling that the electron charge\nis\u0000e) is\n\u000f=@tA\u0003\ne=\u0000~J\n2ev_n\u0002z; (34)\nsuch that, according to Eq. (11),\n\u0011=\u0000~J\n2ev; (35)\nwhich is of opposite sign to Eq. (29). Note that unlike\nthe latter result, Eq. (35) is derived nonperturbatively.\nThe reciprocal torque (12) with this \u0011gives:\n\u001c=\u0011(z\u0002j)\u0002n: (36)\nUsing the helical identity between the current and spin\ndensities,\nj=\u00002ev\n~z\u0002\u001a; (37)\naccording to the velocity operator @p^HD=vz\u0002^\u001b, we\nrecognize in Eq. (36) the torque (27) due to the planar\nexchangeJ. The above relations mimic the structure of\nthe preceding Rashba model. For a vanishing chemical\npotential, the mass term opens a gap, in which case the\nlong-wavelength conductivity tensor is given by the half-\nquantized Hall response:19^g=\u0000sgn(m\u0003)(e2=4\u0019~)z\u0002. In\naddition to the in-plane spin density z\u0002\u001a\u0002zentering\nEq. (36), the out-of-plane component \u001azshould also exert\na torque/J?, according to the exchange coupling (27).\nAt the leading order, the latter contributes to the out-of-\nplane magnetic anisotropy K, which is absorbed by the\nmagnetic free-energy density F(n).20At a \fnite doping,\ntheJ?interaction could in general be also expected to\ngive rise to a dissipative coupling #.\nC. Di\u000busive spin Hall system\nThe previous two models naturally produced the reac-\ntive coupling \u0011between planar charge current and mag-\nnetic dynamics. Here, we recap a di\u000busive spin Hallmodel8,21that results in both \u0011and#, which is based\non a \flm of a featureless isotropic normal-metal conduc-\ntor in contact with ferromagnetic insulator. If electrons\ndi\u000buse through the conductor with weak spin relaxation,\nwe can develop a hydrodynamic description based on con-\ntinuity relations both for spin and charge densities. We\n\frst construct bulk di\u000busion equations and then impose\nspin-charge boundary conditions, which allows us to solve\nfor spin-charge \ruxes in the normal metal and torque on\nthe ferromagnetic insulator.\nThe relevant hydrodynamic quantities in the normal-\nmetal bulk are 3D charge and spin densities, \u001a(r;t) and\n\u001a(r;t), respectively. The associated thermodynamic con-\njugates are the electrochemical potential, \u0016\u0011\u0000e\u000e\u001aF,\nand spin accumulation, \u0016\u0011~\u000e\u001aF, whereF[\u001a;\u001a] is the\nfree-energy functional of the normal metal. Supposing\nonly a weak violation of spin conservation (due to mag-\nnetic or spin-orbit impurities), we phenomenologically\nwrite spin-charge continuity relations as\n@t\u001a=\u0000@{J{; @t\u001a|=\u0000@{J{|\u0000\u0000\u0016|; (38)\nwhere{and|label Cartesian components of real and\nspin spaces, respectively, and the summation over the\nrepeated index {is implied. \u0000 = ~N=2\u001cs, in terms of\nthe (per spin) Fermi-level density of states Nand spin-\nrelaxation time \u001cs.J{are the components of the 3D\nvectorial charge-current density and J{|of the tensorial\nspin-current density, which can be expanded in terms of\nthe thermodynamic forces governed by \u0016and\u0016:\nJ{=\u001b\ne@{\u0016\u0000\u001b0\n2e\u000f{|k@|\u0016k; (39)\n2e\n~J{|=\u0000\u001b+\n2e@{\u0016|\u0000\u001b\u0000\n2e@|\u0016{\u0000\u001b0\ne\u000f{|k@k\u0016; (40)\nwhere\u001bis the (isotropic) electrical conductivity and \u001b0\nthe spin Hall conductivity of the normal-metal bulk. The\nlast terms of Eqs. (39) and (40) are governed by the same\ncoe\u000ecient\u001b0due to the Onsager reciprocity. The bulk\nspin Hall angle \u00120is conventionally de\fned by\ntan\u00120\u0011\u001b0\n\u001b: (41)\nBulk di\u000busion equations (39), (40) are complemented\nby the boundary conditions\nJz= 0 atz=\u0000aN;0 (42)\nfor the charge current, where z=\u0000aNcorresponds to\nthe normal-metal interface with vacuum and z= 0 to\nthe interface with the ferromagnet, and11\nJz=1\n4\u0019(0 at z=\u0000aN \u0010\ng\"#\ni+g\"#\nrn\u0002\u0011\n~\u0016\u0002natz= 0;(43)\nfor the spin current, with Jzstanding for Jz|. Here,\n~\u0016\u0011\u0016\u0000~n\u0002_ncaptures contributions from the spin-\ntransfer torque and spin pumping, respectively.6\nHaving established the general structure of the coupled\nspin and charge di\u000busion, let us calculate the steady-\nstate charge-current density jdriven by a simultaneous\napplication of a uniform electric \feld in the xyplane,\nr\u0016!eE, and magnetic dynamics, _n:\nJ=\u001bE\u0000\u001b0\n2er\u0002\u0016: (44)\nThe spin accumulation \u0016is found by solving\n\u0010\u001b+\n\u001b+\u001b\u0000\n\u001b\u000ezj\u0011\n@2\nz\u0016j=\u0016j\nl2s; (45)\nwherels\u0011p\n~\u001b=4e2\u0000 is the spin-di\u000busion length. Using\nDrude formula for the conductivity \u001b, we get the famil-\niarls=l=p\n3\u000f, wherelis the scattering mean free path\nand\u000f\u0011\u001c=\u001cs\u001c1 is the spin-\rip probability per scatter-\ning (\u001cis the transport mean free time). The boundary\nconditions are\n\u001b0z\u0002E\u0000\u001b+\n2e@z\u0016\u0000\u001b\u0000\n2er\u0016z\n=e\nh(0 at z=\u0000aN \u0010\ng\"#\ni+g\"#\nrn\u0002\u0011\n~\u0016\u0002natz= 0;(46)\nwhereh= 2\u0019~is the Planck's constant.\nIn the limit of vanishing spin-orbit coupling, \u001b+!\u001b,\n\u001b\u0000!0, and\u00120!0. For small but \fnite spin-orbit\ninteraction, we may expect ( \u001b+\u0000\u001b)\u0018\u001b\u0000\u0018O(\u001202). In\nthe following, we will neglect these quadratic terms and\napproximate tan \u00120\u0019\u00120\u001c1, in the spirit of the present\nconstruction.\nIn the limit of ls\u001caN, the spin accumulation decays\nexponentially away from the interface as \u0016(z) =\u00160ez=ls,\nwhere\n\u00160= (\u0018i+\u0018n\u0002) [~_n\u00002els\u00120(z\u0002E)\u0002n] + 2els\u00120z\u0002E:\n(47)\nHere,\u0018\u0011\u001f(1 +\u0010+\u00102\ni) and\u0018i\u0011\u001f\u0010\u0010i, in terms of\n\u0010\u0011\u001b=gQg\"#\nrls,\u0010i\u0011g\"#\ni=g\"#\nr,\u001f\u00001\u0011(1 +\u0010)2+\u00102\ni, and\nthe quantum of conductance gQ\u00112e2=h. The spin accu-\nmulation \u00160consists of the decoupled spin-pumping and\nspin Hall contributions. Integrating the resultant charge-\ncurrent density (44) over the normal-layer thickness aN,\nwe \fnally get for the 2D current density in the \flm:\nj=\u001b\u0012\naNE\u0000\u00120\n2ez\u0002\u00160\u0013\n= ^gfE+ [(\u0011+#n\u0002)_n]\u0002zg;\n(48)\nwhere\n^g\n\u001b= ~aN+ls\u001202\b\n\u0018inz(z\u0002)\u0000\u0018[n2\nz+ (z\u0002n\u0002z)n\u0001]\t\n(49)\nis the anisotropic 2D conductivity tensor (~ aN\u0011aN+\nls\u001202\u0019aN), which is referred in the literature to as the\nspin Hall magnetoconductance,8and\n\u0011\u0019~\n2eaN\u00120\u0018i; #\u0019~\n2eaN\u00120\u0018; (50)neglecting corrections that are cubic in \u00120. If\u0010i\u001c1,\nwhich is typically the case,22we have#\u001d\u0011. It could be\nnoted that restoring \u001b\u0000\u0018O(\u001202) in Eqs. (45) and (46)\nwould a\u000bect ^ gonly at orderO(\u001203).\nThe above spin accumulation can also be used to calcu-\nlate the spin-current density injected into the ferromag-\nnet atz= 0:\nJz=~\u001b\n2e\u0012\n\u00120z\u0002E\u0000\u00160\n2els\u0013\n\u0019\u0000sn\u0002^\u000b_n+ (\u0011+#n\u0002)(z\u0002j)\u0002n; (51)\nwhere\n^\u000b=~2\u001b\n4e2lss(\u0018\u0000\u0018in\u0002); (52)\nand we dropped terms that are cubic in \u00120, as be-\nfore. The corresponding magnetic equation of motion\ns_n=H\u0003\u0002n+Jzreproduces Eq. (10), with the current-\ndriven torque of the form (12) that is Onsager recipro-\ncal to the motive force in Eq. (48). Writing the Gilbert\ndamping/\u0018in Eq. (52) as a\"#=aFidenti\fes the inter-\nfacial damping enhancement in Eq. (14). In the formal\nlimit\u001b!1 (while keeping all other parameters, includ-\ningls, \fxed), which reproduces the perfect spin sink,\nthis givesa\"#=~g\"#\nr=4\u0019S. In the general case, \u0018also\ncaptures the spin back\row from the normal layer.12An\nanisotropic contribution to the Gilbert damping would\nbe produced at the cubic order in \u00120, had we not made\nany approximations in Eq. (51).\nIV. MAGNETIC TEXTURES\nFor completeness, we also provide some rudimentary\nremarks regarding the e\u000bect of directional magnetic inho-\nmogeneities, such as those associated with, for example,\nmagnetic domain walls.23Expanding the 2D magnetic\nfree-energy density to second order in spatial derivatives,\nwe have for a \flm with broken re\rection symmetry in\nthexyplane (see Sec. II for a detailed description of the\nstructure shown in Fig. 1):24\nF(n) =n\u0001H+K\n2n2\nz+ \u0000 (nz@ini\u0000ni@inz) +A\n2(@in)2;\n(53)\nwhere summation over Cartesian coordinates i=x;yis\nimplied and the dot products are in the 3D spin space.\n\u0000 here parametrizes the strength of the Dzyaloshinski-\nMoriya (DM) interaction and Ais the magnetic exchange\nsti\u000bness. A nonzero \u0000 requires macroscopic breaking of\nthe re\rection symmetry as well as a microscopic spin-\norbit interaction that breaks the spin-space isotropicity.\nEquation (53) can be rewritten in a more compact form\nas@xn(y\u0002n)\u0000@yn(x\u0002n) =\u0000nx@xnz+nz@xnx\u0000\nny@ynz+nz@yny\nF(n) =n\u0001H+~K\n2n2\nz+A\n2(Din)2; (54)7\nwhere\nDi\u0011@i+Q(z\u0002ei)\u0002 (55)\nis the so-called chiral derivative,25Q\u0011\u0000=A, and ~K\u0011\nK\u0000\u00002=A.Qis the wave number of the magnetic spiral\nthat minimizes the texture-dependent part of the free\nenergy.\nThe DM interaction of the form (53) arises natu-\nrally from the Rashba Hamiltonian (24). In a min-\nimal model,25where electrons with the single-particle\nHamiltonian (24) magnetically order due to their spin-\nindependent (e.g., Coulombic) interaction, the spin-orbit\nterm/vcan be gauged out at the \frst order in vby a\nposition-dependent rotation in spin space. To see this,\nwe \frst rewrite Eq. (24) as\n^HR=p2\n2m+vp\u0001z\u0002^\u001b=(p+mvz\u0002^\u001b)2\n2m\u0000mv2:(56)\nIt then immediately follows that\n^Uy^HR^U=p2\n2m+O(v2);where ^U=e\u0000iQRr\u0001z\u0002^\u001b=2;\n(57)\nde\fning\nQR\u00112mv\n~: (58)\n^Uis the operator of spin rotation around axis r\u0002z\nby anglerQR(recalling that r2xyplane), such that\nthe electron spin precesses by angle 2 \u0019over distance\nlso\u00112\u0019=QR=h=2mv(the spin-precession length).\nSince the transformed Hamiltonian (57) would describe\nmagnetic order that is spin isotropic, the corresponding\nfree energy is given simply by ( A=2)(@in)2(neglecting ex-\nternal and dipolar \felds). In the original frame of refer-\nence with Rashba Hamiltonian (56), the free-energy den-\nsity is then given by F(n) = (A=2)(@i~n)2, where n=^R~n\nand ^R(r) is the natural SO(3) representation of ^U(r).\nDi\u000berentiating @i~n=^RT(@i+^R@i^RT)n, we \fnally ob-\ntainF(n) = (A=2)(Din)2, where\nDi=@i+^R@i^RT=@i+QR(z\u0002ei)\u0002 (59)\nindeed reproduces Eq. (55) with Q!QR. In Ref. 20, the\nfree-energy density (53) was also obtained for the Dirac\nmodel of Sec. (III B), with the result:\n\u0000D\u0018\u0000~\n8\u0019vJJ?: (60)\nAs was pointed out in Ref. 25, the chiral derivative (55)\nis also expected to govern the nonequilibrium magnetic-\ntexture properties such as the current-driven torque \u001c\nand the spin-motive force \u000f. This can either be derived\nmicroscopically or understood on purely phenomenolog-\nical symmetry-based grounds. For example, the hydro-\ndynamic (advective) spin-transfer torque (along with its\nOnsager-reciprocal motive force)26\n\u001c/(j\u0001r)n; (61)which arises due to spin-current continuity in a model\nwithout any spin-orbit interactions and frozen magnetic\nimpurities, would be modi\fed by replacing r!Din\nthe perturbative treatment of the above Rashba model.\nHowever, while this simpli\fes a phenomenological con-\nstruction of various terms, in general, there is no funda-\nmental reason why the same Qshould de\fne the chiral\nderivatives entering in di\u000berent physical properties (such\nas free energy and spin torque).\nV. CONCLUSIONS\nIn summary, we have developed a phenomenology for\nslow long-wavelength dynamics of conducting quasi-2D\nmagnetic \flms and heterostructures, subject to struc-\ntural symmetries and Onsager reciprocity. The formal-\nism could address both small- and large-amplitude mag-\nnetic precession (assuming it is slow on the characteristic\nelectronic time scales), including, for example, magnetic\nswitching and domain-wall or skyrmion motion. Owing\nto the versatility of available heterostructures, including\nthose based on magnetic and topological insulators, we\nhave focused our discussion on the case of a ferromag-\nnetic/nonmagnetic bilayer, which serves two purposes:\nIt naturally has a broken inversion symmetry, and the\nspin-orbit and magnetic properties could be separately\noptimized and tuned in one of the two layers.\nIn the case when the spin-relaxation length in the nor-\nmal layer is short compared to its thickness, we can asso-\nciate the interplay between spin-orbit and exchange inter-\nactions to a narrow region in the vicinity of the interface,\nfor which we de\fne the kinetic coe\u000ecients such as the\ninterfacially enhanced Gilbert damping parametrized by\na\"#and the spin Hall angle parametrized by #. Such (sep-\narately measurable) phenomenological coe\u000ecients, which\nenter in our theory, must thus be viewed as joint proper-\nties of both of the bilayer materials as well as structure\nand quality of the interface.\nWe demonstrate the emergence of our phenomenology\nout of three microscopic models, based on Rashba, Dirac,\nand di\u000busive normal-metal \flms, all in contact with a\nmagnetic insulator. In addition to Onsager-reciprocal\nspin-transfer torques and electromotive forces, our\nphenomenology also accommodates arbitrary Gilbert-\ndamping and (magneto)resistance anisotropies, which are\ndictated by the same structural symmetries and may mi-\ncroscopically depend on the same exchange and spin-\norbit ingredients as the reciprocal magnetoelectric cou-\npling e\u000bects.\nACKNOWLEDGMENTS\nWe acknowledge stimulating discussions with G. E. W.\nBauer, S. T. B. Goennenwein, and D. C. Ralph. This\nwork was supported in part by FAME (an SRC STAR-\nnet center sponsored by MARCO and DARPA), the NSF8\nunder Grant No. DMR-0840965, and by the Kavli Insti- tute for Theoretical Physics through Grant No. NSF\nPHY11-25915.\n1K. Ando, S. Takahashi, K. Harii, K. Sasage, J. Ieda,\nS. Maekawa, and E. Saitoh, Phys. Rev. Lett. 101, 036601\n(2008); I. M. Miron, G. Gaudin, S. Au\u000bret, B. Rod-\nmacq, A. Schuhl, S. Pizzini, J. Vogel, and P. Gambardella,\nNature Mater. 9, 230 (2010); I. M. Miron, K. Garello,\nG. Gaudin, P.-J. Zermatten, M. V. Costache, S. Au\u000bret,\nS. Bandiera, B. Rodmacq, A. Schuhl, and P. Gambardella,\nNature 476, 189 (2011); L. Liu, T. Moriyama, D. C.\nRalph, and R. A. Buhrman, Phys. Rev. Lett. 106, 036601\n(2011); L. Liu, C.-F. Pai, Y. Li, H. W. Tseng, D. C. Ralph,\nand R. A. Buhrman, Science 336, 555 (2012).\n2Y. Kajiwara, K. Harii, S. Takahashi, J. Ohe, K. Uchida,\nM. Mizuguchi, H. Umezawa, H. Kawai, K. Ando,\nK. Takanashi, S. Maekawa, and E. Saitoh, Nature 464,\n262 (2010); C. W. Sandweg, Y. Kajiwara, A. V. Chumak,\nA. A. Serga, V. I. Vasyuchka, M. B. Jung\reisch, E. Saitoh,\nand B. Hillebrands, Phys. Rev. Lett. 106, 216601 (2011);\nC. Burrowes, B. Heinrich, B. Kardasz, E. A. Montoya,\nE. Girt, Y. Sun, Y.-Y. Song, and M. Wu, Appl. Phys. Lett.\n100, 092403 (2012); C. Hahn, G. de Loubens, O. Klein,\nM. Viret, V. V. Naletov, and J. Ben Youssef, Phys. Rev.\nB87, 174417 (2013).\n3P. M. Haney, H.-W. Lee, K.-J. Lee, A. Manchon, and\nM. D. Stiles, Phys. Rev. B 87, 174411 (2013); A. Brataas\nand K. M. D. Hals, Nature Nanotech. 9, 86 (2014); and\nreferences therein.\n4Y. L. Chen, J.-H. Chu, J. G. Analytis, Z. K. Liu,\nK. Igarashi, H.-H. Kuo, X. L. Qi, S. K. Mo, R. G. Moore,\nD. H. Lu, M. Hashimoto, T. Sasagawa, S. C. Zhang, I. R.\nFisher, Z. Hussain, and Z. X. Shen, Science 329, 659\n(2010); L. A. Wray, S.-Y. Xu, Y. Xia, D. Hsieh, A. V.\nFedorov, Y. S. Hor, R. J. Cava, A. Bansil, H. Lin, and\nM. Z. Hasan, Nature Phys. 7, 32 (2010); J. G. Checkelsky,\nJ. Ye, Y. Onose, Y. Iwasa, and Y. Tokura, ibid.8, 729\n(2012); Y. Fan, P. Upadhyaya, X. Kou, M. Lang, S. Takei,\nZ. Wang, J. Tang, L. He, L.-T. Chang, M. Montazeri,\nG. Yu, W. Jiang, T. Nie, R. N. Schwartz, Y. Tserkovnyak,\nand K. L. Wang, Nature Mater., 13, 699 (2014).\n5L. D. Landau and E. M. Lifshitz, Statistical Physics, Part\n1, 3rd ed., Course of Theoretical Physics, Vol. 5 (Perga-\nmon, Oxford, 1980).\n6E. M. Lifshitz and L. P. Pitaevskii, Statistical Physics, Part\n2, 3rd ed., Course of Theoretical Physics, Vol. 9 (Perga-\nmon, Oxford, 1980); T. L. Gilbert, IEEE Trans. Magn.\n40, 3443 (2004).\n7More precisely, it is only the symmetric part of ^ \u000bthat\nshould be identi\fed with a generalized Gilbert damping.\nIndeed, Onsager reciprocity requires ^ \u000bT(n) = ^\u000b(\u0000n),\nwhile the dissipative (i.e., time-reversal symmetry break-\ning) character dictates ^ \u000b(\u0000n) = ^\u000b(n), which together lead\nto ^\u000bT(n) = ^\u000b(n). The antisymmetric component of ^ \u000b, on\nthe other hand, contributes to the e\u000bective, matrix-valued\ngyromagnetic ratio.\n8H. Nakayama, M. Althammer, Y.-T. Chen, K. Uchida,\nY. Kajiwara, D. Kikuchi, T. Ohtani, S. Gepr ags, M. Opel,\nS. Takahashi, R. Gross, G. E. W. Bauer, S. T. B. Goennen-\nwein, and E. Saitoh, Phys. Rev. Lett. 110, 206601 (2013);Y.-T. Chen, S. Takahashi, H. Nakayama, M. Althammer,\nS. T. B. Goennenwein, E. Saitoh, and G. E. W. Bauer,\nPhys. Rev. B 87, 144411 (2013).\n9When the ferromagnet is insulating, tan \u0012so de\fned de-\nscribes the conversion between 3D current density in the\nnormal metal and the spin-current density absorbed by the\nferromagnetic insulator. In a simple limit of weak spin-\norbit interaction at the interface, such tan \u0012may corre-\nspond to the bulk spin Hall angle of the normal metal.\nWhen the thickness aNis larger than the spin-relaxation\nlength\u0015Nin the normal metal, it is natural to expect \u0012\nde\fned by Eq. (13), as well as \f\u0011#=\u0011, to be essentially\nthicknessaNindependent.\n10V. M. Edelstein, J. Phys.: Condens. Matter 7, 1 (1995).\n11Y. Tserkovnyak, A. Brataas, and G. E. W. Bauer, Phys.\nRev. Lett. 88, 117601 (2002); Y. Tserkovnyak, A. Brataas,\nG. E. W. Bauer, and B. I. Halperin, Rev. Mod. Phys. 77,\n1375 (2005).\n12Y. Tserkovnyak, A. Brataas, and G. E. W. Bauer, Phys.\nRev. B 66, 224403 (2002).\n13When the ferromagnet is metallic, furthermore, \u0012,\f, and\na\"#may also depend on its thickness aFwhen the ferro-\nmagnet is thinner than its spin-relaxation length \u0015F.\n14It would, however, be interesting to experimentally study\nthe dependence of these two coe\u000ecients on the layer thick-\nnesses as well as the substrate and cap materials, in the\nultrathin limit.\n15We remark, however, that Eqs. (9) and (10) can e\u000bectively\nproduce an anisotropic Gilbert damping even if the origi-\nnal ^\u000bis scalar: Solving Eq. (10) for jin the limit L!0\nandE!0, for example, and substituting the resultant\ncurrent into Eq. (9) gives the torque \u001c=n\u0002^a_n, where\n^a=\u00112[g(z\u0002)2\u0000gHz\u0002] (setting, for simplicity, #= 0),\nof which the dissipative term /gcontributes to mag-\nnetic damping (while the Hall term /gHe\u000bectively makes\nthe gyromagnetic ratio anisotropic). An anisotropic and n-\ndependent correction to the resistivity tensor ^ %can simi-\nlarly be constructed, for example, in the limit H\u0003!0.\n16J. C. Slonczewski, J. Magn. Magn. Mater. 159, L1 (1996).\n17Y. Zhou, H. J. Jiao, Y. T. Chen, G. E. W. Bauer, and\nJ. Xiao, Phys. Rev. B 88, 184403 (2013).\n18O. A. Pankratov and B. A. Volkov, in Landau Level Spec-\ntroscopy , edited by G. Landwehr and E. I. Rashba (Elsevier\nScience, 1991) Chap. 14, pp. 817{853; M. Z. Hasan and\nC. L. Kane, Rev. Mod. Phys. 82, 3045 (2010); X.-L. Qi\nand S.-C. Zhang, Rev. Mod. Phys. 83, 1057 (2011).\n19A. N. Redlich, Phys. Rev. Lett. 52, 18 (1984); R. Jackiw,\nPhys. Rev. D 29, 2375 (1984).\n20Y. Tserkovnyak and D. Loss, Phys. Rev. Lett. 108, 187201\n(2012).\n21O. Mosendz, J. E. Pearson, F. Y. Fradin, G. E. W. Bauer,\nS. D. Bader, and A. Ho\u000bmann, Phys. Rev. Lett. 104,\n046601 (2010).\n22A. Brataas, G. E. W. Bauer, and P. J. Kelly, Phys. Rep.\n427, 157 (2006).\n23S. Emori, U. Bauer, S.-M. Ahn, E. Martinez, and\nG. S. D. Beach, Nature Mater. 12, 611 (2013); K.-S. Ryu,9\nL. Thomas, S.-H. Yang, and S. Parkin, Nature Nanotech.\n8, 527 (2013).\n24A. Bogdanov and A. Hubert, J. Magn. Magn. Mater. 138,\n255 (1994).25K.-W. Kim, H.-W. Lee, K.-J. Lee, and M. D. Stiles, Phys.\nRev. Lett. 111, 216601 (2013).\n26Y. Tserkovnyak, A. Brataas, and G. E. W. Bauer, J.\nMagn. Magn. Mater. 320, 1282 (2008)."    },    {        "title": "1406.2491v2.Influence_of_Ta_insertions_on_the_magnetic_properties_of_MgO_CoFeB_MgO_films_probed_by_ferromagnetic_resonance.pdf",        "content": "arXiv:1406.2491v2  [cond-mat.mtrl-sci]  13 Aug 2014Influence ofTa insertions onthe magneticproperties ofMgO /CoFeB/MgOfilms probed by\nferromagnetic resonance\nMariaPatriciaRouelliSabino,Sze TerLim,andMichaelTran\nDataStorage Institute,Agency for Science, Technology and Research, 5Engineering Drive 1,117608 Singapore\n(Dated: April 3,2018)\nAbstract We show by vector network analyzer ferromagnetic resonance measurements that low Gilbert damping α,\ndownto0.006,canbeachievedinperpendicularlymagnetize dMgO/CoFeB/MgOthinfilmswithultrathininsertionsofTa\nintheCoFeBlayer. AlthoughincreasingthenumberofTainse rtionsallowsthickerCoFeBlayerstoremainperpendicular ,\nthe effective areal magnetic anisotropy does not improve withmore insertions, whichcome withan increase in α.\nPerpendicularmagnetic anisotropy (PMA) is the key to furth er downscaling of spin transfer torque magnetoresistive\nrandommemorydevices,asitallowstwokeyrequirementstob esatisfied: lowcriticalcurrent Ic0andhighthermalstability\n∆, the latter of which is proportional to the energy barrier Ebbetween the two stable magnetic states. The spin torque\nswitching efficiency, defined as Eb/Ic0, is commonly used as a metric to account for both requirement s. For a Stoner-\nWohlfarthmodel,itisgivenby[1]( /planckover2pi1/4e)·(η/α),whereαistheGilbertdampingparameter,and ηisthespinpolarization\nfactor, which is related to the tunnel magnetoresistance ra tio (TMR) byη=[TMR(TMR+2)]1/2/[2(TMR+1)]. It thus\nbecomes evident that for high switching e fficiency, one has to decrease αwhile keeping TMR high. Magnetic tunnel\njunctions(MTJs)basedonCoFeB /MgOsystemsarewell knownto providehighTMR[2] andhaverec entlybeenshown\nto possess PMA, which is attributed to the CoFeB /MgO interface.[3] A Ta layer is usually placed adjacent to th e CoFeB\nto induce the proper crystallization necessary for PMA and h igh TMR [4]. In Ta /CoFeB/MgO systems, however, spin\npumping to the Ta increases α. [5] Moreover, the CoFeB layer also needs to be ultrathin (ty pically less than 1.5nm) in\nordertoexhibitPMA.[3]Toimprovethethermalstabilityas devicesarescaleddowntosmallerdiameters,increasingth e\neffectivearealanisotropyenergydensity Kef ftisdesired.\nOne approach to address these issues is the use of double-MgO structures, i.e., those in which both the barrier layer\nand cappinglayer straddlingthe free layerare made of MgO. I mproved Ic0and/or∆have beenreportedin devicesusing\ndoubleMgOfreelayers. [6,7,8,9]Theimprovementintherma lstabilityisattributedtotheadditionalCoFeB /MgOinter-\nface,whereaslower Ic0isassociatedwithlow α. Indeed,αdownto0.005hasbeenmeasuredinin-planeMgO /FeB/MgO\nfilms,[10] which agrees with device measurements.[11] The s tacks investigated in these damping studies, however, did\nnot have the Ta layer used in practical free layers with perpe ndicular anisotropy.[9, 12] In addition, although the inte r-\nfacial anisotropy in the out-of-plane devices measured by T sunegi et al.[11] can be as high as 3.3 mJ /m2, the effective\nperpendicularanisotropywasratherlow ( Kef ft≈0.04mJ/m2) relativetothat ofa Ta /CoFeB/MgOstack[3]. In thiswork,\nweexploretheinfluenceofTainsertionswithintheCoFeBlay erofMgO/CoFeB/MgOfilmsbymagnetometryandvector\nnetwork analyzer ferromagnetic resonance (VNA-FMR) measu rements. The insertion of extremely thin Ta layers (0.3\nnm) inside the CoFeB layer aids crystallization, allowing a larger total CoFeB thickness to remain perpendicular, [13]\nwith aneffectivearealanistropycomparableto thatofTa /CoFeB/MgO.\nTwo sample series were deposited by magnetron sputtering on SiO2substrates with seed layers of Ta 5 /TaN 20/Ta 5\nin an ultrahigh vacuum environment (all thicknesses in nm). The stack configurations of the two sample series are: (1)\nMgO 3/CoFeB 1.0/Ta 0.3/CoFeB 0.5 - 1.5/MgO 3 (“single-insertion”) and (2) MgO 3 /CoFeB 1.0/Ta 0.3/CoFeB 0.5 -\n1.5/Ta0.3/CoFeB1.0/MgO3(“double-insertion”),wheretheCoFeBcompositionis Co40Fe40B20(at%). TheTainsertion\nlayer thickness is in the regime allowing strong ferromagne tic coupling between the CoFeB layers. [16] Two other\nsample serieswere grownasreferences: (a)MgO3 /CoFeB1.0-2.5/MgO3(“zero-insertion”),and(b)seed /CoFeB1.0-\n2.5/MgO3(“single-MgO”).Forallthedouble-MgOsamples,anult rathinCoFeBlayerbelowthebottomMgOlayerwas\nalsodepositedforgoodMgOgrowth. Weconfirmedfromseparat emeasurementsthatthislayerdoesnotcontributetothe\nmagneticsignal. All sampleswerecappedwith15nmofTa forp rotectionandwereannealedpost-growthat 300◦Cfor1\nh in vacuum. Although3 nm MgO is too thick for practical use in MTJs, it was chosen to ensure continuityof the MgO\nlayers and lessen the influence of the layersbeyondit.[10, 1 4, 15] (Measurementsof similar sampleswith 1 nm of MgO\nonbothsidesofthe magneticlayeryieldedthesametrends.)\nMagnetization measurements were performed using an altern ating gradient magnetometer (AGM). The PMA im-\nproveswith doublingof the CoFeB /MgO interface and with increasingnumber of Ta insertions, n, as shown in Fig. 1(a)\nfor samples with a similar total nominal CoFeB thickness tnom≈2.5 nm. We also confirmed that we cannot obtain a\n1perpendicular easy axis in double-MgO structures without T a insertions.[17] The double-insertion sample, on the othe r\nhand,exhibitslargeout-of-planeremanenceas shownin the inset of Fig. 1(a). A coercivefield less than0.01T (inset) is\ntypicalofCoFeBfilmswithPMA[18,16].\n/s49 /s50 /s51 /s52/s49/s50/s51/s52/s53\n/s45/s48/s46/s48/s49 /s48/s46/s48/s49\n/s45/s49/s46/s48 /s45/s48/s46/s53 /s48/s46/s48 /s48/s46/s53 /s49/s46/s48/s32/s115/s105/s110/s103/s108/s101/s32/s77/s103/s79\n/s32/s100/s111/s117/s98/s108/s101/s32/s77/s103/s79/s44/s32/s110/s61/s48\n/s32/s100/s111/s117/s98/s108/s101/s32/s77/s103/s79/s44/s32/s110/s61/s49\n/s32/s100/s111/s117/s98/s108/s101/s32/s77/s103/s79/s44/s32/s110/s61/s50/s77/s47/s65/s32/s40/s65 /s109/s50\n/s47/s109/s50\n/s41/s32/s120/s32/s49/s48/s45/s54\n/s116\n/s110/s111/s109/s32/s40/s110/s109/s41/s48/s40/s98/s41\n/s32/s77/s97/s103/s110/s101/s116/s105/s122/s97/s116/s105/s111/s110/s32/s40/s97/s46/s117/s46/s41\n/s77/s97/s103/s110/s101/s116/s105/s99/s32/s70/s105/s101/s108/s100/s32/s40/s84/s41/s116\n/s110/s111 /s109/s32 /s32/s50/s46/s53/s110/s109/s40/s97/s41\nFigure 1: (Color online) (a) Out-of-planeAGM loops for samp les with single-MgO (red squares), zero-insertion(purple\ninvertedtriangles),single-insertion(bluecircles),an ddouble-insertion(greentriangles),withtotalnominalC oFeBthick-\nnesstnom≈2.5nm. Insetshowsalow-fieldout-of-planeloopforthesame doubleinsertionsample. (b)Magneticmoment\nperunitarea( M/A)asafunctionofthetotalnominalCoFeBthicknessforallsa mpleseries. Linearfitsareshownassolid\nlines.MSandtMDLcanbeextractedfromtheslopeand xintercept,respectively,andaresummarizedinTable 1.\nIt is known that Ta can create a magnetically dead layer (MDL) when it is near a magnetic layer.[19] We plot the\nmagneticmomentperarea against tnom[Fig.1(b)]to obtainthethicknessoftheMDL foreachseries fromthexintercept\nofalinearfit. TheresultsaresummarizedinTable1,alongwi ththeMSvaluesobtainedfromtheslope. WefindanMDL\nthickness tMDLof 0.24±0.09 nm for n=1 and 0.7±0.1 nm for n=2. These thicknesses are similar to the total Ta\ninsertionthicknessintherespectiveseries,andareconsi stentwiththepictureofCoFeBintermixingwithTatoproduc ea\nmagneticallydeadvolume. The tMDLvalueforthesingle-MgOsamples(0.26 ±0.08nm)agreeswithvaluesfoundinthe\nliterature.[14, 20, 21] On the other hand, no dead layer was f ound for the zero-insertionsamples, which is similar to the\nresultsinRef. [19].\nTable1: SummaryofMagneticProperties\nSeries tMDL(nm) MS(MA/m) Ki(mJ/m2) Kv(MJ/m3)\nsingle MgO 0 .26±0.08 1.51±0.08 1.61±0.07−0.29±0.08\ndouble MgO, n=0 0.04±0.06 1.29±0.05 0.91±0.09 0.37±0.05\ndouble MgO, n=1 0.24±0.09 1.12±0.05 2.18±0.08−0.34±0.04\ndouble MgO, n=2 0.7±0.1 1.10±0.04 2.4±0.1−0.25±0.03\nVNA-FMR was used to measure the e ffective anisotropy field and damping parameter of the samples . In the VNA-\nFMR setup, the samples were placed face down on a coplanar wav eguide and situated in a dc magnetic field of up to\n1.2 T applied perpendicular to the film plane. The transmission scattering parameter S21was measured at a specific\nfrequency while the dc field was swept. For each sweep, the rea l and imaginary parts of the resonance response were\nfitted simultaneouslyusingthecomplexsusceptibilityequ ation\nχ(H)=Mef f(H−Mef f+i∆H\n2)\n(H−Mef f)2−/parenleftBig2π/planckover2pi1f\ngµB/parenrightBig2+i∆H(H−Mef f)(1)\nwherefis the frequencyofthe ac field, Mef f=MS−H⊥\nK,∆His the fullwidth at half-maximum, H⊥\nKis the anisotropy\nfield perpendicular to the plane, gis the spectroscopic splitting factor, µBis the Bohr magneton, and /planckover2pi1is the reduced\nPlanck’s constant. Nonmagnetic contributions to the S21parameter and a linear time-dependentdrift of the instrume nts\nwere taken into account during the fit. We note that only one re sonance peak is observed within the range studied. A\nrepresentativefitofthesusceptibilitydataisshowninFig .2(a)foradouble-insertionsamplewith tnom=2.5nm. Inusing\nEq.1,a valueof g=2isfirst assumedtoobtainvaluesfor Mef fand∆H, whichdoesnotaffectthe finalresult.\nForeachfrequency,a resonancefield\nµ0Hres(f)=2π/planckover2pi1\ngµBf+µ0Mef f (2)\naccording to Kittel’s equation is calculated and plotted ag ainst the frequency, as shown in Fig. 2(c). A linear fit, now\nwithgandMef fas fitting parameters, is then performed. The e ffective anisotropyenergydensity Kef fcan be calculated\n2from the effective anisotropy field HKef f(=−Mef f) asKef f=HKef fMS/2, noting that a positive anisotropy constant\ncorrespondsto aperpendiculareasyaxis.\nToobtainα,we performa linearfit ofthe measuredFMRlinewidthasafunc tionofthefrequencyto\nµ0∆H(f)=4π/planckover2pi1α\ngµBf+µ0∆H0 (3)\nwhere∆H0is the inhomogeneous linewidth broadening, and the value of gused is the fitted value from Eq. 2. We\nnote that two-magnon scattering contributions to the linew idth are eliminated owing to the perpendicular measurement\nconfiguration[22]. Such a fit is shown in Fig. 2(c). Only data p oints taken well beyond the saturation field for each\nsamplewereusedinthefit, andasymptoticanalysisasdescri bedinRef. [23]fortheaccessiblefrequencyrangewasalso\nperformed.\nWe define an effective thickness tef f=tnom−tMDLand show the calculated Kef ftef f(to which Ebis proportional)\nfor both sample series in Fig. 3(a). The x-axis error bars originate from the fitting error in obtainin gtMDL. We find that\nfortef f>2 nm,double-insertionsampleshavehigher Kef ftef fthan single-insertionsamplesforthe same tef f. However,\nthe maximum Kef ftef fachieved for both the single- and double-insertion series d oes not significantly exceed Kef ftef f\nmeasuredin ourthinnestsingle-MgOsample( tef f=1.0nm),similar tothatobservedin MTJmeasurements.[7]\nTounderstandthisfurther,we considerthedi fferentcontributionsto Kef ftef f,whichisgivenby\nKef ftef f=Ki+(Kv−µ0M2\nS\n2)tef f (4)\nwhereKiis the total interfacial anisotropy constant, including al l CoFeB/MgO interfaces; Kvis the volume anisotropy\nconstant; and the demagnetizing energy is given by the M2\nSterm. We assume that any interfacial anisotropy from the\nTa/CoFeB interface is negligible.[24] Kiis commonly derived from the yintercept of a linear Kef ftef fversustef ffit,\nwhereasKvcan be calculated from the slope if MSis known. Because it is possible that for CoFeB thicknesses b elow\n1.0 nm,Kiis degraded because of Ta reaching the CoFeB /MgO interface,[25] we consider only the linear region of the\ncurve during the fit. The calculated values, given in Table 1, demonstrate that the absence of a Ta insertion leads to the\nlowest value of Ki(0.91±0.09mJ/m2), explaining why n=0 samples did not exhibit a perpendiculareasy axis. On the\nother hand, Kiforn=1 (2.18±0.08mJ/m2) andn=2 (2.4±0.1mJ/m2) are both larger than the single-MgO series\n(1.61±0.07mJ/m2), as would be expected from the additional PMA from the secon d CoFeB/MgO interface. However,\nthe anisotropy per interface did not double with the additional CoFeB /MgO, which may be attributed to the di fferent\ndegrees of crystallization for single- and double-MgO samp les. An indication of better crystallization into CoFe in th e\nsingle-MgO series is its higher MS.Kvis negative and does not vary appreciably in samples where Ta is present, in\ncontrast to the positive value found for zero-insertion sam ples. The role of Ta with regard to Kvis not yet understood,\nas previouslypointed out by Sinha et al. [20], and a detailed study of the amount, proximity,and profile of Ta would be\nnecessarytoclarifythesee ffects.\nTurningourattentionto α,weidentifyasingle-MgOsample( tef f≈0.8nm)andasingle-insertionsample( tef f≈1.3\nnm)withacomparable Kef ftef f≈0.2mJ/m2. Weimmediatelynoticethat αforthesingle-insertionsampleisaroundtwo\ntimeslowerthanthatforthe single-MgOsample.\nThis dramaticdecrease in αmay be attributedto the suppressionof spin pumpingby the Mg O layers straddlingboth\nsides of the precessing magnet.[10, 26] Indeed, measuremen ts of zero-insertion samples show no thickness dependence\n[purple dashed line in Fig. 3(b)] and a low mean value of α=0.0035±0.0002 comparable to the bulk damping of\nCo40Fe40B20[5,27]. However,adecreasein αwithincreasing tef fcanstillbeseeninboththesingle-anddouble-insertion\nseries. One possible reason is the alloying of CoFeB and Ta, a s Ta is known to readily intermix with CoFeB [21], and\nhigher damping may be expected from CoFeBTa alloys [28]. The relative percentage of CoFeBTa alloy decreases with\nincreasing CoFeB thickness, coinciding with the αdecrease. This picture is also consistent with the jump in αfrom\nsingle-to double-insertionsamples, i.e.,thereismoreCo FeBTa alloybecausetherearemoreTa insertions. [11, 10]\nItmayalsobepossiblethatspinpumpingtotheTainsertionl ayeroccurs,asinthecaseofthePdinterlayerinCoFe /Pd\nmultilayers[29]. The complexityof oursystem, however,pr eventsusfromusinga simple multilayermodel. One reason\nisthatthemiddleCoFeBlayer(inthedouble-insertioncase )mayhavedifferentpropertiesfromtheCoFeBlayersadjacent\nto MgO, because CoFeB crystallizes from the MgO interface, [ 30] with which the middle CoFeB has no contact. The\ndegree of Ta intermixing also depends on the deposition orde r and will vary across the structure.[19] At this point, we\ncannot discriminate the mechanism behind the damping behav ior. It may be worthwhile to study the use of CoFeBTa\nalloysasinterlayerstopossiblyhavemorecontroloverthe amountanddistributionofTa inthestack.[31]\nInconclusion,wehavedemonstratedPMAandlowdampingindo uble-MgOstructures. AthinTainsertionlayerwas\nfound to significantly increase the PMA - no perpendicular ea sy axis was realized in our MgO /Co40Fe40B20/MgO films\nwithout Ta - and adding more insertionsallowed thicker CoFe B layers to remain perpendicular. However, the maximum\nKef ftef findouble-MgOsamplesiscomparableonlywiththatofthesin gle-MgOsampleforthisCoFeBcomposition.[9]\nOn the other hand, αfor double MgO films increases with the number of insertions b ut is still lower than that of single\n3MgO films for the entire range. Considering both trends with n, we find that the optimal stack in the range of samples\nwe studiedis a double-MgO, n=1 sample with tnom=1.75nm,which exhibits Kef ftef f=0.27mJ/m2at a low damping\nvalueof0.006.\nAcknowledgement\nWe expressgratitudeforsupportfromtheA*STARGraduateAc ademySINGAProgram.\nReferences\n[1] J. Z. Sun, S. L. Brown, W. Chen, E. A. Delenia, M. C. Gaidis, J. Harms, G. Hu, Xin Jiang, R. Kilaru, W. Kula,\nG.Lauer,L.Q.Liu,S.Murthy,J.Nowak,E.J.Oâ ˘A´ZSullivan,S.S.P.Parkin,R.P.Robertazzi,P.M.Rice,G.Sa ndhu,\nT.Topuria,andD.C. Worledge: Phys.Rev.B88,104426(2013) .\n[2] S. Yuasa: J. Phys.Soc.Jpn.77,031001(2008).\n[3] S. Ikeda, K. Miura, H. Yamamoto, K. Mizunuma, H. D. Gan, M. Endo, S. Kanai, J. Hayakawa, F. Matsukura, and\nH. Ohno: Nat.Mater. 9,721(2010).\n[4] D. C. Worledge, G. Hu, David W. Abraham, J. Z. Sun, P. L. Tro uilloud, J. Nowak, S. Brown, M. C. Gaidis, E. J.\nO’Sullivan,andR. P.Robertazzi: Appl.Phys.Lett.98,0225 01(2011).\n[5] X. Liu,W.Zhang,M.J. Carter,andG. Xiao: J. Appl.Phys.1 10,033910(2011).\n[6] G. Jan, Y.-J. Wang, T. Moriyama, Y.-J. Lee, M. Lin, T. Zhon g, R.-Y. Tong, T. Torng, and P.-K. Wang: Appl. Phys.\nExpress5,093008(2013).\n[7] J.-H. Park, Y. Kim, W. C. Lim, J. H. Kim, S. H. Park, W. Kim, K . W. Kim, J. H. Jeong, K. S. Kim, H. Kim, Y. J.\nLee, S. C. Oh, J. E. Lee, S. O. Park, S. Watts, D. Apalkov, V. Nik itin, M. Krounbi, S. Jeong, S. Choi, H. K. Kang,\nandC. Chung: VLSITech.Dig.57(2012).\n[8] H. Kubota, S. Ishibashi, T. Saruya, T. Nozaki, A. Fukushi ma, K. Yakushiji, K. Ando, Y. Suzuki, and S. Yuasa: J.\nAppl.Phys.111,07C723(2012).\n[9] H. Sato,M.Yamanouchi,S.Ikeda,S. Fukami,F. Matsukura ,andH.Ohno: Appl.Phys.Lett. 101,022414(2012).\n[10] M.Konoto,H.Imamura,T.Taniguchi,K.Yakushiji,H.Ku bota,A.Fukushima,K.Ando,andS.Yuasa: Appl.Phys.\nExpress6,073002(2013).\n[11] S.Tsunegi,H.Kubota,S.Tamaru,K.Yakushiji,M.Konot o,A.Fukushima,T.Taniguchi,H.Arai,H.Imamura,and\nS. Yuasa: Appl.Phys.Express7,033004(2014).\n[12] K. Yakushiji,A.Fukushima,H.Kubota,M.Konoto,andS. Yuasa: Appl.Phys.Express6,113006(2013).\n[13] V. B. Naik,H.Meng,andR. Sbiaa: AIP Advances,2,042182 (2012).\n[14] D. D. Lam, F. Bonell, S. Miwa, Y. Shiota, K. Yakushiji, H. Kubota, T. Nozaki, A. Fukushima, S. Yuasa, and\nY. Suzuki: J. Kor.Phys.Soc.62,1461(2013).\n[15] M. P.R. G.Sabino,S.T.Lim,andM.Tran: (unpublished).\n[16] V. Sokalski,M.T.Moneck,E.Yang,andJ.-G. Zhu: Appl.P hys.Lett 101,072411(2012).\n[17] H. Sato,M.Yamanouchi,S.Ikeda,S. Fukami,F. Matsukur a,andH.Ohno: IEEETrans.Mag.49,4437(2013).\n[18] G. Malinowski, K. C. Kuiper, R. Lavrijsen, H. J. M. Swagt en, and B. Koopmans: Appl. Phys. Lett. 94, 102501\n(2009).\n[19] S. Y. Jang,C.-Y. You,S. H.Lim,andS. R.Lee.: J. Appl.Ph ys.109,013901(2011).\n[20] J. Sinha, M. Hayashi, A. J. Kellock, S. Fukami, M. Yamano uchi, H. Sato, S. Ikeda, S. Mitani, S.-H. Yang, S. S. P.\nParkin,andH.Ohno: Appl.Phys.Lett.102,242405(2013).\n[21] Y.-H. Wang, W.-C. Chen, S.-Y. Yang, K.-H. Shen, C. Park, M.-J. Kao, and M.-J. Tsai: J. Appl. Phys. 99, 08M307\n(2006).\n[22] N.Mo,J.Hohlfeld,M.ulIslam,C.S.Brown,E.Girt,P.Kr ivosik,W.Tong,A.Rebei,andC.E.Patton: Appl.Phys.\nLett. 92,022506(2008).\n4[23] J. M.Shaw,H.T.Nembach,T.J. Silva,andC.T. Boone: J. A ppl.Phys.114,243906(2013).\n[24] Y. Miura,M. Tsujikawa,andM.Shirai: J. Appl.Phys.113 ,233908(2013).\n[25] N. Miyakawa,D.C. Worledge,andK.Kita: IEEEMagn.Lett .4,2(2013).\n[26] O. Mosendz,J.E. Pearson,F. Y. Fradin,S. D.Bader,andA . Hoffmann: Appl.Phys.Lett. 96,022502(2010).\n[27] M. Oogane,T. Wakitani, S. Yakata, R. Yilgin, Y. Ando, A. Sakuma, and T. Miyazaki: Jpn. J. Appl. Phys. 45, 3889\n(2006).\n[28] J.O.Rantschler,R.D.McMichael,A.Castillo,A.J.Sha piro,W.F. Egelhoff,B.B.Maranville,D.Pulugurtha,A.P.\nChen,andL.M.Connors: J. Appl.Phys.101,033911(2007).\n[29] J. M.Shaw,H.T.Nembach,andT.J.Silva: Phys.Rev.B85, 054412(2012).\n[30] S. Mukherjee,F. Bai, D.MacMahon,C.-L.Lee,S.Gupta,a ndS. Kurinec: J. Appl.Phys.106,033906(2009).\n[31] K. Tsunoda, H. Noshiro,C. Yoshida, Y. Yamazaki, A. Taka hashi, Y. Iba, A. Hatada, M. Nakabayashi,T. Takenaga,\nM. Aoki,andT.Sugii.IEDM,29.1.1(2012).\n5/s50/s48/s48 /s50/s50/s48 /s50/s52/s48/s48/s50/s48/s48/s52/s48/s48/s54/s48/s48\n/s40/s99/s41/s82/s101/s115/s111/s110/s97/s110/s99/s101/s32/s70/s105/s101/s108/s100/s32/s40/s109 /s84/s41/s32\n/s32\n/s53 /s49/s48 /s49/s53 /s50/s48 /s50/s53/s49/s48/s49/s53/s50/s48/s50/s53/s51/s48\n/s76/s105/s110/s101/s119/s105/s100/s116/s104/s32/s40/s109 /s84/s41/s32/s68/s97/s116/s97 /s32\n/s32/s70/s105/s116\n/s70/s114/s101/s113/s117/s101/s110/s99/s121 /s32/s40/s71/s72/s122/s41/s32/s68/s97/s116/s97\n/s32/s70/s105/s116/s83\n/s50/s49/s32/s82/s101/s97/s108/s32/s40/s97/s46/s117/s46/s41\n/s40/s97/s41\n/s40/s98/s41/s83\n/s50/s49/s32/s73/s109/s97/s103/s46/s32/s40/s97/s46/s117/s46/s41\n/s65/s112/s112/s108/s105/s101/s100/s32/s70/s105/s101/s108/s100/s32/s40/s109/s84/s41\nFigure 2: (a) Real and (b) imaginary parts of the S21parameter obtained from VNA-FMR measurements for a double-\ninsertion sample with tnom=2.5 nm at 12 GHz while a perpendiculardc magnetic field is swept. The lines are fits to an\nexpressionusingEq.1, takingnonmagneticcontributionst oS21anda lineardriftintoaccount. (c)Field-sweptlinewidth\nandresonancefieldsforthesamesampleasafunctionoffrequ ency. Thelinearfitsdescribedinthetextareusedtoextract\nHKef f(=−Mef f)andα.\n/s45/s49/s46/s48/s45/s48/s46/s53/s48/s46/s48/s48/s46/s53/s49/s46/s48/s49/s46/s53/s50/s46/s48/s50/s46/s53\n/s48 /s49 /s50 /s51/s50/s52/s54/s56/s49/s48/s49/s50/s49/s52/s115/s105/s110/s103/s108/s101/s32/s77/s103/s79\n/s32/s100/s111/s117/s98/s108/s101/s32/s77/s103/s79/s44/s32/s110/s61/s48\n/s32/s100/s111/s117/s98/s108/s101/s32/s77/s103/s79/s44/s32/s110/s61/s49\n/s32/s100/s111/s117/s98/s108/s101/s32/s77/s103/s79/s44/s32/s110/s61/s50/s75\n/s101/s102/s102/s116\n/s101/s102/s102/s32/s40/s109/s74/s47/s109/s50\n/s41\n/s40/s97/s41/s40/s49/s48/s45/s51\n/s41\n/s116\n/s101/s102/s102/s32/s40/s110/s109/s41/s40/s98/s41\nFigure3: (a) Kef ftef fand(b)αversuseffectiveCoFeBthickness tef fobtainedfromfield-sweptVNA-FMRmeasurements\nfor all sample series. Solidlines in (a)are linear fits. Purp ledashedline in (b) correspondsto the mean αvalueaveraged\noverall zero-insertionsamples,whichwasfoundtobeconst antwithinerroracrosstheentirethicknessrangestudied.\n6"    },    {        "title": "1406.6225v2.Interface_enhancement_of_Gilbert_damping_from_first_principles.pdf",        "content": "arXiv:1406.6225v2  [cond-mat.mtrl-sci]  16 Nov 2014Interface enhancement of Gilbert damping from first-princi ples\nYi Liu,1,∗Zhe Yuan,1,2,†R. J. H. Wesselink,1Anton A. Starikov,1and Paul J. Kelly1\n1Faculty of Science and Technology and MESA+Institute for Nanotechnology,\nUniversity of Twente, P.O. Box 217, 7500 AE Enschede, The Net herlands\n2Institut f¨ ur Physik, Johannes Gutenberg–Universit¨ at Ma inz, Staudingerweg 7, 55128 Mainz, Germany\n(Dated: June 6, 2018)\nThe enhancement of Gilbert damping observed for Ni 80Fe20(Py) films in contact with the non-\nmagnetic metals Cu, Pd, Ta and Pt, is quantitatively reprodu ced using first-principles scattering\ncalculations. The “spin-pumping” theory that qualitative ly explains its dependence on the Py thick-\nness is generalized to include a number of extra factors know n to be important for spin transport\nthrough interfaces. Determining the parameters in this the ory from first-principles shows that inter-\nface spin-flipping makes an essential contribution to the da mping enhancement. Without it, a much\nshorter spin-flip diffusion length for Pt would be needed than the value we calculate independently.\nPACS numbers: 85.75.-d, 72.25.Mk, 76.50.+g, 75.70.Tj\nIntroduction. —Magnetizationdissipation, expressedin\ntermsofthe Gilbert dampingparameter α, is akeyfactor\ndetermining the performance of magnetic materials in a\nhost of applications. Of particular interest for magnetic\nmemorydevicesbasedupon ultrathin magneticlayers[ 1–\n3] is the enhancement of the damping of ferromagnetic\n(FM) materials in contact with non-magnetic (NM) met-\nals [4] that can pave the way to tailoring αfor particu-\nlar materials and applications. A “spin pumping” theory\nhas been developed that describes this interface enhance-\nment in terms of a transverse spin current generated by\nthe magnetization dynamics that is pumped into and ab-\nsorbed by the adjacent NM metal [ 5,6]. Spin pumping\nsubsequently evolved into a technique to generate pure\nspin currents that is extensively applied in spintronics\nexperiments [ 7–9].\nA fundamental limitation of the spin-pumping the-\nory is that it assumes spin conservation at interfaces.\nThis limitation does not apply to a scattering theoret-\nical formulation of the Gilbert damping that is based\nupon energy conservation, equating the energy lost by\nthe spin system through damping to that parametrically\npumped out of the scattering region by the precessing\nspins [10]. In this Letter, we apply a fully relativistic\ndensity functional theory implementation [ 11–13] of this\nscattering formalism to the Gilbert damping enhance-\nment in those NM |Py|NM structures studied experimen-\ntally in Ref. 4. Our calculated values of αas a function\nof the Py thickness dare compared to the experimental\nresults in Fig. 1. Without introducing any adjustable pa-\nrameters, we quantitatively reproduce the characteristic\n1/ddependence aswellasthe dependenceofthe damping\non the NM metal.\nTo interpret the numerical results, we generalize the\nspin pumping theory to allow: (i) for interface [ 14–16]\nas well as bulk spin-flip scattering; (ii) the interface mix-\ning conductance to be modified by spin-orbit coupling;\n(iii) the interface resistance to be spin-dependent. An\nimportant consequence of our analysis is that withoutinterface spin-flip scattering, the value of the spin-flip\ndiffusion length lsfin Pt required to fit the numerical\nresults is much shorter than a value we independently\ncalculate for bulk Pt. A similar conclusion has recently\nbeen drawn for Co |Pt interfaces from a combination of\nferromagnetic resonance, spin pumping and inverse spin\nHall effect measurements [ 17].\nGilbert damping in NM |Py|NM.—We focus on the\nNM|Py|NM sandwiches with NM = Cu, Pd, Ta and Pt\nthat were measured in Ref. 4. The samples were grown\non insulating glass substrates, the NM layer thickness\nwas fixed at l=5 nm, and the Py thickness dwas vari-\nable. To model these experiments, the conventional NM-\nlead|Py|NM-lead two-terminal scattering geometry with\nsemi-infinite ballistic leads [ 10–13] has to be modified\nbecause: (i) the experiments were carried out at room\n0 2 4 6 8 10 \nd (nm) 00.02 0.04 0.06 0.08 0.10 _Pt |Py|Pt \nPd|Py|Pd \nTa|Py|Ta \nCu|Py|Cu Calc. Expt. NM \r\n(l)NM \r\n(l)\nLead \n Lead Py \r\n(d)\nFIG. 1. (color online). Calculated (solid lines) Gilbert da mp-\ning of NM |Py|NM (NM = Cu, Pd, Ta and Pt) compared to\nexperimental measurements (dotted lines) [ 4] as a function of\nthe Py thickness d. Inset: sketch of the structure used in the\ncalculations. The dashed frame denotes one structural unit\nconsisting of a Py film between two NM films.2\ntemperature so the 5 nm thick NM metals used in the\nsamples were diffusive; (ii) the substrate |NM and NM |air\ninterfaces cannot transmit charge or spin and behave ef-\nfectively as “mirrors”, whereas in the conventional scat-\ntering theory the NM leads are connected to charge and\nspin reservoirs.\nWe start with the structural NM( l)|Py(d)|NM(l) unit\nindicated by the dashed line in the inset to Fig. 1that\nconsists of a Py film, whose thickness dis variable, sand-\nwichedbetween l=5nm-thick diffusiveNM films. Several\nNM|Py|NM units are connected in series between semi-\ninfinite leads to calculate the total magnetization dissi-\npation of the system [ 10–13] thereby explicitly assuming\na “mirror” boundary condition. By varying the number\nof these units, the Gilbert damping for a single unit can\nbe extracted [ 18], that corresponds to the damping mea-\nsured for the experimental NM( l)|Py(d)|NM(l) system.\nAs shown in Fig. 1, the results are in remarkably\ngood overall agreement with experiment. For Pt and\nPd, where a strong damping enhancement is observed for\nthin Py layers, the values that we calculate are slightly\nlower than the measured ones. For Ta and Cu where\nthe enhancement is weaker, the agreement is better. In\nthe case of Cu, neither the experimental nor the calcu-\nlated data shows any dependence on dindicating a van-\nishinglysmalldampingenhancement. Theoffsetbetween\nthe two horizontal lines results from a difference between\nthe measured and calculated values of the bulk damping\nin Py. Acareful analysisshowsthat the calculated values\nofαare inversely proportional to the Py thickness dand\napproach the calculated bulk damping of Py α0=0.0046\n[11] in the limit of large dfor all NM metals. However,\nextrapolation of the experimental data yields values of\nα0ranging from 0.004 to 0.007 [ 19]; the spread can be\npartly attributed to the calibration of the Py thickness,\nespecially when it is very thin.\nGeneralized spin-pumping theory. —In spite of the very\ngood agreement with experiment, our calculated re-\nsults cannot be interpreted satisfactorily using the spin-\npumping theory [ 5] that describes the damping enhance-\nment in terms of a spin current pumped through the\ninterface by the precessing magnetization giving rise to\nan accumulation of spins in the diffusive NM metal,\nand a back-flowing spin current driven by the ensuing\nspin-accumulation. The pumped spin current, Ipump\ns=\n(/planckover2pi12A/2e2)Gmixm×˙m, is described using a “mixing con-\nductance” Gmix[20] that is a property of the NM |FM\ninterface [ 21,22]. Here, mis a unit vector in the di-\nrection of the magnetization and Ais the cross-sectional\narea. The theory only takes spin-orbit coupling (SOC)\ninto account implicitly via the spin-flip diffusion length\nlsfof the NM metal and the pumped spin current is con-\ntinuous across the FM |NM interface [ 5].\nWith SOC included, this boundary condition does not\nhold. Spin-flip scattering at an interface is described by\nthe “spin memory loss” parameter δdefined so that thespin-flip probability of a conduction electron crossing the\ninterface is 1 −e−δ[14,15]. It alters the spin accumula-\ntion in the NM metal and, in turn, the backflow into the\nFM material. To take δand the spin-dependence of the\ninterface resistance into account, the FM |NM interface\nis represented by a fictitious homogeneous ferromagnetic\nlayerwithafinitethickness[ 15,16]. Thespincurrentand\nspin-resolved chemical potentials (as well as their differ-\nenceµs, the spin accumulation) are continuous at the\nboundaries of the effective “interface” layer. We impose\nthe boundary condition that the spin current vanishes at\nNM|air or NM |substrate interfaces. Then the spin accu-\nmulation in the NM metal can be expressed as a function\nof the net spin-current Isflowing out of Py [ 23], which\nis the difference between the pumped spin current Ipump\ns\nand the backflow Iback\ns. The latter is determined by the\nspin accumulation in the NM metal close to the inter-\nface,Iback\ns[µs(Is)]. Following the original treatment by\nTserkovnyak et al. [ 5],Isis determined by solving the\nequation Is=Ipump\ns−Iback\ns[µs(Is)] self-consistently. Fi-\nnally, the total damping of NM( l)|Py(d)|NM(l) can be\ndescribed as\nα(l,d) =α0+gµB/planckover2pi1\ne2MsdGmix\neff=α0+gµB/planckover2pi1\ne2Msd\n×/bracketleftbigg1\nGmix+2ρlsfR∗\nρlsfδsinhδ+R∗coshδtanh(l/lsf)/bracketrightbigg−1\n.(1)\nHere,R∗=R/(1−γ2\nR) is an effective interface spe-\ncific resistance with Rthe total interface specific resis-\ntance between Py and NM and its spin polarization,\nγR= (R↓−R↑)/(R↓+R↑) is determined by the con-\ntributions R↑andR↓from the two spin channels [ 16].ρ\nis the resistivity of the NM metal. All the quantities in\nEq. (1) can be experimentally measured [ 16] and calcu-\nlated from first-principles [ 24]. If spin-flip scattering at\nthe interface is neglected, i.e., δ= 0, Eq. ( 1) reduces to\nthe original spin pumping formalism [ 5]. Eq. (1) is de-\nrived using the Valet-Fert diffusion equation [ 25] that is\nstill applicable when the mean free path is comparable\nto the spin-flip diffusion length [ 26].\nMixing conductance. —Assuming that SOC can be\nneglected and that the interface scattering is spin-\nconserving, the mixing conductance is defined as\nGmix=e2\nhA/summationdisplay\nm,n/parenleftbig\nδmn−r↑\nmnr↓∗\nmn/parenrightbig\n, (2)\nin terms of rσ\nmn, the probability amplitude for reflection\nofaNMmetalstate nwithspin σintoaNM state mwith\nthesamespin. UsingEq.( 2), wecalculate GmixforPy|Pt\nand Py|Cu interfaces without SOC and indicate the cor-\nresponding damping enhancement gµB/planckover2pi1Gmix/(e2MsA)\non the vertical axis in Fig. 2with asterisks.\nWhen SOC is included, Eq. ( 2) is no longer applicable.\nWecanneverthelessidentify aspin-pumpinginterfaceen-\nhancement Gmixas follows. We artificially turn off the3\n0 2 4 6 8 10\nd (nm)00.050.100.15αd (nm)Pt\nCuWithout backflow\nWith backflow\nFIG. 2. (color online). Total damping calculated for Pt |Py|Pt\nand Cu|Py|Cu as a function of the Py thickness. The open\nsymbols correspond to the case without backflow while the\nfull symbols are the results shown in Fig. 1where backflow\nwas included. The lines are linear fits to the symbols. The as-\nterisks on the yaxis are the values of Gmixcalculated without\nSOC using Eq. ( 2).\nbackflow by connecting the FM metal to ballistic NM\nleads so that any spin current pumped through the in-\nterface propagatesawayimmediately and there is no spin\naccumulation in the NM metal. The Gilbert damping αd\ncalculated without backflow (dashed lines) is linear in\nthe Py thickness d; the intercept Γ at d= 0 represents\nan interface contribution. As seen in Fig. 2for Cu, Γ\ncoincides with the orange asterisk meaning that the in-\nterface damping enhancement for a Py |Cu interface is,\nwithin the accuracy of the calculation, unchanged by in-\ncluding SOC because this is so small for Cu, Ni and Fe.\nBy contrast, Γ and thus Gmix=e2MsAΓ/(gµB/planckover2pi1) for the\nPy|Pt interface is 25% larger with SOC included, con-\nfirming the breakdown of Eq. ( 2) for interfaces involving\nheavy elements.\nThe data in Fig. 1for NM=Pt and Cu are replotted\nas solid lines in Fig. 2for comparison. Their linearity\nmeans that we can extract an effective mixing conduc-\nTABLE I. Different mixing conductances calculated for\nPy|NM interfaces. Gmixis calculated using Eq. ( 2) without\nSOC.Gmixis obtainedfrom theinterceptofthetotal damping\nαdcalculated as a function of the Py thickness dwith SOCfor\nballistic NM leads. The effective mixing conductance Gmix\neffis\nextracted from the effective αin Fig.1in the presence of 5 nm\nNM on either side of Py. Sharvin conductances are listed for\ncomparison. All values are given in units of 1015Ω−1m−2.\nNM GSh GmixGmixGmix\neff\nCu 0.55 0.49 0.48 0.01\nPd 1.21 0.89 0.98 0.57\nTa 0.74 0.44 0.48 0.34\nPt 1.00 0.86 1.07 0.95tanceGmix\neffwith backflow in the presence of 5 nm dif-\nfusive NM metal attached to Py. For Py |Pt,Gmix\neffis\nonly reduced slightly compared to Gmixbecause there is\nvery little backflow. For Py |Cu, the spin current pumped\ninto Cu is only about half that for Py |Pt. However, the\nspin-flipping in Cu is so weak that spin accumulation in\nCu leads to a backflow that almost exactly cancels the\npumped spin current and Gmix\neffis vanishingly small for\nthe Py|Cu system with thin, diffusive Cu.\nThe values of Gmix,GmixandGmix\neffcalculated for all\nfour NM metals are listed in Table I. Because Gmix(Pd)\nandGmix(Pt) are comparable, Py pumps a similar spin\ncurrent into each of these NM metals. The weaker spin-\nflipping and larger spin accumulation in Pd leads to a\nlarger backflow and smaller damping enhancement. The\nrelatively low damping enhancement in Ta |Py|Ta results\nfrom a small mixing conductance for the Ta |Py interface\nrather than from a large backflow. In fact, Ta behaves\nas a good spin sink due to its large SOC and the damp-\ning enhancement in Ta |Py|Ta can not be significantly\nincreased by suppressing the backflow.\nThickness dependence of NM. —In the following we fo-\ncus on the Pt |Py|Pt system and examine the effect of\nchanging the NM thickness lon the damping enhance-\nment, a procedure frequently used to experimentally de-\ntermine the NM spin-flip diffusion length [ 27–31].\nThe total damping calculated for Pt |Py|Pt is plotted\nin Fig.3as a function of the Pt thickness lfor two thick-\nnessesdof Py. For both d= 1 nm and d= 2 nm,\nαsaturates at l=1–2 nm in agreement with experiment\n0 10 20 30 40 50 l (nm)0.00.51.0\n0 1 2 3 4 5\nl (nm)0.000.050.100.15 αd=1 nm\nd=2 nmPt(l)|Py(d)|Pt( l)G↑↑/G↑\nG↑↓/G↑Pt@RT\nl↑=7.8±0.3 nm\nFIG. 3. αas a function of the Pt thickness lcalculated for\nPt(l)|Py(d)|Pt(l). The dashed and solid lines are the curves\nobtained by fitting without and with interface spin memory\nloss, respectively. Inset: fractional spin conductances G↑↑/G↑\nandG↑↓/G↑when a fully polarized up-spin current is injected\ninto bulk Pt at room temperature. Gσσ′is (e2/htimes) the\ntransmission probability of a spin σfrom the left hand lead\ninto a spin σ′in the right hand lead; G↑=G↑↑+G↑↓. The\nvalue of the spin-flip diffusion length for a single spin chann el\nobtained by fitting is lσ= 7.8±0.3 nm.4\n[17,28–31]. AfitofthecalculateddatausingEq.( 1)with\nδ≡0 requires just three parameters, Gmix,ρandlsf. A\nseparate calculation gives ρ= 10.4µΩcm at T=300 K in\nvery good agreement with the experimental bulk value of\n10.8µΩcm [32]. Using the calculated Gmixfrom Table I\nleaves just one parameter free; from fitting, we obtain\na valuelsf=0.8 nm for Pt (dashed lines) that is consis-\ntent with values between0.5and 1.4nm determined from\nspin-pumping experiments [ 28–31]. However, the dashed\nlines clearly do not reproduce the calculated data very\nwell and the fit value of lsfis much shorter than that\nextracted from scattering calculations [ 11]. By injecting\na fully spin-polarized current into diffusive Pt, we find\nl↑=l↓= 7.8±0.3nm, asshownin theinsettoFig. 3, and\nfrom [25,33],lsf=/bracketleftbig\n(l↑)−2+(l↓)−2/bracketrightbig−1/2= 5.5±0.2 nm.\nThis value is confirmed by examining how the current\npolarization in Pt is distributed locally [ 34].\nIf we allow for a finite value of δand use the in-\ndependently determined Gmix,ρandlsf, the data in\nFig.3(solid lines) can be fit with δ= 3.7±0.2 and\nR∗/δ= 9.2±1.7 fΩm2. The solid lines reproduce the\ncalculateddatamuch better than when δ= 0 underlining\nthe importance of including interface spin-flip scattering\n[17,35]. The large value of δwe find is consistent with a\nlow spin accumulation in Pt and the corresponding very\nweak backflow at the Py |Pt interface seen in Fig. 2.\nConductivity dependence. —Many experiments deter-\nmining the spin-flip diffusion length of Pt have reported\nPt resistivities that range from 4.2–12 µΩcm at low tem-\nperature [ 35–38] and 15–73 µΩcm at room temperature\n[17,39–41]. The large spread in resistivity can be at-\ntributed to different amounts of structural disorder aris-\ning during fabrication, the finite thickness of thin film\nsamples etc. We can determine lsfandρ≡1/σfrom\nfirst principles scattering theory [ 11,12] by varying the\ntemperature in the thermal distribution of Pt displace-\nments in the range 100–500 K. The results are plot-\nted (black solid circles) in Fig. 4(a).lsfshows a lin-\near dependence on the conductivity suggesting that the\nElliott-Yafet mechanism [ 42,43] dominates the conduc-\ntion electron spin relaxation. A linear least squares fit\nyieldsρlsf= 0.61±0.02fΩm2that agrees very well with\nbulk data extracted from experiment that are either not\nsensitive to interface spin-flipping [ 37] or take it into ac-\ncount [17,35,38]. For comparison, we plot values of lsf\nextracted from the interface-enhanced damping calcula-\ntions assuming δ= 0 (empty orange circles). The result-\ning values of lsfare very small, between 0.5 and 2 nm, to\ncompensate for the neglect of δ.\nHaving determined lsf(σ), we can calculate the\ninterface-enhanced damping for Pt |Py|Pt for different\nvalues of σPtand repeat the fitting of Fig. 3using Eq. ( 1)\n[44]. The parameters R∗/δandδare plotted as a func-\ntion of the Pt conductivity in Fig. 4(b). The spin mem-\nory lossδdoes not show any significant variation about0102030R*/δ (fΩ m2)\n0 0.1 0.2 0.3\nσ (108Ω-1m-1)024\nδ1020lsf (nm)Rojas-Sánchez\nNiimi\nNguyen\nKurt50 20 10 7 54ρ (µΩ cm)\nδ=0(a)\n(b)\nFIG. 4. (a) lsffor diffusive Pt as a function of its conductivity\nσ(solid black circles) calculated by injecting a fully polar ized\ncurrent into Pt. The solid black line illustrates the linear\ndependence. Bulk values extracted from experiment that are\neithernotsensitivetointerface spin-flipping[ 37]ortakeitinto\naccount [ 17,35,38] are plotted (squares) for comparison. The\nempty circles are values of lsfdetermined from the interface-\nenhanced damping using Eq. ( 1) withδ= 0. (b) Fit values of\nR∗/δandδas a function of the conductivity of Pt obtained\nusing Eq. ( 1). The solid red line is the average value (for\ndifferent values of σ) ofδ=3.7.\n3.7, i.e., it does not appear to depend on temperature-\ninduced disorder in Pt indicating that it results mainly\nfrom scattering of the conduction electrons at the abrupt\npotential change of the interface. Unlike δ, the effective\ninterfaceresistance R∗decreaseswithdecreasingdisorder\nin Pt and tends to saturate for sufficiently ordered Pt. It\nsuggests that although lattice disorder at the interface\ndoes not dissipate spin angular momentum, it still con-\ntributestotherelaxationofthemomentumofconduction\nelectrons at the interface.\nConclusions. —We have calculated the Gilbert damp-\ning for Py |NM-metal interfaces from first-principles and\nreproduced quantitatively the experimentally observed\ndamping enhancement. To interpret the numerical re-\nsults, we generalized the spin-pumping expression for\nthe damping to allow for interface spin-flipping, a mix-\ning conductance modified by SOC, and spin dependent\ninterface resistances. The resulting Eq. ( 1) allows the\ntwo main factors contributing to the interface-enhanced\ndamping to be separated: the mixing conductance that\ndeterminesthespincurrentpumpedbyaprecessingmag-\nnetization and the spin accumulation in the NM metal\nthat induces a backflow of spin current into Py and low-\ners the efficiency of the spin pumping. In particular, the\nlatter is responsible for the low damping enhancement\nfor Py|Cu while the weak enhancement for Py |Ta arises\nfrom the small mixing conductance.\nWe calculate how the spin-flip diffusion length, spin5\nmemory loss and interface resistance depend on the con-\nductivity of Pt. It is shown to be essential to take ac-\ncount of spin memory loss to extract reasonable spin-\nflip diffusion lengths from interface damping. This has\nimportant consequences for using spin-pumping-related\nexperiments to determine the Spin Hall angles that char-\nacterize the Spin Hall Effect [ 17].\nAcknowledgments. —We are grateful to G.E.W. Bauer\nfor a critical reading of the manuscript. Our work was\nfinancially supported by the “Nederlandse Organisatie\nvoor Wetenschappelijk Onderzoek” (NWO) through the\nresearch programme of “Stichting voor Fundamenteel\nOnderzoek der Materie” (FOM) and the supercomputer\nfacilities of NWO “Exacte Wetenschappen (Physical Sci-\nences)”. It was also partly supported by the Royal\nNetherlands Academy of Arts and Sciences (KNAW). Z.\nYuan acknowledges the financial support of the Alexan-\nder von Humboldt foundation.\n∗Present address: Institut f¨ ur Physik, Johannes\nGutenberg–Universit¨ at Mainz, Staudingerweg 7, 55128\nMainz, Germany\n†zyuan@uni-mainz.de\n[1] See the collection of articles in Handbook of Spin Trans-\nport and Magnetism , edited by E. Y. Tsymbal and\nI.ˇZuti´ c (Chapman and Hall/CRC Press, Boca Raton,\n2011).\n[2] G. E. W. Bauer, E. Saitoh, and B. J. van Wees,\nNature Materials 11, 391 (2012) .\n[3] A. Brataas, A. D. Kent, and H. Ohno,\nNature Materials 11, 372 (2012) .\n[4] S. Mizukami, Y. Ando, and T. Miyazaki,\nJpn. J. Appl. Phys. 40, 580 (2001) ;\nJ. Magn. & Magn. Mater. 226–230 , 1640 (2001) .\n[5] Y. Tserkovnyak, A. Brataas, and G. E. W.\nBauer, Phys. Rev. Lett. 88, 117601 (2002) ;\nPhys. Rev. B 66, 224403 (2002) .\n[6] Y. Tserkovnyak, A. Brataas, G. E. W. Bauer, and B. I.\nHalperin, Rev. Mod. Phys. 77, 1375 (2005) .\n[7] M. V. Costache, M. Sladkov, S. M. Watts,\nC. H. van der Wal, and B. J. van Wees,\nPhys. Rev. Lett. 97, 216603 (2006) .\n[8] F. D. Czeschka, L. Dreher, M. S. Brandt, M. Weiler,\nM. Althammer, I.-M. Imort, G. Reiss, A. Thomas,\nW.Schoch, W.Limmer, H.Huebl, R.Gross, andS. T. B.\nGoennenwein, Phys. Rev. Lett. 107, 046601 (2011) .\n[9] M. Weiler, M. Althammer, M. Schreier, J. Lotze,\nM. Pernpeintner, S. Meyer, H. Huebl, R. Gross,\nA. Kamra, J. Xiao, Y.-T. Chen, H. J. Jiao,\nG. E. W. Bauer, and S. T. B. Goennenwein,\nPhys. Rev. Lett. 111, 176601 (2013) .\n[10] A. Brataas, Y. Tserkovnyak, and G. E. W.\nBauer, Phys. Rev. Lett. 101, 037207 (2008) ;\nPhys. Rev. B 84, 054416 (2011) .\n[11] A. A. Starikov, P. J. Kelly, A. Brataas,\nY. Tserkovnyak, and G. E. W. Bauer,\nPhys. Rev. Lett. 105, 236601 (2010) .\n[12] Y. Liu, A. A. Starikov, Z. Yuan, and P. J. Kelly,Phys. Rev. B 84, 014412 (2011) .\n[13] Z. Yuan, K. M. D. Hals, Y. Liu, A. A. Starikov,\nA. Brataas, and P. J. Kelly, arXiv:1404.1488 (2014).\n[14] A. Fert and S.-F. Lee, Phys. Rev. B 53, 6554 (1996) .\n[15] K. Eid, D. Portner, J. A. Borchers, R. Loloee,\nM. A. Darwish, M. Tsoi, R. D. Slater, K. V.\nO’Donovan, H. Kurt, W. P. Pratt, Jr., and J. Bass,\nPhys. Rev. B 65, 054424 (2002) .\n[16] J. Bass and W. P. Pratt Jr.,\nJ. Phys.: Condens. Matter 19, 183201 (2007) .\n[17] J.-C. Rojas-S´ anchez, N. Reyren, P. Laczkowski,\nW. Savero, J.-P. Attan´ e, C. Deranlot, M. Jamet,\nJ.-M. George, L. Vila, and H. Jaffr` es,\nPhys. Rev. Lett. 112, 106602 (2014) .\n[18] See Supplemental Material at\nhttp://link.aps.org/supplemental/10.1103/PhysRevLet t.???.??????\nfor computational details.\n[19] See §4.3.2 of Y. Liu, Spin relaxation from first-principles ,\nPh.D. thesis , University of Twente, The Netherlands\n(2014).\n[20] A. Brataas, Y. V. Nazarov, and G. E. W. Bauer,\nPhys. Rev. Lett. 84, 2481 (2000) .\n[21] K. Xia, P. J. Kelly, G. E. W. Bauer, A. Brataas, and\nI. Turek, Phys. Rev. B 65, 220401 (2002) .\n[22] M. Zwierzycki, Y. Tserkovnyak, P. J. Kelly, A. Brataas,\nand G. E. W. Bauer, Phys. Rev. B 71, 064420 (2005) .\n[23] The derivation of spin accumulation is different from\nRef.17, where Rojas-S´ anchez et al. treated the inter-\nface region as a non-magnetic medium in their study of a\nCo|Pt interface. Since the resistance of a FM |NM inter-\nface is spin dependent [ 16], this is not strictly correct. In\naddition, their treatment contains an unknown empiri-\ncal parameter, the ratio of the spin-conserved to spin-flip\nrelaxation times.\n[24] A. Brataas, G. E. W. Bauer, and P. J. Kelly,\nPhys. Rep. 427, 157 (2006) .\n[25] T. Valet and A. Fert, Phys. Rev. B 48, 7099 (1993) .\n[26] D. R. Penn and M. D. Stiles,\nPhys. Rev. B 72, 212410 (2005) .\n[27] A. Azevedo, L. H. Vilela-Le˜ ao, R. L. Rodr´ ıguez-\nSu´ arez, A. F. Lacerda Santos, and S. M. Rezende,\nPhys. Rev. B 83, 144402 (2011) .\n[28] L. Liu, T. Moriyama, D. C. Ralph, and R. A. Buhrman,\nPhys. Rev. Lett. 106, 036601 (2011) .\n[29] K. Kondou, H. Sukegawa, S. Mitani, K. Tsukagoshi, and\nS. Kasai, Applied Physics Express 5, 073002 (2012) .\n[30] C. T. Boone, H. T. Nembach, J. M. Shaw, and T. J.\nSilva,J. Appl. Phys. 113, 153906 (2013) .\n[31] W. Zhang, V. Vlaminck, J. E. Pearson,\nR. Divan, S. D. Bader, and A. Hoffmann,\nAppl. Phys. Lett. 103, 242414 (2013) .\n[32] See §12: Properties of Solids - Electrical Resistivity of\nPure Metals in D. R. Lide, ed., CRC Handbook of Chem-\nistry and Physics , 84th ed. (CRC Press, Boca Raton,\n2003).\n[33] M. V. Costache, B. J. van Wees, and S. O. Valenzuela,\ninOne-Dimensional Nanostructures: Principles and Ap-\nplications , edited by T. Zhai and J. Yao (John Wiley &\nSons, Inc., Hoboken, 2013) pp. 473–482.\n[34] R. J. H. Wesselink, Y. Liu, Z. Yuan, A. A. Starikov, and\nP. J. Kelly, unpublished (2014).\n[35] H. Y. T. Nguyen, W. P. Pratt Jr., and J. Bass,\nJ. Magn. & Magn. Mater. 361, 30 (2014) .\n[36] M. Morota, Y. Niimi, K. Ohnishi, D. H. Wei,6\nT. Tanaka, H. Kontani, T. Kimura, and Y. Otani,\nPhys. Rev. B 83, 174405 (2011) .\n[37] Y. Niimi, D. Wei, H. Idzuchi, T. Wakamura, T. Kato,\nand Y. C. Otani, Phys. Rev. Lett. 110, 016805 (2013) .\n[38] H. Kurt, R. Loloee, K. Eid, W. P. Pratt Jr., and J. Bass,\nAppl. Phys. Lett. 81, 4787 (2002) .\n[39] M. Althammer, S. Meyer, H. Nakayama, M. Schreier,\nS. Altmannshofer, M. Weiler, H. Huebl, S. Gepr¨ ags,\nM. Opel, R. Gross, D. Meier, C. Klewe, T. Kuschel, J.-M.\nSchmalhorst, G. Reiss, L. Shen, A. Gupta, Y.-T. Chen,\nG. E. W. Bauer, E. Saitoh, and S. T. B. Goennenwein,\nPhys. Rev. B 87, 224401 (2013) .[40] V. Castel, N. Vlietstra, J. Ben Youssef, and B. J. van\nWees,Appl. Phys. Lett. 101, 132414 (2012) .\n[41] K. Ando, S. Takahashi, K. Harii, K. Sasage,\nJ. Ieda, S. Maekawa, and E. Saitoh,\nPhys. Rev. Lett. 101, 036601 (2008) .\n[42] R. J. Elliott, Phys. Rev. 96, 266 (1954) .\n[43] Y.Yafet,in Solid State Physics ,Vol.14,editedbyF.Seitz\nand D. Turnbull (Academic, New York, 1963) pp. 1–98.\n[44] Experiment [ 8] and theory [ 45] indicate at most a weak\ntemperature dependence of the spin mixing conductance.\n[45] K.Nakata, J. Phys.: Condens. Matter 25, 116005 (2013) ."    },    {        "title": "1407.0635v1.Spin_Waves_in_Ferromagnetic_Insulators_Coupled_via_a_Normal_Metal.pdf",        "content": "Spin Waves in Ferromagnetic Insulators Coupled via a Normal Metal\nHans Skarsv\u0017 ag,\u0003Andr\u0013 e Kapelrud, and Arne Brataas\nDepartment of Physics, Norwegian University of Science and Technology, NO-7491 Trondheim, Norway\n(Dated: May 27, 2022)\nHerein, we study the spin-wave dispersion and dissipation in a ferromagnetic insulator{normal\nmetal{ferromagnetic insulator system. Long-range dynamic coupling because of spin pumping and\nspin transfer lead to collective magnetic excitations in the two thin-\flm ferromagnets. In addition,\nthe dynamic dipolar \feld contributes to the interlayer coupling. By solving the Landau-Lifshitz-\nGilbert-Slonczewski equation for macrospin excitations and the exchange-dipole volume as well as\nsurface spin waves, we compute the e\u000bect of the dynamic coupling on the resonance frequencies and\nlinewidths of the various modes. The long-wavelength modes may couple acoustically or optically.\nIn the absence of spin-memory loss in the normal metal, the spin-pumping-induced Gilbert damp-\ning enhancement of the acoustic mode vanishes, whereas the optical mode acquires a signi\fcant\nGilbert damping enhancement, comparable to that of a system attached to a perfect spin sink. The\ndynamic coupling is reduced for short-wavelength spin waves, and there is no synchronization. For\nintermediate wavelengths, the coupling can be increased by the dipolar \feld such that the modes\nin the two ferromagnetic insulators can couple despite possible small frequency asymmetries. The\nsurface waves induced by an easy-axis surface anisotropy exhibit much greater Gilbert damping\nenhancement. These modes also may acoustically or optically couple, but they are una\u000bected by\nthickness asymmetries.\nPACS numbers: 76.50.+g,75.30.Ds,75.70.-i,75.76.+j\nI. INTRODUCTION\nThe dynamic magnetic properties of thin-\flm fer-\nromagnets have been extensively studied for several\ndecades.1,2Thin-\flm ferromagnets exhibit a rich vari-\nety of spin-wave modes because of the intricate inter-\nplay among the exchange and dipole interactions and the\nmaterial anisotropies. In ferromagnetic insulators (FIs),\nthese modes are especially visible; the absence of disturb-\ning electric currents leads to a clear separation of the\nmagnetic behavior. Furthermore, the dissipation rates\nin insulators are orders of magnitude lower than those\nin their metallic counterparts; these low dissipation rates\nenable superior control of travelling spin waves and facil-\nitate the design of magnonic devices.3\nIn spintronics, there has long been considerable in-\nterest in giant magnetoresistance, spin-transfer torques,\nand spin pumping in hybrid systems of normal met-\nals and metallic ferromagnets (MFs).4{7The experimen-\ntal demonstration that spin transfer and spin pumping\nare also active in normal metals in contact with insu-\nlating ferromagnets has generated a renewed interest in\nand refocused attention on insulating ferromagnets, of\nwhich yttrium iron garnet (YIG) continues to be the\nprime example.8{19In ferromagnetic insulators, current-\ninduced spin-transfer torques from a neighboring normal\nmetal (NM) that exhibits out-of-equilibrium spin accu-\nmulation may manipulate the magnetization of the insu-\nlator and excite spin waves.8,20,21The out-of-equilibrium\nspin accumulation of the normal metal may be induced\nvia the spin Hall e\u000bect or by currents passing through\nother adjacent conducting ferromagnets. Conversely, ex-\ncited spin waves pump spins into adjacent NMs, and this\nspin current may be measured in terms of the inverse spinHall voltages or by other conducting ferromagnets.8{14\nThe magnetic state may also be measured via the spin\nHall magnetoresistance.16{19,23,24Because of these devel-\nopments, magnetic information in ferromagnetic insula-\ntors may be electrically injected, manipulated, and de-\ntected. Importantly, an FI-based spintronic device may\ne\u000eciently transport electric information carried by spin\nwaves over long distances15without any excessive heat-\ning. The spin-wave decay length can be as long as cen-\ntimeters in YIG \flms.22These properties make FI{NM\nsystems ideal devices for the exploration of novel spin-\ntronic phenomena and possibly also important for future\nspintronic applications. Magnonic devices also o\u000ber ad-\nvantages such as rapid spin-wave propagation, frequen-\ncies ranging from GHz to THz, and the feasibility of cre-\nating spin-wave logic devices and magnonic crystals with\ntailored spin-wave dispersions.25\nTo utilize the desirable properties of FI{NM systems,\nsuch as the exceptionally low magnetization-damping\nrate of FIs, it is necessary to understand how the mag-\nnetization dynamics couple to spin transport in adjacent\nnormal metals. The e\u000bective damping of the uniform\nmagnetic mode of a thin-\flm FI is known to signi\f-\ncantly increase when the FI is placed in contact with\nan NM. This damping enhancement is caused by the loss\nof angular momentum through spin pumping.26{30Re-\ncent theoretical work has also predicted the manner in\nwhich the Gilbert damping for other spin-wave modes\nshould become renormalized.31For long-wavelength spin\nwaves, the Gilbert damping enhancement is twice as\nlarge for transverse volume waves as for the macrospin\nmode, and for surface modes, the enhancement can be ten\ntimes stronger or more. Spin pumping has been demon-\nstrated, both experimentally9and theoretically,31to be\nsuppressed for short-wavelength exchange spin waves.arXiv:1407.0635v1  [cond-mat.mes-hall]  2 Jul 20142\nA natural next step is to investigate the magnetization\ndynamics of more complicated FI{NM heterostructures.\nIn ferromagnetic metals, it is known that spin pumping\nand spin-transfer torques generate a long-range dynamic\ninteraction between magnetic \flms separated by normal\nmetal layers.32The e\u000bect of this long-range dynamic in-\nteraction on homogeneous macrospin excitations can be\nmeasured by ferromagnetic resonance. The combined ef-\nfects of spin pumping and spin-transfer torque lead to\nan appreciable increase in the resonant linewidth when\nthe resonance \felds of the two \flms are far apart and\nto a dramatic narrowing of the linewidth when the reso-\nnant \felds approach each other.32This behavior occurs\nbecause the excitations in the two \flms couple acous-\ntically (in phase) or optically (out of phase). We will\ndemonstrate that similar, though richer because of the\ncomplex magnetic modes, phenomena exist in magnetic\ninsulators.\nIn the present paper, we investigate the magnetization\ndynamics in a thin-\flm stack consisting of two FIs that\nare in contact via an NM. The macrospin dynamics in\na similar system with metallic ferromagnets have been\nstudied both theoretically and experimentally.32We ex-\npand on that work by focusing on inhomogeneous mag-\nnetization excitations in FIs.\nFor long-wavelength spin waves travelling in-plane in\na ferromagnetic thin \flm, the frequency as a function\nof the in-plane wave number Qstrongly depends on the\ndirection of the external magnetic \feld with respect to\nthe propagation direction. If the external \feld is in-\nplane and the spin waves are travelling parallel to this\ndirection, the waves have a negative group velocity. Be-\ncause the magnetization precession amplitudes are usu-\nally evenly distributed across the \flm in this geometry,\nthese modes are known as backward volume magneto-\nstatic spin waves (BVMSW). Similarly, spin waves that\ncorrespond to out-of-plane external \felds are known as\nforward volume magnetostatic spin waves (FVMSW),\ni.e., the group velocity is positive, and the precession\namplitudes are evenly distributed across the \flm. When\nthe external \feld is in-plane and perpendicular to the\npropagation direction, the precession amplitudes of the\nspin waves become inhomogeneous across the \flm, ex-\nperiencing localization to one of the interfaces. These\nspin waves are thus known as magnetostatic surface spin\nwaves (MSSW).33,34\nWhen two ferromagnetic \flms are coupled via a normal\nmetal, the spin waves in the two \flms become coupled\nthrough two di\u000berent mechanisms. First, the dynamic,\nnonlocal dipole-dipole interaction causes an interlayer\ncoupling to arise that is independent of the properties\nof the normal metal. This coupling is weaker for larger\nthicknesses of the normal metal. Second, spin pumping\nfrom one ferromagnetic insulator induces a spin accu-\nmulation in the normal metal, which in turn gives rise\nto a spin-transfer torque on the other ferromagnetic in-\nsulator, and vice versa. This dynamic coupling, is in\ncontrast to the static exchange coupling35rather long-ranged and is limited only by the spin-di\u000busion length.\nThis type of coupling is known to strongly couple the\nmacrospin modes. When two ferromagnetic \flms become\ncoupled, the characterization of the spin waves in terms\nof FVMSW, BVMSW, and MSSW still holds, but the\ndispersion relations are modi\fed. It is also clear that the\ndamping renormalization caused by spin pumping into\nthe NM may di\u000ber greatly from that in a simpler FI jN\nbilayer system. To understand this phenomenon, we per-\nform a detailed analytical and numerical analysis of a\ntrilayer system, with the hope that our \fndings may be\nused as a guide for experimentalists.\nThis paper is organized as follows. Section II intro-\nduces the model. The details of the dynamic dipolar\n\feld are discussed, and the boundary conditions associ-\nated with spin pumping and spin transfer at the FI jN\ninterfaces are calculated. Sec. III provides the analyti-\ncal solutions of these equations in the long-wavelength\nregime dominated by the dynamic coupling attributable\nto spin pumping and spin transfer. To create a more\ncomplete picture of the dynamic behavior of this system,\nwe perform a numerical analysis for the entire spin-wave\nspectrum of this system, which is presented in Sec. IV.\nWe conclude our work in Sec. V.\nII. EQUATIONS OF MOTION\nConsider a thin-\flm heterostructure composed of two\nferromagnetic insulators (FI1 and FI2) that are in elec-\ntrical contact via an NM layer. The ferromagnetic in-\nsulators FI1 and FI2 may have di\u000berent thicknesses and\nmaterial properties. We denote the thicknesses by L1,\ndN, andL2for the FI1, NM, and FI2 layers, respectively\n(see Fig. 1(a)). The in-plane coordinates are \u0010;\u0011, and the\ntransverse coordinate is \u0018(see Fig. 1(b)). We will \frst\ndiscuss the magnetization dynamics in isolated FIs and\nwill then incorporate the spin-memory losses and the cou-\npling between the FIs via spin currents passing through\nthe NM.\nA. Magnetization Dynamics in Isolated FIs\nThe magnetization dynamics in the ferromagnetic in-\nsulators can be described by using the Landau-Lifshitz-\nGilbert (LLG) equation,\n_Mi=\u0000\rMi\u0002He\u000b+\u000bMi\u0002_Mi; (1)\nwhere Miis the unit vector in the direction of the mag-\nnetization in layer i= 1;2,\ris the gyromagnetic ratio,\n\u000bis the dimensionless damping parameter, and He\u000bis\nthe space-time-dependent e\u000bective magnetic \feld. The\ne\u000bective magnetic \feld is\nHe\u000b=Hint+hex+hd+hsurface; (2)\nwhere Hintis the internal \feld attributable to an external\nmagnetic \feld and the static demagnetization \feld, hex=3\ndN2+L2\ndN2\n-dN2\n-dN2-L1NFI2\nFI1\nSUBx\n(a)\n (b)\nFIG. 1: (Color online) a) A cross section of the FI1 jNjFI2 het-\nerostructure. The ferromagnetic insulators FI1 and FI2 are\nin contact via the normal metal N. The transverse coordinate\n\u0018is indicated along with the thicknesses L1,dN, andL2of\nFI1, N, and FI2, respectively. b) The coordinate system of\nthe internal \feld (blue) with respect to the coordinate system\nof the FI1jNjFI2 structure (red). \u0012denotes the angle between\nthe \flm normal and the internal \feld, and \u001eis the angle be-\ntween the in-plane component of the magnetic \feld and the\nin-plane wave vector.\n2Ar2M=MSis the exchange \feld ( Ais the exchange\nconstant), hdis the dynamic demagnetization \feld, and\nhsurface =2KS\nM2\nS(Mi\u0001^n)\u000e(\u0018\u0000\u0018i)^n (3)\nis the surface anisotropy \feld located at the FI jN in-\nterfaces. In this work, hsurface is assumed to exist only\nat the FIjN interfaces and not at the interfaces between\nthe FIs and the substrate or vacuum. It is straightfor-\nward to generalize the discussion to include these surface\nanisotropies as well. We consider two scenarios: one with\nan easy-axis surface anisotropy ( KS>0) and one with no\nsurface anisotropy ( KS= 0). Note that a negative value\nofKS\u0018 \u0000 0:03 erg=cm2, which implies an easy-plane\nsurface anisotropy, has also been observed for sputtered\nYIGjAu bilayers.36In general, the e\u000bective \feld He\u000bmay\ndi\u000ber in the two FIs. We assume the two FIs consist of\nthe same material and consider external \felds that are\neither in-plane or out-of-plane. Furthermore, we consider\ndevices in which the internal magnetic \felds in the two\nFI layers are aligned and of equal magnitude.\nIn equilibrium, the magnetization inside the FIs is ori-\nented along the internal magnetic \feld, Mi=M0. In the\nlinear response regime, Mi=M0+mi, where the \frst-\norder correction miis small and perpendicular to M0.The magnetization vanishes outside of the FIs. Because\nthe system is translationally invariant in the \u0011and\u0010di-\nrections, we may, without loss of generality, assume that\nmconsists of plane waves travelling in the \u0010direction,\nmi(\u0010;\u0011;\u0018 ) =miQ(\u0018)ei(!t\u0000Q\u0010): (4)\nLinearizing Maxwell's equations in miimplies that the\ndynamic dipolar \feld must be of the same form,\nhd(\u0010;\u0011;\u0018 ) =hdQ(\u0018)ei(!t\u0000Q\u0010): (5)\nFurthermore, the total dipolar \feld (the sum of the static\nand the dynamic dipolar \felds) must satisfy Maxwell's\nequations, which, in the magnetostatic limit, are\nr\u0001(hd+ 4\u0019MSm) = 0; (6a)\nr\u0002hd= 0; (6b)\nwith the boundary equations\n(hd+ 4\u0019MSm)?;in= (hd)?;out; (7a)\n(hd)k;in= (hd)k;out; (7b)\nwhere the subscript in (out) denotes the value on the FI\n(NM, vacuum or substrate) side of the FI interface and ?\n(k) denotes the component(s) perpendicular (parallel) to\nthe FI{NM interfaces. Solving Maxwell's equations (6)\nwith the boundary conditions of Eq. (7) yields33\nhdQ(\u0018) =Z\nd\u00180^G(\u0018\u0000\u00180)mQ(\u00180); (8)\nwhere ^G(r\u0000r0) is a 3\u00023 matrix acting on min the (\u0011;\u0010;\u0018 )\nbasis,\n^G(\u0018) =0\n@GP(\u0018)\u0000\u000e(\u0018) 0\u0000iGQ(\u0018)\n0 0 0\n\u0000iGQ(\u0018) 0\u0000GP(\u0018)1\nA: (9)\nHere,GP(\u0018) =Qe\u0000Qj\u0018j=2, andGQ(\u0018) =\u0000sign(\u0018)GP.\nNote that the dynamic dipolar \feld of Eq. (8) accounts\nfor both the interlayer and intralayer dipole-dipole cou-\nplings because the magnetization varies across the two\nmagnetic insulator bilayers and vanishes outside these\nmaterials.\nIt is now convenient to perform a transformation from\nthe\u0010-\u0011-\u0018coordinate system de\fned by the sample geome-\ntry to thex-y-zcoordinate system de\fned by the internal\n\feld (see Fig. 1(b)). In the linear response regime, the\ndynamic magnetization milies in thex-yplane, and the\nlinearized equations of motion become33\n\u0014\ni!\u0012\n\u000b\u00001\n1\u000b\u0013\n+11\u0012\n!H+2A\nMS\u0014\nQ2\u0000d2\nd\u00182\u0015\u0013\u0015\nmiQxy(\u0018) =2X\ni=1Z\nd\u00180^Gxy(\u0018\u0000\u00180)miQxy(\u00180): (10)4\nN\nm1,QFI1m2,QFI2\nee\nFIG. 2: (Color online) Two coupled spin waves with ampli-\ntudem1Qin ferromagnet FI1 and amplitude m2Qin ferro-\nmagnet FI2. The spin-waves inject a spin current into the nor-\nmal metal (NM) via spin pumping. In the NM, the spins dif-\nfuse and partially relax, inducing a spin accumulation therein.\nIn turn, the spin accumulation causes spin-transfer torques to\narise on FI1 and FI2. The combined e\u000bect of spin transfer and\nspin pumping leads to a dynamic exchange coupling that, to-\ngether with the dynamic demagnetization \feld, couples the\nspin waves in the two FIs.\nHere, miQxy = (miQx;miQy) is the Fourier transform of\nthe dynamic component of the magnetization in the x-\nyplane and ^Gxy(\u0018) is the 2\u00022 matrix that results from\nrotating ^G(\u0018) into thex-y-zcoordinate system (see Ap-\npendix A), and considering only the xx,xy,yxandyy-\ncomponents.\nB. Boundary Conditions and Spin Accumulation\nThe linearized equations of motion (10) must be sup-\nplemented with boundary conditions for the dynamic\nmagnetization at the FI jN interfaces. A precessing mag-\nnetization at the FI jN boundaries injects a spin-polarized\ncurrent, jSP, into the NM, an e\u000bect known as spin\npumping .8,28{30The emitted spin currents at the lower\nand upper interfaces ( i= 1;2) are\njSP\ni=~\neg?Mi\u0002_Mi\f\f\f\f\n\u0018=\u0018i; (11)\nwhere\u0018i=\u0007dN=2 at the lower and upper interfaces,\nrespectively, and g?is the real part of the transverse spin-\nmixing conductance per unit area.37We disregard the\nimaginary part of the spin-mixing conductance because\nit has been found to be small at FI jN interfaces.38The\nreciprocal e\u000bect of spin pumping is spin transfer into the\nFIs because of a spin accumulation \u0016Sin the NM. In the\nnormal metal at the lower and upper interfaces ( i=1,2),the associated spin-accumulation-induced spin current is\njST\ni=\u00001\neg?Mi\u0002(Mi\u0002\u0016S)\f\f\f\f\n\u0018=\u0018i: (12)\nThe signs of the pumped and spin-accumulation-induced\nspin currents in Eqs. (11) and (12) were chosen such that\nthey are positive when there is a \row of spins from the\nNM toward the FIs.\nThe pumped and spin-accumulation-induced spin cur-\nrents of Eqs. (11) and (12) lead to magnetic torques act-\ning on the FI interfaces. The torques that correspond to\nthe spin pumping and spin transfer localized at the FI jN\ninterfaces are\n\u001cSP\ni=\r~2\n2e2g?\u000e(\u0018\u0000\u0018i)Mi\u0002_Mi; (13a)\n\u001cST\ni=\u0000\r~\n2e2g?Mi\u0002(Mi\u0002\u0016S)\u000e(\u0018\u0000\u0018i);(13b)\nrespectively. In the presence of spin currents to and from\nthe normal metal, the magnetization dynamics in the\nFIs is then governed by the modi\fed Landau-Lifshitz-\nGilbert-Slonczewski (LLGS) equation,\n_M=\u0000\rMi\u0002He\u000b+\u000bMi\u0002_Mi+X\ni=1;2\u001cSP\ni+\u001cST\ni:(14)\nBy integrating Eq. (14) over the FI jN interfaces and the\ninterfaces between the FI and vacuum/substrate, we \fnd5\nthatmimust satisfy the boundary conditions21,31\n\u0012\n\u0006Lidmi\nd\u0018+\u001fi\u0014\n_mi\u00001\n~M0\u0002\u0016\u0015\n+LiKS\nAcos (2\u0012)mi\u0013\nx\f\f\f\f\n\u0018=\u0007dN=2= 0;(15a)\n\u0012\n\u0006Lidmi\nd\u0018+\u001fi\u0014\n_mi\u00001\n~M0\u0002\u0016\u0015\n+LiKs\nAcos2(\u0012)mi\u0013\ny\f\f\f\f\f\n\u0018=\u0007dN=2= 0;(15b)\ndm1\nd\u0018\f\f\f\f\n\u0018=\u0000dN=2\u0000L1= 0;dm2\nd\u0018\f\f\f\f\n\u0018=dN=2+L2= 0:(15c)\nHere, we have introduced the timescale \u001fi=\nLi~2g?=4Ae2. The subscripts xandyin Eqs. (15a) and\n(15b) denote the xandycomponents, respectively. In\nour expressions for the boundary conditions (15), we have\nalso accounted for the possibility of a surface anisotropy\narising from the e\u000bective \feld described by Eq. (3),\nwhereKS>0 indicates an easy-axis surface anisotropy\n(EASA). The boundary conditions of Eq. (15), in combi-\nnation with the transport equations in the NM , which we\nwill discuss next, determine the spin accumulation in the\nNM and the subsequent torques caused by spin transfer.\nIn the normal metal, the spins di\u000buse, creating a spa-\ntially dependent spin-accumulation potential \u0016Q, and\nthey relax on the spin-di\u000busion length scale lsf. The\nspin accumulation for an FI jNjFI system has been cal-\nculated in the macrospin model.39The result of this\ncalculation can be directly generalized to the present\nsituation of spatially inhomogeneous spin waves by re-\nplacing the macrospin magnetization in each layer with\nthe interface magnetization and substituting the spin-\ndi\u000busion length with a wave-vector-dependent e\u000bective\nspin-di\u000busion length lsf!~lsf(Q) such that\n\u0016Q=\u0000~\n2M0\u0002[(_mQ(\u00181) +_mQ(\u00182))\u00001(\u0018)\n\u0000(_mQ(\u00181)\u0000_mQ(\u00182))\u00002(\u0018)]:(16)\nSee Appendix B for the details of the functions \u0000 1and\n\u00002. The e\u000bective spin-di\u000busion length is found by Fouriertransforming the spin-di\u000busion equation (see Appendix\nC), resulting in\n~lsf=lsf=p\n1 + (Qlsf)2: (17)\nWe thus have all the necessary equations to de-\nscribe the linear response dynamics of spin waves in the\nFI1jNjFI2 system. We now provide analytical solutions\nof the spin-wave modes in the long-wavelength limit and\nthen complement these solutions with an extensive nu-\nmerical analysis that is valid for any wavelength.\nIII. ANALYTIC SOLUTIONS FOR THE SPIN\nWAVE SPECTRUM\nThe e\u000bect that the exchange and dipolar \felds have\non the spin-wave spectrum depends on the in-plane wave\nnumberQ. WhenQLi\u001c1, the dipolar \feld dominates\nover the exchange \feld. In the opposite regime, when\nQLi\u001d1, the exchange \feld dominates over the dipo-\nlar \feld. The intermediate regime is the dipole-exchange\nregime. Another length scale is set by the spin-di\u000busion\nlength. When Qlsf\u001d1, the e\u000bective spin-relaxation\nlength ~lsfof Eq. (17) becomes small, and the NM acts\nas a perfect spin sink. In this case, only the relatively\nshort-ranged dipolar \feld couples the FIs. We therefore\nfocus our attention on the dipole-dominated regime, in\nwhich the interchange of spin information between the\ntwo FIs remains active.\nIn the limit QLi\u001c1, the magnetization is homoge-\nneous in the in-plane direction. We may then use the\nansatz that the deviation from equilibrium is a sum of\ntransverse travelling waves. Using the boundary condi-\ntions on the outer boundaries of the stack, Eq. (15c), we\n\fnd\nmiQxy(\u0018) =\u0012\nXi\nYi\u0013\ncos\u001a\nki\u0014\n\u0018\u0006(Li+dN\n2)\u0015\u001b\n;(18)\nwherei= 1 when\u0018is inside FI1 and i= 2 when\u0018is inside\nFI2.k1andk2are the out-of-plane wave vectors of the\nlower and upper \flms, respectively. The eigenfrequencies\nof Eq. (10) depend on ki. To \frst order in the damping\nparameter\u000b, we have\n!(ki) =!M\"\n\u0006s\u0012!H\n!M+A\n2\u0019M2\nSk2\ni\u0013\u0012!H\n!M+A\n2\u0019M2\nSk2\ni+ sin2\u0012\u0013\n+i\u000b\u0012!H\n!M+A\n2\u0019M2\nSk2\ni+1\n2sin2\u0012\u0013#\n: (19)\nWe can, without loss of generality, consider only those frequencies that have a positive real part. The eigen-6\nfrequency!is a characteristic feature of the entire sys-\ntem, so we must require !(k1) =!(k2), which implies\nthatk1=\u0006k2. We will discuss the cases of symmetric\n(L1=L2) and asymmetric ( L16=L2) geometries sepa-\nrately.\nA. Symmetric FI \flms without EASA\nConsider a symmetric system in which the FIs are of\nidentical thickness and material properties. We assume\nthat the e\u000bect of the EASA is negligible, which is the\ncase for thin \flms and/or weak surface anisotropy ener-\ngies such that KSL=A\u001c1, whereL=L1=L2. The\nother two boundary conditions, (15a) and (15b), cou-\nple the amplitude vectors\u0000X1Y1\u0001Tand\u0000X2Y2\u0001Tof\nEq. (18). A non-trivial solution implies that the deter-\nminant that contains the coe\u000ecients of the resulting 4 \u00024\nmatrix equation vanishes. Solving the secular equation,\nwe \fnd the following constraints on k,\ni\u001fA!A=kLtan(kL); (20a)\ni\u001fO!O=kLtan(kL); (20b)\nwhere\n\u001fA=\u001f \n1\u0000\u0014\n1 +2g?lsf\n\u001btanh(dN=2lsf)\u0015\u00001!\n;(21a)\n\u001fO=\u001f \n1\u0000\u0014\n1 +2g?lsf\n\u001bcoth(dN=2lsf)\u0015\u00001!\n;(21b)\nand\u001f=L~2g?=4Ae2. The two solutions correspond\nto a symmetric mode (acoustic) and an antisymmetric\nmode (optical). This result can be understood in terms\nof the eigenvectors that correspond to the eigenvalues of\nEqs. (20), which are m1= +m2andm1=\u0000m2for\nthe acoustic and optical modes, respectively. Typically,\nbecause spin pumping only weakly a\u000bects the magne-\ntization dynamics, the timescale \u001fthat is proportional\nto the mixing conductance g?is much smaller than the\nFMR precession period. In this limit, kLtan(kL)\u001c1.\nThis result allows us to expand the secular equations (20)\naroundkL=n\u0019, wherenis an integral number, which\nyields\ni\u001f\u0017!\u0017;n\u0019(kL+\u0019n)kL; (22)\nwhere\u0017= A;O. This result can be reinserted into the\nbulk dispersion relation of Eq. (19), from which we can\ndetermine the renormalization of the Gilbert damping\ncoe\u000ecient attributable to spin pumping, \u0001 \u000b. We de\fne\n\u0001\u000b=\u000b\u0010\nIm[!(SP)]\u0000Im[!(0)]\u0011\n=Im[!(0)] (23)\nas a measure of the spin-pumping-enhanced Gilbert\ndamping, where !(0)and!(SP)are the frequencies of\nthe same system without and with spin pumping, respec-\ntively.Similar to the case of a single-layer ferromagnetic\ninsulator,31we \fnd that all higher transverse volume\nmodes exhibit an enhanced magnetization dissipation\nthat is twice that of the macrospin mode. The enhance-\nment of the Gilbert damping for the macrospin mode\n(n= 0) is\n\u0001\u000b\u0017;macro =\r~2g?\n2LMSe2\u001f\u0017\n\u001f; (24)\nand for the other modes, we obtain\n\u0001\u000b\u0017;n6=0= 2\u0001\u000b\u0017;macro: (25)\nCompared with single-FI systems, the additional fea-\nture of systems with two FIs is that the spin-pumping-\nenhanced Gilbert damping di\u000bers signi\fcantly between\nthe acoustic and optical modes via the mode-dependent\nratio\u001f\u0017=\u001f. This phenomenon has been explored both\nexperimentally and theoretically in Ref. 32 for the\nmacrospin modes n= 0 when there is no loss of spin\ntransfer between the FIs, lsf!1 . Our results repre-\nsented by Eqs. (24) and (25) are generalizations of these\nresults for the case of other transverse volume modes and\naccount for spin-memory loss. Furthermore, in Sec. IV,\nwe present the numerical results for the various spin-wave\nmodes when the in-plane momentum Qis \fnite. When\nthe NM is a perfect spin sink, there is no transfer of spins\nbetween the two FIs, and we recover the result for a sin-\ngle FIjN system with vanishing back \row, \u001f\u0017!\u001f.31\nNaturally, in this case, the FI jNjFI system acts as two\nindependent FIjN systems with respect to magnetiza-\ntion dissipation. The dynamical interlayer dipole cou-\npling is negligible in the considered limit of this section\n(QL\u001c1).\nIn the opposite regime, when the NM \flm is much thin-\nner than the spin-di\u000busion length and the spin conductiv-\nity of the NM is su\u000eciently large such that g?dN=\u001b\u001c1,\nthen\u001fA!0 and\u001fO!\u001f. This result implies that for\nthe optical mode, the damping is the same as for a sin-\ngle FI in contact with a perfect spin sink, even though\nthe spin-di\u000busion length is very large. The reason for\nthis phenomenon is that when the optical mode is ex-\ncited, the magnetizations of the two \flms oscillate out\nof phase such that one layer acts as a perfect spin sink\nfor the other layer. By contrast, there is no enhance-\nment of the Gilbert damping coe\u000ecient for the acoustic\nmode; when the \flm is very thin and the magnetizations\nof the two layers are in phase, there is no net spin \row or\nloss in the NM \flm and no spin-transfer-induced losses\nin the ferromagnets. Finally, when the NM is a poor con-\nductor despite exhibiting low spin-memory loss such that\ng?dN=\u001b\u001d(lsf=dN)\u001d1, then\u001f\u0017!0 because there is no\nexchange of spin information. For the macrospin modes\nin the absence of spin-memory loss, these results are in\nexact agreement with Ref. 32. Beyond these results, we\n\fnd that regardless of how much spin memory is lost, it\nis also the case that in trilayer systems, all higher trans-\nverse modes experience a doubling of the spin-pumping-\ninduced damping. Furthermore, these modes can still7\nbe classi\fed as optical and acoustic modes with di\u000berent\ndamping coe\u000ecients.\nB. Symmetric Films with EASA\nMagnetic surface anisotropy is important when the\nspin-orbit interaction at the interfaces is strong. In this\ncase, the excited mode with the lowest energy becomes\ninhomogeneous in the transverse direction. For a \fnite\nKS, the equations for the xandycomponents of the\nmagnetization in the boundary condition (15) di\u000ber, re-\nsulting in di\u000berent transverse wave vectors for the two\ncomponents, kxandky, respectively. Taking this situa-\ntion into account, we construct the ansatz\nmiQxy(\u0018) =\u0012\nXicos (kx;i\u0018\u0006kx;i(L+dN=2))\nYicos (ky;i\u0018\u0006ky;i(L+dN=2))\u0013\n;(26)\nwhich, when inserted into the boundary conditions of\nEqs. (15a) and (15b), yields\ni\u001f\u0017!\u0017+LKS\nAcos (2\u0012) =kxdtan (kxd);(27a)\ni\u001f\u0017!\u0017+LKS\nAcos2(\u0012) =kydtan (kyd);(27b)\nwhere\u0017continues to denote an acoustic (A) or optical\n(O) mode, \u0017= A;O. Depending on the sign of KSand\nthe angle\u0012, the resulting solutions kxandkycan be-\ncome complex numbers, which implies that the modes\nare evanescent. Let us consider the case of KS>0 and\nan in-plane magnetization ( \u0012=\u0019=2). Although kyis\nunchanged by the EASA, with LKS=A> 1\u001d\u001f\u0017!\u0017,kx\nis almost purely imaginary, \u0014=ik=KS=A\u0000i!\u0017\u001f\u0017, so\nthat\nmiQx(\u0018) =Xcosh(\u0014\u0018\u0006\u0014(d+dN=2)): (28)\nThe magnetization along the xdirection is exponentially\nlocalized at the FI jN surfaces. Following the same proce-\ndure as in Sec. III A for the KS= 0 case, we insert this\nsolution into the dispersion relation (19) and extract the\nrenormalization of the e\u000bective Gilbert damping:\n\u0001\u000bEASA\n\u0017 =\r~2g?\n2LMSe2\u001f\u0017\n\u001f1 +!H\n!M\u0002\n1 +2LKS\nA\u0003\n\u0000K2\nS\n2\u0019M2\nSA\n1 + 2!H\n!M\u0000K2s\n2\u0019M2\nSA:\n(29)\nIn the presence of EASA, the damping coe\u000ecient is a ten-\nsor; thus, the e\u000bective damping of Eq. (29) is an average,\nas de\fned in Eq. (23). This Gilbert damping enhance-\nment may become orders of magnitude larger than the\n\u0001\u000bmacro of Eq. (24). For thick \flms, \u0001 \u000bmacro\u0018L\u00001,\nwhereas \u0001\u000bEASA\n\u0017 reaches a constant value that is in-\nversely proportional to the localization length at the FI jN\ninterface. Note that for large EASA, the equilibrium\nmagnetization is no longer oriented along the external\n\feld, and Eq. (29) for \u0001 \u000bEASA\n\u0017 becomes invalid.C. Asymmetric FI Films\nLet us now consider an asymmetric system in which\nL16=L2. In this con\fguration, we will \frst consider\nKS= 0, but we will also comment on the case of a \f-\nniteKSat the end of the section. Because the analytical\nexpressions for the eigenfrequencies and damping coe\u000e-\ncients are lengthy, we focus on the most interesting case:\nthat in which the spin-relaxation rate is slow.\nAs in the case of the symmetric \flms, the dispersion\nrelation of Eq. (10) dictates that the wave numbers in the\ntwo layers must be the same. To satisfy the boundary\nequations (15), we construct the ansatz\nmiQxy(\u0018) =\u0012\nXicos (k\u0018\u0006k(L+dN=2))\nYicos (k\u0018\u0006k(L+dN=2))\u0013\n: (30)\nThe di\u000berence between this ansatz and the one for the\nsymmetric case represented by Eq. (26) is that the mag-\nnitudes of the amplitudes, XiandYi, of the two layers,\ni= 1;2, that appear in Eq. (30) is no longer expected to\nbe equal.\nWhen the two ferromagnets FI( L1) and FI(L2) are\ncompletely disconnected, the transverse wave vectors\nmust be equivalent to standing waves, qn;1=\u0019n=L 1and\nqm;2=\u0019m=L 2in the two \flms, respectively, where nand\nmmay be any integral numbers. Because spin pumping\nis weak, the eigenfrequencies of the coupled system are\nclose to the eigenfrequencies of the isolated FIs. This\n\fnding implies that the wave vector kof the coupled sys-\ntem is close to either qn;1orqm;2. The solutions of the\nlinearized equations of motion are then\nk=kn;1=qn;1+\u000ekn;1or (31a)\nk=km;2=qm;2+\u000ekm;2; (31b)\nwhere\u000ekn;1and\u000ekm;2are small corrections attributable\nto spin pumping and spin transfer, respectively. Here,\nthe indices 1 and 2 represent the di\u000berent modes rather\nthan the layers. However, one should still expect that\nmode 1(2) is predominantly localized in \flm 1(2). In\nthis manner, we map the solutions of the wave vectors in\nthe coupled system to the solutions of the wave vectors\nin the isolated FIs. Next, we will present solutions that\ncorrespond to the qn;1of Eq. (31a). The other family of\nsolutions, corresponding to qm;2, is determined by inter-\nchangingL1$L2and making the replacement n!m.\nInserting Eq. (31a) into the boundary conditions of\nEq. (15) and linearizing the resulting expression in the\nweak spin-pumping-induced coupling, we \fnd, for the\nmacrospin modes,\ni!~\u001fA,O\n1;macro = (L1\u000ek0;1)2; (32)\nwhere\n~\u001fA\n1;macro\u00191\n2dN\nlsf\u001b\ng?lsfL1\nL1+L2\u001f1; (33a)\n~\u001fO\n1;macro\u00191\n2L1+L2\nL2\u001f1: (33b)8\nHere,\u001f1=L1~2g?=4Ae2. Inserting this parameter into\nthe dispersion relation of Eq. (19), we obtain the follow-\ning damping renormalizations:\n\u0001\u000bA\nmacro =\r~2g?\n2MSe21\n2dN\nlsf\u001b\ng?lsf1\nL1+L2;(34a)\n\u0001\u000bO\nmacro =\r~2g?\n2MSe21\n2\u00121\nL1+1\nL2\u0013\n: (34b)\nThese two solutions correspond to an acoustic mode\nand an optical mode, respectively. The corresponding\neigenvectors are m1=m2for the acoustic mode and\nL1m1=\u0000L2m2for the optical mode. As in the sym-\nmetric case, the damping enhancement of the acoustic\nmode vanishes in the thin-NM limit. In this limit, the\nbehavior of the acoustic mode resembles that of a single\nFI of thickness L1+L2. It is the total thickness that\ndetermines the leading-order contribution of the damp-\ning renormalization. The optical mode, however, experi-\nences substantial damping enhancement. For this mode,\nthe damping renormalization is the average of two sepa-\nrate FIs that are in contact with a perfect spin sink. The\ncause of this result is as follows. When there is no spin-\nmemory loss in the NM, half of the spins that are pumped\nout from one side return and rectify half of the angular-\nmomentum loss attributable to spin pumping. Because\nthe magnetization precessions of the two \flms are com-\npletely out of phase, the other half of the spin current\ncauses a dissipative torque on the opposite layer. In ef-\nfect, spin pumping leads to a loss of angular momentum,\nand the net sum of the spin pumping across the NM and\nthe back \row is zero. The total dissipation is not a\u000bected\nby spin transfer, and thus, the result resembles a system\nin which the NM is a perfect spin sink.\nFor the higher excited transverse modes, there are two\nscenarios, which we treat separately. I. The allowed wave\nnumber for one layer matches a wave number for the\nother layer. Then, for some integer n > 0,qn;1=qm;2\nfor some integer m. In this case, we expect a coupling\nof the two layers. II. The allowed wave number for one\nlayer does not match any of the wave numbers for the\nother layer, and thus, for some integer n > 0, we have\nqn;16=qm;2for all integers m. We then expect that the\ntwo layers will not couple.\nI. In this case, we \fnd two solutions that correspond\nto acoustic and optical modes. These modes behave very\nmuch like the macrospin modes; however, as in the sym-\nmetric case, the damping renormalization is greater by a\nfactor of 2:\n\u0001\u000bA,O\nn6=0= 2\u0001\u000bA,O\nmacro;Case I: (35)\nThe eigenvectors of these coupled modes have the same\nform as for the macrospin modes, such that m1=m2\nandL1m1=\u0000L2m2for the acoustic and optical modes,\nrespectively.\nII. In this case, the two layers are completely decou-pled. To the leading order in dN=lsf, we \fnd\n\u0001\u000bn6=0=\r~2g?\n2L1MSe2;Case II; (36)\nfor all modes that correspond to excitations in FI1.\nThe damping renormalization is thus half that of the\nFI(L1)jN(lsf= 0) system.31This result can be explained\nby the zero loss of spin memory in the NM. Although half\nof the spins are lost to the static FI2, half of the spins\nreturn and rectify half of the dissipation attributable\nto spin pumping. The amplitudes of these modes are\nstrongly suppressed in FI2 (or FI1, upon the interchange\nof FI1$FI2), such thatjm2j=jm1j\u0018!\u001f2.\nFinally, let us discuss the case in which EASA is\npresent. In the limit KSLi=A\u001d1, the excitation en-\nergies of the surface modes are independent of the FI\nthicknesses. However, the surface modes do not behave\nlike the macrospin modes for the asymmetric stack. The\nexcitation volume of these modes is determined by the\ndecay length A=KSin accordance with Eq. (28). This\n\fnding is in contrast to the result for the macrospin\nmodes, where the excitation volume spans the entire FI.\nThus, the surface modes couple in the same manner as in\nthe symmetric case. With a good experimental control\nof surface anisotropy, the coupling of the surface modes\nis thus robust to thickness variations. The higher ex-\ncited transverse modes, in the presence of EASA, have\nthickness-dependent frequencies, which means that these\nmodes behave similarly to the n>0 modes in the KS= 0\ncase.\nIV. NUMERICAL RESULTS\nWhen the spin-wave wavelength becomes comparable\nto the \flm thickness, the dipolar \feld becomes a compli-\ncated function of the wavelength. We study the proper-\nties of the system in this regime by numerically solving\nthe linearized equations of motion (10) with the bound-\nary conditions (15). We use the method presented in\nRef. 31, which solves the spin-wave excitation spectrum\nfor an FIjN system, and extend this approach to the\npresent trilayer system. The physical parameters used\nin the numerical calculations are listed in Table I. We\ninvestigate two geometries: I. the BWMSW geometry, in\nwhich the spin wave propagates parallel to the external\n\feld, and II. the MSSW geometry, in which the spin wave\npropagates perpendicular to the external \feld.\nTo calculate the renormalization of the Gilbert damp-\ning, we perform one computation without spin pumping\nand one computation with spin pumping, in which the\nintrinsic Gilbert damping is excluded. Numerically, the\nrenormalization can then be determined by calculating\n\u0001\u000b=\u000bIm[!(SP)]\u000b=0=Im[!(0)], where!(0)is the eigenfre-\nquency obtained for the computation without spin pump-\ning and!(SP)is the frequency obtained for the compu-\ntation with spin pumping.319\nTABLE I: Physical parameters used in the numerical calcu-\nlations\nConstant Value Units\ng?a3:4\u00011015cm\u00002e2=h\n\u001bb5:4\u00011017s\u00001\n4\u0019MSc1750 G\nAc3:7\u000110\u00007erg=cm\nHint 0:58\u00014\u0019MS\n\u000bc3\u000110\u00004\nKS 0;d0:05 erg=cm2\na) Ref. [47], b) Ref. [48], c) Ref. [34]\nd) Reported to be in the range of 0 :1\u00000:01 erg=cm2in\nRef. [21]\nA. BVMSW\nFIG. 3: (Color online) FI(100nm) jN(50nm)jFI(101nm): a)\nSpin-pumping-enhanced Gilbert damping \u0001 \u000bas a function\nofQL1of the uniform modes and the n= 1 modes. The inset\npresents the corresponding dispersion relation. b) Relative\nphase and c) amplitude between the out-of-plane magnetiza-\ntions along xat the edges of FI1 jN and FI2jN. The apparent\ndiscontinuity in the green line in c) appears because the phase\nis de\fned on the interval \u0000\u0019to\u0019.\nLet us \frst discuss the BVMSW geometry. The cou-\npling of the uniform modes in the two \flms is robust;it is not sensitive to possible thickness asymmetries. In\ncontrast, at Q= 0, the sensitivity to the ratio between\nthe thickness and the rather weak dynamic coupling at-\ntributable to spin pumping implies that the coupling of\nthe higher transverse modes in the two bilayers is fragile.\nSmall asymmetries in the thicknesses destroy the cou-\npling. This e\u000bect can best be observed through the renor-\nmalization of the damping. However, we will demon-\nstrate that a \fnite wave number Qcan compensate for\nthis e\u000bect such that the higher transverse modes also\nbecome coupled. To explicitly demonstrate this result,\nwe numerically compute the real and imaginary parts\nof the eigenfrequencies of a slightly asymmetric system,\nFI(100nm)jN(50nm)jFI(101nm) with lsf= 350 nm. The\nasymmetry between the thicknesses of the ferromagnetic\ninsulators is only 1%. The surface anisotropy is consid-\nered to be small compared with the ratio Li=A, and we\nsetKS= 0.\nIn Fig. 3, the numerical results for the e\u000bective Gilbert\ndamping, the dispersion of the modes, and the relative\nphase and amplitude between the magnetizations in the\ntwo FIs are presented. As observed in the relative phase\nresults depicted in Fig. 3(c), the two uniform modes in\nwidely separated FIs split into an acoustic mode and\nan optical mode when the bilayers are coupled via spin\npumping and spin transfer. Figure 3(a) also demon-\nstrates that the acoustic mode has a very low renor-\nmalization of the Gilbert damping compared with the\noptical mode. Furthermore, there is no phase di\u000berence\nbetween the two modes with a transverse node ( n= 1) in\nFig. 3(a), which indicates that the modes are decoupled.\nThesen= 1 modes are strongly localized in one of the\ntwo \flms; see Fig. 3(b). For small QL1, Fig. 3(a) demon-\nstrates that these modes have approximately the same\nrenormalization as the optical mode, which is in agree-\nment with the analytical results. Because the magnetiza-\ntion in the layer with the smallest amplitude is only a re-\nsponse to the spin current from the other layer, the phase\ndi\u000berence is \u0019=2 (Fig. 3(b)). When Qincreases, the dipo-\nlar and exchange interactions become more signi\fcant.\nThe interlayer coupling is then no longer attributable\nonly to spin pumping but is also caused by the long-range\ndipole-dipole interaction. This additional contribution to\nthe coupling is su\u000ecient to synchronize the n= 1 modes.\nThe relative amplitude between the two layers then be-\ncomes closer to 1 (see Fig. 3(b)). Again, we obtain an\nacoustic mode and an optical n= 1 mode, which can be\nobserved from the phase di\u000berence between the two lay-\ners in Fig. 3(c). The spin-pumping-induced coupling only\noccurs as long as the e\u000bective spin-di\u000busion length ~lsfis\nlarge or on the order of dN. Once this is no longer the\ncase, the modes rapidly decouple, and the system reduces\nto two separate FI jN systems with a relatively weak in-\nterlayer dipole coupling. In the limit of large QL1, the\nexchange interaction becomes dominant. The energy of\nthe wave is then predominantly attributable to the mo-\nmentum in the longitudinal direction, and the dynamic\npart of the magnetization goes to zero at the FI jN inter-10\nfaces, causing the renormalization attributable to spin\npumping to vanish.31\nWe also note that the dispersion relation depicted in\nthe inset of Fig. 3(a) reveals that the acoustic mode (blue\nline) exhibits a dip in energy at lower QL1than does the\noptical mode (red line). We suggest that this feature\ncan be understood as follows: The shift in the position\nof the energy dip can be interpreted as an increase in\nthe e\u000bective FI thickness for the acoustic mode with re-\nspect to that for the optical mode. When ~lsfis larger\nthan the NM thickness, the uniform mode behaves as\nif the NM were absent and the two \flms were joined.\nThis result indicates that the dispersion relation for the\nacoustic mode exhibits frequency behavior as a function\nofQ~L=2, where the e\u000bective total thickness of the \flm is\n~L=L1+L2. The optical mode, however, \\sees\" the NM\nand thus behaves as if ~L=L1. Consequently, the dip in\nthe dispersion occurs at lower QL1for the acoustic mode\nthan for the optical mode.\nB. MSSW\nFinally, let us study the dynamic coupling of mag-\nnetostatic surface spin waves (MSSWs). We now con-\nsider a perfectly symmetric system, FI(1000 nm) jN(200\nnm)jFI(1000 nm), with lsf= 350 nm. For such thick\n\flms, surface anisotropies may play an important role.\nWe therefore discuss a case in which we include a surface\nanisotropy of KS= 0:05 erg=cm2. According to the an-\nalytical result presented in Eq. (28), the lowest-energy\nmodes with QL1\u001c1 are exponentially localized at the\nFIjN surfaces, with a decay length of A=KS\u0018200 nm.\nWe now compute the eigenfrequencies, !, as a function\nof the wave vector in the range 10\u00004< QL 1<103. In\nFig. 4(a), we present the real part of the frequency for\nthe six lowest-energy modes with a positive real part, and\nin Fig. 4(b), we present the corresponding renormaliza-\ntions of the Gilbert damping for the four lowest-energy\nmodes. The dispersion relations indicate that the mode\npairs that are degenerate at QL1\u001c1 rapidly split in\nenergy when QL1approaches 10\u00002. Strong anticrossings\ncan be observed between the n= 1 andn= 2 modes.\nSuch anticrossings are also present between the surface\nmode and the n= 1 mode; they are almost too strong to\nbe recognized as anticrossings. The enhanced damping\nrenormalizations exhibit very di\u000berent behavior for the\ndi\u000berent modes. We recognize the large-\u0001 \u000bmode of one\npair as the surface optical mode and the low-\u0001 \u000bmode\nas the volume n= 1 acoustic mode. Without EASA,\nthe anticrossings in Fig. 4(a) would become crossings.\nThe lowest-energy modes at QL1\u001c1 would then cut\nstraight through the other modes. In the case considered\nhere, this behavior is now observed only as steep lines at\nQL1\u00180:05 and atQL1\u00180:5.\nWhenQis increased, the e\u000bective spin-di\u000busion\nlength decreases (see Eq. (17)), which reduces the spin-\npumping-induced coupling between the modes at largeQ. WhenQL1\u0018100, the coupling becomes so weak\nthat the two FIs decouple. This phenomenon can be ob-\nserved from the behavior of \u0001 \u000bin Fig. 4(b), where the\ndamping of the acoustic modes become the same as for\nthe optical modes.\nFIG. 4: (Color online) FI(1000nm) jN(200nm)jFI(1000nm)\nlsf= 350 nm, KS= 0:05 erg=cm2: a) The dispersion rela-\ntion as a function of QL1for the six lowest positive-real-part\nmodes. b) The renormalization of the damping attributable\nto spin pumping for the four lowest modes with frequencies\nwith positive real parts as a function of QL1. At largeQL1,\nthe computation becomes increasingly demanding, and the\npoint density of the plot becomes sparse. We have therefore\nindividually marked the plotted points in this region.\nIn the MSSW geometry, an isolated FI has magneto-\nstatic waves that are localized near one of the two sur-\nfaces, depending on the direction of propagation with\nrespect to the internal \feld.34Asymmetries in the exci-\ntation volume are therefore also expected for the trilayer\nin this geometry. In Fig. 5, we present the eigenvectors\nof the surface modes as functions of the transverse co-\nordinate\u0018for increasing values of the wave vector Q.\nAtQL1= 0:5, the modes have already begun to ex-\nhibit some asymmetry. Note that the renormalization\nof the damping observed in Fig. 4(b) is approximately\none order of magnitude larger than the intrinsic Gilbert\ndamping for the optical mode and that the damping of\nany one mode may vary by several orders of magnitude\nas a function of QL1.31Therefore, these e\u000bects should\nbe experimentally observable. The greatest damping oc-\ncurs when the two layers are completely decoupled; see\nFigs. Fig. 4(b) and 5. Because the damping of the opti-\ncal mode is equivalent to that of a system with a perfect\nspin sink, one might expect that the greatest damping11\nFIG. 5: (Color online)FI(1000nm) jN(200nm)jFI(1000nm),\nlsf= 350nm, KS= 0:05 erg=cm2: a) and b) present the\nreal parts of the xcomponents of the out-of-equilibrium mag-\nnetization vectors for the acoustic and optical surface modes,\nrespectively, for several values of QL1. For values of QL1&1,\nthe modes decouple and become localized in one of the two\nlayers. For large values of QL1\u0018100, the two modes are\nstrongly localized at one of the two FI jN interfaces, which\ncorrespond to the peaks in the damping that are apparent in\nFig. 4(b).\nshould occur for this mode. However, the large localiza-\ntion, which is achieved only at large QL1, in combination\nwith the vanishing of the e\u000bective spin-di\u000busion length\nleads to damping that is much greater than that of the\nsynchronized optical mode.\nV. CONCLUSIONS\nWe investigated the dynamic coupling of spin-wave ex-\ncitations, which are present in single FI thin \flms, pri-\nmarily through spin pumping and spin transfer but also\nthrough the dynamic demagnetization \feld created when\ntwo FI thin \flms are in contact via an NM layer. Because\nof this coupling, the modes are split into acoustical and\noptical excitations. When the NM is thin compared with\nlsf, the renormalization of the Gilbert damping vanishes\nfor the acoustic modes, whereas for the optical modes,\nthe renormalization is equally as large as for a single-\nFIjN system in which the NM is a perfect spin sink. A\nspin current pumped by a travelling magnetic wave has a\nwavelength of equal magnitude, which leads to traversal\npaths across the NM that are longer than the thickness\nof the NM. Consequently, the spin-memory loss is greater\nfor short-wavelength spin currents. This phenomenonleads to an e\u000bective spin-di\u000busion length in the NM that\ndecreases for increasing values of Q. As a result, the dy-\nnamic coupling strength is reduced for short-wavelength\nspin waves. At some critical value of Q, the coupling be-\ncomes so weak that the acoustic- and optical-mode con-\n\fgurations are lost in favor of modes that are localized\nin one of the two FIs. At these values of Q, the inter-\nlayer dipole coupling is also dominated by the intralayer\nexchange coupling. For these high-wave-number modes,\nthe system behaves similar to two separate FI jN(lsf= 0)\nsystems.\nWhen the two \flms are of di\u000berent thicknesses, the\nexchange energies of the higher-order transverse n > 1\nmodes di\u000ber between the two layers. Because of the rel-\natively small coupling attributable to spin pumping, the\nsynchronization of these modes at QL1\u001c1 requires that\nthe FI thicknesses be very similar. A small asymmetry\nbreaks the synchronization; however, for larger QL1\u00181,\nthe modes can again become coupled through interlayer\ndipole interaction. This coupling arises in addition to\nthe spin-pumping- induced coupling. For even larger Q,\nthe e\u000bective spin-di\u000busion length becomes small, and the\ncoupling attributable to spin pumping vanishes. The rel-\natively small dipole coupling alone is not su\u000ecient to\ncouple the modes when there is a \fnite di\u000berence in \flm\nthickness , and the synchronization breaks down.\nDepending on the quality of the interface between the\nFIs and the strength of the spin-orbit coupling in the\nNM , additional e\u000bective surface \felds may be present\nbecause of surface anisotropy energies. For the EASA\ncase, the lowest-energy modes are localized at the FI jN\nsurfaces. These modes couple in the same manner as the\nmacrospin modes. For \flms that are much thicker than\nthe decay length A=KS, the energies of the surface modes\ndo not depend on the \flm thickness. Consequently, the\ncoupling of these modes is independent of the thickness\nof the two FIs. Similar to the simpler FI jN system, the\ndamping enhancement may attain values as high as an or-\nder of magnitude larger than the intrinsic Gilbert damp-\ning. However, in the trilayer system, the presence of both\nacoustic and optical modes results in large variations in\nthe e\u000bective damping within the same physical sample.\nBecause of this wide range of e\u000bective damping, which\nspans a di\u000berence in \u0001 \u000bof several orders of magnitude\nas a function of Q, we suggest that trilayer modes should\nbe measurable in an experimental setting.\nWith more complicated FI structures in mind, we be-\nlieve that this work may serve as a guide for experimen-\ntalists. The large variations in e\u000bective damping for dif-\nferent modes make the magnetic properties of the system\ndetectable both with and without EASA. For spin waves,\ndipole-dipole interactions assist spin pumping in inter-\nlayer synchronization, which may facilitate the design of\nfuture spintronic devices.12\nAcknowledgments\nWe acknowledge support from InSpin 612759 and the\nResearch Council of Norway, project number 216700.\n\u0003Electronic address: hans.skarsvag@ntnu.no\n1C. Kittel, Phys. Rev. 73, 155 (1948).\n2R.W. Damon, J.R. Eshbach, J. Phys. Chem. Solids 19, 308\n(1961).\n3V. Cherepanov, I. Kolokolov and V. L'vov, Phys. Rep. 229,\n81 (1993).\n4A. Brataas, A. D. Kent, and H. Ohno, Nature Mat., 373\n(2012).\n5G. Binasch, P. Gr unberg, F. Saurenbach, and W. Zinn,\nPhys. Rev. B 394828 (1989).\n6L. Berger, Phys. Rev. B 54, 9353 (1996).\n7J.C. Slonczewski, J. Magn. Magn. Mater. 159, L1 (1996).\n8Y. Kajiwara, K. Harii, S. Takahashi, J. Ohe, K. Uchida,\nM. Mizuguchi, H. Umezawa, H. Kawai, K. Ando, K.\nTakanashi, S. Maekawa, and E. Saito, Nature (London)\n464, 262 (2010).\n9C. W. Sandweg, Y. Kajiwara, K. Ando, E. Saitoh, and B.\nHillebrands, Appl. Phys. Lett. 97, 252504 (2010).\n10C. W. Sandweg, Y. Kajiwara, A. V. Chumak, A. A. Serga,\nV. I. Vasyuchka, M. B. Jung\reisch, E. Saitoh, and B. Hille-\nbrands, Phys. Rev. Lett. 106, 216601 (2011).\n11B. Heinrich, C. Burrowes, E. Montoya, B. Kardasz, E. Girt,\nY.-Y. Song, Y. Sun, and M. Wu, Phys. Rev. Lett. 107,\n066604 (2011).\n12L. H. Vilela-Leao, C. Salvador, A. Azevedo, S. M. Rezende,\nAppl. Phys. Lett. 99, 102505 (2011).\n13C. Burrowes, B. Heinrich, B. Kardasz, E. A. Montoya, E.\nGirt, Yiyan Sun, Young-Yeal Song, and Mingzhong Wu,\nAppl. Phys. Lett. 100, 092403 (2012).\n14S. M. Rezende, R. L. Rodr\u0013 \u0010guez-Su\u0013 arez, M. M. Soares, L.\nH. Vilela-Le~ ao, D. Ley Dom\u0013 \u0010nguez, and A. Azevedo, Appl.\nPhys. Lett. 102, 012402 (2013).\n15P. Pirro, T. Br acher, A. V. Chumak, B. L agel, C. Dubs,\nO. Surzhenko, P. G ornert, B. Leven and B. Hillebrands,\nAppl. Phys. Lett. 104, 012402 (2014).\n16H. Nakayama, M. Althammer, Y.-T. Chen, K. Uchida, Y.\nKajiwara, D. Kikuchi, T. Ohtani, S. Gepr ags, M. Opel, S.\nTakahashi, R. Gross, G. E. W. Bauer, S. T. B. Goennen-\nwein and E. Saitoh, Phys. Rev. Lett. 110, 206601 (2013).\n17C. Hahn, G. de Loubens, O. Klein, M. Viret, V. V. Naletov,\nand J. Ben Youssef, Phys. Rev. B 87, 174417 (2013).\n18N. Vlietstra, J. Shan, V. Castel, B. J. van Wees, and J.\nBen Youssef, Phys. Rev. B 87, 184421 (2013).\n19A. Brataas, Physics 6, 56 (2013).\n20A. Hamadeh, O. d'Allivy Kelly, C. Hahn, H. Meley, R.\nBernard, A.H. Molpeceres, V. V. Naletov, M. Viret, A.\nAnane, V. Cros, S. O. Demokritov, J. L. Prieto, M. Mu~ noz,\nG. de Loubens and O. Klein, arXiv:1405.7415v1.\n21J. Xiao and G. E. W. Bauer, Phys. Rev. Lett. 108, 217204\n(2012).\n22T. Schneider, A. A. Serga, B. Leven, B. Hillebrands, R. L.\nStamps and M. P. Kostylev, Appl. Phys. Lett. 92, 022505\n(2008).\n23Y.-T. Chen, S. Takahashi, H. Nakayama, M. Althammer,\nS. T. B. Goennenwein, E. Saitoh and G. E. W. Bauer,Phys. Rev. B 87, 144411 (2013).\n24Y. M. Lu, J. W. Cai, S. Y. Huang, D. Qu, B. F. Miao, and\nC. L. Chien, Phys. Rev. B 87, 220409 (2013).\n25V. V. Kruglyak, S. O. Demokritov, and D. Grundler, J.\nPhys. D. Appl. Phys. 43, 264001 (2010).\n26R. Urban, G. Woltersdorf, and B. Heinrich, Phys. Rev.\nLett. 87, 217204 (2001).\n27S. Mizukami and Y. Ando and T. Miyazaki, J. Mag. Mag.\nMater. 239, 42 (2002).\n28Y. Tserkovnyak, A. Brataas, and G. E. W. Bauer, Phys.\nRev. Lett. 88, 117601 (2002).\n29A. Brataas, Y. Tserkovnyak, G. E. W. Bauer, and B. I.\nHalperin, Phys. Rev. B 66, 060404 (2002).\n30Y. Tserkovnyak, A. Brataas, G. E. W. Bauer, and B. I.\nHalperin, Rev. Mod. Phys. 77, 1375 (2005).\n31A. Kapelrud and A. Brataas, Phys. Rev. Lett. 111, 097602\n(2013).\n32B. Heinrich, Y. Tserkovnyak, G. Woltersdorf, A. Brataas,\nR. Urban and G. E. W. Bauer, Phys. Rev. Lett. 90, 187601\n(2003).\n33B. A. Kalinikos and A. N. Slavin, J. Phys. C 19, 7013\n(1986).\n34A. A. Serga, A. V. Chumak and B. Hillebrands, J. Phys.\nD: App. Phys. 43, 264002 (2010).\n35M. Vohl, J. Barna\u0013 s and P. Gr unberg, Phys. Rev. B 39,\n12003 (1989).\n36B. Heinrich, private communication.\n37A. Brataas, Yu. V. Nazarov, and G. E. W. Bauer, Phys.\nRev. Lett. 84, 2481 (2000).\n38M. Althammer, S. Meyer, H. Nakayama, M. Schreier, S.\nAltmannshofer, M. Weiler, H. Huebl, S. Gepr ags, M. Opel,\nR. Gross, D. Meier, C. Klewe, T. Kuschel, J.-M. Schmal-\nhorst, G. Reiss, L. Shen, A. Gupta, Y.-T. Chen, G. E. W.\nBauer, E. Saitoh and S. T. B. Goennenwein, Phys. Rev. B\n87, 224401 (2013).\n39H. Skarsv\u0017 ag, G. E. W. Bauer, and A. Brataas,\narXiv:1405.2267v1.\n40J. Xiao, G. E. W. Bauer, K.-C. Uchida, E. Saitoh, and S.\nMaekawa, Phys. Rev. B 81, 214418 (2010).\n41J. C. Slonczewski, Phys. Rev. B 82, 054403 (2010).\n42J. Xingtao, L. Kai, K. Xia, and G. E. W. Bauer, EPL 96,\n17005 (2011).\n43A. Kapelrud and A. Brataas, (unpublished).\n44Y. Zhou, H.J. Jiao, Y.T. Chen, G. E. W. Bauer, and J.\nXiao, Phys. Rev. B 88, 184403 (2013).\n45Z. Q. Qiu , J. Pearson and S. D. Bader Phys. Rev. B 46,\n8659 (1992).\n46J.-M.L. Beaujour, W. Chen, A. D. Kent and J. Z. Sun, J.\nAppl. Phys. 99, 08N503 (2006).\n47M. B. Jung\reisch, V. Lauer, R. Neb, A. V. Chumak and\nB. Hillebrands, Appl. Phys. Lett. 103, 022411 (2013).\n48D. Giancoli, \"25. Electric Currents and Resistance\". In Jo-\ncelyn Phillips. Physics for Scientists and Engineers with\nModern Physics (4th ed.), (2009) [1984].13\nAppendix A: Dipole Tensor\nThe dipole tensor in the \u0010\u0011\u0018coordinate system, ^G(\u0018)\nfrom Eq. (9) can be rotated by the xyzcoordinate system\nwith the rotation matrix\nR=0\n@s\u0012\u0000c\u0012s\u0012\u0000c\u0012c\u001e\n0c\u001e\u0000s\u001e\nc\u0012s\u0012s\u001es\u0012c\u001e1\nA; (A1)where we have introduced the shorthand notation s\u0012\u0011\nsin\u0012,c\u0012\u0011cos\u0012and so on. We then get that\n(*\n^Gxyz=R^GRT\n=0\n@s2\n\u0012G\u0018\u0018\u0000c\u001es2\u0012G\u0018\u0010+c2\n\u0012c2\n\u001eG\u0010\u0010\u0000s\u001es\u0012G\u0018\u0010+s\u001ec\u001ec\u0012G\u0010\u0010s\u0012c\u0012G\u0018\u0018\u0000s\u0012c\u0012c2\n\u001eG\u0010\u0010+c\u001e(s2\n\u0012\u0000c2\n\u0012)G\u0018\u0010\n\u0000s\u001es\u0012G\u0018\u0010+s\u001ec\u001ec\u0012G\u0010\u0010 s2\n\u001eG\u0010\u0010 \u0000s\u001ec\u0012G\u0018\u0010+s\u001es\u0012c\u001eG\u0010\u0010\ns\u0012c\u0012G\u0018\u0018\u0000s\u0012c\u0012c2\n\u001eG\u0010\u0010+c\u001e(s2\n\u0012\u0000c2\n\u0012)G\u0018\u0010\u0000s\u001ec\u0012G\u0018\u0010+s\u001es\u0012c\u001eG\u0010\u0010c2\n\u0012G\u0018\u0018+s2\u0012c\u001eG\u0018\u0010+c2\n\u001es2\n\u0012G\u0010\u00101\nA:\n(A2)\nBecause we work in the linear respons regime the equilibrium magnetization should be orthogonal to the dynamic\ndeviation, mi\u0001^z= 0, it is therefor su\u000ecient to only keep the xypart of ^Gxyz. We then \fnd\n^Gxy=\u0012s2\n\u0012G\u0018\u0018\u0000c\u001es2\u0012G\u0018\u0010+c2\n\u0012c2\n\u001eG\u0010\u0010\u0000s\u001es\u0012G\u0018\u0010+s\u001ec\u001ec\u0012G\u0010\u0010\n\u0000s\u001es\u0012G\u0018\u0010+s\u001ec\u001ec\u0012G\u0010\u0010 s2\n\u001eG\u0010\u0010\u0013\n: (A3)\nAppendix B: Spin Accumulation\nThe functions \u0000 1(\u0018) and \u0000 2(\u0018) are taken directly from\nRef.39, and modi\fed to cover the more complicated mag-\nnetic texture model. We then have\n\u00001(\u0018)\u0011cosh\u0010\n\u0018=~lsf\u0011\ncosh\u0010\n\u0018=~lsf\u0011\n+\u001bsinh\u0010\n\u0018=~lsf\u0011\n=2g?~lsf;\n\u00002(\u0018)\u0011sinh\u0010\n\u0018=~lsf\u0011\nsinh\u0010\n\u0018=~lsf\u0011\n+\u001bcosh\u0010\n\u0018=~lsf\u0011\n=2g?~lsf:(B1)\nForQlsf\u001d1 the e\u000bective spin di\u000busion length becomes\nshort, \u0000 1!1 and \u0000 2!0 at the FIjN interfaces.\nAppendix C: E\u000bective spin di\u000busion length\nThe di\u000busion in the NM reads\n@t\u0016S=Dr2\u0016S\u00001\n\u001csf\u0016S; (C1)whereDis the di\u000busion constant and \u001csfis the spin \rip\nrelaxation time. We assume that the FMR frequency is\nmuch smaller than the electron traversal time, D=d2\nN, and\nthe spin-\rip relaxation rate, 1 =\u001csf.39This means the LHS\nof Eq. (C1) can be disregarded. In linear response the\nspin accumulation, which is a direct consequence of spin\npumping, must be proportional to the rate of change of\nmagnetization at the FI jN interfaces. We do the same\nFourier transform, as for the magnetization, so that \u0016\u0018\nexpfi(!t\u0000Q\u0010)g. The spin di\u000busion equation then takes\nthe form\n@2\n\u0018\u0016S=\u0012\nQ2+1\nD\u001csf\u0013\n\u0016S: (C2)\nThe spin di\u000busion length is then lsf=pD\u001csf, and\nby introducing the e\u000bective spin di\u000busion length ~lsf=\nlsf=q\n1 + (Qlsf)2one gets\n@2\n\u0018\u0016S=1\n~l2\nsf\u0016S: (C3)"    },    {        "title": "1408.0341v1.Tunnel_magnetoresistance_and_spin_transfer_torque_switching_in_polycrystalline_Co2FeAl_full_Heusler_alloy_magnetic_tunnel_junctions_on_Si_SiO2_amorphous_substrates.pdf",        "content": "1 \n Tunnel magnetoresistance and spin -transfer -torque switching in \npolycrystalline  Co2FeAl full -Heusler  alloy magnetic tunnel junctions  on \nSi/SiO 2 amorphous substrate s \nZhenchao Wen, Hiroaki  Sukegawa, Shinya Kasai, Koichiro Inomata , and Seiji Mitani  \nNational Institute for Materials Science (NIMS), 1 -2-1 Sengen, Tsukuba 305 -0047, Japan  \n \nAbstract:  \nWe studied polycrystalline B2-type Co2FeAl (CFA)  full-Heusler alloy  based magnetic \ntunnel junctions ( MTJs ) fabricated on a Si/SiO 2 amorphous substrate . Polycrystalline  CFA \nfilms with a (001)  orientation, a high  B2 ordering, and  a flat surface  were achieved using a \nMgO buffer layer . A tunnel magnetoresistance (TMR) ratio up to  175% was obtained  for an \nMTJ with a  CFA/MgO/CoFe  structure on a 7.5 -nm-thick  MgO buffer . Spin-transfer  torque \ninduced magnetization  switching  was achieved in the MTJs with a 2-nm-thick \npolycrystalline  CFA film  as a switching  layer. Using a thermal activation model , the \nintrinsic  critical current  density  (Jc0) was determined  to be 8.2 × 106 A/cm2, which is lower  \nthan 2.9 × 107 A/cm2, the value  for epitaxial CFA -MTJs  [Appl. Phys. Lett.  100, 182403 \n(2012) ]. We found that t he Gilbert damping constant  () evaluated using  ferromagnetic  \nresonance measurements  for the polycrystalline CFA film was ~0.015 and was almost \nindependent of the CFA thickness (2~18 nm) . The low Jc0 for the polycrystalline MTJ was \nmainly attributed to the low  of the CFA layer compared with the  value in the  epitaxial \none (~0.04).   \n  \n \n \n  2 \n I. INTRODUCTION  \nHalf-metallic ferromagnets (HMF s) draw  great interest because of the perfect  spin \npolarization  of conduction electrons at the Fermi level , which is considered to enhanc e the  \nspin-dependent transport efficiency  of high-performance spintronic devices . [1–3] Cobalt-\nbased full -Heusler alloys with the chemical formula Co 2YZ (where Y is a transition metal \nand Z is a main group element) , are extensively studied as a type of HMF s owing  to their \nhigh Curie  temperature  of approximately 1000 K , high spin polarizat ion, and low damping \nconstant . [4, 5]  They exhibit great  potential for application s in spintronic s, including  \ncurrent -perpendicular -to-plane giant magnetoresistance ( CPP-GMR ) read head s, [6, 7]  \nmagnetoresistive random access memories (MRAM s), [8] and spin transistors such as  spin-\nfunctional metal -oxide -semiconductor field -effect transistor s (spin -MOSFET s). [9, 10]  In \nparticular , magnetic tunnel junctions (MTJs) with  Co-based full -Heusler alloy  electrodes  \nhave been shown tremendous ly increasing tunnel magnetoresistance ( TMR ) ratios during \nthe last decade  since Inomata et al.  [11] demonstrated a TMR ratio of 16% using \nCo2Cr0.6Fe0.4Al/AlO x/CoFe MTJs  at room temperature (RT) . [12–21] Recently , a \nremarkable  TMR ratio of approximately  2000 % at 4.2 K  (354% at RT ) was achieved  using  \nepitaxial Co2MnSi /MgO /Co2MnSi (001)  MTJs, demonstrating the half -metallicity of Co -\nbased Heusler alloys and strong 1 coherent tunneling effect in the MgO /Heusler MTJ s \n[21].  \nThe Co2FeAl  (CFA)  alloy  is of particular  interest because of  its high spin polarization (a \nhalf-metallic electronic structure) [22] and low effective  damping  constant () ~ 0.001  [23], \nwhich are benefi cial for enhancing the TMR ratio and lowering the magnetization 3 \n switching  current of spin-transfer torque (STT) . CFA fil ms prepared using sputtering \ndeposition generally have a disordered B2 structure  (swapping between Y and Z sites) rather \nthan an ordered L21 structure  owing  to the thermodynamic  stability of CFA  [24]. \nNevertheless , the spin polarization  calculated for the L21 structure is conserved even for the \nB2 structure  [25]. Importantly , a CFA film with a ( 001) orientation  has a large in-plane \nlattice spacing ( d(200)/2 = d(110) = 0.203 nm) compared with other half -metallic Heusler \nalloy s such as Co2FeAl 0.5Si0.5 (d(110) = 0. 201 nm) and Co2MnSi  (d(110) = 0. 198 nm). \nTherefore, a nearly perfect CFA/MgO (001) heterostructure is easily achieved by the \nmagnetron sputtering method , and this is  favorable  for enhancing  the coherent tunneling \neffect  [13, 18]. Recently , TMR  ratios as high as  360% at RT (785% at 10 K) were \ndemonstrated in  epitaxial CFA-based MTJs with a sputter -deposited  MgO barrier  [17–19]. \nThe large  TMR ratio originated  from the high spin polarization of the CFA layer and the \nstrong contribution of the coherent tunneling effect through 1 Bloch states in the CFA and \nthe MgO barrier. Moreover, (001) -textured CFA films can be grown on MgO -buffered \nSi/SiO 2 amorphous substrate s, and a relatively large  TMR ratio of 166% at RT ( 252% at 48 \nK) was achieved in  a (001) -textured CFA/MgO/CoFe MTJ  [20]. Such polycrystalline  full-\nHeusler MTJs on amorphous substrate s are desired  because of  their compatibility with  \npractical industrial  applications of full -Heusler spintronic devices, while s ingle -crystal \nMgO(001) substrates  have a limit ed scope of application.  \nFurthermore, STT -induced magnetization switching  (STT switching) , a key techn ology  \nfor writing information in spintronics devices , was realized using  MTJs  with a n epitaxial  \nCFA ultrathin (~1.5 nm)  layer  as a switching  (free) layer  [26]. However, a large  critical  4 \n switching current density  (Jc0) of 2.9 × 107 A/cm2 was observed  owing  to the enhancement \nof , which was ~0.04,  of the epitaxial CFA film. In addition , STT switching can be \ndisturbed  by the stabilization of intermediate  magnetic states possibly because of  the \npresence of in -plane magnetocrystalline anisotropy , which is generally seen in epitaxial \nmagnetic films (e.g. , 4-fold anisotropy  for cubic  (001) films)  [27]. Therefore, reducing the \nundesirable  magnetic anisotropy using polycrystalline Heusler alloy  films is effective for \nhighly efficient STT switching . \nIn this work, we systematically  studied MTJs with (001) -textur ed polycrystalline CFA \nfilms  on Si/SiO 2 amorphous substrate s. A MgO buffer was introduced  in the MTJs for \nachiev ing (001)  texture with B2-ordering structure  of CFA layers  on the amorphous  \nsubstrates . The (001) -texture, B2 order, and surface morphology  of the polycrystalline  CFA \nfilms and the TMR effect in the entire  MTJ stacks were characterized  for varying MgO \nbuffer  thickness . Also, the MgO -barrier  thickness and resistance -area product ( RA) \ndependence of the TMR ratios were  investigated  for the polycrystalline CFA -MTJs.  \nFurthermore,  STT switching  was examined  in low -RA MTJs with a thin polycrystalline  \nCFA film (2.0 nm)  as a free layer.  Jc0 of 8.2  × 106 A/cm2 was demonstrated by a thermal \nactivation model for switching current ; this is far lower than the value for epitaxial CFA -\nMTJs  [26]. The  values for the polycrystalline  CFA films , obtained  using  a waveguide -\nbased ferromagnetic resonance method , were almost constant against  the CFA thickness , \nand a relatively low  of ~0.015 was demonstrated. We attributed  the reduction in the Jc0 of \nthe STT switching  to the reduced   of the polycrystalline CFA films .  5 \n  \nII. EXPERIMENT  \nAll multilayer stack s were deposited on thermally oxidized Si/SiO 2 amorphous \nsubstrate s at RT  using an ultra-high vacuum magnetron sputtering system with a base \npressure lower than 4  10-7 Pa. MgO  layers were deposited from a sintered MgO target by \nRF sputtering  with an RF power density of 2.19 W/cm2 and an Ar pressure of 10 mTorr . \nCFA layers were deposited from  a stoichiometric Co 50Fe25Al25 (at.%) alloy target  using  DC \npower . The structur al properties  and surface morphology of CFA film s on MgO buffer s \nwere characterized  by out-of-plane  (2θ-ω scan) X-ray diffraction (XRD) with Cu Kα \nradiation  (λ = 0.15418 nm)  and atomic force microscop y (AFM), respectively. MTJ stack s \nwith the structure of CFA/MgO /Co75Fe25/IrMn/Ru (unit: nm)  were deposited on MgO -\nbuffered thermally oxidized Si /SiO 2 amorphous substrate s and patterned into junctions with \nactive area of 5  10 m2 by conventional lithography  methods with Ar ion milling . For \nSTT switching, spin-valve  MTJ s with the structure of MgO  (7.5)/ Cr (40)/CFA  (2)/MgO  \n(0.6–0.8)/Co 75Fe25 (5)/Ru  (0.8)/Co 75Fe25 (5)/IrMn  (15)/Ta  (5)/Ru  (10) (unit: nm)  were \nprepared on the amorphous substrates and nanofabricated into 100 -nm-scale d ellipse s. The \nactual  areas of the MTJ nano -pillars were obtained according to  the ratio of the  RA to the  \njunction resistance ; the RA was characteriz ed by current-in-plane tunneling (CIPT)  \nmeasurement  [28] before patterning . The MTJ  stack s were post-annealed in a vacuum  \nfurnace for 30 minutes  under  a magnetic field of 5 kOe. The magneto -transport  properties \nwere measured  using a DC 2- or 4- probe method . The magnetic damping constant  of the \npolycrystalline CFA film was measured by waveguide -based ferromagnetic resonance  6 \n (FMR) . The films were patterned into rectangular shape elements of 600  × 20 μm2 using \nUV lithography  together with  Ar ion milling, and then coplanar waveguides made of Au \nwere fabricated on them. The FMR signal obtained as change of the real part of S21 signal , \nwas determined  using  a network analyzer. An external magnetic field along the longitudinal \naxis was varied from 0 to  1.9 kOe, while  the excitation power was fixed as 0 dBm. All \nmeasurements were performed at RT. \nIII. RESULTS  \nA. Effect of MgO buffer on polycrystalline CFA films and TMR  \nBefore MTJ multilayer  films  were grown , MgO buffer w as deposited  on a Si/SiO 2 \namorphous  substrate  in order to  establish the (001) -texture of polycrystalline CFA films  by \ntaking advantage of  the unique (001) -texture property of MgO  layer s on an amorphous \nsubstrate . CFA films were subsequently  grown on the  MgO buffer layer, and the structural \nproperties of the CFA films depend ing on the MgO buffer thickness w ere investigated  by \nXRD . The out-of-plane XRD patterns  of 30-nm-thick CFA films are shown  in Fig.  1(a); the \nCFA films were annealed at Ta = 400  °C, and the thickness es of the MgO  buffers ( tMgO) are \n2.5, 5.0, 7.5, and 10.0 nm . In addition to the peaks (denoted  by “s”) from Si/SiO 2 substrates, \nMgO(002), CFA(002) , and (004) peaks were observed  along with the absence of other \noriented peaks, demonstrating  the (001) -texture established  in the  stacks . Figure 1(b) shows  \nthe tMgO dependence of the  integrated  intensity of CFA(002) peaks for as-deposited CFA \nfilms , and CFA films annealed at  400 °C and 480  °C. With  increasing tMgO, the intensity of \nthe peaks initially  increases for all samples  owing to the improved (001) -texture of the \nMgO buffer layer , reaching  a maximum  at tMgO = 7.5 nm. The reduction in intensity at tMgO 7 \n = 10 nm  may be caused by  the degraded surface morphology  of the MgO buffer layer. In \naddition, the XRD intensity increases w ith increasing annealing temperature, which \nindicates  that the increasing temperature  improve s the B2 order and (001) -texture  of the \nCFA films . In the XRD  2θ-ω scan with the diffraction vector along CFA[111], (111) \nreflection was not detected , indicating  that the CFA film s have a B2-ordering structure  with \nswapping between Fe and Al atoms while Co atoms occupy the regular sites.  The degree of \nB2 order ing was estimated according to  the ratio of the integrated  intensity of  the CFA(002) \nand (004) peaks. The peaks were fitted by Voigt profiles , and t he ratio of their integrated  \nintensit ies, i.e., the ratio of  I(002) to I(004), is shown in Fig. 1(c).  The maximum \nI(002)/I(004) value was obtained  at tMgO = 7.5 nm for all  of the samples : as-deposit ed and  \nanneal ed at 400  °C or 480 °C. This value is comparatively large for  CFA film s annealed at \n480 °C, indicating  the improvement of  the B2 ordering and  the mosaicity  of the CFA films  \ndue to  annealing at high temperature . The degree of B2 order ing, SB2, can be  evaluated  \nusing  the ratio s according to the following equation : [29] \n      √                   \n                   ,                   (1) \nwhere  [I(002)/ I(004)] exp. is the ratio of the integrated intensity  of the (002) peak to that of \nthe (004) peak  as determined by experiments, and [I(002)/ I(004)] cal. is the ideal ratio of the \ntwo peaks. For tMgO = 7.5 nm, t he ordering parameter SB2 is calculated to be 0.89, 0.95, and \n0.98 for CFA film s as deposited , anneal ed at 400  °C, and 480  °C, respectively . The results \ndemonstrate  a high B2 order  and an excellent  (001) -texture were established in the  \npolycrystalline  CFA film s on MgO -buffered Si/SiO 2 amorphous  substrates .  8 \n For stacking  MTJ multilayers with the polycrystalline CFA films , the tMgO dependence \nof the surface morphology  of the CFA films  was investigated . Figure 2 shows the average \nsurface roughness (Ra) and peak -to-valley ( P-V) value  as a function of tMgO for 30 -nm-thick \nCFA films annealed at 400  °C and 480  °C, respectively. For the samples annealed at \n400 °C, flat surface s with Ra ~ 0.1 nm and P-V ranging from  1.3 to 1.5 nm were  observed \nfor all values of tMgO. The inset of Fig.  2 shows an example of AFM image s of the samples \n(annealed at 400 °C on a 7.5 -nm-thick MgO buffer ). The results indicate  the feasib ility of \nstacking MTJs with a thin MgO ba rrier. In addition , the samples annealed at 480 °C with \nhigher Ra and P-V values were observed as well as a large tMgO dependence.  \nThe whole MTJ  stacks  with the structure of MgO  (tMgO)/CFA  (30)/MgO  (tbarr)/Co75Fe25 \n(5)/IrMn  (15)/Ru  (10) (unit: nm)  were then fabricated  on Si /SiO 2 substrate s with var ying \nMgO buffer thickness  tMgO (2.510.0 nm) and MgO barrier thickness tbarr (1.5, 1.8, and 2.0 \nnm). The stacks were annealed at 370 °C in the presence of  a magnetic field  of 5 kOe . \nFigure 3 shows TMR  ratios as a function of tMgO for the polycrystalline  CFA -MTJs  \nmeasured a t RT  using CIPT . The 30-nm-thick CFA films were post-annealed at Ta = \n400 °C and 480  °C in order to improve  the B2 ordering . The TMR ratios obtained in MTJs \nwith Ta = 400  °C were higher than those  of MTJs with Ta = 480  °C, which can be attributed  \nto a better  CFA/MgO -barrier  interface due to the flat CFA  surface  annealed at  400 °C, \nalthough a  higher degree of B2 ordering was observed for Ta = 480  °C, as shown  in Fig s. 1 \nand 2 . With increas ing tMgO, the TMR ratio increases and is nearly saturated  at tMgO > 5 nm,  \nindicating  that high-quality CFA films with B2 order and (001) -texture were established \nwith the more than  5-nm-thick MgO buffer layer, which is consistent  with the XRD 9 \n analyses . A slight  reduction in the  TMR ratio was observed at tMgO = 10 nm, which could  \nbe caused by the reduction in the degree of (001) -orientation of the MgO buffer . The MTJs \nwith a 1.8-nm-thick MgO barrier  exhibit  larger  TMR ratio s than those with 1.5- and 2.0 -\nnm-thick MgO barri ers, which could be due to the  plastic relaxation of the MgO barrier  [30] \nand/or the oscillatory behavior of the TMR ratio as a function of MgO thickness  [18].  \nB. MgO barrier thickness and RA dependences of  TMR  \nIn order to realize  STT switching in the polycrystalline  CFA -MTJs, it is expected that  \nintroduction of a conductive underlayer, reduction in the free -layer thickness , and control of \nthe RA of the barrier layer  are required . The MgO barrier thickness and RA dependence s of \nthe TMR  ratio w ere investigated  using spin-valve  MTJs with the structure of MgO  buffer \n(7.5)/Cr  (40)/CFA  (30)/MgO  (tbarr: 1.2-2.0)/Co 75Fe25 (5)/IrMn  (15)/Ta  (5)/Ru  (10) (unit: nm)  \non a Si/SiO 2 substrate . A Cr under layer was selected  as the conductive electrode because  Cr \nhas a very small l attice mismatch  with CFA  (~0.6%)  and can further facilitate  the ordering \nstructure of full -Heusler alloys  [13]. The Cr layer s for the samples  as deposited, annealed at \n400 °C, and 600 °C were prepared  on the 7.5-nm-thick MgO buffer  for the MTJ stacks . The \nentire stacks were annealed at 370 °C in the presence of a magnetic field  of 5000 Oe , and \nthen their TMR ratios and RA values were characterized using  CIPT  measurement .  \nFigure 4 (a) shows t he dependence of the TMR ratios  on the nominal thickness of the \nMgO  barrier ( tbarr) for the MTJs  with different Cr annealing  conditions . For the sample s that \nwere as-deposited and annealed at 400 °C, the TMR ratio increases with tbarr, and TMR \nratios greater than 100% were achieved for the whole range of tbarr. This means  that the 10 \n (001) -texture and  B2 order ing of CFA films can be maintained on the MgO/Cr buffer layers.  \nWe obtained the largest  TMR ratio of 175% for the 400 °C annealed sample  with tbarr = \n1.95 nm ; this TMR ra tio is higher than 16 6%, which was  observed in the MTJ without the \nCr buffer , which  indicates that the Cr buffer with optim al condition s promotes CFA(001) \ngrowth  and improves the effective tunneling spin polarization. On the other hand , the \nsamples annealed at  a high temperature  (600 °C) exhibit ed smaller TMR ratios  (80–120% ). \nThis was  attributed  to the rough  surface  (Ra = 0.4 nm  and P-V = 3.2 nm) of the CFA film \non the Cr layer  annealed at 600 °C, which can lead  to a declined crystalline  orientation of \nthe MgO(001)  barrier . Furthermore, oscillation  behavior  of the TMR ratios as a function of \ntbarr was observed for all of the structure s. The TMR oscillation behavior is typically \nobserved in epitaxial  MTJs such as Fe/MgO /Fe [31], Co 2MnSi/MgO/Co 2MnSi  [32], \nCo2Cr0.6Fe0.4Al/MgO/ Co2Cr0.6Fe0.4Al [33], and CFA/MgO /CoFe  [18] MTJs, while it is \nabsent  in polycrystalline  MTJs such as  CoFeB/MgO /CoFeB  [34] MTJs . More remarkable \noscillation amplitude in epitaxial full-Heusler alloy -based  MTJs than that of epitaxial \nFe/MgO /Fe MTJs was observed, which may be related to the electronic  structures of full -\nHeusler alloy electrode s and the full-Heusler /MgO interface ; however, the origin has not \nbeen understood yet . The unexpected  oscillation behavior  in the polycrystalline CFA -MTJs \nmay be also attributed to the unique electronic  structure of CFA and the interface. In \naddition, the flat buffer layer with a good crystallinity  enabled us to achieve a well -defined \nlayer -by-layer growth for the CFA layer a nd the MgO barrier, which may be advantageous  \nfor observ ing the oscillatory  behavior . The oscillation  period was approximately 0.2 nm  in \nnominal thickness , which seems to be shorter than that for the epitaxial  CFA/MgO/CoFe 11 \n (0.32 nm, short -period)  [18]. Further investigation is needed to clarify the origin of the \nbehavior.  \nThe RA as a function of the MgO barrier thickness  is plotted  in Fig. 4( b). We observed  \na typi cal behavior of an exponential  increase with increasing  tbarr. According to the Wenzel -\nKramer -Brillouin (WKB) approximation , the relationship between RA and tbarr can be \nexpressed as follow s: \n                  √   \n        ,                            (2) \nwhere h, m, and ϕ are Planck’s constant , the effective electron mass  assum ed as a free \nelectron mass (9.11  × 10−31kg) here, and the barrier height energ y of the tunnel barrier , \nrespectively  [31]. A similar barrier height of 0.7 eV was obtained for all three samples by \nthe fitting of RA-tbarr curves. This value is greater  than the reported values for Fe/MgO /Fe \ngrown using molecular -beam-epitaxy  (MBE) (0.39 eV)  [31], sputtered \nCoFeB/MgO /CoFeB  (0.29 –0.39 eV)  [35–37], and CoFeB/MgO  (electron -beam \nevaporated )/CoFeB  (0.48 eV)  [38] MTJs . The reasons  may be due to  the different  densities \nof oxygen vacancy defect s in the MgO barriers and/or the deviation of the actual MgO \nthickness from the nominal one . \nFigure 5 shows TMR ratios in a low -RA regime for the polycrystalline CFA -MTJs with \n2-nm-thick CFA film as a free layer. The MTJs were annealed at 225 ° C for 30 minutes  in \norder to reduce the influence of Cr layer to CFA . A TMR ratio of 40 –60% was achieved \nwith a n RA of 7–20  m2 (nominal MgO thickness: 0.6–0.8 nm) for the 100-nm-scale d 12 \n elliptical MTJs with the thin CFA layer,  which is favorable for achieving STT switching in \nthe MTJ stacks with a thin CFA free layer and the low RA value.  In addition , the TMR ratio \nof the poly crystalline MTJs with 2 -nm-thick CFA film is comparable to that with a thick  \n(30 nm) CFA film at a low RA value, as shown in Fig. 5, which indicates that the (001) -\ntexture and B2 ordering can be maintained in the thin 2 -nm-thick CFA films.  \nC. STT -induced magnetization switching  \nSTT-induced magnetization switching was performed in spin-valve  MTJs with the \nstructure of Si/SiO 2-substr ate/MgO  (7.5)/Cr  (40)/CFA  (2)/MgO  (0.6–0.8)/Co 75Fe25 (5)/Ru  \n(0.8)/Co 75Fe25 (5)/IrMn  (15)/Ta (5)/Ru  (10) (unit: nm). A schematic of the structure of a \npolycrystalline  CFA -MTJ nano pillar  is shown in Fig.  6(a). The synthetic  antiferromagnetic \ncoupling exchange bias of CoFe/Ru/CoFe/IrMn was employed  to reduce the offset \nmagnetic field  of hysteresis  loops . Figure 6(b) indicate s the tunneling resistance  of an MTJ \nnanopillar as a function of the applied magnetic  field ( H) measured with a DC bias voltage \nof 1 mV . A TMR ratio of 43% was observed  in the MTJ with a thin CFA free layer  (2.0 nm) \nand MgO barrie r (0.75 nm). Sharp switching between parallel  (P) and antiparallel (AP) \nmagnetic configurations was observed. The RA of the MTJ was determined using CIPT \nmeasurement s to be 13 m2, and the active area of the MTJ nanopillar was calculated  to \nbe 1.24 ×  10−2 m2. The hysteresis offset field ( Hoffset) and the coerciv ity field ( Hc) were \ndetermined using the R-H loops to be −11 and 2 6 Oe, respectively.  Figure 6(c) shows the \nrepresent ative resistance -current ( R-I) loops of  the CFA -MTJ nanopillar measured by a DC \ncurrent with a sweep rate of 1.2 ×  104 A/s at different magnetic fields of 0, −11 and −20 13 \n Oe, respectively . The positive current indicates that electrons  flow from the bottom \nelectrode to the top electrode . Magnetic switching between P (low-resistance)  and AP \n(high -resistance) states was achieved owing  to the current. When current was applied in the \nnegative ( positive ) direction , the P (AP) state can be  obtained  from A P (P) state , \ncorrespond ing to magnetization reversal of the CFA free layer. Also, the critical switching \ncurrents  (Ic) in both directions shift in the negative  direction  with the decrease of magnetic  \nfield from 0 to −20 Oe . These results indicate  typical behavior s of STT-induced \nmagnetization switching .  \nSince the STT  switching  by the DC current  is a thermally activated process  [39–41], \nwe use a thermal activation model for switching currents deduced from R-I loops to \nevaluate the intrinsic  critical switching current density ( Jc0) and thermal stability  factor  0 \n(= KuV/kBT) for the MTJ , where  Ku is the uniaxial magnetic anisotropy, V is the volume of \nthe free layer, kB is the Boltzmann constant,  and T is the absolute temperature.  In the \nthermal activation model , the sweep  current I(t) is assumed to increase linearly with time t, \ni.e., I(t) = vt, and the cumulative probability distribution function  P(t) of the switching \ncurrent in Hoffset can be expressed  as \n            (      \n  {   [  (    \n   )]        }),     (3) \nwhere f0 is the effective attempt frequency (=109 Hz), Ic0 is the intrinsic switching current,  \nand v is a constant sweep rate  of the sweep current in the measurement of R-I loops  [41]. \nThe distribution of the critical switching current Ic was obtained by repeating the \nmeasurement of the R-I loops for 300 times.  Figure 6(d) shows typical R-I loops at Hoffset = 14 \n −11 Oe for the polycrystalline CFA -MTJ nanopillar. The mean  critical current s in the \npositive  (Ic,PAP) and negative (Ic,APP) directions  are determined  to be 530 an d 400 μA , \ncorresponding  to the critical current density of 4.3 × 106 (Jc,PAP) and 3.2  × 106 A/cm2 \n(Jc,APP), respectively . Figure s 6 (e) and ( f) show the switching probability for Ic,PAP and \nIc, APP as a function of the sweep current.  Using the  constant sweep rate  v = 1.2 ×  10−4 A/s \nin the measurement of R-I loops , the switching probability  was fitted using  Eq. (3) , as \nshown by the solid lines. As a result, the intrinsic  current density of Jc0,PAP (Jc0,APP) = 9.1 \n× 106 A/cm2 (7.3 × 106 A/cm2) and  thermal stability of 0,PAP (0,APP) = 3 0.0 (28.4) were \nachieved for the polycrystalline  CFA -MTJ  nanopillar .  \nD. Gilbert damping  of the polycrystalline CFA film  \nThe Gilbert damping parameter , , is a critical parameter for  determining  the critical \ncurrent density of STT switching. In order to  examine  for the polycrystalline CFA film , \nwaveguide -based FMR  was performe d in a single ferromagnetic layered sample, consisting \nof SiO 2-substrate//MgO(7.5 nm)/Cr(40 nm)/CFA(2 -18 nm) structure . Typical FMR spectra  \nwith varied external magnetic field s (Hext) for a sample of 18-nm-thick  CFA are shown in \nthe inset of Fig. 7(a). A clear shift in the resonant frequency can be seen as Hext increases \nfrom 500 to 1 500 Oe . The peak intensity is relatively small at low magnetic field , possibly \nbecause of  the anisotropy distribution inside the film.  \nFigure s 7(a) and (b) show  the Hext dependence of the resonant frequency  (f0), \ndemagnetization  field (Hd), and magnetic  damping parameter ( H), respectively,  estimated \nby fitting  each spectrum using an analytical solution  [42]. Here, we assume the 15 \n gyromagnetic ratio  () as 2 × 0.00297 GHz/Oe  and neglect the in-plane magnetic \nanisotropic field . Both Hd and H exhibit a weak dependence on Hext possibly due to the \nanisotropy distribution, which become s saturated at a high magnetic field range.  The f0 was \nfitted using  the simpl ified Kittel formula : \nd ext ext H H H f  \n20\n            (4) \nThen, we obtained Hd = 12154  ± 17.9 Oe, which agree s well  with the values  obtained in the \nindividual resonant spectrum  for Hext > 300 Oe . The magnetic field dependence of  can be  \nexcluded by the  fitting  equation , \n0 exp H H kext H \n,             (5) \nwhere k and H0 are fitting parameters , and   is estimated to be 0.01 48 ± 0.0003 . Figure \n7(c) summarize s CFA thickness  dependence of  the saturation  magnetization , Ms, and \nGilbert damping constant , , of polycrystalline CFA  films . The weak thickness dependence \nof Ms and  indicat es that  the CFA film guarantees  a good quality even in a thin thickness \nregime  at around 2 nm .  \nIV. DISCUSSION  \nUsing polycrystalline  CFA full -Heusler thin films , MTJs with the structure of \nCFA /MgO/Co 75Fe25 were successfully fabricated on a SiO 2 amorphous  substrate . The \neffects of the  MgO buffer on the structural properties of the polycrystalline  CFA films and \nthe TMR in the MTJs were investigated. Optimized (001) -texture, B2 order, and surface 16 \n morphology  of the CFA films was demonstrated  on a 7.5-nm-thick MgO buffer , which \ncontributes a large  TMR ratio in the whole polycrystalline MTJ stacks.  In order to achieve \nSTT switching, Cr underlayer s were  utilized  as a conductive electrode on the optimized \nMgO buffer . The Cr layer was known to have very small l attice mismatch (~0.6%) with \nCFA  and facilitate  the ordering structure of full -Heusler alloys.  A TMR ratio of 175% was \nachieved in the  polycrystallin e CFA -MTJs  on MgO/Cr -buffered Si/SiO 2 substrate.  The \nbuffer layer dependence of the structural properties of polycrystalline CFA films and the \nTMR ratios in entire MTJ stacks are significant for practical spintronic applications of full -\nHeusler alloy materials.  A proper buffer layer with minim al diffusion , and enhanc ed (001) -\ntexture and order ing parameter  of polycrystalline  full-Heusler alloys is required for further \nincreasing the TMR ratio s of the MTJs .  \nSTT switching was performed  in the CFA -MTJs with polycrystalline CFA as a free \nlayer.  The average  intrinsic  current density Jc0 = (Jc0,PAP + Jc0,APP)/2 = 8.2 × 106 A/cm2 \nfor the polycrystalline CFA -MTJ  is generally comparable to  that reported for CoFeB -MTJs  \nwith in -plane magnetization  [43, 44]; however, it is much lower than the value of 2.9 × 107 \nA/cm2 for epitaxial  CFA -MTJs  [26]. Based on Slonczewski ’s model of STT switching  [45, \n46], the simplif ied Jc0, ignoring external magnetic field s, is given by , \n         \n      \n     ,                         (6) \nwhere e is the electron charge,   is reduced Planck’s constant , Ms is the saturation \nmagnetization, t is the thickness of the free layer,    is the spin -transfer efficiency , and Heff \nthe effective field acting on the free layer, including the magnetocrystalline  anisotropy field , 17 \n demagneti zation field, stray field . The polycrystalline CFA free layer ( t = 2.0 nm) is thicker  \nthan the epitaxial one  (t = 1.5 nm). A similar  Ms (~1000 emu/cm3) at RT was observed for \nboth polycrystalline  and epitaxial  CFA films. The   of the polycrystalline CFA -MTJ s \nshould be smaller than that of the epitaxial  CFA -MTJ s because  the TMR ratio  of the \npolycrystalline CFA -MTJ s (43%)  is lower than t hat of the epitaxial  CFA -MTJ s (60%)  [26]. \nAccording ly, the low Jc0 in the polycrystalline CFA -MTJ s can be mainly attributed to the \nsmall   of the polycrystalline CFA free layer  (~0.015)  compar ed with that of the epitaxial  \nCFA films  (~0.04)  [26]. For the polycrystalline CFA -MTJ s, annealing was performed with \na low temperature of 225 °C and a short time of 30 minutes , whereas the epitaxial  CFA -\nMTJs  were annealed at 360 °C for 1 hour ; this could be a factor in the reduction in  [47]. \nAs a result, a low er Jc0 was obtained  in the polycrystalline CFA -MTJ s than that in the \nepitaxial  ones. However , the  value is still greater  than that reported for a 50-nm-thick \nCFA film on a MgO  layer  annealed at 600  C (~0.001 ) [23]; this may be attributed  to the \ninter-diffusion of Cr atoms into the CFA layer  and the residual magnetic moments on the \nCr surface . Consequently, a proper buffer material is strongly required for CFA full -\nHeusler MTJs in order to reduce the  of the free layer  and thus the  Jc0 of STT switching.  \nAnother factor  for reduc ing Jc0 is the magnetic anisotropy of the free layer. In perpendicular \nanisotropy CoFeB/MgO/CoFeB tunnel junction s, a low current density for STT switching  \nwas demonstrated owing to the perpendicular magnetic anisotropy (PMA)  [48-53]. To \nevaluate the contribution of interface PMA, magnetization measurements at in -plane and \nout-of-plane of the polycrystalline CFA film was performed.  An effective anisotropy \nenergy density ( Keff) of −5 × 106 erg/cm3 was obtained  for a 2 -nm-thick  CFA/MgO 18 \n structure  where the negative Keff indicates the CFA layer is in -plane magnetized . In general, \nKeff can be simply expressed by the equation of Keff = Kv + Ki/t, where Kv is the volume \nanisotropy energy density which can  be treated as demagnetization energy density  (2πMs2) \nfor simplicity , Ki is the interface anisotropy energy density , and t is the thickness of the \nCFA layer . As a result, the value of Ki can be calculated to be 0.25 erg/cm2, indicating that \nan interface PMA  is induced at the CFA/MgO interface . In the CFA/MgO -MTJs, the \ninterface PMA  can cause a reduce d effective anisotropy , and may also play a role for the \nreduction in Jc0. For further decreasing Jc0, a chosen buffer material, out -of-plane \nmagnetization  of the CFA free layer  [54–56], and/or advanced  fabricating techniques for \nhigh-quality CFA -MTJs are required.  \nV. CONCLUSION  \nIn conclusion, TMR ratios and STT -induced magnetization switching were studied in \n(001) -textur ed polycrystalline CFA full-Heusler based MTJs  on Si /SiO 2 amorphous \nsubstrate s. CFA films with a good  (001) -texture and high B2 order were achieved on MgO -\nbuffer ed Si/SiO 2 amorphous substrate s. The MgO barrier thickness and RA dependence s of \nthe TMR ratio in the polycrystalline CFA -MTJs  were also studied. Moreover, STT \nswitching was achieved in the MTJs with a thin polycrystalline  CFA film  (2.0 nm) as a free \nlayer.  The Jc0 of 8.2 × 106 A/cm2 was demonstrated for the polycrystalline  CFA-MTJs with \nin-plane magnetization using  a thermal activation model for a cumulative switching \nprobability distribution  with a sweep current . The Jc0 is much lower than that reported for \nepitaxial CFA -MTJs , which is mainly attribut ed to the reduced  of the polycrystalline  \nCFA  free layer .  19 \n ACKNOWLEDGMENTS  \nThis work was partly supported by the Japan Science and Technology Agency, CREST,  \nand JSPS  KAKENHI Grant Number 23246006.  \nReference:  \n[1] R. A. de Groot, F. M. Mueller, P. G. vanEngen, and K. H. J. Buschow,  New class of materials: half-\nmetallic ferromagnets , Phys. Rev. Lett.  50, 2024 (1983 ). \n[2] C. Felser, G. H. Fecher, and B. Balke, Spintronics: a challenge for materials science and solid -state \nchemistry , Angew. Chem., Int. Ed. 46, 668 (2007).  \n[3] T. Graf, C. Felser , and S. S. P. Parkin, Simple rules for the understanding of Heusler compounds , \nProg . Solid State Chem . 39, 1 (2011).  \n[4] I. Galanakis, P. Dederichs, and N. Papanikolaou, Slater -Pauling behavior and origin of the half -\nmetallicity of the full -Heusler alloys , Phys. Rev. B 66, 174429 (2002) . \n[5] M. Oogane, T. Kubota, Y. Kota, S. Mizukami, H. Na ganuma , A. Sakuma, and Y. Ando,  Gilbert \nmagnetic damping constant of epitaxially grown Co -based Heusler alloy thin films , Appl. Phys. \nLett. 96, 252501 (2010).  \n[6] T. Furubayashi, K. Kodama, H. Sukegawa, Y. K. Takahashi, K. Inomata , and K. Hono,  Current -\nperpendicular -to-plane giant magnetoresistance in spin -valve structures using epitaxial \nCo2FeAl 0.5Si0.5/Ag/ Co 2FeAl 0.5Si0.5 trilayers , Appl. Phys. Lett. 93, 122507 (2008).  \n[7] Y. Sakuraba, K. Izumi, T. Iwase, S. Bosu, K. Saito, K. Takanashi, Y. Miura, K. Futatsukawa, K. \nAbe, and M. Shirai,  Mechanism of large magnetoresistance in Co2MnSi/Ag/Co2MnSi devices with \ncurrent perpendicular to the plane , Phys. Rev. B 82, 094444 (2010).  \n[8] Z. Bai, L . Shen, G . Han, and Y . P. Feng , Data storage:  review of Heusler compounds, SPIN 02, \n1230006 (2012).  \n[9] Y. Shuto, R. Nakane, W. Wang, H. Sukegawa, S. Yamamoto, M. Tanaka, K. Inomata, and S. \nSugahara, A New Spin -Functional Metal –Oxide –Semiconductor Fiel d-Effect Transistor Based on \nMagnetic Tunnel Junction Technology: Pseudo -Spin-MOSFET , Appl. Phys. Express 3, 013003 \n(2010).  \n[10] Y. Shuto, S. Yamamoto, H. Sukegawa, Z. C Wen, R Nakane, S. Mitani, M. Tanaka, K. Inomata, \nand S. Sugahara, Design and performance of  pseudo -spin-MOSFETs using nano -CMOS devices , \nIEDM,  29.6 (2012).  \n[11] K. Inomata, S. Okamura, R. Goto , and N. Tezuka, Large Tunneling Magnetoresistance at Room \nTemperature Using a Heusler Alloy with the B2 Structure , Jpn. J. Appl. Phys. 42, L419 (2003).  \n[12] S. Tsunegi, Y. Sakuraba, M. Oogane, K. Takanashi, and Yasuo Ando, Large tunnel \nmagnetoresistance in magnetic tunnel junctions using a  Co2MnSi Heusler alloy electrode and a \nMgO barrier , Appl. Phys. Lett. 93, 112506 (2008).  \n[13] H. Sukegawa, W. H. Wang, R. Shan,  T. Nakatani, K. Inomata, and K. Hono, Spin-polarized \ntunneling spectroscopy of fully epitaxial magnetic tunnel junctions using Co 2FeAl 0.5Si0.5 Heusler \nalloy electrodes , Phys. Rev. B 79, 184418 (2009).  \n[14] R. Shan, H. Sukegawa, W. H. Wang, M. Kodzuka, T. Furubayashi, T. Ohkubo, S. Mitani, K. \nInomata, and K. Hono, Demonstration of Half -Metallicity in Fermi -Level -Tuned Heusler Alloy \nCo2FeAl 0.5Si0.5 at Room Temperature , Phys. Rev. Lett. 102, 246601 (2009).  \n[15] N. Tezuka, N. Ikeda, F. Mitsuhashi, and S. Sugimoto, Improved tunnel magnetoresistance of \nmagnetic tunnel junctions with Heusler Co 2FeAl 0.5Si0.5 electrodes fabricated by molecular beam \nepitaxy , Appl. Phys. Lett. 94, 162504 (2009).  20 \n [16] M. Yamamoto, T. Ishikawa,  T. Taira, G. Li, K. Matsuda and T. Uemura, Effect of defects in \nHeusler alloy thin films on spin -dependent tunnelling characteristics of Co 2MnSi/MgO /Co2MnSi \nand Co 2MnGe/MgO /Co2MnGe magnetic tunnel junctions , J. Phys.: Condens. Matter . 22, 164212 \n(2010).  \n[17] W. H. Wang, H. Sukegawa, R. Shan, S. Mitani, and K. Inomata, Giant Tunneling \nMagnetoresistance up to 330% at Room Temperature in Sputter Deposited Co 2FeAl/MgO/CoFe \nMagnetic Tunnel Junctions , Appl. Phys. Lett. 95, 182502 (2009).  \n[18] W. H. Wang, E. Liu, M. Kodzuka, H. Sukegawa, M. Wojcik, E. Jedryka, G. H. Wu, K. Inomata, S. \nMitani, and K. Hono, Coherent tunneling and giant tunneling magnetoresistance in \nCo2FeAl/MgO/CoFe magnetic tunneling junctions , Phys. Rev. B 81, 140402 (R) (2010).  \n[19] W. H. Wang, H. Sukegawa, and K. Inomata, Temperature dependence of tunneling \nmagnetoresistance in epitaxial magnetic tunnel junctions using a Co 2FeAl Heusler alloy electrode , \nPhys. Rev. B 82, 092402 (2010).  \n[20] Z. C. Wen, H. Sukegawa, S. Mitani, and K. Inomata, Tunnel magnetoresistance in textured \nCo2FeAl/MgO/CoFe magnetic tunnel junctions on a Si/SiO 2 amorphous substrate , Appl. Phys. Lett. \n98, 192505 (2011).  \n[21] H. Liu, Y. Honda, T. Taira, K. Matsuda, M. Arita, T. U emura, and M. Yamamotoa, Giant tunneling \nmagnetoresistance in epitaxial Co 2MnSi/MgO/Co 2MnSi magnetic tunnel junctions by half -\nmetallicity of Co 2MnSi and coherent tunneling , Appl. Phys. Lett.  101, 132418 (2012).  \n[22] S. Wurmehl, G. H. Fecher, K. Kroth, F. Kronast, H. A. Dü rr, Y. Takeda, Y. Saitoh, K. Kobayashi, \nH.-J. Lin, G. Schö nhense, and C. Felser, Electronic structure and spectroscopy of the quaternary \nHeusler alloy Co 2Cr1−xFexAl, J. Phys. D: Appl. Phys. 39, 803 (2006).  \n[23] S. Mizukami, D. Watanabe, M. Oogane, Y. Ando, Y. Mi ura, M. Shirai, and T. Miyazaki , Low \ndamping constant for Co 2FeAl Heusler alloy films and its correlation with density of states , J. Appl. \nPhys. 105, 07D306  (2009).  \n[24] R. Y. Umetsu, A. Okubo, and R. Kainuma, Magnetic and chemical order -disorder transformations \nin Co 2Fe(Ga 1−x Si x ) and Co 2Fe(Al 1−y Si y ) Heusler alloys , J. Appl. Phys. 111, 073909 (2012) . \n[25] Y. Miura, K. Nagao, and M. Shirai, Atomic disorder effects on half-metallicity of the full -Heusler \nalloys Co 2(Cr 1-xFex)Al: A first -principles study , Phys. Rev. B 69, 144413 (2004).  \n[26] H. Sukegawa, Z. C. Wen, K. Kondou, S. Kasai, S. Mitani, and K. Inomata, Spin-transfer switching \nin full -Heusler Co 2FeAl -based magnetic tunnel junctions , Appl. Phys. Lett. 100, 182403 (2012).  \n[27] H. Sukegawa, S. Kasai, T. Furubayashi, S. Mitani, and K. Inomata, Spin-transfer switching in an \nepitaxial spin -valve nanopillar with a full -Heusler Co 2FeAl 0.5Si0.5 alloy , Appl. Phys. Lett. 96, \n042508 (2010).  \n[28] D. C. Worledge , and P. L. Trouilloud, Magnetoresistance measurement of unpatterned magnetic \ntunnel junction wafers by current -in-plane tunneling , Appl. Phys. Lett. 83, 84 (2003).  \n[29] S. Ok amura, A. Miyazaki, N. Tezuka, S. Sugimoto, and K. Inomata, Epitaxial Growth of Ordered \nCo2(Cr 1-xFex)Al Full -Heusler Alloy Films on Single Crystal. Substrates , Mater. Trans. 47, 15 \n(2006).  \n[30] J. L. Vassent, M. Dynna, A. Marty, B. Gilles , and G. Patrat , A study of growth and the relaxation of \nelastic strain in MgO on F e(001) , J. Appl. Phys. 80, 5727 (1996) . \n[31] S. Yuasa, T. Nagahama, A. Fukushima, Y. Suzuki, and K. Ando, Giant room -temperature \nmagnetoresistance in single -crystal Fe/MgO/Fe magnetic tunnel junctions , Nature Mater. 3, 868 \n(2004).  \n[32] T. Ishikawa, S. Hakamata , K. Matsuda, T. Uemura, and M. Yamamoto, Fabrication of fully \nepitaxial Co 2MnSi/MgO/Co 2MnSi magnetic tunnel junctions , J. Appl. Phys. 103, 07A919 (2008).  \n[33] T. Marukame, T. Ishikawa, T. Taira, K. Matsuda, T. Uemura, and M. Yamamoto, Giant oscillations \nin spin-dependent tunneling resistances as a function of barrier thickness in fully epitaxial magnetic \ntunnel junctions with a MgO barrier , Phys. Rev. B 81, 134432 (2010).  \n[34] D. D. Djayaprawira, K. Tsunekawa, M. Nagai, H. Maehara, S. Yamagata, N. Watanabe, S.  Yuasa, 21 \n Y. Suzuki, and K. Ando,  230% room -temperature magnetoresistance in CoFeB /MgO /CoFeB \nmagnetic tunnel junctions , Appl. Phys. Lett. 86, 092502 (2005).  \n[35] K. Tsunekawa, D. D. Djayaprawira, M. Nagai, H. Maehara, S. Yamagata, N. Watanabe, S. Yuasa, \nY. Suzuki, a nd K. Ando, Giant tunneling magnetoresistance effect in low -resistance \nCoFeB /MgO(001) /CoFeB magnetic tunnel junctions for read -head applications , Appl. Phys. Lett. \n87, 072503 (2005).  \n[36] J. Hayakawa, S. Ikeda, F. Matsukura, H. Takahashi, and H . Ohno,  Dependence of Giant Tunnel \nMagnetoresistance of Sputtered CoFeB/MgO/CoFeB Magnetic Tunnel Junctions on MgO Barrier \nThickness and Annealing Temperature , Jpn. J. Appl. Phys. 44, L587 (2005).  \n[37] H. Sukegawa, W. H. Wang, R. Shan, and K. Inomata, Tunnel Magnetores istance in Full -Heusler \nCo2FeAl 0.5Si0.5-Based Magnetic Tunnel Junctions , J. Magn. Soc. Jpn. 33, 256 (2009).  \n[38] H. Kurt, K. Oguz, T. Niizeki, and J. M. D. Coey, Giant tunneling magnetoresistance with electron \nbeam evaporated MgO barrier and CoFeB electrodes , J. Appl. Phys. 107, 083920 (2010).  \n[39] R. H. Koch, J. A. Katine, and J. Z. Sun, Time -Resolved Reversal of Spin -Transfer Switching in a \nNanomagnet , Phys. Rev. Lett. 92, 088302 (2004).  \n[40] Z. Li and S. Zhang, Thermally assisted magnetization reversal in the presence of a spin -transfer \ntorque , Phys. Rev. B 69, 134416 (2004).  \n[41] T. Taniguchi , and H. Imamura,  Dependence of Spin Torque Switching Probability on Electric \nCurrent , J. Nanosci. Nanotechnol. 12, 7520 (2012); Theoretical study on dependence of thermal \nswitching tim e of synthetic free layer on coupling field , J. Appl. Phys. 111, 07C901 (2012).  \n[42] N. Toda, K. Saito, K. Ohta, H. Maekawa, M. Mizuguchi, M. Shiraishi, and Y. Suzuki, Highly \nsensitive ferromagnetic resonance measurements using coplanar waveguides , J. Magn. Soc . Jpn . 31, \n435 (2007).  \n[43] Z. Diao, D. Apalkov, M. Pakala, Y. Ding, A. Panchula, and Y. Huai, Spin transfer switching and \nspin polarization in magnetic tunnel junctions with MgO and AlOx barriers , Appl. Phys. Lett. 87, \n232502 (2005).  \n[44] H. Kubota, A. Fukushima, Y. Ootani, S. Yuasa, K. Ando, H. Maehara, K. Tsunekawa, D. D. \nDjayaprawira, N. Watanabe, and Y. Suzuki, Evaluation of Spin -Transfer Switching in \nCoFeB/MgO/CoFeB Magnetic Tunnel Junctions , Jpn. J. Appl. Phys. 44, L1237 (2005).  \n[45] J. Z. Sun, Spin-current interaction with a monodomain magnetic body: A model study , Phys. Rev. \nB 62, 570 (2000).  \n[46] J. C. Slonczewski, Currents, torques, and polarization factors in magnetic tunnel junctions , Phys. \nRev. B 71, 024411 (2005).   \n[47] Y. Cui, B.  Khodadadi, S. Schafer, T. Mewes, J. Lu, and Stuart A. Wolf, Interfacial perpendicular \nmagnetic anisotropy and damping parameter in ultrathin Co 2FeAl films , Appl. Phys. Lett. 102, \n162403 (2013).  \n[48] H. Meng, R . Sbiaa, S . Lua, C . Wang, M . Akhtar, S . K. Wong,  P. Luo, C . Carlberg , and K . S. A. \nAng, Low current density induced  spin-transfer torque switching in  CoFeB –MgO magnetic tunnel \njunctions  with perpendicular anisotropy , J. Phys. D: Appl. Phys. 44, 405001  (2011) . \n[49] W. Kim et al. , Extended scalability of perpendicular STT -MRAM towards sub -20nm MTJ \nnode. IEDM Tech. Dig. 24.1.1 (2011).  \n[50] M. T. Rahman et al. , Reduction of switching current density in perpendicular magnetic tunnel \njunctions by tuning the anisotropy of the CoFeB free layer. J. Appl. Phys. 111, 07C907 (2012).  \n[51] M. Gajek  et al. , Spin torque switching of 20nm magnetic tunnel junctions with perpendicular  \nanisotropy . Appl. Phys. Lett. 100, 132408 (2012) . \n[52] L. Thomas  et al. , Perpendicular spin transfer torque magnetic random access memories with \nhigh spin t orque efficiency and thermal stability for embedded applications (invited) , J. Appl. \nPhys. 115, 172615 (2014) . \n[53] H. Sato, M . Yamanouchi, S . Ikeda, S . Fukami, F . Matsukura, and  H. Ohno , 22 \n MgO/CoFeB/Ta/CoFeB/MgO recording structure in magnetic tunnel  junctions with perpendicular \neasy axis, IEEE T ran. Magn. 49, 4437 ( 2013 ). \n[54] Z. C. Wen, H. Sukegawa, S. Kasai, M. Hayashi, S. Mitani, and K. Inomata,  Magnetic Tunnel \nJunctions with Perpendicular Anisotropy Using a Co2FeAl Full -Heusler Alloy , Appl. Phys. Express \n5, 063003 (2012) . \n[55] K. Chae, D. Lee, T. Shim, J. Hong, and J. Park, Correlation of the structural properties of a Pt seed \nlayer with the perpendicular  magnetic anisotropy features of full Heusler -based \nCo2FeAl/MgO/Co 2Fe6B2 junctions  via a 12 -inch scale Si wafer  process , Appl. Phys. Lett. 103, \n162409 (2013) . \n[56] M. S. Gabor, T. Petrisor Jr., C. Tiusan, and T. Petrisor,  Perpendicular magnetic anisotropy in \nTa/Co 2FeAl/MgO multilayers , J. Appl. Phys. 114, 063905 (2013).  \n \nFigures  and captions : \n \nFIG. 1 . (Color online) (a) Out-of-plane ( 2θ-ω scan) XRD patterns for polycrystalline CFA full -\nHeusler alloy films on MgO -buffered Si/SiO 2 amorphous substrates with varied MgO buffer \nthickness, tMgO: 2.5, 5.0, 7.5, and 10.0 nm. (b) Normalized  integrated  intensity of CFA(002) peak , \nand (c) Ratios of (002) to (004) peaks as a function of tMgO for as -deposited  (as-dep.)  samples and \nthose annealed at  400 °C and 480  °C. \n20 40 60 8010210410610810101012Intensity (Log Scale)\n \n2(degree )2.5 nm5.0 nm7.5 nmTa = 400 oC\nssCFA(004)MgO(002)CFA(002)\n10.0 nm\n2 4 6 8 100.00.10.20.30.4\n  I(002)/I(004)\ntMgO (nm) as-dep.\n 400 oC\n 480 oC(a) (b) (c)\n2 4 6 8 100.20.40.60.81.01.2\n  Normalized intensity\ntMgO (nm) as-dep.\n 400 oC\n 480 oCCFA(002)23 \n  \nFIG. 2. (Color online) Surface morphology  of polycrystalline CFA full -Heusler  alloy films on \nMgO -buffered Si/SiO 2 amorphous substrates with respect to  the thickness of the MgO buffer. T he \nCFA films were 30 -nm-thick  and were deposited at RT and post -annealed at 400 °C and 480 °C, \nrespectively. Inset: AFM image of the surface of the CFA film annealed at 400 °C. \n \n \n \nFIG. 3. (Color online) TMR ratios as a function of MgO buffer thickness tMgO for polycrystalline \nCFA /MgO/CoFe MTJs with different thicknesses of the MgO barrier, tbarr: 1.5 nm, 1.8 nm, and 2 .0 \nnm. The 30 -nm-thick CFA bottom electrodes were directly deposited on the MgO buffer layer and \n2 4 6 8 100.00.20.40.60.81.0\nRa = 0.1 nm, P-V = 1.3 nm Ra, 480 C\n P-V, 480 C\n  Ra (nm)\ntMgO (nm) Ra, 400 C\n P-V, 400 C\n0246\nP-V (nm)\n2 4 6 8 1050100150200\n Ta:  400 C, tbarr:  1.5 nm\n Ta:  480 C, tbarr:  1.8 nm\n  TMR (%)\ntMgO (nm) Ta: 400 C,  tbarr: 1.8 nm\n Ta: 400 C,  tbarr: 2.0 nm24 \n post-anneal ed at 400 °C and 480 °C after deposition at RT. The MTJ  stacks  were annealed at 370 °C \nbefore the CIPT measurement.  \n \n \n \nFIG. 4. (Color online) The thickness of MgO barrier, tbarr, dependence of (a) TMR ratios  and (b) RA \nat RT for polycrystalline CFA /MgO/CoFe MTJs with as -deposited, 400 °C and 600 °C annealed \nMgO(7.5 nm)/Cr(40 nm) buffer layers on SiO 2 amorphous substr ates, characterized using CIPT  \nmeasurement . \n \n \n1.2 1.4 1.6 1.8 2.01011021031041051061.2 1.4 1.6 1.8 2.0050100150200\ntbarr (nm)\n  RA (m2)MgO/Cr(40-nm, as-dep.)\nMgO/Cr(40-nm, 400 C)\nMgO/Cr(40-nm, 600 C)\n  TMR (%)\ntbarr (nm) MgO/Cr(40-nm, as-dep.)\n MgO/Cr(40-nm, 400 C) \n MgO/Cr(40-nm, 600 C)SiO2//MgO/Cr/CFA/MgO/CoFe/IrMn(a)\n(b)25 \n  \nFIG. 5. (Color online) RA dependence of TMR ratios for polycrystalline CFA -MTJs with a 2-nm-\nthick  CFA layer (“thin-CFA ”) as a bottom electrode . The squared symbol  indicates the TMR ratio \nfor a polycrystalline CFA -MTJ with a 30-nm-thick CFA layer ( “thick -CFA ”). \n \n \nFIG. 6. (a) Schematic illustration of the structure of polycrystalline  CFA -MTJ nano pillar.  (b) R-H \nloops  for a polycrystalline CFA -MTJ  nanopillar . Wide arrows show the magnetic configuration s of \nbottom (free) and top (reference) electrodes of the MTJ, and narrow arrows indicate sweep direction \nof the applied magnetic field.  (c) R-I loops for the MTJ at  magnetic fields of 0, –11 and –20 Oe, \nrespectivel y. Arrows indicate sweep direction of the applied current.  (d) Representative R-I loops of \nthe CFA -MTJ at applied magnetic  field of –11 Oe. (e) and ( f) Switching  probabilities for Ic,P->AP and \n0 5 10 15 200204060\n  TMR (%)\nRA (m2) \"thin-CFA\"\n \"thick-CFA\"\n26 \n Ic,AP->P obtained by  repeating R-I measurements for 3 00 times.  Solid lines are fitting curve s given by \nEq. ( 3). All measurements were performed at RT.  \n \nFIG. 7. The Hext dependence of (a) resonant frequency f0, (b) demagnetization  field Hd, and  \nmagnetic damping parameter H estimated by fitting the FMR spectr um at each magnetic field  for \na sample of a 18-nm-thick polycrystalline CFA film. Inset of (a) is t ypical FMR spectra at Hext of \n500, 1000, and 1500 Oe. (c) CFA thickness  t dependence of saturation magnetization  Ms and \ndamping constant 0 of the polycrystalline  CFA film.  \n102030\n  Frequency (GHz)\n0 500 1000 1500 20000.000.020.040.06\n  \n\nHex (Oe)51015\nHd (kOe) 8 12 16-20-100\n0.5 kOe 1 kOe\nFrequency (GHz)\n  Real [S21]\n1.5 kOe\n3 6 912 15 180.000.040.08\n  \n\nt (nm)05001000\nMs (emu/cm3)(b)\n(c)(a)"    },    {        "title": "1408.3499v1.Linear_hyperbolic_equations_with_time_dependent_propagation_speed_and_strong_damping.pdf",        "content": "arXiv:1408.3499v1  [math.AP]  15 Aug 2014Linear hyperbolic equations with time-dependent\npropagation speed and strong damping\nMarina Ghisi\nUniversit` a degli Studi di Pisa\nDipartimento di Matematica\nPISA (Italy)\ne-mail:ghisi@dm.unipi.itMassimo Gobbino\nUniversit` a degli Studi di Pisa\nDipartimento di Matematica\nPISA (Italy)\ne-mail:m.gobbino@dma.unipi.itAbstract\nWe consider a second order linear equation with a time-dependent co efficientc(t) in\nfront of the “elastic” operator. For these equations it is well-know n that a higher space-\nregularity of initial data compensates a lower time-regularity of c(t).\nIn this paper we investigate the influence of a strong dissipation, na mely a friction\nterm which depends on a power of the elastic operator.\nWhat we discover is a threshold effect. When the exponent of the ela stic operator\nin the friction term is greater than 1/2, the damping prevails and the equation behaves\nas if the coefficient c(t) were constant. When the exponent is less than 1/2, the time-\nregularity of c(t) comes into play. If c(t) is regular enough, once again the damping\nprevails. On the contrary, when c(t) is not regular enough the damping might be\nineffective, and there are examples in which the dissipative equation b ehaves as the\nnon-dissipative one. As expected, the stronger is the damping, th e lower is the time-\nregularity threshold.\nWe also provide counterexamples showing the optimality of our result s.\nMathematics Subject Classification 2010 (MSC2010): 35L20, 35L80, 35L90.\nKey words: linear hyperbolic equation, dissipative hyperbolic equation, strong d amp-\ning, fractional damping, time-dependent coefficients, well-posedn ess, Gevrey spaces.1 Introduction\nLetHbe a separable real Hilbert space. For every xandyinH,|x|denotes the norm\nofx, and/a\\}⌊ra⌋ketle{tx,y/a\\}⌊ra⌋ketri}htdenotes the scalar product of xandy. LetAbe a self-adjoint linear\noperator on Hwith dense domain D(A). We assume that Ais nonnegative, namely\n/a\\}⌊ra⌋ketle{tAx,x/a\\}⌊ra⌋ketri}ht ≥0 for every x∈D(A), so that for every α≥0 the power Aαxis defined\nprovided that xlies in a suitable domain D(Aα).\nWe consider the second order linear evolution equation\nu′′(t)+2δAσu′(t)+c(t)Au(t) = 0, (1.1)\nwith initial data\nu(0) =u0, u′(0) =u1. (1.2)\nAs far as we know, this equation has been considered in the literatur e either in the\ncase where δ= 0, or in the case where δ >0 but the coefficient c(t) is constant. Let us\ngive a brief outline of the previous literature which is closely related to our results.\nThe non-dissipative case Whenδ= 0, equation (1.1) reduces to\nu′′(t)+c(t)Au(t) = 0. (1.3)\nThis is the abstract setting of a wave equation in which c(t) represents the square of\nthe propagation speed.\nIf the coefficient c(t) is Lipschitz continuous and satisfies the strict hyperbolicity\ncondition\n0<µ1≤c(t)≤µ2, (1.4)\nthen it is well-know that problem (1.3)–(1.2) is well-posed in the classic e nergy space\nD(A1/2)×H(see for example the classic reference [14]).\nIf the coefficient is not Lipschitz continuous, things are more comple x, even if (1.4)\nstill holds true. This problem was addressed by F. Colombini, E. De Gior gi and S. Spag-\nnolo in the seminal paper [6]. Their results can be summed up as follows ( we refer to\nsection 2 below for the precise functional setting and rigorous sta tements).\n(1) Problem (1.3)–(1.2) has always a unique solution, up to admitting t hat this solu-\ntion takes its values in a very large Hilbert space (ultradistributions) . This is true\nfor initial data in the energy space D(A1/2)×H, but also for less regular data,\nsuch as distributions or ultradistributions.\n(2) If initial data are regular enough, then the solution is regular as well. How much\nregularity is required depends on the time-regularity of c(t). Classic examples are\nthe following. If c(t) is just measurable, problem (1.3)–(1.2) is well-posed in the\nclass of analytic functions. If c(t) isα-H¨ older continuous for some α∈(0,1),\nproblem (1.3)–(1.2) is well-posed in the Gevrey space of order (1 −α)−1.\n1(3) If initial data are not regular enough, then the solution may exh ibit a severe\nderivative loss for all positive times. For example, for every α∈(0,1) there exist a\ncoefficientc(t) which isα-H¨ older continuous, and initial data ( u0,u1) which are in\nthe Gevrey class of order βfor everyβ >(1−α)−1, such that the corresponding\nsolution to (1.3)–(1.2) (which exists in the weak sense of point (1)) is not even a\ndistribution for every t>0.\nIn the sequel we call (DGCS)-phenomenon the instantaneous loss of regularity de-\nscribed in point (3) above.\nThe dissipative case with constant coefficients Ifδ >0 andc(t) is a constant function\n(equal to 1 without loss of generality), equation (1.1) reduces to\nu′′(t)+2δAσu′(t)+Au(t) = 0. (1.5)\nMathematical models with damping terms of this form were proposed in [1], and\nthen rigorously analyzed by many authors from different points of v iew. The first\npapers [2, 3, 4], and the more recent [10], are devoted to analyticity properties of the\nsemigroup associated to (1.5). The classic assumptions in these pap ers are that the\noperatorAis strictly positive, σ∈[0,1], and the phase space is D(A1/2)×H. On a\ndifferent side, the community working on dispersive equations consid ered equation (1.5)\nintheconcretecasewhere σ∈[0,1]andAu=−∆uinRnorspecialclassesofunbounded\ndomains. They proved energy decay and dispersive estimates, but exploiting in an\nessential way the spectral properties of the Laplacian in those do mains. The interested\nreader is referred to [11, 12, 13, 19] and to the references quot ed therein.\nFinally, equation (1.5) was considered in [9] in full generality, namely fo r every\nσ≥0 and every nonnegative self-adjoint operator A. Two different regimes appeared.\nIn the subcritical regime σ∈[0,1/2], problem (1.5)–(1.2) is well-posed in the classic\nenergy space D(A1/2)×Hor more generally in D(Aα+1/2)×D(Aα) withα≥0. In the\nsupercritical regime σ≥1/2, problem (1.5)–(1.2) is well-posed in D(Aα)×D(Aβ) if and\nonly if\n1−σ≤α−β≤σ. (1.6)\nThis means that in the supercritical regime different choices of the p hase space are\npossible, even with α−β/\\e}atio\\slash= 1/2.\nThe dissipative case with time-dependent coefficients As far as we know, the case of a\ndissipative equation with a time-dependent propagation speed had n ot been considered\nyet. The main question we address in this paper is the extent to which the dissipative\nterm added in (1.1) prevents the (DGCS)-phenomenon of (1.3) fro m happening. We\ndiscover a composite picture, depending on σ.\n•In the subcritical regime σ∈[0,1/2], if the strict hyperbolicity assumption (1.4)\nis satisfied, well-posedness results do depend on the time-regularit y ofc(t) (see\nTheorem 3.2). Classic examples are the following.\n2–Ifc(t) isα-H¨ older continuous for some exponent α >1−2σ, then the dis-\nsipation prevails, and problem (1.1)–(1.2) is well-posed in the classic en ergy\nspaceD(A1/2)×Hor more generally in D(Aβ+1/2)×D(Aβ) withβ≥0.\n–Ifc(t) is no more than α-H¨ older continuous for some exponent α <1−2σ,\nthenthedissipationcanbeneglected, sothat(1.1)behavesexact lyasthenon-\ndissipative equation (1.3). This means well-posedness in the Gevrey s pace of\norder (1−α)−1and the possibility to produce the (DGCS)-phenomenon for\nless regular data (see Theorem 3.10).\n–The case with α= 1−2σis critical and also the size of the H¨ older constant\nofc(t) compared with δcomes into play.\n•In the supercritical regime σ >1/2 the dissipation prevails in an overwhelming\nway. In Theorem 3.1 we prove that, if c(t) is just measurable and satisfies just the\ndegenerate hyperbolicity condition\n0≤c(t)≤µ2, (1.7)\nthen (1.1) behaves as (1.5). This means that problem (1.1)–(1.2) is w ell-posed in\nD(Aα)×D(Aβ) if and only if (1.6) is satisfied, the same result obtained in [9] in\nthe case of a constant coefficient.\nThe second issue we address in this paper is the further space-reg ularity of solutions\nfor positive times, since a strong dissipation is expected to have a re gularizing effect\nsimilar to parabolic equations. This turns out to be true provided tha t the assumptions\nof our well-posedness results are satisfied, and in addition σ∈(0,1). Indeed, we prove\nthat in this regime u(t) lies in the Gevrey space of order (2min {σ,1−σ})−1for every\nt>0. We refer to Theorem 3.8 and Theorem 3.9 for the details. This effec t had already\nbeen observed in [15] in the dispersive case.\nWe point out that the regularizing effect is maximum when σ= 1/2 (the only case in\nwhich solutions become analytic with respect to space variables) and disappears when\nσ≥1, meaning that a stronger overdamping prevents smoothing.\nOverview of the technique The spectral theory reduces the problem to an analysis of\nthe family of ordinary differential equations\nu′′\nλ(t)+2δλ2σu′\nλ(t)+λ2c(t)uλ(t) = 0. (1.8)\nWhenδ= 0, a coefficient c(t) which oscillates with a suitable period can produce\na resonance effect so that (1.8) admits a solution whose oscillations h ave an amplitude\nwhich grows exponentially with time. This is the primordial origin of the ( DGCS)-\nphenomenon for non-dissipative equations. When δ >0, the damping term causes\nan exponential decay of the amplitude of oscillations. The competition between the\n3exponential energy growth due to resonance and the exponent ial energy decay due to\ndissipation originates the threshold effect we observed.\nWhenc(t) is constant, equation (1.8) can be explicitly integrated, and the ex plicit\nformulae for solutions led to the sharp results of [9]. Here we need th e same sharp\nestimates, but without relying on explicit solutions. To this end, we int roduce suitable\nenergy estimates.\nIn the supercritical regime σ≥1/2 we exploit the following σ-adapted “Kovaleskyan\nenergy”\nE(t) :=|u′\nλ(t)+δλ2σuλ(t)|2+δ2λ4σ|uλ(t)|2. (1.9)\nIn the subcritical regime σ≤1/2 we exploit the so-called “approximated hyperbolic\nenergies”\nEε(t) :=|u′\nλ(t)+δλ2σuλ(t)|2+δ2λ4σ|uλ(t)|2+λ2cε(t)|uλ(t)|2,(1.10)\nobtained by adding to (1.9) an “hyperbolic term” depending on a suita ble smooth ap-\nproximation cε(t) ofc(t), which in turn is chosen in a λ-dependent way. Terms of this\ntype are the key tool introduced in [6] for the non-dissipative equa tion.\nFuture extensions We hope that this paper could represent a first step in the theory\nof dissipative hyperbolic equations with variable coefficients, both line ar and nonlinear.\nNext steps could be considering a coefficient c(x,t) depending both on time and space\nvariables, and finally quasilinear equations. This could lead to improve t he classic\nresults by K. Nishihara [16, 17] for Kirchhoff equations, whose linear ization has a time-\ndependent coefficient, and finally to consider more general local no nlinearities, in which\ncase the linearization involves a coefficient c(x,t) depending on both variables.\nInadifferent direction, thesubcritical case σ∈[0,1/2]withdegeneratehyperbolicity\nassumptions remains open and could be the subject of further res earch, in the same way\nas [7] was the follow-up of [6].\nOn the other side, we hope that our counterexamples could finally dis pel the dif-\nfuse misconception according to which dissipative hyperbolic equatio ns are more stable,\nand hence definitely easier to handle. Now we know that a friction ter m below a suit-\nable threshold is substantially ineffective, opening the door to patho logies such as the\n(DGCS)-phenomenon, exactly as in the non-dissipative case.\nStructure of the paper This paper is organized as follows. In section 2 we introduce\nthe functional setting and we recall the classic existence results f rom [6]. In section 3 we\nstate our main results. In section 4 we provide a heuristic descriptio n of the competition\nbetween resonance and decay. In section 5 we prove our existenc e and regularity results.\nIn section 6 we present our examples of (DGCS)-phenomenon.\n42 Notation and previous results\nFunctional spaces LetHbe a separable Hilbert space. Let us assume that Hadmits\na countable complete orthonormal system {ek}k∈Nmade by eigenvectors of A. We\ndenote the corresponding eigenvalues by λ2\nk(with the agreement that λk≥0), so that\nAek=λ2\nkekfor everyk∈N. In this case every u∈Hcan be written in a unique way\nin the form u=/summationtext∞\nk=0ukek, whereuk=/a\\}⌊ra⌋ketle{tu,ek/a\\}⌊ra⌋ketri}htare the Fourier components of u. In\nother words, the Hilbert space Hcan be identified with the set of sequences {uk}of real\nnumbers such that/summationtext∞\nk=0u2\nk<+∞.\nWe stress that this is just a simplifying assumption, with substantially no loss of\ngenerality. Indeed, according to the spectral theorem in its gene ral form (see for ex-\nample Theorem VIII.4 in [18]), one can always identify HwithL2(M,µ) for a suitable\nmeasure space ( M,µ), in such a way that under this identification the operator Aacts\nas a multiplication operator by some measurable function λ2(ξ). All definitions and\nstatements in the sequel, with the exception of the counterexamp les of Theorem 3.10,\ncan be easily extended to the general setting just by replacing the sequence {λ2\nk}with\nthe function λ2(ξ), and the sequence {uk}of Fourier components of uwith the element\n/hatwideu(ξ) ofL2(M,µ) corresponding to uunder the identification of HwithL2(M,µ).\nThe usual functional spaces can be characterized in terms of Fou rier components as\nfollows.\nDefinition 2.1. Letube a sequence {uk}of real numbers.\n•Sobolev spaces . For every α≥0 it turns out that u∈D(Aα) if\n/⌊ard⌊lu/⌊ard⌊l2\nD(Aα):=∞/summationdisplay\nk=0(1+λk)4αu2\nk<+∞. (2.1)\n•Distributions . We say that u∈D(A−α) for someα≥0 if\n/⌊ard⌊lu/⌊ard⌊l2\nD(A−α):=∞/summationdisplay\nk=0(1+λk)−4αu2\nk<+∞. (2.2)\n•Generalized Gevrey spaces . Letϕ: [0,+∞)→[0,+∞) be any function, let r≥0,\nand letα∈R. We say that u∈ Gϕ,r,α(A) if\n/⌊ard⌊lu/⌊ard⌊l2\nϕ,r,α:=∞/summationdisplay\nk=0(1+λk)4αu2\nkexp/parenleftbig\n2rϕ(λk)/parenrightbig\n<+∞. (2.3)\n•Generalized Gevrey ultradistributions . Letψ: [0,+∞)→[0,+∞)beanyfunction,\nletR≥0, and letα∈R. We say that u∈ G−ψ,R,α(A) if\n/⌊ard⌊lu/⌊ard⌊l2\n−ψ,R,α:=∞/summationdisplay\nk=0(1+λk)4αu2\nkexp/parenleftbig\n−2Rψ(λk)/parenrightbig\n<+∞. (2.4)\n5Remark 2.2. Ifϕ1(x) =ϕ2(x) for every x >0, thenGϕ1,r,α(A) =Gϕ2,r,α(A) for every\nadmissible value of randα. For this reason, with a little abuse of notation, we consider\nthe spaces Gϕ,r,α(A) even when ϕ(x) is defined only for x >0. The same comment\napplies also to the spaces G−ψ,R,α(A).\nThe quantities defined in (2.1) through (2.4) are actually norms which induce a\nHilbert space structure on D(Aα),Gϕ,r,α(A),G−ψ,R,α(A), respectively. The standard\ninclusions\nGϕ,r,α(A)⊆D(Aα)⊆H⊆D(A−α)⊆ G−ψ,R,−α(A)\nhold true for every α≥0 and every admissible choice of ϕ,ψ,r,R. All inclusions\nare strict if α,randRare positive, and the sequences {λk},{ϕ(λk)}, and{ψ(λk)}are\nunbounded.\nWe observe that Gϕ,r,α(A) is actually a so-called scale of Hilbert spaces with respect\nto theparameter r, withlarger values of rcorresponding to smaller spaces. Analogously,\nG−ψ,R,α(A) is a scale of Hilbert spaces with respect to the parameter R, but with larger\nvalues ofRcorresponding to larger spaces.\nRemark 2.3. Let us consider the concrete case where I⊆Ris an open interval,\nH=L2(I), andAu=−uxx, with periodic boundary conditions. For every α≥0, the\nspaceD(Aα) is actually the usual Sobolev space H2α(I), andD(A−α) is the usual space\nof distributions of order 2 α.\nWhenϕ(x) :=x1/sfor somes>0, elements of Gϕ,r,0(A) withr>0 are usually called\nGevrey functions of order s, the cases= 1 corresponding to analytic functions. When\nψ(x) :=x1/sforsomes>0, elements of G−ψ,R,0(A)withR>0areusually called Gevrey\nultradistributions of order s, the cases= 1 corresponding to analytic functionals. In\nthis case the parameter αis substantially irrelevant because the exponential term is\ndominant both in (2.3) and in (2.4).\nFor the sake of consistency, with a little abuse of notation we use th e same terms\n(Gevrey functions, Gevrey ultradistributions, analytic functions and analytic function-\nals) in order to denote the same spaces also in the general abstrac t framework. To be\nmore precise, we should always add “with respect to the operator A”, or even better\n“with respect to the operator A1/2”.\nContinuity moduli Throughout this paper we call continuity modulus any continuous\nfunctionω: [0,+∞)→[0,+∞) such that ω(0) = 0,ω(x)>0 for every x >0, and\nmoreover\nx→ω(x) is a nondecreasing function , (2.5)\nx→x\nω(x)is a nondecreasing function. (2.6)\nA function c: [0,+∞)→Ris said to be ω-continuous if\n|c(a)−c(b)| ≤ω(|a−b|)∀a≥0,∀b≥0. (2.7)\n6More generally, a function c:X→R(withX⊆R) is said to be ω-continuous if it\nsatisfies the same inequality for every aandbinX.\nPrevious results We are now ready to recall the classic results concerning existence ,\nuniqueness, and regularity for solutions to problem (1.1)–(1.2). We state them using our\nnotations which allow general continuity moduli and general spaces of Gevrey functions\nor ultradistributions.\nProofs are a straightforward application of the approximated ene rgy estimates in-\ntroduced in [6]. In that paper only the case δ= 0 is considered, but when δ≥0 all new\nterms have the “right sign” in those estimates.\nThe first result concerns existence and uniqueness in huge spaces such as analytic\nfunctionals, with minimal assumptions on c(t).\nTheorem A (see [6, Theorem 1]) .Let us consider problem (1.1)–(1.2) under the fol-\nlowing assumptions:\n•Ais a self-adjoint nonnegative operator on a separable Hilbe rt spaceH,\n•c∈L1((0,T))for everyT >0(without sign conditions),\n•σ≥0andδ≥0are two real numbers,\n•initial conditions satisfy\n(u0,u1)∈ G−ψ,R0,1/2(A)×G−ψ,R0,0(A)\nfor someR0>0and someψ: (0,+∞)→(0,+∞)such that\nlimsup\nx→+∞x\nψ(x)<+∞.\nThen there exists a nondecreasing function R: [0,+∞)→[0,+∞), withR(0) =R0,\nsuch that problem (1.1)–(1.2) admits a unique solution\nu∈C0/parenleftbig\n[0,+∞);G−ψ,R(t),1/2(A)/parenrightbig\n∩C1/parenleftbig\n[0,+∞);G−ψ,R(t),0(A)/parenrightbig\n.(2.8)\nCondition (2.8), with the range space increasing with time, simply mean s that\nu∈C0/parenleftbig\n[0,τ];G−ψ,R(τ),1/2(A)/parenrightbig\n∩C1/parenleftbig\n[0,τ];G−ψ,R(τ),0(A)/parenrightbig\n∀τ≥0.\nThis amounts to say that scales of Hilbert spaces, rather than fixe d Hilbert spaces,\nare the natural setting for this problem.\nInthesecondresultweassumestricthyperbolicityand ω-continuityofthecoefficient,\nand we obtain well-posedness in a suitable class of Gevrey ultradistrib utions.\n7Theorem B (see [6, Theorem 3]) .Let us consider problem (1.1)–(1.2) under the fol-\nlowing assumptions:\n•Ais a self-adjoint nonnegative operator on a separable Hilbe rt spaceH,\n•the coefficient c: [0,+∞)→Rsatisfies the strict hyperbolicity assumption (1.4)\nand theω-continuity assumption (2.7) for some continuity modulus ω(x),\n•σ≥0andδ≥0are two real numbers,\n•initial conditions satisfy\n(u0,u1)∈ G−ψ,R0,1/2(A)×G−ψ,R0,0(A)\nfor someR0>0and some function ψ: (0,+∞)→(0,+∞)such that\nlimsup\nx→+∞x\nψ(x)ω/parenleftbigg1\nx/parenrightbigg\n<+∞. (2.9)\nLetube the unique solution to the problem provided by Theorem A.\nThen there exists R>0such that\nu∈C0/parenleftbig\n[0,+∞),G−ψ,R0+Rt,1/2(A)/parenrightbig\n∩C1([0,+∞),G−ψ,R0+Rt,0(A)).\nThe third result we recall concerns existence of regular solutions. The assumptions\nonc(t) are the same as in Theorem B, but initial data are significantly more r egular\n(Gevrey spaces instead of Gevrey ultradistributions).\nTheorem C (see [6, Theorem 2]) .Let us consider problem (1.1)–(1.2) under the fol-\nlowing assumptions:\n•Ais a self-adjoint nonnegative operator on a separable Hilbe rt spaceH,\n•the coefficient c: [0,+∞)→Rsatisfies the strict hyperbolicity assumption (1.4)\nand theω-continuity assumption (2.7) for some continuity modulus ω(x),\n•σ≥0andδ≥0are two real numbers,\n•initial conditions satisfy\n(u0,u1)∈ Gϕ,r0,1/2(A)×Gϕ,r0,0(A)\nfor somer0>0and some function ϕ: (0,+∞)→(0,+∞)such that\nlimsup\nx→+∞x\nϕ(x)ω/parenleftbigg1\nx/parenrightbigg\n<+∞. (2.10)\n8Letube the unique solution to the problem provided by Theorem A.\nThen there exist T >0andr>0such thatrT <r 0and\nu∈C0/parenleftbig\n[0,T],Gϕ,r0−rt,1/2(A)/parenrightbig\n∩C1([0,T],Gϕ,r0−rt,0(A)). (2.11)\nRemark 2.4. The key assumptions of Theorem B and Theorem C are (2.9) and (2.10 ),\nrespectively, representing the exact compensation between spa ce-regularity of initial\ndataandtime-regularityofthecoefficient c(t)requiredinordertoobtainwell-posedness.\nThese conditions do not appear explicitly in [6], where they are replace d by suitable\nspecific choices of ω,ϕ,ψ, which of course satisfy the same relations. To our knowledge,\nthose conditions were stated for the first time in [8], thus unifying se veral papers that in\nthelast 30 years hadbeendevoted tospecial cases (see forexam ple [5] andthe references\nquoted therein).\nRemark 2.5. The standard example of application of Theorem B and Theorem C is\nthe following. Let us assume that c(t) isα-H¨ older continuous for some α∈(0,1),\nnamelyω(x) =Mxαfor a suitable constant M. Then (2.9) and (2.10) hold true with\nψ(x) =ϕ(x) :=x1−α. This leads to well-posedness both in the large space of Gevrey\nultradistributions of order (1 −α)−1, and in the small space of Gevrey functions of the\nsame order.\nRemark 2.6. The choice of ultradistributions in Theorem B is not motivated by the\nsearchforgenerality, butitisinsomesense theonlypossibleonebec auseofthe(DGCS)-\nphenomenon exhibited in [6], at least in the non-dissipative case. When δ= 0, if initial\ndata are taken in Sobolev spaces or in any space larger than the Gev rey spaces of\nTheorem C, then it may happen that for all positive times the solution lies in the space\nof ultradistributions specified in Theorem B, and nothing more. In ot her words, for\nδ= 0 there is no well-posedness result in between the Gevrey spaces o f Theorem C\nand the Gevrey ultradistributions of Theorem B, and conditions (2.9 ) and (2.10) are\noptimal.\nThe aim of this paper is to provide an optimal picture for the case δ >0.\n3 Main results\nIn this section we state our main regularity results for solutions to ( 1.1)–(1.2). To this\nend, we need some further notation. Given any ν≥0, we write Has an orthogonal\ndirect sum\nH:=Hν,−⊕Hν,+, (3.1)\nwhereHν,−is the closure of the subspace generated by all eigenvectors of Arelative to\neigenvalues λk<ν, andHν,+is the closure of the subspace generated by all eigenvectors\nofArelative to eigenvalues λk≥ν. For every vector u∈H, we writeuν,−anduν,+\nto denote its components with respect to the decomposition (3.1). We point out that\n9Hν,−andHν,+areA-invariant subspaces of H, and thatAis a bounded operator when\nrestricted to Hν,−, and a coercive operator when restricted to Hν,+ifν >0.\nIn the following statements we provide separate estimates for low- frequency compo-\nnentsuν,−(t) and high-frequency components uν,+(t) of solutions to (1.1). This is due to\nthe fact that the energy of uν,−(t) can be unbounded as t→+∞, while in many cases\nwe are able to prove that the energy of uν,+(t) is bounded in time.\n3.1 Existence results in Sobolev spaces\nThe first result concerns the supercritical regime σ≥1/2, in which case the dissipation\nalways dominates the time-dependent coefficient.\nTheorem 3.1 (Supercritical dissipation) .Let us consider problem (1.1)–(1.2) under\nthe following assumptions:\n•Ais a self-adjoint nonnegative operator on a separable Hilbe rt spaceH,\n•the coefficient c: [0,+∞)→Ris measurable and satisfies the degenerate hyper-\nbolicity assumption (1.7),\n•σandδare two positive real numbers such that either σ >1/2, orσ= 1/2and\n4δ2≥µ2,\n•(u0,u1)∈D(Aα)×D(Aβ)for some real numbers αandβsatisfying (1.6).\nLetube the unique solution to the problem provided by Theorem A.\nThenuactually satisfies\n(u,u′)∈C0/parenleftbig\n[0,+∞),D(Aα)×D(Aβ)/parenrightbig\n. (3.2)\nMoreover, for every ν≥1such that 4δ2ν4σ−2≥µ2, it turns out that\n|Aβu′\nν,+(t)|2+|Aαuν,+(t)|2≤/parenleftbigg\n2+2\nδ2+µ2\n2\nδ4/parenrightbigg\n|Aβu1,ν,+|2+3/parenleftbigg\n1+µ2\n2\n2δ2/parenrightbigg\n|Aαu0,ν,+|2(3.3)\nfor everyt≥0.\nOur second result concerns the subcritical regime σ∈[0,1/2], in which case the\ntime-regularity of c(t) competes with the exponent σ.\nTheorem 3.2 (Subcritical dissipation) .Let us consider problem (1.1)–(1.2) under the\nfollowing assumptions:\n•Ais a self-adjoint nonnegative operator on a separable Hilbe rt spaceH,\n•the coefficient c: [0,+∞)→Rsatisfies the strict hyperbolicity assumption (1.4)\nand theω-continuity assumption (2.7) for some continuity modulus ω(x),\n10•σ∈[0,1/2]andδ>0are two real numbers such that\n4δ2µ1>Λ2\n∞+2δΛ∞, (3.4)\nwhere we set\nΛ∞:= limsup\nε→0+ω(ε)\nε1−2σ, (3.5)\n•(u0,u1)∈D(A1/2)×H.\nLetube the unique solution to the problem provided by Theorem A.\nThenuactually satisfies\nu∈C0/parenleftbig\n[0,+∞),D(A1/2)/parenrightbig\n∩C1([0,+∞),H).\nMoreover, for every ν≥1such that\n4δ2µ1≥/bracketleftbigg\nλ1−2σω/parenleftbigg1\nλ/parenrightbigg/bracketrightbigg2\n+2δ/bracketleftbigg\nλ1−2σω/parenleftbigg1\nλ/parenrightbigg/bracketrightbigg\n(3.6)\nfor everyλ≥ν, it turns out that\n|u′\nν,+(t)|2+2µ1|A1/2uν,+(t)|2≤4|u1,ν,+|2+2(3δ2+µ2)|A1/2u0,ν,+|2(3.7)\nfor everyt≥0.\nLet us make a few comments on the first two statements.\nRemark 3.3. Inbothresultsweprovedthatasuitablehigh-frequencycompone nt ofthe\nsolution can be uniformly bounded in terms of initial data. Low-frequ ency components\nmight in general diverge as t→+∞. Nevertheless, they can always be estimated as\nfollows.\nLet us just assume that c∈L1((0,T)) for every T >0. Then for every ν≥0 the\ncomponent uν,−(t) satisfies\n|u′\nν,−(t)|2+|A1/2uν,−(t)|2≤/parenleftbig\n|u1,ν,−|2+|A1/2u0,ν,−|2/parenrightbig\nexp/parenleftbigg\nνt+ν/integraldisplayt\n0|c(s)|ds/parenrightbigg\n(3.8)\nfor everyt≥0. Indeed, let F(t) denote the left-hand side of (3.8). Then\nF′(t) =−4δ|Aσ/2u′\nν,−(t)|2+2(1−c(t))/a\\}⌊ra⌋ketle{tu′\nν,−(t),Auν,−(t)/a\\}⌊ra⌋ketri}ht\n≤2(1+|c(t)|)·|u′\nν,−(t)|·ν|A1/2uν,−(t)|\n≤ν(1+|c(t)|)F(t)\nfor almost every t≥0, so that (3.8) follows by integrating this differential inequality.\n11Remark 3.4. The phase spaces involved in Theorem 3.1 and Theorem 3.2 are exactly\nthe same which are known to be optimal when c(t) is constant (see [9]). In particular,\nthe only possible choice in the subcritical regime is the classic energy s paceD(A1/2)×H,\nor more generally D(Aα+1/2)×D(Aα). This “gap1/2” between the powers of Ainvolved\nin the phase space is typical of hyperbolic problems, and it is the same which appears\nin the classic results of section 2.\nOn the contrary, in the supercritical regime there is an interval of possible gaps,\ndescribed by (1.6). This interval is always centered in 1/2, but also d ifferent values are\nallowed, including negative ones when σ>1.\nRemark 3.5. The classic example of application of Theorem 3.2 is the following. Let\nus assume that c(t) isα-H¨ older continuous for some α∈(0,1), namely ω(x) =Mxαfor\nsome constant M. Then problem (1.1)–(1.2) is well-posed in the energy space provided\nthat either α>1−2σ, orα= 1−2σandMis small enough. Indeed, for α>1−2σwe\nget Λ∞= 0, and hence (3.4) is automatically satisfied. For α= 1−2σwe get Λ ∞=M,\nso that (3.4) is satisfied provided that Mis small enough.\nIn all other cases, namely when either α <1−2σ, orα= 1−2σandMis large\nenough, only Theorem B applies to initial data in Sobolev spaces, prov iding global\nexistence just in the sense of Gevrey ultradistributions of order ( 1−α)−1.\nRemark 3.6. Let us examine the limit case σ= 0, which falls in the subcritical regime.\nWhenσ= 0, assumption (3.4) is satisfied if and only if c(t) is Lipschitz continuous\nand its Lipschitz constant is small enough. On the other hand, in the Lipschitz case it\nis a classic result that problem (1.1)–(1.2) is well-posed in the energy s pace, regardless\nof the Lipschitz constant. Therefore, the result stated in Theor em 3.2 is non-optimal\nwhenσ= 0 andc(t) is Lipschitz continuous.\nA simple refinement of our argument would lead to the full result also in this case,\nbut unfortunately it would be useless in all other limit cases in which c(t) isα-H¨ older\ncontinuous with α= 1−2σandσ∈(0,1/2]. We refer to section 4 for further details.\nRemark 3.7. Let us examine the limit case σ= 1/2, which falls both in the subcritical\nand in the supercritical regime, so that the conclusions of Theorem 3.1 and Theorem 3.2\ncoexist. Both of them provide well-posedness in the energy space, but with different\nassumptions.\nTheorem 3.1 needs less assumptions on c(t), which is only required to be measurable\nand to satisfy the degenerate hyperbolicity assumption (1.7), but it requires δto be\nlarge enough so that 4 δ2≥µ2.\nOn the contrary, Theorem 3.2 needs less assumptions on δ, which is only required to\nbe positive, but it requires c(t) to be continuous and to satisfy the strict hyperbolicity\nassumption (1.4). Indeed, inequality (3.4) is automatically satisfied in the caseσ= 1/2\nbecause Λ ∞= 0.\nThe existence of two different sets of assumptions leading to the sa me conclusion\nsuggests the existence of a unifying statement, which could proba bly deserve further\ninvestigation.\n123.2 Gevrey regularity for positive times\nA strong dissipation in the range σ∈(0,1) has a regularizing effect on initial data,\nprovided that the solution exists in Sobolev spaces. In the following t wo statements we\nquantify this effect in terms of scales of Gevrey spaces.\nBoth results can be summed up by saying that the solution lies, for po sitive times,\nin Gevrey spaces of order (2min {σ,1−σ})−1. It is not difficult to show that this order\nis optimal, even in the case where c(t) is constant.\nTheorem 3.8 (Supercritical dissipation) .Let us consider problem (1.1)–(1.2) under\nthe same assumptions of Theorem 3.1, and let ube the unique solution to the problem\nprovided by Theorem A.\nLet us assume in addition that either σ∈(1/2,1), orσ= 1/2and4δ2>µ2. Let us\nsetϕ(x) :=x2(1−σ), and\nC(t) :=/integraldisplayt\n0c(s)ds. (3.9)\nThen there exists r>0such that\n(u,u′)∈C0/parenleftbig\n(0,+∞),Gϕ,α,rC(t)(A)×Gϕ,β,rC(t)(A)/parenrightbig\n, (3.10)\nand there exist ν≥1andK >0such that\n/⌊ard⌊lu′\nν,+(t)/⌊ard⌊l2\nϕ,β,rC(t)+/⌊ard⌊luν,+(t)/⌊ard⌊l2\nϕ,α,rC(t)≤K/parenleftbig\n|Aβu1,ν,+|2+|Aαu0,ν,+|2/parenrightbig\n(3.11)\nfor everyt>0. The constants r,ν, andKdepend only on δ,µ2, andσ.\nOf course, (3.10) and (3.11) are nontrivial only if C(t)>0, which is equivalent to\nsaying that the coefficient c(t) is not identically 0 in [0 ,t]. On the other hand, this weak\nform of hyperbolicity is necessary, since no regularizing effect on u(t) can be expected\nas long asc(t) vanishes.\nTheorem 3.9 (Subcritical dissipation) .Let us consider problem (1.1)–(1.2) under the\nsame assumptions of Theorem 3.2, and let ube the unique solution to the problem\nprovided by Theorem A.\nLet us assume in addition that σ∈(0,1/2](instead of σ∈[0,1/2]), and let us set\nϕ(x) :=x2σ.\nThen there exists r>0such that\nu∈C0/parenleftbig\n(0,+∞),Gϕ,1/2,rt(A)/parenrightbig\n∩C1((0,+∞),Gϕ,0,rt(A)),\nand there exist ν≥1andK >0such that\n/⌊ard⌊lu′\nν,+(t)/⌊ard⌊l2\nϕ,0,rt+/⌊ard⌊luν,+(t)/⌊ard⌊l2\nϕ,1/2,rt≤K/parenleftbig\n|u1,ν,+|2+|A1/2u0,ν,+|2/parenrightbig\n(3.12)\nfor everyt>0. The constants r,ν, andKdepend only on δ,µ1,µ2,σandω.\nTheestimateswhichprovideGevreyregularityofhigh-frequencyc omponentsprovide\nalso the decay of the same components as t→+∞. We refer to Lemma 5.1 and\nLemma 5.2 for further details.\n133.3 Counterexamples\nThe following result shows that even strongly dissipative hyperbolic e quations can ex-\nhibit the (DGCS)-phenomenon, provided that we are in the subcritic al regime.\nTheorem 3.10 ((DGCS)-phenomenon) .LetAbe a linear operator on a Hilbert space\nH. Let us assume that there exists a countable (not necessaril y complete) orthonormal\nsystem{ek}inH, and an unbounded sequence {λk}of positive real numbers such that\nAek=λ2\nkekfor everyk∈N. Letσ∈[0,1/2)andδ>0be real numbers.\nLetω: [0,+∞)→[0,+∞)be a continuity modulus such that\nlim\nε→0+ω(ε)\nε1−2σ= +∞. (3.13)\nLetϕ: (0,+∞)→(0,+∞)andψ: (0,+∞)→(0,+∞)be two functions such that\nlim\nx→+∞x\nϕ(x)ω/parenleftbigg1\nx/parenrightbigg\n= lim\nx→+∞x\nψ(x)ω/parenleftbigg1\nx/parenrightbigg\n= +∞. (3.14)\nThen there exist a function c:R→Rsuch that\n1\n2≤c(t)≤3\n2∀t∈R, (3.15)\n|c(t)−c(s)| ≤ω(|t−s|)∀(t,s)∈R2, (3.16)\nand a solution u(t)to equation (1.1) such that\n(u(0),u′(0))∈ Gϕ,r,1/2(A)×Gϕ,r,0(A)∀r>0, (3.17)\n(u(t),u′(t))/\\e}atio\\slash∈ G−ψ,R,1/2(A)×G−ψ,R,0(A)∀R>0,∀t>0.(3.18)\nRemark 3.11. Due to (3.15), (3.16), and (3.17), the function u(t) provided by Theo-\nrem 3.10 is a solution to (1.1) in the sense of Theorem A with ψ(x) :=x, or even better\nin the sense of Theorem B with ψ(x) :=xω(1/x).\nRemark 3.12. Assumption (3.13) is equivalent to saying that Λ ∞defined by (3.5) is\nequal to + ∞, so that (3.4) can not be satisfied. In other words, Theorem 3.2 giv es\nwell-posedness in the energy space if Λ ∞is 0 or small, while Theorem 3.10 provides\nthe (DGCS)-phenomenon if Λ ∞= +∞. The case where Λ ∞is finite but large remains\nopen. We suspect that the (DGCS)-phenomenon is still possible, bu t our construction\ndoes not work. We comment on this issue in the first part of section 6 .\nFinally, Theorem 3.10 shows that assumptions (2.9) and (2.10) of The orems B and C\nare optimal also in the subcritical dissipative case with Λ ∞= +∞. If initial data are in\nthe Gevrey space with ϕ(x) =xω(1/x), solutions remain in the same space. If initial are\nin a Gevrey space corresponding to some ϕ(x)≪xω(1/x), then it may happen that for\npositive times the solution lies in the space of ultradistributions with ψ(x) :=xω(1/x),\nbut not in the space of ultradistributions corresponding to any give nψ(x)≪xω(1/x).\n144 Heuristics\nThefollowingpicturessummarizeroughlytheresultsofthispaper. I nthehorizontalaxis\nwe represent the time-regularity of c(t). With some abuse of notation, values α∈(0,1)\nmean that c(t) isα-H¨ older continuous, α= 1 means that it is Lipschitz continuous,\nα >1 means even more regular. In the vertical axis we represent the s pace-regularity\nof initial data, where the value sstands for the Gevrey space of order s(so that higher\nvalues ofsmean lower regularity). The curve is s= (1−α)−1.\nα 1s\n1\nδ= 0Potential (DGCS)-phenomenon Well-posedness\nα 1−2σ/Bullets\n1\nδ >0,0<σ<1/2α 1s\n1\nδ >0, σ>1/2\nForδ= 0 we have the situation described in Remark 2.5 and Remark 2.6, name ly\nwell-posedness provided that either c(t) is Lipschitz continuous or c(t) isα-H¨ older con-\ntinuous and initial data are in Gevrey spaces of order less than or eq ual to (1 −α)−1,\nand (DGCS)-phenomenon otherwise. The same picture applies if δ >0 andσ= 0.\nWhenδ >0 and 0< σ <1/2, the full strip with α >1−2σfalls in the well-\nposedness region, as stated in Theorem 3.2. The region with α <1−2σis divided as\nin the non-dissipative case. Indeed, Theorem C still provides well-po sedness below the\ncurve and on the curve, while Theorem 3.10 provides the (DGCS)-ph enomenon above\nthe curve. What happens on the vertical half-line which separates the two regions is\nless clear (it is the region where Λ ∞is positive and finite, see Remark 3.12).\nFinally, when δ >0 andσ>1/2 well-posedness dominates because of Theorem 3.1,\neven in the degenerate hyperbolic case.\nNow we present a rough justification of this threshold effect. As alr eady observed,\nexistence results for problem (1.1)–(1.2) are related to estimates for solutions to the\nfamily of ordinary differential equations (1.8).\nLet us consider the simplest energy function E(t) :=|u′\nλ(t)|2+λ2|uλ(t)|2, whose\ntime-derivative is\nE′(t) =−4δλ2σ|u′\nλ(t)|2+2λ2(1−c(t))uλ(t)u′\nλ(t)\n≤ −4δλ2σ|u′\nλ(t)|2+λ(1+|c(t)|)E(t). (4.1)\n15Sinceδ≥0, a simple integration gives that\nE(t)≤ E(0)exp/parenleftbigg\nλt+λ/integraldisplayt\n0|c(s)|ds/parenrightbigg\n, (4.2)\nwhich is almost enough to establish Theorem A.\nIfinaddition c(t)isω-continuousandsatisfiesthestricthyperbolicitycondition(1.4),\nthen (4.2) can be improved to\nE(t)≤M1E(0)exp(M2λω(1/λ)t) (4.3)\nfor suitable constants M1andM2. Estimates of this kind are the key point in the proof\nof both Theorem B and Theorem C. Moreover, the (DGCS)-phenom enon is equivalent\nto saying that the term λω(1/λ) in (4.3) is optimal.\nLet us assume now that δ >0. Ifσ >1/2, orσ= 1/2 andδis large enough,\nthen it is reasonable to expect that the first (negative) term in the right-hand side of\n(4.1) dominates the second one, and hence E(t)≤ E(0), which is enough to establish\nwell-posedness in Sobolev spaces. Theorem 3.1 confirms this intuition .\nIfσ≤1/2 andc(t) is constant, then (1.8) can be explicitly integrated, obtaining\nthat\nE(t)≤ E(0)exp/parenleftbig\n−2δλ2σt/parenrightbig\n. (4.4)\nIfc(t) isω-continuous and satisfies the strict hyperbolicity assumption (1.4) , then\nwe expect a superposition of the effects of the coefficient, repres ented by (4.3), and the\neffects of the damping, represented by (4.4). We end up with\nE(t)≤M1E(0)exp/parenleftbig\n[M2λω(1/λ)−2δλ2σ]t/parenrightbig\n. (4.5)\nTherefore, it is reasonable to expect that E(t) satisfies an estimate independent of\nλ, which guarantees well-posedness in Sobolev spaces, provided tha tλω(1/λ)≪λ2σ, or\nλω(1/λ)∼λ2σandδis large enough. Theorem 3.2 confirms this intuition. The same\nargument applies if σ= 0 andω(x) =Lx, independently of L(see Remark 3.6).\nOn the contrary, in all other cases the right-hand side of (4.5) dive rges asλ→\n+∞, opening the door to the (DGCS)-phenomenon. We are able to show that it does\nhappen provided that λω(1/λ)≫λ2σ. We refer to the first part of section 6 for further\ncomments.\n5 Proofs of well-posedness and regularity results\nAll proofs of our main results concerning well-posedness and regula rity rely on suitable\nestimates for solutions to the ordinary differential equation (1.8) w ith initial data\nuλ(0) =u0, u′\nλ(0) =u1. (5.1)\nFor the sake of simplicity in the sequel we write u(t) instead of uλ(t).\n165.1 Supercritical dissipation\nLet us consider the case σ≥1/2. The key tool is the following.\nLemma 5.1. Let us consider problem (1.8)–(5.1) under the following ass umptions:\n•the coefficient c: [0,+∞)→Ris measurable and satisfies the degenerate hyper-\nbolicity assumption (1.7),\n•δ,λ,σare positive real numbers such that\n4δ2λ4σ−2≥µ2. (5.2)\nThen the solution u(t)satisfies the following estimates.\n(1) For every t≥0it turns out that\n|u(t)|2≤2\nδ2λ4σu2\n1+3u2\n0, (5.3)\n|u′(t)|2≤/parenleftbigg\n2+µ2\n2\nδ4λ8σ−4/parenrightbigg\nu2\n1+3µ2\n2\n2δ2λ4σ−4u2\n0. (5.4)\n(2) Let us assume in addition that λ≥1andσ≥1/2, and letαandβbe two real\nnumbers satisfying (1.6).\nThen for every t≥0it turns out that\nλ4β|u′(t)|2+λ4α|u(t)|2≤/parenleftbigg\n2+2\nδ2+µ2\n2\nδ4/parenrightbigg\nλ4βu2\n1+3/parenleftbigg\n1+µ2\n2\n2δ2/parenrightbigg\nλ4αu2\n0.(5.5)\n(3) In addition to the assumptions of the statement (2), let u s assume also that there\nexistsr>0satisfying the following three inequalities:\nδλ4σ−2>rµ2,2δr≤1,4δ2λ4σ−2≥(1+2rδ)µ2.(5.6)\nThen for every t≥0it turns out that\nλ4β|u′(t)|2+λ4α|u(t)|2≤/bracketleftbigg\n2/parenleftbigg\n1+2µ2\n2\nδ4+1\nδ2/parenrightbigg\nλ4βu2\n1+3/parenleftbigg\n1+2µ2\n2\nδ2/parenrightbigg\nλ4αu2\n0/bracketrightbigg\n×\n×exp/parenleftbigg\n−2rλ2(1−σ)/integraldisplayt\n0c(s)ds/parenrightbigg\n. (5.7)\n17ProofLet us consider the energy E(t) defined in (1.9). Since\n−3\n4|u′(t)|2−4\n3δ2λ4σ|u(t)|2≤2δλ2σu(t)u′(t)≤ |u′(t)|2+δ2λ4σ|u(t)|2,\nwe easily deduce that\n1\n4|u′(t)|2+2\n3δ2λ4σ|u(t)|2≤E(t)≤2|u′(t)|2+3δ2λ4σ|u(t)|2∀t≥0.(5.8)\nStatement (1) The time-derivative of E(t) is\nE′(t) =−2/parenleftbig\nδλ2σ|u′(t)|2+δλ2σ+2c(t)|u(t)|2+λ2c(t)u(t)u′(t)/parenrightbig\n.(5.9)\nThe right-hand side is a quadratic form in u(t) andu′(t). The coefficient of |u′(t)|2\nis negative. Therefore, this quadratic form is less than or equal to 0 for all values of u(t)\nandu′(t) if and only if\n4δ2λ4σ−2c(t)≥c2(t),\nand this is always true because of (1.7) and (5.2). It follows that E′(t)≤0 for (almost)\neveryt≥0, and hence\nδ2λ4σ|u(t)|2≤E(t)≤E(0)≤2u2\n1+3δ2λ4σu2\n0, (5.10)\nwhich is equivalent to (5.3).\nIn order to estimate u′(t), we rewrite (1.8) in the form\nu′′(t)+2δλ2σu′(t) =−λ2c(t)u(t),\nwhich we interpret as a first order linear equation with constant coe fficients in the\nunknownu′(t), with the right-hand side as a forcing term. Integrating this differ ential\nequation in u′(t), we obtain that\nu′(t) =u1exp/parenleftbig\n−2δλ2σt/parenrightbig\n−/integraldisplayt\n0λ2c(s)u(s)exp/parenleftbig\n−2δλ2σ(t−s)/parenrightbig\nds. (5.11)\nFrom (1.7) and (5.3) it follows that\n|u′(t)| ≤ |u1|+µ2λ2·max\nt∈[0,T]|u(t)|·/integraldisplayt\n0e−2δλ2σ(t−s)ds\n≤ |u1|+µ2λ2\n2δλ2σ/parenleftbigg2\nδ2λ4σu2\n1+3u2\n0/parenrightbigg1/2\n,\nand therefore\n|u′(t)|2≤2|u1|2+µ2\n2λ4\n2δ2λ4σ/parenleftbigg2\nδ2λ4σu2\n1+3u2\n0/parenrightbigg\n,\nwhich is equivalent to (5.4).\n18Statement (2) Exploiting (5.3) and (5.4), with some simple algebra we obtain that\nλ4β|u′(t)|2+λ4α|u(t)|2≤/parenleftbigg\n2+µ2\n2\nδ4·1\nλ4(2σ−1)+2\nδ2·1\nλ4(β+σ−α)/parenrightbigg\nλ4βu2\n1\n+3/parenleftbigg\n1+µ2\n2\n2δ2·1\nλ4(α−β+σ−1)/parenrightbigg\nλ4αu2\n0.\nAll exponents of λ’s in denominators are nonnegative owing to (1.6). Therefore,\nsinceλ≥1, all those fractions can be estimated with 1. This leads to (5.5).\nStatement (3) Let us define C(t) as in (3.9). To begin with, we prove that in this\ncase the function E(t) satisfies the stronger differential inequality\nE′(t)≤ −2rλ2(1−σ)c(t)E(t), (5.12)\nand hence\nE(t)≤E(0)exp/parenleftbig\n−2rλ2(1−σ)C(t)/parenrightbig\n∀t≥0. (5.13)\nComing back to (5.9), inequality (5.12) is equivalent to\nλ2σ/parenleftbig\nδ−rλ2−4σc(t)/parenrightbig\n|u′(t)|2+δλ2σ+2(1−2rδ)c(t)|u(t)|2+λ2(1−2rδ)c(t)u(t)u′(t)≥0.\nAs in the proof of statement (1), we consider the whole left-hand s ide as a quadratic\nform inu(t) andu′(t). Sincec(t)≤µ2, from the first inequality in (5.6) it follows that\nδλ4σ−2>rµ2≥rc(t),\nwhich is equivalent to saying that the coefficient of |u′(t)|2is positive. Therefore, the\nquadratic form is nonnegative for all values of u(t) andu′(t) if and only if\n4λ2σ/parenleftbig\nδ−rλ2−4σc(t)/parenrightbig\n·δλ2σ+2c(t)(1−2rδ)≥λ4c2(t)(1−2rδ)2,\nhence if and only if\n(1−2rδ)c(t)/bracketleftbig\n4δ2λ4σ−2−(1+2rδ)c(t)/bracketrightbig\n≥0,\nand this follows from (1.7) and from the last two inequalities in (5.6).\nNow from (5.13) it follows that\nδ2λ4σ|u(t)|2≤E(t)≤E(0)exp/parenleftbig\n−2rλ2(1−σ)C(t)/parenrightbig\n, (5.14)\nwhich provides an estimate for |u(t)|. In order to estimate u′(t), we write it in the form\n(5.11), and we estimate the two terms separately. The third inequa lity in (5.6) implies\nthat 2δλ4σ−2≥rµ2. SinceC(t)≤µ2t, it follows that\n2δλ2σt≥rλ2−2σµ2t≥rλ2−2σC(t),\n19and hence\n/vextendsingle/vextendsingleu1exp/parenleftbig\n−2δλ2σt/parenrightbig/vextendsingle/vextendsingle≤ |u1|exp/parenleftbig\n−2δλ2σt/parenrightbig\n≤ |u1|exp/parenleftbig\n−rλ2(1−σ)C(t)/parenrightbig\n.(5.15)\nAs for the second terms in (5.11), we exploit (5.14) and we obtain tha t\n/vextendsingle/vextendsingle/vextendsingle/vextendsingle/integraldisplayt\n0λ2c(s)u(s)exp/parenleftbig\n−2δλ2σ(t−s)/parenrightbig\nds/vextendsingle/vextendsingle/vextendsingle/vextendsingle≤λ2µ2/integraldisplayt\n0|u(s)|exp/parenleftbig\n−2δλ2σ(t−s)/parenrightbig\nds\n≤µ2[E(0)]1/2\nδλ2σ−2exp/parenleftbig\n−2δλ2σt/parenrightbig/integraldisplayt\n0exp/parenleftbig\n−rλ2(1−σ)C(s)+2δλ2σs/parenrightbig\nds.\nFrom the first inequality in (5.6) it follows that\n2δλ2σ−rλ2(1−σ)c(s)≥2δλ2σ−rλ2(1−σ)µ2≥δλ2σ,\nhence\n/integraldisplayt\n0exp/parenleftbig\n−rλ2(1−σ)C(s)+2δλ2σs/parenrightbig\nds\n≤1\nδλ2σ/integraldisplayt\n0/parenleftbig\n2δλ2σ−rλ2(1−σ)c(s)/parenrightbig\nexp/parenleftbig\n2δλ2σs−rλ2(1−σ)C(s)/parenrightbig\nds\n≤1\nδλ2σexp/parenleftbig\n2δλ2σt−rλ2(1−σ)C(t)/parenrightbig\n,\nand therefore\n/vextendsingle/vextendsingle/vextendsingle/vextendsingle/integraldisplayt\n0λ2c(s)u(s)exp/parenleftbig\n−2δλ2σ(t−s)/parenrightbig\nds/vextendsingle/vextendsingle/vextendsingle/vextendsingle≤µ2[E(0)]1/2\nδ2λ4σ−2exp/parenleftbig\n−rλ2(1−σ)C(t)/parenrightbig\n.(5.16)\nFrom (5.11), (5.15) and (5.16) we deduce that\n|u′(t)| ≤/parenleftbigg\n|u1|+µ2[E(0)]1/2\nδ2λ4σ−2/parenrightbigg\nexp/parenleftbig\n−rλ2(1−σ)C(t)/parenrightbig\n,\nand hence\n|u′(t)|2≤/parenleftbigg\n2|u1|2+2µ2\n2E(0)\nδ4λ8σ−4/parenrightbigg\nexp/parenleftbig\n−2rλ2(1−σ)C(t)/parenrightbig\n. (5.17)\nFinally, we estimate E(0) as in (5.10). At this point, estimate (5.7) follows from\n(5.17) and (5.14) with some simple algebra (we need to exploit that λ≥1 and assump-\ntion (1.6) exactly as in the proof of statement (2)). /square\n205.1.1 Proof of Theorem 3.1\nLet us fix a real number ν≥1 such that 4 δ2ν4σ−2≥µ2(such a number exists because of\nourassumptions on δandσ). Letusconsiderthecomponents uk(t)ofu(t)corresponding\nto eigenvalues λk≥ν. Sinceλk≥1 and 4δ2λ4σ−2\nk≥µ2, we can apply statement (2) of\nLemma 5.1 to these components. If u0kandu1kdenote the corresponding components\nof initial data, estimate (5.5) read as\nλ4β\nk|u′\nk(t)|2+λ4α\nk|uk(t)|2≤/parenleftbigg\n2+2\nδ2+µ2\n2\nδ4/parenrightbigg\nλ4β\nk|u1,k|2+3/parenleftbigg\n1+µ2\n2\n2δ2/parenrightbigg\nλ4α\nk|u0,k|2.\nSumming over all λk≥νwe obtain exactly (3.3).\nThis proves that uν,+(t) is bounded with values in D(Aα) andu′\nν,+(t) is bounded\nwith values in D(Aβ). The same estimate guarantees the uniform convergence in the\nwhole half-line t≥0 of the series defining Aαuν,+(t) andAβu′\nν,+(t). Since all summands\nare continuous, and the convergence is uniform, the sum is continu ous as well. Since\nlow-frequency components uν,−(t) andu′\nν,−(t) are continuous (see Remark 3.3), (3.2) is\nproved. /square\n5.1.2 Proof of Theorem 3.8\nLet us fix a real number ν≥1 such that 4 δ2ν4σ−2>µ2(such a number exists because of\nour assumptions on δandσ). Then there exists r>0 such that the three inequalities in\n(5.6) hold true for every λ≥ν. Therefore, we can apply statement (3) of Lemma 5.1 to\nall components uk(t) ofu(t) corresponding to eigenvalues λk≥ν. Ifu0kandu1kdenote\nthe corresponding components of initial data, estimate (5.7) read as\n/parenleftBig\nλ4β\nk|u′\nk(t)|2+λ4α\nk|uk(t)|2/parenrightBig\nexp/parenleftbigg\n2rλ2(1−σ)\nk/integraldisplayt\n0c(s)ds/parenrightbigg\n≤K/parenleftBig\nλ4β\nk|u1k|2+λ4α\nk|u0k|2/parenrightBig\nfor everyt≥0, whereKis a suitable constant depending only on µ2andδ. Summing\nover allλk≥νwe obtain exactly (3.11). The continuity of u(t) andu′(t) with values\nin the suitable spaces follows from the uniform convergence of serie s as in the proof of\nTheorem 3.1. /square\n5.2 Subcritical dissipation\nLet us consider the case 0 ≤σ≤1/2. The key tool is the following.\nLemma 5.2. Let us consider problem (1.8)–(5.1) under the following ass umptions:\n•the coefficient c: [0,+∞)→Rsatisfies the strict hyperbolicity assumption (1.4)\nand theω-continuity assumption (2.7) for some continuity modulus ω(x),\n•δ>0,λ>0, andσ≥0are real numbers satisfying (3.6).\n21Then the solution u(t)satisfies the following estimates.\n(1) It turns out that\n|u′(t)|2+2λ2µ1|u(t)|2≤4u2\n1+2/parenleftbig\n3δ2λ4σ+λ2µ2/parenrightbig\nu2\n0∀t≥0.(5.18)\n(2) Let us assume in addition that λ≥1,σ∈[0,1/2], and there exists a constant\nr∈(0,δ)such that\n4(δ−r)(δµ1−rµ2)≥/bracketleftbigg\nλ1−2σω/parenleftbigg1\nλ/parenrightbigg/bracketrightbigg2\n+2δ(1+2r)/bracketleftbigg\nλ1−2σω/parenleftbigg1\nλ/parenrightbigg/bracketrightbigg\n+8rδ3.(5.19)\nThen for every t≥0it turns out that\n|u′(t)|2+2λ2µ1|u(t)|2≤/bracketleftbig\n4u2\n1+2/parenleftbig\n3δ2λ4σ+λ2µ2/parenrightbig\nu2\n0/bracketrightbig\nexp/parenleftbig\n−2rλ2σt/parenrightbig\n.(5.20)\nProofFor everyε>0 we introduce the regularized coefficient\ncε(t) :=1\nε/integraldisplayt+ε\ntc(s)ds∀t≥0.\nIt is easy to see that cε∈C1([0,+∞),R) and satisfies the following estimates:\nµ1≤cε(t)≤µ2∀t≥0, (5.21)\n|c(t)−cε(t)| ≤ω(ε)∀t≥0, (5.22)\n|c′\nε(t)| ≤ω(ε)\nε∀t≥0. (5.23)\nApproximated energy For everyε >0 we consider the approximated hyperbolic\nenergyEε(t) defined in (1.10). Since\n−1\n2|u′(t)|2−2δ2λ4σ|u(t)|2≤2δλ2σu(t)u′(t)≤ |u′(t)|2+δ2λ4σ|u(t)|2,\nwe deduce that\n1\n2|u′(t)|2+µ1λ2|u(t)|2≤Eε(t)≤2|u′(t)|2+(3δ2λ4σ+λ2µ2)|u(t)|2(5.24)\nfor everyε>0 andt≥0. The time-derivative of Eε(t) is\nE′\nε(t) =−2δλ2σ|u′(t)|2−2δλ2σ+2c(t)|u(t)|2\n−2λ2(c(t)−cε(t))u(t)u′(t)+λ2c′\nε(t)|u(t)|2, (5.25)\nhence\nE′\nε(t)≤ −2δλ2σ|u′(t)|2−/parenleftbig\n2δλ2σ+2c(t)−λ2|c′\nε(t)|/parenrightbig\n|u(t)|2\n+2λ2|c(t)−cε(t)|·|u(t)|·|u′(t)|. (5.26)\n22Statement (1) We claim that, for a suitable choice of ε, it turns out that\nE′\nε(t)≤0∀t≥0. (5.27)\nIf we prove this claim, then we apply (5.24) with that particular value o fεand we\nobtain that\n1\n2|u′(t)|2+µ1λ2|u(t)|2≤Eε(t)≤Eε(0)≤2u2\n1+(3δ2λ4σ+λ2µ2)u2\n0,\nwhich is equivalent to (5.18).\nInordertoprove(5.27),weconsiderthewholeright-handsideof( 5.26)asaquadratic\nform in|u(t)|and|u′(t)|. Since the coefficient of |u′(t)|2is negative, this quadratic form\nis less than or equal to 0 for all values of |u(t)|and|u′(t)|if and only if\n2δλ2σ·/parenleftbig\n2δλ2σ+2c(t)−λ2|c′\nε(t)|/parenrightbig\n−λ4|c(t)−cε(t)|2≥0,\nhence if and only if\n4δ2λ4σ−2c(t)≥ |c(t)−cε(t)|2+2δλ2σ−2|c′\nε(t)|. (5.28)\nNow in the left-hand side we estimate c(t) from below with µ1, and we estimate from\nabove the terms in the right-hand side as in (5.22) and (5.23). We obt ain that (5.28)\nholds true whenever\n4δ2µ1≥ω2(ε)\nλ4σ−2+2δω(ε)\nλ2σε.\nThis condition is true when ε:= 1/λthanks to assumption (3.6). This completes\nthe proof of (5.18).\nStatement (2) Let us assume now that λ≥1 and that (5.19) holds true for some\nr∈(0,δ). In this case we claim that, for a suitable choice of ε>0, the stronger estimate\nE′\nε(t)≤ −2rλ2σEε(t)∀t≥0 (5.29)\nholds true, hence\nEε(t)≤Eε(0)exp/parenleftbig\n−2rλ2σt/parenrightbig\n∀t≥0.\nDue to (5.24), this is enough to deduce (5.20). So it remains to prove (5.29).\nOwing to (5.25), inequality (5.29) is equivalent to\n2λ2σ(δ−r)|u′(t)|2+/bracketleftbig\n2λ2σ+2(δc(t)−rcε(t))−λ2c′\nε(t)−4rδ2λ6σ/bracketrightbig\n|u(t)|2\n+2/bracketleftbig\nλ2(c(t)−cε(t))−2rδλ4σ/bracketrightbig\nu(t)u′(t)≥0.\nKeeping (1.4) and (5.21) into account, the last inequality is proved if w e show that\n2λ2σ(δ−r)|u′(t)|2+/bracketleftbig\n2λ2σ+2(δµ1−rµ2)−λ2|c′\nε(t)|−4rδ2λ6σ/bracketrightbig\n|u(t)|2\n23−2/bracketleftbig\nλ2|c(t)−cε(t)|+2rδλ4σ/bracketrightbig\n|u(t)|·|u′(t)| ≥0.\nAs in the proof of the first statement, we consider the whole left-h and side as a\nquadratic form in |u(t)|and|u′(t)|. The coefficient of |u′(t)|is positive because r < δ.\nTherefore, this quadratic form is nonnegative for all values of |u(t)|and|u′(t)|if and\nonly if\n2λ2σ(δ−r)·/bracketleftbig\n2λ2σ+2(δµ1−rµ2)−λ2|c′\nε(t)|−4rδ2λ6σ/bracketrightbig\n≥/bracketleftbig\nλ2|c(t)−cε(t)|+2rδλ4σ/bracketrightbig2.\nNow we rearrange the terms, and we exploit (5.22) and (5.23). We ob tain that the\nlast inequality is proved if we show that\n4(δ−r)(δµ1−rµ2)≥λ2−4σω2(ε)+2δ/parenleftbig\n1+2rελ2σ/parenrightbigω(ε)\nελ2σ+8rδ3\nλ2−4σ.(5.30)\nFinally, we choose ε:= 1/λ, so that (5.30) becomes\n4(δ−r)(δµ1−rµ2)≥/bracketleftbigg\nλ1−2σω/parenleftbigg1\nλ/parenrightbigg/bracketrightbigg2\n+2δ/parenleftbigg\n1+2r\nλ1−2σ/parenrightbigg/bracketleftbigg\nλ1−2σω/parenleftbigg1\nλ/parenrightbigg/bracketrightbigg\n+8rδ3\nλ2−4σ.\nSinceλ≥1 andσ≤1/2, this inequality follows from assumption (5.19). /square\n5.2.1 Proof of Theorem 3.2\nLet us rewrite (3.5) in the form\nΛ∞= limsup\nλ→+∞λ1−2σω/parenleftbigg1\nλ/parenrightbigg\n. (5.31)\nDue to (3.4), there exists ν≥1 such that (3.6) holds true for every λ≥ν. Therefore,\nwe can apply statement (1) of Lemma 5.2 to the components uk(t) ofu(t) corresponding\nto eigenvalues λk≥ν. Ifu0kandu1kdenote the corresponding components of initial\ndata, estimate (5.18) read as\n|u′\nk(t)|2+2λ2\nkµ1|uk(t)|2≤4|u1k|2+2/parenleftbig\n3δ2λ4σ\nk+λ2\nkµ2/parenrightbig\n|u0k|2.\nSinceσ≤1/2 and we chose ν≥1, this implies that\n|u′\nk(t)|2+2λ2\nkµ1|uk(t)|2≤4|u1k|2+2/parenleftbig\n3δ2+µ2/parenrightbig\nλ2\nk|u0k|2.\nSumming over all λk≥νwe obtain exactly (3.7).\nThis proves that uν,+(t) is bounded with values in D(A1/2) andu′\nν,+(t) is bounded\nwith values in H. The continuity of u(t) andu′(t) with values in the same spaces follows\nfrom the uniform convergence of series as in the proof of Theorem 3.1./square\n245.2.2 Proof of Theorem 3.9\nLet us rewrite (3.5) in the form (5.31). Due to (3.4), there exists r >0 andν≥1\nsuch that (5.19) holds true for every λ≥ν. Therefore, we can apply statement (2) of\nLemma 5.2 to the components uk(t) ofu(t) corresponding to eigenvalues λk≥ν. Ifu0k\nandu1kdenote the corresponding components of initial data, estimate (5 .20) reads as\n/parenleftbig\n|u′\nk(t)|2+2λ2\nkµ1|uk(t)|2/parenrightbig\nexp/parenleftbig\n2rλ2σ\nkt/parenrightbig\n≤4|u1k|2+2/parenleftbig\n3δ2λ4σ\nk+λ2\nkµ2/parenrightbig\n|u0k|2.\nSinceσ≤1/2 and we chose ν≥1, this implies that\n/parenleftbig\n|u′\nk(t)|2+2λ2\nkµ1|uk(t)|2/parenrightbig\nexp/parenleftbig\n2rλ2σ\nkt/parenrightbig\n≤4|u1k|2+2/parenleftbig\n3δ2+µ2/parenrightbig\nλ2\nk|u0k|2\nfor everyt≥0. Summing over all λk≥νwe obtain (3.12) with a constant Kdepending\nonly onµ1,µ2, andδ. The continuity of u(t) andu′(t) with values in the suitable spaces\nfollows from the uniform convergence of series as in the proof of Th eorem 3.1. /square\n6 The (DGCS)-phenomenon\nIn this section we prove Theorem 3.10. Let us describe the strateg y before entering into\ndetails. Roughly speaking, what we need is a solution u(t) whose components uk(t) are\nsmall at time t= 0 and huge at time t>0. The starting point is given by the following\nfunctions\nb(ε,λ,t) := (2ελ−δλ2σ)t−εsin(2λt),\nw(ε,λ,t) := sin(λt)exp(b(ε,λ,t)), (6.1)\nγ(ε,λ,t) := 1+δ2\nλ2−4σ−16ε2sin4(λt)−8εsin(2λt). (6.2)\nWith some computations it turns out that\nw′′(ε,λ,t)+2δλ2σw′(ε,λ,t)+λ2γ(ε,λ,t)w(ε,λ,t) = 0 ∀t∈R,\nwhere “primes” denote differentiation with respect to t. As a consequence, if we set\nc(t) :=γ(ε,λ,t) andε:=ω(1/λ), the function u(t) :=w(ε,λ,t) turns out to be a\nsolution to (1.8) which grows as the right-hand side of (4.5). Unfort unately this is not\nenough, because we need to realize a similar growth for countably ma ny components\nwith the same coefficient c(t).\nTo this end, we argue as in [6]. We introduce a suitable decreasing sequ encetk→0+,\nand in the interval [ tk,tk−1] we design the coefficient c(t) so thatuk(tk) is small and\nuk(tk−1) is huge. Then we check that the piecewise defined coefficient c(t) has the\nrequired time-regularity, and that uk(t) remains small for t∈[0,tk] and remains huge\nfort≥tk−1. This completes the proof.\nRoughly speaking, the coefficient c(t) plays on infinitely many time-scales in order\nto “activate” countably many components, but these countably m any actions take place\n25onebyoneindisjointtimeintervals. Ofcoursethismeansthatthelen gthstk−1−tkofthe\n“activationintervals”tendto0as k→+∞. Inordertoobtainenoughgrowth, despiteof\nthe vanishing length of activationintervals, we areforced to assum e thatλω(1/λ)≫λ2σ\nasλ→+∞. In addition, components do not grow exactly as exp( λω(1/λ)t), but just\nmore than exp( ϕ(λ)t) and exp(ψ(λ)t).\nThis is the reason why this strategy does not work when λω(1/λ)∼λ2σandδ\nis small. In this case one would need components growing exactly as ex p(λω(1/λ)t),\nbut this requires activation intervals of non-vanishing length, which are thus forced to\noverlap. In a certain sense, the coefficient c(t) should work once againoninfinitely many\ntime-scales, but now the countably many actions should take place in the same time.\nDefinition of sequences From (3.13) and (3.14) it follows that\nlim\nx→+∞x1−2σω/parenleftbigg1\nx/parenrightbigg\n= +∞, (6.3)\nlim\nx→+∞1\nx1−2σω(1/x)+ϕ(x)\nxω(1/x)+ψ(x)\nxω(1/x)= 0, (6.4)\nand a fortiori\nlim\nx→+∞x1+2σω/parenleftbigg1\nx/parenrightbigg\n= +∞, (6.5)\nlim\nx→+∞x2σ+ϕ(x)+ψ(x)\nxω/parenleftbigg1\nx/parenrightbigg\n= 0. (6.6)\nLet us consider the sequence {λk}, which we assumed to be unbounded. Due to\n(6.5) and (6.4) we can assume, up to passing to a subsequence (not relabeled), that the\nfollowing inequalities hold true for every k≥1:\nλk>4λk−1, (6.7)\nλ1+2σ\nkω/parenleftbigg1\nλk/parenrightbigg\n≥δ4\n210π21\nλ2−8σ\nk−1+4k2\nπ2λ2\nk−1, (6.8)\nλ1+2σ\nkω/parenleftbigg1\nλk/parenrightbigg\n≥4k2\nπ2λ3\nk−1/parenleftbig\nλ2σ\nk−1+ϕ(λk−1)+ψ(λk−1)/parenrightbig\nω/parenleftbigg1\nλk−1/parenrightbigg\n,(6.9)\nλ1+2σ\nkω/parenleftbigg1\nλk/parenrightbigg\n≥λk−1/parenleftbig\nλ2σ\nk−1+ϕ(λk−1)+ψ(λk−1)/parenrightbig\nω/parenleftbigg1\nλk−1/parenrightbigg\n,(6.10)\n1\nλ1−2σ\nkω(1/λk)+ϕ(λk)\nλkω(1/λk)+ψ(λk)\nλkω(1/λk)≤π2\n4k21\nλ2\nk−1. (6.11)\nNow let us set\ntk:=4π\nλk, s k:=π\nλk/floorleftbigg\n2λk\nλk−1/floorrightbigg\n, (6.12)\n26where⌊α⌋denotes the largest integer less than or equal to α, and\nεk:=/braceleftbiggλ2σ\nk+ϕ(λk)+ψ(λk)\nλkω/parenleftbigg1\nλk/parenrightbigg/bracerightbigg1/2\n.\nProperties of the sequences We collect in this section of the proof all the properties\nof the sequences which are needed in the sequel. First of all, it is clear thatλk→+∞,\nhencetk→0 andεk→0 (because of (6.6)). Moreover it turns out that\ntk−1\n4=π\nλk−1≤sk≤2π\nλk−1=tk−1\n2. (6.13)\nKeeping (6.7) into account, it follows that\ntk<sk<tk−1∀k≥1,\nand in particular also sk→0. In addition, it turns out that\nsin(λktk) = sin(λksk) = 0 (6.14)\nand\n|cos(λktk)|=|cos(λksk)|= 1. (6.15)\nSinceσ <1/2,λk→+∞,εk→0,tk→0, keeping (6.3) and (6.4) into account, we\ndeduce that the following seven inequalities are satisfied provided th atkis large enough:\nδ2\nλ2−4σ\nk+16ε2\nk+8εk≤1\n2, (6.16)\nεk≤1\n4, (6.17)\n16πεk+16πδ\nλ1−2σ\nk≤2π, (6.18)\n1\nλ1−2σ\nkω(1/λk)+ϕ(λk)\nλkω(1/λk)+ψ(λk)\nλkω(1/λk)≤1\n52·210·π2, (6.19)\nδ2\n(4π)2−4σ(2tk)1−2σsup/braceleftbiggx1−2σ\nω(x):x∈(0,tk)/bracerightbigg\n≤1\n5, (6.20)\nλ1−2σ\nkω/parenleftbigg1\nλk/parenrightbigg\n≥δ2, (6.21)\n2δ2\nλ2−4σ\nk−1ω(1/λk−1)≤1\n5. (6.22)\n27Letk0∈Nbe a positive integer such that (6.16) through (6.22) hold true for e very\nk≥k0. From (6.21) it follows that\nεkλk≥δλ2σ\nk∀k≥k0. (6.23)\nFrom (6.19) it follows that\n32πεk\nω(1/λk)≤1\n5∀k≥k0. (6.24)\nSincesk≥π/λk−1(see the estimate from below in (6.13)), from (6.8) it follows that\nεkλksk≥δ2\n321\nλ2−4σ\nk−1∀k≥k0, (6.25)\nεkλksk≥2k∀k≥k0, (6.26)\nfrom (6.9) it follows that\nεkλksk≥2kεk−1λk−1∀k≥k0, (6.27)\nand from (6.11) it follows that\nεkλksk≥2k/parenleftbig\nλ2σ\nk+ϕ(λk)+ψ(λk)/parenrightbig\n∀k≥k0. (6.28)\nAs a consequence of (6.26) through (6.28) it turns out that\n2εkλksk≥kεk−1λk−1+2k/parenleftbig\nλ2σ\nk+ϕ(λk)+ψ(λk)/parenrightbig\n+k∀k≥k0.(6.29)\nFinally, from (6.10) it follows that\nεkλk≥εk−1λk−1∀k≥k0. (6.30)\nDefinition of c(t)andu(t) For every k≥1, letℓk:R→Rbe defined by\nℓk(t) :=δ2\ntk−1−sk/parenleftbigg1\nλ2−4σ\nk−1−1\nλ2−4σ\nk/parenrightbigg\n(t−sk)+1+δ2\nλ2−4σ\nk∀t∈R.\nThanks to (6.14), ℓk(t) is the affine function such that\nℓk(sk) =γ(εk,λk,sk) and ℓk(tk−1) =γ(εk−1,λk−1,tk−1).\nLetk0∈Nbe such that (6.16) through (6.22) hold true for every k≥k0. Let us set\nc(t) :=\n\n1 if t≤0,\nγ(εk,λk,t) ift∈[tk,sk] for somek≥k0,\nℓk(t) if t∈[sk,tk−1] for somek≥k0+1,\nγ(εk0,λk0,sk0) ift≥sk0.\n28The following picture describes this definition. The coefficient c(t) is constant for\nt≤0 and fort≥sk0. In the intervals [ tk,sk] it coincides with γ(εk,λk,t), hence it\noscillates, with period of order λ−1\nkand amplitude of order εk, around a value which\ntends to 1. In the intervals [ sk,tk−1] it is just the affine interpolation of the values at\nthe endpoints.\n/Bullet /Bullet /Bullet /Bullet /Bullet /Bullet /Bullet\nsk+2tk+1sk+1tksk tk−1sk−1period∼λ−1\nkperiod∼λ−1\nk−1\n1+δ2\nλ2−4σ\nk∼εk∼εk−1\nFor everyk≥k0, let us consider the solution uk(t) to the Cauchy problem\nu′′\nk(t)+2δλ2σ\nku′\nk(t)+λ2\nkc(t)uk(t) = 0,\nwith “initial” data\nuk(tk) = 0, u′\nk(tk) =λkexp/parenleftbig\n(2εkλk−δλ2σ\nk)tk/parenrightbig\n. (6.31)\nThen we set\nak:=1\nkλkexp(−kϕ(λk)), (6.32)\nand finally\nu(t) :=∞/summationdisplay\nk=k0akuk(t)ek.\nWeclaim that c(t) satisfies (3.15)and(3.16), andthat u(t)satisfies (3.17)and(3.18).\nThe rest of the proof is a verification of these claims.\nEstimate and continuity of c(t) We prove that\n|c(t)−1| ≤1\n2∀t≥0, (6.33)\n29which is equivalent to (3.15), and that c(t) is continuous on the whole real line.\nTo this end, it is enough to check (6.33) in the intervals [ tk,sk], because in the\nintervals [sk,tk−1]the function c(t) isjust aninterpolationofthevalues at theendpoints,\nand it is constant for t≤0 and fort≥sk0.\nIn the intervals [ tk,sk] the function c(t) coincides with γ(εk,λk,t), hence from (6.2)\nit turns out that\n|c(t)−1|=|γ(εk,λk,t)−1| ≤δ2\nλ2−4σ\nk+16ε2\nk+8εk, (6.34)\nso that (6.33) follows immediately from (6.16).\nSince the right-hand side of (6.34) tends to 0 as k→+∞, the same estimate shows\nalso thatc(t)→1 ast→0+, which proves the continuity of c(t) int= 0, the only point\nin which continuity was nontrivial.\nEstimate on c′(t) We prove that\n|c′(t)| ≤32εkλk∀t∈(tk,sk),∀k≥k0, (6.35)\n|c′(t)| ≤32εkλk∀t∈(sk,tk−1),∀k≥k0+1. (6.36)\nIndeed in the interval ( tk,sk) it turns out that\n|c′(t)|=|γ′(εk,λk,t)|=/vextendsingle/vextendsingle−64ε2\nkλksin3(λkt)cos(λkt)−16εkλkcos(2λkt)/vextendsingle/vextendsingle\n≤64ε2\nkλk+16εkλk= 16εkλk(4εk+1),\nso that (6.35) follows from (6.17).\nIn the interval ( sk,tk−1) it turns out that\n|c′(t)|=δ2\ntk−1−sk/parenleftbigg1\nλ2−4σ\nk−1−1\nλ2−4σ\nk/parenrightbigg\n≤δ2\ntk−1−sk·1\nλ2−4σ\nk−1≤δ2\nsk·1\nλ2−4σ\nk−1,\nwhere the last inequality follows from the estimate from above in (6.13 ). At this point\n(6.36) is equivalent to (6.25).\nModulus of continuity of c(t) Let us prove that c(t) satisfies (3.16). Since c(t) is\ncontinuous, and constant for t≤0 andt≥sk0, it is enough to verify its ω-continuity in\n(0,sk0]. In turn, the ω-continuity in (0 ,sk0] is proved if we show that\n|c(ti)−c(tj)| ≤1\n5ω(|ti−tj|)∀i≥k0,∀j≥k0, (6.37)\n|c(a)−c(b)| ≤1\n5ω(|a−b|)∀(a,b)∈[tk,sk]2,∀k≥k0, (6.38)\n30|c(a)−c(b)| ≤1\n5ω(|a−b|)∀(a,b)∈[sk,tk−1]2,∀k≥k0+1. (6.39)\nIndeed, any interval [ s,t]⊆(0,sk0] can be decomposed as the union of at most 5\nintervals whose endpoints fit in one of the 3 possibilities above.\nLet us prove (6.37). From (6.14) it turns out that\n|c(ti)−c(tj)|=δ2/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle1\nλ2−4σ\ni−1\nλ2−4σ\nj/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle≤δ2/vextendsingle/vextendsingle/vextendsingle/vextendsingle1\nλ2\ni−1\nλ2\nj/vextendsingle/vextendsingle/vextendsingle/vextendsingle1−2σ\n,\nwhere the inequality follows from the fact that the function x→x1−2σis (1−2σ)-H¨ older\ncontinuous with constant equal to 1. Now from (6.12) it follows that\nδ2/vextendsingle/vextendsingle/vextendsingle/vextendsingle1\nλ2\ni−1\nλ2\nj/vextendsingle/vextendsingle/vextendsingle/vextendsingle1−2σ\n=δ2\n(4π)2−4σ|t2\ni−t2\nj|1−2σ=δ2\n(4π)2−4σ|ti+tj|1−2σ|ti−tj|1−2σ\nω(|ti−tj|)ω(|ti−tj|).\nSince|ti+tj| ≤2tk0and|ti−tj| ≤tk0, we conclude that\n|c(ti)−c(tj)| ≤δ2\n(4π)2−4σ(2tk0)1−2σsup/braceleftbiggx1−2σ\nω(x):x∈(0,tk0)/bracerightbigg\nω(|ti−tj|),\nso that (6.37) follows from (6.20).\nLet us prove (6.38). Since c(t) isπ/λkperiodic in [ tk,sk], for every ( a,b)∈[tk,sk]2\nthere exists ( a1,b1)∈[tk,sk]2such thatc(a) =c(a1),c(b) =c(b1), and|a1−b1| ≤π/λk.\nThus from (6.35) it follows that\n|c(a)−c(b)|=|c(a1)−c(b1)| ≤32εkλk|a1−b1|= 32εkλk|a1−b1|\nω(|a1−b1|)ω(|a1−b1|),\nso that we are left to prove that\n32εkλk|a1−b1|\nω(|a1−b1|)≤1\n5. (6.40)\nDue to (2.6), (2.5), and the fact that |a1−b1| ≤π/λk, it turns out that\n|a1−b1|\nω(|a1−b1|)≤π/λk\nω(π/λk)≤π\nλkω(1/λk),\nso that now (6.40) follows from (6.24).\nLet us prove (6.39). Since c(t) is affine in [ sk,tk−1], for every aandbin this interval\nit turns out that\n|c(a)−c(b)|=δ2\ntk−1−sk/parenleftbigg1\nλ2−4σ\nk−1−1\nλ2−4σ\nk/parenrightbigg\n|a−b|.\n31Sincesk≤tk−1/2, it follows that\n|c(a)−c(b)| ≤2δ2\ntk−11\nλ2−4σ\nk−1·|a−b|=2δ2\ntk−11\nλ2−4σ\nk−1·|a−b|\nω(|a−b|)·ω(|a−b|).\nDue to (2.6), (2.5), and the fact that |a−b| ≤tk−1, it turns out that\n|a−b|\nω(|a−b|)≤tk−1\nω(tk−1)≤tk−1\nω(1/λk−1),\nso that now (6.39) is a simple consequence of (6.22).\nEnergy functions Let us introduce the classic energy functions\nEk(t) :=|u′\nk(t)|2+λ2\nk|uk(t)|2,\nFk(t) :=|u′\nk(t)|2+λ2\nkc(t)|uk(t)|2.\nDue to (3.15), they are equivalent in the sense that\n1\n2Ek(t)≤Fk(t)≤3\n2Ek(t)∀t∈R.\nTherefore, proving (3.17) is equivalent to showing that\n∞/summationdisplay\nk=k0a2\nkEk(0)exp(2rϕ(λk))<+∞ ∀r>0, (6.41)\nwhile proving (3.18) is equivalent to showing that\n∞/summationdisplay\nk=k0a2\nkFk(t)exp(−2Rψ(λk)) = +∞ ∀R>0,∀t>0. (6.42)\nWe are thus left to estimating Ek(0) andFk(t).\nEstimates in [0,tk] We prove that\nEk(0)≤λ2\nkexp(4π)∀k≥k0. (6.43)\nTo this end, we begin by estimating Ek(tk). From (6.31) we obtain that uk(tk) = 0\nand\n|u′\nk(tk)| ≤λkexp(2εkλktk) =λkexp(8πεk),\nso that\nEk(tk)≤λ2\nkexp(16πεk). (6.44)\n32Now the time-derivative of Ek(t) is\nE′\nk(t) =−4δλ2σ\nk|u′\nk(t)|2−2λ2\nk(c(t)−1)u′\nk(t)uk(t)∀t∈R.\nTherefore, from (3.15) it follows that\nE′\nk(t)≥ −4δλ2σ\nkEk(t)−λk|c(t)−1|·2|u′\nk(t)|·λk|uk(t)| ≥ −/parenleftbigg\n4δλ2σ\nk+λk\n2/parenrightbigg\nEk(t)\nfor everyt∈R. Integrating this differential inequality in [0 ,tk] we deduce that\nEk(0)≤Ek(tk)exp/bracketleftbigg/parenleftbigg\n4δλ2σ\nk+λk\n2/parenrightbigg\ntk/bracketrightbigg\n.\nKeeping (6.44) and (6.12) into account, we conclude that\nEk(0)≤λ2\nkexp/parenleftbigg\n16πεk+16πδ\nλ1−2σ\nk+2π/parenrightbigg\n,\nso that (6.43) follows immediately from (6.18).\nEstimates in [tk,sk] In this interval it turns out that uk(t) :=w(εk,λk,t), where\nw(ε,λ,t) is the function defined in (6.1). Keeping (6.14) and (6.15) into accou nt, we\nobtain that uk(sk) = 0 and\n|u′\nk(sk)|=λkexp(b(εk,λk,sk)) =λkexp/parenleftbig\n(2εkλk−δλ2σ\nk)sk/parenrightbig\n.\nTherefore, from (6.23) it follows that\n|u′\nk(sk)| ≥λkexp(εkλksk),\nand hence\nFk(sk) =Ek(sk)≥λ2\nkexp(2εkλksk). (6.45)\nEstimates in [sk,tk−1] We prove that\nFk(tk−1)≥λ2\nkexp(2εkλksk−4δλ2σ\nktk−1). (6.46)\nIndeed the time-derivative of Fk(t) is\nF′\nk(t) =−4δλ2σ\nk|u′\nk(t)|2+λ2\nkc′(t)|uk(t)|2∀t∈(sk,tk−1).\nSincec′(t)>0 in (sk,tk−1), it follows that\nF′\nk(t)≥ −4δλ2σ\nk|u′\nk(t)|2≥ −4δλ2σ\nkFk(t)∀t∈(sk,tk−1),\nand hence\nFk(tk−1)≥Fk(sk)exp/parenleftbig\n−4δλ2σ\nk(tk−1−sk)/parenrightbig\n≥Fk(sk)exp/parenleftbig\n−4δλ2σ\nktk−1/parenrightbig\n.\nKeeping (6.45) into account, we have proved (6.46).\n33Estimates in [tk−1,+∞) We prove that\nFk(t)≥λ2\nkexp/parenleftbig\n2εkλksk−8δλ2σ\nkt−64εk−1λk−1t/parenrightbig\n∀t≥tk−1.(6.47)\nTo this end, let us set\nIk:= [tk−1,+∞)\\k−1/uniondisplay\ni=k0{ti,si}.\nFirst of all, we observe that\n|c′(t)| ≤32εk−1λk−1∀t∈Ik (6.48)\nIndeed we know from (6.35) and (6.36) that\n|c′(t)| ≤32εiλi∀t∈(ti,si)∪(si,ti−1),\nand of course c′(t) = 0 for every t>sk0. Now it is enough to observe that\nIk= (tk0,sk0)∪(sk0,+∞)∪k−1/uniondisplay\ni=k0+1[(ti,si)∪(si,ti−1)],\nand thatεiλiis a nondecreasing sequence because of (6.30).\nNow we observe that the function t→Fk(t) is continuous in [ tk−1,+∞) and differ-\nentiable in Ik, with\nF′\nk(t) =−4δλ2σ\nk|u′\nk(t)|2+λ2\nkc′(t)|uk(t)|2\n≥ −4δλ2σ\nk|u′\nk(t)|2−|c′(t)|\nc(t)·λ2\nkc(t)|uk(t)|2\n≥ −/parenleftbigg\n4δλ2σ\nk+|c′(t)|\nc(t)/parenrightbigg\nFk(t).\nTherefore, from (6.48) and (3.15) it follows that\nF′\nk(t)≥ −/parenleftbig\n4δλ2σ\nk+64εk−1λk−1/parenrightbig\nFk(t)∀t∈Ik,\nand hence\nFk(t)≥Fk(tk−1)exp/bracketleftbig\n−/parenleftbig\n4δλ2σ\nk+64εk−1λk−1/parenrightbig\n(t−tk−1)/bracketrightbig\n≥Fk(tk−1)exp/bracketleftbig\n−/parenleftbig\n4δλ2σ\nk+64εk−1λk−1/parenrightbig\nt/bracketrightbig\nfor everyt≥tk−1. Keeping (6.46) into account, we finally obtain that\nFk(t)≥λ2\nkexp/parenleftbig\n2εkλksk−4δλ2σ\nktk−1−4δλ2σ\nkt−64εk−1λk−1t/parenrightbig\n,\nfrom which (6.47) follows by simply remarking that t≥tk−1.\n34Conclusion We are now ready to verify (6.41) and (6.42). Indeed from (6.32) an d\n(6.43) it turns out that\na2\nkEk(0)exp(2rϕ(λk))≤1\nk2λ2\nkexp(−2kϕ(λk))·λ2\nkexp(4π)·exp(2rϕ(λk))\n=1\nk2exp(4π+2(r−k)ϕ(λk)).\nThe argument of the exponential is less than 4 πwhenkis large enough, and hence\nthe series in (6.41) converges.\nLet us consider now (6.42). For every t>0 it turns out that t≥tk−1whenkis large\nenough. For every such kwe can apply (6.47) and obtain that\na2\nkFk(t)exp(−2Rψ(λk))\n≥1\nk2exp/parenleftbig\n−2kϕ(λk)−2Rψ(λk)+2εkλksk−8δλ2σ\nkt−64εk−1λk−1t/parenrightbig\n.\nKeeping (6.29) into account, it follows that\na2\nkFk(t)exp(−2Rψ(λk))\n≥1\nk2exp/parenleftbig\n(k−64t)εk−1λk−1+2(k−R)ψ(λk)+(2k−8δt)λ2σ\nk+k/parenrightbig\n≥1\nk2exp(k)\nwhenkis large enough. This proves that the series in (6.42) diverges. /square\nReferences\n[1]G. Chen, D. L. Russell ; A mathematical model for linear elastic systems with\nstructural damping. Quart. Appl. Math. 39(1981/82), no. 4, 433–454.\n[2]S. P. Chen, R. Triggiani ; Proof of extensions of two conjectures on structural\ndamping for elastic systems. Pacific J. Math. 136(1989), no. 1, 15–55.\n[3]S. P. Chen, R. Triggiani ; Characterization of domains of fractional powers of\ncertain operators arising in elastic systems, and applications. J. Differential Equa-\ntions88(1990), no. 2, 279–293.\n[4]S. P. Chen, R. Triggiani ; Gevrey class semigroups arising from elastic systems\nwith gentle dissipation: the case 0 <α<1/2.Proc. Amer. Math. Soc. 110(1990),\nno. 2, 401–415.\n[5]F. Colombini ; Quasianalytic and nonquasianalytic solutions for a class of weakly\nhyperbolic Cauchy problems. J. Differential Equations 241(2007), no. 2, 293–304.\n35[6]F. Colombini, E. De Giorgi, S. Spagnolo ; Sur le ´ equations hyperboliques\navec des coefficients qui ne d´ ependent que du temp. (French) Ann. Scuola Norm.\nSup. Pisa Cl. Sci. (4) 6(1979), no. 3, 511–559.\n[7]F. Colombini, E. Jannelli, S. Spagnolo ; Well-posedness intheGevrey classes\nof the Cauchy problem for a nonstrictly hyperbolic equation with coe fficients de-\npending on time. Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4)10(1983), no. 2, 291–\n312.\n[8]M. Ghisi, M. Gobbino ; DerivativelossforKirchhoffequationswithnon-Lipschitz\nnonlinear term. Ann. Scuola Norm. Sup. Pisa Cl. Sci. (5) 8(2009), no. 4, 613–646.\n[9]M. Ghisi, M. Gobbino, H. Haraux ; Local and global smoothing effects for\nsome linear hyperbolic equations with a strong dissipation. Trans. Amer. Math.\nSoc.To appear. Preprint arXiv:1402.6595 [math.AP] .\n[10]A. Haraux, M. ˆOtani; Analyticity and regularity for a class of second order\nevolution equation. Evol. Equat. Contr. Theor. 2(2013), no. 1, 101–117.\n[11]R. Ikehata ; Decayestimatesofsolutionsforthewaveequationswithstrongd amp-\ning terms in unbounded domains. Math. Methods Appl. Sci. 24(2001), no. 9, 659–\n670.\n[12]R. Ikehata, M. Natsume ; Energy decay estimates for wave equations with a\nfractional damping. Differential Integral Equations 25(2012), no. 9-10, 939–956.\n[13]R. Ikehata, G. Todorova, B. Yordanov ; Wave equations with strong damp-\ning in Hilbert spaces. J. Differential Equations 254(2013), no. 8, 3352–3368.\n[14]J.-L. Lions, E. Magenes , Probl` emes aux limites non homog` enes et applications.\nVol. 3. (French) Travaux et Recherches Mathmatiques, No. 20. D unod, Paris, 1970.\n[15]S. Matthes, M. Reissig ; Qualitative properties of structural damped wave mod-\nels.Eurasian Math. J. 4(2013), no. 3, 84–106.\n[16]K. Nishihara ; Degenerate quasilinear hyperbolic equation with strong damping.\nFunkcial. Ekvac. 27(1984), no. 1, 125–145.\n[17]K. Nishihara ; Decay properties of solutions of some quasilinear hyperbolic equa-\ntions with strong damping. Nonlinear Anal. 21(1993), no. 1, 17–21.\n[18]M. Reed, B. Simon ;Methods of Modern Mathematical Physics, I: Functional\nAnalysis. Second edition . Academic Press, New York, 1980.\n[19]Y. Shibata ; Onthe rate of decay of solutions to linear viscoelastic equation. Math.\nMethods Appl. Sci. 23(2000), no. 3, 203–226.\n36"    },    {        "title": "1408.4861v2.Brownian_motion_of_massive_skyrmions_forced_by_spin_polarized_currents.pdf",        "content": "Brownian motion of massive skyrmions forced by spin polarized currents\nRoberto E. Troncoso1,a)and Alvaro S. Núñez2\n1)Centro para el Desarrollo de la Nanociencia y la Nanotecnología, CEDENNA, Avda. Ecuador 3493,\nSantiago 9170124, Chile\n2)Departamento de Física, Facultad de Ciencias Físicas y Matemáticas, Universidad de Chile, Blanco Encalada 2008,\nSantiago, Chile\nWe report on the thermal effects on the motion of current-driven massive magnetic skyrmions. The reduced\nequation for the motion of skyrmion has the form of a stochastic generalized Thiele’s equation. We propose an\nansatz for the magnetization texture of a non-rigid single skyrmion that depends linearly with the velocity. By\nutilizing this ansatz it is found that the mass of skyrmion is closely related to intrinsic skyrmion parameters,\nsuch as Gilbert damping, skyrmion-charge and dissipative force. We have found an exact expression for the\naverage drift velocity as well as the mean-square velocity of the skyrmion. The longitudinal and transverse\nmobility of skyrmions for small spin-velocity of electrons is also determined and found to be independent of\nthe skyrmion mass.\nIntroduction .- Skyrmions have recently been the focus\nof intense research in spintronics. They are vortex-like\nspin structures that are topologically protected1,2. A\nseries of works report their recent observation in chi-\nral magnets3–8. There is a great interest in their dy-\nnamics due to the potential applications in spintronics\nthat arise from the rather low current densities that\nare necessary to manipulate their location9. Among\nother systems that have been reported hosting skyrmion\ntextures they were observed in bulk magnets MnSi3,4,\nFe1\u0000xCoxSi5,6,10, Mn 1\u0000xFexGe11and FeGe12by means\nof neutron scattering and Lorentz transmission elec-\ntron microscopy. Regarding their dimensions, by the\nproper tuning of external magnetic fields, sizes of the\norder of a few tens of nanometers have been reported.\nSpin transfer torques can be used to manipulate and\neven create isolated skyrmions in thin films as shown\nby numerical simulations13–15. In thin films skyrmions\nhave been observed at low temperatures, however en-\nergy estimates predict the stability of isolated skyrmions\neven at room temperatures16. Under that regime the\nmotion of skyrmions is affected by fluctuating thermal\ntorques that will render their trajectories into stochastic\npaths very much like the Brownian dynamics of a par-\nticle. Proper understanding of such brownian motion is\na very important aspect of skyrmion dynamics. Numer-\nical simulations17,18and experimental results19, suggest\nthat the skyrmion position can be manipulated by expo-\nsure to a thermal gradient and that the skyrmions also\ndisplay a thermal creep motion in a pinning potential20.\nThe thermally activated motion of pinned skyrmions has\nbeen studied in Ref. [21] where it has been reported that\nthermal torques can increase the mobility of skyrmions\nby several orders of magnitude. In this work we present\na study of the random motion of magnetic skyrmions\narising from thermal fluctuations. In our analysis we in-\nclude an assessment of the deformation of the skyrmion\nthat arises from its motion. This deformation induces an\na)Electronic mail: R.E.TroncosoCona@gmail.cominertia-like term into the effective stochastic dynamics of\nthe skyrmion. We present a theory that allows us to es-\ntablish a relation between the fluctuating trajectory of\nthe skyrmion and its effective mass.\nStochastic dynamics .- We begin our analysis from the\nstochastic Landau-Lifschitz-Gilbert (LLG) equation22,23\nthat rules the dynamics of the magnetization direction\n\n. Into this equation we need to include the adiabatic,\ngiven by\u0000vs\u0001r\n, and non-adiabatic, given by \fvs\u0001r\n,\nspin-transfer torques24,25where the strength of the non-\nadiabatic spin-transfer torque is characterized by the pa-\nrameter\f. In those expressions vs=\u0000\u0000\npa3=2eM\u0001\nj\nstands for the spin-velocity of the conduction electrons,\npis the spin polarization of the electric current density\nj,e(>0)the elementary charge, athe lattice constant,\nandMthe magnetization saturation. With those contri-\nbutions the stochastic Landau-Lifshitz-Gilbert equation\nbecomes:\n\u0012@\n@t+vs\u0001r\u0013\n\n=\n\u0002(Heff+h)\n+\u000b\n\u0002\u0012@\n@t+\f\n\u000bvs\u0001r\u0013\n\n;(1)\nwhere Heff=1\n~\u000eE\n\u000e\nis the effective field, with Erepresent-\ning the energy of the system, and \u000bthe Gilbert damping\nconstant. An important aspect of this equation is the in-\nclusion of the white Gaussian fluctuating magnetic field\nh, describing the thermal agitation of the magnetization\nand obeying the fluctuation-dissipation theorem22. The\nstrength of the noise, \u001b= 2\u000bkBTa2=~, is proportional to\nthe thermal energy kBT, the Gilbert damping parameter\n\u000b, and the volume of the finite element grid a2.\nParticle like solutions of the Landau-Lifshitz-Gilbert\nequation, thatrepresentcompactmagnetictexturesmov-\ning as coherent entities with a well defined velocity, have\nknown since long ago. Among other examples we can\nfound the dynamics of domain walls27,28and of Bloch\npoints29,30. The account of the dynamics of skyrmion\ntextures is best handled by the use of the collective\ncoordinates approach. Under this framework the com-\nplex dynamics of the magnetization texture, \n(r;t)isarXiv:1408.4861v2  [cond-mat.mes-hall]  22 Aug 20142\nreduced to the evolution of a small number of degrees\nof freedom given by the skyrmion position and its ve-\nlocity. In this way the magnetization field associated\nto a single-skyrmion moving along the trajectory x(t)\nis represented by a magnetization profile \n(r;t) =\n(r\u0000\nx(t);v(t)). The explicit time-dependence of the magne-\ntization, coming from the dependence on velocity v(t),\nincludes the effects of deformations of the skyrmion31–33.\nThe calculation for the static skyrmion profile, \n0(r),\nhas been addressed elsewhere34, by means of a minimiza-\ntion of the magnetic energy. In this energy the contribu-\ntions from the exchange, perpendicular anisotropy, and\nDzyaloshinskii-Moriya energies must be included. Re-\nplacement of the collective coordinates ansatz and inte-\ngration over space reduce the LLG equation to an equa-\ntion of motion for the collective variables. This equation\nhas the form of a stochastic massive Thiele’s equation\nM_v(t)\u0000g^z\u0002v(t) +\u000bDv(t) =F+\u0011(t):(2)\nNeglecting the contribution of the noise term ( \u0011(t))\nEq. (2) turns into the generalized Thiele’s equation35.\nWe highlight the inertial terms, quantified by the effec-\ntive mass, that correspond to a matrix Mij=Mij+\u0016Mij,\ncomprised by the elements Mij=R\ndr \n\u0001\u0000@\n@xi\u0002@\n@vj\u0001\n,\narising from the conservative dynamics, and \u0016Mij=\n\u000bR\ndr\u0000@\n@xi\u0001@\n@vj\u0001\narising from the dissipative contribu-\ntiontotheLandau-Lifshitz-Gilbertequation. Thesecond\nterm in Eq. (2) describes the Magnus force4exerted by\nthe magnetic texture on the moving skyrmion. In the\ncase of an isolated skyrmion g= 4\u0019WwhereW=\u00001\nstands for the winding number, or skyrmion charge. The\nthird contribution represents the dissipative force which\nis defined through the relation Dij=R\ndr@\n@xi\u0001@\n@xj, that\nbecomesDij=D\u000eijinthe caseofthe highlysymmetrical\ncase of an isolated skyrmion.\nThe dynamics of the skyrmion in Eq. (2) is forced\nby a deterministic term F=\u0000g^z\u0002vs+\fDvs\u0000rV,\nthat contains a contribution from the flowing electrons\nand a force arising from a potential V[x]that reflects\nthe inhomogeneities in the skyrmion‘s path, e.g., mag-\nnetic impurities, local anisotropies or geometric defects.\nWe conclude with the last term of right-hand side of Eq.\n(2), that describes the fluctuating force on the skyrmion.\nThe strength of the Gaussian white noise turns out to\nbe\u001bD, therefore the effective diffusion constant of the\nskyrmion depends not only on the Gilbert damping and\nthe temperature but also on the dissipative parameter\nD. By solving the stochastic Thiele equation Eq. (2),\nfor the homogenous case ( V[x] = 0), we determine the\ntime evolution of both longitudinal and transverse com-\nponents of the velocity of the fluctuating skyrmion (as\nshown in Fig. 1 (a) and (b) respectively) at temperature\nT= 100K. Typical skyrmion speeds of \u00180\u00001m/s are\nreached for spin-velocities on the order of 1m/s. The\nmassive skyrmion dynamics was calculated for a Gilbert-\ndamping parameter \u000b= 0:1, the\fparameter\f= 0:5\u000b,\nthe dissipative force D= 5:577\u0019(from Ref. [13]), and\nwhere the values used for the mass are taken from Ref.[36]. Moreover, it is numerically solved the mean drift\nvelocity, i.e., the steady-state current-induced skyrmion\nmotion, which is displayed in Fig. 1 (c) as a function\nof the spin-velocity vs=vs^xof electrons. In addition,\n0 1 20.01.02.0vxvsHaL\n0 1 2-1.00.01.0\n10-4tt0vyvsHbL\n0 2 4 6 8 100246810\nvs@msD<v>@msDHcL\n<vx>\n<vy>\nFIG. 1. Fluctuating skyrmion velocities in the longitudinal\nand transverse directions, (a) and (b) respectively. The Brow-\nnian dynamics Eq. (2) is solved for a spin velocity of electrons\nvs= 1 m/s along xdirection and for a characteristic time\nscalet0=Mxx\u00196ns (from Ref. [36]). In (c) the average\nvelocities is presented as function of spin-velocity vs=vs^x,\nboth the longitudinal (black line) and transverse (dashed line)\ncomponents. The results are shown for a Gilbert damping\n\u000b= 0:1,\f= 0:5\u000b, and at temperature 100K.\nwe are interested in the probability distribution P[x;v;t]\nassociated to the skyrmion dynamics, which is defined as\nthe probability density that a skyrmion at time t, is in\nthe position xwith a velocity v. The equation of motion\nsatisfied by such distribution is known as Fokker-Planck\nequation and its derivation, as well as its solution in sim-\nple cases, constitutes a standard issue in the analysis of\nstochastic processes37.\nOrigin of the skyrmion mass .- The Brownian skyrmion\nmotion described by Eq. (2) contains as a main ingre-\ndient the inertia term, regarding to Ref. [21], which is\nquantified by the effective mass matrix M. Generally\nspeaking, it is linked to the explicit time-dependence of\nthe magnetization direction. The mass of skyrmions can\nbe determined perturbatively in linear response theory as\nfollows. In the skyrmion center of mass reference frame\nwe see that the magnetization in Eq. (1) is affected by an\nadditional magnetic field \u000eH(r) =\n(r)\u0002(v\u0001r)\n(r).\nWe note that the strength of the effective field is con-\nfined within the perimeter of the skyrmion. However,\nthe effective torque takes a maximum value in the center\nof skyrmion and thus, it suffers a distortion of its shape\ndue to the current-induced motion. This motivates us to\npropose an ansatz for the magnetization texture of a non-\nrigid single skyrmion and its dependence on the velocity\nas\n\n(r;v) =\n0(r) +\u0015\u0018\n0(r)\u0002(v\u0001r)\n0(r);(3)\nwith\n0correspondstothestaticandrigidskyrmiontex-\nture. The deformation of skyrmion size is parameterized3\nby the second term, where \u0015is the dimensionless param-\neter that determines the strength of the velocity induced\ndeformations. In this expression \u0018=~l2\nsk=Ja2withJ\nthe exchange constant, athe lattice constant and lskthe\ncharacteristic skyrmion size. It is worth noting that this\n\u00002\u0000112\u00002\u00001012\nx/lsk0y/lsk\n\u0000101⌦z\n\u00002\u0000112\u00002\u00001012\nx/lsk0y/lsk\n\u0000101\u0000⌦z\nFIG. 2. Top: Pictorial representation of the skyrmion mag-\nnetic texture. The color encodes the out of plane component\nof the magnetization. It changes from being fully aling with\nthe+^zdirection in the center to a complete alignment with\nthe opposite direction in the outer rim. The arrows represent\nthe behavior of the in plane component of the magnetiza-\ntion. For the case displayed those components swirl like a\nvortex. Bottom: Schematic plot for the skyrmion distortion\ngenerated by the motion of the skyrmion. In color we have\nrepresented the out of plane component of the deformation\n\u000e\nz. This contribution is concentrated in the direction trans-\nverse to the motion nearby the perimeter of the skyrmion\n(indicated by a dashed line). The arrows correspond to the\nin plane (\u000e\nx;\u000e\ny).\ncontribution is linear in velocity and conserves the norm\nof the magnetization to leading order in \u0015. In Fig. 2\nwe present schematically the distortion of skyrmions ex-\nerted by the effective field \u000eH. However, without loss of\ngenerality, we assume a motion of the skyrmion along x\ndirection. The deformation in the skyrmion texture con-\nsists of an in plane distortion, that resembles a dipolar\nfield, and an out of plane contribution that is antisym-\nmetric. The nature of the mass of skyrmions can be\ntraced back by using the ansatz given by Eq. (3). By\nreplacing it on the expressions for Mup to linear orderin\u0015, we find that the mass is related both to the dissipa-\ntivematrixandgyrotropictensorby Mxx=Myy=\u0015\u0018D\nandMxy=\u0000Myx=\u0015\u0018\u000bg. We see how the dissipation\nmechanisms encoded by the Gilbert damping \u000bgenerate\nan anti symmetrical contribution to the mass. By com-\nparing our results for Mxxwith earlier theoretical31,32\nand experimental36estimates of the skyrmion mass we\nobtain\u0015\u00180:01. In this case we obtain, using a typical\nskyrmion velocity of 1 m/s in Eq. (3), a characteristic\nstrength of the deformation of the skyrmion in the range\nofj\u000e\nj\u001810\u00003.\nSkyrmion mobility .-Theskyrmiondynamicsisinduced\nby an electric current density via spin-transfer torque\nmechanism. As mentioned previously, the flow of elec-\ntrons exerts a force Fthat drives the skyrmion and its re-\nsponse is described by the stochastic generalized Thiele’s\nequation21. Next, we discuss the role of mass in the dy-\nnamics of skyrmions induced by currents at finite tem-\nperature. The probability distribution for the velocity\nof the skyrmion can be readily found for V(x) = 0by\nsolving the associated Fokker-Planck equation, P[v] =\nNexp\u0000\n\u00001\n2\r(v\u0000\u0016 v)2\u0001\n, where\r=\u0015\u0018\u000b(g2+D2)=\u001bDand\n\u0016 v=\u0016kvs+\u0016?^ z\u0002vs. In this equation \u0016k= (g2+\n\u000b\fD2)=(g2+\u000b2D2)and\u0016?=\u0000(\u000b\u0000\f)gD=(g2+\u000b2D2)\nare the longitudinal and transverse skyrmion mobilities\nrespectively. We are interested in the average drift veloc-\nity, which is determined directly from the probability dis-\ntribution ashvii= \u0016vi. Another relevant quantity as the\nmean-squared velocity is evaluated directly, and is found\ntoobeyhvivji\u0000hviihvji=\u001bD\u0000\n\u0015\u0018\u000b\u0000\ng2+D2\u0001\u0001\u00001\u000eijthat\nrelates the fluctuations on the velocity in terms of tem-\nperature and the skyrmion mass. The mean drift ve-\nlocity scales linearly with the spin-velocity of electrons\nand, unlike the current-induced domain wall dynamics28,\nskyrmions do not exhibit an intrinsic pinning13,14. Note\nthat these values for the spin-velocities correspond to\nelectric current densities on the order of 1010A/m2. As\nwe analytically show, the average velocity of the free\nskyrmion is independent of the mass. However, it is re-\nlated only on intrinsic parameters, such as Gilbert damp-\ning, skyrmion-charge, and dissipative force. Following\nour results, a detailed characterization of the strength of\nvelocity fluctuations can be used to determine the values\nof the skyrmion mass.\nConclusions .- We have investigated the mechanisms by\nwhich thermal fluctuations influence the current-driven\nmassive skyrmion dynamics. Based on the stochas-\ntic Landau-Lifschitz-Gilbert equation we derived the\nLangevin equation for the skyrmion. The equation has\ntheformofastochasticgeneralizedThiele’sequationthat\ndescribes the massive dynamics of a single-skyrmion at\nfinite temperature. We proposed an ansatz for the mag-\nnetization texture of a non-rigid single skyrmion that de-\npends linearly with the velocity. This ansatz has been\nderived based upon an effective field that distorts the\nskyrmion texture. In particular, it implies that the de-\nformation of the skyrmion shape consists of an in plane\ndistortion and an out of plane contribution that is an-4\ntisymmetric. Furthermore, by utilizing this ansatz it is\nfound that the mass of skyrmion is related with intrinsic\nparameters, such as Gilbert damping, skyrmion-charge,\nand dissipative force. This simple results provides a path\nfor a theoretical calculation of the skyrmion mass. We\nhavefoundanexactexpressionfortheaveragedriftveloc-\nity as well as the mean-square velocity of the skyrmion.\nThe longitudinal and transverse mobilities of skyrmions\nfor small spin-velocity of electrons were also determined.\nWe showed that the average velocity of skyrmions, unlike\nthe mean-square velocity, is independent of the mass and\nit varies linearly with the spin-velocity. In future work,\nwe plan to use the formalism developed in this work to\nthe study of the transport of massive skyrmions in disor-\ndered media.\nAcknowledgements .- The authors acknowledge fund-\ning from Proyecto Basal FB0807-CEDENNA, Anillo de\nCiencia y Tecnonología ACT 1117, and by Núcleo Cien-\ntífico Milenio P06022-F.\n1T. H. R. Skyrme, Nucl. Phys. 31, 556 (1962).\n2A. N. Bogdanov, U. K. Rossler and A. A. Shestakov Phys. Rev.\nE67, 016602 (2003).\n3S. Muhlbauer, B. Binz, F. Jonietz, C. Pfleiderer, A. Rosch, A.\nNeubauer, R. Georgii, P. Boni, Science 323, 915 (2009).\n4F. Jonietz , S. Muhlbauer, C. Pfleiderer, A. Neubauer, W. Mun-\nzer, A. Bauer, T. Adams, R. Georgii, P. Boni, R. A. Duine, K.\nEverschor, M. Garst, A. Rosch, Science 330, 1648 (2010).\n5W. Munzer, A. Neubauer, T. Adams, S. Muhlbauer, C. Franz, F.\nJonietz, R. Georgii, P. Boni, B. Pedersen, M. Schmidt, A. Rosch\nand C. Pfleiderer, Phys. Rev. B 81, 041203(R) (2010).\n6P. Milde, D. Kuhler, J. Seidel, L. M. Eng, A. Bauer, A. Chacon,\nJ. Kindervater, S. Muhlbauer, C. Pfleiderer, S. Buhrandt, C.\nSchutte and A. Rosch, Science 340, 1076 (2013).\n7M. Nagao, Y. So, H. Yoshida, M. Isobe, T. Hara, K. Ishizuka and\nK. Kimoto, Nature Nanotechnology 8, 325 (2013).\n8S. Seki, X. Z. Yu, S. Ishiwata and Y. Tokura, Science 336, 198\n(2012).\n9A. Fert, V. Cros and J. Sampaio, Nature Nanotechnology 8, 152\n(2013).\n10X. Z. Yu, Y. Onose, N. Kanazawa, J. H. Park, J. H. Han, Y.\nMatsui, N. Nagaosa and Y. Tokura, Nature Materials 465, 901\n(2010).\n11K. Shibata, X. Z. Yu, T. Hara, D. Morikawa, N. Kanazawa, K.\nKimoto, S. Ishiwata, Y. Matsui and Y. Tokura, Nature Nan-\notechnology 8, 723 (2013).12X. Z. Yu, N. Kanazawa, Y. Onose, K. Kimoto, W. Z. Zhang, S.\nIshiwata, Y. Matsui and Y. Tokura, Nature Materials 10, 106\n(2011).\n13J. Iwasaki, M. Mochizuki and N. Nagaosa, Nature Communica-\ntions 4, 1463 (2013).\n14J. Iwasaki, M. Mochizuki and N. Nagaosa, Nature Nanotechnol-\nogy8, 742 (2013).\n15S. Z. Lin, C. Reichhardt, C. D. Batista, and A. Saxena, Phys.\nRev. Lett. 110, 207202 (2013).\n16S. Heinze, K. von Bergmann, M. Menzel, J. Brede, A. Kubetzka,\nR. Wiesendanger, G. Bihlmayer and S. Blugel, Nature Physics\n7, 713 (2011).\n17L. Kong and J. Zang, Phys. Rev. Lett. 111, 067203 (2013).\n18S. Z. Lin, C. D. Batista, C. Reichhardt, and A. Saxena, Phys.\nRev. Lett. 112, 187203 (2014).\n19M. Mochizuki, X. Z. Yu, S. Seki, N. Kanazawa, W. Koshibae, J.\nZang, M.Mostovoy, Y.TokuraandN.Nagaosa, NatureMaterials\n13, 241 (2014).\n20S. Z. Lin, C. Reichhardt, C. D. Batista and A. Saxena, Phys.\nRev. B 87, 214419 (2013).\n21Roberto E. Troncoso and Alvaro S. Núñez, Phys. Rev. B 89,\n224403 (2014).\n22W. F. Brown, Jr., Phys. Rev. 130, 1677 (1963).\n23R.A. Duine, A. S. Núñez, and A.H. MacDonald, Phys. Rev. Lett.\n98056605 (2007).\n24L. Berger, J. Appl. Phys. 49, 2156 (1978); L. Berger, J. Appl.\nPhys. 55, 1954 (1984); L. Berger, J. Appl. Phys. 71, 2721 (1992).\n25J. C. Slonczewski, J. Magn. Magn. Mater. 159, L1 (1996).\n26A. Rebei and M. Simionato, Phys. Rev. B 71, 174415 (2005).\n27M. Hayashi, L. Thomas, R. Moriya, C. Rettner and S. S. P.\nParkin, Science 320, 209 (2008).\n28P. Landeros and Alvaro S. Núñez, J. Appl. Phys. 108, 033917\n(2010).\n29R. G. Elias and A. Verga, Eur. Phys. J. B 82, 159 (2011).\n30C. Andreas, A. Kakay, and R. Hertel, Phys. Rev. B 89134403\n(2014).\n31C. Moutafis, S. Komineas, and J. A. C. Bland, Phys. Rev. B 79,\n224429 (2009).\n32I. Makhfudz, B. Kruger, and O. Tchernyshyov, Phys. Rev. Lett.\n109, 217201 (2012).\n33K.W. Moon, B.S. Chun, W.K., Z. Q. Qiu, and C. Hwang, Phys.\nRev. B 89, 064413 (2014).\n34A. Bogdanov and A. Hubert, J. Magn. Magn. Mater. 138, 255\n(1994), N.S.Kiselev, A.N.Bogdanov, R.ShaferandU.K.Rossler,\nJ. Phys. D 44, 392001 (2011).\n35A. A. Thiele, Phys. Rev. Lett. 30, 230 (1972).\n36Buttner, F. (2013) Topological mass of magnetic Skyrmions\nprobed by ultrafast dynamic imaging. PhD thesis. Johannes\nGutenberg-Universit at Mainz.\n37H.Risken, TheFokker-PlanckEquation(Springer-Verlag, Berlin,\n1984)."    },    {        "title": "1408.6419v1.Quasi_particle_Lifetime_in_a_Mixture_of_Bose_and_Fermi_Superfluids.pdf",        "content": "Quasi-particle Lifetime in a Mixture of Bose and Fermi Super\ruids\nWei Zheng and Hui Zhai\nInstitute for Advanced Study, Tsinghua University, Beijing, 100084, China\n(Dated: October 13, 2018)\nIn this letter, to reveal the e\u000bect of quasi-particle interactions in a Bose-Fermi super\ruid mixture,\nwe consider the lifetime of quasi-particle of Bose super\ruid due to its interaction with quasi-particles\nin Fermi super\ruid. We \fnd that this damping rate, i.e. inverse of the lifetime, has quite di\u000berent\nthreshold behavior at the BCS and the BEC side of the Fermi super\ruid. The damping rate is a\nconstant nearby the threshold momentum in the BCS side, while it increases rapidly in the BEC\nside. This is because in the BCS side the decay processe is restricted by constant density-of-state of\nfermion quasi-particle nearby Fermi surface, while such a restriction does not exist in the BEC side\nwhere the damping process is dominated by bosonic quasi-particles of Fermi super\ruid. Our results\nare related to collective mode experiment in recently realized Bose-Fermi super\ruid mixture.\nRecently, for the \frst time, ENS group has realized a\nmixture of Bose and Fermi super\ruids [1]. They prepare\na mixture of bosonic7Li atoms and two spin components\nof fermionic6Li atoms nearby an s-wave Feshbach res-\nonance between fermions. At low enough temperature,\nbosonic atoms condense and become a Bose super\ruid,\nwhile fermionic atoms form pairs and become a Fermi su-\nper\ruid. This experimental development generates many\ninteresting questions on interaction e\u000bects between these\ntwo types of super\ruid [2, 3].\nElementary excitations and their interactions play an\nimportant role in quantum many-body system. Here we\ncan compare the low-energy elementary excitations of\nthis super\ruid mixture with other two widely studied\nmixtures, i.e. mixture of a BEC with normal Fermi gas\n[4] and mixture of two BECs [5]. A Bose-Fermi super\ruid\nmixture exhibits two gapless bosonic modes (denoted by\nBbandBfin Fig. 1), corresponding to Goldstone modes\nof Bose super\ruidity ( Bb) and Fermi super\ruidity ( Bf),\nrespectively, and a gapped fermionic excitation that de-\nscribes the Cooper pair breaking (denoted by Ffin Fig.\n1). While in the mixture of a BEC with normal Fermi\ngas, there exists only one bosonic Goldstone mode and\nthe fermionic excitation (particle or hole excitation) is al-\nways gapless at the Fermi surface. Mixture of two BECs\nalso exhibits two bosonic Goldstone modes but there is\nno fermionic excitation in this system.\nMoreover, in the cold atom system the Fermi super-\n\ruid can be continuously tuned from the BCS regime to\nthe BEC regime by utilizing the Feshbach resonance. In\nthe BCS limit, as schematically shown in Fig. 1(a), it is\nknown that the Bfmode has a quite large velocity pro-\nportional to vF=p\n3 [6], while the gap of the Ffmode is\nexponentially small. As approaching the BEC side, as\nshown in Fig. 1(b), the gap of the Ffmode becomes\nlarger and larger, and on the other hand, the velocity of\ntheBfmode becomes smaller and smaller [7].\nTherefore, the interplay between these three modes is\nquite unique in the Bose-Fermi super\ruid mixture, and\nit will lead to di\u000berent behaviors in the BCS and the\nBEC sides of Fermi super\ruid. One manifestation of in-\n0.00.51.01.52.001234\nkEkHaL\n0.00.51.01.52.001234\nkEkHbL\nBb\nBf\nFfBb\nBfFfFIG. 1: Schematic of dispersions of bosonic mode of Bose\nsuper\ruid ( Bb), bosonic mode of Fermi super\ruid ( Bf) and\nfermionic pair breaking mode of Fermi super\ruid ( Ff), respec-\ntively, at the BCS side (a) and at the BEC side (b). In (a),\ndashed line represents value of 2\u0001. Arrows denote that Bb\nmode decays into two Ffmodes. In (b), arrows denote that\nBbmode is scattered by generating an additional Bfmode.\nteraction between elementary excitations is the lifetime\nof quasi-particles. The most well-known e\u000bect is Landau-\nBeliaev damping in the Bose super\ruid [8]. Interaction\nbetween bosonic mode itself gives rise to a \fnite life-\ntime of the bosonic quasi-particle. The damping rate, as\nthe inverse of the lifetime \r= 1=\u001c, is proportional to\nk5at zero-temperature and to T4at \fnite temperature\n[11, 12]. This e\u000bect has been experimentally studied in\natomic BEC by measuring the damping rate of collective\nmodes [13] and theoretically works have also been carried\nout in the content of cold atom systems [14{16]. Landau\ndamping has also been studied for mixture of BEC with\nnormal Fermi gas [9] and dipolar BEC [10].\nIn this letter we present an alternative damping chan-\nnel for bosonic quasi-particle of Bose super\ruid ( Bb\nmode) due to its interaction with quasi-particles in Fermi\nsuper\ruid ( BfandFfmodes). We focus on the typi-\ncal cold atom situation that Fermi super\ruid is in the\nstrongly interacting regime while Bose super\ruid is in\nthe weakly interacting regime. We show that this damp-\ning mechanism will be activated only when momentum of\nBbexcitation exceeds a critical value kc. We investigate\nthe threshold behavior of damping rate \r=C(k\u0000kc)\u000b,\nand the key result is that we \fnd di\u000berent \u000bfor the BCSarXiv:1408.6419v1  [cond-mat.quant-gas]  27 Aug 20142\nside and the BEC side of Fermi super\ruid.\nModel. We consider a mixture of bosons and spin-1 =2\nfermions, whose Hamiltonian is given by\n^Hf=Z\nd3rn\n^cy\n\u001b(r)H0;f^c\u001b(r)\u0000gf^cy\n\"(r)^cy\n#(r)^c#(r)^c\"(r)o\n^Hb=Z\nd3rn\n^by(r)H0;b^b(r) +gb\n2^by(r)^by(r)^b(r)^b(r)o\n^Hbf=gbfZ\nd3r^by(r)^b(r)^cy\n\u001b(r)^c\u001b(r) (1)\nwhereH0;i=\u0000~2r2=(2mi)\u0000\u0016iandi= b;f denotes\nbosons or fermions. Since interaction between fermions\nis nearby a Feshbach resonance, we shall relate gfto scat-\ntering length afas 1=gf=m=(4\u0019~2af) +P\nk1=[2\u000ff(k)]\nwith\u000ff(k) = ~2k2=(2mf). The ground state of ^Hf\nis a super\ruid of fermion pairs. Applying the BCS-\nBEC crossover mean-\feld theory to ^Hfone can ob-\ntain a gapped fermion Ffmode with excitation energy\nEFf=p\n[\u000ff(k)\u0000\u0016f]2+ \u00012. As\u00001=(kFaf) decreases from\nthe BCS side to the BEC side, \u0016fdecreases and \u0001 in-\ncreases [7]. ^Hfalso has a bosonic Bfmode that de-\nscribes center-of-mass motion of Cooper pairs, which\nhas a phonon-like dispersion EBf=~cfk, andcfevolves\nsmoothly from vF=p\n3 top\n\u0019~2amnm=m2\nf[7, 17], where\nam= 0:6afis the scattering length between fermion pairs\n[18] andnmis molecule density. For equal population\ncasenm=n\"=n#=nf.\nWhen magnetic \feld locates nearby a Feshbach reso-\nnance between fermions, generically gbandgbfterms are\nin the weakly interacting regime and can be treated by\nBogoliubov approximation. In the leading order of nb\n(nb=Nb=V,Nbis condensate bosonic atoms), we re-\nplace two of ^byor^boperator withpNbin the interaction\npart. From ^Hbwe obtain a Bogoliubov spectrum for Bose\nsuper\ruidEBb=p\n\u000fb(k)[\u000fb(k) + 2gbnb], where\u000fb(k) =\n~2k2=(2mb) andgb= 4\u0019~2ab=mb. Whenk\u001c1=\u0018=p8\u0019abnb, the excitation is in the phonon regime with a\nlinear dispersion ~cbkandcb=p\n4\u0019~2abnb=m2\nb. When\nk\u001d1=\u0018=p8\u0019abnb, the excitation is in the free-particle\nregime with a quadratic dispersion \u000fb(k) +gbnb. Also\nin the leading order, ^HbfbecomesgbfnbR\nd3rcy\n\u001b(r)c\u001b(r),\nwhich simply provides a constant shift of chemical po-\ntential and will not a\u000bect spectrum and wave function of\nquasi-particles. In the sub-leading order of nb, only one\n^byor^boperator is replaced bypNb, and it describes in-\nteraction between quasi-particles. In this order, ^Hbleads\nto Landau-Beliaev damping discussed before [11, 12]. As\nwe will show later, ^Hbfgives rise to interaction between\nquasi-particles of Bose super\ruid and those of Fermi su-\nper\ruid.\nDamping Threshold. There are two di\u000berent decay\nchannels for bosonic quasi-particle Bbof Bose super\ruid.\nThe \frst is decay into two fermionic quasi-particles Ffof\nFermi super\ruid, i.e. Bb(k)!Ff(k\u0000q) +Ff(q), as\nshown in Fig. 1(a). In this case, the energy-momentum\nFIG. 2: (a-b): Shaded area is a schematic of two-particle\ncontinuum for two di\u000berent damping channels, Ff+Fffor (a)\nandBf+Bbfor (b), corresponding to processes illustrated\nin Fig. 1 (a) and (b), respectively. The red solid line is\ndispersion of bosonic quasi-particle Bbof Bose super\ruid. kc\nmarks the threshold momentum. (c) kc=kFas a function of\n\u00001=(kFaf). A, B and C mark three typical regimes discussed\nin text. Below (above) the dashed line kcis in the phonon\n(free-particle) regime of Bogoliubov dispersion for Bbmode.\nFor A and B, kcis given by (a), and kcis in phonon regime\nfor (A) and is in free-particle regime for (B). For (C), kcis\ngiven by (b) and is in free-particle regime.\nconservation requires EBb(k) =EFf(k\u0000q) +EFf(q).\nSinceEFfis gapped and the minimum of EFf(k) is \u0001 oc-\ncurring atk0withk0=p\n2mf\u0016f=~2for\u0016f>0 andk0= 0\nfor\u0016f<0, a typical two-particle continuum for two Ff\nmodes in the BCS side is shown in Fig. 2(a), which has\na minimum of 2\u0001 for k < 2k0. For this channel, kcis\ndetermined byEBbmeeting this two-particle threshold.\nIn the BCS side of resonance, kccan be determined by\nequationEBb(kc) = 2\u0001 as long as the solution of kcis\nsmaller than 2 k0. Therefore, as\u00001=(kFaf) decreases from\nthe BCS side to unitary regime, kcincreases as shown in\nFig. 2(c). Moreover, when \u0001 \u001c~2=(mb\u00182),kcis in\nthe phonon regime of Bbmode, while on contrary, when\n\u0001\u001d~2=(mb\u00182),kcis in the free-particle regime of Bb\nmode.\nThe second channel is decay into two bosonic quasi-\nparticlesBf, i.e.Bb(k)!Bf(k\u0000q) +Bf(q), or oneBf\nand oneBb, i.e.Bb(k)!Bb(k\u0000q) +Bf(q), as shown\nin Fig. 1(b). Since in the strongly interacting regime\nof Fermi super\ruid, cfis usually much larger than cb\nbecauseam= 0:6af\u001dab, it is easy to show that the\ntwo-particle threshold of Bb+Bfis always lower than\nthat ofBf+Bf. It is also straightforward to show that\nEBb(k) coincides with two-particle threshold of Bb+Bf\nup tokc, as shown in Fig. 2(b). That means for k<k c,\nonly the process with q= 0 can happen which in fact\ndoes not lead to decay of quasi-particle. Thus, damp-3\ning will be activated only when EBb(k) is above two-\nparticle threshold when k > k c, andkcis determined\nby@EBb(k)=@(~k)jk=kc=cf. Also due to cf\u001dcb,kcis\nalways located in the free-particle regime of Bbmode.\nHence our following discussion can be divided into\nthree representative cases, as shown in Fig. 2(c): Case A\nand B are both at the BCS side of Fermi super\ruid, where\ndamping is determined by the \frst process. For Case A,\n\u0001\u001c~2=(mb\u00182) and therefore kcis in the phonon regime\nofBbmode. For Case B, \u0001 \u001d~2=(mb\u00182), and thus kcis\nin the free-particle regime of Bbmode. Case C is at the\nBEC side of Fermi super\ruid, where damping is deter-\nmined by the second process, and kcis in the free-particle\nregime ofBbmode.\nCase A. In this regime we start with BCS mean-\feld\nHamiltonian for ^Hfand Bogoliubov Hamiltonian for ^Hb\ngiven by\n^Hf=X\nkEFf(k)(^\fy\nk^\fk+ ^\ry\nk^\rk); (2)\n^Hb=X\nkEBb(k)^\u000by\nk^\u000bk (3)\nwhere quasi-particle ^ \u000bk,^\fkand ^\rkare related to ^bk\nandck\u001bvia^bk=ub\nk^\u000bk\u0000vb\nk^\u000by\n\u0000k, ^ck\"=uf\nk^\fk+\nvf\nk^\ry\n\u0000kand ^ck#=uf\nk^\rk\u0000vf\nk^\fy\n\u0000k. Hereub\nk(vb\nk) =r\n1\n2\u0010\n\u000fb(k)+gbnb\nEBb(k)\u00061\u0011\nanduf\nk(vf\nk) =r\n1\n2\u0010\n1\u0006\u000ff(k)\u0000\u0016f\nEFf(k)\u0011\n.\nNow we discuss ^Hbfin the order ofpnbby replacing\none of ^bor^byoperator aspNb, which leads to\n^Hbf=gbfrnb\nVX\nkq(^cy\nk+q;\u001b^cq;\u001b^bk+ h.c.): (4)\nWe can further rewrite ^Hbfin term of quasi-particle op-\nerators ^\u000b,^\fand ^\r. Here we focus on zero-temperature\ndamping rate (or lifetime) of bosonic \u000bmode, thus, only\none term retains as [19]\ngbfrnb\nVX\nkqMkq^\fy\nk\u0000q^\ry\nq^\u000bk; (5)\nMkq= (ub\nk\u0000vb\nk)(uf\nk\u0000qvf\nq+vf\nk\u0000quf\nq) (6)\nThis term describes the process that one Bbmode decays\ninto twoFfmodes, as schematically drawn in Fig. 1(a).\nWith Fermi-Golden rule, the damping rate is given by\n\r(k) =2\u0019\n~nbg2\nbf\nVX\nqjMkqj2\u000e[EFf(k\u0000q) +EFf(q)\u0000EBb(k)]:\n(7)\nWhenkcis in the phonon regime, we can approximate\nub\nk(vb\nk) =q\ngbnb\n2~cbk\u00061\n2q\n~cbk\n2gbnb. And since the decay prod-\nucts ofFfmode locate nearby its minimum of dispersion\nEFf(k) atk0, to the leading order we can approximate\n0.00.51.01.52.02.50246810\nkkcgg0H10-5LHaL\n0.00.51.01.52.02.50246810\nkkcgg0H10-4LHbLFIG. 3: Damping rate \rin unit of \r0(\r0=EF=~) as a\nfunction of k=kc. (a): Case A and B in the BCS side are\nshown with 1 =(kFaf) =\u00000:25 for Case A (dashed line) and\n1=(kFaf) =\u00000:5 for Case B (solid line). (b) Case C in the\nBEC side with 1 =(kFaf) = 0:5. For a typical experiment setup\nkF\u00195\u0002106m\u00001and\r0\u00191:3\u0002105Hz.\nuf\nk(vf\nk) = 1=p\n2. ThereforeMkq'q\n~cbk\n2gbnb. Moreover,\nin this regime we can approximate EBb(k) =~cbjkjand\nEFf(k) = \u0001 +~2\n2m\u0003\u00112\nk, where\u0011k=jkj\u0000k0,m\u0003= \u0001=v2\n0\nandv0=~k0=mf. By this approximation, kcis deter-\nmined by 2\u0001 = ~cbkc. With these approximations, the\ndamping rate \r(k) can be simpli\fed as\n\r(k) =g2\nbfcbk\n8\u00192gbZ\nd3q\u000e2\n4~2\u0010\n\u00112\nk\u0000q+\u00112\nq\u0011\n2m\u0003\u0000~cb(k\u0000kc)3\n5:\n(8)\nBasically this integration is to count for the density-\nof-state that satis\fes energy conservation. With quite\nstraightforward calculation [19] we \fnd that\n\r(k) =g2\nbfcb\u0001m2\nf\n2~4gb\u0002(k\u0000kc); (9)\ni.e. the threshold behavior of \r(k) is a constant.\nCase B. In this regime the damping rate is still deter-\nmined by Eq. 7. But since kcis in the free-particle\nregime, we have ub\nk\u00191 andvb\nk\u00190. In this case\nMkq\u00191. Furthermore, EBb(k) is approximated by\n\u000fb(k) +gbnb, and the damping rate \r(k) is given by\n\r(k) =g2\nbfnb\n4\u00192~Z\nd3q\u000e2\n4~2\u0010\n\u00112\nk\u0000q+\u00112\nq\u0011\n2m\u0003\u0000~2\u0000\nk2\u0000k2\nc\u0001\n2mb3\n5;\n(10)\nwhich gives rise to a damping rate\n\r(k) =g2\nbfnb\u0001m2\nf\n~5k\u0002(k\u0000kc)\n'g2\nbfnb\u0001m2\nf\n~5kc\u0012\n1\u0000k\u0000kc\nkc\u0013\n\u0002(k\u0000kc): (11)\nThe leading order is still a constant and the sub-leading\norder gives a slow decreasing of \r(k) asjkjincreases.\nHowever, we shall also note that because the approxima-\ntions implemented, our results are only valid nearby kc\nand cannot be extended to very large momentum.4\nCase C. In this regime the damping is due to coupling\nbetweenBbmode andBfmode. A comprehensive de-\nscription of Bfmode and its coupling to Bbmode can\nbe obtained from \ructuation theory of Fermi super\ruid\n[20]. Here to highlight the essential physics we take a\nsimpler approach by treating the Fermi super\ruid at the\nBEC side as molecular condensate, and we consider a\nHamiltonian of molecular BEC as\n^Hm=Z\nd3rn\n^dy(r)H0;m^d(r) +gm\n2^dy(r)^dy(r)^d(r)^d(r)o\n(12)\nwhereH0;m=\u0000~2r2\n2mm\u0000\u0016m, andgm= 4\u0019~2am=mm.^dy\nrepresents a creation operator for a bosonic molecule.\nThe coupling between the molecular BEC and Bose su-\nper\ruid is due to scattering between bosonic atoms and\nmolecules, which can be e\u000bectively described by\nHbm=gbmZ\nd3r^by(r)^b(r)^dy(r)^d(r) (13)\nwheregbmis determined by atom-molecule scattering\nlength calculated in Ref. [3]. Bogoliubov approximation\ncan be applied to ^Hmwhich gives\n^Hm=X\nkEBf(k)^\u001fy\nk^\u001fk; (14)\nwhereEBf(k) =p\n\u000fm(k) [\u000fm(k) + 2gmnm] with\u000fm(k) =\n~2k2=(2mm). ^\u001fkrelates to ^dkas ^\u001fk=um\nk^dk\u0000vm\nk^dy\n\u0000k,\nwhereum\nk(vm\nk) =r\n1\n2\u0010\n\u000fm(k)+gmnm\nEBf(k)\u00061\u0011\n. Similarly, in\nthe order proportional to nbornm,^Hbmis simply a\nconstant chemical potential shift for both Bose super\ruid\nand molecular condensate.\nSimilar as analysis in case A, by replacing one of ^dy(or\n^d) operator aspNmor one of ^by(or^b) operator aspNb,\nit can be expanded into quite a few terms that describe\nquasi-particle interactions, among which only one term\ncontributes to decay of Bbmode with a lower critical\nvelocity, as discussed above [19]. This term is given by\ngbmrnm\nVX\nkqQkq^\u001fy\nq^\u000by\nk\u0000q^\u000bk; (15)\nQkq= (um\nq\u0000vm\nq)(ub\nk\u0000qub\nk+vb\nk\u0000qvb\nk): (16)\nIn this regime we can approximate um\nk(vm\nk) =q\ngmnm\n2~cfk\u0006\n1\n2q\n~cfk\n2gmnm,ub\nk\u00191 andvb\nk\u00190, thereforeQkqbecomes\nq\n~cfk\n2gmnm. Furthermore, we can approximate EBb(k) by\n\u000fb(k) +gbnb,EBfas~cfjkj, and the damping rate is\n\r(k) =g2\nbmcf\n8\u00192gmZ\nd3qjqj\u000e(\n~cfjqj+~2\u0002\n(k\u0000q)2\u0000k2\u0003\n2mb)\n:\n(17)Straightforward evaluation of this integral gives [19]\n\r(k) =2g2\nbmmbcf\n3\u0019~2gmk(k\u0000kc)3\u0002(k\u0000kc): (18)\nAt leading order \r(k) fast increases as ( k\u0000kc)3oncek\nis above threshold.\nConclusion. The results of damping rate for three\ncases are presented in Fig. 3. We choose nb=k3\nF= 0:1,\n1=(kFab) = 100 and 1 =(kFabf) = 100. For three di\u000ber-\nent cases, we choose 1 =(kFaf) =\u00002:5, 1=(kFaf) =\u00000:5,\nand 1=(kFaf) = 0:5, respectively. We \fnd a di\u000berent\nthreshold behavior \r(k)/(k\u0000kc)\u000bwith\u000b= 0 in the\nBCS regime and \u000b= 3 in the BEC regime. This \fnding,\non one hand, is a unique manifestation of quasi-particle\ninteraction e\u000bect in the Bose-Fermi super\ruid mixture;\non the other hand, reveals fundamental di\u000berent between\nFermi super\ruid in the BCS side and in the BEC side.\nIn the BCS side, the low-energy physics is dominated by\nfermionic quasi-particles nearby the Fermi surface, and\nthe damping processes are also restricted by the constant\ndensity-of-state nearby Fermi surface, which is basically\nthe origin of constant damping rate. While such restric-\ntion does not exist in the BEC side where the low-energy\nphysics is dominated by bosonic mode.\nOur results can be experimentally veri\fed by study-\ning damping rate of collective mode, as done in previous\nBEC experiments [13]. In the recent experiment, ENS\ngroup has \fnd damping of collective oscillation when the\nrelative velocity between Bose and Fermi super\ruid ex-\nceeds a critical velocity. At the unitary regime and in\nthe BEC side the damping rate increases rapidly when\nvelocity is above the critical velocity [1]. They also \fnd a\nnearly constant damping rate at the BCS side [21]. The\nunderlying mechanism of this experimental \fnding may\nbe connected to the physics discussed in this work.\nAcknowledgment : We wish to thank Christophe Sa-\nlomon for helpful discussions. This work is supported by\nTsinghua University Initiative Scienti\fc Research Pro-\ngram, NSFC Grant No. 11174176, and NKBRSFC under\nGrant No. 2011CB921500.\n[1] I. Ferrier-Barbut, M. Delehaye, S. Laurent, A. T. Grier,\nM. Pierce, B. S. Rem, F. Chevy, C. Salomon, arXiv:\n1404.2548v2.\n[2] T. Ozawa, A. Recati, S. Stringari, arXiv:1405.7187v1.\n[3] R. Zhang, W. Zhang, H. Zhai and P. Zhang, to appear;\nX. Cui arXiv:1406.1242;\n[4] G. Modugno,Giacomo Roati, F. Riboli, F. Ferlaino, R.\nJ. Brecha, M. Inguscio, Science 297, 2240 (2002); F.\nSchreck, L. Khaykovich, K. L. Corwin, G. Ferrari, T.\nBourdel, J. Cubizolles, and C. Salomon, Phys. Rev. Lett.\n87, 080403 (2001); F. Ferlaino, R.J. Brecha, P. Han-\nnaford, F. Riboli, G. Roati, G. Modugno, and M. Ingus-\ncio, Journal of Optics B, 5(2), S3 (2003); C. Ospelkaus, S.5\nOspelkaus, K. Sengstock, and K. Bongs, Phys. Rev. Lett.\n96, 020401 (2006); C. A. Stan, M. W. Zwierlein, C. H.\nSchunck, S. M. F. Raupach, and W. Ketterle, Phys. Rev.\nLett. 93, 143001 (2004); S. Inouye, J. Goldwin, M. L.\nOlsen, C. Ticknor, J. L. Bohn, and D. S. Jin, Phys. Rev.\nLett. 93, 183201 (2004); S. Ospelkaus, C. Ospelkaus, L.\nHumbert, K. Sengstock, and K. Bongs, Phys. Rev. Lett.\n97, 120403 (2006); F. Ferlaino, C. D'Errico, G. Roati,\nM. Zaccanti, M. Inguscio, and G. Modugno, Phys. Rev.\nA73, 040702(R) (2006); M. Zaccanti, C. D'Errico, F.\nFerlaino, G. Roati, M. Inguscio, and G. Modugno, Phys.\nRev. A. 74, 041605(R) (2006); B. Deh, C. Marzok, C.\nZimmermann, and P. W. Courteille, Phys. Rev. A 77,\n010701(R) (2008); B. Deh, W. Gunton, B. G. Klappauf,\nZ. Li, M. Semczuk, J. Van Dongen, and K. W. Madison,\nPhys. Rev. A 82, 020701(R) (2010); C.-H. Wu, I. Santi-\nago, J. W. Park, P. Ahmadi, and M. W. Zwierlein, Phys.\nRev. A 84, 011601(R) (2011).\n[5] C. J. Myatt, E. A. Burt, R. W. Ghrist, E. A. Cornell,\nand C. E. Wieman, Phys. Rev. Lett. 78, 586 (1997); D.\nS. Hall, M. R. Matthews, J. R. Ensher, C. E. Wieman,\nand E. A. Cornell, Phys. Rev. Lett. 81, 1539 (1998); P.\nMaddaloni, M. Modugno, C. Fort, F. Minardi, and M.\nInguscio, Phys. Rev. Lett. 85, 2413 (2000).\n[6] P. W. Anderson, Phys. Rev. 112, 1900 (1958).\n[7] S. Giorgini, L. P. Pitaevskii, and S. Stringari, Rev. Mod.\nPhys. 80, 1215 (2008).\n[8] S. T. Beliaev, Soviet Phys. JETP 34, 299 (1958); P.\nC. Hohenberg, P. C. Martin, Ann. Phys. (NY) 34, 291\n(1965); P. Szepfalusy, I. Kondor, Ann. Phys. (NY) 82, 1\n(1974).\n[9] D. H. Santamore, S. Gaudio, E. Timmermans, Phys. Rev.\nLett. 93, 250402 (2004); S. K. Yip, Phys. Rev. A 64,\n023609 (2001); D. H. Santamore, E. Timmermans, Phys.\nRev. A 72, 053601 (2005); X.-J. Liu and H. Hu, Phys.\nRev. A 68, 033613 (2003).\n[10] S. S. Natu, S. Das Sarma, Phys. Rev. A 88,031604 (R)(2013); S. S. Natu, R. M. Wilson, Phys. Rev. A 88,\n063638 (2013).\n[11] C. J. Pethick and H. Smith, Bose-Einstein Condensation\nin Dilute Gases , (Cambridge University Press, New York,\n2002), Chapter 10.\n[12] L. P. Pitaveski and S. Stringari, Bose-Einstein Condensa-\ntion, (Oxford University Press, New York, 2003), Chap-\nter 6.\n[13] D. S. Jin, J. R. Ensher, M. R. Matthews, C. E. Wieman,\nand E. A. Cornell, Phys. Rev. Lett. 77, 420 (1996); D. S.\nJin, M. R. Matthews, J. R. Ensher, C. E. Wieman, and\nE. A. Cornell, Phys. Rev. Lett. 78, 764 (1997); M.-O.\nMewes, M. R. Andrews, N. J. van Druten, D. M. Kurn,\nD. S. Durfee, C. G. Townsend, and W. Ketterle, Phys.\nRev. Lett. 77, 988 (1996); N. Katz, J. Steinhauer, R.\nOzeri, and N. Davidson, Phys. Rev. Letts. 89, 220401\n(2002).\n[14] W. V. Liu, Phys. Rev. Lett. 79, 4056 (1997); W. V. Liu,\nW. C. Schieve, cond-mat/9702122.\n[15] L. P. Pitaveski and S. Stringari, Phys. Lett. A 235, 398\n(1997); S. Giorgini, Phys. Rev. A 57, 2949 (1998).\n[16] P. O. Fedichev, G. V. Shlyapnikov and J. T. M. Walraven,\nPhys. Rev. Lett. 80, 2269 (1998).\n[17] J. Joseph, B. Clancy, L. Luo, J. Kinast, A. Turlapov, and\nJ. E. Thomas, Phys. Rev. Lett. 98, 170401, (2007).\n[18] D. S. Petrov, C. Salomon, and G. V. Shlyapnikov, Phys.\nRev. A 71, 012708 (2005).\n[19] See the supplemental material.\n[20] C. A. R. S\u0013 a de Melo, Mohit Randeria, and Jan R. En-\ngelbrecht, Phys. Pev. Lett. 71, 3202 (1993); Jan R. En-\ngelbrecht, Mohit Randeria, and C. A. R. S\u0013 a de Melo,\nPhys. Rev. B 55, 15153 (1997); Roberto B. Diener, Ra-\njdeep Sensarma, and Mohit Randeria, Phys. Rev. A 77,\n023626 (2008).\n[21] C. Salomon, priviate communication.\nSUPPLEMENTAL MATERIAL\nThe damping rate in the BCS side\nFrom Eq. (1) in the text, the Hamiltonian describing the interaction between the Bosons and Fermions in momentum\nspace is given by\nHbf=gbf\nVX\nq;p;k;\u001b^cy\nq\u0000k;\u001b^cq;\u001b^by\np+k^bp: (19)\nBy replacing one of the ^bor^byoperators aspNb, this Hamiltonian becomes\nHbf=gbfrnb\nVX\nk;q;\u001b\u0010\n^cy\nk+q;\u001b^cq;\u001b^bk+ h.c.\u0011\n: (20)6\nThen we rewrite the Hamiltonian in terms of the quasi-particle operators as\nHbf=gbfrnb\nVX\nk;q\u0000\nub\nk\u0000vb\nk\u0001\u0000\nuf\nk\u0000qvf\nk+vf\nk\u0000quf\nk\u0001^\fy\nk\u0000q^\ry\nq^\u000bk+ h.c.\n+gbfrnb\nVX\nk;q\u0000\nub\nk\u0000vb\nk\u0001\u0000\nuf\nk+quf\nk\u0000vf\nk+qvf\nk\u0001^\fy\nk+q^\fq^\u000bk+ h.c.\n+gbfrnb\nVX\nk;q\u0000\nub\nk\u0000vb\nk\u0001\u0000\nuf\nk+quf\nk\u0000vf\nk+qvf\nk\u0001\n^\ry\nk+q^\rq^\u000bk+ h.c.: (21)\nHere we have ignored the terms such as ^\f^\r^\u000bor^\fy^\ry^\u000by, since they do not conserve the energy, and will not contribute\nto the decay process. At the zero temperature, only the \frst term contribute to the damping of the Bbmode.\nEmploying the approximations discussed in the text, we obtain the integral as Eq. (8) in the text. To calculate\nthis integral, we \frst make the substitution: k\u0000q!k\n2\u0000qandq!k\n2+q, so that the integral becomes\n\r(k) =g2\nbfcbk\n8\u00192gbZ\nd3q\u000e\u0014~2\n2m\u0003\u0010\n\u00112\nk\n2\u0000q+\u00112\nk\n2+q\u0011\n\u0000~cb(k\u0000kc)\u0015\n: (22)\nWe choose the direction of kas theqzaxis, and transform into the cylindrical polar coordinates. The coordinate\ntransformation is given by\nqz=pz;\nqx= (k0sin\u00120+p\u001a) cos\u001e;\nqy= (k0sin\u00120+p\u001a) sin\u001e:\nwhere\u00120is de\fned as cos \u00120=k\n2k0. The Jacobi determinant of this coordinate transformation is dqzdqxdqy=\n(k0sin\u00120+p\u001a)dpzdp\u001ad\u001e. Then the \u0011k\n2\u0006qcan be expanded in the new coordinates as\n\u0011k\n2\u0006q=\f\f\f\fk\n2\u0006q\f\f\f\f\u0000k0\u0019sin\u00120p\u001a\u0006cos\u00120pz;\nwhere the high order terms of p\u001aandpzare ignored. Then the integral becomes\n\r(k) =g2\nbfcbk\n8\u00192gbZ1\n\u00001dpzZ1\n\u0000k0sin\u00120dp\u001aZ2\u0019\n0d\u001e\n\u0002(k0sin\u00120+p\u001a)\u000e\u0014~2\n2m\u0003\u0000\np2\n\u001asin2\u00120+p2\nzcos2\u00120\u0001\n\u0000~cb(k\u0000kc)\u0015\n: (23)\nWe apply a coordinate transformation again as\n\u0000~p\nm\u0003p\u001asin\u00120=rcos\u0010;\n~p\nm\u0003pzcos\u00120=rsin\u0010:\nwhere the corresponding Jacobi determinant is dpzdp\u001a=m\u0003\n~2sin\u00120cos\u00120rdrd\u0010 . So Eq.23 becomes\n\r(k) =g2\nbfcbk\n4\u0019gbZ1\n0drZ2\u0019\u0000\u00100\n\u00100d\u0010m\u0003\n~2sin\u00120cos\u00120r\n\u0002\u0012\nk0sin\u00120\u0000p\nm\u0003\n~sin\u00120rcos\u0010\u0013\n\u000e\u0002\nr2\u0000~cb(k\u0000kc)\u0003\n: (24)\nwhere\u00100is given byjcos\u00100j=~k0sin\u00120p\n2m\u0003~cb(k\u0000kc). Since we are Considering the threshold behavior, we have \u00100= 0. The\ndamping rate is obtained as\n\r(k) =g2\nbfcbk2\n0m\u0003\n2~2gb\u0002 (k\u0000kc) (25)7\nSubstituting the expression of the e\u000bective mass, m\u0003=\u0001m2\nf\n~2k2\n0, to the upper formula, one obtains the Eq. (9) in the\ntext.\nIn the free boson regime, \u0001 \u001d~2=\u0000\nmb\u00182\u0001\n, using the same integral skill, one obtains the damping rate as:\n\r(k) =g2\nbfnbk2\n0m\u0003\n~3k\u0002 (k\u0000kc) (26)\nSubstituting the expression of the e\u000bective mass to the upper formula, we have the Eq. (11) in the text.\nDamping rate in the BEC side\nFrom Eq. (12), in the BEC side the Boson-molecular interaction Hamiltonian in the momentum space is given by\nHbm=gbm\nVX\nq;p;k^dy\nq\u0000k^dq^by\np+k^bp: (27)\nBy replacing one of the ^dor^dyoperators by thepNm, we obtain\nH(1)\nbm=gbmrnm\nVX\nk;q\u0010\n^d\u0000q+^dy\nq\u0011\n^by\nk\u0000q^bk: (28)\nThis term describes the Bbmode scattered by the phonon mode Bfin the Fermi super\ruid. We rewrite this Hamil-\ntonian in terms of the quasi-particle operators\nH(1)\nbm=gbmrnm\nVX\nk;q\u0000\num\nq\u0000vm\nq\u0001\u0000\nub\nk\u0000qub\nk+vb\nk\u0000qvb\nk\u0001\n^\u001fy\nq^\u000by\nk\u0000q^\u000bk+ h.c.\n\u0000gbmrnm\nVX\nk;q\u0000\num\nq\u0000vm\nq\u0001\nvb\nq\u0000kub\nk^\u001fy\nq^\u000bq\u0000k^\u000bk+ h.c.; (29)\nwhere such terms as ^ \u001f^\u000b^\u000bor ^\u001fy^\u000by^\u000byare ignored, since they do not conserve the energy. At the zero temperature,\nonly the \frst term contribute to the damping, which describes the process Bb(k)!Bb(k\u0000q) +Bf(q). The critical\nmomentum of this process can be obtained by energy-momentum conservation as @EBb(k)=@(~k)jk=kc=cf. In the\nfree boson regime, we have ~kc=mfcf.\nBy replacing one of the ^byor^boperators by thepNbin Eq. 27, we obtain\nH(2)\nbm=\u0000gbmrnb\nVX\nk;q\u0000\nub\nk\u0000vb\nk\u0001\num\nk\u0000qvm\nq^\u001fy\nk\u0000q^\u001fy\nq^\u000bk+ h.c.\n+gbmrnb\nVX\nk;q\u0000\nub\nk\u0000vb\nk\u0001\u0000\num\nk+qum\nq+vm\nk+qvm\nq\u0001\n^\u001fy\nk+q^\u001fq^\u000bk+ h.c.; (30)\nAt zero temperature, only the \frst term contribute to the damping, which describes the process Bb(k)!Bf(k\u0000q)+\nBf(q). Using the energy-momentum conservation, one can determine the critical momentum for this process by\nEBb(kc) = 2EBf(kc=2). In the free boson regime, we have ~kc\u00192mfcf, which is larger than the critical momentum of\nthe process discussed above. So we will focus on the threshold behavior of the process Bb!Bb+Bf.\nEmploying the approximations discussed in the text, we obtain damping rate as Eq. (17) in the text:\n\r(k) =g2\nbmcf\n8\u00192gmZ\nd3q\u000e\"\n~cfjqj+~2(k\u0000q)2\n2mb\u0000~2k2\n2mb#\njqj: (31)\nThe Dirac function in this integral gives\n\u000e\"\n~cfjqj+~2(k\u0000q)2\n2mb\u0000~2k2\n2mb#\n=mb\n~2kq\u000e\u0012~q+ 2mbcf\n2~k\u0000cos\u0012\u0013\n; (32)8\nwhere\u0012is the angle between qandk. So the integral becomes\n\r(k) =g2\nbmmbcf\n4\u0019~2gmkZ1\n0q2dqZ0\n\u0019dcos\u0012\u000e\u0012~q+ 2mbcf\n2~k\u0000cos\u0012\u0013\n: (33)\nSince we have\u00001<cos\u0012<1, the integral regime of qis determined as\n\u00002\u0010\nk+mbcf\n~\u0011\n<q< 2\u0010\nk\u0000mbcf\n~\u0011\n: (34)\nThe integral can be simpli\fed into\n\r(k) =g2\nbmmbcf\n4\u0019~2gmkZ2(k\u0000mbcf=~)\n0q2dq:\n=2g2\nbmmbcf\n3\u0019~2gmk(k\u0000mbcf=~)3\u0002 (k\u0000mbcf=~):\n=2g2\nbmmbcf\n3\u0019~2gmk(k\u0000kc)3\u0002 (k\u0000kc): (35)\nHere we can see kc=mbcf=~, which reproduces the critical momentum discussed above."    },    {        "title": "1409.2340v1.Self_similar_solutions_of_the_one_dimensional_Landau_Lifshitz_Gilbert_equation.pdf",        "content": "arXiv:1409.2340v1  [math.AP]  8 Sep 2014Self-similar solutions of the one-dimensional\nLandau–Lifshitz–Gilbert equation\nSusana Gutiérrez1and André de Laire2\nAbstract\nWe consider the one-dimensional Landau–Lifshitz–Gilbert (LLG) equation, a model des-\ncribing the dynamics for the spin in ferromagnetic material s. Our main aim is the analytical\nstudy of the bi-parametric family of self-similar solution s of this model. In the presence of\ndamping, our construction provides a family of global solut ions of the LLG equation which\nare associated to a discontinuous initial data of infinite (t otal) energy, and which are smooth\nand have finite energy for all positive times. Special emphas is will be given to the behaviour\nof this family of solutions with respect to the Gilbert dampi ng parameter.\nWe would like to emphasize that our analysis also includes th e study of self-similar so-\nlutions of the Schrödinger map and the heat flow for harmonic m aps into the 2-sphere as\nspecial cases. In particular, the results presented here re cover some of the previously known\nresults in the setting of the 1d-Schrödinger map equation.\nKeywords and phrases: Landau–Lifshitz–Gilbert equation, Landau–Lifshitz equa tion, ferro-\nmagnetic spin chain, Schrödinger maps, heat-flow for harmon ic maps, self-similar solutions,\nasymptotics.\nContents\n1 Introduction and statement of results 2\n2 Self-similar solutions of the LLG equation 10\n3 Integration of the Serret–Frenet system 12\n3.1 Reduction to the study of a second order ODE . . . . . . . . . . . . . . . . . . . 12\n3.2 The second-order equation. Asymptotics . . . . . . . . . . . . . . . . . . . . . . . 15\n3.3 The second-order equation. Dependence on the parameter s . . . . . . . . . . . . 28\n3.3.1 Dependence on α. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28\n3.3.2 Dependence on c0. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33\n4 Proof of the main results 34\n5 Some numerical results 37\n6 Appendix 41\n1School of Mathematics, University of Birmingham, Edgbasto n, Birmingham, B15 2TT, United Kingdom.\nE-mail:s.gutierrez@bham.ac.uk\n2Laboratoire Paul Painlevé, Université Lille 1, 59655 Ville neuve d’Ascq Cedex, France. E-mail:\nandre.de-laire@math.univ-lille1.fr\n11 Introduction and statement of results\nIn this work we consider the one-dimensional Landau–Lifshi tz–Gilbert equation (LLG)\n∂t/vectorm =β/vectorm×/vector mss−α/vectorm×(/vectorm×/vectormss), s∈R, t>0, (LLG)\nwhere/vectorm = (m 1,m2,m3) :R×(0,∞)−→S2is the spin vector, β≥0,α≥0,×denotes the\nusual cross-product in R3, andS2is the unit sphere in R3.\nHere we have not included the effects of anisotropy or an exter nal magnetic field. The first term\non the right in (LLG) represents the exchange interaction, w hile the second one corresponds to the\nGilbert damping term and may be considered as a dissipative t erm in the equation of motion. The\nparameters β≥0andα≥0are the so-called exchange constant and Gilbert damping coe fficient,\nand take into account the exchange of energy in the system and the effect of damping on the\nspin chain respectively. Note that, by considering the time -scaling/vectorm(s,t)→/vectorm(s,(α2+β2)1/2t),\nin what follows we will assume w.l.o.g. that\nα, β∈[0,1] andα2+β2= 1. (1.1)\nThe Landau–Lifshitz–Gilbert equation was first derived on p henomenological grounds by L. Lan-\ndau and E. Lifshitz to describe the dynamics for the magnetiz ation or spin /vectorm(s,t)in ferromag-\nnetic materials [24, 11]. The nonlinear evolution equation (LLG) is related to several physical\nand mathematical problems and it has been seen to be a physica lly relevant model for several\nmagnetic materials [19, 20]. In the setting of the LLG equati on, of particular importance is to\nconsider the effect of dissipation on the spin [27, 7, 6].\nThe Landau–Lifshitz family of equations includes as specia l cases the well-known heat-flow\nfor harmonic maps and the Schrödinger map equation onto the 2-sphere. Precisely, when β= 0\n(and therefore α= 1) the LLG equation reduces to the one-dimensional heat-flow equation for\nharmonic maps\n∂t/vectorm =−/vectorm×(/vectorm×/vectormss) =/vectormss+|/vectorms|2/vectorm (HFHM)\n(notice that |/vectorm|2= 1, and in particular /vectorm·/vectormss=−|/vectorms|2). The opposite limiting case of the\nLLG equation (that is α= 0, i.e. no dissipation/damping and therefore β= 1) corresponds to\ntheSchrödinger map equation onto the sphere\n∂t/vectorm =/vectorm×/vectormss. (SM)\nBoth special cases have been objects of intense research and w e refer the interested reader to\n[21, 14, 25, 13] for surveys.\nOf special relevance is the connection of the LLG equation wi th certain non-linear Schrödinger\nequations. This connection is established as follows: Let u s suppose that /vectormis the tangent vector\nof a curve in R3, that is/vectorm =/vectorXs, for some curve /vectorX(s,t)∈R3parametrized by the arc-length. It\ncan be shown [7] that if /vectormevolves under (LLG) and we define the so-called filament funct ionu\nassociated to /vectorX(s,t)by\nu(s,t) = c(s,t)ei/integraltexts\n0τ(σ,t)dσ, (1.2)\nin terms of the curvature cand torsion τassociated to the curve, then usolves the following\nnon-local non-linear Schrödinger equation with damping\niut+(β−iα)uss+u\n2/parenleftbigg\nβ|u|2+2α/integraldisplays\n0Im(¯uus)−A(t)/parenrightbigg\n= 0, (1.3)\nwhereA(t)∈Ris a time-dependent function defined in terms of the curvatur e and torsion\nand their derivatives at the point s= 0. The transformation (1.2) was first considered in the\n2undamped case by Hasimoto in [18]. Notice that if α= 0, equation (1.3) can be transformed\ninto the well-known completely integrable cubic Schröding er equation.\nThe main purpose of this paper is the analytical study of self -similar solutions of the LLG\nequation of the form\n/vectorm(s,t) =/vector m/parenleftbiggs√\nt/parenrightbigg\n, (1.4)\nfor some profile /vector m:R→S2, with emphasis on the behaviour of these solutions with resp ect to\nthe Gilbert damping parameter α∈[0,1].\nForα= 0, self-similar solutions have generated considerable inte rest [22, 21, 4, 15, 9]. We are\nnot aware of any other study of such solutions for α >0in the one dimensional case (see [10]\nfor a study of self-similar solutions of the harmonic map flow in higher dimensions). However,\nLipniacki [26] has studied self-similar solutions for a rel ated model with nonconstant arc-length.\nOn the other hand, little is known analytically about the effe ct of damping on the evolution\nof a one-dimensional spin chain. In particular, Lakshmanan and Daniel obtained an explicit\nsolitary wave solution in [7, 6] and demonstrated the dampin g of the solution in the presence\nof dissipation in the system. In this setting, we would like t o understand how the dynamics of\nself-similar solutions to this model is affected by the intro duction of damping in the equations\ngoverning the motion of these curves.\nAs will be shown in Section 2 self-similar solutions of (LLG) of the type (1.4) constitute a\nbi-parametric family of solutions {/vector mc0,α}c0,αgiven by\n/vectormc0,α(s,t) =/vector mc0,α/parenleftbiggs√\nt/parenrightbigg\n, c 0>0, α∈[0,1], (1.5)\nwhere/vector mc0,αis the solution of the Serret–Frenet equations\n\n\n/vector m′=c/vector n,\n/vector n′=−c/vector m+τ/vectorb,\n/vectorb′=−τ/vector n,(1.6)\nwith curvature and torsion given respectively by\ncc0,α(s) =c0e−αs2\n4, τc0,α(s) =βs\n2, (1.7)\nand initial conditions\n/vector mc0,α(0) = (1,0,0), /vector nc0,α(0) = (0,1,0),/vectorbc0,α(0) = (0,0,1). (1.8)\nThe first result of this paper is the following:\nTheorem 1.1. Letα∈[0,1],c0>01and/vector mc0,αbe the solution of the Serret–Frenet system\n(1.6)with curvature and torsion given by (1.7)and initial conditions (1.8). Define/vectormc0,α(s,t)by\n/vectormc0,α(s,t) =/vector mc0,α/parenleftbiggs√\nt/parenrightbigg\n, t> 0.\nThen,\n1The case c0= 0corresponds to the constant solution for (LLG), that is\n/vectormc0,α(s,t) =/vector m/parenleftbiggs√\nt/parenrightbigg\n= (1,0,0),∀α∈[0,1].\n3(i) The function /vectormc0,α(·,t)is a regular C∞(R;S2)-solution of (LLG) fort>0.\n(ii) There exist unitary vectors /vectorA±\nc0,α= (A±\nj,c0,α)3\nj=1∈S2such that the following pointwise\nconvergence holds when tgoes to zero:\nlim\nt→0+/vectormc0,α(s,t) =\n\n/vectorA+\nc0,α,ifs>0,\n/vectorA−\nc0,α,ifs<0,(1.9)\nwhere/vectorA−\nc0,α= (A+\n1,c0,α,−A+\n2,c0,α,−A+\n3,c0,α).\n(iii) Moreover, there exists a constant C(c0,α,p)such that for all t>0\n/bardbl/vectormc0,α(·,t)−/vectorA+\nc0,αχ(0,∞)(·)−/vectorA−\nc0,αχ(−∞,0)(·)/bardblLp(R)≤C(c0,α,p)t1\n2p, (1.10)\nfor allp∈(1,∞). In addition, if α>0,(1.10) also holds for p= 1. Here,χEdenotes the\ncharacteristic function of a set E.\nThe graphics in Figure 1 depict the profile /vector mc0,α(s)for fixedc0= 0.8and the values of\nα= 0.01,α= 0.2, andα= 0.4. In particular it can be observed how the convergence of /vector mc0,α\nto/vectorA±\nc0,αis accelerated by the diffusion α.\nm1m2m3\n(a)α= 0.01\nm1m2m3\n(b)α= 0.2\nm1m2m3\n(c)α= 0.4\nFigure 1: The profile /vector mc0,αforc0= 0.8and different values of α.\nNotice that the initial condition\n/vectormc0,α(s,0) =/vectorA+\nc0,αχ(0,∞)(s)+/vectorA−\nc0,αχ(−∞,0)(s), (1.11)\nhas a jump singularity at the point s= 0whenever the vectors /vectorA+\nc0,αand/vectorA−\nc0,αsatisfy\n/vectorA+\nc0,α/ne}ationslash=/vectorA−\nc0,α.\nIn this situation (and we will be able to prove analytically t his is the case at least for certain ranges\nof the parameters αandc0, see Proposition 1.5 below), Theorem 1.1 provides a bi-para metric\nfamily of global smooth solutions of (LLG) associated to a di scontinuous singular initial data\n(jump-singularity).\n4As has been already mentioned, in the absence of damping ( α= 0), singular self-similar\nsolutions of the Schrödinger map equation were previously o btained in [15], [22] and [4]. In this\nframework, Theorem 1.1 establishes the persistence of a jum p singularity for self-similar solutions\nin the presence of dissipation.\nSome further remarks on the results stated in Theorem 1.1 are in order. Firstly, from the\nself-similar nature of the solutions /vectormc0,α(s,t)and the Serret–Frenet equations (1.6), it follows\nthat the curvature and torsion associated to these solution s are of the self-similar form and given\nby\ncc0,α(s,t) =c0√\nte−αs2\n4t andτc0,α(s,t) =βs\n2√\nt. (1.12)\nAs a consequence, the total energy E(t)of the spin /vectormc0,α(s,t)found in Theorem 1.1 is expressed\nas\nE(t) =1\n2/integraldisplay∞\n−∞|/vectorms(s,t)|2ds=1\n2/integraldisplay∞\n−∞c2\nc0,α(s,t)ds\n=1\n2/integraldisplay∞\n−∞/parenleftbiggc0√\nte−αs2\n4t/parenrightbigg2\nds=c2\n0/radicalbiggπ\nαt, α> 0, t>0. (1.13)\nIt is evident from (1.13) that the total energy of the spin cha in at the initial time t= 0is infinite,\nwhile the total energy of the spin becomes finite for all posit ive times, showing the dissipation\nof energy in the system in the presence of damping.\nSecondly, it is also important to remark that in the setting o f Schrödinger equations, for fixed\nα∈[0,1]andc0>0, the solution /vectormc0,α(s,t)of (LLG) established in Theorem 1.1 is associated\nthrough the Hasimoto transformation (1.2) to the filament fu nction\nuc0,α(s,t) =c0√\nte(−α+iβ)s2\n4t, (1.14)\nwhich solves\niut+(β−iα)uss+u\n2/parenleftbigg\nβ|u|2+2α/integraldisplays\n0Im(¯uus)−A(t)/parenrightbigg\n= 0,withA(t) =βc2\n0\nt(1.15)\nand is such that at initial time t= 0\nuc0,α(s,0) = 2c0/radicalbig\nπ(α+iβ)δ0.\nHereδ0denotes the delta distribution at the point s= 0and√zdenotes the square root of a\ncomplex number zsuch that Im(√z)>0.\nNotice that the solution uc0,α(s,t)is very rough at initial time, and in particular uc0,α(s,0)\ndoes not belong to the Sobolev class Hsfor anys≥0. Therefore, the standard arguments (that\nis a Picard iteration scheme based on Strichartz estimates a nd Sobolev-Bourgain spaces) cannot\nbe applied at least not in a straightforward way to study the l ocal well-posedness of the initial\nvalue problem for the Schrödinger equations (1.15). The exi stence of solutions of the Scrödinger\nequations (1.15) associated to an initial data proportiona l to a Dirac delta opens the question\nof developing a well-posedness theory for Schrödinger equa tions of the type considered here to\ninclude initial data of infinite energy. This question was ad dressed by A. Vargas and L. Vega\nin [29] and A. Grünrock in [12] in the case α= 0and whenA(t) = 0 (see also [2] for a related\nproblem), but we are not aware of any results in this setting w henα >0(see [14] for related\nwell-posedness results in the case α >0for initial data in Sobolev spaces of positive index).\nNotice that when α>0, the solution (1.14) has infinite energy at the initial time, however the\n5energy becomes finite for any t>0. Moreover, as a consequence of the exponential decay in the\nspace variable when α>0,uc0,α(t)∈Hm(R), for allt>0andm∈N. Hence these solutions do\nnot fit into the usual functional framework for solutions of t he Schrödinger equations (1.15).\nAs already mentioned, one of the main goals of this paper is to study both the qualitative and\nquantitative effect of the damping parameter αand the parameter c0on the dynamical behaviour\nof the family {/vectormc0,α}c0,αof self-similar solutions of (LLG) found in Theorem 1.1. Pre cisely, in an\nattempt to fully understand the regularization of the solut ion at positive times close to the initial\ntimet= 0, and to understand how the presence of damping affects the dyn amical behaviour of\nthese self-similar solutions, we aim to give answers to the f ollowing questions:\nQ1: Can we obtain a more precise behaviour of the solutions /vector mc0,α(s,t)at positive times tclose\nto zero?\nQ2: Can we understand the limiting vectors /vectorA±\nc0,αin terms of the parameters c0andα?\nIn order to address our first question, we observe that, due to the self-similar nature of these\nsolutions (see (1.5)), the behaviour of the family of soluti ons/vectormc0,α(s,t)at positive times close to\nthe initial time t= 0is directly related to the study of the asymptotics of the ass ociated profile\n/vector mc0,α(s)for large values of s. In addition, the symmetries of /vector mc0,α(s)(see Theorem 1.2 below)\nallow to reduce ourselves to obtain the behaviour of the profi le/vector mc0,α(s)for large positive values\nof the space variable. The precise asymptotics of the profile is given in the following theorem.\nTheorem 1.2 (Asymptotics) .Letα∈[0,1],c0>0and{/vector mc0,α,/vector nc0,α,/vectorbc0,α}be the solution of\nthe Serret–Frenet system (1.6)with curvature and torsion given by (1.7)and initial conditions\n(1.8). Then,\n(i) (Symmetries). The components of /vector mc0,α(s),/vector nc0,α(s)and/vectorbc0,α(s)satisfy respectively that\n•m1,c0,α(s)is an even function, and mj,c0,α(s)is an odd function for j∈ {2,3}.\n•n1,c0,α(s)andb1,c0,α(s)are odd functions, while nj,c0,α(s)andbj,c0,α(s)are even func-\ntions forj∈ {2,3}.\n(ii) (Asymptotics). There exist an unit vector /vectorA+\nc0,α∈S2and/vectorB+\nc0,α∈R3such that the following\nasymptotics hold for all s≥s0= 4/radicalbig\n8+c2\n0:\n/vector mc0,α(s) =/vectorA+\nc0,α−2c0\ns/vectorB+\nc0,αe−αs2/4(αsin(/vectorφ(s))+βcos(/vectorφ(s)))\n−2c2\n0\ns2/vectorA+\nc0,αe−αs2/2+O/parenleftBigg\ne−αs2/4\ns3/parenrightBigg\n, (1.16)\n/vector nc0,α(s) =/vectorB+\nc0,αsin(/vectorφ(s))+2c0\ns/vectorA+\nc0,ααe−αs2/4+O/parenleftBigg\ne−αs2/4\ns2/parenrightBigg\n, (1.17)\n/vectorbc0,α(s) =/vectorB+\nc0,αcos(/vectorφ(s))+2c0\ns/vectorA+\nc0,αβe−αs2/4+O/parenleftBigg\ne−αs2/4\ns2/parenrightBigg\n. (1.18)\nHere,sin(/vectorφ)andcos(/vectorφ)are understood acting on each of the components of /vectorφ= (φ1,φ2,φ3),\nwith\nφj(s) =aj+β/integraldisplays2/4\ns2\n0/4/radicalbigg\n1+c2\n0e−2ασ\nσdσ, j∈ {1,2,3}, (1.19)\n6for some constants a1,a2,a3∈[0,2π), and the vector /vectorB+\nc0,αis given in terms of /vectorA+\nc0,α=\n(A+\nj,c0,α)3\nj=1by\n/vectorB+\nc0,α= ((1−(A+\n1,c0,α)2)1/2,(1−(A+\n2,c0,α)2)1/2,(1−(A+\n3,c0,α)2)1/2).\nAs we will see in Section 2, the convergence and rate of conver gence of the solutions /vectormc0,α(s,t)\nof the LLG equation established in parts (ii)and(iii)of Theorem 1.1 will be obtained as a con-\nsequence of the more refined asymptotic analysis of the assoc iated profile given in Theorem 1.2.\nWith regard to the asymptotics of the profile established in p art(ii)of Theorem 1.2, it is\nimportant to mention the following:\n(a) The errors in the asymptotics in Theorem 1.2- (ii)depend only on c0. In other words,\nthe bounds for the errors terms are independent of α∈[0,1]. More precisely, we use the\nnotationO(f(s))to denote a function for which exists a constant C(c0)>0depending on\nc0, but independent on α, such that\n|O(f(s))| ≤C(c0)|f(s)|,for alls≥s0. (1.20)\n(b) The terms /vectorA+\nc0,α,/vectorB+\nc0,α,B+\njsin(aj)andB+\njcos(aj),j∈ {1,2,3}, and the error terms in\nTheorem 1.2- (ii)depend continuously on α∈[0,1](see Subsection 3.3 and Corollary 3.14).\nTherefore, the asymptotics (1.16)–(1.18) show how the profi le/vector mc0,αconverges to /vector mc0,0as\nα→0+and to/vector mc0,1asα→1−. In particular, we recover the asymptotics for /vector mc0,0given\nin [15].\n(c) We also remark that using the Serret–Frenet formulae and the asymptotics in Theorem 1.2-\n(ii), it is straightforward to obtain the asymptotics for the der ivatives of/vectormc0,α(s,t).\n(d) Whenα= 0and for fixed j∈ {1,2,3}, we can write φjin (1.19) as\nφj(s) =aj+s2\n4+c2\n0ln(s)+C(c0)+O/parenleftbigg1\ns2/parenrightbigg\n,\nand we recover the logarithmic contribution in the oscillat ion previously found in [15].\nMoreover, in this case the asymptotics in part (ii)represents an improvement of the one\nestablished in Theorem 1 in [15].\nWhenα>0,φjbehaves like\nφj(s) =aj+βs2\n4+C(α,c0)+O/parenleftBigg\ne−αs2/2\nαs2/parenrightBigg\n, (1.21)\nand there is no logarithmic correction in the oscillations i n the presence of damping.\nConsequently, the phase function /vectorφdefined in (1.19) captures the different nature of the\noscillatory character of the solutions in both the absence a nd the presence of damping in\nthe system of equations.\n(e) Whenα= 1, there exists an explicit formula for /vector mc0,1,/vector nc0,1and/vectorbc0,1, and in particular\nwe have explicit expressions for the vectors /vectorA±\nc0,1in terms of the parameter c0>0in the\nasymptotics given in part (ii). See Appendix.\n7(f) At first glance, one might think that the term −2c2\n0/vectorA+\nc0,αe−αs2/2/s2in (1.16) could be\nincluded in the error term O(e−αs2/4/s3). However, we cannot do this because\ne−αs2/2\ns2>e−αs2/4\ns3, for all2≤s≤/parenleftbigg2\n3α/parenrightbigg1/2\n, α∈(0,1/8], (1.22)\nand in our notation the big- Omust be independent of α. (The exact interval where the\ninequality in (1.22) holds can be determined using the so-ca lled Lambert Wfunction.)\n(g) Let/vectorB+\nc0,α,sin= (Bjsin(aj))3\nj=1,/vectorB+\nc0,α,cos= (Bjcos(aj))3\nj=1. Then the orthogonality of\n/vector mc0,α,/vector nc0,αand/vectorbc0,αtogether with the asymptotics (1.16)–(1.18) yield\n/vectorA+\nc0,α·/vectorB+\nc0,α,sin=/vectorA+\nc0,α·/vectorB+\nc0,α,cos=/vectorB+\nc0,α,sin·/vectorB+\nc0,α,cos= 0,\nwhich gives relations between the phases.\n(h) Finally, the amplitude of the leading order term control ling the wave-like behaviour of the\nsolution/vector mc0,α(s)around/vectorA±\nc0,αfor values of ssufficiently large is of the order c0e−αs2/4/s,\nfrom which one observes how the convergence of the solution t o its limiting values /vectorA±\nc0,αis\naccelerated in the presence of damping in the system. See Fig ure 1.\nWe conclude the introduction by stating the results answeri ng the second of our questions. Pre-\ncisely, Theorems 1.3 and 1.4 below establish the dependence of the vectors /vectorA±\nc0,αin Theorem 1.1\nwith respect to the parameters αandc0. Theorem 1.3 provides the behaviour of the limiting\nvector/vectorA+\nc0,αfor a fixed value of α∈(0,1)and “small” values of c0>0, while Theorem 1.4 states\nthe behaviour of /vectorA+\nc0,αfor fixedc0>0andαclose to the limiting values α= 0andα= 1. Recall\nthat/vectorA−\nc0,αis expressed in terms of the coordinates of /vectorA+\nc0,αas\n/vectorA−\nc0,α= (A+\n1,c0,α,−A+\n2,c0,α,−A+\n3,c0,α) (1.23)\n(see part (ii)of Theorem 1.1).\nTheorem 1.3. Letα∈[0,1],c0>0, and/vectorA+\nc0,α= (A+\nj,c0,α)3\nj=1be the unit vector given in\nTheorem 1.2. Then /vectorA+\nc0,αis a continuous function of c0>0. Moreover, if α∈(0,1]the\nfollowing inequalities hold true:\n|A+\n1,c0,α−1| ≤c2\n0π\nα/parenleftbigg\n1+c2\n0π\n8α/parenrightbigg\n, (1.24)\n/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingleA+\n2,c0,α−c0/radicalbig\nπ(1+α)√\n2/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle≤c2\n0π\n4+c2\n0π\nα√\n2/parenleftBigg\n1+c2\n0π\n8+c0/radicalbig\nπ(1+α)\n2√\n2/parenrightBigg\n+/parenleftbiggc2\n0π\n2√\n2α/parenrightbigg2\n,(1.25)\n/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingleA+\n3,c0,α−c0/radicalbig\nπ(1−α)√\n2/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle≤c2\n0π\n4+c2\n0π\nα√\n2/parenleftBigg\n1+c2\n0π\n8+c0/radicalbig\nπ(1−α)\n2√\n2/parenrightBigg\n+/parenleftbiggc2\n0π\n2√\n2α/parenrightbigg2\n.(1.26)\nThe following result provides an approximation of the behav iour of/vectorA+\nc0,αfor fixedc0>0and\nvalues of the Gilbert parameter close to 0and1.\nTheorem 1.4. Letc0>0,α∈[0,1]and/vectorA+\nc0,αbe the unit vector given in Theorem 1.2. Then\n/vectorA+\nc0,αis a continuous function of αin[0,1], and the following inequalities hold true:\n|/vectorA+\nc0,α−/vectorA+\nc0,0| ≤C(c0)√α|ln(α)|,for allα∈(0,1/2], (1.27)\n|/vectorA+\nc0,α−/vectorA+\nc0,1| ≤C(c0)√\n1−α,for allα∈[1/2,1]. (1.28)\nHere,C(c0)is a positive constant depending on c0but otherwise independent of α.\n8As a by-product of Theorems 1.3 and 1.4, we obtain the followi ng proposition which asserts\nthat the solutions /vectormc0,α(s,t)of the LLG equation found in Theorem 1.1 are indeed associate d\nto a discontinuous initial data at least for certain ranges o fαandc0.\nProposition 1.5. With the same notation as in Theorems 1.1 and 1.2, the followi ng statements\nhold:\n(i) For fixed α∈(0,1)there exists c∗\n0>0depending on αsuch that\n/vectorA+\nc0,α/ne}ationslash=/vectorA−\nc0,α for allc0∈(0,c∗\n0).\n(ii) For fixed c0>0, there exists α∗\n0>0small enough such that\n/vectorA+\nc0,α/ne}ationslash=/vectorA−\nc0,α for allα∈(0,α∗\n0).\n(iii) For fixed 0<c0/ne}ationslash=k√πwithk∈N, there exists α∗\n1>0with1−α∗\n1>0small enough such\nthat\n/vectorA+\nc0,α/ne}ationslash=/vectorA−\nc0,α for allα∈(α∗\n1,1).\nRemark 1.6. Based on the numerical results in Section 5, we conjecture th at/vectorA+\nc0,α/ne}ationslash=/vectorA−\nc0,αfor\nallα∈[0,1)andc0>0.\nWe would like to point out that some of our results and their pr oofs combine and extend\nseveral ideas previously introduced in [15] and [16]. The ap proach we use in the proof of the\nmain results in this paper is based on the integration of the S erret–Frenet system of equations\nvia a Riccati equation, which in turn can be reduced to the stu dy of a second order ordinary\ndifferential equation given by\nf′′(s)+s\n2(α+iβ)f′(s)+c2\n0\n4e−αs2\n2f(s) = 0 (1.29)\nwhen the curvature and torsion are given by (1.7).\nUnlike in the undamped case, in the presence of damping no exp licit solutions are known\nfor equation (1.29) and the term containing the exponential in the equation (1.29) makes it\ndifficult to use Fourier analysis methods to study analytical ly the behaviour of the solutions to\nthis equation. The fundamental step in the analysis of the be haviour of the solutions of (1.29)\nconsists in introducing new auxiliary variables z,handydefined by\nz=|f|2, y= Re(¯ff′)andh= Im(¯ff′)\nin terms of solutions fof (1.29), and studying the system of equations satisfied by t hese key\nquantities. As we will see later on, these variables are the “ natural” ones in our problem, in the\nsense that the components of the tangent, normal and binorma l vectors can be written in terms\nof these quantities. It is important to emphasize that, in or der to obtain error bounds in the\nasymptotic analysis independent of the damping parameter α(and hence recover the asymptotics\nwhenα= 0andα= 1as particular cases), it will be fundamental to exploit the c ancellations\ndue to the oscillatory character of z,yandh.\nThe outline of this paper is the following. Section 2 is devot ed to the construction of the family\nof self-similar solutions {/vectormc0,α}c0,αof the LLG equation. In Section 3 we reduce the study of the\nproperties of this family of self-similar solutions to that of the properties of the solutions of the\ncomplex second order complex ODE (1.29). This analysis is of independent interest. Section 4\ncontains the proofs of the main results of this paper as a cons equence of those established in\n9Section 3. In Section 5 we give provide some numerical result s for/vectorA+\nc0,α, as a function of α∈[0,1]\nandc0>0, which give some inside for the scattering problem and justi fy Remark 1.6. Finally,\nwe have included the study of the self-similar solutions of t he LLG equation in the case α= 1\nin Appendix.\nAcknowledgements. S. Gutiérrez and A. de Laire were supported by the British proj ect\n“Singular vortex dynamics and nonlinear Schrödinger equat ions” (EP/J01155X/1) funded by\nEPSRC. S. Gutiérrez was also supported by the Spanish projec ts MTM2011-24054 and IT641-\n13.\nBoth authors would like to thank L. Vega for many enlightening conversations and for his\ncontinuous support.\n2 Self-similar solutions of the LLG equation\nFirst we derive what we will refer to as the geometric represe ntation of the LLG equation. To\nthis end, let us assume that /vectorm(s,t) =/vectorXs(s,t)for some curve /vectorX(s,t)inR3parametrized with\nrespect to the arc-length with curvature c(s,t)and torsion τ(s,t). Then, using the Serret–Frenet\nsystem of equations (1.6), we have\n/vectormss= cs/vectorn+c(−c/vectorn+τ/vectorb),\nand thus we can rewrite (LLG) as\n∂t/vectorm =β(cs/vectorb−cτ/vectorn)+α(cτ/vectorb+cs/vectorn), (2.1)\nin terms of intrinsic quantities c,τand the Serret–Frenet trihedron {/vectorm,/vectorn,/vectorb}.\nWe are interested in self-similar solutions of (LLG) of the f orm\n/vectorm(s,t) =/vector m/parenleftbiggs√\nt/parenrightbigg\n(2.2)\nfor some profile /vector m:R−→S2. First, notice that due to the self-similar nature of /vectorm(s,t)in (2.2),\nfrom the Serret–Frenet equations (1.6) it follows that the u nitary normal and binormal vectors\nand the associated curvature and torsion are self-similar a nd given by\n/vectorn(s,t) =/vector n/parenleftbiggs√\nt/parenrightbigg\n,/vectorb(s,t) =/vectorb/parenleftbiggs√\nt/parenrightbigg\n, (2.3)\nc(s,t) =1√\ntc/parenleftbiggs√\nt/parenrightbigg\nandτ(s,t) =1√\ntτ/parenleftbiggs√\nt/parenrightbigg\n. (2.4)\nAssume that /vectorm(s,t)is a solution of the LLG equation, or equivalently of its geom etric version\n(2.1). Then, from (2.2)–(2.4) it follows that the Serret–Fr enet trihedron {/vector m(·),/vector n(·),/vectorb(·)}solves\n−s\n2c/vector n=β(c′/vectorb−cτ/vector n)+α(cτ/vectorb+c′/vector n), (2.5)\nAs a consequence,\n−s\n2c=αc′−βcτ andβc′+αcτ= 0.\nThus, we obtain\nc(s) =c0e−αs2\n4andτ(s) =βs\n2, (2.6)\n10for some positive constant c0(recall that we are assuming w.l.o.g. that α2+β2= 1). Therefore,\nin view of (2.4), the curvature and torsion associated to a se lf-similar solution of (LLG) of the\nform (2.2) are given respectively by\nc(s,t) =c0√\nte−αs2\n4tandτ(s,t) =βs\n2t, c 0>0. (2.7)\nNotice that given (c,τ)as above, for fixed time t>0one can solve the Serret–Frenet system of\nequations to obtain the solution up to a rigid motion in the sp ace which in general may depend\nont. As a consequence, and in order to determine the dynamics of t he spin chain, we need\nto find the time evolution of the trihedron {/vectorm(s,t),/vectorn(s,t),/vectorb(s,t)}at some fixed point s∗∈R.\nTo this end, from the above expressions of the curvature and t orsion associated to /vectorm(s,t)and\nevaluating the equation (2.1) at the point s∗= 0, we obtain that /vectormt(0,t) =/vector0. On the other\nhand, differentiating the geometric equation (2.1) with res pect tos, and using the Serret–Frenet\nequations (1.6) together with the compatibility condition /vectormst=/vectormts, we get the following relation\nfor the time evolution of the normal vector\nc/vectornt=β(css/vectorb+c2τ/vectorm−cτ2/vectorb)+α((cτ)s/vectorb−ccs/vectorm+csτ/vectorb).\nThe evaluation of the above identity at s∗= 0together with the expressions for the curvature\nand torsion in (2.7) yield /vectornt(0,t) =/vector0. The above argument shows that\n/vectormt(0,t) =/vector0, /vectornt(0,t) =/vector0and/vectorbt(0,t) = (/vectorm×/vectorn)t(0,t) =/vector0.\nTherefore we can assume w.l.o.g. that\n/vectorm(0,t) = (1,0,0), /vectorn(0,t) = (0,1,0)and/vectorb(0,t) = (0,0,1),\nand in particular\n/vector m(0) =/vectorm(0,1) = (1,0,0), /vector n(0) =/vectorn(0,1) = (0,1,0),and/vectorb(0) =/vectorb(0,1) = (0,0,1).(2.8)\nGivenα∈[0,1]andc0>0, from the theory of ODE’s, it follows that there exists a uniq ue\n{/vector mc0,α(·),/vector nc0,α(·),/vectorbc0,α(·)} ∈/parenleftbig\nC∞(R;S2)/parenrightbig3solution of the Serret–Frenet equations (1.6) with\ncurvature and torsion (2.6) and initial conditions (2.8) su ch that\n/vector mc0,α⊥/vector nc0,α, /vector mc0,α⊥/vectorbc0,α, /vector nc0,α⊥/vectorbc0,α\nand\n|/vector mc0,α|2=|/vector nc0,α|2=|/vectorbc0,α|2= 1.\nDefine/vectormc0,α(s,t)as\n/vectormc0,α(s,t) =/vector mc0,α/parenleftbiggs√\nt/parenrightbigg\n. (2.9)\nThen,/vector mc0,α(·,t)∈ C∞/parenleftbig\nR;S2/parenrightbig\nfor allt>0, and bearing in mind both the relations in (2.3)–(2.4)\nand the fact that the vectors {/vector mc0,α(·),/vector nc0,α(·),/vectorbc0,α(·)}satisfy the identity (2.5), a straightfor-\nward calculation shows that /vector mc0,α(·,t)is a regular C∞(R;S2)-solution of the LLG equation for\nallt>0. Notice that the case c0= 0yields the constant solution /vector m0,α(s,t) = (1,0,0). Therefore\nin what follows we will assume that c0>0.\nThe rest of the paper is devoted to establish analytical prop erties of the solutions {/vectormc0,α(s,t)}c0,α\ndefined by (2.9) for fixed α∈[0,1]andc0>0. As already mentioned, due to the self-similar\nnature of these solutions, it suffices to study the properties of the associated profile /vector mc0,α(·)or,\nequivalently, of the solution {/vector mc0,α,/vector nc0,α,/vectorbc0,α}of the Serret–Frenet system (1.6) with curvature\nand torsion given by (2.6) and initial conditions (2.8). As w e will continue to see, the analysis\nof the profile solution {/vector mc0,α,/vector nc0,α,/vectorbc0,α}can be reduced to the study of the properties of the\nsolutions of a certain second order complex differential equ ation.\n113 Integration of the Serret–Frenet system\n3.1 Reduction to the study of a second order ODE\nClassical changes of variables from the differential geomet ry of curves allow us to reduce the nine\nequations in the Serret–Frenet system into three complex-v alued second order equations (see\n[8, 28, 23]). Theses changes of variables are related to ster eographic projection and this approach\nwas also used in [15]. However, their choice of stereographi c projection has a singularity at the\norigin, which leads to an indetermination of the initial con ditions of some of the new variables.\nFor this reason, we consider in the following lemma a stereog raphic projection that is compatible\nwith the initial conditions (2.8). Although the proof of the lemma below is a slight modification\nof that in [23, Subsections 2.12 and 7.3], we have included it s proof here both for the sake of\ncompleteness and to clarify to the unfamiliar reader how the integration of the Frenet equations\ncan be reduced to the study of a second order differential equa tion.\nLemma 3.1. Let/vector m= (mj(s))3\nj=1,/vector n= (nj(s))3\nj=1and/vectorb= (bj(s))3\nj=1be a solution of the Serret–\nFrenet equations (1.6)with positive curvature cand torsion τ. Then, for each j∈ {1,2,3}the\nfunction\nfj(s) =e1\n2/integraltexts\n0c(σ)ηj(σ)dσ,withηj(s) =(nj(s)+ibj(s))\n1+mj(s),\nsolves the equation\nf′′\nj(s)+/parenleftbigg\niτ(s)−c′(s)\nc(s)/parenrightbigg\nf′\nj(s)+c2(s)\n4fj(s) = 0, (3.1)\nwith initial conditions\nfj(0) = 1, f′\nj(0) =c(0)(nj(0)+ibj(0))\n2(1+mj(0)).\nMoreover, the coordinates of /vector m,/vector nand/vectorbare given in terms of fjandf′\njby\nmj(s) = 2/parenleftBigg\n1+4\nc(s)2/vextendsingle/vextendsingle/vextendsingle/vextendsinglef′\nj(s)\nfj(s)/vextendsingle/vextendsingle/vextendsingle/vextendsingle2/parenrightBigg−1\n−1, nj(s)+ibj(s) =4f′\nj(s)\nc(s)fj(s)/parenleftBigg\n1+4\nc(s)2/vextendsingle/vextendsingle/vextendsingle/vextendsinglef′\nj(s)\nfj(s)/vextendsingle/vextendsingle/vextendsingle/vextendsingle2/parenrightBigg−1\n.\n(3.2)\nThe above relations are valid at least as long as mj>−1and|fj|>0.\nProof. For simplicity, we omit the index j. The proof relies on several transformations that are\nrather standard in the study of curves. First we define the com plex function\nN= (n+ib)ei/integraltexts\n0τ(σ)dσ. (3.3)\nThenN′=iτN+ (n′+ib′)ei/integraltexts\n0τ(σ)dσ. On the other hand, the Serret–Frenet equations imply\nthat\nn′+ib′=−cm−iτNe−i/integraltexts\n0τ(σ)dσ.\nTherefore, setting\nψ=cei/integraltexts\n0τ(σ)dσ,\nwe get\nN′=−ψm. (3.4)\nUsing again the Serret–Frenet equations, we also obtain\nm′=1\n2(ψN+ψN). (3.5)\n12Let us consider now the auxiliary function\nϕ=N\n1+m. (3.6)\nDifferentiating and using (3.4), (3.5) and (3.6)\nϕ′=N′\n1+m−Nm′\n(1+m)2\n=N′\n1+m−ϕm′\n1+m\n=−ϕ2ψ\n2−ψ\n2(1+m)(2m+ϕN).\nNoticing that we can recast the relation m2+n2+b2= 1asNN= (1−m)(1+m)and recalling\nthe definition of ϕin (3.6), we have ϕN= 1−m, so that\nϕ′+ϕ2ψ\n2+ψ\n2= 0. (3.7)\nFinally, define the stereographic projection of (m,n,b)by\nη=n+ib\n1+m. (3.8)\nObserve that from the definitions of Nandϕ, respectively in (3.3) and (3.6), we can rewrite η\nas\nη=ϕe−i/integraltexts\n0τ(σ)dσ,\nand from (3.7) it follows that ηsolves the Riccati equation\nη′+iτη+c\n2(η2+1) = 0, (3.9)\n(recall that ψ=cei/integraltexts\n0τ(σ)dσ). Finally, setting\nf(s) =e1\n2/integraltexts\n0c(σ)η(σ)dσ, (3.10)\nwe get\nη=2f′\ncf(3.11)\nand equation (3.1) follows from (3.9). The initial conditio ns are an immediate consequence of\nthe definition of ηandfin (3.8) and (3.10).\nA straightforward calculation shows that the inverse trans formation of the stereographic pro-\njection is\nm=1−|η|2\n1+|η|2, n=2Reη\n1+|η|2, b=2Imη\n1+|η|2,\nso that we obtain (3.2) using (3.11) and the above identities .\nGoing back to our problem, Lemma 3.1 reduces the analysis of t he solution {/vector m,/vector n,/vectorb}of the\nSerret–Frenet system (1.6) with curvature and torsion give n by (2.6) and initial conditions (2.8)\nto the study of the second order differential equation\nf′′(s)+s\n2(α+iβ)f′(s)+c2\n0\n4e−αs2/2f(s) = 0, (3.12)\n13with three initial conditions: For (m1,n1,b1) = (1,0,0)the associated initial condition for f1is\nf1(0) = 1, f′\n1(0) = 0, (3.13)\nfor(m2,n2,b2) = (0,1,0)is\nf2(0) = 1, f′\n2(0) =c0\n2, (3.14)\nand for(m3,n3,b3) = (0,0,1)is\nf3(0) = 1, f′\n3(0) =ic0\n2. (3.15)\nIt is important to notice that, by multiplying (3.12) by ¯f′and taking the real part, it is easy to\nsee that\nd\nds/bracketleftbigg1\n2/parenleftbigg\neαs2\n2|f′|2+c2\n0\n4|f|2/parenrightbigg/bracketrightbigg\n= 0.\nThus,\nE(s) :=1\n2/parenleftbigg\neαs2\n2|f′|2+c2\n0\n4|f|2/parenrightbigg\n=E0,∀s∈R, (3.16)\nwithE0a constant defined by the value of E(s)at some point s0∈R. The conservation of the\nenergyE(s)allows us to simplify the expressions of mj,njandbjforj∈ {1,2,3}in the formulae\n(3.2) in terms of the solution fjto (3.12) associated to the initial conditions (3.13)–(3.1 5).\nIndeed, on the one hand notice that the energies associated t o the initial conditions (3.13)–\n(3.15) are respectively\nE0,1=c2\n0\n8, E 0,2=c2\n0\n4andE0,3=c2\n0\n4. (3.17)\nOn the other hand, from (3.16), it follows that\n/parenleftBigg\n1+4\nc2\n0e−αs2\n2|f′\nj|2(s)\n|fj|2(s)/parenrightBigg−1\n=c2\n0\n8E0,j|fj|2(s), j∈ {1,2,3}.\nTherefore, from (3.17), the above identity and formulae (3. 2) in Lemma 3.1, we conclude that\nm1(s) = 2|f1(s)|2−1, n 1(s)+ib1(s) =4\nc0eαs2/4¯f1(s)f′\n1(s), (3.18)\nmj(s) =|fj(s)|2−1, n j(s)+ibj(s) =2\nc0eαs2/4¯fj(s)f′\nj(s), j∈ {2,3}. (3.19)\nThe above identities give the expressions of the tangent, no rmal and binormal vectors in terms\nof the solutions {fj}3\nj=1of the second order differential equation (3.12) associated to the initial\nconditions (3.13)–(3.15).\nBy Lemma 3.1, the formulae (3.18) and (3.19) are valid as long a smj>−1, which is equivalent\nto the condition |fj| /ne}ationslash= 0. As shown in Appendix, for α= 1there is˜s>0such thatmj(˜s) =−1\nand then (3.18) and (3.19) are (a priori) valid just in a bound ed interval. However, the trihedron\n{/vector m,/vector n,/vectorb}is defined globally and fjcan also be extended globally as the solution of the linear\nequation (3.12). Then, it is simple to verify that the functi ons given by the l.h.s. of formulae\n(3.18) and (3.19) satisfy the Serret–Frenet system and henc e, by the uniqueness of the solution,\nthe formulae (3.18) and (3.19) are valid for all s∈R.\n143.2 The second-order equation. Asymptotics\nIn this section we study the properties of the complex-value d equation\nf′′(s)+s\n2(α+iβ)f′(s)+c2\n0\n4f(s)e−αs2/2= 0, (3.20)\nfor fixedc0>0,α∈[0,1),β >0such thatα2+β2= 1. We begin noticing that in the\ncaseα= 0, the solution can be written explicitly in terms of paraboli c cylinder functions or\nconfluent hypergeometric functions (see [1]). Another anal ytical approach using Fourier analysis\ntechniques has been taken in [15], leading to the asymptotic s\nf(s) =C1ei(c2\n0/2)ln(s)+C2e−is2/4\nse−i(c2\n0/2)ln(s)+O(1/s2), (3.21)\nass→ ∞, where the constants C1,C2andO(1/s2)depend on the initial conditions and c0.\nForα= 1, equation (3.20) can be also solved explicitly and the solut ion is given by\nf(s) =2f′(0)\nc0sin/parenleftbiggc0\n2/integraldisplays\n0e−σ2/4dσ/parenrightbigg\n+f(0)cos/parenleftbiggc0\n2/integraldisplays\n0e−σ2/4dσ/parenrightbigg\n.\nIn the case α∈(0,1), one cannot compute the solutions of (3.20) in terms of known functions\nand we will follow a more analytical analysis. In contrast wi th the situation when α= 0, it is\nfar from evident to use Fourier analysis to study (3.20) when α>0.\nFor the rest of this section we will assume that α∈[0,1). In addition, we will also assume that\ns>0and we will develop the asymptotic analysis necessary to est ablish part (ii)of Theorem 1.2.\nAt this point, it is important to recall the expressions give n in (3.18)–(3.19) for the coordinates\nof the tangent, normal and binormal vectors associated to ou r family of solutions of the LLG\nequation in terms f. Bearing this in mind, we observe that the study of the asympto tic behaviour\nof these vectors are dictated by the asymptotic behaviour of the variables\nz=|f|2, y= Re(¯ff′),andh= Im(¯ff′) (3.22)\nassociated to the solution fof (3.20).\nAs explained in the remark (a) after Theorem 1.2, we need to wo rk with remainder terms that\nare independent of α. To this aim, we proceed in two steps: first we found uniform es timates\nforα∈[0,1/2]in Propositions 3.2 and 3.3, then we treat the case α∈[1/2,1)in Lemma 3.6. In\nSubsection 3.3 we provide some continuity results that allo ws us to take α→1−and give the\nfull statement in Corollary 3.14. Finally, notice that thes e asymptotics lead to the asymptotics\nfor the original equation (3.20) (see Remark 3.9).\nWe begin our analysis by establishing the following:\nProposition 3.2. Letc0>0,α∈[0,1),β >0such thatα2+β2= 1, andfbe a solution of\n(3.20). Define z,yandhasz=|f|2andy+ih=¯ff′. Then\n(i) There exists E0≥0such that the identity\n1\n2/parenleftbigg\neαs2\n2|f′|2+c2\n0\n4|f|2/parenrightbigg\n=E0\nholds true for all s∈R. In particular, f,f′,z,yandhare bounded functions. Moreover,\nfor alls∈R\n|f(s)| ≤√8E0\nc0,|f′(s)| ≤/radicalbig\n2E0e−αs2/4, (3.23)\n|z(s)| ≤8E0\nc2\n0and|h(s)|+|y(s)| ≤8E0\nc0e−αs2/4. (3.24)\n15(ii) The limit\nz∞:= lim\ns→∞z(s)\nexists.\n(iii) Letγ:= 2E0−c2\n0z∞/2ands0= 4/radicalbig\n8+c2\n0. For alls≥s0, we have\nz(s)−z∞=−4\ns(αy+βh)−4γ\ns2e−αs2/2+R0(s), (3.25)\nwhere\n|R0(s)| ≤C(E0,c0)e−αs2/4\ns3. (3.26)\nProof. Part(i)is just the conservation of energy proved in (3.16). Next, us ing the conservation\nlaw in part (i), we obtain that the variables {z,y,h}solve the first-order real system\nz′= 2y, (3.27)\ny′=βs\n2h−αs\n2y+e−αs2/2/parenleftbigg\n2E0−c2\n0\n2z/parenrightbigg\n, (3.28)\nh′=−βs\n2y−αs\n2h. (3.29)\nTo show (ii), plugging (3.27) into (3.29) and integrating from 0to somes>0we obtain\nz(s)−1\ns/integraldisplays\n0z(σ)dσ=−4\nβs/parenleftbigg\nh(s)−h(0)+α\n2/integraldisplays\n0σh(σ)dσ/parenrightbigg\n. (3.30)\nAlso, using the above identity,\nd\nds/parenleftbigg1\ns/integraldisplays\n0z(σ)dσ/parenrightbigg\n=−4\nβs2/parenleftbigg\nh(s)−h(0)+α\n2/integraldisplays\n0σh(σ)dσ/parenrightbigg\n. (3.31)\nNow, since from part (i)|h(s)| ≤8E0\nc0e−αs2/4, bothhandα/integraltexts\n0σh(σ)dσare bounded functions,\nthus from (3.31) it follows that the limit of1\ns/integraltexts\n0zexists, ass→ ∞. Hence (3.30) and previous\nobservations conclude that the limit z∞:= lims→∞z(s)exists and furthermore\nz∞:= lim\ns→∞z(s) = lim\ns→∞1\ns/integraldisplays\n0z(σ). (3.32)\nWe continue to prove (iii). Integrating (3.31) between s>0and+∞and using integration\nby parts, we obtain\nz∞−1\ns/integraldisplays\n0z(σ)dσ=−4\nβ/integraldisplay∞\nsh(σ)\nσ2dσ+4\nβh(0)\ns−2α\nβ/bracketleftbigg1\ns/integraldisplays\n0σh(σ)dσ+/integraldisplay∞\nsh(σ)dσ/bracketrightbigg\n.(3.33)\nFrom (3.30) and (3.33), we get\nz(s)−z∞=−4\nβh(s)\ns+2α\nβ/integraldisplay∞\nsh(σ)dσ+4\nβ/integraldisplay∞\nsh(σ)\nσ2. (3.34)\nIn order to compute the integrals in (3.34), using (3.27) and (3.28), we write\nh=2\nβ/parenleftbiggy′\ns+α\n4z′−2E0\nse−αs2/2+c2\n0\n2sze−αs2/2/parenrightbigg\n.\n16Then, integrating by parts and using the bound for yin (3.24),\n/integraldisplay∞\nsh(σ) =2\nβ/parenleftBigg\n−y\ns+/integraldisplay∞\nsy\nσ2+α\n4(z∞−z)−2E0/integraldisplay∞\nse−ασ2/2\nσ+c2\n0\n2/integraldisplay∞\nsz\nσe−ασ2/2/parenrightBigg\n.(3.35)\nAlso, from (3.27) and (3.34), we obtain\n/integraldisplay∞\nsh(σ)\nσ2=2\nβ/parenleftBigg/integraldisplay∞\nsy′\nσ3+α\n2/integraldisplay∞\nsy\nσ2−2E0/integraldisplay∞\nse−ασ2/2\nσ3+c2\n0\n2/integraldisplay∞\nsz\nσ3e−ασ2/2/parenrightBigg\n.(3.36)\nMultiplying (3.34) by β2, using (3.35), (3.36) and the identity\nα/integraldisplay∞\nse−ασ2/2\nσn=e−αs2/2\nsn+1−(n+1)/integraldisplay∞\nse−ασ2/2\nσn+2,for allα≥0, n≥1,\nwe conclude that\n(α2+β2)(z−z∞) =−4\ns(αy+βh)−8E0\ns2e−αs2/2\n+8α/integraldisplay∞\nsy\nσ2+8/integraldisplay∞\nsy′\nσ3+2c2\n0/integraldisplay∞\nse−ασ2/2z/parenleftbiggα\nσ+2\nσ3/parenrightbigg\n. (3.37)\nFinally, using (3.27) and the boundedness of zandy, an integration by parts argument shows\nthat\n8α/integraldisplay∞\nsy\nσ2+8/integraldisplay∞\nsy′\nσ3=−4αz\ns2−8y\ns3−12z\ns4+8/integraldisplay∞\nsz/parenleftbiggα\nσ3−6\nσ5/parenrightbigg\n. (3.38)\nBearing in mind that α2+β2= 1, from (3.37) and (3.38), we obtain the following identity\nz−z∞=−4\ns(αy+βh)−8E0\ns2e−αs2/2−4αz\ns2−8y\ns3−12z\ns4+8/integraldisplay∞\nsz/parenleftbiggα\nσ3+6\nσ5/parenrightbigg\ndσ\n+2c2\n0/integraldisplay∞\nse−ασ2/2z/parenleftbiggα\nσ+2\nσ3/parenrightbigg\ndσ,(3.39)\nfor alls>0. In order to prove (iii), we first write z=z−z∞+z∞and observe that\n8α/integraldisplay∞\nsz\nσ3= 8α/integraldisplay∞\nsz−z∞\nσ3+4αz∞\ns2,\n/integraldisplay∞\nsz\nσ5=/integraldisplay∞\nsz−z∞\nσ5+z∞\n4s4and\n/integraldisplay∞\nse−ασ2/2z/parenleftbiggα\nσ+2\nσ3/parenrightbigg\n=/integraldisplay∞\nse−ασ2/2(z−z∞)/parenleftbiggα\nσ+2\nσ3/parenrightbigg\n+z∞\ns2e−αs2/2.\nTherefore, we can recast (3.39) as (3.25) with\nR0(s) =−4α(z−z∞)\ns2−8y\ns3−12(z−z∞)\ns4+8/integraldisplay∞\ns(z−z∞)/parenleftbiggα\nσ3+6\nσ5/parenrightbigg\ndσ\n+2c2\n0/integraldisplay∞\nse−ασ2/2(z−z∞)/parenleftbiggα\nσ+2\nσ3/parenrightbigg\ndσ.(3.40)\nLet us take s0≥1to be fixed in what follows. For t≥s0, we denote /bardbl · /bardbltthe norm of\nL∞([t,∞)). From the definition of R0in (3.40) and the elementary inequalities\nα/integraldisplay∞\nse−ασ2/2\nσn≤e−αs2/2\nsn+1,for allα≥0, n≥1, (3.41)\n17and/integraldisplay∞\nse−ασ2/2\nσn≤e−αs2/2\n(n−1)sn−1,for allα≥0, n>1, (3.42)\nwe obtain\n/bardblR0/bardblt≤8/bardbly/bardblt\nt3+4\nt2/parenleftBig\n8+c2\n0e−αt2/2/parenrightBig\n/bardblz−z∞/bardblt.\nHence, choosing s0= 4/radicalbig\n8+c2\n0, so that4\nt2/parenleftBig\n8+c2\n0e−αt2/2/parenrightBig\n≤1/2, from (3.24) and (3.25) we\nconclude that there exists a constant C(E0,c0)>0such that\n/bardblz−z∞/bardblt≤C(E0,c0)\nte−αt2/4,for allα∈[0,1)andt≥s0,\nwhich implies that\n|z(s)−z∞| ≤C(E0,c0)\nse−αs2/4,for allα∈[0,1), s≥s0. (3.43)\nFinally, plugging (3.24) and (3.43) into (3.40) and bearing in mind the inequalities (3.41) and\n(3.42), we deduce that\n|R0(s)| ≤C(E0,c0)e−αs2/4\ns3,∀s≥s0= 4/radicalBig\n8+c2\n0, (3.44)\nand the proof of (iii)is completed.\nFormula (3.25) in Proposition 3.2 gives zin terms of yandh. Therefore, we can reduce our\nanalysis to that of the variables yandhor, in other words, to that of the system (3.27)–(3.29).\nIn fact, a first attempt could be to define w=y+ih, so that from (3.28) and (3.29), we have\nthatwsolves/parenleftBig\nwe(α+iβ)s2/4/parenrightBig′\n=e(−α+iβ)s2/4/parenleftbigg\nγ−c2\n0\n2(z−z∞)/parenrightbigg\n. (3.45)\nFrom (3.43) in Proposition 3.2 and (3.45), we see that the lim itw∗= lims→∞w(s)e(α+iβ)s2/4\nexists (at least when α/ne}ationslash= 0), and integrating (3.45) from some s>0to∞we find that\nw(s) =e−(α+iβ)s2/4/parenleftbigg\nw∗−/integraldisplay∞\nse(−α+iβ)σ2/4/parenleftbigg\nγ−c2\n0\n2(z−z∞)/parenrightbigg\ndσ/parenrightbigg\n.\nIn order to obtain an asymptotic expansion, we need to estima te/integraltext∞\nse(−α+iβ)σ2/4(z−z∞), fors\nlarge. This can be achieved using (3.43),\n/vextendsingle/vextendsingle/vextendsingle/vextendsingle/integraldisplay∞\nse(−α+iβ)σ2/4(z−z∞)dσ/vextendsingle/vextendsingle/vextendsingle/vextendsingle≤C(E0,c0)/integraldisplay∞\nse−ασ2/2\nσdσ (3.46)\nand the asymptotic expansion\n/integraldisplay∞\nse−ασ2/2\nσdσ=e−αs2/2/parenleftbigg1\nαs2−2\nα2s4+8\nα3s6+···/parenrightbigg\n.\nHowever this estimate diverges as α→0. The problem is that the bound used in obtaining\n(3.46) does not take into account the cancellations due to th e oscillations. Therefore, and in\norder to obtain the asymptotic behaviour of z,yandhvalid for all α∈[0,1), we need a more\nrefined analysis. In the next proposition we study the system (3.27)–(3.29), where we consider\nthe cancellations due the oscillations (see Lemma 3.5 below ). The following result provides\nestimates that are valid for s≥s1, for somes1independent of α, ifαis small.\n18Proposition 3.3. With the same notation and terminology as in Proposition 3.2 , let\ns1= max/braceleftBigg\n4/radicalBig\n8+c2\n0,2c0/parenleftbigg1\nβ−1/parenrightbigg1/2/bracerightBigg\n.\nThen for all s≥s1,\ny(s) =be−αs2/4sin(φ(s1;s))−2αγ\nse−αs2/2+O/parenleftBigg\ne−αs2/2\nβ2s2/parenrightBigg\n, (3.47)\nh(s) =be−αs2/4cos(φ(s1;s))−2βγ\nse−αs2/2+O/parenleftBigg\ne−αs2/2\nβ2s2/parenrightBigg\n, (3.48)\nwhere\nφ(s1;s) =a+β/integraldisplays2/4\ns2\n1/4/radicalbigg\n1+c2\n0e−2αt\ntdt,\na∈[0,2π)is a real constant, and bis a positive constant given by\nb2=/parenleftbigg\n2E0−c2\n0\n4z∞/parenrightbigg\nz∞. (3.49)\nProof. First, notice that plugging the expression for z(s)−z∞in (3.25) into (3.28), the system\n(3.28)–(3.29) for the variables yandhrewrites equivalently as\ny′=s\n2(βh−αy)+2c2\n0\nse−αs2/2(βh+αy)+γe−αs2/2+R1(s), (3.50)\nh′=−s\n2(βy+αh), (3.51)\nwhere\nR1(s) =−c2\n0\n2e−αs2/2R0(s)+2c2\n0γe−αs2\ns2, (3.52)\nandR0is given by (3.40).\nIntroducing the new variables,\nu(t) =eαty(2√\nt), v(t) =eαth(2√\nt), (3.53)\nwe recast (3.50)–(3.51) as\n/parenleftbiggu\nv/parenrightbigg′\n=/parenleftbiggαK β(1+K)\n−β0/parenrightbigg/parenleftbiggu\nv/parenrightbigg\n+/parenleftbiggF\n0/parenrightbigg\n, (3.54)\nwith\nK=c2\n0e−2αt\nt, F=γe−αt\n√\nt+e−αt\n√\ntR1(2√\nt),\nwhereR1is the function defined in (3.52). In this way, we can regard (3 .54) as a non-autonomous\nsystem. It is straightforward to check that the matrix\nA=/parenleftbiggαK β(1+K)\n−β0/parenrightbigg\nis diagonalizable, i.e. A=PDP−1, with\nD=/parenleftbiggλ+0\n0λ−/parenrightbigg\n, P=/parenleftbigg−αK\n2β−i∆1/2−αK\n2β+i∆1/2\n1 1/parenrightbigg\n,\n19λ±=αK\n2±iβ∆1/2,and∆ = 1+K−α2K2\n4β2. (3.55)\nAt this point we remark that the condition t≥t1, witht1:=s2\n1/4ands1≥2c0(1\nβ−1)1/2, implies\nthat\n0<K/parenleftbigg1\nβ−1/parenrightbigg\n≤1,∀t≥t1, (3.56)\nso that\n∆ = 1+K−(1−β2)\n4β2K2=/parenleftbigg\n1+K\n2+K\n2β/parenrightbigg/parenleftbigg\n1+K\n2/parenleftbigg\n1−1\nβ/parenrightbigg/parenrightbigg\n≥1\n2,∀t≥t1.(3.57)\nThus, defining\nw= (w1,w2) =P−1(u,v), (3.58)\nwe get/parenleftBig\ne−/integraltextt\nt1Dw/parenrightBig′\n=e−/integraltextt\nt1D/parenleftBig\n(P−1)′Pw+P−1˜F/parenrightBig\n, (3.59)\nwith˜F= (F,0). From the definition of wand taking into account that uandvare real functions,\nwe have that w1= ¯w2and therefore the study of (3.59) reduces to the analysis of t he equation:\n/parenleftBig\ne−/integraltextt\nt1λ+w1/parenrightBig′\n=e−/integraltextt\nt1λ+G(t), (3.60)\nwith\nG(t) =iαK′\n4β∆1/2(w1+ ¯w1)−∆′\n4∆(w1−¯w1)+iF\n2∆1/2.\nFrom (3.60) we have\nw1(t) =e/integraltextt\nt1λ+/parenleftbigg\nw1(t1)+w∞−/integraldisplay∞\nte−/integraltextτ\nt1λ+G(τ)dτ/parenrightbigg\n, (3.61)\nwith\nw∞=/integraldisplay∞\nt1e−/integraltextτ\nt1λ+G(τ).\nSince\nw1=iu\n2∆1/2+v\n2+iαKv\n4β∆1/2, (3.62)\nwe recastGasG=i(G1+G2+G3)with\nG1=αK′v\n4β∆1/2−∆′\n4∆3/2/parenleftbigg\nu+αKv\n2β/parenrightbigg\n, G2=γe−αt\n2t1/2∆1/2andG3=e−αt\n2t1/2∆1/2R1(2t1/2).\nNow, from the definition of Kand∆, we have\nK′=−K/parenleftbigg\n2α+1\nt/parenrightbigg\n, K′′=K/parenleftBigg/parenleftbigg\n2α+1\nt/parenrightbigg2\n+1\nt2/parenrightBigg\n,\n∆′=K′/parenleftbigg\n1−α2K\n2β2/parenrightbigg\nand∆′′=K/parenleftBigg/parenleftbigg\n2α+1\nt/parenrightbigg2\n+1\nt2/parenrightBigg/parenleftbigg\n1−α2K\n2β2/parenrightbigg\n−α2\n2β2K2/parenleftbigg\n2α+1\nt/parenrightbigg2\n.\nAlso, since s1= max{4/radicalbig\n8+c2\n0,2c0(1/β−1)1/2}, for allt≥t1=s2\n1/4, we have in particular\nthatt≥8+c2\n0andt≥c2\n0(1/β−1), hence\nc2\n0\ntβ=c2\n0\nt/parenleftbigg1\nβ−1/parenrightbigg\n+c2\n0\nt≤2 (3.63)\n20and /vextendsingle/vextendsingle/vextendsingle/vextendsingle1−α2K\n4β2/vextendsingle/vextendsingle/vextendsingle/vextendsingle≤1+1\n4β/parenleftbiggc2\n0\ntβ/parenrightbigg\n≤2\nβ. (3.64)\nTherefore\n|K′| ≤c2\n0e−2αt/parenleftbigg2α\nt+1\nt2/parenrightbigg\n,|∆′| ≤2c2\n0e−2αt\nβ/parenleftbigg2α\nt+1\nt2/parenrightbigg\n(3.65)\nand\n|∆′′| ≤24c2\n0\nβe−2αt/parenleftbiggα\nt+1\nt2/parenrightbigg\n. (3.66)\nFrom Proposition 3.2, uandvare bounded in terms of the energy. Thus, from the definition o f\nG1and the estimates (3.56), (3.57) and (3.65), we obtain\n|G1(t)| ≤C(E0,c0)e−2αt\nβ2/parenleftbiggα\nt+1\nt2/parenrightbigg\n.\nSince /vextendsingle/vextendsingle/vextendsinglee±/integraltextτ\nt1λ+/vextendsingle/vextendsingle/vextendsingle≤2, (3.67)\nwe conclude that\n/vextendsingle/vextendsingle/vextendsingle/vextendsingle/integraldisplay∞\nte−/integraltextτ\nt1λ+G1(τ)dτ/vextendsingle/vextendsingle/vextendsingle/vextendsingle≤C(E0,c0)\nβ2/integraldisplay∞\nte−2ατ/parenleftbiggα\nτ+1\nτ2/parenrightbigg\n≤C(E0,c0)e−2αt\nβ2t. (3.68)\nHere we have used the inequality\nα/integraldisplay∞\nte−2ασ\nσndσ≤e−2αt\n2tn, n≥1, (3.69)\nwhich follows by integrating by parts.\nIn order to handle the terms involving G2andG3, we need to take advantage of the oscillatory\ncharacter of the involved integrals, which is exploited in L emma 3.5. From (3.57), (3.65) and\n(3.66), straightforward calculations show that the functi on defined by f=γ/(2t1/2∆1/2)satisfies\nthe hypothesis in part (ii)of Lemma 3.5 with a= 1/2andL=C(E0,c0)/β. Thus invoking this\nlemma with f=γ/(2t1/2∆1/2)and noticing that\n1\n∆1/2= 1+/parenleftbigg1\n∆1/2−1/parenrightbigg\nand that\n/vextendsingle/vextendsingle/vextendsingle/vextendsingle1\n∆1/2−1/vextendsingle/vextendsingle/vextendsingle/vextendsingle=/vextendsingle/vextendsingle/vextendsingle/vextendsingle1−∆\n∆1/2(∆1/2+1)/vextendsingle/vextendsingle/vextendsingle/vextendsingle≤|K|/vextendsingle/vextendsingle/vextendsingle1−α2K\n4β2/vextendsingle/vextendsingle/vextendsingle\n|∆1/2(∆1/2+1)|≤2√\n2c2\n0\nβt,\nwhere we have used (3.57) and (3.64), we conclude that\n/integraldisplay∞\nte−/integraltextτ\nt1λ+G2(τ)dτ=γ\n2(α+iβ)t1/2e−/integraltextt\nt1λ+e−αt+R2(t), (3.70)\nwith\n|R2(t)| ≤C(E0,c0)e−αt\nβ2t3/2.\nForG3, we first write explicitly (recall the definition of R1in (3.52))\nG3(t) =−c2\n0R0(2√\nt)e−3αt\n4t1/2∆1/2+c2\n0γe−5αt\n4t3/2∆1/2:=G3,1(t)+G3,2(t). (3.71)\n21Using (3.44) and (3.57), we see that |G3,1(t)| ≤C(E0,c0)e−4αt/t2, so that we can treat this term\nas we did for G1to obtain\n/vextendsingle/vextendsingle/vextendsingle/vextendsingle/integraldisplay∞\nte−/integraltextτ\nt1λ+G3,1(τ)dτ/vextendsingle/vextendsingle/vextendsingle/vextendsingle≤C(E0,c0)e−4αt\nt. (3.72)\nFor the second term, using (3.57), (3.65) and (3.63), it is ea sy to see that the function fdefined\nbyf= (c2\n0γ)/(4t3/2∆1/2)satisfies\n|f(t)| ≤C(E0,c0)\nt3/2and|f′(t)| ≤C(E0,c0)/parenleftbiggα\nt3/2+1\nt5/2/parenrightbigg\n,\nas a consequence, invoking part (i)of Lemma 3.5, we obtain\n/vextendsingle/vextendsingle/vextendsingle/vextendsingle/integraldisplay∞\nte−/integraltextτ\nt1λ+G3,2(τ)dτ/vextendsingle/vextendsingle/vextendsingle/vextendsingle≤C(E0,c0)e−5αt\nβt3/2. (3.73)\nFrom (3.61), (3.67), (3.68), (3.72) and (3.73), we deduce th at\nw1(t) =e/integraltextt\nt1λ+(w1(t1)+w∞)−γ(β+iα)\n2t1/2e−αt+R3(t)with|R3(t)| ≤C(E0,c0)e−αt\nβ2t.(3.74)\nNow we claim that\ne/integraltextt\nt1λ+=Cα,c0eiβI(t)+H(t),withI(t) =/integraldisplayt\nt1/radicalbig\n1+K(σ)dσ,|H(t)| ≤3c2\n0e−2αt\nt,(3.75)\nand\nCα,c0= exp/parenleftbiggα\n2/integraldisplay∞\nt1Kdσ/parenrightbigg\nexp/parenleftbigg\n−iα2\n4β/integraldisplay∞\nt1K2\n∆1/2+(1+K)1/2dσ/parenrightbigg\n.\nIndeed, recall that λ+=αK\n2+iβ∆1/2so that\ne/integraltextt\nt1λ+=eα/integraltextt\nt1K\n2eiβ/integraltextt\nt1∆1/2\n. (3.76)\nFirst, we notice that\nα/integraldisplayt\nt1K\n2=c2\n0α/integraldisplay∞\nt1e−2ασ\n2σ−c2\n0α/integraldisplay∞\nte−2ασ\n2σ,\nwhere both integrals are finite in view of (3.69). Moreover, b y combining with the fact that\n|1−e−x| ≤x, forx≥0, we can write\nexp/parenleftbigg\n−c2\n0α/integraldisplay∞\nte−2ασ\n2σ/parenrightbigg\n= 1+H1(t),\nwith\n|H1(t)| ≤c2\n0e−2αt\n4t,for allt≥c2\n0/4. (3.77)\nThe above argument shows that\neα/integraltextt\nt1K\n2=eα/integraltext∞\nt1K\n2(1+H1(t)), (3.78)\nwithH1(t)satisfying (3.77).\n22For the second term of the eigenvalue, using the definition of ∆in (3.55), we write\niβ/integraldisplayt\nt1∆1/2=iβ/integraldisplayt\nt1/parenleftBig\n∆1/2−√\n1+K/parenrightBig\n+iβ/integraldisplayt\nt1√\n1+K\n=−iα2\n4β/integraldisplayt\nt1K2\n∆1/2+(1+K)1/2+iβ/integraldisplayt\nt1√\n1+K.\nProceeding as before and using that |1−eix| ≤ |x|, forx∈R, and that\nα/integraldisplay∞\ntK2\n∆1/2+√\n1+K≤α/integraldisplay∞\ntK2(σ)dσ=αc4\n0/integraldisplay∞\nte−4ατ\nτ2≤c4\n0e−4αt\n4t2,\nwe conclude that\neiβ/integraltextt\nt1∆1/2\n=eiβI(t)e−iα2\n4β/integraltext∞\nt1K2\n∆1/2+(1+K)1/2(1+H2), (3.79)\nwith\n|H2(t)| ≤c4\n0e−4αt\n16βt2≤c2\n0e−4αt\n8t,\nbearing in mind (3.63). Therefore, from (3.76), (3.78) and ( 3.79),\ne/integraltextt\nt1λ+=Cα,c0eiβI(t)(1+H1(t))(1+H2(t)).\nThe claim follows from the above identity, the bounds for H1andH2, and the fact that Cα,c0\nsatisfies that |Cα,c0|=|e/integraltext∞\nt1λ+| ≤2(see (3.67)). From (3.74), the claim and writing\nCα,c0(w1(t1)+w∞) = (beia)/2 (3.80)\nfor some real constants aandbsuch thatb≥0anda∈[0,2π), it follows that\nw1(t) =b\n2ei(βI(t)+a)−γ(β+iα)\n2t1/2+Rw1(t)with|Rw1(t)| ≤C(E0,c0)e−αt\nβ2t. (3.81)\nThe above bound for Rw1(t)easily follows from the bounds for R3(t)andH(t)in (3.74) and\n(3.75) respectively, and the fact that\n|w1(t)| ≤C(E0,c0),∀t≥t1. (3.82)\nThis last inequality is a consequence of (3.53), (3.57), (3. 62), (3.63) and the bounds for yandh\nestablished in (3.24) in Proposition 3.2.\nGoing back to the definition of win (3.58), we have (u,v) =P(w1,w2), that is\nu=−αK\n2β(w1+ ¯w1)−i∆1/2(w1−¯w1) = 2Im(w1)+R4(t),\nv= (w1+ ¯w1) = 2Re(w1),(3.83)\nwith\n|R4(t)|=/vextendsingle/vextendsingle/vextendsingle/vextendsingle−αK\nβRe(w1)+2(∆1/2−1)Im(w1)/vextendsingle/vextendsingle/vextendsingle/vextendsingle\n≤K\nβ|Re(w1)|+2|∆−1|\n∆1/2+1|Im(w1)|\n≤2c2\n0e−2αt\nβt(|Re(w1)|+|Im(w1)|)≤C(E0,c0)e−2αt\nβt,\n23where we have used (3.57), (3.64), and (3.82). From (3.81) an d (3.83), we obtain\nu(t) =bsin(βI(t)+a)−αγ\nt1/2e−αt+R5(t),\nv(t) =bcos(βI(t)+a)−βγ\nt1/2e−αt+R6(t),\nwith\n|R5(t)|+|R6(t)| ≤C(E0,c0)e−αt/(β2t).\nThe asymptotics for yandhgiven in (3.47) and (3.48) are a direct consequence of (3.53) and\nthe above identities and bounds.\nFinally, we compute the value of b. In fact, from (3.47) and (3.48)\nlim\ns→∞(y2(s)+h2(s))eαs2/2=b2.\nOn the other hand, since y+ih=¯ff′and using the conservation of energy (3.16)\n/parenleftbig\ny2(s)+h2(s)/parenrightbig\neαs2/2=|y+ih|2(s)eαs2/2=|f′|2|f|2eαs2/2= (2E0−c2\n0\n4|f|2)|f|2,\nso that, taking the limit as s→ ∞ and recalling that z=|f|2, (3.49) follows.\nRemark 3.4. From the definitions of bin(3.49), andbeiain(3.80) (in terms of Cα,c0,w1(t1)\nandw∞in(3.80)), it is simple to verify that bandbeiadepend continuously on α∈[0,1),\nprovided that z∞is a continuous function of α. In Subsection 3.3 we will prove that z∞depends\ncontinuously on α, forα∈[0,1], and establish the continuous dependence of the constants band\nbeiawith respect to the parameter αin Lemma 3.13 above.\nIn the proof of Proposition 3.3, we have used the following ke y lemma that establishes the\ncontrol of certain integrals by exploiting their oscillato ry character.\nLemma 3.5. With the same notation as in the proof of Proposition 3.2.\n(i) Letf∈C1((t1,∞))such that\n|f(t)| ≤L/taand|f′(t)| ≤L/parenleftbiggα\nta+1\nta+1/parenrightbigg\n,\nfor some constants L,a>0. Then, for all t≥t1andl≥1\n/integraldisplay∞\nte−/integraltextτ\nt1λ+e−lατf(τ)dτ=1\n(α+iβ)e−/integraltextt\nt1λ+e−lαtf(t)+F(t),\nwith\n|F(t)| ≤C(l,a,c0)Le−lαt\nβta. (3.84)\n(ii) If in addition f∈C2((t1,∞)),\n|f′(t)| ≤L/ta+1and|f′′(t)| ≤L/parenleftbiggα\nta+1+1\nta+2/parenrightbigg\n, (3.85)\nthen\n|F(t)| ≤C(l,a,c0)Le−lαt\nβta+1. (3.86)\n24HereC(l,a,c0)is a positive constant depending only on l,aandc0.\nProof. Defineλ=λ+. Recall (see proof of Proposition 3.2) that\nλ+=αK\n2+iβ∆1/2and∆ = 1+K−α2K2\n4β2,withK=c2\n0e−2αt\nt.\nSettingRλ= 1/λ−1/(iβ)and integrating by parts, we obtain\n/parenleftbigg\n1+lα\niβ/parenrightbigg/integraldisplay∞\nte−/integraltextτ\nt1λe−lατf(τ)dτ=e−/integraltextt\nt1λe−lαtf(t)/parenleftbigg1\niβ+Rλ/parenrightbigg\n+/integraldisplay∞\nte−/integraltextτ\nt1λe−lατ/parenleftbigg\n−lαfRλ+f′\nλ−fλ′\nλ2/parenrightbigg\ndτ,\nor, equivalently,\n/integraldisplay∞\nte−/integraltextτ\nt1λe−ατf(τ)dτ=1\nlα+iβe−/integraltextt\nt1λe−αtf(t)+F(t),\nwith\nF(t) =iβ\nlα+iβ/parenleftbigg\ne−/integraltextt\nt1λe−lαtRλf+/integraldisplay∞\nte−/integraltextτ\nt1λe−lατ/parenleftbigg\n−lαfRλ+f′\nλ−fλ′\nλ2/parenrightbigg\ndτ/parenrightbigg\n.\nUsing (3.57), (3.63) and (3.65), it is easy to check that for a llt≥t1\n|λ| ≥β√\n2and|λ′| ≤3c2\n0/parenleftbigg2α\nt+1\nt2/parenrightbigg\n. (3.87)\nOn the other hand,\n|Rλ|=/vextendsingle/vextendsingle/vextendsingle/vextendsingleiβ−λ\niβλ/vextendsingle/vextendsingle/vextendsingle/vextendsingle≤√\n2\nβ2/parenleftbigg\nβ|1−∆1/2|+αK\n2/parenrightbigg\n,\nwith, using the definition of ∆in (3.57) and (3.63),\nαK\n2≤c2\n0\n2tand|1−∆1/2|=|1−∆|\n1+∆1/2≤ |1−∆| ≤c2\n0\nt+c2\n0\n4βt/parenleftbiggc2\n0\nβt/parenrightbigg\n≤2c2\n0\nβt.\nPrevious lines show that\n|Rλ| ≤10c2\n0\nβ2t. (3.88)\nThe estimate (3.84) easily follows from the bounds (3.67), ( 3.69), (3.87), (3.88) and the hypothe-\nses onf. To obtain part (ii)we only need to improve the estimate for the term\n/integraldisplay∞\nte−/integraltextτ\nt1λe−lατf′\nλdτ\nin the above argument. In particular, it suffices to prove that\n/vextendsingle/vextendsingle/vextendsingle/vextendsingle/integraldisplay∞\nte−/integraltextτ\nt1λe−lατf′\nλ/vextendsingle/vextendsingle/vextendsingle/vextendsingle≤C(l,c0,a)Le−lαt\nβ2ta+1.\nNow, consider the function g=f′/λ. Notice that from (3.63), (3.87) and the hypotheses on f\nin (3.85), we have\n|g(t)| ≤√\n2L\nβta+1\n25and\n|g′(t)| ≤√\n2\nβL/parenleftbiggα\nta+1+1\nta+2/parenrightbigg\n+6L\nβ/parenleftbiggc2\n0\nβt/parenrightbigg/parenleftbigg2α\nta+1+1\nta+2/parenrightbigg\n≤14L\nβ/parenleftbigg2α\nta+1+1\nta+2/parenrightbigg\n.\nTherefore, from part (i), we obtain\n/vextendsingle/vextendsingle/vextendsingle/vextendsingle/integraldisplay∞\nte−/integraltextτ\nt1λe−lατf′\nλ/vextendsingle/vextendsingle/vextendsingle/vextendsingle≤C(l,c0,a)Le−lαt/parenleftbigg1\nβta+1+1\nβ2ta+1/parenrightbigg\n≤C(l,c0,a)Le−lαt\nβ2ta+1,\nas desired.\nWe remark that if α∈[0,1/2], the asymptotics in Proposition 3.3 are uniform in α. Indeed,\nmax\nα∈[0,1/2]/braceleftBigg\n4/radicalBig\n8+c2\n0,2c0/parenleftbigg1\nβ−1/parenrightbigg1/2/bracerightBigg\n= 4/radicalBig\n8+c2\n0=s0.\nTherefore in this situation we can omit the dependence on s1in the function φ(s1;s), because\nthe asymptotics are valid with\nφ(s) :=φ(s0;s) =a+β/integraldisplays2/4\ns2\n0/4/radicalbigg\n1+c2\n0e−2αt\ntdt. (3.89)\nWe continue to show that the factor 1/β2in the big-Oin formulae (3.47) and (3.48) are due\nto the method used and this factor can be avoided if αis far from zero. More precisely, we have\nthe following:\nLemma 3.6. Letα∈[1/2,1). With the same notation as in Propositions 3.2 and 3.3, we hav e\nthe following asymptotics: for all s≥s0,\ny(s) =be−αs2/4sin(φ(s))−2αγ\nse−αs2/2+O/parenleftBigg\ne−αs2/2\ns2/parenrightBigg\n, (3.90)\nh(s) =be−αs2/4cos(φ(s))−2βγ\nse−αs2/2+O/parenleftBigg\ne−αs2/2\ns2/parenrightBigg\n. (3.91)\nHere, the function φis defined by (3.89) and the bounds controlling the error terms depend on\nc0, and the energy E0, and are independent of α∈[1/2,1)\nProof. Letα∈[1/2,1)and define w=y+ih. From Proposition 3.3 and (1.21), we have that\nfor allα∈[1/2,1)\nlim\ns→∞we(α+iβ)s2/4=bie−i˜a, (3.92)\nwhere˜a:=a+C(α,c0),aandbare the constants defined in Proposition 3.3 and C(α,c0)is the\nconstant in (1.21). Then, since wsatisfies\n/parenleftBig\nwe(α+iβ)s2/4/parenrightBig′\n=e(−α+iβ)s2/4/parenleftbigg\nγ−c2\n0\n2(z−z∞)/parenrightbigg\n, (3.93)\nintegrating the above identity between sand infinity,\nwe(α+iβ)s2/4=ibe−i˜a−/integraldisplay∞\nse(−α+iβ)σ2/4/parenleftbigg\nγ−c2\n0\n2(z−z∞)/parenrightbigg\ndσ.\n26Now, integrating by parts and using (3.41) (recall that 1≤2α), we see that\n/integraldisplay∞\nse(−α+iβ)σ2/4dσ= 2(α+iβ)e(−α+iβ)s2/4\ns+O/parenleftBigg\ne−αs2/4\ns3/parenrightBigg\n,∀s≥s0.\nNext, notice that from (3.43) in Proposition 3.2, we also obt ain\n/integraldisplay∞\nse(−α+iβ)σ2/4(z−z∞)dσ=O/parenleftBigg\ne−αs2/2\ns2/parenrightBigg\n,∀s≥s0.\nThe above argument shows that for all s≥s0\nw(s) =ibe−αs2/4e−i(˜a+βs2/4)−2(α+iβ)γ\nse−αs2/2+O/parenleftBigg\ne−αs2/2\ns2/parenrightBigg\n. (3.94)\nThe asymptotics for yandhin the statement of the lemma easily follow from (3.94) beari ng in\nmind thatw=y+ihand recalling that the function φbehaves like (1.21) when α>0.\nIn the following corollary we summarize the asymptotics for z,yandhobtained in this section.\nPrecisely, as a consequence of Proposition 3.2- (iii), Proposition 3.3 and Lemma 3.6, we have the\nfollowing:\nCorollary 3.7. Letα∈[0,1). With the same notation as before, for all s≥s0= 4/radicalbig\n8+c2\n0,\ny(s) =be−αs2/4sin(φ(s))−2αγ\nse−αs2/2+O/parenleftBigg\ne−αs2/2\ns2/parenrightBigg\n, (3.95)\nh(s) =be−αs2/4cos(φ(s))−2βγ\nse−αs2/2+O/parenleftBigg\ne−αs2/2\ns2/parenrightBigg\n, (3.96)\nz(s) =z∞−4b\nse−αs2/4(αsin(φ(s))+βcos(φ(s)))+4γe−αs2/2\ns2+O/parenleftBigg\ne−αs2/4\ns3/parenrightBigg\n, (3.97)\nwhere\nφ(s) =a+β/integraldisplays2/4\ns2\n0/4/radicalbigg\n1+c2\n0e−2αt\ntdt,\nfor some constant a∈[0,2π),\nb=z1/2\n∞/parenleftbigg\n2E0−c2\n0\n4z∞/parenrightbigg1/2\n, γ= 2E0−c2\n0\n2z∞ andz∞= lim\ns→∞z(s).\nHere, the bounds controlling the error terms depend on c0and the energy E0, and are independent\nofα∈[0,1).\nRemark 3.8. In the case when s<0, the same arguments to the ones leading to the asymptotics\nin the above corollary will lead to an analogous asymptotic b ehaviour for the variables z,hand\nyfors<0. As mentioned at the beginning of Subsection 3.2, here we hav e reduced ourselves to\nthe case of s >0when establishing the asymptotic behaviour of the latter qu antities due to the\nparity of the solution we will be applying these results to.\n27Remark 3.9. The asymptotics in Corollary 3.7 lead to the asymptotics for the solutions fof the\nequation (3.20), at least if |f|∞:=z1/2\n∞is strictly positive. Indeed, this implies that there exist s\ns∗≥s0such thatf(s)/ne}ationslash= 0for alls≥s∗. Then writing fin its polar form f=ρexp(iθ), we\nhaveρ2θ′= Im(¯ff′). Hence, using (3.22), we obtain ρ=z1/2andθ′=h/z. Therefore, for all\ns≥s∗,\nθ(s)−θ(s∗) =/integraldisplays\ns∗h(σ)\nz(σ)dσ. (3.98)\nHence, using the asymptotics for zandhin Corollary 3.7, we can obtain the asymptotics for f.\nIn the case that α∈(0,1], we can also show that the phase converges. Indeed, the asymp totics\nin Corollary 3.7 yield that the integral in (3.98) converges as s→ ∞forα>0, and we conclude\nthat there exists a constant θ∞∈Rsuch that\nf(s) =z(s)1/2exp/parenleftbigg\niθ∞−i/integraldisplay∞\nsh(σ)\nz(σ)dσ/parenrightbigg\n,for alls≥s∗.\nThe asymptotics for fis obtained by plugging the asymptotics in Corollary 3.7 int o the above\nexpression.\n3.3 The second-order equation. Dependence on the parameter s\nThe aim of this subsection is to study the dependence of the f,z,yandhon the parameters\nc0>0andα∈[0,1]. This will allow us to pass to the limit α→1−in the asymptotics in\nCorollary 3.7 and will give us the elements for the proofs of T heorems 1.3 and 1.4.\n3.3.1 Dependence on α\nWe will denote by f(s,α)the solution of (3.20) with some initial conditions f(0,α),f′(0,α)that\nare independent of α. Indeed, we are interested in initial conditions that depen d only onc0(see\n(3.13)–(3.15)). Moreover, in view of (3.17), we assume that the energyE0in (3.16) is a function\nofc0. In order to simplify the notation, we denote with a subindex αthe derivative with respect\ntoαand by′the derivative with respect to s. Analogously to Subsection 3.2, we define\nz(s,α) =|f(s,α)|2, y(s,α) = Re(¯f(s,α)f′(s,α)), h(s,α) = Im(¯f(s,α)f′(s,α)) (3.99)\nand\nz∞(α) = lim\ns→∞|f(s,α)|2.\nObserve that in Proposition 3.2- (ii), we proved the existence of z∞(α), forα∈[0,1). For\nα∈(0,1], the estimates in (3.24) hold true and hence z(s,α)is a bounded function whose\nderivative decays exponentially. Therefore, it admits a li mit at infinity for all α∈[0,1]and\nz∞(1)is well-defined.\nThe next lemma provides estimates for zα,hαandyα.\nLemma 3.10. Letα∈(0,1). There exists a constant C(c0), depending on c0but not onα, such\nthat for all s≥0,\n|zα(s,α)| ≤C(c0)min/braceleftBigg\ns2\n√1−α+s3,s2\n/radicalbig\nα(1−α),1\nα2√1−α/bracerightBigg\n, (3.100)\n|yα(s,α)|+|hα(s,α)| ≤C(c0)e−αs2/4min/braceleftBigg\ns2\n√1−α+s3,s2\n/radicalbig\nα(1−α)/bracerightBigg\n. (3.101)\n28Proof. Differentiating (3.12) with respect to α,\nf′′\nα+s\n2(α+iβ)f′\nα+c2\n0\n4fαe−αs2/2=g, (3.102)\nwhere\ng(s,α) =−/parenleftbigg\n1−iα\nβ/parenrightbiggs\n2f′+c2\n0s2\n8fe−αs2/2.\nAlso, since the initial conditions do not depend on α,\nfα(0,α) =f′\nα(0,α) = 0. (3.103)\nUsing the estimates in (3.23) and that α2+β2= 1, we obtain\n|g| ≤C(c0)/parenleftbiggs\nβe−αs2/4+s2e−αs2/2/parenrightbigg\n,for alls≥0. (3.104)\nMultiplying (3.102) by ¯f′\nαand taking real part, we have\n1\n2/parenleftbig\n|f′\nα|2/parenrightbig′+αs\n2|f′\nα|2+c2\n0\n8/parenleftbig\n|fα|2/parenrightbig′e−αs2/2= Re(g¯f′\nα). (3.105)\nMultiplying (3.105) by 2eαs2/2and integrating, taking into account (3.103),\n|f′\nα|2eαs2/2+c2\n0\n4|fα|2= 2/integraldisplays\n0eασ2/2Re(g¯f′\nα)dσ. (3.106)\nLet us define the real-valued function η=|f′\nα|eαs2/4. Then (3.106) yields\nη2(s)≤2/integraldisplays\n0eασ2/4|g|ηdσ, for alls≥0.\nThus, by the Gronwall inequality (see e.g. [3, Lemma A.5]),\nη(s)≤/integraldisplays\n0eασ2/4|g|,dσ, for alls≥0. (3.107)\nFrom (3.104), (3.106) and (3.107), we conclude that\n(|f′\nα|eαs2/4+c0\n2|fα|)2≤2(|fα|2eαs2/2+c2\n0\n4|fα|2)\n≤4/integraldisplays\n0eασ2/4|g|ηdσ≤4/parenleftBigg\nsup\nσ∈[0,s]η(σ)/parenrightBigg/parenleftbigg/integraldisplays\n0eασ2/4|g|dσ/parenrightbigg\n≤/parenleftbigg/integraldisplays\n0eασ2/4|g|dσ/parenrightbigg2\n.\nThus, using (3.104), from the above inequality it follows\n|f′\nα|eαs2/4+c0\n2|fα| ≤C(c0)/integraldisplays\n0/parenleftbiggσ\nβ+σ2e−ασ2/4/parenrightbigg\ndσ, for alls≥0. (3.108)\nIn particular, for all s≥0,\n|fα(s)| ≤C(c0)min/braceleftBigg\ns2\n√1−α+s3,s2\n/radicalbig\nα(1−α)/bracerightBigg\n,\n|f′\nα(s)| ≤C(c0)e−αs2/4min/braceleftBigg\ns2\n√1−α+s3,s2\n/radicalbig\nα(1−α)/bracerightBigg\n,(3.109)\n29where we have used that\n/integraldisplays\n0σ2e−ασ2/4dσ≤s2/integraldisplays\n0e−ασ2/4dσ≤s2/radicalbig\nπ/α.\nNotice that from (3.103) and (3.109),\n|fα(s)| ≤/integraldisplays\n0|f′\nα|dσ≤C(c0)/radicalbig\nα(1−α)/integraldisplays\n0σ2e−ασ2/4dσ,\nand /integraldisplay∞\n0σ2e−ασ2/4dσ=2√π\nα3/2, (3.110)\nso that\n|fα(s)| ≤C(c0)\nα2√1−α. (3.111)\nOn the other hand, differentiating the relations in (3.99) wi th respect to α,\n|zα| ≤2|fα||f|,|yα+ihα| ≤ |fα||f′|+|f||f′\nα|. (3.112)\nBy putting together (3.23), (3.109), (3.111) and (3.112), we obtain (3.100) and (3.101).\nLemma 3.11. The function z∞is continuous in (0,1]. More precisely, there exists a constant\nC(c0)depending on c0but not onα, such that\n|z∞(α2)−z∞(α1)| ≤C(c0)\nL(α2,α1)|α2−α1|,for allα1,α2∈(0,1], (3.113)\nwhere\nL(α2,α1) :=α2\n1α3/2\n2/parenleftBig\nα3/2\n1√\n1−α2+α3/2\n2√\n1−α1/parenrightBig\n.\nIn particular,\n|z∞(1)−z∞(α)| ≤C(c0)√\n1−α,for allα∈[1/2,1]. (3.114)\nProof. Letα1,α2∈(0,1],α1< α2. By classical results from the ODE theory, the functions\ny(s,α),h(s,α)andz(s,α)are smooth in R×[0,1)and continuous in R×[0,1](see e.g. [5, 17]).\nHence, integrating (3.27) with respect to s, we deduce that\nz∞(α2)−z∞(α1) = 2/integraldisplay∞\n0(y(s,α2)−y(s,α1))ds= 2/integraldisplay∞\n0/integraldisplayα2\nα1dy\ndµ(s,µ)dµds. (3.115)\nTo estimate the last integral, we use (3.101)\n/integraldisplayα2\nα1|dy\ndµ(s,µ)|dµ≤C(c0)s2\n√α1/integraldisplayα2\nα1e−µs2/4\n√1−µdµ. (3.116)\nNow, integrating by parts,\n/integraldisplayα2\nα1e−µs2/4\n√1−µdµ= 2/parenleftBig√\n1−α1e−α1s2/4−√\n1−α2e−α2s2/4/parenrightBig\n−s2\n2/integraldisplayα2\nα1/radicalbig\n1−µe−µs2/4dµ.\nTherefore, by combining with (3.115) and (3.116),\n|z∞(α2)−z∞(α1)| ≤C(c0)√α1/parenleftbigg√\n1−α1/integraldisplay∞\n0s2e−α1s2/4ds−√\n1−α2/integraldisplay∞\n0s2e−α2s2/4ds/parenrightbigg\n,\n30and bearing in mind (3.110), we conclude that\n|z∞(α2)−z∞(α1)| ≤C(c0)√α1/parenleftBigg√1−α1\nα3/2\n1−√1−α2\nα3/2\n2/parenrightBigg\n,\nwhich, after some algebraic manipulations and using that α1,α2∈(0,1], leads to (3.113).\nThe estimate for z∞near zero is more involved and it is based in an improvement of the\nestimate for the derivative of z∞.\nLemma 3.12. The function z∞is continuous in [0,1]. Moreover, there exists a constant\nC(c0)>0, depending on c0but not onαsuch that for all α∈(0,1/2],\n|z∞(α)−z∞(0)| ≤C(c0)√α|ln(α)|. (3.117)\nProof. As in the proof of Lemma 3.11, we recall that the functions y(s,α),h(s,α)andz(s,α)\nare smooth in any compact subset of R×[0,1). From now on we will use the identity (3.39)\nfixings= 1. We can verify that the two integral terms in (3.39) are conti nuous functions at\nα= 0, which proves that z∞is continuous in 0. In view of Lemma 3.11, we conclude that z∞is\ncontinuous in [0,1].\nNow we claim that\n/vextendsingle/vextendsingle/vextendsingle/vextendsingledz∞\ndα(α)/vextendsingle/vextendsingle/vextendsingle/vextendsingle≤C(c0)|ln(α)|√α,for allα∈(0,1/2]. (3.118)\nIn fact, once (3.118) is proved, we can compute\n|z∞(α)−z∞(0)|=/vextendsingle/vextendsingle/vextendsingle/vextendsingle/integraldisplayα\n0dz∞\ndµ(µ)dµ/vextendsingle/vextendsingle/vextendsingle/vextendsingle≤C(c0)/integraldisplayα\n0|ln(µ)|√µdµ= 2C(c0)√α(|ln(α)|+2),\nwhich implies (3.117).\nIt remains to prove the claim. Differentiating (3.39) (recal l thats= 1) with respect to α,\nand using that y(1,·),h(1,·)andz(1,·)are continuous differentiable in [0,1/2], we deduce that\nthere exists a constant C(c0)>0such that\n/vextendsingle/vextendsingle/vextendsingle/vextendsingledz∞\ndα(α)/vextendsingle/vextendsingle/vextendsingle/vextendsingle≤C(c0)+8|I1(α)|+2c2\n0|I2(α)|, (3.119)\nwith\nI1(α) =/integraldisplay∞\n1z\nσ3+α/integraldisplay∞\n1zα\nσ3+6/integraldisplay∞\n1zα\nσ5(3.120)\nand\nI2(α) =−α\n2/integraldisplay∞\n1e−ασ2/2zσ+α/integraldisplay∞\n1e−ασ2/2zα\nσ+2/integraldisplay∞\n1e−ασ2/2zα\nσ3. (3.121)\nBy (3.24) and (3.100), zis uniformly bounded and zαgrows at most as a cubic polynomial,\nso that the first and the last integral in the r.h.s. of (3.120) are bounded independently of\nα∈[0,1/2]. In addition, (3.100) also implies that\n|zα|=|zα|1/2|zα|1/2≤C(c0)(s3)1/2/parenleftbigg1\nα2/parenrightbigg1/2\n=C(c0)s3/2\nα, (3.122)\nwhich shows that the remaining integral in (3.120) is bounde d.\n31Thus, the above argument shows that\n|I1(α)| ≤C(c0)for allα∈[0,1/2]. (3.123)\nThe same arguments also yield that the first two integrals in t he r.h.s. of (3.121) are bounded\nbyC(c0)α−1/2.Using once more that |zα| ≤C(c0)s2α−1/2, we obtain the following bounds for\nthe remaining two integrals in (3.121)\n/vextendsingle/vextendsingle/vextendsingle/vextendsingleα/integraldisplays\n1e−ασ2/2zα\nσ/vextendsingle/vextendsingle/vextendsingle/vextendsingle≤C(c0)√α/integraldisplay∞\n1ασe−ασ2/2dσ=C(c0)√αe−α/2≤C(c0)√α\nand /vextendsingle/vextendsingle/vextendsingle/vextendsingle2/integraldisplay∞\n1e−ασ/2zα\nσ3dσ/vextendsingle/vextendsingle/vextendsingle/vextendsingle≤C(c0)√α/integraldisplay∞\n1e−ασ2/2\nσdσ≤C(c0)|ln(α)|√α.\nIn conclusion, we have proved that\n|I2(α)| ≤C(c0)|ln(α)|√α,\nwhich combined with (3.119) and (3.123), completes the proo f of claim.\nWe end this section showing that the previous continuity res ults allow us to “pass to the limit”\nα→1−in Corollary 3.7. Using the notation b(α) =banda(α) =afor the constants defined for\nα∈[0,1)in Proposition 3.3 in Subsection 3.2, we have\nLemma 3.13. The valueb(α)is a continuous function of α∈[0,1]and the value b(α)eia(α)is\ncontinuous function of α∈[0,1)that can be continuously extended to [0,1]. The function a(α)\nhas a (possible discontinuous) extension for α∈[0,1]such thata(α)∈[0,2π).\nProof. By Lemma 3.12, we have the continuity of z∞in [0,1]. Therefore, in view of Remark 3.4,\nthe function beiais a continuous function of α∈[0,1)and by (3.49) bis actually well-defined\nand continuous in α∈[0,1].\nIt only remains to prove that the limit\nL:= lim\nα→1−b(α)eia(α)(3.124)\nexists. Ifb(1) = 0 , it is immediate that L= 0and we can give any arbitrary value in [0,2π)to\na(1). Let us suppose that b(1)>0. Integrating (3.93), we get\nw(s)e(α+iβ)s2/4=w(s0)e(α+iβ)s2\n0/4+/integraldisplays\ns0e(−α+iβ)σ2/4/parenleftbigg\nγ−c2\n0\n2(z−z∞)/parenrightbigg\ndσ,\nand this relation is valid for any α∈(0,1]. Letα∈(0,1). In view of (3.92), letting s→ ∞, we\nhave\nibei(a+C(α,c0))=w(s0)e(α+iβ)s2\n0/4+/integraldisplay∞\ns0e(−α+iβ)σ2/4/parenleftbigg\nγ−c2\n0\n2(z−z∞)/parenrightbigg\ndσ, (3.125)\nwhereC(α,c0)is the constant in (1.21). Notice that the r.h.s. of (3.125) i s well-defined for any\nα∈(0,1]and by the arguments given in the proof of Lemma 3.11 and the do minated convergence\ntheorem, the r.h.s. is also continuous for any α∈(0,1]. Therefore, the limit Lin (3.124) exists\nand is given by the r.h.s. of (3.125) evaluated in α= 1and divided by ieiC(1,c0). Moreover,\nlim\nα→1−eia(α)=L\nb(1),\nso that by the compactness of the the unit circle in C, there exists θ∈[0,2π)such thateiθ=\nL/b(1)and we can extend aby defining a(1) =θ.\n32The following result summarizes an improvement of Corollar y 3.7 to include the case α= 1\nand the continuous dependence of the constants appearing in the asymptotics on α. Precisely,\nwe have the following:\nCorollary 3.14. Letα∈[0,1],β≥0withα2+β2= 1andc0>0. Then,\n(i) The asymptotics in Corollary 3.7 holds true for all α∈[0,1].\n(ii) Moreover, the values bandbeiaare continuous functions of α∈[0,1]and each term in the\nasymptotics for z,yandhin Corollary 3.7 depends continuously on α∈[0,1].\n(iii) In addition, the bounds controlling the error terms de pend onc0and are independent of\nα∈[0,1].\nProof. Lets≥s0fixed. As noticed in the proof of Lemma 3.11, the functions y(s,α),h(s,α),\nz(s,α)are continuous in α= 1. In addition, by Lemma 3.13 beiais continuous in α= 1, using\nthe definition of φ, it is immediate that bsin(φ(s))andbcos(φ(s))are continuous in α= 1.\nTherefore the big- Oterms in (3.95), (3.96) and (3.97) are also are continuous in α= 1. The\nproof of the corollary follows by letting α→1−in (3.95), (3.96) and (3.97).\n3.3.2 Dependence on c0\nIn this subsection, we study the dependence of z∞as a function of c0, for a fixed value of α.\nTo this aim, we need to take into account the initial conditio ns given in (3.13)–(3.15). More\ngenerally, let us assume that fis a solution of (3.20) with initial conditions f(0)andf′(0)that\ndepend smoothly on c0, for anyc0>0, and that E0>0is the associated energy defined in\n(3.16). To keep our notation simple, we omit the parameter c0in the functions fandz∞. Under\nthese assumptions, we have\nProposition 3.15. Letα∈[0,1]andc0>0. Thenz∞is a continuous function of c0∈(0,∞).\nMoreover if α∈(0,1], the following estimate hold\n/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsinglez∞−/vextendsingle/vextendsingle/vextendsingle/vextendsinglef(0)+f′(0)√π√α+iβ/vextendsingle/vextendsingle/vextendsingle/vextendsingle2/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle≤√2E0c0π\nα/vextendsingle/vextendsingle/vextendsingle/vextendsinglef(0)+f′(0)√π√α+iβ/vextendsingle/vextendsingle/vextendsingle/vextendsingle+/parenleftbigg√2E0c0π\n2α/parenrightbigg2\n. (3.126)\nProof. Since we are assuming that the initial conditions f(0)andf′(0)depend smoothly on c0,\nby classical results from the ODE theory, the functions f,y,handzare smooth with respect to s\nandc0. From (3.39) with s= 1, we have that z∞can be written in terms of continuous functions\nofc0(the continuity of the integral terms follows from the domin ated convergence theorem), so\nthatz∞depends continuously on c0.\nTo prove (3.126), we multiply (3.20) by e(α+iβ)s2/4, so that\n(f′e(α+iβ)s2/4)′=−c2\n0\n4f(s)e(−α+iβ)s2/4.\nHence, integrating twice, we have\nf(s) =f(0)+G(s)+F(s), (3.127)\nwith\nG(s) =f′(0)/integraldisplays\n0e−(α+iβ)σ2/4dσandF(s) =−c2\n0\n4/integraldisplays\n0e−(α+iβ)σ2/4/integraldisplayσ\n0e(−α+iβ)τ2/4f(τ)dτdσ.\n33Since by Proposition 3.2 |f(s)| ≤2√2E0\nc0, we obtain\n|F(s)| ≤√2E0c0\n2/integraldisplays\n0e−ασ2/4/integraldisplayσ\n0e−ατ2/4dτdσ≤√2E0c0\n2·π\nα. (3.128)\nUsing (3.127) and the identity,\n|z1+z2|2=|z1|2+2Re(¯z1z2)+|z2|2, z1,z2∈C,\nwe conclude that z(s) =|f(s)|2satisfies\nz(s) =|f(0)+G(s)|2+2Re(¯F(s)(f(0)+G(s)))+|F(s)|2.\nTherefore, for all s≥0,\n|z(s)−|f(0)+G(s)|2| ≤2|F(s)||f(0)+G(s)|+|F(s)|2.\nHence we can use the bound (3.128) and then let s→ ∞. Noticing that\nlim\ns→∞G(s) =f′(0)/integraldisplay∞\n0e−(α+iβ)σ2/4dσ=f′(0)√π√α+iβ,\nthe estimate (3.126) follows.\n4 Proof of the main results\nIn Section 3 we have performed a careful analysis of the equat ion (3.12), taking also into con-\nsideration the initial conditions (3.13)–(3.15). Therefo re, the proofs of our main theorem consist\nmainly in coming back to the original variables using the ide ntities (3.18) and (3.19). For the\nsake of completeness, we provide the details in the followin g proofs.\nProof of Theorem 1.2 .Letα∈[0,1],c0>0and{/vector mc0,α(·),/vector nc0,α(·),/vectorbc0,α(·)}be the unique\nC∞(R;S2)-solution of the Serret–Frenet equations (1.6) with curvat ure and torsion (2.6) and\ninitial conditions (2.8). In order to simplify the notation , in the rest of the proof we drop the\nsubindexes c0andαand simply write {/vector m(·),/vector n(·),/vectorb(·)}for{/vector mc0,α(·),/vector nc0,α(·),/vectorbc0,α(·)}.\nFirst observe that if we define {/vectorM,/vectorN,/vectorB}in terms of {/vector m,/vector n,/vectorb}by\n/vectorM(s) = (m(−s),−m(−s),−m(−s)),\n/vectorN(s) = (−n(−s),n(−s),n(−s)),\n/vectorB(s) = (−b(−s),b(−s),b(−s)), s∈R,\nthen{/vectorM,/vectorN,/vectorB}is also a solution of the Serret system (1.6) with curvature a nd torsion (2.6).\nNotice also that\n{/vectorM(0),/vectorN(0),/vectorB(0)}={/vector m(0),/vector n(0),/vectorb(0)}.\nTherefore, from the uniqueness of the solution we conclude t hat\n/vectorM(s) =/vector m(s),/vectorN(s) =/vector n(s)and/vectorB(s) =/vectorb(s),∀s∈R.\nThis proves part (i)of Theorem 1.2.\n34Second, in Section 3 we have seen that one can write the compon ents of the Frenet trihedron\n{/vector m,/vector n,/vectorb}as\nm1(s) = 2|f1(s)|2−1, n 1(s)+ib1(s) =4\nc0eαs2/4¯f1(s)f′\n1(s), (4.1)\nmj(s) =|fj(s)|2−1, n j(s)+ibj(s) =2\nc0eαs2/4¯fj(s)f′\nj(s), j∈ {2,3}, (4.2)\nwithfjsolution of the second order ODE (3.12) with initial conditi ons (3.13)-(3.15) respectively,\nand associated initial energies (see (3.17))\nE0,1=c2\n0\n8andEj,1=c2\n0\n8,forj∈ {2,3}. (4.3)\nNotice that the identities (4.1)–(4.2) rewrite equivalent ly as\n\n\nm1,c0,α= 2z1−1, n1,c0,α=4\nc0eαs2/4y1, b1,c0,α=4\nc0eαs2/4h1,\nmj,c0,α=zj−1, nj,c0,α=2\nc0eαs2/4yj, bj,c0,α=2\nc0eαs2/4hj, j∈ {2,3},(4.4)\nin terms of the quantities {zj,yj,hj}defined by\nzj=|fj|2, yj= Re(¯fjf′\nj)andhj= Im(¯fjf′\nj).\nDenote by zj,∞,aj,bj,γjandφjthe constants and function appearing in the asymptotics of\n{yj,hj,zj}proved in Section 3 in Corollary 3.14.\nTaking the limit as s→+∞in (4.1)–(4.2), and since |/vector m(s)|= 1, we obtain that there exists\n/vectorA+= (A+\nj)3\nj=1∈S2with\nA+\n1= 2z1,∞−1, A+\nj=zj,∞−1,forj∈ {2,3}. (4.5)\nThe asymptotics stated in part (ii)of Theorem 1.2 easily follows from formulae (4.1)–(4.2) and the\nasymptotics for {zj,yj,hj}established in Corollary 3.14. Indeed, it suffices to observe that from\nthe formulae for bjandγjin terms of the initial energies E0,jandzj,∞given in Corollary 3.14,\n(4.3) and (4.5) we obtain\nb2\n1=c2\n0\n16(1−(A+\n1)2), b2\n2=c2\n0\n4(1−(A+\n2)2), b2\n3=c2\n0\n4(1−(A+\n3)2), (4.6)\nγ1=−c2\n0\n4A+\n1, γ2=−c2\n0\n2A+\n2, γ3=−c2\n0\n2A+\n3. (4.7)\nSubstituting these constants in (3.95), (3.96) and (3.97) i n Corollary 3.14, we obtain (1.16),\n(1.17) and (1.18). This completes the proof of Theorem 1.2- (ii).\nProof of Theorem 1.1 .Letα∈[0,1], andc0>0. As before, dropping the subindexes, we\nwill denote by {/vector m,/vector n,/vectorb}the unique solution of the Serret–Frenet equations (1.6) wi th curvature\nand torsion (2.6) and initial conditions (2.8). Define\n/vectorm(s,t) =/vector m/parenleftbiggs√\nt/parenrightbigg\n. (4.8)\n35As has been already mentioned (see Section 2), part (i)of Theorem 1.1 follows from the fact\nthat the triplet {/vector m,/vector n,/vectorb}is a regular- (C∞(R;S2))3solution of (1.6)-(2.6)-(2.8) and satisfies the\nequation\n−s\n2c/vector n=β(c′/vectorb−cτ/vector n)+α(cτ/vectorb+c′/vector n).\nNext, from the parity of the components of the profile /vector m(·)and the asymptotics established in\nparts(i)and(ii)in Theorem 1.2, it is immediate to prove the pointwise conver gence (1.9). In\naddition,/vectorA−= (A+\n1,−A+\n2,−A+\n3)in terms of the components of the vector /vectorA+= (A+\nj)3\nj=1.\nNow, using the symmetries of /vector m(·), the change of variables η=s/√\ntgives us\n/bardbl/vectorm(·,t)−/vectorA+χ(0,∞)(·)−/vectorA−χ(−∞,0)(·)/bardblLp(R)=3/summationdisplay\nj=1/parenleftbigg\n2t1/2/integraldisplay∞\n0|mj(η)−A+\nj|pdη/parenrightbigg1/p\n.(4.9)\nTherefore, it only remains to prove that the last integral is finite. To this end, let s0= 4/radicalbig\n8+c2\n0.\nOn the one hand, notice that since /vector mand/vectorA+are unitary vectors,\n/integraldisplays0\n0|mj(s)−Aj|pds≤2ps0. (4.10)\nOn the other hand, from the asymptotics for /vector m(·)in (1.16), (1.20), and the fact that the vectors\n/vectorA+and/vectorB+satisfy|/vectorA+|2= 1and|/vectorB+|2= 2, we obtain\n/parenleftbigg/integraldisplay∞\ns0|mj(s)−A+\nj|pds/parenrightbigg1/p\n≤2√\n2c0(α+β)/parenleftBigg/integraldisplay∞\ns0e−αs2p/4\nsp/parenrightBigg1/p\n+2c2\n0/parenleftBigg/integraldisplay∞\ns0e−αs2p/2\ns2p/parenrightBigg1/p\n+C(c0)/parenleftBigg/integraldisplay∞\ns0e−αs2p/4\ns3p/parenrightBigg1/p\n. (4.11)\nSince the r.h.s. of (4.11) is finite for all p∈(1,∞)ifα∈[0,1], and for all p∈[1,∞)if\nα∈(0,1], inequality (1.10) follows from (4.9), (4.10) and (4.11). T his completes the proof of\nTheorem 1.1.\nProof of Theorem 1.3 .The proof is a consequence of Proposition 3.15. In fact, reca ll the\nrelations (4.5) and (3.17), that is\nA+\n1= 2z1,∞−1,andA+\nj=zj,∞−1,forj∈ {2,3},\nand\nE0,1=c2\n0\n8, E 0,j=c2\n0\n4,forj∈ {2,3},\nThus the continuity of /vectorA+\nc0,αwith respect to c0, follows from the continuity of z∞in Proposi-\ntion 3.15.\nUsing the initial conditions (3.13)–(3.15), the values for the energies E0,jforj∈ {1,2,3}, and\nthe identity√π√α+iβ=√π√\n2/parenleftbig√\n1+α−i√\n1−α/parenrightbig\n,\nwe now compute\n/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsinglefj(0)+f′\nj(0)√π√α+iβ/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle2\n=\n\n1, ifj= 1,\n1+c2\n0π\n4+c0√π√\n2√1+α,ifj= 2,\n1+c2\n0π\n4+c0√π√\n2√1−α,ifj= 3.(4.12)\n36Then, substituting the values (4.12) in (3.126) and using th e above relations together with the\ninequality√1+x≤1+x/2forx≥0, we obtain the estimates (1.24)–(1.26).\nProof of Theorem 1.4 .Recall that the components of /vectorA+\nc0,αare given explicitly in (4.5) in\nterms of the functions zj,∞, forj∈ {1,2,3}. The continuity on [0,1]ofA+\nj,c0,αas a function\nofαforj∈ {1,2,3}follows from that of zj,∞established in Lemma 3.12. Notice also that the\nestimates (1.27) and (1.28) are an immediate consequence of (3.117) in Lemma 3.12 and (3.114)\nin Lemma 3.11, respectively.\nBefore giving the proof of Proposition 1.5, we recall that whe nα= 0orα= 1, the vector\n/vectorA+\nc0,α= (Aj,c0,α)3\nj=1is determined explicitly in terms of the parameter c0(see [15] for the case\nα= 0and Appendix for the case α= 1). Precisely,\nA1,c0,0=e−πc2\n0\n2, (4.13)\nA2,c0,0= 1−e−πc2\n0\n4\n8πsinh(πc2\n0/2)|c0Γ(ic2\n0/4)+2eiπ/4Γ(1/2+ic2\n0/4)|2, (4.14)\nA3,c0,0= 1−e−πc2\n0\n4\n8πsinh(πc2\n0/2)|c0Γ(ic2\n0/4)−2e−iπ/4Γ(1/2+ic2\n0/4)|2(4.15)\nand\n/vectorA+\nc0,1= (cos(c0√π),sin(c0√π),0). (4.16)\nProof of Proposition 1.5 .Recall that (see Theorem 1.1)\n/vectorA−\nc0,α= (A+\n1,c0,α,−A+\n2,c0,α,−A+\n3,c0,α), (4.17)\nwithA+\nj,c0,αthe components of /vectorA+\nc0,α. Therefore /vectorA+\nc0,α/ne}ationslash=/vectorA−\nc0,αiffA+\n1,c0,α/ne}ationslash= 1or−1.\nParts (ii)and(iii)follow from the continuity of A+\n1,c0,αin[0,1]established in Theorem 1.4\nbearing in mind that, from the expressions for A+\n1,c0,0in (4.13) and A+\n1,c0,1in (4.16), we have that\nA+\n1,c0,0/ne}ationslash=±1for allc0>0andA+\n1,c0,1/ne}ationslash=±1ifc0/ne}ationslash=k√πwithk∈N.\nIn order to proof part (i), we will argue by contradiction. Assume that for some α∈(0,1),\nthere exists a sequence {c0,n}n∈Nsuch thatc0,n>0,c0,n−→0asn→ ∞ and/vectorA+\nc0,n,α=/vectorA−\nc0,nα.\nHence from (4.17) the second and third component of /vectorA+\nc0,n,αare zero. Thus the estimate (1.25)\nin Theorem 1.3 yields\nc0,n/radicalbig\nπ(1+α)√\n2≤c2\n0,nπ\n4+c2\n0,nπ\nα√\n2/parenleftBigg\n1+c2\n0,nπ\n8+c0,n/radicalbig\nπ(1+α)\n2√\n2/parenrightBigg\n+/parenleftBigg\nc2\n0,nπ\n2√\n2α/parenrightBigg2\n.\nDividing by c0,n>0and letting c0,n→0asn→ ∞, the contradiction follows.\n5 Some numerical results\nAs has been already pointed out, only in the cases α= 0andα= 1we have an explicit formula\nfor/vectorA+\nc0,α(see (4.13)–(4.16)). Theorems 1.3 and 1.4 give information about the behaviour of /vectorA+\nc0,α\nfor small values of c0for a fixed valued of α, and for values of αnear to 0 or 1 for a fixed valued of\nc0. The aim of this section is to give some numerical results tha t allow us to understand the map\n37(α,c0)∈[0,1]×(0,∞)/ma√sto→/vectorA±\nc0,α∈S2. For a fixed value of α, we will discuss first the injectivity\nand surjectivity (in some appropriate sense) of the map c0/ma√sto→/vectorA±\nc0,αand second the behaviour of\n/vectorA+\nc0,αasc0→ ∞.\nFor fixedα, defineθc0,αto be the angle between the unit vectors /vectorA+\nc0,αand−/vectorA−\nc0,αassociated\nto the family of solutions /vectormc0,α(s,t)established in Theorem 1.1, that is θc0,αsuch that\ncos(θc0,α) = 1−2(A+\n1,c0,α)2. (5.1)\nIt is pertinent to ask whether θc0,αmay attain any value in the interval [0,π]by varying the\nparameterc0>0.\nIn Figure 2 we plot the function θc0,αassociated to the family of solutions /vectormc0,α(s,t)estab-\nlished in Theorem 1.1 for α= 0,α= 0.4andα= 1, as a function of c0>0. The curves θc0,0\nandθc0,1are exact since we have explicit formulae for A+\n1,c0,αwhenα= 0andα= 1(see (4.13)\nand (4.16)). We deduce that in the case α= 0, there is a bijective relation between c0>0and\nthe angles in (0,π). In the case α= 1, there are infinite values of c0>0that allow to reach\nany angle in [0,π]. Ifα∈(0,1), numerical simulations show that there exists θ∗\nα∈(0,π)such\nthat the angles in (θ∗\nα,π)are reached by a unique value of c0, but for angles in [0,θ∗\nα]there are\nat least two values of c0>0that produce them (See θc0,0.4in Figure 2).\nθc0,0\nπ\nc0\nθc0,0.4\nπ\nc0\nθc0,1\nπ\nc0\nFigure 2: The angles θc0,αas a function of c0forα= 0,α= 0.4andα= 1.\nThese numerical results suggest that, due to the invariance of (LLG) under rotations2, for a\nfixedα∈[0,1)one can solve the following inverse problem: Given any disti nct vectors /vectorA+,/vectorA−∈\nS2there exists c0>0such that the associated solution /vectormc0,α(s,t)given by Theorem 1.1 (possibly\nmultiplied by a rotation matrix) provides a solution of (LLG ) with initial condition\n/vectorm(·,0) =/vectorA+χ(0,∞)(·)+/vectorA−χ(−∞,0)(·). (5.2)\nNote that in the case α= 1the restriction /vectorA+/ne}ationslash=/vectorA−can be dropped.\nIn addition, Figure 2 suggests that /vectorA+\nc0,α/ne}ationslash=/vectorA−\nc0,αfor fixedα∈[0,1)andc0>0. Indeed,\nnotice that /vectorA+\nc0,α/ne}ationslash=/vectorA−\nc0,αif and only if A1/ne}ationslash=±1or equivalently cosθc0,α/ne}ationslash=−1, that isθc0,α/ne}ationslash=π,\nwhich is true if α∈[0,1)for anyc0>0(See Figure 2). Notice also that when α= 1, then the\nvalueπis attained by different values of c0.\nThe next natural question is the injectivity of the applicat ionc0−→θc0,α, for fixed α.\nPrecisely, can we generate the same angle using different val ues ofc0? In the case α= 0, the\n2In fact, using that\n(M/vector a)×(M/vectorb) = (det M)M−T(/vector a×/vectorb),for allM∈ M3,3(R), /vector a,/vectorb∈R3,\nit is easy to verify that if /vectorm(s,t)is a solution of (LLG) with initial condition /vectorm0, then/vectormR:=R/vectormis a solution\nof (LLG) with initial condition /vectorm0\nR:=R/vectorm0, for any R∈SO(3).\n38plot ofθc0,0in Figure 2 shows that the value of c0is unique, in fact one has following formula\nsin(θc0,0/2) =A1,c0,0=e−c2\n0\n2π(see [15]). In the case α= 1, we have sin(θc0,1/2) =A1,c0,1=\ncos(c0/radicalbig\nπ), moreover\n/vectorA+\nc0,1=/vectorA+\nc0+2k√π,1,for anyk∈Z. (5.3)\nAs before, if α∈(0,1)we do not have an analytic answer and we have to rely on numeric al\nsimulations. However, it is difficult to test the uniqueness o fc0numerically. Using the command\nFindRoot in Mathematica, we have found such values. For instance, for α= 0.4, we obtain that\nc0≈2.1749andc0≈6.6263give the same value of /vectorA+\nc0,0.4. The respective profiles /vector mc0,0.4(·)are\nshown in Figure 3. This multiplicity of solutions suggests t hat the Cauchy problem for (LLG)\nwith initial condition (5.2) is ill-posed, at least for cert ain values of c0. This interesting problem\nwill be studied in a forthcoming paper.\nm1m2m3\n(a)/vector mc0,0.4(·), withc0≈2.1749\nm1m2m3\n(b)/vector mc0,0.4(·), withc0≈6.6263\nFigure 3: Two profiles /vector mc0,0.4(·), with the same limit vector /vectorA+\nc0,0.4.\nThe rest of this section is devoted to give some numerical res ults on the behaviour of the\nlimiting vector /vectorA+\nc0,α. In particular, the results below aim to complement those es tablished in\nTheorem 1.3 on the behaviour of /vectorA+\nc0,αfor small values of c0, whenαis fixed.\nWe start recalling what it is known in the extremes cases α= 0andα= 1. Precisely, if\nα= 0, the explicit formulae (4.13)–(4.15) for /vectorA+\nc0,0allow us to prove that\nlim\nc0→0+A+\n3,c0,0= 0 andlim\nc0→∞A+\n3,c0,1= 1, (5.4)\nand also that {A+\n3,c0,0:c0∈(0,∞)}= (0,1). Whenα= 1the picture is completely different. In\nfactA+\n3,c0,1= 0for allc0>0, and the limit vectors remain in the equator plane S1×{0}. The\nnatural question is what happens with /vectorA+\nc0,αwhenα∈(0,1)as a function of c0.\nAlthough we do not provide a rigorous answer to this question , in Figure 4 we show some\nnumerical results. Precisely, Figure 4 depicts the curves /vectorA+\nc0,0.01,/vectorA+\nc0,0.4and/vectorA+\nc0,0.8as functions\nofc0, forc0∈[0,1000]. We see that the behaviour of /vectorA+\nc0,αchanges when αincreases in the sense\nthat the first and second coordinates start oscillating more and more as αgoes to 1. In all the\ncases the third component remains monotonically increasin g withc0, but the value of A+\n3,1000,α\nseems to be decreasing with α. At this point it is not clear what the limit value of A+\n3,c0,αas\n39c0→ ∞ is. For this reason, we perform a more detailed analysis of A+\n3,c0,αand we show the\ncurvesA+\n3,1,α,A+\n3,10,α,A+\n3,1000,α(for fixedα∈[0,1]) in Figure 5. From these results we conjecture\nthat{A+\n3,c0,·}c0>0is a pointwise nondecreasing sequence of functions that con verges to 1for any\nα<1asc0→ ∞. This would imply that, for α∈(0,1)fixed,A1,c0,α→0asc0→ ∞, and since\nA1,c0,α→1asc0→0(see (1.24)), we could conclude by continuity (see Theorem 1 .3) that for\nany angleθ∈(0,π)there exists c0>0such thatθis the angle between /vectorA+\nc0,αand−/vectorA+\nc0,α(see\n(5.1)). This provides an alternative way to justify the surj ectivity of the map c0/ma√sto→/vectorA+\nc0,α(in the\nsense explained above).\nA+\n1A+\n2A+\n3\n(a)/vectorA+\nc0,0.01\nA+\n1A+\n2A+\n3\n(b)/vectorA+\nc0,0.4\nA+\n1A+\n2A+\n3\n(c)/vectorA+\nc0,0.8\nFigure 4: The curves /vectorA+\nc0,0.01,/vectorA+\nc0,0.4and/vectorA+\nc0,0.8as functions of c0, forc0∈[0,1000].\n01\n1αA+\n3,1,αA+\n3,10,αA+\n3,1000,α\nFigure 5: The curves A+\n3,1,α,A+\n3,10,α,A+\n3,1000,αas functions of α, forα∈[0,1].\nThe curves in Figure 5 also allow us to discuss further the res ults in Theorem 1.4. In fact,\nwhenαis close to 1 the slope of the functions become unbounded and, roughly speaking, the\nbehaviour of A+\n3,c0,αis in agreement with the result in Theorem 1.4, that is\nA+\n3,c0,α∼C(c0)√\n1−α,asα→1−.\nNumerically, the analysis is more difficult when α∼0, because the number of computations\nneeded to have an accurate profile of A+\n3,c0,αincreases drastically as α→0+. In any case,\nFigure 5 suggests that A+\n3,c0,αconverges to A+\n3,c0,0faster than√α|ln(α)|. We think that this rate\nof convergence can be improved to α|ln(α)|. In fact, in the proof of Lemma 3.10 we only used\nenergy estimates. Probably, taking into account the oscill ations in equation (3.102) (as did in\nProposition 3.3), it would be possible to establish the nece ssary estimates to prove the following\nconjecture:\n|/vectorA+\nc0,α−/vectorA+\nc0,0| ≤C(c0)α|ln(α)|,forα∈(0,1/2].\n406 Appendix\nIn this appendix we show how to compute explicitly the soluti on/vectormc0,α(s,t)of the LLG equation\nin the case α= 1. As a consequence, we will obtain an explicit formula for the limiting vector\n/vectorA+\nc0,1and the other constants appearing in the asymptotics of the a ssociated profile established\nin Theorem 1.2 in terms of the parameter c0in the case when α= 1.\nWe start by recalling that if α= 1thenβ= 0. We need to find the solution {/vector m,/vector n,/vectorb}of the\nSerret–Frenet system (1.6) with c(s) =c0e−s2/4,τ≡0and the initial conditions (1.8). Hence,\nit is immediate that\nm3=n3≡0, b1=b2≡0andb3≡1.\nTo compute the other components, we use the Riccati equation (3.9) satisfied by the stereographic\nprojection of {mj,nj,bj}\nηj=nj+ibj\n1+mj,forj∈ {1,2}, (6.1)\nfound in the proof of Lemma 3.1. For the values of curvature an d torsionc(s) =c0e−s2/4and\nτ(s) = 0 the Riccati equation (3.9) reads\nη′\nj+iβs\n2ηj+c0\n2e−αs2/4(η2\nj+1) = 0. (6.2)\nWe see that when α= 1, and thusβ= 0, (6.2) is a separable equation that we write as:\ndηj\nη2\nj+1=−c0\n2e−αs2/4,\nso integrating, we get\nηj(s) = tan/parenleftBig\narctan(ηj(0))−c0\n2Erf(s)/parenrightBig\n, (6.3)\nwhereErf(s)is the non-normalized error function\nErf(s) =/integraldisplays\n0e−σ2/4dσ.\nAlso, using (1.8) and (6.1) we get the initial conditions η1(0) = 0 andη2(0) = 1 . In particular,\nifc0is small (6.3) is the global solution of the Riccati equation , but it blows-up in finite time if\nc0is large. As long as ηjis well-defined, by Lemma 3.1,\nfj(s) =ec0\n2/integraltexts\n0e−ασ2/4ηj(σ)dσ.\nThe change of variables\nµ= arctan(ηj(0))−c0\n2Erf(s)\nyields/integraldisplays\n0e−ασ2/4ηj(σ)dσ=2\nc0ln/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsinglecos/parenleftbig\narctan(ηj(0))−c0\n2Erf(s)/parenrightbig\ncos(arctan( ηj(0)))/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle,\nand after some simplifications, we obtain\nf1(s) =/vextendsingle/vextendsingle/vextendsinglecos/parenleftBigc0\n2Erf(s)/parenrightBig/vextendsingle/vextendsingle/vextendsingleandf2(s) =/vextendsingle/vextendsingle/vextendsinglecos/parenleftBigc0\n2Erf(s)/parenrightBig\n+sin/parenleftBigc0\n2Erf(s)/parenrightBig/vextendsingle/vextendsingle/vextendsingle.\nIn view of (3.18) and (3.19), we conclude that\nm1(s) = 2|f1(s)|2−1 = cos(c0Erf(s))andm2(s) =|f2(s)|2−1 = sin(c0Erf(s)).(6.4)\n41A priori, the formulae in (6.4) are valid only as long as ηis well-defined, but a simple verification\nshow that these are the global solutions of (1.6), with\nn1(s) =−sin(c0Erf(s))andn2(s) = cos(c0Erf(s)).\nIn conclusion, we have proved the following:\nProposition 6.1. Letα= 1, and thusβ= 0. Then, the trihedron {/vector mc0,1,/vector nc0,1,/vectorbc0,1}solution\nof(1.6)–(1.8)is given by\n/vector mc0,1(s) = (cos(c0Erf(s)),sin(c0Erf(s)),0),\n/vector nc0,1(s) =−(sin(c0Erf(s)),cos(c0Erf(s)),0),\n/vectorbc0,1(s) = (0,0,1),\nfor alls∈R. In particular, the limiting vectors /vectorA+\nc0,1and/vectorA−\nc0,1in Theorem 1.2 are given in\nterms ofc0as follows:\n/vectorA±\nc0,1= (cos(c0√π),±sin(c0√π),0).\nProposition 6.1 allows us to give an alternative explicit pr oof of Theorem 1.2 when α= 1.\nCorollary 6.2. [Explicit asymptotics when α= 1] With the same notation as in Proposition 6.1,\nthe following asymptotics for {/vector mc0,1,/vector nc0,1,/vectorbc0,1}holds true:\n/vector mc0,1(s) =/vectorA+\nc0,1−2c0\ns/vectorB+\nc0,1e−s2/4sin(/vector a)−2c2\n0\ns2/vectorA+\nc0,1e−s2/2+O/parenleftBigg\ne−s2/4\ns3/parenrightBigg\n,\n/vector nc0,1(s) =/vectorB+\nc0,1sin(/vector a)+2c0\ns/vectorA+\nc0,1e−s2/4−2c2\n0\ns2/vectorB+\nc0,1e−s2/2sin(/vector a)+O/parenleftBigg\ne−s2/4\ns3/parenrightBigg\n,\n/vectorbc0,1(s) =/vectorB+\nc0,1cos(/vector a),\nwhere the vectors /vectorA+\nc0,1,/vectorB+\nc0,1and/vector a= (aj)3\nj=1are given explicitly in terms of c0by\n/vectorA+\nc0,1= (cos(c0√π),sin(c0√π),0),/vectorB+\nc0,1= (|sin(c0√π)|,|cos(c0√π)|,1),\na1=/braceleftBigg\n3π\n2,ifsin(c0√π)≥0,\nπ\n2,ifsin(c0√π)<0,a2=/braceleftBigg\nπ\n2,ifcos(c0√π)≥0,\n3π\n2,ifcos(c0√π)<0,anda3= 0.\nHere, the bounds controlling the error terms depend on c0.\nProof. By Proposition 6.1,\n\n\n/vector mc0,1(s) = (cos(c0√π−c0Erfc(s)),sin(c0√π−c0Erfc(s)),0),\n/vector nc0,1(s) =−(sin(c0√π−c0Erfc(s)),cos(c0√π−c0Erfc(s)),0),\n/vectorbc0,1(s) = (0,0,1),(6.5)\nwhere the complementary error function is given by\nErfc(s) =/integraldisplay∞\nse−σ2/4dσ=√π−Erf(s).\nIt is simple to check that\nsin(c0Erfc(s)) =e−s2/4/parenleftbigg2c0\ns−4c0\ns3+24c0\ns5+O/parenleftBigc0\ns7/parenrightBig/parenrightbigg\n,\ncos(c0Erfc(s)) = 1+e−s2/2/parenleftbigg\n−2c2\n0\ns2+8c2\n0\ns4−56c2\n0\ns6+O/parenleftbiggc2\n0\ns8/parenrightbigg/parenrightbigg\n,\n42so that, using (6.5), we obtain that\nm1(s) =n2(s) = cos(c0√π)+2c0\nse−s2/4sin(c0√π)−2c2\n0\ns2e−s2/2cos(c0√π)+O/parenleftBigg\ne−s2/4\ns3/parenrightBigg\n,\nm2(s) =−n1(s) = sin(c0√π)−2c0\nse−s2/4cos(c0√π)−2c2\n0\ns2e−s2/2sin(c0√π)+O/parenleftBigg\ne−s2/4\ns3/parenrightBigg\n.\nThe conclusion follows from the definitions of /vectorA+\nc0,1,/vectorB+\nc0,1and/vector a.\nRemark 6.3. Notice that /vector ais not a continuous function of c0, but the vectors (B+\njsin(aj))3\nj=1\nand(B+\njcos(aj))3\nj=1are.\nReferences\n[1] M. Abramowitz and I. A. Stegun. Handbook of mathematical functions with formulas, graphs,\nand mathematical tables , volume 55 of National Bureau of Standards Applied Mathematics\nSeries . For sale by the Superintendent of Documents, U.S. Governme nt Printing Office,\nWashington, D.C., 1964.\n[2] V. Banica and L. Vega. On the Dirac delta as initial conditi on for nonlinear Schrödinger\nequations. Ann. Inst. H. Poincaré Anal. Non Linéaire , 25(4):697–711, 2008.\n[3] H. Brézis. Opérateurs maximaux monotones et semi-groupes de contract ions dans les espaces\nde Hilbert . North-Holland Publishing Co., Amsterdam, 1973. North-Ho lland Mathematics\nStudies, No. 5. Notas de Matemática (50).\n[4] T. F. Buttke. A numerical study of superfluid turbulence in the self-induction approxima-\ntion.Journal of Computational Physics , 76(2):301–326, 1988.\n[5] E. A. Coddington and N. Levinson. Theory of ordinary differential equations . McGraw-Hill\nBook Company, Inc., New York-Toronto-London, 1955.\n[6] M. Daniel and M. Lakshmanan. Soliton damping and energy l oss in the classical continuum\nheisenberg spin chain. Physical Review B , 24(11):6751–6754, 1981.\n[7] M. Daniel and M. Lakshmanan. Perturbation of solitons in the classical continuum isotropic\nHeisenberg spin system. Physica A: Statistical Mechanics and its Applications , 120(1):125–\n152, 1983.\n[8] G. Darboux. Leçons sur la théorie générale des surfaces. I, II . Les Grands Classiques\nGauthier-Villars. [Gauthier-Villars Great Classics]. Éd itions Jacques Gabay, Sceaux, 1993.\nGénéralités. Coordonnées curvilignes. Surfaces minima. [ Generalities. Curvilinear coordi-\nnates. Minimum surfaces], Les congruences et les équations linéaires aux dérivées partielles.\nLes lignes tracées sur les surfaces. [Congruences and linea r partial differential equations.\nLines traced on surfaces], Reprint of the second (1914) edit ion (I) and the second (1915)\nedition (II), Cours de Géométrie de la Faculté des Sciences. [Course on Geometry of the\nFaculty of Science].\n[9] F. de la Hoz, C. J. García-Cervera, and L. Vega. A numerica l study of the self-similar\nsolutions of the Schrödinger map. SIAM J. Appl. Math. , 70(4):1047–1077, 2009.\n43[10] P. Germain and M. Rupflin. Selfsimilar expanders of the h armonic map flow. Ann. Inst. H.\nPoincaré Anal. Non Linéaire , 28(5):743–773, 2011.\n[11] T. L. Gilbert. A lagrangian formulation of the gyromagn etic equation of the magnetization\nfield.Phys. Rev. , 100:1243, 1955.\n[12] A. Grünrock. Bi- and trilinear Schrödinger estimates in one space dimension with applica-\ntions to cubic NLS and DNLS. Int. Math. Res. Not. , (41):2525–2558, 2005.\n[13] M. Guan, S. Gustafson, K. Kang, and T.-P. Tsai. Global qu estions for map evolution\nequations. In Singularities in PDE and the calculus of variations , volume 44 of CRM Proc.\nLecture Notes , pages 61–74. Amer. Math. Soc., Providence, RI, 2008.\n[14] B. Guo and S. Ding. Landau-Lifshitz equations , volume 1 of Frontiers of Research with the\nChinese Academy of Sciences . World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ,\n2008.\n[15] S. Gutiérrez, J. Rivas, and L. Vega. Formation of singul arities and self-similar vortex motion\nunder the localized induction approximation. Comm. Partial Differential Equations , 28(5-\n6):927–968, 2003.\n[16] S. Gutiérrez and L. Vega. Self-similar solutions of the localized induction approximation:\nsingularity formation. Nonlinearity , 17:2091–2136, 2004.\n[17] P. Hartman. Ordinary differential equations . John Wiley & Sons Inc., New York, 1964.\n[18] H. Hasimoto. A soliton on a vortex filament. J. Fluid Mech , 51(3):477–485, 1972.\n[19] A. Hubert and R. Schäfer. Magnetic domains: the analysis of magnetic microstructure s.\nSpringer, 1998.\n[20] A. M. Kosevich, B. A. Ivanov, and A. S. Kovalev. Magnetic s olitons. Physics Reports ,\n194(3-4):117–238, 1990.\n[21] M. Lakshmanan. The fascinating world of the Landau-Lif shitz-Gilbert equation: an\noverview. Philos. Trans. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. , 369(1939):1280–1300,\n2011.\n[22] M. Lakshmanan, T. W. Ruijgrok, and C. Thompson. On the dy namics of a continuum spin\nsystem. Physica A: Statistical Mechanics and its Applications , 84(3):577–590, 1976.\n[23] G. L. Lamb, Jr. Elements of soliton theory . John Wiley & Sons Inc., New York, 1980. Pure\nand Applied Mathematics, A Wiley-Interscience Publicatio n.\n[24] L. Landau and E. Lifshitz. On the theory of the dispersio n of magnetic permeability in\nferromagnetic bodies. Phys. Z. Sowjetunion , 8:153–169, 1935.\n[25] F. Lin and C. Wang. The analysis of harmonic maps and their heat flows . World Scientific\nPublishing Co. Pte. Ltd., Hackensack, NJ, 2008.\n[26] T. Lipniacki. Shape-preserving solutions for quantum vortex motion under localized induc-\ntion approximation. Phys. Fluids , 15(6):1381–1395, 2003.\n[27] M. Steiner, J. Villain, and C. Windsor. Theoretical and experimental studies on one-\ndimensional magnetic systems. Advances in Physics , 25(2):87–209, 1976.\n44[28] D. J. Struik. Lectures on Classical Differential Geometry . Addison-Wesley Press, Inc.,\nCambridge, Mass., 1950.\n[29] A. Vargas and L. Vega. Global wellposedness for 1D non-l inear Schrödinger equation for\ndata with an infinite L2norm. J. Math. Pures Appl. (9) , 80(10):1029–1044, 2001.\n45"    },    {        "title": "1409.6900v2.Dissipationless_Multiferroic_Magnonics.pdf",        "content": "arXiv:1409.6900v2  [cond-mat.mes-hall]  17 Apr 2015Dissipationless Multiferroic Magnonics\nWei Chen1and Manfred Sigrist2\n1Max-Planck-Institut f ¨ur Festk¨orperforschung, Heisenbergstrasse 1, D-70569 Stuttgart, G ermany\n2Theoretische Physik, ETH-Z¨ urich, CH-8093 Z¨ urich, Switz erland\n(Dated: October 15, 2018)\nWe propose that the magnetoelectric effect in multiferroic i nsulators with coplanar antiferromag-\nnetic spiral order, such as BiFeO 3, enables electrically controlled magnonics without the ne ed of\na magnetic field. Applying an oscillating electric field in th ese materials with frequency as low\nas household frequency can activate Goldstone modes that ma nifests fast planar rotations of spins,\nwhose motion is essentially unaffected by crystalline aniso tropy. Combining with spin ejection mech-\nanisms, such a fast planar rotation can deliver electricity at room temperature over a distance of\nthe magnetic domain, which is free from energy loss due to Gil bert damping in an impurity-free\nsample.\nPACS numbers: 85.75.-d, 72.25.Pn, 75.85.+t\nIntroduction.- A primary goal of spintronic research\nis to seek for mechanisms that enable electric ( E) field\ncontrolled spin dynamics, since, in practice, Efields are\nmuch easier to manipulate than magnetic ( B) fields. As\nspinsdonotdirectlycoupleto Efield, incorporatingspin-\norbit coupling seems unavoidable for this purpose. Along\nthis line came the landmark proposals such as spin field\neffect transistor [1] and spin-orbit torque [2–5], the real-\nizations of which suggest the possibility of spin dynamics\nwith low power consumption. On the other hand, in an-\nother major category of spintronics, namely magnonics,\nwhich aims at the generation, propagation, and detection\nof magnons, a mechanism that enables electrically con-\ntrolled magnonics without the aid of a magnetic field has\nyet been proposed.\nRaman scattering experiments [6, 7] on the room tem-\nperature multiferroic BiFeO 3(BFO) shed light on this\nissue. The magnetic order of BFO is a canted antiferro-\nmagnetic(AF) spiralontheplanespannedbythe electric\npolarization Palong[111]andoneofthethreesymmetry-\nequivalent wave vectors on a rhombohedral lattice [8, 9].\nThe spins have only a very small out-of-plane component\n[10, 11]. Applying a static Efield∼100kV/cm signif-\nicantly changes the cyclon (in-plane) and extra-cyclon\n(out-of-plane) magnons because of the magnetoelectric\neffect [7]. Indeed, spin-orbit coupling induced magneto-\nelectric effects are a natural way to connect Efield to\nthe spin dynamics of insulators [12, 13]. Motivated by\nthe Raman scattering experiments on BFO, in this Let-\nter we propose that applying an oscillating Efield to a\ncoplanar multiferroic insulator (CMI) that has AF spiral\norder can achieve electrically controlled dissipationless\nmagnonics, which can deliver electricity with frequency\nas low as household frequency up to the range of mag-\nnetic domains. Compared to the magnonics that uses B\nfield, microwave, or spin torques to generate spin dynam-\nics in prototype Y 3Fe5O12(YIG) [14–16], the advantage\nof using CMI is that a single domain sample up to mm\nsize is available [17], and Raman scattering data indicatewell-defined magnons in the absence of Bfield [7], so an\nexternal Bfield is not required in the proposed mecha-\nnism.\nSpin dynamics in CMI.- We start from the AF spiral\non a square lattice shown in Fig. 1 (a), described by\nH=/summationdisplay\ni,αJSi·Si+α−Dα·(Si×Si+α) (1)\nwhereα={a,c}are the unit vectors defined on\nthexz-plane,J >0, andDα=Dαˆ y>0 is the\nDzyaloshinskii-Moriya (DM) interaction. The staggered\nmoment ( −1)iSiin the ground state shown in Fig. 1\n(a) is characterized by the angle θα=Q·α=\n−sin−1/parenleftig\nDα/˜Jα/parenrightig\nbetweenneighboringspins,where ˜Jα=\n/radicalbig\nJ2+D2α. The DM interaction\nDα=D0\nα+wE×α (2)\ncan be controlled by an Efield [18], where D0\nαrepresents\nthe intrinsic value due to the lack of in version symmetry\nof theα-bond. In the rotated reference frame S′defined\nby\nS′z\ni=Sz\nicosQ·ri+Sx\nisinQ·ri,\nS′x\ni=−Sz\nisinQ·ri+Sx\nicosQ·ri,(3)\nandS′y\ni=Sy\ni, the Hamiltonian is\nH=/summationdisplay\ni,α˜Jα/parenleftbig\nS′x\niS′x\ni+α+S′z\niS′z\ni+α/parenrightbig\n+JS′y\niS′y\ni+α.(4)\nSince˜Jα>J, the spins have collinear AF order and all\nS′z\ni= (−1)iSlie inxz-plane.\nThespin dynamicsin the absenceof Bfieldisgoverned\nby the Landau-Lifshitz-Gilbert (LLG) equation\ndS′\ni\ndt=∂H\n∂S′\ni×S′\ni+αGS′\ni×/parenleftbigg∂H\n∂S′\ni×S′\ni/parenrightbigg\n(5)2\nexpressed in the S′frame, where αGis the phenomeno-\nlogical damping parameter. Eq. (5) can be solved by the\nspin wave ansatz for the even ( e) and odd ( o) sites [19]\n/parenleftbiggS′x\ne,o\nS′y\ne,o/parenrightbigg\n=/parenleftbiggux\ne,o\nvy\ne,o/parenrightbigg\nei(k·re,o−ωt). (6)\nIgnoringthe dampingterm inEq.(5) yieldseigenenergies\nω±\nk\n2S=\n/parenleftigg/summationdisplay\nα˜Jα±γα−(k)/parenrightigg2\n−/parenleftigg/summationdisplay\nαγα+(k)/parenrightigg2\n1/2\n,(7)\nwhereγα±(k) =/parenleftig\n˜Jα/2±J/2/parenrightig\ncosk·α. Their eigenval-\nues and eigenvectors near k= (0,0) andk= (π,π) are\nsummarized below\n/braceleftig\nω+\nk→(0,0),ω−\nk→(π,π)/bracerightig\n= 2S/radicalbig\n2(D2a+D2c),\n\nue\nve\nuo\nvo\n∝\n0\n1\n0\n∓1\n+O/parenleftbiggD\nJ/parenrightbigg\n.\n/braceleftig\nω−\nk→(0,0),ω+\nk→(π,π)/bracerightig\n= 0,\nue\nve\nuo\nvo\n∝\n1\n0\n∓1\n0\n.(8)\nThe in-plane magnon dS′\ni/dt= (dS′x\ni/dt,0,0) is gapless,\nwhile the out-of-plane magnon dS′\ni/dt= (0,dS′y\ni/dt,0)\ndevelops a gap, as displayed in Fig. 1 (c). Even includ-\ning the damping term in Eq. (5), the in-plane magnons\nvery near the Goldstone modes ω−\nk→(0,0)andω+\nk→(π,π)re-\nmainunchangedanddamping-free. Awayfromthe Gold-\nstone limit, the eigenenergies become complex, hence the\nmagnons are subject to the damping and decay within a\ntime scale set by α−1\nG.\nSpin dynamics induced by oscillating Efield.-We an-\nalyze now the spin dynamics in the damping-free in-\nplane magnonchannel induced bymagnetoelectriceffects\n(Eq. (8)). Unlike the spin injection by using the spin Hall\neffect (SHE) to overcome the damping torque [16], our\ndesign does not require an external Bfield, and is fea-\nsible over a broad range of frequencies. Consider the\ndevice shown in Fig. 2, where an oscillating electric field\nE=E0cosωtis applied parallel to the ferroelectric mo-\nment over a region of length L=Na, such that the DM\ninteraction in Eq. (2) oscillates in this region. Thus, the\nwave length of the spiral changes with time yielding an\noscillation of the number of spirals inside this region,\nnQ=L\n2π/|Q|≈N\n2πJ/bracketleftbig\nD0\na+wE0acosωt/bracketrightbig\n,(9)\nassumingDa=D0\na+wE0a≪J,Dc= 0, and E⊥a.\nSuppose the spin S0at one boundary is fixed by, for\nFIG. 1: (color online) Schematics of 2D AF spiral in the (a)\noriginalS-frame and the (b) rotated S′-frame. Red and blue\narrows indicate the spins on the two sublattices. (c) Spin\nwave dispersion ω+\nk(dashed line) and ω−\nk(solid line) solved\nin theS′frame, with Da/J= 0.14,Dc= 0. Inserts show\ntheir eigen modes in the S′frame near k= (0,0) and (π,π),\nwhere the spin dynamics dS′\ni/dtis indicated by black arrows\nor symbols.\ninstance, surface anisotropy because of specific coating.\nThenSNat the other boundary rotates by\n∂θN\n∂t=−N\nJwE0aωsinωt, (10)\nbecause whenever the number of waves nQchanges by 1,\nSNrotates 2πin orderto to wind or unwind the spin tex-\nture in the Efield region. The significance of this mecha-\nnism is that although the Efield is driven by a very small\nfrequencyω, the spin dynamics ∂tθNat the boundary is\nmanyordersofmagnitude enhanced because of the wind-\ning process. The rotation of SNserves as a driving force\nfor the spin dynamics in the field-free region from SNto\nSN+M. As long as the spin dynamics is slower than the\nenergy scale of the DM interaction ∂tθi<|D0|//planckover2pi1∼THz,\none can safely consider the Efield region as adiabatically\nchanging its wave length but remaining in the ground\nstate. The spins in the field-free region rotate coherently\n∂tθN=∂tθN+1=...=∂tθN+M, synonymous to exciting\ntheω−\nk→(0,0)mode in Eq.(8), hence the spin dynamics\nin the field-free region remains damping-free in an ideal\nsituation.\nIn real materials, crystalline anisotropy and impuri-\nties are the two major sources to spoil the spin rota-\ntional symmetry implicitly assumed here. In the supple-\nmentary material[20], their effects are discussed by draw-\ning analogy with similar situations in the atom absorp-\ntion on periodic substrates and the impurity pinning of\ncharge density wave states. It is found that crystalline3\nquantity symbol magnitude\nlattice constant a nm\ns−dexchange Γ 0.1eV\ns−dexchange time τex 10−14s\nspin relaxation time τsf 10−12s\nspin diffusion length λN 10nm\nspin density n01027/m3\nspin Hall angle θH 0.1\nintrinsic DM D0\nα10−3eV\nsuperexchange J 0.1eV\nEq. (2) w 10−19C\nelectric flux quantum ˜Φ0\nE 1V\nTABLE I: List of material parameters and their order of mag-\nnitude values.\nanisotropy remains idle because of the long spiral wave\nlength and the smallness of crystalline anisotropy com-\npared to exchange coupling. The impurities that tend to\npin the spins alongcertaincrystalline directionopen up a\ngap in the Goldstone mode and cause energy dissipation,\nwhich nevertheless do not obstruct the coherent rotation\nof spins generated by Eq. (10).\nFIG. 2: (color online) Experimental proposal of using oscil -\nlatingEfield to induce spin dynamics in CMI. The AF spiral\norder is shown in the S′frame. The Efield is applied between\nS′\n0andS′\nN, causing dynamics in the whole spin texture. Two\nways for spin ejection out of S′\nN+Mare proposed: (a) Using\nSHE to converted it into a charge current. (b) Using time-\nvarying spin accumulation and inductance.\nSpin ejection and delivery of electricity.- We now ad-\ndress the spin ejection from the CMI to an attached\nnormal metal (NM). A spin current is induced in the\nNM when a localized spin Siat the NM/CMI inter-\nface rotates [16, 21]. Defining the conduction electron\nspinm(r,t) =−∝an}b∇acketle{tσ∝an}b∇acket∇i}ht/2, thes-dcoupling at the interface\nHsd= Γσ·Sidefines a time scale τex=/planckover2pi1/2S|Γ|, withΓ<0 [21]. The Bloch equation in the NM reads\n∂m\n∂t+∇·Js=1\nτexm׈Si−δm\nτsf(11)\nwhereJs=JNM\ns/varotimesσ/planckover2pi1/2 is the spin current tensor, and\nτsfis the spin relaxation time in the NM. In equilib-\nrium, we assume mhybridizes with each Sion the spi-\nral texture locally. If the dynamics of Siis slow com-\npared to 1/τex, which is true for the proposed mechanism\nand also for other usual means such as ferromagnetic\nresonance[16], mfollows−ˆSiat any time with a very\nsmall deviation m=m0+δm=−n0ˆSi+δm, wheren0\nis the local equilibrium spin density. The spin current\ntensorJs=−D0∇δmis obtained from the diffusion of\nδm, whereD0is the spin diffusion constant. Under such\nan adiabatic process, the small deviation is[21]\nδm=τex\n1+ξ2/braceleftigg\n−ξn0∂ˆSi\n∂t−n0ˆSi×∂ˆSi\n∂t/bracerightigg\n,(12)\nwhereξ=τex/τsf<1 so one can drop the first term on\nthe right hand side, and replace ˆSi×∂tˆSi→δ(r)ˆSi×∂tˆSi\nsinceˆSiis located at the NM/CMI interface r= 0 (ras\ncoordinate perpendicular to the interface). The resulting\nequation solves the time dependence of δm. Away from\nr= 0, Eq. (11) yields D0∇2δm=δm/τsf, which solves\nthe spatial dependence of δm. The spin current caused\nby a particular Sithen follows\nJNM\nsδˆm=δmD0\nλN=−τexn0D0\n(1+ξ2)λNˆSi×∂ˆSi\n∂te−r/λN,(13)\nwhereλN=/radicalbig\nD0τsf, similar to results obtained previ-\nously[16]. Ifonlythein-planeGoldstonemodeisexcited,\nas shown in Fig. 2, it is equivalent to a global rotation of\nspinsˆSi= (−1)i(sin(θ(t) +Q·ri),0,cos(θ(t) +Q·ri))\nin the field-free region. Thus the time dependence in\nEq. (13), ˆSi×∂tˆSi=ˆy∂θ/∂t, is that described by\nEq. (10), and is the same for every Siat the NM/CMI\ninterface, even though each Sipoint at a different polar\nangle. In other words, the spin current ejected from each\nSiof the AF spiral, described by Eq. (13), is the same,\nso a uniform spin current flows into the NM.\nWe propose two setups to convert the ejected spin cur-\nrent into an electric signal. The first device uses inverse\nSHE[16]inaNMdepositedatthesideofthespiralplane,\nyieldingδˆmperpendicular to JNM\nsand consequently a\nvoltage in the transverse direction, as shown in Fig. 2\n(a). The second design ejects spin into a NM film de-\nposited on top of the spiral plane, as shown in Fig. 2\n(b), causing δˆmparallel to JNM\ns. A spin accumulation\nin the NM develops and oscillates with time, producing\nan oscillatingmagnetic flux Φ Bthrougha coil that wraps\naround the NM, hence a voltage E=−∂ΦB/∂t.\nExperimental realizations.- TheRamanscatteringdata\non BFO [7] show that applying |E| ∼100kV/cm can4\nchange the spin wave velocity by δv0/v0∼1%. We\ncan make use of this information to estimate the field-\ndependence win Eq. (2). The ω−\nkmode in Eq. (7) near\nk= (0,0) is\nω−\nk→0= 2√\n2SJka/bracketleftbigg\n1+5\n16/parenleftbiggD2\nak2\na+D2\nck2\nc\nJ2k2/parenrightbigg/bracketrightbigg\n= (v0+δv0)k, (14)\nwherev0= 2√\n2SJais the spin wave velocity in the\nabsence of DM interaction. Assuming Da∝ne}ationslash= 0,Dc=\n0, andE⊥a, the Raman scattering data gives w∼\n10−19C∼ |e|. We remark that a coplanar magnetic order\ncan be mapped into a spin superfluid [36, 37] ψiby\n∝an}b∇acketle{tSi∝an}b∇acket∇i}ht=S(sinθi,0,cosθi) =√v(Imψi,0,Reψi),(15)\nwherevis the volumeofthe 3Dunit cell. Within this for-\nmalism, the Efield can induce quantum interference of\nthe spin superfluid via magnetoelectric effect, in which\nthe electric flux vector ΦE=/contintegraltext\nE×dlis quantized\n[24, 25]. The flux quantum is ˜Φ0\nE= 2πJ/w, which is\n˜Φ0\nE∼1V for BFO, close to that ( ∼10V) obtained from\ncurrent-voltage characteristics of a spin field-effect tran-\nsistor [24], indicating that strong spin-orbit interaction\nreduces the flux quantum to an experimentally accessi-\nble regime. For instance, BFO has a spiral wave length\n2π/Q∼100nm, so in a BFO ring of µm size, the num-\nber of spirals at zero field is nQ∼10, and applying\n|E| ∼1kV/cm can change nQby 1. Besides changing\nthe winding number, we remarkthat the magnetoelectric\neffect can also be used to affect the topological proper-\nties of a magnet in a different respect[26]. Table I lists\nthe parameters and their order of magnitude values by\nassuming CMI has similar material properties as other\nmagnetic oxide insulators such as YIG, and we adopt\nlattice constant a∼1nm for both CMI and the NM for\nsimplicity.\nFor the device in Fig. 2, consider the field |E0| ∼\n100kV/cm oscillating with a household frequency ω∼\n100Hz is applied to a range L∼1mm. This region covers\nN=L/a∼106sites with a number of spirals nQ∼104\nat zero field. The Efield changes the number of spi-\nrals tonQ∼105within time period 1 /ω∼0.01s, so\nthe spins at the boundary SNwind with angular speed\n∂tθN∼107sinωtwhich is enhanced by 5 orders of mag-\nnitude from the driving frequency ω. To estimate the\nejected spin current in Eq. (13), we use the typical spin\nrelaxation time τsf∼10−12s and length λN∼10nm\nfor heavy metals [16]. The s-dcoupling can range be-\ntween [16] 0 .01eV to 1eV. We choose Γ ∼0.1eV, which\ngivesτex∼10−14s. The spin Hall angle θH∼0.1 has\nbeen achieved [27, 28]. To estimate n0, we use the fact\nthat thes-dhybridization Γ σ·Siis equivalent to ap-\nplying a magnetic field H= 2ΓSi/µ0gµBlocally at the\ninterface atomic layer of the NM. Given the typical mo-\nlar susceptibility χm∼10−4cm3/mol and molar volumeVm∼10cm3/mol, the interface magnetization of the NM\nisn0µB=χmH/Vm∼104C/sm, thus n0∼1027/m3.\nTheoscillating Efieldgives ˆSi×∂tˆSi=∂tθNˆy∼ˆy107Hz,\nso the ejected spin current is JNM\ns∼1024/planckover2pi1/m2s. Using\nthe design in in Fig. 2(a) to convert JNM\nsinto a charge\ncurrent via inverse SHE yields JNM\nc∼104A/m2, hence\na voltage ∼µV oscillating with ωin a mm-wide sample.\nTo use the setup in Fig. 2(b), a NM film of area ∼1 mm2\nand thickness ∼10nm yields E ∼mV oscillating with ω.\nIn summary, we propose that for multiferroics that\nhave coplanar AF spiral order, such as BFO, applying\nan oscillating Efield with frequency as low as house-\nhold frequency generates a coherent planar rotation of\nthe spin texture whose frequency is many orders of mag-\nnitude enhanced. This coherent rotation activates the\nGoldstone mode of multiferroic insulators that remains\nunaffected by the crystalline anisotropy. The Goldstone\nmode can be used to deliver electricity at room tempera-\nture up to the extensions of magnetic domains, in a way\nthat is free from the energy loss due to Gilbert damping\nif the sample is free from impurities. The needlessness\nofBfield greatly reduces the energy consumption and\nincreases the scalability of the proposed device, pointing\nto its applications in a wide range of length scales.\nWe thank exclusively P. Horsch, J. Sinova, H. Naka-\nmura, Y. Tserkovnyak, D. Manske, M. Mori, C. Ulrich,\nJ. Seidel, and M. Kl¨ aui for stimulating discussions.\nSupplementary material\nI. Crystalline anisotropy in multiferroics\nFirst we demonstrate that because the wave length of\nthe spiral order in multiferroics is typically 1 ∼2 orders\nlonger than the lattice constant, and the exchange cou-\npling is typically few orders larger than the crystalline\nanisotropy energy, the spiral order remains truly incom-\nmensurate and very weakly affected by the crystalline\nanisotropy. For simplicity, we consider a spiral state with\nwave vector Q∝ba∇dbl(1,0) and translationally invariant per-\npendicular to Qsuch that the geometry can be reduced\nto a 1D problem. The classical elastic energy for a 1D\nantiferromagnetic (AF) spiral is\nE0=/summationdisplay\nn−˜JaS2cos(θn+1−θn−θa)\n≈ −N˜JaS2+/summationdisplay\nn1\n2˜JaS2(θn+1−θn−θa)2,(16)\nwhereθn=Q·rnis the angle relative to the staggered\nspin (−1)iSi, andθa=Q·ais the natural pitch an-\ngle between neighboring spins ( a= (a,0)). The square\nlattice symmetry of our model yields a 4-fold degener-\nate crystalline spin anisotropy[38], leading to the total5\nenergy\nE=/summationdisplay\nn1\n2˜JaS2(θn+1−θn−θa)2\n+/summationdisplay\nnVani(1−cos4θn), (17)\nwhereVaniis the anisotropy energy per site. This is the\nwell-known Frenkel-Kontorowa(FK) model[30, 31] that\nhas been discussed extensively owing to its rich physics.\nFIG. 3: (color online) Schematics of mapping the AF spi-\nral order in the presence of crystalline anisotropy into FK\nmodel. The angles θiof staggered spins ( −1)iSi(blue ar-\nrows) are mapped into displacements xiof particles (orange\ndots). The width of the 4-fold degenerate pinning potential\nV(1−cos2πxi/b) isb=π/2, and the spacing of particles in\nthe absence of the pinning potential is a0=Q·a.\nWe consider the limit of weak anisotropy V=\nVani/˜JaS2a2≪1 and the case of long wavelength of\nthe spiral, θa≪π/2 whereπ/2 is the angle between\ntwo minima of the anisotropy potential. In the spirit of\nRef.[32, 33] we assume now that there are prime num-\nbers,MandLwithM˜θa=Lπ/2 andM≫Lwhich is\nthe average pitch in the ground state of Eq.(17). Then\nwe introduce the parametrization\nθn=n˜θa+ϕn\n4(18)\nand the misfit parameter δ= 4(θa−˜θa). Turning to\nthe continuous limit one can derive the effective en-\nergy functional based on expanding the first harmonic\napproximation[32–34],\n˜E[ϕ] =/integraldisplay\ndx/bracketleftigg\n1\n2/parenleftbiggdϕ\ndx−δ/parenrightbigg2\n+VMcos(Mϕ)/bracketrightigg\n(19)\nwithVM∼VMwhich can become extremely small for\nM≫1. The commensurate-incommensurate transition\nhappens ifδis large enough to stabilize the formation of\nsolitonsδ > δc(M)∼4√VM/π. Deep inside the incom-\nmensurate phase, ϕ(x)≈δxsuch thatθn≈θanfollows\nesentially the natural spiral pitch.\nIn our system, BFO, the spiral wave length ℓ≈\n60nm∼100awhich yields M∼100/4 = 25, i.e. every\n25thspin could be pinned along one of the 4 anisotropy\nminima (assuming L= 1). Typical anisotropy ener-\ngies for ferrites[35] lead to Vani∼10−3eV while theexchange energy is Ja∼0.1eV, from which we obtain\nV∼Vani/Ja∼10−2and consequently VM∼10−50is\na negligible number. The misfit parameter may be as\nlarge asδ= 4(θa−˜θa)∼π/M2such thatδ≫δc(M) is\nwell satisfied, even if by an electrical field Mshrinks by\none order of magnitude. Thus, the electric field-driven\noscillations of the spin spiral remains most likely unaf-\nfected by the spin anisotropy. The small VMrenders\nthe energy gap due to the anisotropy energy irrelevant,\nhence the in-plane magnon mode remains essentially un-\ndamped. Another important consequence of this analy-\nsis is that although the concept of spin superfluidity, i.e.,\ntreating the spin texture as a quantum condensate, has\nbeen proposed long ago, its realization in collinear mag-\nnets is problematic because of the crystalline anisotropy\nand subsequently the formation of domain walls. We\ndemonstrate explicitly that multiferroics are not sub-\nject to these problems because of the noncollinear or-\nder, hence a room temperature macroscopic condensate\nof mm size can be realized.\nII. Phase-pinning impurities in multiferroics\nWe proceed to show that dilute, randomly distributed\nimpurities, exist either in the bulk of the multiferroic or\nat the metal/multiferroic interface, do not obstruct the\nproposed electrically controlled multiferroic magnonics.\nDrawing analogyfrom the FK model, impurities that pin\nthe spins along certain crystalline directions, denoted by\nphase-pinningimpurities, arethe impuritiesto be consid-\nered because they tend to impede the coherent motion of\nspins[39]. Since we propose to use an oscillating Efield\nto drive the spin rotation from the boundary, each cross\nsection channel is equivalent, which reduces the problem\nfrom 2D to 1D. This leads us to consider the following\n1D classical model similar to Eq. (17) for the field-free\nregion (S′\nN+1toS′\nN+Min the Fig. 2 of the main text).\nE=/summationdisplay\ni1\n2˜JaS2(θi+a−θi−θa)2−/summationdisplay\ni∈impVimpcos4θi,(20)\nwhereVimp>0 is the pinning potential, and/summationtext\ni∈imp\nsums over impurity sites. The total length of the chain is\nL′=MawithManinteger. Inthepresenceofoscillating\nEfield that causes the winding of boundary spins ( S′\nN\nin the Fig. 2 of the main text), the angle of spins in the\ndisordered field-free region has three contributions\nθi=θ0\ni+∆θi+ηi, (21)\nwhereθ0\nirepresents the spiral texture in the unstretched\ncleanlimitsatisfying θ0\ni+a−θ0\ni−θa= 0,∆θiisthestretch-\ning of the spin texture caused by winding of boundary\nspins, and ηiis the distortion due to impurities. Only\nthe later two contribute to the elastic energy, so Eq. (20)6\nbecomes\nE=/summationdisplay\ni1\n2˜JaS2(∆θi+a−∆θi+ηi+a−ηi)2\n−/summationdisplay\ni∈impVimpcos4θi. (22)\nIn this analysis we consider weak impurities Vimp≪\n˜JaS2, andassumethatthe windingofthe boundaryspins\nis slow such that the winding spreads through the whole\nfield-free region evenly, causing every pair of neighboring\nspins to stretch by the same amount ∆ θi+a−∆θi= ∆θ.\nFor the electrically driven magnonics proposed in the\nmain text, which can achieve winding of boundary spins\nbyθN∼nQ∼105within half-period, a field-free region\nof lengthL′∼mm has ∆θ∼0.1, so our numerics is done\nwith ∆θlimited within this value.\nIn the weak impurity limit, the length scale L0over\nwhichηichanges by O(1) can be calculated in the fol-\nlowing way. The elastic energy part in Eq. (22) within\nL0is, in the continuous limit,\nK(L0) =1\na/integraldisplayL0\n0dx1\n2˜JaS2a2/parenleftbigg∆θ\na+∂xη/parenrightbigg2\n=L0\n2a˜JaS2∆θ2+˜JaS2∆θ\nα1+˜JaS2a\n2α0L0,(23)\nwhereα0andα1are numerical constants of O(1), and\nare set to be unity without loss of generality. Denoting\nimpurity density as nimp=Nimp/L′whereNimpis the\ntotal number of impurities in the sample, the impurity\npotential energy within L0is calculated by\nV(L0) =−VimpRe\n/summationdisplay\ni∈impe4i(θ0\ni+∆θ+η)\n\n=−Vimp/radicalbig\nnimpL0. (24)\nNote that the contribution comes not from the zeroth\norder impurity averaging, but its fluctuation that mimics\na random walk in the complex plane[40]. The phase ηis\nassumed to be constant within L0and chosen to give\nEq. (24) and hence the total energy E(L0) =K(L0) +\nV(L0) withinL0. Minimizing the total energy per site\nE(L0)/L0gives the most probable pinning length L0. In\nthe unstretched case ∆ θ= 0,\nL0=/parenleftigg˜JaS2a\nα0Vimpn1/2\nimp/parenrightigg2/3\n(25)\nis similar to the Fukuyama-Lee-Rice (FLR) length that\ncharacterizes the impurity pinning of a charge density\nwave ground state[40, 41]. Putting Eq. (25) back to\nEqs. (24) and (23), the corresponding E(L0)a/L0<0\ncan be viewed as the pinning energy per site that im-\npedes the coherent rotation of spins, and equivalently\nrepresents the gap opened at the Goldstone mode.00.020.040.060.08 0.10246810\n/CapDΕltaΘLog10/LParen1L0/Slash1a/RParen1/LParen1a/RParen1 Aimp/EΘual10/Minus5\n10/Minus4\n10/Minus3\n10/Minus2\n00.020.040.060.08 0.1/Minus1/Minus0.500.51\n/CapDΕltaΘΕ/Multiply103/LParen1b/RParen1\nAimp/EΘual10/Minus4\n10/Minus3\n5/Multiply10/Minus3\n10/Minus2\nFIG. 4: (color online) (a) The logrithmic of the dimensionle ss\nFLR length log10(L0/a) versus winding angle per site ∆ θ, in\nseveral values of the dirtiness parameter Aimp. Dashed line\nindicates the threshold when L0∼mm. (b) The dimensionless\npinning energy ǫversus winding per site.\nIn the presence of the stretching ∆ θ, the expression\nofL0is rather lengthy. It is convenient to define two\ndimensionless parameters\nAimp=Vimp\n˜JaS2/radicalbigg\nNimpa\nL′,\nǫ=E(L0)\n˜JaS2/parenleftbigga\nL0/parenrightbigg\n, (26)\nwhereAimp(the ”dirtiness parameter”) is the impurity\npotential measured in unit of the elastic constant times\nthe square root of the impurity density, and ǫis the to-\ntal energy per site measured in unit of the elastic con-\nstant. Figure 4 shows the logrithmic of the dimensionless\npinning length L0/aand the dimensionless total energy\nǫ, plotted as functions of the stretching ∆ θ. There are\ntwo evidences showing that the spin texture, originally\npinned by impurities with the pinning length in Eq. (25),\nis depinned by the stretching ∆ θ: Firstly, the pinning\nlengthL0increases as increasing ∆ θ. For a particular\nsample size, for instance L′∼mm, the spin texture is\ndepinned when the pinning length exceeds the sample\nsizeL0> L′, or equivalently when ∆ θis greater than\na certain threshold (intercept of the dashed line and the\ncoloredlinesinFig.4(a)). Secondly, thepinningenergy ǫ\nbecomes positive at large ∆ θ, indicating that the elastic\nenergy from stretching overcomes the impurity pinning\nenergy, so the spin texture is depinned. From Fig. 4, it\nis also evident that the cleaner is the sample, the easier\nit is to depin the spins by stretching, as smaller Aimpre-\nquires smaller threshold value of ∆ θ. We conclude that\nthephasepinning impuritiesdonot hampertheproposed\nelectrically driven multiferroic magnonics as long as the\ndirtiness of the sample is limited, the winding speed of\nthe boundary spin is sufficient, and the sample size is\nshort enough.\n[1] S. Datta and B. Das, Appl. Phys. Lett. 56, 665 (1990).\n[2] A. Manchon and S. Zhang, Phys. Rev. B 78, 212405\n(2008); and Phys. Rev. B 79, 094422 (2009).7\n[3] D. A. Pesin and A. H. MacDonald, Phys. Rev. B 86,\n014416 (2012).\n[4] P. M. Haney and M. D. Stiles, Phys. Rev. Lett. 105,\n126602 (2010).\n[5] P. M. Haney, H.-W. Lee, K.-J. Lee, A. Manchon, and M.\nD. Stiles, Phys. Rev. B 88, 214417 (2013).\n[6] P. Rovillain, M. Cazayous, Y. Gallais, A. Sacuto, R. P.\nS. M. Lobo, D. Lebeugle, and D. Colson, Phys. Rev. B\n79, 180411(R) (2009).\n[7] P. Rovillain et al., Nature Mater. 9, 975 (2010).\n[8] G. Catalan and J. F. Scott, Adv. Mater. 21, 2463 (2009).\n[9] D. Lebeugle, D. Colson, A. Forget, M. Viret, A. M.\nBataille, and A. Goukasov, Phys. Rev. Lett. 100, 227602\n(2008).\n[10] C. Ederer and N. A. Spaldin, Phys. Rev. B 71, 060401\n(2005).\n[11] M. Ramazanoglu et al., Phys. Rev. Lett. 107, 207206\n(2011).\n[12] T. Moriya, Phys. Rev. 120, 91 (1960).\n[13] H. Katsura, N. Nagaosa, and A. V. Balatsky, Phys. Rev.\nLett.95, 057205 (2005).\n[14] V. V. Kruglyak, S. O. Demokritov, and D. Grundler, J.\nPhys. D: Appl. Phys. 43, 264001 (2010).\n[15] A. A. Serga, A. V. Chumak, and B. Hillebrands, J. Phys.\nD: Appl. Phys. 43, 264002 (2010).\n[16] Y. Kajiwara et al., Nature 464, 262 (2010).\n[17] R. D. Johnson, P. Barone, A. Bombardi, R. J. Bean, S.\nPicozzi, P. G. Radaelli, Y. S. Oh, S.-W. Cheong, and L.\nC. Chapon, Phys. Rev. Lett. 110, 217206 (2013).\n[18] K. Shiratori and E. Kita, J. Phys. Soc. Jpn. 48, 1443\n(1980).\n[19] R. de Sousa and J. E. Moore, Phys. Rev. B 77, 012406\n(2008).\n[20] The Supplementary Material is included in this arXiv\nversion.\n[21] S. Zhang and Z. Li, Phys. Rev. Lett. 93, 127204 (2004).\n[22] B. I. Halperin and P. C. Hohenberg, Phys. Rev. 188, 898(1969).\n[23] P. Chandra, P. Coleman, and A. I. Larkin, J. Phys.: Con-\ndens. Matter 2, 7933 (1990).\n[24] W. Chen, P. Horsch, and D. Manske, Phys. Rev. B 87,\n214502 (2013).\n[25] W. Chen and M. Sigrist, Phys. Rev. B 89, 024511 (2014).\n[26] W.-L. You, G.-H. Liu, P. Horsch, and A. M. Ole´ s, Phys.\nRev. B90, 094413 (2014).\n[27] T. Seki, Y. Hasegawa, S. Mitani, S. Takahashi, H. Ima-\nmura, S. Maekawa, J. Nitta, and K. Takanashi, Nature\nMater.7, 125 (2008).\n[28] C.-F. Pai, L. Liu, Y. Li, H. W. Tseng, D. C. Ralph, and\nR. A. Buhrman, Appl. Phys. Lett. 101, 122404 (2012).\n[29] E. B. Sonin, Adv. Phys. 59, 181 (2010).\n[30] Y. I. Frenkel and T. Kontorowa, Zh. Eksp. Theor. Fiz. 8,\n1340 (1938).\n[31] F. C. Frank and J. H. Van der Merwe, Proc. R. Soc. 198,\n205 (1949).\n[32] P. Bak, Rep. Prog. Phys. 45, 587 (1982).\n[33] G. Theodorou and T. M. Rice, Phys. Rev. B 18, 2840\n(1978).\n[34] V. L. Prokovskii and A. L. Talapov, Zh. Eksp. Theor.\nFiz.75, 1151 (1978).\n[35] K. Yosida and M. Tachiki, Prog. Theo. Phys. 17, 331\n(1957).\n[36] B. I. Halperin and P. C. Hohenberg, Phys. Rev. 188, 898\n(1969).\n[37] P. Chandra, P. Coleman, and A. I. Larkin, J. Phys.: Con-\ndens. Matter 2, 7933 (1990).\n[38] E. B. Sonin, Adv. Phys. 59, 181 (2010).\n[39] P. M. Chaikin and T. C. Lubensky, Principles of\ncondensed matter physics , Cambridge University Press\n(1995), p. 616.\n[40] H. Fukuyamaand P. A. Lee, Phys. Rev. B 17, 535 (1978).\n[41] P. A. Lee and T. M. Rice, Phys. Rev. B 19, 3970 (1979)."    },    {        "title": "1410.0439v1.Investigation_of_the_temperature_dependence_of_ferromagnetic_resonance_and_spin_waves_in_Co2FeAl0_5Si0_5.pdf",        "content": "1\n \nInvestigation of the temp erature-dependence of fe rromagnetic resonance and \nspin waves in Co 2FeAl 0.5Si0.5 \n \nLi Ming Loong1, Jae Hyun Kwon1, Praveen Deorani1, Chris Nga Tung Yu2, Atsufumi \nHirohata3,a), and Hyunsoo Yang1,b) \n \n1Department of Electrical and Computer Engine ering, National University of Singapore, 117576 \nSingapore \n2Department of Physics, The Univers ity of York, York, YO10 5DD, UK \n3Department of Electronics, The Unive rsity of York, York, YO10 5DD, UK \n \n \n  Co\n2FeAl 0.5Si0.5 (CFAS) is a Heusler compound th at is of interest for sp intronics applications, due \nto its high spin polarization a nd relatively low Gilbert dampi ng constant. In this study, the \nbehavior of ferromagnetic resonance as a functi on of temperature was investigated in CFAS, \nyielding a decreasing trend of damping constant as  the temperature was increased from 13 to 300 \nK. Furthermore, we studied spin waves in CF AS using both frequency domain and time domain \ntechniques, obtaining group velocities and atte nuation lengths as high as 26 km/s and 23.3 m, \nrespectively, at room temperature. \n  \na) Electronic mail: atsufumi.hirohata@york.ac.uk \nb) Electronic mail: eleyang@nus.edu.sg 2\nHalf-metallic Heusler compounds with  low Gilbert damping constant ( ) are promising \ncandidates for spin transfer torque-based (STT) spintronic devices,1-3 spin-based logic systems,4 \nas well as spin wave-based data comm unication in microelectronic circuits.5 Hence, a deeper \nfundamental understanding of the magnetiza tion dynamics, such as the behavior of \nferromagnetic resonance (FMR) and spin waves in Heusler compounds, could enable better \nengineering and utilization of these compounds fo r the aforementioned applications. In previous \nwork, FMR has been investigated in several Heusler compounds, such as Co 2FeAl (CFA),6 \nCo2MnSi (CMS),7 and Co 2FeAl 0.5Si0.5 (CFAS).8 In addition, the variation of  with temperature \nhas been studied for other material s, such as Co, Fe, Ni, and CoFeB.9-11 However, the \ntemperature-dependence of  in Heusler compounds has not be en reported yet. Furthermore, \nwhile there have been some studies of spin  waves in Heusler compounds, such as CMS and \nCo2Mn 0.6Fe0.4Si (CMFS),7,12 these studies have focused on frequency domain measurements. \nThus, time domain measurements remain scarce, and mainly consist of time-resolved magneto-optic Kerr effect (TR-MOKE) experiments.\n13 In this work, we investigate the temperature-\ndependence of  in CFAS, a half-metallic Heusler compound.14,15 Moreover, we utilize both \nfrequency domain and pulsed inductive micr owave magnetometry (PIMM) time domain \nmeasurements to study the magnetiza tion dynamics in CFAS. We obtain of 0.0025 at room \ntemperature, which is 6 times lower than the va lue at 13 K. In addition, we evaluate the group \nvelocity ( vg) and the attenuation length ( ) in CFAS, leading to values as high as 26 km/s and \n23.3 m respectively, at room temperature. \n CFAS (30 nm thick) was grown by ultrah igh vacuum (UHV) molecular beam epitaxy \n(MBE) on single crystal MgO (001) substrates and capped with 5 nm  of Au. The base pressure \nwas 1.210-8 Pa and the pressure during deposition was typically 1.6 10-7 Pa. The substrates 3\nwere cleaned with acetone, IPA and deionised wate r in an ultrasonic bath before being loaded \ninto the chamber. After the film growth, the samples were in-situ  annealed at 600 °C for 1 hour. \nCFAS alloy and Au pellets were used as targ ets for electron-beam bombardment. Figure 1(a) \nshows the vibrating sample magnetometry (V SM) results, from which the saturation \nmagnetization ( Ms) was extracted. The measurement was also  repeated at different temperatures \nto extract the corresponding values of Ms for subsequent data fitting. The Ms value increases \nfrom 1100 emu/cc at 300 K, to 1160 emu/cc at 13 K.  From the VSM data, we verify a hard axis \nalong [100] and an easy axis along [110] , consistent with earlier reports.3,8 In addition, the -2 \nXRD data shown in Fig. 1(b) verified the presen ce of the characteristic (004) peak, indicating \nthat the CFAS film wa s at least B2-ordered.1,14 As shown in Fig. 1(c), the film was patterned into \nmesas, which were integrated with asymme tric coplanar waveguides (ACPW). The ACPWs \nwere electrically isolated from the mesa by 50 nm of Al 2O3, which was deposited by RF \nsputtering. Vector network analy zer (VNA) and PIMM techniques were used to excite and detect \nferromagnetic resonance (FMR) as well as spin waves in CFAS. The former technique allows \nfrequency domain measurements, while the la tter technique was us ed for time domain \nmeasurements. The experimental setup enable d the excitation of Damon-Eshbach-type (DE) \nmodes, as the external magnetic field was applied along the ACPWs, shown in Fig. 1(c).16  \n A VNA was connected to the AC PWs, and reflection as well as transmission signals were \nmeasured to study the FMR and spin wave pr opagation, respectively. Background subtraction \nwas performed to obtain the resonance peaks.  Figure 2(a) shows th e FMR frequency as a \nfunction of applied magnetic fiel d at different temperatures, with  the corresponding fits using the \nKittel formula,17 \n݂ൌఊ\nଶగඥሺܪܪሻሺܪܪ4ܯߨ ௦ሻ,      (1)4\nwhere f is the resonance frequency,  is the gyromagnetic ratio,  H is the applied magnetic field,  \nand Ha is the anisotropy field. The g factor, which was extr acted using the equation \t\tߛ ൌ\n2ߤ݃ߨ/݄ ,where B is the Bohr magneton and h is Planck’s constant, was found to be 2.03 0.02, \nwhile Ha generally decreased from 130 Oe at  13 K to 70 Oe at 300 K. The ( g – 2) value is lower \nthan those of Co and Ni, but comparable to th ose of other Heusler comp ounds, such as CMS and \nCo2MnAl (CMA).18 The deviation of the g factor from the free electron  value of 2 is correlated \nwith the spin-orbit interaction in a material, where a smaller deviation indicates weaker spin-\norbit interaction, and lower .18 The inset of Fig. 2(a) show s the resonance frequency at H = \n1040 Oe as a function of temperature, with a Bloch fitting, indicating a Curie temperature of \napproximately 1000 K. The Bloch fitting was perf ormed by substituting the following equation19 \ninto Eq. (1): \nܯ௦ൌܽ൫1െܽ ଵܶଷ/ଶെܽଶܶହ/ଶെܽଷܶ/ଶ൯,      ( 2 )  \nwhere T is temperature, and a0, a1, a2, and  a3 are positive coefficients. \n As shown in Fig. 2(b), the extracted FMR field linewidths were fitted with the linear \nequation20 ΔH\tൌ\tΔH0\t\t4αf/, where H is the field linewidth and H0is the extrinsic field \nlinewidth . This enabled the extraction of the intrinsic Gilbert damping ( ) from the fit line slopes. \nFigure 2(c) shows that  increases as the temperature decreases. The value of  at room \ntemperature was found to be 0.0025, which is comp arable with the previously reported room \ntemperature value for CFAS.8 The trend of  with temperature is consistent with previous first-\nprinciple calculations,9 and could be attributed to longe r electron scattering time at lower \ntemperatures, due to a reduction in phonon-elec tron scattering. Consequently, the angular \nmomentum transfer at low temperatures occu rs predominantly by direct damping through \nintraband transitions.11 Similar temperature-dependence of  has also been observed 5\nexperimentally. For example, the of Co 20Fe60B20 has been found to increase by a factor of 3 \nfrom 0.007 at 300 K, to 0.023 at 5 K.11 This is comparable to our results, where  increases by a \nfactor of almost 6 from 0.0025 at 300 K to 0.014 at 13 K. It shoul d be noted that spin pumping \ninto the Au cap layer could have contributed to  the measured resonance linewidth, thus causing \nthe extracted  to be higher than its actual value ( CFAS). Thus,  = CFAS + sp, where sp \ndenotes the spin pumping c ontribution to the damping.21 While an investigation of sp in the \nCFAS/Au system would exceed the scope of this work, sp values for a Fe/Au system21,22 have \nnonetheless been included in Fig. 2(c) to prov ide a gauge of the temperature dependence of sp, \nas well as a rough estimation of the magnitude of sp in the CFAS/Au system. Figure 2(d) shows \nan increase in H0 as temperature increases. This could be due to the effect of temperature on the \ninteraction between magnetic precession and sample inhomogeneities, or on magnon-magnon \nscattering, as these f actors contribute to H0.20,23 In both Fig. 2(c) and 2(d), room temperature \nvalues of  and H0 for sputter-deposited CFAS were in cluded, for comparison with the MBE \nsample. It can be seen that the  is higher for the sputter-deposite d sample, consistent with lower \nhalf-metallic character due to greater structural disorder.1,6 \n We have also measured the time domain PI MM data at 300 K as shown in Fig. 3(a), \nwhere SW15 and SW30 denote edge-to-edge signal line separations of 15 and 30 m, \nrespectively. The width of all the signal lines was fixed at 10 m. Using the temporal positions \nof the centers of the Gaussian wavepackets ( t15 and t30, respectively), the group velocity ( vg) was \ncalculated with the equation5,24 vg\tൌ \t1 5 \tm/ሺt30\t– \tt15ሻ. Fast Fourier transform (FFT) was \nperformed on the PIMM data, as shown in Fig. 3(b), verifying the presence of multiple modes, \nwhere each mode manifested as a dark-light-dark oscillation. The vg decreases from 26 km/s at \n50 Oe to 11 km/s at 370 Oe, as shown in Fig. 3( c). Moreover, from the VNA transmission data, 6\nwhich is another measure of spin wave propagation, attenuation length ( ) and were extracted \nas a function of magnetic fiel d at room temperature, using the method reported elsewhere.24,25 \nThe spin wave amplitude was extracted from  Lorentzian fittings of the VNA transmission \nresonance peaks, which were measured using wa veguides with different center-to-center signal \nline-signal line (S-S) spacings. Then,  was extracted using the equation24 A1expሺ x1/ሻ\t ൌ\t\nA2expሺ x2/ሻ, where A1 and A2 denote the measured spin wave amplitudes, while  x1 and x2 denote \nthe different S-S spacings for the corresponding waveguides. The  decreases from 23.3 m at \n460 Oe, to 12.1 m at 1430 Oe, as shown in Fig. 3(c). Using the following equation,25  was \ncalculated at different magnetic fields, as shown in Fig. 3(d) \n ߙൌఊሺଶగெೞሻమௗషమೖ\nଶగሺுାଶగெ ೞሻ       ( 3 )   \nwhere d is the film thickness and k is a spin wave vector, which can be estimated by 2 /(signal \nline width).5 The  values (0.0026 – 0.0031) are consistent with the room temperature value \n(0.0025) obtained from the FMR measurements.  \n As shown in Fig. 3(c),  and vg decreased as the applie d magnetic field increased, \nconsistent with previous experimental11 and theoretical5 results. This trend can be understood in \nterms of the following equation,5,26 \n ݒൌఊమఓబమெೞమௗ\n଼గ݁ିଶௗ,        ( 4 )  \nwhere μ0 is the permeability of free space. As the a pplied magnetic field increases, the resonance \nfrequency increases, thus vg decreases. In addition,  for a given value of , the magnetic \nprecession will decay within a ce rtain amount of time. Hence, the distance travelled by the \nprecessional disturbance within that amount of time depends on its propagation velocity, vg. \nConsequently, the higher the vg, the longer the distance travelled, and thus, the higher the .The 7\nobtained values of  and vg are comparable to those of other ferromagnetic materials for the \nDamon-Eshbach surface spin wave mode.5,11,12 For example,  of 18.95 m was extracted for \nCFA by micromagnetic simulations,5 while  and vg values as high as 23.9 m and 25 km/s, \nrespectively, were experimentally observed in CoFeB.11 Furthermore,  as high as 16.7 m was \nexperimentally observed in CMFS.12 \n In conclusion, we have found a decreasing trend of  with increasing temperature for \nMBE-grown Co 2FeAl 0.5Si0.5, in the temperature range of 13 – 300 K. The room temperature \nvalue of  was found to be 0.0025, which was approximately 6 times lower than that at 13 K. \nWe have also investigated vg and  in CFAS, obtaining values as high as 26 km/s and 23.3 m \nrespectively, at room temperature. \n \nThis work was supported by the Singa pore NRF CRP Award No. NRF-CRP 4-2008-06. \n  8\nReferences \n1 T. Graf, C. Felser, and S. S. P. Parkin,  Prog. Solid State Chem. 39, 1 (2011). \n2 S.  Ikeda, H. Sato, M. Yamanouchi, H. Ga n, K. Miura, K. Mizunuma, S. Kanai, S. \nFukami, F. Matsukura, N. Kasai, and H. Ohno,  SPIN 2, 1240003 (2012). \n3 H. Sukegawa, Z. C. Wen, K. Kondou, S. Kasai,  S. Mitani, and K. Inomata,  Appl. Phys. \nLett. 100, 042508 (2012). \n4 M. Jamali, J. H. Kwon, S. M. Seo, K. J. Lee, and H. Yang,  Scientific Reports 3, 3160 \n(2013). \n5 J. H. Kwon, S. S. Mukherjee, P. Deorani,  M. Hayashi, and H. Yang,  Appl. Phys. A: \nMater. Sci. Process. 111, 369 (2013). \n6 S. Mizukami, D. Watanabe, M. Oogane, Y. A ndo, Y. Miura, M. Shirai, and T. Miyazaki,  \nJ. Appl. Phys. 105, 07D306 (2009). \n7 H. Pandey, P. C. Joshi, R. P. Pant, R. Pras ad, S. Auluck, and R. C. Budhani,  J. Appl. \nPhys. 111, 023912 (2012). \n8 L. H. Bai, N. Tezuka, M. Kohda, and J. Nitta,  Jpn. J. Appl. Phys. 51, 083001 (2012). \n9 K. Gilmore, Y. U. Idzerda, and M. D. Stiles,  Phys. Rev. Lett. 99, 027204 (2007). \n10 B. Heinrich, D. J. Meredith, and J. F. Cochran,  J. Appl. Phys. 50, 7726 (1979). \n11 H. M. Yu, R. Huber, T. Schwarze, F. Bra ndl, T. Rapp, P. Berberich, G. Duerr, and D. \nGrundler,  Appl. Phys. Lett. 100, 262412 (2012). \n12 T. Sebastian, Y. Ohdaira, T. Kubota, P. Pi rro, T. Bracher, K. Vogt , A. A. Serga, H. \nNaganuma, M. Oogane, Y. Ando, and B. Hillebrands,  Appl. Phys. Lett. 100, 112402 \n(2012). 9\n13 Y. Liu, L. R. Shelford, V. V. Kruglyak, R. J. Hicken, Y. Sakuraba, M. Oogane, and Y. \nAndo,  Phys. Rev. B 81, 094402 (2010). \n14 B. Balke, G. H. Fecher, and C. Felser,  Appl. Phys. Lett. 90, 242503 (2007). \n15 R. Shan, H. Sukegawa, W. H.  Wang, M. Kodzuka, T. Furubayashi, T. Ohkubo, S. Mitani, \nK. Inomata, and K. Hono,  Phys. Rev. Lett. 102, 246601 (2009). \n16 R. W. Damon and J. R. Eshbach,  J. Appl. Phys. 31, S104 (1960). \n17 C. Kittel,  Physical Review 73, 155 (1948). \n18 B. Aktas and F. Mikailov, Advances in Nanoscale Magnetism: Proceedings of the \nInternational Conference on Nanoscale Magnetism ICNM-2 007, June 25 -29, Istanbul, \nTurkey . (Springer, 2008). p. 63. \n19 S. T. Lin and R. E. Ogilvie,  J. Appl. Phys. 34, 1372 (1963). \n20 P. Krivosik, N. Mo, S. Kalarickal , and C. E. Patton,  J. Appl. Phys. 101, 083901 (2007). \n21 E. Montoya, B. Kardasz, C. Burrowes, W. Hu ttema, E. Girt, and B. Heinrich,  J. Appl. \nPhys. 111, 07C512 (2012). \n22 M. Zwierzycki, Y. Tserkovnyak, P. J. Kelly, A. Brataas, and G. E. W. Bauer,  Phys. Rev. \nB 71, 064420 (2005). \n23 M. Farle,  Reports on Progress in Physics 61, 755 (1998). \n24 K. Sekiguchi, K. Yamada, S. M. Seo, K. J. Le e, D. Chiba, K. Kobayashi, and T. Ono,  \nAppl. Phys. Lett. 97, 022508 (2010). \n25 K. Sekiguchi, K. Yamada, S. M. Seo, K. J. Le e, D. Chiba, K. Kobayashi, and T. Ono,  \nPhys. Rev. Lett. 108, 017203 (2012). \n26 D. D. Stancil, Theory of Magnetostatic Waves . (Springer, Berlin, 1993). \n 10\nFigure captions  \n \nFIG. 1. (a) Normalized magnetic hysteresis data along the crystallographic hard [100] and easy \n[110] axes of MBE-grown CFAS. Ms is the saturation magnetization. (b) -2 XRD data of the \nMBE-grown CFAS sample. (c) Optical microscopy image of the CFAS me sa integrated with \nasymmetric coplanar waveguides (ACPW). The or ientation of the in-plane magnetic field ( H) is \nindicated.  \nFIG. 2. (a) FMR frequency at different magnetic fields. Inset: FMR frequency for a fixed field \n(1040 Oe) at different temperat ures. (b) Resonance linewidth as a function of frequency at \ndifferent temperatures (symbols), with correspo nding fit lines. (c) Gilbert damping parameter ( ) \nat different temperatures. The spin  pumping contribution to damping ( sp) for a Fe/Au system \nhas been included, where all sp values were obtained from literature, except those at 13 K and \nroom temperature, which were obtained by extra polating the literature valu es. (d) Extrinsic field \nlinewidth (H0) at different temperatures. \nFIG. 3. (a) PIMM data from two differ ent signal line-signal line spacings at H = 50 Oe for 300 K. \n(b) Fast Fourier transform (FFT) of room temp erature PIMM data. (c) Room temperature group \nvelocity ( vg, axis: left and bottom) and attenuation length ( , axis: top and right) at different \nmagnetic fields. (d) Room temperature Gilbert damping parameter ( ) at different magnetic \nfields.  \n  11\n \n\n \nFIG. 1\n\n\n12\n \nFIG. 2 800 1000 1200 140091011121314\n(d) (c)(b) (a)\n  \n 13 K\n 90 K\n 210 K\n 294 KResonance frequency (GHz)\nMagnetic field (Oe)10 11 12 13 1450100150200250300\n 90 K\n 210 K\n 294 K H (Oe)\nResonance frequency (GHz)\n0 100 200 3000.0000.0050.0100.0150.020\n  \nTemperature (K) MBE\n Sputtered\n sp Au-Fe\n0 100 200 300050100150200 H0 (Oe)\nTemperature (K) MBE\n Sputtered0 200 800 100012000412 fR (GHz)\nTemperature (K)13\n\nFIG. 3 \n\n"    },    {        "title": "1410.1226v1.Ultimate_limit_of_field_confinement_by_surface_plasmon_polaritons.pdf",        "content": "Ultimate limit of field confinement by surface plasmon polaritons  \nJacob B Khurgin \nJohns Hopkins University Baltimore MD 21218 USA  \njakek@jhu.edu   \n \nWe show that electric field confinement in surface plasmon polaritons propagating at the \nmetal/dielectric interfaces enhances the los s due to Landau damping and which effectively \nlimits the degree of confinement itself. We prove that Landau damping and associated with \nit surface collision damping follow directly from Lindhard  formula for the dielectric \nconstant of free electron gas Furthermore, we demonstrate that even if all the conventional \nloss mechanisms, caused by phonons, electron- electron, and interface roughness scattering, \nwere eliminated, the maximum attainable degre e of confinement and the loss \naccompanying it would not change significantly compared to the best existing plasmonic materials, such as silver.  \n \nIntroduction \nSteady advances in nanofabrication made in the last decade had inspired research  in nano-\nplasmonic s, a field that carries many exciting promises in various areas of technology such as \nsub-wavelength imaging, sensing, nano -scale optical interconnects and active devices [1,2].  In \none way or another, these promises are all hinged upon the ability to conc entrate optical field \ninto the sub- wavelength dimensions that is a salient feature of surface plasmon polaritons \n(SPP’ s), whose nature is a combination of electro- magnetic field with charge waves of free \ncarriers in metal (or semiconductor). When the dimen sions  are reduced way below wavelength \nthe magnetic field is greatly diminished  (static limit) , and the energy which is normally stored in \nthe form of magnetic energy is instead stored in the form of kinetic energy of carriers (kinetic \ninductance) which ma kes sub -wavelength oscillation mode sustainable. Unfortunately and \ninevitably, the free carrier oscillations dissipate energy at a very high rate, of the order of \nγ\nm~1014s-1 in noble metals and 1013s-1 in highly doped semiconductors. As a result, the losses in \nthe SPP which are significantly sub -wavelength (in case of propagating SPP’s it means the SPP \npropagation constant β much larger than wave -vector in dielectric k d) are always on the scale of \nγm independent on the shape and exact size as long as it is significantly sub wavelength [3]. Due \nto these high losses the promises of nanoplasmonics, which include miniature efficient sources \nand detectors of radiation, nano- scale optical interconnects , super -resolution imaging, and others  \nhas not been fully realized yet. [4]   \n1 \n The reason the optical losses in metals are high is unfortunately the same one that makes support \ncharge oscillations at optical frequencies sustainable in the first place. High free carri er density \n(Fermi level high in the conduction band) requisite to maintain high plasma frequency also \nassures that the density of states into which these electrons can scatter is also very high, with \nstrong scattering following from this unfortunate fact. In optical range the metal loss is caused by \nmore than one mechanism [5] – scattering by phonons, carrier -carrier Umklapp processes, \nresidual intra -band absorption and scattering on surface imperfections can all contribut e on more \nor less equal scales, hence their reduction is not simple. Nevertheless, some strides in that direction are being made. Most obviously, reduction of metal surface roughness achieved with epitaxial growth ha ve yielded reduced loss in Ag [6], while  using Al in place of Ag [7] reduced \nparasitic interband absorption in the blue part of spectrum. The phonon -assisted absorption can \nbe reduced, but not eliminated by going to cryogenic temperatures because the spontaneous phonon emission is possible even at absolute zero, while the temperature -independent Umklapp \nscattering may be somewhat mitigated in the materials with different (less spherical) shapes of Brillouin zone, although, once again, the numerous efforts with materials as diverse as ITO and \nTiN [ 8] so far have not shown substantial improvement over ubiquitous noble metals.  \nNevertheless, the hope is alive, that sooner or later, a novel material with negative ε and \nsubstantially lower losses in the optical range will emerge, and in anticipation of  these \ndevelopments, it is worthwhile to estimate their practical impact, i.e. what would be the \nmaximum attainable degree of field concentration if all scattering -related loss in the metal in the \noptical range were essentially eliminated. Obviously, this question has been raised before. For \ninstance, it has been long known that since SPP is a combination of field and electronics  \noscillation, maximum SPP wave vector K\nmax cannot possibly exceed Fermi wave vector  \nkF~1.2x108 cm-1 for noble metals , i.e. restricting the degree of fiel d concentration to about 5 A. \nThis is also the  scale at which electron tunneling in the nanogap in the plasmonic dimers \ncommences [ 9] that further assures that electric field cannot be confined to sub- nanometer \ndimensions. Yet  this limit, usually referred to as “quantum limit” has not been achieved \nexperimentally, and mo re recently a different explanation which has put the limit of file \nconcentration in the range of a few nanometers has been put forward. The explanation was bas ed \non the non- locality phenomenon [10], or, in simpler terms on special dispersion of dielectric \nconstant ε(k). Using hydrodynamic model of nonlocality Mortensen et al [11] have shown that \nwhen characteristic dimensions of the sy stem become comparable with  characteristic length \nlc=vF/ω, where vF is a Fermi velocity  and ω is a frequency , the nonlocal effects prevent the light \nfrom tight confinement and broaden resonances, particularly for the dimer structures. Larkin and \nStockman [12] pointed that spatial di spersion effects limit the resolution of the plasmonic \nsuperlens to about 5nm in the visible range, i.e. comparable to a few lc. They also pointed out \nthat spatial dispersion is intimately related to Landau damping –i.e. direct absorption of \nelectromagnetic waves by electrons below the Fermi level. Landau damping in the nanoscale \nmetallic objects can also be interpreted as quantum confinement effect or as abs orption assisted \nby surface collisions as explained by K reibig  and Volmer  [13], who introduced the \n2 \n phenomenological expression for this process as a “surface collision scattering rate” ~/sFva γ  \nwhere vF is  Fermi velocity a is the characteristic dimension of the metallic object.  According to \nthis phenomenological treatment the additional damping is caused by the limited physical \ndimension of the system and is the result of restriction of mean free path of electrons.  \nHowever,  to the best of our knowledge  [14-15], there is no unified treatment of nonlocality (i.e. \nspatial dispersion of real part of ε ) and Landau damping (i.e. spatial dependence of the imaginary \npart of ε ) which would allow one to provide an unambiguous answer on which of two \nphenomena exerts stronger influence on SPP properties. In this work we develop this unified \ntreatment and show that it is Landau damping, i.e. loss induced by the field confinement  \nitself  that is responsible for most dramatic limitations in plasmonics. We show that in case \nof propagating SPP on single metal -dielectric interface, the limitations arise due to the field \n(and not electron) confinement and thus are not influenced by the metal dimensions.  \nLandau damping is shown to become important  when characteristic dimensions are on the scale \nof 10nm which is at least an order of magnitude larger than l c, and, finally, and to some degree \nsurprisingly, it is shown that total elimination of all other loss mechanisms in metals will not \nyield noticea ble improvement in field confinement and power dissipation compared to the best \nplasmonic materials of today.  \n \n \nFig.1 (a) Sketch of the fields in the propagating SPP (b) Phonon or impurity assisted absorption of  a photon with a \nwavevector K~0 and a direct absorption of SPP with large wavevector K (Landau Damping) (c) wavefunctions \ninvolved in the absorption of SPP  \n \nDamping rate of SPP due to surface collisions  \nWe start with the case of SPP propagating at the interface of metal 0mε< and dielectric  0dε> as \nshown in Fig1. a with propagating constant β and two components of electric field, normal  q qdεd>0εm<0\n|Ex|\nXEFE\n∆k\nkk1k2\nE1E2\nSPP K= ∆kPhoton K~0vFq\nEF\nE(x)\nX\n(a) (b)(c)\n3 \n  0\n0cos( ) 0\ncos( ) 0dqx\nx qx m\ndEe z t x\nEEe z t xβω\nεβωε−−>\n=−<  (1) \nthat is subjected to damping by surface collisions, and also the tangential  one  \n 0\n0sin( ) 0\nsin( ) 0dqx\nz\nqx mqEe z t x\nEqEe z t xβωβ\nβωβ−−>\n=\n−<  (2) \nwhich is not da mped by surface collisions.  For the absorption in the metal to take place, the \nelectron with energy 1E and wavevector 1k  below the Fermi level make a transition to a  state \nabove it with energy 21EE ω = + and wavevector 2kresulting in a wavevector mismatch ∆kthat \nmakes transition forbidden (Fig.1b)  The value of mismatch is on the scale of 1~/ ~c Fck vlω−∆  \nwhere Fermi velocity Fv is about 81.4 10 / cm s ×  for noble metals, i.e. for the 1eV photon energy \nthe wave -vector mismatch is about 1nm=1 and is thus much larger than wavevector of \nelectromagnetic wave. As a result, absorption usually involves additional act of scattering, due to \nphonons, electron- electron scattering, or imperfections that occur at the rate 0γwhich occurs on \nthe scale of a few tens of femtoseconds in most plasmonic metals.  At first glance, in the SPP \nboth β and q are much less than 1nm-1 one should not expect `much absorption. But if one takes \na look at the normal direction, one can see that electric field contains all kind of wavevectors –  \nindeed by taking the one dimensional Fourier transform of (1) we obtain the power spectrum of it  \n 2 2\n0 22/()qEK EKqπ=+  (3) \nThus the fraction of power of the wave with wavevectors exceeding ck∆is on the order of \n( ) (2 / )( / )cc FK k q k π >∆ ≈ ∆ . The penetration depth 1~ (2 )pLq− of the SPP is typically on the scale of \na few tens of nanometers , hence a f ew percent of the SPP energy does  wavevector sufficient for \nLandau damping to take place. Landau damping is shown schematically in Fig1.b as direct \n“diagonal” transition between the states 1kand 2kcaused by the electromagnetic wave with wave -\nvector c K=Δk.  As a result of Landau damping imaginary part of the dielectric constant of the \nmetal, describing the energy loss by electromagnetic wave (and hence by SPP) to the individual \nelectronic excitations will increase. The phenomenological “surface collision rate” sγ introduced \nby Kreibig modifies the expression for the effective dielectric constant of the metal as  \n ( )2\n0 \"\n3()ps\neffωγ γεωω+=   (4) \nwhere 0γis the momentum relaxation rate due to phonons, electron- electron scattering and \ndefects, usually ref erred to as bulk scattering rat. , The surface collision rate may be introduced  \n4 \n phenomenologically [ 10,13 ], or quantum -mechanically [16 ], but it is simply added to the \nexpression for dielectric constant using Matthiessen  rule, and is not directly connected to \nnonlocality.  \nWe now evaluate the energy damping rate of SPP polaritons  that is due to localization of the \nfield in the vicinity of the metal interface with dielectric . As one can see from the Fig1.c  the two \nfree electron wavefunctions || 1( 2 ) 1/2\n1(2)x i ikLe e ψ⋅ −=||kr (L being the normalization length) are orthogonal \nand optical tr ansition between them is not allowed. But since the electric field is localized \naccording to (1), the square of the interaction Hamiltonian between two states can be found as  \n 2222 2\n21 20\n12 2 2 22 2 212\n4( )xxe e kk EHm m Lk q ω⋅\n= =∆+pA \n  (5) \nNote that the fact that elect rons get reflected by the surface is not reflected in (5) – the existence \nof transition is strictly due to the confinement of the field. Since both states are  close to Fermi \nenergy, i.e. EF ω<<  , we can make two important approximations Fxvk ω≈∆ where Fxvis \nthe transverse component of velocity  on the Fermi surface , and, furthermore\n22\n12 /2 /2x x Fx k k m mv m ≈  , which upon substitution into (5) yields  \n ( )24 2\n20\n12 42 22 241 /Fx\nFxev EH\nL qvωω=\n+  (6) \nWe now invoke Fermi Golden rule to evaluate the total rate of the field induced upwar d \ntransitions from the state 1,2\n1 12 2 (E )xF R HL πρ= , where  ( )1(E ) 2x F Fxv ρπ==   is one-\ndimensional density of the final states, evaluated under consideration that neither spin nor  \ndirection of propagation change as the transition takes place . We thus obtain  \n \n( )23 2\n0\n12 4 22 241 /Fx\nFxev E\nR\nL qvωω=\n+  (7) \nNext we perform summation over all the states 1 inside Fermi sphere from which transitions into \nunoccupied states can take place. That involves integration over the Fermi surface and then \nmultiplying by ω as well as the normalization length L  to obtain the surface rate of excitation \nof hot electrons per unit of area. Integration over the Fermi surface is  simply multiplication by \nthe three- dimensional  density of sates 2 23\n3 /DFmv ρπ=  and averaging over the angles which \nyields 33/4Fx xvv= , so we obtain  \n5 \n  2 24 2\n20\n24 3 2 216 (1 / )DF\ncdN e m v E\ndt q k πω=+∆  (8) \nSince c qk<< ∆  we can write the expression for the energy loss at surface  \n 2 24 2\n20\n23 216DFdU e m v E\ndt πω−=  (9) \nwhere  2DUis the time -averaged two -dimensional density of kinetic energy of electron s in the \nSPP which can be found as the integral of three -dimensional density of energy  \n 22\n2 0\n2 2\n01( )() ()48me\nD bxN eEU x E x dxqmωεεωω∞∂=−= ∂∫ (10) \nwhere bεis the part of dielectric constant due to bound electrons, and t he electron density in a \nparabolic band is 3 2 33 23/3 /3eF F N k mv ππ = = , hence   \n 2 23 2\n0\n2 2 321\n24F\nDemv EUqπω=  (11) \nComparison of (9) and (11) immediately results in the energy relaxation ra te  \n 2\n21322D\nFs\nDdUqvU dtγ = −= −  (12) \nwhere sγis the momentum scattering rate due to surface collisions  \n 3\n4sF qv γ=  (13) \nthat enters into expression (4) for the dielectric constant. This expression is not much different \nfrom the one in previous works, where the scattering rate for a nanoparticle with radius R is \n/sFAv R γ= but in this work we have obtained this expression using full quantum mechanical \nderivation. The fact that our coefficient  A is less than values for nanoparticles is easy to explain \nby the fact that our problem is one dimensional, hence not every electron contributes to the \nsurface absorption.  \nIt is very important to stress that the “surface collision damping rate”  obtained by  us does \nnot requi re the electrons to be confined, or even reflected – simple confinement of light on \nthe scale of penetration length  1/2penLq=    is sufficient to overcome momentum \nconservation rules and cause substantial absorption even i f the electrons are considered to \n6 \n be free. In other words the confinement of light is what causes the damping, hence the proper \nname for sγshould be “Landau damping rate”, or, better, “ time-of-flight”  broadening . \nNevertheless, we shall use “surface collisions damping” term to conform to the existing \nliterature.  \nSurface collision damping as a non -local effect  \nNext we demonstrate that result (13)  can also be obtained by using a simple, phenomenological \npicture of simply “shifting” the resonance frequency in Drude formula to account for the \npossibility of “diagonal in k- space” transition as shown Fig 1b. In the classical Drude formula, \nobtained in the l ong wavelength limit of Lindhard approximation the energy difference between \ntwo states involved in the optical transition is zero, but for the electromagnetic wave with the \nfinite wavevector K the resonance is shifted by roughly FvK ωα∆=  where α is on a scale of \nunity. Using Klimontovich- Silin -Lindhard approximate formula [17] for the dielectric constant \nof the metal one can obtain 3/5 α=  \n 2\n22\n2 22\n03( , ) 1 ln3 2\n5p F\nbb\nFF F\nFKvKK v Kv KvKv iω ω ωεω ε εωω ωγ +=+ − ≈− −  −+  (14) \nwhere 0γis the “intrinsic” scattering due to phonons, defects and electron- electron scattering, and \nbεis the interband contribution to dielectric constant  ( 4.1bε≈for Ag)  Equation (14) is easy to \ninterpret  as modification of Drude formula. For negligibly small wave vectors the resonant \nenergy of the transition between two free electron states is zero, but as wave- vector increases the \nresonance shifts upward, by roughly 3/5FKv ωω∆ ≈ <<  where Landau  Damping takes place (the \nfactor of 3/5 can be traced to the averaging over the Fermi surface). As shown in Fig2a the whole \ndispersion curve shifts towards higher frequencies resulting in small change of the dielectric constant. Separating dielectric const ant into the real and imaginary parts one can write  \n ( )\n( )22 2 2\n00\n2 22 2 42 2\n0 00\n2 4\n00\n2 22 2 42 2\n0 00(, )\n/\n/(, )\n/p\nrb\np\niKK K\nK\nKK K\nKK\nKK Kωεω εω γω\nω γωεωω γω−\n= −\n−+\n=\n−+  (15) \nwhere 0 5/3c Kk= ∆  is roughly the wave -vector at which “diagonal” absorption of light (Landau \nDamping) commences.   \n7 \n  \nFig.2  (a) Sketch of the resonance shift due to non -locality (Eq (14) . (b) Effective dielectric constant of SPP can be \nobtained by overlapping  spatial power spectrum of SPP |E(k)|2 with wave -vector dependent dielectric constant (real \nand imaginary parts)  \nThe plot of wavevector -dielectric constant is shown in Fig.2b  for 0/ 20ωγ = and for the \nimaginary part of dielectric constant it consists of essentially flat response for 0 KK<<  followed \nby the sharp delta- function like Lorentzian peaks near 0 KK= ±  \n 2\n0\n0 3\n22 2\n00 0\n00 2 2222 2\n0 00      \n(, )/4( )     2 /4p\ni\nppKK\nKKKKK K K\nKK Kωγ\nω\nεωωω γω πδωω γω\n<< \n≈\n ≈ ≈±\n  + \n  (16) \nThe effective dielectric constant can be obtained by integrating (16)  over the normalized power \ndensity spectrum of the electric field inside the metal2 22() /( )EK q K q π = + , also of Lorentzan \nshape as plotted in Fig  2b for 0/10 qK= , which immediately gives us  \n ( )22 22 2\n00 0 0\n, 3 22 2 3 3 3\n03/5(,)pp pp F p s\ni effqv qKqKqωγ ω ωγ ω ω γ γεωωω ω ω ω+=+ ≈+ =+  (17) \nwhere the surface collision broadening is 3 / 5 0.77s FF qv qv γ= ≈  i.e. result that is very close to \nquantum mechanical derivation (13) . Besides providing simple physical interpretation,   equation \n(21) also confirms the Matthiessen’s rule of the strength of absorption induced by surface \nreflection sγbeing added to the strength of absorption induced by all other means0γ.  \nIf we now turn our attention to the real part of the dielectric constant, then one can see that over \nthe region where the 0 KK<< , the power density spectrum 2()EK looks like a delta function, \nwhile the integrating over the real part of Lorentzian  leads to cancellation, and, as a result \n,( ) ( ,0)r eff r ε ω εω ≈ . In other words, the nonlocality effects are dominated by the change in εr\nω\nεr(0)εr(K)∆ω\n-1.5K0 -K0 -0.5K0 0 0.5K0 K0 1.5K0-10-50510152025\nWavevector KArbitrary units\n2q\n(a) (b)\n8 \n imaginary part of the dielectric constant, i.e. surface collision damping (a.k.a. time -of-flight \nbroadening)  \nUltimat e limit of field confinement imposed by surface collision  damping  \nWe now what to see how  the increased damping due to field confinement manifests itself for the \npropagating SPP’s of Fig1.a.   The propagation constant of SPP can be found as  \n ()() ()()m\nD\ndmkεωβω ωε εω=+  (18) \nwhere 1/2() 2 /Ddkcω πε ω= is the  wavevector of the free propagating electromagnetic wave in the \ndielect ric with frequency -independent dielectric constant dε and the metal dielectric constant, \naccording to (14)and  (17) given by  \n 2\n2\n0(,)[ ( )]p\neff b\nsqiqωεω εω ωγ γ≈−++  (19) \nand the collision -induced damping rate being  \n 3() ()4s Fx q vf γ ωω=   (20) \nwhere  \n 22() ()mD qkω βω ε= −   (21) \nand  \n 22\n22 22x\nx\nzxEfq EEβ\nβ= =+ +  (22) \nis a fraction of the energy contained in the normal component of electric field. If we now \nintroduce the frequency of SP resonance, /( )SP p b dω ω εε= +   and normalize the frequencies to it, \n/SP ω ωω= , and introduce effective index of SPP as /Dk ββ=  and normalized decay constant \n/D q qk=  we obtain from (18) and (19)  \n ( )2 1 21\n0\n2 1 21\n0( 1)()(1 )b db bs\nbd siQ i Q\niQ i Qε ω ε εω εωβωεεω ω ω−−\n−−−− + +=+ −+ +   (23) \n    \n9 \n where 00 /SP Q ωγ= and 1/2( ) 4 /3s d F xr Q c v fqωε=  and ()rqω  is the real part of normalized decay \nconstant. Since ()sQωitself is a function of ()βωequation (23) can be solved self -consistently, by \niterations, but prior to that a few important observations can be made.  \nFirst, the impact of surface damping would be come noticeable when 0~sγγ , which will \nhappen not far from the SP resonance, hence when 0~sQQ , and since in the vicinity of SPP \nresonance qβ≈  and 1/2xf≈  we obtain the value of effective index at which the surface dumping \nmast be taken into account,  \n 1/2\n08\n3s\ndFc\nvQβε≈   (24) \nIf we consider the combination of Ag and GaN (\n1/2 13 1 15 1\n00 2.3,  3.2 10 ,  4.5 10 ; 140;d SP s sQ εγ ω−−= = × = ×≈ [18] ) we obtain 1.6sβ≈ while for GaN –Au \ncombination (14 1\n001.2 10 ; 43; sQ γ−= ×≈ [18]) we obtain  5sβ≈  . As expected, it is for good metal, \nlike silver that surface collision role becomes important early on, while for the less perfect metal, \nlike gold the influence of surface collision does not become important until much later.  \nSecond, one can use (23) to find the ultimate value of the effective index, and hence \nconfinement, of SPP which could have been obtained in the hypothetical “ideal” metal with \n00 γ=, i.e. free of defects, phonon scattering, electron -electron interaction, and residual interband \nabsorption. Obviously, such metal does not exist, however it is useful to see what kind of \nimprovement can be achieved by reducing the loss in the metal.  By inserting 1ω= and 1\n00; Q−= \ninto (23)  we obtain  \n ( ) ( )1/2\nmax\nmax,r8\n3ds d\nbd bd FiQ ci\nvεεβεε εε β= ≈++\n  (25) \nTherefore we obtain a rather simple expression for the maximum effective index (real part) \nattainable with the “ideal” metal  \n 1/31/2\nmax,r4\n3d\nb dFc\nvεβεε≈+   (26) \nThen for a wide variety of dielectrics transparent in the visible and near IR with refractive \nindices between 1.5 and 3, max,r 4 4.5β<< , so one arrives at a rather surprising result –  one can \nreduce the wavelength of the SPP propagating on the metal -dielectric interface  by no more than a \nfactor of 4.5 relative to the plane  wave p ropagating in dielec tric, no matter  how low is the l oss in \nthe bulk metal.  \n \n10 \n Numerical Results  \nWe now calculate the dispersion curves    (23) for the SPP propagating on the boundary between \nthe metal with silver -like dispersion ( 4.1,  9.3 ;b SP eV εω= = )  and GaN 1/22.3dε= resulting in SPP \nresonance near 415nm. When it comes to the scattering constant in the metal we shall consider 5 \ndifferent cases:  \nA. The best bulk silver with bulk damping constant13 1\n03.2 10 s γ−= × and no surface collision \ndamping taken into account 0sγ= \nB. The best bulk silver with bulk damping constant13 1\n03.2 10 s γ−= × with  surface collision \ndamping taken into account  \n \nC.  “Dirty silver” with bulk damping constant14 1\n01.2 10 s γ−= × similar to that of gold and no \nsurfa ce collision damping taken into account 0sγ=.  The reason for using “dirty silver” \ninstead of gold is that one cannot observe interface SP resonance in gold in combination with any dielectric d ue to high interband absorption, but to see how the surface collision \ndamping affects metals with reasonably high bulk loss is important.  \nD. “Dirty silver” with bulk damping constant\n14 1\n01.2 10 s γ−= × similar to that of gold with \nsurface collision dam ping taken into account.  \nE. “Ideal metal” with no bulk damping00 γ=, with surface collision damping taken into \naccount.  \n \nFig.3  (a) dispersion curves (b) Figure of Merit vs wavevector (c) Propagation length vs. Confinement width of SPP \non the interface between Ga -N and Ag -like metal for the following cases: (A) The best bulk silver with bulk \ndamping constant13 1\n03.2 10 s γ−= × and no surface collision damping taken into account 0sγ= (B) Same with  \nsurface collision damping taken into account (C)  “Dirty silver” with bulk damping constant14 1\n01.2 10 s γ−= × similar \nto that of gold and no surface col lision damping taken into account 0sγ=.  (D) Same with bulk damping constant\n14 1\n01.2 10 s γ−= × similar to that of gold with surface collision damping taken into account. (E) “Ideal metal” with no \nbulk damping00 γ=, with surface collision damping taken into account.  \nThe results are shown in Fig. 3a, as well as the light line representing propagation of \nplane electromagnetic wave in GaN. As expected, not taking into account surface damping for the best silver (curve A) leads to very large propagation constant, with effective index exceeding 0 50 100 150 200 250100101102103\nSPP wave vector ( µm-1)FOM\nA BC DIdealBest Silver Dirty  Silver\nE\n10-310-210-110-210-1100101102\nConfinement Width ( µm)Propagation Length ( µm)\nA BC DIdeal\nBest Silver\nDirty  Silver\nE(a)(b) (c)\n0 50 100 150 200 2502.62.72.82.933.1\nSPP wave vector ( µm-1)Energy (eV)A\nBCDE (Ideal)\nBest Silver\nDirty  Silver\n11 \n 7, but once surface collisions dam ping have been included (curve B ) the propagation constant is \nreduced almost two -fold. If we now consider the more realis tic silver, full of defects due to \nsurface deposition process, whose damping rate is comparabl e to gold, the curves without (C ) \nand with (D) surface collision damping, the latter’s  impact is less significant, although still \nprominent. But it is the curve ( E) which is most telling – if one starts with the best available \nsilver and then hypothetically gets rid of all damping processes, being it defects, phonons or \nelectron -electron interaction, then, even if the surface is atomically smooth the increase of th e \nattainable propagation constant max, rβ (and hence the degree of confinement q) will be only about \n12%, and the maximum effective index will not exceed roughly 4.4, just as predicted by (26) . \nThat means the minimum confinement depth in the normal direction 11\nmin, ( )/2xdd qq−−= +  will not \nbe less then roughly   / 25Dd n λ  where, 1/2\nddn ε= . Furthermore, if we now consider the imaging \nusing SPP’s, the superlens [ 19,20] and apply the analysis of [ 21], the minimum spot size  \n(confinement in lateral direction) achievable in this configuration would be \nmin,x max, 2 / / 4.4r dn πβ λ≈≈ , i.e, close to the result obtained in [ 12].  \nTo show how surface collision damping affects losses we also plot in Fig.3b the figure of \nmerit, defined as the ratio R e () / 2 I m () FOM ββ = , or as one can say, the phase shift accumulated \nover one absorption length. Once agai n we can see that for the high quality silver (curves A and \nB)  the impact of surface collisions becomes important at large wavevectors with almost an order \nof magnitude difference, achieved at 1150r m βµ−=  while for the higher loss metal (curves C and \nD) the difference is somewhat less . But the most important is the fact that for the “ideal” metal \nFOM improves by only a factor of two relative to the  “best silver”  \nYet another way to show the effect of surface collision damping is to plot the propagation \nlength 1/ 2 Im( )propL β =  vs. the field penetration width in dielectric 1/ 2 Re( )con d Lq= as demonstrated \nin Fig.3c.  Reducing the bulk losses 0γ  helps to increase propagation length by about an order of \nmagnitude for the confinement of wider than 50nm, but for tighter confinement, as surface \ncollision damping becomes dominant, and curves B,D, and E get close to each other, getting rid \nof all bulk losses results in only marginal increase of the propagation length  \n qεd>0 εm<0\n|Ex|\nd\n10-210-1050100150200250\nGap Width ( µm)wave vector ( µm-1)\n10-210-110-210-1100101\nGap Width ( µm)Propagation Length ( µm)Ideal\nSilver\nGoldA\nB\nC\nDE\n(a)(b)(c)\n12 \n Fig.3  (a) sketch of gap SPP (b) Propagation length vs. Confinement width of gap SPP with Ga-N core and metal \ncladding (A) The best bulk silver with bulk damping constant13 1\n03.2 10 s γ−= × and no surface collision damping \ntaken into account 0sγ= (B) Same with  surface collision damping taken into account (C)  Gold with bulk damping \nconstant14 1\n01.2 10 s γ−= × similar to that of gold and no  surface collision damping taken into account 0sγ=.  (D) \nSame with bulk damping constant14 1\n01.2 10 s γ−= × similar to that of gold with surface collision damping taken into \naccount. (E) “Ideal metal” with no bulk damping00 γ=, with surface collision damping taken into account.  \n \nTo further emphasize this point we consider the case of gap plasmon  [22], Fig.4a  with \nGaAs core and metal cladding, made either of Ag or Au.  Once a gain we consider the same 5  \ncases as above, except cases C and D now refer to gold, as well as to “dirty silver”. The \ndispersion curves for all 5 cases, shown in Fig.4b are essentially, identical, but the amount of loss \ndiffer dramatically, as shown in Fig. 4c where the propagation length is plotted versus the gap width.  As one can see inclusion of surface collision damping greatly reduces propagation length for silver (curves A and B) and somewhat less than that for gold. The most significant \nobservation to be made from this figure  is that hypothetically avoiding all bulk loss in the “ideal” \nmetal (curve E) does increase propagation length for the weakly confined gap plasmons with gap \nsize over 100nm, but for the tightly confined ones, with gap size less than 50nm the effect is marg inal. It appears that surface collision damping alone makes propagation length shorter than \n1µm, which makes gap plasmon impractical for application as, say, interconnect.  \nConclusions  \nIn this work we have considered the impact of non- local effects on the properties of \npropagating SPP’s . We have shown that the increase in loss is due to the final extent of the \noptical field, rather than due to collisions with the surface. In other words, surface collision damping /broadening  is better described as time -of-flight broadening. We have obtained full \nquantum -mechanical expression for the damping rate and corresponding change in the imaginary \npart of dielectric constant. W e have confirmed that the increased damping is a non -local effect \nthat follows naturally from the Lindhard theory of the wavevector -dependent dielectric constant \nand does not have to be introduced phenomenologically. We have shown that nonlocality –  \nengendered change in the imaginary part of dielectric constant exerts much stronger inf luence on \nproperties of plasmonic structures than the dispersion of the real part.  \nWe then applied the theory to the case of propagating SPP’s and have shown that not \nonly surface collision damping increases loss, but it actually prevents the field from being \nconcentra ted into the tight regions. As a result, even if the bulk scattering of the metal had been  \ncompletely eliminated, one would have not be able to concentrate the field into the regions that \nare substantially tighter than roughly \n/ 4.5dn λ  i.e exceeding the diffraction limit only by a factor \nof few. One can show that similarly, in case of dimers, not only the line width  will broaden as \ntwo nanoparticles will get closer [14 ], but the  mode itself will expand. This can be simply \n13 \n understood using coupled mode analysis [ 23] in which the gap mode in dimer consist of \nsuperposition of multipole modes. Since higher order modes are strongly confined near surface, \nthey are damped relative to lower order modes, and as a result they do not get excited as easily as \nlower order modes.  \nThese results raise a very important question about the impact of the efforts to reduce \nbulk loss in metals. It appears that eliminating the defects and imperfections of fabrication process, and, hypothetically, reducing phonon and electron- electron scattering will not reduce \nloss significantly relative to today’ s best silver for the substantially (factor of few) sub -\nwavelength structures.  There are of course other compelling reasons for looking at different \nmater ials, such as cost and compatibility with CMOS processes for integration of Plasmonics \nwith electronics. Furthermore, when it comes to the structures that are not tightly confined such as long range SPP’s [Bern] the already relatively low loss can be furth er reducing using material \nwith a smaller bulk loss. But once the confinement gets really tight surface collision damping makes losses high and nearly independent of bulk losses. It seems that as long as there exist two \nelectronic states separated by photo n energy, one occupied and one empty,  there will always be \na transition between two of them, and hence absorption. The only way to avoid all losses, \nincluding Landau damping is to develop materials with narrow conduction bands, such that there \nis no empty  state within \nω from the bottom of the band [24], so that transition cannot take place \nby any means. Perhaps that is where the effort to develop low loss materials should be directed.  \n \nThe author acknowledges fruitful discussions with A. V. Uskov  \n \nReferences  \n1. M. Stockman, \"Nanoplasmonics: past, present, and glimpse into future,\" Opt. Express \n,19, 22029- 22106 (2011). \n2. W. L. Barnes, A. Dereux, and T. W. Ebbesen Surface plasmon subwavelengt h optics, \nNature, London, 424, 824 (2003)  \n3. J. B. Khurgin, G. Sun, Scaling of losses with size and wavelength in Nanoplasmonics”  \nAppl. Phys. Lett, 99, 211106  (2011)  \n4. J. B. Khurgin, A. Boltasseva, “ Reflecting upon the losses in plasmonics  and \nmetamaterials”, MRS Bulletin  ,  37, 768- 779   ( 2012)  \n5. Abeles, ed., Optical Properties of Solids, North- Holland, Amsterdam (1972). \n6. Y. Wu , C. Zhang , N. M.  Estakhri , Y. Zhao , J. Kim ,  M. Zhang , X -X Liu , Greg K. \nPribil , A. Alù ,C -K Shih, X. Li  \"Intrinsic optical properties and enhanced plasmonic \nresponse of epitaxial silver\" Adv. Mater. 26, 6106 –6110 (2014)  \n7. Schwab, P. M. et al.  Linear and nonlinear optical characterization of aluminum \nnanoantennas .Nano Lett. 13, 1535–1540 ( 2013).  \n14 \n 8. A, Boltasseva  and H. A. Atwater Low -Loss Plasmonic Metamaterials Science 331, 290 \n(2011);  \n9. Zuloaga , E. Prodan, P.  Nordlander , Quantum Description of the Plasmon Resonances of a \nNanoparticle Dime Nano Lett.  2009, 9, 887. \n10. F. J. Garcıa de Abajo, “Nonlocal effects in the plasmons of strongly interacting \nnanoparticles, dimers, and waveguides,  J. Phys. Chem. C 112, 17983–17987 (2008).  \n11. N. A. Mortensen, S. Raza, M. Wubs, T. Søndergaard, and S. I. Bozhevolnyi, “A \ngeneralized nonlocal optical response theory for plasmonic nanos tructures,” Nat. \nCommun. 5 (2014). \n12. A. Larkin and M. I. Stockman, “Imperfect perfect lens,” Nano Lett. 5, 339 –343 (2005).  \n13. U. Kreibig and M. Vollmer, Optical Properties of Metal Clusters (Springer -Verlag, \nBerlin, 1995).  \n14. G. Toscano, S. Raza, W. Yan, C. Jeppes en, S. Xiao, M. Wubs, A.- P. Jauho, S. I. \nBozhevolnyi, and N. A. Mortensen, “Nonlocal response in plasmonic waveguiding with \nextreme light confinement,” Nanophotonics 2, 161– 166 (2013).  \n15. N. A. Mortensen, “Nonlocal formalism for nanoplasmonics: phenomenologi cal and semi -\nclassical considerations,” Phot. Nanostr. 11, 302 – 308 (2013)  \n16. A.V. Uskov, I. E. Protsenko, N. A. Mortensen, and E. P. O’Reilly, “Broadening of \nplasmonic  resonance due to electron collisions with nanoparticle boundary: a quantum \nmechanical consideration, ” Plasmonics 9, 185– 192 (2014);  Protsenko IE, Uskov AV \n(2012) Photoemission from metal nanoparticles. Phys -Usp 55:508–518 (2012)  \n17. G. D. Mahan, Many -particl e physics (Kluwer Academics, New York, 2000)  \n18. P. Johnson and R. Christy, “Optical constants of the noble metals,” Phys. Rev. B 6, \n4370–4379 \n19.  J. B. Pendry, Negative Refraction Makes a Perfect Lens  Phys. Rev. Lett. 85, 3966 \n(2000).  \n20. N. Fang, H. Lee, C. Sun, and X. Zhang, Sub- diffraction -limited optical imaging with a \nsilver superlens.  Science 308, 534 (2005)  \n21. B. Zhang, J. B. Khurgin, “ Eigen mode approach to the sub- wavelength imaging with \nsurface plasmon polaritons”   Appl. Phys. Lett, 98, 263102  (2011)  \n22. H. T. Miyazaki and Y. Kurokawa, “Squeezing visible light waves into a 3- nm-thick and \n55-nm-long plasmon  cavity,” Phys. Rev. Lett. 96(9), 097401 (2006).  \n23. G. Sun, J. B. Khurgin, A. Bratkovsky   “Coupled- mode theory of field enhancement in \ncomplex metal nanostructures  “, Phys. Rev, B 84, 045415 (2011)  \n24. J. B. Khurgin, G. Sun, “In search of the elusive lossless metal”, Appl. Phys. Lett., 96, \n181102 (2010)  \n \n15 \n "    },    {        "title": "1410.4796v1.The_fixed_irreducible_bridge_ensemble_for_self_avoiding_walks.pdf",        "content": "The \fxed irreducible bridge ensemble for self-avoiding\nwalks\nMichael James Gilbert\nAbstract\nWe de\fne a new ensemble for self-avoiding walks in the upper half-plane, the \fxed\nirredicible bridge ensemble, by considering self-avoiding walks in the upper half-plane\nup to their n-th bridge height, Yn, and scaling the walk by 1 =Ynto obtain a curve in\nthe unit strip, and then taking n!1 . We then conjecture a relationship between this\nensemble to SLE 8=3in the unit strip from 0 to a \fxed point along the upper boundary\nof the strip, integrated over the conjectured exit density of self-avoiding walk spanning\na strip in the scaling limit. We conjecture that there exists a positive constant \u001bsuch\nthatn\u0000\u001bYnconverges in distribution to that of a stable random variable as n!1 .\nThen the conjectured relationship between the \fxed irreducible bridge scaling limit\nandSLE 8=3can be described as follows: If one takes a SAW considered up to Yn\nand scales by 1 =Ynand then weights the walk by Ynto an appropriate power, then\nin the limit n! 1 , one should obtain a curve from the scaling limit of the self-\navoiding walk spanning the unit strip. In addition to a heuristic derivation, we provide\nnumerical evidence to support the conjecture and give estimates for the boundary\nscaling exponent.\n1arXiv:1410.4796v1  [math-ph]  17 Oct 2014Contents\nContents 2\n1 Introduction 3\n1.1 The in\fnite length upper half-plane self-avoiding walk . . . . . . . . . . . . . 3\n1.2 Summary of Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4\n1.3 Scaling limits and SLE partition functions . . . . . . . . . . . . . . . . . . . 5\n2 The conjecture 7\n2.1 Statement of the conjecture . . . . . . . . . . . . . . . . . . . . . . . . . . . 7\n2.2 The derivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8\n3 SLE predictions of random variables 10\n3.1 The density function \u001a(x) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10\n3.2 The right-most excursion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11\n4 Simulations 13\nA Appendix: Argument that \u001b= 4=3 16\n21 Introduction\n1.1 The in\fnite length upper half-plane self-avoiding walk\nMany important 2-dimensional lattice models arise in the study of statistical mechanics.\nAmong these, the self-avoiding walk is one that has been shown to be a very rich and\ninteresting model full that comes with a plethora of challenging problems.\nThe self-avoiding walk was introduced in 1949 by Paul Flory as a model for polymers. An\nN-step self-avoiding walk (SAW) on a two-dimensional lattice with lattice spacing \u000e >0 is\na sequence of lattice sites\n!= [!(0);!(1);:::;! (N)]\nsuch thatj!(j+ 1)\u0000!(j)j=\u000efor allj= 1;:::;N and such that !(j)6=!(k) for allj6=k.\nLet \nNbe the set of all N-step SAWs !on the lattice Z2=Z+iZwhich begin at the\norigin, i.e. !(0) = 0. We equip \n Nwith the uniform probability measure, i.e. we de\fne\nPN(!) = 1=CN, whereCN=j\nNjis the cardinality of \n N.\nBy concatenating an N-step SAW with an M-step SAW, we can see that\nCN+M\u0014CNCM: (1.1)\nA standard subadditivity argument then shows that there exists a constant \u0016>0 such that\nlim\nN!1logCN\nN= log\u0016; (1.2)\nThe constant \u0016is referred to as the connective constant .\nWe will be considering SAWs !2\nNsuch that Im( !(j))>0 forj= 1;2;:::, equipped with\nthe uniform measure, PH;N. LetHNdenote the set of all N-step upper half-plane SAWs\nwith!(0) = 0, and letH=S1\nn=0HNbe the set of all such upper half-plane SAWs. It was\nshown in [6] that the distributional limit as N!1 of the measures PH;Nexists and gives\na measure on in\fnite length upper-half plane SAWs. Let H1denote the set of all in\fnite\nlength upper half-plane SAWs, and let PH;1denote the weak limit of the measures PH;N\n(considered as measures on H. A SAW!2HNis called a bridge if\nIm(!(0))<Im(!(j))\u0014Im(!(N)) (1.3)\nfor allj= 1;:::;N . Note that the concatenation of any two bridges is still a bridge, while\nnot every bridge can be written as the concatenation of two shorter (non-trivial) bridges.\nA bridge with the latter property will be said to be irreducible . LetBdenote the set of all\nbridges rooted at the origin, i.e. all !2H satisfying (1.3) with !(0) = 0, and letIdenote\nthe set of all irreducible bridges rooted at the origin.\n3An important consequence of the proof of the existence of the limit of the measures PH;Nis\nthat given an !2H1, with probability 1, !can be written as the concatenation of a bridge\nwith another (translated) upper half-plane SAW. The proof of the existence of PH;1utilizes\nKesten's relation :\nX\n!2I\u0016\u0000j!j= 1; (1.4)\nfor the same \u0016as in (1.2), where j!jdenotes the length, or number of steps, of !. Kesten's\nrelation shows that P(!) =\u0016\u0000j!jis a probability measure on I. By concatenating together\nan i.i.d. sequence of irreducible bridges with respect to P, a probability measure is induced\nonH, and [6] shows that this is the only possible candidate for the measure PH;1.\nOne should then think of upper half-plane SAWs as having a renewal structure to them. At\nthe end of an irreducible bridge, the future path of the walk lies entirely in the half-plane\nabove the horizontal line where the bridge ended, and this can be considered as another\nin\fnite length upper half-plane SAW, which can then be written as the concatenation of an\nirreducible bridge with another in\fnite length half-plane SAW.\nGiven any two SAWs !1and!2, we will write !1\b!2to denote the walk obtained by\nconcatenating !1and!2. Then the above result can be stated as follows: With PH;1-\nprobability 1, any !2H1can be written as\n!= ~!\b^!; (1.5)\nwhere ~!2I and ^!2H1. Given a \fnite sequence !1;:::;!j2I, letH1(!1;\u0001\u0001\u0001;!k)\ndenote the set of all !2H such that\n!=!1\b:::\b!k\b^!; (1.6)\nwhere ^!2H1. Then we have PH;1(H1(!1;:::;!k)) =\u0016\u0000Pk\n1j!jj. For a good exposition on\nthe bridge decomposition of self-avoiding walk, the reader is referred to [2],[9],[6].\n1.2 Summary of Results\nConsider the set of all in\fnite upper half-plane SAWs on the lattice Z2=Z+iZrooted at 0,\ndenotedH1, under the weak limit of the uniform measure on HN. Given!2H1,!can be\ndecomposed into the concatenation an i.i.d. sequence of irreducible bridges !1;!2;:::2I.\nLetYn=Yn(!) denote the height of the n-th irreducible bridge in the concatenation, that\nisYn(!) = Im(!(j!1\b\u0001\u0001\u0001\b!nj))\u0000Im(!(0)) for!2H1. We conjecture that there exists\n\u001b>0 such that,\n4lim\nn!1Yn\nn\u001b=Y; (1.7)\nwhereYhas the distribution of a stable random variable, and the convergence here is in\ndistribution. That is, what equation (1.7) is really saying is that we are conjecturing that\nthere exists \u001b >0 such that n\u0000\u001bYn(!) converges in distribution to that of a stable random\nvariable as n!1 . Now given an in\fnite upper half-plane SAW !, scale!by 1=Yn(!)\nto produce a curve in the unit strip and then let n!1 . This should give a probability\nmeasure on curves in the unit strip beginning at 0 and ending anywhere along the upper\nboundary of the strip. We refer to this as the \fxed irreducible bridge scaling limit , or\fxed\nirreducible bridge ensemble . It is then natural to look for some relationship between the \fxed\nirreducible bridge scaling limit and chordal SLE 8=3.\nThe simplest relationship would be the following. Take an in\fnite upper half-plane SAW\n!de\fned on the lattice Z2and \fxn2N, some large number. Let ^ !denote!considered\nup to the (random) height Yn(!). Scale ^!by 1=Yn(!), so as to obtain a curve in the\nunit strip. In the limit n!1 , this gives a probability measure on curves in the unit\nstrip. Since these curves can end anywhere along the upper boundary of the unit strip, it\nis necessary to integrate along the upper boundary of the strip against the conjectured exit\ndensity for the scaling limit of SAW in the unit strip using SLE partition functions. Let\n\u001a(x) be the conjectured exit density for SAW in the scaling limit in the unit strip derived\nin [2] and described in Section 3.1. Chordal SLE 8=3gives a probability measure on curves\nin the unit strip starting at the origin and ending at some prescribed point along the upper\nboundary. Thus, it might be reasonable to ask whether the resulting measure is chordal\nSLE 8=3, integrated along the density \u001a(x). In this paper, we argue that this process of\nscaling the walk to obtain a curve in the unit strip gives chordal SLE 8=3integrated over \u001a(x)\nif before taking the limit n!1 , we \frst weight the walks by Yn(!)p, where the power p\nis conjectured to be \u00001=\u001bfor\u001bde\fned according to (1.7), and then take the limit n!1 .\nThe conjectured value of \u001bis\u001b= 4=3 (see A).\n1.3 Scaling limits and SLE partition functions\nIn this section we review some conjectured scaling limits of self-avoiding walk, along with\nSLE partition functions , which we will use in what is to come. One, which we have already\ndiscussed, is the \fxed irreducible bridge ensemble, which is obtained by considering a self-\navoiding walk up to its n-th bridge height under the measure PH;1, scaling by 1 =Yn(!) and\ntakingn!1 .\nThe next two scaling limits we consider are examples of the Schramm-Loewner evolution ,\nintroduced by Oded Schramm in [10]. Let D\u001aCbe a bounded,simply connected domain\n(other than C) and letz;w2@Dbe boundary points and v2Dbe an interior point. Given\n\u000e >0, let [z];[w];[v] denote the lattice points on \u000eZ2which are a minimum distance from\nz;w andv, respectively. One can then consider all SAWs !in\u000eZ2beginning at [ z] and\n5ending at [w], constrained to stay in D. We weight each walk by \u0016\u0000j!j. The total weight of\nall such walks is then\nZ\u000e(D;z;w) =X\n!\u001aD:z!w\u0016\u0000j!j: (1.8)\nWe then de\fne a probability measure on all such walks !inDfrom [z] to [w] by assigning\nprobability \u0016\u0000j!j=Z\u000e(D;z;w) to each such walk. The scaling limit as \u000e!0+ is believed to\nexist and be equal to chordal SLE 8=3inDfromztow. We will denote the chordal SLE 8=3\nmeasure supported on curves \r: [0;t\r]!Dsuch that\r(0;t\r)\u001aD,\r(0) =z,\r(t\r) =wby\nPchordal\nD;z;w . Of particular interest to us will be the chordal SLE 8=3de\fned as above where D\nis the unit strip S:=fz2H: 0<Imz <1g,z= 0, andw=x+i, wherex2R. We will\ndenote this probability measure by Pchordal\nS;0;x+i.\nOne can also consider self-avoiding walks starting at a boundary point [ z] and ending at an\ninterior point [ v]. The resulting scaling limit is thought to be radial SLE 8=3. However, we\nwill not be concerned with radial SLE 8=3in this paper.\nIn the case that D=H,z= 0, andw=1, in order to obtain the scaling limit, one must \frst\n\fnd a way to de\fne in\fnite length SAWs in H. This was done in [6] and is how the measure\nPH;1was originally de\fned. The scaling limit of PH;1as\u000e!0+ is then conjectured to be\nPchordal\nH;0;1. It is worth mentioning, however, that one can also obtain the probability measure\nPH;1by a method that is similar in spirit to the method for obtaining the scaling limit for\nSAW in bounded domains. If we consider the set of all \fnite length SAWs in Hstarting at\n0 and weight each such walk !by\u0016\u0000j!j, then the total weight of all such walks is in\fnite.\nIf, instead, we weight each such !by\f\u0000j!jfor\f > \u0016 , then the total weight is \fnite. The\nlimit as\f!\u0016+ has been shown to exist and to give the same measure on in\fnite half-plane\nSAWs as the weak limit on the uniform measures [2].\nFinally, let us consider how the normalization factor (1.8) depends on the boundary points\nz;w2@D. It is conjectured that there exists a boundary scaling exponent b >0 and a\nfunctionH(@D;z;w ) such that as \u000e!0+,\nZ\u000e(D;z;w )\u0018\u000e2bH(@D;z;w ); (1.9)\nandH(@D;z;w ) is thought to satisfy the following form of conformal covariance. If \b is a\nconformal transformation from DontoD0, with \b(z) =z0, \b(w) =w0, then\nH(@D;z;w ) =j\b0(z)jbj\b0(z)jbH(@D;z0;w0): (1.10)\n[6, 8, 4]. Note that in [6], the boundary scaling exponent is denoted by a, whereas we are\ndenoting it by b.\nRecently, it has been shown that there are lattice e\u000bects which should persist in the scaling\nlimit for general domains D[3]. Therefore, one cannot expect equations (1.9) and (1.10) to\nprovide a full description of the scaling limit for general domains D\u001aC. However, we will\n6be restricting our attention to curves in the domains HandS, for which there are no lattice\ne\u000bects expected to persist in the scaling limit.\nIn section 3 we will use equation (1.10) to derive the predicted exit density for the scaling\nlimit of self-avoiding walks in the unit strip beginning at the origin and ending anywhere\nalong the upper boundary. We will denote the density by \u001a(x), where we are assuming that\neach walk exits the strip at some point x+iwithx2R.\nIn section 2 we state our conjecture about how to obtain chordal SLE 8=3from the \fxed\nirreducible bridge ensemble precisely and provide a heuristic argument. The conjecture\ninvolves the stability parameter, \u001b, de\fned according to (1.7). In order to test this conjecture\n(section 4), we require a de\fnite value for \u001b. We conjecture that \u001b= 4=3. In the Appendix A,\nwe present a heuristic argument, originally due to Tom Kennedy via private communication,\nin support of this.\n2 The conjecture\n2.1 Statement of the conjecture\nIn order to precisely state our conjecture, we \frst recall some notations introduced in section\n1.PH;Ndenotes the probability measure on N-step upper-half plane SAWs beginning at 0\nde\fned on the lattice Z2, and PH;1denotes the probability measure on in\fnite length SAWs\nin the upper half plane beginning at 0 and ending at 1, de\fned on the lattice Z2.Pchordal\nS;0;x+i\ndenotes chordal SLE 8=3measure in the unit strip Son curves beginning at 0 and ending\natx+i, and\u001a(x) denotes the exit density along the upper boundary Im( z) = 1 of the\nscaling limit for SAW in the unit strip S, starting at 0 and ending anywhere along the upper\nboundary. Then the conjecture can be stated as follows:\nConjecture 2.1. The \fxed irreducible bridge scaling limit of the SAW and chordal SLE 8=3\nin the unit strip Sare related by\nlim\nn!1EH;1\u0002\nYn(!)\u00001=\u001b1 (^!=Yn(!)2E)\u0003\nEH;1[Yn(!)\u00001=\u001b]=Z1\n\u00001dx\u001a(x)Pchordal\nS;0;x+i(E); (2.1)\nwhereEis an event of simple curves in the unit strip Sbeginning at 0and ending anywhere\nalong the upper boundary of the strip, !is an in\fnite upper half-plane SAW, and ^!is the\ncurve!considered up to the time it reaches height Yn(!), thenth bridge height of !.\nSo we can generate chordal SLE 8=3in the unit strip by generating an N-step SAW !for very\nlarge values of N, considered up to height Yn(!) for large values of n, scaled by 1 =Yn(!),\nand then giving it the weight Yn(!)\u00001=\u001b. The conjectured value of \u001bis 4/3.\n72.2 The derivation\nIn order to derive Conjecture 2.1, we \fx two heights y1andy2, which we think of as order\n1, and a large real number L > 0. We will then only consider curves which have a bridge\npoint in the region A=fz2H:y1L\u0014Im(z)\u0014y2Lg. LetIn=I\u0002\u0001\u0001\u0001\u0002I (ntimes) be\nthe set of all !2H1such that!=!1\b\u0001\u0001\u0001\b!n, with!1;:::;!n2I, i.e. the set of all\nconcatenations of nirreducible bridges beginning at the origin. Recall that if ^ !2Inand\nH1(^!) denotes the set of all !2H1such that!= ^!\b~!with ~!2H1, then we have\nPH;1(H1(^!)) =\u0016\u0000j^!j:\nTherefore, the total weight of all SAWs in H1with a bridge pont in Ais\nZ(A) =1X\nn=0X\n^!2In\u0016\u0000j^!j1 (Yn(!)2[y1L;y 2L]): (2.2)\nNow letEbe an event of simple curves in the strip Sstarting at 0 and ending anywhere\nalong the upper boundary of the strip. We de\fne the probability of the event Eto be\nN(E;A)=Z(A), where\nN(E;A) =1X\nn=0X\n^!2In\u0016\u0000j^!j1 (Yn(!)2[y1L;y 2L]) 1 (^!=Yn(!)2E): (2.3)\nAccording to the de\fnition of PH;1, we have\nN(E;A) =1X\nn=0PH;1[1 (Yn(!)2[y1L;y 2L]) 1 (^!=Yn(!)2E)]: (2.4)\nSince we have \fxed Lto be a very large number, this forces each term in the above sum to\nbe zero other than those corresponding to very large values of n. Then, according to 1.7,\nif we \fxN2Nlarge enough, N\u0000\u001bYNshould have approximately the same distribution as\nn\u0000\u001bYnfor allnsu\u000eciently large. Therefore, the condition Yn(!)2[y1L;y 2L] can be replaced\nwith the condition (for very large \fxed N)\ny1Ln\u0000\u001bN\u001b\u0014YN(!)\u0014y2Ln\u0000\u001bN\u001b: (2.5)\nFurthermore, since for large values of n, the distribution of ^ !=Yn(!) approaches the distribu-\ntion of a curve pulled from the \fxed irreducible bridge ensemble, the condition ^ !=Yn(!)2E\ncan be replaced with the condition ^ !=YN(!)2E. This, along with (2.4) and (2.5) lead to\nN(E;A)\u00191X\nn=0EH;1\"\n1 \u0012\nN\u001by1L\nYN(!)\u00131=\u001b\n\u0014n\u0014\u0012\nN\u001by2L\nYN(!)\u00131=\u001b!\n1 (^!=YN(!)2E)#\n:\n(2.6)\n8Now we move the sum on ninside the expectation and consider\n1X\nn=01 \u0012\nN\u001by1L\nYN(!)\u00131=\u001b\n\u0014n\u0014\u0012\nN\u001by2L\nYN(!)\u00131=\u001b!\n:\nThis sum is easy to approximate. We have\n1X\nn=01 \u0012\nN\u001by1L\nYn(!)\u0013\u00001=\u001b\n\u0014n\u0014\u0012\nN\u001by2L\nYn(!)\u0013\u00001=\u001b!\n\u0019L1=\u001bYN(!)\u00001=\u001bN(y2\u0000y1)\n=cNL1=\u001bYn(!)\u00001=\u001b;\nwherec=y2\u0000y1. The factor of cNL1=\u001bwill cancel out of both numerator and denominator,\nand what we are left with is\nlim\nN!1N(E;A)\nZ(A)= lim\nn!1EH;1\u0002\nYn(!)\u00001=\u001b1 (^!=Yn(!)2E)\u0003\nEH;1[Yn(!)\u00001=\u001b]: (2.7)\nNext, we decompose the sum by the value of the bridge heights. Given a SAW !2H1, let\nD=D(!) be the set of bridge heights. That is, D(!) is the set of all y\u00150 such that there\nexistsn= 0;1;:::such thatYn(!) =y. Then we have\nN(E;A) =1X\nn=0X\n^!2In\u0016\u0000j^!j1 (^!=Yn(!)2E) 1 (Yn(!)2[y1L;y 2L])\n=X\ny2Z\\[y1L;y2L]1X\nn=0X\n^!2In\u0016\u0000j^!j1(^!=y2E)1(Yn=y)\n=X\ny2Z\\[y1L;y2L]PH;1(^!=y2E;y2D)\n=X\ny2Z\\[y1L;y2L]PH;1(^!=y2Ejy2D)PH;1(y2D):\nSimilarly, we \fnd that\nZ(A) =X\ny2Z\\[y1L;y2L]PH;1(y2D):\nIn [2], it was shown that conditioning on the event that a SAW !2H1has a bridge height\natyand considering the walk up to height ygives the law for self-avoiding walk in the strip\nfz2H: 0<Imz <yg. Therefore, by taking ylarge enough, and scaling the walk by 1 =y,\none should expect to get a distribution approaching that of the distribution of SAW in the\nunit stripSstarting at 0 and ending anywhere along the upper boundary of the strip, in the\nscaling limit. It follows that if we sum PH;1(^!=y2Ejy2D) over ally2Z\\[y1L;y 2L],\nthen take the limit L!1 , which is e\u000bectively the same as taking the limit N!1 in 2.7,\n9N(E;A)=Z(A) should converge to the law for SLE 8=3in the unit strip, starting at 0 and\nending atx+i,x2R, integrated over the density \u001a(x). In other words, we have\nlim\nL!1N(E;A)\nZ(A)= lim\nL!1P\ny2Z\\[y1L;y2L]PH;1(^!=y2Ejy2D)PH;1(y2D)P\ny2Z\\[y1L;y2L]PH;1(y2D)(2.8)\n\u0019lim\nL!1cPH;1(^!=L2EjL2D)PH;1(L2D)\ncPH;1(L2D)(2.9)\n=Z1\n\u00001dx\u001a(x)Pchordal\nS;0;x+i(E): (2.10)\nThis completes the derivation of conjecture 2.1.\n3 SLE predictions of random variables\n3.1 The density function \u001a(x)\nIn this section we will use SLE partition functions to derive a conjecture for the exit density,\n\u001a(x), for the scaling limit of self-avoiding walk de\fned on the unit strip, along the upper\nboundary. Recall that for a simply connected domain Dand points z;w2@D, the SLE\npartition function H(D;z;w ) satis\fes the conformal covariance property (1.10). It was\npredicted in [6] that the boundary scaling exponent for SAW is b= 5=8. Using this value, the\nconformal covariance property takes the following form: If \b is any conformal transformation,\nthen\nH(D;z;w ) =j\b0(z)\b0(w)j5=8H(\b(D);\b(z);\b(w)): (3.1)\nThis de\fnes H(D;z;w ) up to specifying it for a particular choice of domain Dand boundary\npointszandw. The convention we follow is of taking H(H;0;1) = 1.\nFirst note that if we take \b to be a dilation \b( z) =xz, forx2Rnf0g, then by (3.1), we\nhave\nH(H;0;x) =\u00121\nx2\u00135=8\n=1\nx5=4: (3.2)\nTherefore we can calculate H(S;0;x+i) by considering the conformal map f:S!Hsuch\nthatf(0) = 0,f(x+i) =\u0000e\u0019x\u00001, given by f(z) =e\u0019z\u00001. We havejf0(0)j=\u0019and\njf0(x+i)j=\u0019e\u0019x. Thus,\nH(S;0;x+i) =jf0(0)j5=8jf0(x+i)j5=8H(H;0;\u0000e\u0019x\u00001)\n=\u0014\u00192e\u0019x\n(1 +e\u0019x)2\u00155=8\n=\u0014\u00192\ncosh2(\u0019x=2)\u00155=8\n:\n10According to (1.9), this shows that the probability density function \u001a(x) should be given by\n\u001a(x) =ch\ncosh\u0010\u0019x\n2\u0011i\u00005=4\n; (3.3)\nwherecis a normalization constant.\n3.2 The right-most excursion\nGiven a SAW !de\fned in the unit strip Swith lattice spacing \u000e, letX(!) = max jRe!(j)\ndenote the rightmost excursion of!, i.e. the right-most point on the SAW in the strip. Based\non the results of Section 3.1, we conjecture that, in the scaling limit, Xhas distribution given\nby\nlim\n\u000e!0+P(X <\u0018 ) =Z1\n\u00001Pchordal\nS;0;x+i\u0010\nmax\ntRe\r(t)<\u0018\u0011\n\u001a(x) dx; (3.4)\nwhere\r(t) is an SLE 8=3curve in the unit strip, starting at 0 and ending at x+i, and\u001a(x)\nis given by (3.3). To calculate Pchordal\nS;0;x+i(maxtRe(\r(t))<\u0018), we use the following form of\nconformal invariance : IfDis a simply connected domain, z;w2@D, andf:D!D0is a\nconformal transformation,\nPchordal\nD;z;w (\r[0;1)\\A=;) =Pchordal\nD0;f(z);f(w)(~\r[0;1)\\f(A) =;); (3.5)\nwhere Pchordal\nD;z;w denotes chordal SLE 8=3measure inD, starting at zand ending at w,Pchordal\nD0;f(z);f(w)\ndenotes chordal SLE 8=3measure in D0, starting at f(z) and ending at f(w), andAis a closed\nset such that A\u001aD,z;w =2A,A\\D\u001a@DandDnAis simply connected.\nConformal invariance is built into the de\fnition of every SLE\u0014measure. However, SLE 8=3\nmeasure also satis\fes the following restriction property: If Dis a simply connected domain,\nz;w2@D, andD0\u001aDis another simply connected domain with z;w2@D0, then\nPchordal\nD;z;w (\r[0;1)\\A=;j\r[0;1)\u001aD0) =Pchordal\nD0;z;w(~\r[0;1)\\A=;); (3.6)\nwhereAis as in (3.5), along with the assumption that A\u001aD0. In [7], [5], it is shown that for\nany probability measure on a certain type of random subsets in the plain called restriction\nhulls, which satisfy (3.5), (3.6), the probability in (3.5) can be computed by\nPchordal\nH;0;1(K\\A=;) = \b0\nA(0)\u000b; (3.7)\nfor some real number \u000b, whereKis a restriction hull and \b Ais the unique conformal\ntransformation mapping HnAontoHwith \bA(0) = 0, \b A(1) =1, and \bA(z) =z+o(1)\nasz!1 .\nIn the case of SLE 8=3, it is known that \u000b= 5=8, and therefore we have\nPchordal\nH;0;1(\r[0;1)\\A=;) = \b0\nA(0)5=8: (3.8)\n11It is well known that the map f(z) =e\u0019xde\fnes a conformal transformation from the unit\nstrip to the half-plane Hsatisfyingf(0) = 1,f(x+i) =\u0000e\u0019x. Therefore, the map\n\tx(z) =e\u0019z\u00001\ne\u0019z+e\u0019x(3.9)\nde\fnes a conformal transformation from SontoHwith \tx(0) = 0 and \t x(x+i) =1. It\nfollows then from (3.5) that if x<\u0018 ,\nPchordal\nS;0;x+i\u0010\nmax\ntRe(\r(t))<\u0018\u0011\n=Pchordal\nS;0;x+i(\r[0;1)\\fz2S: Re(z)\u0015\u0018g=;)\n=Pchordal\nH;0;1(~\r[0;1)\\\tx(fz2S: Re(z)\u0015\u0018g) =;):\nLetA= \tx(fz2S: Re(z)\u0015\u0018g). Then we can write A=fz2H:jz\u0000c(x;\u0018)j\u0014a(x;\u0018)g,\nwhere\nc(x;\u0018) =1\n2\u0012e\u0019\u0018+ 1\ne\u0019\u0018\u0000e\u0019x+e\u0019\u0018\u00001\ne\u0019\u0018+e\u0019x\u0013\na(x;\u0018) =1\n2\u0012e\u0019\u0018+ 1\ne\u0019\u0018\u0000e\u0019x\u0000e\u0019\u0018\u00001\ne\u0019\u0018+e\u0019x\u0013\n:\nIn this case we can write down \b Aexplicitly. We have\n\bA(z) = (z\u0000c(x;\u0018)) +a(x;\u0018)2\nz\u0000c(x;\u0018): (3.10)\nEvaluating the derivative of (3.10) at 0 and using (3.8), we \fnd that\nPchordal\nS;0;x+i\u0010\nmax\ntRe(\r(t))<\u0018\u0011\n= \b0\nA(0)5=8\n=\"\n1\u0000\u0012a(x;\u0018)\nc(x;\u0018)\u00132#5=8\n:\nTherefore, we can calculate the distribution of the right most excursion of SAW in the strip\nin the scaling limit by\nlim\n\u000e!0+P(X <\u0018 ) =Z\u0018\n\u00001\"\n1\u0000\u0012a(x;\u0018)\nc(x;\u0018)\u00132#5=8\n\u001a(x) dx;\nwhere\u001a(x) is given by (3.3). Thus, by our Conjecture, 2.1, we should have\nlim\nn!1E\u000e\nH;1\u0002\nYn(!)\u00001=\u001b1(maxjRe(!(j))=Yn(!)<\u0018)\u0003\nE\u000e\nH;1[Yn(!)\u00001=\u001b]=Z\u0018\n\u00001\"\n1\u0000\u0012a(x;\u0018)\nc(x;\u0018)\u00132#5=8\n\u001a(x) dx:\n(3.11)\n124 Simulations\nThe pivot algorithm provides us with a fast chain Monte Carlo algorithm for simulating the\nself-avoiding walk in the full plane or the half-plane. It has also recently been shown in [2]\nthat the pivot algorithm can be used to simulate self-avoiding walks in the strip S. Taking\nlattice e\u000bects into account (see [3]), it should also be possible to simulate the self-avoiding\nwalk in other domains using the pivot algorithm. Recently, Nathan Clisby has developed a\nvery fast implementation of the pivot algorithm, [1], and that is the algorithm that we use\nfor our simulations.\nWe use the pivot algorithm to generate self-avoiding walks in the half-plane with number\nof stepsN= 1 million. Each iteration of the algorithm is highly correlated, so there is no\npoint in sampling each iteration. Instead, we sample every 100 iterations. In this way, we\ngenerated 144 million samples.\nWe \frst test the conjectured density \u001a(x) given by (3.3) against the sampled data. We take\nn= 100, sample self-avoiding walks in the half-plane, considering them up to their 100th\nbridge point, and then scale them by 1 =Ynto get a curve in the unit strip. To test the\nexit density of these curves against \u001a(x), we split the interval [ \u00003;3] into 600 equal parts of\nlength dx= 0:01. We then plot a histogram by summing the weights Y\u00001=\u001b\nn for each curve !\nsampled which satis\fes x\u0014Re(!(s)=Yn)<x+ dx, divided by the sum of the weights Y\u00001=\u001b\nn\nfor every curve sampled. Here we are using sto denote the time at which !reaches height\nYn. We have also plotted a histogram of the exit density of the curves in the strip obtained\nfrom our samples by normalizing by the number of samples generated instead of the sum of\nthe weights Y\u00001=\u001b\nn in order to show that we do not get the conjectured exit density \u001a.\nNext we test the conjecture by making a prediction for the scaling exponent bin (1.9) and\n(1.10). We do this by plotting the log of EN[Y\u00001=\u001b\nn1(x\u0014Re(!(s)=Yn)< x+ dx)] versus\nthe log of cosh\u00002(\u0019(x+ dx=2)=2). We take evenly spaced values of the interval [ \u00001:90;1:90]\nwith spacing d x= 0:01. By Conjecture 2.1, we should have\nlog\u0000\nEN[Y\u00001=\u001b\nn1(x\u0014Re(!(s)=Yn)<x+ dx)]\u0001\n=blog\u0000\ncosh\u00002(\u0019(x+ dx=2)=2)\u0001\n+ const:\n(4.1)\nTherefore, the data points should lie on a line. The slope of the line should be b, which is\nconjectured to be 5 =8 = 0:625.\nThe line shown in Figure 3 was calculated using unweighted least-squares. No attempt was\nmade to \fnd an error estimate. The slope of the least-squares \ft is 0.625303. If we let b\ndenote the slope of our least-squares \ft, then comparing bwith the conjectured value, we\nhave\nb\u00005=8 = 0:000303: (4.2)\nWe have also plotted a log-log graph of the expected value of 1( x\u0014Re(!(s)=Yn)< x+\ndx) versus cosh\u00002(\u0019(x+ dx=2)=2 by calculating the expected value through the number of\nsamples as opposed to summing the weights Y\u00001=\u001b\nn. This should be compared to Figure 3.\n13-3 -2 -1 1 2 30.10.20.30.40.5Figure 1: Histogram of exit points along the upper boundary of the strip for the \fxed\nirreducible bridge ensemble. The conjectured density \u001a(x) is represented by the solid curve,\nwhile the histogram is represented by the data points.\nThe slope of the least-squares \ft in this case is 0.444367.\nNext, we perform numerical tests on the rightmost excursion, which we are denoting by\nX. After generating our self-avoiding walks in the half-plane with N= 1 million steps,\nand considering them up to their 100th bridge point (i.e. taking n= 100), we scale the\nwalks by 1=Ynand weight the probability measure by Y\u00001=\u001b\nn. We denote the probability\nobtained in this manner by PN;n. Of course this depends on the number of steps in the\nwalk, as well as the value of n. But for large enough values of N;n, this measure should\nlook very close to the \fxed irreducible bridge measure. Conjecture 2.1 then states that\nlimn!1limN!1PN;n=Pchordal\nS;0;x+i, integrated against \u001a(x). Given\u0018\u00150, by equation (3.11),\nwe should have (approximately)\nPN;n(X <\u0018 ) =Z\u0018\n\u00001\"\n1\u0000\u0012a(x;\u0018)\nc(x;\u0018)\u00132#5=8\n\u001a(x) dx: (4.3)\nWe use numerical integration to calculate the right hand side of (4.3). Figure 5 shows a plot\nof the cumulative distribution function for Xunder the measure PN;nobtained from our\nsimulations, along with the conjectured cumulative distribution function for Xgiven by the\nright hand side of (4.3), for values of \u0018between 0 and 5. In the scale of the \fgure, the two\ncurves look almost identical. In Figure 6, we plot the di\u000berence between the simulated cdf\nforXand the conjectured cdf for X.\n14-3 -2 -1 1 2 30.10.20.30.40.5Figure 2: Histogram of exit points along the upper boundary of the strip obtained by\nnormalizing by the number of samples generated, as opposed to normalizing by the sum\nof the weights Y\u00001=\u001b\nn. Once again \u001a(x) is represented by the solid curve.\n-5 -4 -3 -2 -1-12.0-11.5-11.0-10.5-10.0-9.5\nFigure 3: log-log plot of EN[Y\u00001=\u001b\nn1(x\u0014Re(!(s)=Yn)< x + dx)] versus cosh\u00002(\u0019(x+\ndx=2)=2). The slope of the least-squares \ft is 0.625303.\n15-5 -4 -3 -2 -1-7.0-6.5-6.0-5.5Figure 4: log-log plot of EN[1(x\u0014Re(!(s)=Yn)<x+ dx)] versus cosh\u00002(\u0019(x+ dx=2)=2).\nWithout taking the weights Y\u00001=\u001b\nn into account, the slope of the least-squares \ft is 0.444367.\nA Appendix: Argument that \u001b= 4=3\nHere we give an argument, originally due to Tom Kennedy via private communication, that\nthe value of \u001bde\fned by (1.7) is 4/3. Let Bh(z) denote the generating function for bridges\nstarting at the 0 with h. That is, if !2Bn, we seth(!) = Im(!(n)) and\nBh(z) =X\n!2Bzj!j1(h(!) =h):\nIn [6], Lawler Schramm and Werner conjecture that we should have Bh(z)\u0010h\u00001=4, where\nhere\u0010means that the ratio of both sides are bounded away from 0 and 1. The argument\ngoes as follows: If we constrain a bridge to have an endpoint at a \fxed point of height\nh, the decay goes like h\u00002b, whereb= 5=8. The number of endpoints that contribute to\nBh(z) is of order h. We begin by running an i.i.d. sequence of irreducible bridges until the\nconcatenation of which has height which strictly exceeds some number L>0. This happens\nwith probability 1, and therefore we have\n16Figure 5: Plot of the conjectured cdf for the rightmost excursion of SAW in the strip in\nthe scaling limit as \u000e!0+ and the simulated rightmost excursion for SAW in the \fxed\nirreducible bridge ensemble. The conjectured cdf is colored in red, while the simulated cdf\nis colored in green. In the scale of the image, it is di\u000ecult to see the di\u000berence.\n1 =1X\nn=1X\n!1;:::;!n2I\u0016\u0000Pn\nj=1j!jj1 nX\nj=1h(!j)>L!\n1 n\u00001X\nj=1h(!j)\u0014L!\n=LX\nh=01X\nn=1X\n!1;:::;!n2I\u0016\u0000Pn\nj=1j!jj1 n\u00001X\nj=1h(!j) =h!\n1 nX\nj=1h(!j)>L!\n=LX\nh=11X\nn=1X\n!1;:::;!n\u000012I\u0016\u0000Pn\u00001\nj=1j!jj1 n\u00001X\nj=1h(!j) =h!X\n!n2I\u0016\u0000j!nj1 (h(!n) +h>L )\n=LX\nh=0Bh(\u0016\u00001)X\n!2I\u0016\u0000j!j1(h(!)>L\u0000h):\nNow we use\n17Figure 6: Plot of the di\u000berence in values for the conjectured cdf for the rightmost excursion\nand the simulated cdf for the rightmost excursion. We subtracted the simulated values for\nthe cdf from the conjectured values. The error is small, but one can see that the error is\nlarger near x= 0. This is because there is a systematic error present due to the fact that\nwe have chosen a \fnite value for n. There is a slight bias for bridge points falling closer to\nx= 0 with a \fnite n. Choosing values of nmuch higher than n= 100 also creates some\nissues, since the number of one-million step SAWs with the given number of bridge points is\ndrastically reduced for larger values of n.\nX\n!2I\u0016\u0000j!j1(h(!)>L\u0000h) =P(h(!)>L\u0000h); (A.1)\nwhere we are using Pto denote the probability measure on Ide\fned by P(!) =\u0016\u0000j!j. We\nwould like to develop a relationship between Bh(z), and the cumulative distribution function\nfor the height of an irreducible bridge. Using (A.1), we have\n1 =LX\nh=0Bh(\u0016\u00001)P(h(!)>L\u0000h): (A.2)\nLet us now assume that Bh(\u0016\u00001)\u0010h\u00001=4andP(h(!)> h)\u0010h\u0000pfor some power p. We\n18will split the above identity into two sums: one from 0 to L=2\u00001, and one from L=2 toL.\nIn the \frst sum, L\u0000h=2 is at least L=2, and so P(h(!)> L\u0000h) is (up to multiplicative\nconstants)L\u0000p. So the \frst sum behaves like\nL=2\u00001X\nh=0h\u00001=4L\u0000p\u0010L\u0000p+3=4:\nIn the second sum, h\u0015L=2 and soBh(\u0016\u00001) is (up to multiplicative constants) L\u00001=4. So\nthe second sum behaves like\nLX\nL=2L\u00001=4(L\u0000h)\u0000p=L=2X\n0L\u00001=4h\u0000p\u0010L3=4\u0000p;\nso both sums behave like L3=4\u0000p. AsL!1 , the identity says that this cannot diverge or\ngo to zero, and so we should have p= 3=4.\nIn conclusion, P(h(!)>h) decays like\nP(h(!)>h)\u0010h\u00003=4:\nThis tells us which stable process the sum of nirreducible bridges converges to in distribution.\nLetYndenote the n-th bridge height. We want to \fnd \u001bso thatYngrows liken\u001b. The cdf\nF(h) of the irreducible bridge heights converges to 1 like 1 \u0000h\u00003=4ash!1 . If there are\nnirreducible bridges, the larges one will roughly have height h, soF(h)\u00191\u00001=n. Thus\nh\u0018n4=3, i.e.\u001b= 4=3.\nReferences\n[1]Clisby, N. E\u000ecient implementation of the pivot algorithm for self-avoiding walks.\nJournal of Statistical Physics 140 , 2 (2010), 349{392.\n[2]Dyhr, B., Gilbert, M., Kennedy, T., Lawler, G., and Passon, S. The self-\navoiding walk spanning a strip. Journal of Statistical Physics 144 , 1 (2011), 1{22.\n[3]Kennedy, T., and Lawler, G. F. Lattice e\u000bects in the scaling limit of the two-\ndimensional self-avoiding walk. arXiv preprint arXiv:1109.3091 (2011).\n[4]Lawler, G. Schramm-loewner evolution (SLE), statistical mechanics, 231 295.\nIAS/Park City Math. Ser 16 .\n[5]Lawler, G. Conformally invariant processes in the plane , vol. 114. Amer Mathematical\nSociety, 2008.\n19[6]Lawler, G., Schramm, O., and Werner, W. On the scaling limit of planar self-\navoiding walk, fractal geometry and applications: a Jubilee of Benoit Mandelbrot, part\n2, 339{364. In Proc. Sympos. Pure Math (2002), vol. 72.\n[7]Lawler, G., Schramm, O., and Werner, W. Conformal restriction: the chordal\ncase. Journal of the American Mathematical Society 16 , 4 (2003), 917{956.\n[8]Lawler, G. F. Partition functions, loop measure, and versions of SLE. Journal of\nStatistical Physics 134 , 5 (2009), 813{837.\n[9]Madras, N., and Slade, G. The self-avoiding walk . Birkh auser, 1993.\n[10]Schramm, O. Scaling limits of loop-erased random walks and uniform spanning trees.\nIsrael Journal of Mathematics 118 , 1 (2000), 221{288.\n20"    },    {        "title": "1411.2857v1.Capturing_of_a_Magnetic_Skyrmion_with_a_Hole.pdf",        "content": "Capturing of a Magnetic Skyrmion with a Hole\nJan M uller1\u0003and Achim Rosch1\n1Institut f ur Theoretische Physik, Universit at zu K oln, D-50937 Cologne, Germany\n(Dated: October 5, 2018)\nMagnetic whirls in chiral magnets, so-called skyrmions, can be manipulated by ultrasmall current\ndensities. Here we study both analytically and numerically the interactions of a single skyrmion in\ntwo dimensions with a small hole in the magnetic layer. Results from micromagnetic simulations are\nin good agreement with e\u000bective equations of motion obtained from a generalization of the Thiele\napproach. Skyrmion-defect interactions are described by an e\u000bective potential with both repulsive\nand attractive components. For small current densities a previously pinned skyrmion stays pinned\nwhereas an unpinned skyrmion moves around the impurities and never gets captured. For higher\ncurrent densities, jc1< j < jc2, however, single holes are able to capture moving skyrmions. The\nmaximal cross section is proportional to the skyrmion radius and top\u000b, where\u000bis the Gilbert\ndamping. For j > jc2all skyrmions are depinned. Small changes of the magnetic \feld strongly\nchange the pinning properties, one can even reach a regime without pinning, jc2= 0. We also show\nthat a small density of holes can e\u000bectively accelerate the motion of the skyrmion and introduce a\nHall e\u000bect for the skyrmion.\nPACS numbers: 12.39.Dc,75.76.+j,74.25.Wx,75.75.-c\nI. INTRODUCTION\nTopologically stable magnetic whirls, so-called\nskyrmions, have recently gained a lot of attention both\ndue to their interesting physical properties and their\npotential for applications. A single skyrmion is shown\nin Fig. 1. A skyrmion is a smooth magnetic con\fgu-\nration where the spin direction winds once around the\nunitsphere. This implies that the spin con\fguration is\ntopological protected and can unwind only by creating\nsingular spin con\fgurations1,2. In bulk chiral magnets,\nlattices of skyrmions are stabilized by Dzyaloshinskii-\nMoriya interactions and thermal \ructuations in a\nsmall temperature and \feld regime3. In \flms of chiral\nmagnets they occur as a stable phase4in a wide range\nof temperatures in the presence of a stabilizing \feld.\nSingle skyrmions are metastable in an even broader\nregime of parameters4. They have been observed in a\nwide range of materials, including insulators5, doped\nsemiconductors4,6and metals3,7,8, with sizes ranging\nfrom a few nanometers up to micrometers and from\ncryogenic temperatures almost up to room temperature8.\nIn bilayer PdFe \flms on Ir substrates, single nanoscale\nskyrmions have been written using the current through\na magnetic tip9.\nDue to their e\u000ecient coupling to electrons by Berry\nphases and the smoothness of the magnetic texture,\nskyrmions can be manipulated by extremely small elec-\ntric current densities10{14. Therefore the potential exists\nto realize new types of memory or logic devices based\non skyrmions2,15. Several studies have therefore investi-\ngated the dynamics of skyrmions in nanostructures and\ntheir creation at defects15,16.\nIn this paper, we investigate how a single defect a\u000bects\nthe dynamics of a single skyrmion in a magnetic \flm, see\ne.g. Fig. 1. As an example of a defect we consider a\nvacancy, i.e., a single missing spin, or more general a\nFIG. 1. Snapshot of a micromagnetic simulation of skyrmion\ndriven by a current ( D= 0:3J=a,\u0016B= 0:09J=a2,vs=\n0:001aJex, and\u000b=\f= 0:4) in the presence of a single\nvacancy: a missing spin (grey sphere). The trajectory of\nthe skyrmion center is indicated by a red line.\nhole in the magnetic \flm with radius small compared to\nthe skyrmion radius. This problem is of interest for at\nleast two reasons. First, this is perhaps the most simple\nexample of a nanostructure which can interact with the\nskyrmion. As we will show, one can use such defects to\ncontrol the capturing and release of skyrmions via the\nmagnetic \feld and the applied current density. Second,\ndefects are always present in real materials. As long as\nthe typical distance of defects is small compared to the\nskyrmion radius, the e\u000bects of a \fnite density of defects\ncan be computed from the solution of the single-defect\nproblem. The in\ruence of a \fnite defect density on a\nlattice of skyrmions has been studied using micromag-\nnetic simulations by Iwasaki, Mochizuki and Nagaosa,\nRef. 12. Interestingly, they observed in their simulations\nthat skyrmions move e\u000eciently around defects. While a\ndi\u000berent type of defect (enhanced easy axis anisotropy)\nwas considered by them, a similar phenomenon will also\nbe of importance for our study.\nBesides the use of micromagnetic simulations, the main\ntheoretical tool will be the analysis of e\u000bective equations\nof motion for the center of the skyrmion. Thiele17pi-arXiv:1411.2857v1  [cond-mat.str-el]  11 Nov 20142\noneered the approach to project the e\u000bective equations\nof motion on the translational mode of a magnetic tex-\nture. This approach has also been successfully applied\nto skyrmions and skyrmion lattices12,16,18,19. Here, we\ncombine this approach with microscopic evaluations of an\ne\u000bective potential describing the defect-skyrmion interac-\ntion. The resulting e\u000bective equation of motion for the\nskyrmion accurately reproduces the results of the micro-\nmagnetic simulation and allows for an analytical analysis\nof the skyrmion dynamics.\nIn the following, we will \frst introduce the model, de-\nrive the e\u000bective potential and resulting equations of mo-\ntion, and use them to investigate how skyrmions are cap-\ntured, released and de\rected by a single defect. Finally,\nwe study the e\u000bects of a \fnite, but low density of defects.\nII. THE MODEL\nTo describe the magnetization of the system we use\nclassical Heisenberg spins M(r) with uniform length\nkM(r)k= 1 on a square lattice. The corresponding free\nenergy functional in the continuum reads\nF[M]=Z\nd2rJ\n2[rM(r)]2+DM(r)\u0001[r\u0002M(r)]\u0000\u0016B\u0001M(r),\n(1)\nincluding the ferromagnetic coupling J, Dzyaloshinskii-\nMoriya interaction Dand the Zeeman interaction with\nthe magnetic \feld B= (0;0;B).\u0016is the magnetiza-\ntion per area. For a single spin 1 =2 per square unit cell\nwith lattice constant a0andgfactorg= 2 one has, for\nexample,\u0016=\u0016B=a2\n0.\nOn a square lattice we use the following discretized\nversion\nF[M] =\u0000JX\nrMr\u0001\u0000\nMr+aex+Mr+aey\u0001\n\u0000DaX\nr\u0000\nMr\u0002Mr+aex\u0001ex+Mr\u0002Mr+aey\u0001ey\u0001\n\u0000B\u0016a2\u0001X\nrMr, (2)\nwhere exandeyare unit vectors in the xandydirection,\nrespectively. The lattice constant aand the interaction\nstrengthJare set to 1 in the following. If not otherwise\nstated, we use D= 0:3J=aand\u0016B= 0:09J=a2. For these\nparameters the ground state is ferromagnetic. Hence the\nsingle skyrmion is a topologically protected, metastable\nexcitation. A vacancy at position Rdis created by setting\nthe magnetization Mat this site to zero.\nThe micromagnetic dynamics of each spin in the pres-\nence of an electric current density jare described by\nthe Landau-Lifshitz-Gilbert (LLG) equation20{22. In the\ncontinuum case the LLG equation reads\n[@t+(vs\u0001r)]M=\u0000\rM\u0002Be\u000b\n+\u000bM\u0002\u0014\n@tM+\f\n\u000b(vs\u0001r)M\u0015\n, (3)wherevsis the drift velocity of spin currents which\nis directly proportional to the current density jand\n\r=g\u0016B=~is the gyromagnetic ratio. Note that we set\nvs;\u000band\fto a constant value, not taking into account\nthat depending on the microscopic realization of the de-\nfect, they might be modi\fed in proximity of the defect.\nAt least for defects small compared to the skyrmion ra-\ndius and su\u000eciently small currents, this approximation\nis justi\fed as the forces on the skyrmions add up from\nall parts of the skyrmion (see below). The (very weak)\ne\u000bects of changes to the current pattern around a notch\nin a nanowire have been studied in Ref. 16. The e\u000bec-\ntive magnetic \feld is given by Be\u000b=\u0000\u000eF[M]\n\u0016\u000eM.\u000band\f\nare phenomenological damping terms. Note that \u000b=\f\nis a special point as in this case the magnetic texture\ndrifts with the current as long as no defects are present,\nM(r;t) =M(r\u0000vst). In our lattice model we rewrite\nEq. (3) using @iM(r) =1\n2a(Mr+aei\u0000Mr\u0000aei).\nIII. EFFECTIVE DYNAMICS OF SKYRMIONS\nA. Generalized Thiele approach\nThe LLG equation describes the movement of every\nmagnetic moment in the system. As we do not want to\ndescribe every spin but the movement of the skyrmion\ncenter, which is a collective movement of spins, we ap-\nply the so-called Thiele approach17. Originally, this ap-\nproach is based on the approximation that the skyrmion\nis a completely rigid object. While this approximation\nfails in the presence of a local defect, we will show that\none can nevertheless use this approach if one performs a\nmicroscopic calculation of the potential V(r) describing\nthe forces between skyrmion and defect.\nOur goal is to derive an equation of motion for the\ncenter Rof the skyrmion ( Ris de\fned below), which\ntakes into account deformations of the skyrmion. If the\nmotion of the skyrmion is su\u000eciently slow, we expect\nthat for each \fxed Rthe skyrmion con\fguration is in a\nlocal minimum of the energy. We therefore approximate\nM(r;t)\u0019M0(R(t)\u0000Rd;r\u0000R(t) ). (4)\nThe magnetic con\fguration M0depends on the distance\nof skyrmion center R(t) and defect position Rdand is\ndetermined from the condition that\nV(R\u0000Rd) =F[M0(R\u0000Rd;r\u0000R)]\u0000F0\n= min\nR\u0000Rd\fxedF[M(r)]\u0000F0 (5)\nis at a local minimum for \fxed distance of skyrmion and\ndefect, R\u0000Rd.V(R\u0000Rd) is the e\u000bective potential\ndescribing the skyrmion-defect interaction. The o\u000bset\nF0is chosen such that V(R!1 ) = 0. Note that the\nstandard Thiele approach neglects the deformation of the\nskyrmion, i.e., the dependence of MonR\u0000Rd.\nTo calculate M0andF[M0] numerically, we have used\ntwo di\u000berent methods. In the case that Ris located3\n0510152025-0.4-0.3-0.2-0.10.00.1\ndistance rapotential VHrLJ\n05101520-0.050.000.05\nFIG. 2. Potential V(R\u0000Rd) of the skyrmion-hole inter-\naction as a function of distance shown for D= 0:3J=a\nand various magnetic \felds from \u0016B= 0:05J=a2(red) to\n\u0016B= 0:12J=a2(blue) in steps of \u0001 \u0016B= 0:01J=a2. Inset:\nRaw data used to calculate the smoothened potential shown\nin the main \fgure. The dark red (light green) data has been\nobtained for \u0016B= 0:09J=a2using the \frst and second algo-\nrithm described in the text. The spread in each curve arises\nas on the square lattice the potential does not only depend\non the distance from the defect but also has a tiny angular\ndependence. For comparison, we also show the estimate for\nthe potential which is obtained when deformations of the\nskyrmion are ignored (dashed line).\non one of the lattice sites, we \fx the position Rof\nthe skyrmion by setting the magnetization at r=R\nto (0;0;\u00001), opposite to the ferromagnetic background.\nThis approach is similar to the method used in Ref. 14\nto numerically calculate the potential of the skyrmion-\nskyrmion interaction. In a second approach, we \frst\ncompute the skyrmion con\fguration Mc(r\u0000R) in the\nclean system without a defect. To determine the en-\nergy minimum in the presence of the defect for \fxed R,\nwe minimize F[M(r)] with the boundary condition that\nM(r) =Mc(r\u0000R) forjr\u0000Rdj>r0. It turns out that this\nprocedure rapidly converges with r0andr0= 4:5agives\naccurate results in the considered parameter range. The\nresults forV(R\u0000Rd) determined from the two methods\nare almost identical, see inset of Fig. 2.\nIn Fig. 2 the resulting potentials V(jR\u0000Rdj) are\nshown. In the continuum model, Eq. (1), the e\u000bective\npotential depends only on the distance of skyrmion and\ndefect,jR\u0000Rdj, whereas in the lattice there is a small\nangular dependence (raw data is shown in the inset of\nFig. 2). For simplicity, we average over this angular\ndependence. We \ft an exponential law for very large\njR\u0000Rdjand interpolate the curve by a polynomial oth-\nerwise. The shape of the potential not only quantita-\ntively but also qualitatively depends on the strenght of\nthe magnetic \feld, which will be important for the fol-\nlowing discussion.\nTo derive an e\u000bective equation of motion for R(t), we\nproceed as follows17. First, both sides of the LLG equa-\ntion are multiplied by\u0016\n\rM\u0002such that\u0016Be\u000b=\u0000\u000eF[M]\n\u000eMis isolated (using that Be\u000bcan be chosen to be perpen-\ndicular to M). Second, Mis replaced by M0de\fned in\nEq. (4). Third, to project onto the translational mode\nin direction ithe resulting equation is multiplied bydM0\ndRiand integrated over space.\nThe resulting di\u000berential equation for R(t), the gen-\neralized Thiele equation, can be written in the following\nform\n\u0000dV\ndR=GR\u0002\u0010\n_R\u0000vs\u0011\n+\u000eGR\u0001vs\n+DR\u0001\u0010\n\u000b_R\u0000\fvs\u0011\n+\f\u000eDR\u0001vs, (6)\nwhere the potential V, the gyrocoupling GRand the ma-\ntrices\u000eGR,DRand\u000eDRare functions of the distance\nfrom the defect, R\u0000Rd.Vis de\fned in Eq. (5), the\nother terms are determined from\n(GR)i=s\u000fijkZ\nd2r1\n2M0\u0001\u0012dM0\ndRj\u0002dM0\ndRk\u0013\n(7)\n(DR)ij=sZ\nd2rdM0\ndRi\u0001dM0\ndRj(8)\n(\u000eGR)ij=sZ\nd2rM0\u0001\u0012dM0\ndRj\u0002\u0012dM0\ndRi+dM0\ndri\u0013\u0013\n(9)\n(\u000eDR)ij=sZ\nd2rdM0\ndRi\u0001\u0012dM0\ndRj+dM0\ndrj\u0013\n, (10)\nwhere we included the spin density\ns=\u0016\n\r, (11)\ne.g.,s=~=(2a2\n0) for a single spin 1/2 in a unit cell of\nlengtha0. Note that some of the derivatives are with\nrespect to the skyrmion position Rand further that the\ncombinationdM0\ndRi+dM0\ndri=\u0000dM0\ndRd;idescribes the change\nof the skyrmion con\fguration when only the position of\nthe defect changes.\nIf the deformation of the skyrmion (and therefore the\nderivativesd\ndRd;i) are ignored, then the correction terms\n\u000eGRand\u000eDRvanish and one can replaced\ndRiby\u0000d\ndrito\nrecover the Thiele equation in the standard form. Within\nthis approximation, the gyrocoupling GR=Gis in the\ncontinuum limit topologically quantized to a multiple of\n4\u0019M(Mis the magnetization per unit cell set to 1 within\nour conventions). This is, however, notthe case if the\ndependence of M0onR\u0000Rdis taken into account.\nThe most important e\u000bect of the deformation is that\nthey strongly modify the e\u000bective potential V(R\u0000Rd),\nas is shown in the inset of Fig. 2. Taking into account\nthe adjustment of the magnetic texture to the defect is\nimportant as it gives rise to corrections of order 1.\nChanges of the gyrocoupling and dissipative tensor\nare, in general, also of importance when nanostructures\nlead to a signi\fcant deformation of the magnetic texture.\nThey are, however, not important for the situation con-\nsidered in our paper. We study the case where the radius\nadof the defect is much smaller than the radius of the4\nskyrmion,as. In this case the deformations a\u000bect only a\nsmall part of the skyrmion and give therefore only small\ncorrections of order ( ad=as)2\u001c1 on the right-hand side\nof the generalized Thiele equation (6). This is shown in\nFig. 3, wherejGRj,Dr\nRandDt\nRare shown as a func-\ntion of the distance from the defect, jR\u0000Rdj. Here\nDr\nR= ^er\u0001DR\u0001^erandDt\nR= ^e\u001e\u0001DR\u0001^e\u001edescribe the\ndissipative tensor projected on the radial and tangential\ndirection, respectively, with ^ er= (R\u0000Rd)=jR\u0000Rdj\nand ^e\u001e= ^z\u0002^er. Far from the defect, one recovers\nthe results predicted by the standard Thiele approach\nwithDr\nR=Dt\nRandjGRj= 4\u0019in the continuum limit,\nwhereas there are small deviations of a few percent when\nthe distance of the defect is of the order of the skyrmion\nradius. Similarly, the corrections arising from \u000eDRand\n\u000eGRare also small. For the following analysis, we will\ntherefore neglect the modi\fcation of the right-hand side\nof the Thiele equation (6) using\n\u0000dV\ndR=G\u0002\u0010\n_R\u0000vs\u0011\n+D\u0001\u0010\n\u000b_R\u0000\fvs\u0011\n, (12)\nwith space-independent G= lim R!1GRandD=\nlimR!1DRwhile the modi\fed potential is fully taken\ninto account.\nDRr/4π\nDRt/4π\n|GR|/4π\n0 2 4 6 8 10 12 141.001.051.101.15\ndistance r/a(DRr,DRt,|GR|) /4π\nFIG. 3.jGRj,Dr\nR, andDt\nRshown as a function of the\ndistance from the defect, r=jR\u0000Rdj, forD= 0:3J=aand\nmagnetic \feld \u0016B= 0:09J=a2.\nB. Comparison of the generalized Thiele approach\nand micromagnetic simulations\nUsing the numerically determined potential, see\nFig. 2, one can directly calculate the trajectories of the\nskyrmions using the Thiele equation, Eq. (12). In Fig. 4\nthe trajectories are shown for two values of the damping\nconstants, i.e., \u000b=\f= 0:4 and 0:04. The properties\nof these solutions will be discussed in Sec. IV. Here we\ncompare them to full micromagnetic simulations of the\nsystem. To track the center of the skyrmion R, we used\nR\u0019P\ni(Mz\n0\u0000Mz\ni)ri=P\ni(Mz\n0\u0000Mz\ni) summing only over\nsites withMz\ni<Mz\n0=\u00000:3.\n-1001020-10-5051015\n-1001020-10-5051015FIG. 4. Comparison of trajectories of the moving skyrmion\nobtained from the full micromagnetic simulation (colored)\nto the results of the e\u000bective potential approach (black).\nThe vacancy is placed in the origin. For better orientation,\nthe green (red) circle indicates the minimum (maximum) of\nthe e\u000bective potential. Parameters are D= 0:3J=a,\u0016B=\n0:09J=a2,vs= 0:001aJex, and\u000b=\f= 0:04 (left),\u000b=\n\f= 0:4 (right).\nComparing the results from the micromagnetic simula-\ntions and the simpli\fed Thiele equation, we \fnd that the\ntwo approaches agree quantitatively with high precision.\nTiny deviations arise partially from using the simpli\fed\nThiele equation (12) instead of (6) neglecting, e.g., the\nspatial variations of GRandDRshown in Fig. 3. The\nother source of error is that the motion of the skyrmion\nleads to internal excitations and emission of spin waves\nnot captured by the generalized Thiele approach. These\ne\u000bects are expected to be larger for smaller \u000bbut even\nfor\u000b= 0:04 they give only small quantitative corrections\nin the considered parameter regime. Most importantly,\nthese corrections do not a\u000bect the physics of capturing\nand depinning discussed in the following paragraphs.\nC. Scaling of the e\u000bective potential and\ndimensionless units\nMost results presented in this paper have been ob-\ntained for a \fxed value of the Dzyaloshinskii-Moriya in-\nteraction,D= 0:3J=a, and for a defect described by\na single missing spin in a two-dimensional lattice. As\nskyrmions are smooth objects, one can use simple scaling\nrelations to determine the e\u000bective potential and there-\nfore also the equation of motion and the skyrmion tra-\njectories for other parameters.\nWithin the continuum theory, the free energy, Eq. (1),\nin the presence of a defect of radius adis invariant under\nthe transformation r!r0=r=\u0015,ad!a0\nd=ad=\u0015,\nJ!J0=J,D!D0=\u0015D, and B!B0=\u00152B. The\nshape of the skyrmion is thereby determined by the scale\ninvariant dimensionless parameter\n\u0010=J\u0016B\nD2, (13)\nwith\u00100=\u0010(called\u00142=Q2in Ref. 23).\nThe scaling invariance implies that the e\u000bective poten-5\ntial in the continuum limit can be written as\nVe\u000b(r) =JV\u0012\n\u0010;ad\u0016B\nD;r\u0016B\nD\u0013\n\u0019a2\ndJ\u00162B2\nD2V\u0010\u0012r\u0016B\nD\u0013\n,\n(14)\nwhereVandV\u0010are dimensionless scaling functions with\ndimensionless arguments and V\u0010(\u0018) =1\n2d2V(\u0010;\r;\u0018 )\nd\r2\f\f\f\n\r=0.\nFor the last expression, we assumed that adis much\nsmaller than the skyrmion radius and used that the term\nlinear inadvanishes.\nStrictly speaking, the above derived equation is fully\nvalid in the limit that the lattice constant ais much\nsmaller than both the skyrmion radius and the defect\nradiusad. But even for a defect given by a single missing\nspin (ad=a), one can use that the skyrmion radius as\nis much larger than a. Therefore the dependence of the\ne\u000bective potential on J;D and\u0016Bcan be described as\nVe\u000b(r)\u0019a2J\u00162B2\nD2~V\u0010\u0012r\u0016B\nD\u0013\n. (15)\nNote that ~V\u0010(\u0018) will in general di\u000ber from V\u0010(\u0018). This\nprediction is con\frmed in Fig. 5 which shows ~V\u0010obtained\nfor a single missing spin and various skyrmion sizes ob-\ntained by changing DandBsuch that\u0010= 1 remains\n\fxed.\n1r0=0.3a\n1r0=0.25a\n1r0=0.2a\n1r0=0.15a\n0123456-0.4-0.20.00.20.40.6\ndistance rr0potential V\nz\nFIG. 5. Rescaled potentials, ~V\u0010=Ve\u000b=E0withE0=\na2J\u00162B2\nD2, see Eq. (15), as a function of the rescaled distance\nr=r0withr0=D=\u0016B . Parameters are a=ad= 1 and\n\u0010= 1.\nThe scaling properties for the skyrmion trajectories\ncan simply be obtained by realizing that both Gand\nDare invariant under the scaling transformation as can\ndirectly be seen from Eqs. (7) and (8). Therefore the\n(generalized) Thiele equations (6) or (12) are invariant if\none uses for \fxed defect radius adthe scaling r!r=\u0015,\nD!\u0015D,B!\u00152Bsuch thatdVe\u000b=dr!\u00153dVe\u000b=dr\nand simultaneously rescales the drift velocity vs!\u00153vs\nand the time t!\u00154t.\nAn important consequence of this analysis is that the\ntypical drift velocity vsof electrons or the corresponding\ncritical current density, jc, to depin a skyrmion from adefect scales with the third power of the inverse skyrmion\nradiusas\u0018D=\u0016B\njc\u0018vs\u0018as\u00003. (16)\nThis is part of the reason why skyrmions can be manip-\nulated by ultrasmall current densities11.\nIt is also an interesting question how the potentials\nchange when instead of a single magnetic layer NL>1\nlayers are considered. For a line defect where all spins\nare removed in a line perpendicular to the surface and\nforNL\u001d1 one can use that away from the surface the\nmagnetic con\fguration is translationally invariant in zdi-\nrection. Therefore, the e\u000bective potential is simply given\nby multiplying Ve\u000bbyNL. As also the gyrocoupling and\ndamping matrix scale linearly in NL, the equation of mo-\ntion for the skyrmion center remains unmodi\fed as long\nas the phase with a single skyrmion in a ferromagnetic\nbackground remains stable. Increasing NLallows to elim-\ninate all e\u000bects of thermal \ructuations. The situation is\nmore complicated when only a few layers NLare con-\nsidered. As the properties of the surface and the inner\nlayers are di\u000berent, Ve\u000bcannot simply be computed from\nthe single-layer result.\nFor the presentation of our results, it is useful to \fnd\nthe minimal set of dimensionless parameters needed to\nparametrize our results. Here it is useful to note, that\nthe dependence on \fin the e\u000bective Thiele equations can\nbe eliminated by parametrizing the e\u000bect of the current\nby the drift velocitiy vdof the skyrmion in the absence\nof any defect. It can be obtained from the equation G\u0002\nvs+\fDvs=G\u0002vd+\u000bDvd. Further we also de\fne the\ndimensionless drift velocity vby\nvd=1\nG2+\u000b2D2\u0000\n(\u000b\u0000\f)DG\u0002vs+ (G2+\u000b\fD2)vs\u0001\nv=vdsD3\na2J\u00163B3. (17)\nIn units where all length scales are measured in units\nofD=\u0016B and all times in units ofsD4\na2J\u00164B4the e\u000bective\nThiele equation (12) takes the form\n\u0000d~V\u0010(R)\ndR=\u00004\u0019^z\u0002\u0010\n_R\u0000v\u0011\n+\u000bD\u0010\u0010\n_R\u0000v\u0011\n, (18)\nwithD\u0010=D=s. Originally, the continuum theory was\nparametrized by J,D,\u0016B,\u000b,\f,vs, and the size of the\ndefect. For a point-like defect, we \fnd that the three di-\nmensionless variables \u0010,v, and\u000bare su\u000ecient to describe\nall regimes.\nIV. SKYRMION DEPINNING, CAPTURING\nAND DEFLECTION\nA. Phase diagram\nWhen studying the qualitative behavior of the\nskyrmions when a current is slowly switched on, it is use-\nful to distinguish two initial states, an initially localized6\nP2 P1CF\n0.6 0.7 0.8 0.9 1.0 1.1 1.20.000.020.040.060.080.100.12\nmagnetic field B/B 0current density j/j a\n0.00.51.01.52.02.53.03.5\nσc/σ0\nFIG. 6. Phase diagram as function of the magnetic \feld\nB=B 0=\u0010and current density j=ja=v\u00103for\u000b= 0:1. Here\nwe use the combination v\u00103=vdJ2s\na2D3as it is independent\nof the magnetic \feld. The colored area encodes the value\nfor the capturing cross section \u001bc=\u001b0with the characteristic\nlength\u001b0=D=\u0016B , see Sec. IV C, which is a measure for\nthe e\u000eciency of capturing.\nskyrmion and a skyrmion approaching the defect from\nfar away.\nIf the skyrmion is initially localized close to the de-\nfect and if the potential has a local minimum, it will re-\nmain there for small current densities and gets depinned\nfor larger current densities. Similarly, a skyrmion ap-\nproaching the defect from far away can either get cap-\ntured (green trajectories in Fig. 4) by the defect or is\njust de\rected (blue trajectories).\nAn overview over these possibilities is given in the\nphase diagram, Fig. 6. The solid lines mark the depin-\nning transition. Below these lines, in the regimes denoted\nby P1, P2 and C, an initially localized skyrmion remains\nlocalized close to the defect when the current is switched\non slowly. In P1 the e\u000bective potential has a local min-\nimum atr= 0 while in P2 and C it has a minimum\nat \fnite skyrmion-defect distance. In the free phase, F,\npinning is not possible and all skyrmions move freely. At\nlow magnetic \felds we \fnd this phase even for zero cur-\nrent density. Note that we consider only \u0010 > 0:56 as\nat this point23the circular symmetric skyrmion becomes\nunstable towards the formation of a bimeron24.\nAn unexpected result is that in the pinning regimes P1\nand P2 a skyrmion approaching the defect from far apart\nisnotcaptured. Instead of getting trapped, it moves\naround the defect and is only de\rected. This is a con-\nsequence of the fact that for long distances the defect-\nskyrmion potential is repulsive. Capturing of approach-\na-6-4-2 024-20246\nb-6-4-2 024-20246\nc-6 -4 -2 0 2 4-20246\nd-6-4-2 024-20246\ne-6-4-2 024-20246\nf-6 -4 -2 0 2 4-20246\ng-6-4-2 024-20246\nh\n-6-4-2 024-20246FIG. 7. Trajectories (black) of the single skyrmion obtained\nfrom the e\u000bective potential approach. The coordinates rare\nde\fned relative to the position of the vacancy in dimension-\nless units r=\u001b0=r\u0016B=D . Parameters are \u0010= 1,\u000b= 0:1 and\ndrift velocities from left to right, top to bottom are v= 0:004,\nv= 0:009,v= 0:015,v= 0:026,v= 0:039,v= 0:060,\nv= 0:091, andv= 0:107. The corresponding drift velocities\nvare also marked in Fig. 8. The orange curve is the separa-\ntrix; the orange area is the capturing area. Red arrows mark\noutgoing \row and green arrows mark ingoing \row at a \fxed\npoint. The green (red) circle indicates the potential minimum\n(maximum).\ning skyrmions is only possible in the region C.\nB. Fixed points and separatices\nFor a quantitative analysis of the qualitatively di\u000berent\ntrajectories and for the construction of the phase diagram\nshown in Fig. 6, an analysis of the stable and unstable\n\fxed points of the Thiele equation (18) is useful.\nIn the continuum limit, the e\u000bective potential depends7\nonly on the relative distance rof skyrmion and defect,\nV(r) =V(r). If we now look for \fxed points of the Thiele\nequation, _R= 0, we \fnd that all \fxed points are on the\nline in the direction ^ eofG\u0002vd+\u000bDvd. At the \fxed\npoint one has\nj~V0\n\u0010(rFP)j=v\r, (19)\nwhere\r=\u0010\n(4\u0019)2+\u000b2D2\n\u0010\u00111\n2. There can be 0 ;2;4 or 6\n\fxed points. To classify the \fxed points, one linearizes\nthe equation of motion around them to obtain a matrix\nequation of the type _R=M\u000eR. It is useful to distin-\nguish 5 di\u000berent types of \fxed points characterized by the\neigenvalues, \u00151;2, of the 2\u00022 matrixM. The eigenvalues\nare either both real or are a pair of complex conjugate\nnumbers. If the real part of an eigenvalue is positive (neg-\native) it describes repulsion (attraction). A \fnite imagi-\nnary part gives an oscillatory behavior around the \fxed\npoint on top of the repulsion or attraction. We therefore\ndistinguish attractive ( \u00151;2<0), repulsive ( \u00151;2>0),\nsemide\fnite ( \u00151>0>\u00152), as well as oscillating attrac-\ntive (Re\u00151= Re\u00152<0;Im\u00151=\u0000Im\u001526= 0) and oscil-\nlating repulsive \fxed points (Re \u00151= Re\u00152>0;Im\u00151=\n\u0000Im\u001526= 0).\nGiven the potential exhibits a local minimum, for suf-\n\fciently small drift velocities, v<vc2, always one attrac-\ntive \fxed point exists. Therefore, for v<vc2, an initially\npinned skyrmion will remain pinned when the current\nand hence the drift velocity is increased slowly (a fast\nincrease is discussed below). In Fig. 7, the trajectories\nof skyrmion centers are shown. All skyrmions starting in\nthe orange-shaded region \fnally end up in the attractive\n\fxed point.\nThis capturing region can either be a bounded re-\ngion or extended to in\fnity. Only in the latter case, a\nskyrmion approaching from far apart can be captured\nby the defect. For small drift velocities, v < vc1, and\na local minimum of the potential, we obtain always\na bounded capturing region where all skyrmions move\naround the defect without being trapped. Only in the\nregimevc1< v < vc2capturing of mobile skyrmions is\npossible.\nOn each separatrix (orange lines in Fig. 7), we \fnd\nat least one semide\fnite \fxed point, where the `critical'\ntrajectories, which de\fne the separatrix, end. In Fig. 7\nthe semide\fnite \fxed points are marked by ingoing green\nand outgoing red arrows. To construct the separatrix nu-\nmerically, it is useful to investigate the time-reversed ver-\nsion of the equation of motion. After time reversal, the\ncritical trajectories start at the metastable \fxed point.\nUsing a small perturbation in the direction of the at-\ntractive eigenvector as a starting point, one can obtain\ndirectly the separatrix by computing the time-reversed\ntrajectories. This procedure is numerically stable as af-\nter time-reversal the repulsive direction of the \fxed point\nbecomes attractive. The time-reversed trajectories either\nreach in\fnity or end at the repulsive direction of a \fxed\npoint. Using this method, we compute for a given drift\nh g f e d cbaα=0.4\nα=0.1\nα=0.04\nα=0.01\n0.00 0.02 0.04 0.06 0.08 0.100123\ndrift velocity vσc/σ0FIG. 8. Cross section for capuring a skyrmion \u001bc=\u001b0as a\nfunction ofvwith the characteristic length \u001b0=D=\u0016B shown\nfor\u0010= 1 and various values of \u000b. Only for a \fnite drift\nvelocity range, vc1< v < vc2, the skyrmion gets trapped by\nthe defect. The vertical lines denote boundaries of various\nregimes (a-h) for \u000b= 0:1. In Fig. 7 typical trajectories for\nthese regimes are shown.\nvelocity the separatrix and identify the orange shaded\ncapturing region. At v=vc1, there is one trajectory\ndirectly connecting the two semide\fnite \fxed points.\nForvclose tovc2the orange-shaded capturing region\nshrinks in width and is displaced. This has the e\u000bect\nthat the way the current is switched on profoundly af-\nfects the pinning properties of skyrmions. For an ad-\nabatic switching on of a current with v<vc2, a skyrmion\nwhich was pinned at v= 0 follows the stable \fxed point\nand hence remains pinned. If, in contrast, the current is\nsuddenly switched on, then the initially pinnied skyrmion\nmay end up outside of the capturing region and thus gets\ndepinned. At v= 0 the position of the trapped skyrmion\nis determined by the location of the minimum of the ef-\nfective potential, shown as a green dashed line in Fig. 7.\nThis line is only partially within the capturing region for\nlarge current densities. Therefore a corresponding frac-\ntion of initially localized skyrmions, Fig. 7e and f, or even\nallskyrmions, Fig. 7g, will be released after a sudden\nswitching on of the current.\nC. Capturing cross section \u001bc\nTo quantify the e\u000eciency for capturing a mobile\nskyrmion by a defect, it is useful to de\fne the 'capturing\ncross-section` \u001bc. As for other scattering experiments, the\ncross section is obtained by dividing the capturing rate\nby the incoming \rux of particles. In the two-dimensional\nsituation discussed here, \u001bcis directly given by the width\nof the orange-shaded capturing region, compare Fig. 7,\nforx!\u00001 . In Fig. 8, \u001bcis shown as a function of\nthe drift velocity vfor a \fxed value of the magnetic \feld\n(\fxed\u0010). As discussed above, mobile skyrmions are only\ntrapped for vc1<v<vc2. The change of \u001bcas a function\nof both \feld and current density (drift velocity) is shown\nin Fig. 6.\nThev-dependence of \u001bcshows a sequence of kinks.8\nThe origin of each of these kinks can be traced back\nto a change of the topological structure de\fned by the\n\fxed points and the separatrices connecting and encir-\ncling them. In Fig. 7 examples of such \fxed point con-\n\fgurations are shown.\nTo trap a skyrmion, damping is needed. Indeed, Fig. 8\nshows that \u001bcgets smaller and smaller for \u000b!0, while\nthe two critical drift velocities vc1andvc2remain \fnite\nin this limit. To prove analytically that the skyrmions\nare not captured for \u000b= 0, one can, for example, rewrite\nEq. (12) in the form _R=\u0000G\u0002dVtot\ndR=jGj2using the\npotentialVtot(R) =V(R)\u0000(G\u0002vs)\u0001R. This implies\nthat skyrmions \row along equipotential lines of Vtotand\na mobile skyrmion thus never gets trapped.\nTaking into account that \u001bcvanishes for \u000b!0 it is\nperhaps surprising that the maximal \u001bcin Fig. 8 appears\nto be of the order of the skyrmion radius even for \u000b=\n0:01. The capturing cross section is maximal for rather\nsmall drift velocities slightly above vc1. A quantitative\nanalysis shows that for \u000b!0 the maximal cross section\nscales with bothp\u000band the typical length scale \u001b0=\nD=(\u0016B) of the order of the skyrmion radius\n\u001bmax\nc\u0019p\u000bc\u0010\u001b0, (20)\nwhere the dimensionless constant c\u0010depends on \u0010with\nc1\u001912. For larger currents (i.e., the regimes f and g in\nFig. 8) we \fnd instead \u001bc/\u000band therefore only tiny\ncapturing cross sections for small \u000b.\nForv!vc1andv!vc2the capturing cross section\n\u001bcvanishes.\nTo analyze the behavior of \u001bcforvclose to the lower\ncritical drift velocity, v&vc1, we use that the proper-\nties of this critical point are governed by the semide\fnite\n\fxed point at the top of Fig. 7a-c and that the sepa-\nratrix can be computed by analyzing the time-reversed\nequation of motion as described above. At v=vc1\nthe time-reversed trajectory coming from the semide\f-\nnite \fxed point at the bottom ends in this \fxed point\nat the top with eigenvalues \u00151>0> \u0015 2and eigenvec-\ntorsb1andb2. Forv=vc1+\u000ev, this trajectory ap-\nproaches the \fxed point from the b1direction and leaves\ninto theb2direction. Close to the \fxed point, one ob-\ntains\u000eR(t) =b1x1(t) +b2x2(t) withxi(t) =xi(0)e\u0000\u0015it.\nThe linearized equation of motion is only valid for small\nx1(t);x2(t)< x 0, wherex0is a cuto\u000b scale. We choose\ntwo timest1andt2such thatx1(t1) =x0andx2(t2) =x0\nin a way that the linearization is valid for t1< t < t 2.\nHeret1(t2) describes a point on the trajectory when\napproaching (leaving) the \fxed point. With these de\f-\nnitions we obtain x1(t2) =x0\u0010\nx0\nx2(t1)\u0011\u00151=\u00152\n. Using that\nthe cross section is approximately proportional to x1(t2)\nand thatx2(t1) depends linearly on v\u0000vc1, we obtain\n\u001bc\u0018(v\u0000vc1)j\u00151=\u00152j. (21)\nWe have checked numerically, that this result is valid\nclose tovc1. Forv!vc2in contrast, we \fnd that thedecay of the capturing cross section can be described by\n\u001bc\u0018(vc2\u0000v)2. (22)\nV. SKYRMIONS AND WEAK DISORDER\nFinally, we will discuss the case of a skyrmion moving\nthrough a weakly disordered medium. The distance of\ndefects is assumed to be much larger than the skyrmion\nradius,nd\u001c(\u0016B=D )2, wherendis the density of defects.\nIn this limit it is interesting to investigate, how the de-\nfects in\ruence the skyrmion Hall e\u000bect and the skyrmion\nmobility.\nIn the absence of any defects, the skyrmions move on\na straight line in a direction set by v. This direction is\nset by the direction of the external current and the dissi-\npation constants, see Eq. 17. When a skyrmion scatters\nfrom a defect, it therefore cannot change its direction.\nThe only net-e\u000bect of scattering is a displacement \u0001 k\nand \u0001?, parallel and perpendicular to v, respectively.\nA parallel displacement \u0001 kimplies that the skyrmion is\ndelayed, \u0001k>0, or has moved faster when passing the\ndefect, \u0001k<0. Therefore \u0001 kleads to changes of the mo-\nbility of the skyrmion. \u0001 ?in contrast, describes a 'side\njump` of the skyrmion due to the defect. Similar to the\nside-jump mechanism of electron scattering25, this leads\nto a contribution to the skyrmion Hall e\u000bect.\n\u0001kand \u0001?are functions of the impact parameter b,\ndescribing the o\u000bset of the incoming skyrmion trajectory\nrelative to the defect position. This dependence is shown\nin Fig. 9 for v < vc1(top \fgure) and v > vc2(bottom\n\fgure). When a skyrmion travels a long distance L, it\nhits several randomly distributed defects with impact pa-\nrameterbi. To calculate the total shift of a skyrmion one\ncan therefore average over all defect positions\n\u0001k=?\nL=1\nLX\ni\u0001k=?(bi)\u00191\nLZ\nnd\u0001k=?(b(r))d2r\n=nd\u0001I\nk=?\n\u0001I\nk=?=Z\ndb\u0001k=?(b). (23)\nThe o\u000bset integrals \u0001I\n?and \u0001I\nkparametrize how e\u000e-\nciently a defect can lead to a displacement of the trajec-\ntory.\nTo linear order in the density of defects nd, the average\nvelocity and the mobility thus change by\n\u0001v\nv\u0019nd\u0001I\nk (24)\nand the average direction of motion of the skyrmions is\nrotated by the angle\n'\u0019nd\u0001I\n?. (25)\nIn Fig. 10 the o\u000bset integrals are shown for v < vc1\nandv>vc2. Forvc1<v<vc2a single skyrmion always\ngets trapped for L! 1 and therefore the discussion9\nv=7.410-3\nv=4.110-3\nv=7.410-4\n-5 0 5-10-50510\nstarting offset bs0DÞ s0,DÈÈs0\nv=0.276\nv=0.158\nv=0.112\n-4-2024-10123456\nstarting offset bs0DÞ s0,DÈÈs0\nFIG. 9. \u0001k(dashed) and \u0001 ?(solid) shown as functions of\nthe impact parameter bfor drift velocities below (top) and\nabove (bottom) the capturing regime. Parameters used here\nare\u0010= 1,\u001b0= 1=0:3a,\u000b= 0:1, and drift velocities as given\nin the \fgures.\ngiven above can only be applied for \fnite systems (not\ndiscussed here).\nFor large drift velocities, v!1 , one can calculate\n\u0001I\nk=?by using that in this limit the potential only induces\nsmall corrections to straight skyrmion trajectories which\ncan be treated perturbatively. In this limit we obtain\n\u0001I\n?\u00192\u0019\u000bGD\nv2(G2+ (\u000bD)2)2Z1\n0drr(V0(r))2+O\u00121\nv3\u0013\n\u0001I\nk\u0019\u000bD\nG\u0001I\n?+O\u00121\nv3\u0013\n. (26)\nNumerically, we \fnd that this formula although derived\nforv! 1 works accurately for \u0001I\n?for allv > vc2\nwhereas for \u0001I\nkcorrections to this formula are of order 1\nforv&vc2, see lower part of Fig. 10.\nBoth o\u000bset integrals are strongly enhanced for v<vc1\nwhen the skyrmion moves around the defect instead of\npassing it. For the parameters shown in Fig. 10, they\nare about two orders of magnitude larger for v.vv1\ncompared to the case v&vc2and also become much\nlarger than the area of the skyrmion. The main reason is\nthat in this regime the skyrmions move around the defect\nDÞIs02\nlog2fit\nlog2fit\nDÈÈIs02\n0.000 0.002 0.004 0.006 vc1-120-100-80-60-40-20\ndrift velocity vDÞIs02,DÈÈIs02\nDÞIs02\nDÞ PTIs02\nDÈÈPTIs02\nDÈÈIs02\nvc20.20.30.40.50.60.70.8-0.8-0.6-0.4-0.20.00.2\ndrift velocity vDÞIs02,DÈÈIs02FIG. 10. O\u000bset integrals shown as functions of the drift ve-\nlocity below (top) and above (bottom) the capturing regime,\ni.e.,v<vc1= 0:0083 andv>vc2= 0:1044. Parameters used\nhere are\u0010= 1,\u001b0= 1=0:3a, and\u000b= 0:1. The logarithmic \fts\nin the upper plot are related to Eq. (27). The perturbative\napproximation in the lower plot is given in Eq. (26).\nin a distance/ln 1=vdue to the exponential tails of the\nskyrmion-defect potential. This sets the relevant length\nscale independent of the damping and hence the e\u000bects\nare not any more suppressed by \u000b. We therefore obtain\n\u0001I\nk;\u0001I\n?/ln21=vforv!0, (27)\nas shown in the upper part of Fig. 10. Note that thermal\n\ructuations, not considered in this study, are expected\nto cut o\u000b the divergency.\nA counter-intuitive result is that for v < vc1, \u0001I\nkis\nnegative implying that defects accelerate the motion of\nskyrmions. This is possible because the speed of the\nskyrmion can grow when the angle \u001ebetween vand _R\ngrows. A simple, analytically solvable limit is the motion\nof the skyrmion parallel to a wall. From the balance of\nforces parallel to the wall ( dV=dRk= 0), one obtains\nusing Eq. (12) that _Rk=vsGcos\u001e\u0000\fDsin\u001e\n\u000bDwhere\u001eis the\nangle between drift velocity and the wall normal. For\nsmall\u000b\u0018\f, obstacles can therefore speed up skyrmion\nmotion by a maximal factor of order 1 =\u000b\u001d1. While\nthe path of the skyrmion which moves around a defect\nincreases, the increased velocity typically overcompen-10\nsates this longer path for v<vc1as shown in Fig. 10 for\n\u000b= 0:1.\nA way to measure \u000b\u0000\fin a weakly damped skyrmion\nsystem experimentally is to determine (in the absence of\ndefects) the skyrmion Hall angle, '0\u0019D(\u000b\u0000\f)\nG, which\nis the angle between the direction of the electric current\nand the skyrmion \row direction. This angle is changed\nby defects as described by Eq. (25). Especially for small\n\u000band\fandv<vc1one can easily reach a regime where\nthe impurity induced contribution to the skyrmion Hall\nangle dominates, '&'0. An alternative point of view is\nthat the defects lead to a strong renormalization of \u000b\u0000\f,\nwith\n(\u000b\u0000\f)e\u000b\u0019\u000b\u0000\f+G\nDnd\u0001I\n?. (28)\nVI. CONCLUSION AND QUANTITATIVE\nESTIMATES FOR SKYRMION DEVICES\nWe found that the Thiele ansatz combined with an\ne\u000bective potential is an e\u000ecient and reliable tool to de-\nscribe the interaction of a skyrmion with a vacancy. Us-\ning this ansatz we determined the phase diagram which\nshows the three phases appearing in this interaction: The\npinning phase where a skyrmion freely moves around va-\ncancies and never gets captured but stays pinned if it\nwas pinned initially, the capturing phase in which a mov-\ning skyrmion can get captured within a certain captur-\ning cross section and the freephase for current densities\nj >jmaxwhere a skyrmion can never be pinned.\nTc\u0015 a m n\nMnSi 29 K 180 \u0017A 4:6\u0017A 0:4\u0016B 3:8\u00011028m\u00003\nFeGe 280 K 700 \u0017A 4:7\u0017A 1\u0016B 2:4\u00011028m\u00003\nTABLE I. Input parameters26{29for the quantitative esti-\nmates.Tc\u0018Jis the transition temperature30,athe lat-\ntice constant for a unit cell containing 4 Mn (or Fe) ions,\n\u0015\u00192\u0019J=D the pitch in the helical phase, m\u0019s\u0016Ba2=(2~) is\nthe magnetization per Mn (or Fe) ion and hence \u0016= 4m=a2,\nandnthe charge density.\nOur results can be used to obtain estimates of neces-\nsary current densities and the depth of the pinning po-\ntential and can, hopefully, be used as a starting point to\ndesign simple skyrmion devices. As an example, we try\nto give estimates of the relevant parameters for MnSi,\nthe perhaps best studied skyrmion material, and for\nFeGe, the skyrmion system with the largest transition\ntemperature8up to now. Input parameters for these es-\ntimates are shown in table I.\nIn table II, we show typical parameters characterizing\na defect with the size of one unit cell for a single-layer of\nMnSi or FeGe for \u0010= 1. Note that the actual numbers\nwill depend on the microscopics of the induced defect andshould therefore be viewed only as order-of-magnitude\nestimates.\nB0E0=kBv0j0 jc2\nMnSi 0.7 T 0.7 K 9m\ns5\u00011010A\nm2 6\u0001109A\nm2\nFeGe 0.2 T 0.5 K 8m\ns3\u0001109A\nm2 3\u0001108A\nm2\nTABLE II. Estimates of typical parameters for the pinning\nof a skyrmion by a single-site defect in a single layer of the\nmaterials MnSi and FeGe at \u0010= 1, i.e., for B=B0=D2\n\u0016J.\nE0=J(a\u0016B)2\nD2is the strength of the pinning potential de-\n\fned by the prefactor in Eq. (15). The typical velocity\nv0=a2J(\u0016B)3\nsD3and the typical current density j0=nev0are\nde\fned such that v=vd=v0=j=j0, whilejc2= 0:11j0is the\ncritical current density for \u0010= 1 atT= 0. Note that the\nparameters depend strongly on the size of the defect and the\nlayer thickness, see text.\nA main result of these estimates is that a single-site va-\ncancy in a monolayer of these materials will not be able\nto pin a skyrmion due to the presence of thermal \ruc-\ntuations,E0\u001ckBT. This shows that indeed skyrmions\nare very insensitive to defects. To build a device with a\nnanostructure which is capable to pin a skyrmion, one\ntherefore needs to consider both larger defects and also\n\flms with a larger number of layers, NL\u001d1, using that\nE0/a2NL, see Sec. III C. The critical current density\nfor depinning, jc2, is independent of NLbut also scales\nwith the area of the defect. For example, using a hole\nwith a diameter of 10 nm for a FeGe \flm with a thick-\nness of 50 nm in a magentic \feld of 0 :2 T, we obtain as\nan order-of-magnitude estimate\nE0=kB\u001920:000 K; jc2\u00191011A=m2, (29)\nclearly su\u000ecient for thermal stability.\nAs we have shown, the shape of the e\u000bective impurity-\nskyrmion potential depends quantitatively and qualita-\ntively on the strength of the magnetic \feld. Changing,\nfor example, the magnetic \feld from 0 :2 T to 0:13 T is\nsu\u000ecient to avoid allpinning, see Fig. 6. By controlling\nboth the magnetic \feld and the current density one can\nvary in a \rexible way not only the capability of a de-\nfect to hold a skyrmion but also its ability to capture a\nskyrmion moving close by, see Fig. 8. We believe that\nthis \rexibility will allow to control skyrmions e\u000eciently\nin devices based on holes and similar nanostructures.\nVII. ACKNOWLEDGEMENTS\nWe would like to thank C. Sch utte for discussions and\nsoftware used within this project. We also acknowl-\nedge useful discussions with M. Garst, M. Kl aui and\nC. P\reiderer. Part of this work was supported by the\nBonn-Cologne Graduate School of Physics and Astron-\nomy (BCGS).11\n\u0003jmueller@thp.uni-koeln.de\n1P. Milde, D. K ohler, J. Seidel, L. M. Eng, A. Bauer,\nA. Chacon, J. Kindervater, S. M uhlbauer, C. P\reiderer,\nS. Buhrandt, C. Sch utte, and A. Rosch, Science 340, 1076\n(2013)\n2N. Nagaosa and Y. Tokura, Nat Nano 8, 899 (dec 2013)\n3S. M uhlbauer, B. Binz, F. Jonietz, C. P\reiderer, A. Rosch,\nA. Neubauer, R. Georgii, and P. B oni, Science 323, 915\n(2009)\n4X. Z. Yu, Y. Onose, N. Kanazawa, J. H. Park, J. H. Han,\nY. Matsui, N. Nagaosa, and Y. Tokura, Nature 465, 901\n(jun 2010)\n5T. Adams, A. Chacon, M. Wagner, A. Bauer, G. Brandl,\nB. Pedersen, H. Berger, P. Lemmens, and C. P\reiderer,\nPhys. Rev. Lett. 108, 237204 (Jun 2012)\n6W. M unzer, A. Neubauer, T. Adams, S. M uhlbauer,\nC. Franz, F. Jonietz, R. Georgii, P. B oni, B. Pedersen,\nM. Schmidt, A. Rosch, and C. P\reiderer, Phys. Rev. B\n81, 041203 (Jan 2010)\n7T. Adams, S. M uhlbauer, C. P\reiderer, F. Jonietz,\nA. Bauer, A. Neubauer, R. Georgii, P. B oni, U. Keider-\nling, K. Everschor, M. Garst, and A. Rosch, Phys. Rev.\nLett. 107, 217206 (Nov 2011)\n8X. Z. Yu, N. Kanazawa, Y. Onose, K. Kimoto, W. Z.\nZhang, S. Ishiwata, Y. Matsui, and Y. Tokura, Nat Mater\n10, 106 (feb 2011)\n9N. Romming, C. Hanneken, M. Menzel, J. E. Bickel,\nB. Wolter, K. von Bergmann, A. Kubetzka, and R. Wiesen-\ndanger, Science 341, 636 (2013)\n10F. Jonietz, S. M uhlbauer, C. P\reiderer, A. Neubauer,\nW. M unzer, A. Bauer, T. Adams, R. Georgii, P. B oni,\nR. A. Duine, K. Everschor, M. Garst, and A. Rosch, Sci-\nence330, 1648 (2010)\n11X. Yu, N. Kanazawa, W. Zhang, T. Nagai, T. Hara, K. Ki-\nmoto, Y. Matsui, Y. Onose, and Y. Tokura, Nat Commun\n3, 988 (aug 2012)\n12J. Iwasaki, M. Mochizuki, and N. Nagaosa, Nat Commun\n4, 1463 (feb 2013)13S.-Z. Lin, C. Reichhardt, C. D. Batista, and A. Saxena,\nPhys. Rev. Lett. 110, 207202 (May 2013)\n14S.-Z. Lin, C. Reichhardt, C. D. Batista, and A. Saxena,\nPhys. Rev. B 87, 214419 (Jun 2013)\n15J. Sampaio, V. Cros, S. Rohart, A. Thiaville, and A. Fert,\nNat Nano 8, 839 (nov 2013)\n16J. Iwasaki, M. Mochizuki, and N. Nagaosa, Nat Nano 8,\n742 (oct 2013)\n17A. A. Thiele, Phys. Rev. Lett. 30, 230 (Feb 1973)\n18K. Everschor, M. Garst, R. A. Duine, and A. Rosch, Phys.\nRev. B 84, 064401 (Aug 2011)\n19K. Everschor, M. Garst, B. Binz, F. Jonietz, S. M uhlbauer,\nC. P\reiderer, and A. Rosch, Phys. Rev. B 86, 054432 (Aug\n2012)\n20R. A. Duine, A. S. N\u0013 u~ nez, J. Sinova, and A. H. MacDonald,\nPhys. Rev. B 75, 214420 (Jun 2007)\n21M. E. Lucassen, H. J. van Driel, C. M. Smith, and R. A.\nDuine, Phys. Rev. B 79, 224411 (Jun 2009)\n22S. Zhang and Z. Li, Phys. Rev. Lett. 93, 127204 (Sep 2004)\n23C. Sch utte and M. Garst, ArXiv e-prints(May 2014),\narXiv:1405.1568 [cond-mat.str-el]\n24M. Ezawa, Phys. Rev. B 83, 100408 (Mar 2011)\n25L. Berger, Phys. Rev. B 2, 4559 (Dec 1970)\n26A. J. Freeman, A. M. Furdyna, and J. O. Dimmock, Jour-\nnal of Applied Physics 37, 1256 (1966)\n27Y. Ishikawa, G. Shirane, J. A. Tarvin, and M. Kohgi, Phys.\nRev. B 16, 4956 (Dec 1977)\n28A. Neubauer, C. P\reiderer, B. Binz, A. Rosch, R. Ritz,\nP. G. Niklowitz, and P. B oni, Phys. Rev. Lett. 102, 186602\n(May 2009)\n29N. A. Porter, J. C. Gartside, and C. H. Marrows, Phys.\nRev. B 90, 024403 (Jul 2014)\n30S. Buhrandt and L. Fritz, Phys. Rev. B 88, 195137 (Nov\n2013)"    },    {        "title": "1412.0688v1.Dissipation_due_to_pure_spin_current_generated_by_spin_pumping.pdf",        "content": "arXiv:1412.0688v1  [cond-mat.mes-hall]  1 Dec 2014Dissipation due to pure spin-current generated by spin pump ing\nTomohiro Taniguchi1,3and Wayne M. Saslow2,3\n1National Institute of Advanced Industrial Science and Tech nology (AIST),\nSpintronics Research Center, Tsukuba, Ibaraki 305-8568, J apan,\n2Department of Physics, Texas A&M University, College Stati on, Texas, 77843-4242, U.S.A.,\n3Center for Nanoscale Science and Technology, National Inst itute of\nStandards and Technology, Gaithersburg, Maryland, 20899- 6202, U.S.A.\n(Dated: June 16, 2021)\nBased on spin-dependent transport theory and thermodynami cs, we develop a generalized theory\nof the Joule heating in the presence of a spin current. Along w ith the conventional Joule heating\nconsisting of an electric current and electrochemical pote ntial, it is found that the spin current and\nspin accumulation give an additional dissipation because t he spin-dependent scatterings inside bulk\nand ferromagnetic/nonmagnetic interface lead to a change o f entropy. The theory is applied to\ninvestigate the dissipation due to pure spin-current gener ated by spin pumping across a ferromag-\nnetic/nonmagnetic/ferromagnetic multilayer. The dissip ation arises from an interface because the\nspin pumping is a transfer of both the spin angular momentum a nd the energy from the ferromagnet\nto conduction electrons near the interface. It is found that the dissipation is proportional to the\nenhancement of the Gilbert damping constant by spin pumping .\nPACS numbers: 72.25.Ba, 72.10.Bg, 85.75.-d, 72.25.Mk\nI. INTRODUCTION\nDissipation due to electron transport in a conductor\nis an important issue for both fundamental and applied\nphysics [1–5]. According to electron transport theory [6],\nthe conductivity of the electron becomes finite because\nof impurity scattering inside the conductor, which leads\nto Joule heating JeE, whereJeandEare the electric\ncurrent density and electric field, respectively. Motivated\nto reduce power consumption due to Joule heating, as\nwell as because of a fundamental interest in its quantum\nmechanical nature, the generation of a pure spin-current\nby spin pumping, spin-Seebeck effect, or spin-Hall effect\nhas been extensively investigated [7–15].\nDissipation is associated with the production of en-\ntropy. Spin-flip processes and spin-dependent scatterings\nwithin a bulk ferromagnet (F) or nonmagnet (N) and at\nan F/N interface mix the spin-up and spin-down states,\nleading to a change of the entropy. Therefore a physi-\ncal system, such as a F/N metallic multilayer, carrying a\npure spin-current, still dissipates energy even in the ab-\nsence of an electric current. A quantitative evaluation\nof the dissipation due to pure spin-current therefore is a\nfundamentally important problem.\nIn 1987, Johnson and Silsbee [1] studied the surface\nand bulk transport coefficients for spin conduction, and\nthe associated entropy production rates, without consid-\nering the rate of interface heating. More recently, Sears\nandSaslow[4]used irreversiblethermodynamicstostudy\ninterface heating due to electric current in a magnetic\nsystem, and Tulapurkar and Suzuki [5] used the Boltz-\nmann equation to investigate bulk and interface heating\nfor spin conduction. Reference [5] shows that, roughly\nspeaking, the dissipation due to spin current is propor-\ntional to the square of the spin polarization of the con-\nduction electrons, indicating that the heating associatedwith the spin current is much smaller than that due to\nthe electric current. However, these works consider only\nacollinearalignmentofthemagnetizationsinaF/Nmul-\ntilayer, so only the longitudinal components of the spin\ncurrent and spin accumulation (i.e., spin chemical poten-\ntial, proportional to the nonequilibrium spin density) ap-\npear. (Longitudinal andtransverse will be used to mean\nthat the direction of the spin polarization is collinear or\nnormal to the local magnetization.) On the other hand,\nin many physical phenomena, such as spin torque switch-\ning[16]andspinpumping[7,8], anon-collinearalignment\nof the magnetizations generally appears, in which trans-\nverse spin current and spin accumulation exist. For ex-\nample, spin pumping is a generation of the transverse\nspin current by the transfer of spin angular momentum\nfrom the ferromagnetic layer to the conduction electrons\n[7,8,17–22]. Bulk heating due to spin pumping in a mag-\nnetic wire within a domain wall (driven by m×H) has\nalso been studied [3], but was not extended to include\ninterface heating. In these works, the main contribu-\ntion to the dissipation arises from the electric current.\nThe present work develops a unified theory of dissipation\nwhich enables the simultaneous evaluation of both bulk\nand interface heating in a ferromagneticsystem, with the\nspin current having arbitrary alignment of the magneti-\nzations. Also, an evaluation of the dissipation due to a\npure spin-current is indispensable for comparison with\nexperiments that determine the rate of heating.\nThis paper develops a general theory of dissipation in\nthe presence of spin current based on the spin-dependent\ntransport theory and thermodynamics. It is found that,\nalong with the conventional Joule heating, the spin cur-\nrentIs(or its density Js) and spin accumulation µcon-\ntribute to the bulk and interfacedissipations, asshownin\nEqs. (17) and (18). We apply the theory to evaluate the\ndissipation due to a pure spin-current generated by spin2\nm1m2\nF1F2Nm1.m1×m1.\nxd2d1\n0Ispump\nIsF1→N \nIsF2→N θ\nFIG. 1: Schematic view of the F 1/N/F2ferromagnetic multi-\nlayer system. The directions of ˙m1andm1×˙m1are indicated\nby arrows.\npumping in the ferromagnetic (F 1) / nonmagnetic (N)\n/ ferromagnetic (F 2) multilayer. Spin pumping provides\nan interesting example to study the dissipation problem\nof pure spin-current. In spin pumping, electric current is\nabsent throughout the system. The electron transport is\ndescribedbyaone-dimensionalequation, andanexternal\ntemperature gradient is absent, which makes evaluation\nof the dissipation simple comparedwith the spin-Seebeck\neffectorspin-Halleffect. Itisfoundthatthedissipationis\nproportional to the enhancement of the Gilbert damping\nby spin pumping. The amount of the dissipation due to\nthe spin pumping is maximized for an orthogonal align-\nment of the two magnetizations. For the conditions we\nstudy, the maximum dissipation is estimated to be two\nto three orders of magnitude smaller than the dissipa-\ntion due to the electric current when there is spin torque\nswitching.\nThe paper is organized as follows. In Sec. II, the sys-\ntem we consider is illustrated. Section III formulates a\ntheory of dissipation of spin-polarized conduction elec-\ntrons, using diffusive spin transport theory and thermo-\ndynamics. Section IV studies the relationship between\nthe dissipation due to spin pumping and the equation de-\nveloped in the previous section. Section V quantitatively\nevaluates the dissipation due to spin pumping. Section\nVI, compares the spin pumping dissipation with the dis-\nsipation in the case of spin torque switching. Section VII\nprovides our conclusions.\nII. SPIN PUMPING IN F/N/F SYSTEM\nFigure 1 shows a schematic view of the F 1/N/F2ferro-\nmagnetic multilayer system, where m1andm2are unit\nvectors pointing along the magnetizations of the F 1and\nF2layers,respectively. Whereneeded, subscripts k= 1,2\ndenote the F klayer. The thickness of the F klayer is de-\nnoted by dk. The F 1and F 2layers lie in the regions\n−d1≤x≤0 and 0≤x≤d2, respectively. We assume\nthat the spincurrentisconservedinthe Nlayer,andthus\nconsider its zero-thickness limit because a typical valuefor the spin diffusion length of an N layer is much greater\nthan its thickness: for example, the spin diffusion length\nfor Cu is on the order of 100 nm, whereas experimental\nthicknesses are less than 5 nm [7,8,23].\nSteady precession of m1with the cone angle θcan be\nexcited by microwave radiation of the angular velocity\nωfor ferromagnetic resonance (FMR) in the F 1layer.\nThen, the F 1layer pumps the pure spin-current\nIpump\ns=/planckover2pi1\n4π/parenleftbigg\ng↑↓\nr(F1)m1×dm1\ndt+g↑↓\ni(F1)dm1\ndt/parenrightbigg\n,(1)\nwherethe realandimaginarypartsofthe mixing conduc-\ntance are denoted by g↑↓\nrandg↑↓\ni, respectively [24,25].\nThe pumped spin current creates spin accumulations in\nthe ferromagnetic ( µF) and nonmagnetic ( µN) layers,\nwhich induce backflow spin current (into N) [20,24–26],\ngiven by\nIF→N\ns=1\n4π/bracketleftbigg(1−γ2)g\n2m·(µF−µN)m\n−g↑↓\nrm×(µN×m)−g↑↓\niµN×m\n+t↑↓\nrm×(µF×m)+t↑↓\niµF×m/bracketrightBig\n.(2)\nThe total interface conductance g=g↑↑+g↓↓and\nthe spin polarization of the interface conductance γ=\n(g↑↑−g↓↓)/(g↑↑+g↓↓)aredefined fromthe interfaceresis-\ntanceofthespin- ν(ν=↑,↓)electrons rνν= (h/e2)S/gνν,\nwhereSis the crosssection area. The real and imaginary\nparts of the transmission mixing conductance at the F/N\ninterfacearedenoted by t↑↓\nr(i). The conditionthat the spin\ncurrent is conserved in the N layer can be expressed as\nIpump\ns+IF1→N\ns+IF2→N\ns=0. (3)\nIII. DISSIPATION FORMULAS\nTo obtain the dissipation due to spin pumping, it\nis necessary to investigate how the spin accumulation\nrelaxes inside the F layers and at the F/N interfaces.\nFor generality we include the terms related to the elec-\ntric current and field, although these are absent in the\nspin-pumped system. The spin accumulation in the\nferromagnetic layer relates to the distribution function\nˆF= (f0+f·σ)/2, which is a 2 ×2 matrix in spin\nspace and satisfies the Boltzmann equation [5,26–33],\nvia [34] µ=/integraltext\nεFTr[σˆF]dε,σbeing the Pauli matri-\nces. The charge and spin distributions are denoted by\nf0andf, respectively. The distributions for spin paral-\nlel,f↑= (f0+m·f)/2, or antiparallel, f↓= (f0−m·f)/2,\nto the local spin, give the longitudinal spin. On the other\nhand, the components of forthogonal to mcorrespond\nto the transverse spin. Below, we introduce the follow-\ning notations to distinguish the longitudinal (”L”) and\ntransverse (”T”) components of the spin current Isand3\nspin accumulation µ:\nIL\ns= (m·Is)m, (4)\nIT\ns=m×(Is×m), (5)\nµL= (m·µ)m, (6)\nµT=m×(µ×m), (7)\nwhereIsequals to Ipump\ns+IF1→N\nsat the F 1/N interface\nand−IF2→N\nsat the F 2/N interface, respectively. The\nspin current density is denoted as Js=Is/S.\nWe first consider the diffusive transport for the longi-\ntudinal spin [27–33]. The longitudinal spin accumulation\nrelates to the electrochemical potential ¯ µν=µ0+δµν−\neV(ν=↑,↓) viaµL= (¯µ↑−¯µ↓)m, whereµ0,δµν, and\n−eVare the chemical potential in equilibrium, its devi-\nation in nonequilibrium, and the electric potential. The\nlongitudinal electron density nν=/integraltext\nd3k/(2π)3fνand its\ncurrent density jν=/integraltext\nd3k/(2π)3vxfνsatisfy [27]\n∂nν\n∂t+∂jν\n∂x=−nν\n2τν\nsf+n−ν\n2τ−ν\nsf, (8)\nwhere the spin-flip scattering time from spin state νto\n−ν(up to down or down to up) is denoted by τν\nsf. The\ncharge density ne=−e(n↑+n↓) and electric current\ndensityJe=−e(j↑+j↓) satisfy the conservation law,\n∂ne/∂t+∂Je/∂x= 0. The electron density nνis related\ntoδµνvianν≃ Nνδµν, whereNνis the density of states\nof the spin- νelectron at the Fermi level. In the diffusive\nmetal,jνcan be expressed as\njν=−σν\ne2∂¯µν\n∂x, (9)\nwheretheconductivityofthespin- νelectronσνrelatesto\nthe diffusion constant Dνand the density of state Nνvia\nthe Einstein law σν=e2NνDν. Detailed balance [35],\nN↑/τ↑\nsf=N↓/τ↓\nsf, is satisfied in the steady state. The\nspin polarizations of the conductivity and the diffusion\nconstantare denoted by β= (σ↑−σ↓)/(σ↑+σ↓) andβ′=\n(D↑−D↓)/(D↑+D↓). FromEq. (8), thelongitudinalspin\naccumulation in the steady state satisfies the diffusion\nequation [27]\n∂2\n∂x2µL=1\nλ2\nsd(L)µL, (10)\nwhereλsd(L)is the longitudinal spin diffusion length de-\nfined as 1 /λ2\nsd(L)= [1/(D↑τ↑\nsf) + 1/(D↓τ↓\nsf)]/2. The lon-\ngitudinal spin current density can be expressed as\nJL\ns=−/planckover2pi1\n2e2∂\n∂x(σ↑¯µ↑−σ↓¯µ↓)m. (11)\nThe issue of whether transport of the transverse spin\nin the ferromagnet is ballistic or diffusive has been dis-\ncussed in [16,25,36] and [29–32]. These two theories are\nsupported by different experiments [26,37–39], and thevalidity of each theory is still controversial. The present\nwork considers the case of diffusive transport for gen-\nerality. Ballistic transport corresponds to the limit of\nλJ,t↑↓\nr(i)→0, where λJis the spin coherence length in-\ntroduced below. In the steady state, the transverse spin\naccumulation µT=µ−µLobeys [26,29]\n∂2\n∂x2µT=1\nλ2\nJµT×m+1\nλ2\nsd(T)µT,(12)\nwhere the first term on the right-hand-side describes the\nprecession of the spin accumulation around the magne-\ntization due to the exchange coupling. The exchange\ncoupling constant Jsdis in relation to the spin coher-\nence length λJviaλJ=/radicalbig\n/planckover2pi1(D↑+D↓)/(2Jsd) [28–33].\nThe spin diffusion length of the transverse spin is λsd(T)\n[29]. The transverse spin current density is related to the\ntransverse spin accumulation via [26,29]\nJT\ns=−/planckover2pi1σ↑↓\n2e2∂\n∂xµT, (13)\nwhereσ↑↓=e2[(N↑+N↓)/2][(D↑+D↓)/2]. The so-\nlutions of the transverse spin accumulation and cur-\nrent are linear combinations of e±x/ℓande±x/ℓ∗with\n1/ℓ=/radicalBig\n(1/λ2\nsd(T))−(i/λ2\nJ).\nIn the nonmagnetic layer, the distinction between the\nlongitudinal and transverse spin is unnecessary. In fact,\nin the limit of zero-spin polarization ( β=β′= 0) and in\nthe absenceofthe exchangecouplingbetweenthe magne-\ntization and electrons’ spin ( Jsd= 0), as for the nonmag-\nnet, Eqs. (10) and (12), or Eqs. (11) and (13), become\nidentical.\nThe relation between the spin accumulation and dis-\nsipation is as follows. The heat density of the longitu-\ndinal spin- νelectrons dqνrelates to the energy density\nuν=/integraltext\nd3k/(2π)3εfν, chemical potential µν=µ0+δµν,\nand the electron density nνvia [40,41]\ndqν=duν−µνdnν. (14)\nThe energydensity uL=u↑+u↓for the longitudinal spin\nsatisfies [6]\n∂uL\n∂t+∂jL\nu\n∂x=JeE, (15)\nwherejL\nu=ju,↑+ju,↓, andju,ν=/integraltext\nd3k/(2π)3εvxfνis\nthe energy current density [6]. Here, the term JeEis\nthe Joule heating due to the electric current. On the\nother hand, the energy current of the transverse spin jT\nu\nsatisfies∂jT\nu/∂x= 0 in the steady state, where the right-\nhand-side is zero because there is no source of the trans-\nverse spin inside the F and N layers. We introduce the\nheat current density by [34]\njq=jL\nu−/summationdisplay\nν=↑,↓µνjν+jT\nu−µT·JT\ns\n/planckover2pi1.(16)4\nIn steady state, the heat current is related to the dis-\nsipation via [42] ∂QV/∂t=T[∂(jq/T)/∂x], where the\ntemperature Tis assumed to be spatially uniform in the\nfollowing calculations. The subscript ” V” is used to em-\nphasize that this is the dissipation per unit volume per\nunit time. Then, ∂QV/∂tis\n∂QV\n∂t=Je\ne∂¯µ\n∂x−∂\n∂xJs\n/planckover2pi1·µ, (17)\nwhere ¯µ= (¯µ↑+ ¯µ↓)/2 is the electrochemical potential.\nThe interface resistance also gives the dissipation, where\nthe dissipation per unit area per unit time is\n∂QA\n∂t=Je\neδ¯µ−Js\n/planckover2pi1·δµ, (18)\nwhereδ¯µandδµarethedifferencesof ¯ µandµattheF/N\ninterface. The subscript ” A” is used to emphasize that\nthis is the dissipation per unit area per unit time. Equa-\ntions(17)and(18)aregeneralizedJouleheatingformulas\nin the presence of spin current, and the main results in\nthis section. The total spin current Jsand spin accu-\nmulation µinclude both the longitudinal and transverse\ncomponents, whereas only the longitudinal components\nappeared in the previous work [5]. The amount of the\ndissipation can be evaluated by substituting the solution\nof the diffusion equation of the spin accumulation into\nEqs. (17) and (18) with accurate boundary conditions\nprovided by Eqs. (1) and (2). We call Eqs. (17) and (18)\nthe bulk and interface dissipations, respectively.\nIV. DISSIPATION DUE TO SPIN PUMPING\nInspin pumping, transversespinangularmomentum is\nsteadily transferred from the magnetic system (F 1layer)\nto the conduction electrons near the F 1/N interface. The\nnetspinangularmomentum, ds= [Ipump\ns+m1×(IF1→N\ns×\nm1)]dt, transferred from the ferromagnet should over-\ncome the potential difference µN−µF1to be pumped\nsteadily from the F 1/N interface to the N layer during\nthe time dt. This means that not only the spin angu-\nlar momentum but also the energy is transferred from\nthe F1layer to the conduction electrons. The trans-\nferred energy per unit area per unit time is given by\n(µN−µF1)·(ds/dt)/(/planckover2pi1S). In terms of the spin cur-\nrent and spin accumulation, this transferred energy is\nexpressed as\n∂QSP\nA\n∂t=1\n/planckover2pi1S/bracketleftbig\nIpump\ns+m1×/parenleftbig\nIF1→N\ns×m1/parenrightbig/bracketrightbig\n·[µN(x= 0)−µF1(x= 0)].(19)\nComparing Eq. (19) with Eq. (18), we find the relation\n/parenleftbigg∂QA\n∂t/parenrightbiggT\nF1/N=−∂QSP\nA\n∂t, (20)where (∂QA/∂t)T\nF1/Nis defined by\n/parenleftbigg∂QA\n∂t/parenrightbiggT\nF1/N=/parenleftbigg∂QA\n∂t/parenrightbigg\nF1/N−/parenleftbigg∂QA\n∂t/parenrightbiggL\nF1/N.(21)\nHere, (∂QA/∂t)F1/Nis the F 1/N interface dissipation de-\nfined by Eq. (18), whereas\n/parenleftbigg∂QA\n∂t/parenrightbiggL\nF1/N=−1\n/planckover2pi1S/parenleftbig\nm1·IF1→N\ns/parenrightbig\nm1\n·[µN(x= 0)−µF1(x= 0)].(22)\nBecause Eq. (22) is defined by the longitudinal compo-\nnents of the spin current and spin accumulation in Eq.\n(18), we call this quantity the longitudinal part of the\nF1/N interface dissipation. On the other hand, Eq. (21)\nis defined by the transverse components of the spin cur-\nrent and spin accumulation at the F 1/N interface. More-\nover, using Eqs. (17), (18) and (21), Eq. (19) can be\nrewritten as\n∂QSP\nA\n∂t=/parenleftbigg∂QA\n∂t/parenrightbigg\nF2/N+/integraldisplayd2\n0dx/parenleftbigg∂QV\n∂t/parenrightbigg\nF2\n+/parenleftbigg∂QA\n∂t/parenrightbiggL\nF1/N+/integraldisplay0\n−d1dx/parenleftbigg∂QV\n∂t/parenrightbigg\nF1,(23)\nwhere the F 2/N interface dissipation, ( ∂QA/∂t)F2/N\nin Eq. (23), and the F 1and F 2bulk dissipations,\n(∂QV/∂t)F1and (∂QV/∂t)F2, are defined from Eqs. (17)\nand (18). As discussed below, Eq. (23) describes the\nenergy dissipation process carried by the spin current.\nTherefore, we define Eq. (23), or equivalently, Eq. (19),\nthe dissipation due to spin pumping.\nWith the help of Figs. 2 (a) and 2 (b) we now discuss\nthe physical interpretation of Eq. (23), which schemati-\ncallyshowtheflowsofspinangularmomentumandofen-\nergy. In spin pumping one usually focuses attention only\non the flow of spin angular momentum, i.e., spin current,\nbut because we are also interested in energy dissipation\nwealsoshowenergyflow. Whenthepumped angularmo-\nmentum reachesthe F 2/Ninterface, partofit isabsorbed\nin the F 2layer, and is depolarized by scattering at the\nF2/N interface and by spin flip and spin diffusion within\nthe F2layer. The remaining part returns to the F 1/N in-\nterface, which we call back flow. The back flow to the F 1\nlayer is relaxed by scattering at the F 1/N interface and\nby spin flip and spin diffusion within the F 1layer, where\nthe transverse component of the back flow at the F 1/N\ninterfacerenormalizesthe pumped spin current. In terms\nof the energy flow shown in Fig. 2 (b), spin absorption\nat the F 2/N interface leads to the interface dissipation\n(∂QA/∂t)F2/Nand bulk dissipation ( ∂QV/∂t)F2due to\nspin depolarization. The back flow at the F 1layer also\ngives the interface dissipation ( ∂QA/∂t)L\nF1/Nand bulk\ndissipation ( ∂QV/∂t)F1. The total dissipation is the sum\nof these dissipations, as indicated by Eq. (23). In other\nwords, the transferred energy from the F 1layer to the5\nT\nLpump\nabsorbed\nback(a)\nF1 F2m1 m2\npump\nback(b)\nF1 F2(∂ ̦V/∂t)F2\n(∂ ̦V/∂t)F1(∂ ̦A/∂t)F2/N \n(∂ ̦A/∂t)F1/N L∂̦ASP /∂t \nFIG. 2: Schematic views of the flows of (a) angular momen-\ntum and (b) energy from the microwave to the ferromagnetic\nmultilayer, in which ”L” and ”T” define the longitudinal and\ntransverse components with respect to m1.\nconduction electrons at the F 1/N interface is not local-\nized, and is dissipated throughout the system. Then,\nEq. (23), or equivalently, Eq. (19), can be regarded as\nthe dissipation due to spin pumping. Also, Eq. (21) is\nregarded as the energy transfer from the F 1layer to the\nconduction electrons near the F 1/N interface. Appendix\nA shows that all terms on the right-hand side of Eq. (23)\nare positive, thus guaranteeingthe second law of thermo-\ndynamics.\nTo conclude this section, it is of interest to compare\nEq. (19) with the dissipation due to electric current.\nLet us assume that an electric current is flowing through\na multilayer, driven by a voltage difference across two\nelectrodes. The total dissipation per unit area per unit\ntime is obtained from Eqs. (17) and (18) as [5]\n∂QEC\nA\n∂t=Je\ne[¯µ(∞)−¯µ(−∞)], (24)\nwhere[¯µ(∞)−¯µ(−∞)]/eisthevoltagedifferencebetween\nthe electrodes. Comparing Eq. (19) with (24), we notice\nthat the net transverse spin current and the difference in\nthe spin accumulation at the F 1/N interface correspond\nto the electric current and applied voltage, respectively,\nand that in spin pumping the F 1/N interface plays the\nrole of the electrode, This is because the angular momen-\ntum and the energy transferred from the magnetization\nof the F 1layer to the conduction electron are pumped\nfrom this interface to the multilayer.V. EVALUATION OF DISSIPATION\nIn this section, we quantitatively evaluate the dissipa-\ntion due to spin pumping, Eq. (19). Substituting the\nsolutions of Eqs. (10) and (12) into Eq. (2), the to-\ntal spin currents at the F 1/N and F 2/N interfaces are,\nrespectively, expressed as\nIpump\ns+IF1→N\ns=/planckover2pi1\n4π/parenleftbigg\n˜g↑↓\nr(F1)m1×dm1\ndt+ ˜g↑↓\ni(F1)dm1\ndt/parenrightbigg\n−1\n4π/bracketleftBig\n˜g∗\nF1(m1·µN)m1+ ˜g↑↓\nr(F1)m1×(µN×m1)\n+˜g↑↓\ni(F1)µN×m1/bracketrightBig\n,\n(25)\nIF2→N\ns=−1\n4π/bracketleftBig\n˜g∗\nF2(m2·µN)m2+ ˜g↑↓\nr(F2)m2×(µN×m2)\n+˜g↑↓\ni(F2)µN×m2/bracketrightBig\n.\n(26)\nThe renormalized conductances, ˜ g∗and ˜g↑↓\nr,i, are defined\nby the following ways:\n1\n˜g∗=2\n(1−γ2)g+1\ngsdtanh(d/λsd(L)),(27)\n/parenleftbigg˜g↑↓\nr\n˜g↑↓\ni/parenrightbigg\n=1\nK2\n1+K2\n2/parenleftbigg\nK1K2\n−K2K1/parenrightbigg/parenleftbiggg↑↓\nr\ng↑↓\ni/parenrightbigg\n,(28)\nwheregsd=h(1−β2)S/(2e2ρλsd(L)), andρ= 1/(σ↑+σ↓)\nis the resistivity. The terms K1andK2are defined as\nK1= 1+t↑↓\nrRe/bracketleftbigg1\ngttanh(d/ℓ)/bracketrightbigg\n+t↑↓\niIm/bracketleftbigg1\ngttanh(d/ℓ)/bracketrightbigg\n,\n(29)\nK2=t↑↓\niRe/bracketleftbigg1\ngttanh(d/ℓ)/bracketrightbigg\n−t↑↓\nrIm/bracketleftbigg1\ngttanh(d/ℓ)/bracketrightbigg\n,\n(30)\nwheregt=hSσ↑↓/(e2ℓ). In the ballistic transport limit\nfor the transverse spin, ˜ g↑↓equals to g↑↓. Then, we ex-\npandµNasµN=/planckover2pi1(ωasinθm1+b˙m1+cm1×˙m1), where\ny=δy/∆ (y=a,b,c) are dimensionless coefficients de-\ntermined by Eq. (3) with Eqs. (25) and (26). In the\nlimit ofg↑↓\nr≫g↑↓\ni[25],δb= 0, and ∆, δa, andδcare\ngiven by\n∆ =/parenleftBig\n˜g↑↓\nr(F1)+ ˜g↑↓\nr(F2)/parenrightBig/bracketleftBig/parenleftBig\n˜g∗\nF1+ ˜g∗\nF2cos2θ+˜g↑↓\nr(F2)sin2θ/parenrightBig\n×/parenleftBig\n˜g↑↓\nr(F1)+ ˜g↑↓\nr(F2)cos2θ+ ˜g∗\nF2sin2θ/parenrightBig\n−/parenleftBig\n˜g↑↓\nr(F2)−˜g∗\nF2/parenrightBig2\nsin2θcos2θ/bracketrightbigg\n,\n(31)6\nδa= ˜g↑↓\nr(F1)/parenleftBig\n˜g↑↓\nr(F1)+ ˜g↑↓\nr(F2)/parenrightBig/parenleftBig\n˜g↑↓\nr(F2)−˜g∗\nF2/parenrightBig\nsinθcosθ,\n(32)\nδc=˜g↑↓\nr(F1)/parenleftBig\n˜g↑↓\nr(F1)+ ˜g↑↓\nr(F2)/parenrightBig\n×/parenleftBig\n˜g∗\nF1+ ˜g∗\nF2cos2θ+ ˜g↑↓\nr(F2)sin2θ/parenrightBig\n.(33)\nEquation (19) in the limit of g↑↓\nr≫g↑↓\niis then given\nby\n∂QSP\nA\n∂t=/planckover2pi1ω2sin2θ˜g↑↓\nr(F1)(1−c)\n4πS\n×/braceleftbigg\nc+ ˜g↑↓\nr(F1)(1−c)Re/bracketleftbigg1\ngttanh(d1/ℓ)/bracketrightbigg/bracerightbigg\n.\n(34)\nIn the ballistic transport limit of the transverse spin, Eq.\n(34) is simplified to /planckover2pi1ω2g↑↓\nr(F1)(1−c)c/(4πS). We empha-\nsize that Eq. (34) is proportional to the enhancement of\nthe Gilbert damping by spin pumping [20,26]:\nα′=γ0/planckover2pi1˜g↑↓\nr(F1)(1−c)\n4πMSd 1, (35)\nwhereγ0is the gyromagnetic ratio. Here, α′is derived\nin the following way. According to the conservation law\nof the total angular momentum, the pumped spin from\nthe F1/N interface per unit time, ds/dt, should equal to\nthe time change of the magnetization in the F 1layer,\ni.e., a torque dm1/dt= [(gµB)/(/planckover2pi1MSd)]ds/dtacts on\nm1, whereM/(gµB) is the number of the magnetic mo-\nments in the F 1layer, and the Land´ e g-factor satisfies\ngµB=γ0/planckover2pi1. This torque, [( gµB)/(/planckover2pi1MSd)]ds/dt, with\nds/dt=Ipump\ns+m1×(IF1→N\ns×m1), can be expressed as\nα′m1×(dm1/dt). Then, α′is identified as the enhance-\nment of the Gilbert damping constant due to the spin\npumping. The present result indicating that the dissi-\npation is proportional to α′represents that the pumped\nspin current at the F 1/N interface carries not only the\nangular momentum but also the energy from the F 1to\nN layer.\nWe quantitatively evaluate Eq. (34) by using parame-\nters taken from experiments for the NiFe/Cu multilayer\nwith the assumption β=β′[23,26,29,43]; ( h/e2)S/[(1−\nγ2)g] = 0.54kΩnm2,γ= 0.7,g↑↓\nr/S= 15nm−2,g↑↓\ni/S=\n1 nm−2,t↑↓\nr/S=t↑↓\ni/S= 4 nm−2,ρ= 241 Ωnm,\nβ= 0.73,λsd(L)= 5.5 nm,λsd(T)=λsd(L)//radicalbig\n1−β2,\nλJ= 2.8 nm,d= 5 nm, γ0= 1.8467×1011rad/(T s),\nM= 605×103A/m, and ω= 2π×9.4×109rad/s, re-\nspectively, where the parameters of the F 1and F2layers\nare assumed to be identical, for simplicity. In Fig. 3 (a),\nwe show the dissipation due to spin pumping, Eq. (34),\nfor an arbitrary cone angle θ. The damping α′, Eq. (35),\nis also shown in Fig. 3 (b). The cone angle θin typi-\ncal FMR experiments [7,8] is small. However, the spin\npumping affects not only the FMR experiment but alsocone angle of magnetization m 1, θ (deg)0 30 60 90 120 150 180 dissipation (fJ/nm 2s) \n080 \n20 \ncone angle of magnetization m 1, θ (deg)0 30 60 90 120 150 18000.0010.0020.004\n0.003damping,  α \n‘40 60 (a)\n(b)\nFIG. 3: Dependencies of (a) the dissipation due to pure spin-\ncurrent, Eq. (19), and (b) the damping, α′, Eq. (35), on the\ncone angle θ.\nspin torque switching [37], in which θvaries from 0◦to\n180◦. Therefore, we show the dissipation and damping\nfor the whole range of θin Fig. 3.\nThe dissipation is zero for θ= 0◦and 180◦because\ndm1/dt=0at these angles. The maximum dissipation\nis about 60 fJ/(nm2s). To understand how large this\ndissipation is, we compare this value with the dissipation\ndue to spin torque switching current in the same system;\nwe discuss this in the next section.\nTo conclude this section, we briefly mention that the\ndissipation due to spin pumping can be evaluated not\nonly from Eq. (19) but also from Eq. (23). Appendix B\ngives explicit forms for each term on the right-hand side\nofEq. (23), from whichthe dissipationcan be calculated.\nVI. COMPARISON WITH SPIN TORQUE\nSWITCHING\nSpinpumpingoccursnotonlyinFMRexperimentsbut\nalso in spin torque switching experiments. An important\nissue in the spin torque switching problem is the reduc-\ntion of power consumption due to heating [44]. Whereas\nheating has usually meant the dissipation due to electric\ncurrent, the results of the previous section indicate that\nspin pumping also contributes to the dissipation. Thus\nit is of interest to quantitatively evaluate the dissipation\ndue to the electric current, and compare it with that due7\nto spin pumping studied in the previous section, which\nwill clarify the ratio of the contribution of spin pumping\nto heating in the spin torque switching experiment.\nWe assume that an electric current Iis injected from\nthe F2layer to the F 1layer. Then, a term\nIFk→N\ns(e)=/planckover2pi1γ\n2eIFk→Nmk, (36)\nshould be added to Eq. (2), which represents a spin cur-\nrent due to the electric current [25]. The current IFk→N\nis the electric current which flows from the F klayer to\nthe N layer, meaning that IF1→N=−IF2→N=−I. As\nin the system studied in the previous section, we assume\nthat the spin current is zero at both ends of the ferro-\nmagnet. Taking into account Eq. (36), Eqs. (25) and\n(26) are replaced by\nIpump\ns+IF1→N\ns=/planckover2pi1\n4π˜g↑↓\nrm1×dm1\ndt\n−1\n4π/bracketleftbigg\n˜g∗(m1·µN)m1+h˜g∗I\n˜geem1+˜g↑↓\nrm1×(µN×m1)/bracketrightbigg\n,\n(37)\nIF2→N\ns=−1\n4π/bracketleftbigg\n˜g∗(m2·µN)m2−h˜g∗I\n˜geem2\n+˜g↑↓\nrm2×(µN×m2)/bracketrightbig\n,(38)\nwhere, as done in the previous section, we assume that\nthe material parameters of two ferromagnets are iden-\ntical, and thus, omit subscripts ”F k” from the conduc-\ntances, for simplicity. We also assume that g↑↓\nr≫g↑↓\ni. A\nnew conductance ˜ geis defined as\n1\n˜ge=2γ\n(1−γ2)g+β\ngsdtanh/parenleftbiggd\n2λsd(L)/parenrightbigg\n.(39)\nA characteristic current of the spin torque switching\nis the critical current of the magnetization dynamics Ic,\nwhich can be defined as the current canceling the Gilbert\ndamping torque of the F 1layer at the equilibrium state\n[38]. The equilibrium state in the present study corre-\nsponds to θ= 0◦. In this limit ( θ→0), Eq. (35) is\nreplaced by\nα′=γ0/planckover2pi1˜g↑↓\nr\n4πMSd 1/parenleftbigg1\n2−π˜g∗I\neω˜g↑↓\nr˜ge/parenrightbigg\n. (40)\nWe assume that the Gilbert damping purely comes from\nthe spin pumping. Then, the critical current is defined\nas the current satisfying α′= 0; i.e.,\nIc=eω˜g↑↓\nr˜ge\n2π˜g∗. (41)\nUsing the same parameter values as in the previous sec-\ntion, the critical current density Jc=Ic/Sis estimated\nas 6.3×106A/cm2. This value is about the same or-\nder of an experimentally observed value [45] ( ∼6×106A/cm2on average) of the critical current having a mag-\nnetic anisotropy field HK, whose magnitude (1-3 kOe) is\nabout the same order of the parameter value, ω/γ0≃3.2\nkOe, used here. The dissipation due to this electric\ncurrent based on the conventional Joule heating for-\nmula,∂QEC\nA/∂t=/summationtext\nk[ρJ2\ncdk+rFk/NJ2\nc], is evaluated\nas 11.8×103fJ/(nm2s), where rF/N= (h/e2)S/gis the\nF/N interface resistance. This value of the dissipation is\ntwo to three orders of magnitude larger than the dissi-\npation due to the spin pumping studied in the previous\nsection.\nWe briefly investigate the origins of a large differ-\nence between the dissipations due to the spin and elec-\ntric currents. Let us assume that the bulk and inter-\nface spin polarizations ( βandγ) are identical, and that\nthe thickness of the ferromagnetic layer is much larger\nthan the spin diffusion length ( d≫λsd(L)), for sim-\nplicity, from which the critical current is simplified as\nIc=eω˜g↑↓\nr/(2πβ). Then the ratio between the dissi-\npations due to spin pumping and electric current be-\ncomes (∂QSP\nA/∂t)/(∂QEC\nA/∂t)∼β2/planckover2pi1/[e2(˜g↑↓\nr/S)(ρd+r)].\nThe square of the spin polarization, β2, is on the or-\nder of 10−1. Also, the orders of [( h/e2)S/(˜g↑↓\nrr)] and\nr/ρdare 1 and 0 .1, respectively. Then, the ratio\n(∂QSP\nA/∂t)/(∂QEC\nA/∂t) is roughly 10−2, which is roughly\nconsistent with the above evaluation. This consideration\nimplies that a largedissipation due to the electric current\ncomes from the smallness of the spin polarization. Also,\na large bulk resistivity ( ρ), in addition to the interface\nresistance ( r), also contributes to the large dissipation\ndue to the electric current, whereas only the interface\nresistance contributes to the spin pumping dissipation\nbecause spin pumping is an interface effect.\nToconclude this section, wemention that the total dis-\nsipation in the FMR consists of that due to spin pump-\ning, Eq. (34), and that due to the intrinsic damping in\nthe F1layer. One can consider the possibility that the\ntotal dissipation in the FMR might become comparable\nto or exceed the dissipation due to the electric current\n(calculated above) when the dissipation due to intrinsic\nmagnetic damping is included, despite the fact the dissi-\npation due to spin pumping is small. However, we found\nthat the intrinsic damping constant α0should be at least\nontheorderof0 .1−1tomakethedissipationinthe FMR\ncomparable with that due to the electric current; see Ap-\npendix C. On the other hand, the experimental value of\nthe intrinsic Gilbert damping constant is on the order of\n0.001−0.01 [46]. Therefore, the dissipation in the FMR\nis still much smaller than that due to the electric current\neven after the dissipation due to the intrinsic damping\nis taken into account. The energy supplied by the mi-\ncrowave to the F 1layer is divided into the power to sus-\ntain the magnetization precessionand that transferredto\nthe conduction electrons near the F 1/N interface, where\ntheir ratio is roughly α0:α′. The former ( ∝α0) is dissi-\npatedbythebulkmagneticdissipationwhereasthe latter\n(∝α′) is dissipated by the spin-flip processes and spin-\ndependent scatterings within bulk and at the interface,8\nas shown by Eq. (23).\nVII. CONCLUSION\nThe dissipation and heating due to a pure spin-current\ngenerated by spin pumping in a ferromagnetic/ nonmag-\nnetic /ferromagneticmultilayerwasquantitativelyinves-\ntigated. Usingspin-dependenttransporttheoryandther-\nmodynamics we generalized the Joule heating formula in\nthe presence of spin current flowing in a ferromagnetic\nmultilayer. The bulk and interface dissipation formulas\nare given by Eqs. (17) and (18), respectively. For spin\npumping, the transferred energy from the ferromagnet to\nthe conduction electrons is not localized at the interface,\nand is dissipated throughout the system by the flow of a\npure spin-current, as shown by Eq. (23). The dissipation\ndue to the spin pumping, Eq. (34), is proportional to the\nenhancement of the Gilbert damping by spin pumping,\nEq. (35). Usingtypicalvaluesofparametersinametallic\nmultilayer system, the amount of the dissipation at max-\nimum is estimated to be twoto three ordersof magnitude\nsmaller than the dissipation due to the electric current\nfor spin torque switching.\nAcknowledgement\nThe authors would like to acknowledge M. D. Stiles, P.\nM. Haney, G. Khalsa, R. Jansen, T. Yorozu, H. Maehara,\nA. Emura, T. Nozaki, H. Imamura, S. Tsunegi, H. Kub-\nota, S. Yuasa, and Y. Utsumi. This work was supported\nby JSPS KAKENHI Grant-in-Aid for Young Scientists\n(B) 25790044.\nAppendix A: Non-negativity of bulk and interface\ndissipations\nIn this Appendix, we prove that all terms on the right-\nhand side of Eq. (23) are positive, which guarantees the\nsecond law of thermodynamics; i.e., the dissipation, or\nrate of the entropy production, is positive [41]. Here, we\nomit the subscript “F k” (k= 1,2) from conductances,\nfor simplicity.\nFirst, we prove the non-negativity of the longitudinal\nand transverse parts of the bulk dissipation. The longi-\ntudinal part of Eq. (17) can be rewritten as\n/parenleftbigg∂QV\n∂t/parenrightbiggL\n=Je\ne∂¯µ\n∂x−∂\n∂xJL\ns\n/planckover2pi1·µL\n=−/summationdisplay\nν=↑,↓jν∂¯µν\n∂x−(¯µ↑−¯µ↓)\n2∂\n∂x(j↑−j↓)\n=/summationdisplay\nν=↑,↓e2\nσν(jν)2+(1−β2)\n4e2ρλ2\nsd(L)(¯µ↑−¯µ↓)2,\n(A1)which is clearly positive. Here, we use the relation ∂(j↑−\nj↓)/∂x=−(1−β2)(¯µ↑−¯µ↓)/(2e2ρλ2\nsd(L)). Also, we can\nconfirm from Eqs. (12) and (13) that the transversepart,\n/parenleftbigg∂QV\n∂t/parenrightbiggT\n=−∂\n∂xJT\ns\n/planckover2pi1·µT\n=2e2\n/planckover2pi12σ↑↓/parenleftbig\nJT\ns/parenrightbig2+σ↑↓\n2e2λ2\nsd(T)/parenleftbig\nµT/parenrightbig2,(A2)\nis positive. Therefore, the bulk dissipation is positive at\nanyx.\nNext, let us prove the non-negativity of the interface\ndissipation by using the solutions of the spin current and\nspin accumulation (see also Appendix B). The longitudi-\nnal part of the F 1/N interface dissipation can be written\nas\n/parenleftbigg∂QA\n∂t/parenrightbiggL\nF1/N=˜g∗\n4π/planckover2pi1S/bracketleftbigg\n1−˜g∗\ngsdtanh(d1/λsd(L))/bracketrightbigg\n(m1·µN)2.\n(A3)\nAccording to Eq. (27), 1 −˜g∗/[gsdtanh(d1/λsd(L))] is\nlarger than zero. Therefore, the longitudinal part of the\nF1/N interface dissipation is positive. The longitudinal\npart of the F 2/N interface dissipation,\n/parenleftbigg∂QA\n∂t/parenrightbigg\nF2/N=JF2→N\ns\n/planckover2pi1·(µF2−µN),(A4)\nis positive because of the same reason. The transverse\npart of the F 2/N interface dissipation,\n/parenleftbigg∂QA\n∂t/parenrightbiggT\nF2/N=˜g↑↓\nr\n4π/planckover2pi1S/braceleftbigg\n1−˜g↑↓\nrRe/bracketleftbigg1\ngttanh(d2/ℓ)/bracketrightbigg/bracerightbigg\n×/bracketleftBig\nµ2\nN−(m2·µN)2/bracketrightBig\n,\n(A5)\nis also positive due to similar reasons, where we use ap-\nproximation ˜ g↑↓\nr≫˜g↑↓\niused in Sec. V for simplicity.\nAppendix B: Theoretical formulas for bulk and\ninterface dissipation\nIn this Appendix, we discuss how to calculate the dis-\nsipation due to spin pumping from Eq. (23). To this\nend, we first show the solutions for the spin current and\nspin accumulation in the F 1and F2layers because each\nterm on the right-hand-side of Eq. (23) consists of spin\ncurrent and spin accumulation, as shown in Eqs. (17)\nand (18). The general solution for the spin current and\nspin accumulation are summarized in our previous work\n[47]. Here, we use these solutions, and express the spin\ncurrentand spin accumulation in terms ofthe coefficients\naandcofµNdefined in Sec. V with the assumptions\n˜g↑↓\nr≫˜g↑↓\ni.\nFirst, we present the theoretical formulas for the spin\ncurrent and spin accumulation within the F 1layer. We9\nintroduce two unit vectors t1=m1×˙m1/|m1×˙m1|and\nt2=−˙m1/|˙m1|, which are orthogonal to the magnetiza-\ntionm1and satisfy t1×t2=m1, because the transverse\ncomponents of the spin current and spin accumulation,\nEqs. (5) and (7), can be projected to these two direc-\ntions. Then, the longitudinal and transverse components\nof the spin current in the F 1layer are given by\nm1·Is(F1)=−/planckover2pi1ω˜g∗asinθ\n4πsinh[(x+d1)/λsd(L)]\nsinh(d1/λsd(L)),(B1)\nt1·Is(F1)=/planckover2pi1ω˜g↑↓\nr(1−c)sinθ\n4πRe/bracketleftbiggsinh[(x+d1)/ℓ]\nsinh(d1/ℓ)/bracketrightbigg\n,\n(B2)\nt2·Is(F1)=/planckover2pi1ω˜g↑↓\nr(1−c)sinθ\n4πIm/bracketleftbiggsinh[(x+d1)/ℓ]\nsinh(d1/ℓ)/bracketrightbigg\n.\n(B3)\nWe can confirm that the sum of these components,\n(m1·I)m1+(t1·Is)t1+(t2·Is)t2, atx= 0 is identical\nto the spin current at the F 1/N interface, Ipump\ns+IF1→N\ns.\nSimilarly, the longitudinal and transverse spin accumu-\nlation in the F 1layer are given by\nm1·µF1=/planckover2pi1ω˜g∗asinθ\ngsdcosh[(x+d1)/λsd(L)]\nsinh(d1/λsd(L)),(B4)\nt1·µF1=−/planckover2pi1ω˜g↑↓\nr(1−c)sinθRe/bracketleftbiggcosh[(x+d1)/ℓ]\ngtsinh(d1/ℓ)/bracketrightbigg\n,\n(B5)\nt2·µF1=−/planckover2pi1ω˜g↑↓\nr(1−c)sinθIm/bracketleftbiggcosh[(x+d1)/ℓ]\ngtsinh(d1/ℓ)/bracketrightbigg\n.\n(B6)\nNext, we present the explicit forms of the spin current\nand spin accumulation in the F 2layer. The magneti-\nzationm2can be expressed in terms of ( t1,t2,m1) as\nm2= cosθm1+ sinθt1. We introduce two unit vec-\ntors,u1=−sinθm1+ cosθt1andu2=t2satisfying\nu1×u2=m2, to decompose the transverse compo-\nnent. In terms of ( u1,u2,m2),µNcan be expressed as\nµN=/planckover2pi1ωsinθ[(acosθ+csinθ)m2+(−asinθ+ccosθ)u1].\nThen, the longitudinal and transverse spin currents are\ngiven by\nm2·Is(F2)=−/planckover2pi1ω˜g∗(asinθcosθ+csin2θ)\n4π\n×sinh[(x−d2)/λsd(L)]\nsinh(d2/λsd(L)),(B7)\nu1·Is(F2)=−/planckover2pi1ω˜g↑↓\nr(−asin2θ+csinθcosθ)\n4π\n×Re/bracketleftbiggsinh[(x−d2)/ℓ]\nsinh(d2/ℓ)/bracketrightbigg\n,(B8)u2·Is(F2)=−/planckover2pi1ω˜g↑↓\nr(−asin2θ+csinθcosθ)\n4π\n×Im/bracketleftbiggsinh[(x−d2)/ℓ]\nsinh(d2/ℓ)/bracketrightbigg\n.(B9)\nWe can confirm that the sum of these components, ( m2·\nI)m2+ (u1·Is)u1+ (u2·Is)u2, atx= 0 is identical\nto the spin current at the F 2/N interface, −IF2→N\ns. The\nlongitudinal and transverse spin accumulations are given\nby\nm2·µF2=/planckover2pi1ω˜g∗(asinθcosθ+csin2θ)\ngsd\n×cosh[(x−d2)/λsd(L)]\nsinh(d2/λsd(L)),(B10)\nu1·µF2=/planckover2pi1ω˜g↑↓\nr(−asin2θ+csinθcosθ)\n×Re/bracketleftbiggcosh[(x−d2)/ℓ]\ngtsinh(d2/ℓ)/bracketrightbigg\n,(B11)\nu2·µF2=/planckover2pi1ω˜g↑↓\nr(−asin2θ+csinθcosθ)\n×Im/bracketleftbiggcosh[(x−d2)/ℓ]\ngtsinh(d2/ℓ)/bracketrightbigg\n.(B12)\nFigures 4 (a) and (b) show the spatial distributions of\nthe spin current density and spin accumulation, respec-\ntively. The spin current density and spin accumulation\nare decomposed into the longitudinal and transverse di-\nrections, where the solid lines correspond to the longitu-\ndinal components whereas the dotted ( ∝bardblt1oru1) and\ndashed (∝bardblt2oru2) correspond to the transverse compo-\nnents. The valuesoftheparametersareidenticalto those\nused in Sec. V with θ= 45◦. Because spin pumping oc-\ncurs at the F 1/N interface, the spin current density and\nspin accumulation are concentrated near this interface.\nWe emphasize that the spatial directions of the longitu-\ndinal and transverse spin are different between the F 1\nand F2layers when the magnetizations, m1andm2, are\nnoncollinear; as a result the spin current in Fig. 4 (a)\nlooks discontinuous at the interface, although Eq. (3) is\nsatisfied.\nWe now consider the dissipation formulas. The lon-\ngitudinal and transverse parts of the bulk dissipation in\nthe F1layer can be expressed as\n/parenleftbigg∂QV\n∂t/parenrightbiggL\nF1=/planckover2pi1ω2\n4πS˜g∗2a2sin2θ\ngsdλsd(L)sinh2(d1/λsd(L))\n×cosh/bracketleftbigg2(x+d1)\nλsd(L)/bracketrightbigg\n,(B13)\n/parenleftbigg∂QV\n∂t/parenrightbiggT\nF1=/planckover2pi1ω2˜g↑↓2\nr(1−c)2sin2θ\n4πS2e2\nhσ↑↓\n×/braceleftBigg\n1\nλ2\nsd(T)/vextendsingle/vextendsingle/vextendsingle/vextendsingleℓcosh[(x+d1)/ℓ]\nsinh(d1/ℓ)/vextendsingle/vextendsingle/vextendsingle/vextendsingle2\n+/vextendsingle/vextendsingle/vextendsingle/vextendsinglesinh[(x+d1)/ℓ]\nsinh(d1/ℓ)/vextendsingle/vextendsingle/vextendsingle/vextendsingle2/bracerightBigg\n.(B14)10\nposition, x (nm)-5 0 5-1010 \n1 2 3 4 -1 -2 -3 -4 30 \n0F1 F2(b)position, x (nm)-5 0 5 spin current density (10 -25  J/nm 2)\n010 \n1 2 3 4 -1 -2 -3 -4 15 \n5F1 F2(a)\nposition, x (nm)-5 0 5 bulk dissipation (fJ/nm 3s) \n010 \n1 2 3 4 -1 -2 -3 -4 15 \n5F1 F2(c)-5  spin accumulation (10 -25  J) \n20 \nFIG. 4: Examples of the distributions of (a) longitudi-\nnal (solid) and transverse (dotted and dashed) spin current\ndensities, (b) longitudinal (solid) and transverse (dotte d and\ndashed) spin accumulations, and (c) bulk dissipations for\nθ= 45◦.\nSimilarly, the longitudinal and transverse parts of the\nbulk dissipation in the F 2layer can be expressed as\n/parenleftbigg∂QV\n∂t/parenrightbiggL\nF2=/planckover2pi1ω2\n4πS˜g∗2(asinθcosθ+csin2θ)2\ngsdλsd(L)sinh2(d2/λsd(L))\n×cosh/bracketleftbigg2(x−d2)\nλsd(L)/bracketrightbigg\n.(B15)\n/parenleftbigg∂QV\n∂t/parenrightbiggT\nF2=/planckover2pi1ω2˜g↑↓2\nr(−asin2θ+csinθcosθ)2\n4πS2e2\nhσ↑↓\n×/braceleftBigg\n1\nλ2\nsd(T)/vextendsingle/vextendsingle/vextendsingle/vextendsingleℓcosh[(x−d2)/ℓ]\nsinh(d2/ℓ)/vextendsingle/vextendsingle/vextendsingle/vextendsingle2\n+/vextendsingle/vextendsingle/vextendsingle/vextendsinglesinh[(x−d2)/ℓ]\nsinh(d2/ℓ)/vextendsingle/vextendsingle/vextendsingle/vextendsingle2/bracerightBigg\n.\n(B16)\nFigure 4 (c) shows the spatial distribution of the bulk\ndissipation, which is also concentrated near the interface.\nThe longitudinal part of the F 1/N interface dissipation\nand the longitudinal and transverse parts of the F 2/Ninterface dissipations are given by\n/parenleftbigg∂QA\n∂t/parenrightbiggL\nF1/N=/planckover2pi1ω2˜g∗a2sin2θ\n4πS/bracketleftbigg\n1−˜g∗\ngsdtanh(d1/λsd(L))/bracketrightbigg\n,\n(B17)\n/parenleftbigg∂QA\n∂t/parenrightbiggL\nF2/N=/planckover2pi1ω2˜g∗(asinθcosθ+csin2θ)2\n4πS\n×/bracketleftbigg\n1−˜g∗\ngsdtanh(d2/λsd(L))/bracketrightbigg\n,(B18)\n/parenleftbigg∂QA\n∂t/parenrightbiggT\nF2/N=/planckover2pi1ω2˜g↑↓\nr(−asin2θ+csinθcosθ)2\n4πS\n×/braceleftbigg\n1−˜g↑↓\nrRe/bracketleftbigg1\ngttanh(d2/ℓ)/bracketrightbigg/bracerightbigg\n.\n(B19)\nForθ= 45◦, we quantitatively evaluate\nthat/integraltext0\n−d1dx(∂QV/∂t)L\nF1= 3.34 fJ/(nm2s),/integraltext0\n−d1dx(∂QV/∂t)T\nF1= 6 .51 fJ/(nm2s),/integraltextd2\n0dx(∂QV/∂t)L\nF2= 18 .15 fJ/(nm2s), and/integraltextd2\n0dx(∂QV/∂t)T\nF2= 4.95 fJ/(nm2s), respectively. Also,\nthe interface dissipations are quantitatively evaluated as\n(∂QA/∂t)L\nF1/N= 0.44 fJ/(nm2s), (∂QA/∂t)L\nF2/N= 2.39\nfJ/(nm2s), and ( ∂QA/∂t)T\nF2/N= 8.03 fJ/(nm2s) for\nθ= 45◦, respectively. We can confirm that the value of\nthe dissipation evaluated from these values as Eq. (23)\nis the same with that evaluated from Eq. (19) with Fig.\n3.\nAppendix C: Dissipation due to intrinsic damping\nIn this Appendix, we briefly evaluate the dissipation\ndue to the magnetization precession in the FMR experi-\nment, whicharisesfromtheintrinsicGilbertdamping. In\nthe FMR, the energysupplied by the microwavebalances\nwith the dissipation due to the damping, and the mag-\nnetization precesses practically on the constant energy\ncurve. The magnetization dynamics with the macrospin\nassumption is described by the Landau-Lifshitz-Gilbert\n(LLG) equation\ndm1\ndt=−γ0m1×H−α0γ0m1×(m1×H),(C1)\nwherethemagneticfield Hrelatestothemagneticenergy\ndensityEviaH=−∂E/∂(Mm1). From Eq. (C1), the\nchange of the energy density averaged on the constant\nenergy curve is given by\ndE\ndt≡1\nτ/contintegraldisplay\ndtdE\ndt\n=−αγ0M\nτ/contintegraldisplay\ndt/bracketleftBig\nH2−(m1·H)2/bracketrightBig\n,(C2)11\nwhere,τ=/contintegraltext\ndtis the precession period on a constant\nenergy curve. Assuming that the ferromagnet has uniax-\nial anisotropy H= (0,0,HKmz) as done in Sec. VI, Eq.\n(C2) is given by\ndE\ndt=−α0γ0MH2\nKsin2θcos2θ. (C3)\nThe microwaveshould supply the energydensity −dE/dt\nto sustain the precession. Then, the energy sup-\nplied by the microwave per unit area per unit time is\nα0γ0MH2\nKd1sin2θcos2θ, whered1is the thickness of the\nferromagnet. Comparing this energywith the dissipation\ndue to the spin pumping carried by the spin current, Eq.\n(34), the ratio of the dissipation between the intrinsicdamping and spin pumping is\n|dE/dt|d1\n∂QSP\nA/∂t∼α0\nα′, (C4)\nwhereα′is given by Eq. (35). The dissipation due to\nthe spin pumping ( ∝α′) is two to three orders of mag-\nnitude smaller than the dissipation due to the electric\ncurrent. Therefore, the intrinsic Gilbert damping con-\nstantα0giving bulk magnetic dissipation of the same\norder of magnitude as the dissipation due to the electric\ncurrent is roughly 102−3×α′. From the value of α′in\nFig. 3 (b), this gives an α0on the order of 0 .1−1.\n1M. Johnson and R. H. Silsbee, Phys. Rev. B 35, 4959\n(1987).\n2J. E. Parrott, IEEE Trans. Electr. Dev. 43, 809 (1996).\n3W. M. Saslow, Phys. Rev. B 76, 184434 (2007).\n4M. R. Sears and W. M. Saslow, Can. J. Phys. 89, 1041\n(2011).\n5A. A. Tulapurkar and Y. Suzuki, Phys. Rev. B 83, 012401\n(2011).\n6J. Rammer, Quantum Transport Theory (Westview Press,\n2008), chap. 5.\n7S. Mizukami, Y. Ando, and T. Miyazaki, J. Magn. Magn.\nMater.239, 42 (2002).\n8S. Mizukami, Y. Ando, and T. Miyazaki, Phys. Rev. B 66,\n104413 (2002).\n9K. Ando, S. Takahashi, K. Harii, K. Sasage, J. Ieda,\nS. Maekawa, and E. Saitoh, Phys. Rev. Lett. 101, 036601\n(2008).\n10K. Ando and E. Saitoh, Nat. Commun. 3, 629 (2012).\n11K. Uchida, S. Takahashi, K. Harii, J. Ieda, W. Koshibae,\nK. Ando, S. Maekawa, and E. Saitoh, Nature 455, 778\n(2008).\n12L. Liu, C.-F. Pai, Y. Li, H. W. Tseng, D. C. Ralph, and\nR. A. Buhrman, Science 336, 555 (2012).\n13L. Liu, O. J. Lee, T. J. Gudmundsen, D. C. Ralph, and\nR. A. Buhrman, Phys. Rev. Lett. 109, 096602 (2012).\n14B. F. Miao, S. Y. Huang, D. Qu, and C. L. Chien, Phys.\nRev. Lett. 111, 066602 (2013).\n15Y. Ando, K. Ichiba, S. Yamada, E. Shikoh, T. Shinjo,\nK. Hamaya, and M. Shiraishi, Phys. Rev. B 88, 140406\n(2013).\n16J. C. Slonczewski, J. Magn. Magn. Mater. 159, L1 (1996).\n17R. H. Silsbee, A. Janossy, and P. Monod, Phys. Rev. B 19,\n4382 (1979).\n18Y. Tserkovnyak, A. Brataas, and G. E. W. Bauer, Phys.\nRev. Lett. 88, 117601 (2002).\n19Y. Tserkovnyak, A. Brataas, and G. E. W. Bauer, Phys.\nRev. B66, 224403 (2002).\n20Y. Tserkovnyak, A. Brataas, and G. E. W. Bauer, Phys.\nRev. B67, 140404 (2003).\n21Y. Tserkovnyak, A. Brataas, G. E. W. Bauer, and B. I.\nHalperin, Rev. Mod. Phys. 77, 1375 (2005).\n22S. Takahashi, Appl. Phys. Lett. 104, 052407 (2014).\n23J. Bass and J. W. P. Pratt, J. Phys.: Condens. Matter 19,183201 (2007).\n24A. Brataas, Y. V. Nazarov, and G. E. W. Bauer, Eur.\nPhys. J. B 22, 99 (2001).\n25A. Brataas, G. E. W. Bauer, and P. J. Kelly, Phys. Rep.\n427, 157 (2006).\n26T. Taniguchi, S. Yakata, H. Imamura, and Y. Ando, Appl.\nPhys. Express 1, 031302 (2008).\n27T. Valet and A. Fert, Phys. Rev. B 48, 7099 (1993).\n28E. Simanek, Phys. Rev. B 63, 224412 (2001).\n29S. Zhang, P. M. Levy, and A. Fert, Phys. Rev. Lett. 88,\n236601 (2002).\n30A. Shpiro, P. M. Levy, and S. Zhang, Phys. Rev. B 67,\n104430 (2003).\n31J. Zhang, P. M. Levy, S. Zhang, and V. Antropov, Phys.\nRev. Lett. 93, 256602 (2004).\n32F. Pi´ echon and A. Thiaville, Phys. Rev. B 75, 174414\n(2007).\n33T. Taniguchi, J. Sato, and H. Imamura, Phys. Rev. B 79,\n212410 (2009).\n34The definition of the spin accumulation obeys Refs.19. A\ncautionary note: Some papers define µ/2 as the spin ac-\ncumulation. According to this latter definition, the longi-\ntudinal spin accumulation is µL= [(¯µ↑−¯µ↓)/2]m, and\nsimilarly, the amount of the transverse spin accumulation\nis a half of that by our definition. Accordingly, /planckover2pi1in Eqs.\n(16), (17), and (18) should be replaced by /planckover2pi1/2.\n35S.HershfieldandH.L.Zhao, Phys.Rev.B 56, 3296(1997).\n36M. D. Stiles and A. Zangwill, Phys. Rev. B 66, 014407\n(2002).\n37W. Chen, M. J. Rooks, N. Ruiz, J. Z. Sun, and A. D. Kent,\nPhys. Rev. B 74, 144408 (2006).\n38T. Taniguchi and H. Imamura, Phys. Rev. B 78, 224421\n(2008).\n39A. Ghosh, S. Auffret, U. Ebels, and W. E. Bailey, Phys.\nRev. Lett. 109, 127202 (2012).\n40N. W. Ashcroft and N. D. Mermin, Solid State Physics\n(Thomson Learning, 1976), chap. 13.\n41D. Kondepudi and I. Prigogine, Mondern Thermodynam-\nics: From Heat Engines to Dissipative Structures (Wiley,\nNew York, 1998), chap. 15.\n42J. M. Ziman, Electrons and Phonons (Oxford University\nPress, New York, 2007), chap. 7.\n43A. Fert and L. Piraux, J. Magn. Magn. Mater. 200, 33812\n(1999).\n44N. Locatelli, V. Cros, and J. Grollier, Nat. Mater. 13, 11\n(2013).\n45K. Yakushiji, A. Fukushima, H. Kubota, M. Konoto, and\nS. Yuasa, Appl. Phys. Express 6, 113006 (2013).\n46M. Oogane, T. Wakitani, S. Yakata, R. Yilgin, Y. Ando,A. Sakuma, and T. Miyazaki, Jpn. J. Appl. Phys. 45, 3889\n(2006).\n47T. Taniguchi and H. Imamura, Mod. Phys. Lett. B 22,\n2909 (2008)."    },    {        "title": "1412.1988v1.Calculating_linear_response_functions_for_finite_temperatures_on_the_basis_of_the_alloy_analogy_model.pdf",        "content": "arXiv:1412.1988v1  [cond-mat.mtrl-sci]  5 Dec 2014Calculating linear response functions for finite temperatu res on the basis of the alloy\nanalogy model\nH. Ebert, S. Mankovsky, K. Chadova, S. Polesya, J. Min´ ar, and D . K¨ odderitzsch\nDepartment Chemie/Phys. Chemie, Ludwig-Maximilians-Uni versit¨ at M¨ unchen,\nButenandtstrasse 5-13, D-81377 M¨ unchen, Germany\n(Dated: 8th December 2014)\nA scheme is presented that is based on the alloy analogy model and allows to account for thermal\nlattice vibrations as well as spin fluctuations when calcula ting response quantities in solids. Various\nmodels to deal with spin fluctuations are discussed concerni ng their impact on the resulting tem-\nperature dependent magnetic moment, longitudinal conduct ivity and Gilbert damping parameter.\nIt is demonstrated that using the Monte Carlo (MC) spin config uration as an input, the alloy ana-\nlogy model is capable to reproduce results of MC simulations on the average magnetic moment\nwithin all spin fluctuation models under discussion. On the o ther hand, response quantities are\nmuch more sensitive to the spin fluctuation model. Separate c alculations accounting for either the\nthermal effect due to lattice vibrations or spin fluctuations show their comparable contributions\nto the electrical conductivity and Gilbert damping. Howeve r, comparison to results accounting for\nboth thermal effects demonstrate violation of Matthiessen’ s rule, showing the non-additive effect of\nlattice vibrations and spin fluctuations. The results obtai ned for bcc Fe and fcc Ni are compared\nwith theexperimental data, showing rather good agreement f or thetemperature dependentelectrical\nconductivity and Gilbert damping parameter.\nI. INTRODUCTION\nFinite temperature has often a very crucial influence\non the response properties of a solid. A prominent ex-\nample for this is the electrical resistivity of perfect non-\nmagnetic metals and ordered compounds that only take\na non-zero value with a characteristic temperature ( T)\ndependence due to thermal lattice vibrations. While the\nHolstein transport equation1,2provides a sound basis for\ncorresponding calculations numerical work in this field\nhas been done so far either on a model level or for sim-\nplified situations.3–6In practice often the Boltzmann-\nformalism is adopted using the constant relaxation time\n(τ) approximation. This is a very popular approach in\nparticular when dealing with the Seebeck effect, as in\nthis case τdrops out.7,8The constant relaxation time\napproximation has also been used extensively when deal-\ning with the Gilbert damping parameter α.9–11Within\nthe description of Kambersky10,12the conductivity- and\nresistivity-like intra- and inter-band contributions to α\nshow a different dependency on τleading typically to\na minimum for α(τ) or equivalently for α(T).10,11A\nscheme to deal with the temperature dependent resistiv-\nity that is formally much more satisfying than the con-\nstant relaxation time approximation is achieved by com-\nbining the Boltzmann-formalism with a detailed calcula-\ntion of the phonon properties. As was shown by various\nauthors,13–16this parameter-free approach leads for non-\nmagneticmetalsingeneraltoaverygoodagreementwith\nexperimental data.\nAs an alternative to this approach, thermal lattice\nvibrations have also been accounted for within various\nstudies by quasi-static lattice displacements leading to\nthermallyinducedstructuraldisorderinthesystem. This\npoint of view provides the basis for the use of the al-\nloy analogy, i.e. for the use of techniques to deal withsubstitutional chemical disorder also when dealing with\ntemperature dependent quasi-static random lattice dis-\nplacements. An example for this are investigations on\nthe temperature dependence of the resistivity and the\nGilbert parameter αbased on the scattering matrix ap-\nproach applied to layered systems.17The necessary aver-\nageovermanyconfigurationsoflatticedisplacementswas\ntakenbymeansofthe supercelltechnique. Incontrastto\nthistheconfigurationalaveragewasdeterminedusingthe\nCoherent Potential Approximation (CPA) within invest-\nigations using a Kubo-Greenwood-like linear expression\nforα.18The same approach to deal with the lattice dis-\nplacements was also used recently within calculations of\nangle-resolved photo emission spectra (ARPES) on the\nbasis of the one-step model of photo emission.19\nAnother important contribution to the resistivity in\nthe case of magnetically ordered solids are thermally in-\nduced spin fluctuations.20Again, the alloy analogy has\nbeen exploited extensively in the past when dealing with\ntheimpactofspinfluctuationsonvariousresponsequant-\nities. Representing a frozen spin configuration by means\nof super cell calculations has been applied for calcula-\ntions of the Gilbert parameter for α17as well as the\nresistivity or conductivity, respectively.17,21,22Also, the\nCPA has been used for calculations of α23as well as the\nresistivity.20,24A crucial point in this context is obvi-\nously the modeling of the temperature dependent spin\nconfigurations. Concerning this, rather simple models\nhave been used,23but also quite sophisticated schemes.\nHere one should mention the transfer of data from Monte\nCarlo simulations based on exchange parameters calcu-\nlated in an ab-initio way25as well as work based on the\ndisordered local moment (DLM) method.24,26Although,\nthe standard DLM does not account for transversal spin\ncomponents it nevertheless allows to represent the para-\nmagnetic regime with no net magnetization in a rigor-2\nous way.Also, for the magnetically ordered regime below\nthe Curie-temperature it could be demonstrated that the\nuncompensated DLM (uDLM) leads for many situations\nstill to goodagreementwith experimentaldata on the so-\ncalled spin disorder contribution to the resistivity.20,24\nIn the following we present technical details and exten-\nsionsofaschemethatwasalreadyused beforewhendeal-\ning with the temperature dependence of response quant-\nities on the basis of Kubo’s response formalism. Various\napplications will be presented for the conductivity and\nGilbert damping parameter accounting simultaneously\nfor various types of disorder.\nII. THEORETICAL FRAMEWORK\nA. Configurational average for linear response\nfunctions\nMany important quantities in spintronics can be\nformulated by making use of linear response formal-\nism. Important examples for this are the electrical\nconductivity,27,28the spin conductivity29or the Gilbert\ndamping parameter.18,30Restricting here for the sake of\nbrevity to the symmetric part of the corresponding re-\nsponse tensor χµνthis can be expressed by a correlation\nfunction of the form:\nχµν∝Tr/angbracketleftbigˆAµℑG+ˆAνℑG+/angbracketrightbig\nc. (1)\nIt should be stressed that this not a real restriction as\nthe scheme described below has been used successfully\nwhen dealing with the impact of finite temperatures on\nthe anomalous Hall conductivity of Ni.31In this case the\nmore complex Kubo-Stˇ reda- or Kubo-Bastin formulation\nfor the full response tensor has to be used.32\nThe vector operator ˆAµin Eq. (1) stands for example\nin case of the electrical conductivity σµνfor the cur-\nrent density operator ˆjµ28while in case of the Gilbert\ndamping parameter αµνit stands for the torque oper-\natorˆTµ.9,18Within the Kubo-Greenwood-like equation\n(1) the electronic structure of the investigated system\nis represented in terms of its retarded Green function\nG+(r,r′,E). Within multiple scattering theory or the\nKKR (Korringa-Kohn-Rostoker)formalism, G+(r,r′,E)\ncan be written as:33–35\nG+(r,r′,E) =/summationdisplay\nΛΛ′Zm\nΛ(r,E)τmn\nΛΛ′(E)Zn×\nΛ′(r′,E)(2)\n−δmn/summationdisplay\nΛZn\nΛ(r,E)Jn×\nΛ′(r′,E)Θ(r′\nn−rn)\n+Jn\nΛ(r,E)Zn×\nΛ′(r′,E)Θ(rn−r′\nn).\nHerer,r′refer to points within atomic volumes around\nsitesRm,Rn, respectively, with Zn\nΛ(r,E) =ZΛ(rn,E) =\nZΛ(r−Rn,E) being a function centered at site Rn. Ad-\nopting a fully relativistic formulation34,35for Eq. (2) one\ngets in a natural way access to all spin-orbit inducedproperties as for example the anomalous and spin Hall\nconductivity29,32,36or Gilbert damping parameter.18In\nthis case, the functions Zn\nΛandJn\nΛstand for the reg-\nular and irregular, respectively, solutions to the single-\nsite Dirac equation for site nwith the associated single-\nsite scattering t-matrix tn\nΛΛ′. The corresponding scat-\ntering path operator τnn′\nΛΛ′accounts for all scattering\nevents connecting the sites nandn′. Using a suitable\nspinor representation for the basis functions the com-\nbined quantum number Λ = ( κ,µ) stands for the relativ-\nistic spin-orbit and magnetic quantum numbers κandµ,\nrespectively.34,35,37\nAs was demonstrated by various authors27,28,38rep-\nresenting the electronic structure in terms of the Green\nfunction G+(r,r′,E) allows to account for chemical dis-\norder in a random alloy by making use of a suitable al-\nloy theory. In this case ∝an}bracketle{t...∝an}bracketri}htcstands for the configura-\ntional average for a substitutional alloy concerning the\nsite occupation. Corresponding expressions for the con-\nductivity tensor have been worked out by Velick´ y27and\nButler28usingthe single-siteCoherentPotentialApprox-\nimation (CPA) that include in particular the so-called\nvertex corrections.\nThe CPA can be used to deal with chemical but also\nwith any other type of disorder. In fact, making use of\nthe different time scales connected with the electronic\npropagation and spin fluctuations the alloy analogy is\nexploited when dealing with finite temperature magnet-\nism on the basis of the disordered local moment (DLM)\nmodel.26,39Obviously, the same approach can be used\nwhen dealing with response tensors at finite temperat-\nures. In connection with the conductivity this is often\ncalled adiabatic approximation.40Following this philo-\nsophy, the CPA has been used recently also when calcu-\nlating response tensors using Eq. ( 1) with disorder in the\nsystem caused by thermal lattice vibrations18,31as well\nas spin fluctuations.20,41\nB. Treatment of thermal lattice displacement\nA way to account for the impact of the thermal dis-\nplacement of atoms from their equilibrium positions, i.e.\nfor thermal lattice vibrations, on the electronic struc-\nture is to set up a representative displacement configura-\ntion for the atoms within an enlargedunit cell (super-cell\ntechnique). In this case one has to use either a very large\nsuper-cell or to take the average over a set of super-cells.\nAlternatively, one may make use of the alloy analogy for\nthe averaging problem. This allows in particular to re-\nstrict to the standard unit cell. Neglecting the correla-\ntion between the thermal displacements of neighboring\natoms from their equilibrium positions the properties of\nthe thermal averaged system can be deduced by making\nuse of the single-site CPA. This basic idea is illustrated\nby Fig.1. To make use of this scheme a discrete set\nofNvdisplacement vectors ∆ Rq\nv(T) with probability xq\nv\n(v= 1,..,Nv) is constructed for each basis atom qwithin3\nFigure 1. Configurational averaging for thermal lattice dis -\nplacements: the continuous distribution P(∆Rn(T)) for the\natomic displacement vectors is replaced by a discrete set of\nvectors ∆ Rv(T) occurring with the probability xv. The con-\nfigurational average for this discrete set of displacements is\nmade using the CPA leading to a periodic effective medium.\nthe standard unit cell that is conform with the local sym-\nmetry and the temperature dependent root mean square\ndisplacement ( ∝an}bracketle{tu2∝an}bracketri}htT)1/2according to:\n1\nNvNv/summationdisplay\nv=1|∆Rq\nv(T)|2=∝an}bracketle{tu2\nq∝an}bracketri}htT. (3)\nIn the general case, the mean square displacement along\nthe direction µ(µ=x,y,z) of the atom ican be either\ntaken from experimental data or represented by the ex-\npression based on the phonon calculations42\n∝an}bracketle{tu2\ni,µ∝an}bracketri}htT=3/planckover2pi1\n2Mi/integraldisplay∞\n0dωgi,µ(ω)1\nωcoth/planckover2pi1ω\n2kBT,(4)\nwhereh= 2π/planckover2pi1the Planck constant, kBthe Boltzmann\nconstant, gi,µ(ω) is a partial phonon density of states.42\nOn the other hand, a rather good estimate for the root\nmean square displacement can be obtained using Debye’s\ntheory. In this case, for systems with one atom per unit\ncell, Eq. ( 4) can be reduced to the expression:\n∝an}bracketle{tu2∝an}bracketri}htT=1\n43h2\nπ2MkBΘD/bracketleftbiggΦ(ΘD/T)\nΘD/T+1\n4/bracketrightbigg\n(5)\nwith Φ(Θ D/T) the Debye function and Θ Dthe Debye\ntemperature43. Ignoring the zero temperature term 1 /4\nand assuming a frozen potential for the atoms, the situ-\nationcanbe dealt with in full analogytothe treatmentof\ndisorderedalloysonthebasisoftheCPA.Theprobability\nxvfor a specific displacement vmay normally be chosen\nas 1/Nv. The Debye temperature Θ Dused in Eq. ( 5) can\nbe either taken fromexperimental data orcalculated rep-\nresenting it in terms of the elastic constants44. In general\nthe latter approach should give more reliable results in\nthe case of multicomponent systems.\nTo simplify notation we restrict in the following to sys-\ntems with one atom per unit cell. The index qnumbering\nsites in the unit cell can therefore be dropped, while the\nindexnnumbers the lattice sites.\nAssuming a rigid displacement of the atomic potential\nin the spirit of the rigid muffin-tin approximation45,46\nthe correspondingsingle-site t-matrix tlocwith respect to\nthe local frame of reference connected with the displaced\natomic position is unchanged. With respect to the globalframe of reference connected with the equilibrium atomic\npositions Rn, however, the corresponding t-matrix tis\ngiven by the transformation:\nt=U(∆R)tlocU(∆R)−1. (6)\nThe so-called U-transformation matrix U(s) is given in\nits non-relativistic form by:45,46\nULL′(s) = 4π/summationdisplay\nL′′il+l′′−l′CLL′L′′jl′′(|s|k)YL′′(ˆs).(7)\nHereL= (l,m) represents the non-relativistic angu-\nlar momentum quantum numbers, jl(x) is a spherical\nBesselfunction, YL(ˆr) a realsphericalharmonics, CLL′L′′\na corresponding Gaunt number and k=√\nEis the\nelectronic wave vector. The relativistic version of the\nU-matrix is obtained by a standard Clebsch-Gordan\ntransformation.37\nThe various displacement vectors ∆ Rv(T) can be used\nto determine the properties of a pseudo-component of a\npseudo alloy. Each of the Nvpseudo-components with\n|∆Rv(T)|=∝an}bracketle{tu2∝an}bracketri}ht1/2\nTis characterized by a corresponding\nU-matrix Uvand t-matrix tv. As for a substitutional\nalloy the configurational average can be determined by\nsolving the multi-component CPA equations within the\nglobal frame of reference:\nτnn\nCPA=Nv/summationdisplay\nv=1xvτnn\nv (8)\nτnn\nv=/bracketleftbig\n(tv)−1−(tCPA)−1+(τnn\nCPA)−1/bracketrightbig−1(9)\nτnn\nCPA=1\nΩBZ/integraldisplay\nΩBZd3k/bracketleftbig\n(tCPA)−1−G(k,E)/bracketrightbig−1,(10)\nwhere the underline indicates matrices with respect to\nthe combined index Λ. As it was pointed out in the pre-\nvious work41, the cutoff for the angular momentum ex-\npansionin these calculations should be taken l≥lmax+1\nwith the lmaxvalue used in the calculations for the non-\ndistorted lattice.\nThe first of these CPA equations represents the re-\nquirement for the mean-field CPA medium that embed-\nding of a component vshould lead in the average to no\nadditional scattering. Eq. ( 9) gives the scattering path\noperator for the embedding of the component vinto the\nCPA medium while Eq. ( 10) gives the CPA scattering\npath operator in terms of a Brillouin zone integral with\nG(k,E) the so-called KKR structure constants.\nHaving solved the CPA equations the linear response\nquantity of interest may be calculated using Eq. ( 1)\nas for an ordinary substitutional alloy.27,28This im-\nplies that one also have to deal with the so-called ver-\ntex corrections27,28that take into account that one\nhas to deal with a configuration average of the type\n∝an}bracketle{tˆAµℑG+ˆAνℑG+∝an}bracketri}htcthat in general will differ from the\nsimpler product ∝an}bracketle{tˆAµℑG+∝an}bracketri}htc∝an}bracketle{tˆAνℑG+∝an}bracketri}htc.4\nC. Treatment of thermal spin fluctuations\nAs for the disorder connected with thermal displace-\nments the impact of disorder due to thermal spin fluc-\ntuations may be accounted for by use of the super-cell\ntechnique. Alternatively one may again use the alloy\nanalogy and determine the necessary configurational av-\nerage by means of the CPA as indicated in Fig. 2. As\nFigure 2. Configurational averaging for thermal spin fluc-\ntuations: the continuous distribution P(ˆen) for the orienta-\ntion of the magnetic moments is replaced by a discrete set of\norientation vectors ˆ efoccurring with a probability xf. The\nconfigurational average for this discrete set of orientatio ns is\nmade using the CPA leading to a periodic effective medium.\nfor the thermal displacements in a first step a set of rep-\nresentative orientation vectors ˆ ef(withf= 1,...,Nf) for\nthelocalmagneticmomentisintroduced(seebelow). Us-\ning the rigid spin approximation the spin-dependent part\nBxcoftheexchange-correlationpotentialdoesnotchange\nfor the local frame of reference fixed to the magnetic mo-\nment when the moment is oriented along an orientation\nvector ˆef. This implies that the single-site t-matrix tloc\nf\nin the local frame is the same for all orientation vectors.\nWith respect to the common global frame that is used\nto deal with the multiple scattering (see Eq. ( 10)) the\nt-matrix for a given orientation vector is determined by:\nt=R(ˆe)tlocR(ˆe)−1. (11)\nHere the transformation from the local to the global\nframe of reference is expressed by the rotation matrices\nR(ˆe) that are determined by the vectors ˆ eor correspond-\ning Euler angles.37\nAgain the configurational average for the pseudo-alloy\ncan be obtained by setting up and solvingCPAequations\nin analogy to Eqs. ( 8) to (10).\nD. Models of spin disorder\nThe central problem with the scheme described above\nis obviously to construct a realistic and representative\nset of orientation vectors ˆ efand probabilities xffor each\ntemperature T. A rather appealing approach is to cal-\nculate the exchange-coupling parameters Jijof a sys-\ntem in an ab-initio way25,47,48and to use them in sub-\nsequent Monte Carlo simulations. Fig. 3(top) shows\nresults for the temperature dependent average reduced\nmagnetic moment of corresponding simulations for bcc-\nFe obtained for a periodic cell with 4096 atom sites. The0 0.2 0.4 0.6 0.8 1 1.2\nT/TC00.20.40.60.81M(T)MC*\nKKR (MC*)\n0 0.2 0.4 0.6 0.8 1 1.2\nT/TC00.20.40.60.81M(T)\nMC\nMF-fit to MC (wMC(T))\nMF-fit to MC (w=const)\nExpt\nMF-fit to Expt (wExpt(T))\n0 0.2 0.4 0.6 0.8 1 1.2\nT/TC00.20.40.60.81M(T)MC\nKKR (MC)\nKKR (DLM)\nFigure 3. Averaged reduced magnetic moment M(T) =\n/angbracketleftmz/angbracketrightT/|/angbracketleftm/angbracketrightT=0|along the z-axis as a function of the tem-\nperature T. Top: results of Monte Carlo simulations using\nscheme MC* (full squares) compared with results of sub-\nsequent KKR-calculations (open squares). Middle: results\nof Monte Carlo simulations using scheme MC (full squares)\ncompared with results using a mean-field fit with a constant\nWeiss field wMC(TC) (open diamonds) and a temperature de-\npendent Weiss field wMC(T) (open squares). In addition ex-\nperimental data (full circles) together with a correspondi ng\nmean-field fit obtained for a temperature dependent Weiss\nfieldwexp(T). Bottom: results of Monte Carlo simulations\nusing scheme MC (full squares) compared with results sub-\nsequent KKR-calculations using the MC (triangles up) and\na corresponding DLM (triangle down) spin configuration, re-\nspectively.\nfull line gives the value for the reduced magnetic mo-\nmentMMC∗(T) =∝an}bracketle{tmz∝an}bracketri}htT/m0projected on the z-axis for\nthe lastMonteCarlostep (ˆ zis the orientationofthetotal\nmoment, i.e. ∝an}bracketle{tm∝an}bracketri}htT∝bardblˆz; the saturated magnetic moment at\nT= 0 K is m0=|∝an}bracketle{tm∝an}bracketri}htT=0|). This scheme is called MC∗\nin the following. In spite of the rather large number of\nsites (4096) the curve is rather noisy in particular when\napproaching the Curie temperature. Nevertheless, the5\nspin configuration of the last MC step was used as an\ninput for subsequent SPR-KKR-CPA calculations using\ntheorientationvectors ˆ efwiththeprobability xf= 1/Nf\nwithNf= 4096. As Fig. 3(top) shows, the temperature\ndependent reducedmagnetic moment MKKR(MC∗)(T) de-\nduced from the electronic structure calculations follows\none-to-one the Monte Carlo data MMC∗(T). This is a\nvery encouraging result for further applications (see be-\nlow) as it demonstrates that the CPA although being a\nmean-field method and used here in its single-site formu-\nlation is nevertheless capable to reproduce results of MC\nsimulations that go well beyond the mean-field level.\nHowever, using the set of vectors ˆ efof scheme MC*\nalso for calculations of the Gilbert damping parameters\nαas a function of temperature led to extremely noisy\nand unreliable curves for α(T). For that reason an av-\nerage has been taken over many MC steps (scheme MC)\nleading to a much smoother curve for MMC(T) as can\nbe seen from Fig. 3(middle) with a Curie temperature\nTMC\nC= 1082 K. As this enlarged set of vectors ˆ efgot\ntoo large to be used directly in subsequent SPR-KKR-\nCPA calculations, a scheme was worked out to get a set\nof vectors ˆ efand probabilities xfthat is not too large\nbut nevertheless leads to smooth curves for M(T).\nThe first attempt was to use the Curie temperature\nTMC\nCtodeduceacorrespondingtemperatureindependent\nWeiss-field w(TC) on the basis of the standard mean-field\nrelation:\nw(TC) =3kBTC\nm2\n0. (12)\nThis leads to a reduced magnetic moment curve MMF(T)\nthat shows by construction the same Curie temperature\nas the MC simulations. For temperatures between T=\n0 K and TC, however, the mean-field reduced magnetic\nmoment MMF(T) is well below the MC curve (see Fig. 3\n(middle) ).\nAs an alternative to this simple approach we intro-\nduced a temperature dependent Weiss field w(T). This\nallows to describe the temperature dependent magnetic\nproperties using the results obtained beyond the mean-\nfield approximation. At the same time the calculation\nof the statistical average can be performed treating the\nmodel Hamiltonian in termsofthe mean field theory. For\nthis reason the reduced magnetic moment M(T), being\na solution of equation (see e.g.49)\nM(T) =L/parenleftbiggwm2\n0M(T)\nkBT/parenrightbigg\n, (13)\nwas fitted to that obtained from MC simulations\nMMC(T)withtheWeissfield w(T)asafittingparameter,\nsuch that\nlim\nw→w(T)M(T) =MMC(T), (14)\nwithL(x) the Langevin function.\nThe corresponding temperature dependent probability\nx(ˆe) for an atomic magnetic moment to be oriented alongˆeis proportionalto exp( −w(T)ˆz·ˆe/kBT) (see, e.g.49). To\ncalculate this value we used NθandNφpoints for a reg-\nular grid for the spherical angles θandφcorresponding\nto the vector ˆ ef:\nxf=exp(−w(T)ˆz·ˆef/kBT)/summationtext\nf′exp(−w(T)ˆz·ˆef′/kBT).(15)\nFig.4shows for three different temperatures the θ-\ndependent behavior of x(ˆe). As one notes, the MF-fit\n0 30 60 90 120 150 180\nθ00.050.10.150.20.250.3P(θ)MC\nMF-fit to MC (wMC(T))T = 200 K\n0 30 60 90 120 150 180\nθ00.050.10.150.2P(θ)MC\nMF-fit to MC (wMC(T))T = 400 K\n0 30 60 90 120 150 180\nθ00.050.1P(θ)MC\nMF-fit to MC (wMC(T))T = 800 K\nFigure 4. Angular distribution P(θ) of the atomic magnetic\nmoment mobtained from Monte Carlo simulations (MC) for\nthe temperature T= 200, 400, and 800 K compared with field\nmean-field (MF) data, xf, (full line) obtained by fitting using\na temperature dependent Weiss field w(T) (Eq.13).\nto the MC-results perfectly reproduces these data for all\ntemperatures. This applies of course not only for the\nangular resolved distribution of the magnetic moments\nshown in Fig. 4but also for the average reduced mag-\nnetic moment recalculated using Eq.( 13), shown in Fig.\n3. Obviously, the MF-curve MMF(MC)(T) obtained using\nthe temperature dependent Weiss field parameter w(T)\nperfectly reproduces the original MMC(T) curve. The\ngreat advantage of this fitting procedure is that it al-\nlows to replace the MC data set with a large number6\nNMC\nfof orientation vectors ˆ ef(pointing in principle into\nany direction) with equal probability xf= 1/NMC\nfby a\nmuch smaller data set with Nf=NθNφwithxfgiven\nby Eq. (15).\nAccordingly, the reduced data set can straight for-\nwardly be used for subsequent electronic structure cal-\nculations. Fig. 3(bottom) shows that the calcu-\nlated temperature dependent reduced magnetic moment\nMKKR−MF(MC)(T) agrees perfectly with the reduced\nmagnetic moment MMC(T) given by the underlying MC\nsimulations.\nThe DLM method has the appealing feature that it\ncombines ab-initio calculations and thermodynamics in\na coherent way. Using a non-relativistic formulation, it\nwas shown that the corresponding averaging over all ori-\nentations of the individual atomic reduced magnetic mo-\nments can be mapped onto a binary pseudo-alloy with\none pseudo-component having up- and downward orient-\nation of the spin moment with concentrations x↑and\nx↓, respectively.24,50For a fully relativistic formulation,\nwith spin-orbitcoupling included, this simplificationcan-\nnot be justified anymore and a proper average has to be\ntaken over all orientations.51As we do not perform DLM\ncalculationsbut use hereonly the DLM picture to repres-\nent MC data, this complication is ignored in the follow-\ning. Having the set of orientation vectors ˆ efdetermined\nby MC simulations the corresponding concentrations x↑\nandx↓can straight forwardly be fixed for each temper-\nature by the requirement:\n1\nNfNf/summationdisplay\nf=1ˆef=x↑ˆz+x↓(−ˆz), (16)\nwithx↑+x↓= 1. Using this simple scheme electronic\nstructure calculations have been performed for a binary\nalloy having collinear magnetization. The resulting re-\nduced magnetic moment MKKR−DLM(MC) (T) is shown in\nFig.3(bottom). As one notes, again the original MC\nresults are perfectly reproduced. This implies that when\ncalculating the projected reduced magnetic moment Mz\nthat is determined by the averaged Green function ∝an}bracketle{tG∝an}bracketri}ht\nthe transversal magnetization has hardly any impact.\nFig.3(middle) gives also experimental data for\ntheM(T).52While the experimental Curie-temperature\nTexp\nC= 1044 K52is rather well reproduced by the MC\nsimulations TMC\nC= 1082 K one notes that the MC-curve\nMMC(T) is well below the experimental curve. In partic-\nular,MMC(T) drops too fast with increasing Tin the\nlow temperature regime and does not show the T3/2-\nbehavior. The reason for this is that the MC simulations\ndo not properly account for the low-energy long-ranged\nspinwaveexcitationsresponsibleforthelow-temperature\nmagnetization variation. Performing ab-initio calcula-\ntions for the spin wave energies and using these data for\nthe calculation of M(T) much better agreement with ex-\nperiment can indeed be obtained in the low-temperature\nregime than with MC simulations.53\nAs the fitting scheme sketched above needs only thetemperature reduced magnetic moment M(T) as input\nit can be applied not only to MC data but also to ex-\nperimental data. Fig. 3shows that the mean field fit\nMMF(exp)(T) again perfectly fits the experimental re-\nduced magnetic moment curve Mexp(T). Based on this\ngood agreement this corresponding data set {ˆef,xf}has\nalso been used for the calculation of responsetensors (see\nbelow).\nAn additional much simpler scheme to simulate the\nexperimental Mexp(T) curve is to assume the individual\natomic moments to be distributed on a cone, i.e. with\nNθ= 1 and Nφ>>1.23In this case the opening angle\nθ(T) of the cone is chosen such to reproduce M(T). In\ncontrasttothestandardDLMpicture,thissimplescheme\nallows already to account for transversal components of\nthe magnetization. Corresponding results for response\ntensor calculations will be shown below.\nFinally, it should be stressed here that the various spin\nconfiguration models discussed above assume a rigid spin\nmoment, i.e. its magnitude does not change with temper-\nature nor with orientation. In contrast to this Ruban et\nal.54usealongitudinalspinfluctuation Hamiltonianwith\nthe corresponding parameters derived from ab-initio cal-\nculations. As a consequence, subsequent Monte Carlo\nsimulations based on this Hamiltonian account in par-\nticular for longitudinal fluctuations of the spin moments.\nA similar approach has been used by Drchal et al.55,56\nleading to good agreement with the results of Ruban et\nal. However, the scheme used in these calculations does\nnot supply in a straightforward manner the necessary\ninput for temperature dependent transport calculations.\nThis is different from the work of Staunton et al.57who\nperformed self-consistent relativistic DLM calculations\nwithout the restriction to a collinear spin configuration.\nThis approach in particular accounts in a self-consistent\nway for longitudinal spin fluctuations.\nE. Combined chemical and thermally induced\ndisorder\nThe various types of disorder discussed above may be\ncombined with each other as well as with chemical i.e.\nsubstitution disorder. In the most general case a pseudo-\ncomponent ( vft) is characterized by its chemical atomic\ntypet, the spin fluctuation fand lattice displacement\nv. Using the rigid muffin-tin and rigid spin approxim-\nations, the single-site t-matrix tloc\ntin the local frame is\nindependent from the orientation vector ˆ efand displace-\nment vector ∆ Rv, and coincides with ttfor the atomic\ntypet. With respect to the common global frame one\nhas accordingly the t-matrix:\ntvft=U(∆Rv)R(ˆef)ttR(ˆef)−1U(∆Rv)−1.(17)\nWith this the corresponding CPA equations are identical\nto Eqs. ( 8) to (10) with the index vreplaced by\nthe combined index ( vft). The corresponding pseudo-\nconcentration xvftcombines the concentration xtof the7\natomic type twith the probability for the orientation\nvector ˆefand displacement vector ∆ Rv.\nIII. COMPUTATIONAL DETAILS\nThe electronic structure of the investigated ferro-\nmagnets bcc-Fe and fcc-Ni, has been calculated self-\nconsistently using the spin-polarized relativistic KKR\n(SPR-KKR) band structure method.58,59For the ex-\nchangecorrelationpotential the parametrizationas given\nby Vosko et al.60has been used. The angular-momentum\ncutoff of lmax= 3 was used in the KKR multiple scatter-\ning expansion. The lattice parameters have been set to\nthe experimental values.\nIn a second step the exchange-coupling parameters\nJijhave been calculated using the so-called Lichten-\nstein formula.25Although the SCF-calculations have\nbeen done on a fully-relativistic level the anisotropy of\nthe exchange coupling due to the spin-orbit coupling has\nbeen neglected here. Also, the small influence of the\nmagneto-crystallineanisotropyfor the subsequent Monte\nCarlo (MC) simulations has been ignored, i.e. these have\nbeen based on a classical Heisenberg Hamiltonian. The\nMC simulations were done in a standard way using the\nMetropolis algorithm and periodic boundary conditions.\nThe theoretical Curie temperature TMC\nChas been de-\nduced from the maximum of the magnetic susceptibility.\nThe temperature dependent spin configuration ob-\ntained during a MC simulation has been used to con-\nstruct a set of orientations ˆ efand probabilities xfac-\ncording to the schemes MC* and MC described in sec-\ntionIIDto be used within subsequent SPR-KKR-CPA\ncalculations (see above). For the corresponding calcu-\nlation of the reduced magnetic moment the potential\nobtained from the SCF-calculation for the perfect fer-\nromagnetic state ( T= 0K) has been used. The calcu-\nlation for the electrical conductivity as well as the Gil-\nbertdampingparameterhasbeenperformedasdescribed\nelsewhere.41,61\nIV. RESULTS AND DISCUSSION\nA. Temperature dependent conductivity\nEq. (1) has been used together with the various\nschemes described above to calculate the temperature\ndependent longitudinal resistivity ρ(T) of the pure fer-\nromagnets Fe, Co and Ni. In this case obviously disorder\ndue to thermal displacements of the atoms as well as spin\nfluctuations contribute to the resistivity.\nTo give an impression on the impact of the thermal\ndisplacementsaloneFig. 5givesthe temperaturedepend-\nent resistivity ρ(T) of pure Cu (Θ Debye= 315 K) that\nis found in very good agreement with corresponding ex-\nperimental data.62This implies that the alloy analogy\nmodel that ignores any inelastic scattering events should0 100 200 300 400 500\nTemperature (K)01234ρxx (10-6Ω⋅cm)Expt\nTheory - alloy analogy\nTheory - LOVA\nCu\nFigure 5. Temperature dependent longitudinal resistivity of\nfcc-Cuρ(T) obtained by accounted for thermal vibrations as\ndescribed in section IIBcompared with corresponding ex-\nperimental data.62In addition results are shown based on\nthe LOVA (lowest order variational approximation) to the\nBoltzmann formalism.14\nin general lead to rather reliable results for the resistivity\ninduced by thermal displacements. Accordingly, com-\nparison with experiment should allow for magnetically\nordered systems to find out the most appropriate model\nfor spin fluctuations.\nFig.6(top) shows theoretical results for ρ(T) of bcc-\nFe due to thermal displacements ρv(T), spin fluctuations\ndescribed by the scheme MC ρMC(T) as well as the com-\nbination of the two influences ( ρv,MC(T)). First of all\none notes that ρv(T) is not influenced within the adop-\ntedmodelbytheCurietemperature TCbutisdetermined\nonly by the Debye temperature. ρMC(T), on the other\nhand, reaches saturation for TCas the spin disorder does\nnot increase anymore with increasing temperature in the\nparamagnetic regime. Fig. 6also shows that ρv(T) and\nρMC(T)arecomparableforlowtemperaturesbut ρMC(T)\nexceedsρv(T) more and more for higher temperatures.\nMost interestingly, however, the resistivity for the com-\nbined influence of thermal displacements and spin fluctu-\nationsρv,MC(T) does not coincide with the sum of ρv(T)\nandρMC(T) but exceeds the sum for low temperatures\nand lies below the sum when approaching TC.\nFig.6(bottom) shows the results of three differ-\nent calculations including the effect of spin fluctuations\nas a function of the temperature. The curve ρMC(T)\nis identical with that given in Fig. 6(top) based on\nMonte Carlo simulations. The curves ρDLM(MC) (T) and\nρcone(MC)(T) are based on a DLM- and cone-like repres-\nentation of the MC-results, respectively. For all three\ncases results are given including as well as ignoring the\nvertex corrections. As one notes the vertex corrections\nplay a negligible role for all three spin disorder models.\nThis is fully in line with the experience for the longitud-\ninal resistivity of disordered transition metal alloys: as\nlong as the the states at the Fermi level have domin-\nantly d-character the vertex corrections can be neglected\nin general. On the other hand, if the sp-character dom-8\n0 0.2 0.4 0.6 0.8 1 1.2\nT/TC020406080100120ρxx  (10-6Ω⋅cm)vib\nfluct (MC)\nvib + fluct (MC)\n0 0.2 0.4 0.6 0.8 1 1.2\nT/TC020406080100120ρxx  (10-6Ω⋅cm)MC (VC)\nMC (NVC)\nDLM (VC)\nDLM (NVC)\ncone (VC)\ncone (NVC)\nFigure 6. Temperature dependent longitudinal resistivity of\nbcc-Feρ(T) obtained by accounted for thermal vibrations\nand spin fluctuations as described in section IIB. Top: ac-\ncounting for vibrations (vib, diamonds), spin fluctuations us-\ning scheme MC (fluct, squares) and both (vib+fluct, circles).\nBottom: accounting for spin fluctuations ˆ ef= ˆe(θf,φf) us-\ning the schemes: MC (squares) with 0 ≤θf≤π;0≤φf≤\n2π, DLM(MC) (triangles up) with θf1= 0,θf2=π, and\ncone(MC) (triangles down) θf=/angbracketleftθf/angbracketrightT;0≤φf≤2π. The\nfull and open symbols represent the results obtained with th e\nvertex corrections included (VC) and excluded (NV), respec t-\nively.\ninates inclusion of vertex corrections may alter the result\nin the order of 10 %.63,64\nComparing the DLM-result ρDLM(MC) (T) with\nρMC(T) one notes in contrast to the results for M(T)\nshown above (see Fig. 3(bottom)) quite an appreciable\ndeviation. This implies that the restricted collinear\nrepresentation of the spin configuration implied by the\nDLM-model introduces errors for the configurational\naverage that seem in general to be unacceptable, For\nthe Curie temperature and beyond in the paramagnetic\nregimeρDLM(MC) (T) andρMC(T) coincide, as it was\nshown formally before.20\nComparing finally ρcone(MC)(T) based on the conical\nrepresentationofthe MCspin configurationwith ρMC(T)\none notes that also this simplification leads to quite\nstrong deviations from the more reliable result. Never-\ntheless, one notes that ρDLM(MC) (T) agrees with ρMC(T)\nfor the Curie temperature and also accounts to some ex-\ntent for the impact of the transversal components of themagnetization.\nThe theoretical results for bcc-Fe (Θ Debye= 420 K)\nbased on the combined inclusion of the effects of thermal\ndisplacementsandspinfluctuationsusingtheMCscheme\n(ρv,MC(T)) are compared in Fig. 7(top) with experi-\nmental data ( ρexp(T)). For the Curie temperature ob-\n0 0.2 0.4 0.6 0.8 1 1.2 1.4\nT/TC020406080100120ρxx  (10-6Ω⋅cm)Expt: J. Bass and K.H. Fischer \nvib + fluct (MC)\nvib + fluct (exp)\n0 0.2 0.4 0.60.8 1 1.2 1.4 1.61.8\nT/TC01020304050ρxx  (10-6Ω⋅cm)Expt.: C.Y. Ho et al. (1983)\nvib\nvib (PM)\nfluct\nvib + fluct\nFigure 7. Top: Temperature dependent longitudinal res-\nistivity of bcc-Fe ρ(T) obtained by accounted for thermal\nvibrations and spin fluctuations using the scheme MC\n(vib+fluct(MC), squares) and a mean-field fit to the experi-\nmental temperature magnetic moment Mexp(vib+fluct(exp),\ndiamonds) compared with experimental data (circles).62Bot-\ntom: corresponding results for fcc-Ni. In addition results are\nshown accounting for thermal displacements (vib) only for\nthe ferromagnetic (FM) as well paramagnetic (PM) regime.\nExperimental data have been taken from Ref. 65.\nviously a very good agreement with experiment is found\nwhile for lower temperatures ρv,MC(T) exceeds ρexp(T).\nThis behavior correlates well with that of the temperat-\nure dependent reduced magnetic moment M(T) shown\nin Fig.3(middle). The too rapid decrease of MMC(T)\ncompared with experiment implies an essentially overes-\ntimated spin disorder at any temperature leading in turn\nto a too large resistivity ρv,MC(T). On the other hand,\nusing the temperature dependence of the experimental\nreducedmagneticmoment Mexp(T)tosetup thetemper-\nature dependent spin configuration as described above a\nvery satisfying agreement is found with the experimental\nresistivity data ρexp(T). Note also that above TCthe\ncalculated resistivity riches the saturation in contrast to\nthe experimental data where the continuing increase of9\nρexp(T) can be attributed to the longitudinal spin fluctu-\nations leading to a temperature dependent distribution\nof local magnetic moments on Fe atoms.54However, this\ncontribution was not taken into account because of re-\nstriction in present calculations using fixed value for the\nlocal reduced magnetic moments.\nFig.7(bottom) shows corresponding results for the\ntemperature dependent resistivity of fcc-Ni (Θ Debye=\n375 K). For the ferromagnetic (FM) regime that the\ntheoretical results are comparable in magnitude when\nonly thermal displacements ( ρv(T)) or spin fluctuations\n(ρMF(T)) are accounted for. In the later case the mean\nfieldw(T) has been fitted to the experimental M(T)-\ncurve. Taking both into account leads to a resistivity\n(ρv,MF(T)) that are well above the sum of the individual\ntermsρv(T) andρMF(T). Comparing ρv,MF(T) with ex-\nperimentaldata ρexp(T)ourfindingshowsthatthetheor-\netical results overshoots the experimental one the closer\none comes to the critical temperature. This is a clear\nindication that the assumption of a rigid spin moment\nis quite questionable as the resulting contribution to the\nresistivity due to spin fluctuations as much too small.\nIn fact the simulations of Ruban et al.54on the basis of\na longitudinal spin fluctuation Hamiltonian led on the\ncase of fcc-Ni to a strong diminishing of the averagelocal\nmagnetic moment when the critical temperature is ap-\nproachedfrom below (about 20% comparedto T= 0K).\nFor bcc-Fe, the change is much smaller (about 3 %) justi-\nfying on the case the assumption of a rigid spin moment.\nTaking the extreme point of view that the spin moment\nvanishescompletely abovethe criticaltemperature orthe\nparamagnetic (PM) regime only thermal displacements\nhave to be considered as a source for a finite resistivity.\nCorresponding results are shown in Fig. 7(bottom) to-\ngether with corresponding experimental data. The very\ngood agreementbetween both obviouslysuggeststhat re-\nmaining spin fluctuations above the critical temperature\nare of minor importance for the resistivity of fcc-Ni.\nB. Temperature dependent Gilbert damping\nparameter\nFig.8shows results for Gilbert damping parameter α\nof bcc-Fe obtained using different models for the spin\nfluctuations. All curves show the typical conductivity-\nlike behaviorfor low temperatures and the resistivity-like\nbehavior at high temperatures reflecting the change from\ndominating intra- to inter-band transitions.66The curve\ndenoted expt isbasedon aspin configurationtoted tothe\nexperimental Mexpt(T) data. Using the conical model to\nfitMexpt(T) as basis for the calculation of α(T) leads\nobviously to a rather good agreement with αM(expt)(T).\nHaving instead a DLM-like representation of Mexpt(T),\non the other hand, transverse spin components are sup-\npressed and noteworthy deviations from αM(expt)(T) are\nfound for the low temperature regime. Nevertheless, the\ndeviations are less pronounced than in the case of the0 200 400 600 800\nTemperature (K)02468α × 103fluct (MC)\nfluct (Expt)\nfluct (DLM)\nfluct (cone)\nFigure 8. Temperature dependent Gilbert damping α(T) for\nbcc-Fe, obtainedbyaccountedfor thermal vibrations andsp in\nfluctuations accounting for spin fluctuations using scheme\nMC (squares), DLM(MC) (triangles up), cone(MC) (triangles\ndown) and a MF fit to the experimental temperature reduced\nmagnetic moment (circles).\nlongitudinal resistivity (see Fig. 6(bottom)), where cor-\nresponding results are shown based on MMC(T) as a ref-\nerence. Obviously, the damping parameter αseems to\nbe less sensitive to the specific spin fluctuation model\nused than the resistivity. Finally, using the spin con-\nfiguration deduced from Monte Carlo simulations, i.e.\nbased on MMC(T) quite strong deviations for the result-\ningαM(MC)(T) fromαM(expt)(T) are found. As for the\nresistivity (see Fig. 6(bottom)) this seems to reflect the\ntoo fast drop of the reduced magnetic moment MMC(T)\nwith temperature in the low temperature regime com-\npared with temperature (see Fig. 3). As found before18\naccountingonly for thermal vibrations α(T) (Fig.6(bot-\ntom)) is found comparableto the casewhen only thermal\nspan fluctuations are allowed. Combing both thermal ef-\nfects does not lead to a curve that is just the sum of the\ntwoα(T) curves. As found for the conductivity (Fig. 6\n(top)) obviously the two thermal effects are not simply\nadditive. As Fig. 9(top) shows, the resulting damping\nparameter α(T) for bcc-Fe that accounts for thermal vi-\nbrationsaswellasspinfluctuationsisfoundinreasonable\ngood agreement with experimental data.18\nFig.9shows also corresponding results for the Gilbert\ndampingoffcc-Niasafunctionoftemperature. Account-\ning only for thermal spin fluctuations on the basis of the\nexperimental M(T)-curveleadsinthis casetocompletely\nunrealistic results while accounting only for thermal dis-\nplacements leads to results already in rather good agree-\nment with experiment. Taking finally both sources of\ndisorder into account again no simple additive behavior\nis found but the results are nearly unchanged compared\nto those based on the thermal displacements alone. This\nimplies that results for the Gilbert damping parameter\nof fcc-Ni hardly depend on the specific spin configura-\ntion model used but are much more governed by thermal\ndisplacements.10\n0 200 400 600 800\nTemperature (K)0246810α × 103vib\nvib + fluct (Expt)\nExpt 1\nExpt 2\n0 100 200 300 400 500\nTemperature (K)00.050.10.150.2αvib\nfluct (Expt)\nvib + fluct (Expt)\nExpt\nFigure 9. Top: Temperature dependent Gilbert damping\nα(T) for bcc-Fe, obtained byaccounted for thermal vibrations\nand spin fluctuations accounting for lattice vibrations onl y\n(circles) and lattice vibrations and spin fluctuations base d on\nmean-field fit to the experimental temperature reduced mag-\nnetic moment Mexpt(diamonds) compared with experimental\ndata (dashed and full lines).67,68Bottom: corresponding res-\nults for fcc-Ni. Experimental data have been taken from Ref.\n67.\nV. SUMMARY\nVarious schemes based on the alloy analogy that al-\nlow to include thermal effects when calculating responseproperties relevant in spintronics have been presented\nand discussed. Technical details of an implementation\nwithin the framework of the spin-polarized relativistic\nKKR-CPA band structure method have been outlined\nthat allow to deal with thermal vibrations as well as spin\nfluctuations. Various models to represent spin fluctu-\nations have been compared with each other concerning\ncorresponding results for the temperature dependence\nof the reduced magnetic moment M(T) as well as re-\nsponse quantities. It was found that response quantities\nare much more sensitive to the spin fluctuation model as\nthe reduced magnetic moment M(T). Furthermore, it\nwas found that the influence of thermal vibrations and\nspin fluctuations is not additive when calculating elec-\ntrical conductivity or the Gilbert damping parameter α.\nUsing experimental data for the reduced magnetic mo-\nmentM(T) to set up realistic temperature dependent\nspin configurations satisfying agreement for the electrical\nconductivity as well as the Gilbert damping parameter\ncould be obtained for elemental ferromagnets bcc-Fe and\nfcc-Ni.\nACKNOWLEDGMENTS\nThis work was supported financially by the Deutsche\nForschungsgemeinschaft (DFG) within the projects\nEB154/20-1, EB154/21-1 and EB154/23-1 as well as\nthe priority program SPP 1538 (Spin Caloric Transport)\nand the SFB 689 (Spinph¨ anomene in reduzierten Dimen-\nsionen). Helpful discussions with Josef Kudrnovsk´ y and\nIlja Turek are gratefully acknowledged.\n1T. Holstein, Annals of Physics 29, 410 (1964) .\n2G. D. Mahan, Many-particle physics , Physics of Solids and\nLiquids (Springer, New York, 2000).\n3P. B. Allen, Phys. Rev. B 3, 305 (1971) .\n4K. Takegahara and S. Wang, J. Phys. F: Met. Phys. 7,\nL293 (1977) .\n5G. Grimvall, Physica Scripta 14, 63 (1976).\n6G. D. Mahan and W. Hansch, J. Phys. F: Met. Phys. 13,\nL47 (1983) .\n7M. Oshita, S. Yotsuhashi, H. Adachi, and H. Akai, J.\nPhys. Soc. Japan 78, 024708 (2009) .\n8K. Shirai and K. Yamanaka, J. Appl. Physics 113, 053705\n(2013).\n9D. Steiauf and M. F¨ ahnle, Phys. Rev. B 72, 064450 (2005) .\n10V. Kambersk´ y, Phys. Rev. B 76, 134416 (2007) .11K. Gilmore, Y. U. Idzerda, and M. D. Stiles, J. Appl.\nPhysics103, (2008).\n12V. Kambersky, Czech. J. Phys. 26, 1366 (1976) .\n13P. B. Allen, T. P. Beaulac, F. S. Khan, W. H. Butler, F. J.\nPinski, and J. C. Swihart, Phys. Rev. B 34, 4331 (1986) .\n14S. Y. Savrasov and D. Y. Savrasov, Phys. Rev. B 54, 16487\n(1996).\n15B. Xu and M. J. Verstraete, Phys. Rev. B 87, 134302\n(2013).\n16B. Xu and M. J. Verstraete, Phys. Rev. Lett. 112, 196603\n(2014).\n17Y. Liu, A. A. Starikov, Z. Yuan, and P. J. Kelly, Phys.\nRev. B84, 014412 (2011) .\n18H. Ebert, S. Mankovsky, D. K¨ odderitzsch, and\nP. J. Kelly, Phys. Rev. Lett. 107, 066603 (2011) ,\nhttp://arxiv.org/abs/1102.4551v1 .11\n19J. Braun, J. Min´ ar, S. Mankovsky, V. N. Strocov, N. B.\nBrookes, L. Plucinski, C. M. Schneider, C. S. Fadley, and\nH. Ebert, Phys. Rev. B 88, 205409 (2013) .\n20J. Kudrnovsk´ y, V. Drchal, I. Turek, S. Khmelevskyi, J. K.\nGlasbrenner, and K. D. Belashchenko, Phys. Rev. B 86,\n144423 (2012) .\n21J. K. Glasbrenner, K. D. Belashchenko, J. Kudrnovsk´ y,\nV. Drchal, S. Khmelevskyi, and I. Turek, Phys. Rev. B\n85, 214405 (2012) .\n22R. Kov´ aˇ cik, P. Mavropoulos, D. Wortmann, andS. Bl¨ ugel,\nPhys. Rev. B 89, 134417 (2014) .\n23S. Mankovsky, D. Koedderitzsch, and H. Ebert, unpub-\nlished (2011).\n24H. AkaiandP. H. Dederichs, Phys. Rev.B 47, 8739 (1993) .\n25A. I. Liechtenstein, M. I. Katsnelson, V. P. Antropov, and\nV. A. Gubanov, J. Magn. Magn. Materials 67, 65 (1987) .\n26B. L. Gyorffy, A. J. Pindor, J. Staunton, G. M. Stocks,\nand H. Winter, J. Phys. F: Met. Phys. 15, 1337 (1985) .\n27B. Velick´ y, Phys. Rev. 184, 614 (1969) .\n28W. H. Butler, Phys. Rev. B 31, 3260 (1985) .\n29S. Lowitzer, M. Gradhand, D. K¨ odderitzsch, D. V. Fe-\ndorov, I. Mertig, and H. Ebert, Phys. Rev. Lett. 106,\n056601 (2011) .\n30A. Brataas, Y. Tserkovnyak, and G. E. W. Bauer, Phys.\nRev. Lett. 101, 037207 (2008) .\n31D. K¨ odderitzsch, K. Chadova, J. Min´ ar, and H. Ebert,\nNew Journal of Physics 15, 053009 (2013) .\n32A.Cr´ epieuxandP.Bruno, Phys.Rev.B 64, 094434 (2001) .\n33J. S. Faulkner and G. M. Stocks, Phys. Rev. B 21, 3222\n(1980).\n34P. Weinberger, Electron Scattering Theory for Ordered\nand Disordered Matter (Oxford University Press, Oxford,\n1990).\n35H. Ebert, in Electronic Structure and Physical Properties\nof Solids , Lecture Notes in Physics, Vol. 535, edited by\nH. Dreyss´ e (Springer, Berlin, 2000) p. 191.\n36S. Lowitzer, D. K¨ odderitzsch, and H. Ebert, Phys. Rev.\nLett.105, 266604 (2010) .\n37M.E.Rose, Relativistic Electron Theory (Wiley,NewYork,\n1961).\n38I. Turek, J. Kudrnovsk´ y, V. Drchal, L. Szunyogh, and\nP. Weinberger, Phys. Rev. B 65, 125101 (2002) .\n39J. B. Staunton and B. L. Gyorffy, Phys. Rev. Lett. 69, 371\n(1992).\n40M.Jonson andG.D.Mahan, Phys.Rev.B 42,9350(1990) .\n41S. Mankovsky, D. K¨ odderitzsch, G. Woltersdorf, and\nH. Ebert, Phys. Rev. B 87, 014430 (2013) .\n42H. B¨ ottger, “Principles of the theory of lattice dynamics, ”\n(Akademie-Verlag, Berlin, 1983).\n43E. M. Gololobov, E. L. Mager, Z. V. Mezhevich, and L. K.\nPan, phys. stat. sol. (b) 119, K139 (1983).\n44E. Francisco, M. A. Blanco, and G. Sanjurjo, Phys. Rev.\nB63, 094107 (2001) .\n45N. Papanikolaou, R. Zeller, P. H. Dederichs, and\nN. Stefanou, Phys. Rev. B 55, 4157 (1997) .46A. Lodder, J. Phys. F: Met. Phys. 6, 1885 (1976) .\n47L. Udvardi, L. Szunyogh, K. Palot´ as, and P. Weinberger,\nPhys. Rev. B 68, 104436 (2003) .\n48H. Ebert and S. Mankovsky, Phys. Rev. B 79, 045209\n(2009).\n49S. Tikadzumi, “Physics of magnetism,” (Willey, Ney York,\n1964).\n50H. Akai, Phys. Rev. Lett. 81, 3002 (1998) .\n51J. B. Staunton, S. Ostanin, S. S. A. Razee, B. L. Gyorffy,\nL. Szunyogh, B. Ginatempo, and E. Bruno, Phys. Rev.\nLett.93, 257204 (2004) .\n52J. Crangle and G. M. Goodman, Proc.\nRoy. Soc. (London) A 321, 477 (1971) ,\nhttp://rspa.royalsocietypublishing.org/content/321/ 1547/477.full.pdf+h\n53S. V. Halilov, H. Eschrig, A. Y. Perlov, and P. M. Oppen-\neer,Phys. Rev. B 58, 293 (1998) .\n54A.V.Ruban, S.Khmelevskyi, P.Mohn, andB. Johansson,\nPhys. Rev. B 75, 054402 (2007) .\n55V. Drchal, J. Kudrnovsk´ y, and I. Turek, EPJ Web of\nConferences 40, 11001 (2013) .\n56V. Drchal, J. Kudrnovsk´ y, and I. Turek, Journal of Su-\nperconductivity and Novel Magnetism 26, 1997 (2013) .\n57J. B. Staunton, M. Banerjee, dos Santos Dias, A. Deak,\nand L. Szunyogh, Phys. Rev. B 89, 054427 (2014) .\n58H.Ebert, D.K¨ odderitzsch, andJ. Min´ ar, Rep.Prog. Phys.\n74, 096501 (2011) .\n59H. Ebert et al., The Munich SPR-KKR package , version\n6.3,\nH. Ebert et al.\nhttp://olymp.cup.uni-muenchen.de/ak/ebert/SPRKKR\n(2012).\n60S. H. Vosko, L. Wilk, and M. Nu-\nsair, Can. J. Phys. 58, 1200 (1980) ,\nhttp://www.nrcresearchpress.com/doi/pdf/10.1139/p80 -159.\n61D. K¨ odderitzsch, S. Lowitzer, J. B. Staunton, and\nH. Ebert, phys. stat. sol. (b) 248, 2248 (2011) .\n62J. Bass, Electrical Resistivity of Pure Metals and Alloys ,\nLandolt-Bornstein New Series, Group III, Part (a), Vol. 15\n(Springer, New York, 1982).\n63J. Banhart, H. Ebert, P. Weinberger, and J. Voitl¨ ander,\nPhys. Rev. B 50, 2104 (1994) .\n64I. Turek, J. Kudrnovsk´ y, V. Drchal, and P. Weinberger,\nJ. Phys.: Cond. Mat. 16, S5607 (2004) .\n65C. Y. Ho, M. W. Ackerman, K. Y. Wu, T. N. Havill, R. H.\nBogaard, R. A. Matula, S. G. Oh, and H. M. James, J.\nPhys. Chem. Ref. Data 12, 183 (1983) .\n66K. Gilmore, Y. U. Idzerda, and M. D. Stiles, Phys. Rev.\nLett.99, 027204 (2007) .\n67S. M. Bhagat and P. Lubitz, Phys. Rev. B 10, 179 (1974) .\n68B. Heinrich and Z. Frait, phys. stat. sol. (b) 16, 1521\n(1966)."    },    {        "title": "1412.2479v1.Magnetization_Dynamics_driven_by_Non_equilibrium_Spin_Orbit_Coupled_Electron_Gas.pdf",        "content": "arXiv:1412.2479v1  [cond-mat.mes-hall]  8 Dec 2014Magnetization Dynamics driven by Non-equilibrium Spin-Or bit Coupled Electron Gas\nYong Wang,1Wei-qiang Chen,2and Fu-Chun Zhang3,1,4\n1Department of Physics, The University of Hong Kong, Hong Kon g SAR, China\n2Department of Physics, South University of Science and Tech nology of China, China\n3Department of Physics, Zhejiang University, China\n4Collaborative Innovation Center of Advanced Microstructu res, Nanjing University, Nanjing, 210093, China\nThe dynamics of magnetization coupled to an electron gas via s-d exchange interaction is investi-\ngated by using density matrix technique. Our theory shows th at non-equilibrium spin accumulation\ninduces a spin torque and the electron bath leads to a damping of the magnetization. For the\ntwo-dimensional magnetization thin film coupled to the elec tron gas with Rashba spin-orbit cou-\npling, the result for the spin-orbit torques is consistent w ith the previous semi-classical theory. Our\ntheory predicts a damping of the magnetization, which is abs ent in the semi-classical theory. The\nmagnitude of the damping due to the electron bath is comparab le to the intrinsic Gilbert damping\nand may be important in describing the magnetization dynami cs of the system.\nI. INTRODUCTION\nIn study of spin transfer torque (STT), it has been\nproposed [1, 2] to manipulate magnetic order parameter\ndynamics by using non-equilibrium electron bath instead\nof external magnetic fields. The proposal has already led\nto commercial products in spintronics engineering. Re-\ncently, there has been much attention on the ”spin-orbit\ntorque”(SOT), which was first proposed in theory[3, 4],\nand later confirmed in experiments[5–8] (see Ref. 9 and\n10 for a comprehensive review). After applying an ex-\nternal electric field to the electron gas with spin-orbit in-\nteraction(SOI), a component ofthe accumulated electron\nspin density mis-aligned with the ferromagnetic ordering\ncan be created[3, 4], which then will induce a field-like\ntorque. The SOT opens the possibility of manipulat-\ning the magnetic order parameter in collinear magnetic\nstructures and may efficiently reduce the critical current\ndensity for magnetization switching[3, 4]. In the theoret-\nical side, a full quantum theory has been proposed and\ndeveloped to describe the dynamics of a single domain\nmagnetunderthecontinuousscatteringbyspin-polarized\nelectrons. The quantum STT theory recovers the results\nof the semiclassical STT theory, and has revealed more\ndetailsaboutthemagnetizationdynamicsintheSTT[11–\n13]. Therefore, it will be natural to apply a full quantum\ntheory to study the magnetization dynamics influenced\nby the SOI electron gas. This may be an extension of\nthe quantum STT theory to SOT. In the full quantum\ntheory, the quantum dynamics of the magnetization can\nbe described by the evolution of its density matrix under\nthe influence of the electron gas, which can be tuned by\nthe external electric field. This treatment will not only\ngive the mean-field effect on the magnetization dynamics\nby the electron bath, but also include the damping of the\nmagnetization due to the fluctuation of the electron spin.\nThe similarstrategyhas been exploited to investigatethe\nphoto-excited dynamics of the order parameter in Peierls\nchain[14].\nThis paper is organized as follows. In section II, we\napply density matrix technique to derive general formal-E\nFIG. 1. (Color online). Schematic diagram for the lattice of\nlocalized spins (orange) coupled to the conductions electr ons\n(blue) through s-d exchange interaction. An external elect ric\nfieldEcan be applied to tune the electron bath.\nism for the magnetization dynamics driven by the elec-\ntron bath through s-d exchange interaction. In section\nIII, we apply the general formalism to the special case\nwhere the spatially uniform magnetization is coupled to\na two dimensional electron gas with Rashba SOI, and\ncalculate the spin-orbit torque and the damping effect of\nthe electron bath. The main results are summarized and\ndiscussed in section IV.\nII. GENERAL FORMALISM\nWe apply density matrix technique to study dynamics\nof the magnetization driven by the electron bath via an\ns-d exchange interaction. The system is schematically\nillustratedinFig.1, wheretheelectronbathcanbe tuned\nbyanexternalelectricfield. TheHamiltonianofthetotal\nsystem is formally written as\nH=HM+He+Hsd. (1)\nHere,HMis the Hamiltonian for the magnetization sub-\nsystem in terms of the local spin operators /hatwideSi,µat sitei\nwith spin directions µ(=x,y,z);Heis the Hamiltonian\noftheelectronsubsystem; Hsddescribesthes-dexchange\ninteraction between the magnetization and the electron,2\nwhere\nHsd=J/summationdisplay\ni,µ/hatwideSi,µ/hatwideσi,µ. (2)\nHere,/hatwideσi,µrepresents the electron spin operator at site i\nwithout /planckover2pi1/2, andJis the exchange coupling strength.\nNote that we have not specified the forms of HMandHe\nyet, thus the results below will be quite general.\nThe effect of the s-d exchange interaction Hsdis\ntwofold. On one hand, the magnetization dynamics is\ndriven by the electron bath via Hsd; on the other hand,\nthe electron states are also affected by the magnetization\nconfiguration in turn due to Hsd. Since the time scale of\nthe electron dynamics is usually much faster than that\nof the magnetization dynamics, we may assume that the\nelectrons under the bias voltage establish a stationary\nnon-equilibrium distribution in a very short time inter-\nval, during which the change of the magnetization con-\nfiguration is negligible and the non-equilirium electron\nbath is approximated to be constant. The validity of\nthis assumption only holds if the spin-lattice interaction\nis stronger than the s-d exchange interaction to relax\nthe electron spin. Consider a short time interval [ t0,t],\nwhere the initial density matrices of the magnetization\nand the electron bath are ρM(t0) andρe(t0) respectively.\nThen the initial magnetization configuration at each site\nisSi,µ(t0) = Tr[/hatwideSi,µρM(t0)], and the initial electron den-\nsity matrix ρe(t0) is determined by the bath Hamiltonian\nHB=He+J/summationtext\ni,µSi,µ(t0)/hatwideσi,µand the open boundary\nconditions.\nIn order to investigate the magnetization dynamics\nduring the time interval [ t0,t] defined above, we rede-\nfine the local spin operators /hatwideSi,µ=Si,µ(t0) +/hatwidesi,µ, then\nthe Hamiltonian Hin Eq. (1) can be rewritten as\nH=HM+HB+Vsd, (3)\nwith the interaction term\nVsd=J/summationdisplay\ni,µ/hatwidesi,µ/hatwideσi,µ. (4)\nDuring this time interval, the electron density matrix ρe\nmay be approximated to be constant because of the neg-\nligible change of the magnetization, and this can be jus-\ntified in the limit t→t0. Assuming the total density\nmatrix as ρ(t) =ρM(t)⊗ρe(t0) and to the second or-\nder of interaction strength, the equation for the density\nmatrix/tildewideρM(t) in the interaction picture is[15]\nd\ndt/tildewideρM(t) =J\ni/planckover2pi1/summationdisplay\ni,µσi,µ(t)[/tildewidesi,µ(t),/tildewideρM(t0)]\n+(J\ni/planckover2pi1)2/summationdisplay\ni,µ;j,ν/integraldisplayt\nt0dτ{Ci,µ;j,ν(t,τ)[/tildewidesi,µ(t),/tildewidesj,ν(τ)/tildewideρM(τ)]\n−Cj,ν;i,µ(τ,t)[/tildewidesi,µ(t),/tildewideρM(τ)/tildewidesj,ν(τ)]}. (5)\nHere,/tildewider···denotes the operators in the interaction\npicture; the electron spin polarization is σi,µ(t) =Tre[/tildewideσi,µ(t)/tildewideρe(t0)]; the electron spin-spin correlation func-\ntion isCi,µ;j,ν(t,τ) = Tr e[/tildewideσi,µ(t)/tildewideσj,ν(τ)/tildewideρe(t0)], which\nis a function of t−τonly and satisfies the relation\nCi,µ;j,ν(t,τ) =C∗\nj,ν;i,µ(τ,t). In priciple, the solution of\nEq. (5) gives the density matrix of the magnetization in\nthe time interval [ t0,t] under the influence of the electron\nbath, and can be applied to study the physical qualities\nthat we are particularly interested in.\nBased on Eq. (5), the dynamical equation for Sl,λ(t) =\nTrM[/tildewideSl,λ(t)/tildewideρM(t)] is obtained as\nd\ndtSl,λ(t) =1\ni/planckover2pi1/an}b∇acketle{t[/hatwideSl,λ,HM]/an}b∇acket∇i}htt+J\n/planckover2pi1/summationdisplay\nµ,νǫλµνσl,µ(t)Sl,ν(t)\n+i(J\ni/planckover2pi1)2/summationdisplay\nj,µ,ν,ξǫλµν/integraldisplayt\nt0dτ{Cl,µ;j,ξ(t,τ)/an}b∇acketle{t/hatwideSl,ν(t)/hatwidesj,ξ(τ)/an}b∇acket∇i}htτ\n−Cj,ξ;l,µ(τ,t)/an}b∇acketle{t/hatwidesj,ξ(τ)/hatwideSl,ν(t)/an}b∇acket∇i}htτ}. (6)\nHere,/an}b∇acketle{t···/an}b∇acket∇i}htt≡TrM[···ρM(t)], and the spin commutation\nrelation [/hatwideSl,λ,/hatwideSi,µ] =iδli/summationtext\nνǫλµν/hatwideSl,νhas been exploited.\nThe first term in the r.h.s.of Eq. (6) gives the intrin-\nsic magnetization dynamics due to HM; the second term\nis the spin torque term due to the accumulation of the\nelectron spin density; the third term gives the damping\neffect of the electron bath. If the operator /hatwideSl,ν(t) in the\ndamping term is approximately replaced by its expecta-\ntion value Sl,ν(t), Eq. (6) becomes\nd\ndtSl,λ(t) =1\ni/planckover2pi1/an}b∇acketle{t[/hatwideSl,λ,HM]/an}b∇acket∇i}htt+J\n/planckover2pi1/summationdisplay\nµ,νǫλµνσl,µ(t)Sl,ν(t)\n+2J2\n/planckover2pi12/summationdisplay\nj,µ,ν,ξǫλµνSl,ν(t)/integraldisplayt\nt0dτKl,µ;j,ξ(t−τ)sj,ξ(τ),(7)\nwhereKl,µ;j,ξ(t−τ) is the imaginary part of Cl,µ;j,ξ(t,τ),\nandsj,ξ(τ) =/an}b∇acketle{t/hatwidesj,ξ/an}b∇acket∇i}htτ. We introduce the kernel func-\ntionγ(t) which satisfies the relation dγl,µ;j,ξ(t)/dt=\nKl,µ;j,ξ(t). The integral in the last term in Eq. (7) is\nrewrittenas/integraltextt\nt0dτγl,µ;j,ξ(t−τ)˙Sj,ξ(τ) afterintegratingby\nparts and neglecting the boundary terms in the limiting\ncaset→t0. It can be further simplified as Γ l,µ;j,ξ˙Sj,ξun-\nder the Markovian approximation ˙Sj,ξ(τ)≈˙Sj,ξ(t), with\nthe coefficient Γ l,µ;j,ξ=/integraltextδt\n0dτγl,µ;j,ξ(τ) forδt=t−t0.\nBased on the discussions above, Eq. (7) can be written\nin a compact form\nd\ndtSl(t) =1\ni/planckover2pi1/an}b∇acketle{t[/hatwideSl,HM]/an}b∇acket∇i}htt+γeBl(t)×Sl(t),(8)\nwhereγeis the gyromagnetic ratio; Blis the effective\nmagnetic field on the the local spin Sloriginating from\nthe electron bath. The µ-component of Blis expressed\nas\nBl,µ(t) =J\nγe/planckover2pi1σl,µ(t)+2J2\nγe/planckover2pi12/summationdisplay\nj,ξΓl,µ;j,ξ(t)˙Sj,ξ(t).(9)\nThe first term in (9) will give the torque term due to the\nelectron spin accumulation, which has been discussed ex-\ntensively in previous studies; the second term will give3\nthe damping effect of the electron bath on the magne-\ntization dynamics, which only emerges in the quantum\ntreatment. The non-local feature of the damping term\ncan be found here, which depends on the spatial correla-\ntion of Γ l,µ;j,ξ.\nSo far we have established a general dynamical equa-\ntion for the magnetization when it is coupled to the elec-\ntron bath via s-d exchange interaction. Here, both the\nHamiltonian for the magnetization subsystem HMand\nthe Hamiltonian for the electron subsystem Hehave not\nbeen specified yet. The treatment is similar to the previ-\nous work on the order parameter dynamics in the photo-\nexcited Peierls chain[14]. In the next section, we apply\nthis general formula to study the magnetization dynam-\nics of a two-dimensional ferromagnetic thin film under\nthe influence of an electron gas with Rashba SOI, i.e. a\nmodel system for “spin-orbit torque”.\nIII. SPIN-ORBIT TORQUE\nA. Electron Bath with Rashba SOI\nWe consider a special system studied by Manchon and\nZhang[3] for the spin-orbit torque. The two-dimensional\nmagnetization thin film in x-y plane consists of N=\nM×Nlattice sites with the lattice constant a, and we\nwill use the discrete notations in both real and reciprocal\nspace. The magnetization is assumed to be uniform due\nto strong exchange interaction. The lack of inversion\nsymmetry in z-direction induces the Rashba spin-orbit\ninteraction in the two-dimensional electron gas. In this\ncase, the Hamiltonian for the electron bath is given as[3]\nHB=/hatwidep2\n2m∗e+αR\n/planckover2pi1(/hatwidep×/hatwidez)·/hatwideσ+JS·/hatwideσ,(10)\nwhere/hatwidepis the electron momentum operator; m∗\neis the\neffective mass of electrons; αRis the Rashba interaction\nstrength; S=Siis the localized spin at each site. For\nS=S(sinθcosφ,sinθsinφ,cosθ), the energy dispersion\nrelation of the electron is\nEk,±=/planckover2pi12k2\n2m∗e±∆k. (11)\nHere, we have denoted the electron wavevector k=\nk(cosϕ,sinϕ), and\n∆k=/radicalBig\nJ2S2+α2\nRk2−2JSαRksinθsin(φ−ϕ).\nThe corresponding electron eigenstates |k,±/an}b∇acket∇i}htare\n|k,±/an}b∇acket∇i}ht=1√\nNeik·r/parenleftBigg\ncosΘk,±\n2e−iΦk\nsinΘk,±\n2/parenrightBigg\n,(12)where the angles Θ k,±and Φ kare determined by\ncosΘk,±\n2=/radicalbig\n∆2\nk−J2S2cos2θ/radicalbig\n2∆2\nk∓2JS∆kcosθ,\nsinΘk,±\n2=±∆k−JScosθ/radicalbig\n2∆2\nk∓2JS∆kcosθ,\ncosΦk=JSsinθcosφ+αRksinϕ/radicalbig\n∆2\nk−J2S2cos2θ,\nsinΦk=JSsinθsinφ−αRkcosϕ/radicalbig\n∆2\nk−J2S2cos2θ.\nThespinpolarizationvectorforthestate |k,±/an}b∇acket∇i}htisPk,±=\n(sinΘk,±cosΦk,sinΘk,±sinΦk,cosΦk,±).\nThe statistical properties of the electron bath are de-\ntermined by the probability distribution function fk,sfor\nthe state |k,s=±/an}b∇acket∇i}ht, which can be tuned by the external\nfield. If an electric field Eis applied, the non-equilibrium\ndistribution of the electron states will be established due\nto the random scattering potential by impurities[3]. The\ndistribution function fk,sis determined by the Boltz-\nmann equation\n−eE\n/planckover2pi1·∇kfk,s=Sc[fk,s]. (13)\nThe collision integral Sc[fk,s] describes the relaxation of\nthe occupied state |k,s/an}b∇acket∇i}htand can be treated by the relax-\nation time approximation, namely,\nSc[fk,s] =−fk,s−f0\nk,s\nτ. (14)\nHere,f0\nk,sis the equilibrium distribution function, and\nan isotropic relaxation time τhas been assumed[3]. To\nthe first orderofthe electric field, the solution of Eq. (13)\nisfk,s=f0\nk,s+gk,s, where the out of equilibrium part\ninduced by the external electric field is\ngk,s=∂f0\nk,s\n∂EeE·vk,sτ, (15)\nwith the electron velocity vk,s=1\n/planckover2pi1∇kEk,s. Such a treat-\nment of the non-equilirium electron distribution was also\nexploited in the previous semiclassical theory[3].\nB. Electron Spin Polarization and Torque\nWith the non-equilibrium distribution function fk,s\ngiven above, the electron spin polarization σl,µat sitel\nand the correlationfunction Cl,µ;j,ξ(t,τ) in Eq. (9) can be\ncalculated, and the torque and damping effect due to the\nelectron bath can be obtained. In the second quantiza-\ntion representation of the basis set {|k,s/an}b∇acket∇i}ht}, the operator\n/hatwideσl,µis expressed as\n/hatwideσl,µ=1\nN/summationdisplay\nk,s;k′,s′χµ\nk,s;k′,s′ei(k′−k)·rl/hatwidec†\nk,s/hatwideck′,s′,4\nwhere the matrix element\nχµ\nk,s;k′,s′= (cosΘk,s\n2eiΦk,sinΘk,s\n2)σµ/parenleftBigg\ncosΘk′,s′\n2e−iΦk′\nsinΘk′,s′\n2/parenrightBigg\n.\nThen the electron spin polarization σl,µis\nσl,µ=1\nN/summationdisplay\nk,sχµ\nk,s;k,sfk,s=1\nN/summationdisplay\nk,sPµ\nk,sfk,s.(16)\nFor the physically relevant case αRk≪JS, the approx-\nimate value of Pk,±to the first order ofαRk\nJSis\nPk,±=±\nSx+αR\nJSSxSykx+αR\nJS(1−S2\nx)ky\nSy−αR\nJS(1−S2\ny)kx−αR\nJSSxSyky\nSz+αR\nJSSySzkx−αR\nJSSxSzky\n.\nHere, the unit vector for the magnetization is denoted as\nS= (sinθcosφ,sinθsinφ,cosφ).\nFor the electric current density je=je(cosϑ,sinϑ,0),\nthe non-equilibrium spin polarization δσlwhich is per-\npendicular to Sis calculated to be (Appendix A)\nδσl=−αRm∗\nejea3\ne/planckover2pi1Ef\ncosϑSxSy+sinϑ(1−S2\nx)\n−cosϑ(1−S2\ny)−sinϑSxSy\ncosϑSySz−sinϑSxSz\n,\nwhereEfdenotes the Fermi energy. The torque Tlis\nthen obtained as\nTl=JSαRm∗\nejea3\ne/planckover2pi12Ef\ncosϑSz\nsinϑSz\n−cosϑSx−sinϑSy\n\n=JαRm∗\nea3\ne/planckover2pi12Ef(/hatwidez×je)×Sl.\nThis result reproduces the form of SOT obtained\nbefore[3], but the magnetization vector is not restricted\nin two-dimensional x-y plane in our derivations. It is\neasily understood from the effective Hamiltonian (10),\nwhere the non-equilibrium distribution of electron states\nwill produce an extra electron spin polarizationalong the\ndirection/hatwidez×je.\nC. Correlation Function and Damping\nWe now calculate the correlation function Cl,µ;j,ξ(t,τ),\nwhich gives the damping term for the magnetization\ndynamics due to the electron bath. Since /hatwideck,s(t) =\n/hatwideck,se−iEk,st//planckover2pi1, the correlation function Cl,µ;j,ξ(t,τ) is for-\nmally written as\nCl,µ;j,ξ(t,τ)\n=1\nN2/summationdisplay\nk,s;k′,s′/summationdisplay\nk′′,s′′;k′′′,s′′′ei(k′−k)·rlei(k′′′−k′′)·rj\n×ei(Ek,s−Ek′,s′)t//planckover2pi1ei(Ek′′,s′′−Ek′′′,s′′′)τ//planckover2pi1\n×χµ\nk,s;k′,s′χξ\nk′′,s′′;k′′′,s′′′/an}b∇acketle{t/hatwidec†\nk,s/hatwideck′,s′/hatwidec†\nk′′,s′′/hatwideck′′′,s′′′/an}b∇acket∇i}ht.(17)We see that Cl,µ;j,ξ(t,τ) is the function of rl−rjand\nt−τ, due to the space and time translation invari-\nance for the investigated system. For simplicity, we es-\ntimateCl,µ;j,ξ(t,τ) with several approximations. Firstly,\nwe assumethat the phase factor ei(k′−k)·(rl−rj)will cause\nthe cancellation of the summations over kandk′if\nrl/ne}ationslash=rj, thusCl,µ;j,ξ=Cµξδlj. Secondly, χµ\nk,s;k′,s′are\ncalculated to the zeroth order ofαRk\nJSfor the relevant\ncaseαRk≪JS, where the electron spin states are k-\nindependent, i.e.\nχ±±=±(sinθcosφ,sinθsinφ,cosθ),\nχ+−= (−cosθcosφ−isinφ,−cosθsinφ+icosφ,sinθ).\nFurthermore, we calculate the correlation function\n/an}b∇acketle{t/hatwidec†\nk,s/hatwideck′,s′/hatwidec†\nk′′,s′′/hatwideck′′′,s′′′/an}b∇acket∇i}htwith the electron bath at equi-\nlibrium, where the effect of the non-equilibrium electric\ncurrent induced by the external field will be neglected.\nThis enable us to apply the Wick contraction[16] to sim-\nplify the calculations. The negligence of the dependence\nof the damping coefficient on the Rashba SOI and the\nnon-equilibrium electric current is valid if the dynamical\nequation (8) is kept to the first order of these two factors.\nWith the above approximations, we get\nCµξ(t)\n=1\nN2/summationdisplay\nk,sχµ\nssχξ\nssfk,s+1\nN2/summationdisplay\nk,s;k′,s′χµ\nssχξ\ns′s′fk,sfk′,s′\n+1\nN2/summationdisplay\nk,s;k′,s′ei(Ek,s−Ek′,s′)t//planckover2pi1χµ\nss′χξ\ns′sfk,s(1−fk′,s′),\nwhere|k,s/an}b∇acket∇i}htand|k′,s′/an}b∇acket∇i}htare different states.\nSince the kernel function γl,µ;j,ξ(t) is given by the\nrelation dγl,µ;j,ξ(t)/dt=Kl,µ;j,ξ(t), where Kl,µ;j,ξ(t) =\nℑ[Cl,µ;j,ξ(t)], their Fourier transformations are related by\nγl,µ;j,ξ(ω) =i\nωKl,µ;j,ξ(ω). The Fourier transformation of\nKl,µ;j,ξ(t) is obtained as (Appendix B)\nKl,µ;j,ξ(ω)\n=δlj(m∗\nea2\n2π/planckover2pi12)2/planckover2pi1\n2i/summationdisplay\ns,s′[χµ\nss′χξ\ns′sgs(ω)−(χµ\nss′χξ\ns′s)∗gs(−ω)],\nwhere the function gs(ω) is defined as\ngs(ω) =\n\n0,/planckover2pi1ω <0;\n/planckover2pi1ω ,0</planckover2pi1ω < E f−sJS;\nEf−sJS , /planckover2pi1ω > E f−sJS.\nThen the damping kernel function γl,µ;j,ξ(t) can be\ncalculated by the inverse Fourier transformation from\nγl,µ;j,ξ(ω), which results in (Appendix B)\nγl,µ;j,ξ(t) =δlj(m∗\nea2\n2π/planckover2pi1)21\n2/summationdisplay\ns(δµξg−\ns(t)+is/summationdisplay\nνǫµξνSνg+\ns(t)).\n(18)\nHere,g±\ns(t) =/integraltext+∞\n−∞dωg±\ns(ω)e−iωtandg±\ns(ω) =\n1\n/planckover2pi1ω(gs(ω)±gs(−ω)), as schematically shown in Fig. 2.5\nThen the coefficient Γ l,µ;j,ξin Eq. (9) is obtained as\nΓl,µ;j,ξ=δlj(m∗\nea2\n2π/planckover2pi1)2(Γ(1)δµξ+Γ(2)/summationdisplay\nνǫµξνSν),(19)\nwith Γ(1)=1\n2/summationtext\ns/integraltextδt\n0dτg−\ns(τ) and Γ(2)=\ni\n2/summationtext\nss/integraltextδt\n0dτg+\ns(τ). Then the damping part in Eq. (8)\ncan be explicitly written as\nDl= 2(Jm∗\nea2\n2π/planckover2pi12)2(Γ(1)˙Sl×Sl+SΓ(2)˙Sl),(20)\nwhich is independent of the Rashba constant and the\nelectric current due to our approximations above.\n0−1−0.500.51\nhωgs+(ω)(a)\nEf+JSs = −s = +\nEf−JS\n000.51\nhωgs−(ω)(b)\ns = +\ns = −Ef−JSEf+JS\nFIG. 2. (Color online). Schematic diagram for g±\ns(ω). Blue\nline fors= +, and red line for s=−. Notice that g+\nsis an\nodd function of ωandg−\ns(ω) is an even function of ω, and\nthey approach to 0 when |ω| → ∞.\nThe first term in (20) will give the damping effect\nwhich drivesthe local spin towardsthe direction with the\nlower energy; while the second term in (20) will give a\nrenormalized factor in Eq. (8). Assuming that J∼1 eV,\nm∗\ne∼me,a∼1˚A, one gets the rough estimation of the\nmagnitudeorderforthe factor(Jm∗\nea2\n2π/planckover2pi12)2∼10−3, thusthe\ndamping effect due to the electron bath is comparable to\nthe intrinsic Gilbert damping of some ferromagnetic ma-\nterials. This damping effect can become important to\nunderstand the dissipative features of the magnetization\ndynamics driven by spin-orbit torque.\nIV. CONCLUSION\nIn conclusion, we have applied density matrix tech-\nnique to formulate the magnetization dynamics of a sys-\ntem consisting of local magnetic moments influenced by\nan electron gas through s-d exchange interaction. In\nthis approach, the magnetic subsystem is treated as an\nopen quantum system and the electron gas acts as a non-\nequilibrium bath tuned by the external electricfield. The\nspin torque due to the non-equilibrium electron spin ac-\ncumulation and the damping effect of the electron bath\nhave been taken into account simultaneously. We ap-\nply the developed formula to the model system for spin-\norbit torque, where the two-dimensional magnetization\nfilm is coupled to the Rashba electron gas through s-dexchange interaction. We have calculated the spin-orbit\ntorque and the results are consistent with the previous\nstudy. However, our method does not require the mag-\nnetization direction to be in the two-dimensional plane\nas in the previous study. Our approach enables us to ob-\ntain the damping effect due to the electron bath, which\nis a new feature absent in the semiclassical theory. The\ndamping caused by the electron bath is estimated to be\ncomparableto the intrinsic Gilbert damping, and may be\nimportanttodescribethemagnetizationdynamicsdriven\nby spin-orbit torque. In brief, this work has extended\nthe previous semiclassical theory for spin-orbit torque\nto a more complete description. Further applications of\nthis approach are expected to understand and to manip-\nulate the magnetization dynamics through electron gas\nin other complex cases.\nACKNOWLEDGMENTS\nThis work was supported in part by the Hong Kong’s\nUniversity Grant Council via grant AoE/P-04/08. This\nwork is also partially supported by National Basic Re-\nsearch Program of China (No. 2014CB921203), NSFC\ngrant (No.11274269), and NSFC grant (No.11204186).\nAppendix A: Electron Spin Polarization\nWe first assume that the electric field is applied along\nx-direction, then\nδσl=1\nN/summationdisplay\nk,sgk,sPk,s=1\nN/summationdisplay\nk(gk,+−gk,−)kxαR\nJSΣx,\nwhereΣx= (SxSy,−(1−S2\ny),SySz). The corresponding\nelectric current density is\nje=−e\nNa3/summationdisplay\nk,sgk,s(vk,s)x≈ −e/planckover2pi1\nm∗e1\nN/summationdisplay\nk,sgk,skx,\nand the spin current density is\njs=/planckover2pi1\n2Na3/summationdisplay\nk,sgk,s(vk,s)xPk,s\n≈/planckover2pi12\n2m∗e1\nNa3/summationdisplay\nk(gk,+−gk,−)kxS.\nThus a rough relation is obtained as\nδσl=−αRm∗\nejea3\ne/planckover2pi1EfΣx,\nwhere the relation js≈ −/planckover2pi1JS\n2eEfjeShas been used here.\nSimilarly, if the electric field is applied along the y-\ndirection, the non-equilibrium spin polarization will be\nδσl=−αRm∗\nejea3\ne/planckover2pi1EfΣy,6\nwithΣy= (1−S2\nx,−SxSy,−SxSz). Therefore, for the\nelectric current density je=je(cosϑ,sinϑ,0), we get\nδσl=−αRm∗\nejea3\ne/planckover2pi1Ef\ncosϑSxSy+sinϑ(1−S2\nx)\n−cosϑ(1−S2\ny)−sinϑSxSy\ncosϑSySz−sinϑSxSz\n.\nAppendix B: Correlation Function and Damping\nKernel\nThe imaginary part of Cµξ(t) is given as\nKµξ(t)\n=ℑ[Cµξ(t)]\n= (m∗\nea2\n2π/planckover2pi12)2/summationdisplay\ns,s′ℑ[χµ\nss′χξ\ns′s/integraldisplayEf\nsJSdǫ/integraldisplay∞\nEfdǫ′ei\n/planckover2pi1(ǫ−ǫ′)t]\n= (m∗\nea2\n2π/planckover2pi12)2/summationdisplay\ns,s′/integraldisplayEf\nsJSdǫ/integraldisplay∞\nEfdǫ′[−i\n2χµ\nss′χξ\ns′sei\n/planckover2pi1(ǫ−ǫ′)t+h.c.].\nHere,fk,sis approximated as the zero-temperature\nFermi distribution function, and the relation1\nN/summationtext\nk→\na2\n(2π)2/integraltext\nd2k=m∗\nea2\n2π/planckover2pi12/integraltext\ndǫhas been used. Its Fourier trans-formation Kµξ(ω) is then\nKµξ(ω)\n=1\n2π/integraldisplay+∞\n−∞dtKµξ(t)eiωt\n= (m∗\nea2\n2π/planckover2pi12)2/summationdisplay\ns,s′/integraldisplayEf\nsJSdǫ/integraldisplay∞\nEfdǫ′\n×[−i\n2χµ\nss′χξ\ns′sδ(ω+ǫ−ǫ′\n/planckover2pi1)+i\n2(χµ\nss′ξξ\ns′s)∗δ(ω+ǫ′−ǫ\n/planckover2pi1)]\n=−(m∗\nea2\n2π/planckover2pi12)2i/planckover2pi1\n2/summationdisplay\ns,s′[χµ\nss′χξ\ns′sgs(ω)−(χµ\nss′χξ\ns′s)∗gs(−ω)],\nwhere the function g(ω) is defined as\ngs(ω) =\n\n0,/planckover2pi1ω <0;\n/planckover2pi1ω ,0</planckover2pi1ω < E f−sJS;\nEf−sJS , /planckover2pi1ω > E f−sJS.\nTherefore,\nγl,µ;j,ξ(ω)\n=δlj(m∗\nea2\n2π/planckover2pi1)21\n2/summationdisplay\ns,s′[ℜ(χµ\nss′χξ\ns′s)g−\ns(ω)+iℑ(χµ\nss′χξ\ns′s)g+\ns(ω)],\nwhereg±\ns(ω) =1\n/planckover2pi1ω(gs(ω)±gs(−ω)), andγl,µ;j,ξ(t) is cal-\nculated as\nγl,µ;j,ξ(t)\n=/integraldisplay+∞\n−∞dωγl,µ;j,ξ(ω)e−iωt\n=δlj(m∗\nea2\n2π/planckover2pi1)21\n2/summationdisplay\ns,s′[ℜ(χµ\nss′χξ\ns′s)g−\ns(t)+iℑ(χµ\nss′χξ\ns′s)g+\ns(t)]\n=δlj(m∗\nea2\n2π/planckover2pi1)21\n2/summationdisplay\ns(δµξg−\ns(t)+is/summationdisplay\nνǫµξνSνg+\ns(t))\n≈δljδµξ(m∗\nea2\n2π/planckover2pi1)2g−(t),\nwhereg±\ns(t) =/integraltext+∞\n−∞dωg±\ns(ω)e−iωtand we have used the\nexpressions\nχµ\n+,+χξ\n+,+=χµ\n−,−χξ\n−,−=SµSξ.\nχµ\n+,−χξ\n−,+= (χµ\n−,+χξ\n+,−)∗=δµξ−SµSξ+i/summationdisplay\nνǫµξνSν.\n[1] J. C. Slonczewski, J. Magn. Magn. Mater. 159, L1\n(1996).\n[2] L. Berger, Phys. Rev. B 54, 9353 (1996).\n[3] A. Manchon and S. Zhang, Phys. Rev. B 78, 212405\n(2008).\n[4] A. Manchon and S. Zhang, Phys. Rev. B 79, 094422\n(2009).\n[5] A. Chernyshov, M. Overby, X. Liu, J. K. Furdyna, Y.\nLyanda-Geller, and L. P. Rokhinson, Nat. Phys. 5, 656(2009).\n[6] I. M. Miron, G. Gaudin, S. Auffret, B. Rodmacq, A.\nSchuhl, S. Pizzini, J. Vogel, and P. Gambardella, Nat.\nMater.9, 230 (2010).\n[7] K. Garello, I. M. Miron, C. O. Avci, F. Freimuth, Y.\nMokrousov, S. Bl¨ ugel, S. Auffret, O. Boulle, G. Gaudin,\nand P. Gambardella, Nat. Nanotechnol. 8, 587 (2013).\n[8] Y.B. Fan, P. Upadhyaya,X.F. Kou, M. R.Lang, S.Takei,\nZ.X. Wang, J. S. Tang, L. He, L.-T. Chang, M. Montaz-7\neri, G.Q. Yu, W. J. Jiang, T. X. Nie, R. N. Schwartz,\nY. Tserkovnyak, and K. L. Wang, Nat. Mater. 13, 699\n(2014).\n[9] P. Gambardella and L. M. Miron, Phil. Trans. R. Soc. A\n369, 3175 (2011).\n[10] A. Brataas and K. M. D. Hals, Nat. Nanotechnol. 9, 86\n(2014).\n[11] Y. Wang and L. J. Sham, Phys. Rev. B 85, 092403\n(2012).\n[12] Y. Wang and L. J. Sham, Phys. Rev. B 87, 174433\n(2013).[13] T. Tay and L. J. Sham, Phys. Rev. B 87, 174407 (2013).\n[14] Y. Wang, W.Q. Chen, and F.C. Zhang, Phys. Rev. B 90,\n205110 (2014).\n[15] K. Blum, Density Matrix Theory and Applications\n(Springer-Verlag, Berlin Heidelberg, 2012).\n[16] A. L. Fetter and J. D. Walecka, Quantum Theory of\nMany-Particle Systems , (McGraw-Hill, New York, 1971)."    },    {        "title": "1412.3783v1.Deviation_From_the_Landau_Lifshitz_Gilbert_equation_in_the_Inertial_regime_of_the_Magnetization.pdf",        "content": "arXiv:1412.3783v1  [cond-mat.mtrl-sci]  11 Dec 2014Deviation From the Landau-Lifshitz-Gilbert equation in th e Inertial regime of the\nMagnetization\nE. Olive and Y. Lansac\nGREMAN, UMR 7347, Universit´ e Fran¸ cois Rabelais-CNRS, Pa rc de Grandmont, 37200 Tours, France\nM. Meyer, M. Hayoun, and J.-E. Wegrowe\nLaboratoire des Solides Irradi´ es, ´Ecole Polytechnique, CEA-DSM, CNRS, F-91128 Palaiseau, Fr ance\n(Dated: February 7, 2018)\nWe investigate in details the inertial dynamics of a uniform magnetization in the ferromagnetic\nresonance (FMR) context. Analytical predictions and numer ical simulations of the complete equa-\ntions within the Inertial Landau-Lifshitz-Gilbert (ILLG) model are presented. In addition to the\nusual precession resonance, the inertial model gives a seco nd resonance peak associated to the nuta-\ntion dynamics provided that the damping is not too large. The analytical resolution of the equations\nof motion yields both the precession and nutation angular fr equencies. They are function of the in-\nertial dynamics characteristic time τ, the dimensionless damping αand the static magnetic field H.\nA scaling function with respect to ατγHis found for the nutation angular frequency, also valid for\nthe precession angular frequency when ατγH≫1. Beyond the direct measurement of the nutation\nresonance peak, we show that the inertial dynamics of the mag netization has measurable effects on\nboth the width and the angular frequency of the precession re sonance peak when varying the applied\nstatic field. These predictions could be used to experimenta lly identify the inertial dynamics of the\nmagnetization proposed in the ILLG model.\nPACS numbers:\nI. INTRODUCTION\nThe Landau-Lifshitz-Gilbert (LLG) equation is a ki-\nnetic equation that does not contain acceleration terms,\ni.e. that does not contain inertia. The corresponding\ntrajectory is reduced to a damped precession around\nthe axis defined by the effective field. The measurement\nof this precession is usually performed by the mean of\nferromagnetic resonance (FMR). The power absorbed by\nthe system is then measured at steady state while adding\nan oscillatory field to the effective field, and tuning the\nfrequency close to the resonance frequency. However,\nthe validity of the LLG equation is limited to large\ntime scales1, or low frequency regimes (similarly to the\nDebye model of electric dipoles2). Indeed, the precession\nwith damping described by the LLG equation is a\ndiffusion process in a field of force, for which the angular\nmomentum has reached equilibrium. Accordingly, if\nthe measurements are performed at fast enough time\nscales, or high enough frequencies, inertial terms should\nbe expected to play a role in the dynamics, which is no\nlonger reduced to a damped precession3–9. A nutation\ndynamics is therefore expected, giving a second resonant\npeak at the nutation frequency, and this new absorption\nshould be measurable with dedicated spectroscopy (e.g.\nusing infrared spectroscopy).\nDespite its fundamental importance, a systematic\nexperimental investigation of possible inertial effects of\nthe uniform magnetization has however been overlooked.\nIn order to evidence experimentally the consequences\nof inertia in the dynamics of a uniform magnetization,\nit is first necessary to establish the characteristics\nthat would allow to discriminate inertia from spuriouseffects in spectroscopy experiments. We propose in this\npaper some simple theoretical and numerical tools than\ncan be used by experimentalists in order to evidence\nunambiguouslytheeffectsofinertiaofthemagnetization.\nThe LLG equation reads :\ndM\ndt=γM×/bracketleftbigg\nHeff−ηdM\ndt/bracketrightbigg\n(1)\nwhereMisthemagnetization, Hefftheeffectivemagnetic\nfield,ηthe Gilbert damping, and γthe gyromagnetic ra-\ntio. If the description is extended to the fast degrees\nof freedom (i.e. the degrees of freedom that includes the\ntime derivative of the angularmomentum), a supplemen-\ntary inertial term should be added with the correspond-\ningrelaxationtime τ. FromthisInertialLandau-Lifshitz-\nGilbert (ILLG) model, the new equation reads3–7:\ndM\ndt=γM×/bracketleftbigg\nHeff−η/parenleftbiggdM\ndt+τd2M\ndt2/parenrightbigg/bracketrightbigg\n(2)\nOne of the main consequences of the new dynami-\ncal equation is the emergence of the second resonance\npeak associated to the nutation at high frequencies, as\nreported in our previous study7. In the literature the\nnutation dynamics of magnetic moments has been in-\nvestigated using various theoretical approaches though\nnot yet evidenced experimentally. B¨ ottcher and Henk\nstudied the significance of nutation in magnetization dy-\nnamics of nanostructures such as a chain of Fe atoms,\nand Co islands on Cu(111)8. They found that the nu-\ntation is significant on the femtosecond time scale with\na typical damping constant of 0.01 up to 0.1. Moreover,2\nthey concluded that nutation shows up preferably in low-\ndimensional systems but with a small amplitude with\nrespect to the precession. Zhu et al.predicted a nuta-\ntion dynamics for a single spin embedded in the tunnel-\ning barrierbetween twosuperconductors10. This unusual\nspin dynamics is caused by coupling to a Josephson cur-\nrent. They argue that this prediction might be directly\ntested for macroscopic spin clusters. The nutation is also\ninvolved in the dynamics of a single spin embedded in\nthe tunnel junction between ferromagnets in the pres-\nence of an alternating current11. In an atomistic frame-\nwork, Bhattacharjee et al.showed that first-principle\ntechniques used to calculate the Gilbert damping factor\nmay be extended to calculate the moment of inertia ten-\nsor associated to the nutation9.\nOur previous work7was focussed on the short time\nnutation dynamics generated by the ILLG equation, and\nwas limited to fixed values of the inertial characteristic\ntime scale τ, the dimensionless damping αand the static\nfieldH. In this paper we present a combined analytical\nand numerical simulation study of the ILLG equation\nwith new results. In particular we derive analytical re-\nsults in the small inclination limit that can be used in\nferromagnetic resonance (FMR) experiments, and which\nallow to predict both the precession and nutation reso-\nnance angular frequencies. We also investigate the ILLG\nequation while varying the three parameters α,τandH,\nand scaling functions are found. Finally, we present im-\nportant indications for experimental investigations of the\ninertial dynamics of the magnetization. Indeed, a conse-\nquence of the ILLG equation is the displacement of the\nwell-known FMR peak combined with a modified shape\nwith respect to that given by the LLG equation. This\ndisplacement could not be without consequences on the\ndetermination of the gyromagnetic factor γby ferromag-\nnetic resonance.\nThe paper is organized as follows. In section II we\nshow analytical solutions of the precession and nutation\ndynamics for the uniform magnetization in a static ap-\nplied field H. The small inclination limit is investigated\nin order to reproduce the usual experimental FMR con-\ntext. In section III we describe the numerical simula-\ntions of the magnetization inertial dynamics in both a\nstatic anda smallperpendicular sinusoidalmagneticfield\n(Heff=H+h⊥(ω)). The resonance curves are computed\nand, provided that the damping is not too large, a nu-\ntation resonance peak appears in addition to the usual\nferromagnetic resonance peak associated to the magne-\ntization precession. In section IV the behavior of the\nILLG equation is investigated in details while varying\nthe characteristic time τof the inertial dynamics, the di-\nmensionless damping αand the static field H. A very\ngood agreement is found between the analytical and nu-\nmerical simulation results, and a scaling function with\nrespect to ατγHis found. In section V we propose ex-\nperiments in the FMR context that should evidence the\ninertial dynamics of the magnetization described in the\nILLGmodel. Inparticular,whenthestaticfieldisvaried,the ILLG precession resonance peak has different behav-\niors compared to the usual LLG precession peak with\nshifted resonance angular frequency and modified shape.\nWe show that the differences between LLG and ILLG\nprecession peaks are more pronounced in large damping\nmaterials and increase with the static field. Finally, we\nderive the conclusions in section VI.\nII. ANALYTICAL SOLUTIONS FOR THE ILLG\nEQUATION\nThe magnetization position is described in spherical\ncoordinates ( Ms,θ,φ), where Msis the radius coordinate\nfixed at a constant value for the uniformly magnetized\nbody,θis the inclination and φis the azimuthal angle.\nIn a static magnetic field Hˆ zapplied in the zdirection,\ni.e.H=H(cosθer−sinθeθ) in the spherical basis\n(er,eθ,eφ), Eq. (2) gives the following system :\n¨θ=−1\nτ˙θ−1\nτ1˙φsinθ+˙φ2sinθcosθ\n−ω2\nτ1sinθ (3a)\n¨φsinθ=1\nτ1˙θ−1\nτ˙φsinθ−2˙φ˙θcosθ (3b)\nwhere the characteristic times are τandτ1=ατ,ω2=\nγHis the Larmor angular frequency, and α=γηMsis\nthe dimensionless damping.\nUsingthedimensionlesstime t′=t/τ, Eqs. (3)become\nθ′′=−θ′−/tildewideτ1φ′sinθ+φ′2sinθcosθ\n−/tildewideω2/tildewideτ1sinθ (4a)\nφ′′sinθ=/tildewideτ1θ′−φ′sinθ−2φ′θ′cosθ(4b)\nwhere\nθ′=dθ/dt′, θ′′=d2θ/dt′2, φ′=dφ/dt′, φ′′=d2φ/dt′2,\nand\n/tildewideτ1=τ\nτ1=1\nα\n/tildewideω2=ω2τ=τγH\nIn the following subsections we extract analytical re-\nsults that can be used to predict the positions in the\nangular frequency domain of the precession and nutation\nresonance peaks. We will consider the small inclination\nlimit which holds in the FMR context.3\nA. Precession : exact and approximate solutions\nTo determine the precession dynamics of the iner-\ntial model we search for the long time scale solution\nφ′(t′) =φ′\nprec, whereφ′\nprecis the constant precession ve-\nlocity. Since the damping progressivelyshifts the magne-\ntization to the zaxis, we investigate the small inclination\nlimit where φ′(t′) =φ′\nprecshould hold. With sin θ∼θ\nand cosθ∼1, Eqs. (4) therefore reads :\nθ′′+θ′+/tildewideω2\n0θ= 0 (5a)\nφ′\nprec=/tildewideτ1θ′\nθ+2θ′(5b)\nwhere the natural angular frequency of the overdamped\nharmonic oscillator θ(t′) defined by Eq. (5a) is given by\n/tildewideω0=/radicalBig\n/tildewideτ1(φ′prec+/tildewideω2)−φ′2prec (6)\nThe characteristic equation associated to the differential\nequation Eq. (5a) is β2+β+/tildewideω2\n0= 0 which gives in the\naperiodic regime the two solutions\nβ±=−1±/radicalbig\n1−4/tildewideω2\n0\n2(7)\nSince|β+|<|β−|, the inclination of the magnetization\nbehaves at long time scales as\nθ(t′)∼eβ+t′,\nwhich inserted in Eq. (5b) gives\nφ′\nprec=/tildewideτ1β+\n1+2β+(8)\nIn original time units, the precession velocity ˙φprecis\ntherefore the solution of\n˙φprec=β+(˙φprec)\nατ/parenleftBig\n1+2β+(˙φprec)/parenrightBig (9)\nwhere the function β+(˙φprec) is given by\nβ+(˙φprec) =−1+/radicalbigg\n1−4τ/parenleftBig˙φprec+γH\nα−τ˙φ2prec/parenrightBig\n2(10)\nEquation 9 may be numerically solved to extract the\nprecession velocity, and therefore the precession reso-\nnance peak when a sinusoidal magnetic field h⊥(ω) is\nsuperimposed perpendicular to the static field Hˆ z.\nForτ≪10−11sandα≤0.1, theprecessionvelocity ˙φprec\nfor small applied static fields may be accurately evalu-\nated from a quadratic equation : in this case /tildewideω2\n0≪1 and\nEq. (7) leads to β+≈ −/tildewideω2\n0. Eq. (8) therefore gives a\ncubic equation in φ′\nprecwhere the cubic term −2αφ′3\nprecis negligeable. In this case the solution of the resulting\nquadratic equation is in original time units\n˙φprec=−b−/radicalbig\nb2+12τγH/α\n6τ(11)\nwithb= 2τγH−α−1/α. We choose the negative so-\nlution of the quadratic equation in order to agree with\nthe negative velocity ˙φLLG=−γH/(1+α2) given by the\nLLG model.\nB. Nutation : angular frequency\nUnlike the precession, the nutation properties should\nbederivedconsideringintermediatetime scaleswherethe\nprecession has not yet reached a constant velocity. Eqs.\n(4) should therefore be reconsidered. To derive the nuta-\ntion properties, it is convenient to examine the angular\nvelocityθ′. For simplicity we note θ′=/tildewideωθandφ′=/tildewideωφ.\nEqs. (4) therefore rewrite\n/tildewideω′\nθ=−/tildewideωθ−/tildewideτ1/tildewideωφsinθ+/tildewideω2\nφsinθcosθ\n−/tildewideω2/tildewideτ1sinθ (12a)\n/tildewideω′\nφsinθ=/tildewideτ1/tildewideωθ−/tildewideωφsinθ−2/tildewideωφ/tildewideωθcosθ(12b)\nWe derive Eq. (12a) with respect to time t′which gives\n/tildewideω′′\nθ=−/tildewideω′\nθ+(2/tildewideωφcosθ−/tildewideτ1)/tildewideω′\nφsinθ−/tildewideτ1/tildewideωφ/tildewideωθcosθ\n+/tildewideω2\nφ/tildewideωθ(cos2θ−sin2θ)−/tildewideω2/tildewideτ1/tildewideωθcosθ\nwhere the term /tildewideω′\nφsinθmay be replaced with the expres-\nsion in Eq. (12b). We therefore obtain\n/tildewideω′′\nθ+/tildewideω′\nθ+/parenleftbig\n/tildewideτ2\n1+/tildewideω2/tildewideτ1cosθ/parenrightbig\n/tildewideωθ=\n/tildewideτ1/tildewideωφsinθ+3/tildewideτ1/tildewideωφ/tildewideωθcosθ−2/tildewideω2\nφcosθsinθ\n−(3cos2θ+sin2θ)/tildewideω2\nφ/tildewideωθ (13)\nEq. (13)shouldbecloselyrelatedtothenutationdynam-\nics since it describes the /tildewideωθoscillator. This assumption\nwill be confirmed in section IVA2 for a broad range of\nparameters. Eq. (13) defines the damped oscillator /tildewideωθ\nwhich is non-linearly coupled to the /tildewideωφoscillator. This\nexpression shows that, in the absence of coupling and in\nthe smallinclination limit θ≪1rad, the/tildewideωθoscillatoros-\ncillatesatthenaturalangularfrequency/radicalbig\n/tildewideτ2\n1+/tildewideω2/tildewideτ1. We\ntherefore deduce an approximate expression for the nu-\ntation angularfrequency in the weak coupling case which\nis given by the expression\n/tildewideωweak\nnu=/radicalBig\n/tildewideτ2\n1+/tildewideω2/tildewideτ1 (14)\nwhich in original time units gives\nωweak\nnu=√1+ατγH\nατ(15)4\nFrom Eq. (15) we deduce the following asymptotic\nbehaviors : when τ≪1/αγHthenωweak\nnu∼1/ατ, and\nwhenτ≫1/αγHthenωweak\nnu∼1/√ατ.\nBecause of the non-linear coupling terms in the right-\nhand side of Eq. (13), the true position of the nuta-\ntion resonancepeak in FMR experiments may differ from\nthe approximate angular frequency defined by Eq. (15).\nHoweverthe simulation of the resonancecurves with a si-\nnusoidal magnetic field h⊥(ω) superimposed perpendic-\nular to the static field Hˆ zwill show in section IVA2\nthat the non-linear coupling terms only slightly shift the\nnutation resonance peak from the approximate angular\nfrequency.\nIII. NUMERICAL SIMULATIONS OF THE\nRESONANCE CURVES IN THE ILLG MODEL\nWe apply a fixed magnetic field H=Hˆ zalong the\nzdirection, and a small sinusoidal magnetic field h⊥=\nh⊥cosωtˆ xin thexdirection. In the spherical basis the\ncomponents of the total magnetic field Heff=H+h⊥in\nEq. (2) are\nHeff\nr=Hcosθ+h⊥sinθcosφcosωt\nHeff\nθ=−Hsinθ+h⊥cosθcosφcosωt\nHeff\nφ=−h⊥sinφcosωt.\nwhich lead to the following dynamical equations for the\nspherical angles ( θ,φ) of the magnetization\n¨θ=−1\nτ˙θ−1\nτ1˙φsinθ+˙φ2sinθcosθ\n−ω2\nτ1sinθ+ω3\nτ1cosθcosφcosωt(16a)\n¨φsinθ=1\nτ1˙θ−1\nτ˙φsinθ−2˙φ˙θcosθ\n−ω3\nτ1sinφcosωt (16b)\nwhereω3=γh⊥is the angular frequency associated to\nthe sinusoidal field.\nUsing the dimensionless time t′=t/τ, Eqs. (16) be-\ncome\nθ′′=−θ′−/tildewideτ1φ′sinθ+φ′2sinθcosθ\n−/tildewideω2/tildewideτ1sinθ+/tildewideω3/tildewideτ1cosθcosφcos/tildewideωt′(17a)\nφ′′sinθ=/tildewideτ1θ′−φ′sinθ−2φ′θ′cosθ\n−/tildewideω3/tildewideτ1sinφcos/tildewideωt′(17b)\nwhere\nθ′=dθ/dt′, θ′′=d2θ/dt′2, φ′=dφ/dt′, φ′′=d2φ/dt′2,and\n/tildewideτ1=τ\nτ1=1\nα\n/tildewideω2=ω2τ=τγH\n/tildewideω3=ω3τ=τγh⊥\n/tildewideω=ωτ\nWe useγ= 1011rad.s−1.T−1, and we vary the charac-\nteristic time τfor three different values of the dimension-\nless damping α= 0.1, 0.01 and 0 .5. We investigate sev-\neralvalues of the static magnetic field from H= 0.2Tup\ntoH= 200T. We numerically integrate Eqs. (17) using\neither a double precision second order Runge-Kutta algo-\nrithm or a double precision five order Gear algorithm12.\nTypically, we use time steps 10−7< δt′<10−3depend-\ning on the values of τandω.\nThe resonance curves are obtained by investigating\nthe magnetization response to the small oscillating field\nh⊥(ω) =h⊥cosωtˆ xapplied perpendicular to the static\nfieldH=Hˆ z. We analyse the permanent dynami-\ncal regime where the magnetization components oscil-\nlate around well defined mean values. For fixed values\nof the oscillating field angular frequency ωand oscil-\nlating field amplitude h⊥, we compute the mean value\n< M⊥>(averaged over time) of the transverse magneti-\nzationM⊥(t) =/radicalBig\nM2x(t)+M2y(t), fromwhichweextract\nfor fixed values of ωthe transverse susceptibility defined\nbyχ⊥=d < M ⊥> /dh ⊥. We choose values of the\noscillating field amplitude h⊥= 10−1,10−2,10−3,10−4\nand 10−5T, and we plot < M⊥>with respect to h⊥\nfor each ω. As an example, we show the case α= 0.1,\nτ= 2×10−10s,H= 2Tandω= 1.2×1011rad.s−1.\nThe inset of Fig. 1 shows that the response is linear\n< M⊥>=χ⊥h⊥wherefrom we extract the transverse\nsusceptibility χ⊥using a linear fitting. We repeat the\nsame procedure for each oscillating field angular fre-\nquencyωwhich gives the resonance curve χ⊥(ω) of the\ntransverse susceptibility shown in Fig. 1. Two peaks\nclearly appear, the usual FMR peak associated to the\nprecession velocity, and the nutation peak associated to\nthe nutation dynamics originatingfrom the inertial term.\nIV. RESULTS\nA. Effects of τ\nWe now examine the ILLG model when varying the\ncharacteristic time τ. For different values of the parame-\nterτ, we show in Fig. 2 the typical profiles of the trans-\nverse susceptibility χ⊥versus the angular frequency ωof\nthe applied oscillating field. The four resonance curves\nplotted in figure 2 are obtained by numerical simulations\nwithH= 2Tandα= 0.1. They show how the nu-\ntation resonance peak position depend on the value of\nτ. Asτis increased, the nutation peak moves towards5\n01×10112×10113×1011\nω (rad.s-1)1234567χ⊥10-510-410-310-2\nh⊥10-510-410-310-210-1< Μ⊥>Precession peak\nNutation peak\nFigure 1: Resonance curves of the transverse susceptibil-\nityχ⊥(ω) with respect to the oscillating field angular fre-\nquencyω. The resonance curves are computed within the\nILLG model with τ= 2×10−10s, for dimensionless damping\nα= 0.1 and for an applied static field H= 2T. Two reso-\nnance peaks are observed : the precession resonance at lower\nangular frequency which is the usual FMR and the nutation\nresonance at higher angular frequency. Inset : Example of\nthe calculation of χ⊥such that < M⊥>=χ⊥h⊥obtained for\nω= 1.2×1011rad.s−1.\nthe precession peak with an increasing intensity which is\nan order of magnitude smaller than the precession one\nforτ= 10−11s. Note that the transverse susceptibil-\nity at the resonance follows a power law of the form\nχ⊥(ωILLG\nnu)∝1/ωILLG\nnu, whereωILLG\nnuis defined as the nu-\ntation resonance angular frequency. A similar power law\nis reported for the precession peak obtained for different\nstatic fields H(see section IVB).\nWe now compare the analytical and numerical sim-\nulation results concerning the positions in the angular\nfrequency domain of both the precession and nutation\nresonance peaks.\n1. Precession peak\nWe define ωprec=|˙φprec|as the angular frequency of\nthe precession. When computed from the exact expres-\nsions (9) and (10) we will refer to ωexact\nprec, and when com-\nputed from the approximate expression (11) we will refer\ntoωapprox\nprec. Finally, we will denote by ωILLG\nprecthe angular\nfrequency of the precession resonance peak obtained in\nthe numerical simulations of the ILLG model. Eq. (9)\nmaybeeasilynumericallysolvedtofindthesolution ˙φprec\nfor several values of αandτ. The behavior with respect\ntoτofωprecobtained either analytically or from the sim-\nulated FMR curves is shown in Fig. 3. There is an excel-10 -4 10 -3 10 -2 10 -1 10 0\n10 11 10 12 10 13 10 14 10 15 χ\n⊥\nω ( rad.s -1 )τ=1×10 -113UHFHVVLRQ \n1utation \nτ=1×10 -12\nτ=1×10 -13\nτ=1×10 -14\nFigure 2: Resonance curves of the transverse susceptibilit y\nshowing the displacement of the nutation peak caused by the\nvariation of τ:τ= 10−11s (open circles), 10−12s (filled\ncircles), 10−13s (crosses), and 10−14s (open squares). These\ncurves are simulated using the ILLG model with α= 0.1,\nandH= 2T. Note that the precession peak positions are\nonly slightly affected. The dotted line shows the power law\nfitted on χ⊥∝1/ωILLG\nnu, whereωILLG\nnuis the resonance angular\nfrequency of the nutation.\nlent agreement between the analytical prediction ωexact\nprec\nand the precession resonance peak ωILLG\nprecobtained in nu-\nmericalsimulations. WealsoshowinFig.3theprecession\nangular frequency ωapprox\nprec. Forτ <10−11sandα= 0.1,\nit nicely agrees with the exact analytical value and with\nthenumericalsimulationresults, but the approximateso-\nlution becomes no longer valid for τ >10−11s. To quan-\ntify the validity of the approximate solution we compute,\nforτ= 10−12sand for three different dampings, the\nrelative difference\nδana\nprec=ωapprox\nprec−ωexact\nprec\nωexactprec×100\nWe show in the inset of Fig. 3 the evolution of δana\nprecwith\nrespect to the applied static field H. ForH <20T\nthe relative difference remains less than 0 .1% for small\ndamping α= 0.01, and remains less than 3% for mod-\nerate damping α= 0.1. For large damping α= 0.5 the\napproximate solution remains valid for small fields, but\nfor 12T < H < 20Tthe error becomes larger than 10%.\n2. Nutation peak\nFigure 4 displays both the analytical prediction of the\nnutation angular frequency ωweak\nnugiven by Eq. (15) and\nthe angular frequency ωIILG\nnuof the nutation resonance6\n10-1610-1510-1410-1310-1210-1110-1010-9\nτ (s)01×10112×1011ωprec (rad.s-1)\n0 5 10 15 20 25 30\nH (T)10-410-310-210-1100101102δprecana (%)α=0.5\nα=0.1\nα=0.01\nFigure 3: (Color online) Comparaison of the analytical and\nnumerical simulation results for the precession angular fr e-\nquency obtained for α= 0.1 andH= 2T. Filled circles\n(black) are the precession angular frequency ωexact\nprec, open cir-\ncles (red) are the position of the precession resonance peak s\nωILLG\nprec, stars (orange) are the approximate precession angular\nfrequencies ωapprox\nprecvalid for small values of τ. Thedashed line\n(black) is the LLG precession angular frequency, i.e.without\ninertial term. Inset : relative difference δana\nprecfor three differ-\nent dampings.\nobtained in the numerical simulations. The agreement\nis excellent for τ <10−11s, and indicates that the non-\nlinearcouplingtermsofEq. (13)donotsignificantlyshift\nthe angular frequency of the nutation resonance from the\napproximate angular frequency ωweak\nnu. On the contrary,\nin the range 10−11s < τ < 10−8s, the simulated nuta-\ntion resonance angular frequency is slightly higher than\nωweak\nnu, as shown in the upper inset of Fig. 4. In the lower\ninset ofFig. 4 we show the relative difference δnubetween\nthe approximate nutation angular frequency ωweak\nnuand\nthe nutation resonance angular frequency ωILLG\nnuof the\nnumerical simulations, i. e.\nδnu=ωILLG\nnu−ωweak\nnu\nωILLGnu×100\nWe therefore see that in the range 10−11s < τ <10−8s,\nthe approximate nutation angular frequency remains\nless than 15% close to the simulated nutation resonance\nangular frequency.\nB. Scaling and overview of the ILLG equation\nIn the preceding section we investigated the behav-\nior of the ILLG model when varying the characteristic\ntime scale τwhich drives the inertial dynamics. We also10-1610-1510-1410-1310-1210-1110-1010-9\nτ (s)1011101210131014101510161017ωnu (rad.s-1)\n10-1110-1010-9\nτ (s)10111012ωnu (rad.s-1)\n10-1310-1210-1110-1010-910-8\nτ (s)010\nδnu (%)\nFigure 4: (Color online) Comparaison of the analytical and\nnumerical simulation results for the nutation angular fre-\nquency obtained for α= 0.1 andH= 2T. Filled cir-\ncles (black) are the approximate nutation angular frequenc ies\nωweak\nnuand open circles (red) are the positions ωILLG\nnuof the\nsimulated nutation resonance peaks. Upper inset : enlarge-\nment showing the effect of the non-linear coupling terms of\nEq. (13). Lower inset : relative difference δnubetween ωweak\nnu\nandωILLG\nnu.\nvary the static field Hand the dimensionless damping\nα. Increasing Hmoves both the precession and nuta-\ntion resonance peaks to higher angular frequencies, with\nsmaller and broadened peaks, while increasing the di-\nmensionless damping moves both peaks to lower angular\nfrequencies with still smaller and broadened peaks. Note\nthat the ILLG precession resonances obtained when the\nstatic field His varied show that the transverse suscep-\ntibility follows a power law χ⊥∝1/ωIILG\nprec(not shown).\nThis law is the same as the one resulting from the LLG\nmodel13.\nEq. (15) suggests a scaling function\nωnu\nγH=√1+x\nx\nwherex=ατγH. Scaling curves obtained for different\nvalues of τ,αandHare shown in the inset of Fig. 5\nwhere both the precession and nutation resonance an-\ngular frequencies are dispayed with respect to ατγH.\nFig. 5 is an enlargement of the intermediate region of\nthe inset where we added the points obtained by the nu-\nmerical simulations for H= 2Tandα= 0.1. The\ntwo asymptotic behaviors of the nutation are highlighted\nwith the dashed lines in agreement with Eq. (15) :\nwhenατγH≪1 thenωweak\nnu/γH= 1/ατγH, and when\nατγH≫1 thenωweak\nnu/γH= 1/√ατγH. Remarquably,\nwe see that the precession peak position divided by γH\nalso scales as ωprec/γH∼1/√ατγHwhenατγH≫1.7\n0.01 0.1 1 10 100\nατγH0.010.1110100ωnu/γH , ωprec/γH\n10-610-410-210010210-2100102104106\n1/ατγH\n1/(ατγH)1/2\nFigure 5: (Color online) Scaling curves : nutation ωnuand\nprecession ωprecpeak positions in the angular frequency do-\nmain divided by γHwith respect to ατγH. Open circles\n(red) are the nutation and precession resonance peak posi-\ntions obtained in the numerical simulations for α= 0.1 and\nH= 2T. Other points are ωweak\nnucomputed from Eq. (15),\nandωexact\npreccomputed from Eq. (9). Different values of the\nstatic field Hand the dimensionless damping αare reported :\nH= 0.2Tandα= 0.1 (red open diamonds), H= 2T(blue\nopen squares for α= 0.1 and blue filled squares for α= 0.01),\nH= 20Tandα= 0.1 (green open triangles), H= 200T\nandα= 0.1 (black crosses). The dashed lines are the two\nasymptotic behaviors of the nutation in agreement with Eq.\n(15). Inset : same scaling curves (without red open circles)\ndisplayed on larger scales.\nThe two asymptotic behaviors intersect at ατγH= 1\nandω/γH= 1. This point corresponds to the max-\nimum value of the LLG precession angular frequency\nωLLG/γH= 1/(1 +α2) which is obtained in the limit\ncase of no damping α= 0.\nThe inset of Figure 5 indicates that only one resonance\npeakisexpectedwhen ατγH→ ∞. Inthiscase,boththe\nnutation and the precession contribute to a unique peak.\nOn the contrary, for finite ατγHthey remain separated.\nThere are two different well-defined peaks in the investi-\ngated range ( ατγH≤100). For ατγH≪1 the preces-\nsion peak is close to the usual LLG precession peak, and\nthe nutation peak shifts rapidly ( ωweak\nnu/γH∼1/ατγH)\nto high angular frequencies. In other words, the nutation\noscillator defined by Eq. (13) is independent of the pre-\ncession for ατγH≪1, whereas both synchronize at the\nsame frequency for ατγH→ ∞.\nAccurate predictions about the precession and nutation\npeak positions in the angular frequency domain can be\nmade, as long as the non-linear coupling terms of Eq.\n(13) remain weak or compensate each other.V. TOWARDS EXPERIMENTAL EVIDENCE\nOF THE INERTIAL DYNAMICS OF THE\nMAGNETIZATION\nThroughout the preceding sections we studied the new\nproperties of the inertial dynamics of the magnetization\nwithin the ILLG model. We specifically considered the\nFMR framework where a small perpendicular sinusoidal\nfield is applied implying that the small inclination limit\nholds. We now focus on possible simple experiments in\nsuch FMR framework that should highlight the inertial\ndynamics of the magnetization.\nThe first direct evidence would of course be the measure\nof the nutation resonance peak at frequencies larger than\nthe precession resonance peak. Since the expected nuta-\ntion resonance peak is given by Eq. (15), the evolution\nwith the static field Hmay be used to discriminate the\nnutation resonnce peak from possible other higher fre-\nquency peaks.\nHowever the amplitude of the nutation resonance peak is\nsmallerthan forthe precessionpeak, and itmaybe tricky\nin unfavorable situations to measure such a peak, for ex-\nample in materials with small characteristic time τ. Fur-\nthermore, for large dimensionless damping αboth peaks\nhave smaller amplitude and are rounded. It may even\nappear that the nutation resonance peak of the magneti-\nzationinthe ILLGmodel disappearsforalargedamping,\nlikethe resonantpeakofthe classicaldrivendamped har-\nmonic oscillator. For exampleFig. 6showsthat for mate-\nrials with a large damping ( α= 0.5) the resonance peaks\nare smaller and rounded compared to smaller damping\n(α= 0.1), and the nutation resonance peak disappears\nforH≤5T.\nIt is therefore necessary to find measurable characteris-\nticsofthemagnetizationinertialdynamicsotherthanthe\ndirect measure of the nutation resonance peak. Actually,\nwe show in the following that beyond the nutation reso-\nnance peak, the inertial dynamics has measurable effects\non the precession resonance peak. Indeed, as shown in\nFig. 7, the shape of the precession peak and its position\nintheangularfrequencydomainaremodifiedbytheiner-\ntial dynamics. And the effects are shown to be more pro-\nnounced for large damping materials and for large static\nmagnetic fields H. To show these effects we compare the\nprecession resonance angular frequencies ωILLG\nprecandωLLG\nprec\nobtained in the numerical simulations of both the ILLG\nand non-inertial LLG models. We use two different di-\nmensionless damping α= 0.1 andα= 0.5, and vary the\namplitude Hof the static magnetic field. For the ILLG\nmodel, we choose, as in Ref. 4, a rough estimation of the\ncharacteristic time scale τ= 10−12s.\nA. Angular frequency of the precession resonance\npeak\nWefirstlookatthepositionoftheprecessionresonance\npeakin the angularfrequencydomain. Fig.8(a)and 8(b)8\n02×10124×10126×1012\nω (rad.s-1)1×10-21×10-11×100χ⊥α=0.5\nFigure 6: (Color online) Resonance curves of the transverse\nsusceptibility χ⊥(ω) with respect to the oscillating field angu-\nlar frequency ω. The resonance curves are computed within\nthe ILLG model with τ= 10−12s, for a large dimension-\nless damping α= 0.5 and for various applied static fields :\nH= 2T(black filled circles), H= 5T(green filled trian-\ngles),H= 20T(blue open circles) and H= 50T(red open\ntriangles). Both resonance peaks clearly appear for H= 20T\nandH= 50T. ForH= 5TandH= 2Tthe nutation reso-\nnance peak dissapears due to the large damping.\n00.10.20.3\n1×10 12 2×10 12 3×10 12 20 T\n00.20.40.6\n0 1×10 12 2×10 12 10 T2 T\n0123\n0 1×10 12 \n00.050.1\n3×10 12 4×10 12 5×10 12 6×10 12 50 T2 T 2 T \nχ\n⊥ILLG LLG ILLG LLG χ\n⊥\nILLG LLG \nILLG LLG \nω (rad.s -1 ) ω (rad.s -1 )\nFigure 7: Precession resonance curves of the transverse sus -\nceptibility simulated for differentvalues ofthestatic fiel dH=\n2T, 10T, 20T, and 50 T. ILLG model (filled circles) and\nnon-inertial LLGmodel (opencircles). γ= 1011rad.s−1.T−1,\nα= 0.1, andτ= 10−12s.\ndisplay the evolution of the resonance angular frequencyωprecwith respect to Hobtained for α= 0.1 andα=\n0.5 within the numerical simulations of both the ILLG\nand LLG models. As expected the resonance angular\n0 10 20 30 40 50 60\nH (T)0123456ωprec (1012 rad.s-1)\n0 10 20 30 40 50 60\nH (T)0123456ωprec (1012 rad.s-1)\n0 10 20 30 40 50 60\nH (T)01020304050δ prec (%)a) b) α=0.1\nc)α=0.5\nα=0.5\nα=0.1LLGLLG\nILLG\nILLG\nFigure 8: (Color online) (a) and (b) Precession resonance an -\ngular frequency with respect to the applied static field. Re-\nsults obtainedin thenumerical simulations of theILLGmode l\n(withτ= 10−12s) and non-inertial LLG model, for dimen-\nsionless damping (a) α= 0.1 (blue open circles for LLG and\nred filled circles for ILLG) and (b) α= 0.5 (blue open squares\nfor LLG and red filled squares for ILLG). (c) Relative differ-\nenceδprecbetween LLG and ILLG precession resonance an-\ngular frequencies for α= 0.1 (green filled circles) and α= 0.5\n(green filled squares).\nfrequency of the LLG precession is linear with Hsince\nωLLG\nprec=γH/(1+α2) whereas the behavior is not linear in\nHfor the ILLG model. In Fig. 8(c) we plot the relative\ndifference\nδprec=ωLLG\nprec−ωILLG\nprec\nωLLGprec×100\nbetweenbothresonanceangularfrequencies. Therelative\ndistance between both precession peaks increases with H\nand with the dimensionless damping α.\nB. Width of the precession resonance peak\nWe now examine the evolution with Hof the shape\nof the precession resonance peak obtained in the sim-\nulations of the ILLG and LLG models. For α= 0.1,\nthe full width at half maximum (FWHM) is shown in\nFig. 9(a) while Fig. 9(b) displays the FWHM divided\nby the resonance angular frequency. For large damping\nα= 0.5 we change the criterion since the reduced\namplitude of the resonant peak does not allow anymore\nto compute the FWHM. We therefore compute the\nbandwith defined by the width of the peak at Amax/√\n29\n0 10 20 30 40 50 60\nH (T)00.10.20.3FWHM / ωprec\n0 10 20 30 40 50 60\nH (T)0.00.51.01.52.0FWHM (1012 rad.s-1)\n0 10 20 30 40 50 60\nH (T)00.51Bandwidth / ωprec\n0 10 20 30 40 50 60\nH (T)012345Bandwidth (1012 rad.s-1)a)\nb)\nd)c)α=0.1\nα=0.1\nα=0.5\nα=0.5LLGLLG\nLLGLLGILLGILLG\nILLGILLG\nFigure 9: (Color online) (a) Full width at half maximum\n(FWHM) for the precession resonance peak for α= 0.1 within\nthe LLG (blue open circles) and the ILLG (red filled circles)\nmodels. (b)FWHMdividedeither by ωLLG\nprec(blueopencircles)\nor byωILLG\nprec(red filled circles). (c) Bandwidth of the preces-\nsion resonance peak for α= 0.5 within the LLG (blue open\nsquares)andILLG(redfilledsquares)models. (d)Bandwidth\ndivided either by ωLLG\nprec(blue open squares) or by ωILLG\nprec(red\nfilled squares).\nThe numerical simulations of the ILLG model are computed\nwithτ= 10−12s\nwhereAmaxis the maximum value of the peak. The\nbandwidth for α= 0.5 is shown in Fig. 9(c) and the\nbandwidth divided by the resonance angular frequency\nis plotted in Fig. 9(d). The numerical simulations of the\nILLG and LLG models lead to different behaviors for\nthe shape of the precession resonance peak. In the LLG\nmodel the FWHM and the bandwidth exhibit a linear\nevolution with the applied static field which results in\na constant evolution when divided by the resonance\nangular frequency. Very different behaviors are observed\nwithin the ILLG model where no linear evolution of the\nFWHM or the bandwidth is measured.\nFigs. 8 and 9 show that high applied static fields in\nlarge damping materials produce large differences be-\ntween the positions and shapes of the precession reso-\nnance peaks originating from the LLG and ILLG mod-\nels. Therefore, applying high static fields in large damp-\ning materials better allows to differentiate the precession\npeak originating from the ILLG and LLG models.\nAlthough the theory is clear and allows in principle to\ndifferentiate inertialfromnon-inertialdynamicswhen ex-\naminingboth precessionresonancepeaks, the experimen-\ntal investigations are rather more complex. Indeed, the\nexperimental demonstration of inertial effects first ne-\ncessitate to identify and control the different contribu-tions to the effective field (anisotropy, dipolar interac-\ntion, magnetostriction, ...) other than the applied static\nfield.\nVI. CONCLUSION\nThe magnetization dynamics in the ILLG model that\ntakes into account inertial effects has been studied from\nboth analytical and numerical points ofview. Within the\nFMR context, a nutation resonance peak is expected in\naddition to the usual precession resonance peak.\nAnalytical solutions of the inertial precession and nuta-\ntion angular frequencies are presented. The analytical\nsolutions nicely agree with the numerical simulations of\nthe resonance curves in a broad range of parameters.\nAt first, we investigated the effects of the time scale τ\nwhich drives the additional inertial term introduced in\nEq. (2)comparedtotheusualLLGequationEq. (1). We\nalso varied the dimensionless damping αand the static\nmagnetic field H, and a scaling function with respect to\nατγHis found for the nutation angular frequency. Re-\nmarquably, the same scaling holds for the precession an-\ngular frequency when ατγH≫1.\nIn the second part of the paper we focussed on the sig-\nnatures of the inertial dynamics which could be detected\nexperimentallywithintheFMRcontext. Weshowedthat\nbeyondthemeasureofthenutationresonancepeakwhich\nwould be a direct signature of the inertial dynamics, the\nprecession is modified by inertia and the ILLG preces-\nsion resonance peak is different from the usual LLG pre-\ncession peak. Indeed, whereas a linear evolution with\nrespect to His expected for the LLG precession reso-\nnance angular frequency, the ILLG precession resonance\nangular frequency is clearly non-linear. Furthermore, the\nshape of the precession resonance peak is different in the\nLLG and ILLG models. Again, the width variation of\nthe precession resonance peak is non-linear in the ILLG\ndynamics as opposed to the linear evolution with Hin\nthe LLG dynamics. We also showed that the difference\nbetween both LLG and ILLG precession peaks is more\npronounced when the damping is increased and when τ\nis increased. For example the discrepancy between the\nLLG and ILLG precession resonance angular frequencies\natH= 20Tforτ= 1psis expected to be of the order\nof 20% for α= 0.1 and 30% for α= 0.5. Therefore, large\ndamping materials are better candidates to experimen-\ntallyevidencetheinertialdynamicsofthemagnetization.\nFinally, a specific behavior of the amplitude of the mag-\nnetic susceptibility as a function of the nutation reso-\nnance angular frequency ωnuis predicted, of the form\nχ⊥(ωnu)∝ω−1\nnu(analogousto that ofthe usual FMR sus-\nceptibility). This law could be a useful criterion in order\nto discriminate the nutation peak among the other exci-\ntations that could also occur close to the infrared region\n(100 GHz up to 100 THz) in a ferromagnetic material.10\n1W.F. Brown Thermal Fluctuations of a Single-Domain\nParticle, Phys. Rev. 130, 1677 (1963).\n2R. Kubo, M. Toda, N. Hashitzume, Statistical physics II,\nNonequilibrium Statistical Mechanics , Springer Series in\nSolid-State Sciences 31, Berlin 1991 (second edition), Ed.\nP. Fulde, Chap 3, Paragraph 3.4.3, p. 131.\n3M.-C. Ciornei, Role of magnetic inertia in damped\nmacrospin dynamics , Ph. D. thesis, Ecole Polytechnique,\nPalaiseau France 2010.\n4M.-C. Ciornei, J. M. Rub´ ı, and J.-E. Wegrowe, Magnetiza-\ntion dynamics in the inertial regime : Nutation predicted\nat short time scales , Phys. Rev. B 83, 020410(R) (2011).\n5M. F¨ ahnle, D. Steiauf, and Ch. Illg, Generalized Gilbert\nequation including inertial damping : Derivation from an\nextended breathing Fermi surface model , Phys. Rev. B 84,\n172403 (2011).\n6J.-E. Wegrowe, C. Ciornei Magnetization dynamics, Gyro-\nmagnetic Relation, and Inertial Effects , Am J. Phys. 80,\n607 (2012).\n7E. Olive, Y. Lansac, and J.-E. wegrowe, Beyond ferromag-\nnetic resonance : the inertial regime of the magnetization ,Appl. Phys. Lett. 100, 192407 (2012).\n8D. B¨ ottcher, and J. Henk Significance of nutation in mag-\nnetization dynamics of nanostructures , Phys. Rev. B 86,\n020404(R) (2012).\n9S. Bhattacharjee, L. Nordstr¨ om, and J. Fransson Atomistic\nspin dynamic method with both damping and moment of\ninertia effects included from first principles , Phys. Rev.\nLett.108, 057204 (2012).\n10J.-X. Zhu, Z. Nussinov, A. Shnirman, and A. V. Balatsky\nNovel spin dynamics in a Josephson junction , Phys. Rev.\nLett.92, 107001 (2004).\n11J. Fransson, and J. Xi. Zhu Spin dynamics in a tunnel\njunction between ferromagnets , New J. Phys. 10, 013017\n(2008).\n12C. W. Gear, Numerical initial value problems in ordinary\ndifferential equations , Prentice Hall, Englewood Cliffs (N.\nJ.) 1971.\n13A. G. Gurevich and G. A. Melkov, Magnetization Oscilla-\ntion and Waves , CRC Press, 1996, p. 19."    },    {        "title": "1412.4032v1.Spin_waves_in_micro_structured_yttrium_iron_garnet_nanometer_thick_films.pdf",        "content": "arXiv:1412.4032v1  [cond-mat.mes-hall]  12 Dec 2014Spin waves in micro-structured yttrium iron garnet nanomet er-thick films\nMatthias B. Jungfleisch,1,a)Wei Zhang,1Wanjun Jiang,1Houchen Chang,2Joseph Sklenar,3Stephen M. Wu,1\nJohn E. Pearson,1Anand Bhattacharya,1John B. Ketterson,3Mingzhong Wu,2and Axel Hoffmann1\n1)Materials Science Division, Argonne National Laboratory, Argonne, Illinois 60439,\nUSA\n2)Department of Physics, Colorado State University, Fort Col lins, Colorado 80523,\nUSA\n3)Department of Physics and Astronomy, Northwestern Univers ity, Evanston, Illinois 60208,\nUSA\n(Dated: 1 May 2018)\nWe investigated the spin-wave propagation in a micro-structured y ttrium iron garnet waveguide of 40 nm\nthickness. Utilizing spatially-resolved Brillouin light scattering microsc opy, an exponential decay of the spin-\nwave amplitude of (10 .06±0.83)µm was observed. This leads to an estimated Gilbert damping constant\nofα= (8.79±0.73)×10−4, which is larger than damping values obtained through ferromagnet ic resonance\nmeasurements in unstructured films. The theoretically calculated s patial interference of waveguide modes\nwas compared to the spin-wave pattern observed experimentally b y means of Brillouin light scattering spec-\ntroscopy.\nI. INTRODUCTION\nMagnonics is an emerging field of magnetism study-\ning the spin dynamics in micro- and nanostructured\ndevices aiming for the development of new spintron-\nics applications.1–3Up to now, ferromagnetic metals\n(for example, Permalloy and Heusler alloys) have been\nwidely used for the investigation of magnetization dy-\nnamicson the nanoscale.4–10However, the Gilbert damp-\ning of Permalloy is two orders of magnitude higher than\nthat of ferrimagnetic insulator yttrium iron garnet (YIG,\nY3Fe5O12). Recent progress in the growth of YIG films\nallows for the fabrication of low-damping nanometer-\nthick YIG films,11–14which are well-suited for patterning\nof micro-structured YIG devices. This enables investiga-\ntions of spin-wave propagation in plain YIG microstruc-\ntures of sub-100 nm thicknesses which are a step forward\nfor future insulator-based magnonics applications.\nInthiswork,weexperimentallydemonstratespin-wave\npropagation in a micro-structured YIG waveguide of\n40 nm thickness and 4 µm width. By utilizing spatially-\nresolvedBrillouinlightscattering(BLS) microscopy4–7,11\nthe exponential decay length of spin waves is deter-\nmined. The corresponding damping parameter of the\nmicro-structured YIG is estimated and compared to that\ndetermined from ferromagnetic resonance (FMR) mea-\nsurements. Furthermore, we show that different spin-\nwave modes quantized in the direction perpendicular to\nthe waveguide lead to a spatial interference pattern. We\ncompare the experimental results to the theoretically ex-\npected spatial interference of the waveguide modes.\na)Electronic mail: jungfleisch@anl.govII. EXPERIMENT\nFigure1shows a schematic illustration of the sam-\nple layout. The YIG film of 40 nm thickness was de-\nposited by magnetron sputtering on single crystal pol-\nished gadolinium gallium garnet (GGG, Gd 3Ga4O12)\nsubstrates of 500 µm thickness with (111) orientation\nunder high-purity argon atmosphere. The film was sub-\nsequently annealed in-situ at 800◦C for 4 hours under an\noxygenatmosphereof1.12Torr. Themagneticproperties\nofthe unstructured film werecharacterizedby FMR: The\npeak-to-peak linewidth µ0∆Has a function of the exci-\ntation frequency fis depicted in Fig. 2(a). The Gilbert\ndamping parameter αFMRcan be obtained from FMR\nmeasurements using15\n√\n3µ0∆H=2αFMR\n|γ|f+µ0∆H0, (1)\nFIG. 1. (Color online) Schematic illustration of the sample\nlayout. The 4 µm wide yttrium iron garnet waveguide is\nmagnetized transversally by the bias magnetic field H. Spin\nwaves are excited by a shortened coplanar waveguide and the\nspin-wave intensity is detected by means of spatially-reso lved\nBrillouin light scattering microscopy. Colorbar indicate s spin-\nwave intensity.2\nwhereµ0is the vacuum permeability, γis the gyromag-\nnetic ratio, fis the resonance frequency and µ0∆H0\nis the inhomogeneous linewidth broadening. We find a\ndamping parameter of αFMR= (2.77±0.49)×10−4[fit\nshown as a red solid line in Fig. 2(a)]. The resonance\nfield,µ0H, as a function of the excitation frequency fis\nshown in Fig. 2(b). A fit to16\nf=µ0|γ|\n2π/radicalbig\nH(H+Meff) (2)\nyields an effective magnetization Meff= (122 ±\n0.30) kA/m [solid line in Fig. 2(b)]. In a subsequent fab-\nrication process, YIG waveguides of 4 µm width were\npatterned by photo-lithography and ion milling with\nan Ar plasma at 600 V for 5 min. In a last step, a\nshortened coplanar waveguide (CPW) made of Ti/Au\n(3 nm/150 nm) is patterned on top ofthe YIG waveguide\n(see Fig. 1). The shortened end of the CPW has a width\nof 5µm. The Oersted field of an alternating microwave\nsignal applied to the CPW exerts a torque on the mag-\nnetic moments in the YIG and forces them to precess.\nThe bias magnetic field is applied perpendicular to the\nshortaxisofthe waveguide(Fig. 1) providingefficient ex-\ncitation of Damon-Eshbach spin waves. The microwave\npowerPMW= 1 mW is sufficiently small to avoid pos-\nsible perturbations of spin-wave propagation caused by\nnonlinearities.\nIII. DISCUSSIONS\nIn order to detect spin-wave propagation in the YIG\nwaveguide spatially-resolved BLS microscopy with a res-\nolution of 250 nm is employed. To characterize the prop-\nagating spin waves, the BLS intensity was recorded at\ndifferentdistancesfromtheantenna. Aspatially-resolved\nBLS intensity map is shown in Fig. 3(b) at an exemplary\nexcitation frequency of f= 4.19 GHz. Spin waves are\nexcited near the antenna and propagate towards the op-\nposite end of the waveguide. To further analyze the data\nand to minimize the influence of multi-mode propagation\nin the YIG stripe (this will be discussed below), the BLS\nintensity is integrated over the width of the waveguide.\nThe correspondingBLS intensity as a function of the dis-\ntance from the antenna is illustrated in Fig. 3(c). The\ndecayofthespin-waveamplitudecanbedescribedby:7,11\nI(z) =I0e−2z\nλ+b, (3)\nwherezis the distance from the antenna, λis the de-\ncay length of the spin-wave amplitude and bis an offset.\nFrom Fig. 3(b), it is apparent that the data-points fol-\nlow an exponential behavior. A fit according to Eq. ( 3)\nyields the decay length λ. For an excitation frequency\noff= 4.19 GHz we find λ= (10.06±0.83)µm. This\nvalue is larger than decay lengths reported for Permalloy\n(<6µm, see Ref. 8, 21, and 22), but it is smaller thanFIG. 2. (Color online) (a) Ferromagnetic resonance peak-to -\npeaklinewidth µ0∆Has afunctionoftheresonance frequency\nfof the unstructured 40 nm YIG film. The red solid line\nrepresents a fit to Eq. (1). A Gilbert damping parameter\nα= (2.77±0.49)×10−4is determined. (b) Ferromagnetic\nresonance field µ0Has a function of f. Error bars are smaller\nthan the data symbols.\nthe largest decay length found for the Heusler-compound\nCo2Mn0.6Fe0.4Si (8.7 – 16.7µm, see Ref. 7). Pirro et al.\nreported a decay length of 31 µm in thicker YIG waveg-\nuides(100nm) grownbyliquidphaseepitaxy(LPE)with\na 9 nm thick Pt capping layer.11In order to understand\nthis discrepancy between our and their results, one has\nto take into account two facts: (1) State-of-the-art LPE\nfabrication technology can not be employed to grow film\nthicknesses below ∼100 nm. To date, sputtering offers\nan alternative approach to grow sub-100 nm thick YIG\nfilms with a sufficient quality.23(2) Taking into account\nthe spin-wave group velocity vg=∂ω/∂kand the spin-\nwave lifetime τ, the theoretically expected decay length\nλcan be calculated from λ=vg·τ. The groupvelocity vg\ncan be derived directly from the dispersion relation (see\nFig.4). A thinner YIG film has flatter dispersion and a\nsmallervg. Consequently, the expected decay length is\nsmaller for thinner YIG samples and so it is natural that\nthe decay length reported here is shorter than the one\nfound in Ref. 11 for 100 nm thick YIG waveguides.\nWe estimate the groupvelocity from the spin-wavedis-\npersion to be vg= 0.35−0.40µm/ns. Using our exper-\nimentally found decay length, the spin-wave lifetime is\ndetermined to be τ= 29 ns. We can use the BLS-data\nto determine the corresponding Gilbert damping param-\neterαBLS. In case of Damon-Eshbach spin waves, the\ndamping is given by243\nFIG. 3. (Color online) (a) Calculated spatial interference\npattern of the first two odd waveguide modes ( n= 1 and\nn= 3. (b) Spatially-resolved BLS intensity map at an ex-\ncitation frequency f= 4.19 GHz, applied microwave power\nPMW= 1 mW, biasing magnetic field µ0∆H= 83 mT. The\nnumbers 1 – 4 highlight the main features of the interference\npattern. (c) Corresponding BLS intensity integrated over t he\nentire width of the YIG waveguide. An exponential decay of\nthe spin-wave amplitude λ= (10.06±0.83)µm is found.\nαBLS=1\nτ(γµ0Meff\n2+2πf)−1. (4)\nA Gilbert damping parameter of the micro-structured\nYIG waveguide obtained by BLS characterization is\nfound to be αBLS= (8.79±0.73)×10−4, which is a\nfactor of 3 times larger than that determined by FMR in\nthe unstructuredfilm [ αFMR= (2.77±0.49)×10−4]. This\ndifference mightbe attributed to the micro-structuringof\nthe YIG waveguide by Ar ion beam etching. The etch-\ning might enhance the roughness of the edges of the YIG\nwaveguides and the resist processing could have an in-\nfluence on the surface quality17,18which could possibly\nlead to an enhancement of the two-magnon scattering\nprocess.19It would be desirable to perform FMR mea-\nsurements on the YIG waveguide. However, since the\nstructured bar is very small, the FMR signal is van-\nishingly small which makes it difficult to determine the\nGilbert damping in this way.\nWhile the discussion above only considered the BLS-\nintensity integrated over the waveguide width, we will\nfocus now on the spatial interference pattern shown in\nFig.3(b). The spin-wave intensity map can be under-\nstood by taking into account the dispersion relation of\nmagnetostatic spin waves in an in-plane magnetized fer-\nromagnetic thin film (see Fig. 4). Due to the lateral con-\nfinement, thewavevectorisquantizedacrossthewidthof\nthe YIG waveguide, ky=nπ/w, wheren∈N. The wave\nvectorkzalong the long axis of the waveguide ( z-axis)is assumed to be non-quantized. We follow the approach\npresented in Ref. 8. The dynamic magnetization is as-\nsumed to be pinned at the edges of the waveguide which\ncan be considered by introducing an effective width of\nthe waveguide.8,20\nFigure4shows the calculated dispersion relations of\ndifferent spin-wave modes quantized across the width of\nthe strip. The dashed line represents a fixed excitation\nfrequency. At a particular frequency different spin-waves\nmodes with different wave vectors kzare excited simulta-\nneously. Thisleadstothe occurrenceofspatiallyperiodic\ninterference patterns. In the present excitation config-\nuration, only modes with an odd quantization number\nncan be excited ( ndetermines the number of maxima\nacross the width of the waveguide). Since the intensity\nof the dynamic magnetization of these modes decreases\nwith increasing nas 1/n2, we only consider the first two\nodd modes n= 1 and n= 3.\nAccording to Ref. 8 the spatial distribution of the dy-\nnamic magnetization of the n-th mode can be expressed\nas\nmn(y,z)∝sin(nπ\nwy)cos(kn\nzz−2πft+φn),(5)\nwherefis the excitation frequency, kn\nzis the longitudinal\nwave vector of the n-th spin-wave mode and φnis the\nphase.25The spin-waveintensity distribution Inof then-\nth mode can be derived by averaging mn(y,z)2over one\noscillation period 1 /f. The entire interference pattern\ncan be obtained from the same procedure using the sum\nm1(y,z) +1\n3m3(y,z). The factor 1/3 accounts for the\nlower excitation efficiency of the n= 3 mode. Thus, the\nintensity is given by8\nIΣ(y,z)∝sin(π\nwy)2+1\n9sin(3π\nwy)2\n+2\n3sin(π\nwy)sin(3π\nwy)cos(∆kzz+∆φ),(6)\nwhere ∆kz=k3\nz−k1\nzand ∆φ=φ3−φ1. This pattern re-\npeatsperiodically. Thephaseshift φshiftsthe entirepat-\nFIG. 4. (Color online) Dispersion relations the first five\nwaveguide modes of a transversally magnetized YIG stripe.26\nOnly modes with a odd quantization number ncan be excited\n(solid lines). fdenotes the excitation frequency.4\ntern along the z-direction and the wave-vector difference\n∆kz= 0.97 rad/µm can be calculated from the disper-\nsion relation (Fig. 4). The calculated spatial interference\npattern is depicted in Fig. 3(a) using ∆ φ= 0 and taking\ninto account for the exponential decay of the spin-wave\namplitude by multiplying Eq. ( 6) withe−2z/λusing the\nexperimentallydetermined λ= 10.03µm. As is apparent\nfrom Fig. 3(a) and (b) a qualitative agreement between\ncalculation and experiment is found. (The numbers 1 –\n4 highlight the main features of the interference pattern\nin experiment and calculation.) The small difference in\nFig.3(a) and (b) can be explained by considering the\nfact that in the calculation a spin-wave propagation at\nan angle of exactly 90◦(Damon-Eshbach configuration)\nwith respect to the antenna/externalmagnetic field is as-\nsumed. However, in experiment small misalignments of\nthe external magnetic field might lead to a small asym-\nmetry in the interference pattern.\nIV. CONCLUSION\nIn summary, we demonstrated spin-wave excitation\nand propagation in micro-fabricated pure YIG wave-\nguides of 40 nm thickness. BLS-characterization re-\nvealed a decay length of the spin-wave amplitude of\n10µm leading to an estimated Gilbert damping pa-\nrameter of αBLS= (8.79±0.73)×10−4. This value\nis a factor 3 larger than the one determined for the\nunstructured YIG film by means of FMR techniques\n[αFMR= (2.77±0.49)×10−4]. The difference might be\nattributed to micro-structuring using ion beam etching.\nThe observed spatial spin-wave intensity distribution is\nexplained by the simultaneous excitation of the first two\nodd waveguide modes. These findings are important for\nthe development of new nanometer-thick magnon spin-\ntronics applications and devices based on magnetic insu-\nlators.\nV. ACKNOWLEDGMENTS\nWork at Argonne was supported by the U.S. Depart-\nment of Energy, Office of Science, Materials Science and\nEngineering Division. Work at Colorado State Univer-\nsity was supported by the U.S. Army Research Office,\nand the U.S. National Science Foundation. Lithography\nwas carried out at the Center for Nanoscale Materials,\nwhich is supported by DOE, Office of Science, Basic En-\nergy Sciences under ContractNo. DE-AC02-06CH11357.\n1S. Neusser and D. Grundler, Adv. Mater. 21, 2927 (2009).2V.V. Kruglyak, S.O. Demokritov, D. Grundler, J. Phys. D: App l.\nPhys.43, 264001 (2010).\n3S.O.Demokritov and A.N.Slavin, Magnonics - From Fundamen-\ntals to Applications , (Springer, 2013).\n4K. Vogt, H. Schultheiss, S. Jain, J.E. Pearson, A. Hoffmann,\nS.D. Bader and B. Hillebrands, Appl. Phys. Lett. 101, 042410\n(2012).\n5K. Vogt, F.Y. Fradin, J.E. Pearson, T. Sebastian, S.D. Bader ,\nB. Hillebrands, A. Hoffmann and H. Schultheiss, Nat. Commun.\n5, 3727 (2014).\n6V.E. Demidov, S.O. Demokritov, K. Rott, P. Krzysteczko and\nG. Reiss, J. Phys. D: Appl. Phys. 41, 164012 (2008).\n7T. Sebastian, Y. Ohdaira, T. Kubota, P. Pirro, T. Br¨ acher,\nK. Vogt, A.A. Serga, H. Naganuma, M. Oogane, Y. Ando, and\nB. Hillebrands, Appl. Phys. Lett. 100, 112402 (2012).\n8V.E. Demidov, S.O. Demokritov, K. Rott, P. Krzysteczko, and\nG. Reiss, Phys. Rev. B 77, 064406 (2008).\n9V.E. Demidov, S.O. Demokritov, D. Birt, B. O’Gorman, M. Tsoi\nand X. Li, Phys. Rev. B 80, 014429 (2009).\n10V.E. Demidov, J. Jersch, S.O. Demokritov, K. Rott, P. Krzys-\nteczko, and G. Reiss, Phys. Rev. B 79, 054417 (2009).\n11P. Pirro, T. Br¨ acher, A.V. Chumak, B. L¨ agel, C. Dubs,\nO. Surzhenko, P. G¨ ornert, B. Leven and B. Hillebrands, Appl .\nPhys. Lett. 104, 012402 (2014).\n12T. Liu, H. Chang, V. Vlaminck, Y. Sun, M. Kabatek, A. Hoff-\nmann, L. Deng, and M. Wu, J. Appl. Phys. 115, 17A501 (2014).\n13O. d’Allivy Kelly, A. Anane, R. Bernard, J. Ben Youssef,\nC. Hahn, A H. Molpeceres, C. Carr´ et´ ero, E. Jacquet, C. Der-\nanlot, P. Bortolotti, R. Lebourgeois, J.-C. Mage, G. de Loub ens,\nO. Klein, V. Cros, and A. Fert, Appl. Phys. Lett. 103, 082408\n(2013).\n14Y. Sun, Y.-Y. Song and M. Wu, Appl. Phys. Lett. 101, 082405\n(2012).\n15S.S.Kalarickal, P.Krivosik,M.Wu, C.E.Patton, M.L.Schne ider,\nP.Kabos, T.J.Silva, andJ.P.Nibarger,J.Appl.Phys. 99,093909\n(2006).\n16A. Azevedo, A.B. Oliveira, F.M de Aguiar, and S.M. Rezende,\nPhys. Rev. B, 62, 5331 (2000).\n17B.J.McMorran, A.C.Cochran, R.K.Dumas, KaiLiu, P.Morrow,\nD.T. Pierce and J. Unguris, J. Appl. Phys. 107, 09D305 (2010).\n18O.D. Roshchupkina, J. Grenzer, T. Strache, J. McCord,\nM. Fritzsche, A. Muecklich, C. Baehtz, and J. Fassbender, J.\nAppl. Phys. 112, 033901 (2012)\n19R. Arias and D.L. Mills, Phys. Rev. B 60, 7395 (1999).\n20K.Yu. Guslienko, S.O. Demokritov, B. Hillebrands, and\nA.N. Slavin, Phys. Rev. B 66, 132402 (2002).\n21M. Madami, S. Bonetti, G. Consolo, S. Tacchi, G. Carlotti,\nG. Gubbiotti, F.B. Mancoff, M.A. Yar, and J. ˚Akerman, Nat.\nNano.6, 635 (2011).\n22P. Pirro, T. Br¨ acher, K. Vogt, Bj¨ orn Obry, H. Schultheiss,\nB. Leven, and B. Hillebrands, Phys. Status Solidi B 238, 2404\n(2011).\n23H. Chang, P. Li, W. Zhang, T Liu, A. Hoffmann, L. Deng, and\nM. Wu, IEEE Magnetic Letters 5, 6700104 (2014).\n24D.D. Stancil and A. Prabhakar, Spin Waves - Theory and Ap-\nplications , (Springer, 2009).\n25T. Schneider, A.A. Serga, T. Neumann, B. Hillebrands, and\nM.P. Kostylev, Phys. Rev. B 77, 2144 (2008).\n26For the calculation of the dispersion relations the followi ng pa-\nrameters have been used: external magnetic field µ0H= 83 mT,\nexchange constant A= 3.6 pJ/m, saturation magnetization\nMS=140 kA/m, effective width of the waveguide weff= 3.5µm,\nYIG-film thickness t= 40 nm."    },    {        "title": "1501.00444v1.Inertia__diffusion_and_dynamics_of_a_driven_skyrmion.pdf",        "content": "Inertia, diffusion and dynamics of a driven skyrmion\nChristoph Sch ¨utte,1Junichi Iwasaki,2Achim Rosch,1and Naoto Nagaosa2, 3,\u0003\n1Institut f ¨ur Theoretische Physik, Universit ¨at zu K ¨oln, D-50937 Cologne, Germany\n2Department of Applied Physics, University of Tokyo, 7-3-1, Hongo, Bunkyo-ku, Tokyo 113-8656, Japan\n3RIKEN Center for Emergent Matter Science (CEMS), Wako, Saitama 351-0198, Japan\n(Dated: January 5, 2015)\nSkyrmions recently discovered in chiral magnets are a promising candidate for magnetic storage devices\nbecause of their topological stability, small size ( \u00183\u0000100nm), and ultra-low threshold current density ( \u0018\n106A/m2) to drive their motion. However, the time-dependent dynamics has hitherto been largely unexplored.\nHere we show, by combining the numerical solution of the Landau-Lifshitz-Gilbert equation and the analysis of a\ngeneralized Thiele’s equation, that inertial effects are almost completely absent in skyrmion dynamics driven by\na time-dependent current. In contrast, the response to time-dependent magnetic forces and thermal fluctuations\ndepends strongly on frequency and is described by a large effective mass and a (anti-) damping depending\non the acceleration of the skyrmion. Thermal diffusion is strongly suppressed by the cyclotron motion and is\nproportional to the Gilbert damping coefficient \u000b. This indicates that the skyrmion position is stable, and its\nmotion responds to the time-dependent current without delay or retardation even if it is fast. These findings\ndemonstrate the advantages of skyrmions as information carriers.\nPACS numbers: 73.43.Cd,72.25.-b,72.80.-r\nI. INTRODUCTION\nMass is a fundamental quantity of a particle determining its\nmechanical inertia and therefore the speed of response to ex-\nternal forces. Furthermore, it controls the strength of quantum\nand thermal fluctuations. For a fast response one usually needs\nsmall masses and small friction coefficients which in turn lead\nto large fluctuations and a rapid diffusion. Therefore, usually\nsmall fluctuations and a quick reaction to external forces are\nnot concomitant. However a “particle” is not a trivial object\nin modern physics, it can be a complex of energy and mo-\nmentum, embedded in a fluctuating environment. Therefore,\nits dynamics can be different from that of a Newtonian parti-\ncle. This is the case in magnets, where such a “particle” can\nbe formed by a magnetic texture1,2. A skyrmion3,4is a rep-\nresentative example: a swirling spin texture characterized by\na topological index counting the number of times a sphere is\nwrapped in spin space. This topological index remains un-\nchanged provided spin configurations vary slowly, i.e., dis-\ncontinuous spin configurations are forbidden on an atomic\nscale due to high energy costs. Therefore, the skyrmion is\ntopologically protected and has a long lifetime, in sharp con-\ntrast to e.g. spin wave excitations which can rapidly decay.\nSkyrmions have attracted recent intensive interest because of\ntheir nano-metric size and high mobility5–14. Especially, the\ncurrent densities needed to drive their motion ( \u0018106A/m2)\nare ultra small compared to those used to manipulate domain\nwalls in ferromagnets ( \u00181011\u000012A/m2)15–19.\nThe motion of the skyrmion in a two dimensional film can\nbe described by a modified version of Newton’s equation. For\nsufficiently slowly varying and not too strong forces, a sym-\nmetry analysis suggests the following form of the equations\nof motion,\nG\u0002_R+\u000bD_R+mR+\u000b\u0000\u0002R=Fc+Fg+Fth:(1)\nHere we assumed translational and rotational invariance of\nthe linearized equations of motion. The ‘gyrocoupling’ G=G^e?is an effective magnetic field oriented perpendicular to\nthe plane,\u000bis the (dimensionless) Gilbert damping of a sin-\ngle spin,\u000bDdescribes the friction of the skyrmion, mits mass\nandRits centre coordinate. \u0000parametrizes a peculiar type of\ndamping proportional to the acceleration of the particle. We\nname this term ‘gyrodamping’, since it describes the damping\nof a particle on a cyclotron orbit (an orbit with R/G\u0002_R),\nwhich can be stronger ( \u0000parallel to G) or weaker (antipar-\nallel to G) than that for linear motion. Our main goal will\nbe to describe the influence of forces on the skyrmion arising\nfrom electric currents ( Fc), magnetic field gradients (Fg) and\nthermal fluctuations (Fth).\nBy analyzing the motion of a rigid magnetic structure\nM(r;t) =M0(r\u0000R(t))forstatic forces, one can obtain\nanalytic formulas for G;\u000bD;FcandFgusing the approach\nof Thiele19–22,24. In Ref. [25], an approximate value for the\nmass of a skyrmion was obtained by simulating the motion of\na skyrmion in a nanodisc and by estimating contributions to\nthe mass from internal excitations of the skyrmion.\nFor rapidly changing forces, needed for the manipulation of\nskyrmions in spintronic devices, Eq. (1) is however not suffi-\ncient. A generalized version of Eq. (1) valid for weak but also\narbitrarily time-dependent forces can be written as\nG\u00001(!)V(!) =Fc(!) +Fg(!) +Fth(!) (2)\n=Sc(!)vs(!) +Sg(!)rBz(!) +Fth(!)\nHereV(!) =R\nei!t_R(t)dtis the Fourier transform of the\nvelocity of the skyrmion, vs(!)is the (spin-) drift velocity\nof the conduction electrons, directly proportional to the cur-\nrent,rBz(!)describes a magnetic field gradient in frequency\nspace. The role of the random thermal forces, Fth(!), is spe-\ncial as their dynamics is directly linked via the fluctuation-\ndissipation theorem to the left-hand side of the equation, see\nbelow. The 2\u00022matrix G\u00001(!)describes the dynam-\nics of the skyrmion; its small- !expansion defines the terms\nwritten on the left-hand side of Eq. (1). One can expectarXiv:1501.00444v1  [cond-mat.str-el]  2 Jan 20152\nFIG. 1: When a skyrmion is driven by a time dependent external\nforce, it becomes distorted and the spins precess resulting in a de-\nlayed response and a large effective mass. In contrast, when the\nskyrmion motion is driven by an electric current, the skyrmion ap-\nproximately flows with the current with little distortion and preces-\nsion. Therefore skyrmions respond quickly to rapid changes of the\nelectric current.\nstrongly frequency-dependent dynamics for the skyrmion be-\ncause the external forces in combination with the motion of\nthe skyrmion can induce a precession of the spin and also ex-\ncite spinwaves in the surrounding ferromagnet, see Fig. 1.\nWe will, however, show that the frequency dependence of the\nright-hand side of the Eq. (2) is at least as important: not only\nthe motion of the skyrmion but also the external forces excite\ninternal modes. Depending on the frequency range, there is\nan effective screening or antiscreening of the forces described\nby the matrices Sc(!)andSg(!). Especially for the current-\ndriven motion, there will be for all frequencies an almost exact\ncancellation of terms from G\u00001(!)andSc(!). As a result\nthe skyrmion will follow almost instantaneously any change\nof the current despite its large mass.\nIn this paper, we study the dynamics of a driven skyrmion\nby solving numerically the stochastic Landau-Lifshitz-Gilbert\n(LLG) equation. Our strategy will be to determine the param-\neters of Eq. (2) such that this equation reproduces the results\nof the LLG equation. Section II introduces the model and\noutlines the numerical implementation. Three driving mecha-\nnisms are considered: section III studies the diffusive motion\nof the skyrmion due to thermal noise, section IV the skyrmion\nmotion due to time-dependent magnetic field gradient and sec-\ntion V the current-driven dynamics. We conclude with a sum-\nmary and discussion of the results in Sec. VI.\nII. MODEL\nOur study is based on a numerical analysis of the stochastic\nLandau-Lifshitz-Gilbert (sLLG) equations27defined by\ndMr\ndt=\rMr\u0002[Be\u000b+b\r(t)]\n\u0000\r\u000b\nMMr\u0002(Mr\u0002[Be\u000b+b\r(t)]):(3)\nHere\ris the gyromagnetic moment and \u000bthe Gilbert damp-\ning;Be\u000b=\u0000\u000eH[M]\n\u000eMris an effective magnetic field created by\nthe surrounding magnetic moments and b\r(t)a fluctuating,stochastic field creating random torques on the magnetic mo-\nments to model the effects of thermal fluctuations, see below.\nThe Hamiltonian H[M]is given by\nH[M] =\u0000JX\nrMr\u0001\u0000\nMr+aex+Mr+aey\u0001\n\u0000\u0015X\nr\u0000\nMr\u0002Mr+aex\u0001ex+Mr\u0002Mr+aey\u0001ey\u0001\n\u0000B\u0001X\nrMr (4)\nWe useJ= 1 ,\r= 1 ,jMrj= 1 ,\u0015= 0:18Jfor the\nstrength of the Dzyaloshinskii-Moriya interaction and B=\n(0;0;0:0278J)for all plots giving rise to a skyrmion with a\nradius of about 15lattice sites, see Appendix A. For this pa-\nrameter set, the ground state is ferromagnetic, thus the single\nskyrmion is a topologically protected, metastable excitation.\nTypically we simulate 100\u0002100spins for the analysis of dif-\nfusive and current driven motion and 200\u0002200spins for the\nforce-driven motion. For these parameters lattice effects are\nnegligible, see appendix B. Typical microscopic parameters\nused, areJ= 1meV (this yields Tc\u001810K) which we use to\nestimate typical time scales for the skyrmion motion.\nFollowing Ref. 27, we assume that the field bfl\nr(t)is gen-\nerated from a Gaussian stochastic process with the following\nstatistical properties\n\nbfl\nr;i(t)\u000b\n= 0\n\nbfl\nr;i(t)bfl\nr0;j(s)\u000b\n= 2\u000bkBT\n\rM\u000eij\u000err0\u000e(t\u0000s) (5)\nwhereiandjare cartesian components and h:::idenotes\nan average taken over different realizations of the fluctuating\nfield. The Gaussian property of the process stems from the in-\nteraction of Mrwith a large number of microscopic degrees\nof freedom (central limit theorem) which are also responsi-\nble for the damping described by \u000b, reflecting the fluctuation-\ndissipation theorem. The delta-correlation in time and space\nin Eq. (5) expresses that the autocorrelation time and length of\nbfl\nr(t)is much shorter than the response time and length scale\nof the magnetic system.\nFor a numerical implementation of Eq. (3) we follow\nRef. 27 and use Heun’s scheme for the numerical integration\nwhich converges quadratically to the solution of the general\nsystem of stochastic differential equations (when interpreted\nin terms of the Stratonovich calculus).\nFor static driving forces, one can calculate the drift veloc-\nity_Rfollowing Thiele20. Starting from the Landau-Lifshitz\nGilbert equations, Eq. (3), we project onto the translational\nmode by multiplying Eq. (3) with @iMrand integrating over\nspace21–23.\nG=~M0Z\ndr n\u0001(@xn\u0002@yn)\nD=~M0Z\ndr(@xn\u0001@xn+@yn\u0001@yn)=2\nFc=G\u0002vs+\fDvs;\nFg=MsrB; M s=M0Z\ndr(1\u0000nz) (6)3\nwhere nis the direction of the magnetization, M0the lo-\ncal spin density, vsthe (spin-) drift velocity of the conduc-\ntion electrons proportional to the electric current, and Ms\nis the change of the magnetization induced by a skyrmion\nin a ferromagnetic background. The ’gyrocoupling vector’\nG= (0;0;G)TwithG=\u0006~M04\u0019is given by the winding\nnumber of the skyrmion, independent of microscopic details.\nIII. THERMAL DIFFUSION\nRandom forces arising from thermal fluctuations play a de-\ncisive role in controlling the diffusion of particles and there-\nfore also the trajectories R(t)of a skyrmion. To obtain R(t)\nand corresponding correlation functions we used numerical\nsimulations based on the stochastic Landau-Lifshitz-Gilbert\nequation27. These micromagnetic equations describe the dy-\nnamics of coupled spins including the effects of damping\nand thermal fluctuations. Initially, a skyrmion spin-texture\nis embedded in a ferromagnetic background. By monitoring\nthe change of the magnetization, we track the center of the\nskyrmion R(t), see appendix A for details.\nOur goal is to use this data to determine the matrix G\u00001(!)\nand the randomly fluctuating thermal forces, Fth(!), which\ntogether fix the equation of motion, Eq. (2), in the presence\nof thermal fluctuations ( rBz=vs= 0). One might worry\nthat this problem does not have a unique solution as both the\nleft-hand and the right-hand side of Eq. (2) are not known\na priori. Here one can, however, make use of the fact that\nKubo’s fluctuation-dissipation theorem26constraints the ther-\nmal forces on the skyrmion described by Fthin Eq. (2) by\nlinking them directly to the dissipative contributions of G\u00001.\nOn averagehFth= 0i, but its autocorrelation is proportional\nto the temperature and friction coefficients. In general it is\ngiven by\nhFi\nth(!)Fj\nth(!0)i=kBT[G\u00001\nij(!) +G\u00001\nji(\u0000!)]2\u0019\u000e(!+!0):\n(7)\nFor small ! one obtainshFx\nth(!)Fx\nth(!0)i =\n4\u0019kBT\u000bD\u000e(!+!0)while off-diagonal correla-\ntions arise from the gyrodamping hFx\nth(!)Fy\nth(!0)i=\n4\u0019i!kBT\u000b\u0000\u000e(!+!0). Using Eq. (7) and demand-\ning furthermore that the solution of Eq. (2) reproduces\nthe correlation function h_Ri(t)_Rj(t0)i(or, equivalently,\nh(Ri(t)\u0000Rj(t0))2i) obtained from the micromagnetic\nsimulations, leads to the condition26\nGij(!) =1\nkBTZ1\n0\u0002(t\u0000t0)h_Ri(t)_Rj(t0)i (8)\nei!(t\u0000t0)d(t\u0000t0):\nWe therefore determine first in the presence of thermal fluc-\ntuations (rBz=vs= 0) from simulations of the stochastic\nLLG equation (3) the correlation functions of the velocities\nand use those to determine Gij(!)using Eq. (8). After a sim-\nple matrix inversion, this fixes the left-hand side of the equa-\ntion of motion, Eq. (2), and therefore contains all information\n0 5 10 15 20 25t ωp00.511.522.5 <ΔR2>α=0.01\nα=0.05\nα=0.1\nα=0.15\nα=0.2FIG. 2: Time dependence of the correlation function\n(Ri(t0+t)\u0000Ri(t0))2\u000b\nforT= 0:1Jand different values\nof the Gilbert damping \u000b(!p=B= 0:0278Jis the frequency for\ncyclotron motion).\non the (frequency-dependent) effective mass, gyrocoupling,\ndamping and gyrodamping of the skyrmion. Furthermore, the\ncorresponding spectrum of thermal fluctuations is given by\nEq. (7).\nFig. 2 showsh(\u0001R)2it=h(Rx(t0+t)\u0000Rx(t0))2i. As\nexpected, the motion of the skyrmion is diffusive: the mean\nsquared displacement grows for long times linearly in time\nh(\u0001R)2it= 2Dt, whereDis the diffusion constant. Usu-\nally the diffusion constant of a particle grows when the fric-\ntion is lowered26. For the skyrmion the situation is opposite:\nthe diffusion constant becomes small for the small friction,\ni.e., small Gilbert damping \u000b. This surprising observation has\nits origin in the gyrocoupling G: in the absence of friction\nthe skyrmion would be localized on a cyclotron orbit. From\nEq. (1), we obtain\nD=kBT\u000bD\nG2+ (\u000bD)2(9)\nThe diffusion is strongly suppressed by G. As in most materi-\nals\u000bis much smaller than unity while D\u0018G , the skyrmion\nmotion is characterized both by a small diffusion constant\nand a small friction. Such a suppressed dynamics has also\nbeen shown to be important for the dynamics of magnetic\nvortices28. For typical parameters relevant for materials like\nMnSi we estimate that it takes 10\u00006sto10\u00005sfor a skyrmion\nto diffusive over an average length of one skyrmion diameter.\nTo analyze the dynamics on shorter time scales we show in\nFig. 3 four real functions parametrizing G\u00001(!): a frequency-\ndependent mass m(!), gyrocouplingG(!), gyrodamping\n\u000b\u0000(!)and dissipation strength \u000bD(!)with\nG\u00001(!) =\u0012\n\u000bD(!)\u0000i!m(!)\u0000G(!) +i\u000b!\u0000(!)\nG(!)\u0000i!\u000b\u0000(!)\u000bD(!)\u0000i!m(!)\u0013\nFor!!0one obtains the parameters of Eq. (1). All pa-\nrameters depend only weakly on temperature, Gandmare ap-\nproximately independent of \u000b, while the friction coefficients4\n0 1 2 3 4 5\nω / ωp04812 α\nthermal diffusion\n0 1 2 3 4 5\nω / ωp00.51-G / 4 π\n0 1 2 3 4 5\nω / ωp0100200\nα Γ0 1 2 3 4 5\nω / ωp050100\nm\ncurrent driven motion\nforce driven motion\nFIG. 3: Dissipative tensor \u000bD, massm, gyrocouplingGand gyro-\ndamping\u000b\u0000as functions of the frequency !for the diffusive motion\natT= 0:1(solid lines). They differ strongly from the “apparent”\ndynamical coefficients (see text) obtained for the force driven (red\ndashed line) and current driven motion (green dot-dashed line). We\nuse\u000b= 0:2,\f= 0:1. The error bars reflect estimates of systematic\nerrors arising mainly from discretization effects, see appendix B.\n00.05 0.1 0.15 0.2α01234 αT=0.15\nT=0.2\n00.05 0.1 0.15 0.2α00.51-G / 4 π\n00.05 0.1 0.15 0.2α010203040\nα Γ00.05 0.1 0.15 0.2α0255075100\nmT=0.05\nT=0.1\nFIG. 4: Dissipative strength \u000bD, massm, gyrocouplingGand gy-\nrodamping\u000b\u0000as functions of the Gilbert damping \u000bfor different\ntemperatures T.\n\u000bDand\u000b\u0000are linear in \u000b, see Fig. 4. In the limit T!0,\nG(!!0)takes the value\u00004\u0019, fixed by the topology of the\nskyrmion15,20.\nBoth the gyrodamping \u0000and and the effective mass m\nhave huge numerical values. A simple scaling analysis of the\nLandau-Lifshitz-Gilbert equation reveals that both the gyro-\ncouplingGandDare independent of the size of the skyrmion,\nwhile \u0000andmare proportional to the area of the skyrmion,\nand frequencies scale with the inverse area, see appendix\nB. For the chosen parameters (the field dependence is dis-cussed in the appendix B), we find m\u00190:3N\ripm0and\n\u000b\u0000\u0019\u000b0:7N\ripm0, wherem0=~2\nJa2is the mass of a sin-\ngle flipped spin in a ferromagnet ( 1in our units) and we have\nestimated the number of flipped spins, N\rip, from the total\nmagnetization of the skyrmion relative to the ferromagnetic\nbackground. As expected the mass of skyrmions grows with\nthe area (consistent with an estimate29formobtained from the\nmagnon spectrum of skyrmion crystals), the observation that\nthe damping rate \u000bDis independent of the size of skyrmions\nis counter-intuitive. The reason is that larger skyrmions have\nsmoother magnetic configurations, which give rise to less\ndamping. For realistic system parameters J= 1meV (which\nyields a paramagnetic transition temperature TC\u001810K, but\nthere are also materials, i.e. FeGe, where the skyrmion lattice\nphase is stabilised near room-temperature16) anda= 5 ˚A and\na skyrmion radius of 200 ˚A one finds a typical mass scale of\n10\u000025kg.\nThe sign of the gyrodamping \u000b\u0000is opposite to that of the\ngyrocouplingG. This implies that \u000b\u0000describes not damp-\ning but rather antidamping: there is less friction for cyclotron\nmotion of the skyrmion than for the linear motion. The nu-\nmerical value for the antidamping turns out to be so large\nthatDm+ \u0000G<0. This has the profound consequence that\nthe simplified equation of motion shown in Eq. (1) cannot be\nused: it would wrongly predict that some oscillations of the\nskyrmion are not damped, but grow exponentially in time due\nto the strong antidamping. This is, however, a pure artifact\nof ignoring the frequency dependence of G\u00001(!), and such\noscillations do not grow.\nFig. 3 shows that the dynamics of the skyrmion has a strong\nfrequency dependence. We identify the origin of this fre-\nquency dependence with a coupling of the skyrmion coordi-\nnate to pairs of magnon excitations as discussed in Ref. 31.\nMagnon emission sets in for ! > 2!pwhere!p=Bis the\nprecession frequency of spins in the ferromagnet (in the pres-\nence of a bound state with frequency !b, the onset frequency\nis!p+!b, Ref. 31). This new damping channel is most ef-\nficient when the emitted spin waves have a wavelength of the\norder of the skyrmion radius.\nAs a test for this mechanism, we have checked that only\nthis high-frequency damping survives for \u000b!0. In Fig. 5\nwe show the frequency dependent damping \u000bD(!)for various\nbare damping coefficients \u000b. For small!it is proportional to\n\u000bas predicted by the Thiele equation. For !>2!p, however,\nan extra dampling mechanism sets in: the skyrmion motion\ncan be damped by the emission of pairs of spin waves. This\nmechanism is approximately independent of \u000band survives\nin the\u000b!0limit. This leads necessarily to a pronounced\nfrequency dependence of the damping and therefore to the ef-\nfective mass m(!)which is related by the Kramers-Kronig\nrelationm(!) =1\n!R1\n\u00001\u000bD(!0)\n!0\u0000!d!0\n\u0019to\u000bD(!). Note also that\nthe large\u000bindependent mass m(!!0)is directly related to\nthe\u000bindependent damping mechanism for large !. Also the\nfrequency dependence of m(!)andG(!)can be traced back\nto the same mechanism as these quantities can be computed\nfrom\u000bD(!)and\u000b\u0000(!)using Kramers-Kronig relations. For\nlarge frequencies, the effective mass practically vanishes and5\n0 1 2 3 4 5\nω/ωp05101520αD(ω)α=0.05\nα=0.1\nα=0.2\nFIG. 5: Effective damping, \u000bD(!)for\u000b= 0:2,0:1and0:05.\n0 1 2 3 4 5\nω / ωp-400-2000200400Mz\ntot600Re/Im SgRe Sg11\nRe Sg21\nIm Sg11\nIm Sg21\nFIG. 6: Dynamical coupling coefficients for the force driven motion\n(\u000b= 0:2). In the static limit everything but the real part of the diago-\nnal vanishes. R eS11\ng(!)however approaches the total magnetization\nMz\ntotas expected. The error bars reflect estimates of systematic er-\nrors, see appendix B.\nthe ‘gyrocoupling’ Gdrops by a factor of a half.\nIV . FORCE-DRIVEN MOTION\nNext, we study the effects of an oscillating magnetic field\ngradient rBz(t)in the absence of thermal fluctuations. As\nthe skyrmion has a large magnetic moment Mz\ntotrelative to the\nferromagnetic background, the field gradient leads to a force\nacting on the skyrmion. In the static limit, the force is exactly\ngiven by\nFg(!!0) =Mz\ntotrBz: (10)\nUsing G\u00001(!)determined above, we can calculate how the\neffective force Sg(!)rBz(!)(see Eq. 2) depends on fre-\nquency. Fig. 6 shows that for !!0one obtains the expectedresultSg(!!0) =\u000eijMz\ntot, while a strong frequency de-\npendence sets in above the magnon gap, for !&!p. This\nis the precession frequency of spins in the external magnetic\nfield.\nIn general, both the screening of forces (parametrized\nbySg(!)) and the internal dynamics (described by\nG\u00001(!)) determines the response of skyrmions, V(!) =\nG(!)Sg(!)rBz(!). Therefore it is in general not possi-\nble to extract, e.g., the mass of the skyrmion as described by\nG\u00001(!)from just a measurement of the response to field gra-\ndients. It is, however, instructive to ask what “apparent” mass\none obtains, when the frequency dependence of Sg(!)is ig-\nnored. We therefore define the “apparent” dynamics G\u00001\na(!)\nbyGa(!)Sg(!= 0) = G(!)Sg(!). The matrix elements\nofG\u00001\na(!)are shown in Fig. 3 as dashed lines. The appar-\nent mass for gradient-driven motion, for example, turns out\nto be more than a factor of three smaller then the value ob-\ntained from the diffusive motion clearly showing the impor-\ntance of screening effects on external forces. The situation\nis even more dramatic when skyrmions are driven by electric\ncurrents.\nV . CURRENT-DRIVEN MOTION\nCurrents affect the motion of spins both via adiabatic and\nnon-adiabatic spin torques30. Therefore one obtains two types\nof forces on the spin texture even in the static limit19–22,24.\nThe effect of a time-dependent, spin-polarized current on\nthe magnetic texture can be modelled by supplementing the\nright hand side of eq. (3) with a spin torque term TST,\nTST=\u0000(vs\u0001r)Mr+\f\nM[Mr\u0002(vs\u0001r)Mr]:(11)\nThe first term is called the spin-transfer-torque term and is\nderived under the assumption of adiabaticity: the conduction-\nelectrons adjust their spin orientation as they traverse the mag-\nnetic sample such that it points parallel to the local magnetic\nmoment Mrowing toJHandJsd. This assumptions is justi-\nfied as the skyrmions are rather large smooth objects (due to\nthe weakness of spin-orbit coupling). The second so called \f-\nterm describes the dissipative coupling between conduction-\nelectrons and magnetic moments due to non-adiabatic effects.\nBoth\u000band\fare small dimensionless constants in typical ma-\nterials. From the Thiele approach one obtains the force\nFc(!!0) =G\u0002vs+\fDvs: (12)\nFor a Galilei-invariant system one obtains \u000b=\f. In this\nspecial limit, one can easily show that an exact solution of the\nLLG equations in the presence of a time-dependent current,\ndescribed by vs(t)is given by M(r\u0000Rt\n\u00001vs(t0)dt0)pro-\nvided, M(~ r)is a static solution of the LLG equation for vs=\n0. This implies that for \u000b=\f, the skyrmion motion exactly\nfollows the external current, _R(t) =vs(t). Using Eq. (2),\nthis implies that for \u000b=\fone has G\u00001(!) =Sc(!). Defin-\ning the apparent dynamics, as above, Ga(!)Sc(!= 0) =\nG(!)Sc(!)one obtains a frequency independent G\u00001\na(!) =6\n0 1 2 3 4 5\nω / ωp-4 π-10-50β D(0)510Re/Im ScIm Sc11\nIm Sc21Re Sc11\nRe Sc21\nFIG. 7: Dynamical coupling coefficients (symbols) for the current-\ndriven motion ( \u000b= 0:2,\f= 0:1,J= 1,\u0015= 0:18J,B= 0:0278 ).\nThese curves follow almost the corresponding matrix elements of\nG\u00001(!)shown as dashed lines. A deviation of symbols and dashed\nline is only sizable for Re S11\nc.\n0 1 2 3 4 5\nω/ωp-5051015 mα=0.2,β=0\nα=0.2,β=0.1\n0 1 2 3 4 5\nω/ωp0510\nα Γα=0.2,β=0.15\nα=0.2,β=0.19\nα=0.2,β=0.3\nFIG. 8: Mass m(!)and gyrodamping \u000b\u0000(!)as functions of the\ndriving frequency !for the current-driven motion. Note that both M\nand\u0000vanish for\u000b=\f.\nSc(!= 0) =\fD 1\u0000i\u001byG: the apparent effective mass and\ngyrodamping are exactly zero in this limit and the skyrmion\nfollows the current without any retardation. For \u000b6=\f, the\nLLG equations predict a finite apparent mass. Numerically,\nwe find only very small apparent masses, ma\nc/\u000b\u0000\f, see\ndot-dashed line in upper-right panel of Fig. 3, where the case\n\u000b= 0:2,\f= 0:1is shown. This is anticipated from the anal-\nysis of the\u000b=\fcase: As the mass vanishes for \u000b=\f= 0,\nit will be small as long as both \u000band\fare small. Indeed\neven for\u000b6=\fthis relation holds approximately as shown\nin Fig. 7. The only sizable deviation is observed for Re S11\nc\nfor which the Thiele equation predicts Re S11\nc(!!0) =\fD\nwhile Re G\u0000111(!!0) =\u000bDas observed numerically.\nA better way to quantify that the skyrmion follows the cur-rent even for \u000b6=\falmost instantaneously is to calculate\nthe apparent mass and gyrodamping for current driven mo-\ntion, where only results for \u000b= 0:2and\f= 0:1have been\nshown. As these quantities vanish for \u000b=\f, one can ex-\npect that they are proportional to \u000b\u0000\fat least for small \u000b;\f.\nThis is indeed approximately valid at least for small frequen-\ncies as can be seen from Fig. 8. Interestingly, one can even\nobtain negative values for \f > \u000b (without violating causal-\nity). Most importantly, despite the rather large values for \u000b\nand\fused in our analysis, the apparent effective mass and\ngyrodamping remain small compared to the large values ob-\ntained for force-driven motion or the intrinsic dynamics. This\nshows that retardation effects remain tiny when skyrmions are\ncontrolled by currents.\nVI. CONCLUSIONS\nIn conclusion, we have shown that skyrmions in chiral mag-\nnets are characterised by a number of unique dynamical prop-\nerties which are not easily found in other systems. First, their\ndamping is small despite the fact that skyrmions are large\ncomposite objects. Second, despite the small damping, the\ndiffusion constant remains small. Third, despite a huge iner-\ntial mass, skyrmions react almost instantaneously to external\ncurrents. The combination of these three features can become\nthe basis for a very precise control of skyrmions by time-\ndependent currents.\nOur analysis of the skyrmion motion is based on a two-\ndimensional model where only a single magnetic layer was\nconsidered. All qualitative results can, however, easily be\ngeneralized to a film with NLlayers. In this case, all terms\nin Eq. (1) get approximately multiplied by a factor NLwith\nthe exception of the last term, the random force, which is en-\nhanced only by a factorpNL. As a consequence, the diffu-\nsive motion is further suppressed by a factor 1=pNLwhile\nthe current- and force-driven motion are approximately unaf-\nfected.\nAn unexpected feature of the skyrmion motion is the an-\ntidamping arising from the gyrodamping. The presence of\nantidamping is closely related to another important property\nof the system: both the dynamics of the skyrmion and the ef-\nfective forces acting on the skyrmion are strongly frequency\ndependent.\nIn general, in any device based on skyrmions a combination\nof effects will play a role. Thermal fluctuations are always\npresent in room-temperature devices, the shape of the device\nwill exert forces13,14and, finally, we have identified the cur-\nrent as the ideal driving mechanism. In the linear regime, the\ncorresponding forces are additive. The study of non-linear\neffects and the interaction of several skyrmions will be impor-\ntant for the design of logical elements based on skyrmions and\nthis is left for future works. As in our study, we expect that\ndynamical screening will be important in this regime.7\n 30\n 40\n 50\n 60\n 70  30 40 50 60 70 0 0.0005 0.001 0.0015 0.002 0.0025 0.003 0.0035 0.004\nFIG. 9: Skyrmion density based on the normalized z-component of\nthe magnetization.\nAcknowledgments\nThe authors are greatful for insightful discussions with K.\nEverschor and Markus Garst. Part of this work was funded\nthrough the Institutional Strategy of the University of Cologne\nwithin the German Excellence Initiative” and the BCGS. C.S.\nthanks the University of Tokyo for hospitality during his re-\nsearch internship where part of this work has been performed.\nN.N. was supported by Grant-in-Aids for Scientific Research\n(No. 24224009) from the Ministry of Education, Culture,\nSports, Science and Technology (MEXT) of Japan, and by the\nStrategic International Cooperative Program (Joint Research\nType) from Japan Science and Technology Agency. J.I. is sup-\nported by Grant-in-Aids for JSPS Fellows (No. 2610547).\nAppendix A: Definition of the Skyrmion’s centre coordinate\nIn order to calculate the Green’s function, Eq. (3), one\nneeds to calculate the velocity-velocity correlation function.\nTherefore it is necessary to track the skyrmion position\nthroughout the simulation. Mostly two methods have been\nused so far for this25: (i) tracking the centre of the topological\ncharge and (ii) tracking the core of the Skyrmion (reversal of\nmagnetization).\nThe topological charge density\n\u001atop(r) =1\n4\u0019^ n(r)\u0001(@x^ n(r)\u0002@y^ n(r)) (A1)\nintegrates to the number of Skyrmions in the system. There-\nfore for our case of a single Skyrmion in the ferromagnetic\nbackground this quantity is normalized to 1. The center of\ntopological charge can therefore be defined as\nR=Z\nd2r\u001atop(r)r (A2)\nFor the case of finite temperature this method can, however,\nnot be used directly. Thermal fluctuations in the ferromagnetic\nbackground far away from the skyrmion lead to a large noise\nto this quantity which diverges in the thermodynamic limit.\nA similar problem arises when tracking the center using the\nmagnetization of the skyrmion.One therefore needs a method which focuses only on the\nregion close to the skyrmion center. To locate the skyrmion,\nwe use thez-component of the magnetization but take into ac-\ncount only points where Mz(r)<\u00000:7(the magnetization of\nthe ferromagnetic background at T= 0is+1). We therefore\nuse\n\u001a(r) = (1\u0000Mz(r)) \u0002[\u0000Mz(r)\u00000:7] (A3)\nwhere \u0002[x]is the theta function. A first estimate, Rest=RV,\nfor the radius is obtained from\nRA=R\nAr\u001a(r)d2rR\nA\u001a(r)d2r(A4)\nby integrating over the full sample volume V.Restis noisy\ndue to the problems mentioned above but for the system\nsizes simulated one nevertheless obtains a good first esti-\nmate for the skyrmion position. This estimate is refined by\nusing in a second step for the integration area only D=\b\nr2R2jjr\u0000Restj<r\t\nwhereris choosen to be larger\nthan the radius of the skyrmion core (we use r= 1:3p\nN<=\u0019,\nwhereN<is the number of spins with Mz<\u00000:7). Thus\nwe obtain a reliable estimate, R=RD, not affected by spin\nfluctuations far away from the skyrmion. From the resulting\nR(t), one can obtain the velocity-velocity correlation function\nhVi(t0+t)VJ(t0)i.\nAppendix B: Scaling invariance\nTo obtain analytic insight into the question of how param-\neters depend on system size and to check the numerics for\nlattice artifacts it is useful to perform a scaling analysis of the\nsLLG equations and the effective equations of motion for the\nskyrmion.\nWe investigate a scaling transformation, where the radius\nof the skyrmion is enlarged by a factor \u0011,M(r)!~M(r) =\nM(r=\u0011).\nFor the scaling analysis, we use the continuum limit of\nEq. (4) (setting a= 1)\nH[M] =Z\nd2r\u0014J\n2(rM)2+\u0015M\u0001r\u0002 M\u0000B\u0001M\u0015\nThe three terms scale with \u00110,\u0011and\u00112, respectively. To ob-\ntain a larger skyrmion, we therefore have to rescale \u0015!\u0015=\u0011\nandB!B=\u00112. This implies that the Be\u000bterm in the sLLG\nequation scales with 1=\u00112and therefore also the time axis has\nto be rescaled, t!\u00112t, implying that all time scales are a\nfactor of\u00112longer and all frequencies a factor 1=\u00112smaller.\nSimiliarly, the driving current, i.e. vsis reduced by a factor\n\u0011\u00001while the temperature remains unscaled. This implies that\nwhen M(r;t)is a solution for a given value of \u0015,Bandvs\nandG(!)the corresponding velocity-correlation function of\nthe skyrmion, then M(r=\u0011;t=\u00112)is a solution for \u0015=\u0011,B=\u00112,\nvs=\u0011with correlation function G(!\u00112).\nAccordingly, the !!0limit and therefore the gyrocou-\nplingG, the friction constant \u000bDand the diffusion constant of8\n0.03 0.035 0.04\nBz00.511.5 α D\n0.03 0.035 0.04\nBz00.51-G / 4 π\n0.03 0.035 0.04\nBz05101520\nα Γ0.03 0.035 0.04\nBz020406080\nm\nFIG. 10: Field dependence of the dissipative tensor \u000bD, the massm,\nthe gyrocouplingGand the gyrodamping \u000b\u0000forJ= 1,\u0015= 0:18J.\nThe main effect is that both mand\u000b\u0000shrink when the size of the\nskyrmion shrinks with increasing Bz.\nthe skyrmion are independent of\u0011, consistent with the analyt-\nical formulas eq. (6). In contrast, the mass of the skyrmion,\nm, and the gyrodamping \u000b\u0000scale with\u00112. They are therefore\nproportional to the number of spins constituting the skyrmion\nconsistent with our numerical findings. When one does, how-\never, change the external magnetic field for fixed strength\u0015\nof the DM interaction, the internal structure of the skyrmion\nand therefore also G(!)change quantitatively in a way not\npredictable by scaling. In Fig. 10 we therefore show the B-dependence of \u000bD,m,Gand\u000b\u0000. For increasing Bthe size of\nthe skyrmion shrinks. From our previous analysis, it is there-\nfore not surprising that both the effective mass mand the gy-\nrodamping\u000b\u0000shrink substantially while \u000bDandGare less\naffected . Quantitatively, this change of mand\u000b\u0000is how-\never not proportional to the number of spins constituting the\nskyrmion.\nWe have tested numerically the scaling properties for \u0011=\n1:2and find that all features are quantitatively reproduced.\nSmall variations on the level of a few percent do, however,\noccur reflecting the typical size of features arising from the\ndiscretization of the continuum theory. A conservative esti-\nmate of such systematic discretization effects for the diffusive\nmotion is given by the error bars in Figs. 3 (all statistical er-\nrors are smaller than the thickness of the line). For the field-\ndriven motion (Fig. 3 and Fig. 6) spatial discretization effects\nlead to a different source of errors. For very small field gradi-\nents and high frequencies the displacement of the skyrmion is\nmuch smaller than the lattice spacing and the response is af-\nfected by a tiny pinning of the skyrmion to the discreet lattice.\nFor larger gradients, however, nonlinear effects set in and for\nsmall frequencies the skyrmion starts to approach the edge of\nthe simulated area. In Fig. 3, we therefore used for the force-\ndriven motionrB= 0:0005 for!<2!pandrB= 0:0015\nfor! >2!p. Error bars have been estimated from variations\nof the numerical values when rBwas varied from 0:0001 to\n0:0015 . For the current-driven motion errors are so tiny that\nthey are not shown.\n\u0003Electronic address: nagaosa@riken.jp\n1Hubert, A. & Sch ¨afer, R. Magnetic Domains: The Analysis of\nMagnetic Microstructures (Springer, Berlin, 1998).\n2Malozemoff, A. P. & Slonczewski, J.C. Magnetic Domain Walls\nin Bubble Materials (Academic Press, New York, 1979).\n3Skyrme, T. H. R. A Non-Linear Field Theory. Proc. Roy. Soc.\nLondon A 260, 127–138 (1961).\n4Skyrme, T. H. R. A unified field theory of mesons and baryons.\nNuc. Phys. 31,556–569 (1962).\n5Bogdanov, A. N. & Yablonskii, D. A. Thermodynamically stable\n”vortices” in magnetically ordered crystals. The mixed state of\nmagnets. Sov. Phys. JETP 68,101–103 (1989).\n6M¨uhlbauer, S. et al. Skyrmion lattice in a chiral magnet. Science\n323, 915–919 (2009).\n7Yu, X. Z. et al. Real-space observation of a two-dimensional\nskyrmion crystal. Nature 465, 901–904 (2010).\n8Heinze, S. et al. Spontaneous atomic-scale magnetic skyrmion lat-\ntice in two dimensions. Nature Phys. 7,713–718 (2011).\n9Seki, S., Yu, X. Z., Ishiwata, S. & Tokura, Y . Observation\nof skyrmions in a multiferroic material. Science 336, 198–201\n(2012).\n10Fert, A., Cros, V . & Sampaio, J. Skyrmions on the track. Nature\nNanotech. 8,152–156 (2013).\n11Lin, S. -Z., Reichhardt, C., Batista, C. D. & Saxena, A. Driven\nskyrmions and dynamical transitions in chiral magnets. Phys. Rev.\nLett. 110, 207202 (2013).\n12Lin, S. -Z., Reichhardt, C., Batista, C. D. & Saxena, A. Particle\nmodel for skyrmions in metallic chiral magnets: Dynamics, pin-ning, and creep. Phys. Rev. B 87,214419 (2013).\n13Iwasaki, J., Mochizuki, M. & Nagaosa, N. Current-induced\nskyrmion dynamics in constricted geometries. Nature Nanotech.\n8,742–747 (2013).\n14Iwasaki, J., Koshibae, W. & Nagaosa, N. Colossal spin transfer\ntorque effect on skyrmion along the edge. Nano letters 14 (8),\n4432–4437 (2014).\n15Jonietz, F. et al. Spin transfer torques in MnSi at ultralow current\ndensities. Science 330, 1648–1651 (2010).\n16Yu, X. Z. et al. Skyrmion flow near room temperature in an ul-\ntralow current density. Nat. Commun. 3,988 (2012).\n17Parkin, S. S. P., Hayashi, M. & Thomas, L. Magnetic domain-wall\nracetrack memory. Science 320, 190–194 (2008).\n18Yamanouchi, M., Chiba, D., Matsukura, F. & Ohno, H. Current-\ninduced domain-wall switching in a ferromagnetic semiconductor\nstructure. Nature 428, 539–542 (2004).\n19Schulz, T. et al. Emergent electrodynamics of skyrmions in a chi-\nral magnet. Nature Phys. 8,301–304 (2012).\n20Thiele, A. A. Steady-state motion of magnetic domains. Phys.\nRev. Lett. 30,230–233 (1973).\n21Everschor, K. et al. Current-induced rotational torques in the\nskyrmion lattice phase of chiral magnets. Phys. Rev. B 86,054432\n(2012).\n22Everschor, K. et al. Rotating skyrmion lattices by spin torques and\nfield or temperature gradients. Phys. Rev. B 86,054432 (2012).\n23He, J. et al. Current-driven vortex domain wall dynamics by mi-\ncromagnetic simulations. Phys. Rev. B 73,184408 (2006).\n24Iwasaki, J., Mochizuki, M. & Nagaosa, N. Universal current-9\nvelocity relation of skyrmion motion in chiral magnets. Nat. Com-\nmun. 4,1463 (2013).\n25Makhfudz, I., Kr ¨uger, B. & Tchernyshyov, O. Inertia and chiral\nedge modes of a skyrmion magnetic bubble. Phys. Rev. Lett. 109,\n217201 (2012).\n26Kubo, R. The fluctuation-dissipation theorem. Rep. Prog. Phys.\n29,255–284 (1966).\n27Garc ´ıa-Palacios, J. L. and L ´azaro, F. J. Langevin-dynamics study\nof the dynamical properties of small magnetic particles. Phys. Rev.\nB58,14937–14958 (1998).28Clarke, D. J. and Tretiakov, O. A. and Chern, G.-W. and Bazaliy,\nY . B. and Tchernyshyov, O., Phys. Rev. B 78, 134412 (2008).\n29Petrova, O. and Tchernyshyov O., Phys. Rev. B 84, 214433\n(2011).\n30Tatara, G., Kohno, H., & Shibata, J. Microscopic approach to\ncurrent-driven domain wall dynamics. Physics Reports 468, 213–\n301 (2008).\n31Sch¨utte, C. and Garst, M. Magnon-skyrmion scattering in chiral\nmagnets. arxiv 1405.1568 (2014)."    },    {        "title": "1501.07731v1.Head_to_Head_Domain_Wall_Structures_in_Wide_Permalloy_Strips.pdf",        "content": "Head-to-Head Domain Wall Structures in Wide Permalloy Strips\nVirginia Est\u0013 evez\u0003and Lasse Laurson\nCOMP Centre of Excellence and Helsinki Institute of Physics,\nDepartment of Applied Physics, Aalto University,\nP.O.Box 11100, FI-00076 Aalto, Espoo, Finland.\nWe analyze the equilibrium micromagnetic domain wall structures encountered in Permalloy strips\nof a wide range of thicknesses and widths, with strip widths up to several micrometers. By per-\nforming an extensive set of micromagnetic simulations, we show that the equilibrium phase diagram\nof the domain wall structures exhibits in addition to the previously found structures (symmetric\nand asymmetric transverse walls, vortex wall) also double vortex and triple vortex domain walls\nfor large enough strip widths and thicknesses. Also several metastable domain wall structures are\nfound for wide and/or thick strips. We discuss the details of the relaxation process from random\nmagnetization initial states towards the stable domain wall structure, and show that our results are\nrobust with respect to changes of e.g. the magnitude of the Gilbert damping constant and details\nof the initial conditions.\nPACS numbers: 75.60.Ch, 75.78.Cd\nI. INTRODUCTION\nDuring the last decade, a lot of e\u000bort has been de-\nvoted to understand static and dynamic properties of\nmagnetic domain walls (DWs) in ferromagnetic nanos-\ntructures such as nanowires and -strips. These studies\nhave been largely driven by promising technological ap-\nplications based on domain walls and their dynamics, in\nparticular memory1,2and logic devices3{5. In typical ex-\nperiments DWs are driven by either applied magnetic\n\felds6,7or spin-polarized electric currents8{11. The re-\nsulting DW dynamics depends crucially on the micro-\nmagnetic DW structure, typically involving various in-\nternal degrees of freedom. These are essential e.g. for\nthe emergence of the Walker breakdown12, an instability\noccurring when the DW internal degrees of freedom get\nexcited by a strong external drive (a magnetic \feld H\nor a spin-polarized current Jexceeding the Walker \feld\nHWor currentJW, respectively), limiting the propaga-\ntion velocity of the DWs.\nTwo main classes of ferromagnetic materials have been\nextensively studied within the strip geometry. Ma-\nterials with a high perpendicular magnetic anisotropy\n(PMA13{16) exhibit simple and narrow DWs of the\nBloch and/or N\u0013 eel type. For H > H WorJ > J W,\nrepeated transitions between these two structures are\nobserved16. The second class of systems includes soft\n(low anisotropy) magnetic materials13,14such as Permal-\nloy, where in-plane domain magnetization along the long\naxis of the strip is induced by shape anisotropy. By us-\ning various experimental techniques17,18and micromag-\nnetic simulations, it has been established that the equi-\nlibrium DW structures separating these in-plane domains\nare more complex, and depend crucially on the sample\ngeometry19{22. Transverse DWs (TWs) and asymmetric\ntransverse DWs (ATWs) are observed for narrow and\nthin strips22{27, while in wider and thicker strips one\nencounters the vortex DW (VW)22,24{26,28,29. In addi-\ntion, various metastable DW structures with higher en-ergy may be found18,28{31. ForH >H WorJ >J W, the\nDW structures exhibit dynamical evolution: for TWs,\nrepeated nucleation and propagation of an antivortex\nacross the strip width takes place19. Similarly, in VWs\nthe vortex core performs oscillatory back and forth per-\npendicular motion19.\nIn Permalloy strips with even larger widths and/or\nthicknesses, one might expect also other, possibly more\ncomplicated equilibrium DW structures. For wider strips\nshape anisotropy is less important, implying that energy\nminima with more complex spin structures closing the\n\rux more e\u000eciently than TWs, ATWs or VWs may ap-\npear. Indeed, e.g. double and triple vortex DWs have\nbeen observed in experiments on wide strips28, but they\nhave been attributed to current-induced vortex nucle-\nation resulting in metastable DW structures. Conse-\nquently, a pertinent and fundamental question is what\nare the possible intermediate equilibrium DW structures\nobservable when the lateral Permalloy strip dimensions\nincrease from those corresponding to the typical nanos-\ntrip geometry (with TW, ATW or VW as the stable DW\nstructure) to strip widths of micrometers and beyond.\nIn this paper we present an extensive numerical study\nof the equilibrium and metastable micromagnetic DW\nstructures in Permalloy strips, with the strip widths up to\nan order of magnitude larger than before19{24. Contrary\nto previous studies focusing on comparing the energies\nof di\u000berent a priori known DW structures19{24, we per-\nform micromagnetic simulations of relaxation dynamics\nfrom random initial states towards the stable DW struc-\ntures. In addition to the previously observed TW, ATW\nand VW DWs, we \fnd also DWs with equilibrium double\nand triple vortex structures for wide and/or thick enough\nstrips. The last structure is encountered only in the very\nlargest system sizes we were able to simulate. Moreover,\nfor wide strips we \fnd a rich variety of metastable DWs\nwith even more complex micromagnetic structures. We\ndemonstrate that our results are robust with respect to\nchanges of the magnitude of the Gilbert damping con-arXiv:1501.07731v1  [cond-mat.mes-hall]  30 Jan 20152\nlength lwidth wthickness Δz\nlr(a)\n(b)\nFIG. 1. (color online) (a) Geometry of the Permalloy strip.\n(b) A top view of the magnetization in the initial state. Mag-\nnetization points along the long axis of the strip within the\ntwo domains (as indicated by the arrows) forming a head-\nto-head con\fguration. In between them, a region of random\nmagnetization (of length lr) has been included.\nstant or using di\u000berent initial conditions for the relax-\nation process. Our results underline the crucial role of\ntopological defects for physics of DWs in soft strips, and\nthat of micromagnetic simulations for \fnding the true\nequilibrium DW structure.\nII. MICROMAGNETIC SIMULATIONS\nThe system studied is a Permalloy strip of width wand\nthickness \u0001 z, satisfying \u0001 z\u001cw, see Fig. 1 (a). In the\nmicromagnetic simulations, magnetic charges are com-\npensated on the left and right ends of the strip, to mimic\nan in\fnitely long strip; the actual simulated length sat-\nis\fesl\u00154wfor all cases considered. The initial state\nfrom which the relaxation towards a stable DW struc-\nture starts is an in-plane head-to-head domain struc-\nture, with a region of random magnetization of length\nlrin the middle of the sample, see Fig. 1 (b). If not\nspeci\fed otherwise, we consider lr= 2w. Material pa-\nrameters of Permalloy are used, i.e. saturation magne-\ntizationMs= 860\u0002103A/m and exchange constant\nAex= 13\u000210\u000012J/m. The typical Gilbert damping\nconstant for Permalloy is \u000b= 0:01, but here we analyze\nalso the in\ruence of \u000bon the relaxation process, and thus\nconsider also other values. For simplicity, we set the tem-\nperatureTto zero, and focus on the ideal case of strips\nfree of any structural disorder or impurities.\nThe simulations are performed using the GPU-\naccelerated micromagnetic code MuMax332{34, o\u000bering\na signi\fcant speedup as compared to CPU codes for the\nlarge system sizes we consider here. To calculate the mag-\nnetization dynamics of the system, the Landau-Lifshitz-\nGilbert equation35,36,\n@m=@t=\rHe\u000b\u0002m+\u000bm\u0002@m=@t; (1)\nis solved numerically. Here, mis the magnetization, \r\nthe gyromagnetic ratio, and He\u000bthe e\u000bective \feld, with\n0 ns 0.2 ns\n1.0 ns 2.0 ns\n3.0 ns 4.0 ns\n6.0 ns 12.0 ns\nFIG. 2. (color online) An example of the temporal evolution\nof the relaxation, with w= 420 nm, \u0001 z= 10 nm,\u000b= 3 and\nlr= 2w. Relaxation towards the equilibrum DW structure\n(here, a VW) takes place via coarsening dynamics of the de-\nfect structure in the magnetic texture. The colorwheel in the\nmiddle shows the mapping between magnetization directions\nand colors.\ncontributions due to exchange, Zeeman, and demagne-\ntizing energies. The size of the discretization cell used\ndepends on the system size, but is always bounded by\nthe exchange length, \u0003 = (2 A=\u0016 0M2\ns)1=2\u00195 nm, in the\nin-plane directions, and equals \u0001 zin the the out-of-plane\ndirection.\nIII. RESULTS\nWe start by considering the e\u000bect of varying \u000bandlr\non the relaxation process. Fig. 2 shows an example of\nthe time evolution of m(r;t) forw= 420 nm, \u0001 z= 10\nnm,\u000b= 3 andlr= 2w. The initially random magne-\ntization evolves via coarsening of the defect structure of\nthe magnetization texture towards the stable DW (here,\na VW). During the relaxation, the total energy Eof the\nsystem decreases in a manner that for a given geometry\n(wand \u0001z) depends on both \u000bandlr, see Fig. 3 (a) and\n(b) wherew= 5120 nm and \u0001 z= 20 nm, is considered.\nFor instance, Edecreases faster for an intermediate \u000b\n[Fig. 3 (a)]. We attribute this behavior to the balance\nbetween inertial e\u000bects related to precession favored by\na small\u000b, helping to overcome energy barriers, and the\nhigher rate of energy dissipation due to a large \u000b. Thus,\nthe relaxation time to reach a (meta)stable DW struc-\nture depends on \u000b. Fig. 3 (b) illustrates that for a \fxed\n\u000b, systems with a larger lrrelax more slowly. Fig. 3\n(c) shows that on average, the early-time relaxation of\nEtowards its \fnal value Efexhibits temporal power-law\ndecay,hE\u0000Efi/t\u0000\fwith\f\u00191:3 for the\u000b= 0:3 case\nshown, possibly related to collective e\u000bects due to inter-\nactions between several topological defects during early\nstages of relaxation (Fig. 2).\nIn general, the \fnal (meta)stable DW structure may3\n02×10-9\nt [s]03×10-136×10-139×10-13E [J]α = 3\nα = 0.3\nα = 0.03\n02×10-9\nt [s]03×10-136×10-13\nE [J]lr = w\nlr = 2w\nlr = 3w\nlr = 2w(a) (b)\nα = 0.3\n10-1010-910-8\nt [s]10-310-210-1100<E(t) - Ef> / <E(0)>\nlr = w\nlr = 2w\nlr = 3w\nt-1.3(c)\nα = 0.3\nFIG. 3. (color online) The energy E(t) as a function of time\ntforw= 5120 nm and \u0001 z= 20 nm. (a) For di\u000berent values\nof\u000bandlr= 2w. (b) For di\u000berent values of lr, and\u000b= 0:3\n[resulting in the fastest relaxation in (a)]. (c) shows that on\naverage, the early time decay of E(t) towards its \fnal value\nEfobeyshE(t)\u0000Efi/t\u0000\f. For the\u000b= 0:3 case shown here,\n\f\u00191:3. Empty (\flled) symbols in (c) correspond to w= 420,\n\u0001z= 10 nm (w= 860, \u0001 z= 20 nm).\ndepend on the realization of the random initial state.\nThus, we consider 21 realizations of the initial random\nmagnetization for each wand \u0001z, and compare the en-\nergies of the resulting relaxed con\fgurations. The struc-\nture with the lowest energy is chosen as the equilibrium\nstructure, while others with higher energy are metastable\nstates. Although, as discussed above, the relaxation\ntimes depend on \u000bandlr, the equilibrium DW struc-\nture is found to be independent of \u000bandlrin the range\nconsidered, i.e. \u000b2[0:01;3] andlr2[w;3w]. Thus, in\nwhat follows, we will use \u000b= 3 andlr= 2w.\nThe main results of this paper are summarized in\nFigs. 4 and 5, showing the phase diagram of the equi-\nlibrium DW structures for wranging from 120 to 5120\nnm, and \u0001 zfrom 5 to 25 nm, and examples of these\nstructures, respectively. For small w, we recover the pre-\nvious results19{22, i.e. phases correspoding to TW, ATW\nand VW, shown in Fig. 5 (a), (b) and (c), respectively.\nFor larger strip widths ( wapproaching or exceeding\n1\u0016m, depending on \u0001 z, see Fig. 4), a new equilibrium mi-\ncromagnetic DW structure, a double vortex wall (DVW),\nis observed. This structure consists of two vortices with\nopposite sense of rotation of the magnetization around\nthe vortex core, see Fig. 5 (d). At the phase boundary\n(blue triangle symbols pointing left in Fig. 4), VW and\nDVW have the same energy. The DVW phase spans a\nrelatively large area within the ( w;\u0001z) space, highlight-\ning the robustness of our results.\nIn addition, a second new phase, with a triple-vortex\nwall (TVW) as the equilibrium structure [see Fig. 5 (e)],\n100 1000\nw [nm]510152025t [nm]V ortex wall\nTransverse wallDouble vortex wall\nAsymmetric transverse wallTriple vortex wallΔzFIG. 4. (color online) Phase diagram of the equilibrium\nDW structure in Permalloy strips of various thicknesses (from\n\u0001z= 5 to 25 nm) and widths ranging from w= 120 nm up to\n5120 nm. The symbols correspond to observations of the vari-\nous equilibrium DW structures, with phase boundaries shown\nas solid lines. Examples of the DW structures corresponding\nto the 5 di\u000berent phases are shown in Fig. 5.\nTransverse Wall\nAsymmetric Transverse Wall V ortex Wall\nDouble V ortex Wall Triple V ortex Wall(a)\n(b) (c)\n(d) (e)\nFIG. 5. (color online) Examples of the di\u000berent equilibrium\nmicromagnetic DW structures: (a) TW for w= 120 nm and\n\u0001z= 5 nm, (b) ATW for w= 160 nm and \u0001 z= 10 nm, (c)\nVW forw= 640 nm and \u0001 z= 15 nm, (d) DVW for w= 2560\nnm and \u0001 z= 20 nm, and (e) TVW for w= 5120 nm and\n\u0001z= 25 nm. The colorwheel (top left) shows the mapping\nbetween magnetization directions and colors.\nis found for the very largest system sizes we have been\nable to simulate. The middle vortex of the TVW has an\nopposite sense of rotation to the other two. For w= 5120\nand \u0001z= 25 nm, DVW and TVW have the same energy\n(the cyan square symbol in the top right corner of Fig. 4),\nsuggesting the presence of a phase boundary between the\ntwo structures. Indeed, by performing a set of 10 ad-\nditional simulations with w= 6144 and \u0001 z= 25 nm\n(i.e. outside the phase diagram in Fig. 4), suggests that\nTVW is the equilibrium DW structure for very large strip\nwidths. This structure has been observed in experiments4\n(a) (b)\n(c) (d)\n(e) (f)\n(g) (h)2V+A V\n3V+A V\n4V+A V\n4V+3A V3V+A V\n3V+2A V\n4V+2A V\n5V+2A V\nFIG. 6. (color online) Examples of metastable DW structures\nobserved for a system with w= 5120 nm and di\u000berent thick-\nnesses \u0001z: (a) Two vortices and an antivortex (2V+AV),\n\u0001z= 25 nm, (b) and (c) three vortices and an antivortex\n(3V+AV), \u0001 z= 5 nm, (d) three vortices and two antivortices\n(3V+2AV), \u0001 z= 10 nm, (e) four vortices and an antivortex\n(4V+AV), \u0001 z= 5 nm, (g) four vortices and three antivor-\ntices (4V+3AV), \u0001 z= 5 nm, and (h) \fve vortices and two\nantivortices (5V+2AV), \u0001 z= 5 nm.\nas a metastable state for smaller systems28,29. Notice also\nthat the middle part of the TVW [Fig. 5 (e)], exhibiting\nfour line-like 90\u000eDWs meeting at a vortex core in the\nmiddle of the TVW, resembles the typical Landau \rux-\nclosure magnetization patterns observed for rectangular\nPermalloy thin \flms37{39.\nFollowing the relaxation from a random magnetization\ninitial state, the system may in general end up into var-\nious metastable states with higher energy than that of\nthe equilibrium DW. Sometimes these metastable states\nhave even a higher probability than the equilibrium one,\nestimated here from the sample of 21 relaxed con\fgura-\ntions. Fig. 6 shows some of the metastable states found\nfor a large strip with w= 5120 nm and di\u000berent values of\n\u0001z; for strips with smaller lateral dimensions, di\u000berent\nmetastable states tend to be less numerous and have a\nsimpler structure. Despite their apparent complexity, all\nthe metastable DW structures shown in Fig. 6 respect the\nbasic principles of topology of DWs. Each of the DWs\nare composed of topological defects, with an associated\nwinding number: +1 for vortices, -1 for antivortices, and\n\u00061=2 for edge defects40. In a DW all the topological\ndefects have to be compensated, i.e. the total winding\nnumber is equal to zero. In the case of the DVW, the\ntwo topological vortex defects are compensated by four\nedge defects [Fig. 5 (d)]. For the metastable state of twovortices (with the same sense of rotation) and an antivor-\ntex [2V+AV, see Fig. 6 (a)], there are two vortices and\nonly two edge defects. Thus, in order to compensate the\ntopological defects, also an antivortex appears. In gen-\neral, we have observed that in a DW with Nvortices with\nthe same sense of rotation, there must be N\u00001 antivor-\ntices to get a zero total winding number, see Fig. 6 (a)\nand (d) for examples with 2V+AV and 3V+2AV con\fg-\nurations, respectively. When some of the vortices have\noppposite sense of rotation, more complex scenarios are\nencountered, with examples shown in Figs. 6 (b), (c), (e),\n(f), (g) and (h). Notice also that two DW structures with\nthe same elements can look very di\u000berent, see e.g. the\ntwo 3V+AV DWs shown in Figs. 6 (b) and (c). All the\nDW structures found, both the equilibrium ones in Fig. 5\nand the metastable states in Fig. 6 obey the principle of\ncompensation of topological defects to yield a total wind-\ning number of zero. The richness of the equilibrium phase\ndiagram and the large collection of metastable states in-\ndicate that for wide/thick strips in particular, the micro-\nmagnetic energy landscape is quite complex, with a large\nnumber of local minima. This is also in agreement with\nour observations of power-law energy relaxation.\nIV. SUMMARY AND CONCLUSIONS\nTo summarize, we have performed an extensive set\nof micromagnetic simulations to study the equilibrium\nand metastable DW structures in Permalloy strips of a\nwide range of widths and thicknesses, as well as the re-\nlaxation dynamics starting from random magnetization\ninitial states. The general trend of our results is that\nboth the equilibrium and metastable DW con\fgurations\nbecome increasingly complex (i.e. they consist of an in-\ncreasing number of topological defects) as the lateral strip\ndimensions increase. We note that somewhat analogous\nbehaviour - i.e. existence of equilibrium magnetization\ncon\fgurations with increasing complexity as the system\nsize increases - is observed also in some other systems\nsuch as three-dimensional cylindrical elements with per-\npendicular anisotropy41,42.\nSeveral remarks are in order: \frst, for strips with\neven larger lateral dimensions one may in principle ex-\npect more complex DW patterns - possibly with four or\nmore vortices with alternating sense of rotation. These,\nhowever, are currently beyond the reach of our available\ncomputing resources. Second, our phase diagram allows\none to check if experimental observations of the various\nDW structures in wider strips are equilibrium con\fgura-\ntions or metastable states. According to our review of\nthe experimental literature, most observations of DVWs\nand TVWs appear to be metastable states28,29. Third,\nwhile the equilibrium structures we \fnd are certainly sta-\nble in the absence of external perturbations such as ap-\nplied magnetic \felds, it remains to be seen how their \feld\ndriven dynamics is like, and whether wide strips with a\nrelatively weak shape ansitropy are able to support the5\nDWs as compact objects also when external perturba-\ntions are being applied43.ACKNOWLEDGMENTS\nWe thank Mikko J. Alava for a critical reading of\nthe manuscript. This work has been supported by\nthe Academy of Finland through its Centres of Excel-\nlence Programme (2012-2017) under project no. 251748,\nand an Academy Research Fellowship (LL, project no.\n268302). We acknowledge the computational resources\nprovided by the Aalto University School of Science\n\\Science-IT\" project, as well as those provided by CSC\n(Finland).\n\u0003virginia.esteveznuno@aalto.\f\n1S. S. P. Parkin, M. Hayashi and L. Thomas, Science 320,\n190 (2008).\n2S. E. Barnes, J. Ieda and S. Maekawa, Appl. Phys. Lett.\n89, 122507 (2006).\n3R. P. Cowburn and M. E. Welland, Science 287, 1466\n(2000).\n4D. A. Allwood, G. Xiong, M. D. Cooke, C. C. Faulkner,\nD. Atkinson, N. Vernier and R. P. Cowburn, Science 296,\n2003 (2002)\n5D. A. Allwood, G. Xiong, C. C. Faulkner, D. Atkinson, D.\nPetit and R. P. Cowburn, Science 309, 1688 (2005).\n6T. Ono, H. Miyajima, K. Shigeto, K. Mibu, N. Hosoito\nand T. Shinjo, Science 284, 468 (1999).\n7G. S. D. Beach, C. Nistor, C. Knutson, M. Tsoi and J. L.\nErskine, Nature Mater. 4, 741 (2005).\n8M. Kl aui, C. A. F. Vaz, J. A. C. Bland, W. Wernsdorfer,\nG. Faini, E. Cambril and L. J. Heyderman, Appl. Phys.\nLett.83, 105 (2003).\n9A. Thiaville, Y. Nakatani, J. Miltat and N. Vernier, J.\nAppl. Phys. 95, 7049 (2004).\n10M. Kl aui, P.-O. Jubert, R. Allenspach, A. Bischof, J. A.\nC. Bland, G. Faini, U. R udiger, C. A. F. Vaz, L. Vila and\nC. Vouille, Phys. Rev. Lett. 95, 026601 (2005).\n11N. Vernier, D. A. Allwood, D. Atkinson, M. D. Cooke and\nR. P. Cowburn, EPL 65, 526 (2004).\n12N. L. Schryer and L. R. Walker. Journal of Applied Physics\n45,5406 (1074).\n13K.H.J. Buschow and F.R. de Boer, Physics of Mag-\nnetism and Magnetic Materials . Kluwer Academic Pub-\nlishers (2003).\n14J.M.D. Coey, Magnetism and Magnetig Materials , Cam-\nbridge University Press.\n15O. Boulle, J. Kimling,P. Warnicke, M. Kl aui, U. R.\nR udiger, G. Malinowski, H. J. M. Swagten, B. Koop-\nmans, C. Ulysse and G. Faini. Phys. Rev. Lett. 101216601\n(2008).\n16E. Martinez. J. Phys.: Condens. Matter 24(2012) 024206.\n17H. Kronm uller and S. S. P. Parkin (eds), Handbook of\nMagnetism and Advanced Materials 3(Chichester, Wiley,\n2007).\n18M. Kl aui, J. Phys.: Condens. Matter 20, 313001 (2008).\n19A. Thiaville and Y. Nakatani, in Spin Dynamics in Con-\n\fned magnetic Structures III , edited by B. Hillebrands and\nA. Thiaville, Topics in Applied Physics 101 (Springer,\nBerlin, 2006).\n20R. D. McMichael and M. J. Donahue, IEEE Trans. Magn.33, 4167 (1997).\n21Y. Nakatani, A. Thiaville and J. Miltat, J. Magn. Magn.\nMater. 290-291 750 (2005).\n22M. Kl aui, C. A. F. Vaz, J. A. C. Bland, L. J. Heyderman,\nF. Nolting, A. Paviovska, E. Bauer, S. Cheri\f, S. Heun,\nand A. Locatelli, Appl. Phys. Lett. 85, 5637 (2004).\n23S. Jamet, N. Rougemaille, J. C. Toussaint, and O.\nFruchart, arXiv preprint arXiv:1412.0679 (2014).\n24Y. Nozaki, K. Matsuyama, T. Ono, and H. Miyajima, Jpn.\nJ. Appl. Phys. Vol. 38(1999) 6282.\n25M. Kl aui, C. A. F. Vaz, J. A. C. Bland, T. L. Monchesky,\nJ. Unguris, E. Bauer, S. Cheri\f, S. Heun, A. Locatelli, L. J.\nHeyderman, and Z. Cui, Phys. Rev. B 68, 134426 (2003).\n26M. Kl aui, U. R udiger, C. A. F. Vaz, J. A. C. Bland, S.\nCheri\f, A. Locatelli, S. Heun, A. Pavlovska, E. Bauer, and\nL. J. Heyderman, Appl. Phys. Lett. 99, 08G308 (2006).\n27D. Backes, C. Schieback, M. Kl aui, F. Junginger, H. Ehrke,\nP. Nielaba, U. R udiger, L. J. Heyderman, C. S. Chen, T.\nKasama, R. E. Dunin-Borkowski, C. A. F. Vaz, and J. A.\nC. Bland, Appl. Phys. Lett. 91, 112502 (2007).\n28M. Kl aui, M. Laufenberg, L. Heyne, D. Backes, U. R udiger,\nC. A. F. Vaz, J. A. C. Bland, L. J. Heyderman, S. Cheri\f,\nA. Locatelli, T. O. Mentes, and L. Aballe, Appl. Phys.\nLett.88232507 (2006).\n29E.-M. Hempe, M. Kl aui, T. Kasama, D. Backes, F. Jungin-\nger, S. Krzyk, L. J. Heyderman, R. Dunin-Borkowski, and\nU. R udiger, phys. stat. sol.(a) 204, 3922 (2007).\n30C. A. F. Vaz, M. Kl aui, L. J. Heyderman, C. David, F.\nNolting, and J. A. C. Bland, Phys. Rev. B. 72, 224426\n(2005).\n31M.H. Park, Y.K. Hong, B.C. Choi, M. J. Donahue, H.Han,\nand S.H. Gee, Phys. Rev. B 73, 094424 (2006).\n32http://mumax.github.io\n33A. Vansteenkiste and B. Van de Wiele, J. Magn. Magn.\nMater. 323, 2585 (2011).\n34A. Vansteenkiste et al. , AIP Advances 4, 107133 (2014).\n35T. L. Gilbert. IEEE Trans. Magn. 40, 3443 (2004).\n36W. F. Brown, Jr., Micromagnetics . (New York, Wiley,\n1963).\n37T. Shinjo, T. Okuno, R. Hassdorf, K. Shigeto, and T. Ono,\nScience 289, 930 (2000).\n38A Wachowiak, J. Wiebe, M. Bode, O. Pietzsch, M. Mor-\ngenstern, and R. Wiesendanger, Science 298, 577 (2002).\n39P. Fischer, M.-Y. Im, S. Kasai, K. Yamada, T. Ono, and\nA. Thiaville, Phys. Rev. B 83, 212402 (2011).\n40O. Tchernyshyov and G.-W. Chern, Phys. Rev. Lett., 95,\n197204 (2005).6\n41C. Mouta\fs, S. Komineas, C. A. F. Vaz, J. A. C. Bland,\nT. Shima, T. Seki, and K. Takanashi, Phys. Rev. B 76,\n104426 (2007).\n42N. Vukadinovic and F. Boust, Phys. Rev. B 78, 184411(2008).\n43C. Zinoni, A. Vanhaverbeke, P. Eib, G. Salis, and R. Al-\nlenspach, Phys. Rev. Lett. 107, 207204 (2011)."    },    {        "title": "1502.00176v1.Bases_and_Structure_Constants_of_Generalized_Splines_with_Integer_Coefficients_on_Cycles.pdf",        "content": "arXiv:1502.00176v1  [math.RA]  31 Jan 2015BASES AND STRUCTURE CONSTANTS OF GENERALIZED\nSPLINES WITH INTEGER COEFFICIENTS ON CYCLES\nNEALY BOWDEN, SARAH HAGEN, MELANIE KING, AND STEPHANIE REIN DERS\nAbstract. Aninteger generalized spline is a set of vertex labels on an edge-\nlabeled graph that satisfy the condition that if two vertices are join ed by an edge,\nthe vertex labels are congruent modulo the edge label. Foundationa l work on these\nobjects comes from Gilbert, Polster, and Tymoczko, who generaliz e ideas from\ngeometry/topology (equivariant cohomology rings) and algebra (a lgebraic splines)\nto develop the notion of generalized splines . Gilbert, Polster, and Tymoczko prove\nthat the ring of splines on a graph can be decomposed in terms of splin es on\nits subgraphs (in particular, on trees and cycles), and then fully an alyze splines\non trees. Following Handschy-Melnick-Reinders and Rose, we analyz e splines on\ncycles, in our case integer generalized splines.\nThe primary goal of this paper is to establish two new bases for the m odule\nof integer generalized splines on cycles: the triangulation basis and t he King ba-\nsis. Unlike bases in previous work, we are able to characterize each b asis element\ncompletely in terms of the edge labels of the underlying cycle. As an ap plication\nwe explicitly construct the multiplication table for the ring of integer g eneralized\nsplines in terms of the King basis.\n1.Introduction\nAninteger generalized spline is a set of vertex labels on an edge-labeled graph that\nsatisfy the condition that if two vertices are joined by an edge, the vertex labels are\ncongruent modulo the edge label. (See Definition 2.1 for a precise sta tement.) Figure\n1 shows examples of splines on a three-cycle.\nTheterm“spline”comesfromthenameofthethinstripsofwooduse dbyengineersto\nmodel larger constructions like ships or cars. Mathematicians later adopted the term\nWe are extremely grateful to Julianna Tymoczko, Elizabeth Drellich, and Yue Cao for their\ninsight and contributions to this paper. We would also like to thank Rut h Haas and Joshua\nBowman for valuable discussions on these topics, and Michael DiPasq uale for his thorough review\nand comments. This work was supported by Smith College and the Nat ional Science Foundation\nthrough the Center for Women in Mathematics [DMS-1143716].\n12 BOWDEN, HAGEN, KING, AND REINDERS\n25\n3\n111\n25\n3\n0212\n25\n3\n0015\nFigure 1. The edge labels are t2,5,3uand the sets of vertex labels\nt1,1,1u,t0,2,12u, and t0,0,15ueach form a spline on the cycle.\ntorefertopiecewisepolynomialsonpolytopeswiththepropertytha tthepolynomials\non the faces agree at their shared edges up to a given degree of sm oothness. These\nmathematical splines are also used for object-modeling purposes, hence the use of\nthe name.\nBillera pioneered the algebraic study of splines, especially looking into q uestions\nregarding thedimension ofthe moduleof splines [2]. Many peoplecont inued Billera’s\nwork, including among others, Rose [12, 13] and Haas [7] who worked on identifying\ndimension and bases for the module of splines.\nSplinetheorydevelopedindependently intopologyandgeometry. Go resky, Kottwitz,\nand MacPherson [6], Payne [11], and Bahri, Franz, and Ray [1] constr ucted equivari-\nant cohomology rings using splines, although they did not use that na me.\nGilbert, Polster, and Tymoczko generalize the notion of splines that we use here to\nwhat they call generalized splines [4] . These generalized splines are built on the\ndual graph of the polytopes found in classical splines. The work of B illera and Rose\nshows that the two constructions (on polytopes or their duals) ar e equivalent in most\ncases, including the cases of classical interest [3].\nCycles turn out to be a particularly important family of graphs to stu dy. Indeed\nGilbert, Polster, and Tymoczko show that the ring of generalized sp lines on a graph\nGcan be decomposed in terms of splines on certain trees and cycles in G[4]. They\ncompletely describe splines on trees, while leaving open the investigat ion of splines\non cycles. Similarly, Rose showed that cycles play a key role in the relat ions defining\nmodules of splines [13].\nHandschy, Melnick, and Reinders begin analysis of integer generalize d splines on\ncycles [9]. They prove the existence of a certain flow-up basis (see D efinition 2.3),\nwhat we call the smallest-value basis, for splines on cycles, and thus prove that suchBASES AND STRUCTURE CONSTANTS OF GENERALIZED SPLINES 3\nspline modules are free. They define their basis for arbitrary cycles , but only have\nformulas for the leading nonzero elements.\nInthispaperwe introducetwo newbases forthemoduleofinteger g eneralized splines\non cycles: the triangulation basis and the King basis. Each of these b ases is fully\nexpressible in terms of the edge labels of the cycle, and each has its o wn strengths.\nThe triangulation basis, so called because it is constructed from tria ngulated cycles,\nis useful because it exists on arbitrary cycles (Theorem 4.2). The a dvantage of the\nKing basis lies in the fact that it is relatively simple to calculate, with the e ntries\nalmost constant (Definition 5.1). Although the King basis only exists o n cycles with\na pair of relatively prime adjacent edge labels, this restriction is not u ncommon in\napplications. Infactanevengreaterrestrictionthatalledgelabe lsberelativelyprime\nis commonly used [5, 10]. The results of our work naturally generalize t o principle\nideal domains, which include classical univariate splines and Pr¨ ufer d omains; see\nforthcoming work [8].\nAs an application we present the multiplication table of splines on cycles where the\nproducts of splines are expressed in terms of the King basis. Finding multiplica-\ntion tables of equivariant cohomology rings in terms of Schubert bas es is the central\nproblem of Schubert calculus. We view this work as a step in that geom etric direc-\ntion.\nThe rest of this paper is organized as follows. In Section 2 we summar ize the im-\nportant definitions and theorems that we use in our work. In Sectio n 3 we provide\na criterion for the existence of flow-up bases. Sections 4 and 5 are dedicated to\nproving the existence of the triangulation basis and King basis respe ctively. In the\nfinal section we give the multiplication table for the King basis and end w ith an open\nquestion.\n2.Preliminaries\n2.1.Results from Handschy, Melnick, and Reinders. Handschy, Melnick, and\nReinders proved a number of results about splines on cycles [9]. Many of their\npropositionsandtheorems play key rolesinour proofsregarding tr iangulationsplines\nand King splines. We also use their notation, which we describe in this se ction.\n2.1.1.Basic Definitions. The foundational combinatorial object we study is an edge-\nlabeled graph, defined here:4 BOWDEN, HAGEN, KING, AND REINDERS\nDefinition 2.1 (Edge-Labeled Graphs) .LetGbe a graph with kedges ordered\ne1,e2,...,e kandnvertices ordered v1,...,vn. Letℓibe a positive integer label on\nedgeeiand letL“ tℓ1,...,ℓkube the set of edge labels. Then pG,L qis an edge-\nlabeled graph.\nWiththisnotationforedge-labeledgraphswehavetheformaldefin itionofsplines:\nDefinition 2.2 (Splines).A spline on the edge-labeled graph pG,L qis a vertex-\nlabeling as follows: if two vertices are connected by an edge eithen the two vertex\nlabels are equivalent modulo ℓi. We denote a spline G“ pg1,...,gnqwheregiis the\nlabel on vertex vifor1ďiďn.\nIn this paper we assume the labels giPZ.\n2.1.2.Flow-Up Classes and the Smallest-Value Basis. Flow-up classes are a partic-\nularly nice class of splines on cycles. They arise geometrically ([5], [10], [1 4]) and are\nan analogue of upper triangular matrices.\nDefinition 2.3 (Flow-Up Classes) .Fix a cycle with edge labels pCn,Lqand fixk\nwith1ďkăn. A flow-up class GkonpCn,Lqis a spline with kleading zeros.\nWe say that a basis whose elements are flow-up classes is a flow-up basis . The\nsimplest flow-up class is the trivial spline; It exists on any edge-labele d cycle.\nProposition 2.4 (Trivial Splines [9, Prop 2.5]) .Fix a cycle with edge labels pCn,Lq.\nThe smallest flow-up class on pCn,LqisG0“ p1,...,1q. Moreover, any multiple of\nG0is also a spline. We call the multiples of G0trivial splines.\nThe following theorem establishes that flow-up classes exist on any e dge-labeled\ncycle.\nTheorem 2.5 (Flow-Up Classes on n-cycles [9, Thrm 4.3]) .Fix a cycle with edge\nlabels pCn,Lq. Letně3and1ďkăn. There exists a flow-up class GkonpCn,Lq.\nThe next definition introduces smallest flow-up classes.\nDefinition 2.6 (Smallest Flow-Up Class) .Fix a cycle with edge labels pCn,Lq. The\nsmallest flow-up class Gk“ p0,...,0,gk`1,...,gnqonpCn,Lqis the flow-up class whose\nnonzero entries are positive and if G1\nk“ p0,...,0,g1\nk`1,...,g1\nnqis another flow-up class\nwith positive entries then g1\niěgifor all entries. By convention we consider\nG0“ p1,...,1qthe smallest flow-up class G0.BASES AND STRUCTURE CONSTANTS OF GENERALIZED SPLINES 5\nThe following theorem gives an explicit formula for the smallest leading e lement of\nflow-up classes.\nTheorem 2.7 (Smallest Leading Element of Gk[9, Thrm 4.5]) .Fix a cycle with edge\nlabels pCn,Lq. Fixně3andksuch that 2ďkăn. LetGk´1“ p0,...,0,gk...,gnqbe a\nflow-up classon pCn,Lq. Theleadingelement gkis amultiple of lcmpℓk´1,gcdpℓk,...,ℓnqq\nand there is a flow-up class Gk´1withgk“lcmpℓk´1,gcdpℓk,...,ℓnqq.\nThe smallest flow-up classes exist and form a basis for the set of splin es given any\nedge-labeled cycle.\nTheorem 2.8 (Basisfor n-Cycles [9, Thrm4.7]) .Fix a cycle with edge labels pCn,Lq.\nThe smallest flow-up classes G0,G1,...,Gn´1exist on pCn,Lqand form a basis over\nthe integers for the Z-module of splines on pCn,Lq.\n2.2.Useful Computational Tool. For reasons related to finding an explicit basis\nfor splines on cycles, we want to find a formula for the value of the va riablexin the\nfollowing pair of congruences:\n#\nx”ymoda\nx”0 modb\nWe note the conditions for when such a solution exists and we give an e xplicit\nformulation for xin terms of y,a, andbprovided a solution does exist.\nProposition 2.9. Consider the system of congruences\n#\nx”ymoda\nx”0 modb.\nIf this system has a solution then one solution is given by the following formula:\n‚Ifa\ngcdpa,bq“1thenx“bis a solution to the system.\n‚Ifa\ngcdpa,bq‰1then\nx“yˆb\ngcdpa,bq˙ ˆb\ngcdpa,bq˙´1\nmodpa\ngcdpa,bqq\nis a solution to the system.6 BOWDEN, HAGEN, KING, AND REINDERS\nProof.The Chinese Remainder Theorem tells us that this system of congrue nces is\nsatisfied if and only if y”0 mod gcd pa,bq. In what follows we will assume that a\nsolution exists, and thus that y”0 mod gcd pa,bq.\nCase 1: Let’s deal first with the case wherea\ngcdpa,bq“1. This condition implies\nthat gcd pa,bq “aand sob“anfor some nPZ. Because y”0 mod gcd pa,bqby\nassumption and gcd pa,bq “awe have y”0 moda. In other words, y“amfor\nsomemPZ. Thenx“bsatisfies the system of congruences because bis congruent\nto zero modulo bandb“anis congruent to y“ammoduloa.\nCase 2:Now supposea\ngcdpa,bq‰1. We can rewrite the system of congruences as\n#\nx“y`as\nx“bt\nEquate both expressions.\nbt“y`as\nRecall that y”0 mod gcd pa,bq. This allows us to divide both sides by gcd pa,bqand\nget an integer as the result.ˆb\ngcdpa,bq˙\nt“y\ngcdpa,bq`ˆa\ngcdpa,bq˙\ns\nPutting this back into modular form we haveˆb\ngcdpa,bq˙\nt“y\ngcdpa,bqmodˆa\ngcdpa,bq˙\n.\nThe integers´\nb\ngcdpa,bq¯\nand´\na\ngcdpa,bq¯\nare relatively prime so we can take the inverse\nof the first modulo the second.\nt”y\ngcdpa,bqˆb\ngcdpa,bq˙´1\nmodˆa\ngcdpa,bq˙\n.\nPlug this expression for tinto the equation x“bt:\nx“yˆb\ngcdpa,bq˙ ˆb\ngcdpa,bq˙´1\nmodpa\ngcdpa,bqq.\nThis value is a solution to the original system of congruences. /square\nNotice that this second case simplifies enormously if gcdpa,bq “1. In this situation\nxreduces to:\nx“ybrb´1smodaBASES AND STRUCTURE CONSTANTS OF GENERALIZED SPLINES 7\n3.Basis Condition\nLetpG,L qbe an arbitrary graph on nvertices with an arbitrary edge-labeling. Con-\nsider a set of flow-up classes G0...Gn´1onpG,L q. In this section we give a necessary\nand sufficient condition for this set to form a basis for the module of t he splines on\npG,L q. Any set G0,...,Gn´1that meets this basis condition is called a flow-up basis .\nSuch a basis is useful because linear independence is trivially verified.\nLetG0...Gn´1be a set of flow-up classes and for each idenote\nGi“ p0,...,0,gpiq\ni`1,...,gpiq\nnq.\nThe subscript of each gpiqindicates the entry-position of gpiqin the spline Gi. The\nsuperscript piqis to keep track of the fact that we are working with the flow-up clas s\nGi. In much of this paper and in previous work the superscript is suppr essed when\nthe flow-up class in question is obvious.\nTheorem 3.1 (Basis Condition) .The following are equivalent:\n‚The set tG0,...,Gn´1uforms a flow-up basis.\n‚For each flow-up spline Ai“ p0,...,0,ai`1,...,a nqthe entry ai`1ofAiis an\ninteger multiple of the entry gpiq\ni`1ofGi.\nProof.Suppose that G0,...,Gn´1forms a flow-up basis for the module of splines on\na graph pG,L q. Suppose that Ai“ p0,...,0,ai`1,...,a nqis a spline on pG,L qwith\nexactlyileading zeros. We will show that ai`1“cgpiq\ni`1for some cPZ.\nSinceG0,...,Gn´1form a basis, we can write Aias a linear combination of the\nsplinesG0,...,Gn´1. The fact that Aihasileading zeros implies that the coeffi-\ncients of G0,...,Gi´1must be 0. Thus we have Ai“ciGi`...`cn´1Gn´1for some\nci,...,c n´1PZ. Consider the pi`1qthentry of the splines on the right-hand side of\nthis equation. Note that Giis the only element of Gi,...,Gn´1with a nonzero entry\nin this position. Considering the pi`1qthentry on each side of the equation, we have\nai`1“cigpiq\ni`1`ci`10`...`cn´10“cigpiq\ni`1.\nNow we prove the converse. Let A“ pa1,...,a nqbe an arbitrary spline on pG,L q.\nWe prove by induction that\nA“A1\nj`j´1ÿ\nk“0ckGk8 BOWDEN, HAGEN, KING, AND REINDERS\nfor all 1 ďjďnwhereA1\njis a spline with (at least) jleading zeros.\nFor our base case, note that by hypothesis we have\nA“¨\n˚˚˚˝an´c0gp0q\nn\n...\na2´c0gp0q\n2\n0˛\n‹‹‹‚`c0G0\nsincea1“c0gp0q\n1. Letting A1\n1“ p0,a2´c0gp0q\n2,...,a n´c0gp0q\nnqgivesA“A1\n1`ř0\nk“0ckGk.\nThus our claim holds for j“1.\nSuppose as our induction hypothesis that we have A“A1\ni`ři´1\nk“0ckGkfor some\n1ďiďn´1. We can write this as\nA“¨\n˚˚˚˚˚˚˚˝a1\nn...\na1\ni`1\n0\n...\n0˛\n‹‹‹‹‹‹‹‚`i´1ÿ\nk“0ckGk.\nBy hypothesis we have that a1\ni`1“cigpiq\ni`1for some ciPZ. So we can write\nA“¨\n˚˚˚˚˚˚˚˚˚˚˝a1\nn´cigpiq\nn\n...\na1\ni`2´cigpiq\ni`2\n0\n0\n...\n0˛\n‹‹‹‹‹‹‹‹‹‹‚`iÿ\nk“0ckGk.\nLettingA1\ni`1“ p0,...,0,0,a1\ni`2´cigpiq\ni`2,...,a1\nn´cigpiq\nnqgivesusA“A1\ni`1`ři\nk“0ckGk.\nBy induction we have A“A1\nj`řj´1\nk“0ckGkfor all 1 ďjďn. In particular we have\nA“A1\nn`řn´1\nk“0ckGk. ButA1\nnis a spline with nleading zeros. So A1\nn“ p0,...,0q.\nThusA“řn´1\nk“0ckGk. We conclude that every spline can be written as a linear\ncombination of G0,...,Gn´1as desired. /squareBASES AND STRUCTURE CONSTANTS OF GENERALIZED SPLINES 9\nOne important observation is that the basis condition is only a conditio n on the first\nnonzero entry of each spline in a set of flow-up classes G0,...,Gn´1. This gives us\nthe following useful corollary:\nCorollary 3.2. Suppose the set of flow-up classes tG0,...,Gn´1uforms a basis for\nthe module of splines. Suppose tG1\n0,...,G1\nn´1uis a set of flow-up classes for which\nfor eachithe first nonzero entry of G1\niequals the first nonzero entry of Gi. Then the\nsettG1\n0,...,G1\nn´1ualso forms a basis for the module of splines.\n4.The Triangulation Splines\nTriangulationsplinesformanotherbasisofflow-upclassesforcycle s. Theyaresimilar\ntoHandschy, Melnick, andReinders’ smallest-valueflow-upclasses inthattheleading\nnonzero elements of both are the same. However we give a formula f or every entry\nof the triangulation splines, unlike the smallest-value flow-up classes .\nDefinition 4.1 (Triangulation Splines) .Fix an edge-labeled cycle pCn,Lq. For\n1ďkďn´1the vector Hk“ p0,...,0,hk`1,...,hnqhas entries as follows:\n‚hk`1“lcmpℓk,gcdpℓk`1,...,ℓnqq\n‚Fork`1ăiďnifℓi´1\ngcdpℓi´1,...,ℓnq“1thenhi“gcdpℓi,...,ℓnq.\n‚Fork`1ăiďnifℓi´1\ngcdpℓi´1,...,ℓnq‰1then\nhi“hi´1ˆgcdpℓi,...,ℓnq\ngcdpℓi´1,...,ℓnq˙ ˆgcdpℓi,...,ℓnq\ngcdpℓi´1,...,ℓnq˙´1\nmodℓi´1\ngcdpℓi´1,...,ℓnq\nThe next theorem establishes that triangulation splines exist on any edge-labeled\ncycle.\nTheorem 4.2 (ExistenceofTriangulationSplines) .Fix an edge-labeledcycle pCn,Lq.\nFor1ďkďn´1the vector Hkis a spline on pCn,Lq.\nProof.Start with an edge-labeled cycle pCn,Lq. For 3 ďkďn´1 add an edge\nbetween vertices v1andvkas shown in Figure 2. Label the edge between v1andvk\nwith gcd pℓk,...,ℓnq. We will show the vector Hksatisfies all of the edge conditions\nrepresented by this graph, which implies it satisfies the cycle’s edge c onditions in\nparticular.10 BOWDEN, HAGEN, KING, AND REINDERS\nℓ1ℓ2ℓ3ℓn´1\nℓn\ngcdpℓn´1,ℓnq\ngcdpℓ4,...,ℓnq\ngcdpℓ3,...,ℓnq\n(a)Add edgesℓ1ℓ2ℓ3ℓn´1\nℓn\ngcdpℓn´1,ℓnq\ngcdpℓ4,...,ℓnq\ngcdpℓ3,...,ℓnq\n0h2h3\n(b)Base case\nFigure 2. Triangulated Cycle\nLabel vertices v1,...,vkzero. Label vertex vk`1with\nhk`1“lcmpℓk,gcdpℓk`1,...,ℓnqq.\nThe integer hk`1satisfies the edge conditions on the downward edges (edges with\nlower-indexed vertices) at vertex vk`1by construction:\n#\nhk`1”0 modℓk\nhk`1”0 mod gcd pℓk`1,...,ℓnq\nThis is our base case, and we will label vertices from hk`2tohn´1inductively.\nOur induction hypothesis is that hk`1,...,hifork`1ďiďn´1 satisfy the edge\nconditions for downward edges. Consider the system of congruen ces at vertex vi`1\nrepresented by the edges labeled ℓiand gcd pℓi`1,...,ℓnq:\n#\nhi`1”himodℓi\nhi`1”0 mod gcd pℓi`1,...,ℓnq\nBy the Chinese Remainder Theorem a solution hi`1exists if and only if hi”0 mod\ngcdpℓi,gcdpℓi`1,...,ℓnqq. In other words a solution exists if and only if hi”0 mod\ngcdpℓi,...,ℓnq. By our induction hypothesis hisatisfies the downward edge conditions\nat vertex viso in particular hi”0 mod gcd pℓi,...,ℓnq. Thus a solution hi`1exists.BASES AND STRUCTURE CONSTANTS OF GENERALIZED SPLINES 11\nThis means\nhi`1“$\n&\n%hi´\ngcdpℓi`1,...,ℓnq\ngcdpℓi,...,ℓnq¯ ´\ngcdpℓi`1,...,ℓnq\ngcdpℓi,...,ℓnq¯´1\nmodℓi\ngcdpℓi`1,...,ℓnqifℓi\ngcdpℓi,...,ℓnq‰1\ngcdpℓi`1,...,ℓnq ifℓi\ngcdpℓi,...,ℓnq“1\nis a solution by Proposition 2.9.\nIn conclusion we can label each vertex vifork`1ăiďn´1 with\nhi“$\n&\n%hi´1´\ngcdpℓi,...,ℓnq\ngcdpℓi´1,...,ℓnq¯ ´\ngcdpℓi,...,ℓnq\ngcdpℓi´1,...,ℓnq¯´1\nmodℓi´1\ngcdpℓi´1,...,ℓnqifℓi´1\ngcdpℓi´1,...,ℓnq‰1\ngcdpℓi,...,ℓnq ifℓi´1\ngcdpℓi´1,...,ℓnq“1\nandhiwill satisfy the edge conditions represented by the edges labeled ℓi´1and\ngcdpℓi,...,ℓnq.\nLastly for an integer hnto satisfy the edge conditions at vertex vnit must satisfy\nthe following system of congruences:\n#\nhn”hn´1modℓn´1\nhn”0 modℓn\nThe Chinese Remainder Theorem tells us that a solution hnexists to this system if\nand only if hn´1”0 mod gcd pℓn´1,ℓnq. We showed by induction that our choice of\nhn´1satisfies the edge conditions of the downward edges at the pn´1q-th vertex. In\nparticular this means hn´1”0 mod gcd pℓn´1,ℓnqbecause this is the edge condition\nrepresented by the edge labeled gcd pℓn´1,ℓnq. Therefore\nhn“$\n&\n%hn´1´\nℓn\ngcdpℓn´1,ℓnq¯ ´\nℓn\ngcdpℓn´1,ℓnq¯´1\nmodℓn´1\ngcdpℓn´1,ℓnqifℓn´1\ngcdpℓn´1,ℓnq‰1\nℓn ifℓn´1\ngcdpℓn´1,ℓnq“1\nsatisfies the vertex vnedge conditions by Proposition 2.9. Choose this integer to\nlabel the n-th vertex.\nAll of the congruences represented by the graph are accounted for so the vector\nHk“ p0,...,0,hk`1,...,hnqis a spline on the graph. In particular Hkis a spline on\nthe cycle pCn,Lqas desired.\n/square12 BOWDEN, HAGEN, KING, AND REINDERS\nThe Corollary to the Basis Condition Theorem allows us to succinctly co nclude that\nthe set of triangulation splines H0,...,Hn´1forms a basis for the set of splines on an\nedge-labeled cycle.\nTheorem 4.3. Fix an edge-labeled cycle pCn,Lq. The set of triangulation splines\nH0,...,Hn´1form a basis for the set of splines on pCn,Lq.\nProof.Thesetofsmallestflow-upclasses G0,...,Gn´1formabasisforthesetofsplines\nonpCn,Lqby Theorem 2.8. The leading entry of Hkequals the leading entry of Gk\nby construction for 0 ďkďn´1. Thus the set of triangulation splines H0,...,Hk\nforms a basis for the set of splines on pCn,Lqby Corollary 3.2. /square\nAs an example, we calculate the triangulation basis for the 4-cycle wit h edge labels\nt2,6,10,15u.\n2615\n10\nThe first basis element H0is, as always, the trivial spline p1,1,1,1q. The nonzero\nentries of the second basis element H1are calculated as follows:\nhp1q\n2“lcmp2,gcdp6,10,15qq “2\nhp1q\n3“2ˆgcdp15,10q\ngcdp6,15,10q˙ ˆgcdp15,10q\ngcdp6,15,10q˙´1\nmod6\ngcdp6,15,10q“2¨5¨ p5q´1\nmod 6 “50\nhp1q\n4“50ˆgcdp10q\ngcdp15,10q˙ ˆgcdp10q\ngcdp15,10q˙´1\nmod15\ngcdp15,10q“50¨2¨ p2q´1\nmod 3 “200\nThe nonzero entries of the third basis element H2are calculated as follows:\nhp2q\n3“lcmp6,gcdp10,15qq “30\nhp2q\n4“30ˆgcdp10q\ngcdp15,10q˙ ˆgcdp10q\ngcdp15,10q˙´1\nmod15\ngcdp15,10q“50¨2¨ p2q´1\nmod 3 “120\nThe only nonzero element of the final basis element H3ishp3q\n4“lcmp15,10q “\n30. Thus we have the following triangulation basis for the 4-cycle with edge la-\nbels t2,6,10,15u:H0“ p1,1,1,1q,H1“ p0,2,15,200q,H3“ p0,0,30,120q, and\nH4“ p0,0,0,30q.BASES AND STRUCTURE CONSTANTS OF GENERALIZED SPLINES 13\n5.The King Splines\nIn this section we define King splines on n-cycles and prove that they form a basis\nfor the set of splines.\nDefinition 5.1 (King splines) .Fix a cycle with edge-labels pCn,Lqand assume ℓn´1\nandℓnrelatively prime. The King splines on pCn,Lqare the vectors\nK0“¨\n˚˚˚˚˚˚˝1\n1\n...\n1\n1\n1˛\n‹‹‹‹‹‹‚,K1“¨\n˚˚˚˚˚˚˝k1\nℓ1...\nℓ1\nℓ1\n0˛\n‹‹‹‹‹‹‚,K2“¨\n˚˚˚˚˚˚˝k2\nℓ2...\nℓ2\n0\n0˛\n‹‹‹‹‹‹‚,...,K n´1“¨\n˚˚˚˚˚˚˝kn´1\n0\n...\n0\n0\n0˛\n‹‹‹‹‹‹‚\nwhere\nki“#\nℓi¨ℓnrℓ´1\nnsmodℓn´1for1ďiďn´2\nℓn´1ℓn fori“n´1.\nBy convention, we call K0the trivial King spline.\nAs our terminology suggests, the King splines are in fact splines.\nTheorem 5.2. Letně3. Fix a cycle with edge-labels pCn,Lqwithℓn´1andℓn\nrelatively prime. The King splines K0,...,K n´1are splines on pCn,Lq.\nProof.First we note that the trivial King spline K0is the same as the trivial spline\nG0which is indeed a spline on pCn,Lqby Proposition 2.4.\nConsideranarbitraryKingspline Ki“ p0,...,0,ℓi,...,ℓ i,kn´1qwhere1 ďiďn´2.\nIt has zero for its first ientries,ℓifor entries i`1 ton´1, andkn´1for its last\nentry. We want to show that Kiis a spline on pCn,Lq. Note that zero is congruent\nto itself modulo any integer, so in particular the following congruence s are satisfied:\n!\n0”0 modℓjfor 1 ďjďi´1 (1)\nAlso, since the integer ℓiis congruent to zero modulo ℓiwe have14 BOWDEN, HAGEN, KING, AND REINDERS\nℓi”0 modℓi (2)\nThe integer ℓiis congruent to itself modulo any integer, so in particular the following\ncongruences are satisfied:\n!\nℓi”ℓimodℓjfori`1ďjďn´2 (3)\nFinally we know ki“ℓi¨ℓnrℓ´1\nnsmodℓn´1satisfies the following two congruences\n#\nki”ℓimodℓn´1\nki”0 modℓn(4)\nby Proposition 2.9. Collect the congruences in 1, 2, 3, and 4 into a sing le system of\ncongruences. This system represents the edge conditions on pCn,Lq. The vector Ki\nsatisfies all of these congruences so Kiis a spline on pCn,Lq.\nNow consider the vector Kn´1“ p0,...,0,kn´1q. Zero is congruent to itself modulo\nany integer, so the following system of congruences is satisfied:\n!\n0”0 modℓjfor 1 ďjďn´2. (5)\nSincekn´1“ℓn´1ℓnwe know\n#\nkn´1”0 modℓn´1\nkn´1”0 modℓn(6)\nCollect the congruences in 5 and 6 into a single system. This system re presents the\nedge conditions on pCn,Lq. The vector Kn´1satisfies all of these congruences so\nKn´1is a spline on pCn,Lq.\nThus we have that Kiis a spline for all 0 ďiďn´1 as desired.\n/square\nNow that we know the King splines are splines, we confirm that they fo rm a ba-\nsis.\nTheorem 5.3. Fix a cycle with edge labels pCn,Lqwithℓn´1andℓnrelatively prime.\nThe set of King splines K0,...,K n´1forms a basis for the set of splines on pCn,Lq.BASES AND STRUCTURE CONSTANTS OF GENERALIZED SPLINES 15\nProof.The set of smallest flow-up classes G0,...,Gn´1form a basis for the set of\nsplines on pCn,Lqby Theorem 2.8. We constructed the King splines so that the\nleading entry Kiequals the leading entry of Gifor 0 ďiďn´1. Thus the set of\nKing splines K0,...,K n´1forms a basis for the set of splines on pCn,Lqby Corollary\n3.2. /square\n6.Multiplication Tables\nThe fact that we have simple explicit formulas for the entries of the K ing basis is\na powerful computational tool. In this section we use the King basis to write the\nproduct of any pair of basis elements as a linear combination of basis e lements. This\nkind of calculation is important in geometry and topology, which use sp lines over\npolynomial rings to describe cohomology rings.\n6.1.Multiplication Tables for n-Cycles on the King Basis. When multiplying\nsplines the operation is performed component-wise. Consider the K ing basis on a\ngiven n-cycle.\nSince the entries in the trivial spline K0are all ones, multiplying any spline Ki(with\n0ďiďn´1) byK0simply yields Ki. The following theorem gives us the product\nof any pair of non-trivial King splines.\nTheorem 6.1. For arbitrary Ki,Kjwithi,j‰0andiďj, we have the product\nKiKj“liKj`kjpki´liq\nkn´1Kn´1.\nProof.We give a proof by construction.\nConsider arbitrary basis elements KiandKjwithi,j‰0 andiďj. Their product\nKiKjhas zeros up to the jthentry. The entries numbered j`1 through n´1 are\nℓi¨ℓj. The last entry is ki¨kj.\nNote that ℓi¨Kjhas zeros for the first jentries,ℓi¨ℓjfrom entries j`1 ton´1,\nandℓi¨kjfor thenthentry. This is almost exactly the product KiKj. However we\nwant this last entry to be ki¨kj. Addingkjpki´liq\nkn´1Kn´1gives the desired result.\nThus for KiKjwithi,j‰0 andiďjwe have16 BOWDEN, HAGEN, KING, AND REINDERS\nKiKj“ℓiKj`kikj´likj\nkn´1Kn´1“ℓiKj`kjpki´liq\nkn´1Kn´1\nSince we are working in the integers, our last step is to prove that th e coefficient\nkjpki´ℓiq\nkn´1\nis indeed an integer. We know ki”ℓimodℓn´1because Kiis a spline. Say\nki´ℓi“pℓn´1for some pPZ. Similarly, we know kj”0 modℓnbecauseKjis\na spline. Say kj“qℓnfor some qPZ. By definition we have kn´1“ℓn´1ℓn.\nPlugging these values into the expressionkikj´likj\nkn´1yields the following:\nkjpki´ℓiq\nkn´1“pqℓnqppℓn´1q\nℓn´1ℓn“pq\nThuskjpki´ℓiq\nkn´1is always an integer.\n/square\nNote that the product KiKn´1for anyiďn´1 simplifies significantly.\nCorollary 6.2. Choose any i‰0. ThenKiKn´1“kiKn´1.\nProof.We apply the formula for the product KiKjto the particular case where\nj“n´1 and simplify:\nKiKn´1“ℓiKn´1`kn´1pki´ℓiq\nkn´1Kn´1“kiKn´1\n/square\nFor example consider the 5-cycle with edge labels t3,4,8,2,5u. The King basis on a\n5-cycle with these labels looks like the following:\nK0\n5348\n2\n1111\n1K1\n5348\n2\n033\n3\n15K2\n5348\n2\n004\n4\n20K3\n5348\n2\n000\n8\n40K4\n5348\n2\n000\n0\n10BASES AND STRUCTURE CONSTANTS OF GENERALIZED SPLINES 17\nLet’s multiply the elements K1andK3. We obtain\nK1K3“ K1\n5348\n2\n033\n3\n15ˆ K3\n5348\n2\n000\n8\n40“\n5348\n2\n000\n24\n600\nBy the formula given above\nK1K3“3K3`40p15´3q\n10K4“3K3`48K4.\nPictorially this solution is shown below.\n3K3`48K4“3K3\n5348\n2\n000\n8\n40`48 K4\n5348\n2\n000\n0\n10“\n5348\n2\n000\n24\n600\nRemark 6.3. The same argument can be used to give the multiplication tabl e for\narbitrarily labeled 3-cycles using the triangulation basi s (Def 4.1, Thrm 4.3). Given\nthe basis elements H0,H1,andH2we have the following table\nH0“¨\n˝1\n1\n1˛\n‚,H1“¨\n˝hp1q\n3\nhp1q\n2\n0˛\n‚,H2“¨\n˝hp2q\n3\n0\n0˛\n‚\nH0H1 H2\nH0H0H1 H2\nH1H1hp1q\n2H1`ΦH2hp1q\n3H2\nH2H2hp1q\n3H2hp2q\n3H2\nwhereΦ“hp1q\n3php1q\n3´hp1q\n2q\nhp2q\n3.\nUnlike with the King basis, we do not have nice formulas for entries of t he triangu-\nlation basis. This leads to the following open question.18 BOWDEN, HAGEN, KING, AND REINDERS\nQuestion 6.4. Is there a positive or combinatorial formula for the multipl ication\ntable of general n-cycles (i.e.not alternating sums from successively correcting each\nspline entry)?\nReferences\n[1] A. Bahri, M. Franz, and N. Ray, The equivariant cohomology ring of weighted projective spa ce,\nMath. Proc. Cambridge Philos. Soc. 146(2009), no. 2, 395-405. MR 2475973\n[2] L. Billera, Homology of smooth splines: generic triangulations and a co njecture of Strang , Trans.\nAmer. Math. Soc. 310(1998), no. 1, 325340. MR 965757\n[3] L. Billera and L. Rose, A dimension series for multivariate splines , Discrete Comput. Geom. 6\n(1991), no. 2, 107-128. MR 1083627\n[4] S.Gilbert, S.Polster,andJ.Tymoczko, Generalized splines on arbitrary graphs , arXiv:1306.0801\n(2013)\n[5] R. Goldin and S. Tolman, Towards generalizing Schubert calculus in the symplectic c ategory. J.\nSymplectic Geom. 7(2009), no. 4, 449-473. MR 2552001\n[6] M. Goresky, R. Kottwitz, and R. MacPherson, Homology of affine Springer fibers in the unram-\nified case . Duke Math. J. 121(2004), no. 3, 509-561. MR 2040285\n[7] R. Haas, Module and vector space bases for spline spaces , J. Approx. Theory 65(1991), no. 1,\n73-89 MR 1098832\n[8] Hagen, S., Tymoczko, J.: A constructive algorithm to find a basis for splines over prin ciple\nrings and Pr¨ ufer domains. In process.\n[9] M. Handschy, J. Melnick, S. Reinders, Integer Generalized Splines on Cycles. arXiv:1409.1481\n(2014)\n[10] A. Knutson and T. Tao, Puzzles and (equivariant) cohomology of Grassmannians . Duke Math.\nJ. 119 (2003), no. 2, 221-260 . MR 1997946\n[11] S. Payne, Equivariant Chow cohomology of toric varieties , Math. Res. Lett. 13(2006), no. 1,\n29-41. MR 2199564\n[12] L. Rose, Combinatorial and topological invariants of modules of pie cewise polynomials . Adv.\nMath.116(1995), no. 1, 3445. MR 1361478\n[13] L. Rose, Graphs, syzygies and multivariate splines , Discrete Comput. Geom, 32(2004), no. 4,\n623637\n[14] J. Tymoczko, An Introduction to Equivariant Cohomology and Homology, Fo llowing Goresky,\nKottwitz, and MacPherson , arXiv:math/0503369 (2005)"    },    {        "title": "1502.00268v2.Nonlocal_Damping_of_Helimagnets_in_One_Dimensional_Interacting_Electron_Systems.pdf",        "content": "Nonlocal damping of helimagnets in one-dimensional interacting electron systems\nKjetil M. D. Hals, Karsten Flensberg and Mark S. Rudner\nNiels Bohr International Academy and the Center for Quantum Devices,\nNiels Bohr Institute, University of Copenhagen, 2100 Copenhagen, Denmark\nWe investigate the magnetization relaxation of a one-dimensional helimagnetic system coupled\nto interacting itinerant electrons. The relaxation is assumed to result from the emission of plas-\nmons, the elementary excitations of the one-dimensional interacting electron system, caused by slow\nchanges of the magnetization pro\fle. This dissipation mechanism leads to a highly nonlocal form\nof magnetization damping that is strongly dependent on the electron-electron interaction. Forward\nscattering processes lead to a spatially constant damping kernel, while backscattering processes pro-\nduce a spatially oscillating contribution. Due to the nonlocal damping, the thermal \ructuations\nbecome spatially correlated over the entire system. We estimate the characteristic magnetization\nrelaxation times for magnetic quantum wires and nuclear helimagnets.\nI. INTRODUCTION\nRecently, intense interest has developed in the helical\nmagnetic ordering of one-dimensional (1D) systems of lo-\ncal moments coupled to itinerant electrons. Such systems\nexhibit a variety of intriguing many-body phenomena,\nsuch as spin-Peierls instabilities1,2and induced topologi-\ncal superconductivity,3{5which result from the details of\nmagnetism, electronic structure, and electron-electron in-\nteractions. These phenomena may be relevant for a wide\nvariety of physical systems, ranging from magnetic atoms\non superconducting3,6{10and normal metal11substrates,\nto single walled carbon nanotubes12and semiconductor-\nbased quantum wires.5,13,14\nWhile much of the work in this area so far has focused\non static and thermodynamic properties of the 1D heli-\nmagnets, a richer understanding may be gained by devel-\noping and employing new dynamical probes for assessing\nthe behaviors of these systems. For example, an inter-\nesting self-tuning e\u000bect was proposed for systems dom-\ninated by a Ruderman-Kittel-Kasuya-Yoshida (RKKY)-\ntype interaction:12{15the local moments are predicted to\norder into a spiral arrangement which, through coherent\nbackscattering, gaps out one spin channel of the itiner-\nant electron system for any value of the electron density.\nThis remarkable phenomenon was even suggested as pro-\nviding a route towards realizing topologically protected\nMajorana bound states in quantum wires.5,8,9However,\nbecause direct probes of magnetization are unavailable\nfor many systems, it can be challenging to positively iden-\ntify this intriguing magnetic state (necessarily via indi-\nrect means).14With further theoretical understanding of\ndynamical responses, such as typical damping or relax-\nation times, additional tests (e.g., density quenches which\nchange the preferred ordering wave vector) could be used\nto clarify the natures of the underlying states.\nMore generally, magnetization relaxation processes de-\ntermine the magnetic response to external perturbations\nas well as to spontaneous thermal \ructuations. Further-\nmore, the nature of the magnetic response is crucially\nimportant for noise and magnetization dynamics in mag-\nnetoelectronic devices.16,17A better understanding of the\nz\nx\nyFIG. 1: (Color online). Helimagnet formed in a 1D conductor.\nspin dynamics in 1D helimagnets may pave the way for\nexploring phenomena such as current-driven magnetiza-\ntion dynamics, with potential practical applications be-\nyond those envisaged so far. Thus, the investigation of\nmicroscopic damping mechanisms is essential for develop-\ning a thorough fundamental and practical understanding\nof these exciting new magnetic systems.\nGiven the motivations above, in this work we investi-\ngate the relaxation of 1D helimagnets via the emission of\ncollective excitations into the interacting itinerant elec-\ntron system. Note that the 1D nature of the itinerant sys-\ntem is important { our theory is meant to describe quasi-\n1D systems with a single transverse mode at the Fermi\nenergy (e.g., semiconductor quantum wires14). The el-\nementary excitations of these 1D electronic systems are\nplasmons, which describe density waves. Interestingly,\nprevious theoretical works have predicted that electron-\nelectron interactions in such 1D systems may play im-\nportant roles both in establishing ordering12,13and in\nthe relaxation dynamics of weakly-coupled (non-ordered)\nnuclear spins.18,19In this work, we use a bosonization ap-\nproach to study the non-perturbative e\u000bects of electron-\nelectron interactions on the damping of ordered spins. We\n\fnd that interactions have a profound e\u000bect on damp-\ning, leading to an enhancement of the damping by sev-\neral orders of magnitude. The damping has a highly\nnon-local character. Consequently, the thermal \ructu-\nations become spatially correlated over the entire sam-\nple. We estimate the characteristic magnetization re-arXiv:1502.00268v2  [cond-mat.mes-hall]  9 Oct 20152\nlaxation times due to this mechanism for two classes of\nsystems: (Ga,Mn)As quantum wires and nuclear heli-\nmagnets formed in GaAs quantum wires.\nII. THEORY AND MODEL\nOur approach is based on the theoretical framework\ndeveloped for magnetization damping in metallic fer-\nromagnets.20,21A key ingredient of the model is that\nthe dynamics of the low-lying collective spin excita-\ntions are parametrized by a classical magnetization order-\nparameter \feld whose magnitude is assumed to be con-\nstant in time and homogeneous in space, while its local\norientation is allowed to \ructuate. In this case, the evolu-\ntion of the spin system can be described by the Landau-\nLifshitz-Gilbert (LLG) phenomenology:16,17,22\n_m(z;t) =\u0000\rm(z;t)\u0002[He\u000b(z;t) +hT(z;t)] +\nm(z;t)\u0002Z\ndz0\u000b(z;z0)_m(z0;t): (1)\nHere, the unit vector m(z;t) parametrizes the local spin-\norder and is oriented parallel to the magnetization vec-\ntorM(z;t) =Msm(z;t),\r=g\u0016B=\u0016his the gyromag-\nnetic ratio in terms of the g-factor of local spins and\nthe Bohr magneton \u0016B, and He\u000b=\u0000\u000eF=\u000eMis the ef-\nfective \feld found by varying the magnetic free energy\nfunctionalF[M] with respect to the magnetization. The\nquantity hT(z;t) in the \frst line is a stochastic magnetic\n\feld induced by the thermal \ructuations (to be discussed\nfurther below). We assume that the free energy func-\ntional stabilizes an equilibrium helimagnetic texture of\nthe form5,8,11{13\nm0(z) = [cos(qz);sin(qz);0]; (2)\nwhereqdepends on the ordering mechanism. Through-\nout this work, we use coordinate axes with the z-axis\noriented along the 1D conductor (see Fig. 1).\nMagnetization relaxation is described by the second-\nrank Gilbert damping tensor \u000bij(z;z0) in Eq. (1). We\nconsider magnetization relaxation via excitations of the\nitinerant electron system. In this case, the Gilbert damp-\ning tensor is given by20,21(see Appendix A for a deriva-\ntion)\n\u000bij(z;z0) =\u00004\rh2\n0\n\u0016h2Mslim\n!!0=m[\u001fij(z;z0;!)]\n!; (3)\nwhere\u001fij(z;z0;!) =R1\n\u00001dt\u001fij(z;z0;t) exp(i!t) is the\nFourier transform of the spin susceptibility of the itiner-\nant electrons, \u001fij(z;z0;t) =\u0000(i=\u0016h)\u0012(t)[^si(z;t);^sj(z0;0)].\nHere, ^s(z;t) = (\u0016h=2) y(z;t)\u001b (z;t) is the spin-density\noperator for itinerant electrons, taken in the interaction\npicture with respect to Hamiltonian (4) below, with the\nstatic magnetization (2). Above, \u001bis the vector of Pauli\nmatrices and  (z) = [ \"(z); #(z)] is the spinor-valued\nfermionic \feld operator.We model the itinerant electrons via the Hamiltonian:\nH=Z\ndz y(z)\u0014^p2\nz\n2m+h0m(z;t)\u0001\u001b\u0015\n (z) + (4)\n1\n2ZZ\ndzdz0 y\n\u001b(z) y\n\u001b0(z0)Vee(z\u0000z0) \u001b0(z0) \u001b(z);\nwhere ^pzis the momentum operator, Veeis the electron-\nelectron interaction potential, and h0is the magnetic cou-\npling. Summation over repeated indices is implied.\nIn the calculation below, we aim to evaluate the Gilbert\ndamping tensor in Eq. (3), using the spin susceptibility\nfor the electronic system described by Eq. (4), with a\n\fxed chemical potential. Linearizing around the Fermi\npoints, we will develop a Luttinger liquid type description\nof the nearly helical system, allowing interactions to be\ntaken into account non-perturbatively.\nIII. RESULTS\nWe now explicitly calculate the Gilbert damping ten-\nsor in Eq. (3). To facilitate calculation of the spin\nsusceptibility, we transform to a non-uniformly rotated\nframe via the unitary transformation  u=U(z) , with\nU(z) =eiqz\u001bz=2. This transformation \\untwists\" the he-\nlix, rendering the free electron part of the transformed\nHamiltonian Hu=UHUytranslationally invariant,\nH(0)\nu=Z\ndz y\nu(z)\u0014^p2\nz\n2m\u0000\u0016hq\n2m\u001bz^pz+h0\u001bx\u0015\n u(z);(5)\nwhile the interaction term is una\u000bected. In this represen-\ntation, the spin susceptibility and the Gilbert damping\ntensor transform to \u001fu(z;z0;t) =R(z)\u001f(z;z0;t)RT(z0)\nand\u000bu(z;z0;t) =R(z)\u000b(z;z0;t)RT(z0), whereR(z) is\nthe SO(3) matrix associated with U(z). The energy dis-\npersion ofH(0)\nuis shown in Fig. 2; its eigenfunctions are\n n;k(z) =\u0011n;k\n n;k(z), wheren2f1;2gis the band in-\ndex,\u0011n;kis the eigenspinor, and  n;k(z) = exp(ikz)=p\nL,\nfor a system of length L.\nIn this work, we set the chemical potential in the gap\nthat separates the bands near k= 0, such that the\nsingle-particle dispersion in Eq. (5) features only a single\nbranch of right and left moving modes at the Fermi en-\nergy. We neglect interband couplings and write an e\u000bec-\ntive description within the lowest band, linearized about\nthe Fermi wavevectors \u0006kF. We \fx the spinor parts of\nthe wave functions to their values at the Fermi energy\n(Fig. 2).23\nTo compute the spin-spin susceptibility in the presence\nof electron-electron interactions, we employ a bosonic de-\nscription. As a \frst step, we express the fermionic \feld\noperator (projected into the lowest band) as a superpo-\nsition of \felds representing right (+) and left (-) movers:\n u(z) = +(z) + \u0000(z). The \felds  r(r2f+;\u0000g)\ntake the form  r=\u0011r\n r(z), where\u0011r=\u00111;rkFand\nthe spatial part (in terms of the destruction operators3\n-k FkF\n2h 0kE\n-v FvFn= 2\nn= 1\nFIG. 2: (Color online). Energy dispersion of the gauge trans-\nformed free-electron Hamiltonian H(0)\nu. The bosonization\nis performed by linearizing the dispersion about the Fermi\nwavevectors k=\u0006kFand \fxing the k-dependent eigenspinors\n\u0011rto their values at the Fermi energy.\nck;r) is r(z) =1p\nLeirkFzP\nkeikzck;r. Substituting the\nfermionic \feld operator into the Hamiltonian Hu, per-\nforming a Fourier transformation to k-space, and evalu-\natingVee(q) at momentum zero and 2 kFfor forward- and\nback- scattering processes, respectively, we obtain\nHu=X\nk;rr\u0016hvFkcy\nk;rck;r+X\nq;r(g2\u001aq;r\u001a\u0000q;\u0000r+g4\u001aq;r\u001a\u0000q;r):\n(6)\nHere,vF= \u0016hkF=m\u0000\u0016hq\u0011y\n+\u001bz\u0011+=2m,g2= (Vee(0)\u0000\nj\u0011y\n+\u0011\u0000j2Vee(2kF))=2L,g4=Vee(0)=2L, and\u001aq;r=P\nkcy\nk\u0000q;rck;ris the Fourier-transformed density opera-\ntor for right/left-movers. Following the standard proce-\ndure,24we write Eq. (6) in the bosonized form\nHu=\u0016h\n2\u0019Z\ndzh\nue\u000bK(@z\u0012)2+ue\u000b\nK(@z\u001e)2i\n;(7)\nwhere\u001eand\u0012are the bosonic \felds, ue\u000bis the density\nwave velocity, and Kis the Luttinger parameter.\nThe bosonic representations of the fermionic \felds are\n r(z) =\u0011r\nUrp\n2\u0019aeirkFze\u0000i[r\u001e(z)\u0000\u0012(z)]; (8)\nwhereais an in\fnitesimal short distance cuto\u000b25and\nfUrgare the Klein factors. The repulsive electron-\nelectron interaction implies that 0 <K\u00141, whereK= 1\nfor non-interacting electrons.\nWe calculate the spin susceptibility following the\nstandard approach for Luttinger liquids, see Ref. 24\nfor details. The resulting (imaginary) time-ordered\nspin-spin correlation function26, ~\u001fu;ij(z;z0;\u001c) =\n\u0000hT\u001c^si(z;\u001c)^sj(z0;0)i, is diagonal and can be written as\n~\u001fu;ii(~z;\u001c) = ~\u001f(0)\nu;ii(~z;\u001c) + ~\u001f(2kF)\nu;ii(~z;\u001c) cos(2kF~z);where\n~\u001f(0)\nu;ii=\u0000\u0012\u0016h\n2\u0019\u00132K(i)\u0003++\nii\n2\u001d2\u0000~z2\n(~z2+\u001d2)2; (9)\n~\u001f(2kF)\nu;ii =\u0000\u0012\u0016h\n2\u0019\u00132\u0003+\u0000\nii\n2a2\u0012a2\n~z2+\u001d2\u0013K\n: (10)Here, ~z=z\u0000z0,\u001d=ue\u000b\u001c+asign(\u001c),K(x)=K(y)=\nK,K(z)=K\u00001, and \u0003rr0\nii=j\u0011y\nr\u001bi\u0011r0j2. The spin\nsusceptibility is \u001fu;ii(~z;t) =\u0000(2=\u0016h)\u0012(t)=m[~\u001fu;ii(~z;t)],\nwhere ~\u001fu;ii(~z;t) is the time-ordered correlation function\ninreal time , which is obtained via the Wick rotation\n\u001c=it+ 0+sign(t). ForK < 1,\u001f(2kF)\nu;ii(!)=!diverges\nin the low-frequency limit. We regularize the divergence\nby evaluating the expression at the low-frequency cut-o\u000b\n!0=ue\u000b2\u0019=L set by the \fnite length of the system.27\nThe analysis above gives the Gilbert damping tensor\n\u000bu;ii(z;z0) =\u000b(0)\nu;ii+\u000b(2kF)\nu;ii cos(2kF~z); (11)\n\u000b(0)\nu;ii=\rh2\n0K(i)\u0003++\nii\n2\u0019\u0016hMsu2\ne\u000b; (12)\n\u000b(2kF)\nu;ii =\rh2\n0\u0003+\u0000\niiFK(\u0010)\n21=2+K\u00193=2\u0016hMsu2\ne\u000b\u0000(K):(13)\nHere, \u0000(K) is the gamma function, \u0010\u0011a!0=ue\u000b,\nandFK(\u0010) =\u0019\u0010K\u00003=2[(IK\u00001=2(\u0010)\u0000L1=2\u0000K(\u0010))\u0000\n2\u0010cos(K\u0019)KK\u00001=2(\u0010)], where I\u0017(\u0010) is the modi\fed\nBessel function of the \frst kind, L\u0017(\u0010) is the modi-\n\fed Struve function, and K\u0017(\u0010) is the modi\fed Bessel\nfunction of the second kind. To obtain the lab-\nframe damping tensor, the transformation \u000b(z;z0;t) =\nRT(z)\u000bu(z;z0;t)R(z0) must be applied. We continue the\nanalysis in the rotated frame , where the expressions are\nmuch simpler.\nIV. DISCUSSION\nEquations (11) - (13) are the central results of this\nwork, and describe magnetization relaxation of 1D heli-\nmagnets via plasmon excitations.\nThe damping consists of two parts with very distinct\nposition dependencies: a homogeneous term \u0018\u000b(0)and\na rapidly oscillating term \u0018\u000b(2kF). The corresponding\nhighly nonlocal magnetization relaxation caused by the\nplasmon excitations di\u000bers markedly from the damping\nof conventional metallic ferromagnets, which is believed\nto be local.28\nConstraints of the model reduce the number of inde-\npendent tensor elements. First, \u0003++\nyy= \u0003+\u0000\nzz= 0 implies\nthat the damping tensor is described by four independent\ncoe\u000ecients: \u000b(0)\nu;xx,\u000b(0)\nu;zz,\u000b(2kF)\nu;xx, and\u000b(2kF)\nu;yy. Second, the\nconstraint _m\u0001m= 0 imposed by normalization implies\nthat _mx= 0 in the rotated reference frame (for small\n\u000em). Thus, the damping is governed by only two coe\u000e-\ncients:\u000b(0)\nu;zzand\u000b(2kF)\nu;yy.\nWhat are the characteristics of these two independent\ndamping coe\u000ecients? The tensor element \u000b(0)\nzzoriginates\nfrom forward scattering processes and governs the damp-\ning of the long wave-length spin-wave modes. Similarly,\n\u000b(2kF)\nyy is associated with electronic backscattering, and\ncontrols the relaxation of short wave-length modes.4\n(10 3 )\n0.2 0.4 0.6 0.8 1.0\nKα*(2k F)\n1.03.05.07.0\nα*(2k F)(10 3 )\n6\n4\n2\n10 -3 10 -2 10 -1 K=0.8K=0.6K=0.4 K=0.2 b a\nFIG. 3: (Color online). (a) The dimensionless damping pa-\nrameter\u000b(2kF)\n\u0003 (K;\u0010) =\u000b(2kF)\nu;yy (K;\u0010)=\u000b(2kF)\nu;yy (1;\u0010) as a func-\ntion of the electron-electron interaction parameter Kwith\n\u0010=a!0=ue\u000b\fxed at\u0010= 3:1\u000210\u00003. (b) The damping pa-\nrameter as a function of \u0010for di\u000berent K.\nInterestingly, \u000b(0)\nzzis proportional to the momentum-\nmomentum correlator h@~z\u0012(~z;\u001c)@~z\u0012(0;0)i, and is thus in-\nversely proportional to the Luttinger parameter K. Con-\nsequently, electron-electron interaction enhances \u000b(0)\nu;zz\nasK\u00001. A much more complex dependency of the\nelectron-electron interactions is seen in the magnetization\ndamping caused by backscattering processes. Remark-\nably, electron-electron interactions may increase \u000b(2kF)\nyy\nby nearly four orders of magnitude compared to its value\nin the non-interacting limit K= 1 (Fig. 3a). The tensor\nelement\u000b(2kF)\nyy reaches a maximum at K\u00190:1 before it\ndrops quickly to zero in the strongly interacting regime\nK!0. In this limit, the potential energy ( ue\u000b=K)(@z\u001e)2\nof the bosonic Hamiltonian (7) completely governs the\nelectron dynamics and density variations of the Luttinger\nliquid become insusceptible to time variations in the mag-\nnetization. The dramatic enhancement of \u000b(2kF)\nyy is re-\nlated to the fact that electron-electron interactions make\nthe damping extremely sensitive to \u0010\u0018a=L, which mea-\nsures the ratio of the short distance cut-o\u000b to the long\ndistance cut-o\u000b (Fig. 3b). In the absence of interac-\ntions,\u000b(2kF)\nyy approaches a constant value in the limit\n\u0010!0. However, with interactions, \u000b(2kF)\nyy is singular\nin this limit and the singularity becomes stronger with\nincreasing strength of the interactions.\nTo investigate the experimental consequences of\nEqs. (11)-(13), we discuss thermal \ructuations and es-\ntimate the characteristic relaxation time for two classes\nof systems which are proposed to hold 1D helimagnetic\nstates. Additionally, we propose a method for probing\nthe relaxation time in nuclear wires via transport mea-\nsurements. The magnetic order is assumed to be to sta-\nbilized by the RKKY interaction, implying q=kF.29\nThe form of the damping tensor has remarkable im-\nplications for the statistical properties of thermal \ruc-\ntuations. While the average of the stochastic magnetic\n\feldhTin Eq. (1) is zero, hhTi= 0, its correlations (inaccordance with the \ructuation-dissipation theorem) are\ngiven in the classical (Maxwell-Boltzmann) limit as20,30\nhhT;i(z;t)hT;j(z0;t0)i=2kBT\n\rMs\u000bij(z;z0)\u000e(t\u0000t0);(14)\nwhereTis the temperature and the average h:::iis taken\nover an ensemble in thermal equilibrium. According to\nEq. (14), the thermal \ructuations are correlated over the\nentire sample. The \ructuations divide into two distinct\nclasses; one class characterized by a spatially constant\ncorrelation and a second class characterized by an oscil-\nlating\u0018cos(2kF~z) correlation. Strongly correlated ther-\nmal \ructuations of this form have not been reported or\ninvestigated before in any magnetic system, and a thor-\nough investigation of how the associated stochastic \feld\nhT(z) in Eq. (1) in\ruences the magnetization dynamics\nshould be an interesting task for future studies.\nFor a magnetization precessing at frequency !, Eq. (1)\nyields two characteristic relaxation times \u001c(0)\nreland\u001c(2kF)\nrel,\nassociated with the homogeneous and oscillatory dissipa-\ntion terms: \u001c(0;2kF)\nrel\u0018[\u000b(0;2kF)L!]\u00001. We now estimate\n\u001c(0;2kF)\nrelfor two classes of systems, which are believed to\nhost 1D helimagnetism:5Ga0:98Mn0:02As quantum wire\nand GaAs wire in which the nuclear spins are hyper\fne\ncoupled to the itinerant system (see Appendix B for ma-\nterial parameters). The (Ga,Mn)As and GaAs wire cross\nsections are assumed to contain 50 \u000250 unit cells. The\ncharacteristic magnon frequency of the \frst excited mode\nis!=kBTc=\u0016hI1=(3\u00002K)\n?,13whereTcis the critical tem-\nperature and \u0016 hI?is the total spin of the cross section.\nWe assume that each unit cell is fully spin polarized. For\nthe magnetic (Ga,Mn)As wire, we \fnd \u001c(0)\nrel= 2:3\u000210\u00008s\nand\u001c(2kF)\nrel= 7:6\u000210\u000011s, while for the nuclear wire the\nrelaxation times are \u001c(0)\nrel= 1:0 s and\u001c(2kF)\nrel= 6:4\u000210\u00003s.\nDynamical probes o\u000ber new routes for characterizing\nthe nature of 1D helimagnetic systems, providing com-\nplementary information to that obtained in static (dc)\nmeasurements. For example, consider the recent exper-\niment of Ref. 14, in which the conductance of a GaAs-\nbased quantum wire was observed to drop from 2 e2=hto\ne2=hwhen the temperature was reduced to below 0.1 K.\nThe appearance of an e2=hplateau hints at the lifting of\nspin-degeneracy, and was interpreted as evidence of the\nformation of a nuclear spin helix. Importantly, within\nthe model used to interpret the experiment, the spatial\nperiod of the helix tunes itself to be equal to half of the\nFermi wavelength, i.e., it is directly linked to the density\nof the electronic system. Furthermore, the appearance\nof ane2=hplateau is directly linked to this commensu-\nration between the ordering and Fermi wavevectors. A\nrapid change of backgate voltage which alters the elec-\ntron density should thus destroy the commensurability;\njust after such a quench, one would then expect to ob-\nserve a conductance of 2 e2=h, which would gradually re-\nturn to a value of e2=has the nuclear system \fnds its\nnew equilibrium order, on a timescale set by the mag-\nnetic relaxation rate. Taking similar assumptions and5\nparameter values to those of the model used to support\nthe conclusions of Ref. 14, we predict very long relax-\nation times for the nuclear magnetic order, on the order\nof milliseconds or more. Nuclear spin di\u000busion, another\npossible important relaxation mechanism, is expected to\nbecome e\u000bective on longer timescales (see Appendix C).\nDynamics on such timescales should in principle be ob-\nservable in experiments, and therefore may provide cru-\ncial independent means for verifying the interpretation of\nthe experiment.\nIn conclusion, we have developed a theoretical formal-\nism for describing magnetization dissipation of 1D heli-\nmagnets via the emission of plasmon excitations. The\ndamping is found to be highly nonlocal and strongly\ndependent on the electron-electron interaction, di\u000bering\nmarkedly from the damping of conventional metallic fer-\nromagnets.\nV. ACKNOWLEDGEMENTS.\nResearch supported the Danish National Research\nFoundation. MR acknowledges support from the Villum\nFoundation and the People Programme (Marie Curie Ac-\ntions) of the European Union's Seventh Framework Pro-\ngramme (FP7/2007-2013) under REA grant agreement\nPIIF-GA-2013-627838.\nAppendix A: Derivation of damping tensor\nWe calculate the magnetization-damping rate by relat-\ning it to the energy absorption rate of the itinerant elec-\ntron system subjected to small \ructuations about a static\nhelimagnetic pro\fle m0(z). In describing the interaction\nbetween the magnetic order parameter \feld and the itin-\nerant electron system, we assume that the magnetiza-tion evolves very little over the characteristic timescales\nof electron dynamics. In this case, the response of the\nelectron system can be calculated using linear response\n(Kubo) theory.\nThe total energy dissipation of the magnetic system is\n_E=Z\ndz_M\u0001\u000eF\n\u000eM: (A1)\nThe time derivative _Mis determined by the LLG equa-\ntion, which yields the energy dissipation\n_E(t) =\u0000Ms\n\rZZ\ndzdz0_m(z;t)\u0001[~\u000b(z;z0)_m(z0;t)]:(A2)\nDue to energy conservation, the energy lost by the\nmagnetic system must be gained by the itinerant elec-\ntron system to which it is coupled. This implies that\n_E=\u0000h_H(t)i, whereH(t) is the Hamiltonian (4) of the\nitinerant system coupled to the magnetization m(z;t) =\nm0(z) +\u000em(z;t).\nWe now use linear response theory to obtain the rate\nof change of the electronic energy due to a slow evolution\nof the magnetization m(z;t):32\nh_H(t)i=\u0000i\n\u0016hZ1\n\u00001dt0\u0012(t\u0000t0)h[_H(t);\u000eH(t0)]i;(A3)\nwhere\u0012(t) is the Heaviside step function, and \u000eH(t0) =\n(2h0=\u0016h)R\ndz0\u000em(z0;t0)\u0001^s(z0;t0) is the perturbing Hamil-\ntonian produced by the small variation \u000em(z0;t0) of the\nhelimagnetic order. Here, ^s(z0;t0) is the spin-density op-\nerator ^s(z) = (\u0016h=2) y(z)\u001b (z), taken in the interaction\npicture with respect to the unperturbed Hamiltonian.\nFourier transforming Eq. (A3) with respect to time and\nusing _H(t) = (2h0=\u0016h)R\ndz_m(z;t)\u0001^s(z;t) yields\n\u0000i!hH(!)i=1\n2\u0019\u00122h0\n\u0016h\u00132ZZ\ndzdz0Z\nd!0i(!\u0000!0)mi(z;!\u0000!0)i\u001fij(z;z0;!0)\n!0i!0\u000emj(z0;!0); (A4)\nwhere\u001fij(z;z0;!) =R1\n\u00001dt\u001fij(z;z0;t) exp(i!t) is the\nFourier transform of the spin susceptibility \u001fij(z;z0;t) =\n\u0000(i=\u0016h)\u0012(t)[^si(z;t);^sj(z0;0)]. To leading order in the pre-\ncession frequency, the behavior of the energy change is\ncaptured by replacing i\u001fij(!)=!by its value in the zero\nfrequency limit. Transforming back to the time domain\nand using\u000e_mj= _mjgives the energy absorption rate\nh_H(t)i=4h2\n0\n\u0016h2ZZ\ndzdz0_mi\u0014\nlim\n!!0i\u001fij(z;z0;!)\n!\u0015\n_mj:(A5)\nComparing Eq. (A5) with Eq. (A2), we identify the fol-lowing expression for the Gilbert damping tensor:\n\u000bij(z;z0) =\u00004\rh2\n0\n\u0016h2Mslim\n!!0=m[\u001fij(z;z0;!)]\n!:(A6)\nAppendix B: Material Parameters\nTable I shows typical material parameters of\n(Ga,Mn)As quantum wires and nuclear helimagnets\nformed in GaAs quantum wires. These parameter values\nare used in the main text to estimate the characteristic6\nTABLE I: Estimates of material parameters for two classes of\nsystems (adapted from Refs. 5 and 13 ).\nMagnetic wire Nuclear wire\nMaterial (Ga,Mn)As GaAs\nL (\u0016m) 20 20\n\u0016F(meV) 20 5.6\na(nm) 5 10\nK 0.5 0.5\nue\u000b(ms\u00001) 3 :2\u00021051:7\u0002105\nMs=\r(Jsm\u00001) 2:3\u000210\u0000232:8\u000210\u000021\nh0(meV) 5 0.07\nI? 500 15000\nTc(K) 2 0.01\nrelaxation times. Here, Lis the length of the quantum\nwire,\u0016Fis the chemical potential, ais the short distance\ncuto\u000b,25Kis the Luttinger parameter, ue\u000bis the den-\nsity wave velocity, Ms=\ris the spin density of the 1D\nmagnetic system, h0describes the coupling between the\nitinerant electrons and the ordered spin system, I?is the\ntotal spin contained in the cross section of the wire, and\nTcis the critical temperature of the spin system.13\nThe short distance cuto\u000b (given by the chemical poten-tial) together with the length of the wire and the density\nwave velocity determine the value of the dimensionless \u0010\nparameter, \u0010=a!0=ue\u000b. Because the damping becomes\nhighly sensitive to the values of \u0010andKin the pres-\nence of interactions (Fig. 3), we believe uncertainties in\nthese two parameters govern the sensitivity in the relax-\nation times estimated from the values in Table I. While\n\u000b(0)\nrelonly depends on K, the backscattering term \u000b(2kF)\nrel\ngrows as\u0010becomes small (see Fig. 3), rising sharply for\nsmallerK(i.e., stronger interactions). For \fxed \u0010, the\nrelaxation is strongly enhanced by interactions up to a\nmax value and then falls o\u000b sharply.\nAppendix C: Relaxation via Nuclear Spin Di\u000busion\nLet\u001c(d)denote the characteristic magnetization relax-\nation time induced by nuclear spin di\u000busion. The spin\ndi\u000busion constant Dof GaAs has been estimated to be\non the order of D\u001810\u000017m2/s.33The relaxation time\n\u001c(d)is the time required to di\u000buse from an internal point\nof the quantum wire to the surrounding nuclei, i.e., a dif-\nfusion length of about the wire diameter d\u001850a0(where\na0is the lattice constant of GaAs). This leads to follow-\ning estimate of the relaxation time\n\u001cd\nr=d2\nD\u001880 s: (C1)\n1P. Pincus, Solid State Comm. 9, 1971 (1971).\n2B. Braunecker, G. I. Japaridze, J. Klinovaja, and D. Loss,\nPhys. Rev. B 82, 045127 (2010).\n3T.-P. Choy, J. M. Edge, A. R. Akhmerov, and C. W. J.\nBeenakker, Phys. Rev. B 84, 195442 (2011).\n4M. Kjaergaard, K. Wolms, and K. Flensberg, Phys. Rev.\nB85, 020503(R) (2012).\n5J. Klinovaja, P. Stano, A. Yazdani, and D. Loss, Phys.\nRev. Lett. 111, 186805 (2013).\n6I. Martin and A. F. Morpurgo, Phys. Rev. B 85, 144505\n(2012).\n7S. Nadj-Perge, I. K. Drozdov, B. A. Bernevig, and A. Yaz-\ndani, Phys. Rev. B 88, 020407(R) (2013).\n8M. M. Vazifeh and M. Franz, Phys. Rev. Lett. 111, 206802\n(2013).\n9B. Braunecker and P. Simon, Phys. Rev. Lett. 111, 147202\n(2013).\n10S. Nadj-Perge et al. , Science 346, 602 (2014).\n11M. Menzel et al. , Phys. Rev. Lett. 108, 197204 (2012).\n12B. Braunecker, P. Simon, and D. Loss, Phys. Rev. Lett.\n102, 116403 (2009).\n13B. Braunecker, P. Simon, and D. Loss, Phys. Rev. B 80,\n165119 (2009).\n14C. P. Scheller, T.-M. Liu, G. Barak, A. Yacoby, L. N. Pfeif-\nfer, K. W. West, and D. M. Zumb uhl, Phys. Rev. Lett. 112,\n066801 (2014).\n15B. Braunecker, C. Bena, and P. Simon, Phys. Rev. B 85,\n035136 (2012).16For a review, see J. A. C. Bland and B. Heinrich, Ultra-\nthin Magnetic Structures III Fundamentals of Nanomag-\nnetism (Springer Verlag, Heidelberg, 2004), and references\ntherein.\n17For reviews, see M. D. Stiles and J. Miltat, Top. Appl.\nPhys. 101, 225 (2006); D. C. Ralph and M. Stiles, J. Magn.\nMat. 320, 1190 (2008); A. Brataas, A. D. Kent, and H.\nOhno, Nature Mat. 11, 372 (2012).\n18A. Kiss et al. , Phys. Rev. Lett. 107, 187204 (2011).\n19A. A. Zyuzin, T. Meng, V. Kornich, D. Loss,\narXiv:1407.2582.\n20A. Brataas, Y. Tserkovnyak, and G. E. W. Bauer, Phys.\nRev. B 84, 054416 (2011).\n21A. Brataas, Y. Tserkovnyak, and G. E. W. Bauer, Phys.\nRev. Lett. 101, 037207 (2008).\n22The continuous order parameter \feld provides a coarse-\ngrained description of the microscopic magnetic impurity\nsystem, which is valid when the spacing between impurities\nis small compared with the inverse ordering wavevector\nand at low temperatures where longitudinal magnetization\n\ructuations can be ignored.\n23Note that in this case the electronic system becomes e\u000bec-\ntively spinless, and the g1andg2processes in the stan-\ndard spinful Tomonaga-Luttinger model take the same\nbosonized form.\n24T. Giamarchi, Quantum Physics in One Dimension (Ox-\nford University Press, New York, 2003).\n25The cuto\u000baparametrizes a \fnite bandwidth. In numerical7\ncalculations, we use a= 1=k0\nFwherek0\nFis the Fermi vector\nin the original (non-rotated) reference frame.\n26Time-ordered quantities are indicated with tildes.\n27For the cut-o\u000b frequency !0= 2\u0019ueft=Lto be considered\nas an in\fnitesimal quantity, we need !0\u001cs<<1, where\n\u001cs\u0018a=ue\u000bis the correlation time of the spin susceptibility.\nUsinga\u00181=k0\nF, this implies that the system length should\nbe much larger than the Fermi wavelength, L>>\u0015 F.\n28Note that a recent \frst-principle study of Permalloy re-\nports signatures of nonlocality in the damping31.29For RKKY systems, the wave vector kFin the rotated\nframe is twice the lab-frame wave vector k0\nF, i.e.,kF= 2k0\nF.\n30W. F. Brown, Phys. Rev. 130, 1677 (1963).\n31Z. Yuan, K. M. D. Hals, Y. Liu, A. A. Starikov, A. Brataas,\nP. J. Kelly, Phys. Rev. Lett. 113, 266603 (2014).\n32J. Rammer, Quantum Field Theory of Non-equilibrium\nStates (Cambridge University Press, New York, 2007).\n33D. Paget, Phys. Rev. B 25, 4444 (1982)."    },    {        "title": "1502.01420v2.Nonlinear_analysis_of_magnetization_dynamics_excited_by_spin_Hall_effect.pdf",        "content": "arXiv:1502.01420v2  [cond-mat.mes-hall]  12 Mar 2015Nonlinear analysis of magnetization dynamics excited by sp in Hall effect\nTomohiro Taniguchi\nNational Institute of Advanced Industrial Science and Tech nology (AIST),\nSpintronics Research Center, Tsukuba, Ibaraki 305-8568, J apan.\n(Dated: October 6, 2018)\nWe investigate the possibility of exciting self-oscillati on in a perpendicular ferromagnet by the\nspin Hall effect on the basis of a nonlinear analysis of the Lan dau-Lifshitz-Gilbert (LLG) equation.\nIn the self-oscillation state, the energy supplied by the sp in torque during a precession on a constant\nenergy curve should equal the dissipation due to damping. Al so, the current to balance the spin\ntorque and the damping torque in the self-oscillation state should be larger than the critical current\nto destabilize the initial state. We find that these conditio ns in the spin Hall system are not satisfied\nby deriving analytical solutions of the energy supplied by t he spin transfer effect and the dissipation\ndue to the damping from the nonlinear LLG equation. This indi cates that the self-oscillation of a\nperpendicular ferromagnet cannot be excited solely by the s pin Hall torque.\nPACS numbers: 75.78.-n, 05.45.-a, 75.78.Jp, 75.76.+j\nI. INTRODUCTION\nNonlinear dynamics such as fast switching and self-\noscillation (limit cycle) has been a fascinating topic in\nphysics1,2. Magnetization dynamics excited by the spin\ntransfer effect3,4in a nanostructured ferromagnet5–12\nprovide fundamentally important examples of such non-\nlinear dynamics. The magnetization switching was first\nobserved in Co/Cu metallic multilayer in 20005. Three\nyears later, self-oscillation was reported in a similar\nsystem6. In these early experiments on the spin transfer\neffect, linear analysis was used to estimate, for exam-\nple, the critical current destabilizing the magnetization\nin equilibrium13,14. However, recently it became clear\nthat nonlinear analysis is necessary to quantitatively an-\nalyze the magnetization dynamics2,15–26. For example,\ncurrent density to excite self-oscillation can be evaluated\nby solvinga nonlinearvectorequation calledthe Landau-\nLifshitz-Gilbert (LLG) equation23,24.\nOriginally, the spin transfer effect was studied by ap-\nplyinganelectriccurrentdirectlytoaferromagneticmul-\ntilayer. Recently, however, an alternative method em-\nploying the spin Hall effect has been used to observe the\nspin transfer effect27–40. The spin-orbit interaction in a\nnonmagnetic heavy metal scatters the spin-up and spin-\ndown electrons to the opposite directions, producing a\npure spin current flowing in the direction perpendicular\nto an applied current. The pure spin current excites the\nspin torque, called spin Hall torque, on a magnetization\nin a ferromagnet attached to a nonmagnet. The direc-\ntionofthespinHalltorqueisgeometricallydetermined27,\nand its magnitude shows a different angular dependence\nthan the spin torque in the ferromagnetic multilayer3.\nTherefore, it is fundamentally unclear whether the phys-\nical phenomena observed in the multilayer5–12can be re-\nproduced in the spin Hall system, and thus, new phys-\nical analysis is necessary. The magnetization switching\nof both in-plane magnetized and perpendicularly mag-\nnetized ferromagnets by spin Hall torque was recently\nreported28–31,36,37. Accordingly, it might be reasonableto expect reports on self-oscillation by spin Hall torque.\nHowever, whereas self-oscillation has been observed in\nthe in-plane magnetized system32, it has not been re-\nported yet in the perpendicularly magnetized system.\nThe purpose of this paper is to investigate the possibil-\nityofexcitingself-oscillationbyspinHalltorquebasedon\na nonlinear analysis of the LLG equation. We argue that\ntwo physical conditions should be satisfied to excite self-\noscillation. The first condition isthat the energythat the\nspintorquesuppliesduringaprecessiononaconstanten-\nergy curve should equal the dissipation due to damping.\nThe second condition is that the current to balance the\nspin torque and the damping torquein the self-oscillation\nstate should be larger than the critical current to desta-\nbilize the initial state. This is because the magnetization\ninitially stays at the minimum energy state, whereas the\nself-oscillation corresponds to a higher energy state. We\nderive exact solutions of the energy supplied by the spin\ntransfer effect and the dissipation due to damping in the\nspin Hall system by solving the nonlinear LLG equation,\nandfindthat theseconditionsarenotsatisfied. Thus, the\nself-oscillation of a perpendicular ferromagnet cannot be\nexcited solely by the spin Hall torque.\nThe paper is organized as follows. The physical condi-\ntions to excite a self-oscillation is summarized in Sec. II.\nThese conditions are applied to the spin Hall system in\nSec. III. Section IV is devoted to the conclusions.\nII. PHYSICAL CONDITIONS TO EXCITE\nSELF-OSCILLATION\nLet us first summarize the physical conditions neces-\nsaryto excite self-oscillation. The magnetization dynam-\nics are described by the LLG equation\ndm\ndt=−γm×H−γHsm×(p×m)+αm×dm\ndt,(1)\nwheremandpare the unit vectors pointing in the\ndirections of the magnetization and the spin polariza-2\ntion of the spin current, respectively. The gyromag-\nnetic ratio and the Gilbert damping constant are de-\nnoted as γandα, respectively. The magnetic field H\nrelates to the energy density of the ferromagnet Evia\nH=−∂E/∂(Mm), where Mis the saturation magneti-\nzation. The strength of the spin torque, Hs, is propor-\ntional to the current density j. Since the LLG equation\nconserves the norm of the magnetization, the magnetiza-\ntion dynamics can be described as a trajectory on a unit\nsphere. The energy density Eshows constant energy\ncurves on this sphere. For example, when the system has\nuniaxial anisotropy, the constant energy curves are lati-\ntudelines. Theself-oscillationisasteadyprecessionstate\non a constant energy curve excited by the field torque,\nthe first term on the right-hand side of Eq. (1). This\nmeans that the second and third terms of Eq. (1), aver-\naged over the constant energy curve, cancel each other.\nIn other words, the energy supplied by the spin trans-\nfer effect during the precession on the constant energy\ncurve equals the dissipation due to the damping. This\ncondition can be expressed as2,24\n/contintegraldisplay\ndtdE\ndt=Ws+Wα= 0, (2)\nwhere the energy supplied by the spin transfer effect and\nthe dissipation due to the damping during the precession\non the constant energy curve of Eare given by2,15–26\nWs(E) =γM/contintegraldisplay\ndtHs[p·H−(m·p)(m·H)],(3)\nWα(E) =−αγM/contintegraldisplay\ndt/bracketleftBig\nH2−(m·H)2/bracketrightBig\n.(4)\nThe time integral is over a precession period on a con-\nstant energy curve. We emphasize that Eqs. (3) and (4)\nare functions of the energy density E. We denote the\nminimum and maximum values of EasEminandEmax,\nrespectively. When the energy density also has saddle\npointsEsaddle,Emaxin the following discussion can be\nreplaced by Esaddle. To excite the self-oscillation, there\nshould be a certain value of the electric current density\nthat satisfies Eq. (2) for Emin< E < E maxin a set of\nreal numbers. Therefore, Eq. (2) can be rewritten as\n∃j∈R,Ws+Wα= 0. (5)\nWe denote the current satisfying the first condition, Eq.\n(2), or equivalently Eq. (5), as j(E).\nAnother condition necessary to excite self-oscillation\nrelatestothefactthatthemagnetizationinitiallystaysat\nthe minimum energy state. To excite any kind of magne-\ntization dynamics, the spin torque should destabilize the\ninitial state, which means that a current density larger\nthan the critical current density, jc=j(Emin), should be\ninjected. Then, the condition\nj(E)> j(Emin), (6)z\nxy\njmspin Hall torque\nHt(a)\n(b)\nspin Hall torquespin Hall torquedamping\ndampingz xyHt // xHt // y\nFIG. 1: (a) Schematic view of system. The current density\njflows in the nonmagnet along the x-axis, exciting the spin\nHall torque pointing in the y-direction on the magnetization\nmin the ferromagnet. The applied magnetic field is denoted\nasHt. (b) Schematic view of the precession trajectory of\nthe magnetization on the constant energy curve. The solid\ncircle is the trajectory in the absence of the magnetic field\nor in the presence of the field along the z-axis, whereas the\ndashed elliptical lines are those in the presence of the field in\nthexandy-axes. The solid and dotted arrows represent the\ndirections of the spin Hall torque and the damping torque,\nrespectively.\nshould be satisfied to excite the self-oscillation. If this\ncondition is not satisfied, the magnetization directly\nmoves to a constant energy curve including the saddle\npoint without showing a stable steady precession, and\nstops dynamics because the spin torque does not balance\nthe damping torquefor Emin< E < E saddle. An example\nof such dynamics is shown below; see Fig. 3. We empha-\nsize that Eqs. (5) and (6) are applicable to any kind of\nphysical system showing a self-oscillation.\nIII. SPIN HALL SYSTEM\nLet us apply the abovediscussions to the spin Hall sys-\ntem schematically shown in Fig. 1 (a), where the electric\ncurrent flows in the nonmagnet along the xdirection,\nwhereas the ferromagnet is attached along the zdirec-\ntion. The spin polarization of the spin current is geomet-\nrically determined as p=ey. In the spin Hall system,\nthe spin torque strength Hsis given by\nHs=/planckover2pi1ϑj\n2eMd, (7)3\nwhereϑanddare the spin Hall angle and the thickness\nof the ferromagnet, respectively. The magnetic field H\nconsists of the applied field Htand the perpendicular\nanisotropy field HKmzez. We can assume that Ht>0\nwithout losing generality because the sign of Htonly af-\nfects the sign of j(E) derived below. Since we are in-\nterested in a perpendicular ferromagnet, we assume that\nHK> Ht>0. Figure 1 (b) schematically shows the\nprecession trajectory of the magnetization on a constant\nenergycurve, where the directions ofthe spin Hall torque\nand the damping torque are represented by the solid and\ndotted arrows, respectively. The spin Hall torque is par-\nallel to the damping torque for my>0, whereas it is\nanti-parallel to the damping torque for my<0. This\nmeans that the spin Hall torque dissipates energy from\nthe ferromagnetwhen my>0, andsuppliesthe energyto\nthe ferromagnet when my<0. Then, due to the symme-\ntry of the trajectory, the net energy supplied by the spin\nHall torque, Ws, is zero when the applied magnetic field\npoints to the x- orz-direction. This means that Eq. (2)\ncannot be satisfied, and thus, self-oscillation cannot be\nexcited in the spin Hall system in the absence of the ap-\nplied magnetic field, or in the presence of the field point-\ning in the x- orz-direction. Therefore, in the following\nwe focus on the applied magnetic field pointing in the\ny-direction. The magnetic field and the energy density\nare given by\nH=Htey+HKmzez, (8)\nE=−MHtmy−MHK\n2m2\nz. (9)\nThe minimum energy of Eq. (9) is\nEmin=−MHK\n2/bracketleftBigg\n1+/parenleftbiggHt\nHK/parenrightbigg2/bracketrightBigg\n,(10)\nwhich corresponds to a point mstable =\n(0,Ht/HK,/radicalbig\n1−(Ht/HK)2). On the other hand,\nEq. (9) has a saddle point at msaddle= (0,1,0),\ncorresponding to the energy density\nEsaddle=−MHt. (11)\nSince the magnetization initially stays at the minimum\nenergy state, and the magnetization dynamics stops\nwhenmreaches the saddle point msaddle, we consider\nthe energy region of Emin< E < E saddle. To calculate\nEqs. (3) and (4), it isnecessarytosolveanonlinearequa-\ntiondm/dt=−γm×H, whichdetermines the precession\ntrajectory of mon the constant energy curve. Since the\nconstant energy curve of Eq. (9) is symmetric with re-\nspect to the yz-plane, it is sufficient for the calculation of\nEqs. (3) and (4) to derive the solutions of mfor half of\nthe trajectory in the region of mx>0, which are exactly\ngiven by\nmx(E) = (r2−r3)sn(u,k)cn(u,k),(12)my(E) =r3+(r2−r3)sn2(u,k),(13)\nmz(E) =/radicalBig\n1−r2\n3−(r2\n2−r2\n3)sn2(u,k),(14)\nwhereu=γ/radicalbig\nHtHK/2√r1−r3t, andrℓare given by\nr1(E) =−E\nMHt, (15)\nr2(E) =Ht\nHK+/radicalBigg\n1+/parenleftbiggHt\nHK/parenrightbigg2\n+2E\nMHK,(16)\nr3(E) =Ht\nHK−/radicalBigg\n1+/parenleftbiggHt\nHK/parenrightbigg2\n+2E\nMHK.(17)\nThe modulus of Jacobi elliptic functions, sn( u,k) and\ncn(u,k), is\nk=/radicalbiggr2−r3\nr1−r3. (18)\nThe derivations of Eqs. (12), (13), and (14) are shown in\nAppendix A. The precession period is\nτ(E) =2K(k)\nγ/radicalbig\nHtHK/2√r1−r3, (19)\nwhereK(k) is the first kind of complete elliptic integral.\nThe work done by spin torque and the dissipation due to\ndamping, WsandWα, are obtained by substituting Eqs.\n(12), (13), and (14) into Eqs. (3) and (4), integrating\nover [0,τ/2], and multiplying a numerical factor 2 be-\ncause Eqs. (12), (13), and (14) are the solution of the\nprecession trajectory for a half period. Then, WsandWα\nforEmin< E < E saddleare exactly given by\nWs=8MHs√r1−r3\n3Ht/radicalbig\nHK/(2Ht)Hs, (20)\nWα=−4αM√r1−r3\n3/radicalbig\nHK/(2Ht)Hα, (21)\nwhereHsandHαare given by\nHs=Ht/parenleftbigg1−r2\n1\nr1−r3/parenrightbigg\nK(k)−/parenleftbiggE\nM+H2\nt\nHK/parenrightbigg\nE(k),(22)\nHα=Ht/parenleftbigg1−r2\n1\nr1−r3/parenrightbigg\nK(k)+/parenleftbigg5E\nM+3HK+2H2\nt\nHK/parenrightbigg\nE(k).\n(23)\nHere,E(k) is the second kind of complete elliptic inte-\ngral. The derivations of Eqs. (20) and (21) are shown in4\nHt/H K=0.1, 0.3, 0.5, 0.7, 0.9current, j(E) \nenergy, E0 0.2 0.4 0.6 0.8 1.000.20.40.60.81.0\nFIG. 2: The dependence of the current j(E), Eq. (24),\nfor several values of Ht/HKon the energy density E. For\nsimplicity, the horizontal and vertical axes are normalize d as\nj(E)/jcandE/(Esaddle−Emin)−[Emin/(Esaddle−Emin)] to\nmakej(Emin) = 1,Emin= 0, and Esaddle= 1.\ntime (μs)0 0.2 0.4 0.6 0.8 1.001.0\n-1.0magnetization mz\nmymx\nFIG. 3: Typical magnetization dynamics excited by the\nspin Hall effect. The parameter values are taken from\nexperiments36–38,42asM= 1500 emu/c.c., HK= 540 Oe,\nα= 0.005,γ= 1.764×107rad/(Oe·s),d= 1 nm, ϑ= 0.1,\nandHt= 50 Oe. The current magnitude is 14 ×106A/cm2,\nwhile the critical current, Eq. (25), is 13 ×106A/cm2.\nAppendix B. The current j(E) forEmin< E < E saddle\nis given by\nj(E) =2αeMd\n/planckover2pi1ϑHtHα\n2Hs. (24)\nThe currents for E→EminandE→Esaddleare41\nj(Emin) =2αeMd\n/planckover2pi1ϑHK\nHt/HK/bracketleftBigg\n1−1\n2/parenleftbiggHt\nHK/parenrightbigg2/bracketrightBigg\n,(25)\nj(Esaddle) =2αeMd\n/planckover2pi1ϑ/parenleftbigg3HK−2Ht\n2/parenrightbigg\n.(26)\nEquation (24) is the current density satisfying Eq. (2),\nor equivalently Eq. (5). Then, let us investigate whetherEq. (24) satisfies Eq. (6). It is mathematically difficult\nto calculate the derivative of Eq. (24) with respect to\nEfor an arbitrary value of E, although we can confirm\nthatj(Emin)> j(Esaddle) forHt< HK. We note that\na parameter determining whether Eq. (6) is satisfied is\nonlyHt/HKbecause the otherparameters, such as αand\nM, are just common prefactors for any j(E). As shown\nin Fig. 2, j(E) is a monotonically decreasing function of\nEfor a wide range of Ht/HK, i.e., Eq. (6) is not satis-\nfied. This result indicates that the magnetization stays\nin the equilibrium state when j < jc=j(Emin), whereas\nit moves to the constant energy curve of Esaddlewithout\nshowing stable self-oscillation when j > jcbecause the\nspin Hall torque does not balance the damping torque\non any constant energy curve between EminandEsaddle.\nThe magnetizationfinally stopsits dynamics at ±msaddle\nbecause all torques become zero at these points. Figure\n3 shows a typical example of such dynamics, in which the\ntime evolution of each component is shown. Therefore,\nself-oscillation solely by the spin Hall torque cannot be\nexcitedin the perpendicularferromagnet. Thisis apossi-\nble reason why the self-oscillation has not been reported\nyet.\nRecently, many kinds of other torques pointing in\ndifferent directions or having different angular depen-\ndencies, such as field-like and Rashba torques, have\nbeen proposed28,29,36,37,40,43–45. These effects might\nchange the above conclusions. Adding an in-plane\nanisotropy21,22, tilting the perpendicular anisotropy40,\nor using higher order anisotropy might be another\ncandidate. Spin pumping is also an interesting phe-\nnomenon because it modifies the Gilbert damping\nconstant46–49. It was shown in Refs.48,50that the en-\nhancement of the Gilbert damping constant in a fer-\nromagnetic/nonmagnetic/ferromagnetic trilayer system\ndepends on the relative angle of the magnetization. This\nmeans that the Gilbert damping constant has an angular\ndependence. In a such case, it might be possible to sat-\nisfy Eqs. (5) and (6) by attaching another ferromagnet\nto the spin Hall system and by choosing an appropriate\nalignment of the magnetizations. The above formulas\nalso apply to these studies. In Appendix C, we briefly\ndiscuss a technical difficulty to include the effect of the\nfield-like torque or Rashba torque.\nIV. CONCLUSION\nIn conclusion, wedevelopedamethod forthe nonlinear\nanalysisofthe LLGequationinthe spinHall systemwith\na perpendicular ferromagnet. We summarized physical\nconditions to excite self-oscillation by the spin transfer\neffect. The first condition, Eq. (2), or equivalently Eq.\n(5), implies that the energy supplied by the spin torque\nduring a precession on a constant energy curve should\nequal the dissipation due to damping. The second con-\ndition, Eq. (6), implies that the current to balance the\nspin torque and the damping torquein the self-oscillation5\nstate should be larger than the critical current to desta-\nbilize the initial state. By solving the nonlinear LLG\nequation, we derived exact solutions of the energy sup-\nplied bythe spintransfereffect andthe dissipationdue to\ndamping, and showed that these conditions are not sat-\nisfied. These results indicate that self-oscillation cannot\nbe excited solely by the spin Hall torque.\nThe author would like to acknowledge T. Yorozu for\nhis great constructive help on this work. The author also\nthanks M. Hayashi, H. Kubota, and A. Emura for their\nkind supports. This work was supported by JSPS KAK-\nENHI Grant-in-Aid for Young Scientists (B) 25790044.\nAppendix A: Precession trajectory on a constant\nenergy curve\nHere, we show the derivation of Eqs. (12), (13), and\n(14). The precession trajectory on a constant energy\ncurve is determined by dm/dt=−γm×H. They-\ncomponent of this equation is dmy/dt=γHKmxmz.\nThus, we find\n/integraldisplay\ndt=1\nγHK/integraldisplaydmy\nmxmz. (A1)\nAs mentioned in Sec. III, since the constant energycurve\nof Eq. (9) is symmetric with respect to the yz-plane, it\nis sufficient to derive the solutions of mfor half of the\ntrajectory in the region of mx>0. Using Eandmy,mx\nandmzare expressed as\nmx=/radicalbigg\n1−m2y+2E\nMHK+2Ht\nHKmy,(A2)\nmz=/radicalbigg\n−2E\nMHK−2Ht\nHKmy. (A3)\nThe initial state of myis chosen as my(0) =r3, where\nr3is given by Eq. (17). Then, myat a certain time tis\ndetermined from Eq. (A1) as\nγ/radicalbig\n2HtHK/integraldisplayt\n0dt\n=/integraldisplaymy\nr3dm′\ny/radicalBig\n(m′y−r1)(m′y−r2)(m′y−r3).(A4)\nWe introduce a new parameter sasmy=r3+(r2−r3)s2.\nThen, we find\nγ/radicalbigg\nHtHK\n2√r1−r3t=/integraldisplays\n0ds′\n/radicalbig\n(1−s′2)(1−k2s′2),(A5)\nwhere the modulus kis given by Eq. (18). The solution\nofsiss= sn(u,k). Therefore, myis given by Eq. (13).\nEquations (12) and (14) are obtained by substituting Eq.\n(13) into Eqs. (A2) and (A3).We note that Eqs. (12), (13), and (14) are periodic\nfunctions with the period given by Eq. (19). On the\nother hand, when E=Esaddle, the magnetization stops\nitsdynamicsfinallyatthesaddlepoint m= (0,1,0). The\nsolution of the constant energy curve of Esaddlewith the\ninitial condition my(0) =r3can be obtained by similar\ncalculations, and are given by\nmx= 2/parenleftbigg\n1−Ht\nHK/parenrightbiggtanh(νt)\ncosh(νt), (A6)\nmy=−1+2Ht\nHK+2/parenleftbigg\n1−Ht\nHK/parenrightbigg\ntanh2(νt),(A7)\nmz= 2/radicalBigg\nHt\nHK/parenleftbigg\n1−Ht\nHK/parenrightbigg1\ncosh(νt),(A8)\nwhereν=γ/radicalbig\nHt(HK−Ht).\nAppendix B: Derivation of Eqs. (20) and (21)\nUsing Eqs. (12), (13), and (14), the explicit form\nof Eq. (3) for the spin Hall system is given by Ws=\nγMHs/integraltext\ndtws, wherewsis given by\nws= (Ht−HKr3)(1−r2\n3)\n+/braceleftbig\n−2Htr3+HK/bracketleftbig\nr3(r2+r3)−(1−r2\n3)/bracketrightbig/bracerightbig\n(r2−r3)sn2(u,k)\n+{−Ht+HK(r2+r3)}(r2−r3)2sn4(u,k).\n(B1)\nSimilarly, Eq. (21) for the spin Hall system is given by\nWα=−αγM/integraltext\ndtwα, wherewαis given by\nwα= (1−r2\n3)(Ht−HKr3)2\n−/bracketleftbig\n2H2\ntr3−H2\nK(r2+r3)(1−2r2\n3)+2HtHK(1−r2r3−2r2\n3)/bracketrightbig\n×(r2−r3)sn2(u,k)\n−[Ht−HK(r2+r3)]2(r2−r3)2sn4(u,k).\n(B2)\nThen, WsandWαare obtained by integrating over\n[0,τ/2], and multiplying a numerical factor 2. The fol-\nlowing integral formulas are useful,\n/integraldisplayu\ndu′sn2(u′,k) =u−E[am(u,k),k]\nk2,(B3)\n/integraldisplayu\ndu′sn4(u′,k) =sn(u,k)cn(u,k)dn(u,k)\n3k2\n+2+k2\n3k4u\n−2(1+k2)\n3k4E[am(u,k),k],(B4)\nwhereE(u,k), am(u,k), and dn( u,k) are the second kind\nofincomplete elliptic integral,Jacobiamplitude function,\nand Jacobi elliptic function, respectively.6\nAppendix C: The effect of the field-like torque or\nRashba torque\nThe direction of the field-like torque or the Rashba\ntorque is given by m×p, wherepis the direction of the\nspin polarization. This means that the effects of these\ntorques can be regarded as a normalization of the field\ntorquem×H. Then, the energy density Eand the\nmagnetic field Hin the calculations of WsandWαshould\nbe replaced with an effective energy density Eand an\neffective field Bgiven by\nE=E−βMHsm·p, (C1)B=H+βHsp, (C2)\nwhere a dimensionless parameter βcharacterizes the\nratio of the field-like torque or Rashba torque to the\nspin Hall torque. We neglect higher order terms of the\ntorque44for simplicity, because these do not change the\nmain discussion here. In principle, j(E) satisfying Eq.\n(5) can be obtained by a similar calculation shown in\nSec. III. However, for example, the right-hand-side of\nEq. (24) now depends on the current through EandH.\nThus, Eq. (24) should be solved self-consistently with\nrespect to the current j, which is technically difficult.\n1S. Wiggins, Introduction to Applied Nonlinear Dynamical\nSystems and Chaos (Springer, 2003), 2nd ed.\n2G. Bertotti, I. Mayergoyz, and C. Serpico, Nonlinear mag-\nnetization Dynamics in Nanosystems (Elsevier, Oxford,\n2009).\n3J. C. Slonczewski, J. Magn. Magn. Mater. 159, L1 (1996).\n4L. Berger, Phys. Rev. B 54, 9353 (1996).\n5J. A. Katine, F. J. Albert, R. A. Buhrman, E. B. Myers,\nand D. C. Ralph, Phys. Rev. Lett. 84, 3149 (2000).\n6S. I. Kiselev, J. C. Sankey, I. N. Krivorotov, N. C. Emley,\nR. J. Schoelkopf, R. A. Buhrman, and D. C. Ralph, Nature\n425, 380 (2003).\n7W. H. Rippard, M. R. Pufall, S. Kaka, S. E. Russek, and\nT. J. Silva, Phys. Rev. Lett. 92, 027201 (2004).\n8H. Kubota, A. Fukushima, Y. Ootani, S. Yuasa,\nK. Ando, H. Maehara, K. Tsunekawa, D. D. Djayaprawira,\nN. Watanabe, and Y. Suzuki, Jpn. J. Appl. Phys. 44,\nL1237 (2005).\n9S. Mangin, D. Ravelosona, J. A. Katine, M. J. Carey, B. D.\nTerris, and E. E. Fullerton, Nat. Mater. 5, 210 (2006).\n10D. Houssameddine, U. Ebels, B. Dela¨ et, B. Rodmacq,\nI. Firastrau, F. Ponthenier, M. Brunet, C. Thirion, J.-\nP. Michel, L. Prejbeanu-Buda, et al., Nat. Mater. 6, 447\n(2007).\n11Z. Zeng, P. K. Amiri, I. Krivorotov, H. Zhao, G. Finocchio,\nJ.-P. Wang, J. A. Katine, Y. Huai, J. Langer, K. Galatsis,\net al., ACS Nano 6, 6115 (2012).\n12H. Kubota, K. Yakushiji, A. Fukushima, S. Tamaru,\nM. Konoto, T. Nozaki, S. Ishibashi, T. Saruya, S. Yuasa,\nT. Taniguchi, et al., Appl. Phys. Express 6, 103003 (2013).\n13J. Z. Sun, Phys. Rev. B 62, 570 (2000).\n14J. Grollier, V. Cros, H. Jaffr´ es, A. Hamzic, J. M. George,\nG. Faini, J. B. Youssef, H. LeGall, and A. Fert, Phys. Rev.\nB67, 174402 (2003).\n15G. Bertotti, I. D. Mayergoyz, and C. Serpico, J. Appl.\nPhys.95, 6598 (2004).\n16G. Bertotti, C. Serpico, I. D. Mayergoyz, A. Magni,\nM. d’Aquino, and R. Bonin, Phys. Rev. Lett. 94, 127206\n(2005).\n17G. Bertotti, I. D. Mayergoyz, and C. Serpico, J. Appl.\nPhys.99, 08F301 (2006).\n18D.M. ApalkovandP.B. Visscher, Phys.Rev.B 72, 180405\n(2005).\n19M. Dykman, ed., Fluctuating Nonlinear Oscillators (Ox-\nford University Press, Oxford, 2012), chap. 6.20K. A. Newhall and E. V. Eijnden, J. Appl. Phys. 113,\n184105 (2013).\n21T. Taniguchi, Y. Utsumi, M. Marthaler, D. S. Golubev,\nand H. Imamura, Phys. Rev. B 87, 054406 (2013).\n22T. Taniguchi, Y. Utsumi, and H. Imamura, Phys. Rev. B\n88, 214414 (2013).\n23T. Taniguchi, H. Arai, S. Tsunegi, S. Tamaru, H. Kubota,\nand H. Imamura, Appl. Phys. Express 6, 123003 (2013).\n24T. Taniguchi, Appl. Phys. Express 7, 053004 (2014).\n25D. Pinna, A. D. Kent, and D. L. Stein, Phys. Rev. B 88,\n104405 (2013).\n26D. Pinna, D. L. Stein, and A. D. Kent, Phys. Rev. B 90,\n174405 (2014).\n27K. Ando, S. Takahashi, K. Harii, K. Sasage, J. Ieda,\nS. Maekawa, and E. Saitoh, Phys. Rev. Lett. 101, 036601\n(2008).\n28I. M. Miron, G. Gaudin, S. Auffret, B. Rodmacq,\nA. Schuhl, S. Pizzini, J. Vogel, and P. Gambardella, Nat.\nMater.9, 230 (2010).\n29I.M. Miron, K.Garello, G. Gaudin, P.-J. Zermatten, M. V.\nCostache, S. Auffret, S. Bandiera, B. Rodmacq, A. Schuhl,\nand P. Gambardella, Nature 476, 189 (2011).\n30L. Liu, O. J. Lee, T. J. Gudmundsen, D. C. Ralph, and\nR. A. Buhrman, Phys. Rev. Lett. 109, 096602 (2012).\n31L. Liu, C.-F. Pai, Y. Li, H. W. Tseng, D. C. Ralph, and\nR. A. Buhrman, Science 336, 555 (2012).\n32L. Liu, C.-F. Pai, D. C. Ralph, and R. A. Buhrman, Phys.\nRev. Lett. 109, 186602 (2012).\n33Y. Niimi, Y. Kawanishi, D. H. Wei, C. Deranlot, H. X.\nYang, M. Chshiev, T. Valet, A. Fert, and Y. Otani, Phys.\nRev. Lett. 109, 156602 (2012).\n34P.M. Haney, H.-W.Lee, K.-J.Lee, A.Manchon, andM. D.\nStiles, Phys. Rev. B 87, 174411 (2013).\n35P.M. Haney, H.-W.Lee, K.-J.Lee, A.Manchon, andM. D.\nStiles, Phys. Rev. B 88, 214417 (2013).\n36J. Kim, J. Sinha, M. Hayashi, M. Yamanouchi, S. Fukami,\nT. Suzuki, S. Mitani, and H. Ohno, Nat. Mater. 12, 240\n(2013).\n37J. Kim, J. Sinha, S. Mitani, M. Hayashi, S. Takahashi,\nS. Maekawa, M. Yamanouchi, and H. Ohno, Phys. Rev. B\n89, 174424 (2014).\n38J. Torrejon, J. Kim, J. Sinha, S. Mitani, M. Hayashi,\nM. Yamanouchi, and H. Ohno, Nat. Commun. 5, 4655\n(2014).\n39Y.TserkovnyakandS.A.Bender, Phys.Rev.B 90, 0144287\n(2014).\n40G. Yu, P. Upadhyaya, Y. Fan, J. Alzate, W. Jiang, K. L.\nWong, S.Takei, S.A.Bender, L.-T. Chang, Y.Jiang, etal.,\nNat. Nanotech. 9, 548 (2014).\n41Equation (26) is applicable to Ht>0. When Ht>0,\nthe magnetization can move the curve except at a point\nmsaddle, and thus, Ws/negationslash= 0, resulting in a finite j(Esaddle).\nOn the other hand, when Ht= 0, the magnetization can-\nnot move at any point on the constant energy curve of\nEsaddle, and thus, Wsis zero.\n42Private communication with M. Hayashi.\n43C. O. Pauyac, X. Wang, M. Chshiev, and A. Manchon,\nAppl. Phys. Lett. 102, 252403 (2013).\n44K. Garello, I. M. Miron, C. O. Avci, F. Freimuth,\nY. Mokrousov, S. Bl¨ ugel, S. Auffret, O. Boulle, G. Gaudin,and P. Gambardella, Nat. Nanotech. 8, 587 (2013).\n45X. Qiu, P. Deorani, K. Narayanapillai, K.-S. Lee, K.-J.\nLee, H.-W. Lee, and H. Yang, Sci. Rep. 4, 4491 (2014).\n46Y. Tserkovnyak, A. Brataas, and G. E. W. Bauer, Phys.\nRev. Lett. 88, 117601 (2002).\n47Y. Tserkovnyak, A. Brataas, and G. E. W. Bauer, Phys.\nRev. B66, 224403 (2002).\n48Y. Tserkovnyak, A. Brataas, and G. E. W. Bauer, Phys.\nRev. B67, 140404 (2003).\n49C. Ciccarelli, K. M. D. Hals, A. Irvine, V. Novak,\nY. Tserkovnyak, H. Kurebayashi, A. Brataas, and A. Fer-\nguson, Nat. Nanotech. 10, 50 (2015).\n50T. Taniguchi and H. Imamura, Phys. Rev. B 76, 092402\n(2007)."    },    {        "title": "1502.02068v1.Microscopic_theory_of_Gilbert_damping_in_metallic_ferromagnets.pdf",        "content": "Microscopic theory of Gilbert damping in metallic ferromagnets\nA. T. Costa and R. B. Muniz\nInstituto de F\u0013 \u0010sica, Universidade Federal Fluminense, 24210-346 Niter\u0013 oi, RJ, Brazil.\nWe present a microscopic theory for magnetization relaxation in metallic ferromagnets of\nnanoscopic dimensions that is based on the dynamic spin response matrix in the presence of spin-\norbit coupling. Our approach allows the calculation of the spin excitation damping rate even for\nperfectly crystalline systems, where existing microscopic approaches fail. We demonstrate that the\nrelaxation properties are notcompletely determined by the transverse susceptibility alone, and that\nthe damping rate has a non-negligible frequency dependence in experimentally relevant situations.\nOur results indicate that the standard Landau-Lifshitz-Gilbert phenomenology is not always ap-\npropriate to describe spin dynamics of metallic nanostructure in the presence of strong spin-orbit\ncoupling.\nMagnetization relaxation in metals is at the heart of\nspin current generation and detection processes currently\nunder investigation, many of them candidates to play\nprotagonist roles in innovative spintronic devices. The\nLandau-Lifshitz-Gilbert (LLG) equation is widely used\nto describe the spin dynamic properties of magnetic ma-\nterials [1, 2]. It includes an important system-dependent\nparameter, called the Gilbert damping constant, usually\ndenoted by \u000bG, that regulates the relaxation of the mag-\nnetization towards stability, after it is driven out of equi-\nlibrium. Recently, a lot of e\u000bort has been put into the\ndetermination of this damping rate [2{8], which charac-\nterizes the pumping and absorption of pure spin currents\nin nanostructures that are of great interest in the \feld of\nspintronic. In most of them spin-orbit interaction is sig-\nni\fcant, and responsible for a desirable interplay between\ncharge spin and angular momentum excitations.\nThere is a general agreement between practitioners in\nthe \feld that a proper microscopic theory of magnetiza-\ntion relaxation in metals requires a good description of\nthe electronic structure of the system including spin-\norbit coupling [3{8]. The conventional approach is to\ncombine a realistic electronic structure with some kind of\nadiabatic approximation to derive expressions that can\nbe directly related to the Landau-Lifshitz-Gilbert phe-\nnomenology. This strategy has been employed by Kam-\nbersk\u0013 y [3] and many others since [4{8]. This conven-\ntional approach has important limitations. It neglects the\ncoupling between transverse spin, longitudinal spin and\ncharge excitations (which is an important consequence of\nthe spin-orbit coupling), and incorrectly pedicts the di-\nvergence of the damping parameter for a perfectly crys-\ntalline system. Actually, for ferromagnets that display\nrotation symmetry in spin space, the Goldstone theorem\nensures that any experiment which measures the total\ntransverse magnetic moment of the sample will produce\na resonant response with zero linewidth [9]. In the pres-\nence of spin-orbit interaction, however, this symmetry is\nexplicitly broken, and the resonant spectrum acquires a\n\fnite linewidth [10].\nWe put forward a more fundamental microscopic ap-\nproach to the calculation of the spin dynamics damp-ing rate that takes fully into account the e\u000bects of SOC\non the spectrum of spin excitations of itinerant sys-\ntems. Namely, we consider the coupling of transverse\nspin excitations to longitudinal spin and charge excita-\ntions, induced by the spin-orbit interaction. We calculate\nthe FMR spectrum at \fnite frequencies and arbitrary\nanisotropy values, without employing any adiabatic ap-\nproximation. We will show that those ingredients are es-\nsential to correctly describe the magnetization relaxation\nin very clean metallic ferromagnets of nanoscopic dimen-\nsions, and that the Landau-Lifshitz-Gilbert phenomenol-\nogy fails to capture essential features of the magnetiza-\ntion dynamics in those systems.\nThis letter is organized as follows: we will present\nbrie\ry our formalism, discuss its main features and\npresent numerical results for two model systems that il-\nlustrate common but qualitatively di\u000berent situations.\nGeneral Formalism - The spectrum of spin excitations\nof a ferromagnet can be obtained from the spectral den-\nsity associated with the transverse spin susceptibility\n\u001f+\u0000(l;l0; \n) =Z\ndtei\nthhS+\nl(t);S\u0000\nl0(0)ii; (1)\nwhere\nhhS+\nl(t);S\u0000\nl0(0)ii\u0011\u0000i\u0012(t)h[S+\nl(t);S\u0000\nl0(0)]i; (2)\nand\nS+\nl=X\n\u0016ay\nl\u0016\"al\u0016#: (3)\nThe operator ay\nl\u0016\u001bcreates one electron in the atomic ba-\nsis state\u0016localized at lattice site lwith spin\u001b. Although\nwe are usually interested in \u001f+\u0000(l;l0; \n) as de\fned above,\nits equation of motion involves the orbital-resolved sus-\nceptibility,\n\u001f+\u0000\n\u0016\u0017\u00160\u00170(l;l0;t)\u0011hhay\nl\u0016\"(t)al\u0017#(t);ay\nl0\u00160#(0)al0\u00170\"(0)ii:(4)\nIn the absence of spin-orbit coupling (SOC) and within\nthe random phase approximation (RPA), the equation ofarXiv:1502.02068v1  [cond-mat.mes-hall]  6 Feb 20152\nmotion for \u001f+\u0000\n\u0016\u0017\u00160\u00170(l;l0;t) is closed and \u001f+\u0000(l;l0; \n) can\nbe expressed in the well-known RPA form,\n\u001f+\u0000(\n) = [1 +U\u001f+\u0000\n0(\n)]\u00001\u001f+\u0000\n0(\n) (5)\nwhere\u001f+\u0000\n0(\n) is the mean-\feld (sometimes called non-\ninteracting, or Hartree-Fock) susceptibility. This expres-\nsion is schematic and must be understood as a matrix in\norbital and site indices, in real space, or a wave-vector\ndependent matrix in reciprocal space. The crucial point,\nhowever, is that, in the absence of spin-orbit coupling,\nwithin the RPA, the transverse spin susceptibility is un-\ncoupled from any other susceptibility. This ceases to\nbe true when SOC is included, as we demonstrated in\nref. 10:\u001f+\u0000becomes coupled to three other susceptibil-\nities, namely\n\u001f(2)\n\u0016\u0017\u00160\u00170(l;l0;t)\u0011hhay\nl\u0016\"(t)al\u0017\"(t);ay\nl0\u00160\"(0)al0\u00170\"(0)ii;(6)\n\u001f(3)\n\u0016\u0017\u00160\u00170(l;l0;t)\u0011hhay\nl\u0016#(t)al\u0017#(t);ay\nl0\u00160#(0)al0\u00170#(0)ii;(7)\n\u001f(4)\n\u0016\u0017\u00160\u00170(l;l0;t)\u0011hhay\nl\u0016#(t)al\u0017\"(t);ay\nl0\u00160\"(0)al0\u00170#(0)ii:(8)\nThe system of equations of motion obeyed by these\nfour susceptibilities can be cast into a form strongly re-\nsembling the RPA result by introducing a block-vector\n~ \u001f\u0011(\u001f(1);\u001f(2);\u001f(3);\u001f(4))T, with\u001f(1)\u0011\u001f+\u0000. With\nan equivalent de\fnition for the mean-\feld susceptibili-\nties\u001f(m)\n0we write\n~ \u001f(\n) =~ \u001f0(\n)\u0000\u0003~ \u001f(\n); (9)\nwhere the \\super-matrix\" \u0003 is proportional to the e\u000bec-\ntive Coulomb interaction strength and involves convolu-\ntions of single particle Green functions. Explicit forms\nfor its matrix elements are found in Ref. 10. The nu-\nmerical analysis of the susceptibilities \u001f(2),\u001f(3)and\u001f(4)\nshow that their absolute values are many orders of mag-\nnitude smaller than those of \u001f(1)=\u001f+\u0000. It is, thus,\ntempting to argue that the transverse susceptibility is ap-\nproximately decoupled from \u001f(2),\u001f(3)and\u001f(4)and that\nit can be calculated via the usual RPA expression with\nthe single particle Green functions obtained with spin-\norbit coupling taken into account. This is not a good\napproximation in general, since the matrix elements of \u0003\nthat couple \u001f(1)to the other susceptibilities are far from\nnegligible. Our numerical calculations indicate that they\nare essential to determine correctly the features of the\nFMR mode around the resonance frequency. Thus, the\nbehaviour of \u001f(1)in the presence of spin-orbit coupling\ncannot be inferred from \u001f(1)\n0in the zero-frequency limit\nalone, as it is usually assumed in the literature on the cal-\nculation of the Gilbert damping parameter [3{5, 8, 11].\nNumerical Results - We start the discussion by pre-\nsenting results for the Gilbert constant \u000bGfor unsup-\nported ultrathin Co \flms. Here we determine \u000bGfrom\nthe ratio between the FMR linewidth \u0001\n and the reso-\nnance frequency \n 0. First we turn o\u000b spin-orbit coupling\n0 10 20 30 40 50\nη (meV)00.010.020.030.040.05αGFIG. 1: Gilbert damping constant \u000bGas a function of the\nimaginary part \u0011added to the real energy, for an ultrathin\n\flm of two atomic layers of Co where SOC has been turned\no\u000b. It is clear that \u000bGvanishes as \u0011!0.\nto check the consistency of our approach. Even with\nSOC turned o\u000b we still \fnd a \fnite linewidth for the\nFMR mode. It comes, as we will shortly demonstrate,\nfrom the small imaginary part \u0011that is usually added\nto the energy in the numerical calculations of the sin-\ngle particle Green functions, in order to move their poles\nfrom the real axis. We calculate \u000bGfor various values of\n\u0011and extrapolate to \u0011!0+, as shown in Fig. 1. It is\nclear that lim \u0011!0+\u000bG= 0. Thus, our approach correctly\npredicts that the Gilbert damping constant vanishes in\nthe absence of SOC, as it should. Indeed, it is easy to\nshow [9] that the FMR mode is a stationary state of the\nmean-\feld hamiltonian and, as such, has in\fnite lifetime\nin the limit \u0011!0+. Now we discuss the dependence\nof\u000bGon\u0011for a \fxed, non-zero value of the spin-orbit\ncoupling strength \u0018. We used LCAO parameters appro-\npriate for bulk Co to describe the electronic structure\nof all Co \flms we investigated. The quantitative details\nof the ferromagnetic ground state and excitation spec-\ntra are sensitive to the LCAO parameters used, but their\nqualitative behaviour is very robust to small changes in\nthe electronic structure. Our strategy is to use the same\nset of LCAO parameters for all \flm thicknesses to avoid\nmodi\fcations in \u000bGcoming directly from changes in the\nLCAO parameters. This allows us to focus on geometric\ne\u000bects and on the \u0011-dependence.\nFigure 2 shows the dependence of the Gilbert damping\nconstant\u000bGon the imaginary part \u0011for Co \flms of var-\nious thicknesses. Clearly \u000bGapproaches \fnite values as\n\u0011!0. Cobalt has a small spin-orbit coupling constant.\nWe would like to investigate the e\u000bect of increasing the\nstrength of the SOC on the damping rate. Instead of\narti\fcially increasing \u0018in Co we consider a more realis-\ntic setting where a double layer of Co is attached to a3\n05 10 15 η\n (meV)00.010.020.030.04αG\nFIG. 2: Gilbert damping constant \u000bGas a function of the\nimaginary part \u0011added to the energy, for Co ultra thin \flms\nof various thicknesses: 1 (circles), 2 (squares), 4 (diamonds)\nand 6 (triangles) atomic layers. The strength of the SOC is\n\u0018= 85 meV. The solid lines are guides to the eye.\nnon-magnetic substrate with high SOC parameter, such\nas Pt. This system has a particularly interesting fea-\nture: the magnetization easy axis is perpendicular to the\nplane. However, we found that, for the LCAO param-\neters we employed, the magnetization in-plane is also\na stable con\fguration, with a small magnetocrystalline\nanisotropy. The damping rate, however, is much larger\nin the 2Co/2Pt system than in the unsupported Co \flms.\nThis is a nice example of how the anisotropy energy is\nstrongly in\ruenced by the system's symmetry, but the\ndamping rate is relatively insensitive to it, depending\nstrongly on the intensity of the spin-orbit coupling. It is\nalso an extremely convenient situation to test an assump-\ntion very frequently found in the literature on Gilbert\ndamping, although sometimes not explicitly stated: that\nthe FMR linewidth \u0001\n is linearly dependent on the res-\nonance frequency \n 0and that \u0001\n!0 as \n 0!0. This\nis not an unreasonable hypothesis, considering the weak\nstatic \felds commonly used in FMR experiments and the\nsmallness of the spin-orbit coupling constant, compared\nto other energy scales of a ferromagnet. Our calculations\nfor the Co \flms con\frm that this relationship is approx-\nimately held. In this case, the Gilbert constant \u000bGmay\nbe extracted from the FMR spectrum by simply \ftting\nit to a Lorentzian and is practically \feld-independent.\nHowever, our results for 2Co/2Pt indicate that the FMR\nlinewidth is \fnite as \n 0!0, leading to a signi\fcantly\nfrequency-dependent \u000bG, as shown in Fig. 3. In order to\nillustrate how the determination of a damping parame-\nter is a\u000bected by the \fnite value of \u0001\n as \n 0!0 we\nextracted the linewidths from the calculated spectra for\nthe 2Co/2Pt system by \ftting Lorentzians to our calcu-\nlated spectral densities. The results are shown in Fig. 3.\nOne of its most important consequences is that, if one\nwishes to de\fne a value of \u000bGfor the system above, it\n00.5 1 Ω\n (meV)05e+051e+061.5e+062e+06FMR spectral density01 2 3 4 B (T)\n00.20.40.60.81Ω0 (meV)(a)\n01 2 3 4 B (T)\n0.10.120.140.160.180.2αG0\n0.2 0.4 0.6 0.8 1 Ω\n0 (meV)00.050.1ΔΩ (meV)(b)FIG. 3: a) Spectral densities of the FMR mode for the\n2Co/2Pt system subjected to various static magnetic \felds\n(from -0.3 T to 4 T). The inset shows the resonance frequency\nas a function of the Zeeman \feld B. b) The Gilbert damping\nparameter\u000bGas a function of applied Zeeman \feld B. The\ninset shows the FMR line width as a function of resonance\nfrequency \n 0. The strengths of the SOC are \u0018Co= 85 meV\nand\u0018Pt= 600 meV.\nmust be de\fned as a function of the Zeeman \feld, as is\nillustrated in Fig. 3. In principle this poses a problem\nfor the procedure usually employed to determine FMR\nspectra experimentally, since there the free variable is\nthe Zeeman \feld, not the frequency of the exciting \feld.\nIn Fig. 4 we illustrate this issue by plotting the FMR\nspectral density as a function of the Zeeman \feld for two\n\fxed pumping frequencies, 24 GHz and 54 GHz. The\ncurves have nice Lorentzian shapes, but the values for\nthe Gilbert damping parameter \u000bGextracted from these\ncurves depend on the pumping frequency ( \u000bG= 0:034\nfor \n 0= 0:10 meV and \u000bG= 0:042 for \n 0= 0:22 meV).\nAlso, they do not correspond to any of the values shown\nin Fig. 3b, although the Zeeman \feld values that de-\ntermine the linewidth in Fig. 4 lie within the range of\nZeeman \feld values showed in Fig. 3b. Thus, if \u000bGis\nde\fned as \u0001\n =\n0, its value for a given sample depends4\n-0.4-0.2 0 0.2 B (T)\n05e+051e+061.5e+062e+06FMR spectral density\nFIG. 4: Spectral densitty of the FMR mode for the 2Co/2Pt\nsystem plotted as a function of the Zeeman \feld Bat a\n\fxed pumping frequencies: \u0017p= 24 GHz (squares) and\n\u0017p= 54 GHz (circles). The solid curves are Lorentzian \fts to\nthe calculated points.\non wether the FMR spectrum is obtained in a \fxed fre-\nquency or \fxed Zeeman \feld set ups. Our results also\nimply that the existing expressions for the damping con-\nstant\u000bGare not valid in general, specially for very clean\nsystems with large spin-orbit coupling materials. The\nconventional approaches express \u000bGas the ratio \u0001\n =\n0\nin the \n 0!0 limit. As we have just shown, this limit\ndoes not exist in general, since \u0001\n approaches a \fnite\nvalue as \n 0!0.\nIn experimental papers [12, 13] the FMR linewidth is\nassumed to have a zero-frequency o\u000bset, just as we de-\nscribed. This is usually attributed to extrinsic broad-\nening mechanisms, such as two-magnon scattering [14],\ndue to the combination between inhomogeneities in the\nmagnetic \flms and dipolar interactions. This is certainly\nthe case in systems with small SOC, such as Fe \flms de-\nposited on GaAs or Au [12]. However, we have shown\nthat there can be zero-frequency o\u000bset of intrinsic ori-\ngin if the SOC is large. The e\u000bect of this intrinsic o\u000bset\nshould be easily separated from that of the two-magnon\nscattering mechanism, since the latter is not active when\nthe magnetization is perpendicular to the plane of the\n\flm [14].\nWe would like to remark that Stoner enhancement in\nPt plays a very important role in the determination of the\ndamping rate. We had shown previously [15] that, in the\nabsence of spin-orbit coupling, Stoner enhancement had\na very mild e\u000bect on the damping rate in the Co/Pd(001)\nsystem. In the presence of SOC, however, the e\u000bect can\nbe very large indeed. Both magnetocrystalline anisotropy\nand damping rate are signi\fcantly di\u000berent in the en-\nhanced and non-enhanced cases, as shown in Fig. 5. The\nGilbert parameter is also very di\u000berent in the two cases:\n\u000benh\nG= 0:11, whereas \u000bnon\u0000enh\nG = 0:33. Thus, proper\n01 2 3 Ω\n (meV)02e+054e+056e+058e+05A(Ω)FIG. 5: a) Spectral densities of the FMR mode for the\n2Co/2Pt system with Stoner enhancement in Pt turned on\n(black line) and o\u000b (red line).\ntreatment of Stoner enhancement in substrates like Pd\nan Pt is essential for the correct determination of spin\nrelaxation features.\nWe presented a microscopic approach to the calcu-\nlation of the Gilbert damping parameter \u000bGfor ultra-\nthin metallic magnetic \flms, illustrated by results for\nCo \flms and Co/Pt bilayers. Our approach is based on\nthe evaluation of the dynamic transverse susceptibility in\nthe presence of spin-orbit coupling, taking into account\nrealistic electronic structures and the coupling between\ntransverse spin, longitudinal spin and charge excitations.\nIt predicts \fnite values of \u000bGin the limit of perfectly\ncrystalline \flms, a regime where methods based on the\ntorque correlation formula \fnd a diverging Gilbert damp-\ning parameter. We showed that the coupling between\ntransverse, longitudinal and charge excitations, due to\nspin-orbit coupling, is of fundamental importance for the\ncorrect determination of FMR spectra in metallic sys-\ntems. We have also shown that the damping rate ex-\ntracted from the FMR spectrum for \fxed pumping fre-\nquency di\u000bers considerably from that extracted from the\nFMR spectrum for \fxed Zeeman \feld. In this case the\nGilbert damping parameter \u000bGbecomes frequency de-\npendent, in contrast to what is assumed in the standard\nLandau-Lifshitz-Gilbert phenomenology. Moreover, we\nhave numerical indications that the Gilbert parameter\nis not well de\fned in the limit of vanishing resonance\nfrequency, a fact that is very relevant to calculational\nschemes based on the adiabatic approximation. Inci-\ndentally, Stoner enhancement in materials like Pt and\nPd also plays an important role in the determination of\nFMR frequencies and damping rates. These results may\nlead to important modi\fcations of the interpretation of\ndamping \\constants\", either calculated or inferred from5\nexperimental results, for systems where spin-orbit cou-\npling is strong. We believe these issues may be crucial\nfor the correct description of relaxation in very clean sys-\ntems of nanoscopic dimensions, specially in the presence\nof relatively weak magnetocrystalline anisotropy.\nThe authors acknowledge partial \fnancial support\nfrom CNPq and FAPERJ. We are grateful to Professor\nCaio Lewenkopf for a critical reading of the manuscript\nand to Dr. Mariana Odashima for enlightening discus-\nsions. RBM acknowledges fruitful discussions with Prof.\nD. M. Edwards and A.Umerski.\n[1] T. Gilbert, Magnetics, IEEE Transactions on 40, 3443\n(2004), ISSN 0018-9464.\n[2] Y. Tserkovnyak, A. Brataas, G. Bauer, and B. Halperin,\nRev. Mod. Phys. 77, 1375 (2005), URL http://link.\naps.org/doi/10.1103/RevModPhys.77.1375 .\n[3] V. Kambersk\u0013 y, Phys. Rev. B 76, 134416 (2007), URL\nhttp://link.aps.org/doi/10.1103/PhysRevB.76.\n134416 .\n[4] I. Garate and A. MacDonald, Phys. Rev. B 79,\n064403 (2009), URL http://link.aps.org/doi/10.\n1103/PhysRevB.79.064403 .\n[5] I. Garate and A. MacDonald, Phys. Rev. B 79,\n064404 (2009), URL http://link.aps.org/doi/10.\n1103/PhysRevB.79.064404 .\n[6] A. Starikov, P. Kelly, A. Brataas, Y. Tserkovnyak, and\nG. Bauer, Phys. Rev. Lett. 105, 236601 (2010), URLhttp://link.aps.org/doi/10.1103/PhysRevLett.105.\n236601 .\n[7] H. Ebert, S. Mankovsky, D. K odderitzsch, and P. J.\nKelly, Phys. Rev. Lett. 107, 066603 (2011), URL\nhttp://link.aps.org/doi/10.1103/PhysRevLett.107.\n066603 .\n[8] E. Barati, M. Cinal, D. M. Edwards, and A. Umerski,\nPhys. Rev. B 90, 014420 (2014), URL http://link.aps.\norg/doi/10.1103/PhysRevB.90.014420 .\n[9] R. B. Muniz and D. L. Mills, Phys. Rev. B 68,\n224414 (2003), URL http://link.aps.org/doi/10.\n1103/PhysRevB.68.224414 .\n[10] A. T. Costa, R. B. Muniz, S. Lounis, A. B. Klautau, and\nD. L. Mills, Phys. Rev. B 82, 014428 (2010), URL http:\n//link.aps.org/doi/10.1103/PhysRevB.82.014428 .\n[11] Y. Liu, Z. Yuan, R. J. H. Wesselink, A. A.\nStarikov, and P. J. Kelly, Phys. Rev. Lett. 113,\n207202 (2014), URL http://link.aps.org/doi/10.\n1103/PhysRevLett.113.207202 .\n[12] R. Urban, G. Woltersdorf, and B. Heinrich, Phys. Rev.\nLett. 87, 217204 (2001), URL http://link.aps.org/\ndoi/10.1103/PhysRevLett.87.217204 .\n[13] E. Montoya, B. Heinrich, and E. Girt, Phys. Rev. Lett.\n113, 136601 (2014), URL http://link.aps.org/doi/\n10.1103/PhysRevLett.113.136601 .\n[14] R. Arias and D. Mills, Phys. Rev. B 60, 7395 (1999),\nURL http://link.aps.org/doi/10.1103/PhysRevB.\n60.7395 .\n[15] D. L. R. Santos, P. Venezuela, R. B. Muniz, and A. T.\nCosta, Phys. Rev. B 88, 054423 (2013), URL http://\nlink.aps.org/doi/10.1103/PhysRevB.88.054423 ."    },    {        "title": "1502.02699v1.Large_amplitude_oscillation_of_magnetization_in_spin_torque_oscillator_stabilized_by_field_like_torque.pdf",        "content": "arXiv:1502.02699v1  [cond-mat.mes-hall]  9 Feb 2015Large amplitude oscillation of magnetization in spin-torq ue oscillator stabilized by\nfield-like torque\nTomohiro Taniguchi1, Sumito Tsunegi2, Hitoshi Kubota1, and Hiroshi Imamura1\n1National Institute of Advanced Industrial Science and Tech nology (AIST),\nSpintronics Research Center, Tsukuba 305-8568, Japan,\n2Unit´ e Mixte de Physique CNRS/Thales and Universit´ e Paris Sud 11, 1 av. A. Fresnel, Palaiseau, France.\n(Dated: July 8, 2021)\nOscillation frequency of spin torque oscillator with a perp endicularly magnetized free layer and\nan in-plane magnetized pinned layer is theoretically inves tigated by taking into account the field-like\ntorque. It is shown that the field-like torque plays an import ant role in finding the balance between\nthe energy supplied by the spin torque and the dissipation du e to the damping, which results in\na steady precession. The validity of the developed theory is confirmed by performing numerical\nsimulations based on the Landau-Lifshitz-Gilbert equatio n.\nSpin torque oscillator (STO) has attracted much at-\ntention as a future nanocommunication device because\nit can produce a large emission power ( >1µW), a high\nquality factor ( >103), a high oscillation frequency ( >1\nGHz), a wide frequency tunability ( >3 GHz), and a nar-\nrowlinewidth ( <102kHz) [1–9]. In particular,STOwith\na perpendicularly magnetized free layer and an in-plane\nmagnetizedpinnedlayerhasbeendevelopedafterthedis-\ncovery of an enhancement of perpendicular anisotropy of\nCoFeB free layer by attaching MgO capping layer [10–\n12]. In the following, we focus on this type of STO. We\nhave investigated the oscillation properties of this STO\nboth experimentally [6, 13] and theoretically [14, 15]. An\nimportant conclusion derived in these studies was that\nfield-like torque is necessary to excite the self-oscillation\nin the absence of an external field, nevertheless the field-\nlike torque is typically one to two orders of magnitude\nsmaller than the spin torque [16–18]. We showed this\nconclusion by performing numerical simulations based on\nthe Landau-Lifshitz-Gilbert (LLG) equation [15].\nThis paper theoretically proves the reason why the\nfield-like torque is necessary to excite the oscillation by\nusing the energy balance equation [19–27]. An effective\nenergy including the effect of the field-like torque is in-\ntroduced. It is shown that introducing field-like torque\nis crucial in finding the energy balance between the spin\ntorque and the damping, and as a result to stabilize a\nsteady precession. A good agreement with the LLG sim-\nulation on the current dependence of the oscillation fre-\nquency shows the validity of the presented theory.\nThesystemunderconsiderationisschematicallyshown\nin Fig. 1 (a). The unit vectorspointing in the magnetiza-\ntion directions of the free and pinned layers are denoted\nasmandp, respectively. The z-axis is normal to the\nfilm-plane, whereas the x-axis is parallel to the pinned\nlayer magnetization. The current Iis positive when elec-\ntrons flow from the free layer to the pinned layer. The\nLLG equation of the free layer magnetization mis\ndm\ndt=−γm×H−γHsm×(p×m)\n−γβHsm×p+αm×dm\ndt,(1)pxz+\n-\nm(a)\n(b)\nmxmy\n1 -1 001\n-1 \nFIG. 1: (a) Schematic view of the system. (b) Schematic\nviews of the contour plot of the effective energy map (dotted) ,\nEq. (2), and precession trajectory in a steady state with I=\n1.6 mA (solid).\nwhereγis the gyromagnetic ratio. Since the external\nfield is assumed to be zero throughout this paper, the\nmagnetic field H= (HK−4πM)mzezconsists of the per-\npendicular anisotropy field only, where HKand 4πMare\nthe crystalline and shape anisotropy fields, respectively.\nSinceweareinterestedintheperpendicularlymagnetized\nfree layer, HKshould be larger than 4 πM. The second\nand third terms on the right-hand-side of Eq. (1) are the\nspin torque and field-like torque, respectively. The spin\ntorque strength, Hs=/planckover2pi1ηI/[2e(1+λm·p)MV], includes\nthe saturation magnetization Mand volume Vof the\nfree layer. The spin polarization of the current and the2\ndependence of the spin torque strength on the relative\nangle of the magnetizations are characterized in respec-\ntive byηandλ[14]. According to Ref. [15], βshould\nbe negative to stabilize the self-oscillation. The values\nof the parameters used in the following calculations are\nM= 1448 emu/c.c., HK= 20.0 kOe,V=π×60×60×2\nnm3,η= 0.54,λ=η2,β=−0.2,γ= 1.732×107\nrad/(Oe·s), andα= 0.005, respectively [6, 15]. The crit-\nical current of the magnetization dynamics for β= 0 is\nIc= [4αeMV/(/planckover2pi1ηλ)](HK−4πM)≃1.2 mA, where Ref.\n[15] shows that the effect of βon the critical current is\nnegligible. Whenthecurrentmagnitudeisbelowthecrit-\nical current, the magnetization is stabilized at mz= 1.\nIn the oscillation state, the energy supplied by the spin\ntorquebalancesthedissipationdue tothedamping. Usu-\nally, the energy is the magnetic energy density defined as\nE=−M/integraltextdm·H[28], which includes the perpendic-\nular anisotropy energy only, −M(HK−4πM)m2\nz/2, in\nthe present model. The first term on the right-hand-side\nof Eq. (1) can be expressed as −γm×[−∂E/∂(Mm)].\nHowever, Eq. (1) indicates that an effective energy den-\nsity,\nEeff=−M(HK−4πM)\n2m2\nz−β/planckover2pi1ηI\n2eλVlog(1+λm·p),\n(2)\nshould be introduced because the first and third terms\non the right-hand-side of Eq. (1) can be summarized as\n−γm×[−∂Eeff/∂(Mm)]. Here, we introduce aneffective\nmagnetic field H=−∂Eeff/∂(Mm) = (β/planckover2pi1ηI/[2e(1 +\nλmx)MV],0,(HK−4πM)mz). Dotted line in Fig. 1 (b)\nschematically shows the contour plot of the effective en-\nergy density Eeffprojected to the xy-plane, where the\nconstant energy curves slightly shift along the x-axis be-\ncause the second term in Eq. (2) breaks the axial sym-\nmetry of E. Solid line in Fig. 1 (b) shows the preces-\nsion trajectory of the magnetization in a steady state\nwithI= 1.6 mA obtained from the LLG equation. As\nshown, the magnetization steadily precesses practically\non a constant energy curve of Eeff. Under a given cur-\nrentI, the effective energy density Eeffdetermining the\nconstant energycurve of the stable precessionis obtained\nby the energy balance equation [27]\nαMα(Eeff)−Ms(Eeff) = 0. (3)\nIn this equation, MαandMs, which are proportional to\nthe dissipation due to the damping and energy supplied\nby the spin torque during a precession on the constant\nenergy curve, are defined as [14, 25–27]\nMα=γ2/contintegraldisplay\ndt/bracketleftBig\nH2−(m·H)2/bracketrightBig\n, (4)\nMs=γ2/contintegraldisplay\ndtHs[p·H−(m·p)(m·H)−αp·(m×H)].\n(5)\nThe oscillation frequency on the constant energy curve(a)\n00.010.02\n-0.01\n-0.020 0.2 0.4 0.6 0.8 1.0\nmzMs, -αM α, Ms-αM αMs\n-αM αMs-αM α\n(b)\n00.010.02\n-0.01\n-0.030 0.2 0.4 0.6 0.8 1.0\nmzMs, -αM α, Ms-αM αMs\n-αM αMs-αM α\n-0.02β=0\nβ=-0.2\nFIG. 2: Dependences of Ms,−αMα, and their difference\nMs−Mαnormalized by γ(HK−4πM) onmz(0≤mz<1)\nfor (a)β= 0, and (b) β=−0.2, where I= 1.6 mA.\ndetermined by Eq. (3) is given by\nf= 1/slashbig/contintegraldisplay\ndt. (6)\nSince we are interested in zero-field oscillation, and from\nthe fact that the cross section of STO in experiment [6]\nis circle, we neglect external field Hextor with in-plane\nanisotropy field Hin−plane\nKmxex. However, the above\nformula can be expanded to system with such effects\nby adding these fields to Hand terms −MHext·m−\nMHin−plane\nKm2\nx/2 to the effective energy.\nIn the absence of the field-like torque ( β= 0), i.e.,\nEeff=E, thereisone-to-onecorrespondencebetween the\nenergy density Eandmz. Because an experimentally\nmeasurable quantity is the magnetoresistance propor-\ntional to ( RAP−RP)max[m·p]∝max[mx] =/radicalbig\n1−m2z,\nit is suitable to calculate Eq. (3) as a function of mz, in-\nstead ofE, whereRP(AP)is the resistance of STO in the\n(anti)parallel alignment of the magnetizations. Figure 2\n(a) shows dependences of Ms,−αMα, and their differ-\nenceMs−αMαonmz(0≤mz<1)forβ= 0, where Ms\nandMαare normalized by γ(HK−4πM). The current is\nset asI= 1.6 mA (> Ic). We also show Ms,−αMα, and\ntheir difference Ms−αMαforβ=−0.2 in Fig. 2 (b),\nwheremxis set as mx=−/radicalbig\n1−m2z. Because −αMα\nis proportional to the dissipation due to the damping,\n−αMαis always −αMα≤0. The implications of Figs.\n2 (a) and (b) are as follows. In Fig. 2 (a), Ms−αMαis\nalways positive. This means that the energy supplied by3\ncurrent (mA)frequency (GHz) \n1.2 1.4 1.6 1.8 2.012\n0345\n: Eq. (6): Eq. (1)\nFIG. 3: Current dependences of peak frequency of |mx(f)|\nobtained from Eq. (1) (red circle), and the oscillation fre-\nquency estimated by using (6) (solid line).\nthe spin torque is always larger than the dissipation due\nto the damping, and thus, the net energy absorbed in\nthe free layer is positive. Then, starting from the initial\nequilibrium state ( mz= 1), the free layer magnetization\nmoves to the in-plane mz= 0, as shown in Ref. [14]. On\nthe other hand, in Fig. 2 (b), Ms−αMαis positive from\nmz= 1to acertain m′\nz, whereasit is negativefrom m′\nzto\nmz= 0 (m′\nz≃0.4 in the case of Fig. 2 (b)). This means\nthat, starting from mz= 1, the magnetization can move\nto a point m′\nzbecause the net energy absorbed by the\nfree layer is positive, which drives the magnetization dy-\nnamics. However, the magnetization cannot move to the\nfilm plane ( mz= 0) because the dissipation overcomes\nthe energy supplied by the spin torque from mz=m′\nztomz= 0. Then, a stable and large amplitude precession\nis realized on a constant energy curve.\nWe confirm the accuracy of the above formula by com-\nparing the oscillation frequency estimated by Eq. (6)\nwiththenumericalsolutionoftheLLGequation, Eq. (1).\nIn Fig. 3, we summarize the peak frequency of |mx(f)|\nforI= 1.2−2.0 mA (solid line), where mx(f) is the\nFourier transformation of mx(t). We also show the oscil-\nlation frequency estimated from Eq. (6) by the dots. A\nquantitatively good agreement is obtained, guaranteeing\nthe validity of Eq. (6).\nIn conclusion, we developed a theoretical formula to\nevaluate the zero-field oscillation frequency of STO in\nthe presence of the field-like torque. Our approach was\nbasedon the energybalance equationbetween the energy\nsuppliedbythe spintorqueandthe dissipationdue tothe\ndamping. An effective energy density was introduced to\ntake into account the effect of the field-like torque. We\ndiscussed that introducing field-like torque is necessary\nto find the energy balance between the spin torque and\nthe damping, which as a result stabilizes a steady preces-\nsion. The validity of the developed theory was confirmed\nby performing the numerical simulation, showing a good\nagreement with the present theory.\nThe authors would like to acknowledge T. Yorozu,\nH. Maehara, H. Tomita, T. Nozaki, K. Yakushiji, A.\nFukushima, K. Ando, and S. Yuasa. This work was sup-\nported by JSPS KAKENHI Number 23226001.\n[1] S. I. Kiselev, J. C. Sankey, I. N. Krivorotov, N. C. Em-\nley, R. J. Schoelkopf, R. A. Buhrman, and D. C. Ralph,\nNature425, 380 (2003).\n[2] W. H. Rippard, M. R. Pufall, S. Kaka, S. E. Russek, and\nT. J. Silva, Phys. Rev. Lett. 92, 027201 (2004).\n[3] D. Houssameddine, U. Ebels, B. Dela¨ et, B. Rodmacq,\nI. Firastrau, F. Ponthenier, M. Brunet, C. Thirion, J.-P.\nMichel, L. Prejbeanu-Buda, et al., Nat. Mater. 6, 447\n(2007).\n[4] S.Bonetti, P.Muduli, F.Mancoff, andJ. Akerman, Appl.\nPhys. Lett. 94, 102507 (2009).\n[5] Z. Zeng, G. Finocchio, B. Zhang, P. K. Amiri, J. A. Ka-\ntine, I. N. Krivorotov, Y. Huai, J. Langer, B. Azzerboni,\nK. L. Wang, et al., Sci. Rep. 3, 1426 (2013).\n[6] H. Kubota, K. Yakushiji, A. Fukushima, S. Tamaru,\nM. Konoto, T. Nozaki, S. Ishibashi, T. Saruya, S. Yuasa,\nT. Taniguchi, et al., Appl. Phys. Express 6, 103003\n(2013).\n[7] H. Maehara, H. Kubota, Y. Suzuki, T. Seki,\nK. Nishimura, Y. Nagamine, K. Tsunekawa,\nA. Fukushima, A. M. Deac, K. Ando, et al., Appl.\nPhys. Express 6, 113005 (2013).\n[8] S. Tsunegi, H. Kubota, K. Yakushiji, M. Konoto,\nS. Tamaru, A. Fukushima, H. Arai, H. Imamura,\nE. Grimaldi, R. Lebrun, et al., Appl. Phys. Express 7,\n063009 (2014).\n[9] A. Dussaux, E. Grimaldi, B. R. Salles, A. S. Jenkins,\nA. V. Khavalkovskiy, P. Bortolotti, J. Grollier, H. Kub-ota, A. Fukushima, K. Yakushiji, et al., Appl. Phys. Lett.\n105, 022404 (2014).\n[10] S. Yakata, H. Kubota, Y. Suzuki, K. Yakushiji,\nA. Fukushima, S. Yuasa, and K. Ando, J. Appl. Phys.\n105, 07D131 (2009).\n[11] S. Ikeda, K. Miura, H. Yamamoto, K. Mizunuma, H. D.\nGan, M. Endo, S. Kanai, J. Hayakawa, F. Matsukura,\nand H. Ohno, Nat. Mater. 9, 721 (2010).\n[12] H. Kubota, S. Ishibashi, T. Saruya, T. Nozaki,\nA. Fukushima, K. Yakushiji, K. Ando, Y. Suzuki, and\nS. Yuasa, J. Appl. Phys. 111, 07C723 (2012).\n[13] S. Tsunegi, T. Taniguchi, H. Kubota, H. Imamura,\nS. Tamaru, M. Konoto, K. Yakushiji, A. Fukushima, and\nS. Yuasa, Jpn. J. Appl. Phys. 53, 060307 (2014).\n[14] T. Taniguchi, H.Arai, S.Tsunegi, S.Tamaru, H.Kubota,\nand H. Imamura, Appl. Phys. Express 6, 123003 (2013).\n[15] T. Taniguchi, S. Tsunegi, H. Kubota, and H. Imamura,\nAppl. Phys. Lett. 104, 152411 (2014).\n[16] J. C. Slonczewski, J. Magn. Magn. Mater. 159, L1\n(1996).\n[17] L. Berger, Phys. Rev. B 54, 9353 (1996).\n[18] A. A. Tulapurkar, Y. Suzuki, A. Fukushima, H. Kub-\nota, H. Maehara, K. Tsunekawa, D. D. Djayaprawira,\nN. Watanabe, and S. Yuasa, Nature 438, 339 (2005).\n[19] D. M. Apalkov and P. B. Visscher, Phys. Rev. B 72,\n180405 (2005).\n[20] G. Bertotti, I. D. Mayergoyz, and C. Serpico, J. Appl.\nPhys.99, 08F301 (2006).4\n[21] M. Dykman, ed., Fluctuating Nonlinear Oscillators (Ox-\nford University Press, Oxford, 2012), chap. 6.\n[22] K. A. Newhall and E. V. Eijnden, J. Appl. Phys. 113,\n184105 (2013).\n[23] D. Pinna, A. D. Kent, and D. L. Stein, Phys. Rev. B 88,\n104405 (2013).\n[24] D. Pinna, D. L. Stein, and A. D. Kent, Phys. Rev. B 90,\n174405 (2014).\n[25] T. Taniguchi, Y. Utsumi, M. Marthaler, D. S. Golubev,and H. Imamura, Phys. Rev. B 87, 054406 (2013).\n[26] T. Taniguchi, Y. Utsumi, and H. Imamura, Phys. Rev.\nB88, 214414 (2013).\n[27] T. Taniguchi, Appl. Phys. Express 7, 053004 (2014).\n[28] E. M. Lifshitz and L. P. Pitaevskii, Statistical Physics\n(part 2), course of theoretical physics volume 9\n(Butterworth-Heinemann, Oxford, 1980), chap. 7, 1st ed."    },    {        "title": "1502.05687v3.Characterization_of_spin_relaxation_anisotropy_in_Co_using_spin_pumping.pdf",        "content": "arXiv:1502.05687v3  [cond-mat.mtrl-sci]  23 Nov 2016Characterization of spin relaxation anisotropy in Co using spin pumping\nY. Li, W. Cao and W. E. Bailey\nMaterials Science & Engineering, Department of Applied Phy sics and Applied Mathematics,\nColumbia University, New York NY 10027, USA\n(Dated: March 27, 2021)\nFerromagnets are believed to exhibit strongly anisotropic spin relaxation, with relaxation lengths\nfor spin longitudinal to magnetization significantly longe r than those for spin transverse to mag-\nnetization. Here we characterize the anisotropy of spin rel axation in Co using the spin pumping\ncontribution to Gilbert damping in noncollinearly magneti zed Py 1−xCux/Cu/Co trilayer structures.\nThe static magnetization angle between Py 1−xCuxand Co, adjusted under field bias perpendicular\nto film planes, controls the projections of longitudinal and transverse spin current pumped from\nPy1−xCuxinto Co. We find nearly isotropic absorption of pure spin curr ent in Co using this tech-\nnique; fits to a diffusive transport model yield the longitudi nal spin relaxation length <2 nm in Co.\nThe longitudinal spin relaxation lengths found are an order of magnitude smaller than those deter-\nmined by current-perpendicular-to-planes giant magnetor esistance measurements, but comparable\nwith transverse spin relaxation lengths in Co determined by spin pumping.\nA key question for spin electronics concerns the relax-\nation mechanisms for spin current injected into a vari-\nety of materials. Spin relaxation in ferromagnets (Fs),\ncentral for spin momentum transfer, is special because of\nthe anisotropyaxis presented by the spontaneousmagne-\ntizationM[1–10]. Longitudinal spin relaxation[1], with\nspin polarization σparallel (antiparallel) to M, causes\nspin accumulation to decrease exponentially with dis-\ntance over a scale greater than the electronic mean free\npath[2]. Transverse spin relaxation, with σorthogonal\ntoM, is governed by the dephasing process of spin-up\nand spin-down eigenmodes due to their different Fermi\nwavevectors, leading to oscillation and decay of spin ac-\ncumulation on a scale shorter than the electronic mean\nfree path[3, 10].\nThe characteristic length scales for the two different\nspin relaxation processes in ferromagnets, λL\nsrfor longi-\ntudinal and λT\nsrfor transverse spin relaxation, have been\nevaluated largely using two separate experimental tech-\nniques: magnetotransport[11] for λL\nsrand ferromagnetic\nresonance (FMR)[4, 5] for λT\nsr. These two measurements\ncharacterizecharge-accompaniedandchargelessspincur-\nrent, respectively[1, 4]. Estimates of λL\nsrcome from the F\nlayer thickness dependence of current-perpendicular-to-\nplanes giant magnetoresistance (CPP-GMR)[11–14]; ex-\ntracted values of λL\nsrrange from 5 nm for Ni 79Fe21up to\n40 nm for Co at room temperature. FMR measurements\nof spin pumping, for collinearly magnetized F 1/Cu/F 2\nstructures, show much shorter penetration depths ( λC)\nto fully absorb transverse spin current[15, 16]. Co has\nthe most anisotropic spin relaxation according to these\nseparate measurements, with λL\nsr/λT\nsr∼16 taking λT\nsr∼\n2λC= 2.4 nm[15, 17].\nIn this manuscript, we demonstrate that the longitu-\ndinal spin relaxation length, in addition to the trans-\nverse spin relaxation length[15], can also be character-\nized using a spin pumping measurement, enabling a mea-\nsurement of the anisotropy of spin relaxation in a givenferromagnetic layer. We present FMR measurements\nof the spin pumping contribution to Gilbert damping\nin noncollinearly magnetized Py 1−xCux/Cu/Co multi-\nlayers (Py=Ni 79Fe21). Using Py 1−xCuxalloys, which\nhave adjustably smaller saturation magnetization Ms\nthan Co, we can change the magnetization alignment of\nPy1−xCuxand Co from collinear for in-plane FMR to\nnear-orthogonalfor perpendicular FMR. As the angle θM\nbetween Py 1−xCuxand Co magnetization tends towards\nπ/2, one component of injected spin from Py 1−xCux\ntends towards the longitudinal direction (Fig. 1), allow-\ning us to probe anisotropy in spin relaxation through the\nlinewidth of the Py 1−xCuxlayer[18, 19]. We find, sur-\nprisingly, that spin relaxation, as measured through the\nspin pumping contribution to Gilbert damping, is mostly\nisotropic. In our Co films we estimate λL\nsr<2 nm for all\ndifferent Py 1−xCux/Cu/Co samples, which is compara-\nble to its transverse counterpart( ∼2.4 nm) but inconsis-\ntent with the much longer ( ∼40 nm) lengths reported\nfrom room-temperature CPP-GMR[11, 14].\nThree types of thin-film heterostructures were pre-\nHBnc \nIs\nωσσ\nF1=Py 1-x Cu x\nF2=Coσ\nσTωσL θMθM\nF1F2\nN\n// m1m1m1m2m2\nm1\nFIG. 1. Left:Noncollinear magnetization alignment of the\nF1/N/F2trilayer at the FMR condition for F 1.Right:m1\nis driven into precession, pumping spin current into m2, with\nspin components both longitudinal ( σL) and transverse ( σT)\nto them2magnetization.\npared by UHV sputtering and characterized by FMR.\nPseudo-spin-valve-type Py 1−xCux(t)/Cu(5 nm)/Co(52\nnm) trilayers were used to characterize the anisotropy\nof spin-current absorption in Co. Their response was\ncompared with two types of Py 1−xCux(t) alloy control\nsamples. Bilayers of Py 1−xCux(5 nm*)/Cu(5 nm) and\ntrilayers of Py 1−xCux(5 nm*)/Cu(5 nm)/Pt(3 nm) were\nused to characterizethe backgrounddamping of the alloy\nand the spin mixing conductance of the alloy/Cu inter-\nface, respectively. Co(5 nm)/Cu(5 nm)/Pt(3 nm) is also\ndeposited. For the alloy Cu contents x= 0 to 0.4 were\nprepared in each case, using confocal sputtering from Py\nand Cu targets[20]; thicker (10 nm*) alloy layers were\nused for x= 0.4. All layers were deposited on Si/SiO 2\nsubstrates, seeded by Ta(5 nm)/Cu(5 nm) and capped\nby Ta(2 nm). See Ref. [21] for details on preparation.\nRoom temperature, variable frequency (3-26 GHz),\nswept-field FMR measurements were used to character-\nize the samples, with instrumentation as described in\n[22]. In order to characterize FMR relaxation of the\nPy1−xCuxlayer under noncollinear magnetization align-\nment with Co, two types of measurements were car-\nried out. First, we compare the frequency-dependent\nlinewidths of Py 1−xCuxand Py 1−xCux/Cu/Co sam-\nples in both in-plane (parallel-condition, pc) and per-\npendicular (normal-condition, nc) FMR[23], for a series\nof four measurements at a given alloy content x; see\nFigs. 2c, 3, and 4. Here we expect the Co magneti-zation of trilayer samples to vary from fully perpendic-\nular to the film plane at high biasing field HB(high\nω/2π) to nearly parallel to the film plane at low HB\n(lowω/2π), while the Py 1−xCuxmagnetization is al-\nways perpendicular to the film plane. Second, we com-\npare the polar angle-dependent linewidths of Py 0.8Cu0.2\nand Py 0.8Cu0.2/Cu/Co samples at a fixed frequency of\nω/2π= 10 GHz; see Fig. 5. Here we expect the mis-\nalignment angle θMto change from zero to maximum as\nwe rotate the biasing field from in-plane (pc) to out-of-\nplane (nc).\nTheoretical models for the spin pumping contribution\nto damping under noncollinear magnetization alignment\nofsymmetricF 1/N/F1structuresweredevelopedinRefs.\n[18, 19]. We have extended these models to consider\nasymmetric F 1/N/F2structures where F 1=Py1−xCux\nand F 2=Co in our samples. In the spin valve structure\nthe spin-pumping damping enhancement ∆ αspof F1is\ncaused by the dissipation of spin current in F 2. If F1and\nF2are misaligned by an angle θM, whereθM=m1·m2\n(Fig. 1), during small-angle precession of F 1, the polar-\nization of spin current pumped into F 2will oscillate from\nfully transverse to maximally longitudinal. The instan-\ntaneous spin-pumping damping will then oscillate from\nαsp(0◦) = ∆α0טg↑↓\n2/(˜g↑↓\n1+˜g↑↓\n2), as given in the standard\ncollinear case[24], to a minimum value given by[25]:\n∆αsp(θM) = ∆α0g∗\n2(Asin2θM−BsinθMcosθM)+ ˜g↑↓\n2(Ccos2θM−BsinθMcosθM)\nAC−B2(1)\nHere ˜g↑↓\niandg∗\ni(i= 1, 2) are the effective trans-\nverse and longitudinal spin conductances, respectively;\n∆α0=γ¯h˜g↑↓\n1/(4πMstF) is the damping enhancement\nwith effective spin mixing conductance of ˜ g↑↓\n1[22];\nin Eq. 1 A(θM) =g∗\n1sin2θM+ ˜g↑↓\n1cos2θM+ ˜g↑↓\n2,\nB(θM) = (˜ g↑↓\n1−g∗\n1)sinθMcosθMandC(θM) =\ng∗\n1cos2θM+ ˜g↑↓\n1sin2θM+g∗\n2. We take the arithmetic\nmean of the two extreme cases as the effective damping\nenhancement, as found to be valid in Ref. [19]. See the\nSupplemental Materials for details.\nTo maximize the spin pumping anisotropy at finite\nθM, we use Co (5 nm) for F 2, where the dimension is\nchosen to be significantly thicker than the transverse\nspin penetration depth, λC= 1.2 nm[15], and thinner\nthan the reported longitudinal relaxation length λL\nsr,\n∼38 nm[11], resulting in a large expected asymmetry\nin spin relaxation. In the analysis of relaxation in non-\ncollinearly magnetized structures, we take spin mixing\nconductances ˜ gi↑↓as parameter inputs, determined from\nthe measurements on the Pt control structures, and\ntake the longitudinal spin relaxation length λL\nsras a fit\nparameter.Fig. 2 summarizes the results of fixed-angle nc-and\npc-FMR measurements for the three sample series. In\nFig. 2(a) we plot resonance fields µ0Hresas a function of\nfrequency for single layers and trilayers in nc-FMR. The\ngood agreement in the µ0Hresof Py1−xCuxmeasured in\nsingle layers and trilayers demonstrates that Py 1−xCux\nproperties are reproducible in deposition. In Fig. 2(b)\nthe effective magnetizations µ0Meff, extracted from fits\nto the linear Kittel equation ω/γ=µ0(Hres−Meff), are\nplotted as a function of x. The data show Slater-Pauling\ndilution of magnetic moment in the Py 1−xCuxlayer\nwith increasing Cu content x[20].\nInFig. 2(c)weplotfull-widthhalf-maximumlinewidth\nµ0∆H1/2as a function of ω/2πatx=0.2. Gilbert-type\ndamping, µ0∆H1/2=µ0∆H0+ 2αω/γ, with negligible\ninhomogeneous broadening µ0∆H0, is observed for both\npc- and nc-FMR in the single layer and for pc-FMR in\nthe trilayer. The linewidths agree closely for pc- and\nnc-FMR in the single layer, showing a negligible role\nfor two-magnon scattering in the linewidth[26]. In the\ntrilayer, nc and pc linewidths agree well for frequencies\nabove 10 GHz. These observations hold for samples3\n(nc)Single layer \n(nc)Trilayer \n(pc)Single layer \n(pc)Trilayer \n= 0.20\n0.1\n0.2\n0.3\n0.4Co satura/g415on: 1.4 T \nSingle layer \nTrilayer \nTrilayer(Co) Single layer \nTrilayer 1/2 (a) (b)\n(c) (d)\nFIG. 2. (a) Perpendicular (nc-FMR) resonance field µ0Hres\nfor Py 1−xCuxsingle layers and Py 1−xCux/Cu/Co trilayers,\nx= 0 - 0.4, as a function of frequency ω/2π. (b) Effective\nmagnetization µ0Meffextracted from (a) as a function of x.\n(c) Resonance linewidths µ0∆H1/2of the Py 0.8Cu0.2single\nlayer and trilayer as a function of frequency ω/2π. The spin\npumping enhancement is clearly visible in the increased slo pe\n(α) of the trilayer data; the low-frequency deviation is dis-\ncussed in Fig. 3. (d) Effective spin mixing conductances g↑↓\neff\nof Py1−xCux/Cu/Co, Py 1−xCux/Cu/Pt and Co/Cu/Pt.\nwith all Cu content 0 ≥x≥0.4; the deviations at\nlow frequency are discussed in Fig. 3. The effective\nspin mixing conductances g↑↓\neffof trilayer samples are\nextracted from ∆ αsp=γ¯hg↑↓\neff/(4πMstM), shown above\nwhere ∆ αspis the difference in αbetween trilayers and\nsingle layers. In Fig. 2(d) we show the extracted g↑↓\neff\nfor Py 1−xCux/Cu/Co and Py 1−xCux/Cu/Pt structures\nas a function of x. We also plot g↑↓\neffof Co/Cu/Pt\nfor reference. The lower level of g↑↓\neff∼7 nm−2for\nPy1−xCux/Cu/Co, compared with ∼15 nm−2measured\nin Ref. [15], is likely to be from a more resistive Cu layer,\nwhich adds an additional resistance of (2 e2/h)tCu/σCu\nto the inverse of total spin mixing conductance where\nσCuis the Cu conductivity. Using these g↑↓\neffvalues,\nwe extract the effective spin mixing conductance of\nPy1−xCux/Cu and Co/Cu interfaces, shown in the\nSupplemental Materials[25]. These parameters will\nbe used to determine the longitudinal spin relaxation\nlengths from the spin pumping data in Figs. 4 and 5.\nIn the measurements presented in Fig. 2(c), the nc-\nFMR linewidths are measured at applied fields below the\nsaturation field for Co, µ0Meff= 1.4 T. The saturation\nfield corresponds to a nc-FMR resonance frequency for\nPy0.8Cu0.2of 25 GHz, as shown in Fig. 2(a). With the\nresultant noncollinear magnetization alignment in the\ntrilayer, we expect to see spin-pumping damping ∆ αsp\nfor Py 0.8Cu0.2reduced in nc-FMR compared with the\nvalues in pc-FMR. Instead, we find that the linewidths\nof the trilayer measured in pc- and nc-FMR agree closely\nwhenω/2π >10 GHz. Furthermore, in nc-FMR there isan additional broadening from 2-10 GHz in the trilayers\nwhich is not predicted by the model.\nIn order to determine whether the low-frequency 1/2 \u001e\u001e\n\u000e\u000e\u001e \u000e = 0\nPy/Cu/Co Py/Cu/MgO/Co Py/Cu/MgO pc nc \nFIG. 3. pc- and nc-FMR linewidths for single (Py) and\ntrilayer (Py/Cu/Co) structures, introducing MgO interlay ers\nto suppress spin pumping. Dashed lines are linear fits to pc-\nFMR linewidths. Solid curves assume (magnetostatic) inter -\nlayer coupling of 10 mT acting on Py and reproduce the low-\nfrequencyupturnin linewidth, seen to be present equally wi th\nand without MgO. Inset:enhancements of nc-FMR linewidth\nover pc-FMR linewidth for the three samples.\nbroadening is related to spin pumping, we have also\nmeasured pc- and nc-FMR linewidths of Py(5 nm)/Cu(5\nnm)/MgO(2 nm) and Py(5 nm)/Cu(3 nm)/MgO(2\nnm)/Co(5 nm) structures, deposited with the same\nseed and capping layers. MgO interlayers are known to\nsuppress spin pumping[27]. Introducing MgO between\nPy and Co, we show in Fig. 3 that the pc linewidths of\nPy in trilayer Py/Cu/Co (blue crosses) are restored to\nthose of single-layerPy/Cu/MgO(overlappinggreen and\nred crosses), demonstrating suppression of spin pumping\nbetween Py and Co. However, we see a very similar\nupturn in low-frequency ( <10 GHz) Py linewidth in\nnc-FMR (red circles), similar to that shown in Fig.\n2(c). We attribute this low-frequency behavior to an\ninterlayer coupling which cants the magnetization of\nPy a few degrees off the film normal when Co is not\nfully saturated along the film normal (i.e. HB< Meff).\nThe solid curves in Fig. 3 assume a coupling field of\n10 mT on Py, parallel to the local Co magnetization,\nwhich reproduce the linewidth broadening of nc-FMR.\nThe peak-like features around 3 GHz show the maximal\nGilbert damping enhancement when the Py magnetiza-\ntion is canted, as demonstrated in Fig. 5 inset.\nFig. 4 shows the central result of the paper. We\ncompare the spin-pumping linewidth enhancements,\nµ0(∆Htri\n1/2−∆Hsingle\n1/2), between pc- and nc-FMR (crosses\nand circles) in Fig. 4(a-d). Here ∆ Hsingle\n1/2and ∆Htri\n1/2\nare the linewidths of Py 1−xCuxin Py1−xCux/Cu single4\nlayers and Py 1−xCux/Cu/Co trilayers, respectively. The\nspin pumping linewidths are quite linear as a function of\nfrequency for the pc-FMR data, as expected. However,\nabove 10 GHz (shaded regions), they are also quite\nlinear in nc-FMR, which is not expected. Collinear\nand noncollinear spin pumping linewidths agree closely.\nThis behavior is in contrast to the predicted behavior\nusingλL\nsr= 38 nm for Co, measured by CPP-GMR[11],\nand calculated in dashed curves according to the theory\nin the Supplemental Materials. From the evident\nagreement between pc- and nc-linewidths above 10 GHz,\nfor all Cu content x, we find no evidence for anisotropy\nin spin relaxation in our Co films. Best fits to the data\nyield longitudinal spin relaxation lengths λL\nsr<2 nm\nin each of the four cases, approximately equal to the\npreviously measured transverse length λT\nsr= 2.4 nm[15].\nOur model has assumed single-domain (macrospin)\n1/2 1/2 1/2 1/2 \n1/2 1/2 1/2 1/2 pc-FMR\npc-FMR(linear fit)nc-FMR\nnc-FMR(theo, λsr L=38 nm)\n(a) x=0.1 (b) x=0.2\n(c) x=0.3 (d) x=0.4(a) x=0.1 (b) x=0.2\n(c) x=0.3 (d) x=0.4\nFIG. 4. Spin pumping contribution to linewidth in pc- and\nnc-FMR. (a-d) Linewidth enhancement of Py 1−xCuxbetween\nsingle layers and trilayers in pc- and nc-FMR, x= 0.1-0.4.\nSolid lines are linear fits to the pc data (crosses); dashed\ncurves are predicted from Eq. (1) using λL\nsr=38 nm. The\nshadows at ω/2π≤10 GHz denote where the low-frequency\nlinewidth broadenings are significant.\nbehavior in both Co and Py 1−xCuxlayers. For\nPy1−xCuxunder field bias well in excess of Ms, the\nmagnetization is well saturated, but for the Co layer,\nwith higher Ms, nonuniform magnetization is possible.\nFor greater control over the Co domain state, we have\nalso carried out angle-dependent, fixed-frequency FMR\nmeasurements on Py 0.8Cu0.2and Py 0.8Cu0.2/Cu/Co.\nHere the Co layer can be saturated more easily be-\ncause the biasing field is canted away from the normal\ncondition. The frequency is set to 10 GHz, where the\nlow-frequency linewidth broadening of Py 0.8Cu0.2is\ninsignificant (Figs. 3 and 4). As the field angle θH\ngoes from 90◦to 0◦(pc to nc), the angle between the\nmagnetizations of Py 0.8Cu0.2and Co changes from zeroto maximum noncollinearity ( ∼50◦) and ∆ αspwould\nbe expected to decrease significantly where the spin\nrelaxation length in Co is markedly anisotropic.\nFig. 5 Insetshows the angular dependence of1/2 1/2 λsr L=38 nmΔH tri/ΔH single\n=1.50 ± 0.02\nθH\n1/2 Single layer \nTrilayer x = 0.2 \nFIG. 5. Angle dependent linewidth ratio ∆ Htri\n1/2/∆Hsingle\n1/2.\nThe shadowed region shows the average with errorbar\n(1.50±0.02).Inset:Angular dependence of µ0∆H1/2for\nPy0.8Cu0.2and Py 0.8Cu0.2/Cu/Co at ω/2π= 10 GHz. Solid\nlines are macrospin calculations.\n∆Hsingle\n1/2(red) and ∆ Htri\n1/2(blue) for Py 0.8Cu0.2.\nThe data can be reproduced through a macrospin\nmodel[28, 29] as shown in the solid curves, using similar\nmagnetizations and isotropic dampings extracted from\nFig. 2(a) and (c) ( µ0Meff= 0.53 T,α1= 0.0114\nfor the single layer, µ0Meff= 0.55 T,α3= 0.0168\nfor the trilayer). The inhomogeneous broadenings are\nnegligible, shown in Fig. 2(c). For small enough θH,\nthe resonance field of the Co starts to fall below the\nexpected macrospin value, as shown in the Supplemental\nMaterials, Section C[25]. We take the angle at which\nthis behavior appears (at θH∼18◦) to be the limit above\nwhich we have the greatest confidence in single-domain\nordering of Co.\nIn the main panel of Fig. 5 we replot the trilayer\nand single-layer linewidths for Py 0.8Cu0.2, shown in\nthe inset, as the ratio ∆ Htri\n1/2/∆Hsingle\n1/2. Because the\ninhomogeneous linewidths are negligible for the struc-\ntures (<0.5 mT), the linewidth ratio for isotropic\nspin pumping would be approximated well through the\nratio of the Gilbert damping for the two configurations,\n∆Htri\n1/2/∆Hsingle\n1/2= 1 + ∆ αsp/α1. We find that the\nlinewidth ratio is in fact constant within experimental\nerror, shown by the shaded region in Fig. 5. The\nblue dashed curve shows the expected behavior for\nanisotropic spin relaxation, assuming λL\nsr= 38 nm, with\na marked decrease in the linewidth ratio for low angles\nθH. A best fit to these data returns λL\nsr<1.1 nm. If\nwe restrict our attention to field angles θH≥18◦, above\nwhich we have confidence in macrospin behavior of the5\nCo layer, the best fit is not changed greatly, with λL\nsr≤4\nnm, within experimental error of the transverse length\nλT\nsr.\nExtrinsic effects, i.e. issues of sample quality, may\nplay some role in the results. First, longitudinal spin\nrelaxation lengths λL\nsr, if equated with the spin diffusion\nlengthλsd, are inversely proportional to (defect-related)\nresistivity[30]. However, four-point probe measure-\nments of the resistivity of our Co (5 nm) films show\n25µΩ·cm, comparable with the 18 µΩ·cm reported\nin the room-temperature CPP-GMR experiment[11],\nand therefore comparably long spin diffusion lengths\nshould be expected. Second, we see that the spin mixing\nconductances g↑↓\neffof Py1−xCux/Cu/Co measured here\nare lower than those measured in Ref. [15], on structures\ndeposited elsewhere. The most plausible source of the\nreduction is a more resistive Cu layer, which adds an\nadditional resistive term[31, 32] (2 e2/h)tCuρCutog↑↓\neff.\nHere however the bulk Cu properties should have little\ninfluence over either spin relaxation length and should\nnot affect the anisotropy of spin relaxation strongly.\nOur estimate of λL\nsrin Co is consistent with a general\nobservation that spin relaxation as measured in spin\npumping/FMR is shorter-ranged than it is as measured\nin magnetotransport. In Pd and Pt, the characteristic\nrelaxation lengths for dynamically pumped spin current\nare measured as 1-5 nm[15, 33–35], whereas in GMR\nthey are closer to 10-20 nm[36, 37]. We suggest therefore\nthat the quantities revealed by the two types of measure-\nments may differ in some respect. For example, robust\nspin-pumping effects have been found in ferrimagnetic\ninsulators such as yttrium iron garnet (YIG). These\neffects clearly have little to do with electronic transport\nin YIG, and their characteristic lengths would refer\nto scattering mechanisms distinct from those involved\nin CPP-GMR. A second possibility, alluded to in the\nreview in Ref. [14], is that the room-temperature spin\ndiffusion length of 38 nm in [11] is an overestimate due\nto technical issues of the CPP-GMR measurement in\nCo multilayers; the majority of such measurements in\nvarious ferromagnets show <10 nm[14]. Our results, in\nthis scenario, may alternately imply that the short spin\ndiffusion length observed in Py is not far away from that\nof Co.\nIn summary, we have experimentally demonstrated\nthat the spin relaxation in Co, as measured by non-\ncollinear spin pumping, is largely isotropic. The\nestimated longitudinal spin relaxation length, <2 nm,\nis an order of magnitude smaller than measured by\nmagnetotransport but comparable to the transverse spin\nrelaxation length. We acknowledge NSF-DMR-1411160\nfor support.[1] T. Valet and A. Fert, Phys. Rev. B 48, 7099 (1993).\n[2] S. Zhang, P. M. Levy and A. Fert, Phys. Rev. Lett. 88,\n236601 (2002).\n[3] M. D. Stiles and A. Zangwill, Phys. Rev. B 66, 014407\n(2002).\n[4] Y. Tserkovnyak, A. Brataas and G. E. W. Bauer, Phys.\nRev. Lett. 88, 117601 (2002).\n[5] B. Heinrich, Y.Tserkovnyak, G. Woltersdorf, A.Brataas ,\nR. Urban and G. E. W. Bauer, Phys. Rev. Lett. 90,\n187601 (2003)\n[6] A. Shpiro, P. M. Levy, and S. Zhang, Phys. Rev. B 67,\n104430 (2003)\n[7] J. ZhangandP. M. Levy, Phys. Rev. B 70, 184442 (2004)\n[8] J. Zhang, P. M. Levy, S. Zhang, and V. Antropov, Phys.\nRev. Lett. 93, 256602 (2004)\n[9] J. ZhangandP. M. Levy, Phys. Rev. B 71, 184426 (2005)\n[10] C. Petitjean, D. Luc and X. Waintal, Phys. Rev. Lett.\n109, 117204 (2012).\n[11] L. Piraux, S. Dubois, A. Fert and L. Belliard, Eur. Phys.\nJ. B4, 413 (1998).\n[12] S. D. Steenwyk, S. Y. Hsu, R. Loloee, J. Bass and W. P.\nPratt Jr, J. Magn. Magn. Mater. 170, L1 (1997).\n[13] S. Dubois, L. Piraux, J. M. George, K. Ounadjela, J. L.\nDuvail and A. Fert, Phys. Rev. B 60, 477 (1999).\n[14] J. Bass and W. P. Pratt Jr, J. Phys.: Condens. Matter\n19, 183201 (2007).\n[15] A. Ghosh, S. Auffret, U. Ebels and W. E. Bailey, Phys.\nRev. Lett. 109, 127202 (2012).\n[16] T. Taniguchi, S. Yakata, H. Imamuraand Y. Ando, Appl.\nPhys. Express 1, 031302 (2008).\n[17] J. Foros, G. Woltersdorf, B. Heinrich, and A. Brataas, J.\nAppl. Phys. 97, 10A714 (2005).\n[18] Y. Tserkovnyak, A. Brataas and G. E. W. Bauer, Phys.\nRev. B.67, 140404(R) (2003).\n[19] T. Taniguchi and H. Imamura Phys. Rev. B. 76, 092402\n(2007).\n[20] Y. Guan and W. E. Bailey J. Appl. Phys. 101, 09D104\n(2007).\n[21] C. Cheng, N. Sturcken, K. Shepard and W. E. Bailey,\nRev. Sci. Instrum. 83, 063903 (2012).\n[22] Y. Li and W. E. Bailey, Phys. Rev. Lett. 116, 117602\n(2016).\n[23] Innc-FMR,thesample normal is aligned carefully (abou t\ntwoaxes, with <0.2◦precision) tomaximize µ0Hresofthe\nPy1−xCuxlayers at 3 GHz. This step is critical to reduce\nthe inhomogeneous broadening.\n[24] Y. Tserkovnyak, A. Brataas, G. E. W. Bauer and B. I.\nHalperin, Rev. Mod. Phys. 77, 1375 (2005).\n[25] See the Supplemental Materials for more discussions.\n[26] M. J. Hurben and C. E. Patton, J. Appl. Phys. 83, 4344\n(1998).\n[27] O. Mosendz, J. E. Pearson, F. Y. Fradin, S. D. Bader\nand A. Hoffmann, Appl. Phys. Lett. 96, 022502 (2010).\n[28] W. Platow, A. N. Anisimov, G. L. Dunifer, M. Farle and\nK. Baberschke, Phys. Rev. B 58, 5611 (1998).\n[29] S. Mizukami, Y. Ando and T. Miyazaki, Phys. Rev. B\n66, 104413 (2002).\n[30] J. Bass and W. P. Pratt Jr, J. Magn. Magn. Mater. 200,\n274 (1999).\n[31] M. Zwierzycki, Y. Tserkovnyak, P. J. Kelly, A. Brataas\nand G. E. W. Bauer, Phys. Rev. B 71, 064420 (2005).6\n[32] A. Ghosh, J. F. Sierra, S. Auffret, U. Ebels and W. E.\nBailey,Appl. Phys. Lett. 98, 052508 (2011).\n[33] W. Zhang, V. Vlaminck, J. E. Pearson, R. Divan, S. D.\nBader and A. Hoffmann, Appl. Phys. Lett. 103, 242414\n(2013).\n[34] V. Vlaminck, J. E. Pearson, S. D. Bader and A. Hoff-\nmann,Phys. Rev. B 88, 064414 (2013).\n[35] M. Caminale, A. Ghosh, S. Auffret, U. Ebels, K. Ollefs,F. Wilhelm, A. Rogalev and W. E. Bailey, Phys. Rev. B\n94, 014414 (2016).\n[36] H. Kurt, R. Loloee, K. Eid, W. P. Pratt Jr. and J. Bass,\nAppl. Phys. Lett. 81, 4787 (2002).\n[37] M. Morota, Y. Niimi, K. Ohnishi, D. H. Wei, T. Tanaka,\nH. Kontani, T. Kimura and Y. Otani, Phys. Rev. B 83,\n174405 (2011).arXiv:1502.05687v3  [cond-mat.mtrl-sci]  23 Nov 2016Supplemental Material to ”Characterization of spin relaxa tion anisotropy in Co using\nspin pumping”\nY. Li, W. Cao and W. E. Bailey1\nDept. of Applied Physics & Applied Mathematics, Columbia Un iversity,\nNew York NY 10027, USA\n(Dated: 27 March 2021)\n1A. Calculation of spin-pumping damping for noncollinearly magnetized,\nasymmetric trilayers\nN\nm1m2\nF1F2θ\nIs1 pump \nθ\nxy\nIs1 pump θ\nxy\nm2 (0,1)\nm1 (-sinθ,cosθ)\nμsN (μ x\nsN ,μ y\nsN )Is1 pump \n(cosθ,sinθ)\nμsN (0, 0, μz\nsN )m1 (-sinθ,cosθ) m2 (0,1)Case 1 Case 2(a)\n(b) (c)Hrf (ωt)\nFIG. 1. (a) Magnetization configuration of the asymmetric F 1/N/F2trilayer. (b) An instant in\nwhich the spin polarization of Ipump\ns1is orthogonal to both m1andm2.µsNis also orthogonal to\nm1andm2. (c) An instant in which the spin polarization of Ipump\ns1is in the same plane of m1and\nm2.µsNis also in the same plane of m1andm2\nConsider an asymmetric ferromagnet / noble metal / ferromagnet (F1/N/F2) spin-valve\ntrilayer structure, shown in Fig. 1(a). The time-averaged magnet ization of F 1is pictured\nalong the film-normal, although it can take any angle with respect to t he film-normal. We\nassume that F 1undergoes small-angle precession. F 2is noncollinearly magnetized with\nrespect to F 1, whereθis the angle of noncollinearity or misalignment; θ= 0 for parallel\nmagnetizations m1=m2, wheremi,i= 1 or 2, is the unit vector of the magnetization Mi\nof Fi. The magnetizaton of F 2is taken to be stationary. The spin current flows from the N\nspacer to each of the F layers F 1, F2are1–4:\nIN→F1\ns=g∗\n1\n4πm1(µsN·m1) +˜g↑↓\n1\n4πm1×µsN×m1 (1)\n2IN→F2\ns=g∗\n2\n4πm2(µsN·m2) +˜g↑↓\n2\n4πm2×µsN×m2 (2)\nwhereµsNis the spin accumulation vector in the N layer, g∗\niand ˜g↑↓\niare the effective\nlongitudinal spin conductance and transverse spin mixing conducta nce for F i/N interface,\nrespectively. Here the spin current vector denotes the direction of spin polarization, the\ndirection of current flow always being normal to interfaces. The co nservation of spin angular\nmomentum, assuming spin-current conservation (negligible dissipat ion) N, gives:\nIN→F1\ns+IN→F2\ns=Ipump\ns1 (3)\nwhereIpump\ns1is the pumped spin current from F 1into N2,5:\nIpump\ns1=¯h\n4π˜g↑↓\n1m1×˙ m1 (4)\nSubstituting Eq. (1), (2) and (4) into the continuity expression (3 ), we obtain a vector\nequation in terms of the vector spin accumulation µsN. To calculate the spin pumping\ndamping enhancement, we seek solutions for µsNin order to find the spin current flow into\nm2, which is absorbed by m2.\nThe vector Ipump\ns1, proportional to m1×˙ m1, rotates in the plane with normal given by m1.\nAssuming a finite misalignment angle θbetween m1andm2,Ipump\ns1will oscillate between\nfully orthogonal to m2(Fig. 1b) and canted away from orthogonality by θ(Fig. 1c). We\nconsider these two extreme cases during the precession of m1. In case 1 (Fig. 1b), Ipump\ns1is\nperpendicular to both m1andm2. In case 2 (Fig. 1c), Ipump\ns1is in the same plane as m1\nandm2and has the largest longitudinal component along m2.\nIncase 1,IN→F1s,IN→F2sandµsNare all parallel to Ipump\ns1. In Eqs. (1) and (2) the first\nterms become zero and only the second terms remain. The solution o f Eqs. (1)-(3) has a\nscalar form along the direction ˆ zorthogonal to both F 1and F 2:\nIN→F1{F2}\ns,z =/parenleftBigg\n˜g↑↓\n1{2}\n˜g↑↓\n1+ ˜g↑↓\n2/parenrightBigg\nIpump\ns1, µz\nsN=/parenleftbigg4π\n˜g↑↓\n1+ ˜g↑↓\n2/parenrightbigg\nIpump\ns1 (5)\nIt has been shown previously that2,5the dissipation of spin angular momentum due to a\ntransverse spin mixing conductance g↑↓leads to an additional Gilbert damping term ∆ α=\nγ¯hg↑↓/4πMsd. With only IN→F2sdissipated, the spin-pumping damping enhancement can be\nexpressed as:\n∆αsp=Iback\ns2·∆α0\nIpump\ns1= ∆α0·˜g↑↓\n2\n˜g↑↓\n1+ ˜g↑↓\n2(6)\n3with ∆α0=γ¯h˜g↑↓\n1/4πMs1d. Eq. (6) is identical to the collinear spin pumping case with an\neffective spin mixing conductance (˜ g↑↓\neff)−1= (˜g↑↓\n1)−1+(˜g↑↓\n2)−1.\nIncase 2,µsNhas only a component coplanar with m1andm2(µx\nsNandµy\nsN). In Eq.\n(1) and (2) both terms need to be considered. The ˆ xand ˆycomponents of Eq. (3) can be\nwritten as:\n4πIpump\ns1/parenleftbiggcosθ\nsinθ/parenrightbigg\n=g∗\n1(−µx\nsNsinθ+µy\nsNcosθ)/parenleftbigg−sinθ\ncosθ/parenrightbigg\n+ ˜g↑↓\n1(µx\nsNcosθ+µy\nsNsinθ)/parenleftbiggcosθ\nsinθ/parenrightbigg\n+g∗\n2µy\nsN/parenleftbigg0\n1/parenrightbigg\n+ ˜g↑↓\n2µx\nsN/parenleftbigg1\n0/parenrightbigg\n(7)\nThe solution of Eq. (7) can be expressed as:\nµx\nsN=4πIpump\ns1(Ccosθ−Bsinθ)\nAC−B2(8a)\nµy\nsN=4πIpump\ns1(Asinθ−Bcosθ)\nAC−B2(8b)\nwhere\nA(θ) =g∗\n1sin2θ+ ˜g↑↓\n1cos2θ+ ˜g↑↓\n2 (9a)\nB(θ) = (˜g↑↓\n1−g∗\n1)sinθcosθ (9b)\nC(θ) =g∗\n1cos2θ+ ˜g↑↓\n1sin2θ+g∗\n2 (9c)\nThe spin torque is equal to the conponent of IN→F2stransverse to m1, or the component\nwhich is parallel to Ipump\ns1. Thus the spin-pumping damping enhancement can be written in\nterms of the defined misalignment-dependent quantities A(θ),B(θ),C(θ) as:\n∆αsp(θ) =IN→F2s·Ipump\ns1\nIpump\ns1·∆α0\nIpump\ns1\n=∆α0·g∗\n2(Asin2θ−Bsinθcosθ)+ ˜g↑↓\n2(Ccos2θ−Bsinθcosθ)\nAC−B2(10)\nIt is easy to verify that at θ= 0◦Eq. (10) recovers Eq. (6), same as the collinear spin\npumping.\nHaving treated the two special spin current orientations, we need to take the average of\nall the orientation possibilities. We refer to the calculation by Taniguc hi, et al.3, that in a\nsymmetric spin valve (F 1=F2) the small-precession limit of averaged spin-pumping damping\nenhancement is equal to the arithmetic mean of damping enhanceme nt with out-of-plane\n4Ipump\ns1(case 1) and in-plane Ipump\ns1(case 2). The Eq. 13 in Ref.3can be simplified, at small\nprecession angle, as:\n∆αsp= ∆α0/bracketleftbigg\n1−(ν/2)sin2θ\n1−ν2cos2θ/bracketrightbigg\n(11)\nwhich is the average of ∆ α0and ∆α0[1−νsin2θ/(1−ν2cos2θ)] (Eq. 5 in Ref.3). We apply\nit to the asymmetric spin valve condition: all the theoretical curves in the main text are\ncalculated from the mean of Eq. (6) and Eq. (10).\nThe theoretical curves in Fig. 4 and 5 of the main text are calculated using the routine,\nassuming λL\nsr= 38 nm for Co. The new estimation of λL\nsr(<2 nm) in the manuscript takes\nthe best value that fits the damping calculation to the experimental data.\nB. Values of g∗and˜g↑↓\nIn this section we calculate the value of the two effective spin conduc tances. The trans-\nverse spin mixing conductance ˜ g↑↓(Sharvin correction includedtserkovnyakRMP2005) of\neachinterfacecanbecalculatedfromtheeffectivespinmixingcondu ctanceofPy 1−xCux/Cu/Co\nstructures and the comparison measurements of Py 1−xCux/Cu/Pt and Co/Cu/Pt (Table I).\nFor Py 1−xCux/Cu/Co, the total spin mixing conductance can be expressed as:\n1\ng↑↓\nPy1−xCux/Cu/Co=1\n˜g↑↓\nPy1−xCux/Cu+1\n˜g↑↓\nCo/Cu(12)\nFor F/Cu/Pt (F=Py 1−xCuxor Co), the effective spin mixing conductance can be formulated\nas:\n1\ng↑↓\nF/Cu/Pt=1\n˜g↑↓\nF/Cu+1\n˜g↑↓\nCu/Pt(13)\nIn the experiment the thicknesses of Pt are kept the same and we c an treat ˜ g↑↓\nCu/Ptas a\nconstant. Solving Eq. (12) and (13) we obtain:\n1\n˜g↑↓\nPy1−xCux/Cu=1\ng↑↓\nPy1−xCux/Cu/Co+1\ng↑↓\nPy1−xCux/Cu/Pt−1\ng↑↓\nCo/Cu/Pt(14a)\n1\n˜g↑↓\nCo/Cu=1\ng↑↓\nPy1−xCux/Cu/Co−1\ng↑↓\nPy1−xCux/Cu/Pt+1\ng↑↓\nCo/Cu/Pt(14b)\nIn Table II we list the calculated values of ˜ g↑↓\nPy1−xCux/Cuand ˜g↑↓\nCo/Cu. Forx= 0.1 and 0.3\nwe take the linear interpolated values to evaluate g↑↓\nPy1−xCux/Cu/Pt. In addition, we also show\n5the values compensating the Sharvin correction, with 1 /g↑↓\ni= 1/˜g↑↓\ni+ 1/2gSh\nCu,gSh\nCu= 15\nnm−2.\nCompared with previous measurements10,11, we find smaller values of g↑↓\nPy1−xCux/Cu/Coand\ng↑↓\nPy1−xCux/Cuforx= 0. However we argue that the spin mixing conductances of Co/Cu/ Pt\nin Table I and Co/Cu interfaces in Table II are reasonable, which ensu res a good Co/Cu\ninterface crucial for the study of spin relaxation anisotropy. It is also possible that a resistive\nCu spacer contributes an additional resistance, (2 e2/h)tCuρCu2, to the right side of Eq. (13).\nTo reduce the spin mixing conductance of Py/Cu/Co from 15.0 nm−2in Ref.10to 7.6 nm−2\nin Table I, one needs to take ρCu= 16.8µΩ·cm. However we point out that this resistive\nscattering will contribute to both transverse and longitudinal spin conductance by the same\namount, and the anisotropy of spin relaxation should not be affecte d. In practice, we take\nthe effective interfacial spin mixing conductance into the model for the estimation of λL\nsrand\nand use the values of λT\nsrfrom Ref.10.\n(Unit: nm−2)x= 0x= 0.1x= 0.2x= 0.3x= 0.4\ng↑↓\nPy1−xCux/Cu/Co7.6 5.6 7.3 6.8 6.8\ng↑↓\nPy1−xCux/Cu/Pt6.0 - 5.0 - 5.0\ng↑↓\nCo/Cu/Pt7.9\nTABLE I. Experimental values of (effective) spin mixing condu ctance of Py1−xCux/Cu/Co,\nPy1−xCux/Cu/Pt and Co /Cu/Pt samples, extracted from spin-pumping linewidth enhance ments.\nThe effective longitudinal spin conductance g∗can be expressed as1:\n1\ng∗=g↑↑+g↓↓\n2g↑↑g↓↓+1\ngsdtanh(tF/λLsr)(15)\nIn the first term, g↑↑{↓↓}\niis the interfacial spin-up {spin-down }conductance. g↑↑{↓↓}can be\ncalculated by 1 /g↑↑{↓↓}= (e2/h)AR↑{↓}\nF/NwhereAR↑{↓}\nF/Nis the electron interface resistance.\nWe use the experimental value from GMR measurements: 2 AR∗= (AR↑+AR↓)/2 = 1.04\nfΩ·m2for Co/Cu6and 1.0 fΩ ·m2for Py/Cu7. We can calculate that 2 g↑↑g↓↓/(g↑↑+g↓↓) = 26\n6(Unit: nm−2)x= 0x= 0.1x= 0.2x= 0.3x= 0.4\n˜g↑↓\nPy1−xCux/Cu11.7 8.6 9.5 9.1 9.1\n˜g↑↓\nCo/Cu21.9 16.2 31.5 27.2 27.2\ng↑↓\nPy1−xCux/Cu8.4 6.7 7.2 7.0 7.0\ng↑↓\nCo/Cu12.7 10.5 15.4 14.3 14.3\nTABLE II. “˜ g↑↓\ni”: Sharvin-corrected spin mixing conductance of Py1−xCux/Cu and Co /Cu in-\nterfaces, calculated from Eq. (14). “ g↑↓\ni”: interfacial spin mixing conductance compensating the\nSharvin conductance of Cu layer. i= Py1−xCux/Cu or Co/Cu.\nnm−2for both interfaces.\nIn the second term, gsdhas been expressed in Ref.1as:\ngsd=h\ne2λL\nsr2σ↑σ↓\nσ↑+σ↓(16)\nwhereσ↑,↓are the spin-up/down electron conductivity in F, his the Planck constant and\neis the electronic charge. Here we simply take σ↑=σ↓=σ/2 (σis the total electrical\nconductivity), which has also been done in Eq. (74) of Ref.2. Following this treatment, the\nterm 2σ↑σ↓/(σ↑+σ↓) is replaced by σ/2. Taking ρCo= 25µΩ·cm andρPy= 30µΩ·cm from\nour four-point probe measurements and λL\nsr= 38 nm for Co8and 4.3 nm for Py9from the\nliteratures, we calculate gsdtanh(tF/λL\nsr) to be 0.18 nm−2for Co and 8.3 nm−2for Py when\nthe F thickness is 5 nm; the large disagreement comes from the expe cted difference in λL\nsr.\nAs a result, g∗= 0.18 nm−2for Co and 6.2 nm−2for Py are obtained from Eq. (15) and\nused to produce the theoretical curves in the manuscript.\nIn the experiment, we do not find the anisotropic response of spin p umping predicted\nabove. According to our model, the lack of anisotropic response ca n be explained best\nthrough a difference in the longitudinal spin conductance g∗for Co/Cu, as this is the most\ndominant terminEq. (10)andsensitive to λL\nsr. This isbecauseintheexperiments wechoose\nthe thickness of Co to be much less than 38 nm in order to examine the spin relaxation\nanisotropy.\nFrom Py to Py 1−xCux, we should expect that both g↑↑{↓↓}andσ↑{↓}will increase due to a\nbetter conducting ability of Cu than Py. λL\nsrmay also vary. However we emphasize that in\nEq. (10), the anisotropy is dominated by g∗\n2and ˜g↑↓\n2and not sensitive to g∗\n1. For example in\n7the angular dependence of linewidth ration for x= 0.2 (Fig. 5 of the main text), increasing\ng∗\nPy0.8Cu0.2/Cuby a factor of two will change the single-domain estimation of λL\nsrfrom 1.8±2.7\nto 2.1±2.8, still much smaller than the GMR measurements. Thus for simplicity we keep\nusing the value of g∗of Py for Py 1−xCuxlayers.\nC. Single-domain limit determination\n= 18 deg = 9 deg\nPy 0.8Cu 0.2 , 10 GHz Co, 14.8 GHz Co, 15.0 GHz \nPy 0.8Cu 0.2 , 10 GHz Py 0.8Cu 0.2 Py 0.8Cu 0.2 Co Theory (f=15 GHz) Theory (f=14.8 GHz)\nFIG. 2. Resonance peak of Co and Py 0.8Cu0.2independently measured in Py 0.8Cu0.2/Cu/Co with\nθH= 18◦(a) and 9◦(b). The resonance frequency of Py 0.8Cu0.2are both 10 GHz. The resonance\nfrequency of Co is adjusted so that the µ0Hresof Co is equal to Py 0.8Cu0.2. Dashed curves show\nthe theoretical prediction of Co resonance signals.\nTo determine whether the Co layer is in a single-domain state in Py 0.8Cu0.2/Cu/Co when\nthe Py 0.8Cu0.2layer is at resonance, we have measured the FMR signal of Co at diffe rentθH.\nFirst we measure the FMR signal of Py 0.8Cu0.2at one angle and determine the resonance\nfieldµ0Hres. Next we adjust the frequency so that the Co FMR signal can be me asured at\nthe same field. Then we compare the lineshape with the macrospin mod el prediction12. In\nFig. 2(a), when θH= 18◦the resonance field µ0Hresfor Py 0.8Cu0.2is 0.53 T at 10 GHz.\nFor Co, the resonance field is located at 0.53 T for 14.8 GHz. The macr ospin model for\nangle-dependent FMR shows ω/2π= 14.8 GHz, identical to the experiment, showing the\nCo can indeed be treated as a macrospin. However for θH= 9◦(Fig. 2b), we find that the\nCo resonance is located at 15.0 GHz, quite different from the macros pin prediction of 12.3\nGHz. To see the difference more clearly, we plot (dashed lines) the ma crospin prediction for\nboth Py 0.8Cu0.2and Co resonances at 15.0 GHz, based on the magnetizations and line widths\n8measured from perpendicular FMR. The Py 0.8Cu0.2peak matches with experiment. The\ncalculated Co peak deviates from experiment in both resonance field and linewidth. Thus\nwe determine the single-domain limit of θHto be somewhere between 9◦and 18◦in the\nsample. The upper bound 18◦is used in the manuscript for the single-domain limit.\nREFERENCES\n1Y. Tserkovnyak, A. Brataas and G. E. W. Bauer, Phys. Rev. B 67, 140404(R) (2003).\n2Y. Tserkovnyak, A. Brataas, G. E. W. Bauer and B. I. Halperin, Rev. Mod. Phys 77, 1375\n(2005).\n3T. Taniguchi and H. Imamura, Phys. Rev. B 76, 092402 (2007).\n4T. Taniguchi, S. Yakata, H. Imamura and Y. Ando, Appl. Phys. Express 1, 031302 (2008).\n5Y. Tserkovnyak, A. Brataas and G. E. W. Bauer, Phys. Rev. Lett. 88, 117601 (2002).\n6A. C. Reilly, W.-C. Chiang, W. Park, S. Y. Hsu, R. Loloee, S. Steenwyk , W. P. Pratt Jr\nand J. Bass, IEEE Trans. Magn. 34, 939 (1998).\n7S. D. Steenwyk, S. Y. Hsu, R. Loloee, J. Bass and W. P. Pratt Jr., J. Magn. Magn. Mater.\n170, L1 (1997).\n8L. Piraux, S. Dubois, A. Fert and L. Belliard, Eur. Phys. J. B 4, 413 (1998)\n9S. Dubois, L. Piraux, J. M. George, K. Ounadjela, J. L. Duvail and A . Fert,Phys. Rev. B\n60, 477 (1999).\n10A. Ghosh, S. Auffret, U. Ebels and W. E. Bailey, Phys. Rev. Lett. 109, 127202 (2012).\n11A. Ghosh, J. F. Sierra, S. Auffret, U. Ebels and W. E. Bailey, Appl. Phys. Lett. 98, 052508\n(2011).\n12S. Mizukami, Y. Ando and T. Miyazaki, Jpn. J. Appl. Phys. 10409, 580 (2001).\n9"    },    {        "title": "1502.06724v2.High_Quality_Yttrium_Iron_Garnet_Grown_by_Room_Temperature_Pulsed_Laser_Deposition_and_Subsequent_Annealing.pdf",        "content": "arXiv:1502.06724v2  [cond-mat.mtrl-sci]  12 Jun 2015Yttrium Iron Garnet Thin Films with Very Low Damping\nObtained by Recrystallization of Amorphous Material\nC. Hauser,1T. Richter,1N. Homonnay,1C. Eisenschmidt,1\nH. Deniz,2D. Hesse,2S. Ebbinghaus,3and G. Schmidt1,4,∗\n1Institute of Physics, Martin-Luther-Universit¨ at Halle- Wittenberg,\nVon-Danckelmann-Platz 3, D-06120 Halle, Germany\n2Max-Planck-Institut fr Mikrostrukturphysik,\nWeinberg 2, D-06120 Halle, Germany\n3Institute of Chemistry, Martin-Luther-Universit¨ at Hall e-Wittenberg,\nKurt-Mothes-Str. 2, D-06120 Halle, Germany\n4Interdisziplinres Zentrum fr Materialwissenschaften,\nMartin-Luther University Halle-Wittenberg, Nanotechnik um Weinberg,\nHeinrich-Damerow-Str. 4, D-06120 Halle, Germany\nAbstract\nWe have investigated recrystallization of amorphous Yttri um Iron Garnet (YIG) by annealing\nin oxygen atmosphere. Our findings show that well below the me lting temperature the material\ntransforms into a fully epitaxial layer with exceptional qu ality, both structural and magnetic.\nIn ferromagnetic resonance (FMR) ultra low damping and extr emely narrow linewidth can be\nobserved. For a 56nm thick layer a damping constant of α=(6.63±1.50)·10−5is found and the\nlinewidth at 9.6GHz is as small as 1.30 ±0.05Oe which are the lowest values for PLD grown thin\nfilms reported so far. Even for a 20nm thick layer a damping con stant ofα=(7.51±1.40)·10−5\nis found which is the lowest value for ultrathin films publish ed so far. The FMR linewidth in this\ncase is 3.49 ±0.10Oe at 9.6GHz. Our results not only present a method of dep ositing thin film\nYIG of unprecedented quality but also open up new options for the fabrication of thin film complex\noxides or even other crystalline materials.I. INTRODUCTION\nYIG can be considered the most prominent material in spin dynamics in thin films and\nrelated areas. It is widely used in ferromagnetic resonance experim ents1–6, research on\nmagnonics7–14and magnon-based Bose-Einstein-condesates15–18because of its exceptionally\nlow damping even in thin films. In research on spin pumping19–23and investigation of the\ninverse spin hall effect1,19,24–27it greatly facilitates experiments because it is an insulating\nmaterial which avoids numerous side effects which occur when ferro magnetic metals are\nused.28,29The field of spin caloritronics30–38also would not have developed that rapidly\nwithout the availability of a non-conducting magnet with long magnon lif etimes.\nThe new fields of applications have resulted in a growing need of high qu ality thin films, for\nexample for integrated magnonics where layers need to be as thin as 100nm or even less.\nWhile formerly only micrometer thick films were used which can be obtain ed by liquid phase\nepitaxy with very high quality39–42ultrathin films are nowadays mostly fabricated by pulsed\nlaser deposition (PLD) of epitaxial films at elevated temperature. E specially for ultra thin\nfilms (20nm or less) grown by PLD quality is high but limited and best resu lts so far show\na linewidth in FMR of 2.1Oe at 9.6GHz.1\nII. SAMPLE FABRICATION\nThe amorphous YIG layers are deposited on (111) oriented gallium ga dolinium garnet\n(GGG) substrates. After deposition the samples are removed fro m the PLD chamber and\ncut into smaller pieces before the subsequent annealing procedure which is done in a quartz\noven under pure (99.998%) oxygen atmosphere at ambient pressu re at 800◦C for 30minutes\n(sampleA, 56nm thick), at 800◦C for three hours (sampleB, 20nm thick), and at 900◦C\nfor four hours (sampleC, 113nm thick). After annealing the sample s are subject to various\nstructural and magnetic characterization experiments.\nIII. STRUCTURAL CHARACTERIZATION\nStructural characterization is done by X-ray diffraction, X-ray r eflectometry transmission\nelectron microscopy, and Reflection high energy electron diffractio n (RHEED).\n2A. X-ray characterization\nX-ray diffraction is performed by doing an ω/2θscan of the (444) reflex and a rocking\ncurve of the YIG layer peak. Before annealing the diffraction patte rn (Figure1a) only shows\nthe peak of the GGG substrate indicating an amorphous or at least h ighly polycrystalline\nYIG film. A truly amorphous nature is confirmed by transmission elect ron microscopy as\ndescribed below. After annealing, the diffraction pattern is complet ely changed. Figure1b\nshows the ω/2θscan for sampleC. Here we clearly observe the diffraction peak of th e YIG\nfilm at an angle corresponding to the small lattice mismatch of YIG on G GG which is only\n0.057%. Even thickness fringes can be observed indicating a very sm ooth layer with low\ninterface and surface roughness. The layer peak is further inves tigated in a rocking curve\n(Figure1c) which shows a full width at half maximum (FWHM) of 0.015◦indicating a fully\npseudomorphic YIGlayer. Roughnessisalsocrosschecked usingX- rayreflectometryshowing\nanRMSvalueofless than0.2nm43. It should, however, benotedthatfornot-annealedlayers\nthe RMS roughness is even smaller than 0.1nm.\nB. Transmission electron microscopy\nFor Transmission electron microscopy (TEM) preparation the samp le surface is protected\nby depositing a thin Pt layer. Then thin lamellae are cut out using focus ed ion beam\npreparation. The orientation of the samples is chosen for cross se ctional TEM along the\ncubic crystalline axis. TEM is performed using a JEOL JEM-4010 electr on microscope\nat an acceleration voltage of 400kV. For the nominally amorphous sa mple the pictures\n(Figure2a)showapurefilmwithoutinclusionsbutalsowithoutanytra ceofpolycristallinity.\nFurtheranalysisusingfastfouriertransformconfirmsthattheY IGlayerisindeedcompletely\namorphous. For an annealed sample (sampleC) the result of the TEM investigation is\nsurprising (Figure2b). The sample is not only monocrystalline but it als o shows no sign of\ninclusions or defects and even the interface to the GGG appears fla wless.\nC. Reflection high energy electron diffraction\nThe atomic order of the layer surface after annealing is further inv estigated by Reflection\nhigh energy electron diffraction (RHEED). For this purpose sampleB is again introduced\n3into the PLD chamber after the annealing process. After evacuat ion a clear RHEED pattern\nis observed. The RHEED image (Figure2c) not only shows the typical pattern for a YIG\nsurface during high temperature growth but also exhibits the so ca lled Kikuchi lines.44We\ndo not observe these lines in high temperature growth of epitaxial Y IG. They are typically\na sign of a surface of excellent two dimensional growth, again indicat ing that the crystalline\nquality of the annealed layers is extremely high.\nIV. MAGNETIC CHARACTERIZATION\nMagnetic characterization is done using SQUID magnetometry and F MR at room tem-\nperature.\nA. SQUID magnetometry\nInSQUID magnetometry hysteresis loopsare taken onsampleC. Th e data iscorrected by\nsubtracting alinear paramagneticcontribution which iscaused by th e GGGsubstrate. After\ncorrection the observed saturation magnetization is (105 ±3)emucm−3which is approx.\n30% below the bulk value45(Figure3). The coercive field is determined as (0.8 ±0.1)Oe.\nB. Ferromagnetic resonance\nFMR is performed by putting the samples face down on a coplanar wav eguide whose\nmagnetic radio frequency (RF) field is used for excitation. The setu p is placed in a homoge-\nnous external magnetic field which is superimposed with a small low fre quency modulation.\nRF absorption is measured using a lock-in amplifier. As expected no sig nal can be detected\nfor unannealed YIG layers. For annealed samples a clear resonance is observed. Figure4a\nshows the resonance signal for sampleA. The linewidth which is obtain ed by multiplying the\npeak to peak linewidth of the derivative of the absorption by a facto r of√\n3/22,3,46,47is only\n1.30±0.05 Oe at 9.6GHz which is the smallest value for thin films reported so fa r.1In Fig-\nure4b the resonance of sampleB is shown. Here the linewidth at 9.6GH z is 3.49±0.10Oe.\nFor sampleC the linewidth is 1.65 ±0.10Oe at 9.6GHz (no figure).\nIn order to determine the damping constant αfrequency dependent measurements are per-\n4formed on sampleA. The excitation frequency is varied between 8 an d 12GHz. Results are\nplotted in Figure4c. As described by Chang et al.46and Liuet al.2we first determine the\ngyromagnetic ratio of γ=(2.92±0.01)MHzOe−1and a linewidth at zero magnetic field of\napprox. 1.11 ±0.05Oe. The damping canthen becalculatedfromthefrequency dep endence\nof the linewidth to α=(6.63±1.50)·10−5. This damping is even lower than the lowest value\nreported by Chang et al.46. It is interesting to note that Chang et al.did not observe a\nsimilarly small linewidth for their layer.46dAllivy Kelly et al.on the other hand do observe\na smaller linewidth for 20nm thick layers of 2.1Oe at 9.6GHz1, however, the damping they\nfind is three times as big as in our case. For sampleB (20nm) we found a gyromagnetic ratio\nofγ=(2.81±0.01)MHzOe−1and a linewidth at zero magnetic field of approx. 3.25 ±0.05\nOe and the damping was determined as α=(7.51±1.40)·10−5which is also lower than any\nother value reported for similarly thin films (Figure4d).\nV. DISCUSSION\nIn conclusion we can state that using high temperature annealing in o xygen atmosphere\nit is possible to transform amorphous YIG layers of tens of nanomet ers of thickness into\nepitaxial thin films with extremely small FMR linewidth and exceptionally lo w damping.\nThe crystalline quality is extremely high. Our findings may thus presen t a new and easy\nroute for thin film fabrication of epitaxial complex oxides.\nVI. METHODS\nThe substrates are prepared by cleaning in acetone and isopropan ol and then introduced\ninto the PLD chamber which is copper sealed and UHV compatible with a b ackground\npressure as low as 10−9mbar. Deposition is done at an oxygen partial pressure of 0.025mba r\nwith the substrate at room temperature. A laser with a typical flue ncy of 2.5Jcm−2and\na wavelength of 248nm is used at a repetition rate of 5Hz. The targe t is a stoichiometric\nYIG target prepared in-house. With these parameters we obtain a growth rate of approx.\n0.5nmmin−1.\n5Acknowledgements\nThis work was supported by the European Commission in the project IFOX under\ngrant agreement NMP3-LA-2010-246102 and by the DFG in the SFB 762. We thank Georg\nWoltersdorf and Sergey Manuilow for fruitful discussion.\n∗Correspondence to G. Schmidt: georg.schmidt@physik.uni- halle.de\n1d’Allivy Kelly, O. et al.Inverse spin hall effect in nanometer-thick yttrium iron garn et/pt\nsystem.Appl. Phys. Lett. 103, 082408 (2013).\n2Liu, T.et al.Ferromagnetic resonance of sputtered yttrium iron garnet n anometer films. J.\nAppl. Phys. 115, 17A501 (2014).\n3Sun, Y.et al.Growth and ferromagnetic resonance properties of nanomete r-thick yttrium iron\ngarnet films. Appl. Phys. Lett. 101, 152405 (2012).\n4Sun, Y.et al.Damping in yttrium iron garnet nanoscale films capped by plat inum.Phys. Rev.\nLett.111, 106601 (2013).\n5Manuilov, S.A.&Grishin,A.M. Pulsedlaserdepositedyigfil ms: Natureofmagneticanisotropy\nii.J. Appl. Phys. 108, 013902 (2010).\n6Manuilov, S. A., Khartsev, S. I. & Grishin, A. M. Pulsed laser deposited yig films: Nature of\nmagnetic anisotropy i. J. Appl. Phys. 106, 123917 (2009).\n7Chumak, A. V., Serga, A. A. & Hillebrands, B. Magnon transist or for all-magnon data process-\ning.Nat. Commun. 5, 4700 (2014).\n8Kurebayashi, H. et al.Controlled enhancement of spin-current emission by three- magnon split-\nting.Nat. Mat. 10, 660–664 (2011).\n9Ustinov, A. B., Drozdovskii, A. V. & Kalinikos, B. A. Multifu nctional nonlinear magnonic\ndevices for microwave signal processing. Appl. Phys. Lett. 96, 142513 (2010).\n10Inoue, M. et al.Investigating the use of magnonic crystals as extremely sen sitive magnetic field\nsensors at room temperature. Appl. Phys. Lett. 98, 132511 (2011).\n11Yu, H.et al.Magnetic thin-film insulator with ultra-low spin wave dampi ng for coherent\nnanomagnonics. Sci. Rep. 4, 6848 (2014).\n12Bi, K.et al.Magnetically tunable mie resonance-based dielectric meta materials. Sci. Rep. 4,\n67001 (2014).\n13Demokritov, S. O. et al.Experimental observation of symmetry-breaking nonlinear modes in\nan active ring. Nature426, 159–162 (2003).\n14Chumak, A. V. et al.All-linear time reversal by a dynamic artificial crystal. Nat. Commun. 1,\n141 (2010).\n15Li, F., Saslow, W. M. & Pokrovsky, V. L. Phase diagram for magn on condensate in yttrium\niron garnet film. Sci. Rep. 3(2013).\n16Nowik-Boltyk, P., Dzyapko, O., Demidov, V. E., Berloff, N. G. & Demokritov, S. O. Spatially\nnon-uniform ground state and quantized vortices in a two-co mponent bose-einstein condensate\nof magnons. Sci. Rep. 2(2012).\n17Demokritov, S. O. et al.Bose-einstein condensation of quassi-equilibrium magnon s at room\ntemperature under pumping. Nature443, 430–433 (2006).\n18Serga, A. A. et al.Bose-einstein condensation in an ultra-hot gas of pumped ma gnons.Nat.\nCommun. 5(2014).\n19Ando, K., Watanabe, S., Mooser, S., Saitoh, E. & Sirringhaus , H. Solution-processed organic\nspin-charge converter. Nat. Mat. 12, 622–627 (2013).\n20Burrowes, C. et al.Enhanced spin pumping at yttrium iron garnet/au interfaces .Appl. Phys.\nLett.100, 092403 (2012).\n21Jungfleisch, M. B., Lauer, V., Neb, R., Chumak, A. V. & Hillebr ands, B. Improvement of the\nyttrium iron garnet/platinum interface for spin pumping-b ased applications. Appl. Phys. Lett.\n103, 022411 (2013).\n22Rezende, S. M. et al.Enhancedspin pumpingdamping in yttrium iron garnet/pt bil ayers.Appl.\nPhys. Lett. 102, 012402 (2013).\n23ˇZuti´ c, I. & Dery, H. Spintronics: Taming spin currents. Nat. Mat. 10, 647–648 (2011).\n24Sakimura, H., Tashiro, T. & Ando, K. Nonlinear spin-current enhancement enabled by spin-\ndamping tuning. Nat. Commun. 5, 5730 (2014).\n25Castel, V., Vlietstra, N., Ben Youssef, J. & van Wees, B. J. Pl atinum thickness dependence\nof the inverse spin-hall voltage from spin pumping in a hybri d yttrium iron garnet/platinum\nsystem.Appl. Phys. Lett. 101, 132414 (2012).\n26Schreier, M. et al.Sign of inverse spin hall voltages generated by ferromagnet ic resonance and\ntemperature gradients in yttrium iron garnet platinum bila yers.J. Phys. D: Appl. Phys. 48,\n7025001 (2015).\n27Kajiwara, Y. et al.Transmission of electrical signals by spin-wave interconv ersion in a magnetic\ninsulator. Nature464, 262–266 (2010).\n28Obstbaum, M. et al.Inverse spin hall effect in ni 81fe19/normal-metal bilayers. Phys. Rev. B\n89(2014).\n29Mecking, N., Gui, Y. S. & Hu, C.-M. Microwave photovoltage an d photoresistance effects in\nferromagnetic microstrips. Phys. Rev. B 76(2007).\n30Bauer, G. E. W., Saitoh, E. & van Wees, B. J. Spin caloritronic s.Nat. Mat. 11, 391–399 (2012).\n31Sinova, J. Spin seebeck effect: Thinks globally but acts local ly.Nat. Mat. 9, 880–881 (2010).\n32Kirihara, A. et al.Spin-current-driven thermoelectric coating. Nat. Mat. 11, 686–689 (2012).\n33Siegel, G., Prestgard, M. C., Teng, S. & Tiwari, A. Robust lon gitudinal spin-seebeck effect in\nbi-yig thin films. Sci. Rep. 4(2014).\n34Qu, D., Huang, S., Hu, J., Wu, R. & Chien, C. Intrinsic spin see beck effect in au/yig. Phys.\nRev. Lett. 110(2013).\n35An, T.et al.Unidirectional spin-wave heat conveyer. Nat. Mat. 12, 549–553 (2013).\n36Uchida, K.-i. et al.Observation of longitudinal spin-seebeck effect in magnetic insulators. Appl.\nPhys. Lett. 97, 172505 (2010).\n37Adachi, H., Ohe, J.-i., Takahashi, S. & Maekawa, S. Linear-r esponse theory of spin seebeck\neffect in ferromagnetic insulators. Phys. Rev. B 83(2011).\n38Agrawal, M. et al.Direct measurement of magnon temperature: New insight into magnon-\nphonon coupling in magnetic insulators. Phys. Rev. Lett. 111(2013).\n39G¨ ornert, P. et al.Growth kinetics and some properties of thick lpe yig layers. J. Cryst. Growth\n87, 331–337 (1988).\n40Wei Hongxu & Wang Wenshu. The growth of lpe yig films with narro w fmr linewidth. IEEE\nT. Magn. 20, 1222–1223 (1984).\n41ˇCerm´ ak, J., Abrah´ am, A., Fabi´ an, T., Kaboˇ s, P. & Hyben, P . Yig based lpe films for microwave\ndevices. J. Magn. Magn. Mater. 83, 427–429 (1990).\n42Yong Choi, D. & Jin Chung, S. Annealing behaviors of lattice m isfit in yig and la-doped yig\nfilms grown on ggg substrates by lpe method. J. Cryst. Growth 191, 754–759 (1998).\n43Lang, M. et al.Proximity induced high-temperature magnetic order in topo logical insulator -\nferrimagnetic insulator heterostructure. Nano Lett. 14, 3459–3465 (2014).\n844Braun, W. Applied RHEED: Reflection high-energy electron diffraction during crystal growth ,\nvol. 154 of Springer tracts in modern physics (Springer, Berlin and New York, 1999).\n45Hansen, P. Saturation magnetization of gallium-substitut ed yttrium iron garnet. J. Appl. Phys.\n45, 2728 (1974).\n46Houchen Chang et al.Nanometer-thick yttrium iron garnet films with extremely lo w damping.\nIEEE Magn. Lett. 5, 6700104 (2014).\n47GeorgWoltersdorf. Spin-Pumping and Two-Magnons Scattering in Magnetic Multil ayers. Ph.D.\nthesis, Simon Fraser University, Vancouver (August 2004).\n9a)\nb)50,0 50,5 51,0 51,5 52,01081091010101110121013\nGGG(444)Intensity [CPS]\n2?[°]\n50,0 50,5 51,0 51,5 52,0103104105106107 GGG (444)Intensity [CPS]\n2???? c)\n24,80 24,85 24,90 24,95 25,00 25,05 25,10107108109Intensity [CPS]\n?[°]FWH M=0.015°\nFIG. 1: X-ray diffraction ( ω/2θscans) for an unannealed (a) and an annealed YIG layer (b).\nBefore annealing only the substrate peak is visible. After a nnealing the YIG peak clearly shows\nup. The position of the peak and the thickness fringes indica te fully pseudomorphic growth and\nsmooth interfaces. (c) shows a rocking curve of the layer pea k shown in (b). The full width at half\nmaximum is only 0.015◦. The dotted line shows a Gaussian fit to the peak.\n10FIG. 2: a) A high resolution TEM (HRTEM) image of an amorphous YIG film on GGG substrate.\nThe inset shows a FFT pattern from the region of interest (dot ted frame) in the amorphous layer.\nb) A HRTEM image of the interface between the annealed YIG film and the GGG substrate of\nsampleC. The insets show FFT patterns from the regions of int erest in the film and the substrate.\nThe YIG film exhibits epitaxial growth with respect to the sub strate and appears monocrystalline.\nc) RHEED image obtained from the surface of an annealed YIG fil m (sampleB). Kikuchi lines44\nindicate a two dimensional highly ordered surface.\n11/s45/s51/s48 /s45/s50/s48 /s45/s49/s48 /s48 /s49/s48 /s50/s48 /s51/s48/s45/s54/s45/s52/s45/s50/s48/s50/s52/s54\n/s32/s32/s77/s97/s103/s110/s101/s116/s105/s99/s32/s77/s111/s109/s101/s110/s116/s32/s91/s49/s48/s45/s53\n/s101/s109/s117/s93\n/s77/s97/s103/s110/s101/s116/s105/s99/s32/s70/s105/s101/s108/s100/s32/s91/s79/s101/s93\nFIG. 3: Hysteresis loop as measured by SQUID magnetometry fo r a 113nm thick YIG sample after\nannealing. The paramagnetic background caused by the GGG su bstrate was subtracted.\n12a)\nb)c)\nd)\n5 6 7 8 9 10 11 12 13 142,83,03,23,43,63,84,0\n/c97= (7.51 ±1.40) /c21510-5Linewidth [Oe]\nFrequency [GHz]8 9 10 11 121,01,11,21,31,41,51,6\n/c97= (6.63 ±1.50) /c21510-5Linewidth [Oe]\nFrequency [GHz]2620 2630 2640 2650-20246\n(1.50±0.05)Oe/c100/c99''//c100H [a.u.]\nMagnetic Field [Oe]\n2680 2690 2700 2710 2720 2730 2740-2-10123/c100/c99''//c100H [a.u.]\nMagnetic Field [Oe](4.02±0.10)Oe\nFIG. 4: a) and b) FMR data obtained at 9.6GHz for a 56nm thick (a , sampleA) and a 20nm\nthick (b, sampleB) YIG layer after annealing. The main reson ance lines have a peak-to-peak\nlinewidth of 1.51 ±0.05 Oe (sampleA) and 4.02 ±0.10 Oe (sampleB). This peak-to-peak linewidth\ncorresponds to a true linewidth of 1.30 ±0.05 Oe and 3.49 ±0.10 Oe, respectively. c) and d)\nFrequency dependence of the FMR linewidth for sampleA and sa mpleB. The fits are a straight\nlinecorrespondingtoadampingof α=(6.63±1.50)·10−5(c, sampleA)and α=(7.51±1.40)·10−5\n(d, sampleB).\n13"    },    {        "title": "1503.01478v2.Critical_current_destabilizing_perpendicular_magnetization_by_the_spin_Hall_effect.pdf",        "content": "arXiv:1503.01478v2  [cond-mat.mes-hall]  1 Aug 2015Critical current destabilizing perpendicular magnetizat ion by the spin Hall effect\nTomohiro Taniguchi1, Seiji Mitani2, and Masamitsu Hayashi2\n1National Institute of Advanced Industrial Science and Tech nology (AIST),\nSpintronics Research Center, Tsukuba 305-8568, Japan\n2National Institute for Materials Science, Tsukuba 305-004 7, Japan\n(Dated: July 5, 2018)\nThe critical current needed to destabilize the magnetizati on of a perpendicular ferromagnet via\nthe spin Hall effect is studied. Both the dampinglike and field like torques associated with the spin\ncurrent generated by the spin Hall effect is included in the La ndau-Lifshitz-Gilbert equation to\nmodel the system. In the absence of the fieldlike torque, the c ritical current is independent of the\ndamping constant and is much larger than that of conventiona l spin torque switching of collinear\nmagnetic systems, as in magnetic tunnel junctions. With the fieldlike torque included, we find that\nthe critical current scales with the damping constant as α0(i.e., damping independent), α, and\nα1/2depending on the sign of the fieldlike torque and other parame ters such as the external field.\nNumerical and analytical results show that the critical cur rent can be significantly reduced when\nthe fieldlike torque possesses the appropriate sign, i.e. wh en the effective field associated with the\nfieldlike torque is pointing opposite to the spin direction o f the incoming electrons. These results\nprovideapathwaytoreducingthecurrentneededtoswitch ma gnetization usingthespin Hall effect.\nPACS numbers: 75.78.-n, 75.70.Tj, 75.76.+j, 75.40.Mg\nI. INTRODUCTION\nThe spin Hall effect1–3(SHE) in a nonmagnetic heavy\nmetal generates pure spin current flowing along the di-\nrection perpendicular to an electric current. The spin\ncurrent excites magnetization dynamics in a ferromagnet\nattached to the nonmagnetic heavy metal by the spin-\ntransfer effect4,5. There have been a number of exper-\nimental reports on magnetization switching and steady\nprecession induced by the spin Hall effect6–9. These dy-\nnamics have attracted great attention recently from the\nviewpoints ofboth fundamental physicsand practicalap-\nplications.\nAn important issue to be solved on the magnetization\ndynamics triggered by the spin Hall effect is the reduc-\ntion of the critical current density needed to destabilize\nthe magnetization from its equilibrium direction, which\ndetermines the current needed to switch the magneti-\nzation direction or to induce magnetization oscillation.\nThe reported critical current density for switching8,10–13\nor precession9is relatively high, typically larger than 107\nA/cm2. One of the reasons behind this may be related\nto the recently predicted damping constant independent\ncritical current when SHE is used14,15. This is in con-\ntrast to spin-transfer-induced magnetization switching in\na typical giant magnetoresistance (GMR) or magnetic\ntunnel junction (MTJ) device where the critical current\nis expected to be proportional to the Gilbert damping\nconstant α. Here the magnetization dynamics is excited\nas a result of the competition between the spin torque\nand the damping torque16. Since the damping constant\nfor typical ferromagnet in GMR or MTJ devices is rela-\ntively small ( α∼10−2−10−3)17,18, it can explain why\nthe critical current is larger for the SHE driven systems.\nThus in particular for device application purposes, it is\ncrucial to find experimental conditions in which the mag-netization dynamics can be excited with lower current.\nAnother factor that might contribute to the reduc-\ntion of the critical current is the presence of the field\nlike torque19. In the GMR/MTJ systems, both the con-\nventional spin torque, often referred to as the damp-\ninglike torque, and the fieldlike torque arise from the\nspin transfer between the conduction electrons and the\nmagnetization4,19–23. Due to the short relaxation length\nof the transverse spin of the conduction electrons24,25,\nthe damping like torque is typically larger than the field-\nlike torque. Indeed, the magnitude of the field like\ntorque experimentally found in GMR/MTJ systems has\nbeen reported to be much smaller than the damping like\ntorque26–29. Because of its smallness, the fieldlike torque\nhad notbeen consideredin estimatingthe criticalcurrent\nintheGMR/MTJsystems16,30–32,althoughitdoesplaya\nkeyrolein particularsystems33,34. In contrast, recentex-\nperiments found that the fieldlike torque associated with\nthe SHE is larger than the damping like torque35–40.\nThe physical origin of the large SHE-induced field-\nlike torque still remains unclear. Other possible sources\ncan be the Rashba effect36,41–44, bulk effect45, and the\nout of plane spin orbit torque46. Interestingly, the field\nlike torque has been reported to show a large angu-\nlar dependence36,37,47(the angle between the current\nand the magnetization), which cannot be explained by\nthe conventional formalism of spin-transfer torque in\nGMR/MTJsystems. Thefieldliketorqueactsasatorque\nduetoanexternalfieldandmodifiestheenergylandscape\nof the magnetization. As a result, a large fieldlike torque\ncan significantly influence the critical current. However,\nthe fieldlike torque had not been taken into account in\nconsidering the current needed to destabilize the magne-\ntization from its equilibrium direction and thus its role\nis still unclear.\nIn this paper, we study the critical current needed to2\ndestabilize a perpendicular ferromagnet by the spin Hall\neffect. The Landau-Lifshitz-Gilbert(LLG) equationwith\nthe dampinglike and fieldlike torques associated with the\nspin Hall effect is solved both numerically and analyti-\ncally. Wefindthatthecriticalcurrentcanbesignificantly\nreduced when the fieldlike torque possesses the appropri-\nate sign with respect to the dampinglike torque. With\nthe fieldlike torque included, the critical current scales\nwith the damping constant as α0(i.e., damping indepen-\ndent),α, andα1/2, depending on the sign of the field-\nlike torque and other parameters. Analytical formulas\nof such damping-dependent critical current are derived\n[Eqs. (19)-(21)], and they show good agreement with the\nnumerical calculations. From these results, we find con-\nditions in which the critical current can be significantly\nreduced compared to the damping-independent thresh-\nold, i.e., systems without the fieldlike torque.\nThe paper is organized as follows. In Sec. II, we\nschematically describe the system under consideration.\nWe discuss the definition of the critical current in Sec.\nIII. Section IV summarizes the dependences of the crit-\nical current on the direction of the damping constant,\nthe in-plane field, and the fieldlike torque obtained by\nthe numerical simulation. The analytical formulas of the\ncritical current and their comparison to the numerical\nsimulations are discussed in Sec. V. The condition at\nwhich damping-dependent critical current occurs is also\ndiscussed in this section. The conclusion follows in Sec.\nVI.\nII. SYSTEM DESCRIPTION\nThe system we consider is schematically shown in Fig.\n1, where an electric current flowing along the x-direction\ninjects a spin current into the ferromagnet by the spin\nHall effect. The magnetization dynamics in the ferro-\nmagnet is described by the LLG equation,\ndm\ndt=−γm×H+αm×dm\ndt\n−γHsm×(ey×m)−γβHsm×ey,(1)\nwhereγandαarethe gyromagneticratioandtheGilbert\ndamping constant, respectively. We assume that the\nmagnetization of the ferromagnet points along the film\nnormal (i.e., along the zaxis), and an external in-plane\nmagnetic field is applied along the xoryaxis. The total\nmagnetic field His given by\nH=HapplnH+HKmzez, (2)\nwhereHapplis the external field directed along the xor\nyaxis and HKis the uniaxial anisotropy field along the\nzaxis.nHandeiare unit vectors that dictate the di-\nrection of the uniaxial anisotropy field and the iaxis,\nrespectively. Here we call the external field along the x\nandydirections the longitudinal and transverse fields,\nrespectively. The third and fourth terms on the right-\nhand side of Eq. (1) are the damping like and fieldlikeHappl // y\nHappl // x\nm\ncurrentxz\ny\nFIG. 1. Schematic view of the spin-Hall system. The x\naxis is parallel to current, whereas the zaxis is normal to the\nfilm plane. The spin direction of the electrons entering the\nmagnetic layer via the spin Hall effect points along the + yor\n−ydirection.\ntorques associated with the spin Hall effect, respectively.\nThetorquestrength Hscanbeexpressedwiththecurrent\ndensityj, the spin Hall angle ϑ, the saturation magneti-\nzationM, and the thickness of the ferromagnet d, i.e.,\nHs=/planckover2pi1ϑj\n2eMd. (3)\nThe ratio of the fieldlike torque to the damping like\ntorque is represented by β. Recent experiments found\nthatβis positive and is larger than 135–40.\nThe magnetization dynamics described by the LLG\nequation can be regarded as a motion of a point particle\non a two-dimensional energy landscape. In the presence\nof the fieldlike torque, the energy map is determined by\nthe energy density given by34\nE=−M/integraldisplay\ndm·H−βMHsm·ey.(4)\nThen, the external field torque and the fieldlike torque,\nwhich are the first and fourth terms on the right-hand-\nside of Eq. (1), can be expressed as −γm×B, where the\neffective field Bis\nB=−∂E\n∂Mm. (5)\nThe initial state of the numerical simulation is chosen to\nbe the direction corresponding to the minimum of the\neffective energy density E. The explicit forms of the ini-\ntial state for the longitudinal and the transverse external\nfields are shown in Appendix A.\nWe emphasize for the latter discussion in Sec. V that,\nusing Eqs. (1), (4), and (5), the time change of the effec-\ntive energy density is described as\ndE\ndt=dEs\ndt+dEα\ndt. (6)3\nHere the first and second terms on the right-hand side\nare the rates of the work done by the spin Hall torque\nand the dissipation due to damping, respectively, which\nare explicitly given by\ndEs\ndt=γMHs[ey·B−(m·ey)(m·B)],(7)\ndEα\ndt=−αγM/bracketleftBig\nB2−(m·B)2/bracketrightBig\n. (8)\nThe sign of Eq. (7) depends on the current direction\nand the effective magnetic field, while that of Eq. (8) is\nalways negative.\nThe magnetic parameters used in this paper mimic the\nconditions achieved in CoFeB/MgO heterostructures48;\nM= 1500 emu/c.c., HK= 540 Oe, ϑ= 0.1,γ=\n1.76×107rad/(Oe s), and d= 1.0 nm. The value of\nβis varied from −2, 0, to 2. Note that we have used a\nreducedHK(Refs.8,49) in ordertoobtain criticalcurrents\nthat are the same order of magnitude with that obtained\nexperimentally. We confirmed that the following discus-\nsions are applicable for a large value of HK(∼1T).\nIII. DEFINITION OF CRITICAL CURRENT\nIn this section, we describe how we determine the crit-\nical current from the numerical simulations. In exper-\niments, the critical current is determined from the ob-\nservation of the magnetization reversal8,12,41,46,48–50. As\nmentioned in Sec. II, in this paper, the initial state for\ncalculation is chosen to be the minimum of the effective\nenergy density. Usually, there are two minimum points\nabove and below the xyplane because of the symmetry.\nThroughout this paper, the initial state is chosen to be\nthe minimum point above the xyplane, i.e., mz(0)>0,\nfor convention.” It should be noted that, once the mag-\nnetization arrives at the xyplane during the current ap-\nplication, it can move to the other hemisphere after the\ncurrent is turned off due to, for example, thermal fluc-\ntuation. Therefore, here we define the critical current as\nthe minimum current satisfying the condition\nlim\nt→∞mz(t)< ǫ, (9)\nwhere a small positive real number ǫis chosen to be\n0.001. The duration of the simulations is fixed to 5 µs,\nlong enough such that all the transient effects due to the\ncurrent application are relaxed. Figures 2(a) and 2(b)\nshow examples of the magnetization dynamics close to\nthe critical current, which are obtained from the numer-\nical simulation of Eq. (1). As shown, the magnetization\nstays near the initial state for j= 3.1×106A/cm2, while\nit moves to the xyplane for j= 3.2×106A/cm2. Thus,\nthe critical current is determined as 3 .2×106A/cm2in\nthis case.\nWe note that the choice of the definition of the criti-\ncal current has some arbitrariness. For comparison, weFIG. 2. Time evolution ofthe zcomponentof themagnetiza-\ntionmzin the presence of the transverse field of Happl= 200\nwith (a) j= 3.1×106A/cm2and (b)j= 3.2×106A/cm2.\nThe value of βis zero.\nshow numerically evaluated critical current with a differ-\nent definition in Appendix B. The main results of this\npaper, e.g., the dependence of the critical current on the\ndamping constant, are not affected by the definition.\nWe also point out that the critical current defined by\nEq. (9) focuses on the instability threshold, and does\nnot guarantee a deterministic reversal. For example,\nin the case of Fig. 2(b), the reversal becomes prob-\nabilistic because the magnetization, starting along + z,\nstops its dynamics at the xyplane and can move back\nto its original direction or rotate to a point along −z\nresulting in magnetization reversal. Such probabilistic\nreversal can be measured experimentally using transport\nmeasurements8,12,41,46,49,50or by studying nucleation of\nmagnetic domains via magnetic imaging48. On the other\nhand, it hasbeen reportedthat deterministicreversalcan\ntake place when a longitudinal in-plane field is applied\nalongside the current41,49. It is difficult to determine the\ncritical current analytically for the deterministic switch-\ning for all conditions since, as in the case of Fig. 2(b),\nthe magnetization often stops at the xyplane during the\ncurrent application. This occurs especially in the pres-\nence of the transverse magnetic field because all torques\nbecome zero at m=±eyand the dynamics stops. Here\nwe thus focus on the probabilistic reversal.4\nFIG. 3. Numerically evaluated mzatt= 5µs for (a)-(c) the longitudinal ( nH=ex) and (d)-(f) the transverse ( nH=ey)\nfields, where the value of βis (a), (d) 0 .0; (b), (e) 2 .0; and (c), (f) −2.0. The damping constant is α= 0.005. The color scale\nindicates the zcomponent of the magnetization ( mz) att= 5µs. The red/white boundary indicates the critical current fo r\nprobabilistic switching, whereas the red/blue boundary gi ves the critical current for deterministic switching.\nIV. NUMERICALLY ESTIMATED CRITICAL\nCURRENT\nIn this section, we show numerically evaluated critical\ncurrent for different conditions. We solve Eq. (1) and\napply Eq. (9) to determine the critical current. Figure\n3 shows the value of mzatt= 5µs in the presence of\n(a)-(c) the longitudinal ( nH=ex) and (d)-(f) the trans-\nverse (nH=ey) fields. The value of βis 0 for Figs.\n3(a) and 3(d), 2 .0 for Figs. 3(b) and 3(e), and −2.0 for\nFigs. 3(c) and 3(f), respectively. The damping constant\nisα= 0.005. The red/white boundary indicates the crit-\nical current for the probabilistic switching, whereas the\nred and blue ( mz=−1) boundary gives the critical cur-\nrent for the deterministic switching. Using these results\nand the definition of the critical current given by Eq. (9),\nand performing similar calculations for different values of\nα, wesummarizethedependenceofthecriticalcurrenton\nthe longitudinal and transverse magnetic fields in Fig. 4.\nThe damping constant is varied as the following in each\nplot:α= 0.005, 0.01, and 0 .02. The solid lines in Fig. 4\nrepresent the analytical formula derived in Sec. V.A. In the presence of longitudinal field\nIn the case of the longitudinal field and β= 0\nshown in Fig. 4(a), the critical current is damping-\nindependent. Such damping-independent critical current\nhas been reported previously for deterministic magneti-\nzation switching14,15. Similarly, in the case of the longi-\ntudinal field and negative β(β=−2.0) shown in Fig.\n4(c), the critical current is damping-independent. In\nthese cases, the magnitude of the critical current is rel-\natively high. In particular, near zero field, the critical\ncurrent exceeds ∼108A/cm2, which is close to the limit\nof experimentally accessible value. These results indicate\nthat the useofthe longitudinal field with zeroornegative\nβis ineffective for the reduction of the critical current.\nOn the other hand, when βis positive, the critical cur-\nrent depends on the damping constant, as shown in Fig.\n4(b). Note that positive βis reported for the torques\nassociated with the spin Hall effect or Rashba effect in\nthe heterostructures studied experimentally35–37,39. The\nmagnitude of the critical current, ∼10×106A/cm2, is\nrelatively small compared with the cases of zero or neg-\nativeβ. In this case, the use of a low damping material\nis effective to reduce the critical current. Interestingly,\nthe critical current is not proportional to the damping\nconstant, while that previously calculated for a GMR or\nMTJ system16is proportional to α. For example, the5\nlongitudinal magnetic field (Oe)critical current density (10 6 A/cm 2)\n0 50 100 150 2000\n-10050 \n-150100150\nβ=0.0\n-50\n-100 -50 -150 -200: α=0.005: α=0.01: α=0.02\ntransverse magnetic field (Oe)critical current density (10 6 A/cm 2)\n0 50 100 150 2000\n-10050 \n-150100150\nβ=0.0\n-50\n-100 -50 -150 -200: α=0.005: α=0.01: α=0.02\ntransverse magnetic field (Oe)critical current density (10 6 A/cm 2)\n0 50 100 150 2000\n-4020 \n-6040 60 \nβ=2.0\n-20\n-100 -50 -150 -200: α=0.005: α=0.01: α=0.02longitudinal magnetic field (Oe)critical current density (10 6 A/cm 2)\n0 50 100 150 2000\n-4020 \n-5040 50 \nβ=2.0\n-20\n-100 -50 -150 -200: α=0.005 : α=0.01 : α=0.02-30-1010 30 \ntransverse magnetic field (Oe)critical current density (10 6 A/cm 2)\n0 50 100 150 2000\n-10050 \n-150100150\nβ=-2.0\n-50\n-100 -50 -150 -200: α=0.005 : α=0.01 : α=0.02(a) (b) (c)\n(d) (e) (f)longitudinal magnetic field (Oe)0 50 100 150 200β=-2.0\n-100 -50 -150 -200: α=0.005: α=0.01: α=0.02critical current density (10 6 A/cm 2)\n0\n-10050 \n-150100150\n-50\nFIG. 4. Numerically evaluated critical currents in the pres ence of (a)-(c) the longitudinal ( nH=ex) and (d)-(f) the transverse\n(nH=ey) fields, where the value of βis (a), (d) 0 .0; (b), (e) 2 .0; and (c), (f) −2.0, respectively. The solid lines are analytically\nestimated critical current in Sec. V.\ncritical current at zero longitudinal field in Fig. 4(b) is\n12.3,17.2, and24 .0×106A/cm2forα= 0.005,0.01, and\n0.02, respectively. These values indicate that the critical\ncurrent is proportional to α1/2. In fact, the analytical\nformula derived in Sec. V shows that the critical current\nis proportional to α1/2for positive β[see Eq. (19)].\nTo summarize the case of the longitudinal field, the\nuse of a heterostructure with positive β, which is found\nexperimentally, has the possibility to reduce the critical\ncurrent if a ferromagnet with low damping constant is\nused. In this case, the critical current is proportional to\nα1/2, which has not been found in previous works.\nB. In the presence of transverse field\nIn the presence of the transverse field with β= 0,\nthe critical current shows a complex dependence on the\ndamping constant α, as shown in Fig. 4(d). When the\ncurrent and the transversefield areboth positive (or neg-\native), the critical current is proportional to the damping\nconstant αexcept near zero field. The numerically cal-\nculated critical current matches well with the analytical\nresult, Eq. (20), shown by the solid lines. In this case,\nthe use of the low damping material results in the reduc-\ntion of the critical current. On the other hand, when the\ncurrent and the transversefield possessthe opposite sign,\nthecriticalcurrentisdampingindependent. Moreover,in\nthis case, thecriticalcurrentisofthe orderof108A/cm2.\nThus, it is preferable to use the current and field having\nthe same sign for the reduction of the critical current. Itshould be noted that, in our definition, the same sign of\ncurrent and field corresponds to the case when the direc-\ntion ofincoming electrons’spin (due to the SHE) and the\ntransverse field are opposite to each other. The reason\nwhy the critical current becomes damping dependent in\nthis situation will be explained in Sec. V.\nWhenβis positive the critical current depends on the\ndamping constant for the whole range of the transverse\nfield, as shown in Fig. 4(e). The critical current is\nroughly proportional to α1/2, in particular, close to zero\nfield. The solid lines display the analytical formula, Eq.\n(21), and showgood agreementwith the numericalcalcu-\nlations. The damping dependence of the critical current\nbecomes complex when the magnitude of the transverse\nfield is increased [see Eq. (21)]. We note that the critical\ncurrentfor the positive βin Fig. 4(e) is smallerthan that\nforβ= 0 in Fig. 4(d) for the whole range of Happl.\nOn the other hand, when βis negative, the critical\ncurrent is almost independent of α, especially near zero\nfield. However, when the transverse field is increased,\nthere is a regime where the critical current depends on\nthe damping constant. Such transition of the critical\ncurrent with the transverse field is also predicted by the\nanalytical solution, Eq. (21).\nTosummarizethe caseofthe transversefield, the αde-\npendence of the critical current can be categorized into\nthe following: α0(damping independent), α,α1/2, or\nother complex behavior. As with the case of the longi-\ntudinal field, the use of a heterostructure with positive β\nallowsreductionofthe criticalcurrentwhen lowdamping\nferromagnet is used. Overall, the most efficient condition6\nto reduce the critical current is to use the transverse field\nwith heterostructures that possess low αand positive β.\nIn this case, the critical current is reduced to the order\nof 106A/cm2.\nV. ANALYTICAL FORMULA OF CRITICAL\nCURRENT\nIn this section, we derive the analytical formula of the\ncritical current from the linearized LLG equation51. The\ncomplex dependences ofthe critical currentonthe damp-\ning constant αdiscussed in Sec. IV are well explained by\nthe analytical formula. We also discuss the physical in-\nsight obtained from the analytical formulas.\nA. Derivation of the critical current\nTo derive the critical current, we consider the stable\ncondition of the magnetization near its equilibrium. It is\nconvenient to introduce a new coordinate XYZin which\ntheZaxis is parallel to the equilibrium direction. The\nrotationfromthe xyz-coordinatetothe XYZcoordinate\nis performed by the rotation matrix\nR=\ncosθ0−sinθ\n0 1 0\nsinθ0 cosθ\n\ncosϕsinϕ0\n−sinϕcosϕ0\n0 0 1\n,(10)\nwhere (θ,ϕ) are the polar and azimuth angles of the\nmagnetization at equilibrium. The equilibrium magne-\ntization direction under the longitudinal and transverse\nmagnetic field is given by Eqs. (A1) and (A2), respec-\ntively. Since we are interested in small excitation of the\nmagnetization around its equilibrium, we assume that\nthe components of the magnetization in the XYZcoor-\ndinate satisfy mZ≃1 and|mX|,|mY| ≪1. Then, the\nLLG equation is linearized as\n1\nγd\ndt/parenleftbigg\nmX\nmY/parenrightbigg\n+M/parenleftbigg\nmX\nmY/parenrightbigg\n=−Hs/parenleftbigg\ncosθsinϕ\ncosϕ/parenrightbigg\n,(11)\nwhere the components of the 2 ×2 matrix Mare\nM1,1=αBX−Hssinθsinϕ, (12)\nM1,2=BY, (13)\nM2,1=BX (14)\nM2,2=αBY−Hssinθsinϕ. (15)\nHere,BXandBYare defined as\nBX=Happlsinθcos(ϕ−ϕH)+βHssinθsinϕ+HKcos2θ,\n(16)BY=Happlsinθcos(ϕ−ϕH)+βHssinθsinϕ+HKcos2θ,\n(17)\nwhereϕHrepresents the direction of the external field\nwithin the xyplane:ϕH= 0 for the longitudinal field\nandπ/2 for the transverse field.\nThe solution of Eq. (11) is mX,mY∝\nexp{γ[±i/radicalbig\ndet[M]−(Tr[M]/2)2−Tr[M]/2]t}, where\ndet[M] and Tr[ M] are the determinant and trace of\nthe matrix M, respectively. The imaginary part of the\nexponent determines the oscillation frequency around\ntheZaxis, whereas the real part determines the time\nevolution of the oscillation amplitude. The critical\ncurrent is defined as the current at which the real part\nof the exponent is zero. Then, the condition Tr[ M] = 0\ngives\nα(BX+BY)−2Hssinθsinϕ= 0,(18)\nFor the longitudinal field, Eq. (18) gives\njLONG\nc=±2e√αMd\n/planckover2pi1ϑ/radicalBig\n2H2\nK−H2\nappl/radicalbig\nβ(2+αβ),(19)\nindicating that the critical current is roughly propor-\ntional to α1/2. This formula works for positive βonly52\nif we assume 0 <2+αβ≃2, which is satisfied for typical\nferromagnets. The critical current when the transverse\nfield is applied reads\njTRANS\nc=2αeMd\n/planckover2pi1ϑ(Happl/HK)HK/bracketleftBigg\n1−1\n2/parenleftbiggHappl\nHK/parenrightbigg2/bracketrightBigg\n,(20)\nwhenβ= 0, indicating that the critical current is pro-\nportional to α. The critical current for finite βis\njTRANS\nc=2eMd\n/planckover2pi1ϑ\n×−(1+αβ)Happl±/radicalBig\nH2\nappl+2αβ(2+αβ)H2\nK\nβ(2+αβ).\n(21)\nEquation (21) works for the whole range of |Happl|(<\nHK) for positive β, while it only works when |Happl|>\n2αβ(2 +αβ)HKfor negative β. For example, when\nβ=−2.0, this condition is satisfied when |Happl|>108\nOe forα= 0.005 and |Happl|>152 Oe for α= 0.01.\nHowever the condition is not satisfied for the present\nrange of Happlforα= 0.02. The solid lines in Fig. 4(f)\nshow where Equation (21) is applicable. The zero-field\nlimits of Eqs. (19) and (21) become identical,\nlim\nHappl→0jc=±2e√αMd\n/planckover2pi1ϑ√\n2HK/radicalbig\nβ(2+αβ),(22)\nindicating that the critical current near zero field is pro-\nportional to α1/2whenβ >0.7\nFIG. 5. Magnetization dynamics under the conditions of (a)\nnH=ey,Happl= 50 Oe, β= 0,α= 0.005, and j= 13.2×106\nA/cm2, and (b) nH=ex,Happl= 50 Oe, β= 0,α= 0.005,\nandj= 90×106A/cm2.\nB. Discussions\nThe solid lines in Fig. 4(b), 4(d), 4(e), and 4(f) show\nthe analytical formulas, Eqs. (19), (20), and (21). As\nevident, these formulas agree well with the numerical re-\nsults in the regions where the critical currents depend on\nthe dampingconstant. In this section, we discussthe rea-\nson why the critical current becomes damping dependent\nor damping independent depending on the field direction\nand the sign of β.\nIt is useful for the following discussion to first study\ntypical magnetization dynamics found in the numerical\ncalculations. Figure 5 shows the time evolution of the\nx,yandzcomponents of the magnetization when the\ncritical current depends on [Fig. 5(a)] or is independent\nof [Fig. 5(b)] the damping constant. For the former,\nthe instability is accompanied with a precession of the\nmagnetization. On the other hand, the latter shows that\nthe instability takes place without the precession.\nWe start with the case when the critical current be-\ncomes damping dependent. To provide an intuitive pic-\nture, we schematically show in Fig. 6(a) the torques ex-\nerted on the magnetization during one precession period\nwhen current is applied. The condition is the same with\nthat described in Fig. 5(a), i.e., the transverse magnetic\nfield is applied with β= 0. In Fig. 6(a), magnetization\nis shown by the large black arrow, while the directions\nof the spin Hall torque, the damping torque and the ex-ternal field torque are represented by the solid, dotted\nand dashed lines, respectively (the external field torque\nis tangent to the precession trajectory). As evident in\nFig. 5(a), the precession trajectory is tilted to the posi-\ntiveydirection due to the transversefield. Depending on\nthe direction of the magnetization the spin Hall torque\nhas a component parallel, antiparallel, or normal to the\ndamping torque. This means that the work done by the\nspin Hall torque, denoted by ∆ Esin Fig. 6 (a), is pos-\nitive, negative, or zero at these positions. This can be\nconfirmed numerically when we calculate the work done\nby the spin Hall torque using Eq. (7). For an infinites-\nimal time ∆ t, the work done by the spin Hall torque\nis equal to the rate of its work ( dEs/dt), given in Eq.\n(7), times ∆ t, i.e. ∆Es= (dEs/dt)∆t. The solid line\nin Fig. 6(b) shows an example of the calculated rate of\nthe work done by the spin Hall torque (solid line), dEs/dt\nin Eq. (7). As shown, dEs/dtis positive, negative, and\nzero, when the magnetization undergoes one precession\nperiod. Similarly, the energy dissipated by the damping\ntorque,dEα/dt, can be calculated using Eq. (8) and is\nshown by the dotted line in Fig. 6(b). The calculated\ndissipation due to damping over a precession period is\nalways negative. Details of how the rates, shown in Fig.\n6, are calculated are summarized in Appendix C.\nNote that the strength of the spin Hall torque for\n∆Es>0 is larger than that for ∆ Es<0 due to the an-\ngular dependence of the spin Hall torque, |m×(ey×m)|.\nAlthough it is difficult to see, thesolid line in Fig. 6(b) is\nslightly shifted upward. Thus the total energy supplied\nby the spin Hall torque during one precession, given by/contintegraltext\ndt(dEs/dt), does not average to zero and becomes posi-\ntive. When the current magnitude, |j|, is larger than |jc|\nin Eq. (20), the energy supplied by the spin Hall torque\novercomes the dissipation due to the damping and con-\nsequently the precession amplitude grows, which leads to\nthe magnetization instability shown in Fig. 5(a). The\nsame picture is applicable when both directions of field\nand current are reversed. For this condition, the insta-\nbility of the magnetization is induced by the competition\nbetween the spin Hall torque and the damping torque.\nTherefore, the critical current depends on the damping\nconstant α. When only the current direction is reversed\nin Figs. 6(a) and 6(b) (i.e., the sign of the magnetic field\nand current is opposite to each other), the sign of ∆ Esis\nreversed and thus the total energy supplied by the spin\nHall torque becomes negative. This means that the spin\nHall torque cannot overcome the damping torque to in-\nduce instability. Therefore, the critical current shown in\nEq. (20) only applies to the case when the sign of the\nfield and current is the same. As described in Sec. IV,\nthe same sign of the current and field in our definition\nmeans that the incoming electrons’ spin direction, due\nto the spin Hall effect, is opposite to the transverse field\ndirection.\nNext, we consider the case when the critical current is\ndamping independent. Figure 6 (c) schematically shows\nthe precession trajectory when the applied field points to8\nFIG. 6. (a) A schematic view of the precession trajectory\nin the presence of the applied field in the positive y-direction.\nThe solid and dotted arrows indicate the directions of the\nspin Hall torque and the damping torque, respectively. The\ndashed line, which is the tangent line to the precession tra-\njectory, shows the field torque. The damping torque always\ndissipates energy from the ferromagnet. On the other hand,\nthe spin Hall torque supplies energy (∆ Es>0) when its di-\nrection is anti-parallel to the damping torque, and dissipa tes\nenergy (∆ Es<0) when the direction is parallel to the damp-\ning torque. When the direction of the spin Hall torque is\northogonal to the damping torque, the spin Hall torque does\nnot change the energy (∆ Es= 0). (b) Typical temporal vari-\nation of the rates of the work done by the spin Hall torque,\nEq. (7), (solid) and the dissipation due to damping, Eq. (8)\n(dotted) in the presence of the transverse field. The time is\nnormalized by the period given by Eq. (C7). (c), (d) Similar\nfigures with the longitudinal field.\nthexdirection and β= 0. The corresponding rate of\nwork done by the spin Hall torque and the dissipation\nrate due to the damping torque are shown in Fig. 6 (d).\nSimilar to the previous case, ∆ Escan be positive, nega-\ntive, or zero during one precession period. However, the\ntotal workdoneby the spin Hall torque,/contintegraltext\ndt(dEs/dt), be-\ncomes zero in this case due to the symmetry of angular\ndependence of the spin Hall torque. This means that the\nspin Hall torque cannot compensate the damping torque,\nand thus, a steady precession assumed in the linearized\nLLG equation is not excited. This is evident in the nu-\nmerically calculated magnetization trajectory shown in\nFig. 5(b). For this case, the linearized LLG equation\ngives|jc| → ∞, indicating that the spin Hall torque can-\nnot destabilize the magnetization. The same picture is\nalsoapplicable, forexample, in the absenceofthe applied\nfield and β= 0.\nHowever, an alternative mechanism can cause destabi-\nlization of the magnetization. As schematically shown in\nFigs. 6(a) and 6(c), there is a component of the damping\nlike spin Hall torque that is orthogonal to the damping\ntorque when ∆ Es= 0. The spin Hall torque at this pointis parallel or antiparallel to the field torque depending on\nthe position of the magnetization. When the spin Hall\ntorqueissufficientlylargerthanthefieldtorque,themag-\nnetization moves from its equilibrium position even if the\ntotal energy supplied by the spin Hall torque is zero or\nnegative. This leads to an instability that occurs before\none precession finishes. In this case, it is expected that\nthe critical current is damping-independent because the\ninstability is induced as a competition between the spin\nHall torque and the field torque, not the damping torque.\nThe time evolution of the magnetization shown in Fig.\n5 (b) represents such instability. The work reported in\nRefs.14,49discusses a similar instability condition.\nThe above physical picture is also applicable in the\npresence of the fieldlike torque. The fieldlike torque,\nwhich acts like a torque due to the transversefield, modi-\nfies the equilibrium direction ofthe ferromagnetand thus\nthe precession trajectory. Consequently, the amount of\nenergy supplied by the spin Hall torque and the dissipa-\ntion due to damping is changed when the fieldlike torque\nis present. Depending on the sign of β, the amount of the\nwork done by the spin Hall torque increases or decreases\ncompared to the case with β= 0. In our definition, posi-\ntiveβcontributes to the increase of the supplied energy,\nresulting in the reduction of the critical current. The\ncomplex dependence of the critical current on αarises\nwhen the fieldlike torque is present.\nTo summarize the discussion, the critical current be-\ncomes damping dependent when the energy supplied by\nthe spin Hall torque during a precession around the equi-\nlibrium is positive. The condition that meets this criteria\ndepends on the relative direction of the spin Hall torque\nand the damping torque, as briefly discussed above. To\nderive an analytical formula that describes the condition\natwhichthe criticalcurrentbecomesdamping dependent\nis not an easy task except for some limited cases53.\nVI. CONCLUSION\nIn summary, we have studied the critical current\nneeded to destabilize a perpendicularly magnetized fer-\nromagnet by the spin Hall effect. The Landau-Lifshitz-\nGilbert (LLG) equation that includes both the damping-\nlike and fieldlike torques associated with the spin Hall\neffect is solved numerically and analytically. The criti-\ncal current is found to have different dependence on the\ndamping constant, i.e., the critical current scales with α0\n(damping-independent), α, andα1/2depending on the\nsign of the fieldlike torque. The analytical formulas of\nthe damping-dependent critical current, Eqs. (19), (20),\nand (21), are derived from the linearized LLG equation,\nwhich explain well the numerical results. We find that\nsystems with fieldlike torque having the appropriate sign\n(β >0 in our definition) are the most efficient way to re-\nduce the criticalcurrent. Fortypicalmaterialparameters\nfound in experiment, the critical current can be reduced\nto the order of 106A/cm2when ferromagnets with rea-9\nsonable parameters are used.\nACKNOWLEDGMENTS\nThe authorsacknowledgeT. Yorozu, Y. Shiota, and H.\nKubota in AIST for valuable discussion sthey had with\nus. This workwassupported by JSPS KAKENHIGrant-\nin-AidforYoungScientists(B),GrantNo. 25790044,and\nMEXT R & D Next-Generation Information Technology.\nAppendix A: Initial state of the numerical\nsimulation\nWe assume that the magnetization in the absence of\nthe applied field points to the positive zdirection. In\nthe presence of the field, the equilibrium direction moves\nfrom the zaxis to the xyplane. Let us denote the zenith\nand azimuth angles of the initial state m(t= 0) asθand\nϕ, i.e.,m(t= 0) = (sin θcosϕ,sinθsinϕ,cosθ). When\nthe applied field points to the x-direction ( nH=ex), the\ninitial state is\n/parenleftbigg\nθ\nϕ/parenrightbigg\nnH=ex=/parenleftBigg\nsin−1[/radicalBig\nH2\nappl+(βHs)2/HK]\ntan−1(βHs/Happl)/parenrightBigg\n,(A1)\nwhere the value of ϕis 0< ϕ < π/ 2 forHappl>0 and\nβHs>0,π/2< ϕ < π forHappl<0 andβHs>0,π <\nϕ <3π/2forHappl<0andβHs<0,and3π/2< ϕ <2π\nforHappl>0 andβHs<0. On the other hand, when\nthe applied field points to the y-direction ( nH=ey), the\ninitial state is\n/parenleftbigg\nθ\nϕ/parenrightbigg\nnH=ey=/parenleftbigg\nsin−1[(Happl+βHs)/HK]\nπ/2/parenrightbigg\n,(A2)\nwhere the range of the inverse sine function is −π/2≤\nsin−1x≤π/2. We note that the choice of the initial\nstate does not affect the evaluation of the critical cur-\nrent significantly, especially in the small field and current\nregimes.\nAppendix B: Numerically evaluated critical current\nwith different definition\nAs mentioned in Sec. III, the definition of the critical\ncurrent has arbitrariness. As an example, we show the\ntime evolution of mzunder the conditions of nH=ex,\nHappl=−30 Oe,β= 0, and j= 110×106A/cm2in\nFig. 7. In this case, the magnetization initially starts at\nmz= cos[sin−1(Happl/HK)]≃0.99, and finally moves to\na pointmz→0.12. Since the final state does not satisfy\nEq. (9), this current, j= 110×106A/cm2, should be\nregarded as the current smaller than the critical current\nin Sec. IV. However, from the analytical point of view,\nthis current can be regarded as the current larger than\nmagnetization 01\n-1 -0.50.5\nj=110×10 6 A/cm2\ntime (ns)0 2 4 6 8 10 Happl=-30 Oe\nFIG. 7. Time evolution of the zcomponent of the mag-\nnetization mzin the presence of the longitudinal field with\nHappl=−30 Oe,β= 0, and j= 110×106A/cm2. The\ndotted line is a guide showing mz= 0.\nthe critical current because the final state of the magne-\ntization is far away from the initial equilibrium.\nRegarding this point, we show the numerically eval-\nuated critical current with a different definition. The\nmagnetic state can be regarded as unstable when it fi-\nnally arrives at a point far away from the initial state54.\nThus, for example, one can define the critical current as\na minimum current satisfying\nlim\nt→∞|mz(t)−mz(0)|> δ, (B1)\nwhere a small positive real number δis chosen to be\n0.1 here. Figure 8 summarizes the numerically evalu-\nated critical current with the definition of Eq. (B1). The\nanalytical formulas, Eqs. (19)-(21), still fit well with the\nnumerical results. The absolute values of the damping-\ndependent critical current are slightly changed when the\ndefinition of the critical current is changed. This is be-\ncause Eq. (B1) is more easily satisfied than Eq. (9),\nand thus the critical current in Fig. 8 is smaller than\nthat shown in Fig. 4. However, the main results of this\npaper, such as the damping dependence of the critical\ncurrent, are not changed by changing the definition of\nthe critical current in the numerical simulations.\nAppendix C: Energy change during a precession\nAs described in Sec. V, the linearized LLG equation\nassumes a steady precession of the magnetization due to\nthe field torque when the current magnitude is close to\nthe critical current. This is because the spin Hall torque\ncompensates with the damping torque. Thus, Figs. 6(b)\nand 6(d) are obtained by substituting the solution of m\nprecessing a constant energy curve of Einto Eqs. (7) and\n(8).\nWhen the transverse field is applied and β= 0, i.e.,\nE=E, whereE=−M/integraltext\ndm·H, the precession trajec-\ntory on the constant energy curve of Eis given by55\nmx(E) = (r2−r3)sn(u,k)cn(u,k),(C1)10\ntransverse magnetic field (Oe)critical current density (10 6 A/cm 2)\n0 50 100 150 2000\n-4020 \n-6040 60 \nβ=2.0\n-20\n-100 -50 -150 -200: α=0.005: α=0.01: α=0.02\ntransverse magnetic field (Oe)critical current density (10 6 A/cm 2)\n0 50 100 150 2000\n-10050 \n-150100150\nβ=-2.0\n-50\n-100 -50 -150 -200: α=0.005 : α=0.01 : α=0.02(a) (b) (c)\n(d) (e) (f)longitudinal magnetic field (Oe)critical current density (10 6 A/cm 2)\n0 50 100 150 2000\n-4020 \n-5040 50 \nβ=2.0\n-20\n-100 -50 -150 -200: α=0.005 : α=0.01 : α=0.02-30-1010 30 \nlongitudinal magnetic field (Oe)critical current density (10 6 A/cm 2)\n0 50 100 150 2000\n-10050 \n-150100150\nβ=0.0\n-50\n-100 -50 -150 -200: α=0.005: α=0.01: α=0.02\nlongitudinal magnetic field (Oe)0 50 100 150 200β=-2.0\n-100 -50 -150 -200: α=0.005: α=0.01: α=0.02\ntransverse magnetic field (Oe)critical current density (10 6 A/cm 2)\n0 50 100 150 2000\n-10050 \n-150100150\nβ=0.0\n-50\n-100 -50 -150 -200: α=0.005: α=0.01: α=0.02\ncritical current density (10 6 A/cm 2)\n0\n-10050 \n-150100150\n-50\nFIG. 8. Numerically evaluated critical currents with a diffe rent definition, Eq. (B1), in the presence of (a)-(c) the long itudinal\n(nH=ex) and (d)-(f) the transverse ( nH=ey) fields, where the value of βis (a), (d) 0 .0; (b), (e) 2 .0; and (c), (f) −2.0. The\nsolid lines are the analytically estimated critical curren t described in Sec. V.\nmy(E) =r3+(r2−r3)sn2(u,k),(C2)\nmz(E) =/radicalBig\n1−r2\n3−(r2\n2−r2\n3)sn2(u,k),(C3)\nwhereu=γ/radicalbig\nHtHK/2√r1−r3t, andrℓare given by\nr1(E) =−E\nMHappl, (C4)\nr2(E) =Happl\nHK+/radicalBigg\n1+/parenleftbiggHappl\nHK/parenrightbigg2\n+2E\nMHK,(C5)\nr3(E) =Happl\nHK−/radicalBigg\n1+/parenleftbiggHappl\nHK/parenrightbigg2\n+2E\nMHK.(C6)The modulus of Jacobi elliptic functions is k=/radicalbig\n(r2−r3)/(r1−r3). The precession period is\nτ(E) =2K(k)\nγ/radicalbig\nHapplHK/2√r1−r3,(C7)\nwhereK(k) is the first kind of complete elliptic inte-\ngral. The initial state is chosen to be my(0) =r3. Fig-\nure 6(b) is obtained by substituting Eqs. (C1), (C2),\nand (C3) into Eqs. (7) and (8). We note that Eqs.\n(C1), (C2), and (C3) are functions of the energy den-\nsityE. Since we are interested in the instability thresh-\nold near the equilibrium, the value of Eis chosen close\nto the minimum energy Emin. In Fig. 6 (b), we use\nE=Emin+ (Emax−Emin)/NwithN= 100, where\nthe minimum energy Eminand the maximum energy\nEmaxareEmin=−(MHK/2)[1 + (Happl/HK)2] and\nEmax=−MHappl, respectively. The value of Happlis\nchosen to be 50 Oe. Figure 6(d) is obtained in a similar\nway.\n1M. I. Dyakonov and V. I. Perel, “Current-induced spin\norientation of electrons in semiconductors,” Phys. Lett. A\n35, 459 (1971).\n2J. E. Hirsch, “Spin Hall Effect,” Phys. Rev. Lett. 83, 1834\n(1999).\n3Y. K. Kato, R. C. Myers, A. C. Gossard, and D. D.\nAwschalom, “Observation of the Spin Hall Effect in Semi-\nconductors,” Science 306, 1910 (2004).4J. C. Slonczewski, “Current-driven excitation of magnetic\nmultilayers,” J. Magn. Magn. Mater. 159, L1 (1996).\n5L. Berger, “Emission of spin waves by a magnetic mul-\ntilayer traversed by a current,” Phys. Rev. B 54, 9353\n(1996).\n6T. Yang, T. Kimura, and Y. Otani, “Giant spin-\naccumulation signal and pure spin-current-induced re-\nversible magnetization switching,” Nat. Phys. 4, 85111\n(2008).\n7K. Ando, S. Takahashi, K. Harii, K. Sasage, J. Ieda,\nS.Maekawa, andE.Saitoh, “Electric Manipulation ofSpin\nRelaxation Using the Spin Hall Effect,” Phys. Rev. Lett.\n101, 036601 (2008).\n8L. Liu, C.-F. Pai, Y. Li, H. W. Tseng, D. C. Ralph, and\nR. A. Buhrman, “Spin-Torque Switching with the Giant\nSpin Hall Effect of Tantalum,” Science 336, 555 (2012).\n9L. Liu, C.-F. Pai, D.C. Ralph, andR. A.Buhrman, “Mag-\nnetization Oscillations Drive by the Spin Hall Effect in 3-\nTerminal Magnetic Tunnel Junctions Devices,” Phys. Rev.\nLett.109, 186602 (2012).\n10C.-F. Pai, L. Liu, Y. Li, H. W. Tseng, D. C. Ralph, and\nR. A. Buhrman, “Spin transfer torque devices utilizing the\ngiant spin Hall effect of tungsten,” Appl. Phys. Lett. 101,\n122404 (2012).\n11M. Yamanouchi, L. Chen, J. Kim, M. Hayashi, H. Sato,\nS. Fukami, S. Ikeda, F. Matsukura, and H. Ohno, “Three\nterminal magnetic tunnel junction utilizing the spin Hall\neffect of iridium-doped copper,” Appl. Phys. Lett. 102,\n212408 (2013).\n12X. Fan, J. Wu, Y. Chen, M. J. Jerry, H. Zhang, and J. Q.\nXiao, “Observation of the nonlocal spin-orbital effective\nfield,” Nat. Commun. 4, 1799 (2013).\n13M. Cubukcu, O. Boulle, M. Drouard, K. Garello, C. O.\nAvci, I. M. Miron, J. Langer, B. Ocker, P. Gambardella,\nand G. Gaudin, “Spin-orbit torque magnetization switch-\ning of a three-terminal perpendicular magnetic tunnel\njunction,” Appl. Phys. Lett. 104, 042406 (2014).\n14K.-S. Lee, S.-W. Lee, B.-C. Min, and K.-J. Lee, “Thresh-\nold current for switching of a perpendicular magnetic layer\ninduced byspin Hall effect,” Appl.Phys. Lett. 102, 112410\n(2013).\n15K.-S. Lee, S.-W. Lee, B.-C. Min, and K.-J. Lee, “Ther-\nmally activatedswitchingofperpendicularmagnet byspin-\norbit spin torque,” Appl. Phys. Lett. 104, 072413 (2014).\n16J. Z. Sun, “Spin-current interaction with a monodomain\nmagnetic body: A model study,” Phys. Rev. B 62, 570\n(2000).\n17S. Ikeda, K. Miura, H Yamamoto, K. Mizunuma, H. D.\nGan, M. Endo, S. Kanai, J. Hayakawa, F. Matsukura, and\nH. Ohno, “A perpendicular-anisotropy CoFeB-MgO mag-\nnetic tunnel junction,” Nat. Mater. 9, 721 (2010).\n18S. Iihama, Q. Ma, T. Kubota, S. Mizukami, Y. Ando,\nand T. Miyazaki, “Damping of Magnetization Precession\nin Perpendicularly Magnetized CoFeB Alloy Thin Films,”\nAppl. Phys. Express 5, 083001 (2012).\n19S. Zhang, P. M. Levy, and A. Fert, “Mechanisms of Spin-\nPolarized Current-DrivenMagnetization Switching,”Phys .\nRev. Lett. 88, 236601 (2002).\n20M. D. Stiles and A. Zangwill, “Anatomy of spin-transfer\ntorque,” Phys. Rev. B 66, 014407 (2002).\n21M. Zwierzycki, Y. Tserkovnyak, P. J. Kelly, A. Brataas,\nand G. E. W. Bauer, “First-principles study of magneti-\nzation relaxation enhancement and spin transfer in thin\nmagnetic films,” Phys. Rev. B 71, 064420 (2005).\n22A. Brataas, G. E. W. Bauer, and P. J. Kelly, “Non-\ncollinear magnetoelectronics,” Phys. Rep. 427, 157 (2006).\n23I. Theodonis, N. Kioussis, A. Kalitsov, M. Chshiev, and\nW. H. Butler, “Anomalous Bias Dependence of Spin\nTorque in Magnetic Tunnel Junctions,” Phys. Rev. Lett.\n97, 237205 (2006).\n24T. Taniguchi, S. Yakata, H. Imamura, and Y. Ando, “De-\ntermination of Penetration Depth of Transverse Spin Cur-rent in Ferromagnetic Metals by Spin Pumping,” Appl.\nPhys. Express 1, 031302 (2008).\n25A. Ghosh, S. Auffret, U. Ebels, and W. E. Bailey, “Pene-\ntration Depth of Transverse Spin Current in Ultrahin Fer-\nromagnets,” Phys. Rev. Lett. 109, 127202 (2012).\n26A. A. Tulapurkar, Y. Suzuki, A. Fukushima, H. Kub-\nota, H. Maehara, K. Tsunekawa, D. D. Djayaprawira,\nN. Watanabe, and S. Yuasa, “Spin-torque diode effect in\nmagnetic tunnel junctions,” Nature 438, 339 (2005).\n27H. Kubota, A. Fukushima, K. Yakushiji, T. Naga-\nhama, S. Yuasa, K. Ando, H. Maehara, Y. Nagamine,\nK. Tsunekawa, D. D. Djayaprawira, N. Watanabe, and\nY. Suzuki, “Quantitative measurement of voltage depen-\ndence of spin-transfer-torque in MgO-based magnetic tun-\nnel junctions,” Nat. Phys. 4, 37 (2008).\n28J. C. Sankey, Y.-T. Cui, J. Z. Sun, J. C. Slonczewski, R. A.\nBuhrman, and D. C. Ralph, “Measurement of the spin-\ntransfer-torque vector in magnetic tunnel junctions,” Nat .\nPhys.4, 67 (2008).\n29S.-C. Oh, S.-Y. Park, A. Manchon, M. Chshiev, J.-H. Han,\nH.-W. Lee, J.-E. Lee, K.-T. Nam, Y. Jo, Y.-C. Kong,\nB. Dieny, and K.-J. Lee, “Bias-voltage dependence of per-\npendicular spin-transfer torque in asymmetric MgO-based\nmagnetic tunnel junctions,” Nat. Phys. 5, 898 (2009).\n30J. Grollier, V. Cros, H. Jaffr´ es, A. Hamzic, J. M. George,\nG. Faini, J. Ben Youssef, H. LeGall, and A. Fert, “Field\ndependence of magnetization reversal by spin transfer,”\nPhys. Rev. B 67, 174402 (2003).\n31H. Morise and S. Nakamura, “Stable magnetization states\nundera spin-polarized current and amagnetic field,” Phys.\nRev. B71, 014439 (2005).\n32D. Gusakova, D. Houssameddine, U. Ebels, B. Dieny,\nL. Buda-Prejbeanu, M. C. Cyrille, and B. Dela¨ et, “Spin-\npolarized current-induced excitations in a coupled mag-\nnetic layer system,” Phys. Rev. B 79, 104406 (2009).\n33T. Taniguchi, S. Tsunegi, H. Kubota, and H. Imamura,\n“Self-oscillation in spin torque oscillator stabilized by field-\nlike torque,” Appl. Phys. Lett. 104, 152411 (2014).\n34T. Taniguchi, S. Tsunegi, H. Kubota, and H. Imamura,\n“Large amplitude oscillation of magnetization in spin-\ntorque oscillator stabilized by field-like torque,” J. Appl .\nPhys.117, 17C504 (2015).\n35J. Kim, J. Sinha, M. Hayashi, M. Yamanouchi, S. Fukami,\nT. Suzuki, S. Mitani, and H. Ohno, “Layer thickness de-\npendence of the current-induced effective field vector in\nTa|CoFeB|MgO,” Nat. Mater. 12, 240 (2013).\n36K. Garello, I. M. Miron, C. O. Avci, F. Freimuth,\nY. Mokrousov, S. Bl¨ ugel, S. Auffret, O. Boulle, G. Gaudin,\nand P. Gambardella, “Symmetry and magnitude of spin-\norbit torques in ferromagnetic heterostructures,” Nat.\nNanotech. 8, 587 (2013).\n37X. Qiu, P. Deorani, K. Narayanapillai, K.-S. Lee, K.-J.\nLee, H.-W. Lee, and H. Yang, “Angular and temperature\ndependence of current induced spin-orbit effective fields in\nTa/CoFeB/MgO nanowires,” Sci. Rep. 4, 4491 (2014).\n38C.-F. Pai, M.-H. Nguyen, C. Belvin, L. H. Viela-Leao,\nD. C. Ralph, and R. A. Buhrman, “Enhancement of per-\npendicular magnetic anisotropy and tranmission of spin-\nHall-effect-induced spin currents by a Hf spacer layer in\nW/Hf/CoFeB/MgO layer structures,” Appl. Phys. Lett.\n104, 082407 (2014).\n39J. Kim, J. Sinha, S. Mitani, M. Hayashi, S. Takahashi,\nS. Maekawa, M. Yamanouchi, and H. Ohno, “Anoma-\nlous temperature dependence of current-induced torques12\nin CoFeB/MgO heterostructures with Ta-based underlay-\ners,” Phys. Rev. B 89, 174424 (2014).\n40J. Torrejon, J. Kim, J. Sinha, S. Mitani, M. Hayashi,\nM. Yamanouchi, and H. Ohno, “Interface control of the\nmagnetic chirality in CoFeB/MgO heterostructures with\nheavy-metal underlayers,” Nat. Commun. 5, 4655 (2014).\n41I.M. Miron, K.Garello, G.Gaudin, P.-J. Zermatten, M.V.\nCostache, S. Auffret, S. Bandiera, B. Rodmacq, A. Schuhl,\nand P. Gambardella, “Perpendicular switching of a single\nferromagnetic layer induced by in-plane current injection ,”\nNature476, 189 (2011).\n42K.-W. Kim, S.-M. Seo, J. Ryu, K.-J. Lee, and H.-W. Lee,\n“Magnetization dynamics induced by in-plane currents in\nultrathin magnetic nanostructures with Rashba spin-orbit\ncoupling,” Phys. Rev. B 85, 180404 (2012).\n43P. M. Haney, H.-W. Lee, K.-J. Lee, A. Manchon, and\nM. D.Stiles, “Currentinducedtorquesandinterfacial spin -\norbit coupling: Semiclassical modeling,” Phys. Rev. B 87,\n174411 (2013).\n44P. M. Haney, H.-W. Lee, K.-J. Lee, A. Manchon, and\nM. D. Stiles, “Current-induced torques and interfacial\nspin-orbit coupling,” Phys. Rev. B 88, 214417 (2013).\n45M. Jamali, K. Narayanapillai, X. Qiu, L. M. Loong,\nA. Manchon, and H. Yang, “Spin-Orbit Torques in Co/Pd\nMultilayer Nanowires,” Phys. Rev. Lett. 111, 246602\n(2013).\n46G. Yu, P. Upadhyaya, Y. Fan, J.G. Alzate, W. Jiang, K. L.\nWong, S. Takei, S. A. Bender, L.-T. Chang, Y. Jiang,\nM. Lang, J. Tang, Y. Wang, Y. Tserkovnyak, P. K. Amiri,\nand K. L. Wang, “Switching of perpendicular magnetiza-\ntion by spin-orbit torques in the absence of external mag-\nnetic fields,” Nat. Nanotech. 9, 548 (2014).47C. O. Pauyac, X. Wang, M. Chshiev, and A. Man-\nchon, “Angular dependence and symmetry of Rashba spin\ntorqueinferromagnetic structures,”Appl.Phys.Lett. 102,\n252403 (2013).\n48J. Torrejon, F. Garcia-Sanchez, T. Taniguchi, J. Shinha,\nS. Mitani, J.-V. Kim, and M. Hayashi, “Current-driven\nasymmetric magnetization switching in perpendicularly\nmagnetized CoFeB/MgO heterostructures,” Phys. Rev. B\n91, 214434 (2015).\n49L. Liu, O. J. Lee, T. J. Gudmundsen, D. C. Ralph, and\nR. A. Buhrman, “Current-Induced Switching of Perpen-\ndicularly Magnetized Magnetic Layers Using Spin Torque\nfrom the Spin Hall Effect,” Phys. Rev. Lett. 109, 096602\n(2012).\n50L. You, O. Lee, D. Bhowmik, D. Labanowski, J. Hong, J.\nBokor, and S. Salahuddin, arXiv:1409.0620.\n51(), a similar approach was recently developed by Yan and\nBazaliy, Phys. Rev. B 91214424 (2015).\n52(),theanalytical formulaofthecritical currentcorrespo nd-\ning the case of β= 0 is discussed in Ref.14. The formula\nof Ref.14is applicable to a large damping limit ( α >0.03),\nwhile we are interested in a low damping limit.\n53G. Bertotti, I. Mayergoyz, and C. Serpico, Nonlinear mag-\nnetization Dynamics in Nanosystems (Elsevier, Oxford,\n2009).\n54S. Wiggins, “Introduction to applied nonlinear dynamical\nsystems and chaos,” (Springer, 2003) Chap. 1.\n55T. Taniguchi, “Nonlinear analysis of magnetization dy-\nnamics excited by spin Hall effect,” Phys. Rev. B 91,\n104406 (2015)."    },    {        "title": "1503.07043v5.Spin_dynamics_and_frequency_dependence_of_magnetic_damping_study_in_soft_ferromagnetic_FeTaC_film_with_a_stripe_domain_structure.pdf",        "content": "Spin dynamics and frequency dependence of magnetic damping study in soft\nferromagnetic FeTaC \flm with a stripe domain structure\nB. Samantaray1a), Akhilesh K. Singh2, A.Perumal2, R. Ranganathan1and P. Mandal1, 2\n1)Saha Institute of Nuclear Physics, 1/AF Bidhannagar, Calcutta 700 064, Indiaa)\n2)Department of Physics, Indian Institute of Technology Guwahati, Guwahati - 781039,\nIndia\n(Dated: 12 November 2021)\nPerpendicular magnetic anisotropy (PMA) and low magnetic damping are the key factors for the free layer\nmagnetization switching by spin transfer torque technique in magnetic tunnel junction devices. The mag-\nnetization precessional dynamics in soft ferromagnetic FeTaC thin \flm with a stripe domain structure was\nexplored in broad band frequency range by employing micro-strip ferromagnetic resonance technique. The\npolar angular variation of resonance \feld and linewidth at di\u000berent frequencies have been analyzed numeri-\ncally using Landau-Lifshitz-Gilbert equation by taking into account the total free energy density of the \flm.\nThe numerically estimated parameters Land\u0013 e g-factor, PMA constant, and e\u000bective magnetization are found\nto be 2.1, 2\u0002105erg/cm3and 7145 Oe, respectively. The frequency dependence of Gilbert damping parameter\n(\u000b) is evaluated by considering both intrinsic and extrinsic e\u000bects into the total linewidth analysis. The value\nof\u000bis found to be 0.006 at 10 GHz and it increases with decreasing precessional frequency.\nSpin transfer torque (STT) has grater credibility com-\npared to other techniques towards ultrafast spin dynam-\nics in ferromagnet by electric current induced magneti-\nzation reversal of spin valves and magnetic tunnel junc-\ntions (MTJ).1The current researchers are more keen to\nfocus on STT technology for its high density magnetic\nrandom access memories (MRAM),2,3STT-driven do-\nmain wall devices4and perpendicular magnetic record-\ning media5applications. In order to make this technol-\nogy more e\u000ecient, lowering the critical current density is\nessential which requires the material speci\fcations with\nlow saturation magnetization ( MS), high spin polariza-\ntion, large uniaxial perpendicular magnetic anisotropy\n(PMA) constant and low magnetic damping.6{8The mag-\nnetic damping parameter ( \u000b) can be described well by\nthe phenomenological Landau-Lifshitz-Gilbert equation\nand is known as the Gilbert damping.9,10Several at-\ntempts have been made for understanding the origin of\nGilbert damping in spin dynamics relaxation in single\nlayer as well as multilayered magnetic alloys, which arises\nfrom both intrinsic and extrinsic parts of the material.\nThe intrinsic contribution to the Gilbert damping pa-\nrameter has been studied by tuning the strength of the\nspin-orbit coupling.8,11,12Recently, Ikeda et al.13have re-\nported that CoFeB-MgO based MTJ with PMA would be\nreliable for high-density non-volatile memory application\ndue to its high thermal stability and e\u000eciency towards\nSTT technology. The investigation on magnetic dynam-\nics, PMA and the apparent magnetic damping have been\nstudied extensively in CoFeB based soft ferromagnetic\nthin \flm by ferromagnetic resonance (FMR) and time-\nresolved magneto-optical Kerr e\u000bect.14,15Malinowski et\nal.16have reported a large increase in Gilbert damping\nwith applied magnetic \feld in perpendicularly magne-\ntized CoFeB thin \flm.\na)Electronic mail: iitg.biswanath@gmail.comIn this letter, we focus on amorphous FeTaC layer due\nto its interesting soft ferromagnetic (FM) properties.17,18\nThe amorphous soft FM layer reduces the number of pin-\nning centers which may lead to the STT-driven domain\nwall motion along with high tunneling magnetoresistance\nratio (TMR). The transcritical loop along with the stripe\ndomain structure, which are the manifestation of PMA\ncomponent were reported on FeTaC thin \flm with thick-\nness of 200 nm.18,19To shed some more light onto its dy-\nnamic magnetic properties, we have further studied this\n\flm by using ferromagnetic resonance technique. Though\nthe magnetic anisotropy and Gilbert damping have been\nstudied by FMR technique in several magnetic thin\n\flms like Heusler alloys, permalloy, soft magnetic ma-\nterials and multilayered (FM/antiferromagnetic or non-\nmagnetic/FM) magnetic \flms for magnetic recording,\nMTJ and TMR reader applications, most of the reports\nare limited to single frequency due to the measurements\nin X-band electron-spin-resonance spectrometer where\nthe cavity resonates at particular frequency.6,8,15,20{23In\nthis report, spin dynamics and magnetic relaxation are\nstudied at di\u000berent magnetization precessional frequen-\ncies.\nSoft ferromagnetic Fe 80Ta8C12single layer \flm with\nthickness 200 nm was deposited by dc magnetron sput-\ntering technique and the details of growing environment\nwere reported earlier.18The static and dynamic magnetic\nproperties were explored by using a vector network ana-\nlyzer (VNA) based homemade micro-strip ferromagnetic\nresonance (MS-FMR) spectrometer. The micro-strip line\nwhich was coupled to VNA and Schottky diode detector\n(Agilent 8473D) through high frequency coaxial cables\nwas mounted in between the pole pieces of the electro-\nmagnet. The magnetic thin \flm with \flm side down-\nward was mounted on the strip line. The frequency of\nthe microwave signal was \fxed by using an Agilent Tech-\nnologies made VNA (Model PNA-X, N-5242A) with a\nconstant microwave power of 5 dBm. The \frst deriva-arXiv:1503.07043v5  [cond-mat.mtrl-sci]  14 May 20152\nFIG. 1. Schematic diagram of MandHvectors in spherical\npolar coordinate system. 'Mand'Hare the in-plane angle\nof magnetization ( M) and external magnetic \feld ( H) with\nrespect toxaxis, while \u0012Mand\u0012Hare out-of-plane angles\nwith respect to zaxis.\ntive of the absorption spectrum with respect to magnetic\n\feld (H) was collected by \feld modulation and lock-in\ndetection technique. The FM thin \flm was treated in-\nplane and out-of-plane orientations. The magnetic \feld\nsweeping FMR spectra were recorded by varying two pa-\nrameters: precessional frequency and the angle between\nHand normal of the \flm. The frequency ( f) and po-\nlar angle (\u0012H) dependence of resonance \feld ( Hr) and\nlinewidth (\u0001 HPP) were extracted from each FMR spec-\ntrum and the numerical calculations were carried out by\nmathematica program for di\u000berent relaxation processes.\nThe precession of magnetization ( M) in the sample\nplane under the in\ruence of microwave and external mag-\nnetic \feld is illustrated in Fig. 1 in a polar coordinate\nsystem.'H('M) is the in-plane angle between H(M)\nandxaxis and\u0012H(\u0012M) is the polar angle between zaxis\nandH(M). The uniform precession of magnetization\ncan be described by the Landau-Lifshitz-Gilbert (LLG)\nequation of motion,9,10\n@\u0000 !M\n@t=\u0000\r\u0010\u0000 !M\u0002\u0000 !Heff\u0011\n+G\n\rM2\nS\"\n\u0000 !M\u0002@\u0000 !M\n@t#\n(1)\nThe \frst term corresponds to the precessional torque in\nthe e\u000bective magnetic \feld and the second term is the\nGilbert damping torque. \r=g\u0016B=~is denoted as gyro-\nmagnetic ratio and written in terms of Land\u0013 e gfactor,\nBohr magneton \u0016B, and Planck constant ~.G=\r\u000bMS\nis related to the intrinsic relaxation rate of the material.\n\u000bis the dimensionless Gilbert damping parameter. The\nfree energy density of a single magnetic thin \flm can be\nwritten as,\nE=\u0000MSH[sin\u0012Hsin\u0012Mcos('H\u0000'M) + cos\u0012Hcos\u0012M]\n\u00002\u0019M2\nSsin2\u0012M+K?sin2\u0012M\n(2)\nwhere the \frst term is analogous to the Zeeman energy,\nthe second term is dipolar demagnetization energy, the\nthird term signi\fes the anisotropy energy, MSis the sat-\nuration magnetization, K?is the PMA constant with\ncorresponding anisotropic \feld H?= 2K?=MS. The\nresonance frequency frof the uniform precession mode is\nFIG. 2. (a) shows typical FMR spectra at di\u000berent frequencies\nin planar orientation, (b) shows fdependence of Hrfor in-\nplane applied magnetic \feld. Experimentally and numerically\ncalculatedHrvalues are shown as open circles and solid line\nrespectively and (c) shows the plot of room temperature M\u0000\nHloop and domain images reproduced from Ref. [19].\ndeduced from the energy density by using the following\nexpression,24\nf2\nr=\u0010\r\n2\u0019\u00112 1\nM2\nSsin2\u0012M\"\n@2E\n@\u00122\nM@2E\n@'2\nM\u0000\u0012@2E\n@\u0012M@'M\u00132#\n(3)\nwhere the derivatives are evaluated at equilibrium posi-\ntions ofMandH.\nFor the in-plane orientation, the typical FMR spectra\nat di\u000berent frequencies are shown in Fig. 2(a). The mea-\nsurements were carried out by varying the frequency from\n1 to 18 GHz with an interval of 0.5 GHz. In the lower\nfrequency range 1-6 GHz, the FMR spectra show two res-\nonance peaks. The low-\feld resonance peaks named as\nsecondary mode and are marked by 2 and 2\u0003for 2.5 and 6\nGHz, respectively in Fig. 2(a). This mode arises from the\nlinear unsaturated zone of the transcritical M(H) loop19\nand is usually observed in stripe-domain structure.25,26\nThe transcritical loop along with the domain structure\nare reproduced from earlier report19and is shown in\nFig. 2(c). The dense stripe-domain structure observed in\nthis \flm con\frms the presence of perpendicular magnetic\nanisotropy. The primary modes usually called uniform\nmode are marked as 1 and 1\u0003for 2.5 and 6 GHz, respec-\ntively. The value of Hrfor secondary resonance peak\nincreases with the increase in fup to 4.5 GHz and then\nfollows the reverse trend as depicted in Fig. 2(b). This3\nFIG. 3. Equilibrium angle of the magnetization, \u0012M, as a\nfunction of the applied \feld direction, \u0012H, in out-of-plane con-\n\fguration at di\u000berent frequencies.\ncould be explained on the basis that the value of Hrof\nthe uniform mode above 4.5 GHz overcomes the parallel\nsaturation \feld, i.e., 280 Oe as observed in M(H) curve.\nAbove 6 GHz, Hrexceeds the parallel saturation \feld in\nlarge extent and this could be the reason for the strong\nattenuation of secondary phase. In planar con\fguration\n(\u0012M=\u0012H=\u0019=2), the solution for the in-plane resonance\nfrequency can be calculated by incorporating the total\nenergy in Eq. 3 and is given by,\nfr=\r\n2\u0019[(4\u0019M+Hcos ('H\u0000'M)) (Hcos ('H\u0000'M))]1\n2\n(4)\nThe value of 'Mcan be calculated by using the solution\nofHat equilibrium condition, i.e.,@E\n@'M=0. However,\nfor the present thin \flm, we could not \fnd any planar\nanisotropy from the 'Hdependence of Hrand hence\nconclude,'H='M. Thefdependence of Hris nu-\nmerically calculated by using Eq. 4 and is shown as a\nsolid line in Fig. 2(b). The numerically calculated values\nyielded a good \ft and the parameters are found to be re-\nliable with 4 \u0019MS= 7791\u000610 Oe and\r= 2.95 MHz/Oe\nwith ag-factor of 2.1. The deduced value of saturation\nmagnetization is very close to earlier reported value from\nM(H) loop measurement.18\nIn out-of-plane con\fguration, the solution for the res-\nonance frequency is deduced from Eq. 3 by employing\nthe conditions, 'H='M=0\u000eand is represented in Eq. 5.\nf2\nr=\u0000\r\n2\u0019\u00012h\nHcos (\u0012M\u0000\u0012H)\u0000\u0010\n4\u0019MS\u00002K?\nMS\u0011\ncos 2\u0012Mi\nh\nHcos (\u0012M\u0000\u0012H)\u0000\u0010\n4\u0019MS\u00002K?\nMS\u0011\ncos2\u0012Mi\n(5)\nThe equilibrium angle \u0012Mis numerically calculated for\nFIG. 4. Angular dependence of resonance \feld Hrin out-\nof-plane con\fguration at di\u000berent frequencies. ( \u000e) shows the\nexperimental points and the line (-) shows the modeled data.\neach value of \u0012Hby minimizing the energy, i.e.,@E\n@\u0012M= 0\nand is depicted in Fig. 3 for di\u000berent frequencies. Fig.\n3 demonstrates that magnetization suddenly attempts to\nalign in planar direction as the magnetic \feld goes away\nfrom the\u0012H=0\u000eand 180\u000e. Fig. 4 shows one complete\nround of\u0012Hdependence of Hrat di\u000berent frequencies.\nUniaxial PMA is found to be observed along with singu-\nlarity at\u0012H= 0\u000eand\u0012H= 180\u000e, which signi\fes that\nin\fnite magnetic \feld is required to turn the Mvector\nparallel toHin perpendicular con\fguration. The depen-\ndence ofHron\u0012His modeled at di\u000berent frequencies\nstarting from 4 to 10 GHz with 2 GHz intervals by using\nEq. 5 and the interpolated values of \u0012Mfrom Fig. 3. The\nmodeled values of Hrare plotted as a solid line in Fig.\n4 and a very good agreement with experimental data is\nobserved. The parameters deduced from this calculation\nare found to be K?=2\u0002105erg/cm3and 4\u0019Meff=7145\nOe.\nFinally, the damping of magnetization precession has\nbeen analyzed from linewidth of FMR spectra. The \u0012H\ndependence of \u0001 HPPas shown in Fig. 5 was extracted\nfrom the polar angle variation of FMR spectrum at di\u000ber-\nent frequencies in the range of 4-10 GHz with an interval\nof 2 GHz. In order to get better clarity of Fig. 5, the\ndata for the 4 GHz frequency are not shown. The total\nlinewidth broadening due to the intrinsic and extrinsic\nparts of the material has been expressed in the following\nequation,8,27\n\u0001HPP= \u0001H(\u000b) + \u0001H(\u00014\u0019Meff) + \u0001H(\u0001\u0012H)\n=2p\n31\nj@!\n@Hrj\u000b\r\nMS\u0010\n@2E\n@\u00122\nM+1\nsin2\u0012M@2E\n@'2\nM\u0011\n+\n1p\n3\u0010\f\f\f@H\n@4\u0019Meff\f\f\f\u00014\u0019Meff\u0011\n+1p\n3\u0010\f\f\f@H\n@\u0012H\f\f\f\u0001\u0012H\u0011\n(6)\nwhere 4\u0019Meff= 4\u0019MS\u00002K?=MS, is the e\u000bective mag-4\nFIG. 5. Out-of-plane angular dependence of total linewidth,\n\u0001Hppat di\u000berent frequencies. ( \u000e) shows the experimental\npoints and the line ( \u0000) shows the modeled data.\nFIG. 6.\u0012Hdependence of \u0001 Hpp, \u0001H(\u000b), \u0001H(\u00014\u0019Meff)\nand \u0001H(\u0001\u0012H) modeled data at 6 GHz frequency.\nnetization. \u0001 H(\u000b) arises from the intrinsic Gilbert type\ndamping and has large contribution towards linewidth\nbroadening. The parameter \u000bsigni\fes how fast the pre-\ncessional energy is dissipated into the lattice. The terms\n\u0001H(4\u0019\u0001Meff) and \u0001H(\u0001\u0012H) represent the linewidth\nbroadening due to the spatial dispersion of the magni-\ntude and direction of Meff, respectively. The \u0012Hdepen-\ndence of \u0001 HPPwas modeled by using Eq. 6 and the\ninterpolated values of \u0012Mfrom Fig. 3 at di\u000berent fre-\nquencies. The numerically calculated values of \u0001 HPP\nare shown as solid lines in Fig. 5. The individual contri-\nbutions towards the total linewidth (\u0001 Hpp) is also shown\nin Fig. 6. The curves are shown for a single frequency\nfor better clarity. The linewidth broadening is observed\nmainly due to the intrinsic Gilbert damping. The ex-trinsic contribution is found to be negligible when \u0012H\nis away from 0\u000eand 180\u000ebut it is large near the per-\npendicular con\fguration. The Gilbert damping param-\neter at di\u000berent frequencies for the FeTaC thin \flm is\nplotted in the inset of Fig. 6. It shows that the \u000bde-\ncrease monotonically with the increase in frequency. The\nlow value of damping parameter observed in the present\nthin \flm can be more relevant towards the STT tech-\nnology or MTJ applications. Such an increase of \u000bby\ndecreasing precessional frequency has been attributed to\nthe inhomogeneous linewidth broadening due to the dis-\npersion of anisotropic \feld.28,29The dispersion in mag-\nnitude and direction of e\u000bective magnetization are found\nto be \u00014\u0019Meff=0.1 KOe, \u0001 \u0012H\u00191\u000210\u00004degree. The\nvalue of Gilbert damping constant for the present Fe-\nTaC thin \flm is found to be comparable to those re-\nported in Fe-based magnetic thin \flms, such as FePd\nternary alloy11, permalloy30, NiFe/CoFeB/CoFe multi-\nlayered sturucture15and (FeCo) 1\u0000xGdx8. However, the\nMn- and Co- based thin \flms6,7have larger damping pa-\nrameter as compared to the present \flm which could be\nunderstood on the basis of spin-orbit coupling.\nIn conclusion, PMA and Gilbert damping which\nare very important and crucial parameters for STT,\nSTT-MRAM and TMR applications, have been analyzed\nin FeTaC soft ferromagnetic thin \flm with a striped\ndomain structure by using MS-FMR technique in broad\nband frequency range. The precise estimation of Land\u0013 e\ng-factor, PMA constant and 4 \u0019Meffwere carried out\nby using total energy density function for magnetic\nthin \flm. Spin dynamics relaxation which is quanti\fed\nby Gilbert damping parameter has been analyzed at\ndi\u000berent frequencies and is found to be 0.006 which\nfalls in the most reliable order from application point\nof view. The values of \u000bare found to be comparable to\nthose reported Fe-based single layer and multilayered\nmagnetic thin \flms.\nAcknowledgement\nThe authors would like to thank Dr. Najmul Haque\nfor helping to write energy minimization calculation in\nmathematica and Mr. Nazir Khan for his help during\nwriting data acquisition software.\n1J. Sankey, Y.-T. Cui, J. Sun, J. Slonczewski, R. Buhrman, and\nD. Ralph, \\Measurement of the spin-transfer-torque vector in\nmagnetic tunnel junctions,\" Nature Physics, 4, 67{71 (2008).\n2T. Devolder, J. Hayakawa, K. Ito, H. Takahashi, S. Ikeda,\nP. Crozat, N. Zerounian, J.-V. Kim, C. Chappert, and H. Ohno,\n\\Single-shot time-resolved measurements of nanosecond-scale\nspin-transfer induced switching: Stochastic versus deterministic\naspects,\" Phys. Rev. Lett., 100, 057206 (2008).\n3Y. Huai, F. Albert, P. Nguyen, M. Pakala, and T. Valet, \\Ob-\nservation of spin-transfer switching in deep submicron-sized and\nlow-resistance magnetic tunnel junctions,\" Applied Physics Let-\nters, 84, 3118{3120 (2004).\n4S. T. e. a. Fukami, S., \\Low-current perpendicular domain wall\nmotion cell for scalable high-speed mram,\" Dig. Tech. Pap.-\nSymp. VLSI Technol., 24, 230 (2009).\n5S. Khizroev and D. Litvinov, \\Perpendicular magnetic record-5\ning: Writing process,\" Journal of Applied Physics, 95, 4521{4537\n(2004).\n6S. Mizukami, A. Sakuma, T. Kubota, Y. Kondo, A. Sugihara,\nand T. Miyazaki, \\Fast magnetization precession for perpendic-\nularly magnetized mnalge epitaxial \flms with atomic layered\nstructures,\" Applied Physics Letters, 103, 142405 (2013).\n7H.-S. Song, K.-D. Lee, J.-W. Sohn, S.-H. Yang, S. S. P. Parkin,\nC.-Y. You, and S.-C. Shin, \\Relationship between gilbert damp-\ning and magneto-crystalline anisotropy in a ti-bu\u000bered co/ni mul-\ntilayer system,\" Applied Physics Letters, 103, 022406 (2013).\n8X. Guo, L. Xi, Y. Li, X. Han, D. Li, Z. Wang, and Y. Zuo,\n\\Reduction of magnetic damping constant of feco \flms by rare-\nearth gd doping,\" Applied Physics Letters, 105, 072411 (2014).\n9L. Landau and E. Lifshitz, Phys. Z. Sowjetunion, 8, 153 (1935).\n10T. L. Gilbert, Phys. Rev., 100, 1243 (1955).\n11P. He, X. Ma, J. W. Zhang, H. B. Zhao, G. L upke, Z. Shi, and\nS. M. Zhou, \\Quad ratic scaling of intrinsic gilbert damping with\nspin-orbital coupling in l1 0 fepdpt \flms: Experiments and ab\ninitio calculat ions,\" Phys. Rev. Lett., 110, 077203 (2013).\n12G. Woltersdorf, M. Kiessling, G. Meyer, J.-U. Thiele, and\nC. H. Back, \\Damping by slow relaxing rare earth impurities\nin ni 80fe20,\" Phys. Rev. Lett., 102, 257602 (2009).\n13S. Ikeda, K. Miura, H. Yamamoto, K. Mizunuma, H. D. Gan,\nM. Endo, S. Kanai, J. Hayakawa, F. Matsukura, and H. Ohno,\n\\A perpendicular-anisotropy cofebmgo magnetic tunnel junc-\ntion,\" Nat. Mater., 9, 721{724 (2010).\n14E. Hirayama, S. Kanai, H. Sato, F. Matsukura, and H. Ohno,\n\\Ferromagnetic resonance in nanoscale cofeb/mgo magnetic tun-\nnel junctions,\" Journal of Applied Physics, 117, 17B708 (2015).\n15L. Lu, Z. Wang, G. Mead, C. Kaiser, Q. Leng, and M. Wu,\n\\Damping in free layers of tunnel magneto-resistance readers,\"\nApplied Physics Letters, 105, 012405 (2014).\n16G. Malinowski, K. C. Kuiper, R. Lavrijsen, H. J. M. Swagten,\nand B. Koopmans, \\Magnetization dynamics and gilbert damp-\ning in ultrathin co48fe32b20 \flms with out-of-plane anisotropy,\"\nApplied Physics Letters, 94, 102501 (2009).\n17K. Tanahashi, A. Kikukawa, Y. Takahashi, and Y. Hosoe, \\Lam-\ninated nanocrystalline soft underlayers for perpendicular record-\ning,\" Journal of Applied Physics, 93, 6766{6768 (2003).\n18A. K. Singh, B. Kisan, D. Mishra, and A. Perumal, \\Thick-\nness dependent magnetic properties of amorphous fetac \flms,\"\nJournal of Applied Physics, 111, 093915 (2012).\n19A. K. Singh, S. Mallik, S. Bedanta, and A. Perumal, \\Spacerlayer and temperature driven magnetic properties in multilayer\nstructured fetac thin \flms,\" Journal of Physics D: Applied\nPhysics, 46, 445005 (2013).\n20K. Zakeri, J. Lindner, I. Barsukov, R. Meckenstock, M. Farle,\nU. von H orsten, H. Wende, W. Keune, J. Rocker, S. S. Kalarickal,\nK. Lenz, W. Kuch, K. Baberschke, and Z. Frait, \\Spin dynamics\nin ferromagnets: Gilbert damping and two-magnon scattering,\"\nPhys. Rev. B, 76, 104416 (2007).\n21Y. Gong, Z. Cevher, M. Ebrahim, J. Lou, C. Pettiford, N. X. Sun,\nand Y. H. Ren, \\Determination of magnetic anisotropies, inter-\nlayer coupling, and magnetization relaxation in fecob/cr/fecob,\"\nJournal of Applied Physics, 106, 063916 (2009).\n22H. Pandey, P. C. Joshi, R. P. Pant, R. Prasad, S. Auluck, and\nR. C. Budhani, \\Evolution of ferromagnetic and spin-wave res-\nonances with crystalline order in thin \flms of full-heusler alloy\nco2mnsi,\" Journal of Applied Physics, 111, 023912 (2012).\n23N. Behera, M. S. Singh, S. Chaudhary, D. K. Pandya, and P. K.\nMuduli, \\E\u000bect of ru thickness on spin pumping in ru/py bi-\nlayer,\" Journal of Applied Physics, 117, 17A714 (2015).\n24O. Acher, S. Queste, M. Ledieu, K.-U. Barholz, and R. Mattheis,\n\\Hysteretic behavior of the dynamic permeability on a ni-fe thin\n\flm,\" Phys. Rev. B, 68, 184414 (2003).\n25O. Acher, C. Boscher, B. Brul, G. Perrin, N. Vukadinovic,\nG. Suran, and H. Joisten, \\Microwave permeability of ferro-\nmagnetic thin \flms with stripe domain structure,\" Journal of\nApplied Physics, 81, 4057{4059 (1997).\n26N. Vukadinovic, M. Labrune, J. B. Youssef, A. Marty, J. C. Tou-\nssaint, and H. Le Gall, \\Ferromagnetic resonance spectra in a\nweak stripe domain structure,\" Phys. Rev. B, 65, 054403 (2001).\n27S. Mizukami, Y. Ando, and T. Miyazaki, \\E\u000bect of spin\ndi\u000busion on gilbert damping for a very thin permalloy layer\nin cu/permalloy/cu/pt \flms,\" Physical Review B, 66, 104413\n(2002).\n28Z. Celinski and B. Heinrich, \\Ferromagnetic resonance linewidth\nof fe ultrathin \flms grown on a bcc cu substrate,\" Journal of\nApplied Physics, 70, 5935{5937 (1991).\n29T. J. Silva, C. S. Lee, T. M. Crawford, and C. T. Rogers,\n\\Inductive measurement of ultrafast magnetization dynamics in\nthin-\flm permalloy,\" Journal of Applied Physics, 85, 7849{7862\n(1999).\n30G. Counil, J.-V. Kim, T. Devolder, C. Chappert, K. Shigeto, and\nY. Otani, \\Spin wave contributions to the high-frequency mag-\nnetic response of thin \flms obtained with inductive methods,\"\nJournal of Applied Physics, 95, 5646{5652 (2004)."    },    {        "title": "1503.07854v2.Thermophoresis_of_an_Antiferromagnetic_Soliton.pdf",        "content": "Brownian thermophoresis of an antiferromagnetic soliton\nSe Kwon Kim,1Oleg Tchernyshyov,2and Yaroslav Tserkovnyak1\n1Department of Physics and Astronomy, University of California, Los Angeles, California 90095, USA\n2Department of Physics and Astronomy, The Johns Hopkins University, Baltimore, Maryland 21218, USA\n(Dated: September 25, 2021)\nWe study dynamics of an antiferromagnetic soliton under a temperature gradient. To this end,\nwe start by phenomenologically constructing the stochastic Landau-Lifshitz-Gilbert equation for an\nantiferromagnet with the aid of the \ructuation-dissipation theorem. We then derive the Langevin\nequation for the soliton's center of mass by the collective coordinate approach. An antiferromagentic\nsoliton behaves as a classical massive particle immersed in a viscous medium. By considering a\nthermodynamic ensemble of solitons, we obtain the Fokker-Planck equation, from which we extract\nthe average drift velocity of a soliton. The di\u000busion coe\u000ecient is inversely proportional to a small\ndamping constant \u000b, which can yield a drift velocity of tens of m/s under a temperature gradient\nof 1 K/mm for a domain wall in an easy-axis antiferromagnetic wire with \u000b\u001810\u00004.\nPACS numbers: 75.78.-n, 66.30.Lw, 75.10.Hk\nIntroduction. |Ordered magnetic materials exhibit\nsolitons and defects that are stable for topological rea-\nsons [1]. Well-known examples are a domain wall (DW)\nin an easy-axis magnet or a vortex in a thin \flm. Their\ndynamics have been extensively studied because of fun-\ndamental interest as well as practical considerations such\nas the racetrack memory [2]. A ferromagnetic (FM) soli-\nton can be driven by various means, e.g., an external\nmagnetic \feld [3] or a spin-polarized electric current [4].\nRecently, the motion of an FM soliton under a temper-\nature gradient has attracted a lot of attention owing to\nits applicability in an FM insulator [5{8]. A temperature\ngradient of 20 K/mm has been demonstrated to drive a\nDW at a velocity of 200 \u0016m/s in an yttrium iron garnet\n\flm [9].\nAn antiferromagnet (AFM) is of a great current inter-\nest in the \feld of spintronics [10{12] due to a few advan-\ntages over an FM. First, the characteristic frequency of\nan AFM is several orders higher than that of a typical\nFM, e.g., a timescale of optical magnetization switching\nis an order of ps for AFM NiO [13] and ns for FM CrO 2\n[14], which can be exploited to develop faster spintronic\ndevices. Second, absence of net magnetization renders\nthe interaction between AFM particles weak, and, thus,\nleads us to prospect for high-density AFM-based devices.\nDynamics of an AFM soliton can be induced by an elec-\ntric current or a spin wave [15{17].\nA particle immersed in a viscous medium exhibits a\nBrownian motion due to a random force that is required\nto exist to comply with the \ructuation-dissipation theo-\nrem (FDT) [18, 19]. An externally applied temperature\ngradient can also be a driving force, engendering a phe-\nnomenon known as thermophoresis [20]. Dynamics of an\nFM and an AFM includes spin damping, and, thus, in-\nvolves thermal \ructuations at a \fnite temperature [21].\nThe corresponding thermal stochastic \feld in\ruences dy-\nnamics of a magnetic soliton [8, 22, 23], e.g., by assisting\na current-induced motion of an FM DW [24].\nFIG. 1. (Color online) A thermal stochastic force caused by\na temperature gradient pushes an antiferromagnetic domain\nwall to a colder region. The di\u000busion coe\u000ecient of the domain\nwall is inversely proportional to a small damping constant,\nwhich may give rise to a sizable drift velocity.\nIn this Rapid Communication, we study the Brown-\nian motion of a soliton in an AFM under a tempera-\nture gradient. We derive the stochastic Landau-Lifshitz-\nGilbert (LLG) equation for an AFM with the aid of the\nFDT, which relates the \ructuation of the staggered and\nnet magnetization to spin damping. We then derive the\nLangevin equation for the soliton's center of mass by em-\nploying the collective coordinate approach [16, 25]. We\ndevelop the Hamiltonian mechanics for collective coordi-\nnates and conjugate momenta of a soliton, which sheds\nlight on stochastic dynamics of an AFM soliton; it can be\nconsidered as a classical massive particle moving in a vis-\ncous medium. By considering a thermodynamic ensemble\nof solitons, we obtain the Fokker-Planck equation, from\nwhich we extract the average drift velocity. As a case\nstudy, we compute the drift velocity of a DW in a quasi\none-dimensional easy-axis AFM.\nThermophoresis of a Brownian particle is a multi-\nfaceted phenomenon, which involves several competing\nmechanisms. As a result, a motion of a particle depends\non properties of its environment such as a medium or\na temperature T[26]. For example, particles in pro-\ntein (e.g., lysozyme) solutions move to a colder region\nforT > 294 K and otherwise to a hotter region [20, 27].arXiv:1503.07854v2  [cond-mat.mes-hall]  8 Jul 20152\nThermophoresis of an AFM soliton would be at least as\ncomplex as that of a Brownian particle. We focus on one\naspect of it in this Rapid Communication; the e\u000bect of\nthermal stochastic force on dynamics of the soliton. We\ndiscuss two other possible mechanisms, the e\u000bects of a\nthermal magnon current and an entropic force [5], later\nin the Rapid Communication.\nMain results .|Before pursuing details of derivations,\nwe \frst outline our three main results. Let us consider\na bipartite AFM with two sublattices that can be trans-\nformed into each other by a symmetry transformation\nof the crystal. Its low-energy dynamics can be devel-\noped in terms of two \felds: the unit staggered spin\n\feldn\u0011(m1\u0000m2)=2 and the small net spin \feld\nm\u0011(m1+m2)=2 perpendicular to n. Here, m1and\nm2are unit vectors along the directions of spin angular\nmomentum in the sublattices.\nStarting from the standard Lagrangian description of\nthe antiferromagnetic dynamics [28], we will show below\nthat the appropriate theory of dissipative dynamics of\nantiferromagnets at a \fnite temperature is captured by\nthe stochastic LLG equation\ns(_n+\fn\u0002_m) =n\u0002(h+hth); (1a)\ns(_m+\fm\u0002_m+\u000bn\u0002_n) =n\u0002(g+gth)\n+m\u0002(h+hth);(1b)\nin conjunction with the correlators of the thermal\nstochastic \felds gthandhth,\nhgth\ni(r;t)gth\nj(r0;t0)i= 2kBT\u000bs\u000e ij\u000e(r\u0000r0)\u000e(t\u0000t0);(2a)\nhhth\ni(r;t)hth\nj(r0;t0)i= 2kBT\fs\u000e ij\u000e(r\u0000r0)\u000e(t\u0000t0);(2b)\nwhich are independent of each other [29]. This is our \frst\nmain result. Here, \u000band\fare the damping constants\nassociated with _nand _m,g\u0011\u0000\u000eU=\u000enandh\u0011\u0000\u000eU=\u000em\nare the e\u000bective \felds conjugate to nandm,U[n;m]\u0011\nU[n]+R\ndVjmj2=2\u001fis the potential energy ( \u001frepresents\nthe magnetic susceptibility), and s\u0011~S=Vis the spin\nangular momentum density ( Vis the volume per spin)\nper each sublattice. The potential energy U[n(r;t)] is\na general functional of n, which includes the exchange\nenergyR\ndVA ij@in\u0001@jnat a minimum [28].\nSlow dynamics of stable magnetic solitons can often\nbe expressed in terms of a few collective coordinates\nparametrizing slow modes of the system. The center of\nmassRrepresents the proper slow modes of a rigid soli-\nton when the translational symmetry is weakly broken.\nTranslation of the stochastic LLG equation (1) into the\nlanguage of the collective coordinates results in our sec-\nond main result, a Langevin equation for the soliton's\ncenter of mass R:\nMR+ \u0000_R=\u0000@U=@R+Fth; (3)\nwhich adds the stochastic force Fthto Eq. (5) of Tveten\net al. [17]. The mass and dissipation tensors are symmet-\nric and proportional to each other: Mij\u0011\u001aR\ndV(@in\u0001@jn) and \u0000 ij\u0011Mij=\u001c;where\u001c\u0011\u001a=\u000bs is the relaxation\ntime,\u001a\u0011\u001fs2is the inertia of the staggered spin \feld\nn. The correlator of the stochastic \feld Fthobeys the\nEinstein relation\nhFth\ni(t)Fth\nj(t0)i= 2kBT\u0000ij\u000e(t\u0000t0): (4)\nA temperature gradient causes a Brownian motion of\nan AFM soliton toward a colder region. In the absence\nof a deterministic force, the average drift velocity is pro-\nportional to a temperature gradient V/kBrTin the\nlinear response regime. The form of the proportional-\nity constant can be obtained by a dimensional analy-\nsis. Let us suppose that the mass and dissipation ten-\nsors are isotropic. The Langevin equation (3) is, then,\ncharacterized by three scalar quantities: the mass M,\nthe viscous coe\u000ecient \u0000, and the temperature T, which\nde\fne the unique set of natural scales of time \u001c\u0011M=\u0000,\nlengthl\u0011pkBTM= \u0000, and energy \u000f\u0011kBT. Using\nthese scales to match the dimension of a velocity yields\nV=\u0000c\u0016(kBrT);where\u0016\u0011\u0000\u00001is the mobility of an\nAFM soliton and cis a numerical constant. The explicit\nsolution of the Fokker-Planck equation, indeed, shows\nc= 1. This simple case illustrates our last main result;\na drift velocity of an AFM soliton under a temperature\ngradient in the presence of a deterministic force Fis given\nby\nV=\u0016F\u0000\u0016(kBrT): (5)\nFor a DW in an easy-axis one-dimensional AFM, the\nmobility is \u0016=\u0015=2\u000bs\u001b, where\u0015is the width of the\nwall and\u001bis the cross-sectional area of the AFM. For\na numerical estimate, let us take an angular momen-\ntum density s= 2~nm\u00001, a width\u0015= 100 nm, and a\ndamping constant \u000b= 10\u00004following the previous stud-\nies [17, 30]. For these parameters, the AFM DW moves\nat a velocity V= 32 m/s for the temperature gradient of\nrT= 1 K/mm.\nStochastic LLG equation. |Long-wave dynamics of an\nAFM on a bipartite lattice at zero temperature can de-\nscribed by the Lagrangian [28]\nL=sZ\ndVm\u0001(n\u0002_n)\u0000U[n;m]: (6)\nWe use the potential energy U[n;m]\u0011R\ndVjmj2=2\u001f+\nU[n] throughout the Rapid Communication, which re-\nspects the sublattice exchange symmetry ( n!\u0000n;m!\nm). Minimization of the action subject to nonlinear con-\nstraintsjnj= 1 and n\u0001m= 0 yields the equations of mo-\ntion for the \felds nandm. Damping terms that break\nthe time reversal symmetry can be added to the equa-\ntions of motion to the lowest order, which are \frst order\nin time derivative and zeroth order in spatial derivative.\nThe resultant phenomenological LLG equations are given3\nby\ns(_n+\fn\u0002_m) =n\u0002h; (7a)\ns(_m+\fm\u0002_m+\u000bn\u0002_n) =n\u0002g+m\u0002h (7b)\n[16, 30, 31]. The damping terms can be derived from the\nRayleigh dissipation function\nR=Z\ndV(\u000bsj_nj2+\fsj_mj2)=2; (8)\nwhich is related to the energy dissipation rate by \u0000_U=\n2R. The microscopic origin of damping terms does not\nconcern us here but it could be, e.g., caused by thermal\nphonons that deform the exchange and anisotropy inter-\naction.\nAt a \fnite temperature, thermal agitation causes \ruc-\ntuations of the spin \felds nandm. These thermal \ruc-\ntuations can be considered to be caused by the stochas-\ntic \felds gthandhthwith zero mean, which are con-\njugate to nandm, respectively; their noise correlators\nare then related to the damping coe\u000ecients by the FDT.\nThe standard procedure to construct the noise sources\nyields the stochastic LLG equation (1). The correlators\nof the stochastic \felds are obtained in the following way\n[18, 32]. Casting the linearized LLG equation (7) into the\nformfh;gg= ^\r\nf_n;_mgprovides the kinetic coe\u000ecients\n^\r. Symmetrizing the kinetic coe\u000ecients ^ \rproduces the\ncorrelators (2) of the stochastic \felds consistent with the\nFDT.\nLangevin equation. |For slow dynamics of an AFM,\nthe energy is mostly dissipated through the temporal\nvariation of the staggered spin \feld ndue toj_mj2'\n(\u000b\u001c)2jnj2\u001cj_nj2(from Eq. (7)), which allows us to set\n\f= 0 to study long-term dynamics of the magnetic soli-\nton [17]. At this point, we switch to the Hamiltonian for-\nmalism of an AFM [33], which sheds light on the stochas-\ntic dynamics of a soliton. The canonical momentum \feld\n\u0019conjugate to the staggered spin \feld nis\n\u0019\u0011\u000eL=\u000e _n=sm\u0002n: (9)\nThe stochastic LLG equations (1) can be interpreted as\nHamilton's equations,\n_n=\u000eH=\u000e \u0019=\u0019=\u001a; _\u0019=\u0000\u000eH=\u000en\u0000\u000eR=\u000e _n+gth;(10)\nwith the Hamiltonian\nH\u0011Z\ndV\u0019\u0001_n\u0000L=Z\ndVj\u0019j2\n2\u001a+U[n]: (11)\nLong-time dynamics of magnetic texture can often be\ncaptured by focusing on a small subset of slow modes,\nwhich are parametrized by the collective coordinates\nq=fq1;q2;\u0001\u0001\u0001g. A classical example is a DW in a one-\ndimensional easy-axis magnet described by the position\nof the wall Xand the azimuthal angle \b [3, 33]. An-\nother example is a skyrmion in an easy-axis AFM \flm,which is described by the position R= (X;Y ) [34, 35].\nTranslation from the \feld language into that of collective\ncoordinates can be done as follows. If the staggered spin\n\feldnis encoded by coordinates qasn(r;t) =n[r;q(t)],\ntime dependence of nre\rects evolution of the coordi-\nnates: _n= _qi@n=@qi. With the canonical momenta p\nde\fned by\npi\u0011@L\n@_qi=Z\ndV@n\n@qi\u0001\u0019; (12)\nHamilton's equations (10) translate into\nM_q=p;_p+ \u0000_q=F+Fth; (13)\nwhere F\u0011\u0000@U=@qis the deterministic force and Fth\ni\u0011R\ndV@ qin\u0001gthis the stochastic force. Hamilton's equa-\ntions (13) can be derived from the Hamiltonian in the\ncollective coordinates and conjugate momenta,\nH\u0011pTM\u00001p=2 +U(q); (14)\nwith the Poisson brackets fqi;pjg=\u000eij;fqi;qjg=\nfpi;pjg= 0. An AFM soliton, thus, behaves as a classi-\ncal particle moving in a viscous medium.\nWe focus on a translational motion of a rigid AFM\nsoliton by choosing its center of mass as the collective\ncoordinates q=R;n(r;t) =n(r\u0000R(t)). Eliminat-\ning momenta from Hamilton's equations (13) yields the\nLangevin equation for the soliton's center of mass:\n\u001cR+_R=\u0016F+\u0011; (15)\nwhere \u0011\u0011\u0016Fthis the stochastic velocity. Here the mo-\nbility tensor of the soliton \u0016\u0011\u0000\u00001relates a deterministic\nforce to a drift velocity h_Ri=\u0016Fat a constant temper-\nature [36]. The mobility is inversely proportional to a\ndamping constant, which can be a small number for an\nAFM, e.g., \u000b\u001810\u00004for NiO [37]. The correlator (2) of\nthermal stochastic \felds is translated into the correlator\nof the stochastic velocity,\nh\u0011i(t)\u0011j(t0)i= 2kBT\u0016ij\u000e(t\u0000t0)\u00112Dij\u000e(t\u0000t0):(16)\nFrom Eq. (16), we see that di\u000busion coe\u000ecient and the\nmobility of the soliton respect the Einstein-Smoluchowski\nrelation:D=\u0016kBT, which is expected on general\ngrounds. It can also be explicitly veri\fed as follows.\nA system of an ensemble of magnetic solitons at ther-\nmal equilibrium is described by the partition function\nZ\u0011R\n\u0005i[dpidxi=2\u0019~] exp(\u0000H=k BT), which provides the\nautocorrelation of the velocity, h_xi_xji=2 =M\u00001\nijkBT=2\n(the equipartition theorem). In the absence of an ex-\nternal force, multiplying \u001cxi+ _xi=\u0011i(15) byxjand\nsymmetrizing it with respect to indices iandjgive the\nequation,\u001cd2hxixji=dt2+dhxixji=dt= 2\u001ch_xi_xji, where\nthe \frst term can be neglected for long-term dynamics\nt\u001d\u001c. This equation in conjunction with the autocorre-\nlation of the velocity allows us to obtain the di\u000busion co-\ne\u000ecientDijin Eq. (16),hxixji= 2kBT\u001cM\u00001\nijt= 2Dijt,4\nwithout prior knowledge about the correlator (2) of the\nstochastic \felds.\nAverage dynamics. |An AFM soliton exhibits Brown-\nian motion at a \fnite temperature. The following Fokker-\nPlanck equation for an ensemble of solitons in an inho-\nmogeneous medium describes the evolution of the density\n\u001a(R;t) at timet\u001d\u001c:\n@\u001a\n@t+r\u0001j= 0;withj\u0011\u0016F\u001a\u0000Dr\u001a\u0000DT(kBrT);(17)\nwhereDT\u0011\u0016\u001ais the thermophoretic mobility (also\nknown as the thermal di\u000busion coe\u000ecient) [20, 38]. A\nsteady-state current density j=\u0016F\u001a0\u0000DT(kBrT) with\na constant soliton density \u001a(r;t) =\u001a0solves the Fokker-\nPlanck equation (17), from which the average drift veloc-\nity of a soliton can be extracted [39]:\nV=\u0016F\u0000\u0016(kBrT): (18)\nLet us take an example of a DW in a quasi one-\ndimensional easy-axis AFM with the energy U[n] =R\ndV(Aj@xnj2\u0000Kn2\nz)=2. A DW in the equilibrium\nisn(0)= (sin\u0012cos \b;sin\u0012sin \b;cos\u0012) with cos \u0012=\ntanh[(x\u0000X)=\u0015], where\u0015\u0011p\nA=K is the width of\nthe wall. The position Xand the azimuthal angle \b\nparametrize zero-energy modes of the DW, which are en-\ngendered by the translational and spin-rotational symme-\ntry of the system. Their dynamics are decoupled, \u0000 X\b=\n0, which allows us to study the dynamics of Xseparately\nfrom \b. The mobility of the DW is \u0016=\u0015=2\u000bs\u001b, where\u001b\nis the cross-sectional area of the AFM. The average drift\nvelocity (18) is given by\nV=\u00001\n2\u000bkB\u0015rT\ns\u001b: (19)\nDiscussion |The deterministic force Fon an AFM\nsoliton can be extended to include the e\u000bect of an elec-\ntric current, an external \feld, and a spin wave [15{17]. It\ndepends on details of interaction between the soliton and\nthe external degrees of freedom, whose thorough under-\nstanding would be necessary for a quantitative theory for\nthe deterministic drift velocity \u0016F. The Brownian drift\nvelocity V(18) is, however, determined by local property\nof the soliton. We have focused on the thermal stochastic\nforce as a trigger of thermophoresis of an AFM soliton in\nthis Rapid Communication. There are two other possi-\nble ingredients of thermophoresis of a magnetic soliton.\nOne is a thermal magnon current, scattering with which\ncould exert a force on a soliton [40]. The other is an\nentropic force, which originates from thermal softening\nof the order-parameter sti\u000bness [5]. E\u000bects of these two\nmechanisms have not been studied for an AFM soliton;\nfull understanding of its thermophoresis is an open prob-\nlem.\nIn order to compare di\u000berent mechanisms of thermally-\ndriven magnetic soliton motion, let us address a closelyrelated problem of thermophoresis of a DW in a quasi\none-dimensional FM wire with an easy- xz-plane easy-\nz-axis [3], which has attracted a considerable scrutiny\nrecently. To that end, we have adapted the approach de-\nveloped in this Rapid Communication to the FM case,\nwhich leads to the conclusion that a DW drifts to a\ncolder region by a Brownian stochastic force at the ve-\nlocity given by the same expression for an AFM DW,\nVB=\u0000kB\u0015rT=2\u000bs\u001b [41]. A thermal magnon cur-\nrent pushes a DW to a hotter region at the velocity\nVM=kBrT=6\u00192s\u0015m, where\u0015m\u0011p\n~A=sT is the\nthermal-magnon wavelength [7]. According to Schlick-\neiser et al. [5], an entropic force drives a DW to a hotter\nregion at the velocity VE=kBrT=4sa, whereais the\nlattice constant. The Brownian stochastic force, there-\nfore, dominates the other forces for a thin wire, \u001b\u001c\u0015a=\u000b\n(supposing rigid motion) [42].\nWithin the framework of the LLG equations that are\n\frst order in time derivative, the thermal noise is white\nas long as slow dynamics of a soliton is concerned,\ni.e., the highest characteristic frequency of the natural\nmodes parametrized by the collective coordinates is much\nsmaller than the temperature scale, ~!\u001ckBT. The\nthermal noise could be colored in general [26], e.g., for\nfast excitations of magnetic systems, which may be ex-\namined in the future. In addition, local energy dissi-\npation (8) allowed us to invoke the standard FDT at\nthe equilibrium to derive the stochastic \felds. It would\nbe worth pursuing to understand dissipative dynamics\nof general magnetic systems, e.g., with nonlocal energy\ndissipation with the aid of generalized FDTs at the out-\nof-equilibrium [43].\nWe have studied dynamics of an AFM soliton in the\nHamiltonian formalism. Hamiltonian's equations (13) for\nthe collective coordinates and conjugate momenta can be\nderived from the Hamiltonian (14) with the conventional\nPoisson bracket structure. By replacing Poisson brackets\nwith commutators, the coordinates and conjugate mo-\nmenta can be promoted to quantum operators. This may\nprovide a one route to study the e\u000bect of quantum \ruc-\ntuations on dynamics of an AFM soliton [44].\nAfter the completion of this work, we became aware\nof two recent reports. One is on thermophoresis of an\nAFM skyrmion [35], whose numerical simulations sup-\nport our result on di\u000busion coe\u000ecient. The other is\non thermophoresis of an FM DW by a thermodynamic\nmagnon recoil [45].\nWe are grateful for useful comments on the manuscript\nto Joseph Barker as well as insightful discussions with\nScott Bender, So Takei, Gen Tatara, Oleg Tretiakov, and\nJiadong Zang. This work was supported by the US DOE-\nBES under Award No. DE-SC0012190 and in part by the\nARO under Contract No. 911NF-14-1-0016 (S.K.K. and\nY.T.) and by the US DOE-BES under Award No. DE-\nFG02-08ER46544 (O.T.).5\n[1] A. Kosevich, B. Ivanov, and A. Kovalev, Phys. Rep. 194,\n117 (1990).\n[2] S. S. P. Parkin, M. Hayashi, and L. Thomas, Science\n320, 190 (2008).\n[3] N. L. Schryer and L. R. Walker, J. Appl. Phys. 45, 5406\n(1974).\n[4] L. Berger, Phys. Rev. B 54, 9353 (1996); J. Slonczewski,\nJ. Magn. Magn. Mater. 159, L1 (1996).\n[5] D. Hinzke and U. Nowak, Phys. Rev. Lett. 107, 027205\n(2011); F. Schlickeiser, U. Ritzmann, D. Hinzke, and\nU. Nowak, Phys. Rev. Lett. 113, 097201 (2014).\n[6] G. E. W. Bauer, E. Saitoh, and B. J. van Wees, Nat.\nMater. 11, 391 (2012).\n[7] P. Yan, X. S. Wang, and X. R. Wang, Phys. Rev. Lett.\n107, 177207 (2011); A. A. Kovalev and Y. Tserkovnyak,\nEurophys. Lett. 97, 67002 (2012).\n[8] L. Kong and J. Zang, Phys. Rev. Lett. 111, 067203\n(2013); A. A. Kovalev, Phys. Rev. B 89, 241101 (2014).\n[9] W. Jiang, P. Upadhyaya, Y. Fan, J. Zhao, M. Wang,\nL.-T. Chang, M. Lang, K. L. Wong, M. Lewis, Y.-T.\nLin, J. Tang, S. Cherepov, X. Zhou, Y. Tserkovnyak,\nR. N. Schwartz, and K. L. Wang, Phys. Rev. Lett. 110,\n177202 (2013); J. Chico, C. Etz, L. Bergqvist, O. Eriks-\nson, J. Fransson, A. Delin, and A. Bergman, Phys. Rev.\nB90, 014434 (2014).\n[10] I. \u0014Zuti\u0013 c, J. Fabian, and S. Das Sarma, Rev. Mod. Phys.\n76, 323 (2004).\n[11] J. Sinova and I. Zutic, Nat. Mater. 11, 368 (2012).\n[12] A. H. MacDonald and M. Tsoi, Philos. Trans. R. Soc.,\nA369, 3098 (2011); E. V. Gomonay and V. M. Loktev,\nLow Temp. Phys. 40, 17 (2014).\n[13] M. Fiebig, N. P. Duong, T. Satoh, B. B. V. Aken,\nK. Miyano, Y. Tomioka, and Y. Tokura, J. Phys. D:\nAppl. Phys. 41, 164005 (2008).\n[14] Q. Zhang, A. V. Nurmikko, A. Anguelouch, G. Xiao, and\nA. Gupta, Phys. Rev. Lett. 89, 177402 (2002).\n[15] K. M. D. Hals, Y. Tserkovnyak, and A. Brataas, Phys.\nRev. Lett. 106, 107206 (2011).\n[16] E. G. Tveten, A. Qaiumzadeh, O. A. Tretiakov, and\nA. Brataas, Phys. Rev. Lett. 110, 127208 (2013).\n[17] E. G. Tveten, A. Qaiumzadeh, and A. Brataas,\nPhys. Rev. Lett. 112, 147204 (2014); S. K. Kim,\nY. Tserkovnyak, and O. Tchernyshyov, Phys. Rev. B\n90, 104406 (2014).\n[18] L. D. Landau, E. M. Lifshitz, and L. P. Pitaevskii, Sta-\ntistical Physics, Part 1 , 3rd ed. (Pergamon Press, New\nYork, 1980).\n[19] M. Ogata and Y. Wada, J. Phys. Soc. Jpn. 55, 1252\n(1986).\n[20] R. Piazza and A. Parola, J. Phys.: Condens. Matter 20,\n153102 (2008).\n[21] W. F. Brown, Phys. Rev. 130, 1677 (1963); R. Kubo\nand N. Hashitsume, Prog. Theor. Phys. Suppl. 46, 210\n(1970); J. L. Garc\u0013 \u0010a-Palacios and F. J. L\u0013 azaro, Phys.\nRev. B 58, 14937 (1998); J. Foros, A. Brataas, G. E. W.\nBauer, and Y. Tserkovnyak, Phys. Rev. B 79, 214407\n(2009); S. Ho\u000bman, K. Sato, and Y. Tserkovnyak, Phys.\nRev. B 88, 064408 (2013); U. Atxitia, P. Nieves, and\nO. Chubykalo-Fesenko, Phys. Rev. B 86, 104414 (2012).\n[22] B. A. Ivanov, A. K. Kolezhuk, and E. V. Tartakovskaya,\nJ. Phys.: Condens. Matter 5, 7737 (1993).[23] C. Sch utte, J. Iwasaki, A. Rosch, and N. Nagaosa, Phys.\nRev. B 90, 174434 (2014).\n[24] R. A. Duine, A. S. N\u0013 u~ nez, and A. H. MacDonald, Phys.\nRev. Lett. 98, 056605 (2007).\n[25] O. A. Tretiakov, D. Clarke, G.-W. Chern, Y. B. Baza-\nliy, and O. Tchernyshyov, Phys. Rev. Lett. 100, 127204\n(2008).\n[26] S. Hottovy, G. Volpe, and J. Wehr, Europhys. Lett. 99,\n60002 (2012).\n[27] S. Iacopini and R. Piazza, Europhys. Lett. 63, 247 (2003).\n[28] A. F. Andreev and V. I. Marchenko, Sov. Phys. Usp. 23,\n21 (1980).\n[29] Employing the quantum FDT would change the corre-\nlator of the stochastic \felds to hhth\ni(r;!)hth\nj(r0;!0)i=\n[2\u0019\u000eij\u000bs~!=tanh( ~!=2kBT)]\u000e(r\u0000r0)\u000e(!\u0000!0) in the fre-\nquency space [18]. We focus on slow dynamics of an AFM\nsoliton in the manuscript, ~!\u001ckBT, which allows us to\nreplace ~!=tanh( ~!=2kBT) with 2kBT, yielding Eq. (2).\n[30] S. Takei, B. I. Halperin, A. Yacoby, and Y. Tserkovnyak,\nPhys. Rev. B 90, 094408 (2014).\n[31] B. A. Ivanov and D. D. Sheka, Phys. Rev. Lett. 72, 404\n(1994); N. Papanicolaou, Phys. Rev. B 51, 15062 (1995);\nH. V. Gomonay and V. M. Loktev, Phys. Rev. B 81,\n144427 (2010); A. C. Swaving and R. A. Duine, Phys.\nRev. B 83, 054428 (2011).\n[32] Y. Tserkovnyak and C. H. Wong, Phys. Rev. B 79,\n014402 (2009).\n[33] H. J. Mikeska, J. Phys. C: Solid St. Phys. 13, 2913 (1980);\nF. D. M. Haldane, Phys. Rev. Lett. 50, 1153 (1983).\n[34] I. Rai\u0014 cevi\u0013 c, D. Popovi\u0013 c, C. Panagopoulos, L. Benfatto,\nM. B. Silva Neto, E. S. Choi, and T. Sasagawa, Phys.\nRev. Lett. 106, 227206 (2011); X. Zhang, Y. Zhou, and\nM. Ezawa, arXiv:1504.01198.\n[35] J. Barker and O. A. Tretiakov, arXiv:1505.06156.\n[36] The relation between the mobility and dissipation tensor\ncan also be understood by equating the (twice) Rayleigh\nfunction, 2R=VT\u0000V, to the dissipation governed by\nthe mobility,\u0000_E=V\u0001F=VT\u0016\u00001V.\n[37] T. Kampfrath, A. Sell, G. Klatt, A. Pashkin, S. Mahrlein,\nT. Dekorsy, M. Wolf, M. Fiebig, A. Leitenstorfer, and\nR. Huber, Nature Photon. 5, 31 (2011).\n[38] N. van Kampen, IBM J. Res. Dev. 32, 107 (1988);\nT. Kuroiwa and K. Miyazaki, J. Phys. A: Math. Theor.\n47, 012001 (2014).\n[39] M. Braibanti, D. Vigolo, and R. Piazza, Phys. Rev. Lett.\n100, 108303 (2008).\n[40] The e\u000bect is, however, negligible at a temperature lower\nthan the magnon gap ( \u001840 K for NiO [37]). Also for our\nexample|a DW in a 1D easy-axis AFM|the conserva-\ntive force and torque exerted by magnons vanish [17].\n[41] Unlike 1D domain walls, Brownian motions are drasti-\ncally distinct between 2D FM and AFM solitons due to\nthe gyrotropic force, which signi\fcantly slows down fer-\nromagnetic di\u000busion [23, 35].\n[42] A DW in a wire with a large crosssection \u001b\u001da2forms\na 2D membrane. Its \ructuations foment additional soft\nmodes of the dynamics, which needs to be taken into\naccount to understand the dynamics of such a DW [25].\n[43] M. Baiesi, C. Maes, and B. Wynants, Phys. Rev. Lett.\n103, 010602 (2009); U. Seifert and T. Speck, Europhys.\nLett. 89, 10007 (2010).\n[44] S.-Z. Lin and L. N. Bulaevskii, Phys. Rev. B 88, 060404\n(2013).\n[45] P. Yan, Y. Cao, and J. Sinova, arXiv:1504.00651."    },    {        "title": "1504.00199v1.Multiscale_modeling_of_ultrafast_element_specific_magnetization_dynamics_of_ferromagnetic_alloys.pdf",        "content": "Multiscale modeling of ultrafast element-specific magnetization dynamics of\nferromagnetic alloys\nD. Hinzke1,∗U. Atxitia1,2, K. Carva3,4, P. Nieves5, O. Chubykalo-Fesenko5, P. M. Oppeneer4, and U. Nowak1\n1Fachbereich Physik, Universit¨ at Konstanz, D-78457 Konstanz, Germany\n2Zukunftskolleg at Universit¨ at Konstanz, D-78457 Konstanz, Germany\n3Faculty of Mathematics and Physics, DCMP, Charles University,\nKe Karlovu 5, CZ-12116 Prague 2, Czech Republic\n4Department of Physics and Astronomy, Uppsala University, Box 516, SE-751 20 Uppsala, Sweden and\n5Instituto de Ciencia de Materiales de Madrid, CSIC, Cantoblanco, 28049 Madrid, Spain\n(Dated: March 31, 2015)\nA hierarchical multiscale approach to model the magnetization dynamics of ferromagnetic ran-\ndom alloys is presented. First-principles calculations of the Heisenberg exchange integrals are linked\nto atomistic spin models based upon the stochastic Landau-Lifshitz-Gilbert (LLG) equation to\ncalculate temperature-dependent parameters (e.g., effective exchange interactions, damping param-\neters). These parameters are subsequently used in the Landau-Lifshitz-Bloch (LLB) model for\nmulti-sublattice magnets to calculate numerically and analytically the ultrafast demagnetization\ntimes. The developed multiscale method is applied here to FeNi (permalloy) as well as to copper-\ndoped FeNi alloys. We find that after an ultrafast heat pulse the Ni sublattice demagnetizes faster\nthan the Fe sublattice for the here-studied FeNi-based alloys.\nI. INTRODUCTION\nExcitation of magnetic materials by powerful femtosec-\nond laser pulses leads to magnetization dynamics on the\ntimescale of exchange interactions. For elemental fer-\nromagnets the emerging dynamics can be probed us-\ning conventional magneto-optical methods1,2. For mag-\nnets composed of several distinct elements, such as fer-\nrimagnetic or ferromagnetic alloys, the individual spin\ndynamics of the different elements can be probed em-\nploying ultrafast excitation in combination with the\nfemtosecond-resolved x-ray magnetic circular dichroism\n(XMCD) technique3,4. An astonishing example of such\nelement-specific ultrafast magnetization dynamics was\nfirst measured on ferrimagnetic GdFeCo alloys5. There,\nit was observed that the rare-earth Gd sublattice demag-\nnetizes in around 1.5 ps whereas the transition metal\nFeCo sublattice has a much shorter demagnetization time\nof 300 fs. Similar element-specific spin dynamics was\nalso observed in CoGd and CoTb alloys6,7. The element-\nselective technique allowed moreover to observe for the\nfirst time the element-specific dynamics of the so-called\n“all-optical switching” (AOS)8in GdFeCo alloys, find-\ning that it unexpectedly proceeds through a transient-\nferromagnetic-like state (TFLS) where the FeCo sublat-\ntice magnetization points in the same direction as that\nof the Gd sublattice before complete reversal5,9. Recent\ntheoretical works supported the distinct demagnetiza-\ntion times observed experimentally10–12and their cru-\ncial role on the TFLS. AOS has been also demonstrated\nfor other rare-earth transition-metal ferrimagnetic alloys\nas TbFe13, TbCo14, TbFeCo15, DyCo16, HoFeCo16, syn-\nthetic ferrimagnets16–18and very recently in the hard-\nmagnetic ferromagnet FePt19.\nAlthough the full theoretical explanation of the\nthermally driven AOS process is still a topic of\ndebate9,12,20–23, the distinct demagnetization rates ofeach of the constituting elements has been suggested as\nthe main driving mechanism for the AOS observed on\nantiferromagnetically coupled alloy9,10,12. These findings\nhave highlighted the question how ultrafast demagneti-\nzation would proceed in ferromagnetically coupled two-\nsublattice materials such as permalloy (Py). Unlike rare-\nearth transition-metal alloys which consists of two intrin-\nsically different metals, Py is composed of Fe (20 %) and\nNi (80 %) which have a rather similar magnetic nature,\ndue to a partially filled 3 dshell. Thus, it is a priori not\nclear if their spin dynamics should be the same or differ-\nent.\nRecent measurements have addressed this question.\nUsing extreme ultraviolet pulses from high-harmonic\ngeneration sources Mathias et al.24probed element-\nspecifically the ultrafast demagnetization in Py and ob-\ntained the same demagnetization rates for each element,\nFe and Ni, but with a 10 to 70 fs delay between them.\nFrom a theoretical viewpoint an important question\nis which materials parameter are defining for the ultra-\nfast demagnetization. Thus far, different criteria have\nbeen suggested25,26. For single-element ferromagnets,\nKazantseva et al.25estimated, based on phenomenolog-\nical arguments, that the timescale for the demagnetiza-\ntion processes is limited by τdemag≈µ/(2λγkBTpulse).\nHere,τdemag depends not only on the elemental atomic\nmagnetic moment, µ, but also on the electron tempera-\nture,Tpulse, and on the damping constant λ. Assuming\nthat the damping constants λand gyromagnetic ratios γ\nare equal for Fe and Ni the demagnetization time would\ntherefore only vary due to the different magnetic mo-\nments of the constituting elements. In that case, the\ndemagnetization time of Fe is larger than the one for Ni\n(sinceµFe>µNi, see Table I below).\nA similar criterion (as in Ref. 25 for single-element fer-\nromagnets) has been suggested by Koopmans et al.26on\nthe basis of the ratio between the magnetic moment andarXiv:1504.00199v1  [cond-mat.mtrl-sci]  1 Apr 20152\nthe Curie temperature, µ/TC. Since for ferromagnetic al-\nloys each element has the same Curie temperature, this\ncriterion would lead to the same conclusions as Kazant-\nseva et al. ; the different atomic magnetic moments of Fe\nand Ni are responsible for the different demagnetization\ntimes. Furthermore, Atxitia et al.10have theoretically\nestimated the demagnetization times in GdFeCo alloys\nproposing that the demagnetization times scale with the\nratio of the magnetic moment to the exchange energy of\neach element and a similar relation is expected for ferro-\nmagnetic alloys. The demagnetization times of Fe and Ni\nin Py were also theoretically investigated by Schellekens\nand Koopmans in Ref. 11 where a modified microscopic\nthree temperature model (M3TM)26was used. Thereby,\nthey obtained a perfect agreement with experimental re-\nsults of Mathias et al. ,24but only assuming an at least\n4 times larger damping constant for Fe. However, this\nwork does not provide a simple general criterion, valid\nfor other ferromagnetic alloys.\nWe have developed a hierarchical multiscale approach\n(cf. Ref. 27) to investigate the element-specific spin dy-\nnamics of ferromagnetic alloys and to obtain a deeper\ninsight into the underlying mechanisms. First, we con-\nstruct and parametrize a model spin Hamiltonian for\nFeNi alloys on the basis of first-principles calculations\n[Sec. II A]. This model spin Hamiltonian in combina-\ntion with extensive numerical atomistic spin dynamics\nsimulations based on the stochastic LLG equation are\nused to calculate the equilibrium properties [Sec. II B]\nas well as the demagnetization process after the appli-\ncation of a step heat pulse. The second step of the pre-\nsented multiscale model links the atomistic spin model to\nthe macroscopic two-sublattices Landau-Lifshitz-Bloch\n(LLB) equation of motion recently derived by Atxitia\net al.28[Sec. III]. The analytical LLB approach allows\nfor efficient simulations, and most importantly, provides\ninsight in the element-specific demagnetization times of\nFeNi alloys.\nII. FROM FIRST PRINCIPLES TO ATOMISTIC\nSPIN MODEL\nA. Building the spin Hamiltonian\nTo start with, we construct an atomistic, classical spin\nHamiltonianHon the basis of first-principles calcula-\ntions. In particular, we consider three relevant alloys:\nFe50Ni50, Fe20Ni80(Py) and Py 60Cu40. The first two al-\nloys will allow us to assess the influence of the Fe and Ni\ncomposition, while the last two alloys will permit us to\nstudy the effect of the inclusion of non-magnetic impuri-\nties on the demagnetization times. This was motivated\nby the work of Mathias et al.24who studied the influence\nof Cu doping on the Fe and Ni demagnetization times in\nan Py 60Cu40alloy.\nTo obtain the spin Hamiltonian we have employed spin-\ndensity functional theory calculations to map the behav-ior of the magnetic material onto an effective Heisenberg\nHamiltonian, which can be achieved in various ways29,30.\nHere we use the two-step approach suggested by Licht-\nenstein et al.31. The first step represents the calculation\nof the self-consistent electronic structure for a collinear\nspin structure at zero temperature. In the second step,\nexchange parameters of an effective classical Heisenberg\nHamiltonian are determined using the one-electron Green\nfunctions. This method has been rather successful in ex-\nplaining magnetic thermodynamic properties of a broad\nclass of magnetic materials32–34.\nThe self-consistent electronic structure was calcu-\nlated using the tight-binding linear muffin-tin orbital\n(TB-LMTO) approach32within the local spin-density\napproximation35to the density functional theory.\nImportantly, the materials we investigate here are al-\nloys. Hence, it is assumed that atoms are distributed\nrandomly on the host fcc lattice. The effect of disor-\nder was described by the coherent-potential approxima-\ntion (CPA)36. The same radii for constituent atoms were\nused in the TB-LMTO-CPA calculations. We have used\naround a million k-points in the full Brillouin zone to\nresolve accurately energy dispersions close to the Fermi\nlevel.\nThe calculations of the Heisenberg exchange constants\nJijin ferromagnets can be performed with a reason-\nable numerical effort by employing the magnetic force\ntheorem29,31. It allows to express the infinitesimal\nchanges of the total energy using changes in one-particle\neigenvalues due to non-self-consistent changes of the ef-\nfective one-electron potential accompanying the infinites-\nimal rotations of spin quantization axes, i.e., without any\nadditional self-consistent calculations besides that for the\ncollinear ground state. The resulting pair exchange in-\nteractions are given by\nJij=1\nπIm/integraldisplayEF\n−∞dE/integraldisplay\nΩidr/integraldisplay\nΩjdr/primeBex(r)G↑\n+Bex(r/prime)G↓\n−,(1)\nwithG↑\n+=G↑(r,r/prime,E+) andG↓\n−=G↓(r/prime,r,E−).EF\ndenotes the Fermi level and Ω ithei-th atomic cell,\nσ=↑,↓is the spin index, E+= limα→0E+ iα,Gσ\nare spin-dependent one-electron retarded Green func-\ntions, andBexis the magnetic field from the exchange-\ncorrelation potential. The validity of this approxima-\ntion has been examined more quantitatively in several\nstudies.37–39Theab initio calculated distance-dependent\nexchange constants for the Fe 20Ni80alloy, i.e., the ex-\nchange within the Fe sublattice (Fe-Fe), the Ni sublat-\ntice (Ni-Ni) as well as between the Fe and Ni sublattices\n(Fe-Ni), are shown in Fig. 1. The calculated magnetic\nmoments for all three alloys considered here are given in\nTable I.\nIn our hierarchical multiscale approach, these com-\nputed material parameters (the exchange constant ma-\ntrix as well as the magnetic moments) are now used as\nmaterial parameters for our numerical simulations based\non an atomistic Heisenberg spin Hamiltonian. We con-\nsider thereto classical spins S/epsilon1\ni=µ/epsilon1\ni/µ/epsilon1\niwith/epsilon1randomly3\nTABLE I. Ab initio calculated magnetic moments µ/epsilon1and experimental lattice constants ∆ used in the atomistic Langevin spin\ndynamics simulations. Effective exchange parameters calculated from ab initio calculations, J/epsilon1,δ\n0=/summationtext\njJ/epsilon1δ\n0j, where the sum is\nhere over all neighbors j. Curie temperatures as calculated from the atomistic simulations, TLLG\nC, and the experimental value,\nTexp\nC.\nalloyµFeµNi∆JNi−Ni\n0JFe−Fe\n0JFe−Ni\n0TLLG\nCTexp\nC\n[µB] [µB] [nm] [J ×10−21] [J×10−21] [J×10−21] [K] [K]\nPy 2.637 0.628 0.3550406.2419 32.3162 26.3654 650 85024\nNi50Fe502.470 0.730 0.3588416.6265 25.3789 25.0656 850\nPy60Cu402.645 0.429 0.3550 2 .6623 56.2789 22.6442 340 40624\ncutoffFe-NiNi NiFe-Fe\nrij[nm]Jij[meV]\n2.5 2 1.5 1 0.514\n8\n4\n2\n0\n-2\n-4\nFIG. 1. (Color online) Ab initio calculated exchange constants\nJijfor the Fe 20Ni80alloy of the distance rijbetween atoms\niandj. Results are given for the three different possible\nsublattice interactions ( JFe−Fe,JNi−Ni, andJFe−Ni). Note\nour hyperbolic scaling. In our atomistic spin simulations the\nexchange constants are taken into account up to a distance rij\n(cutoff) where they are finally small enough to be neglected.\nrepresenting iron ( µ/epsilon1\ni=µFe\ni) or nickel magnetic moments\n(µ/epsilon1\ni=µNi\ni) on the fcc sublattice. For the Cu-doped\nPy60Cu40alloy the calculated magnetic moments on Cu\nvanish, i.e. µCu\ni= 0.\nThe spin Hamiltonian for unit vectors, S/epsilon1\ni, representing\nthe normalized magnetic moments of the i-th atom on\neither the Fe or Ni sublattice reads\nH=−/summationdisplay\nij/parenleftBigJij\n2S/epsilon1\ni·Sδ\nj (2)\n−µ0µ/epsilon1\niµδ\ni\n8π3(S/epsilon1\ni·eij)(eij·Sδ\nj)−S/epsilon1\ni·Sδ\nj\nr3\nij/parenrightBig\n.\nThe first sum represents the exchange energy of mag-\nnetic moments, either on Ni or on Fe sites, distributed\nrandomly with the required concentrations. The ex-\nchange interaction matrices Jij(corresponding to JNi−Ni,\nJFe−Ni, orJNi−Ni) are those from the ab initio calcula-tions (as shown for Py in Fig. 1). These have been taken\ninto account up to a distance of six unit cells (cutoff also\nshown in Fig. 1) until they are finally small enough to\nbe neglected. The second sum describes the magnetic\ndipole-dipole coupling.\nNote, that the exchange interaction given by the ma-\ntricesJijis incorporated in our atomistic spin dynamics\nsimulations via the Fast Fourier transformation method\n(see Ref. 42 for more details). As a side effect, we are able\nto calculate the dipolar interaction without any addi-\ntional computational effort so that we take them into ac-\ncount although they will not influence our results much.\nSince we are interested in thermal properties we\nuse Langevin dynamics, i.e. numerical solutions of the\nstochastic LLG equation of motion\n(1 + (λ/epsilon1\ni)2)µ/epsilon1\ni\nγ/epsilon1\ni˙S/epsilon1\ni=−S/epsilon1\ni×[Hi+λ/epsilon1\ni(S/epsilon1\ni×Hi)],(3)\nwith the gyromagnetic ratio γ/epsilon1\ni, and a dimensionless\nGilbert damping constant λ/epsilon1\nithat describes the coupling\nto the heat-bath and corresponding either to Fe or to Ni.\nThermal fluctuations are included as an additional noise\ntermζiin the internal fields Hi=−∂H\n∂S/epsilon1\ni+ζi(t) with\n/angbracketleftζi(t)/angbracketright= 0,/angbracketleftζiη(0)ζjθ(t)/angbracketright=2kBTλ/epsilon1\niµ/epsilon1\ni\nγ/epsilon1\niδijδηθδ(t),(4)\nwherei,jdenotes lattice sites occupied either by Fe or\nNi andη,θare Cartesian components. All algorithms we\nuse are described in detail in Ref. 43.\nB. Equilibrium properties: element-specific\nmagnetization\nFirst, we investigate the element-specific zero-field\nequilibrium magnetizations for Fe and Ni sublattices.\nThose magnetizations are calculated as the spatial and\ntime average of the sum of local magnetic moments,\nm/epsilon1=/angbracketleftS/epsilon1/angbracketrightwith/epsilon1representing either Fe or Ni. For\nour numerical studies, we assume identical damping con-\nstants (λ=λ/epsilon1\ni) as well as gyromagnetic ratios ( γ=γ/epsilon1\ni\n= 1.76·1011(Ts)−1) for both, Fe or Ni. We perform4\n00.51mǫ\nPy60Cu40\nFe50Ni50Py\n00.51mǫPy60Cu40\nFe50Ni50Py\n00.51\n0 200 400 600 800mǫ\nT[K]Py60Cu40\nFe50Ni50PyFe, MFA\nNi, MFA\nFe, Atomistic\nNi, Atomistic\nFIG. 2. (Color online) Element-specific zero-field equilibrium\nmagnetizations m/epsilon1of either Fe or Ni as a function of tem-\nperature calculated by a rescaled mean-field approximation\n(MFA) (lines) and by the atomistic spin dynamics simulation\n(open symbols). In the MFA the exchange parameters are\nrenormalized by equalizing the Curie temperatures TCcom-\nputed with atomistic simulations with those obtained from\nthe rescaled MFA. System size 128 ×128×128, damping\nparameterλ= 1.0.\nour Langevin spin dynamics simulations for two differ-\nent FeNi alloys, namely Fe 50Ni50and Py, as well as for\npermalloy diluted with copper, Py 60Cu40. All material\nparameters used in our simulations are given in Table I.\nThe temperature dependence of the normalized\nelement-specific magnetizations m/epsilon1are shown in Fig.\n2. The calculated values of the Curie temperatures are\ngiven in Table I together with known experimental val-\nues. Both, the numerical and experimental values, are\nin good agreement. The element-specific magnetizations\nas well as the total magnetization (not shown in Fig. 2)\nof the alloys share the same Curie temperature while in\nthe temperature range below the Curie temperature their\ntemperature dependence is different for the two sublat-\ntices; the normalized magnetization of Ni is lower than\nthat of Fe.\nThe element-specific magnetizations calculated within\nthe framework of a rescaled mean-field approximation\n(MFA) are shown as well. This approach will be dis-\ncussed in detail in Sec. III below where these curves serve\nas material parameters for the simulations based on the\nLLB equation of motion also introduced in the next sec-\ntion.III. FROM ATOMISTIC SPIN MODEL TO\nMACROSCOPIC MODEL\nA. Two-sublattices Landau-Lifshitz-Bloch equation\nWithin the hierarchical multiscale approach, the\nmacroscopic (micromagnetic) equation of motion valid at\nelevated temperatures is the LLB equation27. Initially,\nthe macroscopic LLB equation of motion was derived by\nGaranin for single-species ferromagnets only. Garanin\nfirst calculated the Fokker-Planck equation for a single\nspin coupled to a heat-bath, thereafter a non-equilibrium\ndistribution function for the thermal averaged spin polar-\nization was assumed to drive the non-equilibrium dynam-\nics. Second, the exchange interactions between atomic\nspins were introduced using the mean field approximation\n(MFA) with respect to the spin-spin interactions. This\nlast step reduces to the replacement of the ferromagnetic\nspin Hamiltonian Hwith the MFA Hamiltonian HMFAin\nthe single (macro)spin solution.\nThe LLB formalism was recently broadened to de-\nscribe the distinct dynamics of two-sublattices mag-\nnets, both antiferromagnetically or ferromagnetically\ncoupled28. The derivation of such equations follows sim-\nilar steps as for the ferromagnetic LLB version but con-\nsidering sublattice specific spin-spin exchange interac-\ntions and MFA exchange fields, /angbracketleftH/epsilon1\nMFA/angbracketrightconf. For the ex-\nchange field the random lattice model is used by gener-\nating the random average with respect to disorder con-\nfigurations/angbracketleft.../angbracketrightconf. The corresponding set of coupled\nLLB equations for each sublattice reduced magnetiza-\ntionm/epsilon1=/angbracketleftS/epsilon1/angbracketright=M/epsilon1/M/epsilon1\ns, whereM/epsilon1\nsis the saturation\nmagnetization at 0 K, has the form\n˙m/epsilon1=γ/epsilon1[m/epsilon1×/angbracketleftH/epsilon1\nMFA/angbracketrightconf]−Γ/epsilon1\n⊥[m/epsilon1×[m/epsilon1×m/epsilon1\n0]]\n(m/epsilon1)2\n−Γ/epsilon1\n/bardbl/parenleftbigg\n1−m/epsilon1m/epsilon1\n0\n(m/epsilon1)2/parenrightbigg\nm/epsilon1. (5)\nHere, m/epsilon1\n0=L(ξ/epsilon1\n0)ξ/epsilon1\n0\nξ/epsilon1\n0is the transient (dynamical) magne-\ntization to which the non-equilibrium magnetization m/epsilon1\ntends to relax, and where ξ/epsilon1\n0≡µ/epsilon1\nkBT/angbracketleftH/epsilon1\nMFA/angbracketrightconfis the ther-\nmal reduced field, ξ/epsilon1\n0≡|ξ/epsilon1\n0|, andL(ξ) = coth (ξ)−1/ξis\nthe Langevin function and L/prime(ξ) = dL(ξ)/dξ. The par-\nallel (Γ/epsilon1\n/bardbl) and perpendicular (Γ/epsilon1\n⊥) relaxation rates in Eq.\n(5) are given by\nΓ/epsilon1\n/bardbl= Λ/epsilon1\nN1\nξ/epsilon1\n0L(ξ/epsilon1\n0)\nL/prime(ξ/epsilon1\n0)and Γ/epsilon1\n⊥=Λ/epsilon1\nN\n2/parenleftbiggξ/epsilon1\n0\nL(ξ/epsilon1\n0)−1/parenrightbigg\n.(6)\nΛ/epsilon1\nN= 2kBTγ/epsilon1λ/epsilon1/µ/epsilon1is the characteristic diffusion relax-\nation rate. The damping parameters λ/epsilon1have the same\norigin as those used in the atomistic simulations.\nThe first and the second terms on the right-hand side\nof Eq. (5) describe the transverse motion of the mag-\nnetization. These dynamics are much slower than the\nlongitudinal magnetization dynamics given by the third\nterm in this equation. Therefore, in the following we will5\nFeNi\nFIG. 3. (Color online) Schematics of the magnetic unit cell\nused in the mean-field approximation for the FeNi alloys. The\nunit cell shown by the box contains two spins, one Fe and one\nNi. The only interaction among spins located at the same unit\ncellris defined by JNi−Fe\n0 . The self-interactions are neglected,\nJNi−Ni\n0 (r,r) =JFe−Fe\n0 (r,r) = 0. The rest of the interactions\nare among spins located in neighboring unit cells randr/prime.\nneglect the transverse components (in Eq. (5)) and keep\nonly the longitudinal one,\n˙m/epsilon1=−Γ/epsilon1\n/bardbl(m/epsilon1−m/epsilon1\n0). (7)\nIn spite of the fact that the form of Eq. (7) is similar\nto the well known Bloch equation, the quantity m0=\nm/epsilon1\n0/parenleftbig\nm/epsilon1,mδ/parenrightbig\n(withδthe 2-nd type of element) is not\nthe equilibrium magnetization but changes dynamically\nthrough the dependence of the effective field /angbracketleftH/epsilon1\nMFA/angbracketrightconf\non both sublattice magnetizations. Moreover, the rate\nparameter Γ/epsilon1\n/bardbl= Γ/epsilon1\n/bardbl/parenleftbig\nm/epsilon1\n0,mδ\n0/parenrightbig\ncontains highly non-linear\nterms inm/epsilon1\n0andmδ\n0.\nTherefore, the analytical solution of Eq. (7) and thus\na deeper physical interpretation of the relaxation rates is\ndifficult without any further approximations. However,\nEq. (7) can be easily solved numerically with the aim to\ndirectly compare the solutions to those of the atomistic\nspin simulations. This is discussed in more detail in the\nnext subsections.\nB. From atomistic spin model to\nLandau-Lifshitz-Bloch equation\nNext, to solve Eq. (5) or Eq. (7), one needs to calculate\n/angbracketleftH/epsilon1\nMFA/angbracketrightconffor the here-considered FeNi alloys. An ade-\nquate definition of such a field will allow us to directly\ncompare the magnetization dynamics from our atomistic\nspin simulation with the LLB macroscopic approach.\nHowever, a quantitative comparison between both a\nstandard MFA and atomistic spin model calculations of\nthe equilibrium properties is usually not possible. This is\ndue to the fact that the Curie temperature gained with\nthe MFA approach is overestimated due to the inher-\nent poor approximation of the spin-spin correlations. Al-\nthough, rescaling the exchange parameters conveniently\nin such a way that the Curie temperature calculated\nwith the MFA approach agrees with atomistic simula-\ntions leads to a good agreement of both methods. Hence,we first present the standard MFA for disordered two-\nsublattices magnets, thereafter, we will deal with the\nrescaling of the exchange parameters.\nThe MFA Hamiltonian of the full spin Hamiltonian for\nFeNi alloys (see Eq. (2) introduced in Sec. II) can be\nwritten as\nHMFA=H00−µFe/summationdisplay\niHFe\nMFA·SFe\ni−µNi/summationdisplay\niHNi\nMFA·SNi\ni,(8)\nwhere the dipolar interaction is neglected. The mean\nfield acting on each site ican be separated in two contri-\nbutions; a) the contribution from neighbors of the same\ntypej/epsilon1and b) those of the other type jδ,\nµ/epsilon1/angbracketleftH/epsilon1\nMFA/angbracketrightconf=/summationdisplay\n/epsilon1j/epsilon1J/epsilon1\nj/epsilon1/angbracketleftSj/epsilon1/angbracketright+/summationdisplay\n/epsilon1jδJ/epsilon1\njδ/angbracketleftSjδ/angbracketright,(9)\nwhere sums run over the nearest neighbours. When the\nhomogenous magnetization approximation is applied (i.e.\n/angbracketleftSjFe/angbracketright=mFeand/angbracketleftSjNi/angbracketright=mNifor all sites) one can de-\nfineJ/epsilon1/epsilon1\n0=/summationtext\n/epsilon1j/epsilon1J/epsilon1\nj/epsilon1andJ/epsilon1δ\n0=/summationtext\n/epsilon1jδJ/epsilon1\njδ. A sketch of the\nexchange interaction within the present MFA model is\npresented in Fig. 3. The impurity model is mapped to a\nregular spin lattice where the unit cell (orange box) con-\ntains the two spin species, Fe and Ni, and the exchange\ninteractions among them are weighted in terms of the\nconcentration of each species.\nThe equilibrium magnetization of each sublattice m/epsilon1\ne\ncan be obtained via the self-consistent solution of the\nCurie-Weiss equations m/epsilon1\ne=L(µ/epsilon1\nkBT/angbracketleftH/epsilon1\nMFA/angbracketrightconf).\nFig. 2 shows good agreement of the calculated m/epsilon1\ne(T)\nusing the MFA and the atomistic spin model for the\nthree system studied in the present work. The exchange\ninteractions are rescaled as J/epsilon1δ\n0,MFA/similarequal(1.65/2)J/epsilon1δ\n0, for\nFe50Ni50and Py. For Py 60Cu40it is in agreement with\nJ/epsilon1δ\n0,MFA= (1.78/2)J/epsilon1δ\n0. Here, the atomistic calculations is\nnot as accurate for intermediate temperatures as for the\nother two alloys. This could be because of the increased\ncomplexity introduced by the inclusion of Cu impurities\nwhich cannot be fully described by the MFA.\nC. De- and remagnetization due to a heat pulse\nIn the following, we study the reaction of the element-\nspecific magnetization to a temperature step in Py as well\nas in Py diluted with Cu. In the first part of the temper-\nature step the system is heated up to T= 0.8TCand in\nthe second part it is cooled down to Tpulse = 0.5TC. The\nheat pulse roughly mimics the effect of heating due to a\nshort laser pulse. The first part of the temperature step\ntriggers the demagnetization while the second one trig-\ngers the remagnetization process. We perform atomistic\nas well as LLB simulation of the de- and remagnetization\nof the two sublattices after the application of a step heat\npulse of 500-fs duration.\nThe reaction of the Fe and Ni sublattice magnetiza-\ntions is shown in Fig. 4. While the temperature step6\nFIG. 4. (Color online) Calculated z-component of the nor-\nmalized element-specific magnetization m/epsilon1\nzvs. time for Py\n(top panel) and Py 60Cu40(bottom panel). In both cases the\nquenching of the element-specific magnetizations for Fe and\nNi due to a temperature step of Tpulse = 0.8TCare shown,\ncomputed with atomistic Langevin spin dynamics (open sym-\nbols) as well as LLB simulations (lines). System size 64 ×64\n×64, damping parameter λ= 0.02.\nis switched on, the two sublattices relax to the corre-\nsponding equilibrium value of the sublattice magnetiza-\ntionsm/epsilon1(Tpulse). Note, that these equilibrium values\nare different for the two sublattices in agreement with\nthe temperature-dependent equilibrium element-specific\nmagnetizations shown in Fig. 2.\nBecause of that, the different demagnetization time\nscales are not well distinguishable in Fig. 4. Thus,\nwe use the normalized magnetization, m/epsilon1\nnorm = (m/epsilon1−\nm/epsilon1\nmin)/(m/epsilon1\n(t=0)−m/epsilon1\nmin) of the sublattices, rather than\nm/epsilon1to directly compare the demagnetization times. The\ndemagnetization time after excitation with a tempera-\nture pulse isfaster for Ni than for Fe (Fig. 5 (top panel))\nfor the first 200 fs, while one can see that for times larger\nthan 200 fs both elements demagnetize at the same rate\n(Fig. 5 (bottom panel)). Experiments on Py suggest that\nthe time shift between distinct and similar demagnetiza-\ntion rates in Py is of around 10–70 femtoseconds24.\nD. Understanding relaxation times within the\nLandau-Lifshitz-Bloch formalism\nThe relaxation rates of the Fe and Ni sublattices\ncan be understood by discussing the linearized form of\nEq. (7). Here, the expansion of Γ/epsilon1\n/bardblandm/epsilon1\n0around\n0.010.11∆m/epsilon1z(t)/∆m/epsilon1z(0)τFe/τNi=1.8\n0.40.60.810.1 0.2 0.3 0.4 0.5m/epsilon1ztime [ps]τFe/τNi=1.8τNi=τFeNiFeFIG. 5. (Color online) Top panel: Normalized magnetiza-\ntion dynamics of Fe and Ni sublattices after the application\nof a heat pulse T= 0.8TCas computed with the atomistic\nspin model. The ratio between the Fe and Ni demagneti-\nzation times is 1.8. The intersection of the linear fit to the\nabscissa gives the relaxation time for each sublattice. Bot-\ntom panel: plot of the unnormalized magnetization dynamics\nwhich shows that after the first 0.2 picosecond the element-\nspecific demagnetization proceeds at the same rate.\nτ−τ+\nT/TCτ[ps]\n1 0.75 0.5 0.25 02.0 1.5 1.0 0.5 0.0T/TCτ+/τ−\n1 0.75 0.5 0.25010\n86420\nFIG. 6. (Color online) Relaxation times of the dynamical\nsystem obtained by the LLB equation as a function of tem-\nperature. Inset: The ratio between the relaxation times.\ntheir equilibrium values m/epsilon1\neis considered28and leads\nto∂(∆m)/∂t=A/bardbl∆mwith ∆ m= (∆m/epsilon1,∆mδ) and\nm/epsilon1(δ)=m/epsilon1(δ)\ne+ ∆m/epsilon1(δ). Furthermore, the characteristic\nmatrixA/bardbldrives the dynamics of this linearized equation\nand has the form\nA/bardbl=/parenleftBigg\n−γ/epsilon1α/epsilon1\n/bardbl/Λ/epsilon1/epsilon1γ/epsilon1α/epsilon1\n/bardblJ/epsilon1δ\n0/µ/epsilon1\nγδαδ\n/bardblJδ/epsilon1\n0/µδ−γδαδ\n/bardbl/Λδδ/parenrightBigg\n, (10)\nwith\nΛ/epsilon1δ=J/epsilon1δ\n0\nµ/epsilon1m/epsilon1\ne\nmδeand Λ/epsilon1/epsilon1=/tildewideχ/epsilon1\n/bardbl\n1 +J/epsilon1δ\n0\nµ/epsilon1/tildewideχδ\n/bardbl, (11)7\nwhere/tildewideχ/epsilon1\n/bardblare the longitudinal susceptibilities which can\nbe evaluated in the MFA approximation as\n/tildewideχ/epsilon1\n/bardbl=J/epsilon1δ\n0µδLδL/epsilon1+µ/epsilon1L/epsilon1(kBT−Jδ\n0Lδ)\n(kBT−Jδ\n0Lδ)(kBT−J/epsilon1\n0L/epsilon1)−J/epsilon1δ\n0Jδ/epsilon1\n0LδL/epsilon1,(12)\nwithL/epsilon1=L/prime(ξ/epsilon1\ne) andLδ=L/prime(ξδ\ne). We note that the\nlongitudinal susceptibility in Eq. (12) depends on the ex-\nchange parameter (Curie temperature) and the atomic\nmagnetic moments of both sublattices.\nNext, the longitudinal damping parameter in Eq. (10)\nis defined as α/epsilon1= (2kBTλ/epsilon1m/epsilon1\ne)/µ/epsilon1H/epsilon1\ne,ex, whereH/epsilon1\ne,exis\nthe average exchange field for the sublattice /epsilon1at equi-\nlibrium, defined by the MFA expression (9). The longi-\ntudinal fluctuations are defined by the exchange energy,\naccording to the expression above. However, the longitu-\ndinal relaxation time is not simply inversely proportional\nto the damping parameter. Instead the relaxation pa-\nrameters in Eq. (10) do also depend on the longitudinal\nsusceptibilities which give the main contribution to their\ntemperature dependence.\nIt is important to note that the matrix elements in\nEq. (10) are temperature as well as (sublattice) material\nparameter dependent. The general solution of the char-\nacteristic equation, |A/bardbl−Γ±I|= 0, gives two different\neigenvalues, Γ±= 1/τ±, corresponding to the eigenvec-\ntorsv±. Here,Iis the unit matrix. The computed tem-\nperature dependence of the relaxation times τ±is pre-\nsented in Fig. 6. More interestingly, we observe that the\nratio between relaxation times τ+/τ−[inset Fig. 6] is al-\nmost constant for temperature below 0 .5TCand it has a\nvalue of 1.8 which compares well with atomistic simula-\ntions [Fig. 5]. At elevated temperatures, one relaxation\ntimeτ+will dominate the magnetization dynamics of\nboth sublattices.\nIn Fig. 7(a) we present the temperature dependence of\nthe longitudinal damping parameters and in Fig. 7(b)\nthe temperature dependence of the parameters Γ/epsilon1δ=\nα/epsilon1\n/bardbl/Λ/epsilon1δ. These parameters define the element-specific\nlongitudinal dynamics. In Figs. 7 (c) and (d) the temper-\nature dependent α/epsilon1\n/bardbl/αδ\n/bardbland Λ/epsilon1/epsilon1/Λδδare shown. It can\nbe seen that at least in the range of low temperatures the\nmagnetization dynamics is mainly defined by Γ/epsilon1/epsilon1/greatermuchΓ/epsilon1δ.\nThe general solution of the linearized LLB system for\nthe two sublattices can be written as\n∆mFe(t) =AFeexp (−t/τ+) +BFeexp (−t/τ−)\n∆mNi(t) =ANiexp (−t/τ+) +BNiexp (−t/τ−),(13)\nwhere the coefficients AFe(Ni)andBFe(Ni)will depend\nof the eigenvectors v±and the initial magnetic state,\n∆mFe(0) and ∆mNi(0). For instance\nAFe= ∆mFe(0)/bracketleftBig\n1−∆mNi(0)\n∆mFe(0)x+/bracketrightBig\nx−\nx−−x+, (14)\nwherex+=vFe\n+/vNi\n+andx−=vFe\n−/vNi\n−, is the ratio be-\ntween he eigenvector components. The other coefficients\n00.20.40.6α/bardbl/λ(a) (b)\n(c) (d)\n11.522.53\n0 0.2 0.4 0.6 0.8 1αNi/bardbl/αFe/bardbl\nT/TC(a) (b)\n(c) (d)0.010.1110\nΓ [ps−1] (a) (b)\n(c) (d)\n0 0.2 0.4 0.6 0.8 100.511.522.53\nT/TC(a) (b)\n(c) (d)Fe\nNi\nΓFe−Ni\nΓNi−Fe\nΓFe−Fe\nΓNi−Ni\nΛNi−Ni/ΛFe−Fe\n/tildewideχNi\n/bardbl//tildewideχFe\n/bardblFIG. 7. (Color online) (a) Temperature dependence of the in-\ndividual longitudinal damping parameters for Fe and Ni. (b)\nMatrix elements of the dynamical system defining the magne-\ntization dynamics. (c) Ratio between the individual damping\nparameters. (d) Ratio between the “effective” susceptibilities\nΛ/epsilon1/epsilon1and the actual susceptibilities /tildewideχ/epsilon1\n/bardbl.\nare calculated similarly. This complexity prohibits a gen-\neral analysis of the results. Thus, although the general\nsolution is clearly a bi-exponential decay, one can wonder\nwhen the one exponential decay approximation will give\na good estimate for the individual relaxation dynamics.\nTwo interesting scenarios exist: First, the relaxation\ntimesτ+andτ−could have very different time scales\nand thus one can separate the solution on short and long\ntime scales, defined by τ−andτ+, respectively. This\nis an interesting scenario for ultrafast magnetization dy-\nnamics where only the fast time scale will be relevant.\nFig. 6 shows the ratio τ+/τ−and we can observe that\nthe scenario τ+/τ−/greatermuch1 only happens for temperatures\napproaching TC. As we have seen in the atomistic simula-\ntions, after an initial distinct quenching of each sublattice\nmagnetization, both sublattice demagnetize at the same\nrate but slower than the initial rates (see Fig. 5).\nThe second scenario occurs when AFe≈∆mFe(0) and\nBNi≈∆mNi(0), even if τ+andτ−are of the same or-\nder. This happens, for example, either when the coupling\nbetween sublattices is very weak, or at relatively low tem-\nperatures, see Fig. 6. In this case the system can be con-\nsidered as two uncoupled ferromagnets (although with\nrenormalized parameters), meaning that the matrix in\nEq. (10) defining the dynamics is almost diagonal. Thus,\nwe can approximately associate each eigenvalue of Eq.\n(10) to each sublattice, τ−=τNiandτ+=τFe. The in-\nset in Fig. 6 shows the ratio τ+/τ−for the whole range of\ntemperatures. At low-to-intermediate temperatures we\nfind thatτ+/τ−≈1.8. This is in good agreement with8\nTABLE II. Theoretical results: ab initio calculated ratio be-\ntween the mean exchange interaction at T= 0 K, the ratio\nbetween atomic magnetic moments and the quotient of these\nratios. Results of simulations: atomistic spin model calcu-\nlated ratio between κexponents and relaxation times. The\nratio between the magnetic atomic moments and the expo-\nnentsκis predicted in the main text to give the ratio between\nrelaxation times.\ntheoretical simulations\nalloy/tildewideJFe\n0\n/tildewideJNi\n0µFe\nµNiµFe\nµNi/tildewideJNi\n0\n/tildewideJFe\n0κFe\nκNiτFe\nτNiµFe\nµNiκFe\nκNi\nFe50Ni50 1.592 3.38 2.12 1.492 2.10 2.25\nPy 2.685 4.198 1.563 2.3 1.8 1.8\nPy60Cu404.412 6.17 1.398 2.95 2.1 2.05\natomistic simulations, see Fig. 5(a), and it clearly shows\nthat the relaxation times ratio is not defined by the ratio\nbetween atomic magnetic moments, µFe/µNi≈4.\nIn the case that the longitudinal relaxation rates are\ndefined by the diagonal elements of the matrix (10) and\nTis not close to TCthe longitudinal relaxation time can\nbe estimated as\nτ/epsilon1/similarequal1\n2γ/epsilon1λ/epsilon1m/epsilon1eH/epsilon1e,ex. (15)\nThus the ratio between the relaxation rates of Ni and Fe\n(for the same gyromagnetic ratio value, the same cou-\npling parameter and not too close to TC) is defined by\nτNi\nτFe=/parenleftbiggλFe\nλNiµNi\nµFe/parenrightbigg/tildewideJFe\n0mFe\ne\n/tildewideJNi\n0mNie. (16)\nWe recall that /tildewideJ/epsilon1\n0m/epsilon1\ne=J/epsilon1\n0m/epsilon1\ne+J/epsilon1δ\n0mδ\neis the average ex-\nchange energy for the sublattice /epsilon1at equilibrium. Thus,\nthe interpretation of the ratio of the relaxation times is\nstraightforward. The low temperature value of the ra-\ntio/tildewideJFe\n0//tildewideJNi\n0is presented in Table II for the three alloys\nstudied here. The second column presents the ratio be-\ntween atomic magnetic moments, and the third column\nthe estimated ratio between relaxation times under the\nassumption of equal damping parameter at each sublat-\ntice.\nThe estimated ratios for relaxation times are in rather\ngood agreement with the atomistic simulations (fifth col-\numn) for Fe 50Ni50and Py, however for Py 60Cu40the es-\ntimation is not that good. We have to remember that the\nMFA re-scaling of the exchange parameters did not give\na completely satisfactory result for the shape of m(T) in\nthis alloy (see Fig. 2(a)). Thus, since the re-scaled ex-\nchange parameter does not work completely well at the\nlow-to-intermediate temperature interval, we further in-\nvestigate this case (Py 60Cu40) by relating the obtained\nrelation in Eq. (16) for the ratio τNi/τFeto the slopes of\nthe curvesm(T).This can be easily done by using the linear decrease of\nmagnetization at low temperature, m(T)≈1−κT/T C,\nwhereκ=WkB/J0for classical spin models, here W\nis the Watson integral44. Thus, the ratio between the\nslopes ofm(T) for each sublattice is directly related to\nthe ratio between the exchange values, /tildewideJδ\n0, as follows,\nκFe/κNi=/tildewideJNi\n0//tildewideJFe\n0. It is worth noting that the equilib-\nrium magnetization as a function of temperature can be\nfitted to the power law m(T) = (1−T/T C)κwhich in\nturn gives the low temperature limit m(T) = 1−κT/T C.\nAnd more importantly, it gives a link of the dynamics to\nthe equilibrium thermodynamic properties through the\nratio\nτNi\nτFe=λFe\nλNiµNi\nµFeκNi\nκFe. (17)\nNext, we fit the numerically evaluated m(T) curves to\nthe power law mFe(Ni)(T) = (1−T/T C)κFe(Ni)forT <\n0.5TC. This allows us to directly estimate the ratio be-\ntween the relaxation times for the three alloys, see Table\nII. We can see that the relation in Eq. (17) agrees well\nfor the three alloys even for Py 60Cu40.\nFor a more general case, for instance at elevated tem-\nperatures, where the one-exponential solution is not a\ngood approximation, we have to solve numerically for the\ncoefficients of each exponential decay A/epsilon1andB/epsilon1. Apart\nfrom the exchange interactions and temperature depen-\ndence,A/epsilon1andB/epsilon1also depend on the initial conditions\nδm/epsilon1(0) =m/epsilon1(0)−m/epsilon1\ne.\nE. Effect on distinct local damping parameters on\nthe magnetization dynamics\nThe intrinsic (atomistic) damping parameters λ/epsilon1are\nnot be necessarily the same for both sublattices. To in-\nvestigate the effect of different damping parameters we\nconsider that the magnetic system is initially at equi-\nlibrium at room temperature T= 300 K. Then a heat\npulseTpulse is applied for 1 ps. We define τFe(Ni)the\ntime at which the normalized magnetization, mnorm(t) =\n(m(t)−mmin)/(m(t=0)−mmin) is 1/e. The results for a\nbroad parameter space of λFe/λNiand heat pulse tem-\nperatureTpulse (scaled toTC) are shown in Fig.\nreffig:PhaseDiagramRelaxTimesPy. The line where\nτNi/τFe= 1 lies at low pulse temperature (linear limit\nin the LLB) λFe/λNi= 1.563. The critical ratio/parenleftBig\nλFe\nλNi/parenrightBig\ncris close to the one which could be predicted from Eq.\n(17) assuming τNi/τFe= 1:\n/parenleftbiggλFe\nλNi/parenrightbigg\ncr=µFe\nµNiκFe\nκNi. (18)\nEstimations of this critical ratio at low temperatures can\nbe found in Table II. The ratio is around 2 for all the\nalloys.9\n0.50.7511.25\n12345Tpulse/TC\nλFe/λNi\n00.511.522.53τNi\nτFe10.5\n1.522.5\nFIG. 8. (Color online) Ratio between the relaxation times τ\nof the Fe and Ni sublattices in Py after the application of a\nheat pulse of temperature Tpulse for a range of values of the\nratio of intrinsic damping parameters, λFe/λFe. Black lines\nrepresentλFe/λNivalues where the ratio between relaxation\ntimesτNiandτFeis constant with the value given by the label.\nThe results presented in Fig. 8 show a variety of possi-\nble situations that can be encountered in experiments on\nalloys with two magnetic sublattices. They show that in\nthe case of equal coupling to the heat-bath, the Ni sub-\nlattice demagnetizes faster than the Fe sublattics in all\ntemperature ranges. The situation may be changed if Fe\nis as least twice stronger coupled to the heat-bath than\nNi. This conclusion is not inconsistent with the dispro-\nportional couplings that were assumed in Ref. 11. Thus,\nFe can demagnetize faster than Ni (as reported in Ref.\n24) only if Fe is stronger coupled to the heat-bath.\nIV. DISCUSSION AND CONCLUSION\nElement-specific magnetization dynamics in multi-\nsublattice magnets has attracted a lot of attention\nlately24,45,46. The case of GdFeCo ferrimagnetic al-\nloys is paradigmatic since this was the first material\nwhere the so-called ultrafast all-optical switching (AOS)\nof the magnetization has been observed8. The element-\ndependent magnetization dynamics in GdFeCo alloys has\nmeanwhile been thoroughly studied9,10,12,20–23. From a\nfundamental view point, however, it is also important to\nunderstand the element-specific magnetization dynamics\nin multi-element ferromagnetic alloys. This is challenging\nfrom a modeling perspective and, moreover, contradict-\ning results have been observed in NiFe alloys24,47.\nTo treat such alloys we have developed here a hierar-\nchical multiscale approach for disordered multisublattice\nferromagnets. The electronic structure ab initio calcula-\ntions of the exchange integrals between atomic spins in\nFeNi alloys serves as as an accurate foundation to definea classical Heisenberg spin Hamiltonian which in turn has\nbeen used to calculate the element-specific magnetization\ndynamics of atomic spins through computer simulations\nbased on the stochastic LLG equation. Our simulations\npredict consistently a faster demagnetization of the Ni\nas compared to the Fe. These findings are however in\ncontrast to the dynamics measured by Mathias et al.24\nFrom a modeling perspective, we have linked informa-\ntion obtained from computer simulations of the atom-\nistic Heisenberg Hamiltonian to large scale continuum\ntheory on the basis of the recently derived finite temper-\nature LLB model for two sublattice magnets28. The LLB\nmodel is rather general, it can be applied not only to fer-\nromagnetic alloys, as we have done in the present work,\nbut also to ferrimagnetic alloys.10Thanks to analytical\nexpressions coming from the LLB model we have been\nable to interpret the distinct element-specific dynamics\nin FeNi alloys in terms of the strength of the exchange\ninteraction acting on each sublattice. Assuming equal\ndamping parameters for Fe and Ni, the difference is not\nonly coming from the different atomic moments. Analyt-\nical expressions derived for the ratio between demagneti-\nzation times in Fe and in Ni compare very well to numer-\nical results from computer simulations of the atomistic\nspin model. To investigate the effect of different intrin-\nsic damping parameters we have restrained ourselves to\nuse the LLB approach which is computationally less ex-\npensive than the atomistic spin dynamic simulations on\na large system of atomic spins. Our investigation thus\nprepares a route to an easier characterization, prediction\nand hence, control of the thermal magnetic properties\nof disordered multi-sublattice magnets, something which\nwill be valuable for technological purposes.\nAs for the applicability of our multiscale approach to\nferrimagnetic materials, one would obviously need accu-\nrately calculated exchange integrals as a starting point.\nComputing these for rare-earth transition metals alloys\nmight not straightforward, as the rare-earth ions con-\ntain mostly localized f-electrons with a sizable orbital\ncontribution to the atomic moment. However it is ex-\npected that for ferrimagnetic alloys, or multilayers with\nantiparallel alignment, composed of transition metals\nthis task will be easier. Initial theoretical comparisons\nof the element-specific demagnetization in GdFeCo were\ndone recently by Atxitia el al.10who obtained a good\nagreement with experimental observations. However, in\nthis work the exchange integrals as well as the magnetic\natomic moments were taken from phenomenological con-\nsiderations contrary to the present work where all the pa-\nrameters are obtained from first-principles calculations.\nACKNOWLEDGMENTS\nThis work has been funded through Spanish Min-\nistry of Economy and Competitiveness under the grants\nMAT2013-47078-C2-2-P, the Swedish Research Coun-\ncil (VR), and by the European Community’s Seventh10\nFramework Programme FP7/2007-2013) under grant\nagreement No. 281043, FEMTOSPIN. UA gratefully ac-\nknowledges support from EU FP7 Marie Curie Zukun-\nftskolleg Incoming Fellowship Programme, University ofKonstanz (grant No. 291784). Support from the Swedish\nInfrastructure for Computing (SNIC) is also acknowl-\nedged.\n∗denise.hinzke@uni-konstanz.de\n1E. Beaurepaire, J.-C. Merle, A. Daunois, and J.-Y. Bigot,\nPhys. Rev. Lett. 76, 4250 (1996).\n2A. Kirilyuk, A. V. Kimel, and T. Rasing, Rev. Mod. Phys.\n82, 2731 (2010).\n3J. St¨ ohr and H. C. Siegmann, Magnetism: from fundamen-\ntals to nanoscale dynamics , Vol. 152 (Springer, 2007).\n4I. Radu, K. Vahaplar, C. Stamm, T. Kachel, N. Pon-\ntius, F. Radu, R. Abrudan, H. D¨ urr, T. Ostler, J. Barker,\nR. Evans, R. Chantrell, A. Tsukamoto, A. Itoh, A. Kir-\nilyuk, T. Rasing, and A. Kimel, in SPIE OPTO (In-\nternational Society for Optics and Photonics, 2012) pp.\n82601M–82601M.\n5I. Radu, K. Vahaplar, C. Stamm, T. Kachel, N. Pontius,\nH. A. D¨ urr, T. A. Ostler, J. Barker, R. F. L. Evans, and\nR. W. Chantrell, Nature 472, 205 (2011).\n6V. L´ opez-Flores, N. Bergeard, V. Halt´ e, C. Stamm,\nN. Pontius, M. Hehn, E. Otero, E. Beaurepaire, and\nC. Boeglin, Phys. Rev. B 87, 214412 (2013).\n7N. Bergeard, V. L´ opez-Flores, V. Halt´ e, M. Hehn,\nC. Stamm, N. Pontius, E. Beaurepaire, and C. Boeglin,\nNat. Commun 5, 3466 (2014).\n8C. D. Stanciu, F. Hansteen, A. V. Kimel, A. Kirilyuk,\nA. Tsukamoto, A. Itoh, and T. Rasing, Phys. Rev. Lett.\n99, 047601 (2007).\n9T. A. Ostler, J. Barker, R. F. L. Evans, R. W. Chantrell,\nU. Atxitia, O. Chubykalo-Fesenko, S. El Moussaoui,\nL. Le Guyader, E. Mengotti, L. J. Heyderman, F. Nolt-\ning, A. Tsukamoto, A. Itoh, D. Afanasiev, B. A. Ivanov,\nA. M. Kalashnikova, K. Vahaplar, J. Mentink, A. Kirilyuk,\nT. Rasing, and A. V. Kimel, Nat. Commun. 3, 666 (2012).\n10U. Atxitia, J. Barker, R. W. Chantrell, and O. Chubykalo-\nFesenko, Phys. Rev. B 89, 224421 (2014).\n11A. J. Schellekens and B. Koopmans, Phys. Rev. B 87,\n020407 (2013).\n12S. Wienholdt, D. Hinzke, K. Carva, P. M. Oppeneer, and\nU. Nowak, Phys. Rev. B 88, 020406 (2013).\n13A. Hassdenteufel, B. Hebler, C. Schubert, A. Liebig, M. Te-\nich, M. Helm, M. Aeschlimann, M. Albrecht, and R. Brats-\nchitsch, Adv. Mater. 25, 3122 (2013).\n14S. Alebrand, M. Gottwald, M. Hehn, D. Steil, M. Cinchetti,\nD. Lacour, E. E. Fullerton, M. Aeschlimann, and S. Man-\ngin, Appl. Phys. Lett. 101, 162408 (2012).\n15T. Y. Cheng, J. Wu, M. Willcox, T. Liu, J. W. Cai, R. W.\nChantrell, and Y. B. Xu, IEEE Trans. Magn. 48, 3387\n(2012).\n16S. Mangin, M. Gottwald, C.-H. Lambert, D. Steil, V. Uhl´ ı,\nL. Pang, M. Hehn, S. Alebrand, M. Cinchetti, G. Mali-\nnowski, Y. Fainman, M. Aeschlimann, and E. E. Fullerton,\nNat. Mater. 13, 286 (2014).\n17R. F. Evans, T. A. Ostler, R. W. Chantrell, I. Radu, and\nT. Rasing, Appl. Phys. Lett. 104, 082410 (2014).\n18C. Schubert, A. Hassdenteufel, P. Matthes, J. Schmidt,\nM. Helm, R. Bratschitsch, and M. Albrecht, Appl. Phys.\nLett. 104, 082406 (2014).19C.-H. Lambert, S. Mangin, B. S. D. C. S. Varaprasad,\nY. K. Takahashi, M. Hehn, M. Cinchetti, G. Malinowski,\nK. Hono, Y. Fainman, M. Aeschlimann, and E. E. Fuller-\nton, Science 345, 1337 (2014).\n20J. H. Mentink, J. Hellsvik, D. V. Afanasiev, B. A. Ivanov,\nA. Kirilyuk, A. V. Kimel, O. Eriksson, M. I. Katsnelson,\nand T. Rasing, Phys. Rev. Lett. 108, 057202 (2012).\n21J. Barker, U. Atxitia, T. A. Ostler, O. Hovorka,\nO. Chubykalo-Fesenko, and R. W. Chantrell, Scie. Rep. 3\n(2013), doi:10.1038/srep03262.\n22V. G. Baryakhtar, V. I. Butrim, and B. A. Ivanov, JETP\nLett. 98, 289 (2013).\n23U. Atxitia, T. Ostler, J. Barker, R. F. L. Evans, R. W.\nChantrell, and O. Chubykalo-Fesenko, Phys. Rev. B 87,\n224417 (2013).\n24S. Mathias, C. La-O-Vorakiat, P. Grychtol, P. Granitzka,\nE. Turgut, J. M. Shaw, R. Adam, H. T. Nembach, M. E.\nSiemens, S. Eich, C. M. Schneider, T. J. Silva, M. Aeschli-\nmann, M. M. Murnane, and H. C. Kapteyn, Proc. Natl.\nAcad. Scie. USA 109, 4792 (2012).\n25N. Kazantseva, U. Nowak, R. W. Chantrell, J. Hohlfeld,\nand A. Rebei, Europhysics Lett. 81, 27004 (2008).\n26B. Koopmans, G. Malinowski, F. Dalla Longa, D. Steiauf,\nM. F¨ ahnle, T. Roth, M. Cinchetti, and M. Aeschlimann,\nNat. Mater. 9, 259 (2010).\n27N. Kazantseva, D. Hinzke, U. Nowak, R. W. Chantrell,\nU. Atxitia, and O. Chubykalo-Fesenko, Phys. Rev. B 77,\n184428 (2008).\n28U. Atxitia, P. Nieves, and O. Chubykalo-Fesenko, Phys.\nRev. B 86, 104414 (2012).\n29A. I. Liechtenstein, M. I. Katsnelson, V. P. Antropov, and\nV. A. Gubanov, J. Magn. Magn. Mater. 67, 65 (1987).\n30S. V. Halilov, H. Eschrig, A. Y. Perlov, and P. M. Oppe-\nneer, Phys. Rev. B 58, 293 (1998).\n31A. I. Liechtenstein, M. I. Katsnelson, and V. A. Gubanov,\nJ. Physics F: Metal Phys. 14, L125 (1984).\n32I. Turek, Electronic structure of disordered alloys, sur-\nfaces and interfaces (Kluwer Academic Publishers, Boston,\n1997).\n33O. N. Mryasov, U. Nowak, K. Guslienko, and R. W.\nChantrell, Europhysics Lett. 69, 805 (2005).\n34J. Kudrnovsk´ y, V. Drchal, and P. Bruno, Phys. Rev. B\n77, 224422 (2008).\n35U. von Barth and L. Hedin, J. Phys. C: Solid State Physics\n5, 1629 (1972).\n36P. Soven, Phys. Rev. 156, 809 (1967).\n37P. Bruno, Phys. Rev. Lett. 90, 087205 (2003).\n38M. I. Katsnelson and A. I. Lichtenstein, J. Physics: Con-\ndens. Matter 16, 7439 (2004).\n39I. Turek, J. Kudrnovsk´ y, V. Drchal, and P. Bruno, Phil.\nMag. 86, 1713 (2006).\n40B. Glaubitz, S. Buschhorn, F. Br¨ ussing, R. Abrudan, and\nH. Zabel, J. Physics: Condens. Matter 23, 254210 (2011).\n41I. A. Abrikosov, A. E. Kissavos, F. Liot, B. Alling, S. I.\nSimak, O. Peil, and A. V. Ruban, Phys. Rev. B 76, 01443411\n(2007).\n42U. Nowak and D. Hinzke, J. Magn. Magn. Mat. 221(2000).\n43U. Nowak, Handbook of Magnetism and Advanced Mag-\nnetic Materials (Wiley, 2007).\n44C. Domb, Phase transitions and critical phenomena ,\nVol. 19 (Academic Press, 2000).\n45C. La-O-Vorakiat, M. Siemens, M. M. Murnane, H. C.\nKapteyn, S. Mathias, M. Aeschlimann, P. Grychtol,R. Adam, C. M. Schneider, J. M. Shaw, H. Nembach, and\nT. J. Silva, Phys. Rev. Lett. 103, 257402 (2009).\n46A. R. Khorsand, M. Savoini, A. Kirilyuk, A. V. Kimel,\nA. Tsukamoto, A. Itoh, and T. Rasing, Phys. Rev. Lett.\n110, 107205 (2013).\n47A. Eschenlohr, PhD thesis, Helmholtz Zentrum Berlin\n(2012)."    },    {        "title": "1504.06042v1.Magnetization_damping_in_noncollinear_spin_valves_with_antiferromagnetic_interlayer_couplings.pdf",        "content": "arXiv:1504.06042v1  [cond-mat.mes-hall]  23 Apr 2015Magnetizationdamping innoncollinearspinvalveswithant iferromagnetic interlayer couplings\nTakahiro Chiba1, Gerrit E. W. Bauer1,2,3, and Saburo Takahashi1\n1Institute for Materials Research, Tohoku University, Send ai, Miyagi 980-8577, Japan\n2WPI-AIMR, Tohoku University, Sendai, Miyagi 980-8577, Jap an and\n3Kavli Institute of NanoScience, Delft University of Techno logy, Lorentzweg 1, 2628 CJ Delft, The Netherlands\n(Dated: October 29, 2018)\nWe study the magnetic damping in the simplest of synthetic an tiferromagnets, i.e. antiferromagnetically\nexchange-coupled spin valves in which applied magnetic fiel ds tune the magnetic configuration to become\nnoncollinear. We formulate the dynamic exchange of spin cur rents in a noncollinear texture based on the spin-\ndiffusiontheorywithquantum mechanicalboundaryconditionsa ttheferrromagnet|normal-metal interfacesand\nderive the Landau-Lifshitz-Gilbert equations coupled by t he static interlayer non-local and the dynamic ex-\nchange interactions. We predict non-collinearity-induce d additional damping that can be sensitively modulated\nbyanapplied magnetic field. The theoretical results compar e favorablywithpublished experiments.\nI. INTRODUCTION\nAntiferromagnets (AFMs) boast many of the functionali-\ntiesofferromagnets(FM)thatareusefulinspintroniccirc uits\nanddevices: Anisotropicmagnetoresistance(AMR),1tunnel-\ning anisotropicmagnetoresistance(TAMR),2current-induced\nspintransfertorque,3–8andspincurrenttransmission9–11have\nall been found in or with AFMs. This is of interest because\nAFMshaveadditionalfeaturespotentiallyattractivefora ppli-\ncations. InAFMsthetotalmagneticmomentis(almost)com-\npletely compensated on an atomic length scale. The AFM\norder parameter is, hence, robust against perturbations su ch\nas external magnetic fields and do not generate stray fields\nthemselveseither. AspintronictechnologybasedonAFM el-\nementsisthereforeveryattractive.12,13Drawbacksarethedif-\nficulty to controlAFMs by magnetic fields and much higher\n(THz)resonancefrequencies,14–16whicharedifficulttomatch\nwith conventional electronic circuits. Man-made magnetic\nmultilayers in which the layer magnetizations in the ground\nstate isorderedin anantiparallelfashion,17i.e. so-calledsyn-\nthetic antiferromagnets,donot su ffer fromthis drawbackand\nhave therefore been a fruitful laboratory to study and modu-\nlate antiferromagnetic couplings and its consequences,18but\nalso found applications as magnetic field sensors.19Trans-\nport in these multilayers including the giant magnetoresis -\ntance (GMR)20,21are now well understood in terms of spin\nand charge diffusive transport. Current-induced magnetiza-\ntionswitchinginF|N|Fspinvalvesandtunneljunctions,22has\nbeen a game-changer for devices such as magnetic random\naccess memories(MRAM).23A keyparameterof magnetiza-\ntiondynamicsisthemagneticdamping;asmalldampinglow-\nersthethresholdofcurrent-drivenmagnetizationswitchi ng,24\nwhereasalargedampingsuppresses“ringing”oftheswitche d\nmagnetization.25\nMagnetization dynamics in multilayers generates “spin\npumping”, i.e. spin current injection from the ferromagnet\ninto metallic contacts. It is associated with a loss of an-\ngular momentum and an additional interface-related magne-\ntization damping.26,27In spin valves, the additional damp-\ning is suppressed when the two magnetizations precess in-\nphase, while it is enhanced for a phase di fference ofπ(out-\nof-phase).27–30This phenomenon is explained in terms of a\n“dynamic exchange interaction”, i.e. the mutual exchange o fnon-equilibriumspin currents,which shouldbe distinguis hed\nfrom(butcoexistswith)theoscillatingequilibriumexcha nge-\ncoupling mediated by the Ruderman-Kittel-Kasuya-Yosida\n(RKKY) interaction. The equilibriumcoupling is suppresse d\nwhenthespacerthicknessisthickerthantheelasticmean-f ree\npath,31,32while the dynamiccouplingise ffective onthe scale\noftheusuallymuchlargerspin-flipdi ffusionlength.\nAntiparallel spin valves provide a unique opportunity to\nstudy and control the dynamic exchange interaction between\nferromagnets through a metallic interlayer for tunable mag -\nnetic configurations.33,34An originallyantiparallel configura-\ntionisforcedbyrelativelyweakexternalmagneticfieldsin toa\nnon-collinearconfigurationwith a ferromagneticcomponen t.\nFerromagneticresonance(FMR)andBrillouinlightscatter ing\n(BLS) are two useful experimentalmethodsto investigateth e\nnature and magnitude of exchange interactions and magnetic\ndamping in multilayers.35Both methods observe two reso-\nnances, i.e. acoustic (A) and optical (O) modes, which are\ncharacterizedbytheirfrequenciesandlinewidths.36,37\nTimopheev etal.observedaneffectoftheinterlayerRKKY\ncoupling on the FMR and found the linewidth to be a ffected\nby the dynamic exchange coupling in spin valves with one\nlayerfixed by the exchange-biasof an inert AFM substrate.38\nThey measured the FMR spectrum of the free layer by tun-\ning the interlayer coupling (thickness) and reported a broa d-\nening of the linewidth by the dynamic exchange interaction.\nTaniguchi et al.addressed theoretically the enhancement of\nthe Gilbert damping constant due to spin pumping in non-\ncollinear F|N|F trilayer systems, in which one of the magne-\ntizations is excited by FMR while the other is o ff-resonant,\nbutadoptaroleasspinsink.39Thedynamicsofcoupledspin\nvalvesinwhichbothlayermagnetizationsarefreetomoveha s\nbeencomputedby oneof us29and bySkarsvåg et al.33,49but\nonly for collinear (parallel and antiparallel) configurati ons.\nCurrent-induced high-frequency oscillations without app lied\nmagnetic field in ferromagnetically coupled spin valves has\nbeenpredicted.40\nInthepresentpaper,wemodelthemagnetizationdynamics\nof the simplest of synthetic antiferromagnets, i.e. the ant i-\nferromagnetically exchange-coupled spin valve in which th e\n(in-plane) ground state magnetizations are for certain spa cer\nthicknesses ordered in an antiparallel fashion by the RKKY\ninterlayercoupling.41We focusonthecoupledmagnetization2\nmodes in symmetric spin valves in which in contrast to pre-\nvious studies, both magnetizations are free to move. We in-\nclude static magnetic fields in the film plane that deform the\nantiparallelconfigurationintoacantedone. Microwaveswi th\nlongitudinal and transverse polarizations with respect to an\nexternalmagneticfieldthenexciteAandOresonancemodes,\nrespectively.31,42–46We develop the theory for magnetization\ndynamics and damping based on the Landau-Lifshitz-Gilbert\nequationwithmutualpumpingofspincurrentsandspintrans -\nfer torques based on the spin di ffusion model with quantum\nmechanical boundary conditions.27,47,48We confirm28,49that\nthe additional damping of O modes is larger than that of the\nA modes. We report that a noncollinear magnetization con-\nfigurationinducesadditionaldampingtorquesthat to the be st\nof ourknowledgehavenotbeen discussedin magneticmulti-\nlayers before.50The external magnetic field strongly a ffects\nthe dynamics by modulating the phase of the dynamic ex-\nchange interaction. We compute FMR linewidths as a func-\ntion of applied magnetic fields and find good agreement with\nexperimental FMR spectra on spin valves.31,32The dynam-\nics of magnetic multilayers as measured by ac spin trans-\nfer torque excitation30reveals a relative broadening of the O\nmodes linewidths that is well reproduced by our spin valve\nmodel.\nIn Sec. II we present our model for noncollinear spin\nvalves based on spin-di ffusiontheory with quantum mechan-\nical boundary conditions. In Sec. III, we consider the mag-\nnetization dynamics in antiferromagnetically coupled non -\ncollinear spin valves as shown in Fig. 1(b). We derive the\nlinearized magnetization dynamics, resonance frequencie s,\nand lifetimes of the acoustic and optical resonance modes in\nSec. IV. We discuss the role of dynamicspin torqueson non-\ncollinear magnetization configurations in relation to exte rnal\nmagnetic field dependence of the linewidth. In Sec. V, we\ncompare the calculated microwave absorption and linewidth\nwith published experiments. We summarize the results and\nendwiththeconclusionsinSec. VI.\nII. SPINDIFFUSIONTRANSPORTMODEL\nWe consider F1|N|F2 spin valves as shown in Fig. 1(a), in\nwhichthemagnetizations MjoftheferromagnetsF j(j=1,2)\nare coupled by a antiparallel interlayer exchange interact ion\nand tilted towards the direction of an external magnetic fiel d.\nApplied microwaves with transverse polarizations with re-\nspect to an external magnetic field cause dynamics and, via\nspinpumping,spincurrentsandaccumulationsinthenormal -\nmetal (NM) spacer. The longitudinal component of the spin\naccumulation diffuses into and generates spin accumulations\ninF thatwe showtobesmall later,butdisregardinitially. L et\nusdenotethepumpedspincurrent JP\nj,whileJB\njisthediffusion\n(back-flow)spin currentdensity inducedby a spin accumula-NM zy\nx\nF1 F2 0 dN\nJ1PJ2P\nJ2BJ1Bµsθθ\n(c) Acoustic mode (d) Optical mode (a)\nm1 m2\nH\n*OUFSMBZFS\u0001DPVQMJOH \n(b)\n(b) H\nhx hy\nFIG.1. (a) Sketch of the sample withinterlayer exchange-co uplings\nillustrating the spin pumping and backflow currents. (b) Mag netic\nresonance in an antiferromagnetically exchange-coupled s pin valve\nwith a normal-metal (NM) film sandwiched by two ferromagnets\n(F1,F2)subjecttoamicrowave magneticfield h. Themagnetization\nvectors (m1,m2) are tilted by an angle θin a static in-plane mag-\nneticfield Happliedalongthe y-axis. Thevectors mandnrepresent\nthesum anddifference ofthe twolayer magnetizations, respectively.\n(c) and (d): Precession-phase relations for the acoustic an d optical\nmodes.\ntionµsjin NM,bothat theinterfaceF j, with27,51\nJP\nj=Gr\nemj×/planckover2pi1∂tmj, (1a)\nJB\nj=Gr\ne/bracketleftBig/parenleftBig\nmj·µsj/parenrightBig\nmj−µsj/bracketrightBig\n, (1b)\nwheremj=Mj//vextendsingle/vextendsingle/vextendsingleMj/vextendsingle/vextendsingle/vextendsingleis the unit vector along the magnetic\nmoment of F j(j=1,2). The spin current througha FM |NM\ninterface is governed by the complex spin-mixing conduc-\ntance (per unit area of the interface) G↑↓=Gr+iGi.27The\nreal component Grparameterized one vector component of\nthe transverse spin-currentspumped and absorbed by the fer -\nromagnets. The imaginary part Gican be interpreted as an\neffective exchange field between magnetization and spin ac-\ncumulation, which in the absence of spin-orbit interaction is\nusuallymuchsmallerthantherealpart,forconductingaswe ll\nasinsultingmagnets.523\nThediffusionspin-currentdensityin NMreads\nJs,z(z)=−σ\n2e∂zµs(z), (2)\nwhereσ=ρ−1is the electrical conductivity and µs(z)=\nAe−z/λ+Bez/λthe spin accumulationvectorthat is a solution\nofthespindiffusionequation∂2\nzµs=µs/λ2,whereλ=√Dτsf\nis the spin-diffusion length, Dthe diffusion constant, and τsf\nthe spin-flip relaxation time. The vectors AandBare de-termined by the boundary conditions at the F1 |NM (z=0)\nand F2|NM (z=dN) interfaces: Js,z(0)=JP\n1+JB\n1≡Js1and\nJs,z(dN)=−JP\n2−JB\n2≡−Js2. Theresultingspin accumulation\ninN reads\nµs(z)=2eλρ\nsinh/parenleftBigdN\nλ/parenrightBig/bracketleftBigg\nJs1cosh/parenleftBiggz−dN\nλ/parenrightBigg\n+Js2cosh/parenleftbiggz\nλ/parenrightbigg/bracketrightBigg\n,(3)\nwithinterfacespin currents\nJs1=ηS\n1−η2δJP\n1+η2/parenleftBig\nm2·δJP\n1/parenrightBig\n1−η2(m1·m2)2m1×(m1×m2), (4a)\nJs2=−ηS\n1−η2δJP\n2+η2/parenleftBig\nm1·δJP\n2/parenrightBig\n1−η2(m1·m2)2m2×(m2×m1). (4b)\nHere\nδJP\n1=JP\n1+ηm1×(m1×JP\n2), (5a)\nδJP\n2=JP\n2+ηm2×(m2×JP\n1), (5b)\nS=sinh(dN/λ)/grandη=gr/[sinh(dN/λ)+grcosh(dN/λ)]\nare the efficiency of the back flow spin currents, and gr=\n2λρGris dimensionless. The first terms in Eqs. (4a) and (4b)\nrepresent the mutual pumping of spin currents while the sec-\nondtermsmaybeinterpretedasa spincurrentinducedbythe\nnoncollinear magnetization configuration, including the b ack\nflowfromthe NMinterlayer.\nIII. MAGNETIZATIONDYNAMICSWITH DYNAMIC\nSPINTORQUES\nWe consider the magnetic resonance in the non-collinear\nspin valve shown in Fig. 1. The magnetization dynamics are\ndescribedbytheLandau-Lifshitz-Gilbert(LLG)equation,\n∂tm1=−γm1×Heff1+α0m1×∂tm1+τ1,(6a)\n∂tm2=−γm2×Heff2+α0m2×∂tm2−τ2.(6b)\nThe first term in Eqs. (6a) and (6b) represents the torque in-\nducedbytheeffectivemagneticfield\nHeff1(2)=H+h(t)−4πMsm1(2)zˆz+Jex\nMsdFm2(1),(7)\nwhich consists of an in-plane applied magnetic field H,\na microwave field h(t), and the demagnetization field\n−4πMsm1(2)zˆzwith saturation magnetization Ms. The inter-\nlayer exchange field is Jex/(MsdF)m2(1)with areal density of\ntheinterlayerexchangeenergy Jex<0(forantiferromagnetic-\ncoupling) and F layer thickness dF. The second term is the\nGilbert dampingtorque that governsthe relaxationcharact er-\nized byα0itowards an equilibrium direction. The third term,τ1(2)=γ/planckover2pi1/(2eMsdF)Js1(2), is the spin-transfertorque induced\nby the absorption of the transverse spin currents of Eqs. (4a )\nand (4b), andγandα0are the gyromagnetic ratio and the\nGilbert dampingconstant of the isolated ferromagneticfilm s,\nrespectively. SometechnicaldetailsofthecoupledLLGequ a-\ntionsarediscussedinAppendixA.Introducingthetotalmag -\nnetizationdirection m=(m1+m2)/2andthedifferencevector\nn=(m1−m2)/2,theLLG equationscanbewritten\n∂tm=−γm×(H+h)\n+2πγMs(mzm+nzn)׈z\n+α0(m×∂tm+n×∂tn)+τm, (8a)\n∂tn=−γn×/parenleftBigg\nH+h+Jex\nMsdFm/parenrightBigg\n+2πγMs(nzm+mzn)׈z\n+α0(m×∂tn+n×∂tm)+τn,(8b)\nwhere the spin-transfer torques τm=(τ1+τ2)/2 andτn=\n(τ1−τ2)/2become\nτm/αm=m×∂tm+n×∂tn\n+2ηm·(n×∂tn)\n1−ηCm+2ηn·(m×∂tm)\n1+ηCn,(9a)\nτn/αn=m×∂tn+n×∂tm\n−2ηm·(n×∂tm)\n1+ηCm−2ηn·(m×∂tn)\n1−ηCn,(9b)\nandC=m2−n2,while\nαm=α1gr\n1+grcoth(dN/2λ), (10a)\nαn=α1gr\n1+grtanh(dN/2λ), (10b)\nwithα1=γ/planckover2pi12/(4e2λρMsdF).4\nIV. CALCULATIONANDRESULTS\nWe consider the magnetization dynamics excited by lin-\nearly polarized microwaves with a frequency ωand in-plane\nmagnetic field h(t)=(hx,hy,0)eiωtthat is much smaller than\nthe saturation magnetization. For small angle magnetizati on\nprecession the total magnetization and di fference vector may\nbe separated into a static equilibrium and a dynamic com-\nponent as m=m0+δmandn=n0+δn, respectively,\nwherem0=(0,sinθ,0),n0=(cosθ,0,0),C=−cos2θ,\nands=−ˆzsin2θ. The equilibrium (zero torque) conditions\nm0×H=0 andn0×(H+Jex/(MsdF)m0)=0 lead to the\nrelation\nsinθ=H/Hs, (11)\nwhereHs=−Jex/(MsdF)=|Jex|/(MsdF) is the saturation\nfield. TheLLGequationsread\n∂tδm=−γδm×H−γm0×h\n+2πγMs(δmzm0+δnzn0)׈z\n+α0(m0×∂tδm+n0×∂tδn)+δτm,(12a)\n∂tδn=−γδn×H−γn0×h\n+2πγMs(δnzm0+δmzn0)׈z\n−γHs(m0×δn−n0×δm)\n+α0(m0×∂tδn+n0×∂tδm)+δτn,(12b)\nwithlinearizedspin-transfertorques\nδτm/αm=m0×∂tδm+n0×∂tδn\n−ηsin2θ\n1+ηcos2θ∂tδnzm0+ηsin2θ\n1−ηcos2θ∂tδmzn0,(13a)\nδτn/αn=m0×∂tδn+n0×∂tδm\n+ηsin2θ\n1−ηcos2θδmzm0−ηsin2θ\n1+ηcos2θδnzn0,(13b)\nTo leading order in the small precessing components δmand\nδn,theLLG equationsinfrequencyspace become\nδmx=γhxγ(Hs+4πMs)+iω/parenleftBig\nα0+αm(1+η)\n1−ηcos2θ/parenrightBig\nω2−ω2\nA−i∆Aωsin2θ,(14a)\nδny=−γhxγ(Hs+4πMs)+iω/parenleftBig\nα0+αn(1−η)\n1−ηcos2θ/parenrightBig\nω2−ω2\nA−i∆Aωcosθsinθ,\n(14b)\nδmz=−γhxiω\nω2−ω2\nA−i∆Aωsinθ, (14c)\nδnx=−γhy4πγMs+iω/parenleftBig\nα0+αn(1−η)\n1+ηcos2θ/parenrightBig\nω2−ω2\nO−i∆Oωcosθsinθ,(15a)\nδmy=γhy4πγMs+iω/parenleftBig\nα0+αm(1+η)\n1−ηcos2θ/parenrightBig\nω2−ω2\nO−i∆Oωcos2θ, (15b)\nδnz=γhyiω\nω2−ω2\nO−i∆Oωcosθ. (15c)The A modes (δmx,δny,δmz) are excited by hx, while the O\nmodes (δnx,δmy,δnz) couple to hy. The poles inδm(ω)and\nδn(ω)define the resonance frequencies and linewidths that\ndo not depend on the magnetic field since we disregard\nanisotropyandexchange-bias.\nA. Acoustic andOpticalmodes\nAn antiferromagnetically exchange-coupled spin valves\ngenerallyhave non-collinearmagnetizationconfiguration sby\nthepresenceofexternalmagneticfields. For H<Hs(0<θ<\nπ/2),theacousticmode:\nωA=γH/radicalbig\n1+(4πMs/Hs), (16)\n∆A=α0γ/parenleftBig\nHs+4πMs+Hssin2θ/parenrightBig\n+αmγ(Hs+4πMs)+αA(θ)γHs,(17)\nandthe opticalmode:\nωO=γ/radicalBig\n(4πMs/Hs)(H2s−H2), (18)\n∆O=α0γ/parenleftBig\n4πMs+Hscos2θ/parenrightBig\n+αn4πγMs+αO(θ)γHs, (19)\nwhere\nαA(θ)=α1grsin2θ\n1+grtanh(dN/2λ)+2grsin2θ/sinh(dN/λ),(20)\nαO(θ)=α1grcos2θ\n1+grtanh(dN/2λ)+2grcos2θ/sinh(dN/λ).(21)\nThe additional broadeningin ∆Ais proportional toαmand\nαAwhile that in∆Oscales withαnandαOin. Figure 2 (a)\nshowsαmandαnas a function of spacer layer thickness, in-\ndicating thatαnis always larger than αm, and thatαn(αm)\nstrongly increases (decreases) with decreasing N layer thi ck-\nness, especially for dN<λand large gr. Figure 2(b) shows\nthedependenceofαAandαOonthetiltedangleθfordifferent\nvaluesof dN. Asθincreases,αAincreasesfrom0to αmwhile\nαOdecreasesαmto 0. The additional damping can be ex-\nplained by the dynamic exchange. When two magnetizations\nin spin valves precess in-phase, each magnet receives a spin\ncurrent that compensates the pumped one, thereby reducing\nthe interface damping. When the magnetizations precess out\nof phase, theπphase difference between both spin currents\nmeans that the moduli have to be added, thereby enhancing\nthedamping.\nWhen the magnetizations are tilted by an angle θas\nsketched in Fig. 1, we predict an additional damping torque\nexpressedbythesecondtermsofEqs.(4a)and(4b). Figure3\nshows the ratiosαA(θ)/αmandαO(θ)/αnas a function ofθ\nandgrfor different values ofλ/dN, thereby emphasizing the\nadditional damping in the presence of noncollinear magneti -\nzations. In Fig. 3(a,b)with λ/dN=1, i.e. for a spin-di ffusive\ninterlayer,the additionaldampingof both A- and O-modesis\nsignificant in a large region of parameter space. On the other5\n0.1 1 10 100λ/d N012345αm/α1, αn/α1/α1 /α12\n1\n0.5\n0 30 60 90 \nθ (degree)0.00.10.20.30.40.5αA, αOαO αAλ/d N=1 \n2\n4\n10(b)(a)\nαnαmgr=5\nFIG.2. (a)αm(dashed line)andαn(solidline) as a functionof λ/dN\nfor different values of the dimensionless mixing conductance gr. (b)\nαAC(dashed line)andαOP(solidline)as afunction oftiltangle θfor\ngr=5and different values ofλ/dN.\nhand, in Fig. 3(c,d) with λ/dN=10, i.e. for an almost spin-\nballistic interlayer, the additional damping is more impor tant\nfortheA-mode,whiletheO-modeisa ffectedonlyclosetothe\ncollinear magnetization. In the latter case the intrinsic d amp-\ningα0dominates,however.\nB. In-phaseandOut-of-phasemodes\nWhentheappliedmagneticfieldislargerthanthesaturation\nfield (H>Hs), both magnetizations point in the ˆydirection,\nand theδm(A) andδn(O) modes morph into in-phase and\n180◦out-of-phase (antiphase) oscillations of δm1andδm2,\nrespectively. The resonance frequency53and linewidth of the\nin-phasemodefor H>Hs(θ=π/2)are\nωA=γ/radicalbig\nH(H+4πMs), (22)\n∆A=2(α0+αm)γ(H+2πMs), (23)\nwhilethoseoftheout-of-phasemodeare\nωO=γ/radicalbig\n(H−Hs)(H−Hs+4πMs),(24)\n∆O=2(α0+αn)γ(H−Hs+2πMs).(25)\u0011 \u001a\u0011 \u0017\u0011 \u0014\u0011 \u0011 \u001a\u0011 \u0017\u0011 \u0014\u0011 \u0011\u0017\n\u0015\n\u0013\u0019\u0012\u0011 \n\u0011\u0017\n\u0015\n\u0013\u0019\u0012\u0011 \tB\n \tC\n \n\tD\n \tE\nHS\nHS\nθ (degree) θ (degree)\n\u0011 \u001a\u0011 \u0017\u0011 \u0014\u0011 \nθ (degree)\u0011 \u001a\u0011 \u0017\u0011 \u0014\u0011 \nθ (degree)\u0011\u0017\n\u0015\n\u0013\u0019\u0012\u0011 HS\n\u0011\u0017\n\u0015\n\u0013\u0019\u0012\u0011 HS0.1 0.3 0.5 0.7 0.9 \n0.1 0.2 0.3 0.4 0.5 0.6 \n0.55 0.65 0.75 0.85 0.95 \n0.01 0.02 0.03 0.04 0.05 \nFIG. 3. (a,c)αA(θ)/αmand (b,d)αO(θ)/αnas a function ofθandgr\nfor different values ofλ/dN. (a,b) withλ/dN=1, (c,d) withλ/dN=\n10\nFigure 4(a) shows the calculated resonance frequencies of\nthe A andO modesas a functionof an appliedmagneticfield\nHwhile 4(b) displays the linewidths for α1/α0=1, which\nis representative for ferromagnetic metals, such as permal -\nloy (Py) with an intrinsic magnetic damping of the order of\nα0=0.01andacomparableadditionaldamping α1duetospin\npumping. A value gr=4/5 corresponds toλ=20/200nm,\nρ=10/2.5µΩcmfor N=Ru/Cu,54,55Gr=2/1×1015Ω−1m−2\nfor the N|Co(Py) interface56,57, anddF=1nm, for example.\nThecolorsinthefigurerepresentdi fferentrelativelayerthick-\nnessesdN/λ. The linewidth of the A mode in Fig. 4(b) in-\ncreases with increasing H, while that of the O mode starts\nto decrease until a minimum at the saturation field H=Hs.\nFigure 4(c) shows the linewidths for α1/α0=10, which de-\nscribes ferromagnetic materials with low intrinsic dampin g,\nsuch as Heusler alloys58and magnetic garnets.59In this case,\nthe linewidth of the O modeis much largerthanthat of the A\nmode,especiallyforsmall dN/λ.\nIn the limit of dN/λ→0 is easily established experimen-\ntally. The expressions of the linewidth in Eqs. (17) and (19)\narethengreatlysimplifiedto ∆A=γ(Hs+4πMs+Hssin2θ)α0\nand∆O=γ/parenleftBig\n4πMs+Hscos2θ/parenrightBig\nα0+(4πγMs)grα1,and∆A≪\n∆Owhengrα1≫α0. The additional damping, Eq. (10b) re-\nduces toαm→0 andαn→2[γ/planckover2pi1/(4πMsdF)(h/e2)Gr] when\nthemagnetizationsarecollinearandintheballisticspint rans-\nport limit.27In contrast to the acoustic mode, the dynamic\nexchange interaction enhances damping of the optical mode.\n∆O≫∆AhasbeenobservedinPy |Ru|Pytrilayerspinvalves32\nandCo|Cu multilayers30, consistentwiththepresentresults.\nFor spin valves with ferromagnetic metals,\nthe interface backflow spin-current [(1b)] reads\nJB\nj=(Gr/e)/bracketleftBig\nξF/parenleftBig\nmj·µsj/parenrightBig\nmj−µsj/bracketrightBig\n,whereξF=6\n012ω/(4 πγ Ms)ωAωO\n02468∆/(4 πMsα0) dN/λ=\n0.3\n1(a)\n(b) \n(c)\n0 0.5 1 1.5\nH/H s050dN/λ=0.01\n0.1\n0.3\n1∆/(4 πγ Msα0)0.01γ\nA mode  O mode 0.1\nFIG. 4. (a) Resonance frequencies of the A and O modes as a func -\ntion of magnetic field for Hs/(4πMs)=1. (b), (c) Linewidths of the\nA (dashed line) and the O (solid line) modes for Hs/(4πMs)=1,\ngr=5, and different values of dN/λ. (b)α1/α0=1 and (c)\nα1/α0=10.\n1−(G/2Gr)(1−p2)(1−ηF) (0≤ξF≤1),Gis the\nN|F interface conductance per unit area, and pthe conduc-\ntance spin polarization.51Here the spin diffusion efficiency\nis\n1\nηF=1+σF\nGλFtanh(dF/λF)\ncosh(dF/λF), (26)\nwhereσF,λF, anddFare the conductivity, the spin-flip dif-\nfusion length, and the layer thickness of the ferromagnets,\nrespectively. For the material parameters of a typical fer-\nromagnet with dF=1 nm, the resistivity ρF=10µΩcm,\nG=2Gr=1015Ω−1m−2,λF=10nm,and p=0.7,ξF=0.95,\nwhichjustifiesdisregardingthiscontributionfromtheout set.\nV. COMPARISONWITH EXPERIMENTS\nFMRexperimentsyieldtheresonantabsorptionspectraofa\nmicrowavefield ofa ferromagnet. Themicrowaveabsorptiont\ndP/dH ϕ=90 o\n20 o\n0o×5\n×5ϕHh(t)(a)\n5\nH (kOe) 22 4 6 0 0  0.4  0.8  1.2 H/H s\nCo(3)|Ru(1)|Co(3) Co(3)|Ru(1)|Co(3) \nExperiment (Ref. 11) Calculation \n 0  2  4  6  8  0  0.2  0.4  0.6  0.8  1 ∆/(4 πγ Msα₀)H/H s\nA mode \nO mode \n 0  2000  4000  6000 \nField (Oe) (b)\nExperiment (Ref. 10): [Co(1)|Cu(1)] 10 Co(1) Calculation: Co(1)|Cu(1)|Co(1) \nFIG.5. (a)Derivativeofthemicrowaveabsorptionspectrum dP/dH\nat frequencyω/(2π)=9.22 GHz for different anglesϕbetween the\nmicrowavefieldandtheexternalmagneticfieldfor Hs/(4πMs)=0.5,\nω/(4πγMs)=0.35,dN/λ=0.1,dF/λ=0.3α0=α1=0.02, and\ngr=4. The experimental data have been adopted from Ref. 31.\n(b) Computed linewidths of the A and O modes of a Co |Cu|Co spin\nvalve (dashed line) compared with experiments on a Co |Cu multi-\nlayer (solidline).30\npowerP=2/angbracketlefth(t)·∂tm(t)/angbracketrightbecomesinourmodel\nP=1\n4γ2Ms(Hs+4πMs)∆A\n(ω−ωA)2+(∆A/2)2h2\nxsin2θ\n+1\n4γ2Ms(4πMs)∆O\n(ω−ωO)2+(∆O/2)2h2\nycos2θ. (27)\nPdepends sensitively on the character of the resonance, the\npolarization of the microwave, and the strength of the ap-\nplied magnetic field. In Figure 5(a) we plot the normalized\nderivative of the microwave absorption spectra dP/(P0dH)\nat different anglesϕbetween the microwave field h(t) and\nthe external magnetic field H, where P0=γMsh2and\nh(t)=h(sinϕ,cosϕ,0)eiωt. Here we use the experimen-\ntal values Hs=5kOe, 4πMs=10kOe,dN=1nm,\ndF=3nm, and microwave frequency ω/(2π)=9.22GHz\nas found for a symmetric Co |Ru|Co trilayer.31λ=20nm for7\nRu,α0=α1=0.02, andgr=4 is adopted (correspond-\ning toGr=2×1015Ω−1m−2).Whenh(t) is perpendicularto\nH(ϕ=90◦), only the A mode is excited by the transverse\n(δmx,δmz) component. When h(t) is parallel to H(ϕ=0◦),\nthe O mode couples to the microwave field by the longitudi-\nnalδmycomponent. For intermediate angles ( ϕ=20◦), both\nmodes are excited at resonance. We observe that the opti-\ncal mode signal is broader than the acoustic one, as calcu-\nlated. The theoretical resonance linewidths of the A and O\nmodes as well as the absorption power as a function of mi-\ncrowave polarization reproduce the experimental results f or\nCo(3.2nm)|Ru(0.95nm)|Co(3.2nm)well.31\nFigure 5(b) shows the calculated linewidths of A and\nO modes as a function of an applied magnetic field for a\nCo(1nm)|Cu(1nm)|Co(1nm) spin valve. The experimental\nvaluesλ=200nm andρ=2.5µΩcm for Cu,α0=0.01\nand 4πMs=15kOe for Co, and gr=5 (corresponding to\nGr=1015Ω−1m−2) for the interface have been adopted.57\nWe partially reproduce the experimental data for magnetic\nmultilayers; for the weak-field broadenings of the observed\nlinewidthsagreementisevenquantitative. Theremainingd is-\ncrepanciesintheappliedmagneticfielddependencemightre -\nflect exchange-dipolar49and/or multilayer30spin waves be-\nyondourspinvalvemodelinthe macrospinapproximation.\nVI. CONCLUSIONS\nIn summary, we modelled the magnetization dynamics\nin antiferromagnetically exchange-coupled spin valves as a\nmodel for synthetic antiferromagnets. We derivethe Landau -\nLifshitz-Gilbert equations for the coupled magnetization s in-\ncluding the spin transfer torques by spin pumping based on\nthe spin diffusion model with quantum mechanical boundary\nconditions. We obtain analytic expressionsfor the linewid ths\nof magnetic resonance modes for magnetizations canted byapplied magneticfields and achieve goodagreementwith ex-\nperiments. We findthatthelinewidthsstronglydependonthe\ntype of resonance mode (acoustic and optical) as well as the\nstrength of magnetic fields. The magnetic resonance spectra\nreveal complex magnetization dynamics far beyond a simple\nprecessionevenin the linear responseregime. Our calculat ed\nresults compare favorably with experiments, thereby provi ng\ntheimportanceofdynamicspincurrentsinthesedevices. Ou r\nmodel calculation paves the way for the theoretical design o f\nsyntheticAFMmaterialthatisexpectedtoplayaroleinnext -\ngenerationspin-baseddata-storageandinformationtechn olo-\ngies.\nVII. ACKNOWLEDGMENTS\nTheauthorsthanksK.Tanaka,T.Moriyama,T.Ono,T.Ya-\nmamoto, T. Seki, and K. Takanashi for valuable discussions\nand collaborations. This work was supported by Grants-in-\nAidforScientificResearch(GrantNos. 22540346,25247056,\n25220910,268063)fromtheJSPS,FOM(StichtingvoorFun-\ndamenteel Onderzoek der Materie), the ICC-IMR, EU-FET\nGrant InSpin 612759, and DFG Priority Programme 1538\n“Spin-CaloricTransport”(BA 2954 /2).\nAppendixA: CoupledLandau-Lifshitz-Gilbertequationsin\nnoncollinearspinvalves\nBoth magnets and interfaces in our NM |F|NM spin valves\nare assumed to be identical with saturation magnetization Ms\nandGrthe real part of the spin-mixing conductance per unit\narea (vanishing imaginary part). When both magnetizations\nare allowed to precess as sketched in Fig. 1 (a), the LLG\nequationsexpandedtoincludeadditionalspin-pumpandspi n-\ntransfertorquesread\n∂mi\n∂t=−γmi×Heffi+α0imi×∂mi\n∂t\n+αSPi/bracketleftBigg\nmi×∂mi\n∂t−ηmj×∂mj\n∂t+η/parenleftBigg\nmi·mj×∂mj\n∂t/parenrightBigg\nmi/bracketrightBigg\n+αnc\nSPi(ϕ)mi×/parenleftBig\nmi×mj/parenrightBig\n, (A1)\nαnc\nSPi(ϕ)=αSPiη2\n1−η2(mi·mj)2/bracketleftBigg\nmj·mi×∂mi\n∂t+η/parenleftBigg\nmi·mj×∂mj\n∂t/parenrightBigg\n(mj·mi)/bracketrightBigg\n, (A2)\nwhereγandα0iare the gyromagnetic ratio and the Gilbert\ndamping constant of the isolated ferromagnetic films labele d\nbyiand thickness dFi. Asymmetric spin valves due to the\nthickness differencedFisuppress the cancellation of mutual\nspin-pumpinA-mode,whichmaybeadvantagetodetectboth\nmodesintheexperiment. Thee ffectivemagneticfield\nHeffi=Hi+h(t)+Hdii(t)+Hexj(t) (A3)consistsoftheZeemanfield Hi,amicrowavefield h(t),thedy-\nnamic demagnetization field Hdii(t), and interlayer exchange\nfieldHexj(t). The Gilbert damping torque parameterized\nbyα0igoverns the relaxation towards an equilibrium direc-\ntion. The third term in Eq. (A1) represents the mutual spin\npumping-induced damping-like torques in terms of damping\nparameter8\nαSPi=γ/planckover2pi12Gr\n2e2MsdFiηS\n1−η2, (A4)\nwhere\nη=gr\nsinh(dN/λ)+grcosh(dN/λ)(A5)\nandgr=2λρGrisdimensionless. ThefourthterminEq. (A1)\nis the damping Eq. (A2) that depends on the relative angleϕbetween the magnetizations. When mjis fixed along the\nHidirection, i.e. a spin-sink limit, Eq. (A1) reduces to the\ndynamicstiffnessin spinvalveswithoutanelectricalbias.60\nWhen the magnetizations are noncollinear as in Fig. 1, we\nhave to take into account the additional damping torques de-\nscribedbythe secondtermsin Eqs.(4a) and(4b ,). Inthe bal-\nlistic limit dN/λ→0 and collinear magnetizations, Eq. (A1)\nreduces to the well known LLG equation with dynamic ex-\nchangeinteraction.27,28,38\n1X. Marti, I. Fina, C. Frontera, Jian Liu, P.Wadley, Q. He, R. J .\nPaull, J. D. Clarkson, J. Kudrnovsk´ y, I. Turek, J. Kuneˇ s, D . Yi,\nJ-H. Chu, C. T. Nelson, L. You, E. Arenholz, S. Salahuddin, J.\nFontcuberta, T. Jungwirth, and R. Ramesh, Nat. Mater. 13, 367\n(2014).\n2B. G. Park, J. Wunderlich, X. Mart´ ı, V. Hol´ y, Y. Kurosaki, M .\nYamada, H. Yamamoto, A. Nishide, J. Hayakawa, H. Takahashi,\nA.B. Shick,and T.Jungwirth, Nat.Mater. 10, 347(2011).\n3P.M. Haney, D.Waldron, R.A. Duine, A. S.N´ unez, H. Guo, and\nA.H. MacDonald, Phys.Rev. B 75, 174428 (2007).\n4Z.Wei, A. Sharma, A. S. Nunez, P. M. Haney, R. A. Duine, J.\nBass,A.H.MacDonald,andM.Tsoi,Phys.Rev.Lett. 98,116603\n(2007).\n5S.Urazhdin andN. Anthony, Phys.Rev. Lett. 99, 046602 (2007).\n6Y.Xu,S.Wang, andK.Xia,Phys.Rev. Lett. 100,226602 (2008).\n7K. M. D. Hals, Y. Tserkovnyak, and A. Brataas, Phys. Rev. Lett .\n106, 107206 (2011).\n8H. V. Gomonay, R. V. Kunitsyn, and V. M. Loktev, Phys. Rev. B\n85, 134446 (2012).\n9C. Hahn, G. de Loubens, V. V. Naletov, J. B. Youssef, O. Klein,\nand M. Viret,Eur.Phys.Lett. 108, 57005 (2014).\n10H.Wang,C.Du,P.C.Hammel,andF.Yang,Phys.Rev.Lett. 113,\n097202 (2014).\n11T. Moriyama, M. Nagata, K. Tanaka, K. Kim, H. Almasi,\nW.Wang,and T.Ono, ArXive-prints (2014), arXiv:1411.4100 .\n12R.Duine, Nat Mater 10, 344 (2011).\n13A. H. MacDonald and M. Tsoi, Phil. Trans. Royal Soc. A 369,\n3098 (2011).\n14A. V. Kimel, A. Kirilyuk, P. A. Usachev, R. V. Pisarev, A. M.\nBalbashov, and Th.Rasing, Nature (London) 435, 655 (2005).\n15T. Satoh, S.-J. Cho, R. Iida, T. Shimura, K. Kuroda, H. Ueda, Y .\nUeda, B. A. Ivanov, F. Nori, and M. Fiebig, Phys. Rev. Lett. 105,\n077402 (2010).\n16S. Wienholdt, D. Hinzke, and U. Nowak, Phys. Rev. Lett. 108,\n247207 (2012).\n17P.Gr¨ unberg,R.Schreiber,Y.Pang,M.B.Brodsky,andH.Sow ers,\nPhys.Rev. Lett. 57, 2442 (1986).\n18P.Gr¨ unberg, D.E.B¨ urgler,H.Dassow, A.D.Rata,C.M.Schn ei-\nder, Acta Materialia 55, 1171 (2007).\n19S. S. P. Parkin, X. Jiang, C. Kaiser, A. Panchula, K. Roche, an d\nM. Samant,Proc. IEEE 91, 661 (2003).\n20M. N. Baibich, J. M. Broto, A. Fert, F. Nguyen Van Dau, F.\nPetroff, P. Etienne, G. Creuzet, A. Friederich, and J. Chazelas,\nPhys.Rev. Lett. 61, 2472 (1988).\n21G. Binasch, P. Gr¨ unberg, F. Saurenbach, and W. Zinn, Phys. R ev.\nB39, 4828 (1989).\n22A.Brataas, A.D. Kent,and H.Ohno, Nat.Mater. 11, 372(2012).23E. Chen, D. Apalkov, Z. Diao, A. Driskill-Smith, D. Druist, D .\nLottis, V. Nikitin, X. Tang, S. Watts, S. Wang, S. Wolf, A. W.\nGhosh, J. Lu, S. J. Poon, M. Stan, W. Butler, S. Gupta, C. K. A.\nMewes, T. Mewes, and P. Visscher, IEEE Trans. Magn. 46, 1873\n(2010).\n24D. C. Ralph and M. D. Stiles, J. Magn. Magn. Mater. 320, 1190\n(2008).\n25M. J. Carey, N. Smith, S. Maat, and J. R. Childress, Appl. Phys .\nLett.93, 102509 (2008).\n26Y. Tserkovnyak, A. Brataas, G. E. W. Bauer, Phys. Rev. Lett 88,\n117601 (2002).\n27Y. Tserkovnyak, A. Brataas, G. E. W. Bauer, and B. I. Halperin ,\nRev. Mod. Phys. 77, 1375 (2005).\n28B. Heinrich, Y. Tserkovnyak, G. Woltersdorf, A. Brataas, R. Ur-\nban, and G.E.W.Bauer, Phys.Rev. Lett. 90, 187601 (2003).\n29S.Takahashi, Appl. Phys.Lett. 104, 052407 (2014).\n30K. Tanaka, T. Moriyama, M. Nagata, T. Seki, K. Takanashi, S.\nTakahashi, and T.Ono Appl. Phys.Express 7, 063010 (2014).\n31Z. Zhang, L. Zhou, P. E. Wigen, and K. Ounadjela, Phys. Rev.\nLett.73, 336 (1994).\n32M. Belmeguenai, T. Martin, G. Woltersdorf, M. Maier, and G.\nBayreuther, Phys. Rev. B 76, 104414 (2007).\n33H. Skarsvåg, G. E. W. Bauer, and A. Brataas, Phys. Rev. B 90,\n054401 (2014)\n34R.L.Stamps, Phys.Rev. B 49, 339 (1994).\n35S. O. Demokritov, B. Hillebrands, and A. N. Slavin, Phys. Rep .\n348, 441 (2001).\n36J. F. Cochran, J. Rudd, W. B. Muir, B. Heinrich, and Z. Celinsk i,\nPhys.Rev. B 42, 508 (1990).\n37B. K. Kuanr, M. Buchmeier, D. E. B¨ urgler, and P. Gr¨ unberg, J .\nAppl. Phys. 91, 7209 (2002).\n38A. A. Timopheev, Yu. G. Pogorelov, S. Cardoso, P. P. Freitas, G.\nN.Kakazei, and N.A. Sobolev, Phys.Rev. B 89, 144410 (2014).\n39T.Taniguchi and H.Imamura Phys. Rev. B 76, 092402 (2007).\n40Y. Zhou, J. Xiao, G.E.W. Bauer, and F. C. Zhang, Phys. Rev. B\n87, 020409(R) (2013).\n41A. Layadi and J. O. Artman, J. Magn. Magn. Mate. 176, 175\n(1997).\n42B. K. Kuanr, M. Buchmeier, R. R. Gareev, D. E. B¨ urgler, R.\nSchreiber, andP.Gr¨ unberg, J. Appl.Phys. 93, 3427 (2003).\n43R. F.L.Evans, T. A. Ostler,R. W.Chantrell, I. Radu, and T. Ra s-\ning, Appl. Phys.Lett. 104, 082410 (2014).\n44D. E. Gonzalez-Chavez, R. Dutra, W. O. Rosa, T. L. Marcondes,\nA.Mello, and R.L.Sommer, Phys.Rev. B 88, 104431 (2013).\n45X. M. Liu, Hoa T. Nguyen, J. Ding, M. G. Cottam, and A. O.\nAdeyeye, Phys.Rev. B 90, 064428 (2014).\n46J. J. Krebs, P.Lubitz, A.Chaiken, and G.A. Prinz,J. Appl. Ph ys.\n67, 5920 (1990).9\n47T. Chiba, G. E. W. Bauer, and S. Takahashi, Phys. Rev. Appl. 2,\n034003 (2014)\n48T. Chiba, M. Schreier, G. E. W. Bauer, and S. Takahashi, J. App l.\nPhys.117, 17C715 (2015).\n49H. Skarsvåg, Andr´ e Kapelrud, and A. Brataas, Phys. Rev. B 90,\n094418 (2014).\n50Z.Yuan, K.M. D.Hals, Y.Liu, A.A.Starikov, A.Brataas, andP .\nJ. Kelly,Phys. Rev. Lett. 113, 266603 (2014).\n51H.Jiao andG.E.W.Bauer, Phys.Rev. Lett. 110, 217602 (2013).\n52X. Jia, K. Liu, K. Xia, and G. E. W. Bauer, Europhys. Lett. 96,\n17005 (2011).\n53Z. Zhang and P. E. Wigen, High frequency processes in magnetic\nmaterials ,editedbyG.SrinivasanandA.N.Slavin(WorldScien-\ntific,Singapore, 1995).\n54K. Eid, R. Fonck, M. AlHaj Darwish, W. P. Pratt Jr., and J. Bass ,\nJ. Appl.Phys. 91, 8102 (2002).55S. Yakata, Y. Ando, T. Miyazaki, and S. Mizukami, Jpn.. J. App l.\nPhys.45, 5A (2006).\n56X.Jia,Y.Li,K.Xia,andG.E.W.Bauer,Phys.Rev.B 84,134403\n(2011).\n57K.Xia,P.J.Kelly,G.E.W.Bauer,A.Brataas,andI.Turek,Ph ys.\nRev. B65, 220401 (2002).\n58S. Mizukami, D. Watanabe, M. Oogane, Y. Ando, Y. Miura, M.\nShirai,andT. Miyazaki, J. Appl. Phys. 105, 07D306 (2009).\n59J. M. D. Coey, Magnetism and magnetic materials (Cambridge\nUniversityPress,Cambridge, 2010).\n60Y. Tserkovnyak, A. Brataas, G. E. W. Bauer, Phys. Rev. B 67(R),\n140404(2003)"    },    {        "title": "1505.00522v2.High_topological_number_magnetic_skyrmions_and_topologically_protected_dissipative_structure.pdf",        "content": "High-topological-number magnetic skyrmions and topologically protected dissipative structure\nXichao Zhang,1, 2Yan Zhou,1, 2,\u0003and Motohiko Ezawa3,y\n1Department of Physics, University of Hong Kong, Hong Kong, China\n2School of Electronics Science and Engineering, Nanjing University, Nanjing 210093, China\n3Department of Applied Physics, University of Tokyo, Hongo 7-3-1, 113-8656, Japan\n(Dated: January 6, 2016)\nThe magnetic skyrmion with the topological number of unity ( Q= 1) is a well-known nanometric swirling\nspin structure in the nonlinear \u001bmodel with the Dzyaloshinskii-Moriya interaction. Here, we show that magnetic\nskyrmion with the topological number of two ( Q= 2) can be created and stabilized by applying vertical spin-\npolarized current though it cannot exist as a static stable excitation. Magnetic skyrmion with Q= 2 is a\nnonequilibrium dynamic object, subsisting on a balance between the energy injection from the current and the\nenergy dissipation by the Gilbert damping. Once it is created, it becomes a topologically protected object against\nfluctuations of various variables including the injected current itself. Hence, we may call it a topologically\nprotected dissipative structure. We also elucidate the nucleation and destruction mechanisms of the magnetic\nskyrmion with Q= 2 by studying the evolutions of the magnetization distribution, the topological charge\ndensity as well as the energy density. Our results will be useful for the study of the nontrivial topology of\nmagnetic skyrmions with higher topological numbers.\nPACS numbers: 75.70.-i, 75.78.-n, 85.70.-w, 85.75.-d\nI. INTRODUCTION\nThere is a long history of skyrmions from the particle\nphysics to the condensed matter physics [1–4]. Originally,\nSkyrme introduced \"skyrmions\" in the three-dimensional (3D)\nspace to describe nucleons as elementary particles possess-\ning nontrivial topological numbers [1]. Subsequently, Belavin\nand Polyakov (BP) applied the concept to the two-dimensional\n(2D) ferromagnetic (FM) system and predicted a magnetic ex-\ncitation carrying a nontrivial topological number, that is, the\nPontryagin number [2]. The BP-skyrmion is an exact solution\nof the nonlinear \u001bmodel. However this solution has a scale\ninvariance, and hence the BP-skyrmion has no definite radius.\nIt is necessary for a dynamically stable and physical skyrmion\nto have a definite radius. Consequently, a skyrmion is char-\nacterized by two types of stabilities: the topological stability\nbased on the conservation of a nontrivial Pontryagin number,\nand the dynamical stability having a finite radius.\nSome interactions are required in order to break the scale\ninvariance. For instance, the dipole-dipole interaction breaks\nthe scale invariance. However, since it is a long-distance\nforce, the skyrmion size becomes very large in general [5, 6].\nRecently, the Dzyaloshinskii-Moriya interaction (DMI) has\nattracted much attention to provide the skyrmion with a finite\nradius, where the resultant radius is of the order of nanome-\nters [7–16]. It is called the magnetic skyrmion. Magnetic\nskyrmions might be suitable for building next-generation non-\nvolatile memory devices based on their topological stabil-\nity [10]. Furthermore, it has also been proposed to utilize them\nin logic computing [17].\nStrictly speaking, the topological number is defined only\nin the continuum field theory with an infinitely large space.\n\u0003yanzhou@hku.hk\nyezawa@ap.t.u-tokyo.ac.jpAlthough the underlying system has a finite size and a lat-\ntice structure in the condensed matter physics, the topological\nnumber is well-defined, provided that the skyrmion spin tex-\nture is sufficiently smooth and sufficiently far away from the\nedge. Thus it is an intriguing concept in the condensed matter\nphysics possessing the aspect of the continuum theory and that\nof the lattice theory. We can actually leverage these properties\nfor practical applications. It is possible to create or destroy\nmagnetic skyrmions which are topologically stable [5, 6, 18–\n23].\nAs far as the topological analysis is concerned, a magnetic\nskyrmion with any topological number Qis possible. Let\nus call such a skyrmion with Q\u00152a high-Q skyrmion. It\nwill be realized when the in-plane component of the spin ro-\ntates by 2\u0019Qand the skyrmion acquires a high helicity Q.\nHowever, only the magnetic skyrmion with Q= 1 has so far\nbeen realized. This is because that a static magnetic skyrmion\nwithQ\u00152is unstable since the DMI cannot prevent it from\nshrinking to a point, which we shall prove later. In this paper,\nemploying a dynamical breaking of the scale invariance [21],\nwe create a high-Q skyrmion with Q= 2by applying a spin-\npolarized current perpendicular to the FM nanodisk with the\nDMI. We investigate in details how a high-Q skyrmion with\nQ= 2is created from a magnetic bubble with Q= 0through\nsuccessive creations of two Bloch points.\nOnce the spin texture of a high-Q skyrmion becomes suf-\nficiently smooth with respect to the lattice spacing, the topo-\nlogical protection becomes active. Namely, the resultant spin\ntexture is topologically robust against fluctuations of various\nvariables including the injected current itself. This is due to\nthe fact that the topological number cannot change continu-\nously from its quantized value. However, when the current is\nswitched off, the high-Q skyrmion quickly shrinks to the order\nof the lattice scale and dissipates. The equilibrium and dissi-\npation processes are well-described in terms of the Rayleigh\ndissipation function composed of the energy injection and dis-\nsipation terms [24]. The dissipation is found to spread all overarXiv:1505.00522v2  [cond-mat.mes-hall]  5 Jan 20162\nFIG. 1. (Color online) The ordinary skyrmion ( Q= 1) and the high-Q skyrmion ( Q= 2). (a) Our system consists of a 1-nm-thick FM\nnanodisk with a thickness of 150 nm, where the spin current polarized along the \u0000z-direction is injected into the central 30-nm-diameter\nregion. The out-of-plane magnetization mz, the in-plane magnetization ( mx;my) and the topological charge density qare illustrated for the\nskyrmion (Q= 1) and the high-Q skyrmion ( Q= 2). (b) The topological number Qas a function of time t. We create a skyrmion with\nQ= 1(Q= 2) by applying an external magnetic field perpendicular to the nanodisk plane pointing along the +z-direction with an amplitude\nofBz= 50 mT (Bz= 250 mT), as depicted in blue (red). The DMI constant is D= 2 mJ m\u00002. The spin current with a current density\nofj= 3\u00021012A m\u00002is switched on and off at t= 0 andt= 5 ns. When the DMI is set to be D= 0 mJ m\u00002, no skyrmion is formed\n(Q= 0), as depicted in green. A skyrmion with Q= 1remains stable as it is even after the spin current is switched off when D= 4mJ m\u00002\nandBz= 0 mT, as depicted in cyan, which is consistent with the previous result shown in Ref. 10. (c) A magnetic bubble with Q= 0 has a\npair (two pairs) of blue and red areas, when it develops into a skyrmion with Q= 1 (Q= 2). The cones represent the magnetization, while\nthe color denotes the topological charge density.\nlike a burst at the moment of the skyrmion generation and de-\nstruction [25].\nThe high-Q skyrmion is a topologically protected dissipa-\ntive structure. One might think that the concept is a self-\ncontradictory proposition. However, this is not the case. We\nmay suggest an analogy of the quantum Hall (QH) state. It\nis well-known that the QH state at an integer filling factor is\nrobust against impurities and also against the change of the ex-\nternal magnetic field. Indeed the latter develops QH plateaux.\nThe robustness is due to the fact that the state is protected\nby the conservation of the topological number, that is, the\nChern number. Nonetheless, when the external magnetic field\nis switched off, the topological robustness is lost and the QH\nstate collapses. Clearly, the external magnetic field in the QH\nsystem corresponds to the injected spin current in the topolog-\nically stabilized dissipative structure of this work.\nII. NUMERICAL RESULTS ON THE HIGH-Q SKYRMION\nA. Nucleation of the high-Q skyrmion\nOur system is composed of a FM nanodisk and a spin-\npolarized current injection region with a radius of rc, as il-\nlustrated in Fig. 1(a) (see the Appendix for modeling details).\nThe development of the topological number Qis shown in\nFig. 1(b). Soon after the spin current is injected, Qsuddenly\nincreases to 1or2from 0, and remains stable, in the pres-\nence of the DMI. On the other hand, when the spin current\nis switched off, Qdecreases to 0. It should be noted that the\nhigh-Q skyrmion cannot be created dynamically without theDMI.\nWe are interested in the process how a magnetic bubble\nwithQ= 0 is converted into a magnetic skyrmion with\nQ6= 0. Upon the application of the spin current, there is a\nlarge energy injected into the core through the spin transfer\ntorque (STT). The spins are forced to reverse within the core\nupon the spin-polarized current injection. Due to the DMI, the\nspins are twisted around the core. The topological number is\nzero for such a state. This is a magnetic bubble with Q= 0.\nWe point out that the seed of a magnetic skyrmion is al-\nready present in the magnetic bubble. In Fig. 1(c), we show\nthe densities of the in-plane components of the magnetization\n(mx(x);my(x)) and the topological charge density q(x)of a\nmagnetic bubble before the nucleation to a magnetic skyrmion\nwithQ= 1 or2. We clearly observe a pair (two pairs) of\nblue and red areas indicating negative and positive topologi-\ncal charge densities, respectively.\nIn Fig. 2(a), we show the time evolution of the topological\nnumber, the total energy, the DMI energy, the average mag-\nnetization (mx,my,mz), and the Rayleigh dissipation func-\ntionWfor a high-Q skyrmion with Q= 2 (see Ref. 26 for\nSupplementary Movie 1). The selected top-views are shown\nin Fig. 2(b). The spin component mzmeasures the size of\nthe skyrmion, while mxandmycontribute to the topologi-\ncal charge density. First, mzstarts to decrease, implying that\nthe spins are inverted in the disk region. However, the topo-\nlogical number remains zero, since the in-plane spin compo-\nnents point along the same direction, as shown in Fig. 2(b) at\nt= 0:2ns.\nThe Rayleigh dissipation function gives us a vivid informa-\ntion on how the dissipation occurs in the dissipative system.3\nFIG. 2. (Color online) Time evolution of the high-Q skyrmion ( Q= 2). (a) Time evolution of the total energy Etotal, the DMI energy EDMI,\nthe topological number Q, the in-plane ( mx;my) and out-of-plane mzcomponents of magnetization averaged over the simulation area, and\nthe dissipation functions W. A high-Q skyrmion with Q= 2 is created nearly at t= 0:38ns. The DMI constant D= 2 mJ m\u00002. The spin\ncurrent density j= 3\u00021012A m\u00002. The external magnetic field Bz= 250 mT. (b) Top-views of the magnetization distributions mx,my\nandmzof the FM nanodisk, the corresponding topological charge density distribution qand the Rayleigh dissipation function Wat selected\ntimes. The green circle indicates the spin current injection region. The nucleation of a high-Q skyrmion with Q= 2occurs nearly at t= 0:38\nns, where the dissipation function spreads all over the FM nanodisk, implying that the spin wave propagates. The DMI is turned off at t= 10\nns. It is remarkable that the high-Q skyrmion remains stable even if the DMI is switched off, which demonstrates the topological protection\nagainst the change of a variable, that is, the DMI.\nIn the present system, the energy is injected into the core and\ndissipated in its outer side steadily before and after the nu-\ncleation of a magnetic skyrmion. However, the dissipation\nspreads around all over the FM nanodisk like a burst at the\ntransition moment of the topological number from Q= 0 to\nQ= 2(see Fig. 2(b) at t= 0:38ns).\nWe show a close-up of the change of the topological num-\nberQaroundt= 0:38ns in Fig. 3(a). Clearly, there are two\nsuccessive jumps of Qas0!1!2. We also show the\ntopological charge density, the energy density, and the spin\ndistribution around t= 0:38ns in Fig. 4.\nBased on these we obtain the following picture of the nu-\ncleation process. We focus on the case where there are two\npairs of blue and red areas in the domain boundary region of\nthe magnetic bubble (Fig. 1(c)). Both the topological charge\ndensityq(x)and the energy density \u000f(x)are large in these\nareas. In particular, there is a chance that \u000f(x)becomes large\nin a lattice-scale area so that the area has almost Q=\u00001.\nThis happens when two spins are antiparallel with one down-\nspin site between them, at t= 372:2ps in Fig. 4. A single\nor a few spins make a large rotation in order to decrease the\nenergy of the area. Indeed, the antiparallel spins become par-\nallel att= 372:8ps in Fig. 4. In this process the topological\nnumberQ=\u00001is lost, which is possible in the lattice theory.\nThis phenomenon would be viewed as a generation of a Bloch\npoint in the continuum theory. This makes clearer the role of\nthe Bloch point presented by Sampaio et al. in Ref. 10. Thereexists still a pair of blue and red areas. Now, there is a chance\nthat another lattice-scale area develops which has Q=\u00001in\nthe spin texture with Q= 1. (Note that there are two spins\nantiparallel at t= 406 ps, which become parallel at t= 407:6\nps.) It corresponds to the emergence of another Bloch point.\nAs a result, it turns out that two Bloch points emerge suc-\ncessively in a single magnetic bubble. When the spin texture\nbecomes sufficiently smooth, it yields a high-Q skyrmion with\nQ= 2.\nWe have also numerically observed that a high-Q skyrmion\ncan be successfully created in a wide range of parameters as\nwell as the spin current injection size, as shown for instance\nin Fig. 5. Fig. 5(a) shows the phase diagram of the high-Q\nskyrmion creation with respect to the external magnetic field\nBzand time, where the spin current is injected into a circle\nregion with a radius of rc= 15 nm. Fig. 5(b) shows the phase\ndiagram of the high-Q skyrmion creation with respect to the\nspin current injection region radius rcand the time t, where\nthe spin current is injected into a circle region with a radius of\nrc.\nMoreover, as shown in Fig. 6, the high-Q skyrmion is cre-\nated and maintained even at finite temperature. Fig. 6(a) il-\nlustrates the phase diagram of the high-Q skyrmion creation\nwith respect to the temperature T and time. Fig. 6(b) shows\nthe topological number Qas a function of the time at T = 0\nK and T = 100 K, where the high-skyrmion with Q= 2 is\ncreated shortly after the spin current is switched on. The struc-4\nFIG. 3. (Color online) (a) Nucleation and (b) annihilation pro-\ncesses of the high-Q skyrmion with Q= 2. A sudden change of\nthe topological number Qoccurs twice successively when the high-Q\nskyrmion with Q= 2is created or destroyed. The topological charge\ndensityq(x), the energy density \u000f(x), and the spin-component distri-\nbutionmz(x)of the state indexed by (t;Q)in (a) and (b) are shown\nin Fig. 4 and Fig. 9, respectively.\nture of the high-Q skyrmion is deformed at finite temperature\n(see Fig. 6(b) insets). Its topological number is almost 2but\nfluctuates slightly because the continuity of the spin texture of\na deformed skyrmion is broken at finite temperature.\nB. Evolution of the high-Q skyrmion\nThe continuity of the spin texture is recovered since a\nsmooth texture has a lower energy. Then the topological pro-\ntection becomes active. The dissipation function decreases\nrapidly and oscillates around zero. The system is relaxed to\na steady state around t= 5 ns. Both the topological charge\ndensity and the Rayleigh dissipation function are almost zero\noutside the domain wall encircling the high-Q skyrmion (see\nFig. 2).\nOscillations in various variables occur due to the DMI. It is\ninstructive to switch off the DMI in numerical simulations. In\nFig. 2, we switch off the DMI at t= 10 ns. First of all, the\nhigh-Q skyrmion remains stable. Second, the oscillations in\nthe energy, the magnetization components, and the dissipation\nfunctions disappear. After the relaxation at t= 20 ns, the in-\nplane magnetization components and the dissipation functions\nbecomes exactly zero (see Ref. 26 for Supplementary Movie\n2). The spin texture is described precisely by the magnetiza-\ntion components of the domain wall.We show how the balance holds between the energy injec-\ntion from the spin current and the dissipation by the Gilbert\ndamping in Fig. 7. The energy injection occurs in the vicinity\nof the edge of the current injection region ( r.rc) while the\ndissipation occurs mostly within the domain wall encircling\nit (r\u0018r0) together with energy flow from the inner to outer\nregions. This is due to the fact that spins precess within the\ndomain wall.\nC. Topological protection of the high-Q skyrmion\nIt is important to point out that this energy balance takes\nplace automatically so as to keep the topological number un-\nchanged. To check this, we change the spin current stepwise,\nwhich is shown in Fig. 8. The topological number remains to\nbe2when the spin current intensity changes even more than\ntwice. The energy injection due to the STT becomes larger\nas the spin current density increases. Accordingly, the energy\ndissipation due to the Gilbert damping increases. As a result,\nthe Rayleigh dissipation function remains zero in average al-\nthough it is oscillating. The total energy increases stepwise\nbut remains almost constant for each current strength. The to-\ntalmzdecreases as the spin current density increases, which\nimplies that the magnetic skyrmion expands for larger cur-\nrent density. We have shown that the magnetic skyrmion is\ntopologically robust against a considerable change of the spin\ncurrent injection.\nAs we have stated, once it is created, the high-Q skyrmion\nremains stable even if the DMI is switched off (see Fig. 2).\nFurthermore, it is stable against the fluctuations of various\nvariables. This is because a small change can induce only a\nsmall change of the topological number Q, but this is impos-\nsible sinceQis a quantized quantity. This property is called\nthe topological protection.\nD. Destruction of the high-Q skyrmion\nWhen the spin current is switched off at t= 5ns, the topo-\nlogical number remains as Q= 2 untilt\u00186:4ns and sud-\ndenly decreases to Q= 0, as shown in Fig. 1(b). The radius\nof the magnetic skyrmion shrinks since the skyrmion core is\nfixed by the spin current against the shrinking force due to\nthe kinetic energy as well as the external magnetic field (see\nRef. 26 for Supplementary Movie 3).\nA close examination shows that the collapse of the topo-\nlogical number occurs in two steps as Q= 2!1!0as in\nFig. 3(b). We may understand how the destruction of a mag-\nnetic skyrmion with Q= 2 occurs by investigating the time\nevolution of the topological charge density, the energy den-\nsity, and the spin distribution around t= 6:4ns as shown in\nFig. 9.\nIn the first step the energy density is localized almost on one\nlattice site, where a Bloch point is generated and the topolog-\nical number changes from Q= 2 toQ= 1. The size of the\nmagnetic skyrmion with Q= 1shrinks almost to the order of\nthe lattice site.5\nFIG. 4. (Color online) Nucleation process of the high-Q skyrmion with Q= 2. Snapshots of the topological charge density q(x), the energy\ndensity\u000f(x), the spin-component distribution mz(x), and its close up at sequential times. The DMI constant D= 2 mJ m\u00002. The spin\ncurrent density j= 3\u00021012A m\u00002. The external magnetic field Bz= 250 mT. In the simulation, each cell corresponds to one spin, and\nthe cell size is 1:5nm\u00021:5nm\u00021nm. In the spin distribution panels, each arrow stands for four spins, while it stands for one spin in the\ninsets. The nucleation process of the high-Q skyrmion with Q= 2 is found to occur in two steps. First, it starts when a high-energy-density\npart is localized to a lattice-scale area, which possesses almost Q=\u00001. A few spins rotate by large angles in this area, making the topological\nnumber of the area almost zero. The resultant spin texture has Q= 1. Second, a similar phenomenon occurs, yielding the high-Q skyrmion\nwithQ= 2after the relaxation. See Ref. 26 for Supplementary Movie 4.\nThe second step has some new features. The topologi-\ncal stability of the magnetic skyrmion is guaranteed by the\nfact that the core spin points in the direction opposite to the\nFM background. However, such a core spin disappears at\nt= 6407 ps since there is no lattice site at the core spin.\nAs a result, the spin texture becomes a vortex structure. Ac-\ncordingly, all the spins point along upward direction and the\nmagnetic skyrmion disappears. This is possible since the sys-\ntem is on the lattice and such a transition never happens for\nthe continuum system. In this process, the spiral spin wave is\ngenerated, as seen obviously in the topological density as well\nas the energy density at t= 6410 ps in Fig. 9.\nA comment is in order with respect to the stability of a mag-\nnetic skyrmion with Q= 1when the spin current is turned off\n(see Fig. 1(b)). The stability diagram has been explored in\nRef. 10 in the absence of the spin current. For instance, it is\nstable forD= 4 mJ m\u00002andBz= 0 mT, while it is unsta-\nble forD= 2 mJ m\u00002andBz= 50 mT. Indeed, when the\nspin current is switched off, a magnetic skyrmion with Q= 1\nremains stable or is destroyed according to these parameter\nchoices, as depicted in cyan or in blue in Fig. 1(b).III. THEORETICAL ANALYSIS OF THE HIGH-Q\nSKYRMION\nA. Hamiltonian\nThe Hamiltonian of the system is given by\nH=\u0000JX\nhi;jimi\u0001mj+X\nhi;jiD\u0001(mi\u0002mj)\n+KX\ni[1\u0000(mz\ni)2] +BzX\nimz\ni+HDDI;(1)\nwhere mirepresents the local magnetic moment orientation\nnormalized asjmij= 1 at the sitei, andhi;jiruns over all\nthe nearest neighbor sites in the FM layer. The first term rep-\nresents the FM exchange interaction with the FM exchange\nstiffnessJ. The second term represents the DMI with the\nDMI vector D. The third term represents the perpendicular\nmagnetic anisotropy (PMA) with the anisotropic constant K.\nThe fourth term represents the Zeeman interaction. The fifth\ntermHDDIrepresents the dipole-dipole interaction. Although\nwe have included the dipole-dipole interactions in all numer-\nical calculations, the effect is negligible since the size of a\nmagnetic skyrmion is of the order of nanometers.6\nFIG. 5. (Color online) (a) Phase diagram of the high-Q skyrmion\ncreation with respect to the external magnetic field Bzand timet.\nThe spin current density j= 3\u00021012A m\u00002, which is injected\ninto a circle region with a radius of rc= 15 nm. The DMI constant\nD= 2mJ m\u00002. (b) Phase diagram of the high-Q skyrmion creation\nwith respect to the spin current injection region radius rcand time\nt. The spin current density j= 3\u00021012A m\u00002, which is injected\ninto a circle region with a radius of rc. The DMI constant D= 2\nmJ m\u00002. The external magnetic field Bz= 250 mT. The color scale\nindicates the topological number Q.\nB. Topological number\nThe classical field m(x)is introduced for the spin texture\nin the FM system by considering the zero limit of the lattice\nconstant, that is, a!0. The ground-state spin texture is\nm= (0;0;1). We employ the continuum theory when we\nmake an analytic study of the system.\nA magnetic skyrmion is a spin texture which has a topolog-\nical number. Spins swirl continuously around the core, where\nspins point downward, and approach the spin-up state asymp-\ntotically. The magnetic skyrmion is characterized by the topo-\nlogical number known as the Pontryagin number. It is given\nbybQ=R\nd2xq(x)with the density\nq(x) =\u00001\n4\u0019m(x)\u0001(@xm(x)\u0002@ym(x)): (2)\nThe spin configuration of a magnetic skyrmion is\nparametrized as\nmx= cos\u001e(') sin\u0012(r); my= sin\u001e(') sin\u0012(r);\nmz= cos\u0012(r); (3)\nwhere'is the azimuthal angle and ris the radius in the polar\ncoordinate. The topological charge density q(x)is shown to\nbe a total derivative, and hence the topological number is a\nboundary value. It is explicitly calculated as\nbQ=1\n4\u0019[cos\u0012(1)\u0000cos\u0012(0)][\u001e(2\u0019)\u0000\u001e(0)]; (4)\nwhich does not depend on the detailed profile of cos\u0012(r)\nand\u001e('). The boundary conditions cos\u0012(0) =\u00001and\ncos\u0012(1) = 1 are imposed for any skyrmion at the skyrmion\ncenter (r= 0) and at infinity ( r=1). When\u001e=Q'+\u001f,\nthe topological number is Q, where\u001fstands for the helicity.\nFIG. 6. (Color online) (a) Phase diagram of the high-Q skyrmion\ncreation with respect to the temperature T and the time t. The DMI\nconstantD= 2 mJ m\u00002. The spin current density j= 3\u00021012\nA m\u00002. The external magnetic field Bz= 250 mT. The color scale\nindicates the topological number Q. (b) The topological number Q\nas a function of time tat T = 0 K and T = 100 K. The insets show\nthe snapshots of the high-Q skyrmion at T = 0K and T = 100 K.\nQmust be an integer for the single-valuedness. In general, \u0012\nand\u001fare functions of time t. The latter interpolates the Néel-\ntype (\u001f= 0;\u0019) or Bloch-type ( \u001f=\u0019=2;3\u0019=2) skyrmion.\nWe show the spin configurations of the magnetic skyrmions\nwithQ= 1 andQ= 2 in Fig. 1(a). Spins rotate Qtimes as\n'changes from '= 0to'= 2\u0019for the magnetic skyrmion\nwithQ. That is to say, when going around the spin texture\nof the magnetic skyrmion, the in-plane component of the spin\nrotates by 2\u0019Q.\nC. Dzyaloshinskii-Moriya interaction\nThe DMI is the Néel-type or Bloch-type depending on\nwhether it is introduced from the surface or bulk. We take\nthe interface-induced Néel-type DMI,\nHDM=D?Z\nd2x[nzdivn\u0000(n\u0001r)nz]: (5)\nWe substitute the magnetic skyrmion configuration equa-\ntion (3) into the DMI Hamiltonian, and we find\nHDM=D?Z\nrdrd'1\n2rcos [(Q\u00001)'+\u001f]\n\u0002(Qsin 2\u0012(r) + 2r@r\u0012(r)): (6)\nForQ6= 1, by integrating over ', we findHDM= 0. As a\nresult, the DMI does not prevent a static magnetic skyrmion7\nFIG. 7. (Color online) Profile of mzand the Rayleigh dissipation function Wof the high-Q skyrmion with Q= 2 in the presence and\nabsence of the DMI. (a) The radius of the FM nanodisk equals 75nm, and the spin-polarized current is injected into a circle region with a\nradius ofrc= 15 nm. The simulated skyrmion radius r0is equal to 24:55nm, which is defined as the radius of the circle where mz= 0. The\nDMI constant D= 2 mJ m\u00002. The spin current density j= 3\u00021012A m\u00002. The external magnetic field Bz= 250 mT. (b) The form of\nmz(r)is fitted by the domain wall solution equation (14) with the use of \u0015= 4:54nm, which is in good agreement with the theoretical value\n\u0015=p\nJ=K = 4:33nm. The functions W\u000bandWSTTin (a) are also well fitted by the same domain-wall solution.\nfrom shrinking to a point unless Q= 1. Hence, there is no\nstatic magnetic skyrmion stabilized by the DMI for Q6= 1.\nD. Rayleigh dissipation function\nThe system contains an energy injection by the spin-\npolarized current and an energy dissipation by the Gilbert\ndamping. They cannot be analyzed in the framework of the\nHamiltonian formalism, where the energy is a constant of mo-\ntion. It is described by the generalized Lagrangian formalism\nincluding the Rayleigh dissipation function.\nThe Rayleigh dissipation function consists of two terms,\nW=W\u000b+WSTT, the Gilbert damping term [27],\nW\u000b=~\u000b\u0012dm\ndt\u00132\n=~\u000b\u0010\n_\u00122+_\u001e2sin2\u0012\u0011\n; (7)\nwith the Gilbert damping constant \u000b, and the STT term,\nWSTT=~j\rjuz\u0001(_m\u0002m) =\u0000~j\rju_\u001esin2\u0012; (8)\nwhereudescribes the injection of the spin-polarized current,\nu(r) =j~\n\u00160ejP\n2dMSj(r)withj(r)representing the injected\ncurrent and z= (0;0;1). We takeu(r) =u0forr <rcand\nu(r) = 0 forr >rc. We note that W\u000b>0, whileWSTTcan\nbe positive or negative depending on the direction of the spin\ncurrent. We use the first (second) equations in equation (7)\nand equation (8) for numerical (analytical) calculations.The generalized Lagrange equation reads [24]\nd\ndt\u000eL\n\u000e_Q\u0000\u000eL\n\u000eQ=\u0000\u000eW\n\u000e_Q; (9)\nwhereLis the Lagrangian and Qis the generalized coor-\ndinate. By taking masQ, the generalized Lagrange equa-\ntion yields the Landau-Lifshitz-Gilbert-Slonczewski (LLGS)\nequation,\ndm\ndt=\u0000j\rjm\u0002Beff+\u000bm\u0002dm\ndt+j\rjum\u0002(z\u0002m);\n(10)\nwith ~Beff=\u0000@H=@m.\nThe energyEchanges in the presence of the energy injec-\ntion and dissipation, dE=dt =\u00002R\nd2xW6= 0, in general.\nNevertheless, when we take the time average, we should have\nZ\nd2xhWi= 0; (11)\nbecause this is necessary for a dynamically stabilized mag-\nnetic skyrmion. We may call it the weak stationary condition.\nE. Skyrmion solution\nWe substitute equation (3) in the LLGS equation (10),\nwhich leads to a set of two equations for \u0012(t;r;' )and8\nFIG. 8. (Color online) The high-Q skyrmion with Q= 2 under\nstepwise increasing of the spin-polarized current injection. We show\nhow the topological number Q, the total energy Etotal, the averaged\nout-of-plane magnetization mz, and the dissipation functions ( W\u000b,\nWSTT,W) change, when the injected spin current density jis in-\ncreased. The DMI constant D= 2 mJ m\u00002. The external magnetic\nfieldBz= 250 mT. The topological number Qremains as it is when\nwe change the spin current density. The average of Wis zero, which\nimplies the energy is balanced in average. It demonstrates that the\ntopological protection against the change of a variable, that is, the\nspin current injection.\n\u001f(t;r;' ). They are too complicated to solve, reflecting\ncomplicated behaviors revealed by numerical solutions (see\nRef. 26 for Supplementary Movies 1-3). However, when we\nsetD?= 0, simple behaviors have been revealed by numeri-\ncal simulations. Hence, we solve them by setting D?= 0as\nthe unperturbed system.\nWe search for a solution such that _\u0012= 0. By substituting\nequation (3) in the LLGS equation (10), and by setting _\u0012= 0,\nwe obtain\n\u0000J(@2\nr\u0012+@r\u0012\nr) + (JQ2\n2r2+K) sin 2\u0012\n+Bzcos\u0012+1\n\u000bu(r) sin\u0012= 2D?F(r;\u0012;' );(12)\nwhere\nF=\u0000Q\nrsin\u0012cos[(1\u0000Q)'+\u001f]\n\u00001\n\u000b@r\u0012sin[(1\u0000Q)'+\u001f] sin\u0012: (13)\nThe role of the injected spin-polarized current ( u6= 0) is to\nimpose the boundary condition cos\u0012=\u00001atr= 0.\nWhenQ= 1, sinceF=\u0000sin\u0012=r, equation (12) is numer-\nically solvable with respect to \u0012with the boundary condition\ncos\u0012=\u00001atr= 0andcos\u0012= 1atr=1. The azimuthalangle is given by '0(t) = (\ru=\u000bQ )t. The equations of mo-\ntion are well approximated by J@2\nr\u0012=Ksin 2\u0012for anyQ\nasymptotically. This equation has the domain-wall solution,\ncos\u0012= tanhr\u0000r0\n\u0015; (14)\nwith the domain-wall width \u0015=p\nJ=K and the skyrmion\nradiusr0.\nOur major interest is the case of Q6= 1. Since\u0012and'\nare coupled, it is not straightforward to solve for a magnetic\nskyrmion. Let us require the weak stationary condition by\ntaking the time average of equation (13). We find that hFi= 0\nunlessQ= 1. Then, equation (12) is solvable with respect\nto\u0012with the boundary condition cos\u0012=\u00001atr= 0 and\ncos\u0012= 1atr=1.\nThe profile of mz(r)is given by equation (3) and equa-\ntion (14). The Gilbert damping term W\u000b(r)and the STT term\nWSTT(r)are given by equations (7) - (8) together with _\u0012= 0,\n_\u001e=constant and equation (14) or sin2\u0012= sec2[(r\u0000r0)=\u0015].\nThe theory and numerical simulation lead to identical results\nwhich overlap within the precision of numerical simulation as\nshown in Fig. 7(b).\nOn the other hand, when the injected spin-polarized current\nis switched off, that is, u= 0, there is no skyrmion solution\nand we recover the FM ground state cos\u0012= 1.\nIV . CONCLUSIONS\nWe have analyzed the nucleation, the stability, and the de-\nstruction of the high-Q skyrmion with Q= 2in ferromagnets.\nIt is realized when the in-plane component of the spin rotates\nby2\u0019Qand the magnetic skyrmion has acquired a high helic-\nityQ. In the presence of the DMI, although there exist static\nmagnetic skyrmions with Q= 1, there exist no static high-Q\nskyrmions. Nevertheless, a high-Q skyrmion can be created\nand stabilized dynamically by injecting the spin-polarized cur-\nrent. The DMI plays a crucial role in the creation mechanism\nby twisting spins and generating fluctuations of the energy\ndensity and the topological charge density. Indeed, we have\nnumerically verified that a high-Q skyrmion cannot be created\ndynamically without the DMI.\nWe have also observed numerically that a high-Q skyrmion\ncan be created in a wide range of parameters as well as the\nspin current injection size. Furthermore, the high-Q skyrmion\nhas been found to be created and maintained even at finite\ntemperature, although its structure is deformed due to thermal\nfluctuations.\nThe nucleation process of a high-Q skyrmion is revealed\nby investigating the magnetization distribution, the topologi-\ncal charge density, and the energy density. It occurs in two\nsteps. First, it so happens that the high density part is local-\nized to a lattice-scale area in the boundary of a magnetic bub-\nble withQ= 0. The topological number jumps from Q= 0\ntoQ= 1 by making a large spin rotation in this lattice-scale\narea within a few picoseconds. This phenomenon would be\nviewed as an emergence of a Bloch point in the continuum\ntheory. Second, a similar process occurs to make a jump from9\nFIG. 9. (Color online) Annihilation process of the high-Q skyrmion with Q= 2. Snapshots of the topological charge density q(x), the\nenergy density \u000f(x), the spin-component density mz(x), and its close up at sequential times. The DMI constant D= 2mJ m\u00002. The external\nmagnetic field Bz= 250 mT. The spin current density is switched off at t= 5000 ps. In the simulation, each cell corresponds to one spin,\nand the cell size is 1:5nm\u00021:5nm\u00021nm. In the spin distribution panels, each arrow stands for four spins, while it stands for one spin in\nthe insets. In the first step, the topological number changes from Q= 2toQ= 1, where the skyrmion size remains almost unchanged. Then,\nthe magnetic skyrmion with Q= 1 shrinks to the size of the lattice scale. In the second step, the topological number changes from Q= 1 to\nQ= 0, and the magnetic skyrmion disappears. See Ref. 26 for Supplementary Movie 5.\nQ= 1toQ= 2. The dissipation spreads over the sample like\na burst at the moment of the birth of a magnetic skyrmion.\nThe continuity of the spin texture is recovered since a\nsmooth texture has a lower energy. Once a sufficiently smooth\nhigh-Q skyrmion is generated, it remains stable even if we\nswitch off the DMI or even if we change the current density\nof the injected spin current considerably. We have explained\nits stability as a topologically protected dissipative structure.\nWhen the spin current injection is switched off, the high-Q\nskyrmion is destroyed. The destruction process occurs also\nin two steps as in the case of the nucleation process. How-\never, the detailed mechanism is different. The first step occurs\nby way of a Bloch point, where the continuum picture is still\ngood. In the second step, the lattice structure becomes impor-\ntant since the skyrmion size is so small, where the skyrmion\nspin texture turns into the vortex spin texture and disappears.\nThe dissipation spreads over the sample like a burst at the mo-\nment of the destruction of a magnetic skyrmion.\nIt is a hard problem to solve the nucleation or destruction\nprocess analytically since it is a highly nonlinear process in-\nvolving Bloch points. Furthermore, the lattice structure plays\na key role in these processes at the microscopic level. We hope\nthis work provides useful guidelines in searching new type\nof skyrmions and will afford a new dimension towards fully\nunderstanding the nontrivial topology of magnetic skyrmions\nwith higher topological numbers.ACKNOWLEDGMENTS\nY .Z. acknowledges the support by National Natural Sci-\nence Foundation of China (Project No. 1157040329), the Seed\nFunding Program for Basic Research and Seed Funding Pro-\ngram for Applied Research from the HKU, ITF Tier 3 fund-\ning (ITS/171/13 and ITS/203/14), the RGC-GRF under Grant\nHKU 17210014, and University Grants Committee of Hong\nKong (Contract No. AoE/P-04/08). M.E. thanks the sup-\nport by the Grants-in-Aid for Scientific Research from MEXT\nKAKENHI (Grant Nos. 25400317 and 15H05854). M.E. is\nvery much grateful to N. Nagaosa for many helpful discus-\nsions on the subject. X.Z. was supported by JSPS RONPAKU\n(Dissertation Ph.D.) Program. X.Z. greatly appreciates on-\ngoing discussions with J. Xia.\nAPPENDIX: SIMULATION AND MODELING\nThe micromagnetic simulation is carried out with the\nwell-established Object Oriented MicroMagnetic Framework\n(OOMMF) software (1.2a5 release) [28]. The OOMMF ex-\ntensible solver (OXS) extension module of the interface-\ninduced DMI, that is, the Oxs_DMExchange6Ngbr class, is\nincluded in the simulation [29]. The OXS extension module of\nthe thermal fluctuation, that is, the Xf_ThermSpinXferEvolve10\nclass, is employed to simulate the finite-temperature system.\nThe 3D time-dependent magnetization dynamics at zero tem-\nperature is determined by the LLGS equation [28, 30], while\na highly irregular fluctuating field representing the irregular\ninfluence of temperature is added into the LLGS equation for\nsimulating the magnetization dynamics at finite temperature.\nThe average energy density of the system contains the ex-\nchange energy, the anisotropy energy, the applied field (Zee-\nman) energy, the DMI energy, and the magnetostatic (demag-\nnetization) energy terms.\nIn the simulation, we consider a 1-nm-thick FM nanodisk\nwith a diameter of 150nm, which is attached to a heavy-metal\nsubstrate. The material parameters used by the simulation pro-\ngram are adopted from Refs. 9 and 10: the Gilbert damping\ncoefficient\u000b= 0:01, the gyromagnetic ratio \r=\u00002:211\u0002\n105m A\u00001s\u00001, the saturation magnetization MS= 580 kA\nm\u00001, the exchange stiffness J= 15 pJ m\u00001, the interface-induced DMI constant D= 0\u00183mJ m\u00002, and the PMA\nconstantK= 0:8MJ m\u00003. The polarization rate of the verti-\ncal spin current applied in the simulation is fixed at P= 0:4.\nThe simulated model is discretized into tetragonal cells with\nthe optimum cell size of 1:5nm\u00021:5nm\u00021nm, which gives\na good trade-off between the computational accuracy and effi-\nciency. The finite-temperature simulation is performed with a\nfixed time step of 1\u000210\u000014s, while the time step in the simu-\nlation with zero temperature is adaptive ( \u00186\u000210\u000014s). The\nseed value in the finite temperature simulation is fixed at 100.\nThe output is made with an interval of 1\u000210\u000013\u00181\u000210\u000011\ns. The initial magnetization distribution of the FM nanodisk is\nrelaxed along the +z-direction, except for the tilted magneti-\nzation on the edge due to the DMI. The external magnetic field\nis applied perpendicular to the FM nanodisk pointing along\nthe+z-direction with an amplitude of Bz= 0\u0018500mT.\n[1] T. H. R. Skyrme, Proc. R. Soc. A 260, 1271 (1961).\n[2] A. A. Belavin and A. M. Polyakov, JETP Lett. 22245 (1975).\n[3] R. Rajaraman, Solitons and Instantons (North-Holland, Ams-\nterdam (Netherlands), 1982).\n[4] G. E. Brown and M. Rho, The Multifaced Skyrmions (World\nScientific, Singapore, 2010).\n[5] M. Ezawa, Phys. Rev. Lett. 105, 197202 (2010).\n[6] M. Finazzi, M. Savoini, A. R. Khorsand, A. Tsukamoto, A.\nItoh, L. Duò, A. Kirilyuk, Th. Rasing, and M. Ezawa, Phys.\nRev. Lett. 110, 177205 (2013).\n[7] N. Bogdanov and D. A. Yablonskii, Sov. Phys. JETP 68, 101\n(1989); N. Bogdanov and A. Hubert, J. Magn. Magn. Mater.\n138, 255 (1994); U. K. Roessler, N. Bogdanov, and C. Pflei-\nderer, Nature (London) 442, 797 (2006).\n[8] N. Nagaosa and Y . Tokura, Nat. Nanotech. 8, 899 (2013).\n[9] A. Fert, V . Cros, and J. Sampaio, Nat. Nanotech. 8, 152 (2013).\n[10] J. Sampaio, V . Cros, S. Rohart, A. Thiaville, and A. Fert, Nat.\nNanotech. 8, 839 (2013).\n[11] S. Mühlbauer, B. Binz, F. Jonietz, C. Pfleiderer, A. Rosch, A.\nNeubauer, R. Georgii, and P. Böni, Science 323, 915 (2009).\n[12] X. Z. Yu, Y . Onose, N. Kanazawa, J. H. Park, J. H. Han, Y .\nMatsui, N. Nagaosa, and Y . Tokura, Nature 465, 901 (2010).\n[13] S. Heinze, K. von Bergmann, M. Menzel, J. Brede, A. Kubet-\nzka, R. Wiesendanger, G. Bihlmayer, and S. Blügel, Nat. Phys.\n7, 713 (2011).\n[14] T. Schulz, R. Ritz, A. Bauer, M. Halder, M. Wagner, C. Franz,\nC. Pfleiderer, K. Everschor, M. Garst, and A. Rosch, Nat. Phys.\n8, 301 (2012).\n[15] X. Z. Yu, N. Kanazawa, Y . Onose, K. Kimoto, W. Z. Zhang, S.\nIshiwata, Y . Matsui, and Y . Tokura, Nat. Mat. 10, 106 (2011).[16] W. Munzer, A. Neubauer, T. Adams, S. Mühlbauer, C. Franz,\nF. Jonietz, R. Georgii, P. Böni, B. Pedersen, M. Schmidt, A.\nRosch, and C. Pfleiderer, Phys. Rev. B 81, 041203(R) (2010).\n[17] X. C. Zhang, M. Ezawa, and Y . Zhou, Sci. Rep. 5, 9400 (2015).\n[18] Y . Tchoe and J. H. Han, Phys. Rev. B 85, 174416 (2012).\n[19] J. Iwasaki, M. Mochizuki, and N. Nagaosa, Nat. Nanotech. 8,\n742 (2013).\n[20] Y . Zhou and M. Ezawa, Nat. Commun. 5, 4652 (2014).\n[21] Y . Zhou, E. Iacocca, A. Awad, R. K. Dumas, F. C. Zhang, H. B.\nBraun, and J. Åkerman, Nat. Commun. 6, 8193 (2015).\n[22] N. Romming, C. Hanneken, M. Menzel, J. E. Bickel, B. Wolter,\nK. von Bergmann, A. Kubetzka, and R. Wiesendanger, Science\n341, 636 (2013).\n[23] S.-Z. Lin, C. Reichhardt, C. D. Batista, A. Saxena, Phys. Rev.\nLett. 110, 207202 (2013).\n[24] H. Goldstein, C. Poole, J. Safko, Classical Mechanics, 3rd ed.\n(Addison-Wesley, New York, 2002), Chap. 1, Sec. 5.\n[25] M. Ezawa, Phys. Lett. A 375, 3610 (2011).\n[26] See Supplemental Material at [URL] for supplementary movies\non the nucleation, maintenance, and annihilation of the high-Q\nskyrmion.\n[27] G. Tatara, H. Kohno, and J. Shibata, Phys. Rep. 468, 213\n(2008); J. Phys. Soc. Jpn. 77, 031003 (2008).\n[28] M. J. Donahue and D. G. Porter, OOMMF, User’s Guide, Inter-\nagency Report NISTIR 6376, NIST Gaithersburg, MD (1999)\nhttp://math.nist.gov/oommf.\n[29] S. Rohart and A. Thiaville, Phys. Rev. B 88, 184422 (2013).\n[30] J. Xiao, A. Zangwill, and M. D. Stiles, Phys. Rev. B 70, 172405\n(2004)."    },    {        "title": "1505.06524v4.New_Explicit_Binary_Constant_Weight_Codes_from_Reed_Solomon_Codes.pdf",        "content": "arXiv:1505.06524v4  [cs.IT]  7 Aug 2015New Explicit Binary Constant Weight\nCodes from Reed-Solomon Codes\nLiqing Xu and Hao Chen∗\nSeptember 6, 2018\nAbstract\nBinary constant weight codes have important applications a nd have been\nstudied for many years. Optimal or near-optimal binary cons tant weight\ncodes of small lengths have been determined. In this paper we propose a\nnew construction of explicit binary constant weight codes f romq-ary Reed-\nSolomon codes. Some of our binary constant weight codes are o ptimal or\nnew. Inparticularnewbinaryconstant weight codes A(64,10,8)≥4108 and\nA(64,12,8)≥522 are constructed. We also give explicitly constructed bi -\nnaryconstant weight codes whichimprove GilbertandGraham -Sloane lower\nbounds in some range of parameters. An extension to algebrai c geometric\ncodes is also presented.\nKeywords: Constant weight code, Reed-Solomon codes, algebraic geo-\nmetric codes\n1 Introduction\nA binary contant weight ( n,d,w) code is a set of vectors in Fn\n2such that\n1) every codeword is a vector of Hamming weight w;\n∗L. Xu and H. Chen are with the School of Sciences, Hangzhou Dia nzi Uni-\nversity, Hangzhou 310018, Zhejiang Province, China. e-mai l: lqxu@hdu.edu.cn,\nhaochen@hdu.edu.cn. This research was supported by NSFC Gr ant 11371138.\n12) the Hamming distance wt(x−y) of any two codewords xandyis at\nleastd.\nBinary constant weight codes have important applications ( [3, 9, 10]).\nIn coding theory to determine the maximal possible size A(n,d,w) for a\nbinary constant weight ( n,d,w) code is a classical problem which has been\nstudied by many authors ([15, 8, 18, 1, 2, 4, 7, 19]). For these lowdand\nwand lengths n≤65 orn≤78, the previous best known lower bound for\nA(n,d,w) has been given in [19]. For the upper bounds of A(n,d,w) we\nrefer to the Johnson bound ([15, 7]).\nJohnson upper bound. Ifn≥w >0thenA(n,d,w)≤[n\nwA(n−\n1,d,w−1)]andA(n,d,w)≤[n\nn−wA(n−1,d,w)].\nThe following lower bounds are the most known lower bounds fo r binary\nconstant weight codes ([8]).\nGilbert type lower bound. A(n,2d,w)≥/parenleftBigg\nn\nw/parenrightBigg\nΣd−1\ni=0/parenleftBigg\nw\ni/parenrightBigg\n·/parenleftBigg\nn−w\ni/parenrightBigg.\nGraham-Sloane lower bound. Letqbe the smallest prime power sat-\nisfyingq≥nthenA(n,2d,w)≥1\nqd−1/parenleftBigg\nn\nw/parenrightBigg\n.\nHowever the binary constant weight codes in the Gilbert type lower\nbound is not constructed and the argument is only an existenc e proof. The\nbinary constant weight codes in the Graham-Sloane lower bou nd were not\nexplicitly given, since one has to search at least qdcodes to find the desired\none (see [8], page 38). The Graham-Sloane lower bound was imp roved in\n[18] by the using of algebraic function fields. However the bi nary constant\nweight codes in [18] were not explicitly given, since the con struction there\nwas a generalization of [8].\nIn this paper we propose a general construction of explicit b inary con-\nstant weight codes from general q-ary Reed-Solomon codes. This is a strict\nimprovement on the previous constructions in [5, 6]. Many of our con-\nstructed binary constant weight codes have nice parameters . Some of them\nare new or optimal. In particular two new better binary const ant weight\n2codes are presented. We also give an extension to algebraic g eometric codes.\n2 Constant weight codesfrom Reed-Solomon codes\nIn this section we give explicit binary codes from Reed-Solo mon codes. The\nfirst step construction Proposition 2.1-2.2 is the same as th e DeVore’s about\nrestricted isometry matrices in the compressed sensing ([5 ]). Actually it is\nalso another form of Ericson-Zinoviev construction in [6]. The main results\nTheorem 2.1-2.2 are strcitly better than these previous con structions.\n2.1 Construction\nProposition 2.1. A(q2,2q+2−2r,q)≥qrifr−1< q.\nProof. For a polynomial fwith degree less than or equal to r−1 in\nFa[x], we get a length q2vectorvf= (f(a,b))∈Fq2\n2. It is determined by\nitsq2coordinates f(a,b)for (a,b)∈Fq×Fq. Heref(a,b)= 0 iff(a)/ne}ationslash=b,\nf(a,b)= 1 iff(a) =b. Then we have qrsuch length q2codewords from\nall degree ≤r−1 polynomials, each of these codewords has weight q. For\nany two such codewords from polynomials fandg, the intersection of their\nsupports are exactly the points ( x,f(x) =g(x)). Since there are at most\nr−1 zeros of the polynomial f(x)−g(x) we get the conclusion.\nProposition 2.2. For any positive integer w≤qwhereqis a prime\npower we have A(wq,2w+2−2r,w)≥qrifq≥w > r−1is satisfied.\nProof. We useqrfunctions restriced to a subset WinFqsatisfying\n|W|=w.\nHowever this explaination of codewords as support function s naturally\nleads us to add some new codewords. This will give us new bette r binary\nconstant weight codes which have never been found before.\nTheorem 2.1. A(q2,2q+ 2−2r,q)≥qr+qif2≤r < q+ 1.\nA(wq,2w+2−2r,w)≥qr+wifq≥w > r−1satisfied.\nProof. We add these weight wcodewords supported at the positions\nu×Wwhereucan be any element in the set W. Since the supportsof these\n3codewords of weight ware disjoint and the support of each such codeword\nhas only one common position with the support of each codewor dvf, the\nconclusion is proved.\nTheorem 2.2. Ifr≥3,A(wq,2w+2−2r,w)≥qr+w+w\n[w−r+1\nq−w+r−1]+1\nifr−1< w≤qsatisfied.\nProof. We add the codewords supported at the following sets. Set\nW={P1,...,Pw}. Every support set is included in at most two Pi×Fq’s\nand the intersection of two such support sets have at most r−1 elements.\nInP1×Fqwe takew-element subset and there are q−welements remained.\nInP2×Fqwe only need to take w−(q−w+r−1)-element subset and\nthere are 2( q−w+r−1) elements remained. At the j-th step we have\n(j−1)(q−w+r−1) elements remained in the set Pj×Fq.\nWhen (j−1)(q−w+r−1)≥w−r+1, we add j+1 such codewords\nsatisfying that\n1) The intersection of any two of their supports has at most r−1 elements;\n2) The intersection of any such support with the any image sup port has at\nmost 2≤r−1 elements.\nThus finally we add at least w+w\n[w−r+1\nq−w+r−1]+1such weight wcodewords.\nIn a recent paper [7] of T. Etizon and A. Vardy constructed bin ary con-\nstant weight codes by using the constant dimensional subspa ce codes. They\nprovedA(22m−1,2m+1−4,2m) = 22m−1+ 2m−1. Our this lower bound\nA(22m,2m+1−4,2m)≥23m+2mcan be compared with their result.\nFrom Theorem 2.1-2.2 the following binary constant weight c odes can be\nconstructed, which can be compared with the best known ones i n [19]. The\ncodeA(64,10,8)≥4108 and A(64,12,8)≥522 are new and better than the\npreviously best known ones. Many other codes attain the best known lower\nbounds or optimal values.\nTable 1 Explicit constant weight codes from RS\n4Explicit codes lower bound and upper bound in [19]\nA(25,8,5) = 30 30\nA(35,8,5) = 54 56\nA(40,8,5) = 69 72\nA(42,10,6) = 55 55-56\nA(48,10,6) = 70 72\nA(49,12,7)≥56 56\nA(49,10,7)≥350 385-504\nA(64,10,8)≥4108 4096–8928\nA(64,12,8)≥522 520-720\nA(56,12,7)≥71 71-72\nA(56,10,7)≥519 583-728\nA(81,16,9)≥90 90\nA(64,14,8)≥72 72\nA(63,12,7)≥88 88–90\nA(63,10,7)≥736 831–1116\nA(66,10,6)≥127 143\nA(72,14,8)≥89 89–90\nA(77,12,7)≥128 no\nIn the following table 2 we give some small binary constant we ight codes\nfrom Theorem 2.1- 2.2, which are compared with the closest co des in [19],\nthe Gilbert and Graham-Sloane lower bounds. There is no entr y in the pre-\nvious table [19] for these parameters.\nTable 2 Explicit constant weight codes from RS\nExplicit codes closest codes in [19] GS bound G bound\nA(88,10,8)≥14657 A(64,10,8)≥4096 1071.8 556.99\nA(72,10,8)≥6573A(64,10,8)≥4096 445.4 255.39\nA(88,14,8)≥133 A(72,14,8)≥89 ≤1 6.51\nA(99,16,9)≥133 A(81,16,9)≥90 ≤1 5.29\nA(110,18,10)≥133A(91,18,10)≥91 ≤1 4.44\n52.2 Explicit binary constant weight codes improving Gilber t\nand Graham-Sloane lower bounds\nFrom the Graham-Sloane lower bound A(q2,2(q+1−r),q)≥1\nq2(q−r)/parenleftBigg\nq2\nq/parenrightBigg\n.\nWith the help from the Sterling formula limn!√\n2πn(n\ne)n= 1. We get/parenleftBigg\nq2\nq/parenrightBigg\n≈\n(eq)q\n/radicalbig\n2π(q−1). Thus the Graham-Sloane lower bound in this case n=q2and\nw=qisA(q2,2(q+1−r),q)≥eqq2r−q√\n2π(q−1). On the other hand the binary con-\nstant weight codes staisfying A(q2,2(q+1−2),q)≥qrare explicitly given\nin Proposition 2.1. When r=cq, wherecis a positive constant 0 < c <1,\nit is clear our lower bound is much better than the Graham-Slo ane bound\nwhenqis very large.\nFromasimplecomputationweget/parenleftBigg\nq2\nq/parenrightBigg\nΣq−r\ni=0/parenleftBigg\nq\ni/parenrightBigg\n·/parenleftBigg\nq2−q\ni/parenrightBigg≤/parenleftBigg\nq2\nq/parenrightBigg\n/parenleftBigg\nq\nq−r/parenrightBigg\n·/parenleftBigg\nq2−q\nq−r/parenrightBigg≤\n/parenleftBigg\nq2\nq/parenrightBigg\n/parenleftBigg\nq\nq−r/parenrightBigg\n·(q−1)q−r. From the Sterling formula ifeq/parenleftBigg\nq\nr/parenrightBigg<1 our explicit\nbinary constant weight codes in Proposition 2.1 improve the Gilbert type\nbound. When r=cp,cis a positive constant very close to 1, it is clear\neq/parenleftBigg\nq\nr/parenrightBigg<1 is valid.\nIn [6, 18] binary constant weight codes from algebraic curve s are used\nto improve Gilbert and Graham-Sloane lower bounds. However their codes\ncannot be explicitly constructed. Our codes in Proposition 2.1 are explic-\nitly constructed. As far as our knowledge this is the first imp rovement on\nthesetwolower boundsbyexplicitconstructedbinaryconst ant weight codes.\n63 Extension to algebraic geometric codes\n3.1 First step construction\nProposition 3.1. LetXbe a projective non-singular algebraic curve de-\nfined over a finite field Fqof genus g,P={P1,...,P|P|}be a set of Fq\nrational points on the curve XandGbe aFqrational divisor satisfying\ndegG≥2g−1. Then we have A(q|P|,2|P| −2degG,|P|)≥qdegG−g+1.\nProof.For each f∈L(G), a length q|P|vectorvf= (f(a,b))∈Fq|P|\n2,\nwhere (a,b)∈Fq×P, is defined as follows. f(a,b)is 0 iff(b)/ne}ationslash=aandfa,b)is\n1 iff(b) =a. We have dim(L(G)) =degG−g+1 ifdegG≥2g−1. There\nare at least qdegG−g+1such codewords. On the other hand the intersection\nof two supports of two such codewords associated with functi onsfandg\nare exactly these positions ( x,f(x) =g(x)). Thus it is the zero locus of the\nfunction f−g∈L(G). There are at most degGcommon positions at the\nintersection of supports of two such codewords.\nThe above construction can be generalized to higher dimensi on case. Let\nYbe a non-singular algebraic projective manifold defined ove rFq. The set\nof allFqrational points of this manifold is denoted by Y(Fq). For an effec-\ntive divisor DonY, we will use the function space L(D) which consists of\nall rational functions on Ywith poles at most −D([13]). In many cases the\ndimension of this function space can be computed from the Rie mann-Roch\ntheorem ([13]). For any rational function f∈L(D), a length q·|Y(Fq)−D|\ncodeword v(f)h∈Fq|Y(Fq)−D|\n2 , whereh= (a,b),b∈Y(Fq)−Danda∈Fq,\nis defined as follows. v(f)his zero if f(b)/ne}ationslash=a, andv(f)his 1 iff(b) =a.\nThus the Hamming weight of this codeword is exactly |Y(Fq)−D|. The\ncardinality of the intersection of the supports of two such c odewords v(f1)h\nandv(f2)his at most the number of zero points in Y(Fq) of the function\nf1−f2. That is, the number of common positions in the supports of tw o\nsuch codewords is equal to or smaller than the maximal possib le number of\nFqrational points on members of the linear system Linear(D). We denote\nthis number by N(D).\nProposition 3.2. We have a A(q· |Y(Fq)−D|,2(|Y(Fq)−D| −\nN(D)),|Y(Fq)−D|)≥qdim(L(D)).\nThis part of the construction can be seen as a direct applicat ion of [6]\nto algebraic geometric codes.\n73.2 Adding new codewords\nCurve case: We add the codewords supported at the following sets. Set\nP={P1,...,P|P|}. Every supportset is included in at most several Pi×Fq’s.\nthe intersection of two such support sets have at most degGelements. Sup-\npose|P|=rq+r′wherer′≥0.\nInP1×Fq,...,Pr+1×Fqwe take|P|-element subset and there are q−r′\nelements remained. In Pr+2×Fq,...,P2r+2×Fqwe only need to take\n|P| −(q−r′+degG)-element subset and there are 2( q−r′+degG) el-\nements remained. At the j-th step we have ( j−1)(q−r′+degG) elements\nremained in the set Pj(r+1)×Fq.\nWhen (j−1)(q−r′+degG)≥ |P|−degG, we add j+1 such codewords\nsatisfying that\n1) The intersection of any two of their supports has at most degGelements;\n2) The intersection of any such support with the any image sup port has at\nmostr+1 elements.\nIfr+1≤degG, finally we add at least [|P|\nr+1]+|P|\n[|P|−degG\nq−r′+degG]+1such weight\n|P|codewords.\nTheorem 3.1. LetXbe a projective non-singular algebraic curve de-\nfined over a finite field Fqof genus g,P={P1,...,P|P|}be a set of Fqratio-\nnal points on the curve XandGbe aFqrational divisor satisfying degG≥\n2g−1. Suppose |P|=rq+r′andq > r′≥0. We assume r+1≤degG.\nThen we have A(q|P|,2|P|−2degG,|P|)≥qdegG−g+1+[|P|\nr+1]+|P|\n[P|−degG\nq−r′+degG]+1.\nHigher dimension case: We add the codewords supported at the fol-\nlowing sets. Set Y(Fq)−D={P1,...,PN}. Every support set is included\nin at most several Pi×Fq’s. the intersection of two such support sets have\nat mostN(D) elements. Suppose N=rq+r′wherer′≥0.\nInP1×Fq,...,Pr+1×Fqwe take|P|-element subset and there are q−r′\nelements remained. In Pr+2×Fq,...,P2r+2×Fqwe only need to take\n|P| −(q−r′+N(D))-element subset and there are 2( q−r′+N(D)) el-\nements remained. At the j-th step we have ( j−1)(q−r′+N(D)) elements\n8remained in the set Pj(r+1)×Fq.\nWhen (j−1)(q−r′+N(D))≥N−N(D), we add j+1 such codewords\nsatisfying that\n1) The intersection of any two of their supports has at most N(D) elements;\n2) The intersection of any such support with the any image sup port has at\nmostr+1 elements.\nIfr+1≤N(D), finally we add at least [N\nr+1]+N\n[N−N(D)\nq−r′+N(D)]+1such weight\nNcodewords.\nTheorem 3.2. Suppose |Y(Fq)−D|=N=rq+r′andq > r′≥0.\nWe assume r+1≤N(D). ThenA(q·N,2(N−N(D)),N)≥qdim(L(D))+\n[N\nr+1]+N\n[N−N(D)\nq−r′+N(D)]+1.\n4 Examples: curves\nElliptic curves. LetEbe an elliptic curve over FqwithNrational points\nP={P1,...,PN}. Suppose N=rq+r′as in Theorem 3.1. We have\nA(qN,2(N−s),N)≥qs+N\n[N−s\nq−r′+s]+1if there is a degree srational divisor\nGwhose support satisfying suppG∩Pis empty, 1 < s < N andr+1≤s.\nExample 1. Theellipticcurve y2=x3−2x−3definedover F7has10ra-\ntional points (3 ,2),(2,6),(4,2),(0,5),(5,0),(0,2),(4,5),(2,1),(3,5) and the\nzero element (infinitypoint). Itis clear it hasadegree srational point. Thus\nwehaveA(70,20−2s,10)≥7s+[10\n2]+5(4+s)\n7. Whens= 2,A(70,16,10)≥59\n(A(70,16,9) = 49 in [19]). If only 9 rational points are used, A(63,18−\n2s,9)≥7s+ [9\n2] +9(4+s)\n13when 0 < s < 9. Thus A(63,14,9)≥57\n(A(63,14,8)≥63 in [19]).\nExample 2. There is an elliptic curve over F8with 14 rational points\n(maximal curve, [20]). Thuswehave A(112,28−2s,14)≥8s+[14\n2]+7+7(2+s)\n8\nfor 1< s <14,A(104,26−2s,13)≥8s+ 6 +13(3+s)\n16when 1< s <13,\nA(96,24−2s,12)≥8s+6+3(4+s)\n4when 1< s <12,A(88,22−2s,11)≥\n8s+5+11(5+s)\n16when 1< s <11, andA(80,20−2s,10)≥8s+5+5(6+s)\n8\nwhen 1< s <10.\n9Table 3 Explicit constant weight codes from EC\nExplicit codes closest codes in [19] G-S bound G bound\nA(80,16,10)≥74A(80,16,9)≥80 ≤1 9.43\nA(72,14,9)≥74A(72,14,8)≥89 ≤1 12.76\nA(70,16,10)≥59A(70,16,9)≥49 ≤1 6.80\nA(63,14,9)≥57A(63,14,8)≥63 ≤1 9.07\nA(36,14,9)≥23 A(36,14,8) = 9 ≤1 2.51\nA(36,12,9)≥7266≥A(36,12,8)≥45 ≤1 7.45\nA(36,10,9)≥265A(36,10,8)≥216 ≤1 38.12\nTable 4 Explicit constant weight codes from EC\nExplicit codes G-S bound G bound\nA(104,22,13)≥75 ≤1 4.85\nA(104,20,13)≥523 ≤1 17.86\nA(104,18,13)≥4107 ≤1 98.28\nA(104,16,13)≥32781 93 810.42\nA(96,20,12)≥75 ≤1 5.86\nA(96,14,12)≥52784 798.061557.72\nA(88,18,11)≥73 ≤1 7.30\nA(88,16,11)≥523 ≤1 35.07\nThe above constant weight codes are much better than the code s from\nthe Graham-Sloane lower bound and Gilbert type lower bound.\nExample 3. LetEbethe elliptic curve y2+y=x3+xdefinedover F2r\nwherer≡4mod8. There are N= 2r+2r\n2+1+1 (see [17], Theorem 4.12) ra-\ntional points on this curve. We have A(22r+23r\n2+1+2r,2(2r+2r\n2+1−s),2r+\n2r\n2+1+1)≥2rs+2r−1+2r\n2+(2r+2r\n2+1+1)(2r−2r\n2+1−s)\n2r+1 when 1< s <2r+2r\n2+1.\nHermitian curves. The Hermitian curve over Fq2is defined by xq+\nx=yq+1. It is well-known there are N=q3+ 1 rational points. Thus\nA(q5,2(q3−s),q3)≥q2(s−q2−q\n2+1)+q2−1+q3(q2+s)\nq3+q2whenq2−q−2< s < q3\nfromTheorem3.1. Mostoftheseexplicitconstructedbinary constant weight\ncodes are much better than the Gilbert type lower bound.\nJustastheprevioussectionexpliciltlyconstructedbinar yconstantweight\ncodesfromellipticcurvescanbeusedtoimproveGilbertand Graham-Sloane\n10lower bounds.\n5 Examples: higher dimension case\n5.1. Projective spaces. In Theorem 3.2, we take Pn\nFqandD=rHthe\ndivisor. Then A(qn+1,2(qn−rqn−1−qn−2−···−q−1),qn)≥q/parenleftBigg\nn+r\nr/parenrightBigg\n+\nq−1+q+rqn−1+qn−2+···+q+1\n2from the Segre-Serre-Sorensen bound ([14]). In\nparticular A(q3,2(q2−rq−1),q2)≥q(r+2)(r+1)\n2+q−1+q+rq+1\n1+1/q. We list some\nsuch explicit binary constant weight codes in Table 5. Excep t the second\nA(64,14,16) = 4107 <4603, all others are much better than Gilbert type\nlower bound. Considering Gilbert lower bound is not constru ctive, our this\ncodeA(64,14,16)≥4107 is good.\nTable 5 Explicit constant weight codes from projective surface\nExplicit codes Gilbert type bound\nA(64,22,16)≥72 6.31\nA(64,14,16)≥4107 4603.81\nA(125,38,25)≥135 5.05\nA(125,28,25)≥15639 3015.31\n5.2. Ruled Surface. In Theorem 3.2 we take X=P1\nFq×P1\nFq. The\nset ofFqrational points on P1\nFq×P1\nFqis naturally the disjoint union of\n(q+1) sets of Fqrational points on curves pi×P1\nFq, wherepi,i= 1,...,q+1\nare (q+ 1) rational points of P1\nFq. The divisor Dof type ( d1,d2) con-\nsists of polynomials f(x,y,z,w) which are homogeneous in x,ywith degree\nd1and are homogeneous in z,wwith degree d2. This is a linear system\nwith dimension ( d1+ 1)(d2+ 1). If d1+d2< q+ 1, there are at most\n−d1d2+d1(q+1)+d2(q+1) rational points on any member of this linear\nsystem([12]). Then A(q(q+1)2,2((q+1)2+d1d2−(d1+d2)(q+1)),(q+1)2)≥\nq(d1+1)(d2+1)+q−1 +(q+1)2(q−1+(d1+d2)(q+1)−d1d2)\n(q+1)2+q−1from Theorem 3.2. We\nlist some binary constant weight codes in the following Tabl e 6. Some of\nthem are much better than Gilbert type lower bound.\n11Table 6 Explicit constant weight codes from ruled surface\nExplicit codes Gilbert type bound\nA(100,24,25)≥4111 1771.61\nA(180,40,36)≥15647 39467.85\nA(180,50,36)≥643 38.31\nA(448,84,64)≥117681 616907.85\n5.3. Toric surfaces. Algebraic geometric codes from toric surfaces\nhave been studied in [11]. In this section we give some exlpic it binary con-\nstant weight codes from some toric surfacse in [11].\nLetZ2⊂R2be the set of all integral points . We denote θa primitive\nelement of the finite field Fq. For any integral point m= (m1,m2)∈Z2we\nhave a function e(m) :F∗\nq×F∗\nq→Fqdefined as e(m)(θi,θj) =θm1i+m2jfor\ni= 0,1,...,q−1 andj= 0,1,...,q−1. Let ∆ ⊂R2be a convex polyhedron\nwith vertices in Z2andL(∆) bethefunction space over Fqspannedby these\nfunctions e(m) wheremtakes over all integral points in ∆. In the following\ncases of convex polyhedrons these functions are linearly in dependent from\n[11].\nThe following three cases as in the main results Theorem 1, 2, 3 of [11]\nare considered.\n1) ∆ is the convex polytope with the vertices (0 ,0),(d,0),(0,d) where dis\na positive integer satisfying d < q−1;\n2) ∆ is the convex polytope with the vertices (0 ,0),(d,0),(d,e+rd),(0,e)\nwhered,r,eare positive integers satisfying d < q−1,e < q−1 and\ne+rd < q−1;\n3) ∆ is the convex polytope with the vertices (0 ,0),(d,0),(0,2d) wheredis\na positive integer satisfying 2 d < q−1;\nFor each function f∈L(∆) we have a length q×(q−1)2codeword\nv(f) = (f(a,b)) where ( a,b)∈F∗\nq×F∗\nq×Fqdefined as follows. f(a,b)= 0 if\nf(a)/ne}ationslash=bandf(a,b)= 1 iff(a) =b. The Hamming weight of this codeword\nis exactly ( q−1)2. Then there are qdim(L(∆))such weight ( q−1)2codewords.\nWe have the following result from the main results Theorem 1, 2, 3 of [11]\nand Theorem 3.2.\nProposition 5.1. In the above cases we have\n1)A(q(q−1)2,2((q−1)2−d(q−1)),(q−1)2)≥q(d+1)(d+2)\n2+q−1+(q−1)2(d+1)\nq\n12in the case 1);\n2)A(q(q−1)2,2((q−1)2−min{(d+e)(q−1)−de,(e+rd)(q−1)}),(q−1)2)≥\nq(d+1)(e+1)+rd(d+1)\n2+q−1+(q−1)(q−1+min{(d+e)(q−1)−de,(e+rd)(q−1)})\nqin the case\n2);\n3)A(q(q−1)2,2((q−1)2−2d(q−1)),(q−1)2)≥qd2+2d+1+q−1+(q−1)2(2d+1)\nq\nin the case 3).\nIn the following table we list some explicit binary constant weight codes\nfrom toric surfacse. They are much better than Gilbert type l ower bound.\nTable 7 Explicit constant weight codes from toric surfaces\nExplicit codes Gilbert type bound\nA(80,24,16)≥136 5.57\nA(80,18,16)≥3138 416.62\nA(80,12,16)≥1953141 781764.18\n6 Summary\nExplicit binary constant weight codes have been constructe d from Reed-\nSolomon codes. This is a strict improvement on the previous w orks in [5, 6].\nExamples of nice binary constant weight codes have been give n. Some of\nnew better binary constant weight codes have been explicitl y constructed.\nThe parameters of most of our explicit binary constant weigh t codes are\nmuch better than the Gilbert type lower bound and Graham-Slo ane lower\nbound. Asmptotically our explicit binary constant weight c odes even from\nReed-Solomon codes have parameters better than the non-exp licit Gilbert\nand Graham-Sloane lower boundsin some range of parameters. We also give\nan extension to algebraic geometric codes and many good bina ry constant\nweight codes are explicitly constructed.\nReferences\n[1] E. Agrell, A. Vardy, and K. Zeger, Upper bounds for consta nt-weight\ncodes, IEEE Trans. Inf. Theory, vol. 46 (2000), no. 7, 237323 95.\n[2] A. E. Brouwer and T. Etzion, Some new constant weight code s, Ad-\nvances in Mathematics of Communications, vol. 5 (2011), 417 -424.\n13[3] R. Calderbank, M. A. Herro, and V. Telang, A multilevel ap proach to\nthe design of DC-free line codes, IEEE Trans. Inform. Theory , vol. 35\n(1989) 579-583.\n[4] Y. M. Chee, C. Xing and S. Z. Ling, New constant-weight cod es from\npropagation rules, IEEE Transactions on Information Theor y, vol. 56\n(2010), 1596-1599.\n[5] R. DeVore, Deterministic constructions of compressed s ensingmatrices,\nJournal of Complexity, Vol.23 (2007), no.46, 918-925.\n[6] T. Ericson and V. A. Zinoviev, An improvement of the Gilbe rt bound\nfor constant weight codes, IEEE Transactions on Informatio n Theory,\nvol.33 (1987), 721-723.\n[7] T. Etzion and A. Vardy, A new construction for constant we ight codes,\narXiv:1004.1503v3.\n[8] R. L. Graham and N. J. A. Sloane, Lower bounds for constant weight\ncodes, IEEE Trans. Inf. Theory, vol.26 (1980), no. 1, pp. 37- 43.\n[9] N. Q. A, L. Gyorfri and J. L. Massey, Constructions of bina ry constant-\nweight cyclic codes and cyclically permutable codes, IEEE T rans.\nInform. Theory, vol. 38 (1992), 940-949.\n[10] K. A. Immink, Coding Techniques for Digital Recorders. London:\nPrentice-Hall, 1991.\n[11] J. P. Hansen, Toric surfaces and codes, techniques and e xamples, Cod-\ning theory, cryptography and related areas, ed. J. Bachmann et al.,\nSpringer, 2000.\n[12] S. H. Hansen, Error-correcting codes over higher dimen sional varieties,\nFinite Fields and Their Applications,Vol. 7 (2001), 530-55 2.\n[13] R. Hartshorne, Algebraic geometry, Springer-Verlag, 1977.\n[14] M.Homma and S.J.Kim, An elementary bound for the number of ratio-\nnal points of a hypersurface over finite fields, Finite Fields and Their\nApplications, Vol.20 (2014), 76-83.\n14[15] S. M. Johnson, A new upper bound for error-correcting co des, IRE\nTrans. Inform. Theory, vol. IT-8 (1962), pp. 203-207.\n[16] M. A. Tsfasman and S. G. Vladut, Algebraic-geometric co des, Dor-\ndrecht, Kluwer, 1991.\n[17] L.C.Washington, Elliptic Curves: Number Theory and Cr yptography,\nDiscr. Math. Appl.(series), 2nd ed. Boca Raton, FL: CRC Pres s, 2008.\n[18] C. Xing and J. Ling, A construction of binary constant we ight codes\nfrom algebraic curves over finite fields, IEEE Transactions o n Informa-\ntion Theory, vol.51(2005), 3674-3678.\n[19] http://www.win.tue.nl/ ∼aeb/Andw.html\n[20] http://gerard.vdgeer.net/tables-mathcomp21.pdf\n15"    },    {        "title": "1505.08005v3.Microscopic_Theory_for_Coupled_Atomistic_Magnetization_and_Lattice_Dynamics.pdf",        "content": "Microscopic Theory for Coupled Atomistic Magnetization and Lattice Dynamics\nJ. Fransson,1,\u0003D. Thonig,1P. F. Bessarab,2, 3S. Bhattacharjee,4J. Hellsvik,5, 6and L. Nordstr ¨om1\n1Department of Physics and Astronomy, Box 516, SE-751 20, Uppsala University, Uppsala, Sweden\n2Science Institute of the University of Iceland, 107 Reykjavik, Iceland\n3ITMO University, 197101 St. Petersburg, Russia\n4Indo-Korea Science and Technology Center (IKST), Bangalore, India\n5Nordita, Roslagstullsbacken 23, SE-106 91 Stockholm, Sweden\n6Department of Physics, KTH Royal Institute of Technology, SE-106 91 Stockholm, Sweden\n(Dated: October 8, 2018)\nA coupled atomistic spin and lattice dynamics approach is developed which merges the dynamics of these\ntwo degrees of freedom into a single set of coupled equations of motion. The underlying microscopic model\ncomprises local exchange interactions between the electron spin and magnetic moment and the local couplings\nbetween the electronic charge and lattice displacements. An e \u000bective action for the spin and lattice variables is\nconstructed in which the interactions among the spin and lattice components are determined by the underlying\nelectronic structure. In this way, expressions are obtained for the electronically mediated couplings between the\nspin and lattice degrees of freedom, besides the well known inter-atomic force constants and spin-spin interac-\ntions. These former susceptibilities provide an atomistic ab initio description for the coupled spin and lattice\ndynamics. It is important to notice that this theory is strictly bilinear in the spin and lattice variables and pro-\nvides a minimal model for the coupled dynamics of these subsystems and that the two subsystems are treated\non the same footing. Questions concerning time-reversal and inversion symmetry are rigorously addressed and\nit is shown how these aspects are absorbed in the tensor structure of the interaction fields. By means of these re-\nsults regarding the spin-lattice coupling, simple explanations of ionic dimerization in double anti-ferromagnetic\nmaterials, as well as, charge density waves induced by a non-uniform spin structure are given. In the final\nparts, a set of coupled equations of motion for the combined spin and lattice dynamics are constructed, which\nsubsequently can be reduced to a form which is analogous to the Landau-Lifshitz-Gilbert equations for spin\ndynamics and damped driven mechanical oscillator for the ionic motion. It is important to notice, however,\nthat these equations comprise contributions that couple these descriptions into one unified formulation. Finally,\nKubo-like expressions for the discussed exchanges in terms of integrals over the electronic structure and, more-\nover, analogous expressions for the damping within and between the subsystems are provided. The proposed\nformalism and new types of couplings enables a step forward in the microscopic first principles modeling of\ncoupled spin and lattice quantities in a consistent format.\nI. INTRODUCTION\nThe understanding of how spin and lattice degrees of free-\ndom interact is of fundamental importance [1, 2]. Recently,\nstrong evidence was found for the existence of a significant\ncoupling between magnons and phonons for instance in bcc\nFe [3, 4] and the ferromagnetic semiconductor EuO [5]. Spin-\nlattice coupling is central for seemingly disparate phenomena\nsuch as the mechanical generation of spin currents by spin-\nrotation coupling [6], the spin-Seebeck e \u000bect [7, 8], and the\ndriving of magnetic bubbles with phonons [9]. In the field\nof multiferroic spin-lattice coupling is a central mechanism\nfor the coupling of (anti)ferromagnetic and (anti)ferroelectric\norder parameters (magnetoelectric e \u000bect) [2, 10–12]. Spin-\nlattice coupling also occur in ferroelastic and ferromagnetic\nmaterials (magnetoelastic e \u000bect) [13]. There is also a growing\ninterest in including e \u000bects from mechanical degrees of free-\ndom into theoretical models for ultrafast magnetization dy-\nnamics [14–16], since rapid ionic motion has shown to cause\nnon-trivial temporal fluctuations of the magnetic properties\n[17–20].\nMagnetization dynamics is conventionally understood in\n\u0003Electronic address: Jonas.Fransson@physics.uu.seterms of the phenomenological Landau-Lifshitz-Gilbert [21,\n22] approach. A seminal step towards a formulation of atom-\nistic magnetization dynamics from first principles was taken\nby Antropov et al. [23] who started out from time-dependent\ndensity functional theory and the Kohn-Sham equation and\nconsidered also simultaneous spin and molecular dynamics,\nhowever, incorporating energy dissipation and finite temper-\nature phenomenologically. The equation of motion for local\nspin magnetic moments in the adiabatic limit have also been\nworked out in Refs. [24, 25]. E \u000bects of non-locality in space\nand time were captured in the formalism communicated in\n[26], including a complete basic principle derivation of the\natomistic magnetization dynamics equations of motion.\nRecently, great e \u000bort has been devoted to improve the\nLandau-Lifshitz-Gilbert approach by calculating the damping\ntensor directly from the electronic structure [27–29]. The ad-\ndition of other contributions, as for instance moment of inertia\n[26, 30, 31] observed in Refs. [30, 32, 33], allows for dynam-\nics on shorter time scales. The basic principles of the moment\nof inertia contributions to atomistic magnetization dynamics\nwere derived from a Lagrangian formulation [34]. In the adi-\nabatic limit, the lattice degrees of freedom follow Newton dy-\nnamics [35] and can be derived from the e \u000bective action of\nthe system [36]. Hence, the uncoupled dynamics of spin and\nlattice is well understood [36, 37].\nThere have been in the last years been several simulationsarXiv:1505.08005v3  [cond-mat.mtrl-sci]  23 Jan 20182\nFIG. 1: (Color online) Schematic figure of two atoms (gray balls)\nwith a magnetic moment (green arrows) and lattice vibrations (fading\ngray and transparent balls) in a cloud of electrons (small red balls and\nfoggy environment)\nwith a combined Landau-Lifshitz-Gilbert and lattice dynam-\nics approach [38, 39]. They are based on an atomistic spin\nmodel with position dependent exchange parameters which\nfor instance lead to a spin ordering dependent e \u000bective lattice\ndynamics equations of motion. This spin and lattice coupling\nthen enter through the Taylor expansion of the magnetic ex-\nchange interactions in terms of ionic displacements around the\nequilibrium positions.\nTo put spin and lattice degrees of freedom on the same\nfooting, however, bilinear order of spin-lattice coupling is re-\nquired that seems forbidden from the naive argument of break-\ning the time reversal symmetry in the total energy. Thus, the\nquestion remains about the lowest order in spin-lattice cou-\npling, conserving Newtons third law.\nWe notice that in the past there have been several consid-\nerations of coupling magnetic and elastic degrees of freedom,\nsee for instance Refs. [40–42]. A bi-linear magneto-elastic\ncoupling, which has some similarities to the coupling derived\nin this paper, has also been considered previously [40]. How-\never, all these discussions were based on hydrodynamics ap-\nproaches aiming towards phenomenological descriptions of\nthe macroscopic continuum and mechanisms for coupling be-\ntween magnetic and elastic properties of solids. Such accounts\nare accreditable only in the long wave length limit. We find\nthat there is an apparent lack in the literature of systematic de-\nscriptions addressing the quantum mechanical nature of met-\nals which is responsible for the e \u000bective couplings between\nthe degrees of freedom represented by the spins and lattice at\nan atomistic length scale.\nThe purpose of this Paper is to derive from first principles\na theoretical framework for coupled atomistic magnetization\nand lattice dynamics. In order to treat magnetic and mechani-\ncal degrees of freedom on the same footing, our starting point\nis to formulate the action of the system. From this action\nwe derive, to leading order, bilinear couplings between spins\nand mechanical displacements, couplings which are of three\ndi\u000berent types, namely spin-spin, displacement-displacement\nand the novel bilinear spin-displacement coupling. Further-\nmore we obtain the coupled equations of motion for the me-chanical displacement fQigand velocityfVig, and the mag-\nnetizationfMigdynamics, thus providing a natural extension\nof harmonic lattice dynamics, on the one hand, and the LLG\ndescription of the magnetization dynamics of bilinear spin\nHamiltonians, on the other. The framework is applicable to\ngeneral out-of-equilibrium conditions and includes also retar-\ndation mechanisms.\nIn general terms we address the question whether the elec-\ntrons in a metal that, on the one hand are influenced by the\nionic vibrations, or, phonons, through the electron-phonon\ncoupling and, on the other hand, couple to magnetic mo-\nments via exchange, thereby mediates an interaction between\nthe ionic vibrations and the magnetic moments. With this\nquestion in mind, we derive a general minimal model for the\nmagnetic and mechanical degrees of freedom where the in-\nteractions between the entities are mediated by the underly-\ning electronic structure. We show that the e \u000bective model\ncomprises both the well known bi-linear magnetic indirect ex-\nchange interaction as well as the electronic contribution to\nthe interatomic force constant. However, the derivation also\nshows the existence of a bi-linear coupling between the mag-\nnetic and mechanical entities. The present paper is essentially\nfocused on this derivation and the properties of the bi-linear\nspin-lattice coupling from microscopic theory.\nThe Paper is organized as follows. In Sec. II we derive\na complete and generalized spin-lattice model. The related\nbilinear spin-lattice Hamiltonian and its inherent symmetries\nof the are discussed in Secs. III and IV and a few numerical\nexamples are studied in Sec. V. In Sec. VI we make a brief\ncomparison to expanding the exchange parameters as function\nof spatial coordinates. The dynamics of coupled spin-lattice\nreservoirs are evaluated in Sec.VII and the paper is summa-\nrized in Sec. VIII. Further details are given in the appendix.\nII. DERIVATION OF EFFECTIVE SPIN-LATTICE MODEL\nA. E \u000bective action\nThe e \u000bective action for the coupled spin-lattice system is\nconstructed and analysed. In absence of any ad-hoc coupling\nbetween the spin and lattice subsystems, we address the full\nmicroscopic model of the material through the partition func-\ntion\nZ=eiS; (1)\nwhere the total action Sis given by\nS=S0+Slatt+SWZWN +SB+SE+I\u0010\nHM+Hep\u0011\ndt:(2)\nInstead of expressing all components in mathematical terms\nhere, we discuss the physics involved in each contribution and\nrefer to Appendix A for details.\nAccordingly,S0accounts for the part of the electronic\nstructure that does not directly relate to the localized spin\nmoments Mand lattice displacements Q, whereasSlattpro-\nvides the analogous components for the unperturbed lattice3\nvibrations. As for the latter, we shall not make any assump-\ntions about the model for the lattice dynamics but notice that\nthe mechanism for the coupling between the spin and lat-\ntice subsystems does not depend on the specifics of the lat-\ntice model. Accordingly, the intrinsic lattice vibrations can\nbe treated to any order of accuracy. Furthermore, the Wess-\nZumino-Witten-Novikov component SWZWN accounts for the\nBerry phase accumulated by the spin motion, whereas SBand\nSEcomprise the coupling to the external magnetic and electric\nfields, respectively. Finally, the Hamiltonian HMdescribes\nthe Kondo coupling between the itinerant electron spin ss\u0011\n y\u001b =2 and the localized spin moment MwhileHepprovides\nthe coupling between the electronic charge n=sc\u0011 y\u001b0 and\nthe lattice displacements Q, or in other words, the electron-\nphonon coupling. Here, also  =( \" #)Tis the electron\nspinor,\u001b0is the 2\u00022 identity, and \u001bis the vector of Pauli\nmatrices.\nGiven the above structure we can address both equilibrium\nandnon-equilibrium problems by defining the quantities ap-\npropriately either to a well defined ground state in the former\ncase or by expanding the time integration to the Keldysh con-\ntour and relate the physics to some initial state defined in the\nfar past in the latter. We, therefore, keep the derivation as gen-\neral as possible and choose the latter approach as the generic\none. Despite the additional complexity this route entails, it is\njustified since the equilibrium physics can always be retained\nfrom the non-equilibrium description.\nB. Dynamical bi-linear couplings\nWe obtain the e \u000bective actionSMQfor the coupled magne-\ntization and lattice dynamics through a second order cumulant\nexpansion of the partition function subsequently followed by\ntracing over the electronic degrees of freedom. The resulting\nmodel can be written\nSMQ=\u00001\n2Z\u0010\nQ(x)\u0001[Tcc(x;x0)\u0001Q(x0)+Tcs(x;x0)\u0001M(x0)]\n+M(x)\u0001[Tsc(x;x0)\u0001Q(x0)+Tss(x;x0)\u0001M(x0)]\u0011\ndxdx0;\n(3)\nwhere we have introduced the notation x=(r;t) and defined\nthe interaction tensor\nTpq(x;x0)=Z\n\u0004p(r;\u001a)Kpq(y;y0)\u0004q(\u001a0;r0)d\u001ad\u001a0; (4a)\nKpq(y;y0)=(\u0000i)hTsp(y)sq(y0)i;y=(\u001a;t);p;q=c;s:(4b)\nHere, the parameters \u0004c(r;r0) and \u0004s(r;r0) define the electron-\nphonon and Kondo coupling, respectively, and we have\nadopted the notation where the subscript crefers to charge\nandsto spin.\nThe e \u000bective model given in Eq. (3) can be reduced to an\nanalogous lattice model, the bi-linear Hamiltonian which canbe written as\nHMQ=\u00001\n2X\ni j\u0010\nQi\u0001[Tcc\ni j\u0001Qj+Tcs\ni j\u0001Mj]\n+Mi\u0001[Tsc\ni j\u0001Qj+Tss\ni j\u0001Mj]\u0011\n; (5)\nwhere we denote the magnetic moment centered at the atomic\nposition iasMiand the local atomic displacements as Qi,\nwhere the here instantaneous lattice interactions tensors are\ndenoted as Tpq\ni j.\nThe e \u000bective model presented here, demonstrates the pres-\nence of a bilinear coupling Tsc=csbetween the spin and lat-\ntice subsystems. It also indicates that this coupling is me-\ndiated by the background electronic structure of the material\nin analogous forms as the spin-spin interactions Tssas well\nas the lattice-lattice coupling, or, the electronic contribution\nto the interatomic force constant Tcc. Although this is not\nsurprising, given the set-up of the system, it is nonetheless\nan important observation since it demonstrates the lowest or-\nder of indirect exchange interaction between the spin and lat-\ntice subsystems and, since it is generated by the same inter-\naction field as the spin-spin and lattice-lattice couplings, it is\nexpected to have a non-trivial impact on certain classes of ma-\nterials. It is therefore of utter importance to derive expressions\nfor the spin-lattice couplings in order to both compare to the\nspin-spin /lattice-lattice interactions but also to enable a deeper\nanalysis and understanding of which condition that have to be\nfulfilled to create finite spin-lattice couplings.\nFor the sake of argument we, therefore, decouple the prop-\nagator Kpqinto a product of two single electron Green func-\ntions G, see Appendix A, which are defined by the back-\nground electronic structure, given by the Hamiltonian H0. It\nis then straight forward to derive\nKpq(x;x0)=(\u0000i)sp\u001bpG(x;x0)\u001bqG(x0;x)=2\u000eps+\u000eqs;(6)\nwhere sp denotes the trace over spin space and where \u001bc=\u001b0\nand\u001bs=\u001b.\nNext, since the Hamiltonian can be partitioned into charge\nand spin components according to H0=H(0)\n0\u001b0+H(1)\n0\u0001\u001b, the\nanalogous partitioning can be made for the Green function G\nin terms of charge and spin components G0andG1, respec-\ntively. Thus, we can write G=G0\u001b0+G1\u0001\u001b. Using these two\nobservations, one immediately obtains\nsp\u001bpG\u001bqG=sp\u001bp\u0010\nG0+G1\u0001\u001b\u0011\n\u001bq\u0010\nG0+G1\u0001\u001b\u0011\n:(7)\nBy tracing over the spin degrees of freedom, the nature of the\nlattice-lattice, spin-lattice, and spin-spin interactions can be\nfurther analyzed in terms of the Green function components\nthat constitute the expressions.\nAs one of the purposes with this paper is to construct a co-\nherent formalism for the coupled spin and lattice dynamics,\nwe present the results for all three types of couplings. The\ndetails of the derivations can be found in Sec. III.4\nC. Lattice-lattice coupling\nSetting p=q=cin Eq. (7), the interaction tensor describes\nthe electronic contribution to the interatomic force constant\n\b(x;x0)\u0011T cc(x;x0). Putting the coupling \u0004c(r;r0)=\u0015(r;r0),\nwhere\u0015(r;r0) is the local electron-phonon coupling, see Sec.\nA 2 for more details, the interatomic force constant acquires\nthe form\nTcc(x;x0)=(\u0000i)2Z\n\u0015(r;\u001a)\u0010\nG0(y;y0)G0(y0;y)\n+G1(y;y0)\u0001G1(y0;y)\u0011\n\u0015(\u001a0;r0)d\u001ad\u001a0:(8)\nThe interatomic force constant is therefore a direct measure\nof the total electronic structure to which the lattice vibrations\nare coupled. Moreover, although there is no directionality\ninduced by the spin texture ( G1) in the electronic structure,\nits makes an important contribution to the overall interaction\nstrength. It can also be seen that the tensorial structure of\nthe interactions is governed by the structure factor \u0015of the\nelectron-phonon coupling, as the dyad \u0015\u0015=\u0015i\u0015jˆiˆj.\nD. Spin-spin coupling\nIn case of the spin-spin coupling we put p=q=sin Eq.\n(7), for which we obtain\nM(x)\u0001Tss(x;x0)\u0001M(x0)=J(x;x0)M(x)\u0001M(x0)\n+D(x;x0)\u0001\u0010\nM(x)\u0002M(x0)\u0011\n+M(x)\u0001I(x;x0)\u0001M(x0); (9)\nwhere the three contributions represent the isotropic Heisen-\nberg, and the anisotropic Dzyaloshinskii-Moriya and Ising in-\nteractions, respectively. The order of these contributions is\nnatural since they are the rank 0, 1, and 2 tensors emerging\nfrom the general rank 2 tensor Tss. It should also be noticed\nthat the first (D) and second (I) rank tensors represent the\nanti-symmetric and symmetric contributions to the exchange\n[43]. Similarly as for the interatomic force constant \b, we can\nwrite\nJ(x;x0)=\u0000i\n2Z\n\u0017(r;\u001a)\u0010\nG0(y;y0)G0(y0;y)\n\u0000G1(y;y0)\u0001G1(y0;y)\u0011\n\u0017(\u001a0;r0)d\u001ad\u001a0; (10a)\nD(x;x0)=1\n2Z\n\u0017(r;\u001a)\u0010\nG0(y;y0)G1(y0;y)\n\u0000G1(y;y0)G0(y0;y)\u0011\n\u0017(\u001a0;r0)d\u001ad\u001a0; (10b)\nI(x;x0)=\u0000i\n2Z\n\u0017(r;\u001a)\u0010\nG1(y;y0)G1(y0;y)\n+[G1(y;y0)G1(y0;y)]T\u0011\n\u0017(\u001a0;r0)d\u001ad\u001a0:(10c)\nThese expressions clearly illustrate that the Heisenberg in-\nteraction is finite independently on whether the background\nelectronic structure has a spin texture ( G1) or not, whereasboth the Dzyaloshinskii-Moriya and Ising interactions are fi-\nnite only in materials with non-vanishing spin texture, that is,\neither a simple spin-polarization and /or a non-collinear mag-\nnetic structure. Here, \u0004s(r;r0)=\u0017(r;r0), where\u0017(r;r0) is the\ndirect exchange contribution from the Coulomb integral, see\nSec. A 2 for more details. Eq. (18b) is in agreement with the\nexpression for Dzyaloshinskii-Moriya in Ref. [44].\nThe anti-symmetric properties of Dis also clearly illus-\ntrated by Eq. (18b), since interchanging the spatial coordi-\nnates is accompanied by a sign change, that is, D(r;r0;t;t0)=\n\u0000D(r0;r;t;t0), which signifies the odd property under spa-\ntial reversal. While this property can be obtained, e.g., in\nstructures with finite spin-orbit coupling, it can also be fi-\nnite in general spatially inhomogeneous structures with non-\ncollinear magnetic texture [45]. These observations accord-\ningly suggest that a Dzyaloshinskii-Moriya interaction can be\nengineered in hetero-structures and tunnel junctions [46–49].\nThe Ising interaction, finally, is the symmetric part of the\ntensor and it is finite in materials with a finite spin-polarization\nin the background electronic structure and both for a trivial or\nnon-trivial spin texture [45, 49–51]. Hence, a simple spin-\npolarization along the ˆz-axis generates a finite Izzˆzˆzwhile all\nother components of Ivanish. The contribution to the spin\nmodel then is Izz(x;x0)Sz(x)Sz(x0), which is the usual Ising\nmodel for collinear spins and the reason for calling it the Ising\ninteraction.\nE. Spin-lattice coupling\nHere, we finally discuss the new type of bi-linear interac-\ntion that we propose in this paper, namely, the spin-lattice cou-\npling. Here, we set either p=c,q=sin Eq. (7), or the other\nway around, and for completeness we write both forms given\nby\nTcs(x;x0)=(\u0000i)Z\n\u0015(r;\u001a)\u0010\nG0(y;y0)G1(y0;y)+G1(y;y0)G0(y0;y)\n\u0000iG1(y;y0)\u0002G1(y0;y)\u0011\n\u0017(\u001a0;r0)d\u001ad\u001a0; (11a)\nTsc(x;x0)=(\u0000i)Z\n\u0017(r;\u001a)\u0010\nG0(y;y0)G1(y0;y)+G1(y;y0)G0(y0;y)\n+iG1(y;y0)\u0002G1(y0;y)\u0011\n\u0015(\u001a0;r0)d\u001ad\u001a0: (11b)\nHere, we first notice that the electronically mediated spin-\nlattice coupling exists only in materials with either broken\ntime-reversal symmetry and /or broken inversion symmetry,\nwhich is manifest in the explicit dependence on G1. Sec-\nondly, it can be noticed that the first two contributions to Tcs\nandTscare equal while the third contribution have opposite\nsigns to one another. This structure reflects the composition\nof the tensor into one inversion symmetric and one inversion\nanti-symmetric component.\nIt is, moreover, interesting that the inversion symmetric\ncomponent has an anti-symmetric time-reversal symmetry\nwhile the opposite observation can be made for the inversion\nanti-symmetric component. These properties are necessary\nin order to maintain the even properties of the e \u000bective spin5\nmodel under both inversion and time-reversal symmetry op-\nerations. Hence, the result is that we can interchange the co-\nordinates in, say, the contribution Q(x)\u0001Tcs(x;x0)\u0001M(x0) in\nEq. (3), and from the conclusions in this section it follows\nthat this contribution equals the other spin-lattice contribution,\nsuch that it is only necessary to write 2 Q(x)\u0001Tcs(x;x0)\u0001M(x0)\nin the e \u000bective action. Therefore, the opposite signs of the\ninversion anti-symmetric contributions to TcsandTscensures\nthat the correct symmetries are maintained for the spin-lattice\nmodel.\nFurther aspects regarding the symmetry properties will be\ndiscussed in Sec. IV.\nIII. STATIC BI-LINEAR COUPLINGS\nThe properties of the bilinear couplings Tpq(x;x0) that\nwe have introduced can be further analyzed in the static\nlimit (!!0), that is,Tr\npq(r;r0)\u0011lim!!0Tr\npq(r;r0;!)=\nlim!!0R\nTpq(r;r;t\u0000t0)ei!(t\u0000t0)dt0. Then, the general static in-\nteraction tensor can be written as\nTr\npq(r;r0)=Z\n\u0004p(r;\u001a)Kr\npq(\u001a;\u001a0)\u0004q(\u001a0;r0)d\u001ad\u001a0(12a)\nKr\npq(r;r0)=\u00002\n2\u000eps+\u000eqs\u0019spImZ\nf(\")\u001bpGr(r;r0)\u001bqGr(r0;r)d\";\n(12b)\nwhere the notation Gr(r;r0)\u0011Gr(r;r0;\"). This results is ob-\ntained by noticing that in equilibrium, the retarded suscepti-\nbility Kr\npqcan be written as\nKr\npq(r;r0;!)=1\n2\u000eps+\u000eqsspZf(\")\u0000f(\"0)\n!\u0000\"+\"0+i\u000e\n\u0002\u001bp\u0010\n\u00002ImGr(r;r0)\u0011\n\u001bq\u0010\n\u00002ImGr(r0;r)\u0011d\"\n2\u0019d\"0\n2\u0019:\n(13)\nThen, by application of the Kramers-Kr ¨onig relations, the re-\nsult in Eq. (A9b) follows.\nA tool that is convenient to introduce for further discussion\nis a partitioning of the single electron Green functions accord-\ning to\nG=G0\u001b0+G1\u0001\u001b=(G00+G01)\u001b0+(G10+G11)\u0001\u001b:(14)\nHere, the first superscript 0 (1) refers to charge (spin) quan-\ntities, whereas the second superscript denotes whether the\nGreen function is even, 0, or odd, 1, under space reversal\nor equivalently change of direction r\u001dr0. Then the even\nGreen functions, G00andG10, carry information about the\ncharge and spin densities, respectively, while the odd Green\nfunctions, G01andG11, are related to possible charge and\nspin currents, respectively, that may occur in the system. This\nmeans that only these Green functions may be finite under the\ncurrent operator\u0018r r\u0000rr0in the limit r0!r. In summary,\nthese four Green functions can be characterized in terms of\nbeing even and /or odd under spin and space reversion as is il-\nlustrated in Table I. In this Table we also summarize how they\nTABLE I: Spin dependence and parity properties of the four compo-\nnents in the expansion of the single electron Green function G.\nGreen function spin reversal space reversal time reversal\nG00even even even\nG01even odd odd\nG10odd even odd\nG11odd odd evenbehave under time reversal. Under such an operation not only\nthe spin but also the currents change sign, so G00andG11are\ninvariant under time reversal while G01andG10change sign.\nAn advantage with this formalism is that it becomes straight\nforward to study the e \u000bect of spin-orbit (spin-orbit) coupling.\nThis is because for topologically trivial magnetic systems in\nequilibrium, the odd space reversal Green functions are odd\nin the spin orbit coupling while the even functions are even.\nHence, in the absence of spin-orbit coupling only G00andG10\nwill be finite.\nThis static interaction can for clarity and consistency with\nearlier literature [52, 53] on bi-linear exchange couplings also\nbe expressed in a discrete atomic site or lattice formalism.\nThe\u001aand\u001a0integrations in Eq. (12a) are then taken to be\nover atomic sites iandjand the local interactions \u0004p(r;\u001a) are\nassumed to be on-site only. In order to perform these inte-\ngrals we expand all quantities in local orbitals, e.g. spherical\nor tesseral harmonics, which render all quantities to be matri-\nces in this orbital space, although the local interaction \u0004p\niis\nusually taken to be diagonal. Then we get a lattice represen-\ntation of Eqs. (12a) or (A9b) as\nTpq\ni j=1\n2\u000eps+\u000eqssptr ImZ\nf(\")\u0004p\ni\u001bpGi j\u0004p\nj\u001bqGjid\"; (15)\nwhere the trace is now over both spin (sp) and orbital (tr )\nspace. In this matrix formalism the Green function is a ma-\ntrix over both spin and orbitals. Then when we decompose\nit in the way of Eq. (14) each term is still a matrix over or-\nbitals. This fact lead to that the decomposed Green functions\nare not anymore simply even or odd under change of direction\nor equivalently site exchange. Instead, in case of a real basis\nwe have that\nG00\ni j=fG00\njigT\nG01\ni j=\u0000fG01\njigT\nG10\ni j=fG10\njigT\nG11\ni j=\u0000fG11\njigT; (16)\nwhere the matrix transpose is over the orbitals. For general\ncomplex orbitals this relation will depend on the choice of ba-\nsis, therefore we restrict to real basis in this paper and the ex-\npressions below for the interaction parameters are only valid\nfor this special case.\nA. Lattice-lattice coupling6\nApplying the introduced decomposition of the Green function to the interatomic force constant presented in Eq. (8) we obtain\nthe form\n\bi j=\u00004\n\u0019tr ImZ\nf(\")\u0010\n\u0015iG00\ni j\u0015jG00\nji+\u0015iG01\ni j\u0015jG01\nji+\u0015iG10\ni j\u0015jG10\nji+\u0015iG11\ni j\u0015jG11\nji\u0011\nd\"; (17)\nwhere the products between the Green functions G10andG11\nin the third and fourth term, respectively, should be consid-\nered as scalar products. The presence of the spin-dependent\ncomponents shows that also the spin texture in the material\ncan have a crucial influence on the lattice-lattice coupling in\nthe material, which lead to the well-known fact that the atomic\nforces will be spin dependent for a magnetic system.\nB. Spin-spin coupling\nAs displayed in Eq. (10) the indirect spin-spin exchange can\nbe partitioned into three contributions: isotropic Heisenberg,anisotropic Dzyaloshinskii-Moriya and Ising interactions. By\napplication of the Green function decomposition introduced,\nwe find that these three interactions in the static limit can be\nwritten as\nJi j=\u00001\n2\u0019tr ImZ\nf(\")\u0010\n\u0017iG00\ni j\u0017jG00\nji+\u0017iG01\ni j\u0017jG01\nji\u0000\u0017iG10\ni j\u0001\u0017jG10\nji\u0000\u0017iG11\ni j\u0001\u0017jG11\nji\u0011\nd\"; (18a)\nDi j=\u00002\n\u0019tr ReZ\nf(\")\u0010\n\u0017iG00\ni j\u0017jG11\nji+\u0017iG01\ni j\u0017jG10\nji\u0011\nd\"; (18b)\nIi j=\u00002\n\u0019tr ImZ\nf(\")\u0010\n\u0017iG10\ni j\u0017jG10\nji+\u0017iG11\ni j\u0017jG11\nji\u0011\nd\": (18c)\nFirst, it is important to notice that the three contributions\nare given as a scalar ( J), vector ( D), and a dyad ( I), as would\nbe an expected partitioning of a second rank tensor. These\ninteractions are closely related to other expressions for Jand\nDin the literature [43, 53, 54], now expressed in decomposed\nGreen functions. Second, we notice that since G00orG10\nare always present in a magnetic systems, the Dzyaloshinskii-\nMoriya interaction can be finite only when either G11orG01\ndo not vanish. As mentioned above these two functions vanish\nin the absence of spin-orbit coupling for topologically trivial\nmaterials in equilibrium. Third, it is important to observe thatthe Ising interaction in its most general form, as here, is repre-\nsented by a dyad and due to its first term can be non-vanishing\nalso in the non-relativistic limit without spin-orbit coupling.\nC. Spin-lattice coupling\nFinally, the spin-lattice interactions in the static limit Tcs\ni j\nderived from Eq. (11a), can in terms of the four Green func-\ntions in Table I be written as\nTcs\ni j=\u00004\n\u0019tr ImZ\nf(\")\u0010\n\u0015iG00\ni j\u0017jG10\nji+\u0015iG01\ni j\u0017jG11\nji\u0000i\u0015iG10\ni j\u0002\u0017jG11\nji\u0011\nd\": (19a)\nIt is easily seen that the tensor Tscis related to this tensor by\nthe transpose\nn\nTsc\ni jo\u000b\f=n\nTcs\njio\f\u000b; (20)with the explicit Cartesian tensor components \u000band\f. The\nTcstensor interactions can further be partitioned into two in-7\ndependent terms, Tcs=S+A, with\nSi j=\u00004\n\u0019tr ImZ\nf(\")\u0010\n\u0015iG00\ni j\u0017jG10\nji+\u0015iG01\ni j\u0017jG11\nji\u0011\nd\";\n(21a)\nAi j=\u00004\n\u0019tr ReZ\nf(\")\u0015iG10\ni j\u0002\u0017jG11\njid\": (21b)\nThen it is noteworthy that the Sinteraction is even in the\nspin-orbit coupling strength while the Ain contrast is odd.\nHence, for systems with weak spin-orbit coupling, the first in-\nteraction is expected to dominate if it is allowed by symmetry.\nIt is straight-forward from Eq. (21) to verify that the first term\nis symmetric with respect to site exchange Si j=Sjiwhile the\nsecond is anti-symmetric Ai j=\u0000A ji, by using the relations\nfor the decomposed Green functions of Eq. (16).\nIV . SYMMETRIES\nWe want to study the symmetry of the spin-lattice part of\nthe static interaction Eq. (5), i.e. the heterogenous part\nHsl\nMQ=\u00001\n2X\ni jn\nQi\u0001Tcs\ni j\u0001Mj+Mi\u0001Tsc\ni j\u0001Qjo\n: (22)\nThe fact that the two quantities entering this bi-linear form\nhave di \u000berent symmetries might cause some confusion. The\nlattice distortion Qiis even under time reversal \u0012but odd un-\nder space inversion \u0013while the magnetic moment Miis in-\nvariant with respect to space inversion but change sign underoperation of time reversal. Hence since the interaction en-\nergy is scalar, the interaction coe \u000ecients have to be odd under\nboth space inversion and time reversal which single out the\nheterogenous bi-linear spin-lattice interaction compared to the\nhomogenous bi-linear spin-spin fTssgand lattice-lattice fTccg\ninteractions, that are both invariant under these operations.\nHowever, when accepting this di \u000berence there is nothing that\nforbid such heterogenous interactions, as will be demonstrated\nbelow, first through derivation of explicit expressions for these\ninteraction parameters and then by considering the symmetry\nof the interactions. The odd time reversal property is simply\nstated as\u0012Tcs\ni j=\u0000Tcs\ni jwhile the space inversion has to be dis-\ncussed in more details below.\nFirst we notice for each pair fi jgof sites we have four inter-\naction terms\nQi\u0001Tcs\ni j\u0001Mj+Qj\u0001Tcs\nji\u0001Mi+Mi\u0001Tsc\ni j\u0001Qj+Mj\u0001Tsc\nji\u0001Qi:\n(23)\nThen from the relation (20)\nQi\u0001Tcs\ni j\u0001Mj+Mj\u0001Tsc\nji\u0001Qi=2Qi\u0001Tcs\ni j\u0001Mj; (24)\nso the total spin-lattice interaction written as a sum over pairs\nbecomes\nHsl\nMQ=\u0000X\nfi jgX\n\u000b\f\u0010\nfTcs\ni jg\u000b\fQ\u000b\niM\f\nj+fTcs\njig\u000b\fQ\u000b\njM\f\ni\u0011\n:(25)\nNow we can decompose the pair interaction into a part that is symmetric Si jand one that is antisymmetric Ai jwith respect to\ninterchange of sites, i.e. with Tcs\ni j=Si j+Ai jwe have that Tcs\nji=Si j\u0000A i j. Then Eq. (25) becomes\nHsl\nMQ=\u0000X\nfi jgX\n\u000b\fh\nfSi jg\u000b\f\u0010\nQ\u000b\niM\f\nj+Q\u000b\njM\f\ni\u0011\n+fAi jg\u000b\f\u0010\nQ\u000b\niM\f\nj\u0000Q\u000b\njM\f\ni\u0011i\n=\u0000X\nfi jg\bQi\u0001Si j\u0001Mj+Qj\u0001Si j\u0001Mi+Qi\u0001Ai j\u0001Mj\u0000Qj\u0001Ai j\u0001Mi\t(26)\nIn contrast to the homogeneous bi-linear interactions these interaction parameters Si jandAi jare both general rank two tensors\nin 3D space and can hence both be decomposed into three contributions, scalar ( Si jandAi j), vector ( Si jandAi j) and symmetric\nsecond rank tensor interactions ( S(2)\ni jandA(2)\ni j). By first decomposing these interaction tensors into symmetric and anti-symmetric\npartsSi j=sSi j+aSi jwith respect to exchange of components, the interaction energy can be expressed as\nHsl\nMQ=\u00001\n2X\nfi jg\u000b\fn\nsS\u000b\f\ni j\u0010\nQ\u000b\niM\f\nj+Q\f\niM\u000b\nj+Q\u000b\njM\f\ni+Q\f\njM\u000b\ni\u0011\n+sA\u000b\f\ni j\u0010\nQ\u000b\niM\f\nj+Q\f\niM\u000b\nj\u0000Q\u000b\njM\f\ni\u0000Q\f\njM\u000b\ni\u0011\n+\n+aS\u000b\f\ni j\u0010\nQ\u000b\niM\f\nj\u0000Q\f\niM\u000b\nj+Q\u000b\njM\f\ni\u0000Q\f\njM\u000b\ni\u0011\n+aA\u000b\f\ni j\u0010\nQ\u000b\niM\f\nj\u0000Q\f\niM\u000b\nj\u0000Q\u000b\njM\f\ni+Q\f\njM\u000b\ni\u0011o\n=\n=\u0000X\nfi jgn\nSi j\u0010\nQi\u0001Mj+Qj\u0001Mi\u0011\n+Ai j\u0010\nQi\u0001Mj\u0000Qj\u0001Mi\u0011\n+Si j\u0001\u0010\nQi\u0002Mj+Qj\u0002Mi\u0011\n+Ai j\u0001\u0010\nQi\u0002Mj\u0000Qj\u0002Mi\u0011\n+:::o\n=\u0000X\ni j\u0010\nQi\u0001(Si j+Ai j\u0011\n\u0001Mj=\u0000X\ni j\u0010\nSi j+Ai j\u0011\nQi\u0001Mj\u0000X\ni j\u0010\nSi j+Ai j\u0011\n\u0001Qi\u0002Mj\u0000X\ni jQi\u0001\u0010\nS(2)\ni j+A(2)\ni j\u0011\n\u0001Mj; (27)\nwhere the dots refer to the for moment neglected second rank contributions and note that in the last line we do the full site8\nsum again. Scalar and vector interactions have been intro-\nduced in line with conventions. The scalar interactions are\nrelated to the trace of the symmetric part of the tensors, while\nthe vector interactions are the dual form of the anti-symmetric\npart of the tensors. So for the symmetric tensor Si jwe de-\ncompose it in terms of\nSi j=1\n3TrsSi j; (28)\nand\nS\r\ni j=Si j\u0001ˆ\r=1\n2X\n\u000b\f\u000f\u000b\f\raS\u000b\f\ni j: (29)\nwhere\u000f\u000b\f\ris the anti-symmetric Levi-Civita symbol and ˆ \ris\nthe unit vector along Cartesian axis \r. Finally the second rank\ntensor interactions S(2)\ni jis given as\nS(2)\ni j=sSi j\u0000Si j1; (30)\nwhere 1is the 3D unit matrix.\nIn order to discuss the symmetry under space inversion, let\nus consider that the inversion operation \u0013brings site ito an\nequivalent site i0and correspondingly for site j. In Appendix\nD it is shown that in this case both spin-lattice interaction ten-\nsors are indeed odd under space inversion, i.e.,\n\u0013Si j=\u0000S i0j0\n\u0013Ai j=\u0000A i0j0: (31)\nFor the special case where there exists a center of inversion at\nthe bond center in between sites iandj, inversion brings site\nito site jand\n\u0013Si j=\u0000S ji=\u0000Si j\n\u0013Ai j=\u0000A ji=Ai j; (32)\nHence in this case the interaction tensor Si jhas to vanish. If\ninstead there exist a bond center invariant under the combined\noperation of space inversion and time reversal \u0013\u0012, then instead\n\u0013\u0012Si j=Sji=Si j\n\u0013\u0012Ai j=Aji=\u0000A i j; (33)\ni.e.Ai jhas to vanish.\nThis reminds about the fact that the Dzyaloshinskii-Moriya\ninteraction Di jof Eq. (18b), also vanishes if there is an in-\nversion symmetry at the bond center. However, a di \u000berence\nis that the Dzyaloshinskii-Moriya interaction is even under\nthe inversion per se. It is the asymmetry under site exchange\nwhich makes it vanish, \u0013Di j=Dji=\u0000Di j.\nIn the full magnetic symmetry group the elements generally\nconsist of combined operations, e.g. rotations and inversion\nor rotations and time reversal etc as illustrated in the exam-\nples below. The rotational part of this operation behaves as\nexpected, either on the full interaction tensor or the scalar and\nvector interactions in its decomposition, while as noted both\ninversion and time reversal operations are odd for the spin-\nlattice interaction.Finally it is important to remember that for the heteroge-\nnous spin-lattice interaction the inter-site exchange symme-\ntry is unrelated to the symmetry of the tensor. So the inter-\naction contribution that is symmetric in site exchange, Si j,\ncontributes both to the scalar interaction Qi\u0001Mjas well as\nthe cross product interaction Qi\u0002Mj. This is in contrast to\nthe homogeneous spin-spin interaction where the interaction\nsymmetric in sites, e.g. Heisenberg, only contributes to the\nsymmetric scalar interaction Mi\u0001Mjetc. Anyhow we have\nchosen to di \u000ber between the two contributions as they be-\nhave di \u000berently with the strength of the spin-orbit coupling.\nThe symmetric interaction Si jexists also in absence of spin-\norbit coupling while the anti-symmetric Ai jis linear in a weak\nspin-orbit coupling strength as shown by Eq. (21).\nV . EXAMPLES\nA. Numerical Details\nThe bilinear couplings (A9a) are implemented in our real\nspace tight binding code [92]. Here, we solve the non-\northogonal eigenvalue problem H =\"O where is a linear\ncombination of atomic orbitals (LCAO ansatz) within a sp3d5\norbital basis set. The Hamiltonian H0and the overlap ma-\ntrixOare build up from the Slater-Koster scheme [55], where\nthe Slater-Koster parameter are consider distance dependent\naccording to the formalism of Mehl et al. [56, 57]. The\nfull HamiltonianH=H0+Hsoc+Hmagincludes also spin-\norbit couplingHsoc=\u0018L\u0001Sand magnetic exchange split-\ntingHmag=I\n2M\u0001S, respectively. Both the spin-orbit cou-\npling parameter \u0018and the Stoner excitation energy Iare ob-\ntained from fitting of the electronic structure to ab-initio band\nstructures obtained from a full-relativistic multiple scattering\nGreen’s function method (Korringa-Kohn-Rostoker method,\nKKR)[58].M=mesis the spin magnetic moment. Magnetic\nmoment rotations come from a unitary transformation of the\nHamiltonian with relativistic rotation matrices R, consisting\nof rotations in spin and orbital space [59]. Variations of the\nmagnetic moment es=es(\u0012;\u001e) are addressed by @H=@\u0012iand\n@H=@\u001ei. A local approximation for \u0015iis used by the deriva-\ntive of the Hamiltonian @H=@Qidue to lattice degrees of free-\ndomQi, obtained from Ref. [60]. In the simulations we focus\non low dimensional clusters of Fe, e.g., chains, with periodic\nboundary conditions, where the tight binding parameters are\nfrom Refs. [31, 61, 62].\nSince pure spin and lattice exchange couplings [63–66] are\nalready well understood, we will focus in the following only\non the bilinear spin-lattice coupling mechanism, and then es-\npecially the influence on the lattice from the spin order.\nB. Double anti-ferromagnetic lattice\nIt is discussed in literature [67] that the magnetic ground\nstate in fcc Fe is double anti-ferromagnetic. It is collinear with\nall moments along, say, the ˆ z-direction, where the variations\nalong, say, the ˆ x-direction, is\"\"## and translations of this unit9\ncell (cf. Fig. 2). The symmetry group for this spin structure is\nfe;\u0013;\u0012t2;\u0013\u0012t2g\nT, where T=fnt4;n2Zgis all pure translations\nof the unit cell and t2is a non-trivial translation by two sites.\ne,\u0013, and\u0012are the identity, inversion and the spin (time) rever-\nsal operator, respectively. Note that this choice of symmetry\ngroup is quantization axis free and, consequently, suitable for\nnon-relativistic treatment. The inversion center can be chosen\nas in between atoms 1 and 2 or equivalently in between atoms\n3 and 4 (see Section IV).\nWithout spin-orbit coupling, rotational variation of the\nmagnetic moment \u000e\u0012;\u000e\u001e makes\u0017(r;\u001a) in Eq. (18) propor-\ntional to the Pauli matrices \u001bx;\u001by. Hence, they do not con-\ntribute to spin-lattice coupling due to the spin-diagonal from\nof the Green’s function. It turns out that for the double anti-\nferromagnetic structure Tcs\ni jis related to longitudinal fluctua-\ntions of the magnetic moments, which is proportional to \u001bz\n(Fig. 2). To apply the group symmetry analysis, it is useful\nto treat the couplings to be at the center of the bonds between\natoms (cf. Fig. 2). Here, the symmetric scalar interactions\nSi jvanish at the bond centers 1-2 and 3-4, due to inversion.\nHowever in between 4-1 and 2-3 they can exist and are re-\nlated by\u0012t2, i.e. S41=\u0000S23=s. So there will be forces\nFi=\u0000@Hsl\nMQ=@Qion all four atoms\nF1=\u0000S14m4= +s\nF2=\u0000S13m3=\u0000s\nF3=\u0000S32m2= +s\nF4=\u0000S41m1=\u0000s; (34)\nwhich leads to a dimerization; atoms 1 and 2, respectively,\n3 and 4, move towards each other (cf. Fig. 2 (a) - black\narrow). This was also approved numerically (Fig. 2 (a))\nby comparing di \u000berent collinear magnetic textures, a ferro-\nmagnetic (FM), anti-ferromagnetic (AFM), and double anti-\nferromagnetic (DAFM) structure.\nNote that the magnetic moment length is set to 0 :001\u0016B\nfor a proper ground state description. Tcs\ni jscales linear the\nmoment length; thus the nearest neighbour coupling TNNis\n\u001910eV for the magnetic moment length m=2:3\u0016Bfor Fe.\nIn the first two cases, say FM and AFM, the exchange Tcs\ni j\nis antisymmetric around zero and, consequently, no net-force\nexists. However, it e \u000bects the dynamics of the spin and lattice\ndegree of freedom. Oscillations occur for the AFM structure\nwhich is linked to the alternating spin state. Tcs\ni jin the DAFM\nis not antisymmetric around the origin, but around the bond\ncenter, described in our symmetry analysis. This originates an\nalternating finite force of F=0:39\u0016eV=a:u:(F=1:33eV=a:u:\nfor finite moment) between the atoms and causes dimerization\nof the atoms.\nC. Planar spin density waves\nIn the previous case we kept the crystal and spin structure\nto be simple. If we extend the two magnetic structures to an\ninfinite spiral, represented by\nMi=Mzˆzcos(qxi)+Meˆesin(qxi) (35)\n-40-30-20-10010203040Tcs\nij(µeV/a.u.)\n-10 -5 0 5 10\nrij/aDAFM.\nFM.\nAFM.(a)\n(b)1 2 34FIG. 2: (Color online) (a)Magnetic moment structure (bold arrows)\nand related forces (black arrows) coming from bilinear spin-lattice\ncoupling for the double antiferromagnetic. Atoms are indicated by\ngray balls. The color of the magnetic moments indicate the orienta-\ntion (ˆ z- green arrow;\u0000ˆz- red arrow). (b)Bilinear spin-lattice cou-\npling vs. distance along the variation \"\"## for di \u000berent magnetic\nstates: ferromagnetic (FM, green dots), anti-ferromagnetic (AFM,\nred dotes), and double anti-ferromagnetic (DAFM, blue dotes).\nwhere xiis the x-component of the position of atom i,ri,qis\nthe magnitude of the wave vector q=qˆxand ˆeis either i) ˆ x\norii)ˆy. For Me=0 the magnetic structure would correspond\nto a sinusoidal spin density waves (sSDW) (Fig. 3 a). Here,\ntwo phases are possible, either with a belly or node at x0=0,\nrespectively. We notice that the symmetry groups for the si-\nnusoidal spin density wave is for the bellyn\ne;c2z;\u0012c2x;\u0012c2yo\n\u0002\nfe;\u0013gand for the noden\ne;c2z;\u0012c2x;\u0012c2yo\n\u0002fe;i\u0012g. Here, cn\u0017de-\nfines the n-fold rotation axis along \u0017. Note that the sinusoidal\nmagnetic structure is invariant with respect to \u0013or\u0013\u0012for the\nbelly or nodal type, respectively.\nLet us focus on the symmetric scalar interaction for the\nbelly sinusoidal SDW with a node at qx=0 and a maximum at\nqx=\u0019=2. Thus (not shown here), the symmetric scalar inter-\naction behaves as sj=ssinq(xj+d=2) and the force at atom j\ndue to its nearest neighbour interactions are,\nFj=\u0000\u0010\nSj j\u00001Mj\u00001+Sj j+1Mj+1\u0011\n=\u00002sMsin2qxjcosqd=2; (36)\nwhere dis the distance between two atoms. These forces os-\ncillate with 2 q, however they disappear for qd=2=\u0019=2, i.e. for\nqd=\u0019which correspond to a commensurate AFM, where the\nvariation of magnetic moments disappear, i.e. mj=0. Note\nthat we recounter the double layered AF for qd=\u0019=2 if the10\n-0.4-0.3-0.2-0.10.00.10.20.30.4Fx(eV/Bohr radii)\n0 5 10 15 20 25 30\nr/amag. UC(a)\n(b)\nFIG. 3: (Color online) a) Magnetic moment structure (bold arrows)\nfor the planar spin density wave. b) Lattice forces at di \u000berent po-\nsition in a sinusoidal spin density wave calculated from the force\nrelated to the bilinear spin-lattice coupling (blue triangles) and from\nthe Hellmann-Feynman theorem (red dots). The oscillation period\nextends over 10 atoms, indicated by the vertical dotted line. Lines\nare edit to guide the eye.\nphase shift the sSDW with \u001e=\u00003d=2. This periodicity is also\nrecovered by our numerical method (3 b).\nNote that the calculations are done for a finite magnetic mo-\nment of 2:23\u0016Band all neighbours contribute to the summa-\ntion needed to get the force from the bilinear coupling term.\nThis results in slight variation from the sin-like behavior of\nthe force observed from group symmetric analysis. The peri-\nodicity of 2 q, however, was reproduced. The obtained forces\nare in good agreement with the forces obtained directly from\nHellmann-Feynman theorem. Disparities are due to higher\norders exchange couplings that are included in the Hellmann-\nFeynman force as well as due to long-range exchange.\nD. Cycloidal and helical spin density wave\nContinuing the discussion about spiral spin configurations\n(35), we set Mz=Me=M, which correspond to either a cy-\ncloidal spiral ˆ e=ˆxor helical spiral ˆ e=ˆy, respectively, with\nthe symmetry groupsn\ne;c2z;i\u0012c2x;i\u0012c2yo\nfor the cycloid andn\ne;c2z;\u0012c2x;\u0012c2yo\nfor the helix state.\na. Cycloid For a general position of an atom jatrj,\nthe bond center to the nearest neighbour is conserved by the\ngroupn\ne;\u0013\u0012c2yo\nand allows both a scalar Tcs;s\nj j\u00061and a vectorial\ncoupling along y,Tcs;v\nj j\u00061\u0002ˆy=0. We assume the spiral to be\ncommensurate and point to the atomic position rnsuch that\nqxn=\u0019=2. Caused by the symmetry operation \u0012c2z, the sign\nof the nearest neighbour scalar interaction, Tcs;s\nnn+1=\u0000Tcs;s\nnn\u00001,\nchanges, while the nearest neighbour vector interaction does\nFIG. 4: (Color online) Magnetic moment structure (bold arrows) for\nthe cycloidal spin wave.\nnotTcs;v\nnn+1=Tcs;v\nnn\u00001; it behaves opposite as around the point r0.\nThis can be explained only by an oscillatory behaviour of the\ninteraction parameters along the wave vector.\nThere are two possibilities, either the interactions are sym-\nmetric (see section IV), Sj j+1andSj j+1, or anti-symmetric,\nAj j+1andAj j+1. This gives rise to the following forces on the\natom atrjarising from nearest neighbour interactions due to\nlast part in Eq. (27):\nFS s\nj=\u0000\u0010\nSj j\u00001Mj\u00001+Sj j+1Mj+1\u0011\n=\u0000sMe(qxj)\u00022cos( qd)\u00001\u0003cos(qd=2) (37a)\nFS v\nj=\u0000\u0010\nMj\u00001\u0002Sj j\u00001+Mj+1\u0002Sj j+1\u0011\n=syMn\u00022cos( qd)\u00001\u0003e(qxj)+ˆzcos(qd)o\ncos(qd=2)\n(37b)\nFAs\nj=\u0000\u0010\nAj j\u00001Mj\u00001+Aj j+1Mj+1\u0011\n=\u0000aMe(qxj)\u00022cos( qd)\u00001\u0003cos(qd=2) (37c)\nFAv\nj=\u0000\u0010\nMj\u00001\u0002Aj j\u00001+Mj+1\u0002Aj j+1\u0011\n=\u0000ayMn\ne(qxj)(2cos( qd)+1)+ˆzo\nsin(qd=2);(37d)\nwheree(qxj)=n\nˆzcos(2 qxj)+ˆxsin(2 qxj)o\n. Here, sandaare\nthe magnitude of the oscillating antisymmetric and symmet-\nric, scalar and vectorial couplings. The symmetric scalar force\n(37a) does not vanish in the limit q!0. The symmetric vector\n(37b) and anti-symmetric vector force (37d) vanishes in the\nlimit q!0, but has otherwise in addition to the oscillations\nalso a constant term in ˆ z-direction, where the z-component\ngoes asf(2cos qd+1)cos(2 qxj)+1gsinqd=2.\nb. Helix For the helical spin spiral state, the anti-\nsymmetric interaction has two non-vanishing components,\nsince\u0012c2xAi j=Ai j, and from the symmetry relations at\nqx=0 and qx=\u0019=2 they have to exhibit also an oscillatory\nbehaviour. This leads to a force at atom jas\nFAv\nj=\u0000\u0010\nMj\u00001\u0002Aj j\u00001+Mj+1\u0002Aj j+1\u0011\n=\u0000Mn\n(cosqd+1)(ay+az)\u0000ayo\nsin(qd=2)sin(2 qxj) ˆx;\n(38)\nwhich is also purely oscillatory.\nTo summarize, for the magnetic textures with wave vector\nqdiscussed in Secs. V C and V D, we observe an oscillating\nforce with the double wave vector. In particular for the cy-\ncloidal spin wave, we obtained a constant force in addition to\nthe oscillating force. When the conical wave has xzas the ro-\ntational plane and xas propagating vector, this “o \u000bset force”11\nwill be along the z-direction. This is in good accordance with\nthe inverse Dzyaloshinskii-Moriya e \u000bect discussed by Kat-\nsuraet al. [68], Mostovoy [69], and Sergienko et al. [70], who\ndemonstrated that a cycloidal spiral gives rise to a polarization\nP/(ˆz\u0002ˆe)\u0002qwith contributions both from electronic charge\ndisplacement [68, 70] and from ionic displacement [69, 70].\nThis ferroelectric polarization for cycloidal spirals is unique,\nsince neither a helical spiral nor sinusoidal spin wave states\ngive rise to polarization.\nVI. COMPARISON WITH EXPANSION OF SPIN\nEXCHANGE PARAMETERS\nAs mentioned before this bilinear formulation of spin-\nlattice coupling di \u000ber from the standard approach. In the stan-\ndard formulation the e \u000bective model hamiltonian correspond-\ning to Eq. (5) takes the form\neHMQ=\u00001\n2X\ni j\u0010\nQi\u0001Tcc\ni j\u0001Qj+Mi\u0001˜Tss\ni j[fQg]\u0001Mj\u0011\n;(39)\nwhere ˜Tssdepend on all the ionic displacements fQg. Such\nan expression gives that there is a contribution Fsc\nkto the total\nforce on site kfrom an e \u000bective spin lattice coupling,\nFsc\nk=1\n2X\ni jMi\u0001@˜Tss\ni j\n@Qk\u0001Mj; (40)\nwhich in general involves a double sum and can be fairly\ncumbersome to calculate. However, physically such a deriva-\ntive can be analyzed in some simple limits. First, in case of\npure Heisenberg exchange in nearest neighbor approximation\nwhere the isotropic exchange Jparameter is dependent on the\ndistance between the two atoms, the exchange tensor can be\nwritten as\n˜Tss\ni j[fQg]=J(jRi j+Qi\u0000Qjj)1; (41)\nwith the unit tensor 1. For such a model the force of Eq. (40)\nis only non-vanishing for the two interacting atoms and leads\nto a derivative\n@˜Tss\ni j\n@Qi=\u0000@˜Tss\ni j\n@Qj\u0019J0(jRi jj)1bRi j: (42)\nThe resulting force is in the direction as to gain in Heisenberg\nexchange energy. Such a force give rise to qualitatively similar\nresults as the present method in the examples of double anti-\nferromagnet in V B and sinusoidal spin density wave in V C.\nSecond, in the case of the anisotropic Dzyaloshinskii-Moriya\ninteraction DDMbetween two magnetic sites iand jover a\nbridging ligand site k, the interaction vector can in the super-\nexchange approximation be written as [70]\nDDM\u0019DRi j\u0002Qk; (43)\nwhich gives rise to a force on the ligand atom\nFsc\nk=1\n2DRi j\u0002\u0010\nMi\u0002Mj\u0011\n: (44)This result is in qualitative agreement with the present result\nof the cycloid in V D. In this case Ri jlies in the plane spanned\nbyMiandMjand a resulting non-oscillating force would\nbe in the same plane but perpendicular to the bond direction,\ni.e. what is called ˆ zin the example above.\nTo conclude this section we note that in those insulat-\ning magnets where the spin texture simultaneously breaks\ntime and spatial reversion, third order spin lattice coupling in\nEq. (40) is commonly considered when describing ferroelec-\ntric polarization and multiferroic phases [68–71], and also to\nbe responsible for the dynamic magneto-electric response in\nthe electromagnetic field driven dynamics in the GHz and THz\nregime [11, 12, 72, 73]. Hopefully, we have here made plau-\nsible that the same e \u000bects can also be treated in a bi-linear\nspin-lattice coupling, but a more direct comparison of the two\ndi\u000berent approaches is left for future studies.\nVII. EQUATIONS OF MOTION\nA. General dynamical equations\nHere, we make a brief derivation of the equations of mo-\ntion that can be obtained from the e \u000bective action in Eq. (3).\nHence, in order to access the physics in the spin-lattice sys-\ntem we have to convert the time-integration on the Keldysh\ncontour to real times. While all steps in the conversion are\nshown in Appendix B, we here notice that the transformation\nleads to a natural introduction of slow and fast spin and lattice\nvariables which, in principle, have to be treated coherently\nfor a complete description. Nevertheless, here we will only\naddress the dynamics of the slow variables in presence of a\nmean field generated by the fast variables. Accordingly, by\ndi\u000berentiating the e \u000bective action with respect to the fast vari-\nables we can retain a description solely in the slow variables.\nThe conversion to real times does, however, introduce contri-\nbutions to the model which are quadratic in the fast variables,\nsee Appendix B, such that there remain contributions in the\ndescription explicitly depending on these even after di \u000berenti-\nating. The simplest solution to this problem is to neglect their\nexistence under the assumption that their overall contribution\nto the dynamics is negligible. While this approach is some-\nwhat uncontrolled and non-systematic, the equations of mo-\ntion presented in the main text are obtained in this fashion. A\nmore sophisticated and controlled way to deal with this issue\nis by application of the Hubbard-Stratonovich transformation,\nsee Appendix C, which leads to that the quadratic terms are\nreplaced by linear ones, however, at the cost of introducing\nrandom fields corresponding to quantum fluctuations related\nto the quadratic spin and lattice interactions.\nHere, we adopt the former approach and refer to Appendix\nC for the details concerning inclusion of the quadratic terms.\nOur strategy can be justified from the perspective that we here\naim to address the general structure of the coupled equations\nof motion for the spin-lattice system with focus on the con-\ntribution that arise from the bi-linear coupling between these\nsubsystems. The resulting equations of motion can be gener-\nalized to also include stochastic field of, e.g., Langevin type12\nboth addressing the quadratic interaction but also randomness\ncaused by temperature among others. We refer to Appendix\nC for a discussion of quantum fluctuations caused by rapid\nspin-spin correlations.\nIt should also be noticed that through the conversion into\nreal times the interaction fields Tpqare transformed into re-\ntarded /advanced forms,Tr=a\npq, which are naturally accessible\nfrom electronic structure calculations in terms of the Green\nfunctions, see Sec. II. In this form, we obtain a practical and\nconvenient method to systematically address spin and lattice\ndynamics at the same level of sophistication and approxima-\ntion.\nTaking the saddle point solution of the total e \u000bective spin-\nlattice action with respect to the fast spin and displace-\nment variables, see Appendix B for details, and requiring\n@tjM(x)j2=0 for the spin variable, we derive a set of coupled\nequations of motion given by\n˙M(x)=M(x)\u0002\u0014\n\u0000\rBext(x)\n+Z\u0010\nTr\nsc(x;x0)\u0001Q(x0)+Tr\nss(x;x0)\u0001M(x0)\u0011\ndx0\u0015\n;\n(45a)\nMion¨Q(x)=\rEEext(x)+Z\nVrr0\u0001Q(x0)\u000e(t\u0000t0)dx0\n+Z\u0010\nTr\ncs(x;x0)\u0001M(x0)+Tr\ncc(x;x0)\u0001Q(x0)\u0011\ndx0;\n(45b)\nin the presence of external magnetic and electric fields Bext\nandEext, respectively. Here, ˙M\u0011@tMand ¨Q\u0011@2\ntQ, whereas\nthe dyad Vrr0\u0011rr(rr0V0) represents the ionic contribution to\nthe interatomic force constants . The system in Eq. (45) for\nMandQprovides a general framework for a coupled treat-\nment of magnetization and lattice dynamics. One should note\nthat Eq. (45) emphasizes that the temporal and spatial evolu-\ntion of both QandMdepend non-locally on both the time-\ndependent magnetization and ionic displacements for the en-\ntire structure. The consequence of this non-local description\nis that all retardation e \u000bects within the spin-lattice system that\nare associated with their coupling to the electronic structure\nare included in Eq. (45), despite the seemingly absence of\ncontributions arising from, e.g., damping and moment of iner-\ntia [26]. Conceptually, these and other retardation e \u000bects are\nincluded in the full integration over space and time, however,\nas we shall see in Sec. VII B it can be shown that damping\nand moment of inertia are related to temporal expansion of\nthe spin moments. Analogously, the spin-transfer torques can\nbe related to gradient expansion of the magnetization. In this\ncontext it is interesting to observe that the time evolution of a\nlocal mode [74], is non-locally influenced by the magnetiza-\ntion at di \u000berent points in space and time. Due to the coupling\nit can, moreover, be concluded that the ionic dynamics can\nbe controlled by external magnetic fields, e.g., Bext(x), some-\nthing that was experimentally demonstrated in Ref. [75], and\nreciprocally that magnetic ordering can be driven by electric\nfields, such as for instance when the electric component of a\nTHz electromagnetic pulse couple to a dipole active phononmode and excite electromagnons [11, 72, 73].\nHere it is worth to point out that the uncoupled version of\nEq. (45b), which describes the ionic vibrations, or, phonons\nis related to the linear response equations commonly used for\nsuch calculations [76]. At first glance they look di \u000berent, but\nit easy to show that they are closely connected to one for-\nmulation of linear response, the so-called dielectric approach\n[77, 78].\nThe equations of motion presented in Eq. (45) represent\na generalized form of the equations of motion typically used\nin practical simulations and we will address this issue in Sec.\nVII B. Before entering the next level of approximations, how-\never, it is useful to discuss the general structure of the derived\nequations.\nThe first observation one can make is that one retains\nthe uncoupled equations of motion whenever the interaction\nfieldsTr\nsc=cs!0. In this limit, respective descriptions for\nlattice and spin dynamics are recovered, however, here pro-\nvided in a more generalized form since the full retardation\n(memory) is included in the equations of motion. Secondly,\nwe notice that the coupling termsR\nTr\nsc(x;x0)\u0001Q(x0)dx0andR\nTr\ncs(x;x0)\u0001M(x0)dx0essentially add the e \u000bect of an addi-\ntional magnetic and electric field to the respective equation.\nThese fields are, however, strongly dependent on the proper-\nties contained in the interaction tensors Tr\nsc=csand their cou-\nplings to the ionic displacements Qand magnetic moments\nM. The meaning of the statement lies in the fact that these\nfields may be possible to control through the properties of the\nelectronic structure. In e \u000bect, it also leads to that these in-\nduced fields can be cancelled or amplified by appropriately\nchoosing and controlling the external electromagnetic fields.\nAlong with the first statement then, this should open for op-\nportunities to make continuous transitions between coupled\nand uncoupled dynamics by tuning the external fields [79]. As\na further implication of this transitioning between the coupled\nand uncoupled regimes it should become possible to make di-\nrect measurements of the frequencies of the uncoupled sys-\ntems and frequency shifts associated with the coupled dynam-\nics.\nB. Adiabatic limit\nThe temporal non-locality inherited in the equations of\nmotion, Eq. (45), is of principle value for investigations\nof the dynamics as it carries the full memory of the time-\nevolution. In this sense the equations of motion are non-\nMarkovian. Nonetheless, for practical simulations the non-\nMarkovian character presents undesired complications since\nit requires integrations over all time in addition to keeping\ntrack of the full memory of the past at each evaluation of the\ntime-evolution. Moreover, as the equations of motion given\nin Eq. (45) are opaque regarding the physical interpretation,\nthe physical meaning of the dynamical exchange interactions\nTr\npq(x;x0) is non-trivial to grasp. Therefore, it is meaningful\nto resort to approximations in the time-domain, if not over all\nspace and time. As we remarked in Sec. II B, we shall refer\nto the adiabatic limit in our discussions of slow temporal and13\nspatial variations of the spin and lattice quantities.\nAssuming a slow time-evolution of the spin and displace-\nment variables, we can Taylor expand in the temporal ar-\ngument to linear order f(t0)\u0019f(t)\u0000\u001c˙f(t), where\u001c=t\u0000\nt0. We will, moreover, restrict to the case of small spin\nfluctuations around a ferromagnetic ground state such that\n˙M(r0;t)\u0019˙M(x), as well as slow variations in the displace-\nments such that ˙Q(r0;t)\u0019˙Q(x). Finally, we assume that\nthe interaction tensors have a simple time-dependence, that\nis,Tr\npq(x;x0)=Tr\npq(r;r0;t\u0000t0) which allows to introduce\nTr\npq(r;r0)=lim!!0Tr\npq(r;r0;!)\u0011lim!!0R\nTr\npq(x;x0)ei!\u001cdt0.\nE\u000becting these assumptions into Eq. (45), the result can be\nwritten as\n˙M(x)=M(x)\u0002\u0012\n\u0000\rB(x)+ˆGss(r)\u0001˙M(x)+ˆGsc(r)\u0001˙Q(x)\u0013\n;\n(46a)\nMion¨Q(x)=\rEE(x)+Z\nUrr0\u0001Q(r0;t)dr0\n+ˆGcc(r)\u0001˙Q(x)+ˆGcs(r)\u0001˙M(x): (46b)\nIn this set of coupled equations we have introduced the ef-\nfective magnetic and electric fields BandEwhich both con-\ntain the corresponding external fields and while Balso in-\ncludes both mean fields induced by the surrounding spin and\ndisplacement fields, the e \u000bective electric field Eonly addi-\ntionally includes the mean field of the surrounding spin struc-\nture. The e \u000bective fields are given by\nB(x)=Bext(x)\u00001\n\rZ\u0010\nTr\nss(r;r0)\u0001M(r0;t)\n+Tr\nsc(r;r0)\u0001Q(r0;t)\u0011\ndr0; (47a)\nE(x)=Eext(x)+1\n\rEZ\nTr\ncs(r;r0)\u0001M(r0;t)dr0: (47b)\nIn this sense the e \u000bective magnetic field reduces to the con-\nventional definition in the uncoupled limit while e \u000bects of the\ndisplacement induced pseudo-magnetic field is included in the\ncoupled regime. Simultaneously, the e \u000bective electric field is\nin the coupled regime modified by the induced electric field\nfrom the surrounding spins. Possible displacement induced\nmodifications to the electric field is not included in this con-\ntribution. Instead we redefine the ionic contribution to the in-\nteratomic force constant to include this field in the expression\nUrr0=Vrr0+Tr\ncc(r;r0): (48)\nThe dissipative contributions, comprising the rates of\nchange of the spin and displacement variables, can be col-\nlected into four the di \u000berent damping tensors\nˆGpq(r)=ilim\n!!0@!Z\nTr\npq(r;r0;!)dr0;p;q=s;c: (49a)\nThe properties of the indirect exchange Tr\npq(r;r0) and damp-\ningˆGpq(r;r0) can now be discussed in terms of the dynamical\ninteractionTr\npq(r;r0;!) and employing the decoupling intro-duced the in Sec. II B, we can express it as\nTr\npq(r;r0;!)=\u0000Zf(\")\u0000f(\"0)\n!\u0000\"+\"0+i\u000e\u0004p(r;\u001a)\n2\u000eps\u0004q(\u001a0;r0)\n2\u000eqs\n\u0002sp\u001bpImGr(\u001a;\u001a0;\")\u001bqImGr(\u001a0;\u001a;\"0)d\"\n2\u0019d\"0\n2\u0019d\u001ad\u001a0:\n(50)\nThus, taking the static limit, !!0, we can write the exchange\ninteraction according to (see Appendix III for more details)\nTr\npq(r;r0)=\u00001\n2ImspZ\u0004p(r;\u001a)\n2\u000eps\u0004q(\u001a0;r0)\n2\u000eqs\n\u0002f(\")\u001bpGr(\u001a;\u001a0;\")\u001bqGr(\u001a0;\u001a;\")d\"\n2\u0019d\u001ad\u001a0:\n(51)\nAnalogously, we find the damping tensor given by\nˆGpq(r;r0)=\u00001\n2spZ\u0004p(r;\u001a)\n2\u000eps\u0004q(\u001a0;r0)\n2\u000eqs\n\u0002f0(\")\u001bpImGr(\u001a;\u001a0;\")\u001bqImGr(\u001a0;\u001a;\")d\"\n2\u0019d\u001ad\u001a0:\n(52)\nWritten in these forms it becomes clear that while the indirect\nexchange interaction strongly depends both on the structure\nof the electronic density of states as well as its occupation,\nFermi sea property, the properties of the damping is strongly\ndetermined by the electronic structure near the Fermi surface,\nFermi surface property.\nVIII. SUMMARY AND CONCLUSIONS\nIn summary we have constructed a formalism that merges\nspin and lattice dynamics in a consistent form at the same con-\nceptual level. Starting from a microscopic model of a material,\ncomprising interactions between the delocalized electrons and\nlocal magnetic structure, on the one hand, and the lattice dis-\ntortions, on the other, we derive an e \u000bective model which\nincludes the well-known contributions for bi-linear spin-spin\nand lattice-lattice interactions. The novel aspect of our e \u000bec-\ntive model are contributions that summarize the interactions\nbetween the spin and lattice degrees of freedom in a bi-linear\nform. We, moreover, showed that the interactions are of ten-\nsorial nature which preserve time-reversal and inversion sym-\nmetries between the spin and lattice subsystems.\nOur findings provide a fundamental new and novel perspec-\ntive in the theoretical modelling of coupled spin and lattice\nreservoirs for both, dynamical and static properties. For this\npurpose, multiple achievements were put into practise: i)both\nspin and lattice reservoirs are treated on the same footing by\nmeans of local couplings of the electronic structure with the\nmagnetization on one hand and with lattice distortions on the\nother hand. These local couplings lead to an e \u000bective elec-\ntron mediated spin-lattice coupling. Such type of spin-lattice-\ncoupling was obtained from the e \u000bective action of the system,\nshown not to violate fundamental symmetry operations of the14\ntotal energy. Couplings of this nature were, moreover, numer-\nically determined and analytically corroborated from model\nelectronic structure theory for certain magnetic textures, that\nexists in nature and are already catalogued [67, 80–82]. On\nthe sidelines, a Green function formalism for pure spin-spin\nand lattice-lattice second-order rank couplings in agreement\nwith already established methods [43, 65] was realised. ii)\nThe derived equations of motion account for the most gen-\neral dynamics of the coupled spin-lattice reservoir, including\nspace-time retardation that causes, for instance, energy dissi-\npation through the Gilbert damping [83, 84] as well as higher\norder conservative forces as the moment of inertia [26, 31]. In\nprinciple, also thermal microscopic fields beyond the white-\nnoise and Markovian ansatz [85, 86], due to the fluctuation-\ndissipation theorem, are considered. The Gilbert damping\nˆGss, given in terms of multiple scattering was provided in\nRef. [26], but the corresponding ionic displacement damping\nˆGccand the mixed spin-lattice damping tensors ˆGcsand ˆGsc,\nare provided as generalizations of these expressions.\nThe proposed analytical formalism and first numerical re-\nsults encourage for more detailed theoretical studies. In par-\nticular, it motivates to include bilinear spin-lattice coupling in\ncombined classical atomistic spin-lattice dynamics [38, 87],\nbut also to account for exact energy dissipations caused space-\ntime retardation in the equation of motion. All proposed terms\nfTpqgandfˆGpqg;p;q=s;c, can be implemented in first princi-\nples calculations in a similar manner as for the magnetic ex-\nchange interactions, which is nowadays a standard tool in var-\nious codes. A detailed materials-specific characterization of\nbilinear spin-lattice couplings is necessary to propose classes\nof materials with large fTcsg. The strong hybridisation of\nthe spin and lattice quasiparticle spectra caused by this type\nof coupling and, thus, possibly enhanced group velocities of\nthe quasi-particles could lead to significant improvements in\nmagnonics and phononics applications.\nIn particular, finite temperature phenomena pertaining to\nthe bilinear spin-lattice coupling are highly interesting in,\nfor instance, how critical indices of magnetic or ferroelectric\nphase transitions change or how phonon and spin tempera-\ntures in terms of disorder in the system are a \u000bected.\nOur study requests also novel experiments, as for instance\nneutron scattering measurements, to approve the existence of\na bilinear spin-lattice coupling in the here proposed mag-\nnetic textures. Within the formalism, higher order interac-\ntions, as three and four body interactions including lattice an-\nharmonicity, are accessible in a systematic way, something\nwhich would be of great value for deeper investigations of\nnon-equilibrium dynamics on ultrafast time-scales.\nAcknowledgments\nWe further thank A. V . Balatsky, A. Bergman, J. Lorenzana,\nP. W ¨olfle, and J.-X. Zhu for valuable comments. This work\nwas supported by the Vetenskapsrådet, the Wenner-Gren foun-\ndation, the Icelandic Research Fund (Grant No. 163048-052),\nthe mega-grant of the Ministry of Education and Science of\nthe Russian Federation (grant no. 14.Y26.31.0015), and Stif-telsen Olle Engqvist Byggm ¨astare. J.F. gratefully acknowl-\nedges the generous hospitality shown by the T-Division at Los\nAlamos National Laboratory during his stay in 2012. J.H. is\npartly funded by the Swedish Research Council (VR) through\na neutron project grant (BIFROST, Dnr. 2016-06955).\nAppendix A: Derivation of e \u000bective spin-lattice model\nThroughout Secs. A – C the notation will refer to quantities\nthat are continuous in the spatial dimensions, that is, A=A(r),\nwhere rdenotes the spatial coordinates. While this is made\nfor mathematical convenience it is straight forward to reduce\nto discrete lattice structures by defining the quantity Aon the\nlattice through A(r)=P\nmA\u000e(r\u0000rm), where rmdenotes the\nlattice coordinate.\n1. Microscopic model\nWe model the magnetic interactions by assuming that the\nmagnetization M(r) interacts with the surrounding spin den-\nsityss(r) via the interaction Hamiltonian\nHM=\u0000Z\nv(r;r0)M(r)\u0001ss(r0)drdr0: (A1)\nHere, ss(r)\u0011 y(r)\u001b (r)=2 is defined in terms of the spinor\n (r)=( \"(r) #(r))T, whereas v(r;r0)=v(r0;r) corresponds\nto the direct exchange contribution from the Coulomb inte-\ngral.\nThe charge n(r)=sc(r)\u0011 y(r) (r) is subject to the po-\ntential\u001e(r)=R\n\u001e(r;Q(r0))dr0due to electron-ion interactions,\nwhere Q(r) is the ionic displacement from its equilibrium\nposition. Here, we do not assign any specific nature of the\ndisplacement. For small displacements, we employ the ex-\npansion\u001e(r;Q(r0))\u0019\u001e0(r)+Q(r0)\u0001rr0\u001e0(r), where\u001e0(r)=\nlimQ!0\u001e(r), which gives the interaction between the charge\nand lattice vibrations\nHep=Z\nQ(r0)\u0001\u0015(r;r0)n(r)drdr0; (A2)\nwhere the electron-phonon coupling is denoted by \u0015(r;r0)=\nlimr0!rrr0\u001e0(r0).\n2. E \u000bective action\nGiven the general non-equilibrium conditions in the system,\ne.g. temporal fluctuations and currents, we define the corre-\nsponding action on the Keldysh contour [26, 45, 48, 49, 88–\n90] according to\nS=Z\n(HM+Hep)dt+SB+SWZWN +Slatt+SE:(A3)\nHere,\nSB=\u0000\rZ\nBext(x)\u0001M(x)dx; (A4)15\nx=(r;t), describes the Zeeman coupling to the external mag-\nnetic field Bext(x), whereas\nSWZWN =Z Z1\n0M(x;\u001c)\njM(r)j2\u0001h\n@\u001cM(x;\u001c)\u0002@tM(x;\u001c)i\nd\u001cdx\n(A5)\naccounts for the Berry phase accumulated by the spin. The\nfree lattice is represented by, e.g.,\nSlatt=Z\u001a\u0014\niQ(x)\u0001@tQ(x)\u0000Mion\n2f@tQ(x)g2\u0015\n\u000e(r\u0000r0)\n\u0000Q(x)\u0001Vrr0\u0001Q(r0;t)\u001b\ndr0dx; (A6)\nwith the ionic mass Mionand the dyad Vrr0=rr(rr0V0) is\ntheionic contribution to the interatomic force constant , and\nwhere V0is the ionic potential at equilibrium (vanishing dis-\nplacements). Finally, the coupling between the lattice and the\nexternal electric field Eext(x) is given by\nSE=Z\n\rE(x)Eext(x)\u0001Q(x)dx; (A7)\nwhere\rE(x) essentially comprises the displaced charge at x.\nWe obtain an e \u000bective actionSMQfor the coupled magneti-\nzation and lattice dynamics through a second order cumulant\nexpansion of the partition function Z[M(x);Q(x)]=TrT CeiS\nand tracing over the electronic degrees of freedom (Tr). The\nresult can be written\nSMQ=\u00001\n2Z\u0010\nQ(x)\u0001[Tcc(x;x0)\u0001Q(x0)+Tcs(x;x0)\u0001M(x0)]\n+M(x)\u0001[Tsc(x;x0)\u0001Q(x0)+Tss(x;x0)\u0001M(x0)]\u0011\ndxdx0;\n(A8)\nwhere we have defined the interaction tensor\nTpq(x;x0)=Z\n\u0004p(r;\u001a)Kpq(\u001a;\u001a0;t;t)\u0004q(\u001a0;r0)d\u001ad\u001a0;\n(A9a)\nKpq(\u001a;\u001a0;t;t0)=(\u0000i)hTsp(\u001a;t)sq(\u001a0;t0)i; (A9b)\nwith the notation \u0004c(r;\u001a)=rr\u001ec(\u001a), and \u0004s(r;\u001a)=\u0000v(r;\u001a),\nforp;q=c;s.\nAppendix B: Equations of motion\nThe time integrals in Eq. (3) run over the (Keldysh) con-\ntour, C, in the complex plane and have to be converted into\nreal time integrals. This can be done by the following pro-\ncedure. The contour Chas one branch above and one below\nthe real axis, and we therefore label the involved variables\nMandQwith superscripts uandlfor the upper and lower\nbranches, respectively. Likewise, we introduce the real time\nordered and anti-time ordered propagators Tt=¯t\npq(x;x0) for times\nt;t0both on the upper /lower branch and T<=>\npq(x;x0) for tonthe upper /lower branch and t0on the lower /upper. Using this\nnotation we have, for instance,\nZ\nM(x)\u0001Tss(x;x0)\u0001M(x0)dxdx0\n=Z1\n\u00001\u0010\nMu(x)\u0000Ml(x)\u0011\n\u00010BBBB@Tt\nss(x;x0)T<\nss(x;x0)\nT>\nss(x;x0)T¯t\nss(x;x0)1CCCCA\u00010BBBB@Mu(x0)\n\u0000Ml(x0)1CCCCAdxdx0:(B1)\nHere, the dot (\u0001) between the matrices is retained as a re-\nminder that each contribution to this expression is composed\nof a product of the type M\u0001T\u0001M. Using the unitary rotation\nR=1p\n20BBBB@1 1\n\u00001 11CCCCA; (B2)\nthe above expression becomes\n1\n2Z\u0010\n2Mc(x)Mq(x)\u0011\n\u00010BBBB@0Ta\nss(x;x0)\nTr\nss(x;x0)TK\nss(x;x0)1CCCCA\u00010BBBB@2Mc(x0)\nMq(x0)1CCCCAdxdx0; (B3)\nwhere we have introduced new (slow /fast) variables Mc\u0011\n(Mu+Ml)=2 and Mq=Mu\u0000Ml, requiring Mc\u0001Mq=0, and\nthe retarded /advanced /Keldysh propagators Tr=a=K\nss with\nKr=a\nss(\u001a;\u001a0;t;t0)=(\u0007i)\u0012(\u0006t\u0000\u0007t0)h[s(\u001a;t);s(\u001a0;t0)]i;(B4a)\nKK\nss(\u001a;\u001a0;t;t0)=(\u0000i)hfs(\u001a;t);s(\u001a0;t0)gi: (B4b)\nHere, the brackets, Eq. (B4a), and braces, Eq. (B4b), refer\nto commutation and anti-commutation, respectively. Noticing\nthatR\nMc(x)\u0001Ta\nss(x;x0)\u0001Mq(x0)dxdx0=R\nMq(x)\u0001Tr\nss(x;x0)\u0001\nMc(x0)dxdx0, we can write\n2Z\nMq(x)\u0001\"\nTr\nss(x;x0)\u0001Mc(x0)+1\n4TK\nss(x;x0)\u0001Mq(x0)#\ndxdx0:\n(B5)\nIn this fashion, we obtain one contribution which is linear, and\none which is quadratic, in the fast variables. For now, we will\nomit the quadratic contributions. In App. C we will return to\nthis issue and show how those quadratic contributions can be\nrelated to quantum (spin-spin, lattice-lattice, and spin-lattice)\nfluctuations and included in the formalism through introduc-\ntion of random variables.\nThe equations of motion for the magnetization Mand dis-\nplacement Qare found by variation of the total action Swith\nrespect to fast fluctuations, see e.g. Ref. [26] for details. Re-16\nquiring@tjM(x)j2=0, we obtain\n˙M(x)=M(x)\u0002\u0012\n\u0000\rBext(x)\n+Z\u0010\nTr\nsc(x;x0)\u0001Q(x0)+Tr\nss(x;x0)\u0001M(x0)\u0011\ndx0\u0013\n;\n(B6a)\nMion¨Q(x)=\rE(x)Eext(x)+Z\nVrr0\u0001Q(r0;t)dr0\n+Z\u0012\nTr\ncs(x;x0)\u0001M(x0)+Tr\ncc(x;x0)\u0001Q(x0)\u0013\ndx0;\n(B6b)\nwhere the superscript rrefers to retarded propagators.\nAppendix C: Quantum Fluctuations\nThe expansion of the action on the Keldysh contour leads\nto contributions which are quadratic in the superscript q, and\nhave thus been omitted so far. Here, we shall study the ef-\nfect of those contributions by Bosonization as accomplished\nthrough the Hubbard-Stratonovich transformation.\nFollowing Ref. [91] we notice that e.g. the contribution\ne\u0000i\n4R\nMq(x)\u0001TK(x;x0)\u0001Mq(x0)dxdx0\n=Z\nT\u0018ei\n4R\n\u0018(x)\u0001TK;\u00001(x;x0)\u0001\u0018(x0)dxdx0e\u0000i\n2R\n\u0018(x)\u0001Mq(x)dx;(C1)\nwhere the measure T\u0018=lim\u000f!0Qp\ndet\"TK;\u00001=i2\u0019d\u0018,\nwhereas the random fields \u0018can be related to the spin-\nsusceptibilityTKthrough the following procedure. In Eq.\n(C1), byTK;\u00001we mean the inverse of TK. Assuming that\nthere is a random magnetic field \u0018coupled to the magnetiza-\ntion variable MqthroughH\u0018=\u0000\r\u0018\u0018\u0001Mq. Then, with respect\nto these random fields, the partition function can be written\nZ[\u0018]=tr\u0018e\u0000R\nH\u0018(t)dt\u0019e\u00001\n2R\nhH\u0018(t)H\u0018(t0)idtdt0\n=e\u0000\r2\n\u0018R\nMq(x)\u0001h\u0018(x)\u0018(x0)i\u0001Mq(x0)dxdx0=2:\n(C2)\nInspection of the two equations suggests that the random vari-\nables\u0018have to satisfy the condition\nh\u0018(x)\u0018(x0)i=i\n2\r2\n\u0018TK(x;x0): (C3)\nRecall thatTK(x;x0) denotes the Keldysh field defined in\nterms of the kernel in Eq. (B4b). We also remark that this\nrelation is a clear manifestation that the quantum correlation\ninduced noise is not necessarily of white Gaussian nature. It\nalso shows that the quantum noise strongly depends on the\nelectronic structure. We can generalize this procedure to the\nwhole actionSqsince we can write (omitting the superscriptsqandK)\neiSq=exp8>><>>:\u0000i\n4Z\u0010\nM Q\u0011\n\u00010BBBB@TssTsc\nTcsTcc1CCCCA\u00010BBBB@M\nQ1CCCCAd\u0016(x;x0)9>>=>>;\n=exp\u001a\n\u0000i\n4aiAi jaj\u001b\n; (C4)\nwhere a1=M(x) and a2=Q(x). By means of the Hubbard-\nStratonovich transformation we now obtain\neiSq\n=Z\nT\u001eei\n4R\n\u001e(x)A\u00001(x;x0)\u001e(x0)d\u0016(x;x0)e\u0000i\n2R\n\u001e(x)\u0001a(x)d\u0016(x):(C5)\nDefining the random variables \u0018and\u0010such that\u001e\u0001a=\u0018\u0001M+\n\u0010\u0001Q, we can relate those random variables to the correlation\nfunctionsTpq, through\nh\u001e(x)\u001eT(x0)i=i\n2\u0012\n1\n\r\u00181\n\r\u0010\u00130BBBB@TK\nss(x;x0)TK\nsc(x;x0)\nTK\ncs(x;x0)TK\ncc(x;x0)1CCCCA0BBBBBB@1\n\r\u0018\n1\n\r\u00101CCCCCCA;(C6)\nwhere\u001e(x)=\u0010\n\u0018(x)\u0010(x)\u0011T.\nThe contribution to the spin-lattice coupled system can,\nthus, be written as\nSq=\u00001\n2Z\u0010\n\r\u0018\u0018\u0001Mq(x)+\r\u0010\u0010\u0001Qq(x)\u0011\ndrdt: (C7)\nThis action, which is due to the fast correlations between\nthe magnetization and lattice dynamics, adds correlation ef-\nfects to the equations of motion for MandQvia the random\nfields\u0018and\u0010. The random fields \u0018and\u0010relates to the spin-\nspin, lattice-lattice, and spin-lattice interactions via their cor-\nresponding correlation function. Those new Bosonic degrees\nof freedom represent collective modes that are associated with\nfluctuations in the magnetic and lattice structure, that is, spin\nwaves (magnons) and lattice vibrations (phonons).\nAppendix D: Inversion symmetry\nIf inversion \u0013is a symmetry operation that bring site ito site\ni0as in Figure 5, we can focus on the two pair interactions i j\nrespectively i0j0. In order to study these in detail we introduce\nthe average quantities\nQ=Qi+Qj+Qi0+Qj0=Qii0+Qj j0\nq1=Qi\u0000Qj+Qi0\u0000Qj0=Qii0\u0000Qj j0\nq2=Qi+Qj\u0000Qi0\u0000Qj0=qii0+qj j0\nq3=Qi\u0000Qj\u0000Qi0+Qj0=qii0\u0000qj j0; (D1)\nand\nM=Mi+Mj+Mi0+Mj0=Mii0+Mj j0\nm1=Mi\u0000Mj+Mi0\u0000Mj0=Mii0\u0000Mj j0\nm2=Mi+Mj\u0000Mi0\u0000Mj0=mii0+mj j0\nm3=Mi\u0000Mj\u0000Mi0+Mj0=mii0\u0000mj j0; (D2)17\nijj0i0\nFIG. 5: Interactions (full lines) between two sites iand jthat are\nconnected (dashed lines) to sites i0andj0by inversion.\nthat all have well defined parity properties\n\u0013Q=\u0000Qii0\u0000Qj j0=\u0000Q\n\u0013q1=\u0000Qii0+Qj j0=\u0000q1\n\u0013q2=\u0000qii0+qj j0=q2\n\u0013q3=qii0\u0000qj j0=q3; (D3)\nand\n\u0013M=Mii0+Mj j0=M\n\u0013m1=Mii0\u0000Mj j0=m1\n\u0013m2=\u0000mii0\u0000mj j0=\u0000m2\n\u0013m3=\u0000mii0+mj j0=\u0000m3: (D4)Then since the individual quantities can be obtained by revers-\ning the Eqs. (D1) and (D2)\nQi=1\n4(Q+q1+q2+q3)\nQj=1\n4(Q\u0000q1+q2\u0000q3)\nQi0=1\n4(Q+q1\u0000q2\u0000q3)\nQj0=1\n4(Q\u0000q1\u0000q2+q3); (D5)\nand\nMi=1\n4(M+m1+m2+m3)\nMj=1\n4(M\u0000m1+m2\u0000m3)\nMi0=1\n4(M+m1\u0000m2\u0000m3)\nMj0=1\n4(M\u0000m1\u0000m2+m3) (D6)\nwe can rewrite the pair tensors interactions between sites iandjof Eq. (26) as\nIi j=Si j\u0010\nQiMj+QjMi\u0011\n+Ai j\u0010\nQiMj\u0000QjMi\u0011\n=\n=1\n8Si j\bQM +Qm 2\u0000q1m1\u0000q1m3+q2M+q2m2\u0000q3m1\u0000q3m3\t+\n+1\n8Ai j\b\u0000Qm 1\u0000Qm 3+q1M+q1m2\u0000q2m1\u0000q2m3+q3M+q3m2\t; (D7)\nwhile the corresponding interactions between i0andj0become\nIi0j0=Si0j0\u0010\nQi0Mj0+Qj0Mi0\u0011\n+Ai0j0\u0010\nQi0Mj0\u0000Qj0Mi0\u0011\n=\n=1\n8Si0j0\bQM\u0000Qm 2\u0000q1m1+q1m3\u0000q2M+q2m2+q3m1\u0000q3m3\t+\n+1\n8Ai0j0\b\u0000Qm 1+Qm 3+q1M\u0000q1m2+q2m1\u0000q2m3\u0000q3M+q3m2\t: (D8)\nAs\n\u0013\bQM +Qm 2\u0000q1m1\u0000q1m3+q2M+q2m2\u0000q3m1\u0000q3m3\t=\n=\u0000\bQM\u0000Qm 2\u0000q1m1+q1m3\u0000q2M+q2m2+q3m1\u0000q3m3\t\n\u0013\b\u0000Qm 1+Qm 3+q1M\u0000q1m2+q2m1\u0000q2m3\u0000q3M+q3m2\t=\n=\u0000\b\u0000Qm 1+Qm 3+q1M\u0000q1m2+q2m1\u0000q2m3\u0000q3M+q3m2\t; (D9)\nin order to preserve the inversion symmetry, i.e. that \u0013Ii j=Ii0j0, we can identify that both interaction parameters have to have odd18\nparity\n\u0013Si j=\u0000Si0j0\n\u0013Ai j=\u0000A i0j0: (D10)\nAppendix E: Examples\nWe assume a simple two-dimensional electrons gas, for ex-\nample, surface states on a metallic surface or an analogous\nset-up, in which magnetic defects are embedded. We model\nthis system by the Hamiltonian\nH=X\nk\ty\nk\u000fk\tk+Z\n\ty(r)V(r)\t(r)dr; (E1)\nwhere the spinor \tk=(ck\"ck#)tannihilates electrons with en-\nergy\u000fk=\"k\u001b0at the momentum kand spin\u001b=\";#, whereas\nthe scattering potential V(r)=P\nmVm\u000e(r\u0000rm) defines a col-\nlection of defects Vm=Vm\u001b0+Mm\u0001\u001b.\nIn this model, the unperturbed Green function gkis defined\nfor the first term and is given in reciprocal and real space by\nthe expressions\ngk(!)=\u001b0\n!\u0000\"k+i\u000e;g(r;!)=\u0000iN0\n2H(1)\n0(\u0014r)\u001b0;(E2)\nwhere\u00142=2N0!andN0=me=¯h2, whereas H(1)\nmis the Hankel\nfunction of first kind and order m, and meis the e \u000bective elec-\ntron mass. In this way we have defined gk(!)=g0(k;!)\u001b0\nwhile g1(k;!)\u00110.\nWe calculate the dressed Green function Gin terms of the\nT-matrix expansion of the impurity potential, that is,\nG(k;k0)=\u000e(k\u0000k0)gk+X\nmngke\u0000ik\u0001rmT(Rmn)eik0\u0001rngk0;(E3)\nwhere Rmn=rm\u0000rn, whereas the T-matrix is given by\nT(Rmn)=Vm(t\u00001)mn; (E4a)\ntmn=\u000emn\u001b0+g(Rmn)Vn: (E4b)\nHere, since the scattering potential is partitioned into a non-\nmagnetic and a magnetic component, we can write T(Rmn)=\nT0(Rmn)\u001b0+T1(Rmn)\u0001\u001b.\nFor su \u000eciently large separation between the scattering im-\npurities, the T-matrix reduces to\nT(Rmn)=\u000e(Rmn)h\nV\u00001\nm\u0000g(r=0)i\u00001\n=\u000e(Rmn)h\nt0(rm)\u001b0+t1(rm)\u0001\u001bi\u00001; (E5a)\nt0(rm)=Vm+i(V2\nm\u0000jMmj2)N0=2\n1\u0000(V2m\u0000jMmj2)(N0=2)2+iVmN0; (E5b)\nt1(rm)=Mm\n1\u0000(V2m\u0000jMmj2)(N0=2)2+iVmN0; (E5c)\nwhere the lowercase notation has been used to stress the as-\nsumed simplification.The expansion of the T-matrix into charge and magnetic\ncomponents further allows us to write the corrections \u000eg0and\n\u000eg1to the Green function as, in general,\n\u000eg0(k;k0)=g0(k)X\nmne\u0000ik\u0001rmT0(Rmn)eik0\u0001rng0(k0); (E6a)\n\u000eg1(k;k0)=g0(k)X\nmne\u0000ik\u0001rmT1(Rmn)eik0\u0001rng0(k0);(E6b)\nwhich leads to the corresponding real space expressions\n\u000eg0(r;r0)=X\nmng0(r\u0000rm)T0(Rmn)g0(rn\u0000r0); (E7a)\n\u000eg1(r;r0)=X\nmng0(r\u0000rm)T1(Rmn)g0(rn\u0000r0): (E7b)\nWith the subscripts notation Apqwhere p=0;1 (q=0;1)\nrefers to even or odd time-reversal symmetry (parity), and we\nnotice that\nG00=g0+\u000eg0; G01=0; (E8a)\nG10=\u000eg1; G11=0: (E8b)\nIt can be noticed that the non-magnetic component G00is\nmerely re-normalized by the presence of the magnetic defects,\nhowever, since there is no fundamental change introduced by\nthe correction \u000eg0we shall omit this contribution in the dis-\ncussions below, for simplicity. The components with q=1\nvanish due to the absence of, for instance, spin-orbit coupling\nin the system. The e \u000bect of the defects is, however, to break\nthe translation invariance in the system, something which has\na profound influence on certain magnetic configurations as we\nshall see next.\n1. Double anti-ferromagnetic\nAssume that the magnetic moments are positioned along a\nlinear chain in ˆ x-direction according to Mm\u0011M(xm), where\nrm=xmˆxis the coordinate of the magnetic moment Mmand\nxm+1\u0000xm=a. Analogously, we let Qm\u0011Q(xm). We also as-\nsume the double anti-ferromagnetic structure of the magnetic\nmoments, illustrated in Fig. 6 (a). We wish to calculate the\nnet force exerted on the moment Mmby the nearest neighbor\nmoments Mm\u00061. The procedure is to evaluate the derivative\nF(xm)=\u0000(@=@Qm)hHMQi, given the Hamiltonian\nHMQ=\u00001\n2X\nmn\u0010\nMm\u0001Tss\nmn\u0001Mn+Mm\u0001Tsc\nmn\u0001Qn\n+Qm\u0001Tcs\nmn\u0001Mn+Qm\u0001Tcc\nmn\u0001Qn\u0011\n; (E9)19\nwhich gives\nF(xm)=1\n2X\nn\u0010\nMn\u0001Tsc\nnm+Tcs\nmn\u0001Mn+Qn\u0001Tcc\nnm+Tcc\nmn\u0001Qn\u0011\n:\n(E10)\nIn the following we shall omit the forces by the lattice-lattice\ncoupling since we are mainly interested in the forces induced\nbetween the spin and lattice subsystems. Our interest is con-\ncerned with e \u000bects that may arise from the spin-lattice cou-\nplingsfTscgandfTcsg. According to the theoretical frame-\nworks developed in the main text, we find that we can write\nthese interaction fields in the non-relativistic limit as\nTsc\nnm=\u00004\n\u0019v(xn) \nImZ\nf(!)G00(xn;xm)G10(xm;xn)d!!\n\u0015(xm);\n(E11a)\nTcs\nmn=\u00004\n\u0019\u0015(xm) \nImZ\nf(!)G10(xm;xn)G00(xn;xm)d!!\nv(xn):\n(E11b)\nUsing the results for the Green functions derived above, we\nobtain, for instance,\nG10(xm;xn)G00(xn;xm)\n=\u000eg1(xm;xn)g0(xnm)\n=X\n\u0016\u0017g0(xm\u0016)T(x\u0016\u0017)g0(x\u0017n)g0(xnm); (E12)\n(a)\n(b)xmxm-1\nxm+1\na\naxm=ma\nq=π/4aFm\nF(x)\nFIG. 6: (Color online) Two possible realizations of the collinear,\nor, sinusodal density wave. (a) Double anti-ferromagnetic structure\nwhere pairs of ferromagnetic spins are anti-ferromagnetically config-\nured which leads to a dimerization of the ions. (b) Gradual variation\nof the local moment in a globally anti-ferromagnetic configuration\nleads to a gradual variation of the force (blue – bold) between the\nions with halved period to that of the lattice.where xmn=xm;n=xm\u0000xn.\nFor a simple estimate of the net force we go to the limit\nof large separation between the defects. Then, the correction\n\u000eg1(x;x0)=P\nmng0(x\u0000xm)t1(xmn)g0(xn\u0000x0). We also notice\nthatt1(xm)\u0018Mmand that g0(\u0000r)=g0(r). Considering the\ne\u000bects from the nearest neighbors, we then obtain\nG10(xm;xm\u00061)=X\ns=\u00001;0;1g0(xm;m+s)t1(xm+s)g0(xm+s;m\u00061)\n\u0018g0(a)\u0010\ng0(0)[Mm;m\u00061+Mm]+g0(2a)Mm;m\u00071\u0011\n;\n(E13)\nwhere ais the lattice constant ( jxm\u0000xm\u00061j=a). We also notice\nthatG10(xm;xm\u00061)=G10(xm\u00061;xm) and since G00(xm;xn)=\nG00(xn;xm), it is clear that Tsc\nm\u00061;m=(Tcs\nm;m\u00061)T. Then, summa-\nrizing the force on the mth ion exerted by its two surrounding\nnearest neighbors, assuming that v(xm)=v, for all m, under the\ncondition that, for instance, Mm\u00001=Mm=\u0000Mm+1, we obtain\nX\ns=\u00061Tcs\nm;m+s\u0001Mm+s\u0018\u00002v\u0015(xm)jMmj2\n\u0002ImZ\nf(!)g2\n0(a)\u0012\ng0(0)\u0000g0(2a)\u0013\nd!:\n(E14)\nHence, the finiteness of the force on ion mexerted by the\nnearest neighbors is determined by the real space electronic\nstructure between the ions since g0(0)\u0000g0(2a)\u0018H(1)\n0(0)\u0000\nH(1)\n0(2\u0014a),0, unless a=0. It is therefore clear that there\nis a net force acting on ion m. The sign of the net force de-\npends on the distance between the ions which means that the\ndimerization of the ions can leads to either ferromagnetic or\nanti-ferromagnetic pairs, details that are beyond the scope of\nthe present context.\n2. Sinusodal spin density wave\nNext, we consider planar collinear, or, sinusodal spin den-\nsity waves. Therefore, we assume that the magnetic mo-\nments are positioned along a linear chain according to Mm\u0011\nM(xm)=M0ˆzcosqxm, where xmis the coordinate of the mag-\nnetic moment Mm, as is illustrated in Fig. 6 (b). Following\nthe procedure introduced previously, we obtain the product\nG10(xm;xn)G00(xn;xm)\n=\u000eg1(xm;xn)g0(xnm)\n=X\n\u0016\u0017g0(xm\u0016)T(x\u0016\u0017)g0(x\u0017n)g0(xnm); (E15)\nwhere xmn=xm;n=xm\u0000xn. Again, we go to the limit of large\nseparation between the defects, which leads to that we can\nwrite\nG10(xm;xm\u00061)=X\ns=\u00001;0;1g0(xm;m+s)t1(xm+s)g0(xm+s;m\u00061)\n\u0018M0g0(a)\u0010\ng0(0)[cos qxm\u00061+cosqxm]\n+g0(2a)cosqxm\u00071\u0011\nˆz; (E16)20\nThen, summarizing the force on the mth ion exerted by its two\nsurrounding nearest neighbors, assuming that v(xm\u00061)=v, weobtain\nX\ns=\u00061T(cs)\nmm+s\u0001mm+s\u0018M2\n0v\u0015(xm)ImZ\nf(!)g2\n0(a)\u0012\u0010\ng0(0)[cos qxm\u00001+cosqxm]+g0(2a)cosqxm+1\u0011\ncosqxm\u00001\n+\u0010\ng0(0)[cos qxm+1+cosqxm]+g0(2a)cosqxm\u00001\u0011\ncosqxm+1\u0013\nd!\n=M2\n0v\u0015(xm)ImZ\nf(!)g2\n0(a)\u0012\ng0(0)\u0010\ncos2qxm\u00001+cosqxm[cosqxm\u00001+cosqxm+1]+cos2qxm+1\u0011\n+2g0(2a)cosqxm\u00001cosqxm+1\u0013\nd!: (E17)\nLetting x=xmsuch that we can write xm\u00061=x\u0006a, the trigono-\nmetric expression in the term proportional to g0(0) can be\nrewritten as\n1+cosqa+(cosqa+cos2 qa)cos2 qx; (E18)\nwhereas the corresponding expression in the term proportional\nto 2g0(2a) as\n1\n2\u0010\ncos2 qx+cos2 qa\u0011\n: (E19)\nWith these equalities, we can write the force as proportional\nto\nM2\n0v\u0015(x)ImZ\nf(!)g2\n0(a)\u0012\ng0(0)[1 +cosqa]+g0(2a)cos2 qa\n+\u0010\ng0(0)[cos qa+cos2 qa]+g0(2a)\u0011\ncos2 qx\u0013\nd!:\n(E20)Here, taking q=\u0019=4a, see Fig. 6 (b), this expression reduces\nto\np\n2\n2M2\n0v\u0015(x)ImZ\nf(!)g2\n0(0)\u0014\u0010\n1+p\n2\u0011\ng0(0)\n+\u0010\ng0(0)+p\n2g0(2a)\u0011\ncos\u0019x\n2a\u0015\nd!: (E21)\nThe spatial variation of the resulting forces has a period which\nis half of that of the lattice.\n[1] J. Jensen and A. R. Mackintosh, Rare earth magnetism (Claren-\ndon Press, Oxford, 1991).\n[2] S. Dong, J.-M. Liu, S.-W. Cheong, and Z. Ren, Adv. Phys. 64,\n519 (2015).\n[3] F. K ¨ormann, B. Grabowski, B. Dutta, T. Hickel, L. Mauger,\nB. Fultz, and J. Neugebauer, Phys. Rev. Lett. 113, 165503\n(2014).\n[4] D. Perera, D. M. Nicholson, M. Eisenbach, G. M. Stocks, and\nD. P. Landau, Phys. Rev. B 95, 014431 (2017).\n[5] R. Pradip, P. Piekarz, A. Bosak, D. G. Merkel, O. Waller,\nA. Seiler, A. I. Chumakov, R. R ¨u\u000ber, A. M. Ole ´s, K. Parlin-\nski, M. Krisch, T. Baumbach, and S. Stankov, Phys. Rev. Lett.\n116, 185501 (2016).\n[6] M. Matsuo, J. Ieda, K. Harii, E. Saitoh, and S. Maekawa, Phys.\nRev. B 87, 180402 (2013).\n[7] C. M. Jaworski, J. Yang, S. Mack, D. D. Awschalom, R. C. My-\ners, and J. P. Heremans, Phys. Rev. Lett. 106, 186601 (2011).\n[8] F. B. Wilken, Boltzmann approach to the transversal spin See-\nbeck e \u000bect, Ph.D. thesis, Freien Universit ¨at Berlin (2016).[9] N. Ogawa, W. Koshibae, A. J. Beekman, N. Nagaosa, M. Kub-\nota, M. Kawasaki, and Y . Tokura, Proc. Natl. Acad. Sci. 112,\n8977 (2015).\n[10] T. Kimura, S. Ishihara, H. Shintani, T. Arima, K. Takahashi,\nK. Ishizaka, and Y . Tokura, Phys. Rev. B 68, 060403 (2003).\n[11] A. Pimenov, A. A. Mukhin, V . Y . Ivanov, V . D. Travkin, A. M.\nBalbashov, and A. Loidl, Nat. Phys. 2, 97 (2006).\n[12] A. B. Sushkov, M. Mostovoy, R. Vald ´es Aguilar, S.-W. Cheong,\nand H. D. Drew, J. Phys. Condens. Matter 20, 434210 (2008).\n[13] P. McCorkle, Phys. Rev. 22, 271 (1923).\n[14] A. V . Kimel, B. A. Ivanov, R. V . Pisarev, P. A. Usachev, A. Kir-\nilyuk, and T. Rasing, Nat. Phys. 5, 727 (2009).\n[15] T. A. Ostler, J. Barker, R. F. L. Evans, R. W. Chantrell, U. Atxi-\ntia, O. Chubykalo-Fesenko, S. El Moussaoui, L. Le Guyader,\nE. Mengotti, L. J. Heyderman, F. Nolting, A. Tsukamoto,\nA. Itoh, D. Afanasiev, B. A. Ivanov, A. M. Kalashnikova, K. Va-\nhaplar, J. Mentink, A. Kirilyuk, T. Rasing, and A. V . Kimel,\nNat. Commun. 3, 666 (2012).\n[16] S. L. Johnson, R. A. de Souza, U. Staub, P. Beaud, E. M ¨ohr-21\nV orobeva, G. Ingold, A. Caviezel, V . Scagnoli, W. F. Schlotter,\nJ. J. Turner, O. Krupin, W.-S. Lee, Y .-D. Chuang, L. Patthey,\nR. G. Moore, D. Lu, M. Yi, P. S. Kirchmann, M. Trigo,\nP. Denes, D. Doering, Z. Hussain, Z.-X. Shen, D. Prabhakaran,\nand A. T. Boothroyd, Phys. Rev. Lett. 108, 037203 (2012).\n[17] S. Wall, D. Prabhakaran, A. Boothroyd, and A. Cavalleri, Phys.\nRev. Lett. 103, 097402 (2009).\n[18] K. W. Kim, a. Pashkin, H. Sch ¨afer, M. Beyer, M. Porer, T. Wolf,\nC. Bernhard, J. Demsar, R. Huber, and a. Leitenstorfer, Nat.\nMater. 11, 497 (2012).\n[19] T. F. Nova, A. Cartella, A. Cantaluppi, M. F ¨orst, D. Bossini,\nR. V . Mikhaylovskiy, A. V . Kimel, R. Merlin, and A. Cavalleri,\nNat. Phys. 13, 132 (2016).\n[20] D. M. Juraschek, M. Fechner, A. V . Balatsky, and N. A.\nSpaldin, Phys. Rev. Mater. 1, 014401 (2017).\n[21] L. D. Landau and E. M. Lifshitz, Phys. Z. Sowietunion 8, 153\n(1935).\n[22] T. Gilbert, IEEE Trans. Magn. 40, 3443 (2004).\n[23] V . P. Antropov, M. I. Katsnelson, B. N. Harmon, M. van Schil-\nfgaarde, and D. Kusnezov, Phys. Rev. B 54, 1019 (1996).\n[24] Q. Niu, X. Wang, L. Kleinman, W.-M. Liu, D. M. C. Nicholson,\nand G. M. Stocks, Phys. Rev. Lett. 83, 207 (1999).\n[25] Z. Qian and G. Vignale, Phys. Rev. Lett. 88, 056404 (2002).\n[26] S. Bhattacharjee, L. Nordstr ¨om, and J. Fransson, Phys. Rev.\nLett. 108, 057204 (2012).\n[27] H. Ebert, D. K ¨odderitzsch, and J. Min ´ar, Reports Prog. Phys.\n74, 096501 (2011).\n[28] D. Thonig and J. Henk, New J. Phys. 16, 013032 (2014).\n[29] K. Gilmore, Y . U. Idzerda, and M. D. Stiles, Phys. Rev. Lett.\n99, 027204 (2007).\n[30] M.-C. Ciornei, J. M. Rub ´ı, and J.-E. Wegrowe, Phys. Rev. B\n83, 020410 (2011).\n[31] D. Thonig, O. Eriksson, and M. Pereiro, Sci. Rep. 7, 931\n(2017).\n[32] M. F ¨ahnle, D. Steiauf, and C. Illg, Phys. Rev. B 84, 172403\n(2011).\n[33] M. F ¨ahnle, D. Steiauf, and C. Illg, Phys. Rev. B 88, 219905\n(2013).\n[34] J.-E. Wegrowe and M.-C. Ciornei, Am. J. Phys. 80, 607 (2012).\n[35] J. Forshaw and G. Smith, Dynamics and Relativity (John Wiley\n& Sons, 2009).\n[36] W. Nolting, Theoretical Physics 2 (Springer International Pub-\nlishing, Cham, 2016).\n[37] O. Eriksson, A. Bergman, L. Bergqvist, and J. Hellsvik, Atom-\nistic spin dynamics : foundations and applications (Oxford\nUniversity Press, 2017).\n[38] P.-W. Ma, C. H. Woo, and S. L. Dudarev, Phys. Rev. B 78,\n024434 (2008).\n[39] P.-W. Ma and S. L. Dudarev, Phys. Rev. B 86, 054416 (2012).\n[40] C. Kittel, Phys. Rev. 110, 836 (1958).\n[41] A. I. Akhiezer, V . G. Bar ´ıakhtar, and S. V . Peletminskii, Sov.\nPhys. JETP 35, 157 (1959).\n[42] H. F. Tiersten, Journal of Mathematical Physics 5, 1298 (1964),\nhttp://dx.doi.org /10.1063 /1.1704239 .\n[43] L. Udvardi, L. Szunyogh, K. Palot ´as, and P. Weinberger, Phys.\nRev. B 68, 104436 (2003).\n[44] S. Mankovsky and H. Ebert, arXiv:1706.04165 .\n[45] J. Fransson, Phys. Rev. B 82, 180411 (2010).\n[46] J.-X. Zhu and J. Fransson, J. Phys. Condens. Matter 18, 9929\n(2006).\n[47] J. Fransson, Nanotechnology 19, 285714 (2008).\n[48] J. Fransson and J.-X. Zhu, New J. Phys. 10, 013017 (2008).\n[49] J. Fransson, J. Ren, and J.-X. Zhu, Phys. Rev. Lett. 113, 257201\n(2014).[50] H. Imamura, P. Bruno, and Y . Utsumi, Phys. Rev. B 69, 121303\n(2004).\n[51] S. Lounis, M. Reif, P. Mavropoulos, L. Glaser, P. H. Dederichs,\nM. Martins, S. Bl ¨ugel, and W. Wurth, EPL (Europhysics Lett.\n81, 47004 (2008).\n[52] W. Heisenberg, Zeitschrift f ¨ur Physik 38, 411 (1926).\n[53] A. Liechtenstein, M. Katsnelson, V . Antropov, and V . Gubanov,\nJ. Magn. Magn. Mater. 67, 65 (1987).\n[54] H. Ebert and S. Mankovsky, Phys. Rev. B 79, 045209 (2009).\n[55] J. C. Slater and G. F. Koster, Phys. Rev. 94, 1498 (1954).\n[56] M. J. Mehl and D. A. Papaconstantopoulos, Phys. Rev. B 54,\n4519 (1996).\n[57] R. E. Cohen, M. J. Mehl, and D. A. Papaconstantopoulos, Phys.\nRev. B 50, 14694 (1994).\n[58] J. Zabloudil, R. Hammerling, L. Szunyogh, and P. Weinberger,\neds., Electron Scattering in Solid Matter (Springer, Berlin,\n2005).\n[59] A. Lindner, Drehimpulse in der Quantenmechanik (B. G. Teub-\nner, Stuttgart, 1984).\n[60] J. Dziedzic, TASK QUARTERLY 11, 285 (2006).\n[61] M. J. Mehl and D. A. Papaconstantopoulos, arXiv:9602003\n[mtrl-th] .\n[62] T. Schena, Tight-Binding Treatment of Complex Mag-\nnetic Structures in Low-Dimensional Systems , Master’s the-\nsis, Rheinisch-Westf ¨alische Technische Hochschule Aachen\n(2010).\n[63] M. Pajda, J. Kudrnovsk ´y, I. Turek, V . Drchal, and P. Bruno,\nPhys. Rev. B 64, 174402 (2001).\n[64] D. B ¨ottcher, A. Ernst, and J. Henk, J. Magn. Magn. Mater. 324,\n610 (2012).\n[65] L. E. D ´ıaz-S ´anchez, A. H. Romero, and X. Gonze, Phys. Rev.\nB76, 104302 (2007).\n[66] W. Luo, B. Johansson, O. Eriksson, S. Arapan, P. Souvatzis,\nM. I. Katsnelson, and R. Ahuja, Proc. Natl. Acad. Sci. U. S. A.\n107, 9962 (2010).\n[67] E. Sj ¨ostedt and L. Nordstr ¨om, Phys. Rev. B 66, 014447 (2002).\n[68] H. Katsura, N. Nagaosa, and A. V . Balatsky, Phys. Rev. Lett.\n95, 057205 (2005).\n[69] M. Mostovoy, Phys. Rev. Lett. 96, 067601 (2006).\n[70] I. A. Sergienko and E. Dagotto, Phys. Rev. B 73, 094434\n(2006).\n[71] I. A. Sergienko, C. S ¸en, and E. Dagotto, Phys. Rev. Lett. 97,\n227204 (2006).\n[72] Y . Takahashi, R. Shimano, Y . Kaneko, H. Murakawa, and\nY . Tokura, Nat. Phys. 8, 121 (2011).\n[73] T. Kubacka, J. A. Johnson, M. C. Ho \u000bmann, C. Vicario,\nS. de Jong, P. Beaud, S. Gr ¨ubel, S.-W. Huang, L. Huber,\nL. Patthey, Y .-D. Chuang, J. J. Turner, G. L. Dakovski, W.-\nS. Lee, M. P. Minitti, W. Schlotter, R. G. Moore, C. P. Hauri,\nS. M. Koohpayeh, V . Scagnoli, G. Ingold, S. L. Johnson, and\nU. Staub, Science 343, 1333 (2014).\n[74] A. C. Hewson and D. Meyer, J. Phys. Condens. Matter 14, 427\n(2002).\n[75] H. C. Walker, F. Fabrizi, L. Paolasini, F. de Bergevin,\nJ. Herrero-Martin, A. T. Boothroyd, D. Prabhakaran, and D. F.\nMcMorrow, Science 333, 1273 (2011).\n[76] S. Baroni and S. de Gironcoli, Rev. Mod. Phys. 73, 515 (2001).\n[77] A. A. Quong, A. A. Maradudin, R. F. Wallis, J. A. Gaspar, A. G.\nEguiluz, and G. P. Alldredge, Phys. Rev. Lett. 66, 743 (1991).\n[78] A. Y . Liu and A. A. Quong, Phys. Rev. B 53, R7575 (1996).\n[79] J. H. Mentink, K. Balzer, and M. Eckstein, Nat. Commun. 6,\n6708 (2015).\n[80] E. Fawcett, Rev. Mod. Phys. 60, 209 (1988).\n[81] M. P. Walser, C. Reichl, W. Wegscheider, and G. Salis, Nat.22\nPhys. 8, 757 (2012).\n[82] S. Fust, S. Mukherjee, N. Paul, J. Stahn, W. Kreuzpaintner,\nP. B¨oni, and A. Paul, Sci. Rep. 6, 33986 (2016).\n[83] V . Kambersk ´y, Czechoslov. J. Phys. 34, 1111 (1984).\n[84] K. Gilmore, Y . U. Idzerda, and M. D. Stiles, Phys. Rev. Lett.\n99, 027204 (2007).\n[85] H. Risken, The Fokker-Planck equation : methods of solution\nand applications , 2nd ed. (Springer-Vlg, Berlin, 1989).\n[86] J. Hellsvik, Atomistic Spin Dynamics, Theory and Applications ,\nPh.D. thesis, Uppsala University (2010).\n[87] P. W. Ma, S. L. Dudarev, and C. H. Woo, Phys. Rev. B 85, 30\n(2012).\n[88] U. Eckern, G. Sch ¨on, and V . Ambegaokar, Phys. Rev. B 30,6419 (1984).\n[89] A. I. Larkin and Y . N. Ovchinnikov, Phys. Rev. B 28, 6281\n(1983).\n[90] J.-X. Zhu, Z. Nussinov, A. Shnirman, and A. V . Balatsky, Phys.\nRev. Lett. 92, 107001 (2004).\n[91] J. W. Negele and H. Orland, Quantum Many-Particle Systems\n(Addison-Wesley, Redwood City, 1988).\n[92] CAHMD - classical atomistic Heisenberg magnetization dy-\nnamics, 2013. A computer program package for atomistic mag-\nnetization dynamics simulations. Available from the authors;\nelectronic address: danny.thonig@physics.uu.se."    },    {        "title": "1506.00723v1.Current_Driven_Motion_of_Magnetic_Domain_Wall_with_Many_Bloch_Lines.pdf",        "content": "Journal of the Physical Society of Japan LETTERS\nCurrent-Driven Motion of Magnetic Domain Wall with Many Bloch\nLines\nJunichi Iwasaki1\u0003and Naoto Nagaosa1;2y\n1Department of Applied Physics, University of Tokyo, Tokyo 113-8656, Japan\n2RIKEN Center for Emergent Matter Science (CEMS),Wako, Saitama 351-0198, Japan\nThe current-driven motion of a domain wall (DW) in a ferromagnet with many Bloch lines (BLs) via the\nspin transfer torque is studied theoretically. It is found that the motion of BLs changes the current-velocity\n(j-v) characteristic dramatically. Especially, the critical current density to overcome the pinning force is\nreduced by the factor of the Gilbert damping coe\u000ecient \u000beven compared with that of a skyrmion. This\nis in sharp contrast to the case of magnetic \feld driven motion, where the existence of BLs reduces the\nmobility of the DW.\nDomain walls (DWs) and bubbles1,2)are the spin tex-\ntures in ferromagnets which have been studied inten-\nsively over decades from the viewpoints of both funda-\nmental physics and applications. The memory functions\nof these objects are one of the main focus during 70's, but\ntheir manipulation in terms of the magnetic \feld faced\nthe di\u000eculty associated with the pinning which hinders\ntheir motion. The new aspect introduced recently is the\ncurrent-driven motion of the spin textures.3,4)The \row\nof the conduction electron spins, which follow the direc-\ntion of the background localized spin moments, moves\nthe spin texture due to the conservation of the angu-\nlar momentum. This e\u000bect, so called the spin transfer\ntorque, is shown to be e\u000bective to manipulate the DWs\nand bubbles compared with the magnetic \feld. Magnetic\nskyrmion5,6)is especially an interesting object, which is\na swirling spin texture acting as an emergent particle\nprotected by the topological invariant, i.e., the skyrmion\nnumberNsk, de\fned by\nNsk=1\n4\u0019Z\nd2rn(r)\u0001\u0012@n(r)\n@x\u0002@n(r)\n@y\u0013\n(1)\nwith n(r) being the unit vector representing the direc-\ntion of the spin as a function of the two-dimensional spa-\ntial coordinates r. This is the integral of the solid angle\nsubtended by n, and counts how many times the unit\nsphere is wrapped. The solid angle and skyrmion number\nNskalso play essential role when one derives the equation\nof motion for the center of mass of the spin texture, i.e.,\nthe gyro-motion is induced by Nskin the Thiele equation,\nwhere the rigid body motion is assumed.7,8)\nBeyond the Thiele equation,7)one can derive the equa-\ntion of motion of a DW in terms of two variables, i.e.,\nthe wall-normal displacement q(t;\u0010;\u0011 ) and the wall-\nmagnetization orientation angle  (t;\u0010;\u0011 ) (see Fig. 1)\n\u0003iwasaki@appi.t.u-tokyo.ac.jp\nynagaosa@ap.t.u-tokyo.ac.jp\nψqFig. 1. Schematic magnetization distribution of DW with many\nBloch lines.\nwhere\u0010and\u0011are general coordinates specifying the\npoint on the DW:9)\n\u000e\u001b\n\u000e = 2M\r\u00001h\n_q\u0000\u000b\u0001_ \u0000vs\n?\u0000\f\u0001vs\nk(@k )i\n;(2)\n\u000e\u001b\n\u000eq=\u00002M\r\u00001h\n_ +\u000b\u0001\u00001_q+vs\nk(@k )\u0000\f\u0001\u00001vs\n?i\n;\n(3)\nHere, _ means the time-derivative. kand?indicate\nthe components parallel and perpendicular to the DW\nrespectively. Mis the magnetization, \ris the gyro-\nmagnetic ratio, and \u001b, \u0001 are the energy per area and\nthickness of the DW. vsis the velocity of the conduction\nelectrons, which produces the spin transfer torque. \u000bis\nthe Gilbert damping constant, and \frepresents the non-\nadiabatic e\u000bect. These equations indicate that qand \nare canonical conjugate to each other. This is understood\nby the fact that the generator of the spin rotation nor-\nmal to the DW, which is proportional to sin  in Fig. 1,\ndrives the shift of q. (Note that  is measured from the\n\fxed direction in the laboratory coordinates.)\nIn order to reduce the magnetostatic energy, the spins\nin the DW tend to align parallel to the DW, i.e., Bloch\nwall. When the DW is straight, this structure is coplanar\nand has no solid angle. From the viewpoint of eqs. (2)\n1arXiv:1506.00723v1  [cond-mat.mes-hall]  2 Jun 2015J. Phys. Soc. Jpn. LETTERS\nand (3), the angle  is \fxed around the minimum, and\nslightly canted when the motion of qoccurs, i.e., _ = 0.\nHowever, it often happens that the Bloch lines (BLs)\nare introduced into the DW as shown schematically in\nFig. 1. The angle  rotates along the DW and the N\u0013 eel\nwall is locally introduced. It is noted here that the solid\nangle becomes \fnite in the presence of the BLs. Also with\nmany BLs in the DW, the translation of BLs activates\nthe motion of the angle  , i.e., _ 6= 0, which leads to the\ndramatic change in the dynamics.\nIn the following, we focus on the straight DW which\nextends along x-direction and is uniform in z-direction.\nThus, the general coordinates here are ( \u0010;\u0011) = (x;z).\nq(t;x;z ) is independent of the coordinates q(t;x;z ) =\nq(t), and the functional derivative \u000e\u001b=\u000eq in eq. (3) be-\ncomes the partial derivative @\u001b=@q . In the absence of\nBLs, we set  (t;x;z ) = (t), and\u000e\u001b=\u000e in eq. (2) also\nbecomes@\u001b=@ . Then the equation of motion in the ab-\nsence of BL is\n@\u001b\n@ = 2M\r\u00001h\n_q\u0000\u000b\u0001_ \u0000vs\n?i\n; (4)\n@\u001b\n@q=\u00002M\r\u00001h\n_ +\u000b\u0001\u00001_q\u0000\f\u0001\u00001vs\n?i\n; (5)\nWith many BLs, the sliding motion of Bloch lines along\nDW, which activates _ , does not change the wall energy,\ni.e.,\u000e\u001b=\u000e in eq. (2) vanishes.2)Here, for simplicity, we\nconsider the periodic BL array with the uniform twist\n (t;x;z ) = (x\u0000p(t))=~\u0001 where ~\u0001 is the distance between\nBLs, which leads to\n0 = 2M\r\u00001h\n_q+\u000b\u0001~\u0001\u00001_p\u0000vs\n?\u0000\f\u0001~\u0001\u00001vs\nki\n;(6)\n@\u001b\n@q=\u00002M\r\u00001h\n\u0000~\u0001\u00001_p+\u000b\u0001\u00001_q+~\u0001\u00001vs\nk\u0000\f\u0001\u00001vs\n?i\n;\n(7)\nFirst, let us discuss the magnetic \feld driven motion\nwithout current. The e\u000bect of the external magnetic \feld\nHextis described by the force @\u001b=@q =\u00002MHextin\neqs. (5) and (7). vs\nkandvs\n?are set to be zero. In the\nabsence of BL, as mentioned above, the phase  is static\n_ = 0 with the slight tilt of the spin from the easy-plane,\nand one obtains from eq. (5)\n_q=\u0001\rHext\n\u000b: (8)\nThis is a natural result, i.e., the mobility is inversely\nproportional to the Gilbert damping \u000b. is determined\nby eq. (4) with this value of the velocity _ q.\nIn the presence of many BLs, eqs. (6) and (7) give the\nvelocities of DW and BL sliding driven by the magnetic\n\feld as\n_q=\u000b\n1 +\u000b2\u0001\rHext; (9)_p=\u00001\n1 +\u000b2~\u0001\rHext: (10)\nComparing eqs. (8) and (9), the mobility of the DW is re-\nduced by the factor of \u000b2since\u000bis usually much smaller\nthan unity. We also note that the velocity of the BL slid-\ning _pis larger than that of the wall _ qby the factor of\n\u000b. Physically, this means that the e\u000bect of the external\nmagnetic \feld Hextmostly contributes to the rapid mo-\ntion of the BLs along the DW rather than the motion of\nthe DW itself. These results have been already reported\nin refs.2,9,10)\nNow let us turn to the motion induced by the current\nvs. In the absence of BL, again we put _ = 0 in eqs. (4)\nand (5). Assuming that there is no pinning force or ex-\nternal magnetic \feld, i.e., @\u001b=@q = 0, one obtains from\neq. (5)\n_q=\f\n\u000bvs\n?; (11)\nand eq. (4) determines the equilibrium value of  . When\nthe pinning force @\u001b=@q =Fpinis \fnite, there appears a\nthreshold current density ( vs\n?)cwhich is determined by\nputting _q= 0 in eq. (5) as\n(vs\n?)c=\r\u0001\n2M\fFpin; (12)\nwhich is inversely proportional to \f.11)Since eq. (11) is\nindependent of vs\nk, the threshold current density\u0010\nvs\nk\u0011\nc\nis\u0010\nvs\nk\u0011\nc=1.\nIn the presence of the many BLs, on the other hand,\neqs. (6) and (7) give\n@\u001b\n@q=\u00002M\r\u00001\u00141 +\u000b2\n\u000b\u0001\u00001_q\n\u00001 +\u000b\f\n\u000b\u0001\u00001vs\n?\u0000\f\u0000\u000b\n\u000b~\u0001\u00001vs\nk\u0015\n;\n(13)\nwhich is the main result of this paper. From eq. (13), the\ncurrent-velocity characteristic in the absence of both the\npinning and the external \feld ( @\u001b=@q =0) is\n_q=1 +\u000b\f\n1 +\u000b2vs\n?\u0000\f\u0000\u000b\n1 +\u000b2\u0001~\u0001\u00001vs\nk\n'vs\n?+ (\f\u0000\u000b)\u0001~\u0001\u00001vs\nk; (14)\nwhere the fact \u000b;\f\u001c1 is used in the last step. If we\nneglect the term coming from vs\nk, the current-velocity\nrelation becomes almost independent of \u000band\fin\nsharp contrast to eq. (11). This is similar to the univer-\nsal current-velocity relation in the case of skyrmion,12)\nwhere the solid angle is \fnite and also the transverse\nmotion to the current occurs. Note that vs\nkslightly con-\ntributes to the motion when \u000b6=\f, while it does\nnot in the absence of BL. Even more dramatic is the\ncritical current density in the presence of the pinning\n2J. Phys. Soc. Jpn. LETTERS\n30\n20\n101525\n520\n1015\n5\n3.0\n2.0\n1.01.52.5\n0.50.6\n0.4\n0.20.30.5\n0.10.4 0.2 1.0 0.8 0.6 0.4 0.2 1.0 0.8 0.6\n4000 2000 10000 8000 6000 4000 2000 10000 8000 6000qq\nqq\nt tt tw/o BL\nw/ BLs0.429\n0.707Pinning\nq(a)\n(c) (d)(b)\nFig. 2. The wall displacement qas a fucntion of tfor the DWs\nwithout BL and with BLs. (a) vs\n?= 22 :0. The inset shows the\npinning force Fpin. (b) vs\n?= 21 :0. (c) vs\n?= 0:0043. (d) vs\n?=\n0:0042.\n(@\u001b=@q =Fpin). When we apply only the current per-\npendicular to the DW, i.e., vs\nk= 0, putting _ q= 0 in\neq. (13) determines the threshold current density as\n(vs\n?)c=\r\u0001\n2M\u000b\n1 +\u000b\fFpin; (15)\nwhich is much reduced compared with eq. (12) by the\nfactor of\u000b\f\n1+\u000b\f\u001c1. Note that ( vs\n?)cin eq. (15) is even\nsmaller than the case of skyrmion12)by the factor of\n\u000b. Similarly, the critical current density of the motion\ndriven byvs\nkis given by\n\u0010\nvs\nk\u0011\nc=\r~\u0001\n2M\u000b\nj\f\u0000\u000bjFpin; (16)\nwhich can also be smaller than eq. (12).\nNext we look at the numerical solutions of q(t) driven\nby the current vs\n?perpendicular to the wall under the\npinning force. We assume the following pinning force:\n(\r\u0001=2M)Fpin(q) =v\u0003(q=\u0001) exp\u0002\n\u0000(q=\u0001)2\u0003\n(see the in-\nset of Fig. 2(a)). We employ the unit of \u0001 = v\u0003=\n1 and the parameters ( \u000b;\f) are \fxed at ( \u000b;\f) =\n(0:01;0:02). Here, we compare two DWs without BL\nand with BLs. The maximum value of the pinning force\n(\r\u0001=2M)Fpin\nmax= 0:429 determines the threshold current\ndensity (vs\n?)cas (vs\n?)c= 21:4 and (vs\n?)c= 0:00429 in the\nabsence of BL and in the presence of many BLs, respec-\ntively. In Fig. 2(a), both DWs overcome the pinning at\nthe current density vs\n?= 22:0, although the velocity of\nthe DW without BL is suppressed in the pinning poten-\ntial. At the current density vs\n?= 21:0 below the threshold\nvalue in the absence of BL, the DW without BL is pinned,\nwhile that with BLs still moves easily (Fig. 2(b)). The\nvelocity suppression in the presence of BLs is observed\nat much smaller current density vs\n?= 0:0043 (Fig. 2(c)),\nand \fnally it stops at vs\n?= 0:0042 (Fig. 2(d)).\nAll the discussion above relies on the assumption thatthe wall is straight and  rotates uniformly. When the\nbending of the DW and non-uniform distribution of BLs\nare taken into account, the average velocity and the\nthreshold current density take the values between two\ncases without BL and with many BLs. The situation\nchanges when the DW forms closed loop, i.e., the do-\nmain forms a bubble. The bubble with many BLs and\nlargejNskjis called hard bubble because the repulsive\ninteraction between the BLs makes it hard to collapse\nthe bubble.2)At the beginning of the motion, the BLs\nmove along the DW, which results in the tiny critical cur-\nrent. In the steady state, however, the BLs accumulate\nin one side of the bubble.13,14)Then, the con\fguration\nof the BLs is static and the Thiele equation is justi\fed\nas long as the force is slowly varying within the size of\nthe bubble. The critical current density ( vs)cis given by\n(vs)c/Fpin=Nsk(Nsk(\u001d1): the skyrmion number of\nthe hard bubble), and is reduced by the factor of Nsk\ncompared with the skyrmion with Nsk=\u00061.\nIn conclusion, we have studied the current-induced\ndynamics of the DW with many BLs. The \fnite _ in\nthe steady motion activated by BLs sliding drastically\nchanges the dynamics, which has already been reported\nin the \feld-driven case. In contrast to the \feld-driven\ncase, where the mobility is suppressed by introducing\nBLs, that in the current-driven motion is not necessarily\nsuppressed. Instead, the current-velocity relation shows\nuniversal behavior independent of the damping strength\n\u000band non-adiabaticity \f. Furthermore, the threshold\ncurrent density in the presence of impurities is tiny even\ncompared with that of skyrmion motion by the factor of\n\u000b. These \fndings will stimulate the development of the\nracetrack memory based on the DW with many BLs.\nAcknowledgments We thank W. Koshibae for useful discus-\nsion. This work is supported by Grant-in-Aids for Scienti\fc Re-\nsearch (S) (No. 24224009) from the Ministry of Education, Cul-\nture, Sports, Science and Technology of Japan. J. I. was supported\nby Grant-in-Aids for JSPS Fellows (No. 2610547).\n1) A. Hubert and R. Sch afer, Magnetic Domains: The Analysis\nof Magnetic Microstructures (Springer-Verlag, Berlin, 1998).\n2) A. P. Malozemo\u000b and J.C. Slonczewski, Magnetic Domain\nWalls in Bubble Materials (Academic Press, New York, 1979).\n3) J. C. Slonczewski, J. Magn. Magn. Mater. 159, L1{L7 (1996).\n4) L. Berger, Phys. Rev. B 54, 9353{9358 (1996).\n5) S. M uhlbauer et al., Science 323, 915 (2009).\n6) X. Z. Yu et al., Nature 465, 901 (2010).\n7) A. A. Thiele, Phys. Rev. Lett. 30, 230 (1973).\n8) K. Everschor et al., Phys. Rev. B 86, 054432 (2012).\n9) J. C. Slonczewski, J. Appl. Phys. 45, 2705 (1974).\n10) A. P. Malozemo\u000b and J. C. Slonczewski, Phys. Rev. Lett. 29,\n952 (1972).\n11) G. Tatara et al., J. Phys. Soc. Japan 75, 64708 (2006).\n12) J. Iwasaki, M. Mochizuki and N. Nagaosa, Nat. Commun. 4,\n1463 (2013).\n13) G. P. Vella-Coleiro, A. Rosencwaig and W. J. Tabor, Phys.\nRev. Lett. 29, 949 (1972)\n14) A. A. Thiele, F. B. Hagedorn and G. P. Vella-Coleiro, Phys.\n3J. Phys. Soc. Jpn. LETTERS\nRev. B 8, 241 (1973).\n4"    },    {        "title": "1506.01303v3.Antidamping_spin_orbit_torque_driven_by_spin_flip_reflection_mechanism_on_the_surface_of_a_topological_insulator__A_time_dependent_nonequilibrium_Green_function_approach.pdf",        "content": "Antidamping spin-orbit torque driven by spin-\rip re\rection mechanism on the surface\nof a topological insulator: A time-dependent nonequilibrium Green function approach\nFarzad Mahfouzi,1,\u0003Branislav K. Nikoli\u0013 c,2and Nicholas Kioussis1\n1Department of Physics, California State University, Northridge, CA 91330-8268, USA\n2Department of Physics and Astronomy, University of Delaware, Newark, DE 19716-2570, USA\nMotivated by recent experiments observing spin-orbit torque (SOT) acting on the magnetization\n~ mof a ferromagnetic (F) overlayer on the surface of a three-dimensional topological insulator (TI),\nwe investigate the origin of the SOT and the magnetization dynamics in such systems. We predict\nthat lateral F/TI bilayers of \fnite length, sandwiched between two normal metal leads, will generate\na large antidamping-like SOT per very low charge current injected parallel to the interface. The\nlarge values of antidamping-like SOT are spatially localized around the transverse edges of the F\noverlayer. Our analysis is based on adiabatic expansion (to \frst order in @~ m=@t ) of time-dependent\nnonequilibrium Green functions (NEGFs), describing electrons pushed out of equilibrium both by the\napplied bias voltage and by the slow variation of a classical degree of freedom [such as ~ m(t)]. From it\nwe extract formulas for spin torque and charge pumping, which show that they are reciprocal e\u000bects\nto each other, as well as Gilbert damping in the presence of SO coupling. The NEGF-based formula\nfor SOT naturally splits into four components, determined by their behavior (even or odd) under the\ntime and bias voltage reversal. Their complex angular dependence is delineated and employed within\nLandau-Lifshitz-Gilbert simulations of magnetization dynamics in order to demonstrate capability\nof the predicted SOT to e\u000eciently switch ~ mof a perpendicularly magnetized F overlayer.\nPACS numbers: 72.25.Dc, 75.70.Tj, 85.75.-d, 72.10.Bg\nI. INTRODUCTION\nThe spin-orbit torque (SOT) is a recently discovered\nphenomenon1{4in ferromagnet/heavy-metal (F/HM)\nlateral heterostructures involves unpolarized charge cur-\nrent injected parallel to the F/HM interface induces\nswitching or steady-state precession5of magnetization\nin the F overlayer. Unlike conventional spin-transfer\ntorque (STT) in spin valves and magnetic tunnel junc-\ntion (MTJs),6{8where one F layer acts as spin-polarizer\nof electrons that transfer torque to the second F layer\nwhen its free magnetization is noncollinear to the direc-\ntion of incoming spins, heterostructures exhibiting SOT\nuse a single F layer. Thus, in F/HM bilayers, spin-orbit\ncoupling (SOC) at the interface or in the bulk of the\nHM layer is crucial to spin-polarized injected current\nvia the Edelstein e\u000bect (EE)9,10or the spin Hall e\u000bect\n(SHE),11,12respectively.\nThe SOT o\u000bers potentially more e\u000ecient magnetiza-\ntion switching than achieved by using MTJs underlying\npresent STT-magnetic random access memories (STT-\nMRAM).13Thus, substantial experimental and theo-\nretical e\u000borts have been focused on identifying physi-\ncal mechanisms behind SOT whose understanding would\npave the way to maximize its value by using optimal\nmaterials combinations. For example, very recent ex-\nperiments14{16have replaced HM with three-dimensional\ntopological insulators (3D TIs).17The TIs enhance18{20\n(by a factor ~vF=\u000bR, wherevFis the Fermi velocity on\nthe surface of TI and \u000bRis the Rashba SOC strength22?\nat the F/HM interface) the transverse nonequilibrium\nspin density driven by the longitudinal charge current,\nwhich is responsible for the large \feld-like SOT compo-\nnent20,23observed experimentally.14{16\nFIG. 1. (Color online) Schematic view of F/TI lateral bilayer\noperated by SOT. The F overlayer has \fnite length LF\nxand~ m\nis the unit vector along its free magnetization. The TI layer\nis attached to two N leads which are semi-in\fnite in the x-\ndirection and terminate into macroscopic reservoirs. We also\nassume that F and TI layers, as well as N leads, are in\fnite in\nthey-direction. The unpolarized charge current is injected by\nthe electrochemical potential di\u000berence between the left and\nthe right macroscopic reservoirs which sets the bias voltage,\n\u0016L\u0000\u0016R=eVb. We mention that the results do not change if\nthe TI surface is covered by the F overlayer partially or fully.\nFurthermore, recent experiments have also observed\nantidamping-like SOT in F/TI heterostructures with\nsurprisingly large \fgure of merit (i.e., antidamping\ntorque per unit applied charge current density) that sur-\npasses14{16those measured in a variety of F/HM het-\nerostructures. This component competes against the\nGilbert damping which tries to restore magnetization\nto equilibrium, and its large \fgure of merit is, there-\nfore, of particular importance for increasing e\u000eciency of\nmagnetization switching. Theoretical understanding ofarXiv:1506.01303v3  [cond-mat.mes-hall]  24 Jan 20162\nthe physical origin of antidamping-like SOT is crucial to\nresolve the key challenge for anticipated applications of\nSOT generated by TIs|demonstration of magnetization\nswitching of the F overlayer at room temperature (thus\nfar, magnetization switching has been demonstrated only\nat cryogenic temperature15).\nHowever, the microscopic mechanism behind its large\nmagnitude14{16and ability to e\u000eciently (i.e., using as lit-\ntle dc current density as possible) switch magnetization15\nremains under scrutiny. For example, TI samples used\nin these experiments are often unintentionally doped, so\nthat bulk charge carriers can generate antidamping-like\nSOT via rather large24SHE (but not su\u000ecient to explain\nall reported values14,15). The simplistic picture,14in\nwhich electrons spin-polarized by the EE di\u000buse into the\nF overlayer14to deposit spin angular momentum within\nit, cannot operate in technologically relevant F overlayers\nof'1 nm thickness16or explain complex angular depen-\ndence2,15,25typically observed for SOT. The Berry cur-\nvature mechanism25,26for antidamping-like SOT applied\nto lateral F/TI heterostructures predicts its peculiar de-\npendence on the magnetization orientation,27vanishing\nwhen magnetization ~ mis parallel to the F/TI interface.\nThis feature has thus far not been observed experimen-\ntally,15and, furthermore, it makes such antidamping-like\nSOT less e\u000ecient27(by requiring larger injected currents\nto initiate magnetization switching) than standard SHE-\ndriven3,4antidamping-like SOT.\nWe note that the recent experimental14{16and theo-\nretical14,27studies of SOT in lateral F/TI bilayer have\nfocused on the geometry where an in\fnite F overlayer\ncovers an in\fnite TI layer. Moreover, they assume14,27?\npurely two-dimensional transport where only the top sur-\nface of the TI layer is explicitly taken into account by the\nlow-energy e\u000bective (Dirac) Hamiltonian supplemented\nby the Zeeman term due to the magnetic proximity ef-\nfect. On the other hand, transport in realistic TI-based\nheterostructures is always three-dimensional, with unpo-\nlarized electrons being injected from normal metal con-\ntacts, re\rected from the F/TI edge to \row along the sur-\nface of the TI in the yz-plane and then along the bottom\nTI surface in Fig. 1. In fact, electrons also \row within a\nthin layer (of thickness .2 nm in Bi 2Se3as the prototyp-\nical TI material) underneath the top and bottom surfaces\ndue to top and bottom metallic surfaces of the TI doping\nthe bulk via evanescent wave functions.18Therefore, in\nthis study we consider more realistic and experimentally\nrelevant28F/TI bilayer geometries, illustrated in Fig. 1,\nwhere the TI layer of \fnite length LTI\nxand \fnite thick-\nnessLTI\nzis (partially or fully) covered by the F overlayer\nof lengthLF\nx. The two semi-in\fnite ideal N leads are di-\nrectly attached to the TI layer. we should mention that\nthe result does not depend on the length of TI layer that\nis covered by the FM.\nOur principal results are twofold and are summarized\nas follows:\n(i)Theoretical prediction for SOT: We predict that the\ngeometry in Fig. 1 will generate large antidamping-likeSOT per low injected charge current. By studying spatial\ndependence of the SOT (see Fig. 4), we show that in a\nclean FM/TI interface the electrons exert anti-damping\ntorque on the FM as they enter into the interface and un-\nless interfacial roughness or impurities are included the\ntorque remains mainly concentrated around the edge of\nthe interface. Although the exact results show strong\nnonperturbative features, based on second order pertur-\nbation we present two di\u000berent interpretations showing\nthat the origin of the antidamping SOT relies on the\nspin-\rip re\rection of the chiral electrons injected into\nthe FM/TI interface. Its strong angular dependence (see\nFig. 2), i.e., dependence on the magnetization direction\n~ m, o\u000bers a unique signature that can be used to distin-\nguish it from other possible physical mechanisms. By\nnumerically solving the Landau-Lifshitz-Gilbert (LLG)\nequation in the macrospin approximation, we demon-\nstrate (see Figs. 5 and 6) that the obtained SOT is ca-\npable of switching of a single domain magnetization of a\nperpendicularly magnetized F overlayer with bias voltage\nin the oder of the Magneto-Crystaline Anisotropy (MCA)\nenergy.\n(ii)Theoretical formalism for SOT: The widely used\nquantum (such as the Kubo formula25{27,30) and semi-\nclassical (such as the Boltzmann equation31) transport\napproaches to SOT are tailored for geometries where an\nin\fnite F layer covers an in\fnite TI or HM layer. Due to\ntranslational invariance, the nonequilibrium spin density\n~Sinduced by the EE on the surface of TI or HM layer\nhas uniform orientation ~S= (0;Sy;0) [in the coordinate\nsystem in Fig. 1], which then provides reference direc-\ntion for de\fning \feld-like, \u001cf~ m\u0002^y, and antidamping-\nlike,\u001cad~ m\u0002(~ m\u0002^y), components of SOT. In order to\nanalyze spatial dependence of SOT in the device geome-\ntry of Fig. 1, while not assuming anything a priori about\nthe orientation of \feld-like and antidamping-like compo-\nnents of SOT, we employ adiabatic expansion32of time-\ndependent nonequilibrium Green functions (NEGFs)33,34\nto derive formulas for torque, charge pumping35,36and\nGilbert damping37in the presence of SOC. The NEGF-\nbased formula for SOT naturally splits into four compo-\nnents, determined by their behavior (even or odd) under\nthe time and bias voltage reversal. This gives us a general\nframework in quantum mechanics to analyze the dissi-\npative (antidamping-like) and nondissipative (\feld-like)\nforce (torque) vector \felds for a set of canonical variables\n(magnetization directions). Their angular (see Fig. 2)\nand spatial (see Fig. 4) dependence shows that although\n\feld-like and antidamping-like SOTs are predominantly\nalong the~ m\u0002^yand~ m\u0002(~ m\u0002^y) directions, respectively,\nthey are not uniform and can exhibit signi\fcant devia-\ntion from the trivial angular dependence de\fned by these\ncross products [see Fig. 2(h)].\nThe paper is organized as follows. In Sec. II, we present\nthe adiabatic expansion of time-dependent NEGFs, in a\nrepresentation that is alternative to Wigner representa-\ntion34(usually employed for this type of derivation32),\nand derive expressions for torque, charge pumping and3\nGilbert damping. In Sec. III, we decompose the NEGF-\nbased expression for SOT into four components, deter-\nmined by their behavior (even or odd) under the time and\nbias voltage reversal, and investigate their angular de-\npendence. Section IV discusses the angular dependence\nof the zero-bias transmission function which identi\fes the\nmagnetization directions at which substantial re\rection\noccurs. In Sec. V, we study spatial dependence of SOT\ncomponents and discuss their physical origin. Section VI\npresents LLG simulations of magnetization dynamics in\nthe presence of predicted SOT, as well as a switching\nphase diagram of the magnetization state as a function\nof the in-plane external magnetic \feld and SOT. We con-\nclude in Sec. VII.\nII. THEORETICAL FORMALISM\nWe \frst describe the time-dependent Hamiltonian\nmodel, H(t) =H0+U(t), of the lateral F/TI heterostruc-\nture in Fig. 1. Here H0is the minimal tight-binding\nmodel for 3D TIs like Bi 2Se3on a cubic lattice of spacing\nawith four orbitals per site.38The thickness, LTI\nz= 8a\nof the TI layer is su\u000ecient to prevent hybridization be-\ntween its top and bottom metallic surface states.18The\ntime-dependent potential\nU(t) =\u0000\u0001surf1m~ m(t)\u0001~\u001b=2; (1)\ndepends on time through the magnetization of the F over-\nlayer which acts as the slowly varying classical degree of\nfreedom. Here ~ m(t) is the unit vector along the direc-\ntion of magnetization, \u0001 surf= 0:28 eV is the proximity\ninduced exchange-\feld term and 1mis a diagonal matrix\nwith elements equal to unity for sites within the F/TI\ncontact region in Fig. 1 and zero elsewhere. The semi-\nin\fnite ideal N leads in Fig. 1 are taken into account\nthrough the self-energies33,34\u0006L;Rcomputed for a tight-\nbinding model with one spin-degenerate orbital per site.\nThe details of how to properly couple \u0006L;RtoH0, while\ntaking into account that the spin operators for electrons\non the Bi and Se sublattices of the TI are inequivalent,39\ncan be found in Ref. 40.\nWithin the NEGF formalism33,34the advanced\nand lesser GFs matrix elements of the tight-binding\nHamiltonian, H0, are de\fned by Gii0;oo0;ss0(t;t0) =\n\u0000i\u0002(t\u0000t0)hf^cios(t);^cy\ni0o0s0(t0)gi, andG<\nii0;oo0;ss0(t;t0) =\nih^cy\ni0o0s0(t0)^cios(t)i, respectively. Here, ^ cy\nios(^cios) is the\ncreation (annihilation) operator for an electron on site, i,\nwith orbital, o, and spins, respectively,h:::idenotes the\nnonequilibrium statistical average, and ~= 1 to simplify\nthe notation. These GFs are the matrix elements of the\ncorresponding matrices GandG<used throughout the\ntext.\nUnder stationary conditions, the two GFs depend on\nthe di\u000berence of the time arguments, t\u0000t0, and can\nbe Fourier transformed to energy. In the strictly adi-\nabatic limit one can employ41the same retarded GF,Gt(E) = [E\u0000H(t)\u0000\u0006L\u0000\u0006R]\u00001, as under stationary\nconditions, but where the GF depends parametrically on\ntime (denoted by the subscript t) and is computed for the\nfrozen-in-time con\fguration of U(t). However, even for\nslow evolution of ~ m(t) corrections32to the adiabatic GF\nare needed to describe dissipation e\u000bects such as Gilbert\ndamping or the charge current which can be pumped35\nby the dynamics of ~ m(t).\nThe so-called adiabatic expansion, which yields correc-\ntions beyond the strictly adiabatic limit, is traditionally\nperformed using the Wigner representation34in which\nthe fast and slow time scales are easily identi\fable.32The\nslow motion implies that the NEGFs vary slowly with the\ncentral time tc= (t+t0)=2 while they change fast with\nthe relative time tr=t\u0000t0. By expanding the Wigner\ntransformation of NEGFs\nG(<)\nW(E;tc) =Z1\n\u00001dtreiEtrG(<)\u0012\ntc+tr\n2;tc\u0000tr\n2\u0013\n;\n(2)\nin the central time tcwhile keeping only terms contain-\ning \frst-order derivatives @=@tc(due to the slow varia-\ntion withtc) gives the \frst-order correction beyond the\nstrictly adiabatic limit.32This route requires to han-\ndle complicated expressions resulting from the Wigner\ntransform applied to convolutions of the type C(t1;t2) =R\ndt3C1(t1;t3)C2(t3;t2).\nHere we provide an alternative derivation of the \frst-\norder nonadiabatic correction. Namely, we consider t\n(observation time) and t\u0000t0(relative time) as the nat-\nural variables to describe the time evolution of NEGFs\nand then perform the following Fourier transform42\nG(t;t0) =Z1\n\u00001dE\n2\u0019eiE(t\u0000t0)G(E;t): (3)\nThe standard equations of motion for G(t;t0) and\nG<(t;t0) are cumbersome to manipulate42,43or solve\nnumerically,44so they are usually transformed to some\nother representation.35Here we replace G(t;t0) in the\nstandard equations of motion with the rhs of Eq. (3) to\narrive at:\n\u0014\u0012\nE\u0000i@\n@t\u0013\n1\u0000H0\u0000U(t)\u0000\u0006\u0012\nE\u0000i@\n@t\u0013\u0015\nG(E;t) =1;\n(4)\nand\nG<(t;t0) =ZdE\n2\u0019G(E;t)\u0006<(E)Gy(E;t0)eiE(t\u0000t0):(5)\nFor the two-terminal heterostructure in Fig. 1 in the\nelastic transport regime,33,34\u0006(E) =\u0006L(E) +\u0006R(E)\nand\u0006<(E) =ifL(E)\u0000L(E) +ifR(E)\u0000R(E), where\n\u0000L;R=i(\u0006L;R\u0000\u0006y\nL;R). The Fermi-Dirac distribution\nfunctions of electrons in the macroscopic reservoirs into\nwhich the left and right N leads terminate are fL;R(E) =\nf(E\u0000\u0016L;R), where the di\u000berence between the electro-\nchemical potentials, \u0016L;R=EF+eVL;R, de\fnes the bias\nvoltageeVb=\u0016L\u0000\u0016R.4\nUsing the following identity\nX\n\u000b=L;RiG\u0000\u000bGy= (G\u0000Gy) +i@\n@E\u0012\nG@U\n@tGy\u0013\n+O\u0012@2U\n@t2\u0013\n; (6)\nthe lesser GF to \frst order in small @U(t)=@tand in the low bias Vbregime (i.e., the linear-response transport regime)\ncan be expressed as,\nG<(t;t)'ZdE\n2\u00190\n@[G(E;t)\u0000Gy(E;t)]f(E) +X\n\u000b=L;Rf0eV\u000bGt\u0000\u000bGy\nt+if0Gt@U(t)\n@tGy\nt1\nA; (7)\nwhere, the \frst term corresponds to the density matrix of the equilibrium electrons occupying the time dependent\nsingle particle states, while the second and third terms describe the density matrix of the excited (nonequilibrium)\nelectrons occupying the states close to the Fermi energy due to the bias voltage and time dependent term in the\nHamiltonian, respectively. The retarded GF in the \frst term in Eq. (7) expanded to \frst order in @U(t)=@tis of the\nform\nG(E;t)'Gt+i@Gt\n@E@U(t)\n@tGt: (8)\nThe lesser GF determines the time-dependent nonequilibrium density matrix\n\u001a(t) =1\niG<(t;t); (9)\nfrom which we determine the time-dependent expectation values of physical observables, A(t) = Tr [\u001a(t)^A]. In\nparticular, the relevant quantities for the heterostructure in Fig. 1 are the charge current\nI\u000b(t) =eZdE\n2\u0019iTr\b\n\u0000\u000b(E)G<(t;t) +f\u000b(E)\u0000\u000b(E)[G(E;t)\u0000Gy(E;t)]\t\n=e2\n2\u0019Z\ndEf0(E)8\n<\n:X\n\f(V\f\u0000V\u000b)T\u000b\f(E)\u0000\u0001surf\n2eX\ni@mi\n@tT\u000bi(E)9\n=\n;; (10)\nand the spin density\nsi(t) =ZdE\n2\u0019iTr\u0002\n\u001bi1mG<\u0003\n=ZdE\n2\u00198\n<\n:X\n\u000bf\u000b(E)Ti\u000b(E)\u0000\u0001surf\n2X\njf0(E)@mj\n@tTij(E)9\n=\n;; (11)\nwhere\u000b;\f2fL;Rgandi;j2fx;y;zg.\nThe \\trace-formulas\" in Eqs. (10) and (11)\nT\u000b\f(E) = Trh\n\u0000\u000bGt\u0000\fGy\nti\n; (12a)\nT\u000bi(E) = Trh\n1m\u001biGy\nt\u0000\u000bGti\n; (12b)\nTi\u000b(E) = Trh\n1m\u001biGt\u0000\u000bGy\nti\n; (12c)\nTij(E) = Trh\n1m\u001bi(Gy\nt\u0000Gt)1m\u001bj(Gt\u0000Gy\nt)i\n;(12d)\ndetermine charge current45due toVb, charge current\npumped35by the dynamics of ~ m(t) in the presence of\nSOC, the spin torque, and Gilbert damping tensor, re-\nspectively. In the expression for the spin density we ig-\nnore the antisymmetric part of Tijwhich corresponds to\nthe renormalization of the precession frequency of mag-netization dynamics.32Note thatT\u000biin Eq. (12)(b) and\nits time-reversal Ti\u000bin Eq. (12)(c) reveal a reciprocal46\nrelation between charge pumping by magnetization dy-\nnamics in the presence of SOC and current-driven SOT\nateach instant of time t.\nThe spin density in Eq. (11) enters into the LL equa-\ntion for the magnetization dynamics\n@~ m\n@t=\u0000\u0001surf\n2~ m\u0002~ s(t): (13)\nThe \frst term in Eq. (11) generates the spin torque in\nEq. (13) which has three contributions: ( i) an equilib-\nrium component responsible for the interlayer exchange\ninteraction in the presence of a second F layer and/or\nmagneto-crystalline anisotropy (MCA) in the presence of\nSOC; ( ii) a bias-induced \feld-like torque modifying the5\nequilibrium interlayer exchange and MCA \felds; and ( iii)\na damping (antidamping)-like torque describing angular\nmomentum loss (gain) due to the \rux of Fermi surface\nelectrons.\nIII. ANGULAR DEPENDENCE OF SOT\nCOMPONENTS\nIn order to understand the di\u000berent contributions to\nTi\u000b(E), we decompose it into even (e) or odd (o) terms\nunder time-reversal Gt7!Gy\nt:\nTi\u000b\ne(E) = [Ti\u000b(E) +T\u000bi(E)]=2; (14)\nTi\u000b\no(E) = [Ti\u000b(E)\u0000T\u000bi(E)]=2: (15)\nSinceP\n\u000bTi\u000b\no(E) = 0, the contribution of the odd com-\nponent to the equilibrium spin density in Eq. (11) van-\nishes identically, while its nonzero values appear only for\nEwithin the bias window around EF. This motivates\nfurther splitting of Ti\u000b(E) into four components for the\ncase of two-terminal devices\nTi\ne;\u0017(E) =TiL\n\u0017(E) +TiR\n\u0017(E)\n2; (16a)\nTi\no;\u0017(E) =TiL\n\u0017(E)\u0000TiR\n\u0017(E)\n2; (16b)\nwhere the \frst and second subscripts denote their be-\nhavior (even or odd) under bias reversal Vb7!\u0000Vband\ntime reversal, respectively. The corresponding four com-\nponents of torque are determined by\n~Te;\u0017=ZdE\n2\u0019[fL(E) +fR(E)]~ \u001ce;\u0017(E); (16c)\n~To;\u0017=ZdE\n2\u0019[fL(E)\u0000fR(E)]~ \u001co;\u0017(E);(16d)\nwhere the energy-resolved torque is given by\n~ \u001c\u0016;\u0017(E) =\u0000\u0001surf\n2~ m\u0002~T\u0016;\u0017(E); (17)\nand\u0016;\u00172fe;og. The terms ~To;oand~To;eare non-zero\nonly in nonequilibrium driven by Vb6= 0, and depend on\nelectronic states in the bias voltage window around the\nFermi energy (or on the Fermi surface states in the linear-\nresponse regime where integrals are avoided by multiply-\ning integrand by eVb). The term ~Te;o\u00110 is zero, while\n~Te;eis nonzero also in equilibrium and, therefore, depends\non all occupied electronic states.\nFigures 2(a-c) show the netvector \feld (summed over\nall sites of the F overlayer) of ~ \u001c\u0016;\u0017(EF) at zero temper-\nature for di\u000berent directions of ~ mon the unit sphere.\nTheir angular behavior reveals that: ( i)~ \u001co;oshown in\nFig. 4(a) is the \feld-like SOT generated by the EE, with\npredominant orientation along the ~ m\u0002^y-direction; ( ii)\n~ \u001co;ein Fig. 4(b) is the antidamping-like SOT with pre-\ndominant orientation along the ~ m\u0002(~ m\u0002^y)-direction;\nand ( iii)~ \u001ce;ein Fig. 4(c) along the ~ m\u0002^zdirection isthe \feld-like component whose angular dependence be-\nhaves approximately as 2( ~ m\u0001^z)j~ m\u0002^zj\u0011sin(2\u0012) typical\nfor torque components generated by the MCA \feld.2\nThe corresponding angular dependence of the net\n\u001ci\no;o(i2fx;y;zg),j~ \u001co;ej, and\u001ci\ne;ealong the solid trajecto-\nries shown in Figs. 2(a-c) are plotted in Figs. 2(d-f),\nrespectively. We \fnd that the magnitude of the max-\nimum antidamping-like SOT is about a factor of four\nlarger than that of the \feld-like SOT. Additionally, the\n\feld-like SOT peaks when the magnetization is in-plane.\nIn contrast, the antidamping-like SOT peaks when the\nmagnetization is out of plane, which can be attributed to\nthe gap opening of the Dirac cone which in turn enhances\nthe re\rection (see Sec. IV) at the lateral boundaries of\nthe F overlayer.\nNote that the magnitude of the netSOT components\nshown in Figs. 2(g-i) exhibits strong angular dependence\nbecause of the large SOC on the TI surface similar to that\nfound in F/HM heterostructures when the Rashba SOC\nat the interface is su\u000eciently strong.25,26In particular,\nthe signi\fcant deviation of the angular dependence of the\nantidamping-like SOT from the trivial j~ m\u0002(~ m\u0002^y)jbe-\nhavior in the limit of ~ m!^y, shown quantitatively in\nFig. 2(h), indicates its nonperturbative variation with\nrespect to the magnetization direction. A similar nonper-\nturbative angular behavior (i.e., strong deviation from\nthe standard/sin2\u0012dependence on the precession cone\nangle\u0012) has been found for adiabatic charge pumping\nfrom a precessing F overlayer attached to the edge of 2D\nTIs47,48or to the surface of 3D TIs.36\nIV. ANGULAR DEPENDENCE OF\nTRANSMISSION FUNCTION\nFigure 3 shows the transmission function TRL(EF) in\nEq. (12a) for the heterostructure in Fig. 1 versus the ori-\nentation of ~ mon the unit sphere. We \fnd that the charge\ncurrent determined by TRL(E) is smallest49when~ mk^z\nor~ mk^x. This is due to the re\rection of Dirac elec-\ntrons on the TI surface from the lateral boundaries of\nthe F overlayer. Underneath the F overlayer, the ex-\nchange \feld,\u0000\u0001surf~ m\u0001~\u001b=2, induced by the magnetic\nproximity e\u000bect is superimposed on the Dirac cone sur-\nface dispersion. This opens an energy gap \u0001 surfwhen\n~ mk^z(or smaller gap \u0001 surfcos\u0012formz6= 0) at the DP\nof the TI region underneath the F overlayer, which in\nturn gives rise to strong electronic re\rection when ~ mk^z.\nFor~ mk^x, there is no energy gap at the DP and the\nDirac cone e\u000bectively shifts away from the center of the\nBrillouin zone due to proximity exchange \feld. Neverthe-\nless electrons polarized by the EE along the y-axis re\rect\nfrom the magnetization pointing along the x-axis.6\n(a)                                         (b)                                         (c)\n(d)                                         (e)                                         (f)\n(g)                 (h)                                         ( i)\n-0.20.00.2i\ne,e(EF)(1/a)  x\n y\n z\n0 45 90 135 1800.00.10.20.3|e,e(EF)|/sin  2 (1/a)\nAngle  (deg)\n-5-4-3-2-1010-2i\no,o(EF)(1/a) x\n y\n z\n0 45 90 135 180024681010-2|o,o(EF)|/sin  (1/a)\nAngle  (deg)\n0.00.10.2|o,e(EF)|(1/a) =0o\n =45o\n =90o\n0 45 90 135 1800.00.10.20.3|o,e(EF)|/sin   (1/a)\nAngle  (deg)=90o=45o=0o\nFIG. 2. (Color online) (a){(c) The vector \feld of SOT components ~ \u001c\u0016;\u0017(EF), de\fned by Eq. (16), for di\u000berent directions of\n~ mon the unit sphere. The angular behavior in (a) and (c) shows that ~ \u001co;oand~ \u001ce;ebehave as \feld-like torques, while that in\n(b) shows that ~ \u001co;ebehaves as antidamping-like torque. Cartesian components [(d)-(f)] and magnitude of ~ \u001c\u0016;\u0017[(g){(i)] along\nthe trajectories denoted by solid lines in the corresponding panels (a)-(c). The magnitude of ~ \u001c\u0016;\u0017is divided by:j~ m\u0002^yjin (g);\nj~ m\u0002(~ m\u0002^y)jin (h) and 2( ~ m\u0001^z)j~ m\u0002^zjin (i).\nV. SPATIAL DEPENDENCE OF SOT\nCOMPONENTS AND PHYSICAL ORIGIN OF\nANTIDAMPING-LIKE SOT\nFigure 4(a) demonstrates that large values of\nantidamping-like SOT from Fig. 2 are spatially localized\naround the transverse edges of the F overlayer, for Fermi\nenergy inside and outside of the surface state gap induced\nby out-of-plane magnetic exchange coupling. While con-\nductance in this system is close to zero at the Dirac point,\nwe observe that the anti-damping torque does not de-\npend strongly on the Fermi energy. This suggests a high\ne\u000eciency of SOT per injected current for the Fermi en-\nergy close to the Dirac point. For the \feld-like SOT\nin Fig. 4(b) we see the torque independent of the co-\nordinate in the entire F/TI contact region, as expected\nfrom the phenomenology of the EE. In Fig. 4(c) we plot\nthe contribution of the Fermi energy electrons to the\nFM/TI interface induced MCA \feld. Even though to-\ntal\u001ci\ne;o(EF)\u00110, its spatially-resolved value plotted in\nFig. 4(d) is nonzero which can be removed by perform-\ning a proper gauge transformation.\nTo understand the origin of the anti-damping SOT let\nus \fnd an expression for the average of SOT around a\fxed axes~ m=~ m0+(~ m?cos(\u001e)+~ m0\u0002~ m?sin(\u001e))\u000e\u0012with\nsmall cone angle \u000e\u0012. By applying a rotation operator we\ncan align the \fxed axes along z-axis such that ~ m0= ^ez\nand~ m?= ^ex. In this case we have\nhTzi\u001e=\u0001surf\u000e\u0012\n2=Zd\u001e\n2\u0019ZdE\n2\u0019X\n\u000bf\u000bei\u001eT\u0000;\u000b;(18)\nwhere,T\u0000;\u000b=Tx\u000b\u0000iTy\u000band=refers to the imaginary\npart. Expanding the GFs in Eq. 12(b) to the lowest order\nwith respect to \u000e\u0012e\u0000i\u001e, we obtain,\nT\u0000;\u000b=\u0001surf\n4e\u0000i\u001e\u000e\u0012Trh\n1m\u001b\u0000Gt1m\u001b+Gt\u0000\u000bGy\nt(19)\n+1m\u001b\u0000Gt\u0000\u000bGy\nt1m\u001b+Gy\nti\n:\nPlugging this expression into Eq.(18), and using the iden-\ntity,Gt\u0000Gy\nt=iP\n\u000bGt\u0000\u000bGy\nt, in linear bias voltage\nregime we obtain,\nhTzi\u001e=Vb\u00012\nsurf\u000e\u00122\n16Tr[\u001a\"\"\nL1m\u001a##\nR1m\u0000\u001a\"\"\nR1m\u001a##\nL1m];(20)\nwhere,\u001a\u000b=\u0000iG<\n\u000b=Gt\u0000\u000bGy\nt, corresponds to the\ndensity matrix inside the F overlayer for the electrons7\n(a)\nTRL(EF)×10−2\n4.5\n4\n3.5\n3\n𝑥𝑦𝑧\nFIG. 3. (Color online) The transmission function TRL(EF)\nin Eq. (12a) for two-terminal heterostructure shown in Fig. 1\nat di\u000berent directions of magnetization ~ mon the unit sphere.\nThe Fermi energy is set at EF= 3:1 eV (which is 0 :1 eV above\nthe DP).\n02 04 0-1010\n2 04 0-505-10-\n20τ\nx,ze\n,e(EF)=0×10-4×10-2τye\n,e(EF) (1/a)L\nongitudinal Coordinate xF/TI (a)τxe\n,o(EF) (1/a)m=(0,0,1)τ\ny,ze\n,o(EF)=0τy,zo\n,o(EF)=0τ\nx,zo\n,e(EF)=0 EF=3.3eV \nEF=3.1eV(\nd)( c)(b)( a)×\n10-2τyo\n,e(EF) (1/a)×10-2τxo\n,o(EF) (1/a)\nFIG. 4. (Color online) (a){(d) Spatial dependence of SOT\ncomponents, \u001ci\n\u0016;\u0017(EF) (i2fx;y;zgand\u0016;\u00172fe;og), per\nunit length, for Fermi energy inside the magnetization in-\nduced gap around Dirac point ( EF= 3:1eV) and outside\nthe gap (EF= 3:3eV), for~ mk^zin Fig. 1. Their physical\nmeaning is explained in Fig. 2. The range of x-coordinate\ncorresponds to the length LF\nx= 40aof the F overlayer, while\nthe results are independent of the length of the TI layer un-\nderneath,LTI\nx.\n(holes) at the Fermi surface being injected from the lead\n\u000b(\f6=\u000b). The electron-hole analogy can be understood\nby de\fning the hole density matrix, iG>\n\u000b, from the iden-\ntity\u0000i(G<\n\u000b\u0000G>\n\u000b) = 2=(G) =\u001a\u000b+P\n\f6=\u000b\u001a\f. By con-\nsidering left-lead induced holes instead of right-lead in-\nduced electrons, we can interpret Eq.(20) as spin-resolved\nelectron-hole recombination rate, where opposite spins\nhave opposite contributions to the antidamping-like\nSOT. This picture focuses on the energy anti-dissipative\naspect of the phenomena and, since \u001a\u001b\u001b\nL(\u001a\u001b\u001b\nR) cor-\nFIG. 5. (Color online) SOT-induced magnetization trajecto-\nries~ m(t) under di\u000berent Vband~Bext= 0. Higher color inten-\nsity denotes denser bundle of trajectories which start from all\npossible initial conditions ~ m(t= 0) on the unit sphere. Solid\ncurves show examples of magnetization trajectories, while the\nwhite circles denote attractors of trajectories.\nresponds to the spin- \u001bright (left) moving electrons,\nEq. (20) suggests that spin-momentum locking natu-\nrally has a signi\fcant e\u000bect on the enhancement of\nthe antidamping-like SOT magnitude. In particular,\nin the case of F/TI interface, the enhancement of the\nantidamping-like SOT occurs when the spin-up/down is\nalong they-axis (~ m0k^y) which is the spin-polarization\ndirection of electrons passing through the surface of the\nTI induced by the EE. Additionally, in this case the\nantidamping-like SOT gets smaller away from the F/TI\ntransverse edge because the contribution of both of the\nleads to the spin density become identical. Therefore the\nanti-damping torque in this case is more localized around\nthe edge. This e\u000bect is more signi\fcant when the magne-\ntization is out of the plane and the Fermi energy is inside\nthe \u0001 surfcos\u0012gap on the TI surface.\nA alternative interpretation of the results can be\nachieved by considering Gt\u0000Gy\nt=iP\n\u000bGy\nt\u0000\u000bGt. In\nthis case, the average of the antidamping-like SOT is ex-\npressed by\nhTzi\u001e=Vb\n4Tr[T\"#\nLR\u0000T\"#\nRL]; (21)\nwhere the F overlayer induced spin-\rip transmission ma-\ntrix is de\fned as\nT\"#\n\u000b\f= (t\"#\n\u000b\f)yt\"#\n\u000b\f; (22)\nand\nt\"#\n\u000b\f=\u0001surf\u000e\u0012\n2p\n\u0000\u000bGt\u001b+Gtp\n\u0000\f: (23)\nAlthough Eq. (21) is obtained from perturbative con-\nsiderations, it looks identical to the Eq. (8) of Ref. 48\nwhere a spin-\rip re\rection mechanism at the edge of the\nF/2D-TI interface was recognized to be responsible for\nthe giant charge pumping (i.e., anti-damping torque) ob-\nserved in the numerical simulation.48Eq. (23) describes a\ntransmission event in which electrons injected from lead\n\u000b, get spin-\ripped (from up to down) by the FM and\nthen transmit to the lead \f. The path of the electrons\ndescribing this process is shown in Fig. 1. From the k-\nresolved results of the anti-damping torque (not shown8\nhere) we observe that while for the in-plane magnetiza-\ntion electrons moving in the same transverse direction\n(same sign for ky) on both left and right edges of the\nFM/TI interface contribute to the torque, in the case of\nout-of-plane magnetization for the left (right) edge of the\ninterface the local anti-damping torque is induced mostly\nby the electrons with ky>0 (ky<0).\nIt is worth mentioning that due to nonperturbative na-\nture of the SOT induced by the chiral electrons, the ap-\nproximation presented in this section which can as well\nbe obtained from the self energy corresponding to the\nvacuum polarization Feynman diagrams of the electron-\nmagnon coupled system59, does not capture the phenom-\nena accurately. This is evident in the angular dependence\nof the anti-damping torque which in the current section\nis considered up to second order e\u000bect ( \u000e\u00122), while the\ndivergence-like behavior in Figs.2(h) suggest a linear de-\npendence when the magnetization direction is close to the\ny-axis. This signi\fes the importance of the higher order\nterms with respect to \u000e\u0012that can not be ignored. The ap-\nproximation presented in this section also suggests that\nblocking the lower surface leads to the reduction of the\nanti-damping torque. However, in this case an electron\nexperiences multiple spin-\rip re\rections before transmit-\nting to the next lead and in fact it turns out that the ex-\nact results stay intact even if the lower surface is blocked.\nThis is similar to the conclusion made in Ref. 48 which\nshows the redundancy of blocking the lower edge of the\n2D-TI to obtain a nonzero pumped charge current from\nprecessing FM as proposed in Ref. 47.\nAlthough spin-momentum locking of the surface state\nof the TI resembles the 2D Rashba plane, in the case of\nTI surface state the cones with opposite spin-momentum\nlocking reside on opposite surface sides of the TI slab\nwhile in the case of a Rashba plane they are only sepa-\nrated by the SOC energy. This means one can expect a\nsmaller SOT for a FM on top of a 2D Rashba plane due\nto cancellation of the e\u000bects of the two circles with op-\nposite spin-momentum locking, where the nonzero anti-\ndamping torque originates from the electron-hole asym-\nmetry.\nVI. LLG SIMULATIONS OF MAGNETIZATION\nDYNAMICS IN THE PRESENCE OF SOT\nIn order to investigate ability of predicted\nantidamping-like SOT to switch the magnetization\ndirection of a perpendicularly magnetized F overlayer in\nthe geometry of Fig. 1, we study magnetization dynamics\nin the macrospin approximation by numerically solving\nLLG equation at zero temperature supplemented by\nSOT components analyzed in Sec. III\n@~ m\n@t=1\n2\u0019[~ \u001co;e(~ m;EF) +~ \u001co;o(~ m;EF)]eVb+\r~Bext\u0002~ m\n+~ m\u0002\u0014\n\u000b(~ m)\u0001@~ m\n@t\u0015\n+ (~ m\u0001^z)(~ m\u0002^z)\u0001MCA:(24)\nFIG. 6. (Color online) Phase diagram of the magnetization\nstate in lateral F/TI heterostructure from Fig. 1 as a function\nof an in-plane external magnetic Bextk^xandVb(i.e., SOT\n/Vb). Thick arrows on each of the panels (a){(d) show the\ndirection of sweeping of Bext\nxorVbparameter. The small-\nness of central hysteretic region along the Vb-axis, enclosed\nby white dashed line in panel (b) and (d), shows that low\ncurrents are required to switch magnetization from mz>0\ntomz<0 stable states.\nHere\ris the gyromagnetic ratio, \u000b(~ m)ij =\n\u00012\nsurfTij(~ m;EF)=8\u0019is the dimensionless Gilbert damp-\ning tensor, and \u0001 MCA = \u00010\nMCA +j~Te;ej=j(~ m\u0001^z)(~ m\u0002^z)j,\nwhere \u00010\nMCA represents the intrinsic MCA energy of the\nFM. We solve Eq. (24) by assuming that the Gilbert\ndamping is a constant (its dependence on ~ mis relegated\nto future studies) and ignore the dependence of \u0001 MCA\non~ mandVbwhile retaining its out-of-plane direction.\nFigure 5 shows the magnetization trajectories for all\npossible initial conditions ~ m(t= 0) on the unit sphere\nunder di\u000berent Vb. AtVb= 0, the two attractors are\nlocated as the north and south poles of the sphere. At\n\fniteVb, the attractors shift away from the poles along\nthez-axis within the xz-plane, while additional attractor\nappears on the positive (negative) y-axis under negative\n(positive)Vb. Note that the applied bias voltage Vbdrives\ndc current and SOT proportional to it in the assumed\nlinear-response transport regime.\nFigure 6 shows the commonly constructed3,4,15,27\nphase diagram of the magnetization state in the pres-\nence of an external in-plane magnetic \feld Bextk^xand\nthe applied bias voltage Vb(i.e., SOT/Vb). The thick\narrows in each panel of Fig. 6 denote the direction of the\nsweeping variable|in Fig. 6(a) [6(b)] we increase [de-\ncrease]Vbslowly in time, and similarly in Fig. 6(c) [6(d)]\nwe increase [decrease] the external magnetic \feld gradu-\nally. The size of hysteretic region in the center of these\ndiagrams, enclosed by white dashed line in Figs. 6(b) and\n6(d), measures the e\u000eciency of switching.3,4,15,27Since\nthis region, where both magnetization states mz>0 and\nmz<0 are allowed, is relatively small in Figs. 6(a) and9\n6(b), magnetization can be switched by low Bext\nxand\nsmallVb(or, equivalently, small injected dc current), akin\nto the phase diagrams observed in recent experiments.15\nAlthough we considered the FM a single domain, the\nfact that the anti-damping component of the SOT is\nmainly peaked around the edge of the FM/TI interface\nsuggests that it is be more feasible in realistic cases to\nhave the local magnetic moments at the edge of the FM\nswitch \frst and then the total magnetization switches\nby the propagation of the domain walls formed at the\nedge throughout the FM60,61. Therefore, a micromag-\nnetic simulation of the system is required to investigate\nswitching phenomena in large size systems which we rel-\negate to future works.\nVII. CONCLUSIONS\nIn conclusion, by performing adiabatic expansion of\ntime-dependent NEGFs,33,34we have developed a frame-\nwork which yields formulas for spin torque and charge\npumping as reciprocal e\u000bects to each other connected by\ntime-reversal, as well as Gilbert damping due to SOC. It\nalso introduces a novel way to separate the SOT com-\nponents, based on their behavior (even or odd) under\ntime and bias voltage reversal, and can be applied to\narbitrary systems dealing with classical degrees of free-\ndom coupled to electrons out of equilibrium. For the\ngeometry28proposed in Fig. 1, where the F overlayercovers (either partially or fully) the top surface of the TI\nlayer, we predict that low charge current \rowing solely\non the surface of TI will induce antidamping-like SOT on\nthe F overlayer via a physical mechanism that requires\nspin-momentum locking on the surface of TIs|spin-\rip\nre\rection at the lateral edges of a ferromagnetic island\nintroduced by magnetic proximity e\u000bect onto the TI sur-\nface. This mechanism has been overlooked in e\u000borts to\nunderstand why SO-coupled interface alone (i.e., in the\nabsence of SHE current from the bulk of SO-coupled non-\nferromagnetic materials) can generate antidamping-like\nSOT, where other explored mechanisms have included\nspin-dependent impurity scattering at the interface,55\nBerry curvature mechanism,25,26as well as their com-\nbination.56\nThe key feature for connecting experimentally ob-\nserved SOT and other related phenomena in F/TI het-\nerostructures (such as spin-to-charge conversion28,36,58)\nto theoretical predictions is their dependence2,24on the\nmagnetization direction. The antidamping-like SOT pre-\ndicted in our study exhibits complex angular dependence,\nexhibiting \\nonperturbative\" change with the magnetiza-\ntion direction in Fig. 2(h), which should make it possible\nto easily di\u000berentiate it from other competing physical\nmechanisms.\nACKNOWLEDGMENTS\nF. M. and N. K. were supported by NSF PREM Grant\nNo. 1205734, and B. K. N. was supported by NSF Grant\nNo. ECCS 1509094.\n\u0003farzad.mahfouzi@gmail.com\n1I. M. Miron, K. Garello, G. Gaudin, P.-J. Zermatten, M. V.\nCostache, S. Au\u000bret, S. Bandiera, B. Rodmacq, A. Schuhl,\nand P. Gambardella, \\Perpendicular switching of a single\nferromagnetic layer induced by in-plane current injection,\"\nNature 476, 189 (2011).\n2K. Garello, I. M. Miron, C. O. Avci, F. Freimuth,\nY. Mokrousov, S. Bl ugel, S. Au\u000bret, O. Boulle, G. Gaudin,\nand P. Gambardella, \\Symmetry and magnitude of spin-\norbit torques in ferromagnetic heterostructures,\" Nat.\nNanotechnol. 8, 587 (2013).\n3L. Liu, O. J. Lee, T. J. Gudmundsen, D. C. Ralph, and\nR. A. Buhrman, \\Current-induced switching of perpendic-\nularly magnetized magnetic layers using spin torque from\nthe spin Hall e\u000bect,\" Phys. Rev. Lett. 109, 096602 (2012).\n4L. Liu, C.-F. Pai, Y. Li, H. W. Tseng, D. C. Ralph, and\nR. A. Buhrman, \\Spin-torque switching with the giant spin\nHall e\u000bect of tantalum,\" Science 336, 555 (2012).\n5R. H. Liu, W. L. Lim, and S. Urazhdin, \\Spectral charac-\nteristics of the microwave emission by the spin Hall nano-\noscillator,\" Phys. Rev. Lett. 110, 147601 (2013).\n6D. Ralph and M. Stiles, \\Spin transfer torques,\" J. Magn.\nMagn. Mater. 320, 1190 (2008).\n7M. D. Stiles and A. Zangwill, \\Anatomy of spin-transfer\ntorque,\" Phys. Rev. B 66 , 014407 (2002).\n8I. Theodonis, N. Kioussis, A. Kalitsov, M. Chshiev, andW. H. Butler, \\Anomalous bias dependence of spin torque\nin magnetic tunnel junctions,\" Phys. Rev. Lett. 97, 237205\n(2006).\n9V. M. Edelstein, \\Spin polarization of conduction electrons\ninduced by electric current in two-dimensional asymmetric\nelectron systems,\" Solid State Comm. 73, 233 (1990).\n10A. G. Aronov and Y. B. Lyanda-Geller, \\Nuclear electric\nresonance and orientation of carrier spins by an electric\n\feld,\" JETP Lett. 50, 431 (1989).\n11G. Vignale, \\Ten years of spin Hall e\u000bect,\" J. Supercond.\nNov. Magn. 23, 3 (2010).\n12J. Sinova, S. O. Valenzuela, J. Wunderlich, C. H. Back,\nand T. Jungwirth, \\Spin Hall e\u000bects,\" Rev. Mod. Phys.\n87, 1213 (2015).\n13N. Locatelli, V. Cros, and J. Grollier, \\Spin-torque build-\ning blocks,\" Nat. Mater. 13, 11 (2014); A. D. Kent and\nD. C. Worledge, \\A new spin on magnetic memories,\" Nat.\nNanotechnol. 10, 187 (2015).\n14A. R. Mellnik et al. , \\Spin-transfer torque generated by a\ntopological insulator,\" Nature 511, 449 (2014).\n15Y. Fan et al. , \\Magnetization switching through giant\nspinorbit torque in a magnetically doped topological in-\nsulator heterostructure,\" Nat. Mater. 13, 699 (2014).\n16Y. Wang, P. Deorani, K. Banerjee, N. Koirala, M. Brahlek,\nS. Oh, and H. Yang, \\Topological surface states originated\nspin-orbit torques in Bi 2Se3,\" Phys. Rev. Lett. 114, 25720210\n(2015).\n17M. Z. Hasan and C. L. Kane, \\Colloquium: Topological\ninsulators,\" Rev. Mod. Phys. 82, 3045 (2010); X.-L. Qi\nand S.-C. Zhang, \\Topological insulators and supercon-\nductors,\" Rev. Mod. Phys. 83, 1057 (2011).\n18P.-H. Chang, T. Markussen, S. Smidstrup, K. Stokbro, and\nB. K. Nikoli\u0013 c, \\Nonequilibrium spin texture within a thin\nlayer below the surface of current-carrying topological insu-\nlator Bi 2Se3: A \frst-principles quantum transport study,\"\nPhys. Rev. B 92, 201406(R) (2015).\n19D. Pesin and A. H. MacDonald, \\Spintronics and pseu-\ndospintronics in graphene and topological insulators,\" Nat.\nMater. 11, 409 (2012).\n20I. Garate and M. Franz, \\Inverse spin-galvanic e\u000bect in the\ninterface between a topological insulator and a ferromag-\nnet,\" Phys. Rev. Lett. 104, 146802 (2010).\n21P. Moras, G. Bihlmayer, P. M. Sheverdyaeva, S. K. Ma-\nhatha, M. Papagno, J. S\u0013 anchez-Barriga, O. Rader, L.\nNovinec, S. Gardonio, and C. Carbone, \\Magnetization-\ndependent Rashba splitting of quantum well states at the\nCo/W interface,\" Phys. Rev. B 91, 195410 (2015).\n22A. Manchon, H. C. Koo, J. Nitta, S. M. Frolov, and R. A.\nDuine, \\New perspectives for Rashba spin-orbit coupling,\"\nNat. Mater. 14, 871 (2015).\n23T. Yokoyama, \\Current-induced magnetization reversal on\nthe surface of a topological insulator,\" Phys. Rev. B 84,\n113407 (2011).\n24M. Jamali, J. S. Lee, J. S. Jeong, F. Mahfouzi, Y. Lv,\nZ. Zhao, B. K. Nikoli\u0013 c, K. A. Mkhoyan, N. Samarth, and\nJ.-P. Wang, \\Giant spin pumping and inverse spin Hall\ne\u000bect in the presence of surface and bulk spin-orbit cou-\npling of topological insulator Bi 2Se3,\" Nano Lett. 15, 7126\n(2015).\n25K.-S. Lee, D. Go, A. Manchon, P. M. Haney, M. D. Stiles,\nH.-W. Lee, and K.-J. Lee, \\Angular dependence of spin-\norbit spin-transfer torques,\" Phys. Rev. B 91, 144401\n(2015).\n26H. Li et al. , \\Intraband and interband spin-orbit torques\nin noncentrosymmetric ferromagnets,\" Phys. Rev. B 91,\n134402 (2015); H. Kurebayashi et al. , \\An antidamping\nspinorbit torque originating from the Berry curvature,\"\nNat. Nanotechnol. 9, 211 (2014).\n27P. B. Ndiaye, C. A. Akosa, M. H. Fischer, A. Vaezi, E.-A.\nKim, and A. Manchon, \\Dirac spin-orbit torques at the\nsurface of topological insulators,\" arXiv:1509.06929.\n28J. C. R. S\u0013 anchez, L. Vila, G. Desfonds, S. Gambarelli, J. P.\nAttan\u0013 e, J. M. D. Teresa, C. Mag\u0013 en, and A. Fert, \\Spin-to-\ncharge conversion using Rashba coupling at the interface\nbetween non-magnetic materials,\" Nat. Commun. 4, 2944\n(2013).\n29K. M. M. Habib, R. N. Sajjad, and A. W. Ghosh, \\Chiral\ntunneling of topological states: Towards the e\u000ecient gener-\nation of spin current using spin-momentum locking,\" Phys.\nRev. Lett. 114, 176801 (2015); A. Narayan, I. Rungger, A.\nDroghetti, and S. Sanvito, \\ Ab initio transport across bis-\nmuth selenide surface barriers,\" Phys. Rev. B 90, 205431\n(2014).\n30F. Freimuth, S. Bl ugel, and Y. Mokrousov, \\Spin-orbit\ntorques in Co/Pt(111) and Mn/W(001) magnetic bilayers\nfrom \frst principles,\" Phys. Rev. B 90, 174423 (2014).\n31P. M. Haney, H.-W. Lee, K.-J. Lee, A. Manchon, and\nM. D. Stiles, \\Current induced torques and interfacial spin-\norbit coupling: Semiclassical modeling,\" Phys. Rev. B 87,\n174411 (2013).32N. Bode, L. Arrachea, G. S. Lozano, T. S. Nunner, and\nF. von Oppen, \\Current-induced switching in transport\nthrough anisotropic magnetic molecules,\" Phys. Rev. B 85,\n115440 (2012).\n33G. Stefanucci and R. van Leeuwen, Nonequilibrium Many-\nBody Theory of Quantum Systems: A Modern Introduction\n(Cambridge University Press, Cambridge, 2013).\n34H. Haug and A.-P. Jauho, Quantum kinetics in trans-\nport and optics of semiconductors (Springer-Verlag, Berlin,\n2008).\n35F. Mahfouzi, J. Fabian, N. Nagaosa, and B. K. Nikoli\u0013 c,\n\\Charge pumping by magnetization dynamics in magnetic\nand semimagnetic tunnel junctions with interfacial Rashba\nor bulk extrinsic spin-orbit coupling,\" Phys. Rev. B 85,\n054406 (2012).\n36F. Mahfouzi, N. Nagaosa, and B. K. Nikoli\u0013 c, \\Spin-\nto-charge conversion in lateral and vertical topological-\ninsulator/ferromagnet heterostructures with microwave-\ndriven precessing magnetization,\" Phys. Rev. B 90, 115432\n(2014).\n37T. Yokoyama, J. Zang, and N. Nagaosa, \\Theoretical study\nof the dynamics of magnetization on the topological sur-\nface,\" Phys. Rev. B 81, 241410 (2010).\n38C.-X. Liu, X.-L. Qi, H. Zhang, X. Dai, Z. Fang, and S.-\nC. Zhang, \\Model Hamiltonian for topological insulators,\"\nPhys. Rev. B 82, 045122 (2010).\n39P. G. Silvestrov, P. W. Brouwer, and E. G. Mishchenko,\n\\Spin and charge structure of the surface states in topolog-\nical insulators,\" Phys. Rev. B 86, 075302 (2012); F. Zhang,\nC. L. Kane, and E. J. Mele, \\Surface states of topological\ninsulators,\" Phys. Rev. B 86, 081303 (2012).\n40P.-H. Chang, F. Mahfouzi, N. Nagaosa, and B. K. Nikoli\u0013 c,\n\\Spin-Seebeck e\u000bect on the surface of a topological insula-\ntor due to nonequilibrium spin-polarization parallel to the\ndirection of thermally driven electronic transport,\" Phys.\nRev. B 89, 195418 (2014).\n41S. Salahuddin and S. Datta, \\Self-consistent simulation\nof quantum transport and magnetization dynamics in\nspin-torque based devices,\" Appl. Phys. Lett. 89, 153504\n(2006).\n42L. Arrachea, \\Exact Green's function renormalization ap-\nproach to spectral properties of open quantum systems\ndriven by harmonically time-dependent \felds,\" Phys. Rev.\nB75, 035319 (2007).\n43A.-P. Jauho, N. S. Wingreen, and Y. Meir, \\Time-\ndependent transport in interacting and noninteracting\nresonant-tunneling systems,\" Phys. Rev. B 50, 5528\n(1994).\n44B. Gaury, J. Weston, M. Santin, M. Houzet, C. Groth,\nand X. Waintal, \\Numerical simulations of time-resolved\nquantum electronics,\" Phys. Rep. 534, 1 (2014).\n45C. Caroli, R. Combescot, P. Nozieres, and D. Saint-James,\n\\Direct calculation of the tunneling current,\" J. Phys. C:\nSolid State Phys. 4, 916 (1971).\n46F. Freimuth, S. Bl ugel, and Y. Mokrousov, \\Direct and in-\nverse spin-orbit torques,\" Phys. Rev. B 92, 064415 (2015).\n47X.-L. Qi, T. L. Hughes, and S.-C. Zhang, \\Fractional\ncharge and quantized current in the quantum spin Hall\nstate,\" Nat. Phys. 4, 273 (2008).\n48F. Mahfouzi, B. K. Nikoli\u0013 c, S.-H. Chen, and C.-R. Chang,\n\\Microwave-driven ferromagnet{topological-insulator het-\nerostructures: The prospect for giant spin battery e\u000bect\nand quantized charge pump devices,\" Phys. Rev. B 82,\n195440 (2010).11\n49B. D. Kong, Y. G. Semenov, C. M. Krowne, and K. W.\nKim, \\Unusual magnetoresistance in a topological insula-\ntor with a single ferromagnetic barrier,\" Appl. Phys. Lett.\n98, 243112 (2011).\n50D. Kim, S. Cho, N. P. Butch, P. Syers, K. Kirshenbaum,\nS. Adam, J. Paglione, M. S. Fuhrer, \\Surface conduction\nof topological Dirac electrons in bulk insulating Bi 2Se3,\"\nNat. Phys. 8, 459 (2012).\n51S. Wang, Y. Xu, and K. Xia, \\First-principles study of\nspin-transfer torques in layered systems with noncollinear\nmagnetization,\" Phys. Rev. B 77, 184430 (2008).\n52A. Manchon, N. Ryzhanova, A. Vedyayev, M. Chschiev,\nand B. Dieny, \\Description of current-driven torques in\nmagnetic tunnel junctions,\" J. Phys.: Condens. Matter 20,\n1 (2008).\n53A. Brataas and G. E. W. Bauer and P. J. Kelly, \\Non-\ncollinear magnetoelectronics,\" Phys. Rep. 157, 427 (2006).\n54G. Aut\u0013 es, J. Mathon, and A. Umerski, \\Re\rection mech-\nanism for generating spin transfer torque without charge\ncurrent,\" J. Appl. Phys. 111, 053909 (2012).\n55D. A. Pesin and A. H. MacDonald, \\Quantum kinetic the-\nory of current-induced torques in Rashba ferromagnets,\"\nPhys. Rev. B 86, 014416 (2012).56A. Qaiumzadeh, R. A. Duine, and M. Titov, \\Spin-orbit\ntorques in two-dimensional Rashba ferromagnets,\" Phys.\nRev. B 92, 014402 (2015).\n57M. Lang et al. , \\Competing weak localization and weak\nantilocalization in ultrathin topological insulators,\" Nano\nLett.13, 48 (2013).\n58Y. Shiomi, K. Nomura, Y. Kajiwara, K. Eto, M. Novak,\nK. Segawa, Y. Ando, and E. Saitoh, \\Spin-electricity con-\nversion induced by spin injection into topological insula-\ntors,\" Phys. Rev. Lett. 113, 196601 (2014).\n59F. Mahfouzi and B. K. Nikoli\u0013 c, \\Signatures of electron-\nmagnon interaction in charge and spin currents through\nmagnetic tunnel junctions: A nonequilibrium many-body\nperturbation theory approach,\" Phys. Rev. B 90, 045115\n(2014).\n60G. Yu, P. Upadhyaya, K. L. Wong, W. Jiang, J. G. Alzate,\nJ. Tang, P. Khalili Amiri, and K. L. Wang, \\Magnetization\nswitching through spin-Hall-e\u000bect-induced chiral domain\nwall propagation,\" Phys. Rev. B. 89, 104421 (2014).\n61N. Mikuszeit, O. Boulle, I. M. Miron, K. Garello, P. Gam-\nbardella, G. Gaudin and L. D. Buda-Prejbeanu, \\Spin-\norbit torque driven chiral magnetization reversal in ultra-\nthin nanostructures,\" Phys. Rev. B. 92, 144424 (2015)."    },    {        "title": "1506.05622v2.The_absence_of_intraband_scattering_in_a_consistent_theory_of_Gilbert_damping_in_metallic_ferromagnets.pdf",        "content": "arXiv:1506.05622v2  [cond-mat.str-el]  23 Oct 2015The absence of intraband scattering in a consistent theory o f Gilbert damping in\nmetallic ferromagnets\nD M Edwards\nDepartment of Mathematics, Imperial College London, Londo n SW7 2BZ, United Kingdom\nDamping of magnetization dynamics in a ferromagnetic metal , arising from spin-orbit coupling,\nis usually characterised by the Gilbert parameter α. Recent calculations of this quantity, using\na formula due to Kambersky, find that it is infinite for a perfec t crystal owing to an intraband\nscattering term which is of third order in the spin-orbit par ameterξ. This surprising result conflicts\nwith recent work by Costa and Muniz who study damping numeric ally by direct calculation of\nthe dynamical transverse susceptibility in the presence of spin-orbit coupling. We resolve this\ninconsistency by following the approach of Costa and Muniz f or a slightly simplified model where\nit is possible to calculate αanalytically. We show that to second order in ξone retrieves the\nKambersky result for α, but to higher order one does not obtain any divergent intrab and terms.\nThe present work goes beyond that of Costa and Muniz by pointi ng out the necessity of including\nthe effect of long-range Coulomb interaction in calculating damping for large ξ. A direct derivation\nof the Kambersky formula is given which shows clearly the res triction of its validity to second order\ninξso that no intraband scattering terms appear. This restrict ion has an important effect on the\ndamping over a substantial range of impurity content and tem perature. The experimental situation\nis discussed.\nI. INTRODUCTION\nMagnetization dynamics in a ferromagnetic metal is central to the fi eld of spintronics with its many applications.\nDamping is an essential feature of magnetization dynamics and is usu ally treated phenomenologically by means\nof a Gilbert term in the Landau-Lifshitz-Gilbert equation [1, 2]. For a system with spin-rotational invariance the\nuniform precession mode of the magnetization in a uniform external magnetic field is undamped and the fundamental\norigin of damping in ferromagnetic resonance is spin-orbit coupling (S OC). Early investigations of the effect include\nthose of Kambersky [3–5] and Korenman and Prange [6]. Kambersky ’s [4] torque-correlation formula for the Gilbert\ndamping parameter αhas been used by several groups [7–14], some of whom have given alt ernative derivations.\nHowever the restricted validity of this formula, as discussed below, has not been stressed. In this torque-correlation\nmodel contributions to αof both intraband and interband electronic transitions are usually c onsidered. The theory\nis basically developed for a pure metal with the effect of defects and /or phonons introduced as phenomenological\nbroadening of the one-electron states. Assuming that the electr on scattering-rate increases with temperature T due\nto electron-phonon scattering the intraband and interband tran sitions are found to play a dominant role in low and\nhigh T regimes, respectively. The intraband(interband) term is pre dicted to decrease(increase) with increasing T and\nto be proportional to ξ3(ξ2) whereξis the SOC parameter. Accordingly αis expected to achieve a minimum at an\nintermediate T. This is seen experimentally in Ni and hcp Co [15] but not in Fe [15] and FePt [16]. The ξ2dependence\nofαis well-established at high T [17, 18] but there seems to be no experim ental observation of the predicted ξ3\nbehaviourat lowT. The interband ξ2term in Kambersky’stheory canbe givenaverysimple interpretation in termsof\nsecond-orderperturbation theory [5]. A quite different phenomen ologicalapproach, not applicable in some unspecified\nlow scattering-rate regime, has been adopted to try and find a phy sical interpretation of the intraband term [5, 8].\nNo acceptable theoretical treatment of damping in this low scatter ing regime is available because the intraband term\nof Kambersky’s theory diverges to infinity in the zero-scattering lim it of a pure metal with translational symmetry at\nT=0 [9, 11, 13]. We consider it essential to understand the pure met al limit before introducing impurity and phonon\nscattering in a proper way.\nCosta and Muniz [19] recently studied damping numerically in this limit by direct calculation of the dynamical\nspin susceptibility in the presence of SOC within the random phase app roximation (RPA). They determine αfrom\nthe linewidth of the uniform (wave-vector q= 0) spin-wave mode which appears as a resonance in the transvers e\nsusceptibility. One of the main objects of this paper is to establish so me degree of consistency between the work of\nKambersky and that of Costa and Muniz. We follow the approach of t he latter authors for a slightly simplified model\nwhere it is possible to calculate αanalytically. We show that to second order in ξone retrieves the Kambersky result,\nbut to higher order no intraband terms occur, which removes the p roblem of divergent α. To confirm this point, in\nAppendix A we derive the Kambersky formula directly in a way that mak es clear its restriction to second order in\nξto which order the divergent terms in αarising from intraband transitions do not appear. This throws open the\ninterpretation of the minimum observed in the temperature depend ence ofαfor Ni and Co.\nAt this point we may mention an alternative theoretical approach to the calculation of Gilbert damping using2\nscattering theory [20, 21]. Starikov et al [21] find that, for Ni 1−xFexalloys at T=0, αbecomes large near the pure\nmetal limits x=0,1. They attribute this to the Kambersky intraband c ontribution although no formal correspondence\nis made between the two approaches.\nThe work of Costa and Muniz [19] follows an earlier paper [22] where it is shown that SOC has the effect of\ncoupling the transverse spin susceptibility to the longitudinal spin su sceptibility and the charge response. It is known\nthat a proper calculation of these last two quantities in a ferromagn et must take account of long-range Coulomb\ninteractions [23–27]. The essential role of these interactions is to e nsure conservation of particle number. Costa et\nal [19, 22] do not consider such interactions but we show here that this neglect is not serious for calculating αwith\nsufficiently small SOC. Howeverin the wider frameworkof this paper, where mixed charge-spinresponse is also readily\nstudied, long-rangeinteractions are expected to sometimes play a role. They also come into play, even to second order\ninξ, when inversion symmetry is broken.\nIn section II we establish the structure of spin-density response theory in the presence of SOC by means of exact\nspin-density functional theory in the static limit [28]. In section III w e introduce a spatial Fourier transform and an\napproximation to the dynamical response is obtained by introducing the frequency dependence of the non-interacting\nsusceptibilities. The theory then has the same structure as in the R PA. Section IV is devoted to obtaining an explicit\nexpression for the transverse susceptibility in terms of the non-in teracting susceptibilities. Expressions for these, in\nthe presence of SOC, are obtained within the tight-binding approxim ation in section V. In section VI we consider\nthe damping of the resonance in the q=0 transverse susceptibility a nd show how the present approach leads to the\nKambersky formula for the Gilbert damping parameter αwhere this is valid, namely to second order in the SOC\nparameter ξ. We do not give an explicit formula for αbeyond this order but it is clear that no intraband terms appear.\nIn section VII some experimental aspects are discussed with sugg estions for future work. The main conclusions are\nsummarized in section VIII.\nII. SPIN-DENSITY FUNCTIONAL THEORY WITH SPIN-ORBIT COUPLI NG\nThe Kohn-Sham equation takes the form\n/summationdisplay\nσ′[−δσσ′(/planckover2pi12/2m)∇2+Veff\nσσ′(r)+Hso\nσσ′]φnσ′(r) =ǫnφnσ(r) (1)\nwith the spin index σ=↑,↓corresponding to quantization along the direction of the ground-s tate magnetization in a\nferromagnet. This may be written in 2 ×2 matrix form with eigenvectors ( φn↑,φn↓)T. The density matrix is defined\nin terms of the spin components φnσ(r) of the one-electron orbitals by\nnσσ′=/summationdisplay\nnφnσ(r)φnσ′(r)∗θ(µ0−ǫn) (2)\nwhereθ(x) is the unit step function and µ0is the chemical potential. The electron density is given by\nρ(r) =/summationdisplay\nσnσσ(r) =/summationdisplay\nnσ|φnσ(r)|2θ(µ0−ǫn) (3)\nand the effective potential in (1) is\nVeff\nσσ′(r) =wσσ′(r)+δσσ′/integraldisplay\nd3r′ρ(r′)v(r−r′)+vxc\nσσ′(r) (4)\nwherewσσ′(r) is the external potential due to the crystal lattice and any magn etic fields and v(r) =e2/|r|is the\nCoulomb potential. The exchange-correlation potential vxc\nσσ′(r) is defined as δExc/δnσσ′(r), a functional derivative of\nthe exchange-correlation energy Exc. The term Hso\nσσ′in (1) is the SOC energy. A small external perturbation δwσσ′,\nfor example due to a magnetic field, changes the effective potential toVeff+δVeff, giving rise to new orbitals and\nhence to a change in density matrix δnσσ′. The equation\nδnσσ′(r) =−Ω−1/summationdisplay\nσ1σ′\n1/integraldisplay\nd3r1χ0\nσσ′σ1σ′\n1(r,r1)δVeff\nσ1σ′\n1(r1), (5)\nwhere Ω is the volume of the sample, defines a non-interacting respo nse function χ0and the full response function χ\nis defined by\nδnσσ′(r) =−Ω−1/summationdisplay\nττ′/integraldisplay\nd3r′χσσ′ττ′(r,r′)δwττ′(r′). (6)3\nFrom (4)\nδVeff\nσ1σ′\n1(r1) =δwσ1σ′\n1(r1)+/summationdisplay\nσ2σ′\n2/integraldisplay\nd3r2[v(r1−r2)δσ1σ′\n1δσ2σ′\n2+δvxc\nσ1σ′\n1(r1)\nδnσ2σ′\n2(r2)]δnσ2σ′\n2(r2) (7)\nand we may write\nδvxc\nσ1σ′\n1(r1)\nδnσ2σ′\n2(r2)=δ2Exc\nδnσ2σ′\n2(r2)δnσ1σ′\n1(r1)=Kσ1σ′\n1σ2σ′\n2(r1,r2). (8)\nCombining (5) - (8) we find the following integral equation for the spin -density response function χσσ′ττ′(r,r′):\nχσσ′ττ′(r,r′) =χ0\nσσ′ττ′(r,r′)−(Ω)−1/summationdisplay\nσ1σ′\n1/summationdisplay\nσ2σ′\n2/integraldisplay\nd3r1/integraldisplay\nd3r2χ0\nσσ′σ1σ′\n1(r,r1)[v(r1−r2)δσ1σ′\n1δσ2σ′\n2+Kσ1σ′\n1σ2σ′\n2(r1,r2)]\nχσ2σ′\n2ττ′(r2,r′).\n(9)\nThis equation is a slight generalisation of that given by Williams and von Ba rth [28]. In the static limit it is formally\nexact although the exchange-correlation energy Excis of course not known exactly. In the next section we generalise\nthe equation to the dynamical case approximately by introducing th e frequency dependence of the non-interacting\nresponse functions χ0, and also take a spatial Fourier transform. In the case where SOC is absent the result is directly\ncompared with results obtained using the RPA.\nIII. DYNAMICAL SUSCEPTIBILITIES IN THE PRESENCE OF SPIN-OR BIT COUPLING AND\nLONG-RANGE COULOMB INTERACTION\nIn general the response functions χ(r,r′) are not functions of r−r′and a Fourier representation of (9) for a\nspatially periodic system involves an infinite number of reciprocal latt ice vectors. There are two cases where this\ncomplication is avoided. The first is a homogeneous electron gas and t he second is in a tight-binding approximation\nwith a restricted atomic basis. We may then introduce Fourier trans forms of the form χ(r) =/summationtext\nqχ(q)eiq·ror\nχ(q) = (Ω)−1/integraltext\nd3rχ(r)e−iq·rand write (9) as\nχσσ′ττ′(q,ω) =χ0\nσσ′ττ′(q,ω)−/summationdisplay\nσ1σ′\n1/summationdisplay\nσ2σ′\n2χ0\nσσ′σ1σ′\n1(q,ω)Vσ1σ′\n1σ2σ′\n2(q)χσ2σ′\n2ττ′(q,ω), (10)\nwhere we have also introduced the ωdependence of χas indicated at the end of the last section. Here V(q) is an\nordinary Fourier transform, without a factor (Ω)−1, so that\nVσ1σ′\n1σ2σ′\n2(q) =v(q)δσ1σ′\n1δσ2σ′\n2+Kσ1σ′\n1σ2σ′\n2, (11)\nwherev(q) = 4πe2/q2is the usual Fourier transform of the Coulomb interaction and the s econd term is independent\nofqsinceKis a short-range spatial interaction. In the gas case it is proportio nal to a delta-function δ(r−r′) in the\nlocal-density approximation (LDA) [28] and in tight-binding it can be t aken as an on-site interaction. In both cases\nKmay be expressed in terms of a parameter Uas\nKσ1σ′\n1σ2σ′\n2=−U[δσ1σ′\n1δσ2σ′\n2δσ1σ2+δσ1σ′\n1δσ2σ′\n2δσ′\n1σ2] (12)\nwhereσ=↓,↑forσ=↑,↓. in the tight-binding case this form of Kcorresponds to a simple form of interaction\nwhich leads to a rigid exchange splitting of the bands ( [29], [22]). This is only appropriate for transition metals in a\nmodel with d bands only, hybridization with s and p bands being neglect ed. We adopt this model in order to obtain\ntransparent analytic results as far as possible. Although not as re alistic as ”first-principles” models of the electronic\nstructure it has been used, even with some quantitative success, in treating the related problem of magnetocrystalline\nanisotropy in Co/Pd structures as well as pure metals [30]. In (10) t he response functions χare per unit volume\nin the gas case but, more conveniently, may be taken as per atom in t he tight-binding case with v(q) modified to\nv(q) = 4πe2/(q2Ωa) where Ω ais the volume per atom.4\nTo show how equations (10) - (12) are related to RPA we examine two examples in the absence of SOC. First\nconsider the transverse susceptibility χ↓↑↑↓(q,ω) which is more usually denoted by χ−+(q,ω). Equation (10) becomes\nχ↓↑↑↓=χ0\n↓↑↑↓−χ0\n↓↑↑↓V↑↓↓↑χ↓↑↑↓ (13)\nand, from (11) and (12), V↑↓↓↑=K↑↓↓↑=−U. Hence\nχ↓↑↑↓=χ0\n↓↑↑↓(1−Uχ0\n↓↑↑↓)−1(14)\nwhich is just the RPA result of Izuyama et al [31] for a single-orbital Hubbard model and of Lowde and Windsor [32]\nfor a five-orbital d-band model. Clearly in the absence of SOC the Co ulomb interaction v(q) plays no part in the\ntransverse susceptibility, as is well-known. A more interesting case is the longitudinal susceptibility denoted by χmm\nin the work of Kim et al ( [26], [27]) and in [28]. This involves only the respon se functions χσσττwhich we abbreviate\ntoχστ. In fact [28]\nχmm=χ↑↑+χ↓↓−χ↑↓−χ↓↑. (15)\nWithout SOC χ0\nστtakes the form χ0\nσδστand (10) becomes\nχστ=χ0\nσδστ−/summationdisplay\nσ2χ0\nσVσσ2χσ2τ (16)\nwithVσσ2=v(q)−Uδσσ2. On solving the 2 ×2 matrix equation (16) for χστ, and using (15), we find the longitudinal\nsusceptibility in the form\nχmm=χ0\n↑+χ0\n↓−2[U−2v(q)]χ0\n↑χ0\n↓\n1+(χ0\n↑+χ0\n↓)[v(q)−U]+U[U−2v(q)]χ0\n↑χ0\n↓(17)\nwhich agrees with the RPA result that Kim et al ( [26], [27])found for a s ingle-orbitalmodel. The Coulomb interaction\nv(q) is clearly important, particularly for the uniform susceptibility with q =0, where v→ ∞. It plays an essential\nrole in enforcing particle conservation and hence in obtaining the cor rect result of Stoner theory. In view of the\ncorrespondence between our approach and the RPA method it see ms likely that when SOC is included our procedure\nusing equations (10) - (12) should be almost equivalent to that of Co sta and Muniz [19] in the case of a model with\nd-bands only. However our inclusion of the long-range Coulomb inter action will modify the results.\nIV. AN EXPLICIT EXPRESSION FOR THE TRANSVERSE SUSCEPTIBILI TY\nIn this section we obtain an explicit expression for the transverses usceptibility χ↓↑↑↓in terms of the non-interacting\nresponse functions χ0. We consider equation (10) as an equation between 4 ×4 matrices where σσ′=↓↑,↑↓,↑↑,↓↓\nlabels the rows in that order and ττ′labels columns similarly. The formal solution of (10) is then\nχ= (1+χ0V)−1χ0. (18)\nThis expression could be used directly as the basis of a numerical inve stigation similar to that of Costa and Muniz.\nHowever we wish to show that the present approach leads to a Gilber t damping parameter αin agreement with the\nKambersky formula, to second order in the SOC parameter ξwhere Kambersky’s result is valid. This requires some\nquite considerable analytic development of (18).\nFirst we partition each matrix in (18) into four 2 ×2 matrices. Thus from (11) and (12)\nV=/parenleftbigg\nV10\n0V2/parenrightbigg\n(19)\nwith\nV1=/parenleftbigg\n0−U\n−U0/parenrightbigg\n, V2=/parenleftbigg\nv−U v\nv v−U/parenrightbigg\n. (20)\nAlso\nχ=/parenleftbigg\nχ11χ12\nχ21χ22/parenrightbigg\n(21)5\nand similarly for χ0. If we write\n1+χ0V=/parenleftbigg\n1+χ0\n11V1χ0\n12V2\nχ0\n21V11+χ0\n22V2/parenrightbigg\n=/parenleftbigg\nA B\nC D/parenrightbigg\n(22)\n(18) becomes\nχ=/parenleftbigg\nS−1−S−1BD−1\n−D−1CS−1D−1+D−1CS−1BD−1/parenrightbigg/parenleftbigg\nχ0\n11χ0\n12\nχ0\n21χ0\n22/parenrightbigg\n(23)\nwhere\nS=A+BD−1C. (24)\nThe transverse susceptibility χ↓↑↑↓in which we are interested is the top right-hand element of χ11so this is the\nquantity we wish to calculate. From (23)\nχ11=S−1(χ0\n11−BD−1χ0\n21) (25)\nand, from (24) and (22),\nS= 1+χ0\n11V1−χ0\n12(V−1\n2+χ0\n22)−1χ0\n21V1. (26)\nThe elements of the 2 ×2 matrix S are calculated by straight-forward algebra and\nS11= 1−Uχ0\n↓↑↑↓+(U/Λ)[(X+χ0\n↓↓↓↓)χ0\n↑↑↑↓χ0\n↓↑↑↑−(Y+χ0\n↑↑↓↓)χ0\n↓↓↑↓χ0\n↓↑↑↑\n−(Y+χ0\n↓↓↑↑)χ0\n↑↑↑↓χ0\n↓↑↓↓+(X+χ0\n↑↑↑↑)χ0\n↓↓↑↑χ0\n↓↑↓↓](27)\nwhere\nX= [v−U]/[U(U−2v)], Y=−v/[U(U−2v)] (28)\nand\nΛ = (X+χ0\n↑↑↑↑)(X+χ0\n↓↓↓↓)−(Y+χ0\n↑↑↓↓)(Y+χ0\n↓↓↑↑) (29)\nThe other three elements of Sare given in Appendix B. The transverse susceptibility is obtained fro m (25) as\nχ↓↑↑↓= [S22(χ0\n11−BD−1χ0\n21)12−S12(χ0\n11−BD−1χ0\n21)22]/(S11S22−S12S21) (30)\nand\nBD−1=χ0\n12(V−1\n2+χ0\n22)−1. (31)\nA comparison of the fairly complex equation above for the transver se susceptibility with the simple well-known result\n(14) showsthe extent ofthe new physicsintroducedby SOC.This is due tothe coupling ofthe transversesusceptibility\nto the longitudinal susceptibility and the charge response, both of which involve the long-range Coulomb interaction.\nTo proceed further it is necessary to specify the non-interacting response functions χ0\nσσ′σ1σ′\n1which occur throughout\nthe equations above.\nV. THE NON-INTERACTING RESPONSE FUNCTIONS\nIn the tight-binding approximation the one-electron basis function s are the Bloch functions\n|kµσ∝angb∇acket∇ight=N−1/2/summationdisplay\njeik·Rj|jµσ∝angb∇acket∇ight (32)\nwherejandµare the site and orbital indices, respectively, and Nis the number of atoms in the crystal. The\nHamiltonian in the Kohn-Sham equation now takes the form\nHeff=/summationdisplay\nkµνσ(Tµν(k)+Veff\nσδµν)c†\nkµσckνσ+Hso(33)6\nwhereTµνcorresponds to electron hopping and\nVeff\nσ=−(σ/2)(∆+bex) (34)\nwhereσ= 1,−1 for spin ↑,↓respectively. Here ∆ = 2 U∝angb∇acketleftSz∝angb∇acket∇ight/NwhereSzis the total spin angular momentum, in\nunits of /planckover2pi1, and the Zeeman splitting bex= 2µBBex, whereBexis the external magnetic field and µBis the Bohr\nmagneton. The spin-orbit term Hso=ξ/summationtext\njLj·Sjtakes the second-quantized form\nHso= (ξ/2)/summationdisplay\nkµν[Lz\nµν(c†\nkµ↑ckν↑−c†\nkµ↓ckν↓)+L+\nµνc†\nkµ↓ckν↑+L−\nµνc†\nkµ↑ckν↓] (35)\nwhereLz\nµν,L±\nµνare matrix elements of the atomic orbital angular momentum operat orsLz,L±=Lx±iLyin units\nof/planckover2pi1. Within the basis of states (32) eigenstates of Hefftake the form\n|kn∝angb∇acket∇ight=/summationdisplay\nµσaσ\nnµ(k)|kµσ∝angb∇acket∇ight, (36)\nand satisfy the equation\nHeff|kn∝angb∇acket∇ight=Ekn|kn∝angb∇acket∇ight. (37)\nThus\nc†\nkµσ=/summationdisplay\nnaσ\nnµ(k)∗c†\nkn(38)\nwherec†\nkncreates the eigenstate |kn∝angb∇acket∇ight.\nThenon-interactingresponsefunction χ0\nσσ′σ1σ′\n1(q,ω) isconvenientlyexpressedastheFouriertransformofaretarde d\nGreen function by the Kubo formula\nχ0\nσσ′σ1σ′\n1(q,ω) =/summationdisplay\nk∝angb∇acketleft∝angb∇acketleft/summationdisplay\nµc†\nk+qµσckµσ′;/summationdisplay\nνc†\nkνσ1ck+qνσ′\n1∝angb∇acket∇ight∝angb∇acket∇ight0\nω (39)\nwhere the right-hand side is to be evaluated using the one-electron Hamiltonian Heff. Consequently, using (38), we\nhave\nχ0\nσσ′σ1σ′\n1(q,ω) =/summationdisplay\nkµν/summationdisplay\nmnaσ\nmµ(k+q)∗aσ′\nnµ(k)aσ1nν(k)∗aσ′\n1mν(k+q)∝angb∇acketleft∝angb∇acketleftc†\nk+qmckn;c†\nknck+qm∝angb∇acket∇ight∝angb∇acket∇ight0\nω\n=N−1/summationdisplay\nkµν/summationdisplay\nmnaσ\nmµ(k+q)∗aσ′\nnµ(k)aσ1\nnν(k)∗aσ′\n1mν(k+q)fkn−fk+qm\nEk+qm−Ekn−/planckover2pi1ω+iη.(40)\nThe last step uses the well-known form of the response function pe r atom for a non-interacting Fermi system (e.g. [33])\nandηis a small positive constant which ultimately tends to zero. The occup ation number fkn=F(Ekn−µ0) where\nFis the Fermi function with chemical potential µ0. Clearly for q= 0 the concept of intraband transitions ( m=n),\nfrequently introduced in discussions of the Kambersky formula, ne ver arises for finite ωsince the difference of the\nFermi functions in the numerator of (40) is zero. Equation (40) ma y be written in the form\nχ0\nσσ′σ1σ′\n1(q,ω) =N−1/summationdisplay\nkmnBσσ′\nmn(k,q)Bσ1′σ1\nmn(k,q)∗fkn−fk+qm\nEk+qm−Ekn−/planckover2pi1ω+iη(41)\nwhere\nBσσ′\nmn(k,q) =/summationdisplay\nµaσ\nmµ(k+q)∗aσ′\nnµ(k). (42)7\nVI. FERROMAGNETIC RESONANCE LINEWIDTH; THE KAMBERSKY FORM ULA\nWe now consider the damping of the ferromagnetic resonance in the q= 0 transverse susceptibility. The present\napproach, like the closely-related one of Costa and Muniz [19], is valid f or arbitrary strength of the SOC and can\nbe used as the basis of numerical calculations, as performed by the latter authors. However it is important to show\nanalytically that the present method leads to the Kambersky [4] for mula for the Gilbert damping parameter where\nthis is valid, namely to second order in the SOC parameter ξ. This is the subject of this section.\nIt is useful to consider first the case without SOC ( ξ= 0). The eigenstates nofHeffthen have a definite spin and\nmay be labelled nσ. It follows from (40) that χ0\nσσ′σ1σ′\n1∝δσσ′\n1δσ′σ1. Henceχ0\n12= 0 and, from (31), BD−1= 0. Also,\nfrom Appendix B, S12=S21= 0. Thus, (30) reduces to (14) as it should. Considering χ0\n↓↑↑↓(0,ω), given by (40)\nand (41), we note that state mis pure↓spin, labelled by m↓, andnis pure↑, labelled by n↑. Hence for ξ= 0\nB↓↑\nmn(k,0) =/summationdisplay\nµ∝angb∇acketleftkm|kµ∝angb∇acket∇ight∝angb∇acketleftkµ|kn∝angb∇acket∇ight=δmn (43)\nfrom closure. Thus\nχ0\n↓↑↑↓(0,ω) =N−1/summationdisplay\nknfkn↑−fkn↓\nEkn↓−Ekn↑−/planckover2pi1ω+iη(44)\nand it follows from (34) that Eknσmay be written as\nEknσ=Ekn−(σ/2)(∆+bex). (45)\nHence we find from (14) that for ξ= 0\nχ↓↑↑↓(0,ω) = (2∝angb∇acketleftSz∝angb∇acket∇ight/N)(bex−/planckover2pi1ω+iη)−1. (46)\nThus, as η→0,ℑχ↓↑↑↓(0,ω) has a sharp delta-function resonance at /planckover2pi1ω=bexas expected.\nWhen SOC is included /planckover2pi1ωacquires an imaginary part that corresponds to damping. We now pr oceed to calculate\nthis imaginary part to O(ξ2). To do this we can take ξ= 0 in the numerator of (30) so that\nχ↓↑↑↓(0,ω) =χ0\n↓↑↑↓(0,ω)/(S11−S12S21/S22) (47)\nIn factS12andS21are both O(ξ2) whileS22isO(1). Thus to obtain /planckover2pi1ωtoO(ξ2) we need only solve S11= 0.\nFurthermore all response functions such as χ0\n↑↑↑↓, with all but one spins in the same direction, are zero for ξ= 0 and\nneed only be calculated to O(ξ) in (27). We show below that to this order they vanish, so that to O(ξ2) the last term\ninS11is zero and we only have to solve the equation 1 −Uχ0\n↓↑↑↓= 0 for/planckover2pi1ω. This means that to second order in ξ\nthe shift in resonance frequency and the damping do not depend on the long-range Coulomb interaction.\nTo determine χ0\n↑↑↑↓(0,ω) to first order in ξfrom (40) we notice that states nmust be pure ↑spin, that is |kn∝angb∇acket∇ight=\n|kn↑∝angb∇acket∇ight, while states mmust be calculated using perturbation theory. The latter states m ay be written\n|km1∝angb∇acket∇ight=|km↑∝angb∇acket∇ight−ξ/summationdisplay\npσ∝angb∇acketleftkpσ|hso|km↑∝angb∇acket∇ight\nEkm↑−Ekpσ|kpσ∝angb∇acket∇ight (48)\n|km2∝angb∇acket∇ight=|km↓∝angb∇acket∇ight−ξ/summationdisplay\npσ∝angb∇acketleftkpσ|hso|km↓∝angb∇acket∇ight\nEkm↓−Ekpσ|kpσ∝angb∇acket∇ight, (49)\nwhere we have put Hso=ξhso, and to first order in ξ,\nχ0\n↑↑↑↓=1\nN/summationdisplay\nkµν/summationdisplay\nmn(a↑\nm1µ∗anµa∗\nnνa↓\nm1νfkn↑−fkm↑\nEkm↑−Ekn↑−/planckover2pi1ω+iη+a↑\nm2µ∗anµa∗\nnνa↓\nm2νfkn↑−fkm↓\nEkm↓−Ekn↑−/planckover2pi1ω+iη) (50)\nwithaσ\nmsµ=∝angb∇acketleftkµσ|kms∝angb∇acket∇ight,s= 1,2,andanµ=∝angb∇acketleftkµ|kn∝angb∇acket∇ightis independent of spin. Since a↓\nm1ν∼ξwe takea↑\nm1µ=amµin\nthe first term of (50). Also/summationtext\nµa∗\nmµanµ=δmnby closure so that the first term of (50) vanishes since the differen ce\nof Fermi functions is zero. Only the second term of χ0\n↑↑↑↓remains and this becomes, by use of (49),\nχ0\n↑↑↑↓=−ξ/summationdisplay\nkµν/summationdisplay\nmnp∝angb∇acketleftkp↑ |hso|km↓∝angb∇acket∇ight∗\nEkm↓−Ekp↑a∗\npµanµa∗\nnνamνfkn↑−fkm↓\nEkm↓−Ekn↑−/planckover2pi1ω+iη. (51)8\nAgain using closure only terms with p=m=nsurvive and the matrix element of hsobecomes\n∝angb∇acketleftkn↑ |/summationdisplay\njLj·Sj|kn↓∝angb∇acket∇ight=1\n2∝angb∇acketleftkn|L−|kn∝angb∇acket∇ight= 0 (52)\ndue to the quenching of total orbital angular momentum L=/summationtext\njLj[30]. Thus, to first order in ξ,χ0\n↑↑↑↓(0,ω), and\nsimilar response functions with one reversed spin, are zero. Hence we have only to solve 1 −Uχ0\n↓↑↑↓= 0 to obtain\nℑ(/planckover2pi1ω) toO(ξ2). Here we assume the system has spatial inversion symmetry witho ut which the quenching of orbital\nangular momentum, as expressed by (52), no longer pertains [30]. We briefly discuss the consequences of a breakdown\nof inversion symmetry at the end of this section.\nOn introducing the perturbed states (48) and (49) we write (41) in the form\nχ0\n↓↑↑↓(0,ω) =1\nN/summationdisplay\nkmn(|B↓↑\nm1n1|2fkn1−fkm1\nEkm1−Ekn1−/planckover2pi1ω+iη+|B↓↑\nm1n2|2fkn2−fkm1\nEkm1−Ekn2−/planckover2pi1ω+iη\n+|B↓↑\nm2n1|2fkn1−fkm2\nEkm2−Ekn1−/planckover2pi1ω+iη+|B↓↑\nm2n2|2fkn2−fkm2\nEkm2−Ekn2−/planckover2pi1ω+iη).(53)\nClearlyB↓↑\nm1n1andB↓↑\nm2n2are of order ξ,B↓↑\nm1n2isO(ξ2) andB↓↑\nm2n1isO(1). We therefore neglect the term |B↓↑\nm1n2|2\nand, using (48) and (49), we find\nB↓↑\nm1n1=−B↓↑\nm2n2=ξ\n2∝angb∇acketleftkm|L−|kn∝angb∇acket∇ight\nEkn↓−Ekm↑. (54)\nThe evaluation of |B↓↑\nm2n1|2requires more care. It appears at first sight that to obtain this to O(ξ2) we need to include\nsecond order terms in the perturbed eigenstates given by (48) an d (49). However it turns out that these terms do not\nin fact contribute to |B↓↑\nm2n1|2toO(ξ2) so we shall not consider them further. Then we find\nB↓↑\nm2n1=δmn−ξ∝angb∇acketleftkm|Lz|kn∝angb∇acket∇ight\nEkn−Ekm−ξ2\n4/summationdisplay\np∝angb∇acketleftkm|Lz|kp∝angb∇acket∇ight∝angb∇acketleftkp|Lz|kn∝angb∇acket∇ight\n(Ekm−Ekp)(Ekn−Ekp)(55)\nand hence to O(ξ2)\n|B↓↑\nm2n1|2=δmn(1−ξ2\n2/summationdisplay\np|∝angb∇acketleftkm|Lz|kp∝angb∇acket∇ight|2\n(Ekm−Ekp)2)+ξ2|∝angb∇acketleftkm|Lz|kn∝angb∇acket∇ight|2\n(Ekn−Ekm)2(56)\nThe contribution of this quantity to χ0\n↓↑↑↓(0,ω) in (53) may be written to O(ξ2) as\n1\nN/summationdisplay\nkmn|B↓↑\nm2n1|2fkn1−fkm2\nEkm2−Ekn1−bex+iη(1−bex−/planckover2pi1ω\nEkm2−Ekn1−bex+iη). (57)\nThisisobtainedbyintroducingtheidentity −/planckover2pi1ω=−bex+(bex−/planckover2pi1ω)intherelevantdenominatorin(53),andexpanding\nto first order in bex−/planckover2pi1ωwhich turns out to be O(ξ2). The remaining factors of this second term in (57) may then\nbe evaluated with ξ= 0, as at the beginning of this section, so that this term becomes ( /planckover2pi1ω−bex)/(2U2∝angb∇acketleftSz∝angb∇acket∇ight). By\ncombining equations (53), (54) and (57), and ignoring some real te rms, we find that the equation 1 −Uχ0\n↓↑↑↓(0,ω) = 0\nleads to the relation\nℑ(/planckover2pi1ω) =πξ2/(2∝angb∇acketleftSz∝angb∇acket∇ight)/summationdisplay\nknm[(fkn↑−fkm↓)|∝angb∇acketleftkm|Lz|kn∝angb∇acket∇ight|2δ(Ekm↓−Ekn↑−bex)\n+(1/4)(fkn↑+fkn↓−fkm↑−fkm↓)|∝angb∇acketleftkm|L−|kn∝angb∇acket∇ight|2δ(Ekm−Ekn−bex)](58)\nThe Gilbert damping parameter αis given by ℑ(/planckover2pi1ω)/bex(e.g. [39]) and in (58) we note that\n(fkn↑−fkm↓)δ(Ekm↓−Ekn↑−bex) = [F(Ekn↑−µ0)−F(Ekn↑+bex−µ0)]δ(Ekm↓−Ekn↑−bex)\n=bexδ(Ekn↑−µ0)δ(Ekm↑−µ0)(59)\nto first order in bexat temperature T= 0. Similarly\n(fknσ−fkmσ)δ(Ekm−Ekn−bex) =bexδ(Eknσ−µ0)δ(Ekmσ−µ0). (60)9\nThus from (58)\nα=πξ2/(2∝angb∇acketleftSz∝angb∇acket∇ight)/summationdisplay\nknm|∝angb∇acketleftkm|Lz|kn∝angb∇acket∇ight|2δ(Ekn↑−µ0)δ(Ekm↓−µ0)\n+πξ2/(8∝angb∇acketleftSz∝angb∇acket∇ight)/summationdisplay\nknmσ|∝angb∇acketleftkm|L−|kn∝angb∇acket∇ight|2δ(Eknσ−µ0)δ(Ekmσ−µ0)(61)\ncorrect to O(ξ2). We note that there is no contribution from intraband terms since ∝angb∇acketleftkn|L|kn∝angb∇acket∇ight= 0. It is straight-\nforward to show that to O(ξ2) this is equivalent to the expression\nα=π/(2∝angb∇acketleftSz∝angb∇acket∇ight)/summationdisplay\nknm/summationdisplay\nσσ′|Amσ,nσ′(k)|2δ(Ekmσ−µ0)δ(Eknσ′−µ0) (62)\nwhere\nAmσ,nσ′(k) =ξ∝angb∇acketleftkmσ|[S−,hso]|knσ′∝angb∇acket∇ight (63)\nandS−is the total spin operator/summationtext\njS−\njwithS−\nj=Sx\nj−iSy\nj. This may be written more concisely as\nα=π/(2∝angb∇acketleftSz∝angb∇acket∇ight)/summationdisplay\nknm|Amn(k)|2δ(Ekm−µ0)δ(Ekn−µ0) (64)\nwith\nAmn(k) =ξ∝angb∇acketleftkm|[S−,hso]|kn∝angb∇acket∇ight (65)\nand the understanding that the one-electron states km,knare calculated in the absence of SOC. Equation (64) is the\nstandard form of the Kambersky formula ( [4], [9]) but in the literatur e SOC is invariably included in the calculation\nof the one-electronstates. This means that the intraband terms withm=nno longer vanish. They involve the square\nof a delta-function and this problem is always addressed by invoking t he effect of impurity and/or phonon scattering\nto replace the delta-functions by Lorentzians of width proportion al to an inverse relaxation time parameter τ−1. Then\nas one approaches a perfect crystal ( τ→ ∞) the intraband contribution to αtends to infinity. This behaviour is\nillustrated in many papers ( [7], [10], [13], [14]). In fig. 1 of [14] it is shown c learly that αremains finite if one does\nnot include SOC in calculating the one-electron states. However the effect of not including SOC is not confined to\ntotal removal of the intraband contribution. The remaining interb and contribution is increased considerably in the\nlow scattering rate regime, by almost an order of magnitude in the ca se of Fe. This makes αalmost independent of\nscattering rate in Fe which may relate to its observed temperature independence [15]. The corresponding effect in\nCo is insufficient to produce the increase of αat low scattering rate inferred from its temperature dependence . The\nnon-inclusion of SOC in calculating the one-electron states used in th e Kambersky formula clearly makes a major\nqualitative and quantitative change in the results. This occurs as so on as intraband terms become dominant in\ncalculations where they are included. For Fe, Co and Ni this corresp onds to impurity content and temperature such\nthat the scattering rate 1 /τdue to defects and/or phonons is less that about 1014sec−1( [7, 14]). Typically these\nmetals at room temperature find themselves well into the high scatt ering-rate regime where the damping rate can\nbe reliably estimated from the Kambersky interband term, with or wit hout SOC included in the band structure [35].\nThe physics at room temperature is not particularly interesting. On e needs to lower the temperature into the low\nscattering-rate regime where intraband terms, if they exist, will d ominate and lead to an anomalous ξ3dependence of\nthe damping on spin-orbit parameter ξ( [4], [8], [14]). The origin of this behaviour is explained in [4], [14]. It arises\nin theksum of (64) from a striplike region on the Fermi surface around a line where two different energy bands cross\neach other in the absence of SOC. The strip width is proportional to ξ, or more precisely |ξ|. SinceAnn(k) is of\norderξthe contribution of intraband terms in (64) is proportional to |ξ|3. Thus the intraband terms lead to terms\ninαwhich diverge in the limit τ−1→0 and are non-analytic functions of ξ. The calculation of αin this section can\nbe extended to higher powers of ξthan the second. No intraband terms appear and the result is an an alytic power\nseries containing only even powers of ξ.\nThe interband term in Kambersky’s formula can be given a very simple in terpretation in terms of Fermi’s ”golden\nrule” for transition probability [5]. This corresponds to second orde r perturbation theory in the spin-orbit interaction.\nThe decay of a uniform mode ( q= 0) magnon into an electron-hole pair involves the transition of an ele ctron from\nan occupied state to an unoccupied state of the same wave-vecto r. This is necessarily an interband transition and\nthe states involved in the matrix element are unperturbed, that is c alculated in the absence of SOC. A quite different\napproach has been adopted to try and find a physical interpretat ion of Kambersky’s intraband term ( [5, 8]). This\nemploys Kambersky’s earlier ”breathing Fermi surface” model ( [3, 34]) whose range of validity is uncertain.10\nWe now briefly discuss the consequences of a breakdown of spatial inversion symmetry so that total orbital angular\nmomentum is not quenched. In general response functions such a sχ0\n↑↑↑↓(0,ω) with one reversed spin are no longer\nzero to first order in ξ. HenceS11is not given to order ξ2just by the first two terms of (27) but involves further terms\nwhich depend explicitly on the long-range Coulomb interaction. Conse quentlyαhas a similar dependence which\ndoes not emerge from the torque-correlation approach. In Appe ndix A it is pointed out how the direct proof of the\nKambersky formula breaks down in the absence of spatial inversion symmetry.\nVII. EXPERIMENTAL ASPECTS\nThe inclusion of intraband terms in the Kambersky formula, despite t heir singular nature, has gained acceptance\nbecause they appear to explain a rise in intrinsic damping parameter αat low temperature which is observed in some\nsystems [15]. The calculated intraband contribution to αis proportional to the relaxation time τand it is expected\nthat, due to electron-phonon scattering, τwill increase as the temperature is reduced. This is in qualitative agre ement\nwith data [15] for Ni and hcp Co. Also a small 10% increase in αis observed in Co 2FeAl films as the temperature is\ndecreased from 300 K to 80 K [36]. However in Fe the damping αis found to be independent of temperature down to\n4 K [15]. Very recent measurements [16] on FePt films, with varying an tisite disorder xintroduced into the otherwise\nwell-ordered structure, show that αincreases steadily as xincreases from 3 to 16%. Hence αincreases monotonically\nwith scattering rate 1 /τas expected from the Kambersky formula in the absence of intraba nd terms. Furthermore\nforx= 3% it is found that αremains almost unchanged when the temperature is decreased fro m 200 to 20 K. Ma et\nal [16] therefore conclude that there is no indication of an intraban d term in α. From the present point of view the\norigin of the observed low temperature increase of αin Co and Ni is unclear. Further experimental work to confirm\nthe results of Bhagat and Lubitz [15] is desirable.\nThe second unusual feature of the intraband term in Kambersky’s formula for αis its|ξ|3dependence on the SOC\nparameter ξ. This contrasts with the ξ2dependence of the interband contribution which has been observe d in a\nnumber of alloys at room temperature [17]. Recently this behaviour has been seen very precisely in FePd 1−xPtxalloys\nwhereξcan be varied over a wide range by varying x[18]. Unfortunately this work has not been extended to the\nlow temperature regime where the |ξ|3dependence, if it exists, should be seen. It would be particularly inte resting to\nsee low temperature data for NiPd 1−xPtxand CoPd 1−xPtxsince it is in Ni and Co where the intraband contribution\nhas been invoked to explain the low temperature behaviour of α. From the present point of view, with the intraband\nterm absent, one would expect ξ2behaviour over the whole temperature range.\nVIII. CONCLUSIONS AND OUTLOOK\nIn this paper we analyse two methods which are used in the literature to calculate the damping in magnetization\ndynamics due to spin-orbit coupling. The first common approach is to employ Kambersky’s[4] formula for the Gilbert\ndamping parameter αwhich delivers an infinite value for a pure metal if used beyond second order in the spin-orbit\nparameter ξ. The second approach [19] is to calculate numerically the line-width of the ferromagnetic resonance seen\nin the uniform transverse spin susceptibility. This is always found to b e finite, corresponding to finite α. We resolve\nthis apparent inconsistency between the two methods by an analyt ic treatment of the Costa-Muniz approach for the\nsimplified model of a ferromagnetic metal with d-bands only. It is sho wn that this method leads to the Kambersky\nresult correct to second order in ξbut Kambersky’s intraband scattering term, taking the non-analy tic form |ξ|3, is\nabsent. Higher order terms in the present work are analytic even p owers of ξ. The absence of Kambersky’s intraband\nterm is the main result of this paper and it is in agreement with the conc lusion that Ma et al [16] draw from their\nexperiments on FePt films. Further experimental work on the depe ndence of damping on electron scattering-rate and\nspin-orbit parameter in other systems is highly desirable.\nA secondaryconclusionis that beyond second orderin ξsome additional physics ariseswhich has not been remarked\non previously. This is the role of long-range Coulomb interaction which is essential for a proper treatment of the\nlongitudinal susceptibility and charge response to which the transv erse susceptibility is coupled by spin-orbit interac-\ntion. Costa and Muniz [19] stress this coupling but fail to introduce t he long-range Coulomb interaction. Generally,\nhowever, it seems unnecessary to go beyond second order in ξ[17, 18] and for most bulk systems Kambersky’s for-\nmula, with electron states calculated in the absence of SOC, should b e adequate. However in systems without spatial\ninversion symmetry, which include layered structures of practical importance, the Kambersky formulation may be\ninadequate even to second order in ξ. The long-range Coulomb interaction can now play a role.\nAn important property of ferromagnetic systems without inversio n symmetry is the Dzyaloshinskii-Moriya inter-\naction (DMI) which leads to an instability of the uniform ferromagnet ic state with the appearance of a spiral spin\nstructure or a skyrmion structure. This has been studied extens ively in bulk crystals like MnSi [37] and in layered11\nstructures [38]. The spiral instability appears as a singularity in the t ransverse susceptibility χ(q,0) at a value of q\nrelated to the DMI parameter. The method of this paper has been u sed to obtain a novel closed form expression for\nthis parameter which will be reported elsewhere.\nIn this paper we have analysed in some detail the transverse spin su sceptibility χ↓↑↑↓but combinations of some of\nthe 15 other response functions merit further study. Mixed char ge-spin response arising from spin-orbit coupling is\nof particular interest for its relation to phenomena like the spin-Hall effect.\nAppendix A: A direct derivation of the Kambersky formula\nIn this appendix we give a rather general derivation of the Kambers ky formula for the Gilbert damping parameter\nαwith an emphasis on its restriction to second order in the spin-orbit in teraction parameter ξ.\nWe consider a general ferromagnetic material described by the ma ny-body Hamiltonian\nH=H1+Hint+Hext (A1)\nwhereH1is a one-electron Hamiltonian of the form\nH1=Hk+Hso+V. (A2)\nHereHkis the total kinetic energy, Hso=ξhsois the spin-orbit interaction, Vis a potential term, Hintis the\nCoulomb interaction between electrons and Hextis due to an external magnetic field Bexin thezdirection. Thus\nHext=−Szbexwherebex= 2µBBex, as in (34), and Szis thezcomponent of total spin. Both HsoandVcan contain\ndisorderalthough in this paper we consider a perfect crystal. Follow ingthe general method of Edwardsand Fisher [40]\nwe use equations of motion to find that the dynamical transverse s usceptibility χ(ω) =χ−+(0,ω) satisfies [39]\nχ(ω) =−2∝angb∇acketleftSz∝angb∇acket∇ight\n/planckover2pi1ω−bex+ξ2\n(/planckover2pi1ω−bex)2(χF(ω)−ξ−1∝angb∇acketleft[F−,S+]∝angb∇acket∇ight) (A3)\nwhere\nχF(ω) =/integraldisplay\n∝angb∇acketleft∝angb∇acketleftF−(t),F+∝angb∇acket∇ight∝angb∇acket∇ighte−iωtdt (A4)\nwithF−= [S−,hso]. This follows since S−commutes with other terms in H1and with Hint. For small ω,χis\ndominated by the spin wave pole at /planckover2pi1ω=bext+/planckover2pi1δωwhereδω∼ξ2, so that\nχ(ω) =−2∝angb∇acketleftSz∝angb∇acket∇ight\n/planckover2pi1(ω−δω)−bex. (A5)\nFollowing [39] we compare (A3) and (A5) in the limit /planckover2pi1δω≪/planckover2pi1ω−bexto obtain\n−2∝angb∇acketleftSz∝angb∇acket∇ight/planckover2pi1δω=ξ2(χF(ω)−ξ−1∝angb∇acketleft[F−,S+]∝angb∇acket∇ight) =ξ2[ lim\n/planckover2pi1ω→bexχξ=0\nF(ω)−lim\nξ→0(1\nξ∝angb∇acketleft[F−,S+]∝angb∇acket∇ight)] (A6)\ncorrect to order ξ2. It is important to note that the limit ξ→0 within the bracket must be taken before putting\n/planckover2pi1ω=bex. If we put /planckover2pi1ω=bexfirst it is clear from (A3) that the quantity in brackets would vanish, giving the incorrect\nresultδω= 0. Furthermore it may be shown [M. Cinal, private communication] th at the second term in the bracket\nis real. Hence\nℑ(/planckover2pi1ω) =−ξ2\n2∝angb∇acketleftSz∝angb∇acket∇ightlim\n/planckover2pi1ω→bexℑ[χξ=0\nF(ω)]. (A7)\nKambersky [4] derived this result, using the approach of Mori and K awasaki ( [41] [42]), without noting its restricted\nvalidity to second order in ξ. This restriction is crucial since, as discussed in the main paper, it av oids the appearance\nof singular intraband terms. Oshikawa and Affleck emphasise strong ly a similar restriction in their related work on\nelectron spin resonance (Appendix of [43]).\nEquation (A7) is an exact result even in the presence of disorder in t he potential and spin-orbit terms of the\nHamiltonian. In the following we assume translational symmetry.12\nTo obtain the expression (61) for α=ℑ(/planckover2pi1ω)/bex, which is equivalent to Kambersky’s result (62), it is necessary to\nevaluate the response function χξ=0\nF(ω) in tight-binding-RPA. Using (35)we find\nF−=/summationdisplay\nkµν[Lz\nµνc†\nkµ↓ckν↑+(1/2)L−\nµν(c†\nkµ↓ckν↓−c†\nkµ↑ckν↑)]. (A8)\nHence\nχξ=0\nF=/summationdisplay\nµν/summationdisplay\nαβ[Lz\nµνLz\nβαGµ↓ν↑,β↑α↓+(1/4)L−\nµνL+\nβα(Gµ↓ν↓,β↓α↓+Gµ↑ν↑,β↑α↑−Gµ↓ν↓,β↑α↑−Gµ↑ν↑,β↓α↓)] (A9)\nwhere\nGµσνσ′,βτατ′=∝angb∇acketleft∝angb∇acketleft/summationdisplay\nkc†\nkµσckνσ′;/summationdisplay\nuc†\nuβτcuατ′∝angb∇acket∇ight∝angb∇acket∇ightω. (A10)\nThe Green function Gis to be calculated in the absence of SOC ( ξ= 0). Within RPA it satisfies an equation of the\nform\nGµσνσ′,βτατ′=G0\nµσνσ′,βτατ′−/summationdisplay\nµ1σ1ν1σ′\n1/summationdisplay\nµ2σ2ν2σ′\n2G0\nµσνσ′,µ1σ1ν1σ′\n1Vµ1σ1ν1σ′\n1,µ2σ2ν2σ′\n2Gµ2σ2ν2σ′\n2,βτατ′ (A11)\nwhereG0is the non-interacting (Hartree-Fock) Green function and\nVµ1σ1ν1σ′\n1,µ2σ2ν2σ′\n2=Vσ1σ′\n1σ2σ′\n2(q)δµ1ν1δµ2ν2 (A12)\nwithV(q) given by (11) and (12). Hence\nGµσνσ′,βτατ′=G0\nµσνσ′,βτατ′−/summationdisplay\nµ1σ1σ′\n1/summationdisplay\nµ2σ2σ′\n2G0\nµσνσ′,µ1σ1µ1σ′\n1Vσ1σ′\n1σ2σ′\n2Gµ2σ2µ2σ′\n2,βτατ′. (A13)\nThe form of the interaction Vgiven in (A12) is justified by the connection between (A13) and (10) , withq= 0. To\nsee this connection we note that χσσ′ττ′=/summationtext\nµνGµσµσ′,ντντ′and that (A13) then leads to (10) which is equivalent to\nRPA. On substituting (A13) into (A9) we see that the contributions from the second term of (A13) contain factors\nof the form\n/summationdisplay\nµνµ1Lz\nµνG0\nµ↓ν↑,µ1↑µ1↓,/summationdisplay\nµνµ1L−\nµνG0\nµσνσ,µ 1σ1µ1σ1. (A14)\nWe now show that such factors vanish owing to quenching of orbital angular momentum in the system without SOC\n(ξ= 0). Hence the Green functions Gin (A9) can be replaced by the non-interacting ones G0. The non-interacting\nGreen functions G0are of a similar form to χ0in (40) and for ξ= 0 may be expressed in terms of the quantities\nanµ=∝angb∇acketleftkµ|kn∝angb∇acket∇ightwhere|kn∝angb∇acket∇ightis a one-electron eigenstate as introduced in section VI. Hence we fi nd, in the same way\nthat (44) emerged,\n/summationdisplay\nµνµ1Lz\nµνG0\nµ↓ν↑,µ1↑µ1↓=/summationdisplay\nµν/summationdisplay\nknLz\nµνa∗\nnµanνfkn↑−fkn↓\n∆+bex−/planckover2pi1ω+iη. (A15)\nAlso by closure\n/summationdisplay\nµνLz\nµνa∗\nnµanν=∝angb∇acketleftkn|Lz|kn∝angb∇acket∇ight= 0, (A16)\nthe last step following from quenching of total orbital angular mome ntum. The proof that the second expression\nin (A14) vanishes is very similar.\nHence we can insert the non-interacting Green functions G0in (A9) and straight-forwardalgebra, with use of (A7),\nleads to (58). At the end of section VI this is shown to be equivalent t o the Kambersky formula for α. We emphasize\nagain that the present proof is valid only to order ξ2so that the one-electron states used to evaluate the formula\nshould be calculated in the absence of SOC.\nThis proof relies on the quenching of orbital angular momentum which does not occur in the absence of spatial\ninversion symmetry. When this symmetry is broken it is not difficult to s ee that the second term of (A13) gives a\ncontribution to the first term on the right of (A9) which contains th eq= 0 spin-wave pole and diverges as /planckover2pi1ω→bex.\nHence the proof of the torque-correlationformula (A7) collapses . The method of section VI must be used as discussed\nat the end of that section.13\nAppendix B: Elements of S\nThe element S11of the matrix Sis given in (27). The remaining elements are given below.\nS12=−Uχ0\n↓↑↓↑+(U/Λ)[(X+χ0\n↓↓↓↑)χ0\n↑↑↓↑χ0\n↓↑↑↑−(Y+χ0\n↑↑↓↓)χ0\n↓↓↓↑χ0\n↓↑↑↑\n−(Y+χ0\n↓↓↑↑)χ0\n↑↑↓↑χ0\n↓↑↓↓+(X+χ0\n↑↑↑↑)χ0\n↓↓↓↑χ0\n↓↑↓↓](B1)\nS21=−Uχ0\n↑↓↑↓+(U/Λ)[(X+χ0\n↓↓↓↓)χ0\n↑↑↑↓χ0\n↑↓↑↑−(Y+χ0\n↑↑↓↓)χ0\n↓↓↑↓χ0\n↑↓↑↑\n−(Y+χ0\n↓↓↑↑)χ0\n↑↑↑↓χ0\n↑↓↓↓+(X+χ0\n↑↑↑↑)χ0\n↓↓↑↓χ0\n↑↓↓↓](B2)\nS22= 1−Uχ0\n↑↓↓↑+(U/Λ)[(X+χ0\n↓↓↓↓)χ0\n↑↑↓↑χ0\n↑↓↑↑−(Y+χ0\n↑↑↓↓)χ0\n↓↓↓↑χ0\n↑↓↑↑\n−(Y+χ0\n↓↓↑↑)χ0\n↑↑↓↑χ0\n↑↓↓↓+(X+χ0\n↑↑↑↑)χ0\n↓↓↓↑χ0\n↑↓↓↓](B3)\nAcknowledgement\nMy recent interest in Gilbert damping arose through collaboration wit h O. Wessely, E. Barati, M. Cinal and A.\nUmerski. I am grateful to them for stimulating discussion and corre spondence. The specific work reported here arose\ndirectly from discussion with R.B.Muniz and I am particularly grateful t o him and his colleague A. T. Costa for this\nstimulation.\nReferences\n[1] Landau L D, Lifshitz E M and Pitaevski L P 1980 Statistical Physics part2 (Oxford: Pergamon)\n[2] Gilbert T L 1955 Phys. Rev. 1001243\n[3] Kambersky V 1970 Can. J. Phys. 482906\n[4] Kambersky V 1976 Czech. J. Phys. B261366\n[5] Kambersky V 2007 Phys. Rev. B76134416\n[6] Korenman V and Prange R E 1972 Phys. Rev. B62769\n[7] Gilmore K, Idzerda Y U and Stiles M D 2007 Phys. Rev. Lett. 99027204\n[8] Gilmore K, Idzerda Y U and Stiles M D 2008 J. Appl. Phys. 10307D3003\n[9] Garate I and MacDonald A 2009 Phys. Rev. B79064403\n[10] Garate I and MacDonald A 2009 Phys. Rev. B79064404\n[11] Liu C, Mewes CKA, Chshiev M, Mewes T and Butler WH 2009 Appl. Phys. Lett. 95022509\n[12] Umetsu N, Miura D and Sakuma A 2011 J. Phys. Conf. Series 266012084\n[13] Sakuma A 2012 J. Phys. Soc. Japan 81084701\n[14] Barati E, Cinal M, Edwards D M and Umerski A 2014 Phys.Rev. B90014420\n[15] Bhagat S M and Lubitz P 1974 Phys. Rev. B10179\n[16] Ma X, Ma L, He P, Zhao H B, Zhou S M and L¨ upke G 2015 Phys. Rev. B91014438\n[17] Scheck C, Cheng L, Barsukov I, Frait Z and Bailey WE 2007 Phys. Rev. Lett. 98117601\n[18] He P, Ma X, Zhang JW, Zhao HB, L¨ upke G, Shi Z and Zhou SM 201 3Phys. Rev. Lett. 110077203\n[19] Costa A T and Muniz R B 2015 Phys. Rev. B92014419\n[20] Brataas A, Tserkovnyak Y and Bauer GEW 2008 Phys. Rev. Lett. 101037207\n[21] Starikov A, Kelly PJ, Brataas A, Tserkovnyak Y and Bauer GEW 2010 Phys. Rev. Lett. 105236601\n[22] Costa A T, Muniz R B, Lounis S, Klautau A B and Mills D L 2010 Phys. Rev. B82014428\n[23] Rajagopal A K, Brooks H and Ranganathan N R 1967 Nuovo Cimento Suppl. 5807\n[24] Rajagopal A K 1978 Phys. Rev. B172980\n[25] Low G G 1969 Adv. Phys. 18371\n[26] Kim D J, Praddaude H C and Schwartz B B 1969 Phys. Rev. Lett. 23419\n[27] Kim D J, Schwartz B B and Praddaude H C 1973 Phys. Rev. B7205\n[28] Williams A R and von Barth U 1983, in Theory of the Inhomogeneous Electron Gas eds. Lundqvist S and March N H\n(Plenum)14\n[29] Edwards D M 1984, in Moment Formation in Solids NATO Advanced Study Institute, Series B: Physics Vol. 117, e d.\nBuyers W J L (Plenum)\n[30] Cinal M and Edwards D M 1997 Phys. Rev B553636\n[31] Izuyama T, Kim D J and Kubo R 1963 J. Phys. Soc. Japan 181025\n[32] Lowde R D and Windsor C G 1970 Adv. Phys. 19813\n[33] Doniach S and Sondheimer E H 1998 Green’s Functions for Solid State Physicists Imperial College Press\n[34] Kunes J and Kambersky V 2002 Phys. Rev B65212411\n[35] Gilmore K, Garate I, MacDonald AH and Stiles MD Phys. Rev B84224412\n[36] Yuan HC, Nie SH, Ma TP, Zhang Z, Zheng Z, Chen ZH, Wu YZ, Zha o JH, Zhao HB and Chen LY 2014 Appl. Phys. Lett.\n105072413\n[37] Grigoriev SV, Maleyev SV, Okorokov AI, Chetverikov Yu. O, B¨ oni P, Georgii R, Lamago D, Eckerlebe H and Pranzas K\n2006Phys. Rev. B74214414\n[38] von Bergmann K, Kubetzka A, Pietzsch O and Wiesendanger R 2014J. Phys. Condens. Matter 26394002\n[39] Edwards D M and Wessely O 2009 J. Phys. Condens. Matter 21146002\n[40] Edwards D M and Fisher B 1971 J. Physique 32C1 697\n[41] Mori H 1965 Prog. Theor. Phys. 33423\n[42] Mori H and Kawasaki K 1962 Prog. Theor. Phys. 27529\n[43] Oshikawa M and Affleck I 2002 Phys. Rev. B65134410"    },    {        "title": "1507.03075v1.Realization_of_the_thermal_equilibrium_in_inhomogeneous_magnetic_systems_by_the_Landau_Lifshitz_Gilbert_equation_with_stochastic_noise__and_its_dynamical_aspects.pdf",        "content": "Realization of the thermal equilibrium in inhomogeneous\nmagnetic systems by the Landau-Lifshitz-Gilbert equation with\nstochastic noise, and its dynamical aspects\nMasamichi Nishino1\u0003and Seiji Miyashita2;3\n1Computational Materials Science Center,\nNational Institute for Materials Science, Tsukuba, Ibaraki 305-0047, Japan\n2Department of Physics, Graduate School of Science,\nThe University of Tokyo, Bunkyo-Ku, Tokyo, Japan\n3CREST, JST, 4-1-8 Honcho Kawaguchi, Saitama, 332-0012, Japan\n(Dated: July 14, 2015)\n1arXiv:1507.03075v1  [cond-mat.mtrl-sci]  11 Jul 2015Abstract\nIt is crucially important to investigate e\u000bects of temperature on magnetic properties such as\ncritical phenomena, nucleation, pinning, domain wall motion, coercivity, etc. The Landau-Lifshitz-\nGilbert (LLG) equation has been applied extensively to study dynamics of magnetic properties.\nApproaches of Langevin noises have been developed to introduce the temperature e\u000bect into the\nLLG equation. To have the thermal equilibrium state (canonical distribution) as the steady state,\nthe system parameters must satisfy some condition known as the \ructuation-dissipation relation.\nIn inhomogeneous magnetic systems in which spin magnitudes are di\u000berent at sites, the condition\nrequires that the ratio between the amplitude of the random noise and the damping parameter\ndepends on the magnitude of the magnetic moment at each site. Focused on inhomogeneous mag-\nnetic systems, we systematically showed agreement between the stationary state of the stochastic\nLLG equation and the corresponding equilibrium state obtained by Monte Carlo simulations in\nvarious magnetic systems including dipole-dipole interactions. We demonstrated how violations of\nthe condition result in deviations from the true equilibrium state. We also studied the characteris-\ntic features of the dynamics depending on the choice of the parameter set. All the parameter sets\nsatisfying the condition realize the same stationary state (equilibrium state). In contrast, di\u000berent\nchoices of parameter set cause seriously di\u000berent relaxation processes. We show two relaxation\ntypes, i.e., magnetization reversals with uniform rotation and with nucleation.\nPACS numbers: 75.78.-n 05.10.Gg 75.10.Hk 75.60.Ej\n2|||||||||||||||||||||||||-\nI. INTRODUCTION\nThe Landau-Lifshitz-Gilbert (LLG) equation1has been widely used in the study of dy-\nnamical properties of magnetic systems, especially in micromagnetics. It contains a relax-\nation mechanism by a phenomenological longitudinal damping term. The Landau-Lifshitz-\nBloch (LLB) equation2contains, besides the longitudinal damping, a phenomenological\ntransverse damping and the temperature dependence of the magnetic moment are taken\ninto account with the aid of the mean-\feld approximation. Those equations work well in\nthe region of saturated magnetization at low temperatures.\nThermal e\u000bects are very important to study properties of magnets, e.g., the amount of\nspontaneous magnetization, hysteresis nature, relaxation dynamics, and the coercive force in\npermanent magnets. Therefore, how to control temperature in the LLG and LLB equations\nhas been studied extensively. To introduce temperature in equations of motion, a coupling\nwith a thermal reservoir is required. For dynamics of particle systems which is naturally\nexpressed by the canonical conjugated variables, i.e., ( q;p), molecular dynamics is performed\nwith a Nose-Hoover (NH) type reservoir3{5or a Langevin type reservoir6. However, in the\ncase of systems of magnetic moments, in which dynamics of angular momenta is studied, NH\ntype reservoirs are hardly used due to complexity7. On the other hand, the Langevin type\nreservoirs have been rather naturally applied2,8{18although multiplicative noise19requires the\nnumerical integration of equations depending on the interpretation, i.e., Ito or Stratonovich\ntype.\nTo introduce temperature into a LLG approach by a Langevin noise, a \ructuation-\ndissipation relation is used, where the temperature is proportional to the ratio between\nthe strength of the \ructuation (amplitude of noise) and the damping parameter of the\nLLG equation. For magnetic systems consisting of uniform magnetic moments, the ratio is\nuniquely given at a temperature and it has been often employed to study dynamical prop-\nerties, e.g., trajectories of magnetic moments of nano-particles8, relaxation dynamics in a\nspin-glass system20or in a semiconductor21. The realization of the equilibrium state by\nstochastic LLG approaches by numerical simulations is an important issue, and it has been\ncon\frmed in some cases of the Heisenberg model for uniform magnetic moments.22,23\n3In general cases, however, magnetic moments in atomic scale have various magnitudes of\nspins. This inhomogeneity of magnetization is important to understand the mechanisms of\nnucleation or pinning.24{28To control the temperature of such systems, the ratio between the\namplitude of noise and the damping parameter depends on the magnetic moment at each\nsite. In order to make clear the condition for the realization of the canonical distribution\nas the stationary state in inhomogeneous magnetic systems, we review the guideline of the\nderivation of the condition in the Fokker-Planck equation formalism in the Appendix A.\nSuch a generalization of the LLG equation with a stochastic noise was performed to study\nproperties of the alloy magnet GdFeCo29, in which two kinds of moments exist. They ex-\nploited a formula for the noise amplitude, which is equivalent to the formula of our condition\nA (see Sec II). They found surprisingly good agreements of the results between the stochas-\ntic LLG equation and a mean-\feld approximation. However, the properties in the true\ncanonical distribution is generally di\u000berent from those obtained by the mean-\feld analysis.\nThe LLG and LLB equations have been often applied for continuous magnetic systems or\nassemblies of block spins in the aim of simulation of bulk systems, but such treatment of the\nbulk magnets tend to overestimate the Curie temperature11, and it is still under develop-\nment to obtain properly magnetization curves in the whole temperature region2,11,17,18. The\nin\ruence of coarse graining of block spin systems on the thermal properties is a signi\fcant\ntheme, which should be clari\fed in the future. To avoid such a di\u000eculty, we adopt a lattice\nmodel, in which the magnitude of the moment is given at each magnetic site.\nWithin the condition there is some freedom of the choice of parameter set. In the present\npaper, in particular, we investigate the following two cases of parameter sets, i.e., case A,\nin which the LLG damping constant is the same in all the sites and the amplitude of the\nnoise depends on the magnitude of the magnetic moment at each site, and case B, in which\nthe amplitude of the noise is the same in all the sites and the damping constant depends on\nthe magnitude of the moment. (see Sec II.). We con\frm the realization of the equilibrium\nstate, i.e., the canonical distribution in various magnetic systems including critical region by\ncomparison of magnetizations obtained by the LLG stochastic approach with those obtained\nby standard Monte Carlo simulations, not by the mean-\feld analysis. We study systems\nwith not only short range interactions but also dipole-dipole interactions, which causes\nthe demagnetizing \feld statically. We \fnd that di\u000berent choices of the parameter set which\nsatis\fes the \ructuation-dissipation relation give the same stationary state (equilibrium state)\n4even near the critical temperature. We also demonstrate that deviations from the relation\ncause systematic and signi\fcant deviations of the results.\nIn contrast to the static properties, we \fnd that di\u000berent choices of parameter set cause\nserious di\u000berence in the dynamics of the relaxation. In particular, in the rotation type\nrelaxation in isotropic spin systems, we \fnd that the dependences of the relaxation time on\nthe temperature in cases A and B show opposite correlations as well as the dependences of the\nrelaxation time on the magnitude of the magnetic moment. That is, the relaxation time of\nmagnetization reversal under an unfavorable external \feld is shorter at a higher temperature\nin case A, while it is longer in case B. On the other hand, the relaxation time is longer for\na larger magnetic moment in case A, while it is shorter in case B. We also investigate the\nrelaxation of anisotropic spin systems and \fnd that the metastability strongly a\u000bects the\nrelaxation at low temperatures in both cases. The system relaxes to the equilibrium state\nfrom the metastable state by the nucleation type of dynamics. The relaxation time to the\nmetastable state and the decay time of the metastable state are a\u000bected by the choice of\nthe parameter set.\nThe outline of this paper is as follows. The model and the method in this study are ex-\nplained in Sec II. Magnetization processes as a function of temperature in uniform magnetic\nsystems are studied in Sec III. Magnetizations as a function of temperature for inhomoge-\nneous magnetic systems are investigated in Sec. IV, in which not only exchange interactions\n(short-range) but also dipole interactions (long-range) are taken into account. In Sec. V\ndynamical aspects with the choice of the parameter set are considered, and the dependences\nof the relaxation process on the temperature and on the magnitude of magnetic moments\nare also discussed. The relaxation dynamics via a metastable state is studied in Sec. VI.\nSec. VII is devoted to summary and discussion. In Appendix A the Fokker-Planck equation\nfor inhomogeneous magnetic systems is given both in Stratonovich and Ito interpretations,\nand Appendix B presents the numerical integration scheme in this study.\nII. MODEL AND METHOD\nAs a microscopic spin model, the following Hamiltonian is adopted,\nH=\u0000X\nhi;jiJi;jSi\u0001Sj\u0000X\niDA\ni(Sz\ni)2\u0000X\nihi(t)Sz\ni+X\ni6=kC\nr3\nik\u0010\nSi\u0001Sk\u00003(rik\u0001Si)(rik\u0001Sk)\nr2\nik\u0011\n:(1)\n5Here we only consider a spin angular momentum Sifor a magnetic moment Miat each\nsite (iis the site index) and regard Mi=Siignoring the di\u000berence of the sign between\nthem and setting a unit: g\u0016B= 1 for simplicity, where gis the g-factor and \u0016Bis the Bohr\nmagneton30. Interaction Ji;jbetween the ith andjth magnetic sites indicates an exchange\ncoupling,hi;jidenotes a nearest neighbor pair, DA\niis an anisotropy constant for the ith\nsite,hiis a magnetic \feld applied to the ith site, and the \fnal term gives dipole interactions\nbetween the ith andkth sites whose distance is ri;k, whereC=1\n4\u0019\u00160is de\fned using the\npermeability of vacuum \u00160.\nThe magnitude of the moment Miis de\fned as Mi\u0011jMij, which is not necessarily\nuniform but may vary from site to site. In general, the damping parameter may also have\nsite dependence, i.e., \u000bi, and thus the LLG equation at the ith site is given by\nd\ndtMi=\u0000\rMi\u0002He\u000b\ni+\u000bi\nMiMi\u0002dMi\ndt; (2)\nor in an equivalent formula:\nd\ndtMi=\u0000\r\n1 +\u000b2\niMi\u0002He\u000b\ni\u0000\u000bi\r\n(1 +\u000b2\ni)MiMi\u0002(Mi\u0002He\u000b\ni); (3)\nwhere\ris the gyromagnetic constant. Here He\u000b\niis the e\u000bective \feld at the ith site and\ndescribed by\nHe\u000b\ni=\u0000@\n@MiH(M1;\u0001\u0001\u0001;MN;t) (4)\n, which contains \felds from the exchange and the dipole interactions, the anisotropy, and\nthe external \feld.\nWe introduce a Langevin-noise formalism for the thermal e\u000bect. There have been several\nways for the formulation to introduce a stochastic term into the LLG equation. The stochas-\ntic \feld can be introduced into the precession term and/or damping term8,9,11. Furthermore,\nan additional noise term may be introduced10,12. In the present study we add the random\nnoise to the e\u000bective \feld He\u000b\ni!He\u000b\ni+\u0018iand we have\nd\ndtMi=\u0000\r\n1 +\u000b2\niMi\u0002(He\u000b\ni+\u0018i)\u0000\u000bi\r\n(1 +\u000b2\ni)MiMi\u0002(Mi\u0002(He\u000b\ni+\u0018i)); (5)\nwhere\u0018\u0016\niis the\u0016(=1,2 or 3 for x,yorz) component of the white Gaussian noise applied at\ntheith site and the following properties are assumed:\nh\u0018\u0016\nk(t)i= 0;h\u0018\u0016\nk(t)\u0018\u0017\nl(s)i= 2Dk\u000ekl\u000e\u0016\u0017\u000e(t\u0000s): (6)\n6We call Eq. (5) stochastic LLG equation. We derive a Fokker-Planck equation6,8for\nthe stochastic equation of motion in Eq. (5) in Stratonovich interpretation, as given in\nappendix A,\n@\n@tP(M1;\u0001\u0001\u0001;MN;t) =X\ni\r\n1 +\u000b2\ni@\n@Mi\u0001\u001a\u0014\u000bi\nMiMi\u0002(Mi\u0002He\u000b\ni) (7)\n\u0000\rDiMi\u0002(Mi\u0002@\n@Mi)\u0015\nP(M1;\u0001\u0001\u0001;MN;t)\u001b\n:\nHere we demand that the distribution function at the stationary state ( t!1 ) of the\nequation of motion (Eq. (7)) agrees with the canonical distribution of the system (Eq. (1))\nat temperature T, i.e.,\nPeq(M1;\u0001\u0001\u0001;MN)/exp\u0010\n\u0000\fH(M1;\u0001\u0001\u0001;MN)\u0011\n; (8)\nwhere\f=1\nkBT.\nConsidering the relation\n@\n@MiPeq(M1;\u0001\u0001\u0001;MN) =\fHe\u000b\niPeq(M1;\u0001\u0001\u0001;MN); (9)\nwe \fnd that if the following relation\n\u000bi\nMi\u0000\rDi\f= 0 (10)\nis satis\fed at each site i, the canonical distribution in the equilibrium state is assured.\nWhen the magnetic moments are uniform, i.e., the magnitude of each magnetic moment\nis the same and Mi=jMij=M, the parameters \u000biandDiare also uniform \u000bi=\u000band\nDi=Dfor a given T. However, when Miare di\u000berent at sites, the relation (10) must be\nsatis\fed at each site independently. There are several ways of the choice of the parameters\n\u000biandDito satisfy this relation. Here we consider the following two cases: A and B.\nA: we take the damping parameter \u000bito be the same at all sites, i.e., \u000b1=\u000b2=\u0001\u0001\u0001=\u000bN\u0011\n\u000b. In this case the amplitude of the random \feld at the ith site should be\nDi=\u000b\nMikBT\n\r/1\nMi: (11)\nB: we take the amplitude of the random \feld to be the same at all sites, i.e., D1=D2=\n\u0001\u0001\u0001=DN\u0011D. In this case the damping parameter at the ith site should be\n\u000bi=D\rMi\nkBT/Mi: (12)\n700.20.40.60.81\n0123456m\nTFIG. 1: (color online) Comparison of the temperature dependence of min the stationary state\nbetween the stochastic LLG method and the Langevin function (green circles). Crosses and boxes\ndenotemin case A ( \u000b= 0:05) and case B ( D= 1:0), respectively. In the stochastic LLG\nsimulation \u0001 t= 0:005 was set and 80000 time steps (40,000 steps for equilibration and 40,000\nsteps for measurement) were employed. The system size N=L3= 103was adopted.\nWe study whether the canonical distribution is realized in both cases by comparing data\nobtained by the stochastic LLG method with the exact results or with corresponding data\nobtained by Monte Carlo simulations. We set the parameters \r= 1 andkB= 1 hereafter.\nIII. REALIZATION OF THE THERMAL EQUILIBRIUM STATE IN HOMOGE-\nNEOUS MAGNETIC SYSTEMS\nA. Non-interacting magnetic moments\nAs a \frst step, we check the temperature e\u000bect in the simplest case of non-interacting\nuniform magnetic moments, i.e., Ji;j= 0,DA\ni= 0,C= 0 in Eq. (1) and Mi=M(or\nSi=S), where\u000bandDhave no site i-dependence. In this case the magnetization in a\nmagnetic \feld ( h) at a temperature ( T) is given by the Langevin function:\nm=1\nNhNX\ni=1Sz\nii=M \ncoth\u0010hM\nkBT\u0011\n\u0000kBT\nhM!\n: (13)\nWe compare the stationary state obtained by the stochastic LLG method and Eq. (13).\n8We investigate m(T) ath= 2 forM= 1. Figure 1 shows m(T) when\u000b= 0:05 is \fxed (case\nA) and when D= 1:0 is \fxed (case B). We \fnd a good agreement between the results of\nthe stochastic LLG method and the Langevin function in the whole temperature region as\nlong as the relation (10) is satis\fed. Numerical integration scheme is given in Appendix B.\nThe time step of \u0001 t= 0:005 and total 80000 time steps (40000 steps for equilibration and\n40000 steps for measurement) were adopted.\nB. Homogeneous magnetic moments with exchange interactions\nNext, we investigate homogenous magnetic moments ( Mi=jMij=M) in three di-\nmensions. The following Hamiltonian ( C= 0,Ji;j=J,DA\ni=DA, andh(t) =hin Eq.\n(1)):\nH=\u0000X\nhi;jiJSi\u0001Sj\u0000X\niDA(Sz\ni)2\u0000X\nihSz\ni (14)\nis adopted.\nThere is no exact formula for magnetization ( m) as a function of temperature for this\nsystem, and thus a Monte Carlo (MC) method is applied to obtain reference magnetization\ncurves for the canonical distribution because MC methods have been established to obtain\n\fnite temperature properties for this kind of systems in the equilibrium state. Here we\nemploy a MC method with the Metropolis algorithm to obtain the temperature dependence\nof magnetization.\nIn order to check the validity of our MC procedure, we investigated magnetization\ncurves as functions of temperature (not shown) with system-size dependence for the three-\ndimensional classical Heisenberg model ( DA= 0 andh= 0 in Eq. (14)), and con\frmed that\nthe critical temperature agreed with past studies31, wherekBTc= 1:443Jfor the in\fnite\nsystem size with M= 1.\nWe givem(T) for a system of M= 2 with the parameters J= 1,h= 2 andDA= 1:0\nfor cases A and B in Fig 2. The system size was set N=L3= 103and periodic boundary\nconditions (PBC) were used. Green circles denote mobtained by the Monte Carlo method.\nAt each temperature ( T) 10,000 MC steps (MCS) were applied for the equilibration and\nfollowing 10,000\u000050,000 MCS were used for measurement to obtain m. Crosses and boxes\ndenotemin the stationary state of the stochastic LLG equation in case A ( \u000b= 0:05) and\nin case B ( D= 1:0), respectively. Here \u0001 t= 0:005 was set and 80000 steps (40000 for\n900.511.52\n0 5 10 15 20 25m\nTFIG. 2: (color online) Comparison of temperature ( T) dependence of mbetween the Monte Carlo\nmethod (green circles) and the stochastic LLG method in the homogeneous magnetic system with\nM= 2. Crosses and boxes denote case A with \u000b= 0:05 and case B with D= 1:0, respectively.\ntransient and 40000 for measurement) were used to obtain the stationary state of m. The\nm(T) curves show good agreement between the MC method and the stochastic LLG method\nin both cases. We checked that the choice of the initial state for the MC and the stochastic\nLLG method does not a\u000bect the results. The dynamics of the stochastic LLG method leads\nto the equilibrium state at temperature T.\nIV. REALIZATION OF THE THERMAL EQUILIBRIUM STATE IN INHOMO-\nGENEOUS MAGNETIC SYSTEMS\nA. Inhomogeneous magnetic moments with exchange interactions\nHere we study a system which consists of two kinds of magnitudes of magnetic moments.\nThe Hamiltonian (14) is adopted but the moment Mi=jMijhasi-dependence. We investi-\ngate a simple cubic lattice composed of alternating M= 2 andM= 1 planes (see Fig. 3 (a)),\nwhereJ= 1,h= 2 andDA= 1:0 are applied. We consider two cases A and B mentioned\nin Sec. II.\nThe reference of m(T) curve was obtained by the MC method and is given by green\ncircles in Figs. 3 (b) and (c). In the simulation, at each temperature ( T) 10,000 MCS were\napplied for the equilibration and following 10,000 \u000050,000 MCS were used for measurement.\n10(a)\n00.511.5\n05 1 0 1 5 2 0m\nT(b)\n00.511.5\n05 1 0 1 5 2 0m\nT(c)FIG. 3: (color online) (a) A part of the system composed of alternating M= 2 (red long\narrows) and M= 1 (short blue arrows) layers. (b) Comparison of temperature ( T) dependence of\nmbetween the Monte Carlo method (green circles) and the stochastic LLG method for \u000b= 0:05.\n\u0001t= 0:005 and 80,000 steps (40,000 for transient time and 40,000 for measurement) were employed.\nCrosses denote mwhenDi=D(Mi)\u0011\u000b\nMikBT\n\rwas used. Triangles and Diamonds are mfor\nDi=D(1) =\u000bkBT\n\rfor alliandDi=D(2) =\u000b\n2kBT\n\rfor alli, respectively. (c) Comparison\nof temperature ( T) dependence of mbetween the Monte Carlo method (green circles) and the\nstochastic LLG method for D= 1:0. \u0001t= 0:005 and 80,000 steps (40,000 for transient time and\n40,000 for measurement) were employed. Crosses denote mwhen\u000bi=\u000b(Mi)\u0011D\rMi\nkBTwas used.\nTriangles are mfor\u000bi=\u000b(Mi= 1) =D\r\u00021\nkBTfor alliand Diamonds are mfor\u000bi=\u000b(2) =D\r\u00022\nkBT\nfor alli.\n11The system size N=L3= 103was adopted with PBC. In case A, \u000b(= 0:05) is common for\nall magnetic moments in the stochastic LLG method and Mi(orSi) dependence is imposed\nonDiasDi=D(Mi)\u0011\u000b\nMikBT\n\r. In case B, D= 1:0 is common for all magnetic moments in\nthe stochastic LLG method and \u000bi=\u000b(Mi)\u0011D\rMi\nkBT. Crosses in Figs. 3 (b) and (c) denote\nmby the stochastic LLG method for cases A and B, respectively. For those simulations\n\u0001t= 0:005 and 80,000 steps (40,000 for transient time and 40,000 for measurement) were\nemployed at each temperature. In both Figs. 3 (b) and (c), we \fnd good agreement between\nm(T) by the stochastic LLG method (crosses) and m(T) by the MC method (green circles).\nNext, we investigate how the results change if we take wrong choices of parameters. We\nstudym(T) when a uniform value Di=Dfor case A ( \u000bi=\u000bfor case B) is used for all\nspins, i.e., for both Mi= 1 andMi= 2. IfD(Mi= 2) =\u000b\n2kBT\n\ris used for all spins, m(T)\nis shown by Diamonds in Fig. 3 (b), while if D(Mi= 1) =\u000bkBT\n\ris applied for all spins,\nm(T) is given by triangles in Fig. 3 (b). In the same way, we study m(T) for a uniform\nvalue of\u000b. In Fig. 3 (c) triangles and diamonds denote m(T) when\u000bi=\u000b(Mi= 1) and\n\u000bi=\u000b(Mi= 2) are used, respectively. We \fnd serious di\u000berence in m(T) when we do not\nuse correct Mi-dependent choices of the parameters. The locations of triangle (diamond) at\neach temperature Tare the same in Figs. 3 (a) and (b), which indicates that if the ratio\n\u000b=D is the same in di\u000berent choices, the same steady state is realized although this state is\nnot the true equilibrium state for the inhomogeneous magnetic system. Thus we conclude\nthat to use proper relations of Mi-dependence of Dior\u000biis important for m(T) curves of\ninhomogeneous magnetic systems and wrong choices cause signi\fcant deviations.\nB. Critical behavior of Inhomogeneous magnetic moments\nIn this subsection, we examine properties near the critical temperature. Here we adopt\nthe case ofh= 0 andDA= 0 in the same type of lattice with M= 1 and 2 as Sec. IV A. We\ninvestigate both cases of the temperature control (A and B). The Hamiltonian here has O(3)\nsymmetry and mis not a suitable order parameter. Thus we de\fne the following quantity\nas the order parameter31:\nma=q\nm2\nx+m2\ny+m2\nz; (15)\n1200.511.5\n0123456ma\nTFIG. 4: (color online) Comparison of temperature ( T) dependence of mabetween the MC method\n(green circles) and the stochastic LLG method for the system of inhomogeneous magnetic moments.\nN=L3= 203. PBC were used. In the MC method 10,000 MCS and following 50,000 MCS were\nused for equilibration and measurement at each temperature, respectively. The stochastic LLG\nmethod was performed in case A with \u000b= 0:05 (croses) and in case B with D= 1:0 (diamonds).\nHere \u0001t= 0:005 was applied and 240,000 steps were used (40,000 for transient and 200,000 for\nmeasurement).\nwhere\nmx=1\nNhNX\ni=1Sx\nii; my=1\nNhNX\ni=1Sy\nii;andmz=m=1\nNhNX\ni=1Sz\nii: (16)\nIn Fig. 4, green circles denote temperature ( T) dependence of magiven by the MC\nmethod. The system size N=L3= 203with PBC was adopted and in MC simulations\n10,000 MCS and following 50,000 MCS were employed for equilibration and measurement,\nrespectively at each temperature. The magnetizations of maobtained by the stochastic LLG\nmethod for case A (crosses) and case B (diamonds) are given in Fig. 4. Here \u000b= 0:05 and\nD= 1:0 were used for (a) and (b), respectively. \u0001 t= 0:005 was set and 240,000 steps\n(40,000 for transient and 200,000 for measurement) were applied.\nIn both cases ma(T) curve given by the stochastic LLG method shows good agreement\nwith that obtained by the MC method. Thus, we conclude that as long as the relation (10)\nis satis\fed, the temperature dependence of the magnetization is reproduced very accurately\neven around the Curie temperature, regardless of the choice of the parameter set.\n1300.511.5\n0123456m\nTFIG. 5: (color online) Comparison of temperature ( T) dependence of mbetween the Monte Carlo\nmethod (green circles) and the stochastic LLG method. Crosses and diamonds denote case A\nwith\u000b= 0:05 and case B with D= 1:0, respectively. A reduction of mfrom fully saturated\nmagnetization is observed at around T= 0 due to the dipole interactions. As a reference, mby\nthe MC method without the dipole interactions ( C= 0) is given by open circles.\nC. Inhomogeneous magnetic moments with exchange and dipole interactions\nWe also study thermal e\u000bects in a system with dipole interactions. We use the same\nlattice as in the previous subsections. The system is ( Ji;j=J,DA\ni=DA, andhi(t) =hin\nEq. (1)) given by\nH=\u0000X\nhi;jiJSi\u0001Sj\u0000X\niDA(Sz\ni)2\u0000X\nihSz\ni+X\ni6=kC\nr3\nik\u0010\nSi\u0001Sk\u00003(rik\u0001Si)(rik\u0001Sk)\nr2\nik\u0011\n:(17)\nHere a cubic lattice with open boundary conditions (OBC) is used. Since Jis much larger\nthanC=a3(J\u001dC=a3) for ferromagnets, where ais a lattice constant between magnetic\nsites. However, we enlarge dipole interaction as C= 0:2 witha= 1 forJ= 1 to highlight\nthe e\u000bect of the noise on dipole interactions. We set other parameters as h= 0:1,DA= 0:1.\nStudies with realistic situations will be given separately.\nWe study cases A ( \u000b= 0:05) and B ( D= 1:0) for this system. We depict in Fig. 5\nthe temperature ( T) dependences of mwith comparison between the MC (green circles) and\nstochastic LLG methods. Crosses and diamonds denote m(T) for cases A and B, respectively.\nDipole interactions are long-range interactions and we need longer equilibration steps, and\n14we investigate only a small system with N=L3= 63. In the MC method 200,000 MCS\nwere used for equilibration and 600,000 steps were used for measurement of m, and for\nthe stochastic LLG method \u0001 t= 0:005 was set and 960,000 steps (160,000 and 800,000\ntime steps for equilibration and measurement, respectively) were consumed. A reduction of\nmfrom fully saturated magnetization is observed. As a reference, mby the MC method\nwithout the dipole interactions ( C= 0) is given by open circles in Fig. 5. This reduction of\nmis caused by the dipole interactions.\nWe \fnd that even when dipole interactions are taken into account in inhomogeneous\nmagnetic moments, suitable choices of the parameter set leads to the equilibrium state.\nFinally, we comment on the comparison between the LLG method and the Monte Carlo\nmethod. To obtain equilibrium properties of spin systems, the Monte Carlo method is more\ne\u000ecient and powerful in terms of computational cost. It is much faster than the stochastic\nLLG method to obtain the equilibrium m(T) curves, etc. For example, it needs more than\n10 times of CPU time of the MC method to obtain the data for Fig. 5. However, the MC\nmethod has little information on the dynamics and the stochastic LLG method is used to\nobtain dynamical properties because it is based on an equation of motion of spins. Thus, it\nis important to clarify the nature of stochastic LLG methods including the static properties.\nFor static properties, as we saw above, the choice of the parameter set, e.g., cases A and\nB, did not give di\u000berence. However, the choice gives signi\fcant di\u000berence in dynamical\nproperties, which is studied in the following sections.\nV. DEPENDENCE OF DYNAMICS ON THE CHOICE OF THE PARAMETER\nSET IN ISOTROPIC SPIN SYSTEMS ( DA=0)\nNow we study the dependence of dynamics on the choice of parameter set. The temper-\nature is given by\nkBT=\rDiMi\n\u000bi; (18)\nwhich should be the same for all the sites. In general, if the parameter D(amplitude of\nthe noise) is large, the system is strongly disturbed, while if the parameter \u000b(damping\nparameter) is large, the system tends to relax fast. Therefore, even if the temperature is\nthe same, the dynamics changes with the values of Dand\u000b. When the anisotropy term\nexists, i.e.,DA6= 0, in homogeneous systems ( Mi=M) given by Eq. (14), the Stoner-\n15Wohlfarth critical \feld is hc= 2MDAatT= 0. If the temperature is low enough, the\nmetastable nature appears in relaxation. On the other hand, if Tis rather high orDA= 0,\nthe metastable nature is not observed. In this section we focus on dynamics of isotropic spin\nsystems, i.e.,DA=0.\nA. Relaxation with temperature dependence\nIn this subsection we investigate the temperature dependence of magnetization relaxation\nin cases A and B. We adopt a homogeneous system ( Mi=M= 2) withDA= 0 in Eq. (14).\nInitially all spins are in the spin down state and they relax under a unfavorable external\n\feldh= 2. The parameter set M= 2,\u000b= 0:05,D= 0:05 givesT= 2 by the condition\n(Eq. (10)). Here we study the system at T= 0:2;1;2, and 10. We set \u000b= 0:05 in case A\nand the control of the temperature is performed by D, i.e.D= 0:005;0:025;0:05, and 0:25,\nrespectively. In case B we set D= 0:05, and the control of the temperature is realized by\n\u000b, i.e.,\u000b= 0:5;0:1;0:05, and 0:01, respectively.\nWe depict the temperature dependence of m(t) for cases A and B in Figs. 6 (a) and (b),\nrespectively. Here the same random number sequence was used for each relaxation curve.\nRed dash dotted line, blue dotted line, green solid line, and black dashed line denote T= 0:2,\nT= 1,T= 2 andT= 10, respectively. Relaxation curves in initial short time are given in\nthe insets.\nIn case A, as the temperature is raised, the initial relaxation speed of mbecomes faster\nand the relaxation time to the equilibrium state also becomes shorter. This dependence is\nascribed to the strength of the noise with the dependence D/T, and a noise with a larger\namplitude disturbs more the precession of each moment, which causes faster relaxation.\nOn the other hand, in case B, the relaxation time to the equilibrium state is longer at\nhigher temperatures although the temperature dependence of the initial relaxation speed of\nmis similar to the case A. In the initial relaxation process all the magnetic moments are\nin spin-down state ( Sz\ni'\u00002). There the direction of the local \feld at each site is given\nbyHe\u000b\ni'JP\njSz\nj+h=\u00002\u00026 + 2 =\u000010, which is downward and the damping term\ntends to \fx moments to this direction. Thus, a large value of the damping parameter at a\nlow temperature T(\u000b/1\nT) suppresses the change of the direction of each moment and the\ninitial relaxation speed is smaller. However, in the relaxation process thermal \ructuation\n16-2-1012\n0 50 100 150 200m\ntime(a)\n-2.2-2-1.8-1.6-1.4-1.2-1\n012345678\n-2-1012\n0 50 100 150 200m\ntime-2.2-2-1.8-1.6-1.4-1.2-1\n012345678(b)FIG. 6: (color online) (a) Time dependence of the magnetization ( m(t)) in case A, where \u000b= 0:05\nfor a homogeneous system with M= 2. Red dash dotted line, blue dotted line, green solid line,\nand black dashed line denote T= 0:2,T= 1,T= 2 andT= 10, respectively. Inset shows the time\ndependence of m(t) in the initial relaxation process. (b) Time dependence of the magnetization\n(m(t)) in case B, where D= 0:05 for a homogeneous system with M= 2. Correspondence between\nlines and temperatures is the same as (a).\ncauses a deviation of the local \feld and then a rotation of magnetic moments from \u0000z\ntozdirection advances (see also Fig. 11 ). Once the rotation begins, the large damping\nparameter accelerates the relaxation and \fnally the relaxation time is shorter.\nB. Relaxation with spin-magnitude dependence\nNext we study the dependence of relaxation on the magnitude of magnetic moments\nin cases A and B. Here we adopt a homogeneous system ( Mi=M) without anisotropy(\nDA= 0) atT= 2 andh= 2. The initial spin con\fguration is the same as the previous\nsubsection. Because\nD/T\nM;and\u000b/M\nT; (19)\nraising the value of Mis equivalent to lowering temperature in both cases A and B and it\ncauses suppression of relaxation in case A, while it leads to acceleration of relaxation in case\nB. Because Ma\u000bects the local \feld from the exchange energy at each site, changing the\nvalue ofMunder a constant external \feld his not the same as changing Tand it may show\n17-1.5-1-0.500.511.5\n01 0 2 0 3 0 4 0 5 0m\ntime(a)\n-1.5-1-0.500.511.5\n02468 1 0m\ntime(b)FIG. 7: (color online) Comparison of the time dependence of mbetween cases A and B by the\nstochastic LLG method. Red and blue lines denote cases A and B, respectively. (a) \u000b= 0:05 for\ncase A and D= 1:0 for case B, (b) \u000b= 0:2 for case A and D= 1:0 for case B.\nsome modi\fed features.\nIn the relation (19), T= 0:2, 1, 2, 10 at M= 2 (Fig.6 (a) and (b)) are the same as\nM= 20, 4, 2, 0.4 at T= 2, respectively. We studied the relaxation ratio de\fned as m(t)=M\nwithMdependence at T= 2 for these four values of M, and compared with the relaxation\ncurves of Fig.6 (a) and (b). We found qualitatively the same tendency between relaxation\ncurves with Mdependence and those with 1 =Tdependence in both cases. A di\u000berence was\nfound in the initial relaxation speed (not shown). When M > 2, the initial relaxation at\nT= 2 is slower than that of the corresponding TatM= 2. The downward initial local\n\feld at each site is stronger for larger Mdue to a stronger exchange coupling, which also\nassist the suppression of the initial relaxation.\nIt is found that the relaxation time under a constant external \fled becomes longer as\nthe value of Mis raised in case A, while it becomes shorter in case B. This suggests that\ndi\u000berent choices of the parameter set lead to serious di\u000berence in the relaxation dynamics\nwithMdependence.\n18VI. DEPENDENCE OF DYNAMICS ON THE CHOICE OF THE PARAMETER\nSET IN ANISOTROPIC SPIN SYSTEMS ( DA6= 0)\nA. Di\u000berent relaxation paths to the equilibrium in magnetic inhomgeneity\nIf the anisotropy term exists DA6= 0 but the temperature is relatively high, metastable\nnature is not observed in relaxation. We consider the relaxation dynamics when Mihas\nidependence in this case. We study the system (alternating M= 2 andM= 1 planes)\ntreated in Sec. IV A. We set a con\fguration of all spins down as the initial state and observe\nrelaxation of min cases A and B. In Sec. IV A we studied cases A ( \u000b=0.05) and B ( D=1.0)\nfor the equilibrium state and the equilibrium magnetization is m'0:95 atT= 5. We\ngive comparison of the time dependence of mbetween the two cases in Fig. 7 (a), with the\nuse of the same random number sequence. The red and blue curves denote cases A and\nB, respectively. We \fnd a big di\u000berence in the relaxation time of mand features of the\nrelaxation between the two cases.\nThe parameter values of \u000bandDare not so close between the two cases at this tempera-\nture (T= 5), i.e.,D(M= 1) = 0:25 andD(M= 2) = 0:125 for case A and \u000b(M= 1) = 0:2\nand\u000b(M= 2) = 0:4 for case B. Thus, to study if there is a di\u000berence of dynamics even\nin close parameter values of \u000bandDbetween cases A and B at T= 5, we adopt common\n\u000b= 0:2, whereD(M= 1) = 1 and D(M= 2) = 0:5, as case A and common D= 1:0, where\n\u000b(M= 1) = 0:2 and\u000b(M= 2) = 0:4, as case B. We checked that this case A also gives the\nequilibrium state. In Fig. 7 (b), the time dependence of mfor both cases is given. The red\nand blue curves denote cases A and B, respectively. There is also a di\u000berence (almost twice)\nof the relaxation time of mbetween cases A and B. Thus, even in close parameter region of\n\u000bandD, dynamical properties vary depending on the choice of the parameters.\nB. Relaxation with nucleation mechanism\nIn this subsection we study a system with metastability. We adopt a homogeneous\nsystem (M= 2) withJ= 1,DA= 1 andh= 2. Here the Stoner-Wohlfarth critical \feld\nishc= 2MDA=4, and if the temperature is low enough, the system has a metastable state\nunderh= 2.\nAt a high temperature, e.g., T= 10 (\u000b= 0:05,D= 0:25), the magnetization relaxes\n19(a)\n-2-1012\n0 80 160 240 320m\ntime\n-2-1012\n0 50 100 150 200 250 300 350m\ntime(b)\n-2-1012\n0 50 100 150 200 250 300 350m\ntime(c)FIG. 8: (color online) (a) Dashed line shows m(t) for\u000b= 0:05,D= 0:25, andT= 10. Blue\nand green solid lines give m(t) for\u000b= 0:05 atT= 3:5 (case A) and D= 0:25 atT= 3:5 (case\nB), respectively. These two lines were obtained by taking average over 20 trials with di\u000berent\nrandom number sequences. The 20 relaxation curves for cases A and B are given in (b) and (c),\nrespectively.\nwithout being trapped as depicted in Fig 8(a) with a black dotted line. When the tempera-\nture is lowered, the magnetization is trapped at a metastable state. We observe relaxations\nin cases A and B, where \u000b= 0:05 for case A and D= 0:25 for case B are used. In Figs. 8(b)\nand (c), we show 20 samples (with di\u000berent random number sequences) of relaxation pro-\ncesses atT= 3:5 for case A ( \u000b= 0:05,D= 0:0875) and case B ( D= 0:25,\u000b= 0:143),\nrespectively. The average lines of the 20 samples are depicted in Fig 8(a) by blue and green\nsolid lines for cases A and B, respectively. In both cases, magnetizations are trapped at a\nmetastable state with the same value of m(m'\u00001:55). This means that the metastabil-\nity is independent of the choice of parameter set. Relaxation from the metastable state to\nthe equilibrium is the so-called stochastic process and the relaxation time distributes. The\nrelaxation time in case A is longer. If the temperature is further lowered, the escape time\nfrom the metastable state becomes longer. In Figs. 9 (a) and (b), we show 20 samples of\nrelaxation at T= 3:1 for cases A and B, respectively. There we \fnd the metastable state\nmore clearly.\nHere we investigate the initial relaxation to the metastable state at a relatively low\ntemperature. In Figs. 10 (a) and (b), we depict the initial short time relaxation of 20\nsamples at T= 2 in cases A ( \u000b= 0:05,D= 0:05) and B(D= 0:25,\u000b= 0:25), respectively.\nThe insets show the time dependence of the magnetization in the whole measurement time.\n20-2-1012\n0 200 400 600 800m\ntime(a)\n-2-1012\n0 200 400 600 800m\ntime(b)FIG. 9: (a) and (b) illustrate 20 relaxation curves for \u000b= 0:05 atT= 3:1 (case A) and D= 0:25\natT= 3:1 (case B), respectively. Metastability becomes stronger than T= 3:5. No relaxation\noccurs in all 20 trials in (a), while \fve relaxations take place in 20 trials in (b).\nWe \fnd that the relaxation is again faster in case B.\nThe metastability also depends on Mas well asDAand largeMgives a strong metastabil-\nity. Here we conclude that regardless of the choice of the parameter set, as the temperature\nis lowered, the relaxation time becomes longer due to the stronger metastability, in which\nlargerD(larger\u000b) gives faster relaxation from the initial to the metastable state and faster\ndecay from the metastable state.\nFinally we show typical con\fgurations in the relaxation process. When the anisotropy\nDAis zero or weak, the magnetization relaxation occurs with uniform rotation from \u0000z\ntozdirection, while when the anisotropy is strong, the magnetization reversal starts by a\nnucleation and inhomogeneous con\fgurations appear with domain wall motion. In Figs. 11\nwe give an example of the magnetization reversal of (a) the uniform rotation type (magneti-\nzation reversal for DA= 0 withD= 0:05,T= 2,\u000b= 0:1,M= 4) and of (b) the nucleation\ntype (magnetization reversal for DA= 1 withD= 0:25,T= 3:1,\u000b= 0:161,M= 2 ).\nVII. SUMMARY AND DISCUSSION\nWe studied the realization of the canonical distribution in magnetic systems with the\nshort-range (exchange) and long-range (dipole) interactions, anisotropy terms, and magnetic\n\felds by the Langevin method of the LLG equation. Especially we investigated in detail the\n21-2.2-2-1.8-1.6-1.4-1.2-1.0\n012345678m\ntime(a)\n-2-1012\n0 200 400 600 800\ntime\n-2.2-2-1.8-1.6-1.4-1.2-1\n012345678m\ntime(b)\n-2-1012\n0 200 400 600 800\ntimeFIG. 10: Initial relaxation curves of magnetization. Insets show m(t) in the whole measurement\ntime. (a) and (b) illustrate 20 relaxation curves for \u000b= 0:05 atT= 2 (case A) and D= 0:25 at\nT= 2 (case B), respectively.\n(b)(a)\nFIG. 11: (a) Typical uniform rotation type relaxation observed in the isotropic spin system. (b)\nTypical nucleation type relaxation observed in the anisotropic spin system.\nthermal equilibration of inhomogeneous magnetic systems. We pointed out that the spin-\nmagnitude dependent ratio between the strength of the random \feld and the coe\u000ecient of the\ndamping term must be adequately chosen for all magnetic moments satisfying the condition\n(10). We compared the stationary state obtained by the present Langevin method of the\n22LLG equation with the equilibrium state obtained by the standard Monte Carlo simulation\nfor given temperatures. There are several choices for the parameter set, e.g., A and B. We\nfound that as long as the parameters are suitably chosen, the equilibrium state is realized as\nthe stationary state of the stochastic LLG method regardless of the choice of the parameter\nset, and the temperature dependence of the magnetization is accurately produced in the\nwhole region, including the region around the Curie temperature.\nWe also studied dynamical properties which depend on the choice of the parameters. We\nshowed that the choice of the parameter values seriously a\u000bects the relaxation process to\nthe equilibrium state. In the rotation type relaxation in isotropic spin systems under an\nunfavorable external \feld, the dependences of the relaxation time on the temperature in\ncases A and B exhibited opposite correlations as well as the dependences of the relaxation\ntime on the magnitude of the magnetic moment. The strength of the local \feld in the initial\nstate strongly a\u000bects the speed of the initial relaxation in both cases.\nWe also found that even if close parameter values are chosen in di\u000berent parameter sets\nfor inhomogeneous magnetic systems, these parameter sets cause a signi\fcant di\u000berence of\nrelaxation time to the equilibrium state. In the nucleation type relaxation, the metastability,\nwhich depends on DAandM, strongly a\u000bects the relaxation in both cases A and B. Lowering\ntemperature reinforces the metastability of the system and causes slower relaxation. The\nrelaxation to the metastable state and the decay to the metastable state are a\u000bected by the\nchoice of the parameter set, in which larger Dcauses fast relaxation at a \fxed T.\nIn this study we adopted two cases, i.e., A and B in the choice of the parameter set.\nGenerally more complicated dependence of MiorTon the parameters is considered. How\nto chose the parameter set is related to the quest for the origin of these parameters. It\nis very important for clari\fcation of relaxation dynamics but also for realization of a high\nspeed and a low power consumption, which is required to development of magnetic devices.\nStudies of the origin of \u000bhave been intensively performed32{41. To control magnetization\nrelaxation at \fnite temperatures, investigations of the origin of Das well as\u000bwill become\nmore and more important. We hope that the present work gives some useful insight into\nstudies of spin dynamics and encourages discussions for future developments in this \feld.\n23Acknowledgments\nThe authors thank Professor S. Hirosawa and Dr. S. Mohakud for useful discussions.\nThe present work was supported by the Elements Strategy Initiative Center for Magnetic\nMaterials under the outsourcing project of MEXT and Grant-in-Aid for Scienti\fc Research\non Priority Areas, KAKENHI (C) 26400324.\n24Appendix A: Fokker-Planck equation\nThe LLG equation with a Langevin noise (Eq. (5)) is rewritten in the following form for\n\u0016component ( \u0016= 1;2 or 3 forx;yorz) of theith magnetic moment,\ndM\u0016\ni\ndt=f\u0016\ni(M1;\u0001\u0001\u0001;MN;t) +g\u0016\u0017\ni(Mi)\u0018\u0017\ni(t): (A1)\nHeref\u0016\niandg\u0016\u0017\niare given by\nf\u0016\ni=\u0000\r\n1 +\u000b2\ni\u0014\n\u000f\u0016\u0017\u0015M\u0017\niHe\u000b;\u0015\ni+\u000bi\nMi\u000f\u0016\u0017\u0015\u000f\u0015\u001a\u001bM\u0017\niM\u001a\niHe\u000b;\u001b\ni\u0015\n(A2)\nand\ng\u0016\u0015\ni=\u0000\r\n1 +\u000b2\ni\u0014\n\u000f\u0016\u0017\u0015M\u0017\ni+\u000bi\nMi(\u0000M2\ni\u000e\u0016\n\u0015+M\u0016\niM\u0015\ni)\u0015\n; (A3)\nwhereHe\u000b;\u0015\nican have an explicit time ( t) dependence, and \u000f\u0016\u0017\u0015denotes the Levi-Civita\nsymbol. We employ the Einstein summation convention for Greek indices ( \u0016,\u0017\u0001\u0001\u0001).\nWe consider the distribution function F\u0011F(M1;\u0001\u0001\u0001;MN;t) in the 3N-dimensional\nphase space ( M1\n1;M2\n1;M3\n1;\u0001\u0001\u0001;M1\nN;M2\nN;M3\nN). The distribution function F(M1;\u0001\u0001\u0001;MN;t)\nsatis\fes the continuity equation of the distribution:\n@\n@tF(M1;\u0001\u0001\u0001;MN;t) +NX\ni=1@\n@M\u000b\ni\u001a\u0000d\ndtM\u000b\ni\u0001\nF\u001b\n= 0: (A4)\nSubstituting the relation (A1), the following di\u000berential equation for the distribution func-\ntionFis obtained.\n@\n@tF(M1;\u0001\u0001\u0001;MN;t) =\u0000NX\ni=1@\n@M\u000b\nin\u0000\nfi+g\u000b\f\ni\u0018\f\ni\u0001\nFo\n: (A5)\nRegarding the stochastic equation (A1) as the Stratonovich interpretation, making use\nof the stochastic Liouville approach42, and taking average for the noise statistics (Eq. (6)),\nwe have a Fokker-Planck equation.\n@\n@tP(M1;\u0001\u0001\u0001;MN;t) =\u0000NX\ni=1@\n@M\u000b\ni\u001a\nf\u000b\niP\u0000Dig\u000b\f\ni@\n@M\u001b\ni(g\u001b\f\niP)\u001b\n; (A6)\nwhereP\u0011P(M1;\u0001\u0001\u0001;MN;t) is the averaged distribution function hFi.\nSubstituting the relation\n@\n@M\u001b\nig\u001b\f\ni=\u0000\r\u000bi\nMi(1 +\u000b2\ni)4M\f\ni (A7)\n25and Eq. (A3) into g\u000b\f\ni(@\n@M\u001b\nig\u001b\f\ni), we \fnd\ng\u000b\f\ni(@\n@M\u001b\nig\u001b\f\ni) = 0: (A8)\nThus Eq.(A6) is simpli\fed to\n@\n@tP(M1;\u0001\u0001\u0001;MN;t) =\u0000NX\ni=1@\n@M\u000b\ni\u001a\u0000\nf\u000b\ni\u0000Dig\u000b\f\nig\u001b\f\ni@\n@M\u001b\ni\u0001\nP\u001b\n: (A9)\nSubstituting Eqs. (A2) and (A3), we have a formula in the vector representation.\n@\n@tP(M1;\u0001\u0001\u0001;MN;t) = (A10)\nX\ni\r\n1 +\u000b2\ni@\n@Mi\u0001\u001a\u0014\nMi\u0002He\u000b\ni+\u000bi\nMiMi\u0002(Mi\u0002He\u000b\ni)\n\u0000\rDiMi\u0002(Mi\u0002@\n@Mi)\u0015\nP(M1;\u0001\u0001\u0001;MN;t)\u001b\n:\nSince@\n@Mi\u0001(Mi\u0002He\u000b\ni) = 0, it is written as\n@\n@tP(M1;\u0001\u0001\u0001;MN;t) =X\ni\r\n1 +\u000b2\ni@\n@Mi\u0001\u001a\u0014\u000bi\nMiMi\u0002(Mi\u0002He\u000b\ni) (A11)\n\u0000\rDiMi\u0002(Mi\u0002@\n@Mi)\u0015\nP(M1;\u0001\u0001\u0001;MN;t)\u001b\n:\nIn the case that Eq. (A1) is given under Ito de\fnition, we need Ito-Stratonovich trans-\nformation, and the corresponding equation of motion in Stratonovich interpretation is\ndM\u0016\ni\ndt=f\u0016\ni(M1;\u0001\u0001\u0001;MN;t)\u0000Dig\u0015\u0017\ni(Mi)@g\u0016\u0017\ni(Mi)\n@M\u0015\ni+g\u0016\u0017\ni(Mi)\u0018\u0017\ni(t): (A12)\nThen the Fokker-Planck equation in Ito interpretation is\n@\n@tP(M1;\u0001\u0001\u0001;MN;t) =\u0000NX\ni=1@\n@M\u000b\ni\u001a\u0000\nf\u000b\ni\u0000Dig\u0015\u0017\ni@g\u000b\u0017\ni\n@M\u0015\ni\u0000Dig\u000b\f\nig\u001b\f\ni@\n@M\u001b\ni\u0001\nP\u001b\n:\nSinceg\u0015\u0017\ni@g\u000b\u0017\ni\n@M\u0015\ni=\u00002\r2\n1+\u000b2\niM\u000b\ni, the vector representation is given by\n@\n@tP(M1;\u0001\u0001\u0001;MN;t) =X\ni\r\n1 +\u000b2\ni@\n@Mi\u0001\u001a\u0014\u000bi\nMiMi\u0002(Mi\u0002He\u000b\ni)\n\u00002\rDiMi\u0000\rDiMi\u0002(Mi\u0002@\n@Mi)\u0015\nP(M1;\u0001\u0001\u0001;MN;t)\u001b\n:\n(A13)\n26Appendix B: Numerical integration for stochastic di\u000berential equations\nIn stochastic di\u000berential equations, we have to be careful to treat the indi\u000berentiability\nof the white noise. In the present paper we regard the stochastic equation, e.g., Eq. (5), as\na stochastic di\u000berential equation in Stratonovich interpretation:\ndM\u0016\ni=f\u0016\ni(M1;\u0001\u0001\u0001;MN;t)dt+g\u0016\u0017\ni\u00101\n2\u0000\nMi(t) +Mi(t+dt)\u0001\u0011\ndW\u0017\ni(t); (B1)\nwheredW\u0017\ni(t) =Rt+dt\ntds\u0018\u0017\ni(s), which is the Wiener process. This equation is expressed by\ndM\u0016\ni=f\u0016\ni(M1;\u0001\u0001\u0001;MN;t)dt+g\u0016\u0017\ni(Mi(t))\u000edW\u0017\ni(t); (B2)\nwhere\u000eindicates the usage of the Stratonovich de\fnition.\nA simple predictor-corrector method called the Heun method8,19, superior to the Euler\nmethod, is given by\nM\u0016\ni(t+ \u0001t) =M\u0016\ni(t)\n+1\n2[f\u0016\ni(^M1(t+ \u0001t);\u0001\u0001\u0001;^MN(t+ \u0001t);t+ \u0001t) +f\u0016\ni(M1(t);\u0001\u0001\u0001;MN(t);t)]\u0001t\n+1\n2[g\u0016\u0017\ni(^Mi(t+ \u0001t)) +g\u0016\u0017\ni(Mi(t))]\u0001W\u0017\ni; (B3)\nwhere \u0001W\u0017\ni\u0011W\u0017\ni(t+ \u0001t)\u0000W(t) and ^M\u0016\ni(t+ \u0001t) is chosen in the Euler scheme:\n^M\u0016\ni(t+ \u0001t) =M\u0016\ni(t) +f\u0016\ni(M1(t);\u0001\u0001\u0001;MN(t);t)\u0001t+g\u0016\u0017\ni(Mi(t))\u0001W\u0017\ni: (B4)\nThis scheme assures an approximation accuracy up to the second order of \u0001 Wand \u0001t. Sev-\neral numerical di\u000berence methods19for higher-order approximation, which are often compli-\ncated, have been proposed.\nHere we adopt a kind of middle point method equivalent to the Heun method.\nM\u0016\ni(t+ \u0001t) =M\u0016\ni(t)\n+f\u0016\ni(M1(t+ \u0001t=2);\u0001\u0001\u0001;MN(t+ \u0001t=2);t+ \u0001t=2)\u0001t\n+g\u0016\u0017\ni(Mi(t+ \u0001t=2))\u0001W\u0017\ni; (B5)\nwhereM\u0016\ni(t+ \u0001t=2) is chosen in the Euler scheme:\nM\u0016\ni(t+ \u0001t=2) =M\u0016\ni(t) +f\u0016\ni(M1(t);\u0001\u0001\u0001;MN(t);t)\u0001t=2 +g\u0016\u0017\ni(Mi(t))\u0001~Wi\u0017; (B6)\n27where \u0001 ~Wi\u0017\u0011W\u0017\ni(t+ \u0001t=2)\u0000W\u0017\ni(t). Considering the following relations,\nh\u0001~Wi\u0017\u0001W\u0017\nii=\n[W\u0017\ni(t+ \u0001t=2)\u0000W\u0017\ni(t)][W\u0017\ni(t+ \u0001t)\u0000W\u0017\ni(t)]\u000b\n=Di\u0001t; (B7)\nh\u0001W\u0017\nii= 0 andh\u0001~Wi\u0017i= 0, this method is found equivalent to the Heun method. We can\nformally replace \u0001 ~Wi\u0017by \u0001W\u0017\ni=2 in Eq. (B6) in numerical simulations.\n\u0003Corresponding author. Email address: nishino.masamichi@nims.go.jp\n1H. Kronm ullar and M. F ahnle, \\Micromagnetism and the Microstructure of Ferromagnetic\nSolids\" Cambridge University Press, (2003).\n2D. A. Garanin, Phys. Rev. B 55, 3050 (1997).\n3M. Nishino, K. Boukheddaden, Y. Konishi, and S. Miyashita, Phys. Rev. Lett. 98, 247203\n(2007).\n4S. Nos\u0013 e, J. Chem. Phys. 81, 511 (1984).\n5W. G. Hoover, Phys. Rev. A 31, 1695 (1985).\n6H. Risken, The Fokker-Planck Equation, 2nd ed. Springer, Berlin (1989).\n7A. Bulgac and D. Kusnezov, Phys. Rev. A 42, 5045 (1990).\n8J. L. Garc\u0013 \u0010a-Palacios and F. J. L\u0013 azaro, Phys. Rev. B 58, 14937 (1998)\n9Y. Gaididei, T. Kamppeter, F. G. Mertens, and A. Bishop, Phys. Rev. B 59, 7010 (1999).\n10T. Kamppeter, F. G. Mertens, E. Moro, A. S\u0013 anchez, A. R. Bishop, Phys. Rev. B 59, 11349\n(1999).\n11G. Grinstein and R. H. Koch, R.W. Chantrell, U. Nowak, Phys. Rev. Lett. 90, 207201 (2003).\n12O. Chubykalo, R. Smirnov-Rueda, J.M. Gonzalez, M.A. Wongsam, R.W. Chantrell, U. Nowak,\nJ. Magn. Magn. Mater. 266, 28 (2003).\n13A. Rebei and M. Simionato, Phys. Rev. B 71, 174415 (2005).\n14U. Atxitia, O. Chubykalo-Fesenko, R.W. Chantrell, U. Nowak, and A. Rebei, Phys. Rev. Lett.\n102, 057203 (2009).\n15K. Vahaplar, A. M. Kalashnikova, A.V. Kimel, D. Hinzke, U. Nowak, R. Chantrell, A.\nTsukamoto, A. Itoh, A. Kirilyuk and Th. Rasing, Phys. Rev. Lett. 103, 117201 (2009)\n16D. A. Garanin and O. Chubykalo-Fesenko, Phys. Rev. B 70, 212409 (2004).\n2817O. Chubykalo-Fesenko, U. Nowak, R. W. Chantrell, and D. Garanin, Phys. Rev. B 74, 094436\n(2006).\n18R. F. L. Evans, D. Hinzke, U. Atxitia, U. Nowak, R. W. Chantrell and O. Chubykalo-Fesenko,\nPhys. Rev. B 85, 014433 (2012) .\n19P. E. Kloeden and E. Platen, Numerical Solution of Stochastic Di\u000berential Equations, 3rd corr.\nprinting., Springer (1999).\n20B. Skubic, O. E. Peil, J. Hellsvik, P. Nordblad, L. Nordstr om, and O. Eriksson, Phys. Rev. B\n79, 024411 (2009).\n21J. Hellsvik, B. Skubic, L. L Nordstr om, B. Sanyal, and O. Eriksson, Phys. Rev. B 78, 144419\n(2008).\n22B Skubic, J Hellsvik, L Nordstr om, and O Eriksson, J. Phys. Condens. Matter 20315203 (2008).\n23R. F. L. Evans, W. J. Fan, P. Chureemart, T. A. Ostler, M.O.A. Ellis and R. W. Chantrell, J.\nPhys. Condens. Matter 26103202 (2014).\n24K.-D. Durst and H. Kronm uller, J. Magn. Magn. Mater. 68, 63 (1987).\n25H. Kronm uller and K.-D. Durst, J. Magn. Magn. Mater. 74, 291 (1988).\n26B. Barbara and M. Uehara, Inst. Phys. Conf. Ser. No. 37 Chapter8 (1978).\n27A. Sakuma, S. Tanigawa and M. Tokunaga, J. Magn. Magn. Mater. 84, 52 (1990).\n28A. Sakuma, J. Magn. Magn. Mater. 88, 369 (1990).\n29A. Ostler et al., Phys. Rev. B 84, 024407 (2011).\n30In realistic materials, in particular rare earth systems, the total angular momentum at each\nmagnetic site is given by J=S+L, whereSis a spin angular momentum and Lis an orbital\nangular momentum, and the relation between MandJshould be treated carefully.\n31P. Peczak, A. M. Ferrenberg, and D. P. Landau, Phys. Rev. B 43, 6087 (1991) and references\ntherein.\n32H. J. Skadsem, Y. Tserkovnyak, A. Brataas, and G. E. W. Bauer, Phys. Rev. B 75094416\n(2007).\n33E. Simanek and B. Heinrich, Phys. Rev. B 67, 144418 (2003).\n34V. Kambersk\u0013 y, Phys. Rev. B 76134416 (2007).\n35K. Gilmore, Y. U. Idzerda, and M. D. Stiles, Phys. Rev. Lett. 99, 027204 (2007).\n36K. Gilmore, Y. U. Idzerda, and M. D. Stiles, J. Appl. Phys. 10307D303 (2008).\n37A. Brataas, Y. Tserkovnyak, and G. E. Bauer: Phys. Rev. Lett. 101037207 (2008).\n2938A. A. Starikov, P. J. Kelly, A. Brataas, Y. Tserkovnyak, and G. E. W. Bauer, Phys. Rev. Lett.\n105, 236601 (2010).\n39H. Ebert, S. Mankovsky, D. Kodderitzsch, and P. J. Kelly: Phys. Rev. Lett. 107066603 (2011).\n40A. Sakuma, J. Phys. Soc. Jpn. 4, 084701 (2012).\n41A. Sakuma, J. Appl. Phys. 117, 013912 (2015).\n42R. Kubo, J. Math. Phys. 4, 174 (1963)\n30"    },    {        "title": "1507.06505v2.Nanomagnet_coupled_to_quantum_spin_Hall_edge__An_adiabatic_quantum_motor.pdf",        "content": "Nanomagnet coupled to quantum spin Hall edge: An adiabatic quantum\nmotor\nLiliana Arrachea\nDepartamento de F\u0013 \u0010sica, FCEyN, Universidad de Buenos Aires and IFIBA, Pabell\u0013 on I, Ciudad Universitaria, 1428 CABA\nand International Center for Advanced Studies, UNSAM, Campus Miguelete, 25 de Mayo y Francia, 1650 Buenos Aires,\nArgentina\nFelix von Oppen\nDahlem Center for Complex Quantum Systems and Fachbereich Physik, Freie Universit at Berlin, 14195 Berlin, Germany\nAbstract\nThe precessing magnetization of a magnetic islands coupled to a quantum spin Hall edge pumps charge\nalong the edge. Conversely, a bias voltage applied to the edge makes the magnetization precess. We point\nout that this device realizes an adiabatic quantum motor and discuss the e\u000eciency of its operation based on\na scattering matrix approach akin to Landauer-B uttiker theory. Scattering theory provides a microscopic\nderivation of the Landau-Lifshitz-Gilbert equation for the magnetization dynamics of the device, including\nspin-transfer torque, Gilbert damping, and Langevin torque. We \fnd that the device can be viewed as\na Thouless motor, attaining unit e\u000eciency when the chemical potential of the edge states falls into the\nmagnetization-induced gap. For more general parameters, we characterize the device by means of a \fgure\nof merit analogous to the ZT value in thermoelectrics.\n1. Introduction\nFollowing Ref. [1], Meng et al. [2] recently showed\nthat a transport current \rowing along a quantum\nspin Hall edge causes a precession of the magneti-\nzation of a magnetic island which locally gaps out\nthe edge modes (see Fig. 1 for a sketch of the de-\nvice). The magnetization dynamics is driven by the\nspin transfer torque exerted on the magnetic island\nby electrons backscattering from the gapped region.\nIndeed, the helical nature of the edge state implies\nthat the backscattering electrons reverse their spin\npolarization, with the change in angular momen-\ntum transfered to the magnetic island. This e\u000bect\nis not only interesting in its own right, but may also\nhave applications in spintronics.\nCurrent-driven directed motion at the nanoscale\nhas also been studied for mechanical degrees of free-\ndom, as motivated by progress on nanoelectrome-\nchanical systems. Qi and Zhang [3] proposed that\na conducting helical molecule placed in a homo-\ngeneous electrical \feld could be made to rotate\naround its axis by a transport current and pointedout the intimate relations with the concept of a\nThouless pump [4]. Bustos-Marun et al. [5] devel-\noped a general theory of such adiabatic quantum\nmotors, used it to discuss their e\u000eciency, and em-\nphasized that the Thouless motor discussed by Qi\nand Zhang is optimally e\u000ecient.\nIt is the purpose of the present paper to em-\nphasize that the current-driven magnetization dy-\nnamics is another { perhaps more experimentally\nfeasible { variant of a Thouless motor and that\nthe theory previously developed for adiabatic quan-\ntum motors [5] is readily extended to this device.\nThis theory not only provides a microscopic deriva-\ntion of the Landau-Lifshitz-Gilbert equation for the\ncurrent-driven magnetization dynamics, but also al-\nlows one to discuss the e\u000eciency of the device and\nto make the relation with the magnetization-driven\nquantum pumping of charge more explicit.\nSpeci\fcally, we will employ an extension of the\nLandauer-B uttiker theory of quantum transport\nwhich includes the forces exerted by the electrons\non a slow classical degree of freedom [6, 7, 8, 9].\nPreprint submitted to Elsevier October 11, 2018arXiv:1507.06505v2  [cond-mat.mes-hall]  8 Sep 2015\u0001\u0001\u0001\u0001exeyezeθFigure 1: (Color online) Schematic setup. A nanomagnet\nwith magnetic moment Mcouples to a Kramers pair of edge\nstates of a quantum spin Hall insulator. The e\u000bective spin\ncurrent produces a spin-transfer toque and the magnetic mo-\nment precesses.\nMarkus B uttiker developed Landauer's vision of\nquantum coherent transport as a scattering prob-\nlem into a theoretical framework [10, 11] and ap-\nplied this scattering theory of quantum transport\nto an impressive variety of phenomena. These ap-\nplications include Aharonov-Bohm oscillations [12],\nshot noise and current correlations [11, 13, 14], as\nwell as edge-state transport in the integer Hall ef-\nfect [15] and topological insulators [16]. Frequently,\nB uttiker's predictions based on scattering theory\nprovided reference points with which other theo-\nries { such as the Keldysh Green-function formal-\nism [17, 18, 19, 20] or master equations [21] { sought\nto make contact.\nIn the present context, it is essential that scat-\ntering theory also provides a natural framework to\nstudy quantum coherent transport in systems un-\nder time-dependent driving. For adiabatic driving,\nB uttiker's work with Thomas and Pr^ etre [22] was\ninstrumental in developing a description of adia-\nbatic quantum pumping [4] in terms of scattering\ntheory [23, 24, 25, 26] which provided a useful back-\ndrop for later experiments [27, 28, 29, 30, 31]. Be-\nyond the adiabatic regime, Moskalets and B uttiker\ncombined the scattering approach with Floquet the-\nory to account for periodic driving [32]. These\nworks describe adiabatic quantum transport as a\nlimit of the more general problem of periodic driv-\ning and ultimately triggered numerous studies on\nsingle-particle emitters and quantum capacitors (as\nreviewed by Moskalets and Haack in this volume[33]).\nThe basic idea of the adiabatic quantum mo-\ntor [5] is easily introduced by analogy with the\nArchimedes screw, a device consisting of a screw\ninside a pipe. By turning the screw, water can be\npumped against gravity. This is a classical analog of\na quantum pump in which electrons are pumped be-\ntween reservoirs by applying periodic potentials to\na central scattering region. Just as the Archimedes\npump can pump water against gravity, charge can\nbe quantum pumped against a voltage. In addi-\ntion, the Archimedes screw has an inverse mode\nof operation as a motor : Water pushed through the\ndevice will cause the screw to rotate. The adiabatic\nquantum motor is a quantum analog of this mode\nof operation in which a transport current pushed\nthrough a quantum coherent conductor induces uni-\ndirectional motion of a classical degree of freedom\nsuch as the rotations of a helical molecule.\nThe theory of adiabatic quantum motors [5, 34]\nexploits the assumption that the motor degrees of\nfreedom { be they mechanical or magnetic { are\nslow compared to the electronic degrees of freedom.\nIn this adiabatic regime, the typical time scale of\nthe mechanical dynamics is large compared to the\ndwell time of the electrons in the interaction region\nbetween motor and electrical degrees of freedom. In\nthis limit, the dynamics of the two degrees of free-\ndom can be discussed in a mixed quantum-classical\ndescription. The motor dynamics is described in\nterms of a classical equation of motion, while a\nfully quantum-coherent description is required for\nthe fast electronic degrees of freedom.\nFrom the point of view of the electrons, the mo-\ntor degrees of freedom act as acpotentials which\npump charge through the conductor. Conversely,\nthe backaction of the electronic degrees of freedom\nenters through adiabatic reaction forces on the mo-\ntor degrees of freedom [6, 7, 8, 9]. When there is\njust a single (Cartesian) classical degree of freedom,\nthese reaction forces are necessarily conservative,\nakin to the Born-Oppenheimer force in molecular\nphysics [35]. Motor action driven by transport cur-\nrents can occur when there is more than one mo-\ntor degree of freedom (or a single angle degree of\nfreedom). In this case, the adiabatic reaction force\nneed no longer be conservative when the electronic\nconductor is subject to a bias voltage [6, 7, 8, 9].\nIn next order in the adiabatic approximation,\nthe electronic system also induces frictional and\nLorentz-like forces, both of which are linear in the\nslow velocity of the motor degree of freedom. In-\n2cluding the \ructuating Langevin force which ac-\ncompanies friction yields a classical Langevin equa-\ntion for the motor degree of freedom. This equation\ncan be derived systematically within the Keldysh\nformalism [35] and the adiabatic reaction forces ex-\npressed through the scattering matrix of the coher-\nent conductor [6, 7, 8].\nWhile these developments focused on mechani-\ncal degrees of freedom, it was also pointed out that\nthe scattering theory of adiabatic reaction forces\nextends to magnetic degrees of freedom [9]. In\nthis case, adiabaticity requires that the precessional\ntime scale of the magnetic moment is larger than\nthe electronic dwell time. The e\u000bective classical de-\nscription for the magnetic moment takes the form of\na Landau-Lifshitz-Gilbert (LLG) equation. Similar\nto nanoelectromechanical systems, the LLG equa-\ntion can be derived systematically in the adiabatic\nlimit for a given microscopic model and the coe\u000e-\ncients entering the LLG equation can be expressed\nalternatively in terms of electronic Green functions\nor scattering matrices [36, 37, 38, 39, 9]. In the fol-\nlowing, we will apply this general theory to a mag-\nnetic island coupled to a Kramers pair of helical\nedge states.\nThis work is organized as follows. Section 2\nreviews the scattering-matrix expressions for the\ntorques entering the LLG equation. Section 3 ap-\nplies this theory to helical edge states coupled to\na magnetic island and makes the relation to adia-\nbatic quantum motors explicit. Section 4 de\fnes\nand discusses the e\u000eciency of this device and de-\nrives a direct relation between charge pumping and\nspin transfer torque. Section 5 is devoted to con-\nclusions.\n2. S-matrix theory of spin transfer torques\nand Gilbert damping\n2.1. Landau-Lifshitz-Gilbert equation\nConsider a coherent (Landauer-B uttiker) con-\nductor coupled to a magnetic moment. The latter is\nassumed to be su\u000eciently large to justify a classical\ndescription of its dynamics but su\u000eciently small so\nthat we can treat it as a single macrospin. Then,\nits dynamics is ruled by a Landau-Lifshitz-Gilbert\nequation\n_M=M\u0002[\u0000@MU+Bel+\u000eB]: (1)\nNote that we use units in which Mis an angu-\nlar momentum and for simplicity of notation, Belas well as\u000eBdi\u000ber from a conventional magnetic\n\feld by a factor of gd, the gyromagnetic ratio of\nthe macrospin. The \frst term on the right-hand\nside describes the dynamics of the macrospin in the\nabsence of coupling to the electrons. It is derived\nfrom the quantum Hamiltonian\n^U=\u0000gd^M\u0001B+D\n2^M2\nz; (2)\nwhere M=h^Miis the uncoupled macrospin, Bthe\nmagnetic \feld, and D> 0 the easy-plane anisotropy\nof the macrospin. The coupling to the electrons\nleads to the additional e\u000bective magnetic \feld Bel.\nThis term can be derived microscopically from the\nHeisenberg equation of motion of the macrospin by\nevaluating the commutator of ^Mwith the interac-\ntion Hamiltonian between macrospin and electrons\nin the adiabatic approximation (see, e.g., Ref. [9]).\nKeeping terms up to linear order in the small mag-\nnetization \\velocity\" _M, we can write\nBel=B0(M)\u0000\r(M)_M: (3)\nHere, the \frst contribution B0can be viewed as the\nspin-transfer torque. The second term is a contribu-\ntion to Gilbert damping arising from the coupling\nbetween macrospin and electrons. In general, \rso\nderived is a tensor with symmetric and antisymmet-\nric components. However, it can be seen that only\nthe symmetric part plays a relevant role [9]. Finally,\nby \ructuation-dissipation arguments, the Gilbert\ndamping term is accompanied by a Langevin torque\n\u000eBwith correlator\nh\u000eBl(t)\u000eBk(t0)i=Dlk\u000e(t\u0000t0): (4)\nIts correlations are local in time as a consequence\nof the assumption of adiabaticity. As a result, we\n\fnd the LLG equation\n_M=M\u0002h\n\u0000@MU+B0\u0000\r_M+\u000eBi\n; (5)\nfor the macrospin M.\nThe spin-transfer torque, the Gilbert damping,\nand the correlator Dcan be expressed in terms\nof the scattering matrix of the coherent conductor,\nboth in and out of equilibrium [36, 37, 38, 39, 9].\nBefore presenting the S-matrix expressions, a few\ncomments are in order. First, the expression for the\nGilbert damping only contains the intrinsic damp-\ning originating from the coupling to the electronic\ndegrees of freedom. Coupling to other degrees of\nfreedom might give further contributions to Gilbert\n3damping which could be included phenomenologi-\ncally. Second, in the study of the nanomagnet cou-\npled to the helical modes we will consider the ex-\npressions to lowest order in the adiabatic approxi-\nmation presented in Sec. 2.2. The theory can actu-\nally be extended to include higher order corrections\n[9]. In Sec. 2.3 section, we brie\ry summarize the\nmain steps of the general procedure for complete-\nness.\n2.2. Coe\u000ecients of the LLG equation in the lowest\norder adiabatic approximation\nThis section summarizes the expressions for the\ncoe\u000ecients of the LLG equation that we will use\nto study the problem of the nanomagnet coupled\nto the helical edge states. These correspond to\nthe lowest order in the adiabatic approximation, in\nwhich we retain only terms linear in _MandeV.\nTo this order, we can write the coe\u000ecients of the\nLLG equation in terms of the electronic S-matrix\nfor a static macrospin M. The coupling between\nmacrospin and electronic degrees of freedom enters\nthrough the dependence of the electronic S-matrix\nS0=S0(M) on the (\fxed) macrospin. At this or-\nder, the spin-transfer torque and the Gilbert damp-\ning can be expressed as [36, 37, 38, 39, 9]\nB0(M) =X\n\u000bZd\"\n2\u0019if\u000bTr\u0014\n\u0005\u000b^Sy\n0@S0\n@M\u0015\n(6)\nand\n\rkl(M) =\u0000~X\n\u000bZd\"\n4\u0019f0\n\u000bTr\"\n\u0005\u000b@^Sy\n0\n@Mk@^S0\n@Ml#\ns;(7)\nrespectively. Finally, the \ructuation correlator Dis\nexpressed as [9]\nDkl(M) =~X\n\u000b;\u000b0Zd\"\n2\u0019f\u000b(1\u0000f\u000b0)\n\u0002Tr2\n4\u0005\u000b \n^Sy\n0@^S0\n@Mk!y\n\u0005\u000b0 \n^Sy\n0@^S0\n@Ml!3\n5\ns:(8)\nIn these expressions, \u000b=L;R denotes the reser-\nvoirs with electron distribution function f\u000b, \u0005\u000bis a\nprojector onto the channels of lead \u000b, and Tr traces\nover the lead channels.\n2.3. Corrections to the adiabatic approximation of\nthe S-matrix\nIn order to go beyond linear response in eVand\n_M, we must consider the electronic S-matrix in thepresence of the time-dependent magnetization M(t)\nand expand it to linear order in the magnetization\n\\velocity\" _M(t). This can be done, e.g., by starting\nfrom the full Floquet scattering matrix SF\n\u000b;\f(\"n;\")\nfor a periodic driving with period ![32]. The in-\ndices\u000band\flabel the scattering channels of the\ncoherent conductor and the arguments denote the\nenergies\"of the incoming electron in channel \u000band\n\"n=\"+n~!of the outgoing electron in channel \f.\nFor small driving frequency !, the Floquet scatter-\ning matrix can be expanded in powers of ~!,\n^SF(\"n;\") = ^S0\nn(\") +n~!\n2@^S0\nn(\")\n@\"\n+~!^An(\") +O(\"2): (9)\nHere ^S0\nn(\") is the Fourier transform of the frozen\nscattering matrix S0(M(t)) introduced above,\n^S0(M(t)) =1X\nn=\u00001e\u0000in!t^S0\nn(\"): (10)\nThe matrix ^An(\"), \frst introduced by Moskalets\nand B uttiker, is the \frst adiabatic correction to the\nadiabatic S-matrix and can be transformed in a sim-\nilar way to\n^A(t;\") =1X\nn=\u00001e\u0000in!t^An(\") = _M(t)\u0001^A(t;\"):(11)\nThe matrix ^An(\") can be straightforwardly calcu-\nlated from the retarded Green function of the device\n(see Refs. [20, 9]).\nWe are now in a position to give expressions for\nthe Gilbert damping to next order in the adiabatic\napproximation. (The spin-transfer torque and the\n\ructuation correlator remain unchanged.) To do\nso, we split the Gilbert matrix \rinto its symmetric\nand antisymmetric parts,\n\r=\rs+\ra: (12)\nStrictly speaking, it is only the symmetric part\nwhich corresponds to Gilbert damping. The anti-\nsymmetric part simply renormalizes the precession\nfrequency. One \fnds [9]\n\rkl\ns(M) =\u0000~X\n\u000bZd\"\n4\u0019f0\n\u000bTr\"\n\u0005\u000b@^Sy\n0\n@Mk@^S0\n@Ml#\ns\n+X\n\u000bZd\"\n2\u0019if\u000bTr\"\n\u0005\u000b \n@^Sy\n0\n@Mk^Al\u0000^Ay\nl@^S0\n@Mk!#\ns(13)\n4for the symmetric contribution. It can be seen, the\nsecond line is a pure nonequilibrium contribution\n(/eV~!). Similarly, the antisymmetric part of the\nGilbert damping can be written as [9]\n\rkl\na(M) =\u0000~X\n\u000bZd\"\n2\u0019if\u000b(\")\n\u0002Tr\"\n\u0005\u000b \n^Sy\n0@^Ak\n@Ml\u0000@^Ay\nk\n@Ml^S0!#\na;(14)\nwhich is really a renormalization of the precession\nfrequency as mentioned above.\n3. S-matrix theory of a nanomagnet coupled\nto a quantum spin Hall edge\nWe now apply the above theory to a magnetic\nisland coupled to a quantum spin Hall edge as\nsketched in Fig. 1. The quantum spin Hall edge sup-\nports a Kramers doublet of edge states. The mag-\nnetization M=M?cos\u0012ex+M?sin\u0012ey+Mzez\nof the magnetic island induces a Zeeman \feld JM\nacting on the electrons along the section of length\nLof the edge state which is covered by the magnet.\nThis Zeeman \feld causes backscattering between\nthe edge modes and induces a gap \u0001 = JM?~=2\n[1]. Linearizing the dispersion of the edge modes,\nthe electronic Hamiltonian takes the form [2]\n^H= (vp\u0000JMz)^\u001bz+\u0001(x) (cos\u0012^\u001bx+ sin\u0012^\u001by):(15)\nHere, the\u001bjdenote Pauli matrices in spin space\nand \u0001(x) is nonzero only over the region of length\nLcovered by the magnetic island. We have assumed\nfor simplicity that the spin Hall edge conserves \u001bz.\nThen, a static island magnetization induces a gap\nwhenever it has a component perpendicular to the\nz-direction. Indeed, ^His easily diagonalized for a\nspatially uniform coupling between edge modes and\nmagnet, and the spectrum\nEp=p\n(vp\u0000JMz)2+ \u00012 (16)\nhas a gap \u0001.\nIn the following, we assume that the easy-plane\nanisotropy D > 0 is su\u000eciently large so that the\nmagnetization entering the electronic Hamiltonian\ncan be taken in the xy-plane, i.e., Mz'0. (How-\never, we will have to keep Mzin the LLG equation\nwhen it is multiplied by the large anisotropy D.)\nThe electronic Hamiltonian (15) is equivalent to\nthe electronic Hamiltonian of the Thouless motor\nconsidered in Ref. [5]. Following this reference, wecan readily derive the frozen scattering matrix an-\nalytically [5],\n^S0=1\n\u0003\u0012\u0000iei\u0012\u0015 1\n1\u0000ie\u0000i\u0012\u0015\u0013\n; (17)\nwhere we have de\fned the shorthands\n\u0003 = cos \u001eL\u0000i\"p\n\"2\u0000\u00012sin\u001eL;\n\u0015=\u0001p\n\"2\u0000\u00012sin\u001eL (18)\nwith\n\u001eL(\") =L\n~vp\n\"2\u0000\u00012: (19)\nNote that these expressions are exact for any Land\nvalid for energies \"both inside and outside the gap.\nWe can now use this scattering matrix to eval-\nuate the various coe\u000ecients in the LLG equation,\nemploying the expressions given in Sec. 2.2. As-\nsuming zero temperature, we \fnd\nB0=eV\n2\u0019M\u0018(\u0016)e\u0012; (20)\nfor the spin transfer torque at arbitrary chemical\npotential\u0016. Here, we have de\fned the function\n\u0018(\u0016) =\u00012sin2\u001eL\nj\u00162\u0000\u00012jcos2\u001eL+\u00162sin2\u001eL(21)\nwith\u001eL=\u001eL(\u0016) (see Fig. 2). Below, we will iden-\ntify\u0018with the charge pumped between the reser-\nvoirs during one precessional period of the mag-\nnetization M. The vector B0points in the az-\nimuthal direction in the magnetization plane and\nindeed corresponds to a spin-transfer torque. Sim-\nilarly, we can substitute Eq. (17) into Eq. (13) for\nthe Gilbert damping and \fnd that the only nonzero\ncomponent of the tensor \ris\n\r\u0012\u0012=~\n2\u0019M2\u0018(\u0016): (22)\nSimilarly,\nD\u0012\u0012=~eV\n\u0019M2\u0018(\u0016) (23)\nis the only nonzero component of the \ructuation\ncorrelator. It is interesting to note that this yields\nan e\u000bective \ructuation-dissipation relation D\u0012\u0012=\n2Te\u000b\r\u0012\u0012with e\u000bective temperature Te\u000b=eV.\n5With these results, we can now write the LLG\nequation for the nanomagnet coupled to the helical\nedge state,\n_M =DM\u0002Mzez+\u0018(eV\u0000~_\u0012)\n2\u0019MM\u0002e\u0012\n+M\u0002\u000eB; (24)\nwhere\u0018=\u0018(\u0016), we have expressed _M'M_\u0012e\u0012, and\nassumed zero external magnetic \feld B. This com-\npletes our scattering-theory derivation of the LLG\nequation and generalizes the result obtained in Ref.\n[2] on phenomenological grounds in several respects.\nEquation (24) applies also for \fnite-length magnets\nand chemical potentials both inside and outside the\nmagnetization-induced gap of the edge-state spec-\ntrum. Moreover, the identi\fcation of the _\u0012-term\nas a damping term necessitates the inclusion of the\nLangevin torque \u000eB. Indeed, Ref. [2] refers to the\nentire term involving eV\u0000~_\u0012as the spin-transfer\ntorque. In contrast, our derivation produces the\nterm involving eValready in zeroth order in mag-\nnetization \\velocity\" _M, while the _\u0012term appears\nonly to linear order. Thus, the latter term is re-\nally a conrtribution to damping and related to the\nenergy dissipated in the electron system due to the\ntime dependence of the magnetization.\n4. E\u000eciency of the nanomagnet as a motor\nWhile the electronic Hamiltonian for the edge\nmodes is equivalent to that of the Thouless mo-\ntor discussed in Ref. [5], the LLG equation for the\nmacrospin di\u000ber from the equation of motion of the\nmechanical degrees of freedom discussed in Ref. [5].\nIn this section, we discuss the energetics and the\ne\u000eciency of the magnetic Thouless motor against\nthe backdrop of its mechanical cousin.\nThe dynamics of the macrospin is easily ob-\ntained from the LLG equation (24) [2]. For a large\nanisotropy and thus small Mz, we need to retain\nthez-component of Monly in combination with the\nlarge anisotropy D. Then, the steady-state value of\nMzis \fxed by the \u0012-component of the LLG equa-\ntion,\nMz=\u0000_\u0012\nD: (25)\nThe precessional motion of Mabout thez-axis is\ngoverned by the z-component of the LLG equation,\nwhich yields\n_\u0012=eV\n~(26)and hence Mz=\u0000eV=(~D). It is interesting to\nnote that the angular frequency _\u0012of the preces-\nsion is just given by the applied bias voltage, in-\ndependent of the damping strength. This should\nbe contrasted with the mechanical Thouless motor.\nHere, the motor degree of freedom satis\fes a New-\nton equation of motion which is second order in\ntime. Thus, the frequency of revolution is inversely\nproportional to the damping coe\u000ecient.\nIn steady state, the magnetic Thouless motor bal-\nances the energy provided by the voltage source\nthrough the spin-transfer torque B0against the dis-\nsipation through Gilbert damping due to the intrin-\nsic coupling between magnetic moment and elec-\ntronic degrees of freedom. It is instructive to look at\nthese contributions independently. The work per-\nformed by the spin-transfer torque per precessional\nperiod is given by\n\u0001Wspin\u0000transfer =Z2\u0019=_\u0012\n0dtB0\u0001_M: (27)\nWriting this as an integral over a closed loop of\nthe magnetization Mand inserting the S-matrix\nexpression (6), we \fnd\n\u0001Wspin\u0000transfer =X\n\u000bZd\"\n2\u0019if\u000b\n\u0002I\ndM\u0001Tr\"\n\u0005\u000b^Sy\n0@^S0\n@M#\n: (28)\nWithout applied bias, the integrand is just the gra-\ndient of a scalar function and the integral vanishes.\nThus, we expand to linear order in the applied bias\nand obtain\n\u0001Wspin\u0000transfer =ieV\n4\u0019\n\u0002X\n\u000bI\ndM\u0001Tr\"\n(\u0005L\u0000\u0005R)^Sy\n0@^S0\n@M#\n:(29)\nComparing Eq. (29) with the familiar S-matrix ex-\npression for the pumped charge [23], the right-hand\nside can now be identi\fed as the bias voltage multi-\nplied by the charge pumped between the reservoirs\nduring one revolution of the magnetization,\n\u0001Wspin\u0000transfer =QpV: (30)\nWith every revolution of the magnetization, a\nchargeQpis pumped between the reservoirs. The\ncorresponding gain QpVin electrical energy is driv-\ning the magnetic Thouless motor. This result can\nalso be written as\n_Wspin\u0000transfer =QpV\n2\u0019_\u0012 (31)\n6for the power provided per unit time by the voltage\nsource.\nThe relation between spin-transfer torque and\npumped charge also allows us to identify the func-\ntion\u0018(\u0016) appearing in the LLG equation as the\ncharge in units of epumped between the reservoirs\nduring one precessional period of the macrospin,\nQp=e\u0018: (32)\nThis can be obtained either by deriving the pumped\ncharge explicitly from the S-matrix expression or by\nevaluating Eq. (27) using the explicit expression Eq.\n(20).\nThe electrical energy gain is compensated by the\nenergy dissipated through Gilbert damping. The\ndissipated energy per period is given by\n\u0001WGilbert =Z2\u0019=_\u0012\n0dt_MT\r_MT\n= 2\u0019M2\r\u0012\u0012_\u0012: (33)\nUsing Eq. (22), this yields the dissipated energy\n\u0001WGilbert =\u0018~_\u0012 (34)\nper precessional period or\n_WGilbert =\u0018~\n2\u0019_\u00122(35)\nper unit time. These expressions have a simple\ninterpretation. Due to the \fnite frequency of the\nmagnetization precession, each pumped charge ab-\nsorbs on average an energy ~_\u0012which is then dissi-\npated in the reservoirs.\nArmed with these results, we can \fnally discuss\nthe e\u000eciency of a magnetic Thouless motor and fol-\nlow the framework introduced in Ref. [40] to de\fne\nan appropriate \fgure of merit (analogous to the ZT\nvalue of thermoelectrics). Imagine the same setup\nas in Fig. 1, but with an additional load coupled\nto the magnetization. We can now de\fne the e\u000e-\nciency of the magnetic Thouless motor as the ratio\nof the power delivered to the load and the electri-\ncal powerIVprovided by the voltage source. In\nsteady state, the power delivered to the load has\nto balance against the power provided by the elec-\ntrons, i.e., Bel\u0001_M. Thus, we can write the e\u000eciency\nas\n\u0011=_W\nIV; (36)\n00.20.40.60.81ξ,ηmax0\n0.5 1 1.5 2 2.5 3 µ/Δ\n00.20.40.60.81ξ,ηmaxFigure 2: (Color online) The parameter \u0018(dashed lines) en-\ntering the coe\u000ecients of the LLG equation and the maximal\ne\u000eciency\u0011max(solid lines) of the motor for a \fxed voltage\nV. Upper and lower panels correspond to nanomagnets of\nlengthL=~v=\u0001 andL= 10 ~v=\u0001, respectively.\nwhere\n_W= _Wspin\u0000torque\u0000_WGilbert\n=\u0018\n2\u0019eV_\u0012\u0000\u0018~\n2\u0019_\u00122: (37)\nThe total charge current \rowing along the topolog-\nical insulator edge averaged over the cycle is the\nsum of the dccurrentGVdriven by the voltage,\nwhereGis the dcconductance of the device, and\nthe pumping current Qp_\u0012=(2\u0019),\nI=GV+e\u0018\n2\u0019_\u0012: (38)\nWe can now optimize the e\u000eciency of the motor\nat a given bias Vas function of the frequency _\u0012of\nthe motor revolution. Note that due to the load,\nthe latter is no longer tied to the bias voltage eV.\nThis problem is analogous to the problem of the\noptimal e\u000eciency of a thermoelectric device which\nleads to the de\fnition of the important ZT value.\nThis analogy was discussed explicitly in Ref. [40].\nApplying the results of this paper to the present\ndevice yields the maximal e\u000eciency\n\u0011max=p1 +\u0010\u00001p1 +\u0010+ 1; (39)\nwith a \fgure of merit \u0010analogous to the ZT value\nde\fned by\n\u0010=e2\u0018(\u0016)\nhG(\u0016); (40)\n7where\u0018(\u0016) is de\fned in Eq. (21) and the conduc-\ntance reads\nG(\u0016) =e2\nhj\u00162\u0000\u00012j2\nj\u00162\u0000\u00012jcos2\u001eL+\u00162sin2\u001eL(41)\nas obtained from the Landauer-B uttiker equation.\nAs in thermoelectrics, the maximum e\u000eciency is\nrealized for \u0010!1 which requires a \fnite pumped\ncharge at zero conductance. Unlike thermoelectrics,\nthe motor e\u000eciency is bounded by \u0011= 1 instead of\nthe Carnot e\u000eciency. This re\rects the fact that\nelectrical energy can be fully converted into mag-\nnetic energy. Speci\fcally, unit e\u000eciency is reached\nin the limit of a true Thouless motor with zero\ntransmission when the Fermi energy falls into the\ngap and nonzero and quantized pumped charge per\nperiod. This can be realized to a good approxi-\nmation for a su\u000eciently long magnet, as seen from\nthe lower panel in Fig. 2. For chemical potentials\noutside the gap, the conductance and the pumped\ncharge exhibit Fabry-Perot resonances. This yields\na distinct sequence of maxima and minima in the\ne\u000eciency. For shorter magnets, the conductance\nremains nonzero within the gap, leading to lower\ne\u000eciencies. This is shown in the upper panel of\nFig. 2. Moreover, the Fabry-Perot resonances are\nwashed out, so that there is only a feature at the gap\nedge where the conductance vanishes while \u0018!1=2\nfor arbitrary L.\n5. Conclusions\nImplementing directional motion of a mechani-\ncal or magnetic degree of freedom is a fundamental\nproblem of nanoscale systems. An attractive gen-\neral mechanism relies on running quantum pumps\nin reverse. This is the underlying principle of adia-\nbatic quantum motors which drive periodic motion\nof a classical motor degree of freedom by applying a\ntransport current. In this paper, we emphasize that\na magnetic island coupled to a quantum spin Hall\nedge, recently discussed by Meng et al. [2], is just\nsuch an adiabatic quantum motor. We derive the\nLandau-Lifshitz-Gilbert equation for the magneti-\nzation dynamics from a general scattering-theory\napproach to adiabatic quantum motors, providing\na microscopic derivation of spin-transfer torque,\nGilbert damping, and Langevin torque. This ap-\nproach does not only provide a detailed microscopic\nunderstanding of the operation of the device but\nalso allows one to discuss its e\u000eciency. We \fndthat the device naturally approaches optimal e\u000e-\nciency when the chemical potential falls into the\nmagnetization-induced gap and the conductance is\nexponentially suppressed. This makes this system\na Thouless motor and possibly its most experimen-\ntally feasible variant to date.\nSeveral issues are left for future work. While we\nderived microscopic expressions for the Langevin\ntorque, we have not explored its consequences for\nthe motor dynamics. It should also be interesting\nto consider thermal analogs driven by a tempera-\nture gradient instead of a bias voltage. Inducing the\nmagnetization precession by a temperature gradient\nwould realize a quantum heat engine. Conversely,\nforcing a magnetic precession can be used to pump\nheat against a temperature gradient. Setups with\nseveral magnetic islands could be engineered to ef-\nfect exchange of charge and energy without employ-\ning a dcbattery. These devices have been explored\nin the literature on quantum pumps [41, 42, 43] and\ntheir e\u000eciencies could be analyzed in the thermo-\nelectric framework of Ref. [40].\nAcknowledgement\nWe thank Gil Refael and Ari Turner for dis-\ncussions. This work was supported by CON-\nICET, MINCyT and UBACyT (L.A.) as well\nas the Deutsche Forschungsgemeinschaft and the\nHelmholtz Virtual Institute New States of Matter\nand Their Excitations (F.v.O.). L.A. thanks the\nICTP Trieste for hospitality and the Simons Foun-\ndation for support. F.v.O. thanks the KITP Santa\nBarbara for hospitality during the \fnal preparation\nof this manuscript. This research was supported\nin part by the National Science Foundation under\nGrant No. NSF PHY11-25915.\nReferences\n[1] Qi, X-L., Hughes, T.L., and Zhang, S.-C., 2008, Frac-\ntional charge and quantized current in the quantum spin\nHall state, Nature Phys. 4, 273 - 276 (2008).\n[2] Meng, Q., Vishveshwara, S., and Hughes, T.L., 2014,\nSpin-transfer torque and electric current in helical edge\nstates in quantum spin Hall devices, Phys. Rev. B 90,\n205403.\n[3] Qi X.-L. and Zhang S.-C., 2009, Field-induced gap and\nquantized charge pumping in a nanoscale helical wire,\nPhys. Rev. B 79, 235442.\n[4] Thouless, D.J., 1983, Quantization of particle trans-\nport, Phys. Rev. B 27, 6083-6087.\n[5] Bustos-Marun, R., Refael, G. and von Oppen, F., 2013,\nAdiabatic Quantum Motors, Phys. Rev. Lett. 111,\n060802.\n8[6] Bode, N., Viola Kusminskiy, S., Egger, R., and von\nOppen, F., 2011, Scattering Theory of Current-Induced\nForces in Mesoscopic Systems, Phys. Rev. Lett. 107,\n036804.\n[7] Bode, N., Viola Kusminskiy, S., Egger, R., and von\nOppen, F., 2012, Scattering Theory of Current-Induced\nForces in Mesoscopic Systems, Beilstein J. Nanotechnol.\n3, 144.\n[8] Thomas, M., Karzig, T., Viola Kusminskiy, S., Zarand,\nG., von Oppen, F., 2012, Scattering theory of adiabatic\nreaction forces due to out-of-equilibrium quantum en-\nvironments, Phys. Rev. B 86, 195419.\n[9] Bode, N., Arrachea, L., Lozano, G.S., Nunner, T.S.,\nand von Oppen, F., 2012, Current-induced switching\nin transport through anisotropic magnetic molecules,\nPhys. Rev. B 85, 115440.\n[10] B uttiker, M., 1986, Four-Terminal Phase-Coherent\nConductance, Phys. Rev. Lett. 57, 1761\n[11] B uttiker, M., 1990, Scattering theory of thermal and\nexcess noise in open conductors, Phys. Rev. Lett. 65,\n2901-2904.\n[12] B uttiker, M., Imry Y., Landauer, R., and Pinhas, S.,\n1985, Generalized many-channel conductance formula\nwith application to small rings, Phys. Rev. B 31, 6207-\n6215.\n[13] B uttiker, M., 1992, Scattering theory of current and\nintensity noise correlations in conductors and wave\nguides, Phys. Rev. B 46, 12485-12507.\n[14] B uttiker, M. and Blanter, Y., 2000, Shot Noise in Meso-\nscopic Conductors, Phys. Reports 336, 1 - 166.\n[15] B uttiker, M., 1988, Absence of backscattering in the\nquantum Hall e\u000bect in multiprobe conductors, Phys.\nRev. B 38, 9375-9389.\n[16] Delplace, P., Li, J., and B uttiker, M., 2012, Magnetic-\nField-Induced Localization in 2D Topological Insula-\ntors, Phys. Rev. Lett. 109, 246803.\n[17] Pastawski H.M., 1992, Classical and quantum transport\nfrom generalized Landauer-B ttiker equations II. Time-\ndependent resonant tunneling, Phys. Rev. B 46, 4053-\n4070.\n[18] Jauho, A-P., Wingreen, N.S., and Meir, Y., 1994, Time-\ndependent transport in interacting and noninteracting\nresonant-tunneling systems, Phys. Rev. B 50, 5528 -\n5544.\n[19] Arrachea, L., 2005, Green-function approach to trans-\nport phenomena in quantum pumps, Phys. Rev. B 72,\n125349.\n[20] Arrachea, L. and Moskalets, M., 2006, Relation between\nscattering-matrix and Keldysh formalisms for quantum\ntransport driven by time-periodic \felds, Phys. Rev. B\n74, 245322.\n[21] Kohler, S., Lehmann, J., and H anggi, P., 2005, Driven\nquantum transport on the nanoscale, Phys. Rep. 406,\n379-443.\n[22] B uttiker, M., Thomas, H., and Pr^ etre, A., 1994, Cur-\nrent partition in multiprobe conductors in the presence\nof slowly oscillating external potentials, Z. Physik B\nCondensed Matter 94, 133-137.\n[23] Brouwer, P.W., 1998, Scattering approach to paramet-\nric pumping, Phys. Rev. B 58, 10135-10138.\n[24] Avron, J.E., Elgart, A., Graf, G.M., and Sadum, L.,\n2001, Optimal Quantum Pumps, Phys. Rev. B 87,\n236601.\n[25] Vavilov, M.G., Ambegoakar, V., and Aleiner, I.L., 2001,\nCharge pumping and photovoltaic e\u000bect in open quan-tum dots, Phys. Rev. B 63, 195313.\n[26] Moskalets, M. and B uttiker, M., 2004, Adiabatic quan-\ntum pump in the presence of external acvoltages Phys.\nRev. B 69, 205316.\n[27] Switkes, M., Marcus, C.M., Campman, M.K., and\nGossard, A.C., 1999, An Adiabatic Quantum Electron\nPump, Science 283, 1905 -1909.\n[28] Leek, P.J., Buitelaar, M.R., Talyanskii, V.I., Smith,\nC.G., Anderson, D., Jones, G.A.C., Wei, J., and Cob-\nden, D.H., 2005, Charge Pumping in Carbon Nan-\notubes, Phys. Rev. Lett. 95, 256802.\n[29] Geerligs, L.J., Anderegg, V.F., Holweg, P.A.M., Mooij,\nJ.E., Pothier, H., Esteve, D., Urbina, C., and Devoret,\nM.H., 1990, Frequency-locked turnstile device for single\nelectrons, Phys. Rev. Lett. 64, 2691 - 2694.\n[30] DiCarlo, L., Marcus, C.M., and Harris, J.S. Jr., 2003,\nPhotocurrent, Recti\fcation, and Magnetic Field Sym-\nmetry of Induced Current through Quantum Dots,\nPhys. Rev. Lett. 91, 246804.\n[31] Blumenthal, M.D., Kaestner, B., Li, L., Giblin, S.,\nHanssen, T.J.B.M., Pepper, M., Anderson, D., Jones,\nG., and Ritchie, D.A., 2007, Gigahertz quantized charge\npumping, Nature Phys. 3, 343 -347.\n[32] Moskalets. M. and B uttiker, M., 2002, Floquet scatter-\ning theory of quantum pumps, Phys. Rev. B 66, 205320.\n[33] Moskalets, M. and Haack G., 2015, Single-electron\ncoherence: \fnite temperature versus pure dephasing,\narXiv:1506.09028.\n[34] Fernandez-Alcazar, L.J., Bustos-Marun, R.A.,\nPastawski, H.M., 2015, Decoherence in current\ninduced forces: Application to adiabatic quantum\nmotors, Phys. Rev. B 92, 075406.\n[35] Pistolesi, F., Blanter, Y.M., and Martin, I., 2008, Self-\nconsistent theory of molecular switching, Phys. Rev. B\n78.\n[36] Tserkovnyak, Y., Brataas, A., and Bauer, G.E.W.,\n2002, Enhanced Gilbert Damping in Thin Ferromag-\nnetic Films, Phys. Rev. Lett. 88, 117601.\n[37] Brataas, A., Tserkovnyak, Y., and Bauer, G.E.W.,\n2008, Scattering Theory of Gilbert Damping, Phys.\nRev. Lett. 101, 037207.\n[38] Brataas, A., Tserkovnyak, Y., and Bauer, G.E.W.,\n2011, Magnetization dissipation in ferromagnets from\nscattering theory, Phys. Rev. B 84, 054416.\n[39] Hals, K.M.D., Brataas, A., Tserkovnyak, Y., 2010,\nScattering theory of charge-current-induced magnetiza-\ntion dynamics, Europhys Lett. 90, 47002.\n[40] Ludovico, M.F., Battista, F., von Oppen, F., Ar-\nrachea, L., 2015, Adiabatic response and quan-\ntum thermoelectrics for acdriven quantum systems,\narXiv:1506.08617.\n[41] Arrachea, L., Moskalets, M. and Martin-Moreno, L.,\n2007, Heat production and energy balance in nanoscale\nengines driven by time-dependent \felds, Phys. Rev. B\n75, 245420.\n[42] Juergens, S., Haupt, F., Moskalets, M., and\nSplettstoesser, J., 2013, Thermoelectric performance of\na driven double quantum dot, Phys. Rev. B 87, 245423.\n[43] Moskalets, M. and B uttiker, M., 2009, Heat production\nand current noise for single- and double-cavity quantum\ncapacitors, Phys. Rev. B 80, 081302.\n9"    },    {        "title": "1507.06748v1.Boosting_Domain_Wall_Propagation_by_Notches.pdf",        "content": "arXiv:1507.06748v1  [cond-mat.mes-hall]  24 Jul 2015Boosting Domain Wall Propagation by Notches\nH. Y. Yuan and X. R. Wang1,2,∗\n1Physics Department, The Hong Kong University of Science and Technology, Clear Water Bay, Kowloon, Hong Kong\n2HKUST Shenzhen Research Institute, Shenzhen 518057, China\nWereportacounter-intuitivefindingthatnotchesinanothe rwise homogeneousmagnetic nanowire\ncan boost current-induced domain wall (DW) propagation. DW motion in notch-modulated wires\ncan be classified into three phases: 1) A DW is pinned around a n otch when the current density\nis below the depinning current density. 2) DW propagation ve locity is boosted by notches above\nthe depinning current density and when non-adiabatic spin- transfer torque strength βis smaller\nthan the Gilbert damping constant α. The boost can be manyfold. 3) DW propagation velocity is\nhindered when β > α. The results are explained by using the Thiele equation.\nPACS numbers: 75.60.Ch, 75.78.-n, 85.70.Ay, 85.70.Kh\nI. INTRODUCTION\nMagnetic domain wall (DW) motion along a nanowire\nunderpins many proposals of spintronic devices1,2. High\nDW propagation velocity is obviously important because\nit determines the device speed. In current-driven DW\npropagation,many efforts havebeen devoted to high DW\nvelocity and low current density in order to optimize de-\nvice performance. The issue of whether notches can en-\nhance current-induced DW propagation is investigated\nhere.\nTraditionally, notches are used to locate DW\npositions1–4. Common wisdom expects notches to\nstrengthen DW pinning and to hinder DW motion. In-\ndeed, in the field-driven DW propagation, intentionally\ncreated roughness slows down DW propagation although\nthey can increase the Walker breakdown field5. Unlike\nthe energy-dissipation mechanism of field-induced DW\nmotion6, spin-transfer torque (STT)7–10is the driven\nforce behind the current-driven DW motion. The torque\nconsists of an adiabatic STT and a much smaller non-\nadiabatic STT9,10. In the absence of the non-adiabatic\nSTT, there exists an intrinsic pinning even in a homoge-\nneous wire, below which a sustainable DW motion is not\npossible11,12. Interestingly, there are indications13that\nthe depinningcurrentdensityofaDWtrappedinanotch\nis smaller than the intrinsic threshold current density in\nthe absence of the non-adiabatic STT. Although there is\nno intrinsic pinning1,10in the presence ofa non-adiabatic\nSTT, It is interesting to ask whether notches can boost\nDW propagation in the presence of both adiabatic STT\nand non-adiabatic STT.\nIn this paper, we numerically study how DW propa-\ngates along notch-modulated nanowires. Three phases\nare identified: pinning phase when current density is be-\nlow depinning current density ud; boosting phase and\nhindering phase when the current density is above ud\nandthe non-adiabaticSTT strength βissmallerorlarger\nthan the Gilbert damping constant α, respectively. The\naverage DW velocity in boosting and hindering phases\nis respectively higher and lower than that in the wire\nwithout notches. It is found that DW depinning is facili-tated by antivortex nucleation. In the case of β < α, the\nantivortexgenerationis responsiblefor velocityboost be-\ncause vortices move faster than transverse walls. In the\nother case of β > α, the longitudinal velocity of a vor-\ntex/antivortex is slower than that of a transverse wall in\nahomogeneouswallandnotcheshinderDWpropagation.\nII. MODEL AND METHOD\nWe consider sufficient long wires (with at least 8\nnotches) of various thickness and width. It is well known\nthat14narrowwiresfavoronlytransversewallswhilewide\nwires prefer vortex walls. Transverse walls are the main\nsubjects of this study. A series of identical triangular\nnotches of depth dand width ware placed evenly and\nalternately on the two sides of the nanowires as shown in\nFig. 1a with a typical clockwise transverse wall pinned\nat the center of the first notch. The x−,y−andz−axis\nare along length, width, and thickness directions, respec-\ntively. The magnetization dynamics of the wire is gov-\nerned by the Landau-Lifshitz-Gilbert (LLG) equation\n∂m\n∂t=−γm×Heff+αm×∂m\n∂t−(u·∇)m+βm×(u·∇)m,\nwherem,γ,Heff, andαare respectively the unit vec-\ntor of local magnetization, the gyromagnetic ratio, the\neffective field including exchange and anisotropy fields,\nand the Gilbert damping constant. The third and fourth\nterms on the right hand side are the adiabatic STT and\nnon-adiabatic STT10. The vector uis along the electron\nflow direction and its magnitude is u=jPµB/(eMs),\nwherej,P,µB,e, andMsare current density, current\npolarization,the Bohrmagneton, the electronchargeand\nthe saturation magnetization, respectively. For permal-\nloy ofMs= 8×105A/m,u= 100 m/s corresponds\ntoj= 1.4×1012A/m2. In this study, uis lim-\nited to be smaller than both 850 m/s (corresponding to\nj≃1.2×1013A/m2!) and the Walkerbreakdowncurrent\ndensity because current density above the values gener-\natesintensivespinwavesaroundDWsandnotches,which\nmakes DW motion too complicated to be even described.2\nxy\nz\n(b) (a) \nL\nw\ndj\nFIG. 1. (color online) (a) A notch-modulated nanowire. L\nis the separation between two adjacent notches. The color\ncodes the y−component of mwith red for my= 1, blue for\nmy=−1 and green for my= 0. The white arrows denote\nmagnetization direction. (b) The phase diagram in β−u\nplane. A is the pinning phase; B is the boosting phase; and C\nis the hindering phase. Vortices are (are not) generated nea r\nnotches by a propagating DW in C1 (C2). Inset: The notch\ndepth dependence of depinning current udwhen notch width\nis fixed at w= 48 nm.\nDimensionless quantity βmeasures the strength of non-\nadiabatic STT and whether βis larger or smaller than α\nis still in debate10,15,16. The LLGequation isnumerically\nsolved by both OOMMF17andMUMAX18packages19. The\nelectric current density is modulated according to wire\ncross section area while the possible change of current\ndirection around notch is neglected. The material pa-\nrameters are chosen to mimic permalloy with exchange\nstiffness A= 1.3×10−11J/m,α= 0.02 andβvarying\nfrom 0.002 to 0.04. The mesh size is 4 ×4×4 nm3.\nIII. RESULTS\nA. Transverse walls in wide wires: boosting and\nhindering\nThis is the focus of this work. Our simulations on\nwires of 4 nm thick and width ranging from 32 nm to\n128 nm and notches of d= 16 nm and wvarying from\n16 nm to 128 nm show similar behaviors. Domain walls\nin these wires are transverse. Results presented below\nare on a wire of 64 nm wide and notches of w= 48 nm.\nThree phases can be identified. A DW is pinned at a\nnotch when uis below a depinning current density ud.\nThis pinning phase is denoted as A (green region) in Fig.\n1b. Surprisingly, udincreases slightly with β, indicatingthat the β-term actually hinders DW depinning out of\na notch although it is responsible for the absence of the\nintrinsic pinning in a uniform wire (see discussion below\nfor possible cause). When uis above ud, a DW starts to\npropagate and it can either be faster or slower than the\nDW velocity in the corresponding uniform wire, depend-\ning on relative values of βandα.\nWhenβ < α, DW velocity is boosted through antivor-\ntexgenerationat notches. Thisphaseisdenoted asphase\nB. When β > α, the boosting of DW propagation is sup-\npressed no matter vortices are generated (phase C1) or\nnot (phase C2). The upper bound of the phase plane\nis determined by the Walker breakdown current density\nandu= 850 m/s. If the current density is larger than\nthe upper bound, spin waves emission from DW20and\nnotches are so strong that new DWs may be created.\nAlso, the Walker breakdown is smaller than the depin-\nning value udforβ >0.04. Thus the phase plane in Fig.\n1b is bounded by β= 0.04. Although the general phase\ndiagram does not change, the phase boundaries depend\non the wire and notch specificities. The inset is notch\ndepth dependence of the depinning current when w= 48\nnm andβ= 0.0121.\nBoosting phase: The boost of DW propagation for β < α\ncan be clearly seen in Fig. 2. Figure 2a is the average\nDW velocity ¯ vas a function of notch separation Lfor\nu= 600 m/s > ud. ¯vis maximal around an optimal\nnotch separation Lp, which is close to the longitudinal\ndistance that an antivortex travels in its lifetime. Lp\nincreases with βand it is respectively about 1.5 µm, 2\nµm, and 4 µm forβ= 0.005 (squares), 0.01 (circles) and\n0.015 (up-triangles). This result suggests that the an-\ntivortex generation and vortex dynamics are responsible\nfor the DW propagation boost. Filled symbols in Fig. 2b\nare ¯vfor various current density when Lpis used. For a\ncomparison, DW velocities in the corresponding homoge-\nneous wires are also plotted as open symbols which agree\nperfectlywith ¯ v=βu/αdiscussedbelow. Take β= 0.005\nas an example, ¯ vis zero below ud= 550 m/s and jumps\nto an average velocity ¯ v≃550 m/s at ud, which is about\nfour times of the DW velocity in the homogeneous wire.\nAs the current density further increases, the average ve-\nlocityalsoincreasesandisapproximatelyequalto u. The\ninset of Fig. 2b shows the instantaneous DW velocity for\nβ= 0.005 and u= 600 m/s. Blue dots denote the mo-\nments at which the DW is at notches. Right after the\ncurrent is turned on at t= 0 ns, the instantaneous DW\nvelocity is very low until an antivortex of winding num-\nberq=−123,24is generatednear the notch edge at 0.5ns\n(see discussion and Fig. 9 below). The motion of the an-\ntivortex core drags the whole DW to propagate forward\nat a velocity around 600 m/s. The antivortex core anni-\nhilatesitselfatthebottomedgeofthewireaftertraveling\nabout 1.5 µm and the initial transverse wall reverses its\nchirality at the same time24. Surprisingly, the reversal of\nDW chiralityleads to a significantincreasesofDW veloc-\nity as shown by the peaks of the instantaneous velocity\nat about 2.0ns in the inset. Another antivortex of wind-3\n(a) \n(b) \nFIG. 2. (color online) (a) L−dependence of average DW ve-\nlocity ¯vforu= 600 m/s, α= 0.02, andβ= 0.005 (squares),\n0.01 (circles), 0.015 (up-triangles). The dash lines are βu/α.\n(b)u−dependence of ¯ vforβ= 0.005 (squares), 0.01 (circles),\n0.015 (up-triangles). Open symbols are DW velocity in the\ncorresponding homogeneous wires. Straight lines are βu/α. ¯v\nis above βu/αwhenu > u d. Inset: instantaneous DW speed\nforu= 600 m/s, β= 0.005, and L= 1.5µm. The blue dots\nindicate the moments when the DW is at notches.\ning number q=−1 is generated at the second notch and\nDW propagation speeds up again. Once the antivortex\ncore forms, it pulls the DW out of notch. This process\nthen repeats itself and the DW propagates at an average\nlongitudinal velocity of about 600 m/s. A supplemental\nmovie corresponding to the inset is attached25.\nHindering phase: Things are quite different for β > α.\nFigure 3a shows that ¯ vincreases monotonically with L\nforβ= 0.025, 0.03 and 0.035, which are all larger than\nα. In order to make a fair comparison with the results of\nβ < α, Fig. 3b is the current density dependence of ¯ vfor\nL= 2µm andβ= 0.025 (filled squares), 0.03 (filled cir-\ncles)and0.035(filledup-triangles). Again,DWvelocities\nin the corresponding homogeneouswires are presented as\nopen symbols. Take β= 0.025 as an example, although\nthe average velocity jumps at the depinning current den-\nsity 565 m/s, it’s still well below the DW velocity in the\ncorresponding uniform wire. The inset of Fig. 3b shows(b) (a) \nFIG. 3. (color online) (a) L−dependence of ¯ vforu= 600\nm/s and β= 0.025 (squares), 0.03 (circles), and 0.035 (up-\ntriangles), all larger than α= 0.02. The dash lines are βu/α.\n(b)u−dependence of ¯ vforL= 2µm. Fill symbols (squares\nforβ= 0.025, circles for β= 0.03, and up-triangles for\nβ= 0.035) are numerical data in notched wire of w= 48\nnm andd= 16 nm. Open circles are DW velocity of the cor-\nrespondinghomogeneous wire. Straight lines are βu/α. Inset:\ninstantaneous DW velocity for u= 600 m/s and β= 0.025.\nThebluedotsdenotethemomentswhentheDWisatnotches.\nthe instantaneous DW velocity for u= 600 m/s. An\nantivortex is generated at the first notch. In contrast\nto the case of β < α, the antivortex slows down DW\npropagation velocity below the value in the correspond-\ning uniform wire. Moreover, the transverse wall keeps its\noriginal chirality unchanged when the antivortex is anni-\nhilated at wire edge, and no vortex/antivortex is gener-\nated at the second notch. However, another antivortex\nis generated at the third notch. This is the typical cycle\nof phase C1. As uincreases above 640 m/s, phase C1\ndisappears and the DW passes all the notches without\ngenerating any vortices. This motion is termed as phase\nC2. For β >0.025, only phase C2 is observed. In C2,\nDW profile is not altered, and the average DW velocity\nis slightly below that in a uniform wire.4\n(b) (a) \nFIG. 4. (color online) (a) u−dependence of ¯ vforβ=\n0.01 (filled circles) and 0.015 (filled up-triangles). (b)\nu−dependence of ¯ vforβ= 0.03 (filled squares) and 0.035\n(filled up-triangles). Open symbols are DW velocity in the\ncorresponding homogeneous wires. Straight lines are βu/α. ¯v\nis below βu/αwhenu > u d. The nanowire is 8 nm wide and\n1 nm thick while the notch size is 10 nm wide and 2 nm deep\nfor (a) and 50 nm wide and 2 nm deep for (b). The separation\nof adjacent notches is 100 nm.\nB. Transverse walls in very narrow wires\nOne interesting question is whether notches can boost\nDW propagation in very narrow wires such that the nu-\ncleation of a vortex/antivortex is highly unfavorable. To\naddress this issue, Fig. 4a are u−dependence of the av-\nerage DW velocity on a 8 nm wide wire for β < α(circles\nforβ= 0.01 and up-triangles for β= 0.015) with (filled\nsymbols) and without (open symbols) notches. When\nnotches are placed, notch depth is 2 nm, L= 100 nm,\nw= 10 nm. DW velocity in the corresponding ho-\nmogeneous wire (open symbols) follows perfectly with\n¯v=βu/α(straight lines). It is clear that averaged DW\nvelocity in the notched wire (filled symbols) is below the\nvalues of the DW velocity in the corresponding homo-\ngeneous wire. Take β= 0.015 as an example, ¯ vis zero\nbelowud= 310 m/s and jumps to an average velocity\n¯v≃168 m/s at ud, which is below the DW velocity in\nthe corresponding uniform wire.\nThings are similar for β > α. Figure 4b is the cur-\nrent density dependence of ¯ vforβ= 0.03 (filled squares)\nand 0.035 (filled up-triangles). Again, DW velocities in\nhomogeneous wire are presented as open symbols for a\ncomparison. The averaged DW velocity in the notched\nwire (filled symbols) is below the values of the DW ve-\nlocity in the corresponding homogeneous wire.\nC. Vortex walls in very wide wires\nAlthough our main focus is on transverse walls, it\nshould be interesting to ask whether DW propagation\nboost can occur for vortex walls. It is well-known that\na vortex/antivortex wall is more stable for a much wider\nwire in the absence of a field and a current14. One\nmay expect that DW propagation boost would not oc-\ncur in such a wire because the boost comes from vor-\ntex/antivortex generation near notches and a such vor-\ntex/antivortex exists already in a wider wire even in the\nmy\n+1 -1 0200 nm 1.5 ns 13.5 ns \n18.0 ns 26.0 ns 0 ns \n47.5 ns 14.5 ns 0 ns (c) \n(d) (a) (b) \nFIG. 5. (color online) (a) u−dependence of ¯ vforβ=\n0.01 (filled circles) and 0.015 (filled up-triangles). (b)\nu−dependence of ¯ vforβ= 0.025 (filled squares) and 0.03\n(filled circles). Open symbols are DW velocity in the corre-\nsponding homogeneous wires. Straight lines are βu/α. ¯vis\nabove (below) βu/αwhenu > u dandβ < α(β > α). The\nnanowire is 520 nm wide and 10 nm thick while the rectan-\ngular notch is 160 nm wide and 60 nm deep. The separation\nof adjacent notches is 8 µm. (c) and (d) The spin configu-\nrations in a uniform wire (a) and in a notched wire (b) at\nvarious moments for β= 0.01 andu= 650 m/s. The time\nis indicated on the bottom-right corner of each configuratio n.\nThe color codes the value of myand color bar is shown in the\nbottom-right corner.\nabsence of a current. However, DW propagation boost\nwas still observed as shown in Fig. 5 for a wire of 520\nnm wide and 10 nm thick. Rectangular notches of 60\nnm deep and 160 nm wide are separated by L= 8µm.\nWhenβ < α(Fig. 5a: circles for β= 0.01 and up-\ntriangles for β= 0.015), the average DW propagation\nvelocities in the notched wire (filled symbols) is higher\nthan the DW velocity in the corresponding homogeneous\nwire (open symbols) when u > ud. Figure 5b shows that\nthe average DW propagation velocities in a notched wire\n(filled symbols) is lower than that in the corresponding\nhomogeneous wire (open symbols) for β > α(squares for\nβ= 0.025 and circles for β= 0.03). Figure 5c shows the\nspin configurations of the DW in the homogeneous wire\nofβ= 0.01before a current is applied (the left configura-\ntion) and during the current-driven propagation (middle\nand right configurations). When a current u= 650 m/s\nis applied at 0 ns, a vortex wall moves downward. The\nvortex was annihilated at wire edge, and the vortex wall\ntransformintoatransversewall. TheDWkeepsitstrans-\nverse wall profile and propagates with velocity of βu/α\n(solid lines in Fig. 5a and 5b). The middle and right\nconfigurations are two snapshots at 14.5 ns and 47.5 ns.\nTime is indicated in the bottom-right corner. Figure 5d\nare snapshots of DW spin configurations in the notched\nwire ofβ= 0.01 when a current u= 650 m/s is applied5\nu\n    m\nxxy\nzu\n    m\nx(a) (b)\nFIG. 6. (color online) Directions of vortex core magnetizat ion\n(red symbols) and non-adiabatic torque (blue symbols) for a\nclockwise transverse wall (a) and a counterclockwise trans -\nverse wall (b). The dots (crosses) represent ±z-direction.\natt= 0 ns. At t= 0 ns, a vortex wall is pinned near\nthe first notch. Right after the current is turned on, the\nvortex wall starts to depin and complicated structures\nmay appear during the depinning process as shown by\nthe snapshot at t= 1.5 ns. At t= 13.5 ns, the DW\ntransforms to a transverse wall and propagates forward.\nWhen the transverse wall reaches the second notch at\naboutt= 18.0 ns, new vortex core nucleates near the\nnotch and drags the whole DW to propagate forward. In\ncontrast to the case of homogeneous wire where a prop-\nagating DW prefers a transverse wall profile, DW with\nmore than one vortices can appear as shown by the snap-\nshot att= 26.0 ns. The vortex core in this structure\nboosts DW velocity above the average DW velocity of a\nuniform wire. This finding may also explain a surprising\nobservation in an early experiment4that depinning cur-\nrent does not depend on DW types. A vortex wall under\na current transforms into a transverse wall before depin-\nning from a notch. Thus both vortex wall and transverse\nwall have the same depinning current.\nIV. DISCUSSION\nA. Depinning process analysis\nEmpirically, we found that vortex/antivortex polarity\nis uniquely determined by the types of transverse wall\nand current direction. This result is based on more than\ntwenty simulations that we have done by varying var-\nious parameters like notch geometry, wire width, mag-\nnetic anisotropy, damping etc. Within the picture that\nDW depinning starts from vortex/antivortex nucleation,\ntheβ−dependence of depinning current density udcan\nbe understood as follows. For a clockwise (counter-\nclockwise) transverse wall and current in −xdirection,\np= +1 (p=−1), as shown in Fig. 6. If one as-\nsumes that vortex/antivortex formation starts from the\nvortex/antivortex core, it means that the core spin ro-\ntates into + z-direction for a vortex of p= 1. For a clock-\nwise wall, β-torque ( βm×∂m\n∂x) tends to rotate core spin\nin−z-direction, as shown in Fig. 6a, so the presence of\na smallβ-torque tries to prevent the nucleation of vor-\ntices. Thus, the larger βis, the higher udwill be. This\nmay be the reason why the depinning current density ud\nincreases as βincreases.(a) (b)\nFIG. 7. (a) Depinning current density as a function of an\nexternal field. A 0.4 ns field pulse in the x-direction is turned\non simultaneously with the current. The shape of a pulse of\nH= 100 Oe is shown in the inset. Since the depinning field\nof the wire (64 nm wide and 4 nm thick) is 150 Oe, the field\namplitude is limited to slightly below 150 Oe in the curve. (b )\nDepinningcurrentdensityas afunctionof nanowire thickne ss.\nOur simulations suggest that DW depinning starts\nfrom vortex/antivortexnucleation. Adiabatic spin trans-\nfertorquetendstorotatethespinsattheedgedefectnear\na notch out of plane and to form a vortex/antivortex\ncore. Thus, any mechanisms that help (hinder) the\ncreation of a vortex/antivortex core shall decrease (in-\ncrease) the depinning current density ud. To test this\nhypothesis, we use a magnetic field pulse of 0.4 ns along\n±x−direction (shown in the inset of Fig. 7a) such that\nthe field torque rotates spins out of plane. Figure 7a\nis the numerical results of the magnetic field depen-\ndence of the depinning current density for a 64 nm wide\nwire with triangular notches of 48 nm wide and 16 nm\ndeep. The non-adiabatic coefficient is β= 0.01. As\nexpected, uddecreases (increases) with field when it is\nalong -x−direction (+ x−direction) so that spins rotate\ninto +z−direction (- z−direction). All other parameters\nare the same as those for Fig. 2.\nIf the picture is correct, one should also expect the\ndepinning current density depends on the wire thick-\nness. The shape anisotropy impedes vortex core for-\nmation because it does not favor a spin aligning in the\nz−direction. The shape anisotropy decreases as the\nthickness increases. Thus, one should expect the depin-\nning current density decreases with the increase of wire\nthickness. Indeed, numerical results shown in Fig. 7b\nverifiestheconjecture. All otherparametersarethe same\nas that in Fig. 7a ( H= 0).\nB. Width effects on the depinning current density\nThe DW propagating boost shown above is from the\nwire in which the notch depth (16 nm) is relatively big in\ncomparisonwith wire width (64 nm). Naturally, one may\nask whether the DW propagation boost exists also in a\nwire when the notch depth is much smaller than the wire\nwidth. To address the issue, we fix the notch geometry\nand vary the wire width. Figure 8 is the nanowire width\ndependence of depinning current density when the notch6\n(a) (b) \n(c) (d) 50 nm 50 nm \nFIG. 8. (color online) (a) and (b) are nanowire width de-\npendence of depinning current density for β= 0.005 (a) and\nβ= 0.01 (b), respectively. The wire thickness is 4 nm and\nnotch size is fixed at 48 ×16 nm2. (c) and (d) are the real\nconfigurations of initial domain walls pinned at the notch fo r\n64 nm and 160 nm wide wires, respectively. The color coding\nis the same as that of Fig. 5. The blue jagged lines indicate\nthe profiles of triangular notches.\nsize is fixed at 48 ×16 nm2. Figures 8a and 8b show the\nphase boundary between vortex-assisted boosting phase\nand the pinning phase. DW propagation boost exists\nwhen nanowire width is one order of magnitude larger\nthan the notch depth. The top view of the wire and spin\nconfigurations for 64 nm wide and 160 nm wide wires are\nshown in Fig. 8c and Fig. 8d, respectively.\nC. DW Propagation and vortex dynamics\nDW propagation boost and slow-down by vortices can\nbe understood from the Thiele equation10,26,27,\nF+G×(v−u)+D·(αv−βu) = 0,(1)\nwhereFis the external force related to magnetic field\nthat is zero in our case, Gis gyrovector that is zero for a\ntransverse wall and G=−2πqplM s/γˆ zfor a 2D vortex\nwall, where qis the winding number (+1 for a vortex and\n-1foranantivortex), pisvortexpolarity( ±1forcorespin\nin±zdirection) and lis the thickness of the nanowire. D\nis dissipation dyadic, whose none zero elements for a vor-\ntex/antivortex wall are Dxx=Dyy=−2MsWl/(γ∆)27,\nwhereWis nanowire width and ∆ is the Thiele DW\nwidth26.vis the DW velocity.\nFor a transverse wall, v=βu/α(solid lines) agrees\nperfectly with numerical results (open symbols) in ho-\nmogeneous wires as shown in Figs. 2b and 3b without\nany fitting parameters. For a vortex wall, the DW veloc-\nity is\nvy=1\n1+α2W2/(π2∆2)W\nπqp∆(α−β)u,(2)\nvx=u\n1+α2W2/(π2∆2)/parenleftbigg\n1−β\nα/parenrightbigg\n+βu\nα.(3)vydepends on DW width, αas well as β/α. For a given\nvortexwall, vyhas opposite sign for β < αandβ > α. In\nterms of topological classification of defects23, the edge\ndefect of the transverse DW at the first notch (Fig. 1a)\nhas winding number q=−1/2, and this edge defect can\nonlygivebirthtoanantivortexof q=−1andp= 1while\nitself changes to an edge defect of q= 1/2 as shown in\nFig. 9a. Empirically, we found that antivortexpolarityis\nuniquely determined by the types of transverse wall and\ncurrent direction. A movie visualizing the DW propa-\ngation in boosting phase is shown in the Supplemental\nMovie25. All the parameters are the same as the inset of\nFig. 2b. The three segments of identical length 1200 nm\nare connected in series to form a long wire. When β < α,\nthe antivortex moves downward ( vy<0) to the lower\nedge defect of winding number of q= 1/2. The lower\nedge defect changes its winding number to q=−1/2\nand the transverse DW reverses its chirality24when the\nvortex merges with the edge defect. Then another an-\ntivortex of winding number q=−1 andp=−1 is gen-\nerated at the second notch on the lower wire edge and\nit moves upward ( vy>0). The DW reverses its chiral-\nity again at upper wire edge when the antivortex dies.\nThen this cycle repeats itself. The spin configurations\ncorresponding to various stages are shown in the lower\npanels of Fig. 9a. When β > α, as shown in Fig. 9b,\nthe antivortex of q=−1 andp= +1 moves upward\nsincevy>0. The chirality of the original transverse\nwall shall not change when the antivortex is annihilated\nat the upper edge defect because of winding number con-\nservation. No antivortex is generated at the even number\nnotches and same type of the antivortex is generated at\nodd number notches, hence the transverse wall preserves\nits chirality throughout propagation. The corresponding\nspin configurations are shown in the lower panels of Fig.\n9b.\nThe second term in Eq. (3) (for vx) isβu/α, the same\nas the transverse DW velocity in a homogeneous wire\n(straight lines in Figs. 2b and 3b). The first term de-\npends on DW properties as well as βandα. It changes\nsign atβ=α.vxis larger than βu/αin the presence of\nvortices if β < α. Therefore, in this case vortex genera-\ntions and vortex dynamics boost DW propagation. For\nsmallαand to the leading order correction in αandβ,\nEq. (3) becomes vx=u−(α2−αβ)uW2/(π2∆2). Thus,\nthe longitudinal velocity equals approximately uand de-\npends very weakly on β. This is what was observed in\nFig. 2b. vx=ucorresponds to the complete conversion\nof itinerant electron spins into local magnetic moments.\nAlthough the Thiele equation cannot explain why a DW\ngenerates vortices around notches in phase B, it explains\nwell DW propagation boost for β < α. This result is in\ncontrast to the field-driven DW propagation where vor-\ntex/antivortexgenerationreduces the Walker breakdown\nfield and inevitably slows down DW motion5,24.\nBefore conclusion, we would also like to point out that\nit is possible to realize both β < α(boosting phase) and\nβ > α(hindering phase) experimentally in magnetic ma-7\n(a) \n(b) +1/2 +1/2 +1/2 +1/2 \n+1/2 +1/2 \n+1/2 +1/2 \n+1/2 +1/2 \n-1/2 -1/2 -1/2 \n+1/2 +1/2 +1/2 +1/2 +1/2 +1/2 +1/2 +1/2 -1/2 -1/2 +1/2 -1/2 -1/2 -1/2 \n-1 \n-1 -1 \n-1 -1 \n-1/2 \n(s1) (s2) (s3) (s4) (s5) \ns1 \ns1 (s1) (s2) (s3) (s4) (s5) s2 \ns2 s3 \ns3 s4 \ns4 s5 \ns5 (s6) \n(s6) 50 nm \ns6 \ns6 \nFIG. 9. (color online) (a) Illustrations of changes of topol og-\nical defects (transverse DW edge defects and vortices) duri ng\nthebirthanddeathofvortices inPhase Bas aDWpropagates\nfrom the left to the right and the corresponding spin config-\nurations at various moments. Lines represent DWs. Big blue\ndots for vortices and open circles for edge defects of wind-\ning number −1/2 and filled black circles for edge defects of\nwinding number 1 /2. The color coding is the same as that of\nFig. 5. The blue jagged lines indicate the profiles of trian-\ngular notches. The nanowire is 64 nm wide and 4 nm thick.\nThe notch dimensions are 48 ×16 nm3. The interval between\nadjacent notches is L= 1500 nm. u= 600 m/s, β= 0.005.\n(b) Illustrations of changes of topological defects in Phas e C1\nand the the corresponding spin configurations at various mo-\nments. The nanowire is 64 nm wide and 4 nm thick. The\nnotch dimensions are 48 ×16 nm2. The interval between ad-\njacent notches is L= 2000 nm. u= 600 m/s, β= 0.025.terials like permalloy with damping coefficient engineer-\ning. A recent study28demonstrated that αof permalloy\ncan increaseby four times througha dilute impurity dop-\ning of lanthanides (Sm, Dy, and Ho).\nV. CONCLUSIONS\nIn conclusion, notches can boost DW propagation\nwhenβ < α. The boost is facilitated by antivortex\ngeneration and motion, and boosting effect is optimal\nwhen two neighboring notches is separated by the dis-\ntance that an antivortex travels in its lifetime. In the\nboosting phase, DW can propagate at velocity uthat\ncorresponds to a complete conversion of itinerant elec-\ntron spins into local magnetic moments. When β > α,\nthe notches always hinder DW propagation. According\nto Thiele’s theory, the generation of vortices increases\nDW velocity for β < αand decreases DW velocity when\nβ > α. This explains the origin of boosting phase and\nhindering phase. Furthermore, it is found that a vortex\nwall favored in a very wide wire tends to transform to a\ntransverse wall under a current. This may explain exper-\nimental observation that the depinning current density is\nnot sensitive to DW types.\nVI. ACKNOWLEDGMENTS\nWe thank Gerrit Bauer for useful comments. HYY ac-\nknowledges the support of Hong Kong PhD Fellowship.\nThis work was supported by NSFC of China (11374249)\nas well as Hong Kong RGC Grants (163011151 and\n605413).\n∗Corresponding author: phxwan@ust.hk\n1S. S. P. Parkin, M. Hayashi, and L. Thomas, Science 320,\n190 (2008).\n2D. A. Allwood, G. Xiong, C. C. Faulkner, D. Atkinson, D.\nPetit, and R. P. Cowburn, Science 309, 1688 (2005).\n3M. Kl¨ aui, C. A. F. Vaz, J. A. C. Bland, W. Wernsdorfer,\nG. Faini, E. Cambril, L. J. Heyderman, F. Nolting, and U.\nR¨ udiger, Phys. Rev. Lett. 94, 106601 (2005).\n4M. Hayashi, L. Thomas, C. Rettner, R. Moriya, X. Jiang,\nand S. S. P. Parkin, Phys. Rev. Lett. 97, 207205 (2006).\n5Y. Nakatani, A. Thiaville, and J. Miltat, Nat. Mater. 2,\n521 (2003).\n6X. R. Wang, P. Yan, J. Lu and C. He, Ann. Phys. (N.Y.)\n324, 1815 (2009); X. R. Wang, P. Yan, and J. Lu, Euro-\nphys. Lett. 86, 67001 (2009).\n7L. Berger, J. Appl. Phys. 55, 1954 (1984).\n8J. C. Slonczewski, J. Magn. Magn. Mater. 159, L1 (1996).\n9S. Zhang and Z. Li, Phys. Rev. Lett. 93, 127204 (2004).10A. Thiaville, Y. Nakatani, J. Miltat, and Y. Suzuki, Euro-\nphys. Lett. 69, 990 (2005).\n11Z. Li and S. Zhang, Phys. Rev. B 70, 024417 (2004).\n12G. Tatara and H. Kohno, Phys. Rev. Lett. 92, 086601\n(2004).\n13H. Y. Yuan and X. R. Wang, European Physical Journal\nB (in press); arXiv:1407.4559 [cond-mat.mes-hall]\n14R. D. McMichael and M. J. Donahue, IEEE Trans. Magn.\n33, 4167 (1997).\n15G. S. D. Beach, C. Knutson, C. Nistor, M. Tsoi, and J. L.\nErskine, Phys. Rev. Lett. 97, 057203 (2006).\n16L. Thomas, R. Moriya, C. Rettner, and S. S. P. Parkin,\nScience330, 1810 (2010).\n17http://math.nist.gov/oommf.\n18A. Vansteenkiste, J. Leliaert, M. Dvornik, M. Helsen, F.\nGarcia-Sanchez, F. B. V. Waeyenberge, AIP Adbances 4,\n107133 (2014).8\n19OOMMF package was used in the early stage of this re-\nsearch. In order to simulate a long and wide wire, we\nswitched to MUMAX package. Two packages give almost\nidentical results on shorter wires, and the results present ed\nhere were generated from MUMAX.\n20B. Hu and X. R. Wang, Phys. Rev. Lett. 111, 027205\n(2013); X. S. Wang, P. Yan, Y. H. Shen, G. E.W. Bauer,\nand X. R. Wang, Phys. Rev. Lett. 109, 167209 (2012).\n21Notch geometry affects depinning current because of\nthe change of current density and perpendicular shape\nanisotropy (see Ref. 22) in notch area. Both effects help to\ngenerate vortices and thus reduce the depinning current.This may explain the result.\n22A. Aharoni, J. Appl. Phys. 83, 3432 (1998).\n23O. Tchernyshyov and G. -W. Chern, Phys. Rev. Lett. 95,\n197204 (2005).\n24H. Y. Yuan and X. R. Wang, J. Magn. Magn. Mater. 368,\n70 (2014).\n25See Supplemental Material at [URL] for DW propagation\nin boosting phase.\n26A. A. Thiele, Phys. Rev. Lett. 30, 230 (1973).\n27D. L. Huber, Phys. Rev. B 26, 3758 (1982).\n28S. G. Reidy, L. Cheng, and W. E. Bailey, Appl. Phys. Lett.\n82, 1254 (2003)."    },    {        "title": "1507.08227v2.Spin_dynamics_and_relaxation_in_the_classical_spin_Kondo_impurity_model_beyond_the_Landau_Lifschitz_Gilbert_equation.pdf",        "content": "Spin dynamics and relaxation in the classical-spin Kondo-impurity model beyond the\nLandau-Lifschitz-Gilbert equation\nMohammad Sayad and Michael Pottho\u000b\nI. Institut f ur Theoretische Physik, Universit at Hamburg, Jungiusstra\u0019e 9, 20355 Hamburg, Germany\nThe real-time dynamics of a classical spin in an external magnetic \feld and locally exchange\ncoupled to an extended one-dimensional system of non-interacting conduction electrons is studied\nnumerically. Retardation e\u000bects in the coupled electron-spin dynamics are shown to be the source\nfor the relaxation of the spin in the magnetic \feld. Total energy and spin is conserved in the\nnon-adiabatic process. Approaching the new local ground state is therefore accompanied by the\nemission of dispersive wave packets of excitations carrying energy and spin and propagating through\nthe lattice with Fermi velocity. While the spin dynamics in the regime of strong exchange coupling\nJis rather complex and governed by an emergent new time scale, the motion of the spin for\nweakJis regular and qualitatively well described by the Landau-Lifschitz-Gilbert (LLG) equation.\nQuantitatively, however, the full quantum-classical hybrid dynamics di\u000bers from the LLG approach.\nThis is understood as a breakdown of weak-coupling perturbation theory in Jin the course of time.\nFurthermore, it is shown that the concept of the Gilbert damping parameter is ill-de\fned for the\ncase of a one-dimensional system.\nPACS numbers: 75.78.-n, 75.78.Jp, 75.60.Jk, 75.10.Hk, 75.10.Lp\nI. INTRODUCTION\nThe Landau-Lifshitz-Gilbert (LLG) equation1{3has\noriginally been considered to describe the dynamics of\nthe magnetization of a macroscopic sample. Nowadays it\nis frequently used to simulate the dynamics of many mag-\nnetic units coupled by exchange or magnetostatic interac-\ntions, i.e., in numerical micromagnetics.4The same LLG\nequation can be used on an atomistic level as well.5{9For\na suitable choice of units and for several spins Sm(t) at\nlattice sites m, it has the following structure:\ndSm(t)\ndt=Sm(t)\u0002B+X\nnJmnSm(t)\u0002Sn(t)\n+X\nn\u000bmnSm(t)\u0002dSn(t)\ndt: (1)\nIt consists of precession terms coupling the spin at site\nmto an external magnetic \feld Band, via exchange\ncouplingsJmn, to the spins at sites n. Those pre-\ncession terms typically have a clear atomistic origin,\nsuch as the Ruderman-Kittel-Kasuya-Yoshida (RKKY)\ninteraction10{12which is mediated by the magnetic po-\nlarization of conduction electrons. The non-local RKKY\ncouplingsJmn=J2\u001fmnare given in terms of the ele-\nments\u001fmnof the static conduction-electron spin suscep-\ntibility and the local exchange Jbetween the spins and\nthe local magnetic moments of the conduction electrons.\nOther possibilities comprise direct (Heisenberg) exchange\ninteractions, intra-atomic (Hund's) couplings as well as\nthe spin-orbit and other anisotropic interactions. The re-\nlaxation term, on the other hand, is often assumed as lo-\ncal,\u000bmn=\u000emn\u000b, and represented by purely phenomeno-\nlogical Gilbert damping constant \u000bonly. It describes the\nangular-momentum transfer between the spins and a usu-\nally unspeci\fed heat bath.On the atomistic level, the Gilbert damping must be\nseen as originating from microscopic couplings of the\nspins to the conduction-electron system (as well as to\nlattice degrees of freedom which, however, will not be\nconsidered here). There are numerous studies where the\ndamping constant, or tensor, \u000bhas been computed nu-\nmerically from a more fundamental model including elec-\ntron degrees of freedom explicitly13{15or even from \frst\nprinciples.16{21All these studies rely on two, partially\nrelated, assumptions: (i) The spin-electron coupling J\nis weak and can be treated perturbatively to lowest or-\nder, i.e., the Kubo formula or linear-response theory is\nemployed. (ii) The classical spin dynamics is slow as\ncompared to the electron dynamics. These assumptions\nappear as well justi\fed but they are also necessary to\nachieve a simple e\u000bective spin-only theory by eliminat-\ning the fast electron degrees of freedom.\nThe purpose of the present paper is to explore the\nphysics beyond the two assumptions (i) and (ii). Us-\ning a computationally e\u000ecient formulation in terms of\nthe electronic one-particle reduced density matrix, we\nhave set up a scheme by which the dynamics of classi-\ncal spins coupled to a system of conduction electrons can\nbe treated numerically exactly. The theory applies to ar-\nbitrary coupling strengths and does not assume a separa-\ntion of electron and spin time scales. Our approach is a\nquantum-classical hybrid theory22which may be charac-\nterized as Ehrenfest dynamics, similar to exact numerical\ntreatments of the dynamics of nuclei, treated as classical\nobjects, coupled to a quantum system of electrons (see,\ne.g., Ref. 23 for an overview). Some other instructive ex-\namples of quantum-classical hybrid dynamics have been\ndiscussed recently.24,25\nThe obvious numerical advantage of an e\u000bective spin-\nonly theory, as given by LLG equations of the form (1),\nis that in solving the equations of motion there is only\nthe time scale of the spins that must be taken care of. AsarXiv:1507.08227v2  [cond-mat.mes-hall]  28 Nov 20152\ncompared to our hybrid theory, much larger time steps\nand much longer propagation times can be achieved. Op-\nposed to ab-initio approaches16,17,26we therefore con-\nsider a simple one-dimensional non-interacting tight-\nbinding model for the conduction-electron degrees of free-\ndom, i.e., electrons are hopping between the nearest-\nneighboring sites of a lattice. Within this model ap-\nproach, systems consisting of about 1000 sites can be\ntreated easily, and we can access su\u000eciently long time\nscales to study the spin relaxation. An equilibrium state\nwith a half-\flled conduction band is assumed as the ini-\ntial state. The subsequent dynamics is initiated by a\nsudden switch of a magnetic \feld coupled to the classical\nspin. The present study is performed for a single spin,\ni.e., we consider a classical-spin Kondo-impurity model\nwith antiferromagnetic local exchange coupling J, while\nthe theory itself is general and can be applied to more\nthan a single or even to a large number of spins as well.\nAs compared to the conventional (quantum-spin)\nKondo model,27,28the model considered here does not\naccount for the Kondo e\u000bect and therefore applies to sit-\nuations where this is absent or less important, such as\nfor systems with large spin quantum numbers S, strongly\nanisotropic systems or, as considered here, systems in a\nstrong magnetic \feld. To estimate the quality of the\nclassical-spin approximation a priori is di\u000ecult.29{31For\none-dimensional systems, however, a quantitative study\nis possible by comparing with full quantum calculations\nand will be discussed elsewhere.32\nThere are di\u000berent questions to be addressed: For\ndimensional reasons, one should expect that linear-\nresponse theory, even for weak J, must break down at\nlong times. It will therefore be interesting to compare\nthe exact spin dynamics with the predictions of the LLG\nequation for di\u000berent J. Furthermore, the spin dynamics\nin the long-time limit can be expected to be sensitively\ndependent on the low-energy electronic structure. We\nwill show that this has important consequences for the\ncomputation of the damping constant \u000band that\u000bis\neven ill-de\fned in some cases. An advantage of a full\ntheory of spin and electron dynamics is that a precise\nmicroscopic picture of the electron dynamics is available\nand can be used to discuss the precession and relaxation\ndynamics of the spin from another, namely from the elec-\ntronic perspective. This information is in principle exper-\nimentally accessible to spin-resolved scanning-tunnelling\nmicroscope techniques33{36and important for an atom-\nistic understanding of nano-spintronics devices.37,38We\nare particularly interested in the physics of the system\nin the strong- Jregime or for a strong \feld Bwhere the\ntime scales of the spin and the electron dynamics become\ncomparable. This has not yet been explored but could\nbecome relevant to understand real-time dynamics in re-\nalizations of strong- JKondo-lattice models by means of\nultracold fermionic Yb quantum gases trapped in optical\nlattices.39,40\nThe paper is organized as follows: We \frst introduce\nthe model and the equations of motion for the exactquantum-classical hybrid dynamics in Sec. II and discuss\nsome computational details in Sec. III. Sec. IV provides\na comprehensive discussion of the relaxation of the clas-\nsical spin after a sudden switch of a magnetic \feld. The\nreversal time as a function of the interaction and the \feld\nstrength is analyzed in detail. We then set the focus on\nthe conduction-electron system which induces the relax-\nation of the classical spin by dissipation of energy. In Sec.\nV, the linear-response approach to integrate out the elec-\ntron degrees of freedom is carefully examined, including a\ndiscussion of the additional approximations that are nec-\nessary to re-derive the LLG equation and the damping\nterm in particular. Sec. VI summarizes the results and\nthe main conclusions.\nII. MODEL AND THEORY\nWe consider a classical spin SwithjSj= 1=2, which is\ncoupled via a local exchange interaction of strength Jto\nthe local quantum spin si0at the sitei0of a system of N\nitinerant and non-interacting conduction electrons. The\nconduction electrons hop with amplitude \u0000T(T > 0)\nbetween non-degenerate orbitals on nearest-neighboring\nsites of aD-dimensional lattice, see Fig. 1. Lis the\nnumber of lattice sites, and n=N=L is the average\nconduction-electron density.\nThe dynamics of this quantum-classical hybrid\nsystem22is determined by the Hamiltonian\nH=\u0000TX\nhiji;\u001bcy\ni\u001bcj\u001b+Jsi0S\u0000BS: (2)\nHere,ci\u001bannihilates an electron at site i= 1;:::;L with\nspin projection \u001b=\";#, andsi=1\n2P\n\u001b\u001b0cy\ni\u001b\u001b\u001b\u001b0ci\u001b0is\nthe local conduction-electron spin at i, where\u001bdenotes\nthe vector of Pauli matrices. The sum runs over the\ndi\u000berent ordered pairs hijiof nearest neighbors. Bis\nan external magnetic \feld which couples to the classical\nspin.\nTo be de\fnite, an antiferromagnetic exchange coupling\nJ > 0 is assumed. If Swas a quantum spin with\nS= 1=2, Eq. (2) would represent the single-impurity\nKondo model.27,28However, in the case of a classical spin\nconsidered here, there is no Kondo e\u000bect. The semiclas-\nsical single-impurity Kondo model thus applies to sys-\ntems where a local spin is coupled to electronic degrees\nof freedom but where the Kondo e\u000bect absent or sup-\npressed. This comprises the case of large spin quantum\nnumbersS, or the case of temperatures well above the\nKondo scale, or systems with a ferromagnetic Kondo cou-\nplingJ <0 where, for a classical spin, we expect a qual-\nitatively similar dynamics as for J >0.\nWe assume that initially, at time t= 0, the clas-\nsical spinS(t= 0) has a certain direction and that\nthe conduction-electron system is in the corresponding\nground state, i.e., the conduction electrons occupy the\nlowestNone-particle eigenstates of the non-interacting3\nHamiltonian Eq. (2) for the given S=S(t= 0) up to\nthe chemical potential \u0016. A non-trivial time evolution is\ninitiated if the initial direction of the classical spin and\nthe direction of the \feld Bare non-collinear.\nTo determine the real-time dynamics of the electronic\nsubsystem, it is convenient to introduce the reduced one-\nparticle density matrix. Its elements are de\fned as ex-\npectation values,\n\u001aii0;\u001b\u001b0(t)\u0011hcy\ni0\u001b0ci\u001bit; (3)\nin the system's state at time t. Att= 0 we have\u001a(0) =\n\u0002(\u0016\u0000T(0)). The elements of \u001a(0) are given by\n\u001ai\u001b;i0\u001b0(0) =X\nkUi\u001b;k\u0002(\u0016\u0000\"k)Uy\nk;i0\u001b0; (4)\nwhere \u0002 is the step function and where Uis the uni-\ntary matrix diagonalizing the hopping matrix T(0), i.e.,\nUyT(0)U=\"with the diagonal matrix \"given by the\neigenvalues of T(0). The hopping matrix at time tis\ncan be read o\u000b from Eq. (2). It comprises the physical\nhopping and the contribution resulting from the coupling\nterm. Its elements are given by\nTi\u001b;i0\u001b0(t) =\u0000T\u000ehii0i\u000e\u001b\u001b0+\u000eii0\u000ei0i0J\n2(S(t)\u001b)\u001b\u001b0:(5)\nHere\u000ehii0i= 1 ifi;i0are nearest neighbors and zero else.\nThere is a closed system of equations of motion for the\nclassical spin vector S(t) and for the one-particle density\nmatrix\u001a(t). The time evolution of the classical spin is\ndetermined via ( d=dt)S(t) =fS;Hclass:gby the classical\nHamilton function Hclass:=hHi. This equation of mo-\ntion is the only known way to consistently describe the\ndynamics of quantum-classical hybrids (see Refs. 22,41,42\nand references therein for a general discussion). The\nPoisson bracket between arbitrary functions AandBof\nthe spin components is given by,43,44\nfA;Bg=X\n\u000b;\f;\r\"\u000b\f\r@A\n@S\u000b@B\n@S\fS\r; (6)\nwhere the sums run over x;y;z and where \"\u000b\f\ris the\nfully antisymmetric \"-tensor. With this we \fnd\nd\ndtS(t) =Jhsi0it\u0002S(t)\u0000B\u0002S(t): (7)\nThis is the Landau-Lifschitz equation where the expec-\ntation value of the conduction-electron spin at i0is given\nby\nhsi0it=1\n2X\n\u001b\u001b0\u001ai0\u001b;i0\u001b0(t)\u001b\u001b0\u001b; (8)\nand where Jhsi0itacts as an e\u000bective time-dependent\ninternal \feld in addition to the external \feld B.\nS(t)JTi0BFIG. 1: (Color online) Classical spin S(t) coupled via an an-\ntiferromagnetic local exchange interaction of strength Jto a\nsystem of conduction electrons hopping with nearest-neighbor\nhopping amplitude Tover the sites of a one-dimensional lat-\ntice with open boundaries. The spin couples to the central\nsitei0of the system and is subjected to a local magnetic \feld\nof strength B.\nThe equation of motion for hsiitreads as\nd\ndthsiit=\u000eii0JS(t)\u0002hsiit\n+Tn:n:X\nj1\n2iX\n\u001b\u001b0(hcy\ni\u001b\u001b\u001b\u001b0cj\u001b0it\u0000c.c.);(9)\nwhere the sum runs over the nearest neighbors of i. The\nsecond term on the right-hand side describes the coupling\nof the local conduction-electron spin to its environment\nand the dissipation of spin and energy into the bulk of\nthe system (see below). Apparently, the system of equa-\ntions of motion can only be closed by considering the\ncomplete one-particle density matrix Eq. (3). It obeys a\nvon Neumann equation of motion,\nid\ndt\u001a(t) = [T(t);\u001a(t)] (10)\nas is easily derived, e.g., from the Heisenberg equation of\nmotion for the annihilators and creators.\nAs is obvious from the equations of motion, the real-\ntime dynamics of the quantum-classical Kondo-impurity\nmodel on a lattice with a \fnite but large number of sites L\ncan be treated numerically exactly (see also below). Nev-\nertheless, the model comprises highly non-trivial physics\nas the electron dynamics becomes e\u000bectively correlated\ndue to the interaction with the classical spin. In addi-\ntion, the e\u000bective electron-electron interaction mediated\nby the classical spin is retarded: electrons scattered from\nthe spin at time twill experience the e\u000bects of the spin\ntorque exerted by electrons that have been scattered from\nthe spin at earlier times t0<t.\nIII. COMPUTATIONAL DETAILS\nEqs. (5), (7), (8) and (10) represent a coupled non-\nlinear system of \frst-order ordinary di\u000berential equations\nwhich can be solved numerically. By blocking up the\nvon Neumann equation (10), the di\u000berential equations\nare written in a standard form _y=f(y(t);t), wherey(t)\nis a high-dimensional vector, such that an explicit Runge-\nKutta method can be applied. A high-order propagation4\ntechnique is used45which provides the numerically exact\nsolution up to 6-th order in the time step \u0001 t. For a typ-\nical system consisting of about L= 103sites this implies\nthat\u0018106coupled equations are solved.\nWe consider a one-dimensional system with open\nboundaries consisting of L= 1001 sites and a local per-\nturbation at the central site i0of the system, see Fig. 1.\nFor a half-\flled tight-binding conduction band the Fermi\nvelocityvF= 2Troughly determines the maximum speed\nof the excitations and de\fnes a \\light cone\".46,47This\nmeans that \fnite-size e\u000bects due to scattering at the sys-\ntem boundaries become relevant after a propagation time\ntmax\u0018500 (in units of 1 =T). A time step \u0001 t= 0:1 is\nusually su\u000ecient for reliable numerical results up to tmax,\ni.e., about 5000 time steps are performed. The compu-\ntational cost is moderate, and calculations can be per-\nformed in a few hours on a standard desktop computer.\nAssuming, for example, that B= (0;0;B), the Hamil-\ntonian is invariant under rotations around the zaxis. It\nis then easily veri\fed that not only the length of the spin\njSj= 1=2 is conserved but also the total number of con-\nduction electrons,\nNtot=X\ni\u001bhcy\ni\u001bci\u001bi; (11)\nthez-component of the total spin,\nStot;z=Sz+X\nihsizi; (12)\nas well as the total energy,\nEtot=hHi= tr (\u001a(t)T(t))\u0000BS(t): (13)\nDespite the fact that the model does not include a di-\nrect (e.g., Coulomb) interaction among the conduction\nelectrons, the average occupation numbers of the basis of\none-particle states in which the hopping matrix T(t) is\ndiagonal at time tarenotconserved. This is due to the\ne\u000bective retarded interaction mediated by the classical\nspin. Hence, the system is not integrable, unlike a free\nfermion gas. The conservation of the above-mentioned\nglobal observables serves as a sensitive check for the ac-\ncuracy of the numerical procedure.\nIV. COUPLED SPIN AND ELECTRON\nDYNAMICS\nA. Spin relaxation\nFig. 2 shows the real-time dynamics of the classical\nspin forJ= 1. Energy and time units are \fxed by the\nnearest-neighbor hopping T= 1 throughout the paper.\nInitially, for t= 0, the spin is oriented (almost) antipar-\nallel to the external local \feld B= (0;0;B) withB= 1,\ni.e., initially Sx(0) =1\n2sin#,Sy(0) = 0,Sz(0) =\u00001\n2cos#\n-0.5-0.3-0.10.10.30.5-0.5-0.3-0.10.10.30.5-0.5-0.3-0.10.10.30.5\nxyz\n10−1100101102\ntimet-0.50-0.250.000.250.50/vectorSSx\nSzFIG. 2: (Color online) Real-time dynamics of the classical\nspin. Upper panel: Bloch sphere representation. Lower panel:\nxandzcomponent of S(t) (jSj= 1=2). Calculations for\nexchange coupling J= 1 and \feld strength B= 1 and for\na system of L= 1001 sites. ( L= 1001 is kept \fxed for the\nrest of the paper). Energy and time units are \fxed by the\nnearest-neighbor hopping T= 1.\nwhere a non-zero but small polar angle #=\u0019=50 is nec-\nessary to slightly break the symmetry of the initial state\nand to start the dynamics.\nFor the same setup, the Landau-Lifschitz-Gilbert equa-\ntion would essentially predict two e\u000bects: \frst, a preces-\nsion of the classical spin around the \feld direction with\nLarmor frequency !L=Bz, and second, a relaxation\nof the spin to the equilibrium state with Sparallel to\nB=Bezfort!1 . Both e\u000bects are also found in\nthe full dynamics of the quantum-classical hybrid model.\nThe frequency of the oscillation of Sx(t) that is seen in\nFig. 2 is!L, andSzis reversed after a few hundred time\nunits. The precessional motion is easily explained by the\ntorque on the spin exerted by the \feld according to Eq.\n(7). The explanation of the damping e\u000bect is more in-\nvolved:\nEven for the high \feld strength considered here, the\nspin dynamics is slow as compared to the characteristic\nelectronic time scale such that it could be reasonable to\nassume the electronic system being in its instantaneous5\n10\u0000210\u00001100101102timet3.083.103.123.14~S(t)h~si0it\u0000(t)~B\u0000(t)J=46810J=20xy\nFIG. 3: (Color online) Time dependence of the angle \r(t)\nenclosed by S(t) andhsi0itforB= 0:1 and di\u000berent Jas\nindicated.\nground state at any instant of time and corresponding\nto the con\fguration of the classical spin. This, however,\nwould imply that the expectation value hsi0itof the local\nconduction-electron moment at i0is always strictly par-\nallel toS(t) and, hence, there would not be any torque\nmediated by the exchange coupling JonS(t).\nIn fact, the direction of hsi0itis always somewhat be-\nhind the \\adiabatic direction\", i.e., behind \u0000S(t): This\nis shown in Fig. 3 for a \feld of strength B= 0:1 where\nthe spin dynamics is by a factor 10 slower, compared to\nFig. 2, and for di\u000berent stronger exchange couplings J.\nEven in this case the process is by no means adiabatic,\nand the angle \r(t) betweenS(t) andhsi0itis close to but\nclearly smaller than \r=\u0019at any instant of time. This\nnon-adiabaticity results from the fact that the motion\nof the classical spin a\u000bects the conduction electrons in\na retarded way, i.e., it takes a \fnite time until the local\nconduction-electron spin hsi0itati0reacts to the motion\nof the classical spin.\nThis retardation e\u000bect results in a torque Jhsi0it\u0002\nS(t)6= 0 exerted on the classical S(t) in the +zdirection\nwhich adds to the torque due to Band which drives the\nspin to its new equilibrium direction. Hence, retardation\nis the physical origin of the Gilbert spin damping.\nWith increasing time, the deviation of \r(t) from the\ninstantaneous equilibrium value \r=\u0019increases in mag-\nnitude, i.e., the direction of hsi0itis more and more be-\nhind the adiabatic direction, and the torque increases. Its\nmagnitude is at a maximum at the same time when the\noscillatingx;ycomponents of S(t) are at a maximum\n(see Fig. 2). The z-component of the torque does not\nvanish before the spin has reached its new equilibrium\npositionS(t)/ez.\nFig. 3 shows results for di\u000berent J. Generally, non-\nadiabatic e\u000bects show up if the typical time scale of the\ndynamics is faster than the relaxation time, i.e., the time\nnecessary to transport the excitation away from the lo-\ncationi0where it is created initially. Roughly this time\nscale is set by the inverse hopping 1 =T. One therefore\nexpects that, for \fxed T, a stronger Jimplies a stronger\n0 20 40 60 80 100\nexchange coupling J050100150200250300350400reversal time τ100∗τ1/τ2\nτ1\nτ2FIG. 4: (Color online) Time for a spin reversal Sz=1\n2cos(\u0019\u0000\n#) =\u00001\n2cos(#)!Sz=1\n2cos(#) for#=#1=\u0019=50 (reversal\ntime\u001c1) and for#=#2=\u0019=25 (reversal time \u001c2). Calcula-\ntions as a function of Jfor \fxedB= 0:1.\nretardation of the conduction-electron dynamics. The re-\nsults for di\u000berent Jshown in Fig. 3 in fact show that the\nmaximum deviation of \r(t) from the adiabatic direction\n\r=\u0019increases with increasing J(for very strong Jthe\ndynamics becomes much more complicated, see below).\nThis results in a stronger torque on S(t) inzdirection\nand thus in a stronger damping. The picture is also qual-\nitatively consistent with the LLG equation as the Gilbert\ndamping constant \u000bincreases with J.\nThe spin (almost) reverses its direction after a \fnite\nreversal time \u001cwhich is shown in Fig. 4 as a function\nofJ. Calculations have been performed for an initial\ndirection of the classical spin with Sz(0) =\u0000(1=2) cos#\nwith two di\u000berent polar angles #1;2. If#is su\u000eciently\nsmall, the results for di\u000berent #are expected to di\u000ber by\naJandBindependent constant factor only. As Fig. 4\ndemonstrates, the ratio \u001c1=\u001c2is in fact nearly constant.\nFor weakJand up to coupling strengths of about J.30,\nwe \fnd that the reversal time decreases with increasing\nJ. With increasing J, the retardation e\u000bect increases, as\ndiscussed above, and the stronger damping results in a\nshorter reversal time.\n0 2 4 6 8 10\n1/B0100200300400reversal time τ\nJ= 2\nJ= 4\nFIG. 5: (Color online) Reversal time \u001c1as function of Bfor\n\fxedJ= 2 andJ= 4 as indicated.6\nThe prediction of the LLG equation for the reversal\ntime of a single spin \u001ccan be derived analytically48and\nis given by\n\u001c/1 +\u000b2\n\u000b1\nBln\f\f\f1=2\u0000Sz(0)\n1=2 +Sz(0)\f\f\f: (14)\nHowever, down to the smallest Jfor which\u001ccan be cal-\nculated reliably, our results for the full spin dynamics do\nnot scale as \u001c/1=J2as one would expect for weak J\nassuming that \u000b/J2(see discussion in Sec. V B).\nTheBdependence of the reversal time is shown in Fig.\n5. With increasing \feld strength, the classical spin S(t)\nprecesses with a higher Larmor frequency !L\u0019Baround\nthezaxis. Hence, non-adiabatic e\u000bects increase. The\nstronger the \feld, the more delayed is the precessional\nmotion of the local conduction-electrons spin hsi0it. This\nresults in a stronger torque in + zdirection exerted on\nthe classical spin. Therefore, the relaxation is faster and\nthe reversal time \u001csmaller. For weak and intermediate\n\feld strengths, \u001cis roughly proportional to 1 =B. This is\nconsistent with the prediction of the LLG equation, see\nEq. (14).\nIn the limit of very strong \felds one would expect an\nincrease of the reversal time with increasing Bsince the\n\feld term will eventually dominate the dynamics, i.e.,\nonly the precessional motion survives which implies a di-\nverging reversal time. In fact, for a \feld strength exceed-\ning a critical strength Bc, which depends on J, there is\nno full relaxation any longer, and \u001c=1. This strong-\nBregime cannot be captured by the LLG equation and\ndeserves further studies.\nThe strong- Jregime is interesting as well. For cou-\npling strengths exceeding J\u001930 the reversal time in-\ncreases withJ(see Fig. 4). Eventually, the reversal time\nmust even diverge. This is obvious as the dynamics is\ndescribed by a simple two-spin model in the limit J=1\nwhich cannot show spin relaxation. The corresponding\nequations of motion are obtained by from Eqs. (7) and\n(9) by setting t= 0 andB= 0:\nd\ndtS(t) =Jhsi0it\u0002S(t);\nd\ndthsi0it=JS(t)\u0002hsi0it (15)\nNote that we have jhsi0itj= 1=2 forJ!1 . The equa-\ntions are easily solved by exploiting the conservation of\nthe total spin Stot=S(t) +hsi0it. Both spins precess\nwith constant frequency !0=JStotaroundStot. Their\ncomponents parallel to Stotare equal, and their compo-\nnents perpendicular to Stotare anti-parallel and of equal\nlength.\nHowever, the two-spin dynamics of the J=1limit\nis not stable against small perturbations. Fig. 6 shows\nthe classical spin dynamics of the full model (with T=\n1 andB= 0:1) for a very strong but \fnite coupling\nJ= 100. Here, the motion of the classical spin gets\nvery complicated as compared with the highly regular\n-0.50.00.5\n/vectorSx\ny\nz\n-2.00.02.0\nJ/angbracketleft/vector si0/angbracketright×/vectorS\n-0.50.00.5\n−/vectorB×/vectorS\n10−1100101102\ntimet3.103.153.20\nγ=/negationslash(/vectorS/vector s0)FIG. 6: (Color online) Real-time dynamics in the strong- J\nregime:J= 100 and B= 0:1.First panel: Classical spin\nS(t).Second panel: Torque on S(t) due to the exchange\ninteraction. Third panel: Torque on S(t) due to the \feld\nterm. Fourth panel: Angle enclosed by S(t) andhsi0it\nbehavior in the weak- Jregime (cf. Fig. 2). In particular,\nthez-component of S(t) is oscillating on nearly the same\nscale as the xandycomponents. This characteristic\ntime scale \u0001 t\u00194 of the oscillation corresponds to a\nfrequency!\u00191:5 which di\u000bers by more than an order\nof magnitude from both, the Larmor frequency B= 0:1\nand from the exchange-coupling strength J= 100. Note\nthat the oscillation of Sz(t) is actually the reason for the\nambiguity in the determination of the precise reversal\ntime and gives rise to the error bars in Fig. 4 for strong\nJ.\nWe attribute the complexity of the dynamics to the\nfact that the torque due to the \feld term and the torque\ndue to the exchange coupling are of comparable magni-\ntudes, see second and third panel in Fig. 6. It is interest-\ning that even for strong J, where one would expect S(t)\nandhsi0itto form a tightly bound local spin-zero state,\nthere is actually a small deviation from perfect antiparal-\nlel alignment, i.e., \r(t)6=\u0019, as can be seen in the fourth\npanel of Fig. 6. This results in a \fnite z-component of the\ntorque onS(t) which leads to a very fast reversal with\nSz(t)\u0019+0:5 at timet\u00192:1. Contrary to the weak-\ncoupling limit, however, the z-component of the torque\nchanges sign at this point and drives the spin back to the7\n\u0000z-direction. At t\u00193:2, however, the z-component of\nS(t) once more reverses its direction. Here, the torque\ndue to the exchange coupling vanishes completely as S(t)\nandhsi0itare perfectly antiparallel (see the \frst zero of\n\r(t) in the fourth panel). The motion continues due to\nthe non-zero \feld-induced torque. This pattern repeats\nseveral times. S(t) mainly oscillates within a plane in-\ncluding and slowly rotating around the z-axis.\nEventually, there is a perfect relaxation of the classical\nspin for large tbut in a very di\u000berent way as compared to\nthe weak-coupling limit. While the deviation from \r=\u0019\nis small at any instant of time as for weak J, the most\napparent di\u000berence is perhaps that the new ground state\nis approached with an oscillating behavior of \r(t) around\n\r=\u0019, i.e.,hsi0itmay run behind or ahead ofS(t) as\nwell.\nLet us summarize at this point the main di\u000berences\nbetween the quantum-classical hybrid and the e\u000bective\nLLG dynamics of the classical spin: For weak JandB,\nthe qualitative behavior, precessional motion and relax-\nation, is the same in both approaches. Quantitatively,\nhowever, the LLG equation is inconsistent with the ob-\nservedJdependence of the reversal time when assuming\n\u000b/J2. TheBdependence of 1 =\u001cis linear as expected\nfrom the LLG approach. For strong J, the spin dynamics\nqualitatively di\u000bers from LLG dynamics and gets more\ncomplicated with a new time scale emerging. Absence of\ncomplete relaxation, as observed in the strong- Blimit, is\nalso not accessible to an e\u000bective spin-only theory.\nB. Energy dissipation\nTo complete the picture of the relaxation dynamics\nof the classical spin, the discussion should also comprise\nthe dynamics of the electronic degrees of freedom. The\nspin relaxation must be accompanied by a dissipation of\nenergy and spin into the bulk of the electronic system\nsince the total energy and the total spin are conserved\nquantities, see Eqs. (12) and (13), while conservation of\nthe total particle number, Eq. (11), is trivially ensured by\nthe particle-hole symmetric setup considered here where\nthe average conduction-electron number at every site is\ntime-independent:P\n\u001bhni\u001bit= 1.\nThe total energy is given by Etot=hHi, see Eq. (13),\nand is a sum over di\u000berent contributions, Etot=EB(t)+\nEhop(t)+Eint(t), namely the interaction energy with the\n\feld\nEB(t) =\u0000BS(t); (16)\nthe kinetic (hopping) energy of the conduction-electron\nsystem\nEhop(t) =\u0000TX\nhijiX\n\u001bhcy\ni\u001bcj\u001bit; (17)\nand the exchange-interaction energy\nEint(t) =Jhsi0itS(t): (18)\n-0.50.00.5\nEB\n-1.2724-1.2718-1.2712energyEhop/L\nEtot/L\n-0.68-0.67-0.66\nEint\n100101102\ntimet−1.30−1.25−1.20−1.15\nei0±1\nei0±3\nei0±2ei0±50ei0±100FIG. 7: (Color online) Di\u000berent contributions to the total\nenergy as functions of time for J= 5 andB= 1. Top panel :\nEB, energy of the classical spin in the external \feld. Second\npanel :Ehop=L, kinetic (hopping) energy of the conduction-\nelectron system per site, and Etot=L, total energy per site.\nThird panel :Eint, exchange-interaction energy. The fourth\npanel shows the time-dependence of the total-energy density\nat di\u000berent distances from the site i0.\nThe time dependence of those contributions is shown in\nFig. 7 forJ= 5 andB= 1.\nThe top panel of Fig. 7 shows that the system re-\nleases the interaction energy j2BSjof the classical spin\nin the external \feld by aligning the spin to the \feld di-\nrection. In the long-time limit, this energy is stored in\nthe conduction-electron system: The average kinetic en-\nergy per site ( L= 1001) increases by the same amount\nas shown in the second panel (note the di\u000berent scales).\nThe exchange-interaction energy changes with time but\nis the same for t= 0 andt!1 (see third panel). The\ntotal energy is constant (second panel).\nThe relaxation of the classical spin in the external \feld\nBimplies an energy \row away from the site i0into the\nbulk of the conduction-electron system such that locally ,\nin the vicinity of i0the system is in its new ground state.\nTo discuss this energy \row, it is convenient to consider\nthe total energy as a lattice sum Etot=P\niei(t) over the8\ni0−500 i0 i0+ 500\nsites0255075100125150175200225250time\n-1.3-1.2-1.1-1.0-0.9-0.8\ntotal energy density ei\nFIG. 8: (Color online) Spatiotemporal evolution of the total-\nenergy density ei(t), as de\fned in Eq. (19). Calculation for\nJ= 5,B= 1.\ntotal energy \\density\" de\fned as\nei(t) =\u0000Tn:n:(i)X\njX\n\u001bhcy\ni\u001bcj\u001bit+\u000eii0(Jhsi0it\u0000B)S(t);\n(19)\nwhere the sum over jruns over the nearest neighbors of\nsitei. The time dependence of the energy density in the\nvicinity ofi0and at distances 50 and 100 is shown in the\nfourth panel of Fig. 7.\nAt any site in the conduction-electron system, the\nenergy density increases from its ground-state value,\nreaches a maximum and eventually relaxes to the en-\nergy density of the new ground state. Since the latter\nis just the ground state with the reversed classical spin,\nthe new ground-state energy density is the same as in\nthe initial state at t= 0. As can be seen in the fourth\npanel of Fig. 7, there is also a slight spatial oscillation\nof the ground-state energy density which just re\rects the\nFriedel oscillations around the impurity at i0.\nComplete relaxation means that the excitation energy\nis completely removed from the vicinity of i0and trans-\nported into the bulk of the system. That this is in fact\nthe case can be seen by comparing the energy density at\ndi\u000berent distances from i0. It is also demonstrated by\nFig. 8 which visualizes the energy-current density which\nsymmetrically points away from i0: The total energy of\nthe excitation \rowing through each pair of sites i0\u0006\u0001iis\nconstant, i.e., the time-integrated energy \rux,R\ndtei(t)\nis the same for all i.\nAs is seen in Fig. 7 (fourth panel) and Fig. 8, there is\na considerable dispersion of the excitation wave packet\ncarrying the energy. For example, at i0+ 100 it takes\nmore than four times longer, as compared to i0+ 1, un-\ntil most of the excitation has passed through (note the\nlogarithmic time scale).\nThe broadening of the wave packet, due to dispersion,\nis asymmetric and bound by an upper limit for the speed\nof the excitation which is roughly set by the Fermi veloc-\nityvF= 2T. This Lieb-Robinson bound46,47determines\ni0\u0000500i0i0+5 0 0sites050100150200250300350400450500time \u0000 \u0000 \u0000 \u0000 \u0000-0.06-0.030.000.030.06hszii\ni0\u0000500i0i0+5 0 0sites050100150200250300350400450500time \u0000 \u0000 \u0000 \u0000 \u0000-0.04-0.020.000.020.04hsxiisx\nszFIG. 9: (Color online) Spatiotemporal evolution of the total-\nspin densityhsiit(upper panel: x-component, lower panel z-\ncomponent) for J= 5 andB= 0:1. See Fig. 10 for snapshots\nat times indicated by the arrows.\nthe \\light cone\" seen in Fig. 8.\nC. Spin dissipation\nThe same upper speed limit, given by the Fermi ve-\nlocity of the conduction-electron system, is also seen in\nthe spatiotemporal evolution of the conduction-electron\nspin densityhsiit. This is shown in Fig. 9 for a di\u000berent\nmagnetic \feld strength B= 0:1 where the classical spin\ndynamics is slower. Apparently, the wave packet of exci-\ntations emitted from the impurity not only carries energy\nbut also spin. It symmetrically propagates away from i0\nand, att\u0019300, reaches the system boundary where it\nis re\rected perfectly. Up to t= 500 there is hardly any\ne\u000bect visible in the local observables close to i0that is\na\u000bected by the \fnite system size.\nSnapshots of the conduction-electron spin dynamics\nare shown in Fig. 10 for the initial state at t= 0 and for\nstates at four later times t >0 which are also indicated\nby the arrows in Fig. 9. At t= 0 the conduction-electron\nsystem is in its ground state for the given initial direction\nof the classical spin. The latter basically points into the\n\u0000zdirection, apart from a small positive x-component\n(#=\u0019=50) which is necessary to break the symmetry9\n-0.0030.0000.003\n-0.0030.0000.003\n-0.0030.0000.003/angbracketleft/vector si/angbracketright\n-0.0030.0000.003sx sz\ni0−100i0−60i0−20-0.0030.0000.003\nt= 0 t= 80t= 60 t= 100sz sx\ni0i0+ 100 i0+ 500\nsites\nt= 250\nFIG. 10: (Color online) Snapshots the of conduction-electron magnetic moments hsiitat di\u000berent times tas indicated on the\nright and by the corresponding arrows in Fig. 9. Red lines: z-components ofhsiit. Blue lines: xcomponents. The pro\fles are\nperfectly symmetric to the impurity site i=i0but displayed up to distances ji\u0000i0j\u0014100 on the left-hand side and up to the\nsystem boundary, ji\u0000i0j\u0014500, on the right-hand side. Parameters J= 5;B= 0:1.\nof the problem and to initiate the dynamics. This tiny\ne\u000bect will be disregarded in the following.\nFrom the perspective of the conduction-electron sys-\ntem, the interaction term JSsi0acts as a local external\nmagnetic \feld JSwhich locally polarizes the conduction\nelectrons at i0. SinceJis antiferromagnetic, the local\nmomenthsi0ipoints into the + zdirection. At half-\flling,\nthe conduction-electron system exhibits pronounced an-\ntiferromagnetic spin-spin correlations which give rise to\nan antiferromagnetic spin-density wave structure aligned\nto thezaxis att= 0, see \frst panel of Fig. 10.\nThe total spin Stot= 0 att= 0, i.e., the classical\nspinSis exactly compensated by the total conduction-\nelectron spinhstoti=P\nihsii=\u0000Sin the ground state.\nThis can be traced back to the fact that for a D= 1-\ndimensional tight-binding system with an odd number of\nsitesL, withN=Land with a single static magnetic\nimpurity, there is exactly one localized state per spin pro-\njection\u001b, irrespective of the strength of the impurity po-\ntential (here given by JS= 0:5Jez). The number of \"\none-particle eigenstates therefore exceeds the number of\n#states by exactly one.\nSince the energy of the excitation induced by the ex-\nternal \feldBis completely dissipated into the bulk, the\nstate of the conduction-electron system at large t(but\nshorter than t\u0019500 where \fnite-size e\u000bects appear)\nmust locally, close to i0, resemble the conduction-electron\nground state for the reversed spin S= +0:5ez. This\nimplies that locally all magnetic moments hsiitmust re-\nverse their direction. In fact, the last panel in Fig. 10(left) shows that the new spin con\fguration is reached\nfort= 250 at sites with distance ji\u0000i0j.100, see\ndashed line, for example. For later times the spin con-\n\fguration stays constant (until the wave packet re\rected\nfrom the system boundaries reaches the vicinity of i0).\nThe reversal is almost perfect, e.g., hsi0it=0= 0:2649!\nhsi0it\u0015250=\u00000:2645. Deviations of the same order\nof magnitude are also found at larger distances, e.g.,\ni=i0\u0000100. We attribute those tiny e\u000bects to a weak de-\npendence of the local ground state on the non-equilibrium\nstate far from the impurity at t= 250, see right part of\nthe last panel in Fig. 10.\nThe other panels in Fig. 10 demonstrate the mecha-\nnism of the spin reversal. At short times (see t= 60,\nsecond panel) the perturbation of the initial equilibrium\ncon\fguration of the conduction-electron moments is still\nweak. For t= 80 andt= 100 one clearly notices the\nemission of the wave packet starting. Locally, the an-\ntiferromagnetic structure is preserved (see left part) but\nsuperimposed on this, there is an additional spatial struc-\nture of much longer size developing. This \fnally forms\nthe wave packet which is emitted from the central re-\ngion. Its spatial extension is about \u0001 \u0019300 as can be\nestimated for t= 250 (last panel on the right) where\nit covers the region 200 .i.500. The same can be\nread o\u000b from the upper part of Fig. 9. Assuming that\nthe reversal of each of the conduction-electron moments\ntakes about the same time as the reversal of the classi-\ncal spin, \u0001 is roughly given by the reversal time times\nthe Fermi velocity and therefore strongly depends on J10\nandB. For the present case, we have \u001c1\u0019150=Twhich\nimplies \u0001\u0019150\u00022 = 300 in rough agreement with the\ndata.\nIn the course of time, the long-wave length structure\nsuperimposed on the short-range antiferromagnetic tex-\nture develops a node. This can be seen for t= 100 and\ni\u001940 (fourth panel, see dashed line). The node marks\nthe spatial border between the new (right of the node,\ncloser toi0) and the original antiferromagnetic structure\nof the moments and moves away from i0with increasing\ntime.\nAt a \fxed position i, the reversal of the conduction-\nelectron moment hsiittakes place in a similar way as\nthe reversal of the classical spin (see both panels in Fig.\n9 for a \fxed i). During the reversal time, its xandy\ncomponents undergo a precessional motion while the z\ncomponent changes sign. Note, however, that during the\nreversaljhsiijgets much larger than its value in the initial\nand in the \fnal equilibrium state.\nV. EFFECTIVE CLASSICAL SPIN DYNAMICS\nA. Perturbation theory\nEqs. (7) and (9) do not form a closed set of equations of\nmotion but must be supplemented by the full equation of\nmotion (10) for the one-particle conduction-electron den-\nsity matrix. This implies that the fast electron dynamics\nmust be taken into account explicitly even if the spin\ndynamics is much slower. Hence, there is a strong mo-\ntivation to integrate out the conduction-electron degrees\nof freedom altogether and to take advantage from a much\nlarger time step within a corresponding spin-only time-\npropagation method. Unfortunately, a simple e\u000bective\nspin-only action can be obtained in the weak-coupling\n(small-J) limit only.13,14This weak-coupling approxima-\ntion is also implicit to all e\u000bective spin-only approaches\nthat consider the e\u000bect of conduction electrons on the\nspin dynamics.49\nIn the weak- Jlimit the electron degrees of freedom can\nbe eliminated in a straightforward way by using standard\nlinear-response theory:50We assume that the initial state\natt= 0 is given by the conduction-electron system in its\nground state or in thermal equilibrium and an arbitrary\nstate of the classical spin. This may be realized formally\nby suddenly switching on the interaction J(t) at time\nt= 0, i.e.,J(t) =J\u0002(t) and by switching the local \feld\nfrom some initial value Biniatt= 0 to a \fnal value B\nfort >0. The response of the conduction-electron spin\nati0and timet >0 (hsi0it= 0 fort= 0) due to the\ntime-dependent perturbation J(t)S(t) is\nhsi0it=JZt\n0dt0\u0005(ret)(t;t0)\u0001S(t0) (20)\nup to linear order in J. Here, the free ( J= 0) local\nretarded spin susceptibility of the conduction electrons\u0005(ret)(t;t0) is a tensor with elements\n\u0005(ret)\n\u000b\f(t;t0) =\u0000i\u0002(t\u0000t0)h[s\u000b\ni0(t);s\f\ni0(t0)]i; (21)\nwhere\u000b;\f =x;y;z . Using this in Eq. (7), we get an\nequation of motion for the classical spins only,\nd\ndtS(t) =S(t)\u0002B\n\u0000J2S(t)\u0002Zt\n0dt0\u0005(ret)(t\u0000t0)\u0001S(t0) (22)\nwhich is correct up to order J2.\nThis represents an equation of motion for the classical\nspin only. It has a temporally non-local structure and\nincludes an e\u000bective interaction of the classical spin at\ntimeS(t) with the same classical spin at earlier times\nt0< t. In the full quantum-classical theory where the\nelectronic degrees of freedom are taken into account ex-\nactly, this retarded interaction is mediated by a non-\nequilibrium electron dynamics starting at site i0and time\nt0and returning back to the same site i0at timet > t0.\nHere, for weak J, this is replaced by the equilibrium and\nhomogeneous-in-time conduction-electron spin suscepti-\nbility \u0005(ret)(t\u0000t0). Compared with the results of the\nfull quantum-classical theory, we expect that the pertur-\nbative spin-only theory breaks down after a propagation\ntimet\u00181=Jat the latest.\nUsing Wick's theorem,50the spin susceptibility is eas-\nily expressed in terms of the greater and the lesser equi-\nlibrium one-particle Green's functions, G>\nii;\u001b\u001b0(t;t0) =\n\u0000ihci\u001b(t)cy\ni\u001b0(t0)iandG<\nii;\u001b\u001b0(t;t0) =ihcy\ni\u001b0(t0)ci\u001b(t)i, re-\nspectively:\n\u0005(ret)\n\u000b\u000b0(t\u0000t0) = \u0002(t\u0000t0)1\n2\n\u0002Im tr 2\u00022h\n\u001b\u000bG>\ni0i0(t;t0)\u001b\u000b0G<\ni0i0(t0;t)i\n:(23)\nAssuming that the conduction-electron system is charac-\nterized by a real, symmetric and spin-independent hop-\nping matrix Tij(as given by the \frst term of Eq. (2)),\nG>andG<are unit matrices with respect to the spin\nindices. They are easily expressed as explicit functions\nofT(see Ref. 51, for example). With tr ( \u001b\u000b\u001b\u000b0) = 2\u000e\u000b\u000b0,\nwe \fnd\n\u0005(ret)\n\u000b\u000b0(t\u0000t0) = \u0002(t\u0000t0)\u000e\u000b\u000b0Im\n\u0002 \ne\u0000iT(t\u0000t0)\n1 +e\u0000\f(T\u0000\u0016)!\ni0i0 \ne\u0000iT(t0\u0000t)\ne\f(T\u0000\u0016)+ 1!\ni0i0(24)\nfor a conduction-electron system at inverse temperature\n\fand chemical potential \u0016.\nUsing Eq. (24) we have computed the spin suscepti-\nbility for the ground state ( \f=1) of the conduction-\nelectron system. This \fxes the kernel in the integro-\ndi\u000berential equation Eq. (22) which is solved numeri-\ncally by standard techniques.52We again consider the11\n10−1100101102\ntimet-0.50-0.250.000.250.50/vectorSJ= 1.0Sz\nSx\nFIG. 11: (Color online) Components Sx(t) andSz(t) of the\nclassical spin after a sudden switch of the \feld from xto\nzdirection at t= 0. Calculations for B= 1 andJ= 1.\nSolid lines: results of the linear-response dynamics, Eq. (22).\nDashed lines: results of the exact quantum-classical dynamics\nforL= 1001.\nsystem displayed in Fig. 1 with a single classical spin\ncoupled via Jto the central site i0of a chain consisting\nofL= 1001 sites. Finite-size artifacts do not show up\nbeforetmax= 500.\nFigs. 11 and 12 show the resulting linear-response dy-\nnamics of the classical spin after preparing the initial\nstate of the system with the classical spin pointing into\n+xdirection while B= (0;0;B). The external magnetic\n\feld induces a precessional motion of the classical spin:\nthere is a rapid oscillation of its xcomponent (and of its\nycomponent, not shown) with frequency !\u0019B(blue\nlines). Damping is induced by dissipation of energy and\nspin: for large times, the zcomponent aligns to the ex-\nternal \feld (red lines).\nFor weak coupling, up to J= 1 (Fig. 11), there is an al-\nmost perfect agreement between the results of the exact\nquantum-classical dynamics (full lines) and the linear-\nresponse theory (dashed lines) up to the maximum prop-\nagation time tmax= 500. We note that, compared to\nthe full theory, there is a tiny deviation of the linear-\nresponse result for the zcomponent of S(t) visible in\nFig. 11 for times t&10. Hence, on this level of accuracy,\nt1\u001910 sets the time scale up to which the linear-response\ntheory is valid. This may appear surprising as this im-\npliest1J= 10 for the \\small\" dimensionless parameter\nof the perturbation theory. One has to keep in mind,\nhowever, that even if the perturbation is \\strong\", its\ne\u000bects can be rather moderate since only non-adiabatic\nterms\u0018S(t)\u0002S(t0) contribute in Eq. (22).\nForJ= 3, see Fig. 12, damping of the classical spin\nsets in much earlier. Visible deviations of the linear-\nresponse theory from the full dynamics already appear\non a time scale that is almost two orders of magnitude\nsmaller as compared to the case J= 1. A simple rea-\nsoning based on the argument that the dimensionless\nexpansion parameter is t1Jfails as this disregards the\nstrong enhancement of retardation e\u000bects with increas-\n10−1100101\ntimet-0.50-0.250.000.250.50/vectorSJ= 3.0Sz\nSxFIG. 12: (Color online) The same as Fig. 11 but for a di\u000berent\nexchange coupling constant J= 3.\ningJ, which have been discussed in Sec. IV A. These\ne\u000bects make the perturbation much more e\u000bective, i.e.,\nlead to a torque, which is exerted by the conduction elec-\ntrons on the classical spin, growing stronger than linear\ninJ.\nWe conclude that linear-response theory is highly at-\ntractive formally as it provides a tractable spin-only e\u000bec-\ntive theory. On the other hand, substantial discrepancies\ncompared to the full (non-perturbative) theory show up\nas soon as damping e\u000bects become stronger. Note that,\nwith increasing time, these deviations must diminish and\ndisappear eventually since both, the full and the e\u000bec-\ntive theory, predict a fully relaxed spin state for t!1\n{ see lower panel of Fig. 12, for example. At least for\nsimple systems with a single classical spin, as considered\nhere, this implies that the e\u000bective theory provides qual-\nitatively reasonable results.\nB. Landau-Lifschitz-Gilbert equation\nTo derive the LLG equation, the linear-response the-\nory must be further simpli\fed:14,53As is obvious from\nEq. (22), the spin-susceptibility \u0005(ret)(t\u0000t0) can be inter-\npreted as an e\u000bective retarded self-interaction of the spin.\nWe assume that the electron dynamics is much faster\nthan the spin dynamics. On the time scale of the spin\ndynamics, the self-interaction then takes place almost in-\nstantaneously, i.e., the memory kernel \u0005(ret)\n\u000b\u000b0(t\u0000t0) =\n\u000e\u000b\u000b0\u0005(ret)(t\u0000t0) in Eq. (22) is peaked at t0\u0019t. We can\ntherefore approximate S(t0)\u0019S(t) + (t0\u0000t)_S(t) under\nthe integral in Eq. (22). This immediately yields:\ndS(t)\ndt=S(t)\u0002B+\u000b(t)S(t)\u0002dS(t)\ndt; (25)\nwhere\n\u000b(t) =J2Zt\n0d\u001c\u001c\u0005(ret)(\u001c) (26)12\nafter substituting t07!\u001c=t\u0000t0.\nEq. (25) takes the form of the standard LLG equation\nfor a single classical spin [cf. Eq. (1)] if \u000b(t) is replaced\nby\n\u000b\u0011lim\nt!1\u000b(t) =J2Z1\n0dtt\u0005(ret)(t): (27)\nNote that this is a necessary step to arrive at a constant\ndamping parameter which can again be justi\fed by not-\ning that \u0005(ret)(t) is peaked at t= 0.\nBefore proceeding, let us stress that Eq. (27) is, or\nis equivalent to, the standard expression used for com-\nputing the Gilbert damping constant in various studies:\nAfter Fourier transformation,\n\u0005(ret)(!) =Z\ndtei!t\u0005(ret)(t); (28)\none ends up with\n\u000b=\u0000iJ2@\n@!\u0005(ret)(!)\f\f\f\n!=0=J2@\n@!Im \u0005(ret)(!)\f\f\f\n!=0;\n(29)\nwhich has also been derived, e.g., in Refs. 14,54 in dif-\nferent contexts. The frequency-dependent spin correla-\ntion \u0005(ret)(!) in Eq. (29) can be obtained explicitly as\nthe Fourier transform of \u0005(ret)(t) given by Eq. (24). A\nstraightforward calculation yields:\n\u000b=\u0019\n2J2Z\nd!df(!)\nd!Aloc(!)Aloc(!); (30)\nwheref(!) = 1=(exp(\f!)+1) is the Fermi function, and\nAloc(!) =L\u00001X\nk\u000e(!+\u0016\u0000\"(k)) (31)\nforL!1 is the local one-particle spectral function.\nHere we have assumed periodic boundary conditions, i.e.,\nspatial homogeneity, such that the hopping matrix Tis\ndiagonalized by Fourier transformation:\nTij=X\nkUik\"(k)Uy\nkj(32)\nwithUik=L\u00001=2exp(ikRi). The eigenvalues of T\nare given by the tight-binding dispersion \"(k) of the\nconduction-electron Bloch band.\nWithin our simple tight-binding model for the\nconduction-electron system, Eqs. (29) and (30) are equiv-\nalent with Kambersky's breathing Fermi-surface the-\nory, related torque-correlation models and scattering\ntheory and have frequently been used for ab initio as\nwell as model computations of the Gilbert damping\nconstant.15,18,20,21,55{58\nLet us remark that Eq. (30) demonstrates that \u000b<0.\nThis results from the convention for the coupling of the\nmagnetic \feld to the spin, namely H=H(B= 0)\u0000BS\n[see Eq. 2)], which has been adopted here. As a conse-\nquence, the precession of S(t) aroundBis described by\naleft-hand helix, _S=S(t)\u0002B, and thus \u000bmust be\nnegative to describe damping.\n10−1100101102103\ntimet−0.10−0.050.000.050.10t·Πret(t)\nΠret(t)FIG. 13: (Color online) Local retarded spin susceptibility of\nthe conduction electrons \u0005(ret)(t) (red line) and t\u0005(ret)(t)\n(blue) as obtained from Eq. (33) for a one-dimensional\nconduction-electron system with L= 10000 sites and peri-\nodic boundary conditions.\nC. Ill-de\fned Gilbert damping\nThe above discussion shows that Eq. (27) represents\nthe fundamental de\fnition of the Gilbert damping con-\nstant\u000band that the limit t!1 is crucial to recover the\nLLG equation in its standard form. The existence of the\nlong-time limit, however, decisively depends on the long-\ntime behavior of the retarded spin-correlation function\n\u0005(ret)(t). Starting from Eq. (24), this is easily computed\nas\n\u0005(ret)(t) = \u0002(t)Im1\nL2X\nk;pe\u0000i\"(k)t\n1 +e\u0000\f(\"(k)\u0000\u0016)ei\"(p)t\ne\f(\"(p)\u0000\u0016)+ 1:\n(33)\nFig. 13 gives an example for the time-dependence of\n\u0005(ret)(t). The calculations have been done at half-\flling,\n\f=1forL= 104sites. We note that the susceptibility\nis in fact peaked at t= 0. For long times, it oscillates\nwith frequency !\u0005= 4 and decays as 1 =t. This is an im-\nportant observation as it implies that the limit in Eq. (27)\ndoes not exist and that, therefore, the damping constant\n\u000bis ill-de\fned.\nTo analyze the physical origin of the divergent integral,\nwe rewrite the spin susceptibility in Eq. (33) as\n\u0005(ret)(t) = \u0002(t)Im[A(unocc)\nloc(\u0000t)A(occ)\nloc(t)]: (34)\nIts long-time behavior is governed by the long-time be-\nhavior of the Fourier transform\nA(occ;unocc)\nloc(t) =Z\nd!ei!tA(occ;unocc)\nloc(!) (35)\nof the occupied, A(occ)\nloc(!)\u0011f(!)Aloc(!), and of the un-\noccupied,A(unocc)\nloc(!)\u0011f(!)Aloc(!), part of the spectral\ndensity, see Eq. (31).\nFor functions with smooth !dependence, the Fourier\ntransform generically drops to zero exponentially fast if13\nt!1 . A power-law decay, however, is obtained if there\nare singularities of A(occ;unocc)\nloc(!). We can distinguish\nbetween van Hove singularities, which are, e.g., of the\nform/\u0002(!\u0000!0)(!\u0000!0)k(withk>\u00001), and the step-\nlike singularity/\u0002(!\u0000!0) (i.e.k= 0), arising in the\nzero-temperature limit at !0= 0 due to the Fermi func-\ntion. Generally, a singularity of order kgives rise to the\nasymptotic behavior A(occ;unocc)\nloc(t)/t\u00001\u0000k, apart from\na purely oscillatory factor ei!0t. For the present case,\nthe van Hove singularities of A(occ;unocc)\nloc(!) at\u0006!0= 2\nexplain, via Eq. (34) the oscillation of \u0005(ret)(t) with fre-\nquency!\u0005= 2!0= 4.\nGenerally, the location of the van Hove singularity\non the frequency axis, i.e. !0, determines the oscilla-\ntion period while the decay of \u0005(ret)(t) is governed by\nthe strength of the singularity. Consider, as an exam-\nple, the zero-temperature case and assume that there are\nno van Hove singularities. The sharp Fermi edge im-\npliesA(occ;unocc)\nloc(t)/t\u00001, and thus \u0005(ret)(t)/t\u00002. The\nGilbert-damping constant is well de\fned in this case.\nThe strength of van Hove singularities depends on\nthe lattice dimension D.59For a one-dimensional lat-\ntice, we have van Hove singularities with k=\u00001=2, and\nthus \u0005(ret)(t)/t\u00001, consistent with Fig. (13). Here,\nthe strong van Hove singularity dominates the long-time\nasymptotic behavior as compared to the weaker Fermi-\nedge singularity. For D= 3, we have k= 1=2 and\n\u0005(ret)(t)/t\u00003if\f <1while for\f=1the Fermi-\nedge dominates and \u0005(ret)(t)/t\u00002. TheD= 2 case is\nmore complicated: The logarithmic van Hove singularity\n/lnj!jleads to \u0005(ret)(t)/t\u00002. This, however, applies\nto cases o\u000b half-\flling only. At half-\flling the van Hove\nand the Fermi-edge singularity combine to a singularity\n/\u0002(!) lnj!jwhich gives \u0005(ret)(t)/ln2(t)=t2. For \fnite\ntemperatures, we again have \u0005(ret)(t)/t\u00002.\nThe existence of the integral Eq. (27) depends on the\nt!1 behavior and either requires a decay as \u0005(ret)(t)/\nt\u00003or faster, or an asymptotic form \u0005(ret)(t)/ei!0t=t2\nwith an oscillating factor resulting from a non-zero po-\nsition!06= 0 of the van Hove singularity. For the one-\ndimensional case, we conclude that the LLG equation\n(with a time-independent damping constant) is based on\nan ill-de\fned concept. Also, the derivation of Eqs. (29)\nand (30) is invalid in this case as the !derivative and the\ntintegral do not commute. This conclusion might change\nfor the case of interacting conduction electrons. Here one\nwould expect a regularization of van Hove singularities\ndue to a \fnite imaginary part of the conduction-electron\nself-energy.\nVI. CONCLUSIONS\nHybrid systems consisting of classical spins coupled\nto a bath of non-interacting conduction electrons rep-\nresent a class of model systems with a non-trivial real-\ntime dynamics which is numerically accessible on longtime scales. Here we have considered the simplest vari-\nant of this class, the Kondo-impurity model with a clas-\nsical spin, and studied the relaxation dynamics of the\nspin in an external magnetic \feld. As a fundamental\nmodel this is interesting of its own but also makes con-\ntact with di\u000berent \felds, e.g., atomistic spin dynamics in\nmagnetic samples, spin relaxation in spintronics devices,\nfemto-second dynamics of highly excited electron systems\nwhere local magnetic moments are formed due to electron\ncorrelations, and arti\fcial Kondo systems simulated with\nultracold atoms in optical lattices.\nWe have compared the coupled spin and electron dy-\nnamics with the predictions of the widely used Landau-\nLifshitz-Gilbert equation which is supposed to cover the\nregime of weak local exchange Jand slow spin dynamics.\nFor the studied setup, the LLG equation predicts a rather\nregular time evolution characterized by spin precession,\nspin relaxation and eventually reversal of the spin on a\ntime scale\u001cdepending on J(and the \feld strength B).\nWe have demonstrated that this type of dynamics can be\nrecovered and understood on a microscopic level in the\nmore fundamental quantum-classical Kondo model. It is\ntraced back to a non-adiabatic dynamics of the electron\ndegrees of freedom and the feedback of the electronic sub-\nsystem on the spin. It turns out that the spin dynamics\nis essentially a consequence of the retarded e\u000bect of the\nlocal exchange. Namely, the classical spin can be seen as\na perturbation exciting the conduction-electron system\nlocally. This electronic excitation propagates and feeds\nback to the classical spin, but at a later time, and thereby\ninduces a spin torque.\nWe found that this mechanism drives the relaxation of\nthe system to its local ground state irrespective of the\nstrength of the local exchange J. As the microscopic dy-\nnamics is fully conserving, the energy and spin of the\ninitial excitation which is locally stored in the vicinity of\nthe classical spin, must be dissipated into the bulk of the\nsystem in the course of time. This dissipation could be\nuncovered by studying the relaxation process from the\nperspective of the electron degrees of freedom. Dissipa-\ntion of energy and spin takes place through the emission\nof a dispersive spin-polarized wave packet propagating\nthrough the lattice with the Fermi velocity. In this pro-\ncess the local conduction-electron magnetic moment at\nany given distance to the impurity undergoes a reversal,\ncharacterized by precession and relaxation, similar to the\nmotion of the classical spin.\nThe dynamics of the classical spin can be qualitatively\nvery di\u000berent from the predictions of the LLG equation\nfor strongJ. In this regime we found a complex mo-\ntion characterized by oscillations of the angle between\nthe classical spin S(t) and the local conduction-electron\nmagnetic moment at the impurity site hsi0iaround the\nadiabatic value \r=\u0019which takes place on an emergent\nnew time scale.\nIn the weak- Jlimit, the classical spin dynamics is qual-\nitatively predicted correctly by the LLG equation. At\nleast partially, however, this must be attributed to the14\nfact that the LLG approach, by construction, recovers\nthe correct \fnal state where the spin is parallel to the\n\feld. In fact, quantitative deviations are found during\nthe relaxation process. The LLG approach is based on\n\frst-order perturbation theory in Jand on the additional\nassumption that the classical spin is slow. To pinpoint\nthe source of the deviations, we have numerically solved\nthe integro-di\u000berential equation that is obtained in \frst-\norder-in-Jperturbation theory and compared with the\nfull hybrid dynamics. The deviations of the perturbative\napproach from the exact dynamics are found to gradually\nincrease with the propagation time (until the proximity\nto the \fnal state enforces the correct long-time asymp-\ntotics). This is the expected result as the dimensionless\nsmall parameter is Jt. However, with increasing Jthe\ntime scale on which perturbation theory is reliable de-\ncreases much stronger than 1 =Jdue to a strong enhance-\nment of retardation e\u000bects which make the perturbation\nmore e\u000bective and produce a stronger torque.\nGenerally, the perturbation can be rather ine\u000bective in\nthe sense that it produces a torque /S(t)\u0002S(t0) which\nis very weak if the process is nearly adiabatic. This ex-\nplains that \frst-order perturbation theory and the LLG\nequation is applicable at all for couplings of the order of\nhoppingJ\u0018T. For the present study this can also be\nseen as a fortunate circumstance since the regime of very\nweak couplings J\u001cTis not accessible numerically. In\nthis case the spin-reversal time scale gets so large that\nthe propagation of excitations in the conduction-electron\nsubsystem would by a\u000bected by backscattering from the\nedges of the system which necessarily must be assumed\nas \fnite for the numerical treatment.\nFor the one-dimensional lattice studied here, a di-\nrect comparison between LLG equation and the exact\nquantum-classical theory is not meaningful as the damp-\ning constant \u000bis ill-de\fned in this case. We could argue\nthat the problem results from the strength of the vanHove singularities in the conduction-electron density of\nstates which dictates the long-time behavior of the mem-\nory kernel of the integro-di\u000berential equation which is\ngiven by the equilibrium spin susceptibility. As the type\nof the van Hove singularity is characteristic for all sys-\ntems of a given dimension, we can generally conclude that\nthe LLG approach reduces to a purely phenomenological\nscheme in the one-dimensional case. However, it is an\nopen question, which will be interesting to tackle in the\nfuture, if this conclusion is still valid for systems where\nthe Coulomb interaction among the conduction electrons\nis taken into account additionally.\nThere are more interesting lines of research which are\nbased on the present work and could be pursued in the\nfuture. Those include systems with more than a single\nspin where, e.g., the e\u000bects of a time-dependent and\nretarded RKKY interaction can be studied additionally.\nWe are also working on a tractable extension of the the-\nory to account for longitudinal \ructuations of the spins\nto include time-dependent Kondo screening, and the\ncompetition with RKKY coupling, on a time-dependent\nmean-\feld level. Finally, lattice rather than impurity\nvariants of the quantum-classical hybrid model are\nhighly interesting to address the time-dependent phase\ntransitions.\nAcknowledgments\nWe would like to thank M. Eckstein, A. Lichtenstein,\nR. Rausch, E. Vedmedenko and R. Walz for instruc-\ntive discussions. Support of this work by the Deutsche\nForschungsgemeinschaft within the SFB 668 (project B3)\nand within the SFB 925 (project B5) is gratefully ac-\nknowledged.\n1L. D. Landau and E. M. Lifshitz, Phys. Z. Sow. 153(1935).\n2T. Gilbert, Phys. Rev. 100, 1243 (1955).\n3T. Gilbert, Magnetics, IEEE Transactions on 40, 3443\n(2004).\n4A. Aharoni, Introduction to the Theory of Ferromagnetism\n(Oxford University Press, Oxford, 1996).\n5G. Tatara, H. Kohno, and J. Shibata, Physics Reports 468,\n213 (2008).\n6B. Skubic, J. Hellsvik, L. Nordstr om, and O. Eriksson, J.\nPhys.: Condens. Matter 20, 315203 (2008).\n7G. Bertotti, I. D. Mayergoyz, and C. Serpico, Nonlinear\nMagnetization Dynamics in Nanosystemes (Elsevier, Am-\nsterdam, 2009).\n8M. F ahnle and C. Illg, J. Phys.: Condens. Matter 23,\n493201 (2011).\n9R. F. L. Evans, W. J. Fan, P. Chureemart, T. A. Ostler,\nM. O. A. Ellis, and R. W. Chantrell, J. Phys.: Condens.\nMatter 26, 103202 (2014).\n10M. A. Ruderman and C. Kittel, Phys. Rev. 96, 99 (1954).11T. Kasuya, Prog. Theor. Phys. 16, 45 (1956).\n12K. Yosida, Phys. Rev. 106, 893 (1957).\n13M. Onoda and N. Nagaosa, Phys. Rev. Lett. 96, 066603\n(2006).\n14S. Bhattacharjee, L. Nordstr om, and J. Fransson, Phys.\nRev. Lett. 108, 057204 (2012).\n15N. Umetsu, D. Miura, and A. Sakuma, J. Appl. Phys. 111,\n07D117 (2012).\n16V. P. Antropov, M. I. Katsnelson, M. van Schilfgaarde,\nand B. N. Harmon, Phys. Rev. Lett. 75, 729 (1995).\n17V. P. Antropov, M. I. Katsnelson, B. N. Harmon, M. van\nSchilfgaarde, and D. Kusnezov, Phys. Rev. B 54, 1019\n(1996).\n18J. Kune\u0014 s and V. Kambersk\u0013 y, Phys. Rev. B 65, 212411\n(2002).\n19K. Capelle and B. L. Gyor\u000by, Europhys. Lett. 61, 354\n(2003).\n20H. Ebert, S. Mankovsky, D. K odderitzsch, and P. J. Kelly,\nPhys. Rev. Lett. 107, 066603 (2011).15\n21A. Sakuma, J. Phys. Soc. Jpn. 81, 084701 (2012).\n22H.-T. Elze, Phys. Rev. A 85, 052109 (2012).\n23D. Marx and J. Hutter, Ab initio molecular dynamics:\nTheory and Implementation, In:Modern Methods and Al-\ngorithms of Quantum Chemistry , NIC Series, Vol. 1, Ed.\nby J. Grotendorst, p. 301 (John von Neumann Institute\nfor Computing, J ulich, 2000).\n24J. Dajka, Int. J. Theor. Phys. 53, 870 (2014).\n25L. Fratino, A. Lampo, and H.-T. Elze, Phys. Scr. T163 ,\n014005 (2014).\n26M. R. Mahani, A. Pertsova, and C. M. Canali, Phys. Rev.\nB90, 245406 (2014).\n27J. Kondo, Prog. Theor. Phys. 32, 37 (1964).\n28A. C. Hewson, The Kondo Problem to Heavy Fermions\n(Cambridge University Press, Cambridge, 1993).\n29M. Sayad, D. G utersloh, and M. Pottho\u000b, Eur. Phys. J. B\n85, 125 (2012).\n30J. P. Gauyacq and N. Lorente, Surf. Sci. 630, 325 (2014).\n31F. Delgado, S. Loth, M. Zielinski, and J. Fern\u0013 andez-\nRossier, Europhys. Lett. 109, 57001 (2015).\n32M. Sayad, R. Rausch, and M. Pottho\u000b (to be published).\n33R. Wiesendanger, Rev. Mod. Phys. 81, 1495 (2009).\n34G. Nunes and M. R. Freeman, Science 262, 1029 (1993)..\n35S. Loth, M. Etzkorn, C. P. Lutz, D. M. Eigler, and A. J.\nHeinrich, Science 329, 1628 (2010).\n36M. Morgenstern, Science 329, 1609 (2010).\n37S. A. Wolf, D. D. Awschalom, R. A. Buhrman, J. M.\nDaughton, S. von Molnar, M. L. Roukes, A. Y. Chtchelka-\nnova, and D. M. Treger, Science 294, 1488 (2001).\n38A. A. Khajetoorians, J. Wiebe, B. Chilian, and R. Wiesen-\ndanger, Science 332, 1062 (2011).\n39F. Scazza, C. Hofrichter, M. H ofer, P. C. D. Groot,\nI. Bloch, and S. Flling, Nature Physics 10, 779 (2014).\n40G. Cappellini, M. Mancini, G. Pagano, P. Lombardi,\nL. Livi, M. S. de Cumis, P. Cancio, M. Pizzocaro,D. Calonico, F. Levi, et al., Phys. Rev. Lett. 113, 120402\n(2014).\n41A. Heslot, Phys. Rev. D 31, 1341 (1985).\n42M. J. W. Hall, Phys. Rev. A 78, 042104 (2008).\n43K.-H. Yang and J. O. Hirschfelder, Phys. Rev. A 22, 1814\n(1980).\n44M. Lakshmanan and M. Daniel, J. Chem. Phys. 78, 7505\n(1983).\n45J. H. Verner, Numerical Algorithms 53, 383 (2010).\n46E. H. Lieb and D. W. Robinson, Commun. Math. Phys.\n28, 251 (1972).\n47S. Bravyi, M. B. Hastings, and F. Verstraete, Phys. Rev.\nLett. 97, 050401 (2006).\n48R. Kikuchi, J. Appl. Phys. 27, 1352 (1956).\n49S. Zhang and Z. Li, Phys. Rev. Lett. 93, 127204 (2004).\n50A. L. Fetter and J. D. Walecka, Quantum Theory of Many-\nParticle Systems (McGraw-Hill, New York, 1971).\n51M. Balzer and M. Pottho\u000b, Phys. Rev. B 83, 195132\n(2011).\n52W. Press, S. A. Teukolsky, W. T. Vetterling, and B. Flan-\nnery, Numerical Recipes (Cambridge, Cambridge, 2007),\n3rd ed.\n53J. Fransson, Nanotechnology 19, 285714 (2008).\n54E. Simanek and B. Heinrich, Phys. Rev. B 67, 144418\n(2003).\n55V. Kambersk\u0013 y, Phys. Rev. B 76, 134416 (2007).\n56A. Brataas, Y. Tserkovnyak, and G. E. W. Bauer, Phys.\nRev. Lett. 101, 037207 (2008).\n57S. Mankovsky, D. K odderitzsch, G. Woltersdorf, and\nH. Ebert, Phys. Rev. B 87, 014430 (2013).\n58D. Thonig and J. Henk, New J. Phys. 16, 013032 (2014).\n59N. W. Ashcroft and N. D. Mermin, Solid State Physics\n(Holt, Rinehart and Winston, New York, 1976)."    },    {        "title": "1508.00629v2.A_Critical_Analysis_of_the_Feasibility_of_Pure_Strain_Actuated_Giant_Magnetostrictive_Nanoscale_Memories.pdf",        "content": " 1 A Critical Analysis  of the Feasibility of Pure Strain -Actuated Giant Magnetostrictive \nNano scale Memories  \nP.G. Gowtham1, G.E. Rowlands1, and R.A. Buhrman1 \n1Cornell University, Ithaca, New York, 14853, USA \n \nAbstract  \n  Concepts for memories  based on the  manipulation  of giant magnetostrictive \nnano magnets  by stress pulses  have garnered recent attention  due to their potential for ultra -low \nenergy operation in the high storage density limit . Here we discuss the feasibility of making such \nmemories in light of the fact that the Gilbert damping of such materials is typically quite  high. \nWe report the results of numerical simulations for  several classes of  toggle precessional and non -\ntoggle dissipative  magnetoelastic  switching modes. M aterial candidates  for each o f the several \nclasses are analyzed  and f orms for the anisotropy energy density and range s of material \nparameters appropriate for each material class are employed.   Our study indicates that the Gilbert \ndamping  as well as the anisotropy and demagnetization energies are all crucial for  determining \nthe feasibility of  magnetoelastic toggle -mode precessional switching schemes.  The role s of \nthermal stability  and thermal fluctuations for stress -pulse switching of gia nt magnetostrictive \nnanomagnets are also discussed in detail and are shown to be important in the viability, design, \nand footprint of magnetostrictive switching schemes.  \n \n \n \n  2 I. Introduction  \n \nIn recent years pure electric -field based control of magnetization has become a subject of \nvery active research. It has been demonstrated in a variety of systems ranging from multiferroic \nsingle phase materials, gated dilute ferromagnetic semiconductors 1–3, ultra -thin metallic \nferromagnet/oxide interfaces 4–10 and piezoelectric /magnetoelastic composites 11–15. Beyond the \ngoal of establishing an understanding of the physics involved in each of these systems, this work \nhas been strongly motivated by the fact that  electrical -field based manipulation of magnetization \ncould form the basis for a new generation of ultra -low power, non -volatile memories.  Electric -\nfield based magnetic devices are not necessarily limited by Ohmic losses during the write cycle \n(as can be the case in current based memories such as spin -torque magnetic random access \nmemory (ST -MRAM) ) but rather by the capacitive charging/decharging energies incurred per \nwrite cycle. As the capacitance of these devices scale with area the write energies have the \npotential to be as low as 1 aJ per write cycle or less.  \nOne general approach to the electrical control of magnetism utilizes a magnetostrictive \nmagnet/piezoelectric transducer hybrid as the active component of a nanoscale memory element. \nIn this appro ach a mechanical strain is generated by an electric field within the piezoelectric \nsubstrate or film and is then transferred to a thin, nanoscale magnetostrictive magnet that is \nformed on top of the piezoelectric. The physical interaction driving the write  cycle of these \ndevices is the magnetoelastic interaction that describes the coupling between strain in a magnetic \nbody and the magnetic anisotropy energy. The strain imposed upon the magnet creates an \ninternal effective magnetic field via the magnetoelast ic interaction that can exert a direct torque \non the magnetization. If successfully implemented this torque can switch the magnet from one \nstable configuration to another, but whether imposed stresses and strains can be used to switch a  3 magnetic element be tween two bi -stable states depend s on the strength of the magnetoelastic \ncoupling (or the magnetostriction). Typical values of the magnetostriction ( = 0.5 -60 ppm) in \nmost ferromagnets yield strain and stress scales that make the process of strain -induced \nswitching inefficient or impossible. However, considerable advances have been made in \nsynthesizing materials both in bulk and in thin film form that have magnetostrictions that are one \nto two orders of magnitude larger than standard transition metal ferrom agnets.  These giant \nmagnetostrictive materials allow the efficient conversion of strains into torque on the \nmagnetization.  However it is important to note that a large magnetostrictive (or magnetoelastic) \neffect tends to also translate into very high magnetic damping by virtue of the strong coupling \nbetween magnons and the phonon thermal bath, which has important implicati ons, both positive \nand negative, for piezoelectric based magnetic devices.   \nIn this paper we provide an analysis of the switching modes  of several different \nimplementations of piezoelectric/magnetostrictive devices.  We discuss how the high damping \nthat is generally associated with giant magnetoelasticity affects the feasibility of different \napproaches, and we also take other key material  properties into consideration, including the \nsaturation magnetization of the magnetostrictive element, and the form and magnitude of its \nmagnetic anisotropy. Th e scope of th is work excludes device concepts and physics \ncircumscribed by magneto -elastic mani pulation of domain walls in magnetic films, wires, and \nnanoparticle arrays 11,12,16. Instead we focus  here on analyzing various magnetoelastic reversal \nmodes, principally within the single domain approximation, but we do extend this work to \nmicromagnetic modeling in cases where it is not clear that the macrospin approximation prov ides \na fully successful description of the essential physics. We enumerate potential material \ns 4 candidates for each of the modes evaluated and discuss the various challenges inherent in \nconstructing reliable memory cells based on each of the reversal modes t hat we consider.  \nII. Toggle -Mode Precessional Switching  \n \nStress pulsing of a magnetoelastic element can be used to construct a toggle mode \nmemory. The toggling mechanism between two stable states relies on transient dynamics of the \nmagnetization that are initi ated by an abrupt change in the anisotropy energy that is of fixed and \nshort duration. This change in the anisotropy is created by the stress pulse and under the right \nconditions can generate precessional dynamics about a new effective field. This effectiv e field \ncan take the magnetization on a path such that when the pulse is turned off the magnetization \nwill relax to the other stable state. This type of switching mode is referred to as toggle switching \nbecause the same sign of the stress pulse will take t he magnetization from one state to the other \nirrespective of the initial state. We can divide the consideration of the toggle switching modes \ninto two cases; one that utilizes a high \nsM  in-plane magnetized element, and the other that \nemploys perpendicular magnetic anisotro py (PMA) materials with a lower  \nsM. We make this \ndistinction largely because of differences in the structure of the torques and stress fields required \nto induce a switch in these two class es of systems.  The switching of in -plane giant \nmagnetostrictive nanomagnets with sizeable out -of-plane demagnetization fields relies on the use \nof in-plane uniaxial stress -induced effective fields that overcome the in -plane anisotropy (~O( 102 \nOe)). The mo ment will experience a torque canting the moment out of plane and causing \nprecession about the large demagnetization field. Thus the precessional time scales for toggling \nbetween stable in -plane states will be largely determined by the d emagnetization fiel d (and thus \nsM\n). The dynamics of this mode bears striking resemblance to the dynamics in hard -axis field  5 pulse switching of nanomagnets 17. On the other hand, the dominant energy scale in PMA giant \nmagnetostrictive materials is the perpendicular anisotropy energy. This energy scale can vary \nsubstantially (anywhere from \nuK  ~ 105-107 ergs/cm3) depending on the materials utilized and the \ndetails of their growth. The anisotropy energy scale in these materials can be tuned into a region \nwhere stress -induced anisotropy energies can be comparable to it. A biaxial stress -induced \nanisotropy energy, i n this geometry, can induce switching by cancelling and/or overcoming the \nperpendicular anisotropy energy. As we shall see, this fact and the low\nsM  of these systems \nimply dynamical time scales that are substantially different from the case where in -plane \nmagnetized materials are employed.  \nA. In-Plane Magnetized Magnetostrictive Materials  \n \nWe first treat the macrospin switching dynamics of an in -plane magnetized \nmagnetostrictive nanomagnet with uniaxial anisotropy under a simple rectangular uniaxial stress \npulse. Giant magnetostriction in in -plane magnetized systems have been demonstrated for \nsputtered polycrystalline Tb 0.3Dy0.7Fe2 (Terfenol -D) 18, and more recently in quenched Co xFe1-x  \nthin film systems 19. We assume that the uniaxial anisotropy is defined completely by the shape \nanisotropy of the elliptical element and that any magneto -crystalline anisotropy in the film is \nconsiderably weaker. This is a reasonabl e assumption for the materials considered here in the \nlimit where the grain size is considerably smaller than the nanomagnet’s dimensions. The stress \nfield is applied by voltage pulsing an anisotropic piezoelectric film that is in contact with the \nnanomagn et. The proper choice of the film orientation of a piezoelectric material such as <110> \nlead magnesium niobate -lead titanate (PMN -PT) can ensure that an effective uniaxial in -plane \nstrain develops along a particular crystalline axis after poling the piezo in the z -direction. We  6 assume that the nanomagnet major axis lies along such a crystalline direction (the <110> -\ndirection of PMN -PT) so that the shape anisotropy is coincident with the strain axis  (see Figure 1 \nfor the relevant geometry) . For the analysis below we use material values appropriate to \nsputtered, nanocrystalline Tb0.3Dy0.7Fe2 18 (\nsM= 600 emu/cm3, \ns = 670 ppm is the saturation \nmagnetostriction). Nanocrystalline Tb0.3Dy0.7Fe2  films, with a mean crystalline grain diameter \ngraind\n < 10 nm, can  have an extremely high magnetostriction while being relatively magnetica lly \nsoft with coercive fields,  \ncH  ~ 50-100 Oe, results which can be achieved by thermal processing \nduring sputter growth at T  ~ 375 ºC 20. The nanomagnet dimensions were as sumed to be 80 nm \n(minor axis) × 135 nm (major axis) × 5 nm (thickness) yielding a shape anisotropy field \n4 ( )k y x sH N N M\n = 323 Oe and \n4 ( )demag z y sH N N M = 5.97 kOe. We use \ndemagnetization factors that are correct for an elliptical cylinder 21. \nThe value of the Gilbert damping parameter \n  for the magnetostrictive element is quite \nimportant in determining its dynamical behavior during in -plane stress -induced toggle switching. \nPrevious simulation results 22–24 used a value (\n0.1  for Terfenol -D) that, at least arguably, is \nconsid erably lower than is reasonable since that value was extracted from spin pumping in a Ni \n(2 nm) /Dy(5 nm) bilayer 25. However, that bilayer material is not a good surrogate for a rare -\nearth transition -metal alloy (especially for \n0L  rare earth ions). In the latter case the loss \ncontribution from direct magnon to short w avelength phonon conversion is important, as has \nbeen directly confirmed by studies of \n0L  rare earth ion doping into transition metals 26,27. For \nexample in -plane magnetized nanocrystalline 10% Tb -doped Py shows \n~ 0.8  when magnetron \nsputtered at 5 mtorr Ar pressure, even though the magnetostriction is small within this region of \nTb doping 27. We contend that a substantial increase in the magnetoelastic interaction in alloys  7 with higher Tb content is likely to make \n even larger. Magnetization rotation in a highly \nmagnetostrictive magnet will efficiently generate longer wavelength acoustic  phonons as well \nand heat loss will be generated when these phonons thermalize. Unfortunately, measurements of \nthe magnetic damping parameter in polycrystalline Tb0.3Dy0.7Fe2 do not appear to be available in \nthe literature. However, some results on the amo rphous Tb x[FeCo] 1-x system, achieved by using \nrecent ultra -fast demagnetization techniques, have extracted  \n~ 0.5 for compositions (x  ~ 0.3) \nthat have high magnetostriction 28. We can also estimate the scale for the Gilbert damping by \nusing a formalism that takes into account direct magnon to long wavelength phonon conversion  \nvia the magnetoelastic interaction  and subsequent phonon relaxation to the thermal phonon \nbath29. The damping can be estimated by the following formula:  \n \n2\n2236 1 1\n22s\nsT s L s\neff ex eff exMc M c M\nAA\n\n\n\n \n   \n            \n(1) \n \nUsing \nsM  = 600 emu/cm3, the exchange stiffness \nexA = 0.7x10-6 erg/cm, a mass density ρ \n= 8.5 g/cm3, Young’s modulus of 65 GPa 30, Poisson ratio \n0.3 , and an acoustic damping time \n\n= 0.18 ps 29 the result is an estimate of  \n~1 . Given the uncertainties in the various  parameter s \ndetermining the Gilbert damping , we examine the magnetization dynamics for values of  \n\nranging from 0.3 to 1.0.  \n We simulate the switching dynamics of the magnetic moment of a Terfenol -D \nnanomagnet at T=300 K using the Landau -Lifshitz -Gilbert form of the equation  describing the \nprecession of a magnetic moment  \nm:  8  \n( ) ( )eff eff eff Langevinddttdt dt       mmm H m H m   \n(2) \n \nwhere\neff  is the gyromagnetic ratio. As Tb0.3Dy0.7Fe2 is a rare earth – transition metal (RE-TM) \nferrimagnet (or more accurately a speromagnet), the gyromagnetic ratio cannot simply be \nassumed to be the free electron value. Instead we use the value\neff  = 1.78 107 Hz/Oe as \nextracted from a spin wave resonance study in the TbFe 2 system 31 which appears appropriate \nsince Dy and Tb are similar in magnetic moment/atom (10\nB  and 9\nB  respectively) and g factor \n( ~4/3 and ~3/2 respectively).  \nThe first term in Equation (2) represents the torque on the magnetization from any \napplied fields, the effective stress field, and any anisotropy and demagnetization fields that might \nbe present. The third term in the LLG represents the damping torque that acts to relax the \nmagnetization towards the direction of the effective field and hence damp out precessional \ndynamics. The second term is the Gaussian -distributed Langevin field that takes into account the \neffect thermal fluctuations on the magnetization dynamics. From the fluctuation -dissipation \ntheorem, \n2RMS B\nLangevin\neff skTHM V t\n  where \nt  is the simulation time -step 32. Thermal fluc tuations \nare also accounted for in our modeling by assuming that the equilibrium azimuthal and polar \nstarting angles (\n0  and \n0 /2  respectively) have a random mean fluctuation given by \nequipartition as  \n00 2\n2RMS BkT\nEV\n\n\n\n and \n0 24 ( )RMS B\nz y skT\nN N M V  . A \nbiasH   of 100 Oe was \n 9 used for our simulations which creates two stable energy minima at  \n0arcsin ~ 18bias\nkH\nH\n\n and \n1162\n symmetric about \n/2 . This non -zero starting angle ensures that \n00RMS  . \nThis field bias is essential as the initial torque from a stress pulse depends on the initial starting \nangle. This angular dependence generates much larger thermally -induced fluctu ations in the \ninitial torque than a hard -axis field pulse. The hard axis bias field also reduces the energy barrier \nbetween the two stable states. For Hbias = 100 Oe the energy barrier between the two states is Eb \n= 1.2 eV yielding a room temperature \n/bBE k T = 49. This ensures the long term thermal \nstability required for a magnetic memory.  \nTo incorporate the effect of a stress pulse in Equation (2) we employ a free energy form \nfor the effective field,  \n( ) /efftE  Hm that expresses the effect of a stress pulse along the x -\ndirection of our in -plane nanomagnet with a uniaxial shape anisotropy in the x -direction. The \nstress enters the energy as an effective in -plane anisotropy term that adds to the shape anisotropy \nof the  magnet (first term in Equation (3) below). The sign convention here is such that \n0\nimplies a tensile stress on the x -axis while \n0 implies a compressive strain. We also include \nthe possibility of a bias field applied along the hard axis in the final term in Equation (3).     \n \n22\n223( , , ) [2 ( ) ( )]2\n2 ( )x y z y x s s x\nz y s z bias s yE m m m N N M t m\nN N M m H M m \n  \n     \n(3) \n \n The geometry that we have assumed allows only for fast compressive -stress pulse based \ntoggle mode switching. The application of a DC compressive stress along the x -axis only reduces \nthe magnitude of the anisotropy and changes the position of the equilibriu m magnetic angles \n0 10 and \n10180\n  while keeping the potential wells associated with these states symmetric as \nwell. Adiabatically increasing the value of the compressive stress moves the angles toward \n/2\n until \n3()2sutK  but obviously can never induce a magnetic switch.  \n Thus the magnetoelastic memory in this geometry must make use of the transient \nbehavior of the magnetization under a stress pulse as opposed to re lying on quasistatic changes \nto the energy landscape. A compressive stress pulse where \n3()2sutK  creates a sudden \nchange in the effective field. The resultant effective field\n32ˆsu\neff y bias\nsKmHM  \nHy  \npoints in the y -direction and causes a  torque that brings the magnetization out of plane. At this \npoint the magnetization rotates rapidly about the very large perpendicular demagnetization field\nˆ 4demag s z Mm Hz\n and if the pulse is turned off at the right time will relax down to the \nopposite state at \n1  = 163. Such a switching trajectory for our simulated nanomagnet is shown in \nthe red curve in  Figure 2. This mode of switching is set by a minimum characteristic time scale\n1~ 7.54sw\nspsM\n, but the precession time will in general be longer than \nsw  for moderate \nstress pulse amplitudes,  \n( ) 2 / 3us tK , as the magnetization then cants out of plane enough to \nsee only a fraction of the maximum possible  \ndemagH . Larger stress pulse amplitudes result in \nshorter pulse duratio ns being required as the magnetization has a larger initial excursion out of \nplane. For pulse durations that are longer than required for a rotation (blue and green curves \nin Figure 2) \nm will exhibit damped elliptical precession about  \n/2 . If the stress is released \nduring the correct portion of any of these subsequent precessional cycles the magnetization \n180\n 11 should relax down to the \n1  state [blue curve in  Figure 2], but otherwise it will relax down to the \noriginal state [green curve in  Figure 2].  \nThe prospect of a practical device working reliably in the long pulse regime appears to be \nrather poor. The high damping of giant magnetostrictive magnets and the large field scale of the \ndemagnetization field yield very stringent pulse timing requirements  and fast damping times for \nequilibration to \n/2 . The natural time scale for magnetization damping in the in -plane \nmagnetized thin film case is  \n1\n2d\nsM  , which ranges from 50 ps down to 15 ps for\n0.3 1\n with \nsM = 600 emu/cm3. This high damping also results in the influence of thermal \nnoise on the magnetization dynamics being quite strong since  \nLangevinH  . Thus large stress \nlevels with extremely short pulse durations are required in order to rotate the magnetization \naround the \n/2 minimum within the damping time, and to keep the precession amplitude \nlarge enough that the magnetization  will deterministically relax to the reversed state. Our \nsimulation results for polycrystalline Tb0.3Dy0.7Fe2 show  that a high stress pulse amplitude of\n85 MPa\nwith a pulse duration ~ 65  ps is required if  \n0.5  (Figure 3a). However, the \npulse duration window for which the magnetization will deterministically switch is extremely \nsmall in this case (<5 ps). This is due to the fact that the precession amplitude about the \n/2  \nminimum at this damping gets small enough that thermal fluctuations allow only a very small \nwindow for which switching is reliable. For the lowest damping that we consider reasonable to \nassume,  \n0.3 , reliable switching is possible between \npulse ~ 30-60 ps at \n85 MPa . At a \nlarger damping \n0.75  we find that the switching is non -deterministic for all pulse widths as \nthe magnetization damps too quickly; instead very high stresses , \n200 MPa are required to \n1 12 generate deterministic switching of the magnetization with a pulse duration w indow \npulse ~ 25-\n45 ps ( Figure 3b).  \nGiven the high value of the expected damping we have also simulated the magnetization \ndynamics in the Landau Lifshitz (LL) form:  \n \n2(1 ) ( ( ) ( ))LL eff Langevinddttdt dt       mmm H H m   \n(4) \n \nThe LL form and the LLG form are equivalent in low damping limit (\n1 ) but they \npredict different dynamics at higher damping values.  Which of these norm -preserving forms for \nthe dynamics has the right damping form is still a subject of debate 33–37. As one increases α in \nthe LL form the precessional speed is kept the same while the damping is assumed to affect only \nthe rate of decay of the precession amplitude. The damping in the LLG dynamics, on the other \nhand, is a viscosity term and retards the pre cessional speed. The effect of this retardation can be \nseen in the LLG dynamics as the precessional cycles move to longer times as a function of \nincreasing damping. Our simulations show that the LL form (for fixed \n ) predicts highe r \nprecessional speeds than the LLG and hence an even shorter pulse duration window for which \nswitching is deterministic than the LLG, ~12 ps for LL as opposed to ~ 30 ps for LLG ( Figure \n3c). \nThe damping clearly plays a crucial role in the stress amplitude scale and pulse duration \nwindows for which deterministic switching is possible, regardless of the form used to describe \nthe dynamics. Even though the magnetostriction of Tb 0.3Dy0.7Fe2 is high and the stress required \nto entirely overcome the anisotropy energy is only 9.6 MPa, the fast damping time scale and \nincreased thermal noise (set by the large damping and the out -of-plane demagnetization) means  13 that the stress -amplitude that is required to achieve deterministic toggle switching is 10 -20 times \nlarger. In addition, the pulse duration for in -plane toggling must be extremely short, with typical \npulse durations of 10 -50 ps with tight time windows of 20 -30 ps within which the acoustic pulse \nmust be turned off. Given ferroelectric switching rise times on the order of ~50 ps extracted from \nexperiment38 and considering the acoustical resonant response of the entire piezoelectric / \nmagnetostrictive nanostructure and acoustic ringing and inertial terms in the lattice dynamics, \ngeneration of such large stresses with the strict pulse time requirem ents needed for switching in \nthis mode is likely unfeasible.  In addition, the stress scales  required to successfully toggle switch \nthe giant magnetostrictive nanomagnet in this geometry are nearly as high or even higher than \nthat for transition metal ferromagnets such as Ni (\n~ 38 ppms with \n0.045 ). For example, \nwith a 70 nm × 130 nm elliptical Ni nanomagnet with a thickness of 6 nm and a hard axis bias \nfield of 120 Oe we should obtain  switching at stress values \n = +95 MPa and \npulse  = 0.75 ns. \nTherefore  the use of giant magnetostrictive nanomagnets with high damping in this toggle mode \nscheme confers no clear advantage over the use of a more conventional transition metal \nferromagnet, and in neither case does this approach appear particularly viable for t echnological \nimplementation.  \nB. Magneto -Elastic Materials with PMA: Toggle Mode Switching  \n \nCertain amorphous sputtered RE/TM alloy films with perpendicular magnetic anisotropy \nsuch as a -TbFe 2 39–42 and a - Tb0.3Dy0.7Fe2 43 have  properties that may make these materials \nfeasible for use in  stress -pulse toggle switching. In certain composition ranges they exhibit large \nmagnetostriction (\ns  > 270 ppm for a -TbFe 2, and both  \ns  and the effective out of plane  14 anisotropy can be tuned over fairly wide ranges by varying the process gas pressure during \nsputter deposition, the target atom -substrate incidence angle, and the substrate temperature.  \nWe consider the energy of such an out -of-plane magnetostrictive material under the \ninfluence of a magnetic field  \nbiasH  applied in the \nˆx  direction and a pulsed biaxial stress:  \n \n223( , , ) [ 2 ( )]2u\nx y z s s biaxial z s bias xE m m m K M t m M H m        \n(5) \n \nSuch a biaxial stress could be applied to the magnet if it is part of a patterned [001] -poled PZT \nthin film/ferromagnet bilayer. A schematic of this device geometry is depicted in Figure 4.When\n0biasH\n, it is straightforward to see the stress pulse will not result in reliable switching since, \nwhen the tensile biaxial stress is large enough, the out of plane anisotropy becomes an easy -plane \nanisotropy and the equator presents a zero -torque condition on t he magnetization, resulting in a \n50%, or random, probability of reversal when the pulse is removed. However, reliable switching \nis possible for \n0biasH  since that results in a finite canting of \nm towards the x -axis. This \ncanting is required for the same reasons a hard -axis bias field was needed for the toggle \nswitching of an in -plane magnetized element as discussed previously.  A pulsed biaxial stress \nfield can then in principle lead to deterministic precessional toggle switching between  the +z and \n–z energy minima . This mode of pulsed switching is analogous to voltage pulse switching in the \nultra-thin CoFeB|MgO using the voltage -controlled magnetic anisotropy effect.5,8 Previous \nsimulation results have also di scussed  this class of  macrospin  magnetoelast ic switc hing in the \ncontext  of a Ni|Barium -Titatate  multilayer44 and a zero -field, biaxial stress -pulse induced  toggle \nswitching  scheme  taking advantage of micromagnetic  inhomogeneities has recently appeared in \nthe literature45. Here we discuss biaxial stress -pulse  switching for a broad class of  giant  15 magnetostrictive  PMA  magnets where we argue  that the monodomain limit strictly applies \nthroughout the switching process and extend past previous macrospin modeling by \nsystematically think ing about  how pulse -timing requirements  and critical write stress amplitudes \nare determined  by the damping, the PMA strength, and \nsM  for values  reasonable for these \nmaterials.  \nFor our simulation study of stress -pulse toggle switching of a PMA magnet, we \nconsidered a Tb 33Fe67 nanomagnet with an \nsM  = 300 emu/cm3, \neffK  = 4.0×105 ergs/cm3 and \ns  \n= 270 ppm.  To estimate the appropriate value for the damping parameter we noted that ultrafast \ndemagnetization measurements on Tb 18Fe82 have yielded  \n0.27 . This 18 -82 composition lies \nin a region where the magnetostriction is moderate (\ns ~50 ppm) 43 so we assumed that the \ndamping will be on the same order or higher for a -TbFe 2 due to its high magnetostriction. \nTherefore we ran simulations for the range of  \n= 0.3 -1. For the gyromagnetic ratio  we used\neff  \n= 1.78×107 s-1G-1 which  is appropriate for a -TbFe 2 31. We assumed an effective exchange \nconstant \n611 10effA erg cm   46 implying an exchange length  \nexeff no stress\neffAlK\n = 15.8 nm (in \nthe absences of an applied str ess) and \n22exeff pulse\nsAlM = 13.3 nm (assuming that the stress pulse \namplitude is just enough to cancel the out of plane anisotropy). A monodomain crossover \ncriterion of \ncd ~  ~ 56 nm  (with the pulse off) and \ncd ~\n22ex\nsA\nM ~ 47 nm  (with the pulse \non) can be calculated by considering the minimum length -scale associated with supporting \nthermal λ/2 confined spin wave modes 47. The important point here is that the low \nsM  of these \nsystems ensures that the exchange length is still fairly long even during the switching process, \n4ex\nuA\nK 16 which suggests that the macrospin approximation should be valid for describing the switching \ndynamics of this system for reasonably sized nanomagnets.  \nWe simulated a circular element with a diameter of 60 nm and a thickness of 10 nm, \nunder an x -axis bias field, \nbiasH = 500 Oe which creates an initial canting angle of 11 degrees \nfrom the  vertical  (z-axis).  This starting angle is sufficient to enable deterministic toggle \nprecessional switching between the +z and –z minima via biaxial stress pulsing.  The assumed \ndevice geometry, anisotropy energy density and bias field corresponded to an energy barrier \nbE   \n= 4.6  eV for thermally activated reversal, and hence a room temperature thermal stability factor  \n\n = 185.  \nWe show selected results of the macrospin simulations of stress -pulse toggle switching of \nthis modeled TbFe 2 PMA nanomagnet. Typical switching trajectories are shown in Figure 5a. The \nswitching transition can be divided into two stages (see Figure 5b): the precessional stage that \noccurs when the stress field is applied, during which the dynamics of the magnetization are \ndominated by precession about the effective field that arises from the sum of the bias field and \nthe easy -plane anisotropy field \n3 ( ) 2eff\ns\nz\nstKmM , and the dissipative stage that begins when the \npulse is turned off and where the large \neffK and the large \n  result in a comparatively quick \nrelaxation to the other energy minimum. Thus most of the switching process is spent in the \nprecessional phase and the entire switching process is not much longer than the actual stress \npulse duration.  For pulse amplitudes a t or not too far above the critical stress for reversal,\n2 / 3eff\ns K\n the two relevant timescales for the dynamics are set approximately by the \nprecessional period\n1/ 100 pssw bias H of the nanomagnet and the damping time 17 \n~ 2 /d bias H  . Both of these timescales are much longer than the timescales set by precession \nand damping about the demagnetization field in the in -plane magnetized toggle switching case. \nThe result is that even with quite high damping one can have reliable s witching over much \nbroader pulse width windows, 200 -450 ps . (Figure 6a,b). The relatively large pulse duration \nwindows within which reliable switching is possible (as compared to the in -plane toggle mode) \nhold for both the LL and LLG damping. However, the diffe rence between the two forms is \nevident in the PMA case ( Figure 6c). At fixed \n , the LLG damping predicts a larger pulse \nduration window than the LL damping. Also the effective viscosity implicit within the LLG \nequation ensures that the switching time scales are slower than in the LL case as can also be seen \nin Figure 6c. \nAn additional and important point concerns the factors that determine the critical \nswitching amplitude. In the in -plane toggle mode switching of the previous section, it was found \nthat the in-plane anisotropy field was not the dominant factor in determining the stress scale \nrequired to transduce a deterministic toggle switch. Instead, we found that the stress scale was \nalmost exclusively dependent on the need to generate a high enough preces sion \namplitude/precession speed during the switching trajectory so as to not be damped out to the \ntemporary equilibrium at \n/2  (at least within the damping range considered). This means \nthat the critical stress scale to transduce a deterministic switch is essentially determined by the \ndamping. We find that the situation is fundamentally different for the PMA based toggle \nmemories.  The critical amplitude \nc  is nearly independent of the damping from a range of \n0.3 0.75\n up until \n~1 where the damping is sufficiently high (i.e. damping times equaling \nand/or exceeding the p recessional time scale) that at \n85  MPa the magnetization traverses \ntoo close to the minimum at \n/2 ,\n0 . The main reason for this difference between the  18 PMA toggle based memories and the in-plane toggle based memory lies in the role that the \napplication of stress plays in the dynamics. First, in the in -plane case, the initial elliptical \namplitude and the initial out of plane excursion of the magnetization is set by the stress pulse \nmagnitu de. Therefore the stress has to be high to generate a large enough amplitude such that the \ndamping does not take the trajectory too close to the minimum at which point Langevin \nfluctuations become an appreciable part of the total effective field. This is n ot true in the PMA \ncase where the initial precession amplitude about the bias field is large and the effective stress \nscale for initiating this precession about the bias field is the full cancellation of the perpendicular \nanisotropy.  \nSince the minimum stre ss-pulse amplitude required to initiate a magnetic reversal in out -\nof-plane toggle switching scales with \neffK in the range of damping values considered, lowering \nthe PMA of the nanomagnet is a straightforward way to reduce the stress and write energy \nrequirements for this type of memory cell. Such reductions can be achieved by strain engineering \nthrough the choice of substrate, base electrode and transducer layers, by the choice of deposition \nparameters, and/or by post -growth annealing  protocols. For example growing a TbFe 2 film with a \nstrong tensile biaxial strain can substantially lower \neffK . If the P MA of such a nanomagnet can \nbe reliably r educed to \neffK = 2105 ergs/cm3 our simulations indicate that this would result in \nreliable pulse toggle switching at \n ~ -50 MPa (corresponding to a strain amplitude on the TbFe 2 \nfilm of less than 0.1%) with \npulse  ≈ 400 ps, for 0.3 ≤ \n  ≤  0.75 and \nbiasH ~ 250  Oe . Electrical \nactuation of this level of stress/strain in the sub -ns regime, while challenging, may  be possible to \nachieve.48 If we again assume \nsM  =300 emu/cm3, a diameter of 60 nm and a thickness of  10 nm, \nthis low PMA nanomagnet would still have a high thermal stability with  \n92 . The challenge, \n 19 of course, is to consistently and uniformly control the residual strain  in the magnetostrictive \nlayer. It is important to note that no  such tailoring (short of systematically lowering the damping) \ncan exist in the in -plane toggle mode case.  \nIII. Two -State Non -Toggle Switching  \n \nSo far we have discussed toggle mode switching where the same polarity strain pulse is \napplied to reverse the magnetization between two bi -stable states. In this case the strain pulse \nacts to create a temporary field around which the magnetization precesse s and the pulse is timed \nso that the energy landscape and magnetization relax the magnetization to the new state with the \ntermination of the pulse.  Non -toggle mode magneto -elastic switching differs fundamentally \nfrom the precessional dynamics of toggle -mode switching, being an example of dissipative \nmagnetization dynamics where a strain pulse of one sign destabilizes the original state (A) and \ncreates a global energy minimum for the other state (B). The energy landscape and the damping \ntorque completely de termine the trajectory of the magnetization and the magnetization \neffectively “rolls” down to its new global energy minimum. Reversing the sign of the strain pulse \ndestabilizes state B and makes state A the global energy minimum – thus ensuring a switch ba ck \nto state A. There are some major advantages to this class of switching for magneto -elastic \nmemories over toggle mode memories. Precise acoustic pulse timing is no longer an issue. The \nswitching time scales, for reasonable stress values, can range  from q uasi-static to nanoseconds. \nIn addition, the large damping typical of magnetoelastic materials does not present a challenge \nfor achieving robust switching trajectories in deterministic switching as it does in toggle -mode \nmemories. Below we will discuss det erministic switching for magneto -elastic materials that have \ntwo different types of magnetic anisotropy.   20 C. The Case of Cubic Anisotropy  \n \nWe first consider magneto -elastic materials with cubic anisotropy under the influence of a \nuniaxial stress field pulse. T here are many epitaxial Fe -based magnetostrictive materials that \nexhibit a dominant cubic anisotropy when magnetron -sputter grown on oriented C u underlayers \non Si or on MgO,  GaAs , or PMN -PT substrates. For example, Fe 81Ga19 grown on MgO [100] or \non GaAs ex hibit a cubic anisotropy 49–51. Given the low cost of these Fe -based materials \ncompared to rare -earth alloys, it is worth investigating whether such films can be used to \nconstruct a two state memory. Fe 81Ga19 on MgO exhibits easy axes along <100>. In ad dition, \nepitaxial Fe 81Ga19 films have been found to have a reasonably high magnetostriction λ100=180 \nppm making them suitable for stress induced switching. If we assume that the  cubic \nmagnetoelastic thin-film nanomagnet has circular cross section, that the  stress field is applied by \na transducer along the [100] direction , and that a bias field  is applied  at \n4  degrees, the  \nmagnetic free energy is :  \n \n2 2 2 2 2\n11\n2( , ) (1 ) 2 ( )\n3( ) ( )2 2x y x y z z z s z\ns bias\nx y s xE m m K m m K m m N N M m\nMHm m t m\n    \n  \n   \n(6) \n \nEquation (6) shows that, in the absence of a bias field, the anisotropy energy is 4 -fold \nsymmetric in the film -plane. It is rather easy to see that it is im possible to make a two -state non -\ntoggle switching with a simple cubic anisotropy energy and uniaxial stress field along [100].  \nFigure 7a shows the free energy landscape described by Equation (6) without stress applied. To \ncreate a two -state deterministic magnetostrictive device ,  \nbiasH  needs to be strong enough to \neradicate the energy minima at \n and \n3 / 2  which strictly requires that  \n1 0.5 /bias sH K M .  21 Finite temperature considerations can lower this minimum bias field requirement considerably. \nThis is due to the fact that the bias field can make the lifetime to escape the energy minima in th e \nthird quadrant and fourth qua drant small and the energy bar rier to return them from the energy \nminima in the first quadrant extremely large.  We arbitrarily set this requirement for the bias \nfield to correspond to a lifetime of 75 μs. The typical energy barriers to hop from back to the \nmetastable minima in the thi rd and fourth quadrant for device volumes we will consider are on \nthe order of several eV.  \nThe requirement for thermal stability of the two minima in the first quadrant , given a \ndiameter\nd  and a thickness \nfilmt  for the nanomagnet,  sets an upper  bound on  \nbiasH  as we require \n/ 40bbE k T \n at room temp erature between the two states (see Figure 7c). It is desirable that \nthis upper bound is high enough that there is some degree of tolerance to the value of the bias \nfield at device dimensions that are employed. This sets requirement s on the minimum volume of \nthe cylindical  nanomagnet that are dependent on\n1K . \nFor a circular element with \nd  = 100 nm, \nfilmt  = 12.5 nm and  \n1K= 1.5 105 ergs/cm3, two -\nstate non -toggle switching with the required thermal stability can only occur for \nbiasH  between \n50 - 56 Oe. This is too small a range of acceptable bias fields. However , by increasing \nfilmt  to 15 \nnm the bias field range grows to \nbiasH = 50 - 90 Oe wh ich is an acceptable range. For\n1K = \n2.0×105 erg/cm3 with \nd= 100 nm and \nfilmt = 12.5 nm , there is an appreciable region of bias field \n(~65-120 Oe) for which \n/barrier BE k T  > 42. For\n1K  = 2.5 105 ergs/cm3, the bias range goes from \n90 – 190 Oe for the same volume.  The main po int here is that, given the scale  for the cubic \nanisotropy in Fe 81Ga19, careful attention must be paid to the  actual  values of the anisotropy \n\n 22 constants, device lateral dimensions, film thickness, and the exchange bias strength in order  to \nensure device stability  in the sub -100 nm diameter  regime .  \n We now discuss the dynamics for a simulated case where  \nd = 100 nm,  \nfilmt= 12.5 nm,  \n1K\n= 2.0×105 ergs/cm3, \nbiasH  = 85 Oe, and \nsM  = 1300 emu/cm3.  Two stable minima exist at \n\n=10o and \n = 80o. Figure 7b shows the effect of the stress pulse on the energy landscape.  When \na compressive stress \nc  is applied, the potential minimum at \n =10o is rendered unstable \nand the magnetization follows the free energy gradient to \n = 80o (green curve). Since the stress \nfield is applied along [100] the magnetization first switches to a minima very close to but greater \nthan \n = 80o and when the stress is released it gently relaxes down to the zero stress minimum at \n\n= 80o. In order to switch from \n = 80o  to \n = 10o we need to reverse the sign of the applied \nstress field to tensile (red curve). A memory constructed on these principles is thus non -toggle.  \nThe magnetization -switching trajectory is simple and follows the dissipative dynamics \ndictated by the free energy landscape (see Figure 8a). We have assumed a damping of \n0.1  \nfor the Fe 81Ga19 system,  based on previous measurements52 and as confirmed by our own. Higher \ndamping only ends up speeding up the sw itching and ri ng-down process. Figure 8b shows the \nsimulated stress amplitude and pulse switching probability phas e diagram at room temperature.  \nUltimately, we must take the macrospin estimates for device parameters as  only a roug h \nguide. The macrospin dynamics approximate the true micromagnetics less and less well as the \ndevice diameter gets larger. The mai n reason for this is the large\nsM  of Fe 81Ga19 and the \ntendency of the magnetization to curl at the sample edges. Accordingly we have  performed T  = 0 \nºK micromagnetic simulations in  OOMMF.53 An exchange bias field \nbiasH  = 85 Oe was applied  23 at \n  = 45º and we assume \n1K  = 2.0×105 ergs/cm3,  \nsM  = 1300 emu/cm3, and \nexA = 1.9 × 10-6 \nerg/cm. Micromagnetics show that the macrospin picture quantitatively captures the switching \ndynamics, the angular positions of the stables states (\n0~ 10 and\n1~ 80 ) and the critical \nstress amplitude at (\n ~ 30 MPa) when the device diameter \nd < 75 nm. The switching is \nessentially a rigid in -plane rotation of the magnetization from  \n0 to \n1 . However, we cho se to \nshow the switching for  an element with \nd  = 100 nm because it allowed for thermal stability of \nthe devices in a region of thicknes s (\nfilmt  = 12-15 nm) where \nbiasH ~ 50-100 Oe at room \ntemperature could be reasonably expected. The initial average magnetization angle is larger (\n0~ 19\nand \n1~ 71 ) than would b e predicted by macrospin for a \nd = 100 nm element. \nThis is due to the magnetization c urling at the devices edges at\nd = 100 nm (see Figure 8c). \nDespite the fact that magnetization profile differs from the macrospin picture we find that there \nis no appreciable difference between the stress scales required for switching , or the basi c \nswitching mechanism.  \nThe stress amplitude scale for writing the simulated Fe 81Ga19 element at ~ 30 MPa is not \nexcessively high and  there are essentially no demands on the acoustic pulse width requirements.  \nThese memories can thus be written at pulse amplitudes of ~ 30 MPa with acoustical pulse \nwidths of ~ 10 ns. These numbers do not represent a major challenge from the acoustical \ntransduction point of view.  The drawback s to this scheme are  the necessity of growing high \nquality single crystal thin film s of Fe 81Ga19 on a piezoelectric substrate that can generate large \nenough strain to switch the magnet (e.g. PMN -PT) and difficulties  associated with  tailoring the \nmagnetocrystalline anisotropy \n1K  and ensuring thermal stability at low lateral device \ndimensions.   24 D. The Case of Uniaxial Anisotropy   \n \nLastly we discuss deterministic (non -toggle) switching of an in -plane giant \nmagnetostrictive magnet with uniaxial anisotropy. In -plane magnetized polycrystalline TbDyFe \npatterned into ellipti cal nanomagnets could serve as a potential candidate material in  such a \nmemory scheme. To implement deterministic switching in this geometry a bias field  \nbiasH  is \napplied along the hard axis of the nanomagnet. This generates two stable minima at \n0 and \n0 180\n symmetric about the hard axis. The axis of the stress pulse then needs to be non -\ncollinear with respect to the e asy axis in order to break the symmetry of the potential wells and \ndrive the transition to the selected equilibrium position. Figure 9 below shows a schematic of the \nsituation. When a stress pulse is applied in the direction that makes an angle\n with respect to the \neasy axis of the nanom agnet,  \noo0 90 , the free energy within the macrospin approximation \nbecomes:  \n \n2 2 2 2\n2( , , ) [2 ( ) 2 ( )\n3( ) (cos( ) sin( ) )2x y z y x s x z y s z bias s y\ns y x\nsE m m m N N M m N N M m H M m\nt m mM\n      \n     \n(7) \n \nFrom Equation (7) it can be seen that a sufficiently strong compressive stress pulse can switch \nthe magnetization between \n0  and \no\n0 180 , but only if \n0 is between\n and . To see why \nthis condition is necessary, we look at the magnetization dynamics in the high stress limit when \n0 0\n.  During such a strong pulse the magnetization will s ee a hard axis appear at\n  \nand hence will rotate towards the new easy axis at \n90 , but when the stress pulse is \no90 25 turned off the magnetization will equilibrate back to \n0 . This situation is represented by the \ngreen trajectory shown in  Figure 11a.   \nBut when \no\n090 , a sufficiently strong compressive stress pulse defines a new easy \naxis close to \no90  and when the pulse is turned off the magnetization will relax to\n0 180\n (blue trajectory in Figure 11a). Similarly the possibility of switching from \no180  \nto \nwith a tensile  strain depends on whether \no o o90 180 90     . Thus\no45 is the \noptimal situation as then the energy landscape becomes mirror symmetric about the hard axis and \nthe amplitude of the required switching stress (voltage) are equal. This scheme is quite similar to \nthe case of deterministic switching in biaxial anisotropy systems (with the coordinate system \nrotated by ). We note that a set of papers54–56 have previously proposed this particular case as \na candidate for non -toggle magnetoelectric memory  and have experimentally demonstrated \noperation of such a memory in the large feature -size (i.e. extended film ) limit .55  \nWe argue here that in-plane giant magnetostrictive magnets operated in the non -toggle \nmode could be a good candidate for  construct ing memories with  low write stress amplitude, and \nnanosecond -scale write time operation. However , as we will discuss , the prospects of this type of \nswitching mode  being suitable  for implementation in ultrahigh density memory appear to be  \nrather poor. The m ain reason for this lies in the hard axis bias field requirements for maintaining \nlow write error rates and the effect that such a hard axis bias field will have on  the long term \nthermal stability  of the element . At T = 0 ºK the requirement on  \nbiasH  is only that it be strong \nenough that  \n0 > 45º. However, this is no longer sufficient  at finite temperature where thermal \nfluctuations impl y a thermal, Gaussian distribution of the initial orientation of the magnetization \no45 26 direction \n0 about  \n0. If a significant componen t of this angular distribution falls below 45 \ndegrees there will be a high write error rate. Thus we must ensure that \nbiasH  is high enough that \nthe probability of \n  < 45º is extremely low. We have selected the re quirement that \n  < 45º  is a \n8\n event  where \n  is the standard deviation of \n about  \n0 and is  given by the relation\n. However,  \nbiasH  must be low enough to be technologically feasible,  but also \nmust not exceed a value that compromises the energy barrier between the two potential minima – \nthus rendering the nanomagnet thermally unstable . These minimum and maximum  requirement s \non \nbiasH  puts significant constraints on the minimum size of the nanomagnet that can be used in \nthis device approach. It also sets some rather tight requirements on the hard axis bias field, as we \nshall see.  \nWe first disc uss the effects of these requirements  in the case of a relatively large \nmagnetostrictive  device.  We assume the use  of a polycrystalline Tb 0.3Dy0.7Fe2 element having \nsM\n = 600 emu/cm3 and an elliptical cross section of 400×900 nm2 and a thickness \nfilmt  = 12.5 \nnm. This results in a  shape anisotropy field \nkH  ≈ 260 Oe. We find that for an applied hard axis \nbias field \nbiasH  ~ 200 Oe, a field strength that can be reasonably engineered on -chip, the \nequilibrium angle of the  element is \n0 ≈ 51º and its root mean square (RMS)  angular fluctuation  \namplitude is \nRMS ≈ 0.75º. Thus element ’s anisotropy field  and the assumed hard axis biasing \ncondition s just satisfy  the assumed requirement that  \n08RMS   > 45º (see Figure 10b). The \nmagnetic energy barrier to thermal energy ratio for the element at \nbiasH  = 200 Oe is \n/bBE k T\n02\n2BkT\nEV\n\n\n\n 27 ≈ 350, which  easily  satisf ies the long-term thermal  stability  requirement  (see Figure 10a), and \nwhich also provides some latitude for the use of a slightly higher\nbiasH  if desired to further reduce \nthe write error rate .  \nIt is straightforward to see from these numbers that if the area of the magnetostrictive \nelement is substantially reduced below 400 ×900 nm2 there must be a corresponding increase in \nkH\n  and hence in\nbiasH if the write error rate for the device is to remain acceptable.  Of course an \nincrease in the thickness of the element can partially reduce the increase in fluctuation amplitude \ndue to the decrease in the magnetic a rea, but the feasible range of thickness variation cannot \nmatch the effect of, for example, reducing the cross -sectional area by a factor of 10 to 100, with \nthe latter, arguably, being the minimum required for high density memory applications.    While \nperhaps a strong shape anisotropy and an increased \nfilmt  can yield the required \nkH  ≥ 1 kOe, the \nfact that in this deterministic mode of magnetostrictive switching we must also have \nbiasH  ~ \nkH  \nresults in a bias field requirement that is not technologically feasible.  We could of course allow \nthe write error rate to be much larger than indicated by an 8\nfluctuation probability, but this \nwould only relax the requirement on \nbiasH  marginally, which always must be such that \n0  > \n45o.Thus the deterministic  magneto strictive  device  is not a viable  candidate for ultra -high density \nmemory.  Instead this approach is only feasible for device s with lateral area  ≥ 105 nm2 . \nWhile the requir ement of a large footprint is a limitation of the deterministic  \nmagneto strictive  memory element , this device does have the significant advantage that the stress \nscale required to switch the memory is quite low. We have simulated T = 300 ºK macrospin \nswitching dynamics for a 400×900 nm2 ellipse with thickness \nfilmt  = 12.5 nm with \nbiasH = 200 Oe \nsuch that \n0 ~ 51º.  The Gilbert damping parameter was set to  \n0.5 and magnetostriction  \ns =  28 670 ppm. The magnetization switches by simple rotation from \n0  = 51º to \n1129\n that is \ndriven by the stress pulse induced change in the energy landscape (see Figure 11a). Phase \ndiagram results are provided in Figure 11b where the switching from  \n0 = 51º to \n1 = 129 º \nshows a 100% switching probability for stresses as low as \n  = - 5 MPa for pulse widths as short \nas 1 ns.  \nSince the dimensions of the ellipse are large enough that t he macrospin picture is  not strictly \nvalid, we have also conducted T = 0 K micromagnetic simulations of the stress -pulse induced \nreversal in this geometry. We find that the trajectories are essentially well described by a quasi -\ncoherent rotation with  non-uniformities in the magnetization being more pronounced at the \nellipse edges (see Figure 11c). The minimum stress pulse amplitude for swi tching is even lower \nthan that predicted by macrospin at \n  = - 3 MPa. This stress scale for switching is substantially \nlower than any of the switching mode schemes discussed before. Despite the fact that this \nscheme is not scalable down into the 100 -200 nm size regime, it can be  appropriate for larger \nfootprint memori es that can be written at very low write stress pulse amplitudes.  \nIV. CONCLUSION  \n   \nThe physical properties of giant magnetostrictive magnets (particularly of the rare -earth \nbased TbFe 2 and Tb 0.3Dy0.7Fe2 alloys) place severe restrictions on the viability of such materials \nfor use in fast, ultra -high density , low energy consumption  data storage. We have enumerated the \nvarious potential problems that might arise from the characteristically high damping of giant \nmagnetostrictive nanoma gnets in toggle -mode switch ing. We have also discussed  the rol e that \nthermal fluctuation s have on the various switching modes and the challenges involved in  29 maintaining long -time device thermal stability that arise mainly from the necessity of employing \nhard axis bias fields .  \nIt is clear that the task of constructing a reliable memory using pure stress induced \nreversal of g iant magnetostrictive magnets will be , when pos sible, a question of trade -offs and \ncareful engineering . PMA based giant magnetostrictive nanomagnets can be made extremely \nsmall (\nd  < 50 nm) while still maintaining thermal stability.  The small diameter and low cross -\nsectional area of these PMA giant magnetostrictive devices could , in principle,  lead to very low \ncapacitive write energies.   The counterpoint is that  the stress fields required to switch the device \nare not  necessarily  small and the acoustical pulse timing  requirements  are demanding.  However, \nit might  be possible t o tune the magnetostriction  \ns ,\nK , and \nsM  (either by adjustment of the \ngrowth conditions of the magnetostrictive magnet or by engineering the RE-TM multilayers \nappropriately) in order to significantly reduce the pulse amplitudes required f or switching (down \ninto the 20-50 MPa  range) and reduce th e required in -plane bias field – without compromising \nthermal stability of the bit . Such tuning must be carried out carefully. As we have discussed , the \nGilbert dampi ng \n, \ns ,\nK , and \nsM  can all affect the pure stress -driven switching process  and \ndevice thermal stability  in ways that are certainly interlinked and not necessarily complementary.   \nTwo state  non-toggle memories  such as we described in Section  III D  could have  extremely low \nstress write amplitudes and non-restrictive pulse requirements . However, the trade -off arises \nfrom thermal stability considerations and such a switching scheme is not scala ble down into the \n100-200 nm size regime . Despite this limitation there  may well be a place for  durable  memories \nwith very low write stress pulse amplitudes  and low write energies that operate reliably in the \nnanosecond regime .  30 ACKNOWLEDGEMENTS  \nWe thank  R.B. van Dover,  W.E. Bailey, C. Vittoria,  J.T. Heron, T. Gosavi, and S. Bhave \nfor fruitful discussions.  We also thank D.C. Ralph and T. Moriyama for comments and \nsuggestions on the manuscript. This work was supported by the Office of Naval Research and the \nArmy Research Office.   \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n  31 REFERENCES  \n \n1 H. Ohno, D. Chiba, F. Matsukura, T. Omiya, E. Abe, T. Dietl, Y. Ohno, and K. Ohtani, Nature \n408, 944 (2000).  \n2 D. Chiba, M. Sawicki, Y. Nishitani, Y. Nakatani, F. Matsukura, and H. Ohno, Nature 455, 515 \n(2008).  \n3 D. Chiba, M. Yamanouchi, F. Matsukura, and H. Ohno, Science 301, 943 (2003).  \n4 M. Weisheit, S. Fähler, A. Marty, Y. Souche, C. Poinsignon, and D. Givord,  Science 315, 349 \n(2007).  \n5 T. Maruyama, Y. Shiota, T. Nozaki, K. Ohta, N. Toda, M. Mizuguchi, A.A. Tulapurkar, T. \nShinjo, M. Shiraishi, S. Mizukami, Y. Ando, and Y. Suzuki, Nat. Nanotechnol. 4, 158 (2009).  \n6 M. Endo, S. Kanai, S. Ikeda, F. Matsukura, and H. Ohno, Appl. Phys. Lett. 96, 212503 (2010).  \n7 W.-G. Wang, M. Li, S. Hageman, and C.L. Chien, Nat. Mater. 11, 64 (2012).  \n8 Y. Shiota, T. Nozaki, F. Bonell, S. Murakami, T. Shinjo, and Y. Suzuki, Nat. Mater. 11, 39 \n(2012).  \n9 T. Nozaki, Y. Shiota, S. Miwa, S. Murakami, F. Bonell, S. Ishibashi, H. Kubota, K. Yakushiji, \nT. Saruya, A. Fukushima, S. Yuasa, T. Shinjo, and Y. Suzuki, Nat. Phys. 8, 492 (2012).  \n10 P. Khalili Amiri, P. Upadhyaya, J.G. Alzate, and K.L. Wang, J. Appl. Phys. 113, 013912 \n(2013).  \n11 T. Br intlinger, S. -H. Lim, K.H. Baloch, P. Alexander, Y. Qi, J. Barry, J. Melngailis, L. \nSalamanca -Riba, I. Takeuchi, and J. Cumings, Nano Lett. 10, 1219 (2010).  \n12 D.E. Parkes, S.A. Cavill, A.T. Hindmarch, P. Wadley, F. McGee, C.R. Staddon, K.W. \nEdmonds, R.P. Campion, B.L. Gallagher, and A.W. Rushforth, Appl. Phys. Lett. 101, 072402 \n(2012).  \n13 M. Weiler, A. Brandlmaier, S. Geprägs, M. Althammer, M. Opel, C. Bihler, H. Huebl, M.S. \nBrandt, R. Gross, and S.T.B. Goennenwein, New J. Phys. 11, 013021 (2009).  \n14 M. We iler, L. Dreher, C. Heeg, H. Huebl, R. Gross, M.S. Brandt, and S.T.B. Goennenwein, \nPhys. Rev. Lett. 106, 117601 (2011).  \n15 M. Weiler, H. Huebl, F.S. Goerg, F.D. Czeschka, R. Gross, and S.T.B. Goennenwein, Phys. \nRev. Lett. 108, 176601 (2012).   32 16 V. Novosad,  Y. Otani, A. Ohsawa, S.G. Kim, K. Fukamichi, J. Koike, K. Maruyama, O. \nKitakami, and Y. Shimada, J. Appl. Phys. 87, 6400 (2000).  \n17 H. Schumacher, C. Chappert, P. Crozat, R. Sousa, P. Freitas, J. Miltat, J. Fassbender, and B. \nHillebrands, Phys. Rev. Lett.  90, 017201 (2003).  \n18 P.I. Williams, D.G. Lord, and P.J. Grundy, J. Appl. Phys. 75, 5257 (1994).  \n19 D. Hunter, W. Osborn, K. Wang, N. Kazantseva, J. Hattrick -Simpers, R. Suchoski, R. \nTakahashi, M.L. Young, A. Mehta, L.A. Bendersky, S.E. Lofland, M. Wuttig, and I. Takeuchi, \nNat. Commun. 2, 518 (2011).  \n20 K. Ried, M. Schnell, F. Schatz, M. Hirscher, B. Lu descher, W. Sigle, and H. Kronmüller, \nPhys. Status Solidi 167, 195 (1998).  \n21 M. Beleggia, M. De Graef, Y.T. Millev, D.A. Goode, and G. Rowlands, J. Phys. D. Appl. \nPhys. 38, 3333 (2005).  \n22 K. Roy, S. Bandyopadhyay, and J. Atulasimha, Phys. Rev. B 83, 2244 12 (2011).  \n23 K. Roy, S. Bandyopadhyay, and J. Atulasimha, J. Appl. Phys. 112, 023914 (2012).  \n24 M. Salehi Fashami, K. Roy, J. Atulasimha, and S. Bandyopadhyay, Nanotechnology 22, \n309501 (2011).  \n25 J. Walowski, M.D. Kaufmann, B. Lenk, C. Hamann, J. McCord,  and M. Münzenberg, J. Phys. \nD. Appl. Phys. 41, 164016 (2008).  \n26 W. Bailey, P. Kabos, F. Mancoff, and S. Russek, IEEE Trans. Magn. 37, 1749 (2001).  \n27 S.E. Russek, P. Kabos, R.D. McMichael, C.G. Lee, W.E. Bailey, R. Ewasko, and S.C. \nSanders, J. Appl. Phys. 91, 8659 (2002).  \n28 Y. Ren, Y.L. Zuo, M.S. Si, Z.Z. Zhang, Q.Y. Jin, and S.M. Zhou, IEEE Trans. Magn. 49, 3159 \n(2013).  \n29 C. Vittoria, S.D. Yoon, and  A. Widom, Phys. Rev. B 81, 014412 (2010).  \n30 Q. Su, J. Morillo, Y. Wen, and M. Wuttig, J. Appl. Phys. 80, 3604 (1996).  \n31 S.M. Bhagat and D.K. Paul, AIP Conf. Proc. 29, 176 (1976).  \n32 W.F. Brown, Phys. Rev. 130, (1963).  \n33 W.M. Saslow, J. Appl. Phys. 105, 07D315 (2009).   33 34 M. Stiles, W. Saslow, M. Donahue, and A. Zangwill, Phys. Rev. B 75, 214423 (2007).  \n35 N. Smith, Phys. Rev. B 78, 216401 (2008).  \n36 S. Iida, J. Phys. Chem. Solids 24, 625 (1963).  \n37 T.L. Gilbert, IEEE Trans. Magn. 40, 3443 (2004).  \n38 J. Li, B. Nagaraj, H. Liang, W. Cao, C.H. Lee, and R. Ramesh, Appl. Phys. Lett. 84, 1174 \n(2004).  \n39 R.B. van Dover, M. Hong, E.M. Gyorgy, J.F. Dillon, and S.D. Albiston, J. Appl. Phys. 57, \n3897 (1985).  \n40 F. Hellman, Appl. Phys. Lett. 64, 1947 (1994).  \n41 F. Hellman, R.B. van Dover, and E.M. Gyorgy, Appl. Phys. Lett. 50, 296 (1987).  \n42 T. Niihara, S. Takayama, and Y. Sugita, IEEE Trans. Magn. 21, 1638 (1985).  \n43 P.I. Williams and P.J. Grundy, J. Phys. D Appl. Phys. 27, 897 (1994).  \n44 M. Ghidini, R. Pellicel li, J.L. Prieto, X. Moya, J. Soussi, J. Briscoe, S. Dunn, and N.D. \nMathur, Nat. Commun. 4, 1453 (2013).  \n45 J.-M. Hu, T. Yang, J. Wang, H. Huang, J. Zhang, L. -Q. Chen, and C. -W. Nan, Nano Lett. 15, \n616 (2015).  \n46 F. Hellman, A.L. Shapiro, E.N. Abarra, R.A. Robinson, R.P. Hjelm, P.A. Seeger, J.J. Rhyne, \nand J.I. Suzuki, Phys. Rev. B 59, 408 (1999).  \n47 J.Z. Sun, R.P. Robertazzi, J. Nowak, P.L. Trouilloud, G. Hu, D.W. Abraham, M.C. Gaidis, \nS.L. Brown, E.J. O’Sullivan, W.J. Gallagher, and D.C. Worledge, Phys. Re v. B 84, 064413 \n(2011).  \n48 A. Grigoriev, D.H. Do, D.M. Kim, C.B. Eom, P.G. Evans, B. Adams, and E.M. Dufresne, \nAppl. Phys. Lett. 89, 1 (2006).  \n49 A. Butera, J. Gómez, J.L. Weston, and J.A. Barnard, J. Appl. Phys. 98, 033901 (2005).  \n50 A. Butera, J. Gómez,  J.A. Barnard, and J.L. Weston, Phys. B Condens. Matter 384, 262 \n(2006).  \n51 A. Butera, J.L. Weston, and J.A. Barnard, J. Magn. Magn. Mater. 284, 17 (2004).   34 52 J. V. Jäger, A. V. Scherbakov, T.L. Linnik, D.R. Yakovlev, M. Wang, P. Wadley, V. Holy, \nS.A. Cavill, A. V. Akimov, A.W. Rushforth, and M. Bayer, Appl. Phys. Lett. 103, 032409 \n(2013).  \n53 M.J. Donahue and D.G. Porter, OOMMF User’s Guide, Version 1.0, Inter agency Report \nNISTIR 6376, National Institute of Standards and Technology, Gaithersburg, MD  (1999).  \n54 N. Tiercelin, Y. Dusch, V. Preobrazhensky, and P. Pernod, J. Appl. Phys. 109, 07D726 (2011).  \n55 Y. Dusch, N. Tiercelin, A. Klimov, S. Giordano, V. Preobrazhensky, and P. Pernod, J. Appl. \nPhys. 113, 17C719 (2013).  \n56 S. Giordano, Y. Dusch, N. Tiercelin, P. Pernod, and V. Preobrazhensky, J. Phys. D. Appl. \nPhys. 46, 325002 (2013).  \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n  35  \n \n \nFigure 1. Magnetoelastic elliptical memory element schematic with associated coordinate system for in -\nplane stress -pulse induced toggle switching.  Here  \nM is the magnetization  vector  with \n and \n being \npolar and azimuthal angles . For the in -plane t oggle switching case, the initial normalized magnetization  \n0 0 0ˆˆ cos sinm x y\n and is in the film plane with \n0arcsin[ / ]bias kHH   and \nˆbias bias H Hy . \n \n \n \nFigure 2. Toggle switching trajectory for an in -plane magnetized polycrystalline Tb 0.3Dy 0.7Fe2 element \nwith \nLLG  = 0.3, \n  = -120 MPa, and \npulse  = 50 ps (red) and 125 ps (blue) and 160 ps (green).  \n 36  \nFigure 3. a) Effect of the Gilbert damping on pulse switching probability statistics for\n  = -85 MPa. b) \nEffect of increasing stress pulse amplitude for high damping  \nLLG  = 0.75. Very high stress pulses ( >200 \nMPa) are required to allow precession to be fast enough to cause a switch before dynamics are damped \nout. c) Comparison of switching statistics for the LL and LLG dynamics at \n  = -200 MPa, \n  = 0.75. \nThe LL dynamics exhibits faster precession than the LLG for a given torque implying shorter windows of \nreliability and requirements for faster pulses.  \n \n \nFigure 4.  Schematic of TbFe 2 magnetic element under biaxial stress generated by a PZT layer.  \nHere the initial normalized magnetization  \n0 0 0ˆˆ cos sinm z x   is predominantly out of the \nfilm plane with a cant  \n0arcsin[ / ]bias kHH   in the x -direction provided by  \nˆbias bias H Hx . \n  \n 37  \n \nFigure 5. a) Switching trajectories for a TbFe 2 nanomagnet under a pulsed biaxial stress \n  = -85 MPa, \npulse\n = 400 ps ( green ) and \n  = -120 MPa and\npulse  = 300 ps (blue ) b) Switching trajectory time \ntrace for {m x,my,mz} for \n  = -85 MPa . The pulse is initiated at t  = 500 ps. The blue region \ndenotes when precession about \nbiasH  dominates (i.e. while the pulse is on) and the red when the \ndissipative dynamics rapidly damp the system down to the other equilibrium point.  \n \n 38 Figure 6. a) Dependence of the simulated pulse switching probability on \n  for \n  = -85 MPa . b) \nDependence of pulse switching probability on stress amplitude. Stress -induced switching is possible even \nfor \n  = 1.0. c) Comparison of pulse switching probability  for LL and LLG dynamics for \n = -85 MPa \nand \n  = 0.75. Here the difference between the LL and LLG dynamics has a significant effect on the \nwidth of the pulse window where reliable switching is predicted by the simulations (\nLL  = 200 ps and \nLLG\n=320 ps.)  \n \n \nFigure 7. a) Energy (normalized to \n1K ) landscape as a function of angle for various values of exchange \nbias energy. b)  \n= 80º (\n= 10 º) is the only stab le equilibrium for compressive ( tensi le) stress. \nDissipative dynamics and the free energy landscape then dictate the non -toggle switching dynamics. c) \nShows the energy barrier dependence  on the [110] bias field for a \nd   = 100 nm, \nfilmt = 12.5 nm circular \nelement with (curve 1) \n1K = 2.5x105 ergs/cm3, (curve 2) \n1K = 2.0×105 ergs/cm3, and ( curve 4) \n1K\n=1.5×105 ergs/cm3. Curve 3 shows the energy barrier dependence for  \n1K=1.5x105 ergs/cm3 and \nd  = 100 \nnm & \nfilmt  = 15 nm . \n \n \n \n \n 39  \nFigure 8. a) Magnetoelastic switching trajectory for Fe 81Ga19 with \n  = -45 MPa and \npulse = 3 ns. The \nmain part of the switching occurs within 200 ps. The magnetization relaxes to the equilibrium defined \nwhen the pulse is on and then relaxes to the final equilibrium when the pulse is turned off. b) Switchin g \nprobability phase diagram for Fe 81Ga19 with biaxial anisotropy at T  = 300 ºK. c)  T = 0 ºK OOMMF \nsimulations showing the equilibrium m icromagnetic configuration for \n1K  = 2×105 ergs/cm3 and \nsM  = \n1300 emu/cm3. Subsequent shots show the rotational switching mode for a 45 MPa uniaxial compressive \nstress along [100].  Color scale is blue -white -red indicating the local projection \n1xm  (blue), \n0xm\n(white), \n1xm (red).  \n  \n 40  \n \n \nFigure 9. Schematic of magnetostrictive device geometry that utilizes uniaxial anisotropy to achieve \ndeterministic switching. Polycrystalline Tb 0.3Dy 0.7Fe2 on PMN -PT with 1 axis oriented at angle \n  with \nrespect to the easy axis.  In this geometry, \nM  lies in the x -y plane (film -plane) with the normalized  \nˆˆ cos sinm x y\n.  \n  \n 41  \n \nFigure 10. a) In-plane shape anisotropy field (\nkH ) and hard axis bias field (\nbiasH ) for a 400×900 nm2 \nellipse as a function of film thickness required to ensure \n0  = 51º . Thermal stability parameter\n  plotted \nversus film thickness with\nkH , \nbiasH  such that  \n0 = 51º .  b) Eight  times the RMS angle fluctuation \nabout three different  average \n0 > 45º versus film thickness for a 400×900 nm2 ellipse at T = 300 ºK. \n \n \n 42 Figure 11. a) Magnetization trajectories for\n = 45º, \n= -5 MPa ,\npulse = 3 ns, with ~ 200 Oe \nyielding \n0  = 51º  ( red) and\n = 45º,\n = -20 MPa with \nbiasH  = 120 Oe yielding  \n0 = 28º ( green). b) T = \n300 ºK stress pulse (compressive) switching prob ability phase diagram for a 400×90 0 nm2 ellipse with \nfilmt\n = 12.5 nm , \n= 45º, \n0 = 51º c) Micromagneti c switching trajectory of a 400×90 0 nm2 ellipse under \na DC compressive stress of -3 MPa transduced along 45 degrees.  Color scale is blue -white -red indicating \nthe local projection \n1xm  (blue), \n0xm (white), \n1xm (red).  \n \n \n \nbiasH"    },    {        "title": "1508.01427v1.Large_spin_wave_bullet_in_a_ferrimagnetic_insulator_driven_by_spin_Hall_effect.pdf",        "content": "arXiv:1508.01427v1  [cond-mat.mes-hall]  6 Aug 2015Large spin-wave bullet in a ferrimagnetic insulator driven by spin Hall effect\nM. B. Jungfleisch,1,∗W. Zhang,1J. Sklenar,1,2J. Ding,1W. Jiang,1H. Chang,3\nF. Y. Fradin,1J. E. Pearson,1J. B. Ketterson,2V. Novosad,1M. Wu,3and A. Hoffmann1\n1Materials Science Division, Argonne National Laboratory, Argonne IL 60439, USA\n2Department of Physics and Astronomy, Northwestern Univers ity, Evanston IL 60208, USA\n3Department of Physics, Colorado State University, Fort Col lins CO 80523, USA\n(Dated: June 25, 2021)\nDue to its transverse nature, spin Hall effects (SHE) provide the possibility to excite and detect\nspin currents and magnetization dynamics even in magnetic i nsulators. Magnetic insulators are out-\nstanding materials for the investigation of nonlinear phen omena and for novel low power spintronics\napplications because of their extremely low Gilbert dampin g. Here, we report on the direct imaging\nof electrically driven spin-torque ferromagnetic resonan ce (ST-FMR) in the ferrimagnetic insulator\nY3Fe5O12based on the excitation and detection by SHEs. The driven spi n dynamics in Y 3Fe5O12\nis directly imaged by spatially-resolved microfocused Bri llouin light scattering (BLS) spectroscopy.\nPreviously, ST-FMR experiments assumed a uniform precessi on across the sample, which is not\nvalid in our measurements. A strong spin-wave localization in the center of the sample is observed\nindicating the formation of a nonlinear, self-localized sp in-wave ‘bullet’.\nMagneticmemoryandlogicdevicesrelyontheefficient\nmanipulation of the orientation of their magnetization\nusing low power1,2. Recently, there has been revitalized\ninterest in the ferrimagnetic insulator yttrium iron gar-\nnet (YIG, Y 3Fe5O12) motivated by the discovery of spin-\ntronic effects by combining this material and heavy met-\nals such as Pt3–7. Its extremely small magnetic damping\nenables low power data transmission and processing on\nthe basis of magnons, the elementary quanta of magnetic\nexcitations.3–5,8–12. In addition the low damping YIG\nalso enables nonlinear phenomena where the superposi-\ntion principle breaks down11. Previous work reported on\nthe formation of spin-wave caustics13, Bose Einstein con-\ndensationofmagnons14andnonlinearmode conversion15\nto name only a few. Recently, it has become possible to\ngrow nanometer-thick YIG films, which allow the prepa-\nration of micro- and nanostructured devices5,8,9,12,16.\nTherefore,the studyofnonlinearspin dynamicsin minia-\nturized YIG systems has only just begun.\nIndependent of the progress of the YIG film growth,\nthe development in employing spin-orbit interaction in\nheavy metals17,18and their alloys19in contact with a\nferromagnet (FM) has flourished. The SHE20,21can be\nused for the generation of strong current-driven torques\non the magnetization in the FM layer. The resultant\nspin current can drive spin-torque ferromagnetic reso-\nnance (ST-FMR) in bilayers consisting of ferromagnetic\nand nonmagnetic metals and be detected by a hom-\ndyne mixing of the microwavesignal with the anisotropic\nmagnetoresistance22. Recent theories propose that ST-\nFMR can be extended to insulating FM/normal metal\nbilayers. Here, the detection of magnetization precession\noccurs by spin pumping and a rectification of the spin\nHall magnetoresistance23,24. We showed recently that\nthis rectification process is indeed possible in YIG/Pt\nbilayers25. All previousanalysisof electric measurements\nassume uniform precession across the sample22,26. In or-\nder to validate this assumption it is highly desirable to\nimageaccurrent-driven spin dynamics spatially-resolvedand frequency-resolved. These investigations provide\ninteresting insights in the underlying physics, such as\nwhether bulk or edge modes are preferably excited by\nST-FMR or nonlinear spin dynamics may occur.\nIn this letter, we show experimentally the excitation of\nspin dynamics in microstructured magnetic insulators by\nthe SHE of an adjacent heavy metal and observe the for-\nmation of a nonlinear, self-localized spin-wave intensity\nin the center of the sample27–29. The magnetization dy-\nnamics in a nanometer-thick YIG layer is driven simulta-\nneouslybythe Oerstedfield andaspin torqueoriginating\nfrom a spin current generated by the SHE of an attached\nPt layer. The dynamics is detected in two complemen-\ntary ways: (1) Electrically, by a rectification mechanism\n(a) (b)\n(c) (d)\n-2.0-1.5-1.0-0.50.00.5DC voltage VDC (µV) \n12001000800600\nMagnetic field H (Oe) Data\n Total fit\n SMR\n Spin pumping1.5\n1.0\n0.5\n0.0DC voltage VDC (µV) \n12001000800600\nMagnetic field H (Oe)\nFIG. 1. (a) Schematic of the ST-FMR experimental setup (b)\nST-FMR mechanism in the YIG/Pt bilayer. The alternat-\ningrfcurrent drives an Oersted field hrfexerting a field-like\ntorqueτHon the magnetization M. At the same time a oscil-\nlatorytransverse spinaccumulationat theYIG/Ptinterfac e is\ngenerated by the SHE which results in a damping-like torque\nτSTT. (c) and (d) Typical dcvoltage spectra recorded at in-\nplane angles of φ= 30◦andφ= 240◦andP= +10 dBm.2\n-4-2024 VDC (µV)\n-1000 -500 0500 1000\n Magnetic field H (Oe)+15 dBm\n+12 dBm\n+10 dBmf = 4 GHz(a)\n(b)\n-3-2-10123SMR voltage VSMR (µV)\n250 200 150 10050 0\nIn-plane angle φ (°)-400-2000200400Spin-pumping\nvoltage VSP (nV)-3-2-1VDC (µV)subsidiary \n   mode main\nmode\nFIG. 2. (a) Typical VDCspectra at a constant frequency\nf= 4 GHz for various applied microwave powers. The in-\nset shows the resonance peak at P= +15 dBm. Two modes\nare detected. (b) In-plane angular dependence of the SMR,\nVSMR, and of the spin-pumping contribution, VSP, to the dc\nvoltage. The solid lines represent fits ∝cosφsin 2φ.\nof the spin Hall magnetoresistance (SMR)30–32as well as\nby spin pumping3–5,33–35and (2) Optically, by spatially-\nresolved Brillouin light scattering (BLS) microscopy36.\nThe experimental findings are further validated by mi-\ncromagnetic simulations37.\nYIG(40 nm)/Pt bilayers were fabricated by in-situ\nmagnetron sputtering under high-purity argon atmo-\nsphere on single crystal gadolinium gallium garnet\n(GGG, Gd 3Ga5O12) substrates of 500 µm thickness with\n(111) orientation16. For the electrical measurements a\nPt thickness of 2 nm was used, while for the optical in-\nvestigations the thickness was 5 nm in order to minimize\nthe influence of additional heating effects by the laser.\nIn a subsequent fabrication process, stripes in the shape\nof 30×5µm2(electrical measurements) and 5 ×5µm2\n(optical measurements) were patterned by photolithog-\nraphy and ion milling5. A coplanar waveguide (CPW)\nmade of Ti/Au (3 nm/120 nm) was structured on top of\nthe bar allowing the signal line to serve as a lead for the\nYIG/Pt bar as illustrated in Fig. 1(a). In this ST-FMR\nconfiguration a bias-T is utilized to allow for simultane-\nous transmission of a microwave signal with dcvoltage\ndetection via lock-in technique across the Pt. For this\npurpose the amplitude of the rfcurrent is modulated at\n3 kHz. We use a BLS microscope with a spatial reso-\nlution of 250 nm, where the laser spot is focused onto\nthe sample and the frequency shift of the back reflected\nlight is analyzed by a multi-pass tandem Fabry P´ erot\ninterferometer36. The detected BLS intensity is propor-\ntional to the square of the dynamic magnetization, i.e.,\nthe spin-wave intensity.In order to excite a dynamic response by ST-FMR in\ntheYIGsystema rfsignalispassedthroughthePtlayer.\nThe magnetization dynamics is governed by a modified\nLandau-Lifshitz-Gilbert equation23,24:\ndM\ndt=−|γ|M×Heff+αM×dM\ndt+|γ|/planckover2pi1\n2eMsdFJs,(1)\nwhereγis the gyromagnetic ratio, Heff=hrf+HD+H\nis the effective magnetic field including the microwave\nmagnetic field hrf, demagnetization fields HD, and the\nbias magnetic field H.αis the Gilbert damping param-\neter [the second term describes the damping torque τα,\nFig.1(b)] and Jsis a transverse spin current at the inter-\nface generated by the SHE from the alternating charge\ncurrent in the Pt layer23,24:\nJs=Re(g↑↓\neff)\neM×(M×µs)+Im(g↑↓\neff)\neM×µs.(2)\nHere,g↑↓\neffis the effective spin-mixing conductance and µs\nis the spin accumulation at the YIG/Pt interface. The\nfirstterminEq.( 2)describesananti-damping-liketorque\nτSTTand the second term is a field-like torque τH. As\nillustrated in Fig. 1(b) and described by Eq. ( 1) the mag-\nnetizationis drivenbythe independent torquetermscon-\ntaininghrfandJs.\nFirst, we describe the electrical characterization of the\nYIG/Pt bars by means of ST-FMR. Figure 1(c) and\n(d) illustrate typical dcvoltage spectra; exemplarily, we\nshow spectra recorded at in-plane angles of φ= 30◦and\nφ= 240◦, with applied rfpowerP= +10 dBm. A sig-\nnalisobservedwhen thesystemisdrivenresonantly. The\ndata is analyzed using the model proposed by Chiba et\nal. (supplementary information)23,24. According to the\nmodel, two signals contribute to the dcvoltage: (1) Spin\npumping which manifests in a symmetric contribution to\nthe Lorentzian lineshape. (2) Spin Hall magnetoresis-\ntance which is a superimposed symmetric and antisym-\nmetric Lorentzian curve [Fig. 1(c,d)].\nFIG. 3. Color-coded dispersion relation measured by BLS\nmicroscopy. The laser spot was focused onto the center of\nthe sample while the rffrequency as well as magnetic field\nwere varied. As for the electrical measurements two modes\nare detected by BLS. The inset shows corresponding field de-\npendence of the resonance measured by electrical means.3\nFIG. 4. Spatially-resolved BLS map of the 5 ×5µm2large\nYIG/Pt sample. The magnetic field H= 665 Oe is applied\natφ∼45◦. (a) - (d) Driving microwave frequency increases\nfrom 3.7 GHz to 3.85 GHz, microwave power P= +17 dBm.\nFigure2(a) illustrates dcvoltage spectra at a fixed\nmicrowave frequency f= 4 GHz for three different ap-\nplied powers. The offset is due to the longitudinal spin\nSeebeck effect6,7(see supplementary information) and\ndoes not affect the conclusions drawn from the resonance\nsignal7. The inset in Fig. 2(a) shows the resonance peak\natP= +15 dBm. Clearly, a less intense, secondary\nmode in addition to the main mode is detected. Accord-\ning to the Chiba model23,24thedcvoltage signal can be\ndeconvoluted into a spin-pumping and a SMR contribu-\ntion as also shown in Fig. 1(c) and (d). To analyze the\ndata employing the model we use a spin-mixing conduc-\ntance ofg↑↓\neff= 3.36×1014Ω−1m−2and a spin-Hall angle\nofγSHE= 0.0938. A fit to the angular-dependent data\nyields a phasedifference between Oersted field and the ac\ncurrent of δ= 64±5◦[see Fig. 1(c,d)]. Figure 2(b) shows\nthe angular dependences of the fitted spin-pumping and\nthe SMR signals. The model predicts the same angular\ndependent behavior ∝cosφsin2φfor spin pumping and\nSMR. As seen in Fig. 2(b), we find a good agreement be-\ntweentheory(solidlines)andexperimentforbothcurves.\nPlease note that we observe a small, non-vanishing volt-\nage at angles φ=n·90◦,n∈N, where the model sug-\ngests zero voltage23–25. In this angular range the model\nbreaks down and the experimental data cannot be fitted\n(see supplementary information).\nIn the following we compare the electrical measure-\nmentswiththeresultsobtainedbyBLSimaging. Theop-\ntical measurements wereperformed on YIG(40 nm)/Pt(5\nnm) bars having a lateral size of 5 ×5µm2. The ex-\nternal magnetic field is applied at an angle of φ∼45◦\nwhere the dcvoltage detection is maximized [Fig. 2(b)].\nFigure3shows the dispersion relation measured by BLS\nin a false color-coded image where red indicates a highspin-wave intensity and the blue area shows the ab-\nsence of spin waves. The measured dispersion is in\nagreement with the electrical measurements as shown in\nthe inset: As the field increases the resonance shifts to\nhigher frequencies as is expected from the Kittel equa-\ntion,f=|γ|\n2π/radicalbig\nH(H+4πMeff), where Meffis the effec-\ntive magnetization.\nAs is apparent from Fig. 2(a) and Fig. 3magnetization\ndynamics can be excited in a certain bandwidth around\nthe resonance which is determined by the specific de-\nvice characteristics. Furthermore, both figures (electrical\nand optical detection) suggest that there is an additional\nmode below the main mode. At first, one might identify\nthis mode as an edge mode39,40. However, this is not the\ncase as it will be discussed below.\nIn order to spatially map the spin-wave intensity, the\napplied magnetic field is kept fixed at H= 665 Oe. Fig-\nure4illustrates the experimental observations in false\ncolor-coded images. At an excitation frequency below\nthe resonance frequency, e.g., f= 3.7 GHz no magneti-\nzation dynamics is detected [Fig. 4(a)]. As the frequency\nincreases the system is driven resonantly and a strong\nspin-wave intensity is observed from f= 3.725 GHz\ntof= 3.8 GHz, Fig. 4(b,c). Increasing the frequency\neven further results in a diminished signal, Fig. 4(d) for\nf= 3.85 GHz. At even larger frequencies no magnetiza-\ntion dynamics is detected as it is also apparent from the\ndispersion illustrated in Fig. 3. In conventional electrical\nST-FMR measurements, a uniform spin-wave intensity\ndistribution across the lateral sample dimensions is as-\nsumed. However, as our experimental results show, this\nassumption is not fulfilled: A strong spin-wave signal is\nlocalized in the center of the YIG/Pt bar. It is desir-\nable to experimentally investigate at what minimum ex-\ncitation power the formation of the localization occurs.\nHowever, in the investigated range of powers we always\nobserve a localization in the center of the sample (see\nsupplementary information). For rfpowers of less than\n+11 dBm the signal is below our noise-floor.\nIn spite of this experimental limitation, we also car-\nried out micromagnetic simulations in order to gain fur-\nther insight into the underlying magnetization dynamics.\nThe simulations confirmed qualitatively the experimen-\ntal observations as is depicted in Fig. 5: Two modes can\nbe identified in the simulations, Fig. 5(a). In the low\npower regime, which is not accessible experimentally, we\nfind that the spatial magnetization distribution of the\nmain mode is almost uniform and the less intense sub-\nsidiary mode is localized at the edges ( hrf= 0.25 Oe,\nnot shown). With increasing rfpower, the spatial dis-\ntributions of both modes transform and at a threshold\nofhrf≈1 Oe a localization of both modes in the center\nof the sample is observed. Figure 5(b,c) show the corre-\nsponding spatial dynamic magnetization distributions at\nhrf= 5 Oe and agreewell with the experimental findings,\nFig.4.\nThis spatial profile can be understood asthe formation\nof a nonlinear, self-localized ‘ bullet’-like spin-wave inten-4\n1.0 \n0.8 \n0.6 \n0.4 \n0.2 \n0.0 Intensity  (a.u.) \n3.8 3.7 3.6 3.5 3.4 3.3 \nFrequency f (GHz)(a) \nmain mode subsidiary mode \n1.0 \n0.8 \n0.6 \n0.4 \n0.2 Normalized integrated \n BLS intensity  (a.u.) 50 40 30 20 \nrf  power P (mW) 1.0 \n0.8 \n0.6 \n0.4 \n0.2 Normalized integrated \nintensity simulation (a.u.) \n20 15 10 5 0rf magnetic field h rf (Oe) (d) \n1.0 \n0.8 \n0.6 \n0.4 \n0.2 \n0.0 (b) \n (c) subsidiary mode main mode \n1 µm \nFIG. 5. Micromagnetic simulations: (a) The spectrum reveal s\ntwo modes. Spatially-resolved magnetization distributio n of\nthe main mode, (b), and the less intense, subsidiary mode,\n(c). (d) The normalized integrated BLS intensity saturates\nat high excitation powers P, which is validated by micromag-\nnetic simulations at large driving rfmagnetic fields hrf.\nsity caused by nonlinear cross coupling between eigen-\nmodes in the system15. This process is mainly deter-\nmined by nonlinear spin-wave damping which transfers\nenergy from the initially excited ferromagnetic resonance\ninto other spin-wave modes rather than into the crys-\ntalline lattice15. To check this assumption, we plot-\nted in Fig. 5(d) the normalized integrated BLS intensity\nas well as the integrated spatial magnetization distribu-\ntion as a function of the applied microwave power and\ntherfmagnetic field, respectively. Both integrated sig-\nnals demonstrate a nonlinear behavior and saturate at\nhigh powers/microwave magnetic fields. This observa-\ntion is a direct manifestation of nonlinear damping: en-\nergy is absorbed by the ferromagnetic resonance and re-\ndistributedtosecondaryspin-wavemodesmoreandmore\neffectively15.\nUntil now, ST-FMR experiments assumed a uni-\nform magnetization precession22–24,26. However, as our\nspatially-resolved BLS results demonstrate and con-\nfirmed by micromagnetic simulations, the driven lateral\nspin-wave intensity distribution in insulating FMs devi-ates from this simple model at higher excitation powers\nwhich are common in ST-FMR measurements. The for-\nmation of a localized spin-wave mode was not considered\nin previous ST-FMR experiments neither in metals nor\nin insulators. Our findings have direct consequences on\nthe analysis and interpretation of ST-FMR experiments.\nThe precession amplitude is not uniform across the sam-\nple implying that the effective spin-mixing conductance\ng↑↓\neffis actually an average over the sample cross sec-\ntion. In areas where the precession amplitude is large,\ng↑↓\neffis underestimated, whereas it is over estimated in\nlow-intensityareas. This also complicates the determina-\ntion of the spin-Hall angle from ST-FMR measurements.\nMicromagnetic simulations show phase inhomogeneity,\nspecifically aroundthe perimeter ofthe mode. The phase\ninhomogeneity tends to equally lag and lead the main\nuniform phase of the center mode; effectively the phase\ninhomogeneity then leads to no significant change to the\nlineshape. However,assuming the phaseat the perimeter\nto be uniformly leading the bulk phase results in a cor-\nrectionto the lineshape that is still negligiblebecause the\neffective areaand amplitude where the phase is deviating\nis significantly smaller than the bulk area. Nevertheless,\nin general the issue of inhomogeneous phase distribution\nmay complicate the analysis of electrical ST-FMR spec-\ntra, especially in smaller samples.\nInconclusion,wedemonstratedthattheconceptofST-\nFMR can be extended to magnetic insulators where the\nformation of a nonlinear, self-localized spin-wave inten-\nsity driven by an accurrent was observed. We adopted\nan electrically-driven ST-FMR excitation and detection\nscheme in magnetic insulator (YIG)/heavy normal metal\n(Pt)bilayersthatwasoriginallydevelopedforall-metallic\nsystems. A dcvoltage in YIG/Pt bilayers was observed\nunder resonance condition by a SMR-mediated spin-\ntorque diode effect in agreement with theoretical pre-\ndictions. Spatially-resolved BLS microscopy revealed a\nstrong ‘bullet’-like spin-wave localization in the center\nof the sample due to nonlinear cross coupling of eigen-\nmodes in the system. Since the observed electrical signal\nis sufficiently large and the signal-to-noise ratio is rea-\nsonably good, down-scaling of sample dimensions to the\nnanometer-scale is feasible.\nACKNOWLEDGMENTS\nWe thank Stephen Wu for assistance with ion milling.\nThe work at Argonne was supported by the U.S. De-\npartment of Energy, Office of Science, Materials Science\nand Engineering Division. Lithography was carried out\nat the Center for Nanoscale Materials, an Office of Sci-\nence user facility, which is supported by DOE, Office of\nScience, Basic Energy Science under Contract No. DE-\nAC02-06CH11357. The work at Colorado State Univer-\nsity was supported by the U. S. Army Research Office\n(W911NF-14-1-0501),theU.S.NationalScienceFounda-\ntion (ECCS-1231598),C-SPIN(one ofthe SRCSTARnet5\nCenters sponsored by MARCO and DARPA), and the U. S. Departme nt of Energy (DE-SC0012670).\n∗jungfleisch@anl.gov\n1D.C. Ralph and M.D. Stiles, J. Magn. Magn. Mater. 320,\n1190 (2008).\n2A. Hoffmann and H. Schultheiß, Curr. Opin. Solid State\nMater. Sci. (2014), doi:10.1016/j.cossms.2014.11.004.\n3Y. Kajiwara, K. Harii, S. Takahashi, J. Ohe, K. Uchida,\nM. Mizuguchi , H. Umezawa, H. Kawai, K. Ando,\nK. Takanashi, S. Maekawa, and E. Saitoh, Nature 464,\n262 (2010).\n4M.B. Jungfleisch, A.V. Chumak, V.I. Vasyuchka,\nA.A. Serga, B. Obry, H. Schultheiss, P.A. Beck,\nA.D. Karenowska, E. Saitoh, and B. Hillebrands, Appl.\nPhys. Lett. 99, 182512 (2011).\n5M.B. Jungfleisch, W. Zhang, W. Jiang, H. Chang, J. Skle-\nnar, S.M. Wu, J.E. Pearson, A. Bhattacharya, J.B. Ketter-\nson, M. Wu, andA. Hoffmann, J. Appl. Phys. 117, 17D128\n(2015).\n6K. Uchida, S. Takahashi, K. Harii, J. Ieda, W. Koshibae,\nK. Ando, S. Maekawa and E. Saitoh, Nature 455, 778\n(2008).\n7M.B. Jungfleisch, T.An,K.Ando,Y.Kajiwara, K.Uchida,\nV.I. Vasyuchka, A.V. Chumak, A.A. Serga, E. Saitoh and\nB. Hillebrands, Appl. Phys. Lett. 102, 062417 (2013).\n8A. Hamadeh, O. d’Allivy Kelly, C. Hahn, H. Meley,\nR. Bernard, A.H. Molpeceres, V.V. Naletov, M. Viret,\nA. Anane, V. Cros, S.O. Demokritov, J.L. Prieto,\nM. Mu˜ noz, G. de Loubens, and O. Klein, Phys. Rev. Lett.\n113, 197203 (2014).\n9O. d’Allivy Kelly, A. Anane, R. Bernard, J. Ben Youssef,\nC. Hahn, A.H. Molpeceres, C. Carr´ et´ ero, E. Jacquet,\nC. Deranlot, P. Bortolotti, R. Lebourgeois, J.-C. Mage,\nG. de Loubens, O. Klein, V. Cros, and A. Fert, Appl.Phys.\nLett.103, 082408 (2013).\n10M. Wu and A. Hoffmann, Recent advances in magnetic\ninsulators - From spintronics to microwave applications,\nSolid State Physics 64, (Academic Press , 2013).\n11A.G. Gurevich and G.A. Melkov, Magnetization oscilla-\ntions and waves, CRC Press (1996).\n12P. Pirro, T. Br¨ acher, A.V. Chumak, B. L¨ agel, C. Dubs,\nO. Surzhenko, P. G¨ ornert, B. Leven and B. Hillebrands,\nAppl. Phys. Lett. 104, 012402 (2014).\n13T. Schneider, A.A. Serga, A.V. Chumak, C.W. Sandweg,\nS. Trudel, S. Wolff, M.P. Kostylev, V.S. Tiberkevich,\nA.N. Slavin, and B. Hillebrands, Phys. Rev. Lett. 104,\n197203 (2010).\n14S.O. Demokritov, V.E. Demidov, O. Dzyapko,\nG.A. Melkov, A.A. Serga, B. Hillebrands, and A.N. Slavin,\nNature443, 430-433 (2006).\n15M. Kostylev, V.E. Demidov, U.-H. Hansen, and\nS.O. Demokritov, Phys. Rev. B 76, 224414 (2007).\n16H. Chang, P. Li, W. Zhang, T. Liu, A. Hoffmann, L. Deng,\nand M. Wu, IEEE Magn. Lett. 5, 6700104 (2014).\n17A. Hoffmann, IEEE Trans. Magn. 49, 5172 (2013).\n18W. Zhang, M.B. Jungfleisch, W. Jiang, J. Sklenar,\nF.Y. Fradin, J.E. Pearson, J.B. Ketterson, and A. Hoff-\nmann, J. Appl. Phys. 117, 172610 (2015).\n19W. Zhang, M.B. Jungfleisch, W. Jiang, J.E. Pearson,\nA. Hoffmann, F. Freimuth, and Y. Mokrousov, Phys. Rev.Lett.113, 196602 (2014).\n20M.I. D’yakonov and V.I. Perel’, Sov. Phys. JETP Lett. 13,\n467 (1971).\n21J.E. Hirsch, Phys. Rev. Lett. 83, 1834 (1999).\n22L. Liu, T. Moriyama, D.C. Ralph, and R.A. Buhrman,\nPhys. Rev. Lett. 106, 036601 (2011).\n23T. Chiba, G.E.W. Bauer, and S. Takahashi, Phys. Rev.\nApplied 2, 034003 (2014).\n24T. Chiba, M. Schreier, G.E.W. Bauer, and S. Takahashi,\nJ. Appl. Phys. 117, 17C715 (2015).\n25J. Sklenar, W. Zhang, M.B. Jungfleisch, W. Jiang,\nH. Chang, J.E. Pearson, M. Wu, J.B. Ketterson, and\nA. Hoffmann, ArXiv e-prints (2015), arXiv:1505.07791\n[cond-mat.meshall].\n26A.R. Mellnik, J.S. Lee, A. Richardella, J.L. Grab,\nP.J. Mintun, M.H. Fischer, A. Vaezi, A. Manchon, E.-\nA. Kim, N. Samarth, and D.C. Ralph, Nature 511, 449\n(2014).\n27V.E. Demidov, S. Urazhdin, H. Ulrichs, V. Tiberkevich,\nA. Slavin, D. Baither, G. Schmitz, and S.O. Demokritov,\nNat. Mater. 11, 1028 (2012).\n28A. Slavin and V. Tiberkevich, Phys. Rev. Lett. 95, 237201\n(2005).\n29R.H. Liu, W.L. Lim, and S. Urazhdin, Phys. Rev. Lett.\n110, 147601 (2013).\n30J.C. Sankey, P.M. Braganca, A.G.F. Garcia, I.N. Krivoro-\ntov, R.A. Buhrman, and D.C. Ralph, Phys. Rev. Lett. 96,\n227601 (2006).\n31H. Nakayama, M. Althammer, Y.-T. Chen, K. Uchida,\nY. Kajiwara, D. Kikuchi, T. Ohtani, S. Gepr¨ ags, M. Opel,\nS. Takahashi, R. Gross, G.E.W. Bauer, S.T.B. Goennen-\nwein, and E. Saitoh, Phys. Rev. Lett. 110, 206601 (2013).\n32M. Schreier, T. Chiba, A. Niedermayr, J. Lotze,\nH. Huebl, S. Gepr¨ ags, S. Takahashi, G.E.W. Bauer,\nR. Gross, and S.T.B. Goennenwein, ArXiv e-prints (2014),\narXiv:1412.7460 [cond-mat.meshall].\n33Y.Tserkovnyak,A.Brataas andG.E.W. Bauer, Phys.Rev.\nLett.88, 117601 (2002).\n34A. Azevedo, L.H. Vilela Le˜ ao, R.L. Rodr´ ıguez-Su´ arez,\nA.B. Oliveira, and S.M. Rezende, J. Appl. Phys. 97,\n10C715 (2005).\n35O. Mosendz, J.E. Pearson, F.Y. Fradin, G.E.W. Bauer,\nS.D. Bader, and A. Hoffmann, Phys. Rev. Lett. 104,\n046601 (2010).\n36B. Hillebrands, Brillouin light scattering spectroscopy, in\nNovel techniques for characterising magnetic materials,\nedited by Y. Zhu, Springer (2005).\n37A. Vansteenkiste, J. Leliaert, M. Dvornik, M. Helsen,\nF. Garcia-Sanchez, and B. Van Waeyenberge, AIP Ad-\nvances4, 107133 (2014).\n38W. Zhang, V. Vlaminck, J.E. Pearson, R. Divan,\nS.D. Bader, and A. Hoffmann, Appl. Phys. Lett. 103,\n242414 (2013).\n39J. Jorzick,S.O. Demokritov, B. Hillebrands, M. Bailleul,\nC. Fermon, K.Y. Guslienko, A.N. Slavin, D.V. Berkov, and\nN.L. Gorn, Phys. Rev. Lett. 88, 047204 (2002).\n40S. Neusser and D. Grundler, Advanced Materials 21, 2927\n(2009)."    },    {        "title": "1508.07118v3.The_inviscid_limit_for_the_Landau_Lifshitz_Gilbert_equation_in_the_critical_Besov_space.pdf",        "content": "arXiv:1508.07118v3  [math.AP]  25 Aug 2016THE INVISCID LIMIT FOR THE LANDAU-LIFSHITZ-GILBERT\nEQUATION IN THE CRITICAL BESOV SPACE\nZIHUA GUO AND CHUNYAN HUANG\nAbstract. We prove that in dimensions three and higher the Landau-Lifshitz-\nGilbert equation with small initial data in the critical Besov space is glob ally well-\nposed in a uniform way with respect to the Gilbert damping parameter . Then we\nshow that the global solution converges to that of the Schr¨ oding er maps in the\nnatural space as the Gilbert damping term vanishes. The proof is ba sed on some\nstudies on the derivative Ginzburg-Landau equations.\n1.Introduction\nInthis paperwe study theCauchy problemfortheLandau-Lifshitz -Gilbert (LLG)\nequation\n∂ts=as×∆s−εs×(s×∆s), s(x,0) =s0(x), (1.1)\nwheres(x,t) :Rn×R→S2⊂R3,×denotes the wedge product in R3,a∈R\nandε >0 is the Gilbert damping parameter. The equation (1.1) is one of the\nequations of ferromagnetic spin chain, which was proposed by Land au-Lifshitz [19]\nin studying the dispersive theory of magnetisation of ferromagnet s. Later on, such\nequations were also found in the condensed matter physics. The LL G equation has\nbeen studied extensively, see [17, 7] for an introduction on the equ ation.\nFormally, if a= 0, then (1.1) reduces to the heat flow equations for harmonic\nmaps\n∂ts=−εs×(s×∆s), s(x,0) =s0(x), (1.2)\nand ifε= 0, then (1.1) reduces to the Schr¨ odinger maps\n∂ts=as×∆s, s(x,0) =s0(x). (1.3)\nBoth special cases have been objects of intense research. The p urpose of this paper\nis to study the inviscid limit of (1.1), namely, to prove rigorously that t he solutions\nof (1.1) converges to the solutions of (1.3) as ε→0 under optimal conditions on\nthe initial data.\nThe inviscid limit is an important topic in mathematical physics, and has b een\nstudied in various settings, e.g. for hyperbolic-dissipative equation s such as Navier-\nStokes equation to Euler equation (see [11] and references ther ein), for dispersive-\ndissipative equations such as KdV-Burgers equation to KdV equatio n (see [9]) and\nGinzburg-Landau equation to Schr¨ odinger equations (see [23, 12 ]). The LLG equa-\ntion (1.1) is an equation with both dispersive and dissipative effects. T his can be\n2010Mathematics Subject Classification. 35Q55.\nKey words and phrases. Landau-Lifshitz-Gilbert equation, Schr¨ odinger maps, Inviscid limit ,\nCritical Besov Space.\n12 Z. GUO AND C. HUANG\nseen from the stereographic projection transform. It was know n that (see [18]) let\nu=P(s) =s1+is2\n1+s3, (1.4)\nwheres= (s1,s2,s3) is a solution to (1.1), then usolves the following complex\nderivative Ginzburg-Landau type equation\n(i∂t+∆−iε∆)u=2a¯u\n1+|u|2n/summationdisplay\nj=1(∂xju)2−2iε¯u\n1+|u|2n/summationdisplay\nj=1(∂xju)2\nu(x,0) =u0.(1.5)\nOn the other hand, the projection transform has an inverse\nP−1(u) =/parenleftbiggu+ ¯u\n1+|u|2,−i(u−¯u)\n1+|u|2,1−|u|2\n1+|u|2/parenrightbigg\n. (1.6)\nTherefore, (1.1) is equivalent to (1.5) assuming PandP−1is well-defined, and we\nwill focus on (1.5). The previous works [13, 14, 1, 2, 3] on the Schr¨ odinger maps\n(ε= 0) were also based on this transform. Note that (1.5) is invariant u nder the\nfollowing scaling transform: for λ >0\nu(x,t)→u(λx,λ2t), u0(x)→u0(λx).\nThus the critical Besov space is ˙Bn/2\n2,1in the sense of scaling.\nTo study the inviscid limit, the crucial task is to obtain uniform well-pos edness\nwith respect to the inviscid parameter. Energy method was used in [1 1]. For\ndispersive-dissipative equations, one needs to exploit the dispersiv e effect uniformly.\nStrichartz estimates and energy estimates were used in [23] for Gin zburg-Landau\nequations, andBourgainspacewasused in[9]forKdV-Burgersequ ations. In[24,10]\nthe inviscid limit for the derivative Ginzburg-Landau equations were s tudied by us-\ning the Strichartz estimates, local smoothing estimates and maxima l function esti-\nmates. However, these results requires high regularities when app lied to equation\n(1.5). In this paper we will use Bourgain-type space and exploit the n ull structure\nthat are inspired by the latest development for the Schr¨ odinger m aps (ε= 0) (see\n[4, 3, 1, 2, 13, 14, 8]) to study (1.5) with small initial data in the critica l Besov\nspace. In [2] and [14] it was proved independently that global well-p osedness for\n(1.3) holds for small data in the critical Besov space. We will extend t heir results\nto (1.1) uniformly with respect to ε. We exploit the Bourgain space in a differ-\nent way from both [2] and [14]. One of the novelties is the use of X0,1-structure\nthat results in many simplifications even for the Schr¨ odinger maps. The presence\nof dissipative term brings many technical difficulties, e.g. the lack of s ymmetry\nin time and incompatibility with Xs,bstructure. We need to overcome these dif-\nficulties when extending the linear estimates for the Schr¨ odinger e quation to the\nSchr¨ odinger-dissipative equation uniformly with respect to ε.\nBy scaling we may assume a=±1. From now on, we assume a= 1 since the\nother case a=−1 is similar. For Q∈S2, the space ˙Bs\nQis defined by\n˙Bs\nQ=˙Bs\nQ(Rn;S2) ={f:Rn→R3;f−Q∈˙Bs\n2,1,|f(x)| ≡1 a.e. in Rn},\nwhere˙Bs\n2,1is the standard Besov space. It was known the critical space is ˙Bn/2\nQ.\nThe main result of this paper isLANDAU-LIFSHITZ EQUATION 3\nTheorem 1.1. Assumen≥3. The LLG equation (1.1)is globally well-posed for\nsmall data s0∈˙Bn/2\nQ(Rn;S2),Q∈S2in a uniform way with respect to ε∈(0,1].\nMoreover, for any T >0, the solution converges to that of Schr¨ odinger map (1.3)\ninC([−T,T] :˙Bn/2\nQ)asε→0.\nAs we consider the inviscid limit in the strongest topology (same space as the\ninitial data), no convergence rate is expected. This can be seen fr om linear solutions\nfor (1.5). However, if assuming initial data has higher regularity, on e can have\nconvergence rate O(εT) (see (5.3) below).\n2.Definitions and Notations\nForx,y∈R,x/lessorsimilarymeans that there exists a constant Csuch that x≤Cy, and\nx∼ymeans that x/lessorsimilaryandy/lessorsimilarx. We use F(f),ˆfto denote the space-time Fourier\ntransform of f, andFxi,tfto denote the Fourier transform with respect to xi,t.\nLetη:R→[0,1] be an even, non-negative, radially decreasing smooth function\nsuch that: a) ηis compactly supported in {ξ:|ξ| ≤8/5}; b)η≡1 for|ξ| ≤5/4. For\nk∈Zletχk(ξ) =η(ξ/2k)−η(ξ/2k−1),χ≤k(ξ) =η(ξ/2k),/tildewideχk(ξ) =/summationtext9n\nl=−9nχk+l(ξ),\nand then define the Littlewood-Paley projectors Pk,P≤k,P≥konL2(Rn) by\n/hatwidestPku(ξ) =χk(|ξ|)/hatwideu(ξ),/hatwideP≤ku(ξ) =χ≤k(|ξ|)/hatwideu(ξ),\nandP≥k=I−P≤k−1,P[k1,k2]=/summationtextk2\nj=k1Pj. We also define /tildewidePku=F−1/tildewideχk(|ξ|)/hatwideu(ξ)\nLetSn−1be the unit sphere in Rn. Fore∈Sn−1, define /hatwidePk,eu(ξ) =/tildewideχk(|ξ·\ne|)χk(|ξ|)/hatwideu(ξ). Since for |ξ| ∼2kwe have\n5n/summationdisplay\nl=−5nχk+l(ξ1)+···+5n/summationdisplay\nl=−5nχk+l(ξn)∼1,\nthen let\nβj\nk(ξ) =/summationtext5n\nl=−5nχk+l(ξj)\n/summationtextn\nj=1/summationtext5n\nl=−5nχk+l(ξj)·1/summationdisplay\nl=−1χk+l(|ξ|), j= 1,···,n.\nDefine the operator Θj\nkonL2(Rn) by/hatwidestΘj\nkf(ξ) =βj\nk(ξ)ˆf(ξ), 1≤j≤n. Lete1=\n(1,0,···,0),···,en= (0,···,0,1). Then we have\nPk=n/summationdisplay\nj=1Pk,ejΘj\nk. (2.1)\nFor anyk∈Z, we define the modulation projectors Qk,Q≤k,Q≥konL2(Rn×R) by\n/hatwidestQku(ξ,τ) =χk(τ+|ξ|2)/hatwideu(ξ,τ),/hatwideQ≤ku(ξ,τ) =χ≤k(τ+|ξ|2)/hatwideu(ξ,τ),\nandQ≥k=I−Q≤k−1,Q[k1,k2]=/summationtextk2\nj=k1Qj.\nFor anye∈Sn−1, we can decompose Rn=λe⊕He, whereHeis the hyperplane\nwith normal vector e, endowed with the induced measure. For 1 ≤p,q <∞, we\ndefineLp,q\nethe anisotropic Lebesgue space by\n/ba∇dblf/ba∇dblLp,q\ne=/parenleftBigg/integraldisplay\nR/parenleftbigg/integraldisplay\nHe×R|f(λe+y,t)|qdydt/parenrightbiggp/q\ndλ/parenrightBigg1/p4 Z. GUO AND C. HUANG\nwith the usual definition if p=∞orq=∞. We write Lp,q\nej=Lp\nxjLq\n¯xj,t. We use\n˙Bs\np,qto denote the homogeneous Besov spaces on Rnwhich is the completion of the\nSchwartz functions under the norm\n/ba∇dblf/ba∇dbl˙Bsp,q= (/summationdisplay\nk∈Z2qsk/ba∇dblPkf/ba∇dblq\nLp)1/q.\nTo exploit the null-structure we also need the Bourgain-type space associated to\ntheSchr¨ odinger equation. Inthis paperwe use themodulation-ho mogeneousversion\nas in [2, 8]. We define X0,b,qto be the completion of the space of Schwartz functions\nwith the norm\n/ba∇dblf/ba∇dblX0,b,q= (/summationdisplay\nk∈Z2kbq/ba∇dblQkf/ba∇dblq\nL2\nt,x)1/q. (2.2)\nIfq= 2 we simply write X0,b=X0,b,2. By the Plancherel’s equality we have\n/ba∇dblf/ba∇dblX0,1=/ba∇dbl(i∂t+ ∆)f/ba∇dblL2\nt,x. SinceX0,b,qis not closed under conjugation, we also\ndefine the space ¯X0,b,qby the norm /ba∇dblf/ba∇dbl¯X0,b,q=/ba∇dbl¯f/ba∇dblX0,b,q, and similarly write ¯X0,b=\n¯X0,b,2. It’s easy to see that X0,b,qfunction is unique modulo solutions of the homo-\ngeneous Schr¨ odinger equation. For a more detailed description of theX0,b,pspaces\nwe refer the readers to [21] and [20]. We use X0,b,p\n+to denote the space restricted to\nthe interval [0 ,∞):\n/ba∇dblf/ba∇dblX0,b,p\n+= inf\n˜f:˜f=font∈[0,∞)/ba∇dbl˜f/ba∇dblX0,b,p.\nIn particular, we have\n/ba∇dblf/ba∇dblX0,1\n+∼ /ba∇dbl˜f/ba∇dblX0,1 (2.3)\nwhere˜f=f(t,x)1t≥0+f(−t,x)1t<0.\nLetL=∂t−i∆ and¯L=∂t+i∆. We define the main dyadic function space. If\nf(x,t)∈L2(Rn×R+) has spatial frequency localized in {|ξ| ∼2k}, define\n/ba∇dblf/ba∇dblFk=/ba∇dblf/ba∇dblX0,1/2,∞\n++/ba∇dblf/ba∇dblL∞\ntL2x+/ba∇dblf/ba∇dbl\nL2\ntL2n\nn−2\nx\n+2−(n−1)k/2sup\ne∈Sn−1/ba∇dblf/ba∇dblL2,∞\ne+2k/2sup\n|j−k|≤20sup\ne∈Sn−1/ba∇dblPj,ef/ba∇dblL∞,2\ne,\n/ba∇dblf/ba∇dblYk=/ba∇dblf/ba∇dblL∞\ntL2x+/ba∇dblf/ba∇dbl\nL2\ntL2n\nn−2\nx+2−(n−1)k/2sup\ne∈Sn−1/ba∇dblf/ba∇dblL2,∞\ne\n+2−kinf\nf=f1+f2(/ba∇dblLf1/ba∇dblL2\nt,x+/ba∇dbl¯Lf2/ba∇dblL2\nt,x),\n/ba∇dblf/ba∇dblZk=2−k/ba∇dblLf/ba∇dblL2\nt,x\n/ba∇dblf/ba∇dblNk= inf\nf=f1+f2+f3(/ba∇dblf1/ba∇dblL1\ntL2x+2−k/2sup\ne∈Sn−1/ba∇dblf2/ba∇dblL1,2\ne+/ba∇dblf3/ba∇dblX0,−1/2,1\n+)+2−k/ba∇dblf/ba∇dblL2\nt,x.\nThen we define the space Fs,Ys,Zs,Nswith the following norm\n/ba∇dblu/ba∇dblFs=/summationdisplay\nk∈Z2ks/ba∇dblPku/ba∇dblFk,/ba∇dblu/ba∇dblYs=/summationdisplay\nk∈Z2ks/ba∇dblPku/ba∇dblYk,\n/ba∇dblu/ba∇dblZs=/summationdisplay\nk∈Z2ks/ba∇dblPku/ba∇dblZk,/ba∇dblu/ba∇dblNs=/summationdisplay\nk∈Z2ks/ba∇dblPku/ba∇dblNk.\nObviously, Fk∩Zk⊂Yk, and thus Fs∩Zs⊂Ys. In the end of this section, we\npresent a standard extension lemma (See Lemma 5.4 in [22]) giving the r elation\nbetween Xs,band other space-time norm.LANDAU-LIFSHITZ EQUATION 5\nLemma 2.1. Letk∈ZandBbe a space-time norm satisfying with some C(k)\n/ba∇dbleit0eit∆Pkf/ba∇dblB≤C(k)/ba∇dblPkf/ba∇dbl2\nfor anyt0∈Randf∈L2. Then\n/ba∇dblPku/ba∇dblB/lessorsimilarC(k)/ba∇dblPku/ba∇dblX0,1/2,1.\n3.Uniform linear estimates\nIn this section we prove some uniform linear estimates for the equat ion (1.5) with\nrespect to the dissipative parameter. First we recall the known line ar estimates for\nthe Schr¨ odinger equation, see [16] and [14].\nLemma 3.1. Assumen≥3. For any k∈Zwe have\n/ba∇dbleit∆Pkf/ba∇dbl\nL2\ntL2n\nn−2\nx∩L∞\ntL2x/lessorsimilar/ba∇dblPkf/ba∇dbl2, (3.1)\nsup\ne∈Sn−1/ba∇dbleit∆Pkf/ba∇dblL2,∞\ne/lessorsimilar2(n−1)k\n2/ba∇dblPkf/ba∇dbl2, (3.2)\nsup\ne∈Sn−1/ba∇dbleit∆Pk,ef/ba∇dblL∞,2\ne/lessorsimilar2−k\n2/ba∇dblPkf/ba∇dbl2. (3.3)\nLemma 3.2. Assumen≥3,u,Fsolves the equation: for ε >0\nut−i∆u−ε∆u=F(x,t), u(x,0) =u0.\nThen for b∈[1,∞]\n/ba∇dblPku/ba∇dblX0,1/2,b\n+/lessorsimilar/ba∇dblPku0/ba∇dblL2+/ba∇dblPkF/ba∇dblX0,−1/2,b\n+, (3.4)\n/ba∇dblPku/ba∇dblZk/lessorsimilarε1/2/ba∇dblPku0/ba∇dblL2+2−k/ba∇dblPkF/ba∇dblL2\nt,x, (3.5)\nwhere the implicit constant is independent of ε.\nProof.First we show the second inequality. We have\nu=eit∆+εt∆u0+/integraldisplayt\n0ei(t−s)∆+ε(t−s)∆F(s)ds (3.6)\nand thus\n/ba∇dblPku/ba∇dblZk/lessorsimilarε2k/ba∇dblPku/ba∇dblL2\nt,x+2−k/ba∇dblPkF/ba∇dblL2\nt,x\n/lessorsimilarε1/2/ba∇dblPku0/ba∇dblL2+2−k/ba∇dblPkF/ba∇dblL2\nt,x.\nNow we show the first inequality. We only prove the case b=∞since the other\ncases aresimilar. First we assume F= 0. Then u=eit∆+εt∆u0. Let ˜u=eit∆+ε|t|∆u0,\nthen ˜uis an extension of u. Then\n/ba∇dblPku/ba∇dblX0,1/2,∞\n+/lessorsimilarsup\nj2j/2/ba∇dblFt(e−it|ξ|2−ε|t|·|ξ|2)(τ)ˆu0(ξ)χk(ξ)χj(τ+|ξ|2)/ba∇dblL2\nτ,ξ\n/lessorsimilarsup\nj2j/2/ba∇dbl(ε|ξ|2)−1/parenleftbigg\n1+|τ+|ξ|2|2\n(ε|ξ|2)2/parenrightbigg−1\nˆu0(ξ)χk(ξ)χj(τ+|ξ|2)/ba∇dblL2\nτ,ξ\n/lessorsimilar/ba∇dblPku0/ba∇dbl2.\nNext we assume u0= 0. Fix an extension ˜FofFsuch that\n/ba∇dbl˜F/ba∇dblX0,−1/2,∞≤2/ba∇dblF/ba∇dblX0,−1/2,∞\n+.6 Z. GUO AND C. HUANG\nThen define ˜ u=F−11\nτ+|ξ|2+iε|ξ|2F˜F. We see ˜ uis an extension of uand then\n/ba∇dblPku/ba∇dblX0,1/2,∞\n+/lessorsimilar/ba∇dbl˜u/ba∇dblX0,1/2,∞\n/lessorsimilarsup\nj2j/2/ba∇dblχk(ξ)χj(τ+|ξ|2)1\nτ+|ξ|2+iε|ξ|2F˜F/ba∇dblL2\nτ,ξ\n/lessorsimilar/ba∇dbl˜F/ba∇dblX0,−1/2,∞/lessorsimilar/ba∇dblF/ba∇dblX0,−1/2,∞\n+.\nThus we complete the proof. /square\nLemma 3.3. Letn≥3,k∈Z,ε≥0. Assume u,Fsolves the equation\nut−i∆u−ε∆u=F(x,t), u(x,0) =u0.\nThen for any e∈Sn−1we have\n/ba∇dblPku/ba∇dblL2,∞\ne/lessorsimilar2k(n−1)/2/ba∇dblu0/ba∇dblL2+2k(n−2)/2sup\ne∈Sn−1/ba∇dblF/ba∇dblL1,2\ne, (3.7)\n/ba∇dblPk,eu/ba∇dblL∞,2\ne/lessorsimilar2−k/2/ba∇dblu0/ba∇dblL2+2−ksup\ne∈Sn−1/ba∇dblF/ba∇dblL1,2\ne, (3.8)\nwhere the implicit constant is independent of ε.\nProof.Ifε= 0, then the inequalities were proved in [14]. By the scaling and\nrotational invariance, we may assume k= 0 and e= (1,0,···,0). Then the second\ninequality follows from Proposition 2.5, 2.7 in [10]. We prove the first ineq uality by\nthe following two steps.\nStep 1: F= 0.\nFrom the fact\n|eεt∆f(·,t)(x)| ≤(εt)−n/2/integraldisplay\ne−|x−y|2\n2εt|f(y,t)|dy\n/lessorsimilar(εt)−n/2/integraldisplay\ne−|x−y|2\n2εt/ba∇dblf(y,t)/ba∇dblL∞\ntdy\nwe get\n/ba∇dbleit∆+εt∆P0u0/ba∇dblL2x1L∞\n¯x1,t/lessorsimilar/ba∇dbleit∆P0u0/ba∇dblL2x1L∞\n¯x1,t/lessorsimilar/ba∇dblu0/ba∇dbl2.\nStep 2: u0= 0.\nDecompose P0u=U1+···Unsuch that FxUiis supported in {|ξ| ∼1 :|ξi| ∼\n1}×R. Thus it suffices to show\n/ba∇dblUi/ba∇dblL2x1L∞\n¯x,t/lessorsimilar/ba∇dblF/ba∇dblL1,2\nei. (3.9)\nWe only show the estimate for U1. We still write u=U1. We assume FxFis\nsupported in {|ξ| ∼1 :ξ1∼1}×R. We have\nu(t,x) =/integraldisplay\nRn+1eitτeixξ\nτ+|ξ|2+iε|ξ|2/hatwideF(ξ,τ)dξdτ\n=/integraldisplay\nRn+1eitτeixξ\nτ+|ξ|2+iε|ξ|2/hatwideF(ξ,τ)(1{−τ−|¯ξ|2∼1}c+1−τ−|¯ξ|2∼1,|τ+|ξ|2|/lessorsimilarε\n+1−τ−|¯ξ|2∼1,|τ+|ξ|2|≫ε)dξdτ\n:=u1+u2+u3.\nForu1, we simply use the Plancherel equality and get\n/ba∇dbl∆u1/ba∇dblL2+/ba∇dbl∂tu1/ba∇dbl2≤ /ba∇dblF/ba∇dbl2,LANDAU-LIFSHITZ EQUATION 7\nand thus by Sobolev embedding and Bernstein’s inequality we obtain th e desired\nestimate. For u2, using the Lemma 2.1, Lemma 3.1 and Bernstein’s inequality we\nget\n/ba∇dblu2/ba∇dblL2x1L∞\n¯x1,t/lessorsimilar/ba∇dblu2/ba∇dbl˙X0,1/2,1/lessorsimilarε−1/2/ba∇dbl/hatwideF/ba∇dblL2/lessorsimilar/ba∇dblF/ba∇dblL1,2\ne1.\nNow we estimate u3. Since|τ+|ξ|2| ≫ε, we have\n1\nτ+|ξ|2+iε|ξ|2=1\nτ+|ξ|2+∞/summationdisplay\nk=1(−iε|ξ|2)k\n(τ+|ξ|2)k+1.\nMoreover, let s= (−τ−|¯ξ|2)1/2. Thenτ+|ξ|2=−(s−ξ1)(s+ξ1), and thus we get\n|s−ξ1| ≫ε,|s+ξ1| ∼1\n(τ+|ξ|2)−1=−1\n2s(1\ns−ξ1+1\ns+ξ1) =−1\n2s(s−ξ1)(1+s−ξ1\ns+ξ1).\nHence\n1\nτ+|ξ|2+iε|ξ|2=1\nτ+|ξ|2+∞/summationdisplay\nk=1(−iε|ξ|2)k\n(2s(ξ1−s))k+1\n+∞/summationdisplay\nk=1(−iε|ξ|2)k\n(2s(ξ1−s))k+1[(1+s−ξ1\ns+ξ1)k+1−1]\n:=a1(ξ,τ)+a2(ξ,τ)+a3(ξ,τ).\nInserting this identity into the expression of u3, then we have u3=u1\n3+u2\n3+u3\n3,\nwhere\nuj\n3=/integraldisplay\nRn+1eitτeixξaj(ξ,τ)/hatwideF(ξ,τ)1−τ−|¯ξ|2∼1,|τ+|ξ|2|≫εdξdτ, j = 1,2,3.\nForu1\n3, this corresponds to the case ε= 0 which is proved in [14]. For u3\n3, we can\ncontrol it similarly as u1, since\n|a3(ξ,τ)|/lessorsimilar∞/summationdisplay\nk=1εk|ξ|2k\n(2|s(s−ξ1)|)k+1(k+1)|s−ξ1|\n|s+ξ1|/lessorsimilar1.\nIt remains to control u2\n3. LetGk(x1,¯ξ,τ) =F−1\nx11−τ−|¯ξ|2∼11|ξ|∼1|ξ|2k/hatwideF. Note that\n/ba∇dblGk/ba∇dblL1,2\ne1/lessorsimilarck/ba∇dblF/ba∇dblL1,2\ne1.\nThen\nu2\n3=∞/summationdisplay\nk=1(−iε)k/integraldisplay\nR/integraldisplay\nRn+1eitτeixξ1|s−ξ1|≫ε\n(2s(ξ1−s))k+1[e−iy1ξ1Gk(y1,¯ξ,τ)]dξdτdy 1\n=∞/summationdisplay\nk=1(−iε)k/integraldisplay\nRTk\ny1(G(y1,·))(t,x)dy1\nwhere\nTk\ny1(f)(t,x) =/integraldisplay\nRn+1eitτeixξ1|s−ξ1|≫ε\n(2s(ξ1−s))k+1[e−iy1ξ1f(y1,¯ξ,τ)]dξdτ.8 Z. GUO AND C. HUANG\nWe have\nTk\ny1(f)(t,x) =/integraldisplay\nRn/integraldisplay\nRei(x1−y1)ξ11|s−ξ1|≫ε\n(ξ1−s)k+1dξ1·(2s)−k−1eitτei¯x·¯ξ[f(y1,¯ξ,τ)]d¯ξdτ\n=/integraldisplay\nRei(x1−y1)ξ11|ξ1|≫ε\n(ξ1)k+1dξ1·/integraldisplay\nRnei(x1−y1)s(2s)−k−1eitτei¯x·¯ξ[f(y1,¯ξ,τ)]d¯ξdτ.\nThen we get\n|Tk\ny1(f)(t,x)|/lessorsimilarM−kε−k/vextendsingle/vextendsingle/vextendsingle/vextendsingle/integraldisplay\nRnei(x1−y1)s·(2s)−k−1eitτei¯x·¯ξ[f(y1,¯ξ,τ)]d¯ξdτ/vextendsingle/vextendsingle/vextendsingle/vextendsingle.\nMaking a change of variable η1=s=/radicalbig\n−τ−|¯ξ|2,dτ=−2η1dη1, we obtain\n/integraldisplay\nRnei(x1−y1)s·(2s)−k−1eitτei¯x·¯ξ[f(y1,¯ξ,τ)]d¯ξdτ\n=/integraldisplay\nRnei(x1−y1)η1·(2η1)−keit(η2\n1+|¯ξ|2)ei¯x·¯ξ[f(y1,¯ξ,η2\n1+|¯ξ|2)]d¯ξdτ.\nThus, by the linear estimate (see Lemma 3.1) we get\n/ba∇dblTk\ny1(f)/ba∇dbl\nL2x1L∞\n¯x,t/lessorsimilarM−kε−k/ba∇dblf/ba∇dbl2,\nwhich suffices to give the estimate for u2\n3. We complete the proof of the lemma. /square\nLemma 3.4. Letn≥3,k∈Z,ε≥0. Assume u,Fsolves the equation\nut−i∆u−ε∆u=F(x,t), u(x,0) =u0.\nThen for any e∈Sn−1we have\n/ba∇dblPku/ba∇dbl\nL∞\ntL2x∩L2\ntL2n\nn−2\nx/lessorsimilar/ba∇dblu0/ba∇dblL2+/ba∇dblF/ba∇dblNk, (3.10)\nwhere the implicit constant is independent of ε.\nProof.Since we have for t >0\n/ba∇dbleit∆+εt∆u0/ba∇dblL∞x/lessorsimilart−n/2/ba∇dblu0/ba∇dblL1x\n/ba∇dbleit∆+εt∆u0/ba∇dblL2x/lessorsimilar/ba∇dblu0/ba∇dblL2x\nwhere the implicit constant is independent of ε, then by the abstract framework of\nKeel-Tao [16] we get the Strichartz estimates\n/ba∇dblPku/ba∇dbl\nL∞\ntL2x∩L2\ntL2n\nn−2\nx/lessorsimilar/ba∇dblu0/ba∇dblL2+/ba∇dblF/ba∇dblL1\ntL2x,\nwith the implicit constant independent of ε.\nBy the same argument as in Step 2 of the proof of Lemma 3.3, we get\n/ba∇dblPku/ba∇dbl\nL∞\ntL2x∩L2\ntL2n\nn−2\nx/lessorsimilar/ba∇dblu0/ba∇dblL2+2−k/2sup\ne∈Sn−1/ba∇dblF/ba∇dblL1,2\ne.\nOn the other hand, by Lemma 2.1, Lemma 3.1 and Lemma 3.2, we get\n/ba∇dblPku/ba∇dbl\nL∞\ntL2x∩L2\ntL2n\nn−2\nx/lessorsimilar/ba∇dblPku/ba∇dblX0,1/2,1\n+/lessorsimilar/ba∇dblu0/ba∇dblL2+/ba∇dblF/ba∇dblX0,−1/2,1\n+.\nThus we complete the proof. /square\nGathering the above lemmas, we can get the following linear estimates :LANDAU-LIFSHITZ EQUATION 9\nLemma 3.5 (Linear estimates) .Assumen≥3,u,Fsolves the equation: for ε >0\nut−i∆u−ε∆u=F(x,t), u(x,0) =u0.\nThen for s∈R\n/ba∇dblu/ba∇dblFs∩Zs/lessorsimilar/ba∇dblu0/ba∇dbl˙Bs\n2,1+/ba∇dblF/ba∇dblNs, (3.11)\nwhere the implicit constant is independent of ε.\n4.Nonlinear estimates\nIn this section we prove some nonlinear estimates. The nonlinear ter m in the\nLandau-Lifshitz equation is\nG(u) =¯u\n1+|u|2n/summationdisplay\nj=1(∂xju)2.\nBy Taylor’s expansion, if /ba∇dblu/ba∇dbl∞<1 we have\nG(u) =∞/summationdisplay\nk=0¯u(−1)k|u|2kn/summationdisplay\nj=1(∂xju)2.\nSo we will need to do multilinear estimates.\nLemma 4.1. (1) Ifj≥2k−100andXis a space-time translation invariant Banach\nspace, then Q≤jPkis bounded on Xwith bound independent of j,k.\n(2) For any j,k,Q≤jPk,eis bounded on Lp,2\neandQ≤jis bounded on Lp\ntL2\nxfor\n1≤p≤ ∞, with bound independent of j,k.\nProof.See the proof of Lemma 5.4 in [8]. /square\nLemma 4.2. Assumen≥3,k1,k2,k3∈Z. Then\n/ba∇dblPk1uPk2v/ba∇dblL2\nt,x/lessorsimilar2(n−1)k1/22−k2/2/ba∇dblPk1u/ba∇dblYk1+Fk1/ba∇dblPk2v/ba∇dblFk2,(4.1)\n/ba∇dblPk3(Pk1uPk2v)/ba∇dblL2\nt,x/lessorsimilar2(n−2)min( k1,k2,k3)\n2/ba∇dblPk1u/ba∇dblYk1/ba∇dblPk2v/ba∇dblYk2. (4.2)\nProof.For the first inequality, we have\n/ba∇dblPk1uPk2v/ba∇dblL2\nt,x/lessorsimilarn/summationdisplay\nj=1/ba∇dblPk1uPk2,ejΘj\nk2v/ba∇dblL2\nt,x/lessorsimilarn/summationdisplay\nj=1/ba∇dblPk1u/ba∇dblL2,∞\nej/ba∇dblPk2,ejv/ba∇dblL∞,2\nej\n/lessorsimilar2(n−1)k1/22−k2/2/ba∇dblPk1u/ba∇dblYk1+Fk1/ba∇dblPk2v/ba∇dblFk2.\nFor the second inequality, if k3≤min(k1,k2), then\n/ba∇dblPk3(Pk1uPk2v)/ba∇dblL2\nt,x/lessorsimilar2k3(n−2)/2/ba∇dblPk1uPk2v/ba∇dbl\nL2\ntL2n\n2n−2\nx\n/lessorsimilar2k3(n−2)/2/ba∇dblPk1u/ba∇dbl\nL2\ntL2n\nn−2\nx/ba∇dblPk2v/ba∇dblL∞\ntL2x.\nIfk1≤min(k2,k3), then\n/ba∇dblPk3(Pk1uPk2v)/ba∇dblL2\nt,x/lessorsimilar/ba∇dblPk1u/ba∇dblL2\ntL∞x/ba∇dblPk2v/ba∇dblL∞\ntL2x\n/lessorsimilar2k1(n−2)/2/ba∇dblPk1u/ba∇dbl\nL2\ntL2n\nn−2\nx/ba∇dblPk2v/ba∇dblL∞\ntL2x.\nIfk2≤min(k1,k3), the proof is identical to the above case. /square10 Z. GUO AND C. HUANG\nLemma 4.3 (Algebra properties) .Ifs≥n/2, then we have\n/ba∇dbluv/ba∇dblYs/lessorsimilar/ba∇dblu/ba∇dblYs/ba∇dblv/ba∇dblYn/2+/ba∇dblu/ba∇dblYn/2/ba∇dblv/ba∇dblYs.\nProof.We only show the case s=n/2. By the embedding ˙Bn/2\n2,1⊂L∞\nxwe get\n/ba∇dblu/ba∇dblL∞\nx,t≤ /ba∇dblu/ba∇dblL∞\nt˙Bn/2\n2,1/lessorsimilar/ba∇dblu/ba∇dblYn/2.\nThe Lebesgue component can be easily handled by para-product de composition and\nH¨ older’s inequality. Now we deal with Xs,b-type space. By (2.3) it suffices to show\n/summationdisplay\nk2nk/22−k/ba∇dblPk(fg)/ba∇dblX0,1+¯X0,1/lessorsimilar/ba∇dblf/ba∇dblYn/2/ba∇dblg/ba∇dblYn/2, (4.3)\nWe have\n/summationdisplay\nk2nk/22−k/ba∇dblPk(fg)/ba∇dblX0,1+¯X0,1\n/lessorsimilar/summationdisplay\nki2nk3/22−k3/ba∇dblPk3(Pk1fPk2g)/ba∇dblX0,1+¯X0,1\n/lessorsimilar(/summationdisplay\nki:k1≤k2+/summationdisplay\nki:k1>k2)2k3n/22−k3/ba∇dblPk3(Pk1fPk2g)/ba∇dblX0,1+¯X0,1\n:=I+II.\nBy symmetry, we only need to estimate the term I.\nAssumePk1f=Pk1f1+Pk1f2,Pk2g=Pk2g1+Pk2g2such that\n/ba∇dblPk1f1/ba∇dblX0,1+/ba∇dblPk1f2/ba∇dbl¯X0,1/lessorsimilar/ba∇dblPk1f/ba∇dblX0,1+¯X0,1,\n/ba∇dblPk2g1/ba∇dblX0,1+/ba∇dblPk2g2/ba∇dbl¯X0,1/lessorsimilar/ba∇dblPk2g/ba∇dblX0,1+¯X0,1.\nThen we have\nI/lessorsimilar/summationdisplay\nki:k1≤k22/summationdisplay\nj=12nk3/22−k3/ba∇dblPk3(Pk1fjPk2g1)/ba∇dblX0,1\n+/summationdisplay\nki:k1≤k22/summationdisplay\nj=12nk3/22−k3/ba∇dblPk3(Pk1fjPk2g2)/ba∇dbl¯X0,1\n:=I1+I2.\nWe only estimate the term I1since the term I2can be estimated in a similar way.\nWe have\nI1/lessorsimilar/summationdisplay\nki:k1≤k22/summationdisplay\nj=12nk3/22−k3/ba∇dblPk3(Pk1fjPk2g1)/ba∇dblX0,1. (4.4)\nFirst we assume k3≤k1+5 in the summation of (4.4). We have\nI1/lessorsimilar/summationdisplay\nki:k1≤k22/summationdisplay\nj=12nk3/22−k3(/ba∇dblPk3Q≤k1+k2+9(Pk1fjPk2g1)/ba∇dblX0,1\n+/ba∇dblPk3Q≥k1+k2+10(Pk1fjPk2g1)/ba∇dblX0,1)\n:=I11+I12.LANDAU-LIFSHITZ EQUATION 11\nFor the term I11we have\nI11/lessorsimilar/summationdisplay\nki:k1≤k22/summationdisplay\nj=12k3n/22−k32k1+k2/ba∇dblPk1fj/ba∇dblL∞\ntLnx/ba∇dblPk2g1/ba∇dbl\nL2\ntL2n\nn−2\nx\n/lessorsimilar/summationdisplay\nki:k1≤k22/summationdisplay\nj=12k3n/22−k32k22k1n/2/ba∇dblPk1fj/ba∇dblL∞\ntL2x/ba∇dblPk2g1/ba∇dbl\nL2\ntL2n\nn−2\nx\n/lessorsimilar/ba∇dblf/ba∇dblYn/2/ba∇dblg/ba∇dblYn/2.\nFor the term I12, we need to exploit the nonlinear interactions as in [8]. We have\nFPk3Q≥k1+k2+10(Pk1fjPk2g1)\n=χk3(ξ3)χ≥k1+k2+10(τ3+|ξ3|2)/integraldisplay\nξ3=ξ1+ξ2,τ3=τ1+τ2χk1(ξ1)/hatwidefj(τ1,ξ1)χk2(ξ2)/hatwideg1(τ2,ξ2).\nWe assume j= 1 since j= 2 is similar. On the plane {ξ3=ξ1+ξ2,τ3=τ1+τ2}we\nhave\nτ3+|ξ3|2=τ1+|ξ1|2+τ2+|ξ2|2−H(ξ1,ξ2) (4.5)\nwhereHis the resonance function in the product Pk3(Pk1fjPk2g1)\nH(ξ1,ξ2) =|ξ1|2+|ξ2|2−|ξ1+ξ2|2. (4.6)\nSince|H|/lessorsimilar2k1+k2, then one of Pk1fj,Pk2g1has modulation larger than the output\nmodulation, namely\nmax(|τ1+|ξ1|2|,|τ2+|ξ2|2|)/greaterorsimilar|τ3+|ξ3|2|.\nIfPk1fjhas larger modulation, then\nI12/lessorsimilar/summationdisplay\nki:k1≤k22nk3/22−k3/ba∇dbl2j3/ba∇dblPk3Qj3(Pk1fjPk2g1)/ba∇dblL2\nt,x/ba∇dbll2\nj3≥k1+k2+10\n/lessorsimilar/summationdisplay\nki:k1≤k22/summationdisplay\nj=12nk3/22nk3/22−k3(/summationdisplay\nj3≥k1+k2+1022j3/ba∇dblQ≥j3Pk1fj/ba∇dbl2\nL2\nt,x)1/2/ba∇dblPk2g1/ba∇dblL∞\ntL2x\n/lessorsimilar/summationdisplay\nki:k1≤k22nk32−k3(/ba∇dblPk1f1/ba∇dblX0,1+/ba∇dblPk1f2/ba∇dbl¯X0,1)/ba∇dblPk2g1/ba∇dblYk2\n/lessorsimilar/ba∇dblf/ba∇dblYn/2/ba∇dblg/ba∇dblYn/2.\nIfPk2g1has larger modulation, then\nI12/lessorsimilar/summationdisplay\nki:k1≤k22/summationdisplay\nj=12nk3/22−k3/ba∇dblPk1fj/ba∇dblL∞\nt,x(/summationdisplay\nj3≥k1+k222j3/ba∇dblPk2Q≥j3g1/ba∇dbl2\nL2\ntL2x)1/2\n/lessorsimilar/summationdisplay\nki:k1≤k22/summationdisplay\nj=12nk3/22−k32nk1/2/ba∇dblPk1fj/ba∇dblYk1/ba∇dblPk2g1/ba∇dblX0,1\n/lessorsimilar/ba∇dblf/ba∇dblYn/2/ba∇dblg/ba∇dblYn/2.12 Z. GUO AND C. HUANG\nNow we assume k3≥k1+6 in the summation of (4.4) and thus |k2−k3| ≤4. We\nhave\nI1/lessorsimilar/summationdisplay\nki:k1≤k22/summationdisplay\nj=12nk3/22−k3(/ba∇dblPk3Q≤k1+k2+9(Pk1fjPk2g1)/ba∇dblX0,1\n+/ba∇dblPk3Q≥k1+k2+10(Pk1fjPk2g1)/ba∇dblX0,1)\n:=˜I11+˜I12.\nBy Lemma 4.2 we get\n˜I11/lessorsimilar/summationdisplay\nki:k1≤k22/summationdisplay\nj=12nk3/22−k3(/ba∇dblPk3Q≤k1+k2+9(Pk1fjPk2g1)/ba∇dblX0,1\n/lessorsimilar/summationdisplay\nki:k1≤k22nk3/22k12(n−2)k1/2/ba∇dblPk1fj/ba∇dblYk1/ba∇dblPk2g1/ba∇dblYk2/lessorsimilar/ba∇dblf/ba∇dblYn/2/ba∇dblg/ba∇dblYn/2.\nFor the term ˜I12, similarly as the term I12, one ofPk1fj,Pk2g1has modulation larger\nthan the output modulation. If Pk1fjhas larger modulation, then\n˜I12/lessorsimilar/summationdisplay\nki:k1≤k22/summationdisplay\nj=12nk3/22−k3(/summationdisplay\nj322j3/ba∇dblPk1Q≥j3fj/ba∇dbl2\nL2\ntL∞x)1/2/ba∇dblPk2g1/ba∇dblL∞\ntL2x\n/lessorsimilar/ba∇dblf/ba∇dblYn/2/ba∇dblg/ba∇dblYn/2.\nIfPk2g1has larger modulation, then\n˜I12/lessorsimilar/summationdisplay\nki:k1≤k22/summationdisplay\nj=12nk3/22−k3/ba∇dblPk1fj/ba∇dblL∞\ntL∞x(/summationdisplay\nj322j3/ba∇dblPk2Q≥j3g1/ba∇dbl2\nL2\ntL2x)1/2\n/lessorsimilar/ba∇dblf/ba∇dblYn/2/ba∇dblg/ba∇dblYn/2.\nThus, we complete the proof. /square\nLemma 4.4. We have\n/summationdisplay\nkj2k3(n−2)/2/ba∇dblPk3[un/summationdisplay\ni=1(∂xiPk1v∂xiPk2w)]/ba∇dblL2\nt,x/lessorsimilar/ba∇dblu/ba∇dblYn/2/ba∇dblv/ba∇dblFn/2/ba∇dblw/ba∇dblFn/2.(4.7)\nProof.We have\nLHS of (4.7) /lessorsimilar/summationdisplay\nkj2k3(n−2)/2/ba∇dblPk3[P≥k3−10un/summationdisplay\ni=1(∂xiPk1v∂xiPk2w)]/ba∇dblL2\nt,x\n+/summationdisplay\nkj2k3(n−2)/2/ba∇dblPk3[P≤k3−10un/summationdisplay\ni=1(∂xiPk1v∂xiPk2w)]/ba∇dblL2\nt,x\n:=I+II.LANDAU-LIFSHITZ EQUATION 13\nBy symmetry, we may assume k1≤k2in the above summation. For the term II,\nsincen≥3, then we have\nII/lessorsimilar/ba∇dblu/ba∇dblYn/2/summationdisplay\nkj2k3(n−2)/2/ba∇dbl˜Pk3n/summationdisplay\ni=1(∂xiPk1v∂xiPk2w)]/ba∇dblL2\nt,x\n/lessorsimilar/ba∇dblu/ba∇dblYn/2/summationdisplay\nk1,k3≤k2+52k3(n−2)/22k1+k22[(n−1)k1−k2]/2/ba∇dblPk1v/ba∇dblFk1/ba∇dblPk2w/ba∇dblFk2\n/lessorsimilar/ba∇dblu/ba∇dblYn/2/ba∇dblv/ba∇dblFn/2/ba∇dblw/ba∇dblFn/2.\nFor the term I, ifk3≤k2+20, then we get from Lemma 4.2 that\nI/lessorsimilar/summationdisplay\nkj2nk3/22k3(n−2)/2/ba∇dblP≥k3−10u/ba∇dblL∞\ntL2x/ba∇dbln/summationdisplay\ni=1(∂xiPk1v∂xiPk2w)]/ba∇dblL2\nt,x\n/lessorsimilar/summationdisplay\nkj2nk3/22k3(n−2)/22(n+1)k1/22k2/2/ba∇dblP≥k3−10u/ba∇dblL∞\ntL2x/ba∇dblPk1v/ba∇dblFk1/ba∇dblPk2w/ba∇dblFk2\n/lessorsimilar/ba∇dblu/ba∇dblYn/2/ba∇dblv/ba∇dblFn/2/ba∇dblw/ba∇dblFn/2.\nIfk3≥k2+20, then uhas frequency ∼2k3, and thus we get\nI/lessorsimilar/summationdisplay\nkj2k3(n−2)/2/ba∇dblPk3u/ba∇dblL∞\ntL2x/ba∇dbln/summationdisplay\ni=1(∂xiPk1v∂xiPk2w)]/ba∇dblL2\ntL∞x\n/lessorsimilar/summationdisplay\nkj2k3(n−2)/2/ba∇dblPk3u/ba∇dblL∞\ntL2x2nk2/2/ba∇dbln/summationdisplay\ni=1(∂xiPk1v∂xiPk2w)]/ba∇dblL2\ntL2x\n/lessorsimilar/summationdisplay\nkj2k3(n−2)/22k1+k22(n−1)k1/22(n−1)k2/2/ba∇dblPk3u/ba∇dblL∞\ntL2x/ba∇dblPk1v/ba∇dblFk1/ba∇dblPk2w/ba∇dblFk2\n/lessorsimilar/ba∇dblu/ba∇dblYn/2/ba∇dblv/ba∇dblFn/2/ba∇dblw/ba∇dblFn/2.\nTherefore we complete the proof. /square\nLemma 4.5. We have\n/ba∇dblun/summationdisplay\ni=1(∂xiv∂xiw)/ba∇dblNn/2/lessorsimilar/ba∇dblu/ba∇dblYn/2/ba∇dblv/ba∇dblFn/2∩Zn/2/ba∇dblw/ba∇dblFn/2∩Zn/2. (4.8)\nProof.By the definition of Nn/2, theL2component was handled by the previous\nlemma. We only need to control\n/summationdisplay\nki2k4n/2/ba∇dblPk4[Pk1un/summationdisplay\ni=1(Pk2∂xiv∂xiPk3w)]/ba∇dblNk4. (4.9)14 Z. GUO AND C. HUANG\nBy symmetry we may assume k2≤k3in the above summation. If in the above\nsummation we assume k4≤k1+40, then\n(4.9)/lessorsimilar/summationdisplay\nki2k4n/2/ba∇dblPk4[Pk1uPk2∂xiv∂xiPk3w]/ba∇dblL1\ntL2x\n/lessorsimilar/summationdisplay\nki2k1n/2/ba∇dblPk1u/ba∇dblL∞\ntL2x2k2/ba∇dblPk2v/ba∇dblL2\ntL∞x2k3/ba∇dblPk3w/ba∇dblL2\ntL∞x\n/lessorsimilar/summationdisplay\nki2k1n/2/ba∇dblPk1u/ba∇dblL∞\ntL2x2k2n/2/ba∇dblPk2v/ba∇dbl\nL2\ntL2n\nn−2\nx2k3n/2/ba∇dblPk3w/ba∇dbl\nL2\ntL2n\nn−2\nx\n/lessorsimilar/ba∇dblu/ba∇dblYn/2/ba∇dblv/ba∇dblFn/2/ba∇dblw/ba∇dblFn/2.\nThus from now on we assume k4≥k1+ 40 in the summation of (4.9). We bound\nthe summation case by case.\nCase 1: k2≤k1+20\nIn this case we have k4≥k2+20 and hence |k4−k3| ≤5. By Lemma 4.2 we get\n(4.9)/lessorsimilar/summationdisplay\nki2k4n/22−k4/2/ba∇dblPk4[Pk1uPk2∂xiv∂xiPk3w]/ba∇dblL1,2\ne\n/lessorsimilar/summationdisplay\nki2k4n/22−k4/2/ba∇dblPk1uPk3∂xiw/ba∇dblL2\nx,t/ba∇dblPk2∂xiv/ba∇dblL2,∞\ne\n/lessorsimilar/summationdisplay\nki2k4n/22−k4/22(n−1)k1/22−k3/22(n−1)k2/2/ba∇dblPk1u/ba∇dblYk1/ba∇dblPk3∂xiw/ba∇dblFk3/ba∇dblPk2∂xiv/ba∇dblFk2\n/lessorsimilar/ba∇dblu/ba∇dblYn/2/ba∇dblv/ba∇dblFn/2/ba∇dblw/ba∇dblFn/2.\nCase 2: k2≥k1+21\nIn this case we have k4≤k3+40. Let g=/summationtextn\ni=1(Pk2∂xiv·Pk3∂xiw). Then we have\n(4.9)/lessorsimilar/summationdisplay\nki2k4n/2/ba∇dblPk4[Pk1uQ≤k2+k3g]/ba∇dblNk4+/summationdisplay\nki2k4n/2/ba∇dblPk4[Pk1uQ≥k2+k3g]/ba∇dblNk4\n:=I+II.\nFirst we estimate the term II. We have\nII/lessorsimilar/summationdisplay\nki2k4n/2/ba∇dblPk4[Pk1Q≥k2+k3−10u·Q≥k2+k3g]/ba∇dblNk4\n+/summationdisplay\nki2k4n/2/ba∇dblPk4[Pk1Q≤k2+k3−10u·Q≥k2+k3g]/ba∇dblNk4\n:=II1+II2.LANDAU-LIFSHITZ EQUATION 15\nFor the term II1we have\nII1/lessorsimilar/summationdisplay\nki2k4n/2/ba∇dblPk4[Pk1Q≥k2+k3−10u·Q≥k2+k3g]/ba∇dblL1\ntL2x\n/lessorsimilar/summationdisplay\nki2k4n/2/ba∇dblPk1Q≥k2+k3−10u/ba∇dblL2\ntL∞x/ba∇dblQ≥k2+k3g]/ba∇dblL2\ntL2x\n/lessorsimilar/summationdisplay\nki2k4n/22k1n/2/ba∇dblPk1Q≥k2+k3−10u/ba∇dblL2\ntL2x/ba∇dblQ≥k2+k3g]/ba∇dblL2\ntL2x\n/lessorsimilar/summationdisplay\nki2k4n/22k1n/22−(k2+k3)/ba∇dblPk1Q≥2k1+10u/ba∇dblX0,1\n·2[(n−1)k2−k3]/22k2+k3/ba∇dblPk2v/ba∇dblFk2/ba∇dblPk3w/ba∇dblFk3\n/lessorsimilar/ba∇dblu/ba∇dblYn/2/ba∇dblv/ba∇dblFn/2/ba∇dblw/ba∇dblFn/2.\nFor the term II2, sincek4≥k1+ 40, then we may assume ghas frequency of size\n2k4. The resonance function in the product Pk1u·Pk4gis of size /lessorsimilar2k1+k4. Thus the\noutput modulation is of size /greaterorsimilar2k2+k3. Then we get\nII2/lessorsimilar/summationdisplay\nki2k4n/22−(k2+k3)/2/ba∇dblPk4[Pk1Q≤k2+k3−10u·Q≥k2+k3g]/ba∇dblL2\nt,x\n/lessorsimilar/summationdisplay\nki2k4n/22−(k2+k3)/22k1n/2/ba∇dblPk1u/ba∇dblL∞\ntL2x·/ba∇dblg/ba∇dblL2\nt,x\n/lessorsimilar/summationdisplay\nki2k4n/22−(k2+k3)/22k1n/22[(n−1)k2−k3]/22k2+k3/ba∇dblPk1u/ba∇dblYk1/ba∇dblPk2v/ba∇dblFk2/ba∇dblPk3w/ba∇dblFk3\n/lessorsimilar/ba∇dblu/ba∇dblYn/2/ba∇dblv/ba∇dblFn/2/ba∇dblw/ba∇dblFn/2.\nNow we estimate the term I. We have\nI/lessorsimilar/summationdisplay\nki2k4n/2/ba∇dblPk4[Pk1u·Q≤k2+k3n/summationdisplay\ni=1(Pk2∂xiQ≥k2+k3+40v·Pk3∂xiw)]/ba∇dblNk4\n+/summationdisplay\nki2k4n/2/ba∇dblPk4[Pk1u·Q≤k2+k3n/summationdisplay\ni=1(Pk2∂xiQ≤k2+k3+39v·Pk3∂xiw)]/ba∇dblNk4\n:=I1+I2.\nFor the term I1, since the resonance function in the product Pk2v·Pk3wis of size\n/lessorsimilar2k2+k3, then we may assume Pk3whas modulation of size /greaterorsimilar2k2+k3. Then we get\nI1/lessorsimilar/summationdisplay\nki2k4n/22−k4/2/ba∇dblPk4[Pk1u·Q≤k2+k3n/summationdisplay\ni=1(Pk2∂xiQ≥k2+k3+40v·Pk3∂xiQ≥k2+k3−5w)]/ba∇dblL1,2\ne\n/lessorsimilar/summationdisplay\nki2k4n/22−k4/2/ba∇dblPk1u/ba∇dblL∞\nt,x2k2+k3/ba∇dblPk2v/ba∇dblL2,∞\ne/ba∇dblPk3Q≥k2+k3−5w/ba∇dblL2\nt,x\n/lessorsimilar/summationdisplay\nki2k4n/22−k4/22(k2+k3)/22(n−1)k2/22k1n/2/ba∇dblPk1u/ba∇dblYk1/ba∇dblPk2v/ba∇dblFk2/ba∇dblPk3w/ba∇dblFk3\n/lessorsimilar/ba∇dblu/ba∇dblYn/2/ba∇dblv/ba∇dblFn/2/ba∇dblw/ba∇dblFn/2.16 Z. GUO AND C. HUANG\nFinally, we estimate the term I2. For this term, we need to use the null structure\nobserved by Bejenaru [1]. We can rewrite\n−2∇u·∇v= (i∂t+∆)u·v+u·(i∂t+∆)v−(i∂t+∆)(u·v).(4.10)\nThen we have\nI2=/summationdisplay\nki2k4n/2/ba∇dblPk4[Pk1u·Q≤k2+k3(Pk2LQ≤k2+k3+39v·Pk3w)]/ba∇dblNk4\n+/summationdisplay\nki2k4n/2/ba∇dblPk4[Pk1u·Q≤k2+k3(Pk2Q≤k2+k3+39v·Pk3Lw)]/ba∇dblNk4\n+/summationdisplay\nki2k4n/2/ba∇dblPk4[Pk1u·Q≤k2+k3L(Pk2Q≤k2+k3+39v·Pk3w)]/ba∇dblNk4\n:=I21+I22+I23.\nFor the term I21, we have\nI21/lessorsimilar/summationdisplay\nki2k4n/2/ba∇dblPk4[Pk1u·Q≤k2+k3(Pk2LQ≤k2+k3+39v·Pk3w)]/ba∇dblL1\ntL2x\n/lessorsimilar/summationdisplay\nki2k4n/22k1n/2/ba∇dblPk1u/ba∇dblL∞\ntL2x2k2(n−2)/2/ba∇dblPk2Lv/ba∇dblL2\nt,x/ba∇dblPk3w/ba∇dbl\nL2\ntL2n\nn−2\nx\n/lessorsimilar/ba∇dblu/ba∇dblYn/2/ba∇dblv/ba∇dblZn/2/ba∇dblw/ba∇dblFn/2.\nFor the term I22, we may assume whas modulation /lessorsimilar2k2+k3. Then we get\nI22/lessorsimilar/summationdisplay\nki2k4n/22−k4/2/ba∇dblPk4[Pk1u·Q≤k2+k3(Pk2Q≤k2+k3+39v·Pk3Q≤k2+k3+100Lw)]/ba∇dblL1,2\ne\n/lessorsimilar/summationdisplay\nki2k4n/22−k4/22k1n/2/ba∇dblPk1u/ba∇dblL∞\ntL2x/ba∇dblPk2v/ba∇dblL2,∞\ne/ba∇dblPk3Q≤k2+k3+100Lw)]/ba∇dblL2\nt,x\n/lessorsimilar/summationdisplay\nki2k4n/22−k4/22nk1/22(n−1)k2/22(k2+k3)/2/ba∇dblPk1u/ba∇dblYk1/ba∇dblPk2v/ba∇dblFk2/ba∇dblPk3w/ba∇dblX0,1/2,∞\n/lessorsimilar/ba∇dblu/ba∇dblYn/2/ba∇dblv/ba∇dblFn/2/ba∇dblw/ba∇dblFn/2.\nNext we estimate the term I23. We have\nI23/lessorsimilar/summationdisplay\nki2k4n/2/ba∇dblPk4[Pk1u·Q[k1+k4+100,k2+k3]L(Pk2Q≤k2+k3+39v·Pk3w)]/ba∇dblNk4\n+/summationdisplay\nki2k4n/2/ba∇dblPk4[Pk1u·Q≤k1+k4+99L(Pk2Q≤k2+k3+39v·Pk3w)]/ba∇dblNk4\n:=I231+I232.\nFor the term I232we have\nI232/lessorsimilar/summationdisplay\nki2k4n/22−k4/2/ba∇dblPk4[Pk1u·Q≤k1+k4+99L(Pk2Q≤k2+k3+39v·Pk3w)]/ba∇dblL1,2\ne\n/lessorsimilar/summationdisplay\nki2k4n/22−k4/2/ba∇dblPk1u/ba∇dblL2,∞\ne2k1+k42[(n−1)k2−k3]/2/ba∇dblPk2v/ba∇dblFk2/ba∇dblPk3w/ba∇dblFk3\n/lessorsimilar/ba∇dblu/ba∇dblYn/2/ba∇dblv/ba∇dblFn/2/ba∇dblw/ba∇dblFn/2.LANDAU-LIFSHITZ EQUATION 17\nFor the term I231we have\nI231/lessorsimilar/summationdisplay\nkik2+k3/summationdisplay\nj2=k1+k4+1002k4n/2/ba∇dblPk4Q≤j2−10[Pk1u·Qj2L(Pk2Q≤k2+k3+39v·Pk3w)]/ba∇dblNk4\n+/summationdisplay\nkik2+k3/summationdisplay\nj2=k1+k4+1002k4n/2/ba∇dblPk4Q≥j2−9[Pk1u·Qj2L(Pk2Q≤k2+k3+39v·Pk3w)]/ba∇dblNk4\n:=I2311+I2312.\nFor the term I2312we have\nI2312/lessorsimilar/summationdisplay\nkik2+k3/summationdisplay\nj2=k1+k4+100/summationdisplay\nj3≥k2−92k4n/22−j3/2\n·/ba∇dblPk4Qj3[Pk1u·Qj2L(Pk2Q≤k2+k3+39v·Pk3w)]/ba∇dblL2\nt,x\n/lessorsimilar/summationdisplay\nki2k4n/22k1n/22(k2+k3)/22[(n−1)k2−k3]/2/ba∇dblPk1u/ba∇dblYk1/ba∇dblPk2v/ba∇dblFk2/ba∇dblPk3w/ba∇dblFk3\n/lessorsimilar/ba∇dblu/ba∇dblYn/2/ba∇dblv/ba∇dblFn/2/ba∇dblw/ba∇dblFn/2.\nFor the term I2311we have\nI2311/lessorsimilar/summationdisplay\nkik2+k3/summationdisplay\nj2=k1+k4+1002k4n/2/ba∇dblPk4Q≤j2−10[Pk1˜Qj2u·Qj2L(Pk2Q≤k2+k3+39v·Pk3w)]/ba∇dblL1\ntL2x\n/lessorsimilar/summationdisplay\nkik2+k3/summationdisplay\nj2=k1+k4+1002k4n/22k1n/2/ba∇dblPk1˜Qj2u/ba∇dblL2\nt,x2j22[(n−1)k2−k3]/2/ba∇dblPk2v/ba∇dblFk2/ba∇dblPk3w/ba∇dblFk3\n/lessorsimilar/ba∇dblu/ba∇dblYn/2/ba∇dblv/ba∇dblFn/2/ba∇dblw/ba∇dblFn/2.\nTherefore, we complete the proof. /square\nCombining all the estimates above we get\nLemma 4.6 (Nonlinear estimates) .Assumeu∈Fn/2∩Zn/2with/ba∇dblu/ba∇dblYn/2≪1.\nThen/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble¯u\n1+|u|2n/summationdisplay\nj=1(∂xju)2/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble\nNn/2/lessorsimilar/ba∇dblu/ba∇dblYn/2\n1−/ba∇dblu/ba∇dbl2\nYn/2/ba∇dblu/ba∇dblFn/2∩Zn/2/ba∇dblu/ba∇dblFn/2∩Zn/2.\nProof.SinceYn/2⊂L∞, then\n¯u\n1+|u|2n/summationdisplay\nj=1(∂xju)2=∞/summationdisplay\nk=0¯u(−1)k|u|2kn/summationdisplay\nj=1(∂xju)2.\nThe lemma follows from Lemma 4.5, Lemma 4.4 and Lemma 4.3. /square\n5.The limit behaviour\nIn this section we prove Theorem 1.1. It is equivalent to prove\nTheorem 5.1. Assumen≥3,ε∈[0,1]. There exists 0< δ≪1such for any\nφ∈˙Bn/2\n2,1with/ba∇dblφ/ba∇dbl˙Bn/2\n2,1≤δ, there exists a unique global solution uεto(1.5)such that\n/ba∇dbluε/ba∇dblFn/2∩Zn/2/lessorsimilarδ,18 Z. GUO AND C. HUANG\nwhere the implicit constant is independent of ε. The map φ→uεis Lipshitz from\nBδ(˙Bn/2\n2,1)toC(R;˙Bn/2\n2,1)and the Lipshitz constant is independent of ε. Moreover,\nfor anyT >0,\nlim\nε→0+/ba∇dbluε−u0/ba∇dblC([0,T];˙Bn/2\n2,1)= 0.\nFortheuniformglobalwell-posedness, wecanproveitbystandard Picarditeration\nargument by using the linear and nonlinear estimates proved in the pr evious section.\nIndeed, define\nΦu0(u) :=eit∆+εt∆u0\n−i/integraldisplayt\n0ei(t−s)∆+ε(t−s)∆/bracketleftbigg2¯u\n1+|u|2n/summationdisplay\nj=1(∂xju)2−2iε¯u\n1+|u|2n/summationdisplay\nj=1(∂xju)2/bracketrightbigg\nds.\nThenusingtheLemma3.5andLemma4.6wecanshowΦ u0isacontractionmapping\nin the set\n{u:/ba∇dblu/ba∇dblFn/2∩Zn/2≤Cδ}\nif/ba∇dblu0/ba∇dbl˙Bn/2\n2,1≤δwithδ >0sufficiently small. Thus wehaveexistence anduniqueness.\nMoreover, by standard arguments we immediately have the persist ence of regularity,\nnamely if u0∈˙Bs\n2,1for some s > n/2, thenu∈Fs∩Zsand\n/ba∇dblu/ba∇dblFs∩Zs/lessorsimilar/ba∇dblu0/ba∇dbl˙Bs\n2,1(5.1)\nuniformly with respect to ε∈(0,1].\nNow we prove the limit behaviour. Assume uεis a solution to the Landau-Lifshitz\nequation with small initial data φ1∈˙Bn/2\n2,1, anduis a solution to the Schr¨ odinger\nmap with small initial data φ2∈˙Bn/2\n2,1. Letw=uε−u,φ=φ1−φ2, thenwsolves\n(i∂t+∆)w=iε∆uε+/bracketleftbigg2¯uε\n1+|uε|2n/summationdisplay\nj=1(∂xjuε)2−2¯u\n1+|u|2n/summationdisplay\nj=1(∂xju)2/bracketrightbigg\n−2iε¯uε\n1+|uε|2n/summationdisplay\nj=1(∂xjuε)2, (5.2)\nw(0) =φ.\nFirst we assume in addition φ1∈˙B(n+4)/2\n2,1. By the linear and nonlinear estimates,\nfor anyT >0 we get\n/ba∇dblw/ba∇dblFn/2∩Zn/2/lessorsimilar/ba∇dblφ/ba∇dbl˙Bn/2\n2,1+εT/ba∇dbluε/ba∇dblL∞\nt˙B(n+4)/2\n2,1+δ2/ba∇dblw/ba∇dblFn/2∩Zn/2+ε/ba∇dbluε/ba∇dbl3\nFn/2∩Zn/2.\nThen we get by (5.1)\n/ba∇dblw/ba∇dblFn/2∩Zn/2/lessorsimilar/ba∇dblφ/ba∇dbl˙Bn/2\n2,1+εT/ba∇dblφ1/ba∇dbl˙B(n+4)/2\n2,1+εδ3. (5.3)\nNow we assume φ1=φ2=ϕ∈˙Bn/2\n2,1with small norm. For fixed T >0, we need to\nprove that ∀η >0, there exists σ >0 such that if 0 < ε < σthen\n/ba∇dblSε\nT(ϕ)−ST(ϕ)/ba∇dblC([0,T];˙Bn/2\n2,1)< η (5.4)LANDAU-LIFSHITZ EQUATION 19\nwhereSε\nTis the solution map corresponding to (5.2) and ST=S0\nT. We denote\nϕK=P≤Kϕ. Then we get\n/ba∇dblSε\nT(ϕ)−ST(ϕ)/ba∇dblC([0,T];˙Bn/2\n2,1)\n≤/ba∇dblSε\nT(ϕ)−Sε\nT(ϕK)/ba∇dblC([0,T];˙Bn/2\n2,1)\n+/ba∇dblSε\nT(ϕK)−ST(ϕK)/ba∇dblC([0,T];˙Bn/2\n2,1)+/ba∇dblST(ϕK)−ST(ϕ)/ba∇dblC([0,T];˙Bn/2\n2,1).\nFrom the uniform global well-posedness and (5.3), we get\n/ba∇dblSǫ\nT(ϕ)−ST(ϕ)/ba∇dblC([0,T];˙Bn/2\n2,1)/lessorsimilar/ba∇dblϕK−ϕ/ba∇dbl˙Bn/2\n2,1+εC(T,K,/ba∇dblϕ/ba∇dbl˙Bn/2\n2,1).(5.5)\nWe first fix Klarge enough, then let εgo to zero, therefore (5.4) holds.\nAcknowledgment. Z. Guo is supported in part by NNSF of China (No.11371037),\nand C. Huang is supported in part by NNSF of China (No. 11201498).\nReferences\n[1] I. Bejenaru, On Schr¨ odinger maps, Amer. J. Math. 130 (2008 ), 1033-1065.\n[2] I. Bejenaru, Global results for Schr¨ odinger maps in dimensions n≥3, Comm. Partial Differ-\nential Equations 33 (2008), 451-477.\n[3] I. Bejenaru, A. D. Ionescu, and C. E. Kenig, Global existence a nd uniqueness of Schr¨ odinger\nmaps in dimensions d≥4, Adv. Math. 215 (2007), 263-291.\n[4] I. Bejenaru, A. D. Ionescu, C. E. Kenigand D. Tataru, Global S chr¨ odingermapsin dimensions\nd≥2: small data in the critical Sobolevspaces, Annals ofMathematics1 73(2011), 1443-1506.\n[5] N.H. Chang,J.Shatah, K.Uhlenbeck, Schr¨ odingermaps, Com m.PureAppl. Math.53(2000),\n590-602.\n[6] W. Ding and Y. Wang, Local Schr¨ odinger flow into K¨ ahler manifold s, Sci. China Ser. A, 44\n(2001), 1446-1464.\n[7] B. Guo and S. Ding. Landau-Lifshitz equations, volume 1 of Front iers of Research with the\nChinese Academy of Sciences. World Scientific Publishing Co. Pte. Ltd ., Hackensack, NJ,\n2008.\n[8] Z. Guo, Spherically averaged maximal function and scattering fo r the 2D cubic derivative\nSchr¨ odinger equation, to appear Int. Math. Res. Notices.\n[9] Z. Guo and B. Wang, Global well posedness and inviscid limit for the K orteweg-de Vries-\nBurgers equation, J. Differential Equations 246 (2009) 3864-390 1.\n[10] L. Han, B. Wang and B. Guo, Inviscid limit for the derivative Ginzbu rg-Landau equation with\nsmall data in modulation and Sobolev spaces, Appl. Comput. Harmon. Anal. 32(2012),no.2,\n197-222.\n[11] T. Hmidi and S. Keraani, Inviscid limit for the two-dimensional N-S system in a critical Besov\nspace, Asymptot. Anal., 53 (3) (2007), 125-138.\n[12] C. Huang, B. Wang, Inviscid limit for the energy-critical complex Ginzburg-Landau equation,\nJ. Funct. Anal., 255 (2008), 681-725.\n[13] A. D. Ionescu and C. E. Kenig, Low-regularity Schr¨ odinger ma ps, Differential Integral Equa-\ntions 19 (2006), 1271-1300.\n[14] A. D. Ionescu and C. E. Kenig, Low-regularity Schr¨ odinger ma ps, II: global well-posedness in\ndimensions d≥3, Comm. Math. Phys., 271 (2007), 523-559.\n[15] C. E. Kenig, G. Ponce, and L. Vega, Smoothing effects and local existence theory for the\ngeneralized nonlinear Schr¨ odinger equations, Invent. Math., 134 (1998), 489-545.\n[16] M. Keel and T. Tao, Endpoint Strichartz estimates, Amer. J. M ath., 120 (1998), 360–413.\n[17] M. Lakshmanan. The fascinating world of the Landau-Lifshitz- Gilbert equation: an overview.\nPhilos. Trans. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 369 (193 9):1280-1300, 2011.\n[18] M. Lakshmanan and K. Nakamura, Landau-Lifshitz Equation of Ferromagnetism: Exact\nTreatment of the Gilbert Damping, Phy. Rev. Let. 53 (1984), NO. 2 6, 2497-2499.\n[19] L. D. Landau and E. M. Lifshitz, On the theory of the dispersion of magnetic permeability in\nferromagnetic bodies, Phys. Z. Sovietunion. 8 (1935) 153-169.20 Z. GUO AND C. HUANG\n[20] T. Tao, Global regularity of wave maps II. Small energy in two dim ensions, Commun. Math.\nPhys. 224 (2001), 443-544.\n[21] D. Tataru, Local andglobalresults forthe wavemaps I, Comm . PartialDifferential Equations,\n23 (1998), no. 9-10, 1781-1793.\n[22] B. Wang, Z. Huo, C. Hao and Z. Guo, Harmonic Analysis Method fo r Nonlinear Evolution\nEquations, I, World Scientific Press, 2011.\n[23] B. Wang, The limit behavior of solutions for the Cauchy problem of the Complex Ginzburg-\nLandau equation, Commu. Pure. Appl. Math., 55 (2002), 0481-050 8.\n[24] B. Wang and Y. Wang, The inviscid limit for the derivative Ginzburg- Landau equations, J.\nMath. Pures Appl., 83 (2004), 477-502.\nSchool of Mathematical Sciences, Monash University, VIC 38 00, Australia &\nLMAM, School of Mathematical Sciences, Peking University, Beijing 100871, China\nE-mail address :zihua.guo@monash.edu\nSchool of Statistics and Mathematics, Central University o f Finance and Eco-\nnomics, Beijing 100081, China\nE-mail address :hcy@cufe.edu.cn"    },    {        "title": "1509.01807v1.Study_of_spin_dynamics_and_damping_on_the_magnetic_nanowire_arrays_with_various_nanowire_widths.pdf",        "content": "1 \n Study of spin dynamics  and damping  on the magnetic nanowire  arrays   \nwith various nanowire widths  \n \nJaehun Cho a, Yuya  Fujii b, Katsunori  Konioshi b, Jungbum Yoon c, Nam -Hui Kim a, Jinyong Jung a, \nShinji Miwa b, Myung -Hwa Jung d, Yoshishige Suzuki b, and Chun -Yeol You  a,* \n \na Department of Physics, Inha University , Inch eon, 402-751, South Korea  \nb Graduate School of Engineering Science,  \nOsaka University,  Toyonaka, Osaka 560 -8531,  Japan  \nc Department of Electrical and Computer  Engineering ,  \nNational University of Singapore , Singapore  117576  \nd Department of Physics, Sogang University, Seoul, 121 -742, South Korea  \n \nAbstract  \nWe investigate the spin dynamics including Gilbert damping in the ferromagnetic  nanowire \narray s. We have measured  the ferromagnetic resonance of ferromagnetic nanowire arrays \nusing vector -network analyzer ferromagnetic resonance  (VNA -FMR)  and analyzed the results \nwith the micromagnetic si mulations . We find excellent agree ment  between the experimental \nVNA -FMR spectra and micromagnetic simulations result  for various applied magnetic fields . \nWe find that  the demagnetization factor for longitudinal conditions, Nz (Ny) increases  \n(decreas es) as decreasing the nanowire width  in the micromagnetic simulation s. For the \ntransverse magnetic field , Nz (Ny) increas es (decreas es) as increasing the nanowire width . We \nalso find that t he Gilbert damping constant increases  from 0.018 to 0.051 as the incr easing \nnanowire width  for the transverse case , while it is almost constant as 0.021  for the \nlongitudinal case . \n 2 \n * Corresponding author.  FAX: +82 32 872 7562.  \nE-mail address:  cyyou@inha.ac.kr  \nKeywords : Nanowires , Ferromagnetic Resonance , Micromagnetic  Simulations , Gilbert \ndamping  \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n 3 \n  \nFerromagnetic nanostructures have recently attracted much interest for the wide potential \napplications in high density spintronic information storage , logic devices and various spin \norbit torque phenomena .1,2,3,4,5 It is well known that the detail spin dynamics of nanostructure \nis far from the one of the bulk’s because of many reasons, different boundary conditions, \nchanges of the magnetic properties including the saturat ion magnetization, anisotropy energy, \nand exchange stiffness constant, etc. Since the magnetic properties are usually sensitive \nfunctions of the sample fabrication conditions, it has been widely accepted that the detail \nsample fabrications are also importa nt in the study of spin dynamics. However, the relatively \nless caution has been made for the boundary conditions of the spin dynamics in the \nnanostructure.  \nIn the spin transfer torque magnetic random access memory  (STT -MRAM), the magnetic \ndamping constant is important because the switching current density  is proportional to the \ndamping constant .6 In the nanowire, damping constant also plays crucial role in the spin \ndynamics including domain wall motion with magnetic field7 and spin transfer torque .8 \nFurthermore, it is the most important material parameter in spin wave (SW) dynamics .9 \nDespite of the importance of the damping constant, many studies about spin dynamics in \nferromagnetic nanowires  have not taken into account the damping constant  properly .10,11,12 \nOnly a few studies paid attention to the magnetic  damping in the nanowires  spin 4 \n dynamics .13,14 \nIn this study, arrays of CoFeB nanowire s are prepared by e -beam lithography , and they are  \ncovered coplanar  wave -guide  for the ferromagnetic resonance  (FMR) measurement  as shown \nin Fig. 1 . We measured FMR signal with longitudinal (wire direction) and transverse \nmagnetic field s in order to investigate the spin dynamics with different boundary conditions. \nAlso w e extract Gilbert damping constant  using micromagnetic  simulations  with the different \napplied magnetic field directions in various nanowire arrays . We find the damping constant \ndecreas es with increasing the nanowire width for the transverse magnetic field  with constant \ninput damping consta nt in micromagnetic simulations, while we obtain almost constant \ndamping constant for the longitudinal field.  \nThe film s were  prepared using DC magnetron sputtering. The stack s consist of Ta (5  \nnm)/Co 16Fe64B20 (30 nm)/Ta (5  nm) on single crystal MgO (001) substrate s. The film s are \npatterned as 100 -nm-width wire array s with 200 -nm-space each wires using e -beam \nlithography and an Ar ion milling technique  as shown in Fig. 1. The width is determined with \na scanning electron microscope (SEM).  These nanowire arrays are covered by coplanar wave \nguide in order to characterized with the Vector Network Analyzer ( VNA )-FMR technique \ndescribed elsewhere .15 We prepare nanowire arrays  as shown in Fig. 1 , and external DC \nmagnetic field direction  for FMR measurement  is also depicted.  \nWe use  VNA -FMR spectra to measure  imaginary parts of the susceptibility of the samples.16 5 \n The measured imaginary parts of the susceptibility raw data are calibrated with the careful \ncalibration procedures .16 The calibrated  imaginary parts of the susceptibility are shown in Fig \n2(a) and (b)  for an applied magnetic field at 0.194  T for the nanowire arrays . The un -\npatterned thin film is also examined for the reference. We find two resonance frequencies, \n17.2 and 26 .4 GHz, as shown in Fig. 2(a) for the nanowire array, while  there is only one peak \nat 16.8 GHz  for the  un-patterned  thin film  as shown in Fig 2(b). We believe that the smaller \npeak  (17.2 GHz) in Fig. 2(a) is originated  from the un-patterned part of the nanowire array, \nbecause the frequency is closed to the un -patterned thin film’s peak (16.8 GHz). Probably, the \nun-patterned part of the nanowire is formed due to poor e -beam lithography processes. On the \nother hand, t he resonance f requency  near 26.0 GHz is calculated from micromagnetic \nsimulation  at an applied  magnetic  field at 0.200 T , as shown in Fig. 2 (c).  We clarif y the \nsource of the main  peak (26.4 GHz) is nanowire  arrays  by using micromagnetic simulation.  \nThese two peaks name d as the uniform FMR  mode (smaller peak position) and nanowire \nmode (higher peak position).  \nIn order to determine the saturation magnetization, the resonance frequencies are measured \nas a function of the applied magnetic field, and the results are fitted with the Kittel ’s \nequation .17 This equation employs the corresponding demagnetization factors of Nx = 0, Ny = \n0 and Nz = 1 for un -patterned film, when applied magnetic field H is x- direction  with \nfollowing equations , 6 \n  \n    2y x s z x s f H N N M H N N M\n    \n.   (1) \n \n Here,  is the gyromagnetic ratio, H is the applied magnetic field, Ms is saturated \nmagnetization, Nx, Ny, and Nz are the demagnetization factor s applying the cyclic permutation \nfor the  applied magnetic field direction .  \nThe micromagnetic simulations are performed by using the Objective -Oriented -\nMicroMagnetic Framework (OOMMF)18 with 2-dimensional  periodic boundary  condition \n(PBC ).19 We select a square slat of 100 nm × 100 nm × 30 nm  nanowire separated 200 nm in \ny- direction with a cell size of 5 nm × 5 nm ×  30 nm. The material parameters of CoFeB used \nin our simulation are summarized as follows: Ms = 15.79 × 105 A/m, the exchange stiffness \n1.5 × 1011 J/m, the gyromagnetic ratio 2.32  × 1011 m/(A ∙s) and we ignore the magneto -\ncrystalline anisotropy. In this simulation, the Gilbert damping constant  of 0.0 27 is fixed. The \nsaturation magnetization and Gilbert damping constant are determined by using VNA -FMR  \nmeasurement for un -patterned thin film . For t he exchange  stiffness constant, experimentally \ndetermined values are range of 0.98 to 2.84 × 1011 J/m which value  has dependence on the \nfabrication processes20 and composition of ferromagnetic materials ,21 while we have picked \n1.5 × 1011 J/m as the exchange  stiffness  constant . The determination  method of Gilbert \ndamping constant will be described later.  7 \n In order to mimic FMR experiments in the micromagnetic simulations , a “sinc”  function\n0 0 0 ( ) sin 2 / 2y H HH t H f t t f t t    \n, with H0 = 10 mT, and field frequency fH = 45 \nGHz, is applied the whole nanowire  area.22 We obtain the FMR spectra in the corresponding \nfrequency range from 0 to 45 GHz . The FMR spectra due to the RF -magnetic field are \nobtained by the fast Fourier  transform  (FFT) of stored My(x) (x, y, t ) in longitudinal  (transverse)  \nH0 field. More details can be described elsewhere .23 \nThe closed blue circles  in Fig. 3 is the calculated values with the fitting parameter using Eq. \n(1) which are fitted with the experimental data of un -patterned thin film. The obt ained Ms is \n15.79 × 105 A/m while gyromagnetic ratio is fixed as 2.32 × 1011 m/(A∙s) . The obtained Ms \nvalue is similar  with vibrating sample magnetometer  method24 which CoFeB structure has Ta \nbuffer layer. The resonance frequencies of uniform FMR mode in nanowire arrays are plotted \nas open red circles in Fig. 3. The resonance frequencies of uniform FMR mode is measured \nby VNA -FMR are agreed well with resonance freq uency of un-patterned thin film measured \nby VNA -FMR. In Fig. 3, the applied field dependences of  the resonance frequencies \nMeasured by VNA -FAM for the nanowire are plotted as open black  rectangular , along with \nthe result of micromagnetics calculated with E q. (1)  as depicted  closed black rectangular . It is \nalso well agreed with the experimental result in nanowire mode and micromagnetic \nsimulation result  in the nanowire arrays.  \nIn order to reveal the effect s of spin dynamics properties with various  nanowire width s, we 8 \n perform micromagnetic simulat ions. The nanowire width s are varied from 50 to 150 nm in \n25-nm step for fixed 200 -nm-space with PBC , it causes changes of the demagnetization \nfactor  of the nanowire . In Fig. 4 (a) shows  the nanowire width  dependences of the resonance \nfrequencies for the longitudinal magnetic field (open symbols) along with the resonance \nfrequencies calculated with Eq. (1)  (solid lines) . The demagnetization factors can be \ndetermined by fitting Eq. (1) while Nx is fixed as 0  to represent infinitely long wire . The \nagreements between the results of micromagnetic simulations  (open circles)  and Eq. (1)  \n(solid lines)  are excellent.  \nFor the transverse  magnetic  field, the direction of applied magnetic field is y - axis, Eq. (1) \ncan be rewritten as follows:  \n \n    2x y s z y s f H N N M H N N M\n    \n.    (2) \n \nIn this equation, we use the relation of demagnetization factors , \n1x y zN N N   , in \norder to remove uncertainty in the fitting procedure . In the transverse field, the \ndemagnetization factors are determined by Eq. (2). The resonance frequencies  for transverse \nmagnetic field  which are obtained by micromagnetic simulation  (open circles)  and \ncalculated  by Eq. (2)  (solid lines)  as a function of the appl ied magnetic field with various \nnanowire width  are displayed in Fig. 4(b). The longitudinal case, when the field direction is 9 \n easy axis, they are saturated with small field. However, the transverse case, when the field \ndirection is hard axis, certain amoun t of field is necessary to saturate along the transverse \ndirections. The narrower nanowire, the larger field is required as shown in Fig. 4 (b).  \nFig. 5(a) and (b) show the changes of demagnetization factors in longitudinal and transverse \nmagnetic  fields as a functi on of  the nanowire width , respectively.  The demagnetization \nfactors play important role in the domain wall dynamics, for example the Walker breakdown \nis determined by the demagnetization factors .25 Furthermore, they are essential physical \nquantities to analyze the details of the spin dynamics. It is clear ly shown  that the Nz (Ny) \nincrease s (decrease s) with increasing the nanowire width  in longitudinal magnetic  field. For \nthe transverse magnetic  field, Nz (Ny) increase s (decrease s) with increasing the nanowire \nwidth , during Nx is almost zero value.  The demagnetization factors both longitudinal and \ntransverse have similar tendency with the effective demagnetization factors of dynamic \norigin26 and the static demagnetization factors for the prism geometry.27  \nNow, let us discuss about the Gilbert damping constant . The relation of the full width and \nhalf maxim a (f) of a resonance peak s as a function of applied field are shown in Fig. 6 for \nlongitudinal (a) and transverse (b). The f is given by15: \n \n,\n,2\n22xy\ns z ex yxN\nf H M N N f\n         \n. (3)  \n 10 \n where, fex is the extrinsic line width  contributions , when  the applied magnetic field  is x-(y-\n)axis for longitudinal (transverse) case . The symbol s are the result s of the micromagnetic \nsimulations and the solid lines are the fitting result of Eq. (3) . We use pre -determined \ndemagnetization factors (Fig. 5) during fitting procedures, and the agree ments are excellent.  \nWe have plotted the Gilbert damping constant as a function of the wavevector in nanowire \nwidth  (\n/ qa , a is the nanowire width ) in Fig. 7. The black open rectangles are data \nextracted from the transverse field and the red open circles are longitudinal field data. We \nfind that the Gilbert damping constant varied from 0.051 to 0.018 by changing  the \nwavevector  in nanowire width  in transverse field. On the other hand, lon gitudinal field case \nthe damping constant is almost constant as 0.021. Let us discuss about the un -expected \nbehavior of the damping constant of transverse case. T he wire width acts as a kind of cut -off \nwavelength of the SW excitations in the confined geome try. SWs whose wavelength are \nlarger than 2 a are not allowed in the nanowire. Therefore, only limited SW can be excited for \nthe narrower wire, while more SW can be existed in the wider wire. For example, we show \ntransverse standing SW as profiled  in the inset of Fig. 6 for 150 -nm width nanowire in our \nmicromagnetic simulations. More possible SW excitations imply more energy dissipation \npaths, it causes larger damping constant. For narrower nanowire (50 -nm), only limited SWs \ncan be excited, so that the damping constant is smaller. However, for the limit case of infinite \na case, it is the same with un -patterned thin films, there is no boundar y so that only uniform 11 \n mode can be excited, the obtained damping constant must be the input  value.  \nIn summary, the VNA -FMR experiments is employed  to investigate the magnetic properties \nof CoFeB nanowire  arrays  and the micromagnetic simulations is proposed to understand the \nmagnetic properties including Gilbert damping constant of various CoFeB nanowire  arrays  \nwidth. We f ind that the demagnetization factors are similar with the dynamic origin and static \nfor the prism geometry. The wire width or SW wavevector dependent damping constants can \nbe explained with number of SW excitation modes.  \n \nACKNOWLEDGMENTS  \nThis work  was supported by the National Research Foundation of Korea  (NRF) Grants  (Nos. \n616-2011 -C00017  and 2013R1A12011936 ). \nReferences  \n                                            \n1 S. S. P. Parkin,  M. Hayashi, and L. Thomas, Science 320, 190 (2008) . \n2 D. A. Allwood,  G. Xiong, C. C. Faulkner, D. Atkinson, D. Petit, and R. P. Cowburn,  \nScience 309, 1688 (2005) . \n3 I. M. Miron,  G. Gaudin, S. Aufftet, B. Rodmacq, A. Schuhl, S. Pizzini, J. V ogel and P. \nGambardella, Nature Materials 9, 230 (2010) . \n4 H-R Lee , K. Lee, J. Cho, Y . -H. Choi, C. -Y . You, M. -H. Jung, F. Bonell, Y. Shiota, S. Miwa \nand Y . Suzuki, Sci. Rep.  4, 6548 (2014).  \n5 J. Cho, et al.  Nat. Commun. 6, 7635 (2015).  \n6 S. Ikeda , K. Miura, H. Yamamoto, K. Mizunuma, H. D. Gan, M. Endo, S. Kanai, J. \nHayakawa, F. Matsukura and H. Ohno , Nat. Mater. 9, 721 (2010).  \n7 J.-S. Kim et al. , Nat. Commun . 5, 3429 (2014).  \n8 L. Thomas, R. Moriya, C. Rettner and S. S. P. Parkin , Science  330, 1810 (2010).  12 \n                                                                                                                                         \n9 J.-S. Kim, M. Stark, M. Klaui, J. Yoon, C. -Y . You, L. Lopez -Diaz and E. Martinez , Phys. \nRev. B 85, 174428 (2012).  \n10 B. Kuanr, R. E. Camleym, and Z. Celinski, Appl. Phys. Let t. 87, 012502 (2005) . \n11 J. P. Park, P. Eames, D. M. Engebretson, J. Berezovsky, and P. A. Crowell, Phys. Rev. Le tt. \n89, 277201 (2002).  \n12 L. Kraus, G. Infante, Z. Frait and M. Vazquez, Phys. Rev. B 83, 174438 (2011) . \n13 C. T. Boone , J. A. Katine, J. R. Childress, V . Tiberkevich, A. Slavin, J. Zhu, X. Cheng and \nI. N. Krivorotov , Phys. Rev. Let t. 103, 167 601 (2009) . \n14 M. Sharma, B. K. Kuanr, M. Sharma and Ananjan Nasu, IEEE Trans. Magn 50, 4300110 \n(2014).  \n15 D.-H. Kim, H .-H. Kim, and Chun -Yeol You, Appl. Phys. Lett. 99, 072502 (2011).  \n16 D.-H. Kim, H. -H. Kim, Chun -Yeol You and Hyungsuk Kim, J. Magn etics 16, 206 (2011).  \n17 C. Kittel, Introduction to Solid State Physics, 7th ed., pp. 504, (1996) . \n18 M. J. Donahue and D. G. Porter: OOMMF User’ s Guide : Ver. 1.0, NISTIR 6376  (National \nInstitute of Standards and Technology, Gaithersburg, Maryland, United States, 1999 ). \n19 W. Wang, C. Mu, B. Zhang, Q. Liu, J. Wang, D. Xue, Comput. Mater. Sci. 49, 84 (2010) . \nSee: http://oommf -2dpbc.sourceforge.net.  \n20 J. Cho, J . Jung, K .-E. Kim, S .-I. Kim, S .-Y. Park, M .-H. Jung, C .-Y. You, J. Magn. Magn. \nMater. 339, 36 (2013).  \n21 C. Bilzer, T. Devolder, J -V . Kim, G. Counil, C. Chappert, S. Cardoso and P. P. Feitas , J. \nAppl. Phys. 100, 053903 (2006).  \n22 K.-S. Lee, D. -S. Han , S.-K. Kim, Phys. Rev. Let t. 102, 127202 (2009).  \n23 J. Yoon, C. -Y . You, Y . Jo, S. -Y. Park, M. H. Jung , J. Korean Phys. Soc. 57, 1594 (2010) .  \n24 Y . Shiota, F. Bonell, S. Miwa, N. Muzuochi, T. Shinjo and Y . Suzuki , Appl. Phys. Le tt. 103. \n082410 (2013),  \n25 I. Purnama, I. S. Kerk, G. J. Lim  and W. S. Lew , Sci. Rep. 5, 8754 (2015).  \n26 J. Ding, M. Kostylev, and A. O. Adeyeye, Phys. Rev. B. 84, 054425 (2011) . \n27 A. Aharoni, J. Appl. Phys. 83, 3432 (2011) . 13 \n Figure  Captions  \n \nFig. 1. Measurement geometry with SEM image s of the 100 -nm-width nanowires with a gap \nof 200 nm between nanowires . The longitudinal nanowire arrays are shown. After the \nnanowire patterns have been  defined by e -beam lithography, they are covered by co -planar \nwave guides.  \n \nFig. 2. (a) The measured FMR  spectrum of the CoFeB nanowire with H =0.194 T. The red  (lower \npeak)  and blue  (higher peak)  arrows indicate t he resonance frequencies of the  uniform FMR  mode and \nthe nanowire mode, respectively. (b) The measured FMR spectrum of the CoFeB thin  film with H \n=0.194 T. (c) Simulated FMR spectrum of the CoFeB nanowire with H= 0.200 T.  \n \nFig. 3. Measured and calculated FMR frequencies with the applied magnetic field for 100 -\nnm-width nanowire. The open black rectangles  are nanowire mode and open red circles are \nthe uniform FMR mode for CoFeB thin film. The closed black rectangles  are calculated by \nOOMMF and the closed blue circles  are theoretical ly calculated by Eq. (1)  using fitted \nparameters form un -patterned film . \n \nFig. 4. Variation of resonanc e frequencies with the applied magnetic field for the different \nPBC wire width for (a) longitudinal field and (b) transverse field.  Inset: The geometry of 2 -\ndimensional PBC micromagnetic simulation with nanowire width a and a gap of 200 nm \nbetween nanowire s. The black open rectangles, red open circles, green open upper triangles, \nblue open down triangles, cyan open diamonds represent as nanowire width as 50 nm, 75nm, \n100 nm, 125nm, and 150 nm, respectively.  \n \nFig. 5. Demagnetization factor with PBC wire widt h for (a) longitudinal and (b) transverse \nfield.  The black open circles, red open rectangles, blue open upper triangles represent as \ndemagnetization factors, Ny, Nz, and Nx, respectively.  \n \nFig. 6. Full width and half maxim a with the applied magnetic field for (a) longitudinal and (b) \ntransverse field.  The black open rectangles, red open circles, green open upper triangles, blue \nopen down triangles, cyan open diamonds represent as nanowire width as 50 nm, 75nm, 100 \nnm, 125nm, and 150 nm, respectively.  \n \nFig. 7. Damping constants with wavevector for transverse ( the black open rectangles ) and \nlongitudinal ( the red open circles) field with errors. The black line is the input value which is \ndetermined from un -patterned film. Inset presents the profile of the trans verse spin density as SWs.  \n Fig. 1 \n \n \n    \n \n      Fig. 2. \n \n \n \nFig. 3. \n \n \n \n  \nFig. 4 \n \n` \n \n \n \nFig. 5 \n \n \n \nFig. 6 \n \n \n \nFig. 7 \n \n \n"    },    {        "title": "1510.01894v1.Tunable_damping__saturation_magnetization__and_exchange_stiffness_of_half_Heusler_NiMnSb_thin_films.pdf",        "content": "Tunable damping, saturation magnetization, and exchange sti\u000bness of half-Heusler\nNiMnSb thin \flms\nP. D urrenfeld,1F. Gerhard,2J. Chico,3R. K. Dumas,1, 4M. Ranjbar,1A. Bergman,3\nL. Bergqvist,5, 6A. Delin,3, 5, 6C. Gould,2L. W. Molenkamp,2and J. \u0017Akerman1, 4, 5\n1Department of Physics, University of Gothenburg, 412 96 Gothenburg, Sweden\n2Physikalisches Institut (EP3), Universit at W urzburg, 97074 W urzburg, Germany\n3Department of Physics and Astronomy, Uppsala University, Box 520, 752 20 Uppsala, Sweden\n4NanOsc AB, 164 40 Kista, Sweden\n5Materials and Nano Physics, School of ICT, KTH Royal Institute of Technology, Electrum 229, 164 40 Kista, Sweden\n6Swedish e-Science Research Centre (SeRC), 100 44 Stockholm, Sweden\nThe half-metallic half-Heusler alloy NiMnSb is a promising candidate for applications in spin-\ntronic devices due to its low magnetic damping and its rich anisotropies. Here we use ferromagnetic\nresonance (FMR) measurements and calculations from \frst principles to investigate how the com-\nposition of the epitaxially grown NiMnSb in\ruences the magnetodynamic properties of saturation\nmagnetization MS, Gilbert damping \u000b, and exchange sti\u000bness A.MSandAare shown to have a\nmaximum for stoichiometric composition, while the Gilbert damping is minimum. We \fnd excellent\nquantitative agreement between theory and experiment for MSand\u000b. The calculated Ashows the\nsame trend as the experimental data, but has a larger magnitude. Additionally to the unique in-\nplane anisotropy of the material, these tunabilities of the magnetodynamic properties can be taken\nadvantage of when employing NiMnSb \flms in magnonic devices.\nI. INTRODUCTION\nInterest in the use of half-metallic Heusler and half-\nHeusler alloys in spintronic and magnonic devices is\nsteadily increasing,1{3as these materials typically exhibit\nboth a very high spin polarization4{8and very low spin-\nwave damping.9{12One such material is the epitaxially\ngrown half-Heusler alloy NiMnSb,13,14which not only has\none of the lowest known spin-wave damping values of\nany magnetic metal, but also exhibits an interesting and\ntunable combination of two-fold in-plane anisotropy15\nand moderate out-of-plane anisotropy,10all potentially\ninteresting properties for use in both nanocontact-\nbased spin-torque oscillators16{22and spin Hall nano-\noscillators23{27. To successfully employ NiMnSb in such\ndevices, it is crucial to understand, control, and tailor\nboth its magnetostatic and magnetodynamic properties,\nsuch as its Gilbert damping ( \u000b), saturation magnetiza-\ntion (MS), and exchange sti\u000bness ( A).\nHere we investigate these properties in Ni 1-xMn1+xSb\n\flms using ferromagnetic resonance (FMR) measure-\nments and calculations from \frst principles for compo-\nsitions of -0.1\u0014x\u00140.4.MSandAare shown experi-\nmentally to have a maximum for stoichiometric compo-\nsition, while the Gilbert damping is minimum; this is\nin excellent quantitative agreement with calculations of\nand experiment on MSand\u000b. The calculated Ashows\nthe same trend as the experimental data, but with an\noverall larger magnitude. We also demonstrate that the\nexchange sti\u000bness can be easily tuned over a wide range\nin NiMnSb through Mn doping, and that the ultra-low\ndamping persists over a wide range of exchange sti\u000b-\nnesses. This unique behavior makes NiMnSb ideal for\ntailored spintronic and magnonic devices. Finally, by\ncomparing the experimental results with \frst-principlescalculations, we also conclude that the excess Mn mainly\noccupies Ni sites and that interstitial doping plays only\na minor role.\nII. METHODS\nA. Thin Film Growth\nThe NiMnSb \flms were grown by molecular beam\nepitaxy onto InP(001) substrates after deposition of a\n200 nm thick (In,Ga)As bu\u000ber layer.15The \flms were\nsubsequently covered in situ by a 10 nm thick mag-\nnetron sputtered metal cap to avoid oxidation and sur-\nface relaxation.28The Mn content was controlled dur-\ning growth via the temperature, and hence the \rux, of\nthe Mn e\u000busion cell. Six di\u000berent samples (see table I)\nwere grown with increasing Mn concentration, sample 1\nhaving the lowest and sample 6 the highest concentra-\ntion of Mn. High-resolution x-ray di\u000braction (HRXRD)\nmeasurements give information on the structural proper-\nties of these samples, con\frming the extremely high crys-\ntalline quality of all samples with di\u000berent Mn concentra-\nSamplevertical\nlattice\nconstant ( \u0017A)thickness\n(nm)uniaxial\neasy axis2K1\nMS(Oe)\n1 5.94 38 [110] 170\n2 5.97 38 [110] 8.4\n3 5.99 40 [110] 0\n4 6.02 45 [1 \u001610] 9.0\n5 6.06 45 [1 \u001610] 14.2\n6 6.09 38 [1 \u001610] 25.5\nTable I. Overview of NiMnSb \flms investigated in this study.arXiv:1510.01894v1  [cond-mat.mtrl-sci]  7 Oct 20152\ntion, even in the far from stoichiometric cases (samples 1\nand 6).15The vertical lattice constant is found to increase\nwith increasing Mn concentration and, assuming a linear\nincrease,29we estimate the di\u000berence in Mn concentra-\ntion across the whole set of samples to be about 40 at. %.\nWe will thus represent the Mn concentration in the fol-\nlowing experimental results by the measured vertical lat-\ntice constant. Stoichiometric NiMnSb exhibits vertical\nlattice constants in the range of 5.96{6.00 \u0017A, leading to\nthe expectation of stoichiometric NiMnSb in samples 2\nand 3.15Finally, the layer thicknesses are also determined\nfrom the HRXRD measurements, giving an accuracy of\n\u00061 nm.\nB. Ferromagnetic Resonance\nBroadband \feld-swept FMR spectroscopy was per-\nformed using a NanOsc Instruments PhaseFMR system\nwith a coplanar waveguide for microwave \feld excitation.\nMicrowave \felds hrfwith frequencies of up to 16 GHz\nwere applied in the \flm plane, perpendicularly oriented\nto an in-plane dc magnetic \feld H. The derivative of\nthe FMR absorption signal was measured using a lock-\nin technique, in which an additional low-frequency mod-\nulation \feld Hmod<1 Oe was applied using a pair of\nHelmholtz coils parallel to the dc magnetic \feld. The\n\feld directions are shown schematically in Fig. 1(a) and\na typical spectrum measured at 13.6 GHz is given in the\ninset of Fig. 1(b). In addition to the zero wave vector\nuniform FMR mode seen at about H=2.1 kOe, an addi-\ntional weaker resonance is observed at a much lower \feld\nof about 500 Oe, and is identi\fed as the \frst exchange-\ndominated perpendicular standing spin wave (PSSW)\nmode. The PSSW mode has a nonzero wave vector point-\ning perpendicular to the thin \flm plane and a thickness-\ndependent spin-wave amplitude and phase.30,31This can\nbe e\u000eciently excited in the coplanar waveguide geome-\ntry due to the nonuniform strength of the microwave \feld\nacross the \flm thickness.32\nThe \feld dependence of the absorption spectra (inset\nof Fig. 1(b)) can be \ft well (red line) by the sum of a sym-\nmetric and an antisymmetric Lorentzian derivative:33,34\ndP\ndH(H) =\u00008C1\u0001H(H\u0000H0)\nh\n\u0001H2+ 4 (H\u0000H0)2i2\n+2C2\u0000\n\u0001H2\u00004(H\u0000H0)2\u0001\nh\n\u0001H2+ 4 (H\u0000H0)2i2; (1)\nwhereH0is the resonance \feld, \u0001 Hthe full width at\nhalf maximum (FWHM), and C1andC2\ftted param-\neters representing the amplitude of the symmetric and\nantisymmetric Lorentzian derivatives, respectively. Both\nthe FMR and the PSSW peaks can be \ftted indepen-\ndently, as they are well separated by the exchange \feld\n\u00160Hex/(\u0019=d)2, wheredis the thickness of the layer.\neasy axis\nH, Hmod\nhard axis\nhrf p = 0\n(FMR)p = 1\n(PSSW)z\nHexx\ny\nFMRPSSW(a)\n(b)\n0\n500\n1000\n1500\n2000\n2500\n0\n5\n10\n15\nf(GHz)\nField (Oe)\n400\n600\n2000\n2200\n-0.5\n0.0\n0.5\n1.0\n1.5\ndP/dH (a. u.)\nField (Oe)\nf= 13.6 GHzFigure 1. (a) Schematic diagram of the FMR measurement\nshowing \feld directions. In our setup, the FMR mode and the\n\frst PSSW mode are excited. (b) Frequency vs. resonance\n\felds of the PSSW (red) and uniform FMR (black) mode for\nsample 2. The solid lines are \fts to the Kittel equation, and\nboth modes are o\u000bset horizontally by Hex. Inset: Resonance\ncurves forf=13.6 GHz. The \frst PSSW mode on the left and\nthe FMR mode on the right were \ft with Eq. 1\nFor our chosen sample thicknesses, the di\u000berences in res-\nonance \felds are always much larger than the resonance\nlinewidths.\nThe \feld dependence of both resonances is shown in\nFig. 1(b) and can now be used to extract information\nabout the magnetodynamic properties and anisotropies\nof the \flms. The curves are \fts to the Kittel equation,\nincluding internal \felds from the anisotropy and the ex-\nchange \feld for the PSSW excitation:15,35\nf=\r\u00160\n2\u0019\u0014\u0012\nH0+2KU\nMS\u00002K1\nMS+Hex\u0013\n\u0002\u0012\nH0+2KU\nMS+K1\nMS+Hex+Me\u000b\u0013\u00151=2\n;(2)\nwhereH0is the resonance \feld, \r=2\u0019the gyromagnetic\nratio, and\u00160the permeability of free space. Me\u000bis the ef-\nfective magnetization, which has a value close to the satu-\nration magnetization MS. 2KU=MSand 2K1=MSstands\nfor the internal anisotropy \felds coming from the uniaxial\n(KU) and biaxial ( K1) anisotropy energy densities in the\nhalf-Heusler material. The e\u000bective magnetic \feld also\nincludes an exchange \feld \u00160Hex= (2A=M S)(p\u0019=d )2,3\nwhich is related to the exchange sti\u000bness A, the \flm\nthicknessd, and the integer order of the PSSW mode\np, wherep= 0 denotes the uniform FMR excitation and\np= 1 the \frst PSSW mode. This mode numbering re-\n\rects the boundary conditions with no surface pinning of\nthe spins, which is expected for the in-plane measurement\ngeometry.36\nWe stress that the expression for the anisotropy con-\ntribution in Eq. 2 is only valid for the case in which the\nmagnetization direction is parallel to the uniaxial easy\naxis and also parallel to the applied \feld. A full angular-\ndependent formulation of the FMR condition is described\nin Ref. 15. To ful\fll the condition of parallel alignment\nfor all resonances, we perform the FMR measurements\nwith the dc magnetic \feld being applied along the domi-\nnant uniaxial easy axis of each \flm, which changes from\nthe [110] crystallographic direction to the [1 \u001610]-direction\nwith increasing Mn concentration (see Table I).\nThe values of the biaxial anisotropy2K1\nMShave been de-\ntermined in a previous study by \fxed-frequency in-plane\nangular dependent FMR measurements,15and were thus\ntaken as constant values in the \ftting process for Eq. 2;\na simultaneous \ft of both contributions can yield arbi-\ntrary combinations of anisotropy \felds due to their great\ninterdependence. The values for the uniaxial anisotropy\n2KU\nMSobtained from the frequency-dependent \ftting are\nin very good agreement with the previously obtained val-\nues in Ref. 15. The gyromagnetic ratio was measured\nto be\r=2\u0019= (28.59\u00060.20) GHz/T for all investigated\nsamples, and was therefore \fxed for all samples to allow\nbetter comparison of the e\u000bective magnetization values.\nThe Gilbert damping \u000bof the \flms is obtained by\n\ftting the FMR linewidths \u0001 Hwith the linear depen-\ndence:37\n\u00160\u0001H=\u00160\u0001H0+4\u0019\u000b\n\rf; (3)\nwhere \u0001H0is the inhomogeneous linewidth broaden-\ning of the \flm. The parallel alignment between mag-\nnetization and external magnetic \feld ensures that the\nlinewidth is determined by the Gilbert damping process\nonly.38\nC. Calculations from First Principles\nThe electronic and magnetic properties of the NiMnSb\nhalf-Heusler system were studied via \frst-principles cal-\nculations. The material was assumed to be ordered in a\nface-centered tetragonal structure with an in-plane lat-\ntice parameter ak\nlat= 5.88 \u0017A, close to the lattice con-\nstant of the InP substrate, and an out-of-plane lattice\nconstant of a?\nlat= 5.99 \u0017A, matching the value for the\nstoichiometric composition. Fixed values for the lattice\nparameters were chosen since an exact relation between\nthe o\u000b-stoichiometric composition and the experimen-\ntally measured vertical lattice constants cannot be es-\ntablished. Moreover, calculations with a varying verticallattice parameter for a constant composition showed only\na negligible e\u000bect on M S,A, and\u000b. The calculations\nwere performed using the multiple scattering Korringa-\nKohn-Rostocker (KKR) Green's function formalism as\nimplemented in the SPRKKR package.39Relativistic ef-\nfects were fully taken into account by solving the Dirac\nequation for the electronic states, the shape of the poten-\ntial was considered via the Atomic Sphere Approximation\n(ASA), and the local spin density approximation (LSDA)\nwas used for the exchange correlation potential. The co-\nherent potential approximation (CPA) was used for the\nchemical disorder of the system.\nThe Gilbert damping \u000bof the material was calculated\nusing linear response theory40, including the temperature\ne\u000bects from interatomic displacements and spin \ructua-\ntions.41,42\nThe exchange interactions Jijbetween the atomic\nmagnetic moments were calculated using the magnetic\nforce theorem, as considered in the LKAG formalism.43,44\nThe interactions were calculated for up to 4.5 times the\nlattice constant in order to take into account any long-\nrange interactions. Given the interatomic exchange in-\nteractions, the spin-wave sti\u000bness Dcan be calculated.\nDue to possible oscillations in the exchange interactions\nas a function of the distance, it becomes necessary to in-\ntroduce a damping parameter, \u0011, to assure convergence\nof the summation. Dcan then be obtained by evaluating\nthe limit\u0011!0 of\nD=2\n3X\nijJijp\nMiMjr2\nijexp\u0012\n\u0000\u0011rij\nalat\u0013\n; (4)\nas described in [45]. Here, MiandMjare the local mag-\nnetic moments at sites iandj,Jijis the exchange cou-\npling between the magnetic moments at sites iandj,\nandrijis the distance between the atoms iandj. This\nformalism can be extended to a multisublattice system46.\nTo calculate the e\u000bect of chemical disorder on the ex-\nchange sti\u000bness of the system, the obtained exchange in-\nteractions were summed over a supercell with a random\ndistribution of atoms in the chemically disordered sub-\nlattice. The e\u000bect that distinct chemical con\fgurations\ncan have over the calculation of the exchange sti\u000bness\nwas treated by taking 200 di\u000berent supercells. The re-\nsults were then averaged and the standard deviation was\ncalculated. The cells were obtained using the atomistic\nspin dynamics package UppASD.47\nFinally, with the spin-wave sti\u000bness determined as de-\nscribed above, the exchange sti\u000bness Acan be calculated\nfrom:48\nA=DM S(T)\n2g\u0016B: (5)\nHere,gis the Land\u0013 e g-factor of the electron, \u0016Bthe Bohr\nmagneton, and MS(T) the magnetization density of the\nsystem for a given temperature T, which for T= 0 K\ncorresponds to the saturation magnetization.\nFrom the \frst-principles calculations, the magnetic\nproperties for ordered NiMnSb and chemically disordered4\n0.60.70.80.95\n.956 .006 .056 .100123(b) m0MS \nm0Meffm0M (T)(a)4  µB/u.f.KS (mJ/m2)v\nertical lattice constant (Å)\nFigure 2. (a) MSandMe\u000bas functions of vertical lattice\nconstant. The theoretical value of 4.0 \u0016B=u.f. is shown by\nthe blue dashed line. (b) The calculated surface anisotropy\ndensity follows from the di\u000berence between MSandMe\u000b.\nNi1-xMn1+xSb were studied. To obtain the values of the\nexchange sti\u000bness AforT= 300 K, the exchange interac-\ntions from the ab initio calculations were used in conjunc-\ntion with the value of the magnetization at T= 300 K\nobtained from Monte Carlo simulations.\nIII. RESULTS\nA. Magnetization\nThe values of \u00160Me\u000bare plotted in Fig. 2(a) as red\ndots. The e\u000bective magnetization is considerably lower\nthan the saturation magnetization \u00160MS, which was in-\ndependently assessed using SQUID measurements and al-\nternating gradient magnetometry (AGM). The values for\n\u00160MScorrespond to a saturation magnetization between\n3.5\u0016B=unit formula and 3.9 \u0016B=u.f., with the latter\nvalue being within the error bars of the theoretically ex-\npected value of 4.0 \u0016B=u.f. for stoichiometric NiMnSb.49\nA reduction of MSis expected in Mn-rich NiMnSb alloys,\ndue to the antiferromagnetic coupling of the Mn Nidefects\nto the Mn lattice in the C1 bstructure of the half-Heusler\nmaterial.29An even stronger reduction is observed for\nthe Ni-rich sample 1, which is in accordance with the\nformation of Ni Mnantisites.50\nWhile the measurement error for MSis comparatively\nlarge due to uncertainties in the volume determination,\nthe error bars for Me\u000b, as obtained from ferromagnetic\nresonance, are negligible. NiMnSb \flms have been shown\nto possess a small but substantial perpendicular mag-\nnetic anisotropy, which can arise from either interfacial\nanisotropy or lattice strain.10,12To quantify the di\u000ber-\nence observed between MSandMe\u000b, we assume a uniax-\nial perpendicular anisotropy due to a surface anisotropy\n2\n4\n6\n8\n10\n5.95\n6.00\n6.05\n6.10\n0\n2\n4\n(b)\nA (pJ/m)\n(a)\nα(10-3)\nvertical lattice constant (Å)\n2\n4\n6\n8\n10\n-0.1\n0.0\n0.1\n0.2\n0.3\n0.4\n0\n2\n4\n(d)\n(c)\nxantisitesFigure 3. (a) and (b) show respectively the exchange sti\u000bness\nand Gilbert damping constant obtained from FMR measure-\nments, plotted as a function of the vertical lattice constant.\n(c) and (d) show the corresponding values obtained from \frst-\nprinciple calculations for T= 300 K. Negative values for x\nimply the introduction of Ni Mnantisites and positive values\nare related to Mn Niantisite defects. The error bars in (c) are\nthe standard deviations from repeated \frst-principles calcu-\nlations with 200 randomized supercells.\nenergy density KS, which is known to follow the rela-\ntion:51\n\u00160Me\u000b=\u00160MS\u00002KS\nMSd: (6)\nTheKScalculated in this way has values between\n0.5mJ=m2and 1.5mJ=m2, as shown in Fig. 2(b); these\nare comparable to the surface anisotropies obtained in\nother crystalline thin \flm systems.52. Although the \flm\nthicknesses in our set vary unsystematically, we can ob-\nserve systematic behavior of KSwith the vertical lat-\ntice constant, with an apparent minimum under the con-\nditions where stoichiometric NiMnSb is expected|that\nis, for samples 2 and 3. The increasing values for o\u000b-\nstoichiometric NiMnSb can be thus attributed to the con-\ncomitant increase in lattice defects, and thus of surface\ndefects, in these \flms.\nB. Exchange Sti\u000bness and Gilbert Damping\nThe experimentally determined exchange sti\u000bness, as\na function of the vertical lattice constant, and the Gilbert\ndamping parameter are shown in Fig. 3(a) and (b), re-\nspectively. The minimum damping observed in our mea-\nsurements is 1 :0\u000210\u00003for sample 3, and so within sto-\nichiometric composition. Sample 1, with a de\fciency\nof Mn atoms, showed nonlinear linewidth behavior at\nlow frequencies, which vanished for out-of-plane measure-\nments (not shown). This is typical with the presence of\ntwo-magnon scattering processes.52However, the damp-\ning is considerably lower in all samples than in a permal-\nloy \flm of comparable thickness.\nThe exchange sti\u000bness and Gilbert damping ob-\ntained from the \frst-principles calculations are shown in5\nFig. 3(c) and (d), respectively. For both parameters, the\nexperimental trends are reproduced quantitatively, with\nAhaving a maximum and \u000ba minimum value at stoi-\nchiometry.\nAs the concentration of both Mn or Ni antisites in-\ncreases, the exchange sti\u000bness decreases. This behavior\ncan be explained by analyzing the terms in the expres-\nsion for the spin-wave sti\u000bness, Eq. 4. It turns out that\nthe new exchange couplings Jij, which appear when an-\ntisites are present, play a major role, whereas changes in\nthe atomic magnetic moments or the saturation magne-\ntization appear to be relatively unimportant. Mn anti-\nsites in the Ni sublattice (i.e., excess Mn) have a strong\n(2 mRy) antiferromagnetic coupling to the Mn atoms in\nthe adjacent Mn layers. This results in a negative contri-\nbution toDcompared to the stoichiometric case, where\nthis interaction is not present. On the other hand, Ni\nantisites in the Mn sublattice have a negative in-plane\nexchange coupling of 0.3 mRy to their nearest-neighbor\nMn atoms, with a frustrated antiferromagnetic coupling\nto the Ni atoms in the adjacent Ni plane. The net e\u000bect is\na decreasing spin-wave sti\u000bness as the composition moves\naway from stoichiometry. The calculated values of Aare\naround 30 % larger than the experimental results, which\nis the same degree of overestimation we recently observed\nin a study of doped permalloy \flms53. It thus seems to\nbe inherent in our calculations from \frst principles.\nThe calculated Gilbert damping also agrees well with\nthe experimental values. The damping has its minimum\nvalue of 1.0\u000210\u00003at stoichiometry and increases with\na surplus of Ni faster than with the same surplus of\nMn. Both Mn and Ni antisites will act as impurities and\nit is thus reasonable to attribute the observed increase\nin damping at o\u000b-stoichiometry to impurity scattering.\nWhile the damping at stoichiometry also agrees quanti-\ntatively, the increase in damping is underestimated in the\ncalculations compared to the experimental values.\nDespite the fact that the calculations here focus purely\non the formation of Mn Nior Ni Mnantisites, they are\nnonetheless capable of reproducing the experimental\ntrends well. However, interstitials|that is, Mn or Ni sur-\nplus atoms in the vacant sublattice|may also be a possi-\nble o\u000b-stoichiometric defect in our system.50We have cal-\nculated their e\u000bects and can therefore discuss about the\nexistence of interstitials in our samples. A large fraction\nof Mn interstitials seems unlikely, as an increase in the\nsaturation magnetization can be predicted through calcu-\nlations, contrary to the experimental trend; see Fig. 2(a).On the other hand, the existence of Ni interstitials may\nbe compatible with the observed experimental trend, as\nthey decrease the saturation magnetization|albeit at a\nslower rate than Ni antisites and slower than experimen-\ntally observed. Judging from the measured data, it is\ntherefore likely that excess Ni exists in the samples as\nboth antisites and interstitials.\nIV. CONCLUSIONS\nIn summary, we have found that o\u000b-stoichiometry in\nthe epitaxially grown half-Heusler alloy NiMnSb has a\nsigni\fcant impact on the material's magnetodynamic\nproperties. In particular, the exchange sti\u000bness can be\naltered by a factor of about 2 while keeping the Gilbert\ndamping very low ( \u00195 times lower than in permalloy\n\flms). This is a unique combination of properties and\nopens up for the use of NiMnSb in, e.g., magnonic cir-\ncuits, where a small spin wave damping is desired. At the\nstoichiometric composition, the saturation magnetization\nand exchange sti\u000bness take on their maximum values,\nwhereas the Gilbert damping parameter is at its mini-\nmum. These experimentally observed results are repro-\nduced by calculations from \frst principles. Using these\ncalculations, we can also explain the microscopic mecha-\nnisms behind the observed trends. We also conclude that\ninterstitial Mn is unlikely to be present in the samples.\nThe observed e\u000bects can be used to \fne-tune the mag-\nnetic properties of NiMnSb \flms towards their speci\fc\nrequirements in spintronic devices.\nACKNOWLEDGMENTS\nWe acknowledge \fnancial support from the G oran\nGustafsson Foundation, the Swedish Research Council\n(VR), Energimyndigheten (STEM), the Knut and Alice\nWallenberg Foundation (KAW), the Carl Tryggers Foun-\ndation (CTS), and the Swedish Foundation for Strate-\ngic Research (SSF). F.G. acknowledges \fnancial support\nfrom the University of W urzburg's \\Equal opportunities\nfor women in research and teaching\" program. This work\nwas also supported initially by the European Commission\nFP7 Contract ICT-257159 \\MACALO\". A.B acknowl-\nedges eSSENCE. The computer simulations were per-\nformed on resources provided by the Swedish National\nInfrastructure for Computing (SNIC) at the National Su-\npercomputer Centre (NSC) and High Performance Com-\nputing Center North (HPC2N).\n1R. Okura, Y. Sakuraba, T. Seki, K. Izumi, M. Mizuguchi,\nand K. Takanashi, \\High-power rf oscillation induced\nin half-metallic Co 2MnSi layer by spin-transfer torque,\"\nAppl. Phys. Lett. 99, 052510 (2011).\n2T. Yamamoto, T. Seki, T. Kubota, H. Yako, and\nK. Takanashi, \\Zero-\feld spin torque oscillation in Co 2(Fe,\nMn)Si with a point contact geometry,\" Appl. Phys. Lett.106, 092406 (2015).\n3P. D urrenfeld, F. Gerhard, M. Ranjbar, C. Gould, L. W.\nMolenkamp, and J. \u0017Akerman, \\Spin Hall e\u000bect-controlled\nmagnetization dynamics in NiMnSb,\" J. Appl. Phys. 117,\n17E103 (2015).\n4R. A. de Groot, F. M. Mueller, P. G. van Engen, and\nK. H. J. Buschow, \\New Class of Materials: Half-Metallic6\nFerromagnets,\" Phys. Rev. Lett. 50, 2024{2027 (1983).\n5D. Ristoiu, J. P. Nozi\u0012 eres, C. N. Borca, T. Komesu,\nH. k. Jeong, and P. A. Dowben, \\The surface composi-\ntion and spin polarization of NiMnSb epitaxial thin \flms,\"\nEurophys. Lett. 49, 624 (2000).\n6T. Kubota, S. Tsunegi, M. Oogane, S. Mizukami,\nT. Miyazaki, H. Naganuma, and Y. Ando,\n\\Half-metallicity and Gilbert damping constant in\nCo2FexMn1\u0000xSi Heusler alloys depending on the \flm\ncomposition,\" Appl. Phys. Lett. 94, 122504 (2009).\n7G. M. M uller, J. Walowski, M. Djordjevic, G.-X. Miao,\nA. Gupta, A. V. Ramos, K. Gehrke, V. Moshnyaga,\nK. Samwer, J. Schmalhorst, A. Thomas, A. H utten,\nG. Reiss, J. S. Moodera, and M. M unzenberg, \\Spin po-\nlarization in half-metals probed by femtosecond spin exci-\ntation,\" Nature Mater. 8, 56{61 (2009).\n8M. Jourdan, J. Min\u0012 ar, J. Braun, A. Kronenberg,\nS. Chadov, B. Balke, A. Gloskovskii, M. Kolbe, H.J.\nElmers, G. Sch onhense, H. Ebert, C. Felser, and M. Kl aui,\n\\Direct observation of half-metallicity in the Heusler com-\npound Co 2MnSi,\" Nat. Commun. 5, 3974 (2014).\n9B. Heinrich, G. Woltersdorf, R. Urban, O. Mosendz,\nG. Schmidt, P. Bach, L. Molenkamp, and E. Rozenberg,\n\\Magnetic properties of NiMnSb(001) \flms grown on In-\nGaAs/InP(001),\" J. Appl. Phys. 95, 7462{7464 (2004).\n10A. Koveshnikov, G. Woltersdorf, J. Q. Liu, B. Kardasz,\nO. Mosendz, B. Heinrich, K. L. Kavanagh, P. Bach, A. S.\nBader, C. Schumacher, C. R uster, C. Gould, G. Schmidt,\nL. W. Molenkamp, and C. Kumpf, \\Structural and mag-\nnetic properties of NiMnSb/InGaAs/InP(001),\" J. Appl.\nPhys. 97, 073906 (2005).\n11S. Trudel, O. Gaier, J. Hamrle, and B. Hillebrands, \\Mag-\nnetic anisotropy, exchange and damping in cobalt-based\nfull-Heusler compounds: an experimental review,\" J. Phys.\nD: Appl. Phys. 43, 193001 (2010).\n12Andreas Riegler, Ferromagnetic resonance study of the\nHalf-Heusler alloy NiMnSb , Ph.D. thesis, Universit at\nW urzburg (2011).\n13P. Bach, A. S. Bader, C. R uster, C. Gould, C. R. Becker,\nG. Schmidt, L. W. Molenkamp, W. Weigand, C. Kumpf,\nE. Umbach, R. Urban, G. Woltersdorf, and B. Heinrich,\n\\Molecular-beam epitaxy of the half-Heusler alloy NiMnSb\non (In,Ga)As/InP (001),\" Appl. Phys. Lett. 83, 521{523\n(2003).\n14P. Bach, C. R uster, C. Gould, C. R. Becker, G. Schmidt,\nand L. W. Molenkamp, \\Growth of the half-Heusler alloy\nNiMnSb on (In,Ga)As/InP by molecular beam epitaxy,\"\nJ. Cryst. Growth 251, 323 { 326 (2003).\n15F. Gerhard, C. Schumacher, C. Gould, and L. W.\nMolenkamp, \\Control of the magnetic in-plane anisotropy\nin o\u000b-stoichiometric NiMnSb,\" J. Appl. Phys. 115, 094505\n(2014).\n16M. Tsoi, A. G. M. Jansen, J. Bass, W.-C. Chiang, M. Seck,\nV. Tsoi, and P. Wyder, \\Excitation of a Magnetic Multi-\nlayer by an Electric Current,\" Phys. Rev. Lett. 80, 4281{\n4284 (1998).\n17J. Slonczewski, \\Excitation of spin waves by an electric\ncurrent,\" J. Magn. Magn. Mater. 195, 261{268 (1999).\n18T. J. Silva and W. H. Rippard, \\Developments in nano-\noscillators based upon spin-transfer point-contact devices,\"\nJ. Magn. Magn. Mater. 320, 1260{1271 (2008).\n19M. Madami, S. Bonetti, G. Consolo, S. Tacchi, G. Carlotti,\nG. Gubbiotti, F. B. Manco\u000b, Yar M. A., and J. \u0017Akerman,\n\\Direct observation of a propagating spin wave induced byspin-transfer torque,\" Nature Nanotech. 6, 635{638 (2011).\n20S. Bonetti, V. Tiberkevich, G. Consolo, G. Finocchio,\nP. Muduli, F. Manco\u000b, A. Slavin, and J. \u0017Akerman, \\Ex-\nperimental Evidence of Self-Localized and Propagating\nSpin Wave Modes in Obliquely Magnetized Current-Driven\nNanocontacts,\" Phys. Rev. Lett. 105, 217204 (2010).\n21R. K. Dumas, E. Iacocca, S. Bonetti, S. R. Sani,\nS. M. Mohseni, A. Eklund, J. Persson, O. Heinonen,\nand J. \u0017Akerman, \\Spin-Wave-Mode Coexistence on the\nNanoscale: A Consequence of the Oersted-Field-Induced\nAsymmetric Energy Landscape,\" Phys. Rev. Lett. 110,\n257202 (2013).\n22R. K. Dumas, S. R. Sani, S. M. Mohseni, E. Iacocca,\nY. Pogoryelov, P. K. Muduli, S. Chung, P. D urren-\nfeld, and J. \u0017Akerman, \\Recent Advances in Nanocontact\nSpin-Torque Oscillators,\" IEEE Trans. Magn. 50, 4100107\n(2014).\n23V. E. Demidov, S. Urazhdin, H. Ulrichs, V. Tiberkevich,\nA. Slavin, D. Baither, G. Schmitz, and S. O. Demokritov,\n\\Magnetic nano-oscillator driven by pure spin current,\"\nNature Mater. 11, 1028{1031 (2012).\n24R. H. Liu, W. L. Lim, and S. Urazhdin, \\Spectral Charac-\nteristics of the Microwave Emission by the Spin Hall Nano-\nOscillator,\" Phys. Rev. Lett. 110, 147601 (2013).\n25V. E. Demidov, H. Ulrichs, S. V. Gurevich, S. O. Demokri-\ntov, V. S. Tiberkevich, A. N. Slavin, A. Zholud, and\nS. Urazhdin, \\Synchronization of spin Hall nano-oscillators\nto external microwave signals,\" Nat. Commun. 5, 3179\n(2014).\n26Z. Duan, A. Smith, L. Yang, B. Youngblood, J. Lindner,\nV. E. Demidov, S. O. Demokritov, and I. N. Krivoro-\ntov, \\Nanowire spin torque oscillator driven by spin orbit\ntorques,\" Nat. Commun. 5, 5616 (2014).\n27M. Ranjbar, P. D urrenfeld, M. Haidar, E. Iacocca,\nM. Balinskiy, T. Q. Le, M. Fazlali, A. Houshang, A. A.\nAwad, R. K. Dumas, and J. \u0017Akerman, \\CoFeB-Based\nSpin Hall Nano-Oscillators,\" IEEE Magn. Lett. 5, 3000504\n(2014).\n28C. Kumpf, A. Stahl, I. Gierz, C. Schumacher, S. Ma-\nhapatra, F. Lochner, K. Brunner, G. Schmidt, L. W.\nMolenkamp, and E. Umbach, \\Structure and relaxation\ne\u000bects in thin semiconducting \flms and quantum dots,\"\nPhys. Status Solidi C 4, 3150{3160 (2007).\n29M. Ekholm, P. Larsson, B. Alling, U. Helmersson, and\nI. A. Abrikosov, \\Ab initio calculations and synthesis of\nthe o\u000b-stoichiometric half-Heusler phase Ni 1-xMn1+xSb,\"\nJ. Appl. Phys. 108, 093712 (2010).\n30A. M. Portis, \\Low-lying spin wave modes in ferromagnetic\n\flms,\" Appl. Phys. Lett. 2, 69{71 (1963).\n31B. Lenk, G. Eilers, J. Hamrle, and M. M unzenberg, \\Spin-\nwave population in nickel after femtosecond laser pulse ex-\ncitation,\" Phys. Rev. B 82, 134443 (2010).\n32I. S. Maksymov and M. Kostylev, \\Broadband stripline fer-\nromagnetic resonance spectroscopy of ferromagnetic \flms,\nmultilayers and nanostructures,\" Physica E 69, 253 { 293\n(2015).\n33Georg Woltersdorf, Spin-pumping and two-magnon scat-\ntering in magnetic multilayers , Ph.D. thesis, Simon Fraser\nUniversity (2004).\n34N. Mecking, Y. S. Gui, and C.-M. Hu, \\Microwave pho-\ntovoltage and photoresistance e\u000bects in ferromagnetic mi-\ncrostrips,\" Phys. Rev. B 76, 224430 (2007).\n35A. Conca, E. Th. Papaioannou, S. Klingler, J. Greser,\nT. Sebastian, B. Leven, J. L osch, and B. Hillebrands,7\n\\Annealing in\ruence on the Gilbert damping parameter\nand the exchange constant of CoFeB thin \flms,\" Appl.\nPhys. Lett. 104, 182407 (2014).\n36M. van Kampen, C. Jozsa, J. T. Kohlhepp, P. LeClair,\nL. Lagae, W. J. M. de Jonge, and B. Koopmans, \\All-\nOptical Probe of Coherent Spin Waves,\" Phys. Rev. Lett.\n88, 227201 (2002).\n37S. S. Kalarickal, P. Krivosik, M. Wu, C. E. Patton, M. L.\nSchneider, P. Kabos, T. J. Silva, and J. P. Nibarger,\n\\Ferromagnetic resonance linewidth in metallic thin \flms:\nComparison of measurement methods,\" J. Appl. Phys. 99,\n093909 (2006).\n38Kh. Zakeri, J. Lindner, I. Barsukov, R. Meckenstock,\nM. Farle, U. von H orsten, H. Wende, W. Keune, J. Rocker,\nS. S. Kalarickal, K. Lenz, W. Kuch, K. Baberschke, and\nZ. Frait, \\Spin dynamics in ferromagnets: Gilbert damp-\ning and two-magnon scattering,\" Phys. Rev. B 76, 104416\n(2007).\n39H. Ebert, D. K odderitzsch, and J. Min\u0013 ar, \\Calculating\ncondensed matter properties using the KKR-Green's func-\ntion method - recent developments and applications,\" Rep.\nProg. Phys. 74, 096501 (2011).\n40I. Garate and A. MacDonald, \\Gilbert damping in con-\nducting ferromagnets. II. Model tests of the torque-\ncorrelation formula,\" Phys. Rev. B 79, 064404 (2009).\n41S. Mankovsky, D. K odderitzsch, G. Woltersdorf, and\nH. Ebert, \\First-principles calculation of the Gilbert damp-\ning parameter via the linear response formalism with ap-\nplication to magnetic transition metals and alloys,\" Phys.\nRev. B 87, 014430 (2013).\n42H. Ebert, S. Mankovsky, K. Chadova, S. Polesya, J. Min\u0013 ar,\nand D. K odderitzsch, \\Calculating linear-response func-\ntions for \fnite temperatures on the basis of the alloy anal-\nogy model,\" Phys. Rev. B 91, 165132 (2015).\n43A. I. Liechtenstein, M. I. Katsnelson, and V. A. Gubanov,\n\\Exchange interactions and spin-wave sti\u000bness in ferro-\nmagnetic metals,\" J. Phys. F: Met. Phys. 14, L125 (1984).\n44A. I. Liechtenstein, M. I. Katsnelson, V. P. Antropov, andV. A. Gubanov, \\Local spin density functional approach to\nthe theory of exchange interactions in ferromagnetic metals\nand alloys,\" J. Magn. Magn. Mater. 67, 65 { 74 (1987).\n45M. Pajda, J. Kudrnovsk\u0013 y, I. Turek, V. Drchal, and\nP. Bruno, \\ Ab initio calculations of exchange interactions,\nspin-wave sti\u000bness constants, and Curie temperatures of\nFe, Co, and Ni,\" Phys. Rev. B 64, 174402 (2001).\n46J. Thoene, S. Chadov, G. Fecher, C. Felser, and J. K ubler,\n\\Exchange energies, Curie temperatures and magnons in\nHeusler compounds,\" J. Phys. D: Appl. Phys. 42, 084013\n(2009).\n47B. Skubic, J. Hellsvik, L. Nordstr om, and O. Eriksson,\n\\A method for atomistic spin dynamics simulations: im-\nplementation and examples,\" J. Phys. Condens. Matter.\n20, 315203 (2008).\n48C. A. F. Vaz, J. A. C. Bland, and G. Lauho\u000b, \\Magnetism\nin ultrathin \flm structures,\" Rep. Prog. Phys. 71, 056501\n(2008).\n49T. Graf, C. Felser, and S. S. P. Parkin, \\Simple rules\nfor the understanding of Heusler compounds,\" Prog. Solid\nState Chem. 39, 1 { 50 (2011).\n50B. Alling, S. Shallcross, and I. A. Abrikosov, \\Role of\nstoichiometric and nonstoichiometric defects on the mag-\nnetic properties of the half-metallic ferromagnet NiMnSb,\"\nPhys. Rev. B 73, 064418 (2006).\n51Y.-C. Chen, D.-S. Hung, Y.-D. Yao, S.-F. Lee, H.-P. Ji,\nand C. Yu, \\Ferromagnetic resonance study of thickness-\ndependent magnetization precession in Ni 80Fe20\flms,\" J.\nAppl. Phys. 101, 09C104 (2007).\n52X. Liu, W. Zhang, M. J. Carter, and G. Xiao, \\Ferro-\nmagnetic resonance and damping properties of CoFeB thin\n\flms as free layers in MgO-based magnetic tunnel junc-\ntions,\" J. Appl. Phys. 110, 033910 (2011).\n53Y. Yin, F. Pan, M. Ahlberg, M. Ranjbar, P. D urrenfeld,\nA. Houshang, M. Haidar, L. Bergqvist, Y. Zhai, R. K.\nDumas, A. Delin, and J. \u0017Akerman, \\Tunable permalloy-\nbased \flms for magnonic devices,\" Phys. Rev. B 92, 024427\n(2015)."    },    {        "title": "1510.03571v2.Nonlocal_torque_operators_in_ab_initio_theory_of_the_Gilbert_damping_in_random_ferromagnetic_alloys.pdf",        "content": "arXiv:1510.03571v2  [cond-mat.mtrl-sci]  19 Nov 2015Nonlocal torque operators in ab initio theory of the Gilbert damping in random\nferromagnetic alloys\nI. Turek∗\nInstitute of Physics of Materials, Academy of Sciences of th e Czech Republic, ˇZiˇ zkova 22, CZ-616 62 Brno, Czech Republic\nJ. Kudrnovsk´ y†and V. Drchal‡\nInstitute of Physics, Academy of Sciences of the Czech Repub lic,\nNa Slovance 2, CZ-182 21 Praha 8, Czech Republic\n(Dated: July 5, 2018)\nWe present an ab initio theory of theGilbert dampingin substitutionally disorder ed ferromagnetic\nalloys. The theory rests on introduced nonlocal torques whi ch replace traditional local torque\noperators in the well-known torque-correlation formula an d which can be formulated within the\natomic-sphereapproximation. Theformalism is sketchedin asimpletight-bindingmodel andworked\nout in detail in the relativistic tight-binding linear muffin -tin orbital (TB-LMTO) method and\nthe coherent potential approximation (CPA). The resulting nonlocal torques are represented by\nnonrandom, non-site-diagonal and spin-independent matri ces, which simplifies the configuration\naveraging. The CPA-vertex corrections play a crucial role f or the internal consistency of the theory\nand for its exact equivalence to other first-principles appr oaches based on the random local torques.\nThis equivalence is also illustrated by the calculated Gilb ert damping parameters for binary NiFe\nand FeCo random alloys, for pure iron with a model atomic-lev el disorder, and for stoichiometric\nFePt alloys with a varying degree of L1 0atomic long-range order.\nPACS numbers: 72.10.Bg, 72.25.Rb, 75.78.-n\nI. INTRODUCTION\nThe dynamics of magnetization of bulk ferromagnets,\nutrathin magnetic films and magnetic nanoparticles rep-\nresents an important property of these systems, espe-\ncially in the context of high speed magnetic devices for\ndata storage. While a complete picture of magnetization\ndynamics including, e.g., excitation ofmagnonsand their\ninteraction with other degrees of freedom, is still a chal-\nlenge for the modern theory of magnetism, remarkable\nprogresshas been achieved during the last years concern-\ning the dynamics of the total magnetic moment, which\ncan be probed experimentally by means of the ferromag-\nnetic resonance1or by the time-resolved magneto-optical\nKerr effect.2Time evolution of the macroscopic magne-\ntization vector Mcan be described by the well-known\nLandau-Lifshitz-Gilbert (LLG) equation3,4\ndM\ndt=Beff×M+M\nM×/parenleftbigg\nα·dM\ndt/parenrightbigg\n,(1)\nwhereBeffdenotes an effective magnetic field (with the\ngyromagnetic ratio absorbed) acting on the magnetiza-\ntion,M=|M|, and the quantity α={αµν}denotes\na symmetric 3 ×3 tensor of the dimensionless Gilbert\ndamping parameters ( µ,ν=x,y,z). The first term in\nEq. (1) defines a precession of the magnetization vector\naround the direction of the effective magnetic field and\nthe second term describes a damping of the dynamics.\nThe LLG equation in itinerant ferromagnets is appropri-\nate for magnetization precessions very slow as compared\nto precessions of the single-electron spin due to the ex-\nchange splitting and to frequencies of interatomic elec-\ntron hoppings.A large number of theoretical approaches to the\nGilbert damping has been workedout during the last two\ndecades; here we mention only schemes within the one-\nelectron theory of itinerant magnets,5–20where the most\nimportant effects of electron-electron interaction are cap-\ntured by means of a local spin-dependent exchange-\ncorrelation (XC) potential. These techniques can be\nnaturally combined with existing first-principles tech-\nniques based on the density-functional theory, which\nleads to parameter-freecalculations of the Gilbert damp-\ning tensor of pure ferromagnetic metals, their ordered\nand disordered alloys, diluted magnetic semiconductors,\netc. One part of these approaches is based on a static\nlimit of the frequency-dependent spin-spin correlation\nfunction of a ferromagnet.5–8,15,16Other routes to the\nGilbert damping employ relaxations of occupation num-\nbers of individual Bloch electron states during quasi-\nstatic nonequilibrium processes or transition rates be-\ntween different states induced by the spin-orbit (SO)\ninteraction.9–12,14,20The dissipation of magnetic energy\naccompanying the slow magnetization dynamics, evalu-\nated within a scattering theory or the Kubo linear re-\nsponse formalism, leads also to explicit expressions for\nthe Gilbert damping tensor.13,17–19Most of these formu-\nlations yield relations equivalent to the so-called torque-\ncorrelation formula\nαµν=−α0Tr{Tµ(G+−G−)Tν(G+−G−)},(2)\ninwhich thetorqueoperators Tµareeither duetothe XC\nor SO terms of the one-electron Hamiltonian. In Eq. (2),\nwhich has a form of the Kubo-Greenwood formula and is\nvalid for zero temperature of electrons, the quantity α0is\nrelated to the system magnetization (and to fundamental2\nconstants and units used, see Section IIB), the trace is\ntaken over the whole Hilbert space of valence electrons,\nandthesymbols G±=G(EF±i0)denotetheone-particle\nretarded and advanced propagators (Green’s functions)\nat the Fermi energy EF.\nImplementation of the above-mentioned theories in\nfirst-principles computational schemes proved opposite\ntrends of the intraband and interband contributions to\nthe Gilbert damping parameter as functions of a phe-\nnomenological quasiparticle lifetime broadening.7,11,12\nThese qualitative studies have recently been put on a\nmore solid basis by considering a particular mechanism\nof the lifetime broadening, namely, a frozen temperature-\ninduced structural disorder, which represents a realistic\nmodel for a treatment of temperature dependence of the\nGilbert damping.21,22This approach explained quanti-\ntatively the low-temperature conductivity-like and high-\ntemperature resistivity-like trends of the damping pa-\nrameters of iron, cobalt and nickel. Further improve-\nmentsofthemodel, includingstatictemperature-induced\nrandomorientationsoflocalmagneticmoments, haveap-\npeared recently.23\nTheab initio studies have also been successful in re-\nproduction and interpretation of values and concentra-\ntion trends of the Gilbert damping in random ferromag-\nnetic alloys, such as the NiFe alloy with the face-centered\ncubic (fcc) structure (Permalloy)17,22and Fe-based al-\nloys with the body-centered cubic (bcc) structure (FeCo,\nFeV,FeSi).19,22,24Otherstudiesaddressedalsotheeffects\nof doping the Permalloy and bcc iron by 5 dtransition-\nmetal elements19,20,22and of the degree of atomic long-\nrange order in equiconcentration FeNi and FePt alloys\nwith the L1 0-type structures.20Recently, an application\nto halfmetallic Co-based Heusler alloys has appeared as\nwell.25The obtained results revealed correlations of the\ndamping parameter with the density of states at the\nFermienergyandwiththesizeofmagneticmoments.22,24\nIn a one-particle mean-field-like description of a ferro-\nmagnet, the total spin is not conserved due to the XC\nfield and the SO interaction. The currently employed\nformsofthetorqueoperators Tµinthe torque-correlation\nformula (2) reflect these two sources; both the XC- and\nthe SO-induced torques are local and their equivalence\nfor the theory of Gilbert damping has been discussed\nby several authors.15,16,26In the case of random alloys,\nthis equivalence rests on a proper inclusion of vertex cor-\nrections in the configuration averaging of the damping\nparameters αµνas two-particle quantities.\nThe purpose of the present paper is to introduce an-\nother torque operator that can be used in the torque-\ncorrelationformula(2) andto discussits properties. This\noperatoris due to intersiteelectronhopping andit is con-\nsequently nonlocal; in contrast to the local XC- and SO-\ninduced torques which are random in random crystalline\nalloys, the nonlocal torque is nonrandom, i.e., indepen-\ndent on the particular configuration of a random alloy,\nwhich simplifies the configuration averaging of Eq. (2).\nWe show that a similar nonlocal effective torque appearsin the fully relativistic linear muffin-tin orbital (LMTO)\nmethod in the atomic-sphere approximation (ASA) used\nrecently for calculations of the conductivity tensor in\nspin-polarized random alloys.27,28Here we discuss theo-\nretical aspects of the averaging in the coherent-potential\napproximation (CPA)29,30and illustrate the developed\nab initio scheme byapplicationsto selected binaryalloys.\nWe also compare the obtained results with those of the\nLMTO-supercell technique17and with other CPA-based\ntechniques, the fully relativisticKorringa-Kohn-Rostoker\n(KKR) method19,22and the LMTO method with a sim-\nplified treatment of the SO interaction.20\nThe paper is organized as follows. The theoretical for-\nmalism is contained in Section II, with a general discus-\nsion of various torque operators and results of a simple\ntight-binding model presented in Section IIA. The fol-\nlowingSection IIB describes the derivation of the LMTO\ntorque-correlation formula with nonlocal torques; tech-\nnical details are left to Appendix A concerning linear-\nresponse calculations with varying basis sets and to Ap-\npendix B regarding the LMTO method for systems with\na tilted magnetization direction. Selected formal proper-\nties of the developed theory are discussed in Section IIC.\nApplications of the developed formalism can be found\nin Section III. Details of numerical implementation are\nlisted in Section IIIA followed by illustrating examples\nforsystemsofthreedifferent kinds: binarysolidsolutions\nof 3dtransition metals in Section IIIB, pure iron with a\nsimple model of random potential fluctuations in Section\nIIIC, and stoichiometric FePt alloys with a partial long-\nrange order in Section IIID. The main conclusions are\nsummarized in Section IV.\nII. THEORETICAL FORMALISM\nA. Torque-correlation formula with alternative\ntorque operators\nThe torque operators Tµentering the torque-\ncorrelation formula (2) are closely related to compo-\nnents of the time derivative of electron spin. For spin-\npolarized systems described by means of an effective\nSchr¨ odinger-Paulione-electronHamiltonian H, actingon\ntwo-componentwavefunctions, thecompletetimederiva-\ntive of the spin operator is given by the commutation re-\nlationtµ=−i[σµ/2,H], where ¯ h= 1 is assumed and σµ\n(µ=x,y,z) denote the Pauli spin matrices. Let us write\nthe Hamiltonian as H=Hp+Hxc, whereHpincludes all\nspin-independent terms and the SO interaction (Hamil-\ntonian of a paramagnetic system) while Hxc=Bxc(r)·σ\ndenotes the XC term due to an effective magnetic field\nBxc(r). The complete time derivative (spin torque) can\nthen be written as tµ=tso\nµ+txc\nµ, where\ntso\nµ=−i[σµ/2,Hp], txc\nµ=−i[σµ/2,Hxc].(3)\nAs discussed, e.g., in Ref. 15, the use of the complete\ntorquetµinthetorque-correlationformula(2)leadsiden-3\ntically to zero; the correct Gilbert damping coefficients\nαµνfollow from Eq. (2) by using either the SO-induced\ntorquetso\nµ, or the XC-induced torque txc\nµ. Note that only\ntransverse components (with respect to the easy axis of\nthe ferromagnet)of the vectors tsoandtxcare needed for\nthe relevant part of the Gilbert damping tensor (2).\nThe equivalence of both torque operators (3) for the\nGilbert damping can be extended. Let us consider a sim-\nple system described by a model tight-binding Hamilto-\nnianH, written now as H=Hloc+Hnl, where the first\ntermHlocis a lattice sum of local atomic-like terms and\nthe nonlocal second term Hnlincludes all intersite hop-\nping matrix elements. Let us assume that all effects of\nthe SO interaction and XC fields are contained in the\nlocal term Hloc, so that the hopping elements are spin-\nindependent and [ σµ,Hnl] = 0. (Note that this assump-\ntion, often used in model studies, is satisfied only ap-\nproximatively in real ferromagnets with different widths\nof the majority and minority spin bands.) Let us write\nexplicitly Hloc=/summationtext\nR(Hp\nR+Hxc\nR), whereRlabelsthe lat-\ntice sites and where Hp\nRcomprises the spin-independent\npart and the SO interaction of the Rth atomic poten-\ntial while Hxc\nRis due to the local XC field of the Rth\natom. The operators Hp\nRandHxc\nRact only in the sub-\nspace of the Rth site; the subspaces of different sites\nare orthogonal to each other. The total spin operator\ncan be written as σµ/2 = (1/2)/summationtext\nRσRµ, where the local\noperator σRµis the projection of σµon theRth sub-\nspace. Let us assume that each term Hp\nRis spherically\nsymmetric and that Hxc\nR=Bxc\nR·σR, where the effec-\ntive field Bxc\nRof theRth atom has a constant size and\ndirection. Let us introduce local orbital-momentum op-\neratorsLRµand their counterparts including the spin,\nJRµ=LRµ+ (σRµ/2), which are generators of local\ninfinitesimal rotations with respect to the Rth lattice\nsite, and let us define the corresponding lattice sums\nLµ=/summationtext\nRLRµandJµ=/summationtext\nRJRµ=Lµ+(σµ/2). Then\nthe local terms Hp\nRandHxc\nRsatisfy, respectively, commu-\ntation rules [ JRµ,Hp\nR] = 0 and [ LRµ,Hxc\nR] = 0. By using\nthe above assumptions and definitions, the XC-induced\nspin torque (3) due to the XC term Hxc=/summationtext\nRHxc\nRcan\nbe reformulated as\ntxc\nµ=−i/summationdisplay\nR[σRµ/2,Hxc\nR] =−i/summationdisplay\nR[JRµ,Hxc\nR] (4)\n=−i/summationdisplay\nR[JRµ,Hp\nR+Hxc\nR] =−i[Jµ,Hloc]≡tloc\nµ.\nThe last commutator defines a local torque operator tloc\nµ\ndue to the local part of the Hamiltonian Hlocand the op-\neratorJµ,incontrasttothespinoperator σµ/2inEq.(3).\nLet us define the complementary nonlocal torque tnl\nµdue\nto the nonlocal part of the Hamiltonian Hnl, namely,\ntnl\nµ=−i[Jµ,Hnl] =−i[Lµ,Hnl], (5)\nand let us employ the fact that the complete time deriva-\ntive of the operator Jµ, i.e., the torque ˜tµ=−i[Jµ,H] =\ntloc\nµ+tnl\nµ, leads identically to zero when used in Eq. (2).This fact implies that the Gilbert damping parame-\nters can be also obtained from the torque-correlation\nformula with the nonlocal torques tnl\nµ. These torques\nare equivalent to the original spin-dependent local XC-\nor SO-induced torques; however, the derived nonlocal\ntorques are spin-independent, so that commutation rules\n[tnl\nµ,σν] = 0 are satisfied.\nInordertoseetheeffect ofdifferent formsofthe torque\noperators, Eqs. (3) and (5), we have studied a tight-\nbinding model of p-orbitals on a simple cubic lattice with\nthe ground-state magnetization along zaxis. The local\n(atomic-like) terms of the Hamiltonian are specified by\nthe XC term bσRzand the SO term ξLR·σR, which\nare added to a random spin-independent p-level at en-\nergyǫ0+DR, whereǫ0denotes the nonrandom center of\nthep-band while the random parts DRsatisfy configu-\nration averages /an}bracketle{tDR/an}bracketri}ht= 0 and /an}bracketle{tDR′DR/an}bracketri}ht=γδR′Rwith\nthe disorder strength γ. The spin-independent nonlocal\n(hopping) part of the Hamiltonian has been confined to\nnonrandom nearest-neighbor hoppings parametrized by\ntwoquantities, W1(ppσhopping) and W′\n1(ppπhopping),\nsee, e.g., page 36 of Ref. 31. The particular values have\nbeen set to b= 0.3,ξ= 0.2,EF−ǫ0= 0.1,γ= 0.05,\nW1= 0.3 andW′\n1=−0.1 (the hoppings were chosen\nsuch that the band edges for ǫ0=b=ξ=γ= 0 are±1).\nTheconfigurationaverageofthe propagators /an}bracketle{tG±/an}bracketri}ht=¯G±\nand of the torque correlation (2) was performed in the\nself-consistentBornapproximation(SCBA)includingthe\nvertex corrections. Since all three torques, Eqs. (3) and\n(5), are nonrandom operators in our model, the only rel-\nevant component of the Gilbert damping tensor, namely\nαxx=αyy=α, could be unambiguously decomposed in\nthe coherent part αcohand the incoherent part αvcdue\nto the vertex corrections.\nThe results are summarized in Fig. 1 which displays\nthe torque correlation α/α0as a function of the SO cou-\nplingξ(Fig. 1a) and the XC field b(Fig. 1b). The total\nvalueα=αcoh+αvcis identical for all three forms of\nthe torque operator, in contrast to the coherent parts\nαcohwhich exhibit markedly different values and trends\nas compared to each other and to the total α. This re-\nsult is in line with conclusions drawn by the authors of\nRef. 15, 16, and 26 proving the importance of the ver-\ntex corrections for obtaining the same Gilbert damping\nparameters from the SO- and XC-induced torques. The\nonly exception seems to be the case of the SO splitting\nmuch weaker than the exchange splitting, where the ver-\ntex corrections for the SO-induced torque can be safely\nneglected, see Fig. 1a. This situation, encountered in\n3dtransition metals and their alloys, has been treated\nwith the SO-induced torque on an ab initio level with ne-\nglectedvertexcorrectionsinRef. 11and12. Onthe other\nhand, the use of the XC-induced torque calls for a proper\nevaluation of the vertex corrections; their neglect leads\ntoquantitativelyandphysicallyincorrectresultsasdocu-\nmented by recent first-principles studies.19,22The vertex\ncorrectionsareindispensablealsoforthe nonlocaltorque,\nin particular for correct vanishing of the total torque cor-4\n 0 2 4\n 0  0.1  0.2torque  correlation\nspin-orbit  coupling(a)\ntotcoh-nl\ncoh-xc\ncoh-so\n 0 2 4\n 0  0.1  0.2  0.3torque  correlation\nexchange  field(b)\ntotcoh-nl\ncoh-xc\ncoh-so\nFIG. 1. (Color online) The torque correlation α/α0, Eq. (2),\nin a tight-binding p-orbital model treated in the SCBA as\na function of the spin-orbit coupling ξ(a) and of the ex-\nchange field b(b). The full diamonds display the total torque\ncorrelation (tot) and the open symbols denote the coherent\ncontributions αcoh/α0calculated with the SO-induced torque\n(coh-so), the XC-induced torque (coh-xc), Eq. (3), and the\nnonlocal torque (coh-nl), Eq. (5).\nrelation both in the nonrelativistic limit ( ξ→0, Fig. 1a)\nand in the nonmagnetic limit ( b→0, Fig. 1b).\nFinally, let us discuss briefly the general equivalence of\nthe SO- and XC-induced spin torques, Eq. (3), in the\nfully relativistic four-component Dirac formalism.32,33\nThe Kohn-Sham-Dirac Hamiltonian can be written as\nH=Hp+Hxc, whereHp=cα·p+mc2β+V(r) and\nHxc=Bxc(r)·βΣ, wherecis the speed of light, mde-\nnotes the electron mass, p={pµ}refers to the momen-\ntum operator, V(r) is the spin-independent part of the\neffective potential and the α={αµ},βandΣ={Σµ}\narethe well-known4 ×4matricesofthe Diractheory.34,35Then the XC-induced torque is txc=Bxc(r)×βΣ,\nwhich is currently used in the KKR theory of the Gilbert\ndamping.19,22The SO-induced torque is tso=p×cα,\ni.e., it is given directly by the relativistic momentum ( p)\nand velocity ( cα) operators. One can see that the torque\ntsoislocalbutindependent oftheparticularsystemstud-\nied. A comparison of both alternatives, concerning the\ntotal damping parameters as well as their coherent and\nincoherent parts, would be desirable; however, this task\nis beyond the scope of the present study.\nB. Effective torques in the LMTO method\nIn ourab initio approach to the Gilbert damping, we\nemploy the torque-correlation formula (2) with torques\nderived from the XC field.15,19,22The torque operators\nare constructed by considering infinitesimal deviations of\nthe direction of the XC field of the ferromagnet from its\nequilibrium orientation, taken asa reference state. These\ndeviations result from rotations by small angles around\naxesperpendiculartothe equilibrium direction ofthe XC\nfield; componentsofthetorqueoperatorarethengivenas\nderivatives of the one-particle Hamiltonian with respect\nto the rotation angles.36\nFor practical evaluation of Eq. (2) in an ab initio tech-\nnique (such as the LMTO method), one has to consider\na matrix representation of all operators in a suitable or-\nthonormal basis. The most efficient techniques of the\nelectronic structure theory require typically basis vectors\ntailored to the system studied; in the present context,\nthis leads naturally to basis sets depending on the angu-\nlar variables needed to define the torque operators. Eval-\nuation of the torque correlation using angle-dependent\nbases is discussed in Appendix A, where we prove that\nEq. (2) can be calculated solely from the matrix ele-\nments of the Hamiltonian and their angular derivatives,\nsee Eq. (A7), whereas the angular dependence of the ba-\nsis vectorsdoes not contribute directly to the final result.\nThe relativistic LMTO-ASA Hamiltonian matrix for\nthe reference system in the orthogonal LMTO represen-\ntation is given by37–39\nH=C+(√\n∆)+S(1−γS)−1√\n∆, (6)\nwhere the C,√\n∆ andγdenote site-diagonal matrices\nof the standard LMTO potential parameters and Sis\nthe matrix of canonical structure constants. The change\nof the Hamiltonian matrix Hdue to a uniform rotation\nof the XC field is treated in Appendix B; it is sum-\nmarized for finite rotations in Eq. (B7) and for angu-\nlar derivatives of Hin Eqs. (B8) and (B9). The resol-\nventG(z) = (z−H)−1of the LMTO Hamiltonian (6)\nfor complex energies zcan be expressed using the auxil-\niary resolvent g(z) = [P(z)−S]−1, which represents an\nLMTO-counterpart of the scattering-path operator ma-\ntrix of the KKR method.32,33The symbol P(z) denotes\nthe site-diagonal matrix of potential functions; their an-\nalytic dependence on zand on the potential parameters5\ncan be found elsewhere.27,37The relation between both\nresolvents leads to the formula28\nG+−G−=F(g+−g−)F+, (7)\nwhere the same abbreviation F= (√\n∆)−1(1−γS) as in\nEq. (B8) was used and g±=g(EF±i0) .\nThe torque-correlation formula (2) in the LMTO-ASA\nmethod follows directly from relations (A7), (B8), (B9)\nand (7). The components of the Gilbert damping tensor\n{αµν}in the LLG equation (1) can be obtained from a\nbasic tensor {˜αµν}given by\n˜αµν=−α0Tr{τµ(g+−g−)τν(g+−g−)},(8)\nwhere the quantities\nτµ=−i[Jµ,S] =−i[Lµ,S] (9)\ndefine components of an effective torque in the LMTO-\nASA method. The site-diagonal matrices JµandLµ\n(µ=x,y,z) are Cartesian components of the total and\norbital angular momentum operator, respectively, see\ntext aroundEqs.(B8) and (B9). The tracein (8) extends\nover all orbitals of the crystalline solid and the prefactor\ncan be written as α0= (2πMspin)−1, where Mspinde-\nnotes the spin magnetic moment of the whole crystal in\nunits of the Bohr magneton µB.15,19,22\nLet us discuss properties of the effective torque (9).\nIts form is obviously identical to the nonlocal torque (5).\nThe matrix τµis non-site-diagonal, but—for a random\nsubstitutional alloy on a nonrandom lattice—it is non-\nrandom (independent on the alloy configuration). More-\nover, it is given by a commutator of the site-diagonal\nnonrandom matrix Jµ(orLµ) and the LMTO structure-\nconstantmatrix S. Thesepropertiespointtoacloseanal-\nogy between the effective torque and the effective veloci-\nties in the LMTO conductivity tensor based on a concept\nof intersite electron hopping.27,28,40Let us mention that\nexisting ab initio approaches employ random torques,\neither the XC-induced torque in the KKR method19,22\nor the SO-induced torque in the LMTO method.20An-\nother interesting property of the effective torque τµ(9)\nis its spin-independence which follows from the spin-\nindependence of the matrices LµandS.\nThe explicit relation between the symmetric tensors\n{αµν}and{˜αµν}canbeeasilyformulatedfortheground-\nstate magnetization along zaxis; then it is given simply\nbyαxx= ˜αyy,αyy= ˜αxx, andαxy=−˜αxy. These\nrelations reflect the fact that an infinitesimal deviation\ntowards xaxis results from an infinitesimal rotation of\nthe magnetization vector around yaxis and vice versa.\nNote that the other components of the Gilbert damp-\ning tensor ( αµzforµ=x,y,z) are not relevant for the\ndynamics of small deviations of magnetization direction\ndescribed by the LLG equation (1). For the ground-\nstate magnetization pointing along a general unit vector\nm= (mx,my,mz), one has to employ the Levi-Civita\nsymbolǫµνλin order to get the Gilbert damping tensorαas\nαµν=/summationdisplay\nµ′ν′ηµµ′ηνν′˜αµ′ν′, (10)\nwhereηµν=/summationtext\nλǫµνλmλ. The resultingtensor(10) satis-\nfies the condition α·m= 0 appropriate for the dynamics\nof small transverse deviations of magnetization.\nThe application to random alloys requires configura-\ntion averaging of ˜ αµν(8). Since the effective torques τµ\nare nonrandom, one can write a unique decomposition\nof the average into the coherent and incoherent parts,\n˜αµν= ˜αcoh\nµν+ ˜αvc\nµν, where the coherent part is expressed\nby means of the averaged auxiliary resolvents ¯ g±=/an}bracketle{tg±/an}bracketri}ht\nas\n˜αcoh\nµν=−α0Tr{τµ(¯g+−¯g−)τν(¯g+−¯g−)}(11)\nand the incoherent part (vertex corrections) is given as a\nsum of four terms, namely,\n˜αvc\nµν=−α0/summationdisplay\np=±/summationdisplay\nq=±sgn(pq)Tr/an}bracketle{tτµgpτνgq/an}bracketri}htvc.(12)\nIn this work, the configuration averaging has been done\nin the CPA. Details concerning the averaged resolvents\ncan be found, e.g., in Ref. 39 and the construction of the\nvertex corrections for transport properties was described\nin Appendix to Ref. 30.\nC. Properties of the LMTO torque-correlation\nformula\nThe damping tensor (8) has been formulated in the\ncanonical LMTO representation. In the numerical im-\nplementation, the well-known transformation to a tight-\nbinding(TB)LMTOrepresentation41,42isadvantageous.\nThe TB-LMTO representation is specified by a diag-\nonal matrix βof spin-independent screening constants\n(βR′ℓ′m′s′,Rℓms=δR′Rδℓ′ℓδm′mδs′sβRℓin a nonrelativis-\ntic basis) and the transformation of all quantities be-\ntween both LMTO representations has been discussed in\nthe literature for pure crystals42as well as for random\nalloys.28,39,43The same techniques can be used in the\npresent case together with an obvious commutation rule\n[Jµ,β] = [Lµ,β] = 0. Consequently, the conclusions\ndrawn are the same as for the conductivity tensor:28the\ntotal damping tensor (8) as well as its coherent (11) and\nincoherent (12) parts in the CPA are invariant with re-\nspect to the choice of the LMTO representation.\nIt should be mentioned that the central result, namely\nthe relations(8) and (9), is not limited to the LMTO the-\nory, but it can be translatedinto the KKRtheory aswell,\nsimilarly to the conductivity tensor in the formalism of\nintersite hopping.40The LMTO structure-constant ma-\ntrixSandtheauxiliaryGreen’sfunction g(z)willbethen\nreplacedrespectivelybytheKKRstructure-constantma-\ntrix and by the scattering-path operator.32,33Note, how-6\never, that the total ( Jµ) and orbital ( Lµ) angular mo-\nmentum operatorsin the effective torques (9) will be rep-\nresented by the same matrices as in the LMTO theory.\nLet us mention for completeness that the present\nLMTO-ASA theory allows one to introduce effective lo-\ncal (but random) torques as well. This is based on the\nfact that only the Fermi-level propagators g±defined by\nthe structure constant matrix Sand by the potential\nfunctions at the Fermi energy, P=P(EF), enter the\nzero-temperature expression for the damping tensor ˜ αµν\n(8). Since the equation of motion ( P−S)g±= 1 implies\nimmediately S(g+−g−) =P(g+−g−) and, similarly,\n(g+−g−)S= (g+−g−)P, one can obviously replace the\nnonlocal torques τµ(9) in the torque-correlation formula\n(8) by their local counterparts\nτxc\nµ= i[P,Jµ], τso\nµ= i[P,Lµ].(13)\nThese effective torques are represented by random, site-\ndiagonal matrices; the τxc\nµandτso\nµcorrespond, respec-\ntively, to the XC-induced torque used in the KKR\nmethod22and to the SO-induced torque used in the\nLMTO method with a simplified treatment of the SO-\ninteraction.20In the case of random alloys treated in the\nCPA, the randomness of the local torques (13) calls for\nthe approach developed by Butler44for the averaging of\nthe torque-correlationcoefficient (8). One can provethat\nthe resulting damping parameters ˜ αµνobtained in the\nCPA with the local and nonlocal torques are fully equiv-\nalent to each other; this equivalence rests heavily on a\nproper inclusion of the vertex corrections45and it leads\nto further important consequences. First, the Gilbert\ndamping tensor vanishes exactly for zero SO interaction,\nwhich follows from the use of the SO-induced torque τso\nµ\nand from the obvious commutation rule [ P,Lµ] = 0 valid\nfor the spherically symmetric potential functions (in the\nabsence of SO interaction). This result is in agreement\nwith thenumericalstudyofthe toymodel inSectionIIA,\nsee Fig. 1a for ξ= 0. On an ab initio level, this prop-\nerty has been obtained numerically both in the KKR\nmethod22and in the LMTO method.26Second, the XC-\nandSO-inducedlocaltorques(13)withintheCPAareex-\nactly equivalent as well, as has been indicated in a recent\nnumerical study for a random bcc Fe 50Co50alloy.26In\nsummary, the nonlocaltorques(9) andboth localtorques\n(13) can be used as equivalent alternatives in the torque-\ncorrelation formula (8) provided that the vertex correc-\ntions are included consistently with the CPA-averaging\nof the single-particle propagators.\nIII. ILLUSTRATING EXAMPLES\nA. Implementation and numerical details\nThe numerical implementation of the described the-\nory and the calculations have been done with similar\ntools as in our recent studies of ground-state46and 3 6 9\n 0  10  20  30103 α\nε  (µRy)fcc Ni80Fe20bcc Fe80Co20\n(x 10)\nFIG. 2. The Gilbert damping parameters αof random fcc\nNi80Fe20(full circles) and bcc Fe 80Co20(open squares) alloys\nas functions of the imaginary part of energy ε. The values of\nαfor the Fe 80Co20alloy are magnified by a factor of 10.\ntransport27,28,47properties. The ground-state magne-\ntization was taken along zaxis and the selfconsistent\nXC potentials were obtained in the local spin-density ap-\nproximation (LSDA) with parametrization according to\nRef. 48. The valence basis comprised s-,p-, andd-type\norbitalsand the energyargumentsforthe propagators¯ g±\nand the CPA-vertex corrections were obtained by adding\na tiny imaginary part ±εto the real Fermi energy. We\nhave found that the dependence of the Gilbert damping\nparameter on εis quite smooth and that the value of\nε= 10−6Ry is sufficient for the studied systems, see\nFig. 2 for an illustration. Similar smooth dependences\nhavebeenobtainedalsoforotherinvestigatedalloys,such\nas Permalloy doped by 5 delements, Heusler alloys, and\nstoichiometric FePt alloys with a partial atomic long-\nrange order. In all studied cases, the number Nofk\nvectors needed for reliable averaging over the Brillouin\nzone (BZ) was properly checked; as a rule, N∼108in\nthe full BZ was sufficient for most systems, but for di-\nluted alloys (a few percent of impurities), N∼109had\nto be taken.\nB. Binary fcc and bcc solid solutions\nThe developed theory has been applied to random bi-\nnary alloys of 3 dtransition elements Fe, Co, and Ni,\nnamely, to the fcc NiFe and bcc FeCo alloys. The most\nimportant results, including a comparison to other exist-\ningab initio techniques, are summarized in Fig. 3. One\ncan see a good agreement of the calculated concentration\ntrends of the Gilbert damping parameter α=αxx=αyy\nwith the results of an LMTO-supercell approach17and\nof the KKR-CPA method.22The decrease of αwith in-7\n 0 4 8 12\n 0  0.2  0.4  0.6103 α\nFe  concentrationfcc  NiFe(a)\nthis work\nLMTO-SC\n 0 2 4 6\n 0  0.2  0.4  0.6103 α\nCo  concentrationbcc  FeCo(b)\nthis work\nKKR-CPA\nFIG. 3. (Color online) The calculated concentration depen-\ndences of the Gilbert damping parameter αfor random fcc\nNiFe (a) and bcc FeCo (b) alloys. The results of this work\nare marked by the full diamonds, whereas the open circles\ndepict the results of other approaches: the LMTO supercell\n(LMTO-SC) technique17and the KKR-CPA method.22\ncreasingFecontentintheconcentratedNiFealloyscanbe\nrelatedto the increasingalloymagnetization17andto the\ndecreasing strength of the SO-interaction,20whereas the\nbehaviorinthedilutelimitcanbeexplainedbyintraband\nscattering due to Fe impurities.11,12,14In the case of the\nFeCo system, the minimum of αaround 20% Co, which\nis also observed in room-temperature experiments,49,50\nis related primarily to a similar concentration trend of\nthe density of states at the Fermi energy,22though the\nmaximum of the magnetization at roughly the same alloy\ncomposition51might partly contribute as well.\nA more detailed comparison of all ab initio results is\npresented in Table I for the fcc Ni 80Fe20random alloy\n(Permalloy). The differences in the values of αfrom the\ndifferenttechniquescanbe ascribedtovarioustheoretical\nfeatures and numericaldetails employed, such asthe sim-TABLE I. Comparison of the Gilbert damping parameter α\nfor the fcc Ni 80Fe20random alloy (Permalloy) calculated by\nthe present approach and by other techniques using the CPA\nor supercells (SC). The last column displays the coherent pa rt\nαcohof the total damping parameter according to Eq. (11).\nThe experimental value corresponds to room temperature.\nMethod α αcoh\nThis work, ε= 10−5Ry 4 .9×10−31.76\nThis work, ε= 10−6Ry 3 .9×10−31.76\nKKR-CPAa4.2×10−3\nLMTO-CPAb3.5×10−3\nLMTO-SCc4.6×10−3\nExperimentd8×10−3\naReference 22.\nbReference 20.\ncReference 17.\ndReference 49.\nplified treatment of the SO-interaction in Ref. 20 instead\nof the fully relativistic description, or the use of super-\ncells in Ref. 17 instead of the CPA. Taking into account\nthat calculated residual resistivities for this alloy span a\nwide interval between 2 µΩcm, see Ref. 27 and 52, and\n3.5µΩcm, see Ref. 17, one can consider the scatter of the\ncalculated values of αin Table I as little important. The\ntheoretical values of αare smaller systematically than\nthe measured values, typically by a factor of two. This\ndiscrepancy might be partly due to the effects of finite\ntemperatures as well as due to additional structural de-\nfects of real samples.\nA closer look at the theoretical results reveals that the\ntotal damping parameters αareappreciablysmallerthan\nthemagnitudesoftheircoherentandvertexparts,seeTa-\nble I for the case of Permalloy. This is in agreement with\ntheresultsofthemodelstudyinSectionIIA;similarcon-\nclusions about the importance of the vertex corrections\nhave been done with the XC-induced torques in other\nCPA-based studies.19,22,26The present results prove that\nthis unpleasant feature of the nonlocal torques does not\nrepresent a serious obstacle in obtaining reliable values\nof the Gilbert damping parameter in random alloys. We\nnote that the vertex corrections can be negligible in ap-\nproaches employing the SO-induced torques, at least for\nsystems with the SO splittings much weaker than the XC\nsplittings,12such as the binary ferromagnetic alloys of 3 d\ntransition metals,26see also Section IIA.\nC. Pure iron with a model disorder\nAs it has been mentioned in Section I, the Gilbert\ndamping of pure ferromagnetic metals exhibits non-\ntrivial temperature dependences, which have been re-\nproduced by means of ab initio techniques with vari-\nous levels of sophistication.11,12,21,23In this study, we\nhave simulated the effect of finite temperatures by intro-8\n 0 3 6 9\n0123402040103 α\nρ  (µΩcm)\n103 δ2  (Ry2)bcc  Fe\nFIG. 4. (Color online) The calculated Gilbert damping pa-\nrameter α(full squares) and the residual resistivity ρ(open\ncircles) of pure bcc iron as functions of δ2, where δis the\nstrength of a model atomic-level disorder.\nducing static fluctuations of the one-particle potential.\nThe adopted model of atomic-level disorderassumes that\nrandom spin-independent shifts ±δ, constant inside each\natomic sphere and occurring with probabilities 50% of\nbothsigns,areaddedtothenonrandomselfconsistentpo-\ntential obtained at zero temperature. The Fermi energy\niskeptfrozen,equaltoitsselfconsistentzero-temperature\nvalue. This model can be easily treated in the CPA; the\nresulting Gilbert damping parameter αof pure bcc Fe as\na function of the potential shift δis plotted in Fig. 4.\nThecalculateddependence α(δ) isnonmonotonic, with\na minimum at δ≈30 mRy. This trend is in a qualita-\ntive agreement with trends reported previously by other\nauthors, who employed phenomenological models of the\nelectron lifetime11,12as well as models for phonons and\nmagnons.21,23The origin of the nonmonotonic depen-\ndenceα(δ) has been identified on the basis of the band\nstructure of the ferromagnetic system as an interplay be-\ntween the intraband contributions to α, dominating for\nsmall values of δ, and the interband contributions, domi-\nnating for large values of δ.7,11,12Since the present CPA-\nbased approach does not use any bands, we cannot per-\nform a similar analysis.\nThe obtained minimum value of the Gilbert damping,\nαmin≈10−3(Fig. 4), agrees reasonably well with the\nvalues obtained by the authors of Ref. 11, 12, 21, and\n23. This agreement indicates that the atomic-level dis-\norder employed here is equivalent to a phenomenological\nlifetime broadening. For a rough quantitative estimation\nof the temperature effect, one can employ the calculated\nresistivity ρof the model, which increases essentially lin-\nearly with δ2, see Fig. 4. Since the metallic resistivity\ndue to phonons increases linearly with the temperature\nT(for temperatures not much smaller than the Debye\ntemperature), one can assume a proportionality between 10 20 30 40\n0 0.5 1152535103 α\nDOS(EF)  (states/Ry)\nLRO  parameter  SL10  FePt\nFIG. 5. (Color online) The calculated Gilbert damping pa-\nrameter α(full squares) and the total DOS (per formula unit)\nat the Fermi energy (open circles) of stoichiometric L1 0FePt\nalloys as functions of the LRO parameter S.\nδ2andT. The resistivity of bcc iron at the Curie tem-\nperature TC= 1044 K due to lattice vibrations can be\nestimated around 35 µΩcm,23,53which sets an approx-\nimate temperature scale to the data plotted in Fig. 4.\nHowever, a more accurate description of the temperature\ndependence of the Gilbert damping parameter cannot be\nobtained, mainly due to the neglected true atomic dis-\nplacements and the noncollinearity of magnetic moments\n(magnons).23\nD. FePt alloys with a partial long-range order\nSince important ferromagnetic materials include or-\ndered alloys, we address here the Gilbert damping in sto-\nichiometric FePt alloys with L1 0atomic long-range order\n(LRO). Their transport properties47and the damping\nparameter20have recently been studied by means of the\nTB-LMTO method in dependence on a varying degree\nof the LRO. These fcc-based systems contain two sublat-\nticeswith respectiveoccupationsFe 1−yPtyandPt 1−yFey\nwherey(0≤y≤0.5) denotes the concentration of anti-\nsite atoms. The LRO parameter S(0≤S≤1) is then\ndefined as S= 1−2y, so that S= 0 corresponds to the\nrandom fcc alloy and S= 1 corresponds to the perfectly\nordered L1 0structure.\nThe resulting Gilbert damping parameter is displayed\nin Fig. 5 as a function of S. The obtained trend with a\nbroadmaximumat S= 0andaminimumaround S= 0.9\nagrees very well with the previous result.20The values of\nαin Fig. 5 are about 10% higher than those in Ref. 20,\nwhich can be ascribed to the fully relativistic treatment\nin the present study in contrast to a simplified treatment\nof the SO interaction in Ref. 20. The Gilbert damping9\nin the FePt alloys is an order of magnitude stronger than\nin the alloys of 3 delements (Section IIIB) owing to the\nstronger SO interaction of Pt atoms. The origin of the\nslow decrease of αwith increasing S(for 0≤S≤0.9)\ncan be explained by the decreasing total density of states\n(DOS) at the Fermi energy, see Fig. 5, which represents\nan analogy to a similar correlation observed, e.g., for bcc\nFeCo alloys.22\nAll calculated values of αshown in Fig. 5, correspond-\ning to 0 ≤S≤0.985, are appreciably smaller than\nthe measured one which amounts to α≈0.06 reported\nfor a thin L1 0FePt epitaxial film.54The high measured\nvalue of αmight be thus explained by the present cal-\nculations by assuming a very small concentration of an-\ntisites in the prepared films, which does not seem too\nrealistic. Another potential source of the discrepancy\nlies in the thin-film geometry used in the experiment.\nMoreover, the divergence of αin the limit of S→1\n(Fig. 5) illustrates a general shortcoming of approaches\nbased on the torque-correlation formula (2), since the\nzero-temperature Gilbert damping parameter of a pure\nferromagnet should remain finite. A correct treatment\nof this case, including the dilute limit of random alloys\n(Fig. 3), must take into account the full interacting sus-\nceptibility in the presence of SO interaction.15,55Pilotab\ninitiostudies in this direction have recently appeared for\nnonrandom systems;56,57however, their extension to dis-\nordered systems goes far beyond the scope of this work.\nIV. CONCLUSIONS\nWe have introduced nonlocal torques as an alterna-\ntive to the usual local torque operators entering the\ntorque-correlation formula for the Gilbert damping ten-\nsor. Within the relativistic TB-LMTO-ASA method,\nthis idea leads to effective nonlocal torques as non-site-\ndiagonal and spin-independent matrices. For substitu-\ntionally disordered alloys, the nonlocal torques are non-\nrandom, which allows one to develop an internally con-\nsistent theory in the CPA. The CPA-vertex corrections\nproved indispensable for an exact equivalence of the non-\nlocal nonrandom torques with their local random coun-\nterparts. The concept of the nonlocal torques is not lim-\nited to the LMTO method and its formulation both in\na semiempirical TB theory and in the KKR theory is\nstraightforward.\nThe numerical implementation and the results for bi-\nnary solid solutions show that the total Gilbert damping\nparameters from the nonlocal torques are much smaller\nthan magnitudes of the coherent parts and of the ver-\ntex corrections. Nevertheless, the total damping param-\neters for the studied NiFe, FeCo and FePt alloys compare\nquantitatively very well with results of other ab initio\ntechniques,17,20,22which indicates a fair numerical sta-\nbility of the developed theory.\nThe performed numerical study of the Gilbert damp-\ning in pure bcc iron as a function of an atomic-level dis-order yields a nonmonotonic dependence in a qualitative\nagreementwith the trends consisting of the conductivity-\nlike and resistivity-like regions, obtained from a phe-\nnomenological quasiparticle lifetime broadening7,11,12or\nfrom the temperature-induced frozen phonons21,22and\nmagnons.23Future studies should clarify the applicabil-\nity of the introduced nonlocal torques to a full quanti-\ntative description of the finite-temperature behavior as\nwell as to other torque-related phenomena, such as the\nspin-orbit torques due to applied electric fields.58,59\nACKNOWLEDGMENTS\nThe authors acknowledge financial support by the\nCzech Science Foundation (Grant No. 15-13436S).\nAppendix A: Torque correlation formula in a matrix\nrepresentation\nIn this Appendix, evaluation of the Kubo-Greenwood\nexpression for the torque-correlation formula (2) is dis-\ncussed in the case of the XC-induced torque operators\nusing matrix representations of all operators in an or-\nthonormal basis that varies due to the varying direc-\ntion of the XC field. All operators are denoted by a\nhat, in order to be distinguished from matrices repre-\nsenting these operators in the chosen basis. Let us con-\nsider a one-particle Hamiltonian ˆH=ˆH(θ1,θ2) depend-\ning on two real variables θj,j= 1,2, and let us denote\nˆT(j)(θ1,θ2) =∂ˆH(θ1,θ2)/∂θj. In our case, the variables\nθjplay the role of rotation angles and the operators ˆT(j)\nare the corresponding torques. Let us denote the resol-\nvents of ˆH(θ1,θ2) at the Fermi energy as ˆG±(θ1,θ2) and\nlet us consider a special linear response coefficient (argu-\nmentsθ1andθ2are omitted here and below for brevity)\nc= Tr{ˆT(1)(ˆG+−ˆG−)ˆT(2)(ˆG+−ˆG−)} (A1)\n= Tr{(∂ˆH/∂θ1)(ˆG+−ˆG−)(∂ˆH/∂θ2)(ˆG+−ˆG−)}.\nThis torque-correlation coefficient equals the Gilbert\ndamping parameter (2) with the prefactor ( −α0) sup-\npressed. For its evaluation, we introduce an orthonormal\nbasis|χm(θ1,θ2)/an}bracketri}htand represent all operators in this ba-\nsis. This leads to matrices H(θ1,θ2) ={Hmn(θ1,θ2)},\nG±(θ1,θ2) ={(G±)mn(θ1,θ2)}andT(j)(θ1,θ2) =\n{T(j)\nmn(θ1,θ2)}, where\nHmn=/an}bracketle{tχm|ˆH|χn/an}bracketri}ht,(G±)mn=/an}bracketle{tχm|ˆG±|χn/an}bracketri}ht,\nT(j)\nmn=/an}bracketle{tχm|ˆT(j)|χn/an}bracketri}ht=/an}bracketle{tχm|∂ˆH/∂θj|χn/an}bracketri}ht,(A2)\nand, consequently, to the response coefficient (A1) ex-\npressed by using the matrices (A2) as\nc= Tr{T(1)(G+−G−)T(2)(G+−G−)}.(A3)\nHowever, in evaluation of the last expression, atten-\ntion has to be paid to the difference between the ma-\ntrixT(j)(θ1,θ2) and the partial derivative of the matrix10\nH(θ1,θ2) with respect to θj. This difference follows from\nthe identity ˆH=/summationtext\nmn|χm/an}bracketri}htHmn/an}bracketle{tχn|, which yields\nT(j)\nmn=∂Hmn/∂θj+/summationdisplay\nk/an}bracketle{tχm|∂χk/∂θj/an}bracketri}htHkn\n+/summationdisplay\nkHmk/an}bracketle{t∂χk/∂θj|χn/an}bracketri}ht, (A4)\nwhere we employed the orthogonality relations\n/an}bracketle{tχm(θ1,θ2)|χn(θ1,θ2)/an}bracketri}ht=δmn. Their partial derivatives\nyield\n/an}bracketle{tχm|∂χn/∂θj/an}bracketri}ht=−/an}bracketle{t∂χm/∂θj|χn/an}bracketri}ht ≡Q(j)\nmn,(A5)\nwhere we introduced elements of matrices Q(j)={Q(j)\nmn}\nforj= 1,2. Note that the matrices Q(j)(θ1,θ2) reflect\nexplicitlythe dependenceofthebasisvectors |χm(θ1,θ2)/an}bracketri}ht\nonθ1andθ2. The relation (A4) between the matrices\nT(j)and∂H/∂θ jcan be now rewritten compactly as\nT(j)=∂H/∂θ j+[Q(j),H]. (A6)\nSince the last term has a form of a commutator with the\nHamiltonianmatrix H, theuseofEq.(A6)intheformula\n(A3) leads to the final matrix expression for the torque\ncorrelation,\nc= Tr{(∂H/∂θ 1)(G+−G−)(∂H/∂θ 2)(G+−G−)}.(A7)\nThe equivalence of Eqs. (A3) and (A7) rests on the rules\n[Q(j),H] = [EF−H,Q(j)] and (EF−H)(G+−G−) =\n(G+−G−)(EF−H) = 0 and on the cyclic invariance of\nthe trace. It is also required that the matrices Q(j)are\ncompatible with periodic boundary conditions used in\ncalculations of extended systems, which is obviously the\ncase for angular variables θjrelated to the global changes\n(uniform rotations) of the magnetization direction.\nThe obtained result means that the original response\ncoefficient (A1) involving the torques as angular deriva-\ntives of the Hamiltonian can be expressed solely by us-\ning matrix elements of the Hamiltonian in an angle-\ndependent basis; theangulardependence ofthebasisvec-\ntors does not enter explicitly the final torque-correlation\nformula (A7).\nAppendix B: LMTO Hamiltonian of a ferromagnet\nwith a tilted magnetic field\nHere we sketch a derivation of the fully relativis-\ntic LMTO Hamiltonian matrix for a ferromagnet with\nthe XC-field direction tilted from a reference direction\nalong an easy axis. The derivation rests on the form of\nthe Kohn-Sham-Dirac Hamiltonian in the LMTO-ASA\nmethod.37–39The symbols with superscript 0 refer to the\nreferencesystem,thesymbolswithoutthissuperscriptre-\nfer to the system with the tilted XC field. The operators\n(Hamiltonians, rotation operators) are denoted by sym-\nbols with a hat. The spin-dependent parts of the ASApotentials due to the XC fields are rigidly rotated while\nthe spin-independent parts are unchanged, in full anal-\nogy to the approach employed in the relativistic KKR\nmethod.19,22\nThe ASA-Hamiltonians of both systems are given by\nlattice sums ˆH0=/summationtext\nRˆH0\nRandˆH=/summationtext\nRˆHR, where\nthe individual site-contributions are coupled mutually by\nˆHR=ˆURˆH0\nRˆU+\nR, whereˆURdenotesthe unitaryoperator\nof a rotation (in the orbital and spin space) around the\nRth lattice site which brings the local XC field from its\nreference direction into the tilted one. Let |φ0\nRΛ/an}bracketri}htand\n|˙φ0\nRΛ/an}bracketri}htdenote, respectively, the phi and phi-dot orbitals\nof the reference Hamiltonian ˆH0\nR, then\n|φRΛ/an}bracketri}ht=ˆUR|φ0\nRΛ/an}bracketri}ht,|˙φRΛ/an}bracketri}ht=ˆUR|˙φ0\nRΛ/an}bracketri}ht(B1)\ndefine the phi and phi-dot orbitals of the Hamiltonian\nˆHR. The orbital index Λ labels all linearly indepen-\ndentsolutions(regularattheorigin)ofthespin-polarized\nrelativistic single-site problem; the detailed structure of\nΛ can be found elsewhere.37–39Let us introduce further\nthe well-known empty-space solutions |K∞,0\nRN/an}bracketri}ht(extending\nover the whole real space), |Kint,0\nRN/an}bracketri}ht(extending over the\ninterstitial region), and |K0\nRN/an}bracketri}htand|J0\nRN/an}bracketri}ht(both trun-\ncated outside the Rth sphere), needed for the definition\nof the LMTOs of the reference system.41,42,60Their in-\ndexN, which defines the spin-spherical harmonics of the\nlarge component of each solution, can be taken either in\nthe nonrelativistic ( ℓms) form or in its relativistic ( κµ)\ncounterpart. We define further\n|ZRN/an}bracketri}ht=ˆUR|Z0\nRN/an}bracketri}htforZ=K∞, K, J. (B2)\nIsotropyofthe emptyspaceguaranteesrelations(for Z=\nK∞,K,J)\n|ZRN/an}bracketri}ht=/summationdisplay\nN′|Z0\nRN′/an}bracketri}htUN′N,\n|Z0\nRN/an}bracketri}ht=/summationdisplay\nN′|ZRN′/an}bracketri}htU+\nN′N, (B3)\nwhereU={UN′N}denotes a unitary matrix represent-\ning the rotation in the space of spin-spherical harmonics\nand where U+\nN′N≡(U+)N′N= (UNN′)∗= (U−1)N′N;\nthe matrix Uis the same for all lattice sites Rsince we\nconsider only uniform rotations of the XC-field direction\ninside the ferromagnet. The expansion theorem for the\nenvelope orbital |K∞,0\nRN/an}bracketri}htis\n|K∞,0\nRN/an}bracketri}ht=|Kint,0\nRN/an}bracketri}ht+|K0\nRN/an}bracketri}ht\n−/summationdisplay\nR′N′|J0\nR′N′/an}bracketri}htS0\nR′N′,RN,(B4)\nwhereS0\nR′N′,RNdenote elements of the canonical\nstructure-constant matrix (with vanishing on-site ele-\nments,S0\nRN′,RN= 0) of the reference system. The use\nof relations (B3) in the expansion (B4) together with an\nabbreviation\n|Kint\nRN/an}bracketri}ht=/summationdisplay\nN′|Kint,0\nRN′/an}bracketri}htUN′N (B5)11\nyields the expansion of the envelope orbital |K∞\nRN/an}bracketri}htas\n|K∞\nRN/an}bracketri}ht=|Kint\nRN/an}bracketri}ht+|KRN/an}bracketri}ht\n−/summationdisplay\nR′N′|JR′N′/an}bracketri}ht(U+S0U)R′N′,RN,(B6)\nwhereUandU+denote site-diagonal matrices with el-\nementsUR′N′,RN=δR′RUN′Nand (U+)R′N′,RN=\nδR′RU+\nN′N. Note the same form of expansions (B4) and\n(B6), with the orbitals |Z0\nRN/an}bracketri}htreplaced by the rotated or-\nbitals|ZRN/an}bracketri}ht(Z=K∞,K,J), with the interstitial parts\n|Kint,0\nRN/an}bracketri}htreplacedbytheirlinearcombinations |Kint\nRN/an}bracketri}ht, and\nwith the structure-constant matrix S0replaced by the\nproduct U+S0U.\nThe non-orthogonal LMTO |χ0\nRN/an}bracketri}htfor the reference\nsystem is obtained from the expansion (B4), in which all\norbitals|K0\nRN/an}bracketri}htand|J0\nRN/an}bracketri}htare replaced by linear com-\nbinations of |φ0\nRΛ/an}bracketri}htand|˙φ0\nRΛ/an}bracketri}ht. A similar replacement of\nthe orbitals |KRN/an}bracketri}htand|JRN/an}bracketri}htby linear combinations of\n|φRΛ/an}bracketri}htand|˙φRΛ/an}bracketri}htin the expansion (B6) yields the non-\northogonal LMTO |χRN/an}bracketri}htfor the system with the tilted\nXC field. The coefficients in these linear combinations—\nobtained from conditions of continuous matching at the\nsphere boundaries and leading directly to the LMTO po-\ntentialparameters—areidenticalforboth systems, asfol-\nlows from the rotationrelations (B1) and (B2). For these\nreasons, the only essential difference between both sys-\ntems in the construction of the non-orthogonal and or-\nthogonal LMTOs (and of the accompanying Hamiltonian\nand overlap matrices in the ASA) is due to the difference\nbetween the matrices S0andU+S0U.\nAs a consequence, the LMTO Hamiltonian matrix in\nthe orthogonalLMTO representationfor the system with\na tilted magnetizationis easilyobtained fromthat forthe\nreference system, Eq. (6), and it is given by\nH=C+(√\n∆)+U+SU(1−γU+SU)−1√\n∆,(B7)\nwhere the C,√\n∆ andγare site-diagonal matrices of\nthe potential parameters of the reference system and\nwhere we suppressed the superscript 0 at the structure-\nconstant matrix Sof the reference system. Note that\nthe dependence of Hon the XC-field direction is con-\ntained only in the similarity transformation U+SUof\nthe original structure-constant matrix Sgenerated by\nthe rotation matrix U. For the rotation by an angle\nθaround an axis along a unit vector n, the rotation\nmatrix is given by U(θ) = exp( −in·Jθ), where the\nsite-diagonal matrices J≡(Jx,Jy,Jz) with matrix\nelements Jµ\nR′N′,RN=δR′RJµ\nN′N(µ=x,y,z) reduce\nto usual matrices of the total (orbital plus spin) angu-\nlar momentum operator. The limit of small θyields\nU(θ)≈1−in·Jθ, which leads to the θ-derivative of\nthe Hamiltonian matrix (B7) at θ= 0:\n∂H/∂θ= i(F+)−1[n·J,S]F−1, (B8)\nwhere we abbreviated F= (√\n∆)−1(1−γS) andF+=\n(1−Sγ)[(√\n∆)+]−1. Since the structure-constant matrixSis spin-independent, the total angular momentum op-\neratorJin (B8) canbe replacedbyits orbitalmomentum\ncounterpart L≡(Lx,Ly,Lz), so that\n∂H/∂θ= i(F+)−1[n·L,S]F−1.(B9)\nThe relations (B8) and (B9) are used to derive the\nLMTO-ASA torque-correlation formula (8).\nAppendix C: Equivalence of the Gilbert damping in\nthe CPA with local and nonlocal torques\n(Supplemental Material)\n1. Introductory remarks\nThe problem of equivalence of the Gilbert damping\ntensor expressed with the local (loc) and nonlocal (nl)\ntorques can be reduced to the problem of equivalence of\nthese two expressions:\nαloc=α0Tr/an}bracketle{t(g+−g−)[P,K](g+−g−)[P,K]/an}bracketri}ht\n=α0/summationdisplay\np=±/summationdisplay\nq=±sgn(pq)Tr/an}bracketle{tgp[P,K]gq[P,K]/an}bracketri}ht\n=α0/summationdisplay\np=±/summationdisplay\nq=±sgn(pq)βloc\npq, (C1)\nand\nαnl=α0Tr/an}bracketle{t(g+−g−)[K,S](g+−g−)[K,S]/an}bracketri}ht\n=α0/summationdisplay\np=±/summationdisplay\nq=±sgn(pq)Tr/an}bracketle{tgp[K,S]gq[K,S]/an}bracketri}ht\n=α0/summationdisplay\np=±/summationdisplay\nq=±sgn(pq)βnl\npq. (C2)\nThe symbols Tr and /an}bracketle{t.../an}bracketri}htand the quantities α0,g±,P\nandShavethe samemeaning as in the main text and the\nquantity Ksubstitutes any of the operators (matrices)\nJµorLµ. Note that owing to the symmetric nature of\ntheoriginaldampingtensors,theanalysiscanbeconfined\nto scalar quantities αlocandαnldepending on a general\nsite-diagonal nonrandom operator K. The choice of K=\nKµin (C1) and (C2) produces the diagonal elements of\nboth tensors, whereas the choice of K=Kµ±Kνfor\nµ/ne}ationslash=νleads to all off-diagonal elements. The quantities\nβloc\npqandβnl\npqare expressions of the form\nβloc= Tr/an}bracketle{tg1(P1K−KP2)g2(P2K−KP1)/an}bracketri}ht,\nβnl= Tr/an}bracketle{tg1[K,S]g2[K,S]/an}bracketri}ht, (C3)\nwhere the g1andg2replace the gpandgq, respectively.\nFor an internal consistency of these and following expres-\nsions, we have also introduced P1=P2=P.\nThis supplement contains a proof of the equivalence\nofβlocandβnland, consequently, of αlocandαnl. The\nCPA-average in βnlwith a nonlocal nonrandom torque\nhas been done using the theory by Velick´ y29as worked\nout in detail within the present LMTO formalism by\nCarva et al.30whereas the averaging in βlocinvolving\na local but random torque has been treated using the\napproach by Butler.4412\n2. Auxiliary quantities and relations\nSincethenecessaryformulasoftheCPAinmultiorbital\ntechniques30,44are little transparent, partly owing to the\ncomplicated indices of two-particle quantities, we employ\nhere a formalism with the lattice-site index Rkept but\nwith all orbital indices suppressed.\nThe Hilbert spaceis a sum ofmutually orthogonalsub-\nspaces of individual lattice sites R; the corresponding\nprojectors will be denoted by Π R. A number of rele-\nvant operators are site-diagonal, i.e., they can be written\nasX=/summationtext\nRXR, where the site contributions are given\nbyXR= ΠRX=XΠR= ΠRXΠR. Such operators\nare, e.g., the random potential functions, Pj=/summationtext\nRPj\nR,\nand the nonrandom coherent potential functions Pj=/summationtext\nRPj\nR, wherej= 1,2. The operator Kin (C3) is site-\ndiagonal as well, but its site contributions KRwill not\nbe used explicitly in the following.\nAmong the number ofCPA-relationsfor single-particle\nproperties, we will use the equation of motion for the\naverage auxiliary Green’s functions ¯ gj(j= 1,2),\n¯gj(Pj−S) = (Pj−S)¯gj= 1, (C4)\nas well as the definition of random single-site t-matrices\ntj\nR(j= 1,2) with respect to the effective CPA-medium,\ngiven by\ntj\nR= (Pj\nR−Pj\nR)[1+ ¯gj(Pj\nR−Pj\nR)]−1.(C5)\nThe operators tj\nRare site-diagonal, being non-zero only\nin the subspace of site R. The last definition leads to\nidentities\n(1−t1\nR¯g1)P1\nR=P1\nR+t1\nR(1−¯g1P1\nR),\nP2\nR(1−¯g2t2\nR) =P2\nR+(1−P2\nR¯g2)t2\nR,(C6)\nwhich will be employed below together with the CPA-\nselfconsistency conditions /an}bracketle{ttj\nR/an}bracketri}ht= 0 (j= 1,2).\nFor the purpose of evaluation of the two-particle aver-\nages in (C3), we introduce several nonrandom operators:\nf12= ¯g1K−K¯g2, ζ12= ¯g1[K,S]¯g2,(C7)\nand a site-diagonal operator γ12=/summationtext\nRγ12\nR, where\nγ12=P1K−KP2, γ12\nR=P1\nRK−KP2\nR.(C8)\nBy interchanging the superscripts 1 ↔2 in (C7) and\n(C8), one can also get quantities f21,ζ21,γ21andγ21\nR;\nthis will be implicitly understood in the relations below\nas well. The three operators f12,ζ12andγ12satisfy a\nrelation\nf12+ζ12+¯g1γ12¯g2= 0, (C9)\nwhich can be easily proved from their definitions (C7)\nand (C8) and from the equation of motion (C4). An-\nother quantityto be used in the followingis a nonrandomsite-diagonal operator ϑ12related to the local torque and\ndefined by\nϑ12\nR=/an}bracketle{t(1−t1\nR¯g1)(P1\nRK−KP2\nR)(1−¯g2t2\nR)/an}bracketri}ht,\nϑ12=/summationdisplay\nRϑ12\nR. (C10)\nIts site contributions can be rewritten explicitly as\nϑ12\nR=γ12\nR+/an}bracketle{tt1\nR(f12+ ¯g1γ12\nR¯g2)t2\nR/an}bracketri}ht.(C11)\nThe last relation follows from the definition (C10), from\ntheidentities(C6)andfromtheCPA-selfconsistencycon-\nditions. Moreover,the site contributions ϑ12\nRandγ12\nRsat-\nisfy a sum rule\nγ12\nR=/summationdisplay\nR′′/an}bracketle{tt1\nR¯g1γ12\nR′¯g2t2\nR/an}bracketri}ht+/an}bracketle{tt1\nRζ12t2\nR/an}bracketri}ht+ϑ12\nR,(C12)\nwhere the prime at the sum excludes the term with R′=\nR. This sum rule can be proved by using the definitions\nofζ12(C7) and γ12\nR(C8) and by employing the previous\nrelation for ϑ12\nR(C11) and the equation of motion (C4).\nThe treatment of two-particle quantities requires the\nuse of a direct product a⊗bof two operators aandb.\nThis is equivalent to the concept of a superoperator, i.e.,\na linear mapping defined on the vector space of all linear\noperators. In this supplement, superoperators are de-\nnoted by an overhat, e.g., ˆ m. In the present formalism,\nthe direct product of two operators aandbcan be iden-\ntified with a superoperator ˆ m=a⊗b, which induces a\nmapping\nx/ma√sto→ˆmx= (a⊗b)x=axb, (C13)\nwherexdenotes an arbitrary usual operator. This defi-\nnition leads, e.g., to a superoperator multiplication rule\n(a⊗b)(c⊗d) = (ac)⊗(db). (C14)\nIn the CPA, the most important superoperators are\nˆw12=/summationdisplay\nR/an}bracketle{tt1\nR⊗t2\nR/an}bracketri}ht (C15)\nand\nˆχ12=/summationdisplay\nRR′′\nΠR¯g1ΠR′⊗ΠR′¯g2ΠR(C16)\nwhere the prime at the double sum excludes the terms\nwithR=R′. The quantity ˆ w12represents the irre-\nducible CPA-vertex and the quantity ˆ χ12corresponds to\narestrictedtwo-particlepropagatorwithexcludedon-site\nterms. By using these superoperators, the previous sum\nrule (C12) can be rewritten compactly as\n(ˆ1−ˆw12ˆχ12)γ12= ˆw12ζ12+ϑ12,(C17)\nwhereˆ1 = 1⊗1 denotes the unit superoperator.13\nLet us introduce finally a symbol {x;y}, wherexand\nyare arbitrary operators, which is defined by\n{x;y}= Tr(xy). (C18)\nThissymbolissymmetric, {x;y}={y;x}, linearinboth\narguments and it satisfies the rule\n{(a⊗b)x;y}={x;(b⊗a)y},(C19)\nwhich follows from the cyclic invariance of the trace. An\nobvious consequence of this rule are relations\n{ˆw12x;y}={x; ˆw21y},\n{ˆχ12x;y}={x; ˆχ21y}, (C20)\nwhere ˆw21and ˆχ21are defined by (C15) and (C16) with\nthe superscript interchange 1 ↔2.\n3. Expression with the nonlocal torque\nThe configuration averaging in βnl(C3), which con-\ntains the nonrandom operator [ K,S], leads to two terms\nβnl=βnl,coh+βnl,vc, (C21)\nwhere the coherent part is given by\nβnl,coh= Tr{¯g1[K,S]¯g2[K,S]}(C22)\nand the vertex corrections can be compactly written as30\nβnl,vc={(ˆ1−ˆw12ˆχ12)−1ˆw12ζ12;ζ21},(C23)\nwith all symbols and quantities defined in the previous\nsection. The coherent part can be written as a sum of\nfour terms,\nβnl,coh=βnl,coh\nA+βnl,coh\nB+βnl,coh\nC+βnl,coh\nD,\nβnl,coh\nA= Tr{S¯g1KS¯g2K},\nβnl,coh\nB= Tr{¯g1SK¯g2SK},\nβnl,coh\nC=−Tr{¯g1KS¯g2SK},\nβnl,coh\nD=−Tr{S¯g1SK¯g2K}, (C24)\nwhich can be further modified using the equation of mo-\ntion (C4) and its consequences, e.g., S¯gj=Pj¯gj−1. For\nthe first term βnl,coh\nA, one obtains:\nβnl,coh\nA= Tr{P1¯g1KP2¯g2K}+Tr{KK}\n−Tr{KP2¯g2K}−Tr{P1¯g1KK}.(C25)\nThe last three terms do not contribute to the sum over\nfour pairs of indices ( p,q), where p,q∈ {+,−}, in\nEq. (C2). For this reason, they can be omitted for the\npresent purpose, which yields expressions\n˜βnl,coh\nA= Tr{P1¯g1KP2¯g2K},\n˜βnl,coh\nB= Tr{¯g1P1K¯g2P2K}, (C26)where the second relation is obtained in the same way\nfrom the original term βnl,coh\nB. A similar approach can\nbe applied to the third term βnl,coh\nC, which yields\nβnl,coh\nC=−Tr{¯g1KP2¯g2P2K}\n+Tr{¯g1KP2K}+Tr{¯g1KSK}.(C27)\nThe last term does not contribute to the sum over four\npairs (p,q) in Eq. (C2), which leads to expressions\n˜βnl,coh\nC= Tr{¯g1KP2K}−Tr{¯g1KP2¯g2P2K},\n˜βnl,coh\nD= Tr{P1K¯g2K}−Tr{P1¯g1P1K¯g2K},(C28)\nwhere the second relation is obtained in the same way\nfrom the original term βnl,coh\nD. The sum of all four con-\ntributions in (C26) and (C28) yields\n˜βnl,coh=˜βnl,coh\nA+˜βnl,coh\nB+˜βnl,coh\nC+˜βnl,coh\nD\n= Tr{¯g1KP2K}+Tr{P1K¯g2K}\n+Tr{¯g1γ12¯g2γ21}, (C29)\nwhere weused the operators γ12andγ21defined by (C8).\nThe total quantity βnl(C21) is thus equivalent to\n˜βnl=˜βnl,coh+βnl,vc\n= Tr{¯g1KP2K}+Tr{P1K¯g2K}\n+Tr{¯g1γ12¯g2γ21}+βnl,vc,(C30)\nwhere the tildes mark omission of terms irrelevant for the\nsummation over ( p,q) in Eq. (C2).\n4. Expression with the local torque\nThe configuration averagingin βloc(C3), involving the\nrandom local torque, leads to a sum of two terms:44\nβloc=βloc,0+βloc,1, (C31)\nwhere the term βloc,0is given by a simple lattice sum\nβloc,0=/summationdisplay\nRβloc,0\nR,\nβloc,0\nR= Tr/angbracketleftbig\n¯g1(1−t1\nR¯g1)(P1\nRK−KP2\nR)\nׯg2(1−t2\nR¯g2)(P2\nRK−KP1\nR)/angbracketrightbig\n,(C32)\nsee Eq. (76) of Ref. 44, and the term βloc,1can be written\nin the present formalism as\nβloc,1={ˆχ12(ˆ1−ˆw12ˆχ12)−1ϑ12;ϑ21},(C33)\nwhich corresponds to Eq. (74) of Ref. 44. The definitions\nof ˆw12and ˆχ12aregivenby(C15)and(C16), respectively,\nand ofϑ12andϑ21by (C10).\nThe quantity βloc,0\nR(C32) gives rise to four terms,\nβloc,0\nR=QR,A+QR,B+QR,C+QR,D, (C34)\nQR,A= Tr/an}bracketle{t¯g1(1−t1\nR¯g1)P1\nRK¯g2(1−t2\nR¯g2)P2\nRK/an}bracketri}ht,\nQR,B= Tr/an}bracketle{tP1\nR¯g1(1−t1\nR¯g1)KP2\nR¯g2(1−t2\nR¯g2)K/an}bracketri}ht,\nQR,C=−Tr/an}bracketle{tP1\nR¯g1(1−t1\nR¯g1)P1\nRK¯g2(1−t2\nR¯g2)K/an}bracketri}ht,\nQR,D=−Tr/an}bracketle{t¯g1(1−t1\nR¯g1)KP2\nR¯g2(1−t2\nR¯g2)P2\nRK/an}bracketri}ht,14\nwhich will be treated separately. The term QR,Acan\nbe simplified by employing the identities (C6) and the\nCPA-selfconsistency conditions. This yields:\nQR,A=UR,A+VR,A, (C35)\nUR,A= Tr{¯g1P1\nRK¯g2P2\nRK},\nVR,A= Tr/an}bracketle{t¯g1t1\nR(1−¯g1P1\nR)K¯g2t2\nR(1−¯g2P2\nR)K/an}bracketri}ht\n=VR,A1+VR,A2+VR,A3+VR,A4,\nVR,A1= Tr/an}bracketle{t¯g1t1\nRK¯g2t2\nRK/an}bracketri}ht,\nVR,A2= Tr/an}bracketle{t¯g1t1\nR¯g1P1\nRK¯g2t2\nR¯g2P2\nRK/an}bracketri}ht,\nVR,A3=−Tr/an}bracketle{t¯g1t1\nR¯g1P1\nRK¯g2t2\nRK/an}bracketri}ht,\nVR,A4=−Tr/an}bracketle{t¯g1t1\nRK¯g2t2\nR¯g2P2\nRK/an}bracketri}ht.\nA similar procedure applied to QR,Byields:\nQR,B=UR,B+VR,B, (C36)\nUR,B= Tr{P1\nR¯g1KP2\nR¯g2K},\nVR,B= Tr/an}bracketle{t(1−P1\nR¯g1)t1\nR¯g1K(1−P2\nR¯g2)t2\nR¯g2K/an}bracketri}ht\n=VR,B1+VR,B2+VR,B3+VR,B4,\nVR,B1= Tr/an}bracketle{tt1\nR¯g1Kt2\nR¯g2K/an}bracketri}ht,\nVR,B2= Tr/an}bracketle{tP1\nR¯g1t1\nR¯g1KP2\nR¯g2t2\nR¯g2K/an}bracketri}ht,\nVR,B3=−Tr/an}bracketle{tt1\nR¯g1KP2\nR¯g2t2\nR¯g2K/an}bracketri}ht,\nVR,B4=−Tr/an}bracketle{tP1\nR¯g1t1\nR¯g1Kt2\nR¯g2K/an}bracketri}ht.\nThe term QR,Crequires an auxiliary relation\nP1\nR¯g1(1−t1\nR¯g1)P1\nR=P1\nR(¯g1P1\nR−1)\n+P1\nR−(1−P1\nR¯g1)t1\nR(1−¯g1P1\nR),(C37)\nthat follows from a repeated use of the identities (C6).\nThis relation together with the CPA-selfconsistency lead\nto the form:\nQR,C=UR,C+VR,C, (C38)\nUR,C= Tr{P1\nR(1−¯g1P1\nR)K¯g2K}\n−Tr/an}bracketle{tP1\nRK¯g2(1−t2\nR¯g2)K/an}bracketri}ht,\nVR,C=−Tr/an}bracketle{t(1−P1\nR¯g1)t1\nR(1−¯g1P1\nR)K¯g2t2\nR¯g2K/an}bracketri}ht\n=VR,C1+VR,C2+VR,C3+VR,C4,\nVR,C1=−Tr/an}bracketle{tt1\nRK¯g2t2\nR¯g2K/an}bracketri}ht,\nVR,C2=−Tr/an}bracketle{tP1\nR¯g1t1\nR¯g1P1\nRK¯g2t2\nR¯g2K/an}bracketri}ht,\nVR,C3= Tr/an}bracketle{tt1\nR¯g1P1\nRK¯g2t2\nR¯g2K/an}bracketri}ht,\nVR,C4= Tr/an}bracketle{tP1\nR¯g1t1\nRK¯g2t2\nR¯g2K/an}bracketri}ht.\nA similar procedure applied to QR,Dyields:\nQR,D=UR,D+VR,D, (C39)\nUR,D= Tr{¯g1KP2\nR(1−¯g2P2\nR)K}\n−Tr/an}bracketle{t¯g1(1−t1\nR¯g1)KP2\nRK/an}bracketri}ht,\nVR,D=−Tr/an}bracketle{t¯g1t1\nR¯g1K(1−P2\nR¯g2)t2\nR(1−¯g2P2\nR)K/an}bracketri}ht\n=VR,D1+VR,D2+VR,D3+VR,D4,\nVR,D1=−Tr/an}bracketle{t¯g1t1\nR¯g1Kt2\nRK/an}bracketri}ht,\nVR,D2=−Tr/an}bracketle{t¯g1t1\nR¯g1KP2\nR¯g2t2\nR¯g2P2\nRK/an}bracketri}ht,\nVR,D3= Tr/an}bracketle{t¯g1t1\nR¯g1KP2\nR¯g2t2\nRK/an}bracketri}ht,\nVR,D4= Tr/an}bracketle{t¯g1t1\nR¯g1Kt2\nR¯g2P2\nRK/an}bracketri}ht,Let us focus now on U-terms in Eqs. (C35 – C39). The\nsecond terms in UR,C(C38) and UR,D(C39) do not con-\ntribute to the sum over four pairs ( p,q) in Eq. (C1),\nso that the original UR,CandUR,Dcan be replaced by\nequivalent expressions\n˜UR,C= Tr{P1\nR(1−¯g1P1\nR)K¯g2K},\n˜UR,D= Tr{¯g1KP2\nR(1−¯g2P2\nR)K}.(C40)\nThe sum of all U-terms for the site Ris then equal to\n˜UR=UR,A+UR,B+˜UR,C+˜UR,D\n= Tr{P1\nRK¯g2K}+Tr{¯g1KP2\nRK}\n+Tr{¯g1γ12\nR¯g2γ21\nR}, (C41)\nwhereγ12\nRandγ21\nRare defined in (C8), and the lattice\nsum of all U-terms can be written as\n/summationdisplay\nR˜UR= Tr{P1K¯g2K}+Tr{¯g1KP2K}\n+/summationdisplay\nRTr{¯g1γ12\nR¯g2γ21\nR}. (C42)\nThe summation of V-terms in Eqs. (C35 – C39) can be\ndone in two steps. First, we obtain\nVR,1=VR,A1+VR,B1+VR,C1+VR,D1\n= Tr/an}bracketle{tt1\nRf12t2\nRf21/an}bracketri}ht,\nVR,2=VR,A2+VR,B2+VR,C2+VR,D2\n= Tr/an}bracketle{tt1\nR¯g1γ12\nR¯g2t2\nR¯g2γ21\nR¯g1/an}bracketri}ht,\nVR,3=VR,A3+VR,B3+VR,C3+VR,D3\n= Tr/an}bracketle{tt1\nR¯g1γ12\nR¯g2t2\nRf21/an}bracketri}ht,\nVR,4=VR,A4+VR,B4+VR,C4+VR,D4\n= Tr/an}bracketle{tt1\nRf12t2\nR¯g2γ21\nR¯g1/an}bracketri}ht, (C43)\nwhere the operators f12andf21have been defined in\n(C7). Second, one obtains the sum of all V-terms for the\nsiteRas\nVR=VR,1+VR,2+VR,3+VR,4 (C44)\n= Tr/an}bracketle{tt1\nR(f12+ ¯g1γ12\nR¯g2)t2\nR(f21+ ¯g2γ21\nR¯g1)/an}bracketri}ht.\nThe lattice sums of all U- andV-terms lead to an expres-\nsion equivalent to the original quantity βloc,0(C32):\n˜βloc,0=/summationdisplay\nR˜UR+/summationdisplay\nRVR\n= Tr{P1K¯g2K}+Tr{¯g1KP2K}\n+/summationdisplay\nRTr{¯g1γ12\nR¯g2γ21\nR}\n+/summationdisplay\nRTr/angbracketleftbig\nt1\nR(f12+¯g1γ12\nR¯g2)\n×t2\nR(f21+ ¯g2γ21\nR¯g1)/angbracketrightbig\n,(C45)\nwhere the tildes mark omission of terms not contributing\nto the summation over ( p,q) in Eq. (C1).\nLet us turn now to the contribution βloc,1(C33). It\ncan be reformulated by expressing the quantity ϑ12(and15\nϑ21) in terms of the quantities γ12andζ12(andγ21and\nζ21) from the sum rule (C17) and by using the identities\n(C20). The resultingformcan be written compactlywith\nhelp of an auxiliary operator ̺12(and̺21) defined as\n̺12= ˆχ12γ12+ζ12. (C46)\nThe result is\nβloc,1=βnl,vc+{ˆχ12γ12;γ21}\n−{ˆw12̺12;̺21}, (C47)\nwhere the first term has been defined in (C23). For the\nsecond term in (C47), we use the relation\nˆχ12γ12=/summationdisplay\nRΠR¯g1(γ12−γ12\nR)¯g2ΠR,(C48)\nwhich follows from the site-diagonal nature of the opera-\ntorγ12(C8) and from the definition of the superoperator\nˆχ12(C16). This yields:\n{ˆχ12γ12;γ21}= Tr{¯g1γ12¯g2γ21}\n−/summationdisplay\nRTr{¯g1γ12\nR¯g2γ21\nR}.(C49)\nFor the third term in (C47), only the site-diagonalblocks\nof the operator ̺12(and̺21), Eq. (C46), are needed be-\ncauseofthesite-diagonalnatureofthesuperoperator ˆ w12\n(C15). These site-diagonal blocks are given by\nΠR̺12ΠR= ΠR/bracketleftbig\n¯g1(γ12−γ12\nR)¯g2+ζ12/bracketrightbig\nΠR\n=−ΠR(f12+ ¯g1γ12\nR¯g2)ΠR,(C50)\nwhichfollowsfromthepreviousrelations(C48)and(C9).This yields:\n{ˆw12̺12;̺21}=/summationdisplay\nRTr/angbracketleftbig\nt1\nR(f12+ ¯g1γ12\nR¯g2)\n×t2\nR(f21+ ¯g2γ21\nR¯g1)/angbracketrightbig\n.(C51)\nThe term βloc,1(C47) is then equal to\nβloc,1=βnl,vc+Tr{¯g1γ12¯g2γ21}\n−/summationdisplay\nRTr{¯g1γ12\nR¯g2γ21\nR}\n−/summationdisplay\nRTr/angbracketleftbig\nt1\nR(f12+¯g1γ12\nR¯g2)\n×t2\nR(f21+ ¯g2γ21\nR¯g1)/angbracketrightbig\n.(C52)\nThe total quantity βloc(C31) is thus equivalent to the\nsum of (C45) and (C52):\n˜βloc=˜βloc,0+βloc,1\n= Tr{P1K¯g2K}+Tr{¯g1KP2K}\n+Tr{¯g1γ12¯g2γ21}+βnl,vc,(C53)\nwhere the tildes mark omission of terms irrelevant for the\nsummation over ( p,q) in Eq. (C1).\n5. Comparison of both expressions\nA comparison of relations (C30) and (C53) shows im-\nmediately that\n˜βnl=˜βloc, (C54)\nwhich means that the original expressions βlocandβnl\nin (C3) are identical up to terms not contributing to the\n(p,q)-summations in (C1) and (C2). This proves the ex-\nact equivalence of the Gilbert damping parameters ob-\ntained with the local and nonlocal torques in the CPA.\n∗turek@ipm.cz\n†kudrnov@fzu.cz\n‡drchal@fzu.cz\n1B. Heinrich, in Ultrathin Magnetic Structures II , edited by\nB. Heinrich and J. A. C. Bland (Springer, Berlin, 1994) p.\n195.\n2S. Iihama, S. Mizukami, H. Naganuma, M. Oogane,\nY. Ando, and T. Miyazaki, Phys. Rev. B 89, 174416\n(2014).\n3L. P. Pitaevskii and E. M. Lifshitz, Statistical Physics,\nPart 2: Theory of the Condensed State (Butterworth-\nHeinemann, Oxford, 1980).\n4T. L. Gilbert, Phys. Rev. 100, 1243 (1955).\n5V. Kambersk´ y, Czech. J. Phys. B 26, 1366 (1976).\n6E.ˇSim´ anek and B. Heinrich, Phys. Rev. B 67, 144418\n(2003).7J. Sinova, T. Jungwirth, X. Liu, Y. Sasaki, J. K. Furdyna,\nW. A. Atkinson, and A. H. MacDonald, Phys. Rev. B 69,\n085209 (2004).\n8Y. Tserkovnyak, G. A. Fiete, and B. I. Halperin, Appl.\nPhys. Lett. 84, 5234 (2004).\n9D. Steiauf and M. F¨ ahnle, Phys. Rev. B 72, 064450 (2005).\n10M. F¨ ahnle and D. Steiauf, Phys. Rev. B 73, 184427 (2006).\n11K. Gilmore, Y. U. Idzerda, and M. D. Stiles, Phys. Rev.\nLett.99, 027204 (2007).\n12V. Kambersk´ y, Phys. Rev. B 76, 134416 (2007).\n13A. Brataas, Y. Tserkovnyak, and G. E. W. Bauer, Phys.\nRev. Lett. 101, 037207 (2008).\n14K. Gilmore, Y. U. Idzerda, and M. D. Stiles, J. Appl.\nPhys.103, 07D303 (2008).\n15I. Garate and A. H. MacDonald, Phys. Rev. B 79, 064403\n(2009).16\n16I. Garate and A. H. MacDonald, Phys. Rev. B 79, 064404\n(2009).\n17A. A. Starikov, P. J. Kelly, A. Brataas, Y. Tserkovnyak,\nand G. E. W. Bauer, Phys. Rev. Lett. 105, 236601 (2010).\n18A. Brataas, Y. Tserkovnyak, and G. E. W. Bauer, Phys.\nRev. B84, 054416 (2011).\n19H. Ebert, S. Mankovsky, D. K¨ odderitzsch, and P. J. Kelly,\nPhys. Rev. Lett. 107, 066603 (2011).\n20A. Sakuma, J. Phys. Soc. Japan 81, 084701 (2012).\n21Y. Liu, A. A. Starikov, Z. Yuan, and P. J. Kelly, Phys.\nRev. B84, 014412 (2011).\n22S. Mankovsky, D. K¨ odderitzsch, G. Woltersdorf, and\nH. Ebert, Phys. Rev. B 87, 014430 (2013).\n23H. Ebert, S. Mankovsky, K. Chadova, S. Polesya, J. Min´ ar,\nand D. K¨ odderitzsch, Phys. Rev. B 91, 165132 (2015).\n24I. Barsukov, S. Mankovsky, A. Rubacheva, R. Mecken-\nstock, D. Spoddig, J. Lindner, N. Melnichak, B. Krumme,\nS. I. Makarov, H. Wende, H. Ebert, and M. Farle, Phys.\nRev. B84, 180405 (2011).\n25A. Sakuma, J. Phys. D: Appl. Phys. 48, 164011 (2015).\n26A. Sakuma, J. Appl. Phys. 117, 013912 (2015).\n27I. Turek, J. Kudrnovsk´ y, and V. Drchal, Phys. Rev. B 86,\n014405 (2012).\n28I. Turek, J. Kudrnovsk´ y, and V. Drchal, Phys. Rev. B 89,\n064405 (2014).\n29B. Velick´ y, Phys. Rev. 184, 614 (1969).\n30K. Carva, I. Turek, J. Kudrnovsk´ y, and O.Bengone, Phys.\nRev. B73, 144421 (2006).\n31A.Gonis, Green Functions for Ordered and Disordered Sys-\ntems(North-Holland, Amsterdam, 1992).\n32J. Zabloudil, R. Hammerling, L. Szunyogh, and P. Wein-\nberger, Electron Scattering in Solid Matter (Springer,\nBerlin, 2005).\n33H.Ebert, D. K¨ odderitzsch, andJ. Min´ ar, Rep.Prog. Phys.\n74, 096501 (2011).\n34M. E. Rose, Relativistic Electron Theory (John Wiley &\nSons, New York, 1961).\n35P. Strange, Relativistic Quantum Mechanics (Cambridge\nUniversity Press, 1998).\n36K. Carva and I. Turek, Phys. Rev. B 80, 104432 (2009).\n37I. V. Solovyev, A. I. Liechtenstein, V. A. Gubanov, V. P.\nAntropov, and O. K. Andersen, Phys. Rev. B 43, 14414\n(1991).\n38A.B. Shick, V.Drchal, J. Kudrnovsk´ y, andP.Weinberger,\nPhys. Rev. B 54, 1610 (1996).\n39I. Turek, V. Drchal, J. Kudrnovsk´ y, M. ˇSob, and P. Wein-\nberger,Electronic Structure of Disordered Alloys, Surfacesand Interfaces (Kluwer, Boston, 1997).\n40I. Turek, J. Kudrnovsk´ y, V. Drchal, L. Szunyogh, and\nP. Weinberger, Phys. Rev. B 65, 125101 (2002).\n41O. K. Andersen and O. Jepsen, Phys. Rev. Lett. 53, 2571\n(1984).\n42O. K. Andersen, Z. Pawlowska, and O. Jepsen, Phys. Rev.\nB34, 5253 (1986).\n43I. Turek, J. Kudrnovsk´ y, and V. Drchal, in Electronic\nStructure and Physical Properties of Solids , Lecture Notes\ninPhysics, Vol.535,editedbyH.Dreyss´ e(Springer, Berli n,\n2000) p. 349.\n44W. H. Butler, Phys. Rev. B 31, 3260 (1985).\n45See Supplemental Material in Appendix C for a proof of\nequivalence of the local and nonlocal torques in the CPA.\n46I. Turek, J. Kudrnovsk´ y, and K. Carva, Phys. Rev. B 86,\n174430 (2012).\n47J. Kudrnovsk´ y, V. Drchal, and I. Turek, Phys. Rev. B 89,\n224422 (2014).\n48S. H. Vosko, L. Wilk, and M. Nusair, Can. J. Phys. 58,\n1200 (1980).\n49M. Oogane, T. Wakitani, S. Yakata, R. Yilgin, Y. Ando,\nA. Sakuma, andT. Miyazaki, Jpn. J. Appl. Phys. 45, 3889\n(2006).\n50F. Schreiber, J. Pflaum, Z. Frait, T. M¨ uhge, and J. Pelzl,\nSolid State Commun. 93, 965 (1995).\n51I. Turek, J. Kudrnovsk´ y, V. Drchal, and P. Weinberger,\nPhys. Rev. B 49, 3352 (1994).\n52J. Banhart, H. Ebert, and A. Vernes, Phys. Rev. B 56,\n10165 (1997).\n53R. J. Weiss and A. S. Marotta, J. Phys. Chem. Solids 9,\n302 (1959).\n54S. Mizukami, S. Iihama, N. Inami, T. Hiratsuka, G. Kim,\nH.Naganuma, M.Oogane, andY.Ando,Appl.Phys.Lett.\n98, 052501 (2011).\n55V. Korenman and R. E. Prange, Phys. Rev. B 6, 2769\n(1972).\n56M. dos Santos Dias, B. Schweflinghaus, S. Bl¨ ugel, and\nS. Lounis, Phys. Rev. B 91, 075405 (2015).\n57S. Lounis, M. dos Santos Dias, and B. Schweflinghaus,\nPhys. Rev. B 91, 104420 (2015).\n58A.ManchonandS.Zhang,Phys.Rev.B 79,094422(2009).\n59F. Freimuth, S. Bl¨ ugel, and Y. Mokrousov, Phys. Rev. B\n90, 174423 (2014).\n60O.K.Andersen, O.Jepsen, andD.Gl¨ otzel, in Highlights of\nCondensed-Matter Theory , edited by F. Bassani, F. Fumi,\nand M. P. Tosi (North-Holland, New York, 1985) p. 59."    },    {        "title": "1510.06793v1.Laser_induced_THz_magnetization_precession_for_a_tetragonal_Heusler_like_nearly_compensated_ferrimagnet.pdf",        "content": "arXiv:1510.06793v1  [cond-mat.mtrl-sci]  23 Oct 2015Laser-induced THz magnetization precession for a tetragon al Heusler-like nearly\ncompensated ferrimagnet\nS. Mizukami,1,a)A. Sugihara,1S. Iihama,2Y. Sasaki,2K. Z. Suzuki,1and T. Miyazaki1\n1)WPI Advanced Institute for Materials Research, Tohoku Univ ersity,\nSendai 980-8577, Japan\n2)Department of Applied Physics, Tohoku University, Sendai 9 80-8579,\nJapan\n(Dated: 23 July 2018)\nLaser-inducedmagnetizationprecessional dynamicswasinvestiga tedinepitaxialfilms\nof Mn 3Ge, which is a tetragonal Heusler-like nearly compensated ferrimag net. The\nferromagnetic resonance (FMR) mode was observed, the preces sion frequency for\nwhich exceeded 0.5 THz and originated from the large magnetic anisot ropy field of\napproximately 200 kOe for this ferrimagnet. The effective damping c onstant was\napproximately 0.03. The corresponding effective Landau-Lifshitz c onstant of approx-\nimately 60 Mrad/s and is comparable to those of the similar Mn-Ga mate rials. The\nphysical mechanisms for the Gilbert damping and for the laser-induc ed excitation of\nthe FMR mode were also discussed in terms of the spin-orbit-induced damping and\nthe laser-induced ultrafast modulation of the magnetic anisotropy , respectively.\na)Electronic mail: mizukami@wpi-aimr.tohoku.ac.jp\n1Among the various types of magnetization dynamics, coherent mag netization precession,\ni.e.,ferromagneticresonance(FMR),isthemostfundamentaltype, andplaysamajorrolein\nrf spintronics applications based on spin pumping1–5and the spin-transfer-torque (STT).6,7\nSpin pumping is a phenomenon through which magnetization precessio n generates dc and rf\nspin currents in conductors that are in contact with magnetic films. The spin current can be\nconverted into anelectric voltage throughthe inverse spin-Hall eff ect.8The magnitude of the\nspin current generatedvia spinpumping is proportionaltothe FMRf requency fFMR;4,5thus,\nthe output electric voltage is enhanced with increased fFMR. In the case of STT oscillators\nand diodes, the fFMRvalue for the free layer of a given magnetoresistive devices primarily\ndetermines the frequency range for those devices.9,10An STT oscillator and diode detector\nat a frequency of approximately 40 GHz have already been demonst rated;11–13therefore, one\nof the issues for consideration as regards practical applications is the possibility of increasing\nfFMRto hundreds of GHz or to the THz wave range (0.1-3 THz).11,14\nOnesimple methodthroughwhich fFMRcanbeincreased utilizes magneticmaterials with\nlarge perpendicular magnetic anisotropy fields Heff\nkand small Gilbert damping constants\nα.13,15,16This is because fFMRis proportional to Heff\nkand, also, because the FMR quality\nfactor and critical current of an STT-oscillator are inversely and d irectly proportional to α,\nrespectively. The Heff\nkvalue is determined by the relation Heff\nk= 2Ku/Ms−4πMsfor thin\nfilms, where KuandMsare the perpendicular magnetic anisotropy constant and saturat ion\nmagnetization, respectively. Thus, materials with a small Ms, largeKu, and low αare\nvery favorable; these characteristics are similar to those of mate rials used in the free layers\nof magnetic tunnel junctions integrated in gigabit STT memory applic ation.17We have\npreviously reported that the Mn-Ga metallic compound satisfies the above requirements,\nand that magnetization precession at fFMRof up to 0.28 THz was observed in this case.18\nA couple of research groups have studied magnetization precessio n dynamics in the THz\nwave range for the FePt films with a large Heff\nk, and reported an αvalue that is a factor of\nabout 10 larger than that of Mn-Ga.19–21Thus, it is important to examine whether there are\nmaterialsexhibiting properties similartothoseofMn-Gaexist, inorde r tobetter understand\nthe physics behind this behavior.\nIn this letter, we report on observed magnetization precession at fFMRof more than 0.5\nTHz for an epitaxial film of a Mn 3Ge metallic compound. Also, we discuss the relatively\nsmall observed Gilbert damping. Such THz-wave-range dynamics ca n be investigated by\n2means of a THz wave22or pulse laser. Here, we use the all-optical technique proposed\npreviously;23therefore, the mechanism of laser-induced magnetization preces sion is also dis-\ncussed, because this is not very clearly understood.\nMn3Ge has a tetragonal D0 22structure, and the lattice constants are a= 3.816 and\nc= 7.261˚A in bulk materials [Fig. 1(a)].24,25The Mn atoms occupy at two non-equivalent\nsites in the unit-cell. The magnetic moment of Mn I(∼3.0µB) is anti-parallel to that of\nMnII(∼1.9µB), because of anti-ferromagnetic exchange coupling, and the net magnetic\nmoment is ∼0.8µB/f.u. In other words, this material is a nearly compensated ferrima gnet\nwith a Curie temperature Tcover 800 K.26The tetragonal structure induces a uniaxial\nmagnetic anisotropy, where the c-axis is the easy axis.24The D0 22structure is identical to\nthat of tetragonally-distorted D0 3, which is a class similar to the L2 1Heusler structure;\nthus, D0 22Mn3Ge is also known as a tetragonal Heusler-like compound, as is Mn 3Ga.27\nThe growth of epitaxial films of D0 22Mn3Ge has been reported quite recently, with these\nfilms exhibiting a large Kuand small Ms, similar to Mn-Ga.28–30Note that Mn 3Ge films\nwith a single D0 22phase can be grown for near stoichiometric compositions.29,30Further, an\nextremely large tunnel magnetoresistance is expected in the magn etic tunnel junction with\nMn3Ge electrodes, owing to the fully spin-polarized energy band with ∆ 1symmetry and the\nBloch wave vector parallel to the c-axis at the Fermi level.29,31These properties constitute\nthe qualitative differences between the Mn 3Ge and Mn 3Ga compounds from the material\nperspective.\nAll-optical measurement for the time-resolved magneto-optical K err effect was employed\nusing a standard optical pump-probe setup with a Ti: sapphire laser and a regenerative\namplifier. The wavelength and duration of the laser pulse were appro ximately 800 nm and\n150 fs, respectively, while the pulse repetition rate was 1 kHz. The p ulse laser beam was\ndivided into an intense pump beam and a weaker probe beam; both bea ms weres-polarized.\nThe pump beam was almost perpendicularly incident to the film surface , whereas the angle\nof incidence of the probe beam was ∼6◦with respect to the film normal [Fig. 1(b)].\nBoth laser beams were focused on the film surface and the beam spo ts were overlapped\nspatially. The probe and pump beams had spot sizes with 0.6 and 1.3 mm, respectively.\nThe Kerr rotation angle of the probe beam reflected at the film surf ace was analyzed using\na Wollaston prism and balanced photodiodes. The pump beam intensity was modulated\nby a mechanical chopper at a frequency of 360 Hz. Then, the volta ge output from the\n3FIG. 1. (a) Illustration of D0 22crystal structure unit cell for Mn 3Ge. (b) Diagram showing\ncoordinate system used for optical measurement and ferroma gnetic resonance mode of magnetiza-\ntion precession. The net magnetization (= MII−MI) precesses about the equilibrium angle of\nmagnetization θ, whereMI(MII) is the magnetization vector for the Mn I(MnII) sub-lattice. (b)\nOut-of-plane normalized hysteresis loop of the Kerr rotati on angle φkmeasured for the sample.\nphotodiodes was detected using a lock-in amplifier, as a function of d elay time of the pump-\nprobe laser pulses. The pump pulse fluence was ∼0.6 mJ/cm2. Note that the weakest\npossible fluence was used in order to reduce the temperature incre ase while maintaining the\nsignal-to-noise ratio. A magnetic field Hof 1.95 T with variable direction θHwas applied\nusing an electromagnet [Fig. 1 (b)].\nThec-axis-oriented Mn 3Ge epitaxial films were grown on a single-crystalline (001) MgO\nsubstrate with a Cr seed layer, and were capped with thin MgO/Al lay ers at room tempera-\nture using a sputtering method with a base pressure below 1 ×10−7Pa. The characteristics\nof a 130-nm-thick film with slightly off-stoichiometric composition (74 a t% Mn) deposited\nat 500◦C are reported here, because this sample showed the smallest coer civity (less than\n1 T) and the largest saturation magnetization (117 emu/cm3) of a number of films grown\nwith various thicknesses, compositions, and temperatures. Thes e properties are important\nto obtaining the data of time-resolved Kerr rotation angle φkwith a higher signal-to-noise\nratio, because, as noted above, Mn 3Ge films have a large perpendicular magnetic anisotropy\n4field and a small Kerr rotation angle.30Figure 1(c) displays an out-of-plane hysteresis loop\nofφkobtained for a sample without pump-beam irradiation. The loop is norm alized by the\nsaturation value φk,sat 1.95 T. The light skin depth is considered to be about 30 nm for the\nemployed laser wavelength, so that the φkvalue measured using the setup described above\nwas almost proportional to the out-of-plane component of the ma gnetization Mzwithin the\nlight skin depth depth. The loop shape is consistent with that measur ed using a vibrating\nsample magnetometer, indicating that the film is magnetically homogen eous along the film\nthickness and that value of φk/φk,sapproximates to the Mz/Msvalue.\nFigure 2(a) shows the pump-pulse-induced change in the normalized Kerr rotation angle\n∆φk/φk,s(∆φk=φk−φk,s) as a function of the pump-probe delay time ∆ twith an applied\nmagnetic field Hperpendicular to the film plane. ∆ φk/φk,sdecreases quickly immediately\nafter the pump-laser pulse irradiation, but it rapidly recovers within ∼2.0 ps. This change\nis attributed to the ultrafast reduction and ps restoration of Mswithin the light skin depth\nregion, and is involved in the process of thermal equilibration among t he internal degrees of\nfreedom, i.e., the electron, spin, and lattice systems.32. After the electron system absorbs\nlight energy, the spin temperature increases in the sub-ps timesca le because of the heat\nflow from the electron system, which corresponds to a reduction in Ms. Subsequently, the\nelectron and spin systems are cooled by the dissipation of heat into t he lattices, which have\na high heat capacity. Then, all of the systems reach thermal equilib rium. This process is\nreflected in the ps restoration of Ms. Even after thermal equilibrium among these systems is\nreached, the heat energy remains within the light skin depth region a nd the temperature is\nslightly higher than the initial value. However, this region gradually co ols via the diffusion\nof this heat deeper into the film and substrate over a longer timesca le. Thus, the remaining\nheat causing the increased temperature corresponds to the sma ll reduction of ∆ φk/φk,safter\n∼2.0 ps.\nWith increasing θHfrom out-of-plane to in-plane, a damped oscillation becomes visible\nin the ∆φk/φk,sdata in the 2-12 ps range [Fig. 2(b)]. Additionally, a fast Fourier tran sform\nof this data clearly indicates a single spectrum at a frequency of 0.5- 0.6 THz [Fig. 2(c)].\nThese damped oscillations are attributed to the temporal oscillation ofMz, which reflects\nthe damped magnetization precession,23because the zcomponent of the magnetization\nprecession vector increases with increasing θH. Further, the single spectrum apparent in\nFig. 2(c) indicates that there are no excited standing spin-waves ( such as those observed in\n5thick Ni films), even though the film is thicker than the optical skin de pth.23\nFerrimagnets generally have two magnetization precession modes, i.e., the FMR and\nexchange modes, because of the presence of sub-lattices.33In the FMR mode, sub-lattice\nmagnetization vectors precess while maintaining an anti-parallel dire ction, as illustrated in\nFig. 1(b), such that their frequency is independent of the exchan ge coupling energy between\nthe sub-lattice magnetizations. On the other hand, the sub-lattic e magnetization vectors\nare canted in the exchange mode; therefore, the precession fre quency is proportional to the\nexchange coupling energy between them and is much higher than tha t of the FMR mode.\nAs observed in the case of amorphous ferrimagnets, the FMR mode is preferentially excited\nwhen the pump laser intensity is so weak that the increase in tempera ture is lower than the\nferrimagnet compensation temperature.34No compensation temperature is observed in the\nbulk Mn 3Ge.25,26Also, the temperature increase in this experiment is significantly sma ller\nthanTcbecause the reduction of Msis up to 4 %, as can be seen in Fig. 2(a). Therefore,\nthe observed magnetization precession is attributed to the FMR mo de. Further, as the\nmode excitation is limited to the light skin depth, the amplitude, freque ncy, and etc., for\nthe excited mode are dependent on the film thickness with respect t o the light skin depth.\nThis is because the locally excited magnetization precession propaga tes more deeply into\nthe film as a spin wave in cases where fFMRis in the GHz range.23Note that it is reasonably\nassumed that such a non-local effect is negligible in this study, becau se the timescale of the\ndamped precession discussed here ( ∼1-10 ps) is significantly shorter than that relevant to a\nspin wave with wavelength comparable to the light skin depth ( ∼100 ps).\nThe FMR mode in the THz-wave range is quantitatively examined below. When the ex-\nchangecouplingbetween thesub-latticemagnetizationsissufficient ly strongandthetemper-\nature is well below both Tcand the compensation temperature, the magnetization dynamics\nfor a ferrimagnet can be described using the effective Landau-Lifs hitz-Gilbert equation35\ndm\ndt=−γeffm×/bracketleftbig\nH+Heff\nk(m·z)z/bracketrightbig\n+αeffm×dm\ndt, (1)\nwheremis the unit vector of the net magnetization parallel (anti-parallel) to the magnetiza-\ntion vector MII(MI) for the Mn II(MnI) sub-lattice [Fig. 1(b)]. Here, the spatial change of\nmis negligible, as mentioned above. Heff\nkis the effective value of the perpendicular magnetic\nanisotropyfieldincluding thedemagnetizationfield, even thoughthe demagnetizationfieldis\nnegligibly small for thisferrimagnet (4 πMs= 1.5 kOe). Further, γeffandαeffaretheeffective\n6FIG. 2. Change in Kerr rotation angle ∆ φknormalized by the saturation value φk,sas a function\nof pump-probe delay time ∆ t: (a) for a short time-frame at θH= 0◦and (b) for a relatively long\ntime-frame and different values of θH. The solid curves in (a) and (b) are a visual guide and values\nfitted to the data, respectively. The data in (b) are plotted w ith offsets for clarity. (c) Power\nspectral density as a function of frequency fand magnetic field angle θH.\n7values of the gyromagnetic ratio and the damping constant, respe ctively, which are defined\nasγeff= (MII−MI)/(MII/γII−MI/γI) andαeff= (αIIMII/γII−αIMI/γI)/(MII/γII−MI/γI),\nrespectively, using the gyromagnetic ratio γI(II)and damping constant αI(II)for the sub-\nlattice magnetization of Mn I(II). In the case of Heff\nk≫H,fFMRand the relaxation time of\nthe FMR mode τFMRare derived from Eq. (1) as\nfFMR=γeff/2π/parenleftbig\nHeff\nk+Hz/parenrightbig\n, (2)\n1/τFMR= 2παefffFMR. (3)\nHere,Hzis the normal component of H. Figure 3(a) shows the Hzdependence of the\nprecession frequency fp. This is obtained using the experimental data on the oscillatory\npart of the change in ∆ φk/φk,svia least-square fitting to the damped sinusoidal func-\ntion, ∆φk,p/φk,sexp(−t/τp)sin(2πfp+φp), with an offset approximating the slow change\nof ∆φk/φk,s[solid curves, Fig. 2(b)]. Here, ∆ φk,p/φk,s,τp, andφpare the normalized am-\nplitude, relaxation time, and phase for the oscillatory part of ∆ φk/φk,s, respectively. The\nleast-square fitting of Eq. (2) to the fpvs.Hzdata yields γeff/2π= 2.83 GHz/kOe and\nHeff\nk= 183 kOe [solid line, Fig. 3(a)]. The γeffvalue is close to 2.80 GHz/kOe for the free\nelectron. The value of Heff\nkis equal to the value determined via static measurement (198\nkOe)30within the accepted range of experimental error. Thus, the analy sis confirms that\nthe THz-wave range FMR mode primarily results from the large magne tic anisotropy field in\nthe Mn 3Ge material. The αeffvalues, which are estimated using the relation αeff= 1/2πfpτp\nfollowing Eq. (3), are also plotted in Fig. 3(a). The experimental αeffvalues are indepen-\ndent ofHzwithin the accepted range of experimental error, being in accorda nce with Eq.\n(3); the mean value is 0.03. This value of αefffor D0 22Mn3Ge is slightly larger than the\npreviously reported values for for D0 22Mn2.12Ga (∼0.015) and L1 0Mn1.54Ga (∼0.008).18\nIn the case of metallic magnets, the Gilbert damping at ambient tempe rature is primarily\ncaused by phonon and atomic-disorder scattering for electrons a t the Fermi level in the\nBloch states that are perturbed by the spin-orbit interaction. Th is mechanism, the so-\ncalled Kambersky mechanism,36,37predicts α∝M−1\ns, so that it is more preferable to use\nthe Landau-Lifshitz constants λ(≡αγMs) for discussion of the experimental values of α\nfor different materials. Interestingly, λeff(≡αeffγeffMs) for Mn 3Ge was estimated to be 61\nMrad/s, which is almost identical to the values for D0 22Mn2.12Ga (∼81 Mrad/s) and L1 0\nMn1.54Ga(∼66Mrad/s). The λfortheKamberkymechanism isapproximatelyproportional\n8FIG. 3. (a) Normal component of magnetic field Hdependence on precession frequency fpand\neffective dampingconstant αeffforMn 3Gefilm. (b)Oscillation amplitudeoftheKerrrotation angle\n∆φk,p/φk,scorresponding to the magnetization precession as a functio n of the in-plane component\nofH. The solid line and curve are fit to the data. The dashed line de notes the mean value of αeff.\ntoλ2\nSOD(EF), whereλSOisthespin-orbitinteractionconstant and D(EF)isthetotaldensity\nof states at the Fermi level.37The theoretical values of D(EF) for the above materials are\nroughly identical, because of the similar crystal structures and co nstituent elements, even\nthough the band structures around at the Fermi level differ slight ly, as mentioned at the\nbeginning.18,29Furthermore, the spin-orbit interactions for Ga or Ge, depending on the\natomic number, may not differ significantly. Thus, the difference in αefffor these materials\ncan be understood qualitatively in terms of the Kambersky mechanis m. Further discussion\nbased on additional experiments is required in order to obtain more p recise values for αeff\nand to examine whether other relaxation mechanisms, such as extr insic mechanisms (related\nto the magnetic inhomogeneities), must also be considered.\nFinally, the excitation mechanism of magnetization precession in this s tudy is discussed\nbelow, in the context of a previously proposed scenario for laser-in duced magnetization\n9precession in Ni films.23The initial equilibrium direction of magnetization θis determined\nby thebalance between HandHeff\nk[Fig. 1(b)]. Duringtheperiodinwhich thethree internal\nsystems are not in thermal equilibrium for ∆ t <∼2.0 ps after the pump-laser irradiation\n[Fig. 2(a)], not only the value of Ms, but also the value of the uniaxial magnetic anisotropy,\ni.e.,Heff\nk, is altered. Thus, the equilibrium direction deviates slightly from θand is restored,\nwhich causes magnetization precession. This mechanism may be exam ined by considering\nthe angular dependence of the magnetization precession amplitude . Because the precession\namplitudemaybeproportionaltoanimpulsive torquegeneratedfro mthemodulationof Heff\nk\nin Eq. (1), the torque has the angular dependence |m0×(m0·z)z|, wherem0is the initial\ndirection of the magnetization. Consequently, the z-component of the precession amplitude,\ni.e., ∆φk,p/φk,s, is expressed as ∆ φk,p/φk,s=ζcosθsin2θ∼ζ/parenleftbig\nHx/Heff\nk/parenrightbig2, whereζis the\nproportionalityconstant and Hxisthe in-plane component of H. The experimental values of\n∆φk,p/φk,sare plotted as a function of Hxin Fig. 3(b). The measured data match the above\nrelation, which supports the above-described scenario. Although ζcould be determined via\nthe magnitude and the period of modulation of Heff\nk, it is necessary to consider the ultrafast\ndynamics of the electron, spin, and lattice in the non-equilibrium stat e in order to obtain a\nmore quantitative evaluation;38,39this is beyond the scope of this report.\nIn summary, magnetization precessional dynamics was studied in a D 022Mn3Ge epitaxial\nfilm using an all-optical pump-probe technique. The FMR mode at fFMRup to 0.56 THz\nwas observed, which was caused by the extremely large Heff\nk. A relatively small damping\nconstant of approximately 0.03 was also obtained, and the corresp onding Landau-Lifshitz\nconstant for Mn 3Ge were shown to be almost identical to that for Mn-Ga, being in quali-\ntatively accordance with the prediction of the Kambersky spin-orb it mechanism. The field\ndependence of the amplitude of the laser-induced FMR mode was qua litatively consistent\nwith the model based on the ultrafast modulation of magnetic anisot ropy.\nThis work was supported in part by a Grant-in-Aid for Scientific Rese arch for Young\nResearchers (No. 25600070) and for Innovative Area (Nano Spin Conversion Science, No.\n26103004), NEDO, and the Asahi Glass Foundation.\nREFERENCES\n1R. H. Silsbee, A. Janossy, and P. Monod, Phys. Rev. B 19,4382 (1979).\n102S. Mizukami, Y. Ando, and T. Miyazaki, Jpn. J. Appl. Phys. 40,580 (2001).\n3R. Urban, G. Woltersdorf, and B. Heinrich, Phys. Rev. Lett. 87,217204 (2001).\n4S. Mizukami, Y. Ando, and T. Miyazaki, Phys. Rev. B 66,104413 (2002).\n5Y. Tserkovnyak, A. Brataas, and G. Bauer, Phys. Rev. Lett. 88,117601 (2002).\n6J. C. Slonczewski, J. Magn. Magn. Mater. 159,L1 (1996).\n7L. Berger, Phys. Rev. B 54,9353 (1996).\n8E. Saitoh, M. Ueda, H. Miyajima, and G. Tatara, Appl. Phys. Lett. 88,182509 (2006).\n9S.I. Kiselev, J.C. Sankey, I. N.Krivorotov, N. C.Emley, R.J. Sch oelkopf, R.A. Buhrman,\nand D. C. Ralph, Nature 425,380 (2003).\n10A. A. Tulapurkar, Y. Suzuki, A. Fukushima, H. Kubota, H. Maehara , K. Tsunekawa, D.\nD. Djayaprawira, N. Watanabe, and S. Yuasa, Nature 438,339 (2005).\n11S. Bonetti, P. Muduli, F. Mancoff, and J. Akerman, Appl. Phys. Lett .94,102507 (2009).\n12H. Maehara, H. Kubota, Y. Suzuki, T. Seki, K. Nishimura, Y. Nagamin e, K. Tsunekawa,\nA. Fukushima, H. Arai, T. Taniguchi, H. Imamura, K. Ando, and S. Yu asa, Appl. Phys.\nExpress7,23003 (2014).\n13H. Naganuma, G. Kim, Y. Kawada, N. Inami, K. Hatakeyama, S. Iiham a, K. M. Nazrul\nIslam, M. Oogane, S. Mizukami, and Y. Ando, Nano Lett. 15,623 (2015).\n14M. A. Hoefer, M. J. Ablowitz, B. Ilan, M. R. Pufall, and T. J. Silva, Phy s. Rev. Lett. 95,\n267206 (2005).\n15W. H. Rippard, A. M. Deac, M. R. Pufall, J. M. Shaw, M. W. Keller, S. E. Russek, G. E.\nW. Bauer, and C. Serpico, Phys. Rev. B 81,014426 (2010).\n16T. Taniguchi, H. Arai, S. Tsunegi, S. Tamaru, H. Kubota, and H. Ima mura, Appl. Phys.\nExpress6,123003 (2013).\n17K. Yamada, K. Oomaru, S. Nakamura, T. Sato, and Y. Nakatani, Ap pl. Phys. Lett. 106,\n042402 2015.\n18S. Mizukami, F. Wu, A. Sakuma, J. Walowski, D. Watanabe, T. Kubota , X. Zhang, H.\nNaganuma, M. Oogane, Y. Ando, and T. Miyazaki, Phys. Rev. Lett. 106,117201 (2011).\n19P. He, X. Ma, J. W. Zhang, H. B. Zhao, G. Lpke, Z. Shi, and S. M. Zho u, Phys. Rev. Lett.\n110,077203 (2013).\n20S. Iihama, S. Mizukami, N. Inami, T. Hiratsuka, G. Kim, H. Naganuma, M. Oogane, T.\nMiyazaki, and Y. Ando, Jpn. J. Appl. Phys. 52,073002 (2013).\n21J. Becker, O. Mosendz, D. Weller, A. Kirilyuk, J. C. Maan, P. C. M. Ch ristianen, T.\n11Rasing, and A. Kimel, Appl. Phys. Lett. 104,152412 (2014).\n22M. Nakajima, A. Namai, S. Ohkoshi, and T. Suemoto, Opt. Express 18,18260 (2010).\n23M. van Kampen, C. Jozsa, J. Kohlhepp, P. LeClair, L. Lagae, W. de J onge, and B.\nKoopmans, Phys. Rev. Lett. 88,227201 (2002).\n24G. Kren and E. Kader, Int. J. Magn. 1,143 (1971).\n25T. Ohoyama, J. Phys. Soc. Jpn. 16,1995 (1961).\n26N. Yamada, J. Phys. Soc. Jpn. 59,273 (1990).\n27B. Balke, G.H. Fecher, J. Winterlik, C. Felser, Appl. Phys. Lett. 90,(2007) 152504.\n28H. Kurt, N. Baadji, K. Rode, M. Venkatesan, P. Stamenov, S. San vito, and J. M. D. Coey,\nAppl. Phys. Lett. 101,132410 (2012).\n29S. Mizukami, A. Sakuma, A. Sugihara, T. Kubota, Y. Kondo, H. Tsuc hiura, and T.\nMiyazaki, Appl. Phys. Express 6,123002 (2013).\n30A. Sugihara, S. Mizukami, Y. Yamada, K. Koike, and T. Miyazaki, Appl. Phys. Lett. 104,\n132404 (2014).\n31Y. Miura and M. Shirai, IEEE Trans. Magn. 50,1 (2014).\n32E.Beaurepaire, J.-C.Merle, A.Daunois, andJ.-Y.Bigot, Phys. Rev .Lett.76,4250(1996).\n33R. Wangsness, Phys. Rev. 91,1085 (1953).\n34A. Mekonnen, M. Cormier, A. V. Kimel, A. Kirilyuk, A. Hrabec, L. Rann o, and T. Rasing,\nPhys. Rev. Lett. 107,117202 (2011).\n35M. Mansuripur, The Physical Principles of Magneto-optical Record ing, Cambridge Uni-\nversity Press, Cambridge, UK, 1995.\n36V. Kambersky, Can. J. Phys. 48,2906 (1970).\n37V. Kambersky, Czech. J. Phys. B 34,1111 (1984).\n38B.Koopmans, G.Malinowski, F.DallaLonga,D.Steiauf, M.Fhnle, T.R oth, M.Cinchetti,\nand M. Aeschlimann, Nat. Mater. 9,259 (2010).\n39U. Atxitia, P. Nieves, and O. Chubykalo-Fesenko, Phys. Rev. B 86,104414 (2012).\n12"    },    {        "title": "1511.04227v1.Magnified_Damping_under_Rashba_Spin_Orbit_Coupling.pdf",        "content": "Magnified Damping under Rashba Spin Orbit Coupling  \nNovember 1, 2015  \n \n \n1 \n Magnified Damping  under Rashba Spin Orbit Coupling  \n \nSeng  Ghee  Tan† 1,2 ,Mansoor  B.A.Jalil1,2   \n(1) Data Storage Institute, Agency for Science, Technology and Research (A*STAR)  \n2  Fusionopolis Way , #08-01 DSI , Innovis , Singapore 138634  \n \n(2) Department of Electrical  Engineering, National University of Singapore,  \n             4 Engineering Drive 3, Singapore 117576  \n \n \nAbstract  \nThe spin orbit coupling spin torque consists of the field -like [REF: S.G. Tan et al.,  \narXiv:0705.3502, (2007). ] and the damping -like terms [REF:  H. Kurebayas hi et al., Nature \nNanotechnology 9, 211 (2014). ] that have been widely studied for applications in magnetic \nmemory. We focus , in this article,  not on the spin orbit effect producing the above spin \ntorques, but on its magnifying the damping constant of all field like spin torques. As first \norder precession leads to second order damping, the Rashba constant is naturally co -opted, \nproducing a magnified field -like dam ping effect. The Landau -Liftshitz -Gilbert equations are \nwritten separately for the local magnet ization and the itinerant spin, allowing the \nprogression of magnetization to be self -consistently locked to  the spin.   \n \n  \n \n \n \nPACS: 03.65.Vf, 73.63. -b, 73.43. -f \n† Correspondence author:  \nSeng Ghee Tan  \nEmail: Tan_Seng_Ghee@dsi.a -star.edu.sg  \n Magnified Damping under Rashba Spin Orbit Coupling  \nNovember 1, 2015  \n \n \n2 \n 1. Introduction  \n      In spintronic and magnetic physics, magnetization switching and spin torque [1] have \nbeen well -studied. The advent of the Rashba spin-orbit coupling ( RSOC) [2,3]  due to \ninversion asymmetry at the i nterface of the ferromagnetic/heavy atom (FM/HA) \nheterostructure introduces new spin torque to the FM magnetization. The field -like [4-6] \nand the damping -like [7] SOC spin torque had been  theoretically derived  based on the gauge \nphysics and the Pancharatna m-Berry’s phase , as well as  experimentally verified  and resolved . \nThe numerous  observation s of spin -orbit generation of spin torque [8-10], are all related to \nthe experimental resolutions [6,7] of their field -like and damping -like nature, thus ushering \nin the possibility of spin -orbit based magnetic memory. While the damping -like spin torque  \ndue to Kurebayashi et al. [7] is dissipative in nature, the field -like due to Tan et al. [4,11], is \nnon-dissipative , and precession causing . Recent studies have even mo re clearly \ndemonstrated the physics and application promises of both the field -like and the damping -\nlike SOC spin torque [12-14]. Besides , similar SOC spin torque have also been studied \ntheoretically  in FM/3D -Rashba [15] and FM -topological -insulator materi al [16,17] , and \nexperimentally shown [18, 19 ] in topological insulator materials.  \n      The dissipative physics of all field -like magnetic torque terms have been  derived in \nsecond -order manifestation in a manner introduced by Gilbert  in the 1950 ’s. Conven tional \nstud y of magnetization dynamics is based on a Gilbert damping constant  which is  \nincorporated manually into the Landau -Lifshit z-Gilbert  (LLG)  equation. In this paper, we will \nfocus our attention not so much on the spin-orbit effect producing the SOC spin torque, as \non the spin -orbit effect magnifying the damping constant of all field -like spin torques. As \nfield -like spin torques,  regardless of origin s, generate first-order precession , the Rashba \nconstant will be co -opted in to the second -order damping effect, producing a mag nified \ndamping  constant . On the other hand, c onventional incorporation of the dissipative \ndamping physics into the LLG would fail t o account for the spin-orbit magnification of  the \ndamping strength . It would therefore be necessary to  deriv e the LLG equations from a \nHamiltonian  which  describes electron  due to  the local  FM magnetization (𝒎), and those \nitinerant (𝒔) and injected from external parts . We present a set of modified LLG  equation s \nfor the 𝒎 and the 𝒔. This will be necessary  for a more precise modeling of the 𝒎 trajectory  \nthat simultaneously tracks the  𝒔 trajectory. In summary, the two central themes of this Magnified Damping under Rashba Spin Orbit Coupling  \nNovember 1, 2015  \n \n \n3 \n paper is our presentation of  a self-consistent set of LLG equations under the Rashba SOC \nand the  derivation of the Ra shba -magnified d amping constant in the  second -order damping -\nlike spin torque .  \n \n2. Theory of Magnified Damping  \n    The system under consideration is a FM/HA hetero -structure  with inversion asymmetry \nprovided by the interface. F ree electron denoted by  𝒔, is injected in an in -plane manner into \nthe device . The FM equilibrium  electron  is denoted by  𝒎. One considers the external \nsource -drain bias  to inject  electron  of free-electron  nature 𝒔 into the FM with kinetic, \nscattering , magnetic, and spin -orbit  energ ies.  The Hamiltonian is \n𝐻𝑓=𝑝2\n2𝑚+𝑉𝑖𝑚𝑝𝑠+𝐽𝑠𝑑𝑺.𝑴 +𝜇0𝑴.𝑯𝒂𝒏𝒊+(2(𝜆+𝜆′)\nℏ)(𝒔+𝒎).(𝒑×𝑬𝒕)\n−𝑖(𝜆+𝜆′)(𝒔+𝒎).(∇×𝑬𝒕) \n(1) \nwhere  𝒔,𝒎  have  the unit s of angular momentum i.e.  𝑛ℏ\n2 ,  while 𝑴=(𝑔𝑠𝜇𝐵\nℏ)𝒎 has the unit \nof magnetic moment , and 𝜇𝐵=𝑒ℏ\n2𝑚 is the Bohr magneton . Note that  (2𝜆\nℏ) is the vacuum SOC \nconstant, while  (2𝜆′\nℏ=2𝜂𝑅\nℏ2𝐸𝑖𝑛𝑣) is the Rashba  SOC constant. The SOC part of the Hamiltonian \nillustrates th e simultaneous presence of vacuum and Rashba SOC. The proportion of the \nnumber of electron subject to each coupling would depend on the degree of hybridization.  \nBut s ince 𝜆′≫𝜆, the above can be written with just the Rashba SOC effect. Care is taken t o \nensure  𝜆,𝜆′ share the same dimension of 𝑇𝑒𝑠𝑙 𝑎−1, and  𝑬𝒕 is the total electric field ,  𝐽𝑠𝑑 is \nthe s -d coupling constant, 𝑉𝑖𝑚𝑝𝑠 denotes the spin flip scattering potential, 𝑯𝒂𝒏𝒊 denotes the \naniso tropy field of the FM  material. On the other hand, one needs to be aware that the \nabove is an e xpanded SOC expression that c omprises a momentum part as well as a \ncurvature part [20]. One can then consider  the physics of the electric  curvature as related to  \nthe time dynamic of the spin moment , which bears  a similar origin to the Faraday effect.  In \nthe modern context of  Rashba physics  [21], one considers electron spin to lock to the orbital \nangular momentum  𝑳 due to intrinsic spin orbit coupling at the  atomic level. Due to  broken  Magnified Damping under Rashba Spin Orbit Coupling  \nNovember 1, 2015  \n \n \n4 \n inversion symmetry , electric field (𝑬𝒊𝒏𝒗) points perpendicular to the plane of the FM/HA \nhost . Because of hybridization,  the 𝒔,𝑳,𝒑 of an electron is coupled in a complic ated way by \nthe electric field. In a simple way, one first considers 𝑳 to be  coupled as 𝐻=(2𝜆\nℏ)𝑳.(𝒑×\n𝑬𝒊𝒏𝒗). As spin 𝒔 is coupled via atomic spin orbit locking to 𝑳, an effective coupling of 𝒔 to \n𝑬𝒊𝒏𝒗 can be expected  to occur with strength as determined by the atomic electric field.   We \nwill now take things a step further to make an assumption that 𝒔 is also coupled  via 𝑳  to \nother sources of electric f ields  e.g. those  arising from spin dynamic  (𝒅 𝑴\n𝒅𝒕,𝒅 𝑺\n𝒅𝒕), in the same \nway that it is coupled to 𝑬𝒊𝒏𝒗 . The actual extent of coupling will , however,  be an \nexperimental parameter that measures  the efficiency of Rashba coupling to 𝑬𝒊𝒏𝒗 as opposed \nto electric fields (𝑬𝒎 ,𝑬𝒔) arising due to spin dynamic . The total electric field  in the system \nis now  𝑬𝒕=𝑬𝒊𝒏𝒗 +𝑬𝒎+𝑬𝒔 , where 𝑬𝒎 ,𝑬𝒔 arise due to 𝒅 𝑴\n𝒅𝒕,𝒅 𝑺\n𝒅𝒕, respectively. On the \nmomentum part of the Hamiltonian  2 𝜆′𝒔.(𝒌×𝑬𝒕), we only need to  consider  that \n𝑬𝒕=𝑬𝒊𝒏𝒗 as one can, for simplicity, consider 𝑬𝒎 and  𝑬𝒔 to simply vanish on average. Thus \nin this renewed treatment, the momentum part is : \n2𝜆′\nℏ𝒔.(𝒑×𝑬𝒊𝒏𝒗)=𝜂𝑅𝝈.(𝒌×𝒆𝒊𝒏𝒗) \n(2) \nwhere 𝜂𝑅=𝜆′ℏ𝐸𝑖𝑛𝑣 is the Rashba constant that has been vastly measured  in many material \nsystems  with  experimental values ranging from 0.1 to 2 𝑒𝑉𝐴̇. On the curvature part, one \nconsiders   𝑬𝒕=𝑬𝒎+𝑬𝒔 without the 𝑬𝒊𝒏𝒗 as 𝑬𝒊𝒏𝒗  is spatially uniform and thus would have \nzero curvature.  In summary, the theory of this paper has it that the time -dynamic of the \nspin in a Rashba system produces a curvature part o f  𝑖𝜆′(𝒔+𝒎).(∇×𝑬𝒕). Without the \nRashba effect, this energy term  would just  take on the vacuum constant of (2𝜆\nℏ) instead of \nthe magnified (2𝜆′\nℏ).  The key physics is that in a Rashba FM/HA system , curvature \n𝑖𝜆′(𝒔+𝒎).(∇×𝑬𝒕) is satisfied by the first-order precession due to 𝒅𝑴\n𝒅𝒕,𝒅𝑺\n𝒅𝒕  which provide \nthe electric field curvature  in the form of −𝜇0(1+𝜒𝑚)𝑑 𝑴\n𝑑𝑡=∇×𝑬𝒎 , and  −𝜇0(1+\n𝜒𝑠)𝑑𝑺\n𝑑𝑡=∇×𝑬𝒔 , where we remind reader again that 𝑴,𝑺 have the unit of magnetic \nmoment.  This results in  spin becoming couple d to its own time dynamic, producing a spin -Magnified Damping under Rashba Spin Orbit Coupling  \nNovember 1, 2015  \n \n \n5 \n orbit second -order damping -like spin torque.  The electric field effect is illustrated in Fig. 1 \nbelow:  \n         \n \n \n \n \n \n \n \n \nFig.1 . Magnetic precession under the effect of electric fields due to inv ersion asymmetry, self -dynamic of  𝑑𝑴\n𝑑𝑡 \nand the spin dynamic of  𝑑𝑺\n𝑑𝑡 . Projecting 𝑑𝑀 to the heterostructure surface, one could visualize the emergence \nof an induced electric field in the form of 𝛻𝑋𝐸 in such orientation as to satisfy the law of electromagnetism.  \n \n \n    One notes that the LLG equation is normally derived by letting 𝑺 satisfy the physical \nrequirements of spin transport . One example  of these requirements is assumed and \ndiscussed in REF 1 , with definitions  contained  therein :   \n𝑺(𝒓,𝑡)=𝑆0𝒏+𝜹𝑺 \n𝑱(𝒓,𝑡)=−𝜇𝐵𝑃\n𝑒 𝑱𝒆⊗𝒏−𝐷0∇𝜹𝑺 \n(3) \nwhere 𝒏  is the unit vector of 𝑴, and 𝐷0 is the spin diffusion constant.  Thus 𝑺=𝑺𝟎+𝜹𝑺 \nwould be the total spin density that contains , respectively,  the equilibrium, the non-\nequilibrium adiabatic,  non-adiabatic , and Rashba field -like terms , i.e. 𝜹𝑺=𝜹𝑺𝒂+𝜹𝑺𝒏𝒂+\n𝜹𝑺𝑹.  One notes that 𝑺𝟎 is the equilibrium part of 𝒔 that is aligned to 𝒎, meaning 𝒔𝟎 could  \nexist  in the absence of external field and current in the system. The conditions to satisfy are \nrepresented explicitly by the equations  of:   𝑑𝑀 \n𝐸 𝑓𝑖𝑒𝑙𝑑  𝑑𝑢𝑒 𝑡𝑜 \n𝑖𝑛𝑣𝑒𝑟𝑠𝑖𝑜𝑛  𝑎𝑠𝑦𝑚𝑚𝑒𝑡𝑟𝑦  \n𝛻𝑋𝐸 \n𝑑𝑀 Magnified Damping under Rashba Spin Orbit Coupling  \nNovember 1, 2015  \n \n \n6 \n 𝜕𝜹𝑺 \n𝜕𝑡=0, 𝐷0∇2𝜹𝑺=0,−𝜇𝐵𝑃\n𝑒 𝛁.𝑱𝒆𝑴\n𝑀𝑠=0, 𝑠0𝑴(𝒓,𝑡)\n𝑡𝑓𝑀𝑠=0 \n (4) \nIn the steady state treatment where  𝜕  𝜹𝑺\n𝜕𝑡=0, one recover s the adiabatic component of \n𝜹𝑺𝒂=𝒏×𝒋𝒆.𝛁𝒏 , and the non -adiabatic component of 𝜹𝑺𝒏𝒂=𝒋𝒆.𝛁𝒏. We also take the \nopportunity  here  to reconcile this with  the gauge physics of spin torque, in which case , the \nspin potential 𝐴𝜇𝑠𝑚=𝑒 [𝛼 𝑈𝐸𝑖𝜎𝑗𝜀𝑖𝑗𝜇𝑈†+𝑖ℏ\n𝑒𝑈𝜕𝜇𝑈†] would  correspond , respectively, to  \n𝜹𝑺𝑹+ 𝜹𝑺𝒂. In fact, t he emergent spin p otential [22, 23] can be considered to encapsulate  \nthe physics of electron interaction with the local magnetization under the effect of SOC [4, 5 , \n24-26]. Here  we caution that 𝜹𝒔𝑹 is restricted to the field -like spin -orbit effect  only .  \n    However, in this paper ,  𝑺 is defined to satisfy the transport equations in Eq.(4) except for \n𝜕𝜹𝑺 \n𝜕𝑡=0. Keeping the dynamic property of 𝑺 here allows  a self -consistent equation set \n𝑑𝑺\n𝑑𝑡,𝑑𝑴\n𝑑𝑡 to be introduced . The energy as experienced by the 𝑺,𝑴 electron are, respectively,  \n𝐻𝑓𝑠=𝑺.𝛿𝐻𝑓\n𝛿𝑺 ,       𝐻𝑓𝑚=𝑴.𝛿𝐻𝑓\n𝛿𝑴 \n(5) \nwith caution that  𝐻𝑓𝑠≠𝐻𝑓𝑚 . Upon rearrangement, the 𝒔,𝒎 centric energ ies are, \nrespectively,   \n𝐻𝑓𝑠=(𝑝2\n2𝑚+𝑉𝑖𝑚𝑝𝑠+𝐽𝑠𝑑𝑺.𝑴+𝑺.𝑩𝑹−𝒊𝜆′𝒔.(∇×𝑬𝒕)) \n𝐻𝑓𝑚=(𝐽𝑠𝑑𝑴.𝑺+𝜇0𝑴.𝑯𝒂−𝒊𝜆′𝒎.(∇×𝑬𝒕) ) \n(6) \nwhere 2𝜆′\nℏ𝒔.(𝒑×𝑬𝒕)=𝑺.𝑩𝑹, while  2𝜆′\nℏ𝒎.(𝒑×𝑬𝒕)  vanishes . We particularly note that \nthere have been recent discussions on the field -like [4,6,11 ] spin orbit torque as well as the \ndamping [7] version. With 𝑑𝒔\n𝑑𝑡=−𝟏\n𝒊ℏ[𝒔,𝐻𝑓𝑠] ,𝑑𝒎\n𝑑𝑡=−𝟏\n𝒊ℏ[𝒎,𝐻𝑓𝑚], one would now  have four \ndissipative torque t erms experienced by electron 𝒔,𝒎 as shown below : Magnified Damping under Rashba Spin Orbit Coupling  \nNovember 1, 2015  \n \n \n7 \n ( 𝝉𝑺𝑺 𝝉𝑺𝑴\n𝝉𝑴𝑺 𝝉𝑴𝑴)=𝑖𝜆′𝜇0(𝒔×(1+𝜒𝑠−1)𝑑𝑺\n𝑑𝑡𝒔×(1+𝜒𝑚−1)𝑑𝑴\n𝑑𝑡\n𝒎×(1+𝜒𝑠−1)𝑑𝑺\n𝑑𝑡𝒎×(1+𝜒𝑚−1)𝑑𝑴\n𝑑𝑡) \n(7) \nTo be consistent with conventional necessity to preserve magnetization norm in the physics \nof the LLG equation, we will drop the  off-diagonal terms which are  norm -breaking (non -\nconservation) . This is in order to keep the LLG equation in its conventional norm -conserving  \nform, simplifying physics and calculation therefrom. Nonetheless, the non -conserving parts \nrepresent new dynamic physics that can be analysed in the future with techniques other \nthan the familiar LLG  equations.  The self -consistent pair of spin torque equations in their \nopen forms are: \n𝜕𝑺\n𝜕𝑡=−(𝑺× 𝑩𝑹+𝑺\n𝑡𝑓)−1\n𝑒𝛻𝑎(𝑗𝑎𝒔 𝑺)−(𝑺×𝑴\n𝑚𝑡𝑒𝑥)−𝝉𝑺𝑺 \n𝜕𝑴\n𝜕𝑡=−𝛾𝑴×𝜇0𝑯𝒂−𝑴×𝑺\n𝑚 𝑡𝑒𝑥−𝝉𝑴𝑴  \n(8) \nwhere 𝐽𝑠𝑑=1\n𝑚𝑡𝑒𝑥 has been applied, 𝛾 is the gyromagnetic ratio, 𝜒𝑚 is the susceptibility.  For \nthe stud y of Rashba -magnified damping i n this paper, we only need to keep  the most \nrelevant term which is 𝝉𝑴𝑴=𝑖 𝜂𝑅\nℏ𝐸𝑖𝑛𝑣𝜇0(1+𝜒𝑚−1) 𝒎×𝑑 𝑴\n𝑑𝑡. In the phenomenological physics \nof Gilbert,  the first-order precession leads inevitably to the second -order dissipative terms \nvia  𝒔.𝒅𝑺\n𝒅𝒕 ,𝒎.𝒅𝑴\n𝒅𝒕. But in this paper, the general SOC physics had been expanded  as shown in \nearlier sections,  so that the dissipative terms  are to naturally arise fr om such expansion. The \nadvantage of the non -phenomenological approach is  that, as said earlier, the Rashba \nconstant will be co -opted into the second -order damping effect, resulting in the \nmagnification of the damping constant associated with all field -like spin torque.   \n \n Magnified Damping under Rashba Spin Orbit Coupling  \nNovember 1, 2015  \n \n \n8 \n 3. Conclusion  \n     The im portant result in this paper is that the damping constants have been magnified by \nthe Rashba effect. This would not be possible if the damping constant was incorporated \nmanually by standard means of Gilbert. As the Rashba constant is larger than the vacuum \nSOC constant  as can be deduced from Table 1 and shown below  \n𝛼𝑅=𝛼𝜆′\n𝜆 ,  \n(9) \nmagnetization dynamics in FM/HA hetero -structure with inversion asymmetry (interface, or \nbulk) might have to be modelled with the new equations. It is important to remind  that all \npreviously measured 𝜂𝑅 has had 𝐸𝑖𝑛𝑣 captured in the measured value.  But w hat is needed in \nour study is the coupling of 𝑺 to a dynamic electric field, and that requires the value of just \nthe coupling strength (𝜆′). As most measurement is carr ied out for 𝜂𝑅, the exact knowledge \nof 𝐸𝑖𝑛𝑣 corresponding to a specific 𝜂𝑅 will have a direct impact on the actual value of 𝜆′.  We \nwill, nonetheless, provide a quick, possibly exaggerated estimate. Noting that 𝜆=𝑒ℏ\n4𝑚2𝑐2  \nand 𝜆′=𝜂𝑅\nℏ𝐸𝑖𝑛𝑣, and taking one measured value  of 𝜂𝑅=1×10−10𝑒𝑉𝑚 , corresponding to a \n𝐸𝑖𝑛𝑣=1010𝑉/𝑚, the magnification of 𝛼 works out to 104 times in magnitude , which may \nseem unrealistically strong . The caveat lies  in the exact correspondence of 𝜂𝑅 to 𝐸𝑖𝑛𝑣, which \nremains to be determined experimentally.  For example, if an experimentally determined   \n𝜂𝑅 actual ly corresponds to a much larger 𝐸𝑖𝑛𝑣, that would mean that 𝜆′=𝜂𝑅\nℏ𝐸𝑖𝑛𝑣 which \nmagnifies the damping constant through  𝛼𝑅=𝛼𝜆′\n𝜆 might actually be much lower than \npres ent estimate.  Therefore, it is worth remembering, for simplicity sake  that 𝛼𝑅 actually \ndepends on the ratio of 𝜂𝑅\n𝐸𝑖𝑛𝑣 but not 𝜂𝑅.  It has  also been assumed that  𝑳 couples to  𝑬𝒔,𝑬𝒎 \nwith the same efficiency that it couples to 𝑬𝒊𝒏𝒗.  This is still uncertain as th e Rashba \nconstant with respect to 𝑬𝒔,𝑬𝒎 might actually be lower than  those  𝜂𝑅 values that have \nbeen experimentally measured mostly with respect to 𝑬𝒊𝒏𝒗. Last, we note that as damping \nconstant has been magnified here, and as increasingly high -precision, live monitoring of \nsimult aneous 𝒔,𝒎 evolution is no longer redundant in smaller devices, care has been taken Magnified Damping under Rashba Spin Orbit Coupling  \nNovember 1, 2015  \n \n \n9 \n to present the LLG equations  in the form of a self-consistent  pair of dynamic equations \ninvolving 𝑴 and 𝑺. This will be necessary for the accurate modeling of the simultaneous \ntrajectory of both 𝑴 and 𝑺.   \n \nTable 1. Summary of damping torque and damping con stant with and without Rashba effects.   \n Hamiltonian  Torque  Damping constant  \n1. 𝐻=(2𝜆\nℏ)𝒔.(𝒑×𝑬𝒕) \n𝜆=𝑒ℏ\n4𝑚2𝑐2  \n𝜕𝒎\n𝑑𝑡=𝑖𝜆𝜇0𝒎×(1+𝜒𝑚−1)𝜕𝑴\n𝑑𝑡  \n𝛼=𝑖𝜆\n2𝜇0𝑀𝑠(1+𝜒𝑚−1) \n2. 𝐻𝑅=(2𝜆′\nℏ)𝒔.(𝒑×𝑬𝒕) \n𝜆′=𝜂𝑅\nℏ𝐸𝑖𝑛𝑣  \n𝜕𝒎\n𝑑𝑡=𝑖𝜆′𝜇0𝒎×(1+𝜒𝑚−1)𝑑𝑴\n𝑑𝑡  \n𝛼𝑅=𝑖𝜆′\n2𝜇0𝑀𝑠(1+𝜒𝑚−1) \n    \n \n \n \n \nREFERENCES  \n \n \n[1] S. Zhang & Z. Li, “Roles of Non -equilibrium conduction electrons on the  magnetization dynamics \nof ferromagnets”, Phys. Rev. Letts 93, 127204 (2004).  \n[2] F.T. Vasko, “Spin splitting in the spectrum of two -dimensional electrons due to the surface \npotential”, Pis’ma Zh.  Eksp. Teor. Fiz.  30, 574 (1979) [ JETP Lett. , 30, 541].  \n[3] Y.A. Bychkov  & E.I. Rashba, “Properties of a 2D electron gas with lifted spectral degeneracy”, \nPis’ma Zh. Eksp. Teor. Fiz. , 39, 66 (1984) [ JETP Lett. , 39, 78].  \n[4] S. G. Tan, M. B. A. Jalil, and Xiong -Jun Liu, ”Local spin dynamic arising from the non -perturbative \nSU(2) gauge field of the spin orbit effect”,  arXiv:0705.3502, (2007).  \n[5] K. Obata and G. Tatara, “Current -induced domain wall motion in Rashba spin -orbit system”, Phys. \nRev. B 77, 214429 (2008).  \n[6] Jun Yeon Kim et al., “Layer thickness depende nce of the current -induced effective field vector in  \nTa/CoFeB/MgO”, Nature Materials 12, 240 (2013).   \n[7] H. Kurebayashi et al., “An antidamping spin –orbit torque originating from the Berry curvature”, \nNature Nanotechnology 9, 211 (2014).  \n[8] Ioan Mihai M iron et al., “Current -driven spin torque induced by the Rashba effect in a \nferromagnetic metal layer”, Nat. Mater. 9, 230 (2010).  \n[9] Luqiao Liu et al., “ Spin -Torque Switching with the Giant Spin Hall Effect of Tantalum”, Science 336, \n555 (2012).  \n[10] Xin Fan et al., “Observation of the nonlocal spin -orbital effective field”, Nature Communications \n4, 1799 (2013).  \n[11] S. G. Tan, M. B. A. Jalil, T. Fujita, and X. J. Liu, “Spin dynamics under local gauge fields in chiral \nspin–orbit coupling systems”, Ann. Phy s. (NY) 326, 207 (2011).  Magnified Damping under Rashba Spin Orbit Coupling  \nNovember 1, 2015  \n \n \n10 \n [12] T.I. Mahdi Jamali et al., “Spin -Orbit Torques in Co/Pd Multilayer Nanowires”, Phys. Rev. Letts. \n111, 246602 (2013)  \n[13] Xuepeng Qiu et al., “Angular and temperature dependence of current induced spin -orbit \neffective fields in Ta/CoFeB/MgO nanowires”, Scientific Report 4, 4491 (2014).  \n[14] Junyeon Kim, Jaivardhan Sinha, Seiji Mitani, Masamitsu Hayashi, Saburo Takahashi, Sadamichi \nMaekawa,  Michihiko Yamanouchi, and Hideo Ohno, “Anomalous temperature dependence of \ncurrent -induced  torques in CoFeB/MgO  heterostructures with Ta -based underlayers “, Physical Rev. \nB 89, 174424 (2014) . \n[15] Kazuhiro Tsutsui & Shuichi Murakami, “Spin -torque efficiency enhanced by Rashba spin splitting \nin three dimensions”, Phys. Rev. B 86, 115201 (2012) . \n[16] T. Yokoyama, J. D. Zhang, and N. Nagaosa, “Theoretical study of the dynamics of magnetization \non the topological surface”,  Phys. Rev. B 81, 241410(R) (2010).   \n[17] Ji Chen, Mansoor Bin Abdul Jalil, and Seng Ghee Tan, “Current -Induced Spin Torque on  \nMagnetization Textures Coupled to the Topological Surface States of Three -Dimensional Topological \nInsulators”, J. Phys. Soc. Japan 83, 064710 (2014).  \n[18] Yabin Fan et al., “Magnetization switching through giant spin –orbit torque in a magnetically \ndoped t opological insulator heterostructure”,  Nature Material 13, 700 (2014).  \n[19] A. R. Mellnik at al., “Spin -transfer torque generated by a topological insulator”, Nature 511, 449 \n(2014).  \n[20] M.C. Hickey, J. S. Moodera, “Origin of Intrinsic Gilbert Damping”,  Phys. Rev. Letts. 102, 137601 \n(2009).  \n[21] Jin -Hong Park, Choong H. Kim, Hyun -Woo Lee, and Jung Hoon Han, “Orbital chirality and Rashba \ninteraction in magnetic bands”, Phys. Rev. B 87, 041301(R) (2013)  \n[22] Naoto Nagaosa  & Yoshinori Tokura, “Emergent ele ctromagnetism in solids”, P hys. Scr. T146 \n014020 (2012).  \n[23] Mansoor B. A. Jalil & Seng Ghee Tan, “Robustness of topological Hall effect of nontrivial spin \ntextures”, Scientific Reports 4, 5123 (2014).  \n[24] Y.B. Bazaliy, B.A. Jones, S. -C. Zhang, “Modific ation of the Landau -Lifshitz equation in the \npresence of a spin -polarized current in colossal - and giant -magnetoresistive materials”, Phys. Rev. B \n57 (1998) 3213(R).  \n[25] Takashi Fujita, MBA Jalil, SG Tan, Shuichi Murakami, “Gauge Field in Spintronics”, J.  Appl. Phys. \n[Appl. Phys. Rev.] 110, 121301 (2011).  \n[26] Seng Ghee Tan & Mansoor B.A. Jalil  “Introduction to the Physics of Nanoelectronic”, Woodhead \nPublishing Limited, Cambridge, U.K . Chapter -5 (2012).  \n \n  "    },    {        "title": "1511.04802v1.Determination_of_intrinsic_damping_of_perpendicularly_magnetized_ultrathin_films_from_time_resolved_precessional_magnetization_measurements.pdf",        "content": "1 \n Determination of intrinsic damping of perpendicularly magnetized \nultrathin films from time resolved precessional magnetization \nmeasurements  \n \nAmir Capua1,*, See -hun Yang1, Timothy Phung1, Stuart S. P. Parkin1,2 \n \n1 IBM Research Division, Almaden Research Center, 650 Harry Rd., San Jose,  California \n95120, USA  \n2 Max Planck  Institute for Microstructure Physics, Halle (Saale), D -06120, Germany  \n \n*e-mail: acapua@us.ibm.com  \nPACS number(s) : 75.78. -n \n \n \nAbstract:  \nMagnetization dynamics are strongly influenced by damping, namely the loss of spin \nangular momentum from the magnetic system to the lattice. An “effective” damping \nconstant  αeff is often determined experimentally from the spectral linewidth of the free \ninduction decay of the magnetization after the system  is excited to its non -equilibrium state . \nSuch an  αeff, however, reflects both intrinsic damping as well as inhomogeneous \nbroadening that arises , for example,  from spatial variations of the anisotropy field.  In this \npaper we compare measurements of the m agnetization dynamics in ultrathin non -epitaxial \nfilms having perpendicular magnetic anisotropy using  two different techniques,  time-\nresolved magneto optical Kerr effect (TRMOKE ) and hybrid  optical -electrical  \nferromagnetic resonance  (OFMR) . By using a n external  magnetic field that is applied at \nvery small angles  to the film plane  in the TRMOKE studies , we develop  an explicit closed -\nform analytical expression  for the TRMOKE spectral linewidth and show how this can be \nused to reliably extract the intrinsic Gilbert damping constant. The damping constant \ndetermined in this way is in exc ellent agreement with that determined from the  OFMR \nmethod  on the same samples. Our studies indicate  that the asymptotic high -field approach \nthat is often used in the TRMOKE method to distinguish the  intrinsic damping  from the 2 \n effective damping may result in significant error , because such high external  magnetic \nfields are required  to make this approach valid that they  are out of reach . The error becomes \nlarger the lower is the intrinsic damping  constant, and thus may account for the \nanomalously high damping constants that are  often  reported  in TRMOKE studies . In \nconventional ferromagnetic resonance ( FMR ) studies , inhomogeneous contributions can \nbe readily distinguished from intrinsic damping contributions from the magnetic field \ndependence of the FMR linewidth. Using the analogous approach, w e show how reliable \nvalues of the intrinsic damping can be extracted from  TRMOKE in two distinct magnetic  \nsystems with significant perpendicular magnetic anisotropy: ultrathin CoFeB layers and \nCo/Ni/Co trilayers.  \n \n  3 \n I. Introduction  \nSpintronic nano -devices have been identified in recent years as one of the most \npromisin g emerging technologies for future low power microelectronic circuits1, 2. In the \nheart of the dynamical spin -state transi tion stands the energy loss  parameter of the Gilbert \ndamping . Its accurate dete rmination is of paramount importance  as it determines the \nperformance of key building blocks required for  spin manipulation  such as t he switching  \ncurrent threshold of the spin transfer torq ue magnetic tunnel junction  (MTJ)  used in \nmagnetic random access memory (MRAM) as w ell as the skyrmion velocities  and the \ndomain wall motion current threshold . Up-scaling for high logic and data capacities while \nobtaining stability with high retention energies  require in addition that large  magnetic  \nanisotropies be ind uced. T hese cannot be achieved simply by engineering the geometrical \nasymmetries in the nanometer -scale range , but rather  require harnessing the induced spin-\norbit interaction a t the interface of the  ferromagnet ic film to obtain perpendicular  magnetic  \nanisotropy  (PMA)2. Hence an increasing  effort is invested in the quest for perpendicular \nmagnetized materials  having large anisotropies with low Gilbert damping3-11. \nTwo distinct families of experimental methods  are typically used for measurement \nof Gilbert damping , namely,  time-resolved pump -probe and continuous microwave \nstimulated ferromagnetic resonance ( FMR ), either of which can be implemented using \noptical and/or electrical methods . While in some cases good agreement between these \ndistinct techniques have been reported12, 13, there is often significant disagreement between \nthe methods14, 15.  4 \n When the time resolved pump -probe  method  is implemented using the magneto \noptical Kerr effect ( TRMOKE), a clear advantage over the FMR  method  is gained  in the \nability to  operate at  very high fields and frequencies16, 17. On the other hand , the FMR \nmethod a llows operation over a wide r range of geometrical configurations . The \nfundamental geometrical restriction of the TRMOKE comes from the fact that  the \nmagnetization  precession s are initiated from  the perturbation of the  effective  anisotropy \nfield by the pump  pulse , by momentarily  increasing the lattice temperature18, 19. In cases \nwhere  the torque exerted by the  effective  anisotropy field is n egligible , the pump pulse \ncannot sufficiently perturb the magnetization . Such a case occurs  for example  whenever \nthe magnetization lays in the plane of the sample in uniaxial  thin films having \nperpendicular magneti c anisotropy . Similar limitations exist if the magnetic field i s applied  \nperpendicular to the film . Hence in TRMOKE experiments , the external field is usually \napplied at  angles  typically not smaller than about \n10\n  from  either the film plane or its \nnormal.  This fact has however the consequence that the steady state magnetization \norientation , determined by the bal ancing condition for  the torques , cannot be described  \nusing a n explicit -form algebraic expression , but rather a numerical approach  should be \ntaken5. Alternatively , the dynamics  can be  described using an effective damping  from \nwhich the intrinsic damping , or at least an upper bound o f its value , is estimated at the high \nmagnetic field  limit with the limit being  undetermined . These approaches are hence less \nintuitive while the latter does not indicate directly on the energy losses but rather on the  \ncombination of the energy loss rate , coherence  time of the spin ensemble  and geometry of \nthe measurement .  5 \n In this paper , we present an approach where the TRMOKE system is operated while \napplying the  magnetic field at  very sm all angles with respect to the sample plane. This \nenables  us to use explicit closed -form analytical expressions derived for a perfectly in -\nplane external magnetic field as an approximate  solution. Hence , extraction of the intrinsic \nGilbert damping using an analytical model becomes possible without the need to drive the \nsystem to the high magnetic field limit providing at the same time an intuitive \nunderstanding of the measured responses. The validity of t he method  is verified using a \nhighly sensitive hybrid optical -electrical FMR system (OFMR) capable of operating with \na perfectly in -plane magnetic field where the analytical expressions hold. In particular, we \nbring to test the high -field asymptotic approa ch used for evaluation of the intrinsic damping \nfrom the effective damping and show that in order for it to truly indicate the intrinsic \ndamping, extremely high fields need to be applied. Our analysis reveals the  resonance \nfrequency dispersion relation as well as the inhomogeneous broadening to be the source of \nthis requirement which becomes more difficult  to fulfill the smaller the intrinsic damping \nis. The presented method is applied  on two distinct families of technologically relevant \nperpendicularly mag netized systems; CoFeB4, 6 and Co/Ni/Co20-23. Interestingly, the results \nindicate  that the Ta seed layer  thickness used in  CoFeB  films  strongly affects the intrinsic \ndamping , while t he static characteristics of the  films remain intact . In the Co/Ni/Co trilayer \nsystem which has in contrast a large effective anisotropy field, unexpected ly large spectral \nlinewidth s are measured when  the external  magnetic field is comparable to the  effective  \nanisotropy field, which cannot be explained by the  conventional model  of no n-interacting \nspins  describing the inhomogeneous broadening . This  suggest s that under the low stiffness  6 \n conditions associated with such bias fi elds, cooperative  exchange  interactions, as two \nmagnon scattering, become relevant8, 24.  \nII. EXPERIMENT  \nThe experiments present ed were carried out on three PMA samples: two samples \nconsist ed of Co36Fe44B20 which differed by the thickness of the underlayer  and a third \nsample consisting of Co/Ni/Co trilayer . The CoFeB samples were characterized by low \neffective anisotropy  (Hkeff) values as well as by small distribution of its value  in contrast to \nthe Co /Ni/Co trilayer system . We define here Hkeff as 2Ku/Ms-4πMs where Ku is the \nanisotropy energy constant and Ms being the saturation magnetization.   \nThe structure s of the two CoFeB samples were 50Ta|11CoFeB |11MgO |30Ta, and \n100Ta|11CoFeB |11MgO |30Ta (units are in Å)  and had  similar Ms value s of 1200 emu/cc  \nand Hkeff of 1400 Oe and 1350 Oe respectively.  The t hird system studied was  \n100AlO x|20TaN |15Pt|8Pt 75Bi25|3Co|7Ni|1.5Co |50TaN  with Ms of 600 emu/cc  and Hkeff \nvalue  of about 4200 Oe . All samples were grown on oxidized Si substrates  using DC \nmagnetron sputtering  and exhibited sharp perpendicular switching characteristics . The \nsamples consisting of CoFeB were annealed for  30 min  at \n275\n  C in contrast to the \nCo/Ni/Co which was measured as deposited.  Since the resultant film has a polycrystalline  \ntexture , the in -plane anisotropy is averaged out and the films are regarded as uniaxial \ncrystals with the symmetry axis being perpendicular to the film plane.  7 \n The t wo configurations of the experimental setup were driven by a Ti:Sapphire  laser \nemitting 70 fs pulses at 800 nm  having energy of 6 nJ. In the first configuration a standard  \npolar  pump -probe TRMOKE was implemented with the probe  pulse being a ttenuated by \n15 dB compared to the  pump pulse. Both  beams were focused on the sample to an estimated \nspot size of 10.5 m defined by the full width at half maximum (FWHM) . In the hybrid \noptical -electrical  OFMR system , the Ti:Sapphire laser  served to pro be the magnetization \nstate via  the magneto -optical  Kerr effect after being attenuated  to pulse energies of about \n200 pJ  and was phase -locked with a microwave oscillator in a similar configuration to the \none reported in Ref. [ 25]. For this measurement , the film was patterned into a  20 m x 20 \nm square island with a Au wire  deposited in proximity  to it, which  was driven by the \nmicrowave signal. Prior to reaching the sample, the probing laser beam traversed the \noptical delay line that enabled mapping of the time axis and in particular the  out of plane -\nmz component of the magnetization  as in the  polar TRMOKE  experiment . With this \nconfiguration the OFMR realizes  a conventional FMR system where the magnetization \nstate is read in the time-domain using the magneto optical Kerr effect  and hence its high \nsensitivity . The OFMR  system therefore  enables operation  even when the external field is \napplied fully in the sample plane.  \nIII. RESULTS AND DISCUSSION  \nA. TRMOKE measurements on 50 Å-Ta CoFeB  film  \nThe first experiments we present  were performed  on the 50  Å-Ta CoFeB  system  \nwhich is similar to the one studied in Ref. [4]. The TRMOKE measurement was carried 8 \n out at two angles  of applied magnetic field,  \nH, of \n4\n and \n1\n  measured from the surface \nplane as indicated in Fig. 1. We de fine here in addition the comple mentary angle measured \nfrom the surface normal, \n2HH   . Having i ts origin in the effective anisotropy, the \ntorque  generated by the optical pump  is proportional to \n cos( )sins keffMH   with θ being \nthe angle of the magnetization relative to the normal of the sample plane.  Hence, f or \n1H\n, the angle θ becomes close to \n/2 , and the resultant torque generated by the optical pump \nis not strong enough to  initiate reasonable precessions . For the same reason, the maximum \nfield measureable for the \n1H\n  case is significantly lower than for the \n4H\n  case. This \nis clearly demonstrated in the m easured MOKE signals for the two \nH  angles in Fig. 2 (a). \nWhile for \n4H\n  the precessional motion  is clearly seen even at a bias field of 12 kOe, \nwith \n1H\n  the precessions are hardly observable already at a bias field of 5.5 kOe.  \nAdditionally, it is also possible that the  lower signal to noise ratio observed for  \n1H\n may \nbe due to a breakdown into domains with the almost in -plane applied magnetic field26. \nAfter reduction of the background signal, the measured data can be fitted to a decaying \nsinusoidal response from which t he frequency and decay time can be extracted in the usual \nmanner 6 (Fig. 2(b)) . The measured precession frequency as a function of the applied \nexternal field , H0, is plotted in Fig. 3(a). Significant differences near Hkeff are observed for \nmerely a change of three degrees in the angle of the applied magnetic field . In particular, \nthe trace for \n1H\n  exhibits a minimum point at approximately  Hkeff in contrast to the \nmonotonic behavior of the \n4H\n  case. The theoretical dependence of the resonance 9 \n frequency  on the magnetic  bias field expressed in normalized units, \n/keffH , with \n  \nbeing the resonance angular frequency and \n  the gyromagnetic ratio, is presented in Fig. \n3(b) for several  representative  angles of the applied  field. The resonance frequency at the \nvicinity of Hkeff is very sensitive to slight changes in the angle of the applied field  as \nobserved also in the experiment . Actually the derivative of the resonance frequency with \nrespect to the applied field  at the vicinity of Hkeff is even more sensitive where it diverges \nfor \n90\n  but reaches a value of zero for the slightest angle divergence.  A discrepancy \nbetween the measurement and the theoretical solution exists however.  At field values much \nhigher than Hkeff the precession frequenc y should be identical for all angles (Fig. 3(b)) but \nin practice the resonance frequency measured for \nH  of \n4\n  is consistently higher by nearly \n2 GHz  than at \n1\n . The t heory  also predicts that for the case of \n4\n , the resonance frequencies \nshould  exhibit a minimum point as well which is not observed  in the measurement . The \norigin of the difference is not clear and may be related to the inhomogeneities in the local \nfields  or to the higher orders of the  interface induced anisotropy  which were neglected in \nthe theoretical calculation .  \nIn Fig. 3(c), we plot the effective Lorentzian resonance linewidth in the frequency \ndomain , \neff , defined by \n2/eff eff  with \neff being the measured decay time extracted \nfrom the measured responses. Decompos ing the measured linewidth t o an  intrinsic \ncontribution that represent s the energy loss es upon precession  and an extrinsic contribution \nwhich represent s the inhomogeneities in the local fields and is not related  to energy loss of 10 \n the spin system , we express  the linewidth as : \nint eff IH     . \nint  is given by the \nSmit -Suhl formula27, 28 and equals \n2/  with \n  denoting  the intrinsic spin  precession  decay \ntime where as \nIH  represents the dispersion in the resonance frequencies due to the \ninhomogeneities. If the variations  in the resonance frequency  are assumed to be primarily \ncaused by variations in the local effective anisotrop y field \nkeffH , \nIH  may be given by : \n/IH keff keff d dH H  \n. For the case of \n/2H  or \n0H , \neff  has a closed  \nmathematical  form.  In PMA  films  with bias field applied in the sample plane , the \nexpression  for \neff  becomes : \n      \n0\n002\n00\n0\n0022\n002         for        H\n2\n2     for         Heff keff keff keff\nkeff\nkeff keff\neff keff keff\nkeffkeffHH H H H\nH H H\nHH HH H HHH HH \n      \n\n\n              ,    (1) \nwith \n denot ing the Gilbert damping . The first term s in Eq. (1) stem from  the intrinsic \ndamping , while the second term s stem  from the inhomogeneous broadening . Eq. (1)  shows \nthat while the contribution of the intrinsic part to the total spectral linewidth  is finite, as the \nexternal field approaches  Hkeff either from higher or lower field values, the inhomogeneous \ncontribution diverges. Equation  (1) further shows that for H0 >> Hkeff , the slope of \neff  \nbecomes \n2  with a constant offset given by \n/2keffH . Although Eq. (1) is valid only \nfor\n/2H , it is still  instructi ve to apply it on the measured linewidth for the  \n4H\n  case.  11 \n The theoretical intrinsic linewidth for \n/2H , inhomogeneous contribution and the sum \nof the two a fter fitting \n  and \nkeffH  in the range H 0 > 5000 Oe  are plotted in Fig. 3(c). The \nresul tant fitting values were 0.023 ±0.002 for the Gilbert damping and 175 Oe for \nkeffH . At \nexternal fields comparable to Hkeff the theoretical expression derived for the  \ninhomogeneous broadening for a perfectly in -plane field does not describe properly the \nexperiment . In the theoretical analysis , at fields comparable to Hkeff, the derivative \n0/d dH\ndiverges  and therefore also the derivative \n/keff d dH  as understood from Fig. 3(b).  In the \nexperiment however , \n/2H  and the actual derivative \n/keff d dH  approache s zero. \nHence any variation in Hkeff result s in minor  variation of the frequency . This mean s that the \ncontribution of the inhomogeneous broadening to the total linewidth is suppressed near \nHkeff in the experiment as opposed to being expanded in  the theoretical calculation which \nwas carried out for  \n/2H . The result  is an overestimate d theoretical linewidth  near \nHkeff. After reduction of the inhomogeneous broadening , the extracted  intrinsic measured \nlinewidth is  presented in Fig. 3(c) as well  showing the deviation from  the theor etical \nintrinsic contribution as the field approaches  Hkeff.  \nTo further investigate the e ffect of tilt ing the magnetic field , we study the TRMOKE \nresponses for the \n1H\n  case.  The measured linewidth for this  case is presented in Fig. \n3(d). In contrast to the \n4H\n  case, the measured linewidth now increases at fields near \nHkeff as expected theoretically . Furthermore,  the measured linewidth  for the \n1H\n  case is 12 \n well describe d by Eq. (1) even in the vicinity of Hkeff as well as for bias  fields smaller than \nHkeff. The fitting result s in the same damping value of 0.023 ±0.0015 as with the \n4H\n  \ncase,  and a variation in \nkeffH  of 155 Oe, which is 20 Oe smaller  than the value fitted for \nthe \n4H\n  case.  \nWe next turn to examine  the G ilbert damping. In the absence of the demagnetization \nand crystalline anisotrop y fields, the expression  for the intrinsic Gilbert damping is given \nby:  \n                                                     \n1 .                  (2) \nOnce the anisotropy and the demagnetization field s are included , the expression for the \nintrinsic Gilbert damping becomes :  \n          \n 0\n0\n0\n0\n001                                         for           \n21       for           \n2keff\nkeff\nkeff keffdHHHd\ndHHHd H H H H \n   \n   \n  ,          (3) \nand is valid  only for \n2H  and for  crystals having uniaxial symmetry. At oth er angles \na numerical method5 should be used  to relate the precession decay time to the Gilbert \ndamping. Eq. (3) is merely the intrinsic contribution in Eq. (1)  written in the form \nresembling Eq. (2) . At high fields both Eq s. (2) and (3) converge  to the same result since  13 \n \n1 0dH\nd. As seen in Fig. 3(b), at bias field s comparable to Hkeff the additional derivative \nterm of Eq. (3) becomes very significant . When substituting the measured  decay time,\neff\n, for \n , Eq. (2) gives what is often interpreted as  the “effective ” damping , αeff, from which \nthe intrinsic damping is measured by evaluating it at high fields when the damping becomes  \nasymptotically field independent. Additionally, t he asymptotic limit should be reached \nwith respect to  the inhomogeneous contribution of  Eq. (1). In Fig. 3(e), we plot the effective \ndamping  using \neff  and Eq. (2) . We further show the intrinsic damping value after  \nextracting the intrinsic linewidth  and using Eq. (3). Examining first the effective damping \nvalues, we see that for the two angles , the values are distinctively  different at low fields  \nbut converge at approximately 41 00 Oe  (Beyond  5500 Oe the data  for the \n1H\n  case \ncould not be measured).  In fact , the behavior of the effective damping seems to be related \nto the dependence of the resonance frequency on H0 (Fig. 3(a)) in which  for the \n1H\n  \ncase reaches an extremum while  the \n4H\n  case exhibits a monotonic behavior . Since Eq. \n(2) lacks  the derivative term \n0/dH d , near Hkeff the effective damping is related to the \nGilbert damping by the relation: \n01\neffd\ndH  for H0 > Hkeff. Furthermore, since  \n does not \ndepend on the magnetic field to the first order, the dependence  of the effective damping,\neff , on \nthe bias field stems from the derivative  term \n0 d dH which becomes larger and eventually \ndiverges to infinity  when the magnetic field reaches Hkeff as can be inferred from Fig. 3(b) \nfor the case of \n0H\n  for which Eq. (3) was derived . Hence the increase in \neff at bias 14 \n fields near Hkeff. The same considerations apply also for H0 < Hkeff. As the angle \nH  increases , \nthis analysis becomes  valid only for bias fields which are large enough or small enough \nrelative to Hkeff. When examined  separately, each effective damping trace may give the \nimpression that  at the higher fields  it has become bias field independent and reached its \nasymptotic value from which two very distinct  Gilbert  damping  values of ~0.027 and \n~0.039 are extracted at field values of 12 kOe and 5.5 kOe  for the \n4H\n and \n1H\nmeasurements , respectively . These values are also rather different from the intrinsic \ndamping value of 0.023  extracted using  the analytical model . In contrast  to the effective \ndamping , the intrinsic damping obtained from the analytical model  reveal s a constant and \ncontinuous behavior  which is field and angle independent.  The presumably negative values \nmeasure d for the \n4H\n case stem of course  from the fac t that the expressions in Eqs. (1) \nand (3)  are derived for the  \n2H  case. The error in using the effective damping in \nconjunction with the asymptotic approximation compared to using  the analytical model is \ntherefore 17% and 70% for the \n4H\n and \n1H\n  measurements respectively.  \nIt is important in addition to understand th e conse quence of using Eq. (2) rathe r than \nEq. (3) . In Fig. 3(f) we present the error  in the damping value  after accounting for the \ninhomogeneous broadening using Eq. (2) instead of the complete  expression of  Eq. (3) . As \nexpected , the error increases as the applied field approaches Hkeff. For the measurement \ntaken with \n4H\n  the error is significantly smaller due to the smaller value of the \nderivative\n0/d dH .  15 \n As mentioned previously, i n order to evaluate the intrinsic damping from the total \nmeasured linewidth , the asymptotic limit should be reached with respect to the \ninhomogeneous broadening as well  (Eq. (1) ). In Fig s. 3(c) and 3(d) we see that this is not \nthe case  where the contribution  of the inhomogeneous linewidth is still large compared to \nthe intrinsic l inewidth . Examining Figs. 3(d) and 3(f) for the case of \n1H\n , we see that \nthe overall error of 70% resulting in the asymptotic evaluation stems from both the  \ncontribution of  inhomogeneous broadening as well as from the use of Eq. ( 2) rather than \nEq. (3) while for \n4H\n  (Figs. 3(c) and 3(f)) the error of 17% is solely due to contribution \nof the inhomogeneous broadening which was not as negligible as conceived when applying \nthe asymptotic approximation .  \nB. Comparison of TRMOKE and OFMR measurements in 100  Å-Ta CoFeB  \nfilm \nWe next turn to  study the magnetization dynamics using the OFMR system where \nthe precession s are driven with the microwave signal . Hence, the external magnetic field \ncan be applied perfectly in the sample plane. The 100 Å-Ta CoFeB sample  was used for \nthis experiment. Before patterning the film for the OFMR measurement, a TRMOKE \nmeasurement was carried out at \n4H\n  which  exhibited a similar  behavior to that observed \nwith the sample having  50 Å Ta  as a seeding layer . The  dependence of the  resonance \nfrequency on the magnetic  field as well  as the measured linewidth and its  different \ncontributions  are presented in Figs. 4(a) and 4(b). Before reduction of the inhomogeneous 16 \n broadening the asymptotic effective damping was measured to be ~0.0168 while after \nextraction  of the intrinsic damping a value of 0.0109 ±0.0015 was measured marking a \ndifference of 54%  (Fig. 4(c)). The fitted \nkeffH  was 205 Oe. Fig. 4(b) shows that the origin \nof the error stems from significan t contribution of the inhomogeneous broadening \ncompared to the intrinsi c contribution which plays a mor e significant role when the \ndamping is low.  By us ing the criteria  for the  minimum  field that results in  \n10IH eff   \nto estimat e the point where the asymptotic approximation would be valid , we arrive to a \nvalue of at least 4.6 T which is rather impractic al. The threshold of this minimal f ield is \nhighly dependent on the damping so that for a lower damping an even higher field would \nbe required.  \nAn example of a measured trace using the OFMR system  at a low microwave \nfrequency of 2.5 GHz  is presented in Fig. 4(d). The square root of the magn etization \namplitude  (out of plane mz component)  while preserving its sign  is plotted to show detail . \nThe high sensitivity of the OFMR system enable s operation  at very low frequencies and \nbias fields. For every frequency  and DC  magnetic  field value , several  cycles of the \nmagnetization precession  were recorded  by scanning the optical delay  line. The magnetic \nfield was then swept  to fully capture the resonance . The trace should be examined  \nseparately in two sections, be low Hkeff and above Hkeff (marked in the figure by black dashed \nline). For frequencies of up to \nkeffH  two resonances are crossed as indicated by the guiding \nred dashed line which represents the out -of-phase component of the magnetization, namely \nthe imaginary part of the magnetic susceptibility . Hence the cross section along this line 17 \n gives  the field dependent absorption spectrum  from which the resonance frequency and \nlinewi dth can be identified. This spectrum is show n in Fig. 4( e) together with the fitted \nlorentzian lineshapes  for bias fields below and above Hkeff. The resultant resonance \nfrequencies  of all measurements are plotted in addition in Fig. 4(a).  \nThe resonance linewidth s extracted  for bias fields larger than Hkeff, are presented in \nFig. 4(f). Here the effective magnetic field linewidth , ΔHeff, that includes the contribution \nof the inhomogeneous broadening derived  from the same principles that led to Eq. (1) with \n/2H\n is given by: \n              \n02\n2\n0\n00\n2 2\n0211                     for       2\n4\n2\n                   \n        with       keff\neff keff keff\nkeff\nkeff keff\neff keff\nkeff\nkeffHH H H H\nH\nHH HHHH H H\nHH\n\n\n\n\n\n\n      \n      \n \n       \n 0      for       keff HH   (5).  \nThe second terms in Eq. (5) denote  the contribution of the inhomogeneous broadening , \nIHH\n, and are frequency dependent as opposed to the case where the field is applied out \nof the sample plane9. The dispersion in the effective anisotropy , \nkeffH , and the intrinsic \nGilbert damping  were found by fitting the linewidth in the  seemingly  linear range at \nfrequencies larger th an 7.5 GHz . The contribution s of the intrinsic  and inhomogeneous \nparts  and the ir sum are presented as well in Fig. 4(f).  18 \n It is apparent that the measured linewidth at the lower frequencies is much broader \nthan the theoretical one.  The reason for that lies in the fact that in practice the bias field is \nnot applied perfectly in the sample plane as well as in the fact that there migh t be locally  \ndifferent orientation s of the polycrystalline  grains  due to the natural imperfections of the \ninterfaces that further result  in angle distribution of \nH . Since  the measured  field linewidth \nis a projection of the spectral  linewidth into the  magnetic  field domain, the relation between \nthe frequency and the field  intrinsic  linewidth s is given by: \n1\nint int\n0dHdH   \n . The \nintrinsic linewidth , \nint , in the frequency domain near Hkeff is finite , as easily seen from \nEq. (1)  while the derivative term near Hkeff is zero for even the slightest angle misalignment  \nas already seen. H ence the field-domain  linewidth  diverg es to infinity as observed \nexperimentally. The inhomogeneous broadening component does not diverge in that \nmanner  but is rather suppressed . To show that the excessive linewidth at low field s is \nindeed related to the derivative of \n0/d dH  we empirically multiply the total theoretical \nlinewidth by the factor \n0 / ( )d d H  which  turns out to fit  the data surprisingly well (Fig. \n4(f)). This  is merely a phenomenological qualitative description, and a rigorous description \nshould still be derived.   \nThe fitted linewidth of Fig. 4(f) results in  the intrinsic damping value of \n0.011 ±0.0005  and is identical to the value obtained by the TRMOKE  method . Often \nconcerns regarding the differences between the TRMOKE and FMR measurements such \nas spin wave emission away from the pump laser spot in the TRMOKE29, increase of 19 \n damping due to thermal heating  by the pump pulse  as well as differences in the  nature of \nthe inhomogeneous broadening  are raised. Such effects do not seem to be significant here . \nAdditionally , it is worth noting that s ince the linewidth seems to reach a  linear  dependence \nwith respect to  the field  at high fields , it may be naively fitted using a constant frequency -\nindependent inhomogeneous broadening factor . In that case an underestimated value of \n~0.0096 would have been obtained . The origin of this misinterpretation is seen clearly by \nexamining the  inhomogeneous broadening contribu tion in Fig. 4(f) that show s it as well to \nexhibit a seemingly linear dependence at the high fields. Regarding the  inhomogeneous \nbroadening , the anisotropy field dispersion,  \neffKH , obtained with the TRMOKE was 205 \nOe while the value  obtained from the OFMR system was 169 Oe . Although these values  \nare of the same order of magnitude , the difference is rather significant. It is possible that \nthe discrepancy  is related to the differences in the measurement  techniques. For instance, \nthe fact that both  the pump and probe beams have the same spot size may cause an uneven \nexcitation across the probed region in the case of the TRMOKE measurement  while in the \ncase of the OFMR measurement the amplitude of the microwave field decays at increasin g \ndistances away from the microwire. These effects may  be reflect ed in the measurements as \ninhomogeneous broadening. Nevertheless , the measured intrinsic damping values are \nsimilar.  \nFinally , we compare the effective damping of the OFMR and the TRMOKE \nmeasu rements without correcting for the inhomogeneous broadening in Fig. 4(g). The 20 \n figure shows a deviation in the low field values which  is by now understood to  be unrelated \nto the energy losses  of the system .  \nFurthermore, we observe that the thickness of the Ta underlayer affects the \ndamping. The comparison  of the 50 Å -Ta CoFeB and the 100 Å -Ta CoFeB samples  shows \nthat the increase by merely 50 Å of Ta, reduced significantly the damping while leaving \nthe anisot ropy field unaffected.  \nC. TRMOKE and OFMR measurements in Co /Ni/Co film  \nIn the last set of measurements we study the Co /Ni/Co film which has distinctively \ndifferent static properties compared to the CoFeB samples . The sample was studied  using \nthe TRMOKE setup at two \nH  angles of \n1\n  and \n4\n  and using the OFMR system at\n0H\n. The resultant  resonance frequency traces are depicted in Fig. 5(a). The spectral linewidth \nmeasure d for \n4H\n using  the TRMOKE  setup  is presented in Fig . 5(b). A linear fit at the \nquasi linear high field range results in a large damping value of 0.081 ±0.015 and in a very \nlarge \neffKH  of 630 Oe . The large damping is attributed to the efficient spin pumping into \nPtBi30 layer  having large spin -orbit coupling . When the angle of the  applied  magnetic f ield \nis reduced to  \n1H\n  a clearer picture of the  contribution  of the inhomogeneous broadening \nto the total linewidth is obtained (Fig. 5(c)) revealing  that it cannot explain solely the \nmeasured  spectral  linewidth s. While the theoretical model predicts that  the increase in \nbandwidth spans  a relatively  narrow  field range around Hkeff, the measurement shows an \nincrease over a much larger range around Hkeff. The linewidth broadening originating from 21 \n the anisotropy dispersion was theoretically calculated under the assumption of a small \nperturbation of the resonance frequency. A large \nkeffH  value was measured however  from \nthe TRMOKE measurement taken at \n4H\n . Calculating numerically the exact variation \nof the resonance frequency improved slightly the fit but definitely  did not resolve the \ndiscrepancy  (not presented) . From this fact we understand that  there should be an  additional  \nsource  contributing  to the line broadening  at least near Hkeff. A possible explanation may \nbe related to the low stiffness27 associated with the \n0 keff HH  conditions . Under such \nconditions weaker torques which are usually neglected may become relevant24, 31. These  \ntorques could possibly originate from  dipolar or exchange coupling resulting in two \nmagnon scattering processes  or even in a breakdown into magnetic domains as described \nby Grolier et al.26. From the limited data range at this angle, the damping could not be \nmeasured.  \nThe OFMR system  enabl ed a wider range of fields and frequencies than the  ones \nmeasured with the TRMOKE  for \n1H\n  (Fig. 5(a)). Fig. 5(d) presents t he measured  \nOFMR linewidth . The quasi -linear regime of the linewidth seems to be reached  at \nfrequencies of 12 GHz corresponding to bias field values which are larger than 7500 Oe . \nThe resultant intrinsic damping after fitting to this range was 0.09±0.005  with a \nkeffH  of \n660 Oe  which differ  by approximately 10% from the  values obtained from the  TRMOKE \nmeasurement . The effective measured damping is plotted in Fig. 5(e). The asymptotic \ndamping value , though not fully reached  for this high damping sample , would be about 0.1.  22 \n This represents an error of about 10% which is smaller compared to the errors of 17% and \n54% encountered in the CoFeB samples because of the larger damping of the Co /Ni/Co \nsample.   \nD. Considerations of two -magnon scattering  \nIn general, two -magnon spin wave scattering by impurities may exist in our \nmeasurements  at all field ranges32, 33, not only near Hkeff as suggested in the discussion of \nthe previous section32, 33. The resultant additional linewidth broadening would then be \nregarded as an extr insic contribution to the damping34-36. While in isotropic films which \nexhibit low crystalline anisotropy or in films having in -plane  crystalline anisotropy, two -\nmagnon scattering is maximized when the external field is applied in the film plane, in \nPMA films this is not necessarily the case and the highest efficiency of two -magnon \nscattering may be obtained at some oblique angle35.  \nIn films where two -magnon scatt ering is significant , the measured linewidth should \nexhibit  an additional  nonlinear dependence on the external field which  cannot be accounted \nfor by the present model . In such case, a s trong dependence on the external  field would be  \nobserved  for fields below Hkeff  due to the variation in the  orientation of the magnet ization \nwith the external magnetic field. At higher fields the dependence on the external field is \nexpected to be moderate35.  \nWhile at bias field values below Hkeff our data is relatively limited, at external \nmagnetic fields that are larger than Hkeff, the observed linewidth seems to be described well 23 \n by our model  resulting in a field independent Gilbert damping coefficient . This seems to \nsupport our model that the scattering of spin waves does not have a prominent effect. It is \npossible however that a moderate dependence on the bias field, especially at high field \nvalues, may have been “linearized” and classified as intrinsic damping.  \n \nIV. CONCLUSION  \nIn conclusion, in this paper we studied the time domain magnetization dynamics  in \nnon-epitaxial  thin films  having perpendicular magnetic  anisotropy  using the TRMOKE  and \nOFMR  system s. The analytical model used to interpret the magnetization dynamics from \nthe TRMOKE  responses indicated that the asymptotic high -field approach often used to \ndistinguish the intrinsic damping from the effective damping may result in significant error  \nthat increases the lower the damping is . Two sources for the error  were  identified  while \nvalidity of th e asymptotic  approach was shown to require very high magnetic fields. \nAdditionally, the effective damping was shown to be highly affected by the derivative of \nthe resonance frequency with respect to  the magnetic field  \n0/d dH . The analytical \napproach  developed here was verified by use of the OFMR measurement showing excellent \nagreement whenever the intrinsic damping was compared and ruled out the possibility of \nthermal heating by the laser or  emission of spin waves away from  the probed  area.  \n  24 \n As to the systems studied, a  large impact of the seed layer  on the intrinsic damping \nwith minor effect on the static characteristics  of the CoFeB system was observed  and may \ngreatly aid in engineering the proper materials for the MTJ. Interestingly, the use of the \nanalytical model enabled identification of  an additional exchange torque when low stiffness \nconditions prevailed. While  effort still remains  to understand th e limits on the angle of the \napplied magnetic field to which the analytical solution  is valid , the approach  presented is \nbelieved to accelerate  the discovery of novel materials for new applications .  25 \n Acknowledgments:  \nA.C. thanks the Viterbi foundation and the Feder Family foundation for supporting this \nresearch.  \nReferences:  \n1 Z. Yue, Z. Weisheng, J. O. Klein, K. Wang, D. Querlioz, Z. Youguang, D. Ravelosona, and C. \nChappert, in Design, Automation and Test in Europe Conference and Exhibition (DATE), 2014 , \np. 1. \n2 A. D. Kent and D. C. Worledge, Nat Nano 10, 187 (2015).  \n3 M. Shigemi, Z. Xianmin, K. Takahide, N. Hiroshi, O. Mikihiko, A. Yasuo, and M. Terunobu, \nApplied Physics Express 4, 013005 (2011).  \n4 S. Iihama, S. Mizukami, H. Nagan uma, M. Oogane, Y. Ando, and T. Miyazaki, Physical Review \nB 89, 174416 (2014).  \n5 S. Mizukami, Journal of the Magnetics Society of Japan 39, 1 (2015).  \n6 G. Malinowski, K. C. Kuiper, R. Lavrijsen, H. J. M. Swagten, and B. Koopmans, Applied \nPhysics Letters 94, 102501 (2009).  \n7 A. J. Schellekens, L. Deen, D. Wang, J. T. Kohlhepp, H. J. M. Swagten, and B. Koopmans, \nApplied Physics Letters 102, 082405 (2013).  \n8 J. M. Beaujour, D. Ravelosona, I. Tudosa, E. E. Fullerton, and A. D. Kent, Physical Review B 80, \n180415  (2009).  \n9 J. M. Shaw, H. T. Nembach, and T. J. Silva, Applied Physics Letters 105, 062406 (2014).  \n10 T. Kato, Y. Matsumoto, S. Okamoto, N. Kikuchi, O. Kitakami, N. Nishizawa, S. Tsunashima, \nand S. Iwata, Magnetics, IEEE Transactions on 47, 3036 (2011).  \n11 H.-S. Song, K. -D. Lee, J. -W. Sohn, S. -H. Yang, S. S. P. Parkin, C. -Y. You, and S. -C. Shin, \nApplied Physics Letters 103, 022406 (2013).  \n12 S. S. Kalarickal, P. Krivosik, M. Wu, C. E. Patton, M. L. Schneider, P. Kabos, T. J. Silva, and J. \nP. Nibarger, Journ al of Applied Physics 99, 093909 (2006).  \n13 M. Shigemi, A. Hiroyuki, W. Daisuke, O. Mikihiko, A. Yasuo, and M. Terunobu, Applied \nPhysics Express 1, 121301 (2008).  \n14 I. N. Krivorotov, N. C. Emley, J. C. Sankey, S. I. Kiselev, D. C. Ralph, and R. A. Buhrman , \nScience 307, 228 (2005).  \n15 I. Neudecker, G. Woltersdorf, B. Heinrich, T. Okuno, G. Gubbiotti, and C. H. Back, Journal of \nMagnetism and Magnetic Materials 307, 148 (2006).  \n16 A. Mekonnen, M. Cormier, A. V. Kimel, A. Kirilyuk, A. Hrabec, L. Ranno, and T. Rasing, \nPhysical Review Letters 107, 117202 (2011).  \n17 S. Mizukami, et al., Physical Review Letters 106, 117201 (2011).  \n18 J.-Y. Bigot, M. Vomir, and E. Beaurepaire, Nat Phys 5, 515 (2009).  \n19 B. Koopmans, G. Malinowski, F. Dalla Longa, D. Steiauf, M. Fahnle, T. Roth, M. Cinchetti, and \nM. Aeschlimann, Nat Mater 9, 259 (2010).  \n20 K.-S. Ryu, L. Thomas, S. -H. Yang, and S. Parkin, Nat Nano 8, 527 (2013).  \n21 S. Parkin and S. -H. Yang, Nat Nano 10, 195  (2015).  \n22 K.-S. Ryu, S. -H. Yang, L. Thomas, and S. S. P. Parkin, Nat Commun 5 (2014).  \n23 S.-H. Yang, K. -S. Ryu, and S. Parkin, Nat Nano 10, 221 (2015).  \n24 E. Schlomann, Journal of Physics and Chemistry of Solids 6, 242 (1958).  26 \n 25 I. Neudecker, K. Perzlma ier, F. Hoffmann, G. Woltersdorf, M. Buess, D. Weiss, and C. H. Back, \nPhysical Review B 73, 134426 (2006).  \n26 V. Grolier, J. Ferré, A. Maziewski, E. Stefanowicz, and D. Renard, Journal of Applied Physics \n73, 5939 (1993).  \n27 J. Smit and H. G. Beljers, phili ps research reports 10, 113 (1955).  \n28 H. Suhl, Physical Review 97, 555 (1955).  \n29 Y. Au, M. Dvornik, T. Davison, E. Ahmad, P. S. Keatley, A. Vansteenkiste, B. Van \nWaeyenberge, and V. V. Kruglyak, Physical Review Letters 110, 097201 (2013).  \n30 Y. Tserkovny ak, A. Brataas, and G. E. W. Bauer, Physical Review Letters 88, 117601 (2002).  \n31 C. E. Patton, Magnetics, IEEE Transactions on 8, 433 (1972).  \n32 K. Zakeri, et al., Physical Review B 76, 104416 (2007).  \n33 J. Lindner, K. Lenz, E. Kosubek, K. Baberschke, D. Spoddig, R. Meckenstock, J. Pelzl, Z. Frait, \nand D. L. Mills, Physical Review B 68, 060102 (2003).  \n34 H. Suhl, Magnetics, IEEE Transactions on 34, 1834 (1998).  \n35 M. J. Hurben and C. E. Patton, Journal of Applied Physics 83, 4344 (1998).  \n36 D. L. Mills and S. M. Rezende, in Spin Dynamics in Confined Magnetic Structures II , edited by \nB. Hillebrands and K. Ounadjela (Springer Berlin Heidelberg, 2003).  \n \n \n  27 \n Figure 1  \n \nFIG 1. Illustration of the angles \nH , \nH and \n . M and H0 vectors denote the magnetization \nand external magnetic field , respectively.  \n  \n28 \n Figure 2 \n \nFIG. 2. Measured TRMOKE  responses  at \nH  angles of \n4\n and \n1\n . (a) TRMOKE signal at \nlow and high external magnetic field values. Traces are shifted for clarity. (b) Measured \nmagnetization responses after reduction of background signal (open circles) \n29 \n superimposed  with the fitted decaying sine  wave  (solid lines). Traces are shifted and \nnormalized to have the same peak  amplitude. Data presented for  low and high external \nmagnetic field values.  \n  30 \n Figure 3 \n  \nFIG. 3. TRMOKE measurements at \n4H\n  and \n1H\n . (a) Measured resonance \nfrequency versus magnetic field. (b) Theoretical dependence of resonance frequency on \nmagnetic field  presented in normalized units . (c) & (d)  Measured linewidth (blue) , fitted \ntheoretical con tributions to l inewidth (green, cyan, magenta) and extracted intrinsic \nlinewidth from measurement (red) for \n4H\n  and \n1H\n , respectively. (e) Intrinsic and \neffective damping. (f) Error in damping value when using Eq. (2) instead  of Eq. (3 ).  \n31 \n Figure 4 \n \nFIG. 4. TRMOKE and OFMR measurements at \n4H\n  and \n0H\n , respectively. (a) \nMeasured resonance frequency versus magnetic field. (b)  Measured linewidth (blue), fitted \ntheoretical contributions to linewidth (green, cyan, magenta) and extracted intrinsic \n32 \n linewidth from measurement (red) using the TRMOKE with  \n4H\n . (c) Intrinsic and \neffective damping  using TRMOKE . (d) Representative OFMR trace at 2.5 GHz.  The \nfunction sign( mz)(mz)1/2 is plotted.  (e) Field dependent absorption spectrum  (blue)  \nextracted from the cross section along the red dashed lined of (d) together with fitted \nlorentzian lineshapes (red).  (f) Measured linewidth (blue), fitted theoretical contributions \nto linewidth (green, cyan, black) and empirical fit that describes the angle misalignment \n(magenta) using the OFMR with \n0H\n . (g) Effective damping using the OFMR and \nTRMOKE .    33 \n Figure 5 \n \nFIG. 5. TRMOKE at \n4H\n  and \n1H\n  and OFMR measurement at \n0H\n  for Co/Ni/Co \nsample . (a) Measured resonance frequency versus magnetic field. (b ) Measured linewidth \n(blue), fitted theoretical contributions to linewidth (green, cyan, magenta) and extracted \nintrinsic linewidth from measurement (red) using the TRMOKE with \n4H\n . (c) \nMeasured linewidth (blue), fitted theoretical c ontributions to linewidth (green, cyan, \nmagenta) using the TRMOKE with \n1H\n . (d) Measured linewidth (blue), fitted \ntheoretical contributions to linewidth (green, cyan, black) using the OFMR with \n0H\n . \n34 \n (e) Effecti ve (blue)  and intrinsic  (black ) damping using the TRMOKE at \n4H\n  and \neffective damping measured with the OFMR at \n0H\n  (red).  "    },    {        "title": "1512.00557v1.Bose_Einstein_Condensation_of_Magnons_Pumped_by_the_Bulk_Spin_Seebeck_Effect.pdf",        "content": "Bose-Einstein Condensation of Magnons Pumped by the Bulk Spin Seebeck E\u000bect\nYaroslav Tserkovnyak,1Scott A. Bender,1Rembert A. Duine,2and Benedetta Flebus2\n1Department of Physics and Astronomy, University of California, Los Angeles, California 90095, USA\n2Institute for Theoretical Physics and Center for Extreme Matter and Emergent Phenomena,\nUtrecht University, Leuvenlaan 4, 3584 CE Utrecht, The Netherlands\nWe propose inducing Bose-Einstein condensation of magnons in a magnetic insulator by a heat\n\row oriented toward its boundary. At a critical heat \rux, the oversaturated thermal gas of magnons\naccumulated at the boundary precipitates the condensate, which then grows gradually as the thermal\nbias is dialed up further. The thermal magnons thus pumped by the magnonic bulk (spin) Seebeck\ne\u000bect must generally overcome both the local Gilbert damping associated with the coherent magnetic\ndynamics as well as the radiative spin-wave losses toward the magnetic bulk, in order to achieve\nthe threshold of condensation. We quantitatively estimate the requisite bias in the case of the\nferrimagnetic yttrium iron garnet, discuss di\u000berent physical regimes of condensation, and contrast\nit with the competing (so-called Doppler-shift) bulk instability.\nPACS numbers: 72.25.Mk,72.20.Pa,75.30.Ds,85.75.-d\nIntroduction .|The rapidly developing thermoelectric\ntransport capabilities to probe nonconducting materials\nare instigating a shift in the \feld of spintronics toward in-\nsulating magnets [1{3]. While allowing for seamless spin\ninjection and detection at their boundaries [4{6], insulat-\ning magnets (including ferromagnets, antiferromagnets,\nand spin liquids) may o\u000ber also e\u000ecient spin propaga-\ntion owing to the lack of the electronic channels for dis-\nsipation of angular momentum. Recent measurements of\nspin signals mediated by thick layers of antiferromagnetic\nnickel oxide [7] and, especially, long di\u000busion lengths of\nmagnons in ferrimagnetic yttrium iron garnet (YIG) [8{\n10], even at room temperature, bear this view out.\nThe bosonic nature of magnons, furthermore, naturally\nlends itself to condensation instabilities when driven by\nlarge biases into a nonlinear response [11{13]. While the\nelectric spin Hall driving of magnetic insulators [14, 15]\nclosely mimics the familiar spin-transfer torque insta-\nbilities of conducting ferromagnets [16], the possibility\nof inducing magnonic (Bose-Einstein) condensation also\nby a heat \rux [12] o\u000bers new exciting opportunities\nthat are unique to the insulating heterostructures. The\nkey physics here is played out in the framework of the\nspin Seebeck/Peltier phenomenology [17], according to\nwhich the heat and spin currents carried by magnons\nare intricately intertwined [18]. While the problem of\nthe thermoelectrically-driven magnon condensation has\nbeen systematically addressed previously in thin-layer\nheterostructures [12, 13], the more basic regime of an\ninterfacial condensation induced by a bulk heat \rux re-\nmains unexplored. This concerns the standard geometry\nof the (longitudinal) spin Seebeck e\u000bect, which is suit-\nable for complex lateral heterostructures that could ulti-\nmately give rise to useful devices [19].\nApplying a large heat \rux from a ferromagnet toward\nits interface with another material (either conducting or\ninsulating), which can carry heat but blocks spin \row,\nleads to a nonequilibrium pile up of magnons at the\nT(x)x0µ(x)µ0.B\u0000n(x, t)nonmagnetic substratemagnetic insulator\njx✓(!,k)zFIG. 1. A monodomain ferromagnet with uniform equilibrium\nspin density pointing in the \u0000zdirection (in the presence of\na magnetic \feld Bpointing up along z). A positive ther-\nmal gradient, @xT >0, induces magnonic \rux jxtowards the\ninterface, where an excess of thermal magnons is accumulat-\ning over their spin-di\u000busion length \u0015. When the correspond-\ning nonequilibrium interfacial chemical potential \u00160reaches a\ncritical value (in excess of the magnon gap), the magnetic or-\nder undergoes a Hopf bifurcation toward a steady precessional\nstate, whose Gilbert damping and radiative spin-wave losses\nare replenished by the thermal-magnon pumping /\u00160. The\ncoherent transverse magnetic dynamics decays away from the\ninterface as nx\u0000iny/ei(kx\u0000!t), where Imk>0.\nboundary. See Fig. 1 for a schematic. When the associ-\nated chemical potential of magnons exceeds the lowest-\nmode frequency of the magnet, the latter gets pumped\nby the magnonic thermal gas, leading to its condensation\nat a critical bias. The problem of \fnding the threshold\nfor this phenomenon as well as considering detrimental\nand competing e\u000bects are the main focus of this Letter.\nOnce experimentally established, such pumped conden-\nsates should provide a fertile platform for studying and\nexploiting spin super\ruidity [20].\nTwo-\ruid magnon hydrodynamics .|The interplay be-\ntween thermal-magnon transport and coherent order-\nparameter dynamics is naturally captured within thearXiv:1512.00557v1  [cond-mat.mes-hall]  2 Dec 20152\ntwo-\ruid formalism developed in Ref. [21]. Namely, we\nstart with a generic long-wavelength spin Hamiltonian\nH=Z\nd3r\u0012\n\u0000A\n2s^s\u0001r2^s+B^sz+K\n2s^s2\nz\u0013\n; (1)\nwhere ^sis the spin-density operator (in units of ~),A\nis the magnetic sti\u000bness, Bthe external \feld along the\nzaxis,Kthe quadratic anisotropy in the same direc-\ntion (with K > 0 corresponding to the easy xyplane\nandK < 0 easyzaxis), andsthe saturation spin den-\nsity. We then perform the Holstein-Primako\u000b transfor-\nmation [22] to the bosonic \feld ^\b\u0019(^sx\u0000i^sy)=p\n2s,\nwhich is composed of the super\ruid order parameter\n\b\u0011h^\biand the quantum-\ructuating piece ^\u001e:^\b = \b+ ^\u001e.\nThese relate to the original spin variables as s\u0011h^si\u0019\n(p\n2sRe\b;\u0000p\n2sIm\b;nc+nx\u0000s), where \b =pnce\u0000i'\nandnx=h^\u001ey^\u001ei, withncandnxbeing respectively the\ncondensed and thermal magnon densities. It is clear that\n'is the azimuthal angle of the coherent magnetic pre-\ncession in the xyplane.\nFollowing the Landau-Lifshitz-Gilbert (LLG) phe-\nnomenology [23] of long-wavelength spin-wave dynamics,\nthe following hydrodynamic equations are obtained [21]:\n_nx+r\u0001jx+\u001b\u0016=\u00152= 2\u0011(!\u0000\u0016=~)nc; (2)\nfor the normal dynamics, where jx=\u0000\u001br\u0016\u0000&rT(\u001b\nbeing the magnon conductivity, &the bulk Seebeck co-\ne\u000ecient, and \u0016the chemical potential) is the thermal\nmagnon \rux and \u0015is the magnon di\u000busion length, and\n_nc+r\u0001jc+ 2\u000b!nc= 2\u0011(\u0016=~\u0000!)nc; (3a)\n~(!\u0000\n)\u0000Knc\ns=A\u0014\n(r')2\u0000r2pncpnc\u0015\n;(3b)\nfor the condensate, where jc=\u0000(2A=~)ncr'and~\n =\nB\u0000K(1\u00002nx=s) is the magnon gap (where we take for nx\nto be the equilibrium cloud density at the ambient tem-\nperatureTand self-consistently suppose that \n >0, so\nthat the ferromagnet is in the normal state with n=\u0000z\nin equilibrium [21]). Furthermore, \u0011\u0018(K=T )2(T=Tc)3\nis the dimensionless constant parametrizing the rate of\nthe thermal-cloud|condensate scattering [21], in terms\nof the Curie temperature Tc.\nFor our present purposes, it will be convenient to re-\ncast the condensate dynamics (3) in the form of the LLG\nequation, as discussed in Ref. [13]:\n~(1 +\u000bn\u0002)_n\u0000[~\n +K(1 +n\u0001z)]z\u0002n\n=An\u0002r2n+\u0011n\u0002(\u0016z\u0002n\u0000~_n);(4)\nwhere the second term on the right-hand side is the local\nthermomagnonic torque parametrized by \u0011. Rewriting\nEq. (2) in the same spirit, we have:\n_nx+r\u0001jx+\u001b\u0016=\u00152=\u0011sz\u0001n\u0002(_n\u0000\u0016z\u0002n=~):(5)Spin Seebeck-driven instability .|For the boundary\nconditions at the interface, x= 0, we will take the sim-\nplest scenario of a hard wall, for which both the thermal\nand coherent spin currents vanish, leading to\n\u001b@x\u0016+&@xT= 0 and @xn= 0; (6)\nwith the latter corresponding to the usual exchange\nboundary condition for classical ferromagnetic dynam-\nics. Below or near the onset of magnetic instability (con-\ndensation in the language of Ref. [12]), we can neglect\nthe right-hand side of Eq. (5). This produces the spin-\ndi\u000busion equation, which is solved by\n\u0016(x) =\u00160e\u0000x=\u0015;where\u00160=\u0015&@xT=\u001b; (7)\nin the steady state (established in response to a uniform\nthermal gradient @xT) and subject to the boundary con-\ndition (6). The magnon chemical potential \u0016is, natu-\nrally, maximized at the interface.\nFor the remainder of this section, we analyze Eq. (4)\nsubject to the magnonic torque induced by \u0016(x) in\nEq. (7). Let us \frst solve the problem in the limit \u0015!1\n(relative to other relevant lengthscales, to be identi\fed\nbelow) resulting in homogeneous dynamics. Rewriting\nthe corresponding LLG equation (4) as\n~(_n\u0000~\nz\u0002n) =n\u0002(\u0011\u00160z\u0002n\u0000~\u000b~_n)\n\u0019(\u0011\u00160\u0000~\u000b~~\n)n\u0002z\u0002n;(8)\nwhere ~\u000b\u0011\u000b+\u0011and~~\n\u0011~\n+K(1+n\u0001z). Here, we as-\nsumed ~\u000b;\u0011\u001c1 and thus approximated _n\u0019~\nz\u0002nin the\nGilbert damping term in going to the second line. It is\nnow easy to see that when the antidamping torque /\u0011\novercomes net damping ~ \u000b, the static equilibrium state\nn=\u0000zbecomes unstable [16]. In the case of the easy-\naxis anisotropy, K < 0, this leads to magnetic switch-\ning toward the stable n=zstate when \u00160>(~\u000b=\u0011)~\n.\nIn the more interesting easy-plane case, K > 0 (corre-\nsponding to repulsive magnon-magnon interactions), the\nanisotropy stabilizes magnetic dynamics at a limit cycle\n(realizing a Hopf bifurcation). The corresponding pre-\ncession angle \u0012is then found to be\n\u0012= 2 sin\u00001r\n\u0011\u00160\u0000~\u000b~\n2~\u000bK; (9)\neventually saturating at \u0012!\u0019when\u00160\u0015(~\u000b=\u0011)(~\n +\n2K).\nLet us estimate the thermal gradient necessary to reach\nthe critical heat \rux for condensation, \u00160= (~\u000b=\u0011)~\n, in\nthe case of yttrium iron garnet. The critical thermal\ngradient is given by\n@xT(c)=~\u000b\n\u0011\u001b\n\u0015&~\n: (10)\nFollowing the magnon-transport theory of Ref. [21] (Sup-\nplemental Material), \u001b=&\u00181 [24]. Taking conservatively3\n~\u000b=\u0011\u0018100 [13] and \u0015\u001810\u0016m [9] at room temperature\n(which is consistent with theoretical estimates based on\nRef. [13]), we get @xT(c)\u00181 K/\u0016m, for \n=2\u0019\u00182 GHz\n(corresponding to a kG \feld). Achieving such thermal\ngradients should be experimentally feasible [2, 8].\nCondensate out\row .|More generally, for \fnite \u0015, the\ncondensate is driven near the interface (where \u00166= 0)\nand should eventually decay su\u000eciently deep into the\nferromagnet. This causes spin super\row away from the\ninterface, furnishing radiative spin-wave losses into the\nbulk, which should suppress condensation and raise the\nheat-\rux threshold. The corresponding instability is de-\nscribed by the LLG equation (4), which we rewrite more\ncompactly as\n~(_n\u0000~\nz\u0002n) =An\u0002@2\nxn+n\u0002(\u0011\u0016z\u0002n\u0000~\u000b~_n);(11)\nwhere both ~\n and\u0016become position dependent (both\ndecreasing away from the interface toward \n and 0, re-\nspectively, in the bulk). Supposing a smooth onset of\ninstability, we will look for the thermal threshold by set-\nting ~\n!\n.\nTaking, furthermore, the opposite extreme of \u0015!0\n(relative to the absolute value of the condensate wave\nnumber, to be checked for internal consistency later), we\ncan integrate Eq. (11) over a distance \u0019\u0015near the inter-\nface [noting that n\u0002@2\nxn\u0011@x(n\u0002@xn)] to obtain the\nboundary condition,\n~\u0015(_n\u0000\nz\u0002n)\u0019An\u0002@xn+\u0015n\u0002(\u0011\u00160z\u0002n\u0000~\u000b~_n);\n(12)\nfor the intrinsic bulk dynamics,\n~(_n\u0000\nz\u0002n) =An\u0002@2\nxn\u0000~\u000b~n\u0002_n; (13)\nin the ferromagnet. In order to \fnd the steady-state\nlimit-cycle solution at the onset of the condensation, we\nlinearize these equations with respect to small deviations\nmaway from equilibrium, n\u0011\u0000z+m, and solve for\nthe ansatz m\u0011mx\u0000imy/ei(kx\u0000!t)(requiring that\nImk>0 and!is real valued), to obtain\n~(!\u0000\n) =Ak2\u0000i~\u000b~!; (14)\nsubject to the boundary condition\ni~(!\u0000\n) =Ak=\u0015\u0000\u0011\u00160+ ~\u000b~!: (15)\nThe \frst term on the right-hand side of this equation\ndescribes coherent spin out\row into the bulk, the sec-\nond term magnonic pumping, and the last term Gilbert\ndamping. The critical chemical potential is correspond-\ningly raised as\n\u00160=~\u000b~!+ARek=\u0015\n\u0011: (16)\nThe spin Seebeck-induced magnonic pumping /\u0011thus\nneeds to overcome the condensate out\row /Ain addi-\ntion to the Gilbert damping /~\u000b. We proceed to solveEqs. (14), (15) supposing that Im k\u001c\u0015\u00001, for internal\nconsistency, and \fnd\nImk= \n~\u000bp\n\u0015\n2\u00152s!2=3\n;Rek=r\nImk\n\u0015=\u0012~\u000b\n2\u0015\u00152s\u00131=3\n;\n(17)\nwhere\u0015s\u0011p\nA=~\n (\u001810 nm, using \n =2\u0019\u00182 GHz\nand typical YIG parameters [25]). In deriving Eqs. (17),\nwe have assumed that ~ \u000b\u001c\u0015=\u0015s, which should not be\nan issue in practice. The \fnal internal consistency check\nis Imk\u001c\u0015\u00001, which thus boils down to ~ \u000b(\u0015=\u0015s)2\u001c1.\nFor YIG with ~ \u000b\u001810\u00004, this would be borderline when\n\u0015=\u0015s\u0018100 (which should be relevant in practice for a\nshorter\u0015and/or lower \n). The frequency according to\nEq. (15) is found as != \n(1+Imk\u00152\ns=\u0015)\u0019\n, so that the\ninstability threshold is \fnally found according to Eq. (16)\nas\n@xT(c)\u0019~\u000b\n\u0011\u001b\n\u0015&~\n\"\n1 +\u0012\u00152\nsp\n2~\u000b\u00152\u00132=3#\n; (18)\nwhich is the central result of this Letter.\nNote that Eq. (18) naturally captures also the \u0015!1\nlimit (10) obtained above (thus indicating its general va-\nlidity for extrapolating between both small and large \u0015\nregimes), which we now understand as corresponding to\n~\u000b(\u0015=\u0015s)2\u001d1. In the case of YIG at room temperature,\nwe thus expect Eq. (10) to give a good quantitative esti-\nmate for the threshold bias. The details of the magnetic\npro\fle beyond the instability threshold can in general be\nexpected to be quite complex, as described by the non-\nlinear Eq. (11), especially if one takes into account the\nfeedback of coherent dynamics on the magnon di\u000busion\naccording to Eq. (5). This nonlinear regime is outside\nthe scope of this work.\nDiscussion and outlook .|At a su\u000eciently large\nmagnon \rux in the bulk of the ferromagnet, the trans-\nverse dynamics exhibit also the Doppler-shift instabil-\nity [26], according to the bulk thermomagnonic torque\n/jx@xn[27]. We \fnd the corresponding threshold to be\ngiven byjx\u0018s\n\u0015s, which translates into @xT\u0018s\n\u0015s=&.\nDividing it by the threshold (10), we get @xT=@xT(c)\u0018\n(\u0011=~\u000b)(s\u0015s\u0015=~\u001b). Taking [21] \u001b\u0018(T=Tc)(s2=3l)=~, where\nlis the magnon mean free path, we thus get for this ratio\n\u0018(\u0011=~\u000b)(Tc=T)(s1=3\u0015s\u0015=l). Performing, once again, an\nestimate for YIG at room temperature by taking \u0011=~\u000b\u0018\n10\u00002,Tc=T\u00182,s1=3\u00182/nm,l\u00181\u0016m,\u0015s\u001810 nm,\nand\u0015\u001810\u0016m, we \fnd that @xT=@xT(c)&1, so that\nboth instability scenarios are in fact viable and could po-\ntentially be competing. This could of course be easily\nchecked as the Doppler-shift instability is independent of\nthe heat-\rux direction, while the BEC of magnons dis-\ncussed here is unipolar, corresponding to the heat \rux\ntowards the interface, as sketched in Fig. 1.\nIt needs also be stressed that the ratio \u0011=~\u000b\u001810\u00002\nemployed in this Letter for our estimates corresponds4\nonly to thermal magnons and disregards low-energy\nmagnons that are beyond the Bose-Einstein thermaliza-\ntion description [13, 21]. When \u00160approaches and ul-\ntimately exceeds the magnon gap ~\n, the overpopula-\ntion of magnons pumped at the bottom of the spec-\ntrum could e\u000bectively enhance this factor, approaching\n\u0011=~\u000b!1 in the extreme case (realizing the limit of the\nstrong condensate-cloud coupling studied in Ref. [12]).\nThis innately nonequilibrium regime, which would yield\na lower threshold for magnonic condensation, is, however,\nbeyond our present formalism.\nOnce established, the interfacial condensate of\nmagnons can be readily detected by monitoring the spin\naccumulation (utilizing, for example, the magneto-optic\nKerr e\u000bect) in the adjacent metallic (nonmagnetic) sub-\nstrate or detecting the associated spin pumping by the\ninverse spin Hall e\u000bect (as in the conventional spin See-\nbeck geometry [2]). In the latter case, the theory would\nhave to be complemented with the appropriate treatment\nof spin leakage into and relaxation in the normal metal\n[21]. The condensate can also be used as a starting point\nto study and exploit collective \\conveyor-belt\" heat and\nspin \row [21] tangential to the interface, which would\nre\rect its super\ruid nature.\nThe authors thank Joseph P. Heremans and Roberto\nC. Myers for helpful discussions and experimental moti-\nvation for this work. This work is supported by the Army\nResearch O\u000ece under Contract No. 911NF-14-1-0016,\nNSF-funded MRSEC under Grant No. DMR-1420451,\nUS DOE-BES under Award No. de-sc0012190, and in\npart by the Stichting voor Fundamenteel Onderzoek der\nMaterie (FOM).\n[1] K. Uchida, J. Xiao, H. Adachi, J. Ohe, S. Takahashi,\nJ. Ieda, T. Ota, Y. Kajiwara, H. Umezawa, H. Kawai,\nG. E. W. Bauer, S. Maekawa, and E. Saitoh, Nat. Mater.\n9, 894 (2010).\n[2] K. Uchida, H. Adachi, T. Ota, H. Nakayama,\nS. Maekawa, and E. Saitoh, Appl. Phys. Lett. 97, 172505\n(2010).\n[3] H. Nakayama, M. Althammer, Y.-T. Chen, K. Uchida,\nY. Kajiwara, D. Kikuchi, T. Ohtani, S. Gepr ags,\nM. Opel, S. Takahashi, R. Gross, G. E. W. Bauer, S. T. B.\nGoennenwein, and E. Saitoh, Phys. Rev. Lett. 110,\n206601 (2013).\n[4] X. Jia, K. Liu, K. Xia, and G. E. W. Bauer, Europhys.\nLett.96, 17005 (2011).\n[5] S. Takei, B. I. Halperin, A. Yacoby, and Y. Tserkovnyak,\nPhys. Rev. B 90, 094408 (2014).\n[6] S. Chatterjee and S. Sachdev, Phys. Rev. B 92, 165113\n(2015).\n[7] T. Moriyama, S. Takei, M. Nagata, Y. Yoshimura,\nN. Matsuzaki, T. Terashima, Y. Tserkovnyak, and\nT. Ono, Appl. Phys. Lett. 106, 162406 (2015).\n[8] B. L. Giles, Z. Yang, J. Jamison, and R. C. Myers, \\Long\nrange pure magnon spin di\u000busion observed in a non-localspin-Seebeck geometry,\" arXiv:1504.02808.\n[9] L. J. Cornelissen, J. Liu, R. A. Duine, J. Ben\nYoussef, and B. J. van Wees, Nature Phys. (2015),\narXiv:1505.06325.\n[10] S. T. B. Goennenwein, R. Schlitz, M. Pernpeintner,\nM. Althammer, R. Gross, and H. Huebl, \\Non-\nlocal magnetoresistance in YIG/Pt nanostructures,\"\narXiv:1508.06130.\n[11] S. O. Demokritov, V. E. Demidov, O. Dzyapko, G. A.\nMelkov, A. A. Serga, B. Hillebrands, and A. N. Slavin,\nNature 443, 430 (2006).\n[12] S. A. Bender, R. A. Duine, and Y. Tserkovnyak, Phys.\nRev. Lett. 108, 246601 (2012); S. A. Bender, R. A.\nDuine, A. Brataas, and Y. Tserkovnyak, Phys. Rev. B\n90, 094409 (2014).\n[13] S. A. Bender and Y. Tserkovnyak, \\Thermally-\ndriven spin torques in layered magnetic insulators,\"\narXiv:1511.04104.\n[14] Y. Tserkovnyak and S. A. Bender, Phys. Rev. B 90,\n014428 (2014).\n[15] A. Hamadeh, O. d'Allivy Kelly, C. Hahn, H. Meley,\nR. Bernard, A. H. Molpeceres, V. V. Naletov, M. Viret,\nA. Anane, V. Cros, S. O. Demokritov, J. L. Prieto,\nM. Mu~ noz, G. de Loubens, and O. Klein, Phys. Rev.\nLett. 113, 197203 (2014); M. Collet, X. de Milly,\nO. d'Allivy Kelly, V. V. Naletov, R. Bernard, P. Bor-\ntolotti, V. E. Demidov, S. O. Demokritov, J. L. Prieto,\nM. Mun~ oz, V. Cros, A. Anane, G. de Loubens, and\nO. Klein, \\Generation of coherent spin-wave modes in\nyttrium iron garnet microdiscs by spin-orbit torque,\"\narXiv:1504.01512.\n[16] D. C. Ralph and M. D. Stiles, J. Magn. Magn. Mater.\n320, 1190 (2008).\n[17] J. P. Heremans and S. R. Boona, Physics 7, 71 (2014).\n[18] G. E. W. Bauer, E. Saitoh, and B. J. van Wees, Nat.\nMater. 11, 391 (2012).\n[19] A. Kirihara, K. Uchida, Y. Kajiwara, M. Ishida, Y. Naka-\nmura, T. Manako, E. Saitoh, , and S. Yorozu, Nat.\nMater. 2012 , 686 (2012).\n[20] E. B. Sonin, Adv. Phys. 59, 181 (2010).\n[21] B. Flebus, S. A. Bender, Y. Tserkovnyak, and R. A.\nDuine, \\Two-\ruid theory for spin super\ruidity in mag-\nnetic insulators,\" arXiv:1510.05316.\n[22] T. Holstein and H. Primako\u000b, Phys. Rev. 58, 1098\n(1940).\n[23] E. M. Lifshitz and L. P. Pitaevskii, Statistical Physics,\nPart 2 , 3rd ed., Course of Theoretical Physics, Vol. 9\n(Pergamon, Oxford, 1980); T. L. Gilbert, IEEE Trans.\nMagn. 40, 3443 (2004).\n[24] This was obtained for a di\u000busive transport of magnons\nsubject to disorder scattering and Gilbert damping.\nAccounting for the magnon-phonon drag, furthermore,\nwould increase the bulk spin Seebeck coe\u000ecient &, thus\nreducing the ratio \u001b=&and lowering thermal threshold\n(10).\n[25] S. Bhagat, H. Lesso\u000b, C. Vittoria, and C. Guenzer, Phys.\nStatus Solidi 20, 731 (1973).\n[26] Y. B. Bazaliy, B. A. Jones, and S.-C. Zhang, Phys. Rev.\nB57, R3213 (1998); Y. Tserkovnyak, H. J. Skadsem,\nA. Brataas, and G. E. W. Bauer, ibid.74, 144405 (2006).\n[27] A. A. Kovalev and Y. Tserkovnyak, Europhys. Lett. 97,\n67002 (2012); S. K. Kim and Y. Tserkovnyak, Phys. Rev.\nB92, 020410(R) (2015)."    },    {        "title": "1512.05408v2.Parity_time_symmetry_breaking_in_magnetic_systems.pdf",        "content": "Parity-time symmetry breaking in magnetic systems\nAlexey Galda1and Valerii M. Vinokur1\n1Materials Science Division, Argonne National Laboratory,\n9700 South Cass Avenue, Argonne, Illinois 60439, USA\n(Dated: July 1, 2016)\nThe understanding of out-of-equilibrium physics, especially dynamic instabilities and dynamic phase transi-\ntions, is one of the major challenges of contemporary science, spanning the broadest wealth of research areas that\nrange from quantum optics to living organisms. Focusing on nonequilibrium dynamics of an open dissipative\nspin system, we introduce a non-Hermitian Hamiltonian approach, in which non-Hermiticity reflects dissipation\nand deviation from equilibrium. The imaginary part of the proposed spin Hamiltonian describes the e \u000bects of\nGilbert damping and applied Slonczewski spin-transfer torque. In the classical limit, our approach reproduces\nLandau-Lifshitz-Gilbert-Slonczewski dynamics of a large macrospin. We reveal the spin-transfer torque-driven\nparity-time symmetry-breaking phase transition corresponding to a transition from precessional to exponen-\ntially damped spin dynamics. Micromagnetic simulations for nanoscale ferromagnetic disks demonstrate the\npredicted e \u000bect. Our findings can pave the way to a general quantitative description of out-of-equilibrium phase\ntransitions driven by spontaneous parity-time symmetry breaking.\nINTRODUCTION\nA seminal idea of parity-time ( PT)-symmetric quantum\nmechanics [1,2], that has stated that the condition of Hermitic-\nity in standard quantum mechanics required for physical ob-\nservables and energy spectrum to be real can be replaced by\na less restrictive requirement of invariance under combined\nparity and time-reversal symmetry, triggered an explosive de-\nvelopment of a new branch of science. The interpretation of\nPTsymmetry as “balanced loss and gain” [3] connected PT\nsymmetry breaking to transitions between stationary and non-\nstationary dynamics and established its importance to under-\nstanding of the applied field-driven instabilities. Experiments\non a diverse variety of strongly correlated systems and phe-\nnomena including optics and photonics [4–10], superconduc-\ntivity [11–13], Bose-Einstein condensates [14], nuclear mag-\nnetic resonance quantum systems [15], and coupled electronic\nand mechanical oscillators [16–18] revealed PT symmetry-\nbreaking transitions driven by applied fields. These observa-\ntions stimulated theoretical focus on far-from-equilibrium in-\nstabilities of many-body systems [12,13,19] that are yet not\nthoroughly understood.\nHere we demonstrate that the non-Hermitian extension\nof classical Hamiltonian formalism provides quantitative de-\nscription of dissipative dynamics and dynamic phase transi-\ntions in out-of-equilibrium systems. Focusing on the case of\nspin systems, we consider the zero-temperature spin dynamics\nunder the action of basic nonconservative forces: phenomeno-\nlogical Gilbert damping [20] and Slonczewski spin-transfer\ntorque [21] (STT). The latter serves as the most versatile way\nof directly manipulating magnetic textures by external cur-\nrents. We propose a general complex spin Hamiltonian, in\nwhich Slonczewski STT emerges from an imaginary magnetic\nfield. ThePT-symmetric version of the Hamiltonian is shown\nto exhibit a phase transition associated with inability of the\nsystem to sustain the balance between loss and gain above a\ncertain threshold of external nonconservative field.\nIn the classical limit of a large spin, our formalism repro-\nduces the standard Landau-Lifshitz-Gilbert-Slonczewski [20–22] (LLGS) equation of spin dynamics and predicts the PT\nsymmetry-breaking phase transition between stationary (con-\nservative) and dissipative (nonconservative) spin dynamics. In\nthis Letter we focus on a single spin, yet our theory can be ex-\ntended to coupled spin systems in higher dimensions. More-\nover, as spin physics maps onto a wealth of strongly corre-\nlated systems and phenomena ranging from superconductiv-\nity to cold-atom and two-level systems, our results provide\nquantitative perspectives on the nature of phase transitions as-\nsociated withPTsymmetry breaking in a broad class of far-\nfrom-equilibrium systems.\nWe introduce the non-Hermitian Hamiltonian for a single\nspin operator ˆS:\nˆH=E\u0000ˆS\u0001+ij\u0001ˆS\n1\u0000i\u000b; (1)\nwhere E\u0000ˆS\u0001denotes the standard Hermitian spin Hamil-\ntonian determined by the applied magnetic field Hand\nmagnetic anisotropy constants kiin the x;y;zdirections:\nE\u0000ˆS\u0001=P\nikiˆS2\ni+\rH\u0001ˆS. A schematic system setup is shown\nin Fig. 1. The phenomenological constant \u000b>0 in Eq. (1)\ndescribes damping; the imaginary field ijis responsible for\nthe applied Slonczewski STT, with jS=ep(~=2e)\u0011Jbeing\nthe spin-angular momentum deposited per second in the di-\nrection epwith spin polarization \u0011=(J\"\u0000J#)=(J\"+J#) of the\nincident current J; and\r=g\u0016B=~is the absolute value of the\ngyromagnetic ratio; g'2,\u0016Bis the Bohr magneton, ~is the\nPlanck’s constant, and eis the elementary charge. We conjec-\nture that Eq. (1) serves as a fundamental generalization of the\nHamiltonian description of both quantum and classical spin\nsystems, which constitutes one of our core results. This form\nof the Hamiltonian proves extremely useful for the general un-\nderstanding of STT-driven dissipative spin dynamics. In this\nwork we focus primarily on the classical limit of spin dynam-\nics, while the semiclassical limit of finite spin will be consid-\nered elsewhere.\nSpin dynamics in the classical limit is conveniently\nobtained by studying expectation value of the Hamilto-\nnian (1) with respect to SU(2) spin-coherent states [23,24]:arXiv:1512.05408v2  [cond-mat.other]  30 Jun 20162\nFIG. 1. Schematic representation of the system setup. Ferromagnetic\ncylinder (blue) is placed in magnetic field Happlied along the xaxis,\nand STT-inducing electric current Jis polarized in the direction ep\nalong theyaxis. Spin-polarized current passes through a nonmag-\nnetic metallic spacer and induces torque (Slonczewski STT, shown\nby the small red arrow) on the total spin S.\njzi=ezˆS+jS;\u0000Si, where ˆS\u0006\u0011ˆSx\u0006iˆSy, and z2Cis the stan-\ndard stereographic projection of the spin direction on a unit\nsphere, z=(sx+isy)=(1\u0000sz), with si\u0011Si=S. Note that such\nparametrization of the phase space for a classical single-spin\nsystem (i.e., in the limit S!1 ) guarantees the invariance of\nthe traditional equation of motion [24] under generalization to\nnon-Hermitian Hamiltonians (see Appendix A):\n˙z=i(1+¯zz)2\n2S@H\n@¯z; (2)\nwhere zand ¯zform a complex conjugate pair of stereographic\nprojection coordinates, and\nH(z;¯z)=hzjˆHjzi\nhzjzi(3)\nis the expectation value of the Hamiltonian (1) in spin-\ncoherent states (for a detailed review see, e.g., Ref. [25]). In\nthis formulation, the eigenstates of ˆHcorrespond to the fixed\npoints ziof the equation of motion for H, while the eigen-\nvalues (i.e., energy values) are equal to Hevaluated at the\ncorresponding fixed points, Ei=H(zi;¯zi).\nAssuming a constant magnitude of the total spin, ˙S=0,\nEq. (2) reduces to the following equation of spin dynamics in\nthe classical limit:\n˙S=rS(ReH)\u0002S+1\nS\u0002rS(ImH)\u0002S\u0003\u0002S: (4)\nHere we refer to the real and imaginary parts of the Hamil-\ntonian functionHwritten in the spin Srepresentation. For\nthe non-Hermitian Hamiltonian (1), Eq. (4) reproduces theLLGS equation describing dissipative STT-driven dynamics\nof a macrospin:\n\u0010\n1+\u000b2\u0011˙S=\rHe\u000b\u0002S+\u000b\r\nS[\rHe\u000b\u0002S]\u0002S+1\nSS\u0002[S\u0002j]\n+\u000bS\u0002j; (5)\n\rHe\u000b=rSE(S): (6)\nThe first two terms in Eq. (5) describe the standard Landau-\nLifshitz torque and dissipation, while the last two are respon-\nsible for the dissipative (antidamping) and conservative (e \u000bec-\ntive field) Slonczewski STT contributions, correspondingly,\nboth of which appear naturally from the imaginary magnetic\nfield term in the Hamiltonian (1).\nPT-SYMMETRIC HAMILTONIAN\nSlonczewski STT turns the total spin-angular momentum,\nS, in the direction of spin-current polarization, ep, without\nchanging its magnitude. On the S-sphere this can be repre-\nsented by a vector field converging in the direction of epand\noriginating from the antipodal point. It is the imaginary mag-\nnetic field ijthat produces exactly the same e \u000bect on spin dy-\nnamics, according to Eq. (2). The action of STT is invari-\nant under the simultaneous operations of time reversal and\nreflection with respect to the direction ep, which is the un-\nderlying reason behind the inherent PTsymmetry of certain\nSTT-driven magnetic systems, including the one considered\nbelow.\nBefore turning to the PT-symmetric form of Hamilto-\nnian (1), we note that PT-symmetric systems play an im-\nportant role in the studies of nonequilibrium phenomena\nand provide a unique nonperturbative tool for examining the\nphase transition between stationary and nonstationary out-\nof-equilibrium dynamics. We show that despite being non-\nHermitian, such systems can exhibit both of the above types\nof behavior, depending on the magnitude of the external non-\nconservative force. In the parametric regime of unbroken\nPTsymmetry, systems exhibit physical properties seemingly\nequivalent [26] to those of Hermitian systems: real energy\nspectrum, existence of integrals of motion (see Appendix C),\nand, notably, the validity of the quantum Jarzynski equal-\nity [27]. However, in the regime of brokenPT symme-\ntry, one observes complex energy spectrum and nonconser-\nvative dynamics. Therefore, the true transition between sta-\ntionary and nonstationary dynamics can be identified as the\nPTsymmetry-breaking phase transition.\nSpin systems are generally subject to various non-\nlinear magnetic fields including ones originating from\nshape, exchange, and magnetocrystalline and magnetoelastic\nanisotropies. Restricting ourselves for simplicity to a second-\norder anisotropy term, we arrive at the following Hamiltonian\nfor a nonlinear magnetic system with uniaxial anisotropy and\napplied Slonczewski STT:\nˆHPT=\rH0\u0010\nkzˆS2\nz+hxˆSx+i\fˆSy\u0011\n; (7)3\nFIG. 2. Real (a) and imaginary (b) parts of energy spectrum of the Hamiltonian (7) as a function of the STT parameter \fforhx=1 and\nD=20. Blue and red lines correspond to the eigenvalues E1;2andE3\u00006, respectively. The first PTsymmetry-breaking transition occurs at\nj\fj=\f1\u00194:5.\nwhere the applied magnetic field hxis measured in units of\nsome characteristic magnetic field H0, and\fis a dimension-\nless STT parameter determining the relative to Samount of\nangular momentum transfered in time \u001c\u0011(\rH0)\u00001(charac-\nteristic timescale of the dynamics, used as a unit of dimen-\nsionless time in what follows). The Hamiltonian (7) modeling\nthe dynamics of the free magnetic layer in a typical nanopil-\nlar device with fixed polarizer layer (see Fig. 1) is PTsym-\nmetric: It is invariant under simultaneous action of parity and\ntime-reversal operators ( y!\u0000y,t!\u0000t,i!\u0000i). Because\nthe Hamiltonian ˆHPTcommutes with an antilinear operator\nPT, its eigenvalues are guaranteed to appear in complex con-\njugate pairs. Notice that PT-symmetric Hamiltonian (7) does\nnot contain damping, which is assumed to be negligibly small,\nas is the case in many experimental systems.\nCLASSICAL SPIN SYSTEM\nIn order to best illustrate the mechanism of PT symme-\ntry breaking, we focus on the classical limit, S!1 and\nkzS!D=2, where Dis the dimensionless uniaxial anisotropy\nconstant. Formula (2) then yields the following equation of\nmotion for the Hamiltonian (7):\n˙z(t)=\u0000i(hx+\f)\n2 \nz2\u0000hx\u0000\f\nhx+\f!\n\u0000i D z1\u0000jzj2\n1+jzj2; (8)\nwith up to six fixed points zk,k=1;:::; 6.\nShown in Fig. 2 are the real and imaginary parts of the en-\nergy spectrum E1\u00006as functions of the STT amplitude \f. It\nreveals that in a system with strong anisotropy, D\u001d1,PT\nsymmetry breaking occurs in three separate transitions, with\nthe first one atj\fj=\f1=jhxjq\u00021+p\n1+(2D=jhxj)2\u0003=2,\nwhich corresponds to the smallest amplitude of STT at which\nIm(E),0. Therefore,PTsymmetry is not broken in the en-\ntire phase space of initial spin directions simultaneously, at\nvariance to the linear spin system with D=0 (see AppendixB). Instead, the regions of broken and unbroken PTsymme-\ntry may coexist in the phase diagram of a nonlinear spin sys-\ntem.\nIn what follows we consider a system described by\nthe Hamiltonian (7) with hx=1 and D=20. For all\nj\fj<\f 1\u00194:5,PT symmetry is unbroken and the character\nof spin (magnetization) dynamics is oscillatory in the entire\nphase diagram, i.e., for all possible initial conditions z. At\nj\fj=\f1the phase transition (first of the three, see Fig. 2) oc-\ncurs sharply in a wide region around the easy plane, jzj=1,\ni.e. near the equator of the unit S-sphere, shown in gray in\nFigs. 3(a) and 3(b) in Cartesian and stereographic projection\ncoordinates, respectively. It this region the nature of spin dy-\nnamics becomes fundamentally di \u000berent—all spin trajectories\nfollow the lines connecting the fixed points z1andz2, where\nz1;2=\u0000\u0000Dhx\u0006ip\n\f4\u0000\f2h2x\u0000D2h2x\u0001=(hx+\f)\f, and no closed\ntrajectories are possible; see Fig. 3(b).\nAsj\fjis increased further, the region of broken PTsymme-\ntry expands until it eventually closes around the fixed point z5\nat\f2\u00199:3 (second bifurcation in Fig. 2) and, eventually, the\nlast region of unbroken PT symmetry near z3disappears at\n\f3\u001910:8. The second and third phase transitions are less rel-\nevant experimentally as they occur in the vicinity of the least\nfavorable spin directions (parallel and antiparallel to the hard\naxisz) and at considerably higher applied currents.\nThe predicted transition from precessional dynamics (un-\nbrokenPTsymmetry) to exponentially fast saturation in the\ndirection z1(hx;\f) for any initial spin position around the easy\nplane (brokenPT symmetry) occurs in the setup with mu-\ntually perpendicular applied magnetic field and Slonczewski\nSTT. Such a transition in nanoscale magnetic structures can\nbe used for STT- or magnetic-field-controlled magnetization\nswitching in spin valves and a variety of other experimental\nsystems. This e \u000bect can further be used for direct measure-\nments of the amplitude of the applied STT, which, unlike the\napplied current, can be hard to quantify experimentally.4\nFIG. 3. (a, b) Spin dynamics described by Eq. (8) with hx=1,\f=4:7, and D=20.PTsymmetry is broken in the shaded region around\nthe easy planejzj=1 (dashed line), encompassing two fixed points, z1;2(blue dots), appearing as source and sink nodes. The green line\ndepicts a typical nonoscillatory spin trajectory in the region of broken PTsymmetry. Red dots represent the fixed points z3\u00006. (c) Results of\nmicromagnetic simulations for \f\u0003as a function of stereographic projection of the initial spin direction z. In the blue region, 4 :6.\f\u0003.4:8, the\nPTsymmetry is broken at all j\fj<\f\u0003, and the spin takes under 0 :5 ns to saturate in the direction of z1, which is in full agreement with the\nanalytical result.\nNUMERICAL SIMULATIONS OF PTSYMMETRY\nBREAKING\nHere we present the results of numerical simulations con-\nfirming thePT symmetry-breaking phase transition in the\nclassical single-spin system (7) by modeling magnetization\ndynamics of a ferromagnetic disk 100 nm in diameter and\nd=5 nm thick, which is consistent with the anisotropy con-\nstant D=20 in Eq. (8). We used the following typical\npermalloy material parameters: damping constant \u000b=0:01,\nexchange constant Aex=13\u000210\u000012J/m and saturation mag-\nnetization Msat=800\u0002103A/m . The simulations were car-\nried out using the open-source GPU-accelerated micromag-\nnetic simulation program MuMax3 [28] based on the LLGS\nequation (5) discretized in space. We used a cubic discretiza-\ntion cell of 5 nm in size, which is smaller than the exchange\nlength in permalloy, lex=(2Aex=\u00160M2\nsat)1=2\u00195:7 nm.\nThe permalloy disk was simulated in an external magnetic\nfield applied along the xaxis, H0=400 Oe, which corre-\nsponds to the characteristic time \u001c\u00190:14 ns. The STT was\nproduced by applying electric current perpendicular to the\ndisk in the zdirection with spin polarization \u0011=0:7 along\nep=ˆy(see Fig. 1) and current density \fmeasured in dimen-\nsionless units of 2 eH0Msatd=\u0011~\u00190:7\u0002108A/cm2. While\nsuch current density is comparable to typical switching cur-\nrent densities in STT-RAM devices [29,30], its magnitude can\nbe optimized for various practical applications by changing\nH0and adjusting the size, shape, and material of the ferro-\nmagnetic element.\nFor all possible initial spin directions z, we calculated the\ncritical amplitude of the applied STT, \f\u0003, for which the char-\nacter of spin dynamics changes from oscillatory (at j\fj< \f\u0003)\nto exponential saturation. Shown in Fig. 3(c) is the color map\nof\f\u0003as a function of zin complex stereographic coordinates.\nThe region shown in the shades of blue corresponds to the\ninitial conditions z, for which the minimum values of \fthat\nwould guarantee saturation of spin dynamics in the directionofz1in under 0:5 ns are between 4.6 and 4.8. This is in full\nagreement with the region of broken PTsymmetry at \f=4:7\ncalculated analytically, i.e., the shaded area in Fig. 3(b) [the\noutline is repeated in Fig. 3(c) for comparison]. Outside of\nthis region, a considerably larger magnitude of the applied\nSTT is required to break PTsymmetry.\nThe agreement between theoretical results and micromag-\nnetic simulations is remarkable considering the non-zero\nGilbert damping parameter ( \u000b=0:01) and nonlinear e \u000bects\n(demagnetizing field, finite size and boundary e \u000bects, etc.)\ninherently present in the micromagnetic simulations but not\nincluded in the model Hamiltonian (7).\nCONCLUSION\nThe presented non-Hermitian Hamiltonian formulation of\ndissipative nonequilibrium spin dynamics generalizes the pre-\nvious result [31], where the classical Landau-Lifshitz equa-\ntion was derived from a non-Hermitian Hamilton operator,\nto open STT-driven spin systems. The introduction of Slon-\nczewski STT in the imaginary part of the Hamiltonian re-\nvealed the possibility of STT-driven PTsymmetry-breaking\nphase transition. Micromagnetic simulations confirm the\nPTsymmetry-breaking phenomenon in realistic mesoscopic\nmagnetic systems and its robustness against weak dissipation,\nindicating high potential for impacting spin-based informa-\ntion technology. The way STT enters the complex Hamilto-\nnian (1), i.e. as imaginary magnetic field, provides a unique\ntool for studying Lee-Yang zeros [32] in ferromagnetic Ising\nand Heisenberg models and, more generally, dynamics and\nthermodynamics in the complex plane of physical parame-\nters. We envision further realizations of the PT symmetry-\nbreaking phase transitions in diverse many-body condensed-\nmatter systems and the expansion of practical implementa-\ntions of thePT symmetry beyond the present realm of op-\ntics [33] and acoustics [34].5\nFIG. 4. Real (a) and imaginary (b) parts of energy spectrum of the linear Hamiltonian ˆH0PTas functions of \fforhx=1.PTsymmetry-\nbreaking transition occurs at j\fj=1.\nACKNOWLEDGEMENTS\nWe thank Alex Kamenev for critical reading of the\nmanuscript and valuable suggestions. This work was sup-\nported by the U.S. Department of Energy, O \u000ece of Science,\nBasic Energy Sciences, Materials Sciences and Engineering\nDivision.\nAPPENDIX A. GENERALIZED EQUATION OF MOTION\nFOR NON-HERMITIAN SPIN HAMILTONIANS IN THE\nCLASSICAL LIMIT\nThe remarkable simplicity of the equation of motion (2)\nfor an arbitrary non-Hermitian spin Hamiltonian function\nHstems from the choice of parametrization of the phase\nspace, i.e., the complex stereographic projection coordinates\nfz;¯zg. The extension of classical equations of motion to non-\nHermitian Hamiltonians in terms of canonical coordinates\nfq;pghas the following generalized form [35]:\n \nq\np!\n= \n\u00001rq;p(ReH)\u0000G\u00001rq;p(ImH); (A1)\nwhere \nandGare the symplectic structure and metric of\nthe underlying classical phase space, respectively, which must\nsatisfy the compatibility condition [36] written in the matrix\nform as\n\n\u00001=\u0010\n\u0000\u00001\n \u0000\u00001\u0011T: (A2)\nIn the stereographic projection coordinates, one obtains the\nfollowing symplectic structure and metric:\n\n =2\n\u00001+jzj2\u00012 \n0i\n\u0000i0!\n; G=2\n\u00001+jzj2\u00012 \n0 1\n1 0!\n:(A3)\nIt is the form of these matrices that leads to Eq. (2), where the\nreal and imaginary parts (as written in the Srepresentation)\nof the Hamiltonian combine naturally into a single complexfunctionH. Therefore, when written in stereographic projec-\ntion coordinates, the generalized classical equation of motion\nfor non-Hermitian Hamiltonians coincides with that for tradi-\ntional Hermitian Hamiltonians.\nAPPENDIX B.PTSYMMETRY BREAKING IN LINEAR\nSPIN SYSTEM\nIn the absence of magnetic anisotropy fields, the Hamilto-\nnian (1) from the main text becomes linear:\nˆH0= \rH+ij\n1\u0000i\u000b!\n\u0001ˆS; (B1)\nwith e \u000bects of applied magnetic field, damping and Slon-\nczewski STT contributions all incorporated in the complex\nmagnetic field (in parentheses). The PT-symmetric version\nof this Hamiltonian has mutually perpendicular real and imag-\ninary parts of the complex magnetic field:\nˆH0PT=\rH0\u0010\nhxˆSx+i\fˆSy\u0011\n: (B2)\nThe quantum spin-1\n2version of this Hamiltonian describes\na two-level quantum system with balanced loss and gain and\nis known [37,38] to exhibit PTsymmetry-breaking transition\nathx=\u0006\f. Whenjhxj>j\fj, the Hamiltonian ˆH0PThas real\neigenvalues, \u00151;2=\u0006p\nh2x\u0000\f2, while in the parametric region\njhxj<j\fjeigenvalues are imaginary, see Fig. 4.\nThe generalized Hamilton’s equation of motion (2) for this\nHamiltonian takes the form\n˙z(t)=f(t)=\u0000i(hx+\f)\n2 \nz2\u0000hx\u0000\f\nhx+\f!\n: (B3)\nIt has two fixed points: z1;2=\u0006p\n(hx\u0000\f)=(hx+\f), which\ncorrespond to stereographic coordinates of the spin equi-\nlibria directions on a unit Bloch sphere. The character\nof spin dynamics around the fixed points is fully deter-\nmined by the eigenvalues of the complex Jacobian matrix6\nFIG. 5. Spin trajectories for classical linear Hamiltonian H0PTin the regime of unbroken, (a) and (c), (at \f=0:8) and broken, (b) and (d), (at\n\f=1:2)PTsymmetry for hx=1.\nJC=@(Ref;Imf)=@(Rez;Imz)in their vicinity. The solu-\ntion of Eq. (B3) takes the form of a M ¨obius transformation:\nz(t)=z(0)+ihx+\fp\nh2x\u0000\f2tan p\nh2x\u0000\f2\n2t!\nihx\u0000\fp\nh2x\u0000\f2tan p\nh2x\u0000\f2\n2t!\nz(0)+1: (B4)\nThe parametric region jhxj>j\fjdefines the regime of un-\nbrokenPT symmetry with real Hamiltonian eigenvalues,\nE1;2=\u0006p\nh2x\u0000\f2. In the classical approximation, the spin\nperforms persistent oscillations along circular orbits about the\nfixed points z1;2situated on the real axis, see Figs. 5(a) and\n5(c). The eigenvalues of JCatz1;2are purely imaginary, iden-\ntifying the fixed points are centers , according to the stan-\ndard classification [39]. Closed trajectories represent PT-\nsymmetric dynamics with balanced loss and gain: the spin\nsystem gains and loses equal amounts of energy from the non-\nconservative term i\fSyon they <0 andy >0 segments of\ntrajectories, respectively.As the driving parameter j\fjis increased, z1;2move towards\neach other until they eventually collide at j\fj=jhxj, which\nmarks the point of PT symmetry breaking. In the regime\nofbrokenPT symmetry,jhxj<j\fj, the energy eigenvalues\nare imaginary, E1;2=\u0006ip\n\f2\u0000h2x, and no closed trajectories\nare possible. The eigenvalues of the Jacobian JC(z1;2) in this\nregime are real and of the same sign, defining z1;2assink\nandsource nodes [39]. All trajectories follow circle arcs con-\nnecting the fixed points with coordinates in three-dimensional\nspace\u0010\n0;\u0006p\n1\u0000(hx=\f)2;\u0000hx=\f\u0011\n, which are now out of the xz\nplane, see Figs. 5(b) and 5(d).\nWe emphasize that in the linear Hamiltonian (B1) the ef-\nfects of damping can always be fully compensated by the\nappropriate choice of the applied STT. For instance, the\nPT-symmetric Hamiltonian (B2) describes a single spin\nplaced in external magnetic field H=H0(hx;\u000b\f; 0)and STT\nj=\rH0(\u0000\u000bhx;\f;0). Note that in the general case of nonlin-\near spin Hamiltonian, dissipation cannot be completely can-\nceled by STT. However, in many magnetic systems dissipative\nforces are extremely weak, which justifies the approximation\nof zero damping.7\nAPPENDIX C. INTEGRALS OF MOTION\nThere exists a remarkable similarity between spin dynam-\nics in the regime of unbroken PT symmetry and that of a\nfully Hermitian system. Indeed, it follows from Eq. (B3) that\nall spin trajectories are circular with the precession frequency\nequal top\nh2x\u0000\f2. The equivalent [26] to (B2) Hermitian\nHamiltonian reads:\nH0=\rH0q\nh2x\u0000\f2S0\nx; (B5)for which the equation of motion is obtained from Eq. (B3) by\nthe circle-preserving M ¨obius transformation\nz0=s\nhx+\f\nhx\u0000\fz: (B6)\nThe relation between time evolution of complex linear spin\nHamiltonians and M ¨obius transformations will be described\nelsewhere. The existence of this transformation leads to the\nnon-Hermitian system (B2) having an integral of motion:\nI[z;¯z]=(hx+\f)z+¯z\n1+hx+\f\nhx\u0000\fjzj2; (B7)\nwhich is nothing but the magnetic energy conserved by the\nHermitian Hamiltonian (B5):\nI\u0002z0;¯z0\u0003=q\nh2x\u0000\f2s0\nx=q\nh2x\u0000\f2z0+¯z0\n1+jz0j2(B8)\nafter the transformation (B6).\n1C. M. Bender and S. Boettcher. Real spectra in non-Hermitian\nHamiltonians having PT symmetry. Phys. Rev. Lett. 80, 5243\n(1998).\n2C. M. Bender, S. Boettcher, and P. N. Meisinger. PT-symmetric\nquantum mechanics. J. Math. Phys. 40, 2201-2229 (1999).\n3A. Ruschhaupt, F. Delgado, and J. G. Muga. Physical realization\nof PT-symmetric potential scattering in a planar slab waveguide.\nJ. Phys. A 38, L171 (2005).\n4S. Klaiman, U. G ¨unther, and N. Moiseev. Visualization of branch\npoints in PT-symmetric waveguides. Phys. Rev. Lett. 101, 080402\n(2008).\n5S. Longhi. Bloch oscillations in complex crystals with PT symme-\ntry.Phys. Rev. Lett. 103, 123601 (2009).\n6S. Longhi. Optical realization of relativistic non-Hermitian quan-\ntum mechanics. Phys. Rev. Lett. 105, 013903 (2010).\n7H. Schomerus. Quantum noise and self-sustained radiation of PT-\nsymmetric systems. Phys. Rev. Lett. 104, 233601 (2010).\n8H. Ramezani, T. Kottos, R. El-Ganainy, and\nD. N. Christodoulides. Unidirectional nonlinear PT-symmetric\noptical structures. Phys. Rev. A 82, 043803 (2010).\n9C. E. R ¨uteret al. Observation of paritytime symmetry in optics.\nNature Phys. 6, 192-195 (2010).\n10J. Sheng, M. A. Miri, D. N. Christodoulides, and M. Xiao. PT-\nsymmetric optical potentials in a coherent atomic medium. Phys.\nRev. A 88, 041803 (2013).\n11J. Rubinstein, P. Sternberg, and Q. Ma. Bifurcation diagram and\npattern formation of phase slip centers in superconducting wires\ndriven with electric currents. Phys. Rev. Lett. 99, 167003 (2007).\n12J. Rubinstein, P. Sternberg, and K. Zumbrun. The resistive state in\na superconducting wire: bifurcation from the normal state. Arch.\nRation. Mech. Anal. 195, 117 (2010).\n13N. M. Chtchelkatchev, A. A. Golubov, T. I. Baturina, and\nV . M. Vinokur. Stimulation of the fluctuation superconduc-\ntivity by PT symmetry. Phys. Rev. Lett. 109, 150405 (2012).\n14E. M. Graefe, H. J. Korsch, and A. E. Niederle. Mean-field dynam-\nics of a non-Hermitian Bose-Hubbard dimer. Phys. Rev. Lett. 101,\n150408 (2008).15C. Zheng, L. Hao, and G. L. Long. Observation of fast evolu-\ntion in paritytime-symmetric system. Phil. Trans. R. Soc. A 371,\n20120053 (2013).\n16C. M. Bender, B. K. Berntson, D. Parker, and E. Samuel. Observa-\ntion of PT phase transition in a simple mechanical system. Am. J.\nPhys. 81, 173 (2013).\n17C. M. Bender et al. Twofold transition in PT-symmetric coupled\noscillators. Phys. Rev. A. 88, 062111 (2013).\n18J. Schindler et al. . Experimental study of active LRC circuits with\nPT symmetries. Phys. Rev. A 84, 040101(R) (2011).\n19N. Poccia et al. Critical behavior at a dynamic vortex insulator-to-\nmetal transition. Science 349, 1202-1205 (2015).\n20T. L. Gilbert. A phenomenological theory of damping in ferro-\nmagnetic materials. IEEE Trans. Magn. 40, 3443 (2004).\n21J. C. Slonczewski. Current-driven excitation of magnetic multilay-\ners.J. Magn. Magn. Matter ,159, L1-L7 (1996).\n22L. D. Landau and E. M. Lifshitz. On the theory of the dispersion\nof magnetic permeability in ferromagnetic bodies. Phys. Z. Sow-\njetunion 8, 101-114 (1935).\n23E. H. Lieb. The classical limit of quantum spin systems. Commun.\nMath. Phys. 34, 327-340 (1973).\n24M. Stone, K.-S. Park, and A. Garg. The semiclassical propagator\nfor spin coherent states. J. Math. Phys. 41, 8025-8049 (2000).\n25J. M. Radcli \u000be. Some properties of coherent spin states. J. Phys. A\n4, 313 (1971).\n26A. Mostafazadeh. Exact PT-symmetry is equivalent to Hermitic-\nity.J. Math. Phys. 36, 7081 (2003).\n27S. De \u000bner and A. Saxena. Jarzynski equality in PT-symmetric\nquantum mechanics. Phys. Rev. Lett. 114, 150601 (2015).\n28A. Vansteenkiste et al. . The design and verification of MuMax3.\nAIP Adv. 4, 107133 (2014).\n29K. L. Wang, J. G. Alzate, and P. K. Amiri. Low-power non-volatile\nspintronic memory: STT-RAM and beyond. J. Phys. D: Appl.\nPhys. 46, 074003 (2013).\n30A. D. Kent, B. ¨Ozyilmaz, and E. Del Barco. Spin-transfer-induced\nprecessional magnetization reversal. Appl. Phys. Lett. 84, 3897\n(2004).8\n31R. Wieser. Comparison of quantum and classical relaxation in spin\ndynamics. Phys. Rev. Lett. 110, 147201 (2013).\n32C. N. Yang and T. D. Lee. Statistical theory of equations of state\nand phase transitions. II. Lattice gas and Ising model. Phys. Rev.\n87, 410 (1952).\n33B. Peng et al. Parity-time-symmetric whispering-gallery micro-\ncavities. Nature Phys. 10, 394 (2014).\n34R. Fleury, D. Sounas, and A. Al `u. An invisible acoustic sensor\nbased on parity-time symmetry. Nat. Commun. 6, 5905 (2015).\n35E. M. Graefe, M. H ¨oning, and H. J. Korsch. Classical limit of non-\nHermitian quantum dynamics - a generalized canonical structure.\nJ. Phys. A 43, 075306 (2010).36D. C. Brody and L. P. Hughston. Geometric quantum mechanics.\nJ. Geom. Phys. 38, 19 (2001).\n37C. M. Bender, P. N. Meisenger, and Q. Wang. Calculation of the\nhidden symmetry operator in PT-symmetric quantum mechanics.\nJ. Phys. A 36, 6791 (2003).\n38D. C. Brody. Biorthogonal quantum mechanics. J. Phys. A 47,\n035305 (2014).\n39D. V . Anosov, V . I. Arnold, and S. Kh. Aranson. Ordinary di \u000ber-\nential equations and smooth dynamical systems, Vol. 1 (Springer\nScience & Business Media, 1997)."    },    {        "title": "1601.02210v3.Interfacial_Dzyaloshinskii_Moriya_interaction__surface_anisotropy_energy_and_spin_pumping_at_spin_orbit_coupled_Ir_Co_interface.pdf",        "content": "1 Interfacial Dzyaloshinskii -Moriya interaction, surface anisotropy  energy,  \nand spin pumping at spin orbit  coupled Ir/Co interface  \n \nNam -Hui Kim,1 Jinyong Jung,1 Jaehun Cho,1 Dong -Soo Han,3 Yuxiang Yin,3 June-Seo \nKim,3,a) Henk J. M. Swagten,3 and Chun -Yeol You1,2,b) \n1Department of Physics, Inha University, Incheon , 22212 , South Korea  \n2Department of Emerging Materials Science, DGIST, Daegu , 42988 , South  Korea   \n3Department of Applied Physics,  Center for Nano materials, Eindhoven University of \nTechnology, PO Box 513,  5600 MB Eindhoven, The Netherlands  \n \nThe interfacial Dzyaloshinskii -Moriya interaction  (iDMI) , surface anisotropy  energy,  and \nspin pumping at the Ir/Co interface  are experimentally investigated  by performing Brillouin \nlight scattering.  Contrary to previous reports, we suggest that the sign of  the iDMI at the \nIr/Co interface is the same as in  the case of the Pt/Co  interface . We also find  that the \nmagnitude of the iDM I energy density is relatively smaller than in the case  of the Pt/Co \ninterface , despite the large strong  spin-orbit coupling (SOC)  of Ir . The saturation \nmagnetization and the perpendicular magnetic anisotropy (PMA) energy are significantly \nimproved due to a strong  SOC . Our finding s suggest  that a n SOC in an Ir/Co system behaves \nin different ways for iDMI and PMA. Finally , we determine the spin pumping effect at the \nIr/Co interface , and it increases  the Gilbert damping constant from 0.01 2 to 0.024  for 1.5 nm-\nthick Co.   \n  \n                                          \na) E-mail: spin2mtj@gmail.com   \nb) E-mail: cyyou@inha.ac.kr  2 Spin-orbit coupling (SOC)  plays a crucial role  in various basic magnetic phenomena such as  \nmagnetic crystalline anisotrop ies, magnetostriction  effect s, magneto -optical Kerr effect s, \nanomalous Hall effects, anisotropic magnetoresistance s, and magnetic damping processes .1 \nRecently the  SOC has been of growing interest due to  novel physics and  emerging \ntechnolog ies of “spin  orbitronics” such as  the spin Hall effect,2,3,4 the Rashba effect,5,6 the \nDzyaloshinskii -Moriya interaction (DMI) ,7,8,9 and spin pumping effects .10,11 Therefore, an \nunderstanding  the SOC , in particular  at the interface between ferromagnetic (FM) material \nand heavy metals (HM) , is of great importance for an investigation of the underlying physics \nof this exotic phenomena . Among these SOC -related phenomena, a n interfacial DMI (iDMI) \nhas drawn recent interest . The iDMI  is known to arise at the interface due to inversion \nsymmetry breaking and large SOC in heavy metals (HM). The iDMI  manifest s itself by \nforming spiral spin configurations with a preferred chirality . Therefore, it plays  an important \nrole in the dynamics of the chiral domain wall (DW)12,13\n and the skyrmion formation.14,15 \nExperimental  determination of the iDMI energy density is needed. H owever, general \napproach es for the iDMI measurements (i.e., asymmetric domain wall motion or bubble \nexpansion ) have been difficul t to obtain  the exact value of  the DMI energy density.16,17,18  \nRecently, direct  measurement s of the iDMI via Brillouin light scattering (BLS)  have been \ndemonstrated in Pt/Co/AlO x, Pt/CoFeB/AlO x,19 and Ta/Pt/Co/AlO x20 systems . Based on the \nnon-reciprocal spin wave (SW) theorem under the iDMI,21 BLS can provide a direct value of \nthe iDMI energy density from a measurement of the frequency difference between negative \n(Stokes) and positive (anti -Stokes) SW frequencies . Therefore, BLS is considered to be a \nsuperior approach for the study of iDMI.22,23,24 Since BLS is able to  measure intrinsic \nmaterial parameters such as the  saturation magnetization  (MS), the effective magnetic \nanisotropy energy, and the Gilbert damping constant simultaneously, it is suitable for the \nmeasurement of complex SOC -related phenomena accompanying perpendicul ar magnetic \nanisotropy (PMA), the enhancement of Gilbert damping by  spin pumping, and the \nenhancement of saturat ion magnetization by the proximity effect. For example, it is well \nknown that PMA is proportional to the square of SOC strength ,25,26 whereas spin pumping is \nproportional to the SOC strength.10,11 Furthermore, the enhancement of proximity -induced \nmagnetization  (PIM) is also related to SOC.27  \n  3 In this Letter , we experimentally investigate the interfac ial magnetic properties of an \nIr/Co( tCo)/AlO x system by BLS .19,20 Iridium, which is a  heavy metal , was chosen to compare \nthe SOC at the interface between Cobalt and HM. From systematic BLS measurements, a \nlarge  value of  MS due to  a proximity  effect  and an enhanced  value of  KS due to  SOC at  the \nIr/Co interface were  observed. We measure d iDM I energy densities  from the magnetic field-\ndependent frequency difference measurements , and f ound  that they are  smaller than in the \ncase of the Pt/Co interface. Contrary to  previous reports,28,29,30 the measured sign of the \niDMI is the same with Pt/Co /AlO x systems . Furthermore, a noticeable enhancement of the \nGilbert  damping constant  due to a strong  spin pumping  effect  was observed . This fact implies \nthat the role s of SOC for iDMI, PMA, proximity -induced magnetization, and  a spin pumping  \neffect  are not the same based on our experimental observations.   \nThe inset in Fig. 1(a) shows the sample geometry . A wedge -shaped Ta(4 nm)/Ir(4 \nnm)/Co( tCo)/AlO x(2 nm) sample was deposited o n a thermally -oxidized Si  wafer . A \nmagnetron sputtering system was used to deposit all the layers. Especially, t he Co layer was \ndeposited wedge d shape in the range of 1 to 3 nm. In order to break the inversion symmetry, a \n2-nm AlO x capping was used on the top of the Co layer. In our BLS measurement s, a p-\npolarized laser  with 300 mW of power and a 532 nm wavelength was used to create and \nannihilate magnons at the surfaces of the Co layer, which is called the Damon -Eshbach \nsurface mode.31 More  detailed descriptions about BLS measurements are given  in the \nreferences .19,32,33  \nFigure 1(a) indicates the frequency differences ( f) between Stokes and anti -Stokes peak \npositions in BLS spectra as a function of the Co thickness. Due to the non -reciprocal SW \npropagation properties with finite iDMI, f is directly proportional to the iDM I energy \ndensity  (D). The correlation between f and the D is given by21 \n∆𝑓=2𝛾𝐷\n𝜋𝑀s𝑘x                                                    (1) \nwhere  𝑘x and D are the propagating spin wave  (SW)  wave  vector along the x-direction and \nthe iDM I energy density, respectively.  In this study, the in-plane k-vector is fixed at 𝑘x = \n0.0167 nm-1, corresponding to the back -scattered light by thermal excitation with an angle of \nincidence of 45o. 4  In Fig. 1(b), the deduced iDM I energy densities based on  Eq. (1) are plotted as a function \nof 𝑡Co−1. The figure  shows inverse proportionality, which is direct evidence that the iDMI at \nthe Ir/Co bilayer is generated  at the interface .19,20 We note  that the iDM I energy densities are \nmuch smaller than in the case of the Pt/Co interface , as shown in Fig. 1(b) . \nNext, we consider ed the magnitude and sign of iDMI for  an Ir/Co /AlO x system. Recently, \nusing  the Vienna ab initio  simulation package (V ASP), Yang et al. 30 reported that the iDMI \nof an Ir/Co system is much smaller than those of Pt/Co systems . They also found that the \nmagnitude and sign of  the iDMI at Ir/Co is very small and has a direction  opposite to the  case \nof the  Pt/Co  interface , respectively. This is consistent with reported experimental \nobservation s of a Pt(111)/Ir( tIr)/[Ni/Co] N and Ta/Pt/Co/Ir( tIr)/Pt system.17,28 Chen et al.28 \nobserved real space images by spin -polarized low -energy electron microscopy (SPLEEM), \nand they found  the transition of DW chirality to be a function of  an Ir inserting layer between \nPt and [N i/Co] N. This implies that the iDMI of the interface of Ir/[Ni/Co] N is opposite to that \nof Pt/[Ni/Co] N. Hrabec et al.17 used a magneto -optical Kerr effect (MOKE) micr oscope to \nobserve asymmetric magnetic domain expansion due to the iDMI , and they inserted  an Ir \nlayer in to the  Ta/Pt/Co/Ir( tIr)/Pt system.  Based on the asymmetric DW velocities , they \nconclude d that the sign of Co/Ir is opposite that of the Co/Pt interfaces.  \nContrary to previous theoretical and experimental result s,17,28,30 our experimental data \nindicate s that the sign of iDMI at the Ir/Co /AlO x interface  is the same with the Pt/Co/AlO x, \nPt/CoFeB/AlO x, and Ta/Pt/Co/AlO x system s.19,20 In order to cross -check  our field dependence \nf measurements  (see Fig. 1(b) ), SW propagation angle  dependence measurements  were \nperformed . The relation between f and 𝜃α, the angle b etween the direction of the \npropagating SWs and the direction of the applied field , is given by34\n∆𝑓(𝜃α)=∆𝑓0sin𝜃α,          (2) \nwhere f0 is the maximum  f when𝜃α equals  ±π/2; this  is a typical measurement geometry  \nfor our magnetic field and SW wave  vector dependence measurements . The angle 𝜃α-\ndependent f measurements from  -90o to +90o with a 22.5o interval  for the Ir/Co ( tCo = 1.75 \nnm and 2.0 nm) and Pt/Co ( tCo = 1.8 nm) cases are shown  in Fig. 2.  All sinusoidal fitting \ncurves are well matched with the measured f as a function of 𝜃α. The green area  showing  5 that f is less than about 0. 3 GHz i ndicates the limit of our BLS experiment s. Consequently , \nwe suggest  that the signs of iDM I are the same at Ir/Co and Pt/Co interfaces.  \nCurrent ly, the physical origin of  the sign of iDMI at Ir/Co /AlO x is not fully unde rstood . \nHowever, there are several possible reasons for the opposite sign reported in previous \nexperimental results .17, 28 Our measurements were conducted on Ir(4 nm) /Co/AlO x, whereas \nthe previous results had different sample stacks ; i.e., Ta/Pt/Co/Ir/Pt17 and \nPt(111)/Ir/[Ni/Co] N.28 For example, Hrabec et al. 17 inserted an Ir layer on the top of a Co \nlayer and covered it with  Pt capping . Chen et al. investigate d the Ir/Ni and Co/Ir interface,28,35 \nnot the Ir/Co interface. Based on our experimental results, we suggest that the iDM I energy \ndensity is quite sensitive to the detail s of the multilayer structures. Moreover, the iDMI is not  \nonly determined by the nearest interfaces , it may have long range characteristics , as predicted \nby Fert and Levy.36  \nWe investigat e other important and SOC -related quantities in  a Ta/Ir/Co/AlO x system by \nusing BLS measurement s. We ignored the effect of iDMI on the SW excitation frequency, \nand took the  average of the Stokes and anti -Stokes SW frequencies without  the iDMI \ncontribution . This can be expressed as37 \n𝑓SW=𝛾\n2𝜋√𝐻ex(𝐻ex−2𝐾eff\n𝜇0𝑀S)            (3) \nHere, we ignored the exchange energy and the bulk PMA contribution . 𝐾eff=2𝐾s\n𝑡Co−1\n2𝜇0𝑀S2, \nKS is the surface anisotropy energy , is the gyromagnetic ratio , and Hex is the applied \nmagnetic field.  We can extract the effective anisotropy energy Keff from the external field -\ndependent  fsw using Eq. (3), and 𝐾eff×𝑡Co versus tCo is shown in Fig. 3. From the slope and \ny-intercept, the saturation magnetization MS (=1.68±0.02×106 A/m) and the surface \nanisotropy energy  KS (=1.36 ±0.01 mJ/m2) were  determined , respectively. For tCo > 1.5 nm, \nthe effective anisotropy becomes negative such that the easy axis of the sample is in -plane.  \nSurprisingly, the obtained  values of  KS and MS at the Ir/Co interface are significantly \nenhanced in comparison with the case of  the Pt/Co  (MS= 1.42±0.02×106 A/m)  interface .20 \nEspecially, the measured MS (=1.68 ×106 A/m) is 20% greater than the bulk magnetization of \nCo (MS = 1.4×106 A/m). Experimental evidence that a large PIM exists at  the Ir/Co interfa ce 6 has recently been reported .27 The reported PIM in an  Ir/Co/Ni/Co system is 19%, which is \nquite similar to our observed value.  \nThe KS value (1.36 mJ/m2) of the Ir/Co system is noticeably enhanced compared to the \nvalues for  Pt/Co/AlO x (KS = 0.54 mJ/m2) and Ta/Pt/Co/AlO x (KS = 1.1 mJ/m2) in our previous \nreports.19,20 On a theoretical basis, it  has been reported that Ir monolayer capping induces the \nstrongest surface PMA of an Fe(001) layer .38 They found that the PIM and the PMA of Ir is \neven larger than  that of  Pt. This gives a clue regarding the observed enhancement s of the \nvalues of  MS and KS in our Ir/Co/AlO x system. Regardless , their study is about 5 d transition \nmetals with Fe , and  not Co . Furthermore, Broeder et al.  reported that KS for Ir/Co (~  0.8 \nmJ/m2) is larger than  that of  Pt/Co (0.5  ~ 0.58 mJ/m2).39 \nAnother important effect of a strong SOC is the enhancement of Gilbert damping due to a \nstrong  spin pumping  effect .10,11 The precess ion of spins  in a ferromagnetic layer induces a \nspin current in the adjacent layer and a loss of angular momentum , and causes additional \ndamping. The amount of spin pumping is closely related to the SOC through the spin flip \nrelaxation time and the interface mixing conductance. As a result, spin pumping is an \nimportant  path for the magnetic damping of HM/FM structure s. It has been reported that spin \npumping can be suppressed by int erface engineering , or by introducing a nano -oxide layer \nbetween HM and FM  by using vector -network analyzer ferromagnetic resonance (VNA -\nFMR) .11 \nWe obtain a full -width at half maximum (FWHM) from each resonance frequency spectrum  \nfrom BLS SW spectra , similar to  the VNA -FMR experiment . To extract the Gilbert damping \nconstant, we applied a modified equation which used from  FMR system  as the condition of \nthe applied large  in-plain magnetic field in a PMA system . The FWHM (∆𝑓res) has the \nfollowing relation  with the Gilbert damping constant 𝛼:  \n∆𝑓res= 2𝛼𝜇0𝛾\n𝜋𝐻ex+∆𝑓resextrinsic      (4) \nwhere ∆𝑓resextrinsic is the additional linewidth at the resonance frequency by an extrinsic \nsource . Figure 4(a) clearly shows the linear relation s between linewidths  (∆𝑓res) and the  \napplied mag netic fields  (𝐻ex) for tCo = 1.5 and 3.0 nm.  7  We examined another possible mechanism of FWHM broadening: the two -magnon \nscattering (TMS) process. It is well known that TMS occurs when 𝜃α(𝑘⃗ ∥), the angle between \nthe SW propagation direction and the in -plane field (or magnetization direction), is smaller \nthan the critical angle 𝜃c.40 Here, the critical angle 𝜃c=sin−1[𝐻ex\n𝐵0+𝐻𝑆]1\n2 = 41.2o when the in -\nplane external field Hex is 0.2 T. B0 = Hex+4πMS; HS = 2KS/Msd; and MS, KS, and d are the \nsaturation magnetization, the surface anisotropy, and the thickness of the cobalt, respectively.  \nThe necessary condition of TMS is 𝜃α(k||) < 𝜃c for k|| > 0, which corresponds to the anti -\nStokes case in our experimental geometry. Since we extracted the FHWM from the anti -\nStokes peaks and  𝜃α(𝑘⃗ ∥)=90o configuration, there is no contribution from TMS. Furthermore, \nif TMS contributes to the linewidth, we require a non -zero f for the 𝜃α(𝑘⃗ ∥)=0 case. \nHowever, we already show a negligible f for 𝜃α(𝑘⃗ ∥)=0 in Fig. 2. Therefore, we can exclude \nthe TMS contribution in the observed FWHM broadening.  \nFrom the slope s of the linear fit tings, the magnetic damping constant s 𝛼 for each Co \nthickness  were  deduced  and are shown  in Fig. 4(b). Normally, the energy dissipation  in a \nmagnetic system depends on the imaginary part of the susceptibility  of the magn etic system. \nIt has been claimed that the imaginary part of the eigenfrequency is modified by a factor of \n(1+𝑓DM(𝑘)/𝑓0(𝑘)) due to the iDMI .23 Here, 𝑓DM  and 𝑓0 are the resonance frequency with \niDMI and without iDMI, respectively. For general cases  (𝑓DM\n𝑓0≪1), the enhancement of \ndamping due to the iDMI  is not significant. However, since  the observed damping constant \nenhancement in our experiment is about double  at tCo = 1.2 nm compared with Bulk, the \nenhancement due to the iDMI is about 10%. Therefore, we can rule out a contribution due to \niDMI in the enhancement of the damping constant.  \n Figure 4(b) shows that  the 𝑡Co−1 dependen ce of the damping constant  is mainly due to  the \nspin pumping effect . Consequently, d ue to spin pumping at the Ir/Co interface, an \nenhancement of the damping constant  (= 0.02) is observed at tCo = 1.5 nm. With increasing \ntCo, the measured  decreases as shown in Fig. 4( b), since spin pumping is a kind of interface \neffect . Therefore, the spin pumping  is being s meared away when the thickness of the FM \nlayer  increases . In Fig. 4(b), t he measured versus 𝑡Co−1 and a linear dependency with a \nfinite y-intercept is seen. The physical meaning of the damping at 𝑡Co−1=0 (tCo→∞) is the 8 damping constant ( bulk) of bulk c obalt . In these  measurement s, we determined that bulk ~ \n0.012, which is in good agreement  with the magnetic damping constant for bulk Co ( 𝛼Cobulk= \n0.011).41  \nIn conclusion, we used BLS to observe  SOC -related physical quantities such as  the \ninterfacial Dzyaloshins kii-Moriya interaction , surface magnetic  anisotropy , and the magnetic \ndamping constant  accompanying the spin pumping effect  at the Ir/Co interface . From \nsystematic BLS measurement s, we suggest that the measured iDM I energy density is \nrelatively smaller than in the case of the Pt/Co interface . On the other hand, the saturation \nmagnetization and the surface magnetic anisotropy are significantly improved due to a higher \nproximity -induced magnetization. From this result, we  believe  that the iDMI and PMA \nbehave  in different way s at the Ir/Co interface . Based on the results in  previous reports , the \nsign of the iDMI at  the Ir/Co interface is the same as that of the Pt/Co interface . 9 Acknowledgement  \nThis work is supported by the Research Programme  of the Foundation for Fundamental \nResearch on Matter (FOM), which is part of the Netherlands Organisation for Scientific \nResearch (NWO) and the National Research Foundation of Korea (Grant nos. \n2015M3D1A1070467, 2013R1A1A2011936, and 2015M2A2A6021171 ). 10 Figure Captions  \nFig. 1. (a) iDM I induced spin -wave frequency differences ( △f) as fun ction of tCo. (b) iDM I \nenergy density as a function of 𝑡Co−1 for the external magnetic field dependence ( DH) in Ir/Co  \n(black squares) , Pt/Co/AlO x (blue circles) , and Ta/Pt/Co/AlO x (red triangles), respectively.  \nThe iDM I energy density  of the Ir/Co  interface has a maximum value  (0.7 mJ/m2) that is \nmuch smaller than that of Pt/Co/AlO x (1.3 mJ/m2) and Ta/Pt/Co/AlO x (1.7 mJ/m2). \n \nFig. 2. The frequency differences ( △f) between Stokes and anti -Stokes fr om 𝜃α= -90∘ to +90∘ \nin Ta/Ir/Co/AlO x (tCo=1.75 nm and  2.0 nm) and Ta/Pt/Co/AlO x (tCo=1.8 nm). The measured f \nas a function of the  𝜃α. The solid line is the fitting curve from  Eq. (2).  \n \nFig. 3. Keff × tCo versus  tCo plot with a linear fitting. KS and MS were extracted from the slope \nand y-intercept . For tCo > 1.54 nm, the effective uniaxial anisotropy becomes negative , which \nmeans the direction of the easy axis changes  perpendicular to  the in-plane.  \n \nFig. 4. (a) The linewidths as a function of Hex for Ta  (4 nm)/Ir  (4 nm)/Co( tCo)/AlO x (2 nm). \nThe black and red open symbols  are experimental values from each of spectr um, and the solid \nlines are results of linear  fitting (tCo = 1.5, 3.0 nm ). The error bars were obtained from \nLorentzian fitting of SW peaks . (b) The Gilbert damping parameters  as function of inverse Co \nthickness. The extrapolated  for infinitely -thick Co is approximately 0.01 2, which is well-\nmatched with the damping constant of bulk cobalt ( 𝛼CoBulk= 0.011 ). \n  11 References  \n                                          \n1 R. C. O’Handley , “Modern Magnetic Materials -Principles and Applications ”, John Wiley & \nSon, pp86 (2000).  \n2 J. Kim, J. Sinha, M. Hayashi, M. Yamanouchi, S. Fukami, T. Suzuki, S. Mitani, and H. \nOhno, Nat. Mater. 12, 240 (2013).  \n3 K.-W. Kim and H. -W. Lee, Nat. Phys. 10, 549 (2014).   \n4 L. Liu, C. -F. Pai, Y . Li, H. W. Tseng, D. C. Ralph, and R. A. Buhrman, Science 336, 555 \n(2012).  \n5 I. M. Miron, T. Moore, H. Szambolics, L. D. Buda -Prejbeanu, S. Auffret, B. Rodmacq, S. \nPizzini, J. V ogel, M. Bonfim, A. Schuhl, and G. Gaudin, Nat. Mater. 10, 419 (2011).  \n6 I. M. Miron, K. Garello, G. Gaudin, P. -J. Zermatten, M. V Costache, S. Auffret, S. Bandiera, \nB. Rodmacq, A. Schuhl, and P. Gambardella, Nature 476, 189 (2011).  \n7 T. Moriya, Phys. Rev. 120, 91 (1960).  \n8 I. E. Dzyaloshins kii, Sov. Phys. JETP, 5, 1259  (1957), I. E. Dzyaloshins kii, Sov. Phys. \nJETP, 20, 665 (1965).  \n9 S. Emori, U. Bauer, S. -M. Ahn, E. Martinez, and G. S. D. Beach, Nat. Mater. 12, 611 \n(2013).  \n10 Y . Tserkovnyak, A. Brataas, and G. E. W. Bauer, Phys. Rev. Lett. 88, 117601 (2002 ). \n11 D.-H. Kim, H .-H. Kim, and C .-Y. You, Appl. Phys. Lett. 99, 072502 (2011).  \n12 A. Thiaville, S. Rohart, É. Jué, V . Cros, and A. Fert, Europhysics Lett. 100, 57002 (2012).  \n13 K.-S. Ryu, L. Thomas, S. -H. Yang, and S. Parkin, Nat. Nanotechnol. 8, 527 (2013).  \n14 A. Fert, V . Cros, and J. Sampaio, Nat. Nanotechnol. 8, 152 (2013).  \n15 J. Sampaio, V . Cros, S. Rohart, a Thiaville, and a Fert, Nat. Nanotechnol. 8, 839 (2013).  \n 12                                                                                                                                  \n16 S.-G. Je, D. -H. Kim, S. -C. Yoo, B. -C. Min, K. -J. Lee, and S. -B. Choe, Phys. Rev. B 88, \n214401 (2013).  \n17 A. Hrabec, N. A. Porter, A. Wells, M. J. Benitez, G. Burnell, S. McVitie, D. McGrouther, T. \nA. Moore, and C. H. Marrows, Phys. Rev. B 90, 020402 (2014).   \n18 R. Lavrijsen, D. M. F. Hartmann, A. van den Brink, Y . Yin, B. Barcones, R. A. Duine, M. \nA. Verheijen, H. J. M. Swagten, and B. Koopmans, Phys. Rev. B 91, 104414 (2015).  \n19 J. Cho, N .-H. Kim, S . Lee, J .-S. Kim, R . Lavrijsen, A . Solignac, Y . Yin, D .-S. Han, N . J. J. \nvan Hoof, H . J. M. Swagten, B . Koopmans, and C .–Y. You, Nat. Comm. 6, 7635 (2015).  \n20 N.-H. Kim, D. -S. Han, J. Jung, J. Cho, J. -S. Kim, H. J. M. Swagten, and C. -Y . You, Appl. \nPhys. Lett. 107, 142408 (2015).  \n21 J. H. Moon, S. M. Seo, K. J. Lee, K. W. Kim, J. Ryu, H. W. Lee, R. D. McMichael, and M. \nD. Stiles, Phys. Rev. B 88, 184404 (2013).  \n22 M. Belmeguenai, J. -P. Adam, Y . Roussign é, S. Eimer, T. Devolder, J. -V . Kim, S. M. Cherif, \nA. Stashkevich, and A. Thiaville, Phys. Rev. B 91, 180405(R) (2015).  \n23 K. Di, V . L. Zhang, H . S. Lim, S . C. Ng, M . H. Kuok, J . Yu, J . Yoon, X . Qiu, and H . Yang , \nPhys. Rev. Lett. 114, 047201 (2015).  \n24 H. T. Nembach, J. M. Shaw, M. Weiler, E. Ju é, and T. J. Silva, Nat. Phys. 11, 825 (2015).  \n25 P. Bruno, Phys. Rev. B 39, 865 (1989).  \n26 D. S. Wang, R. Wu, and A. J. Freeman, Phys. Rev. B 47, 14932 (1993).  \n27 K.-S. Ryu , S.-H. Yang, L. Thomas, and S. S. P. Parkin, Nat. Commun. 5, 3910 (2014).  \n28 G. Chen, T. Ma, A. T. N ’Diaye, H. Kwon, C. Won, Y . Wu and A. K. Schmid, Nat. Commun.  \n4, 2671 (2013).  \n 13                                                                                                                                  \n29 H. Yang, S. Rohart, A. Thiaville, A. Fert, and M. Chshiev, in APS March Me eting 2015, \nSan Antonio, Texas, USA, 2 March –6 March 2015, Bulletin of the American Physical Society, \n60 (2015).  \n30 H. Yang, A. Thiaville, S. Rohart, A. Fert, and M. Chshiev, arXiv:1501.05511 (2015).  \n31 R.W. Damon, J.R. Eshbach, J. Phys. Chem. Solids 19, 308  (1961).  \n32 S.-S. Ha, N. -H. Kim, S. Lee, C. -Y . You, Y . Shiota, T. Maruyama, T. Nozaki, Y . Suzuki, \nAppl. Phys. Lett. 96, 142512 (2010).  \n33 J. Cho, J. Jung, K. -E. Kim, S. -I. Kim, S. -Y . Park, M. -H. Jung, C. -Y . You, J. Mag. Mag. \nMat. 339, 36 (2013).  \n34 D. Cortés -Ortuño and P. Landeros, J. Phys. Cond. Matter 25, 156001 (2013).  \n35 G. Chen, A. T. N ’Diaye, Y . Wu, and A. K. Schmid , Appl. Phys. Lett. 106, 062402 (2015) . \n36 A. Fert and P. M. Levy, Phys. Rev. Lett. 44, 1538 (1980).  \n37 J. R. Dutcher, B. Heinrich, J. F. Cochran, D. A. Steigerwald, and W. F. Egelhoff Jr., J. Appl. \nPhys. 63, 3464 (1988).  \n38 D. Odkhuu, S. H. Rhim, N. Park, and S. C. Hong , Phys. Rev. B 88, 184405 (2013) . \n39 F. J. A. den Broeder, W. Hoving, and P. J. H. Bloemen , J. Magn. Magn. Mater. 93, 562 \n(1991 ). \n40 R. Arias and D. L. Mills , Phys. Rev. B 60, 7395 (1999) . \n41 S. Pal, B. Rana, O. Hellwig, T. Thomson, and A. Barman, Appl. Phys. Lett. 98, 082501 \n(2011).  "    },    {        "title": "1601.06213v1.Nonlinear_magnetization_dynamics_of_antiferromagnetic_spin_resonance_induced_by_intense_terahertz_magnetic_field.pdf",        "content": " 1Nonlinear magnetization dyna mics of antiferromagnetic \nspin resonance induced by inte nse terahertz magnetic field \n \nY Mukai1,2,4,6, H Hirori2,3,4,7, T Yamamoto5, H Kageyama2,5, and K Tanaka1,2,4,8 \u001f \n1 Department of Physics, Graduate School of  Science, Kyoto University, Sakyo-ku, Kyoto \n606-8502, Japan  \n2 Institute for Integrated Cell-Material Scien ces (WPI-iCeMS), Kyoto University, Sakyo-ku, \nKyoto 606-8501, Japan \n3 PRESTO, Japan Science and Technology Agency, Kawaguchi, Saitama 332-0012, Japan \n4 CREST, Japan Science and Technology Agency, Kawaguchi, Saitama 332-0012, Japan \n5 Department of Energy and Hydrocarbon Chemistr y, Graduate School of Engineering, Kyoto \nUniversity, Nishikyo-ku, Kyoto 615-8510, Japan \n \nE-mail: \n6 mukai@scphys.kyoto-u.ac.jp \n7 hirori@icems.kyoto-u.ac.jp \n8 kochan@scphys.kyoto-u.ac.jp \n We report on the nonlinear magnetization dynamics of a HoFeO\n3 crystal induced by a strong \nterahertz magnetic field resonantly enhanced with a split ring resonator and measured with \nmagneto-optical Kerr effect microscopy. The terahertz magnetic field induces a large change (~40%) in the spontaneous magnetization. The frequency of the antife rromagnetic resonance \ndecreases in proportion to the square of the magnetization change. A modified \nLandau-Lifshitz-Gilbert equation with a phenomenological nonlinear damping term \nquantitatively reproduced the nonlinear dynamics. \nPACS: 75.78.Jp, 76.50.+g, 78.47.-p, 78.67.Pt \n    21. Introduction \nUltrafast control of magnetization dynamics by a femtosecond optical laser pulse has \nattracted considerable attention from the persp ective of fundamental physics and technological \napplications of magnetic recording and inform ation processing [1]. The first observation of \nsubpicosecond demagnetization of a fe rromagnetic nickel film demonstrated that a femtosecond \nlaser pulse is a powerful stimulus of ultrafast magnetization dynamics [2], and it has led to numerous theoretical and experimental inves tigations on metallic and semiconducting magnets \n[3-8]. The electronic state created by the laser pulse has a strongly nonequilibrium distribution \nof free electrons, which consequently leads to  demagnetization or even magnetic reversal \n[1,2,9-11]. However, the speed of the magnetizat ion change is limited by the slow thermal \nrelaxation and diffusion, and an alternative t echnique without the limits of such a thermal \ncontrol and without excessive thermal energy would be desirable.  \nIn dielectric magnetic media, carrier heating hardly occurs, since no free electrons are present \n[12]. Consequently, great effort has been devoted  to clarifying the spin dynamics in magnetic \ndielectrics by means of femtosecond laser pulses. A typical method for nonthermal optical \ncontrol of magnetism is the inverse Faraday effect, where circularly polarized intense laser \nirradiation induces an effective magnetic field in  the medium. Recently, new optical excitation \nmethods avoiding the thermal effect such as the ma gneto-acoustic effect is also reported [13,14]. \nIn particular, these techniques have been used in  many studies on antiferromagnetic dielectrics \nbecause compared with ferromagne ts, antiferromagnets have inhe rently higher spin precessional \nfrequencies that extend into the terahertz (THz) regime [12,15]. Additionally, ultrafast \nmanipulation of the antiferromagnetic order parame ter may be exploited in order to control the \nmagnetization of an adjacent ferromagnet through the exchange interaction [16]. The THz wave \ngeneration technique is possibly a new way of optical spin control through direct magnetic \nexcitation without undesirable thermal effects [17-19]. As yet however, no technique has been \nsuccessful in driving magnetic motion excited directly by a magnetic field into a nonlinear \ndynamics regime that would presumably be fo llowed by a magnetization reversal [20-22]. \n \nIn our previous work [23], we demonstrated that the THz magnetic field can be resonantly \nenhanced with a split ring resonator (SRR) and may become a tool for the efficient excitation of \na magnetic resonance mode of antiferromagnetic dielectric HoFeO\n3. We applied a Faraday \nrotation technique to detect the magnetization ch ange but the observed Faraday signal averaged  3the information about inhomogeneous magnetiza tion induced by localized THz magnetic field \nof the SRR over the sample thickness [23]. In th is Letter, we have developed a time-resolved \nmagneto-optical Kerr effect (MOKE) micr oscopy in order to access the extremely \nfield-enhanced region, sample surface near th e SRR structure. As a result, the magnetic \nresponse deviates from the linear response in the strong THz magnetic field regime, remarkably \nshowing a redshift of the antiferromagnetic r esonance frequency that is proportional to the \nsquare of the magnetization change. The observe d nonlinear dynamics could be reproduced with \na modified Landau-Lifshitz-Gilbert (LLG) e quation having an additional phenomenological \nnonlinear damping term.  \n2. Experimental setup \nFigure 1 shows the experimental setup of MOKE microscopy with a THz pump pulse \nexcitation. Intense single-cycle THz pulses were  generated by optical rectification of \nnear-infrared (NIR) pulses in a LiNbO\n3 crystal [24-26]; the maximum peak electric field was \n610 kV/cm at focus. The sample was a HoFeO 3 single crystal polished to a thickness of 145 µm, \nwith a c-cut surface in the Pbnm  setting [27]. (The x-, y-, and z-axes are parallel to the \ncrystallographic a-, b-, and c-axes, respectively. ) Before the THz pump excitation, we applied a \nDC magnetic field to the sample to saturate its magnetization along the crystallographic c-axis. We fabricated an array of SRRs on the crystal surface by using gold with a thickness of 250 nm. \nThe incident THz electric field, parallel to the metallic arm with the SRR gap (the x-axis), drove \na circulating current that resulted in a strong  magnetic near-field normal to the crystal surface \n[23,28,29]. The SRR is essentially subwavelength LC circuit, and the current induces magnetic \nfield B\nnr oscillating with the LC resonance frequency (the Q-factor is around 4). The right side \nof the inset in figure 1 shows the spatial distribut ion of the magnetic field of the SRR at the LC \nresonance frequency as calculated by the fin ite-difference time-domain (FDTD) method. \nAround the corner the current density in the metal is very high, inducing the extremely \nenhanced magnetic field in the HoFeO 3 [29]. \n \nAt room temperature, the two magnetizations mi (i=1,2) of the different iron sublattices in \nHoFeO 3 are almost antiferromagnetically aligned along the x-axis with a slight canting angle \n0(=0.63°) owing to the Dzyaloshinskii fiel d and form a spontaneous magnetization MS along \nthe z-axis [30]. In the THz region, ther e are two antiferromagnetic resonance modes \n(quasiantiferromagnetic (AF) and quasiferroma gnetic (F) mode [31]). The magnetic field Bnr  4generated along the z-axis in our setup causes AF-mode motion; as illustrated in figure 2(a), the \nZeeman torque pulls the spins along the y-ax is, thereby triggering precessional motions of mi \nabout the equilibrium directions. The precessional motions cause the macroscopic magnetization M=m\n1+m2 to oscillate in the z-direction [32,33]. The resultant magnetization \nchange Mz(t) modulates the anti-symmetric off-diagonal element of the dielectric tensor \nεxyaሺൌ െεyxaሻ and induces a MOKE signal (Kerr ellipticity change \u001f [34,35] (see Appendix A \nfor the detection scheme of the MOKE measuremen t). The F-mode oscillation is also excited by \nTHz magnetic field along the x or y-axis. Howe ver, the magnetization deviations associated \nwith the F-mode, Mx and My, do not contribute to the MOKE in our experimental geometry, \nwhere the probe light was incident no rmal to the c-cut surface of HoFeO 3 (the xy-plane) [34,35]. \nIn addition, the amplitude of the F-mode is much smaller than AF-mode because the F-mode \nresonance frequency ( F~0.37 THz) differs from the LC resonance frequency ( LC~0.56 THz). \n-10010x position (µm)\n-10 0 10\ny position (µm)\nTHz pump HoFeO3 \nz y \nx \nSRR  \nObjective lens \nNonpolarized \nbeam splitter Quarter \nwave plateWollaston  \nprism \nLens  Balanced photodiodes \nVisible probe Bin Bnr \nEin \n-10 0 10\ny position (µm)\n120\n80\n40\n0\nFigure 1. Schematic setup of THz pump-visible MOKE measurement. The left side\nof the inset shows the photograph of SRR fabricated on the c-cut surface of the\nHoFeO 3 crystal and the white solid line indicat es the edge of the SRR. The red soli d\nand blue dashed circles indicate the probe spots for the MOKE measurement. The\nright side of the inset shows the spatia l distribution of the enhancement facto r\ncalculated by the FDTD method, i.e., the ratio between the Fourier amplitude at LC\nof the z-component of Bnr (at z=0) and the incident THz pulse Bin.  5To detect the magnetization change induced onl y by the enhanced magnetic field, the MOKE \nsignal just around the corner of the SRR (indicated by the red circle in figure 1’s inset), where \nthe magnetic field is enhanced 50-fold at the LC  resonance frequency, was measured with a 400 \nnm probe pulse focused by an objective lens (spot diameter of ~1.5 µm). Furthermore, although \nthe magnetic field reaches a maximum at the surface and decreases along the z-axis with a \ndecay length of lTHz~5 µm, the MOKE measurement in refl ection geometry, in contrast to the \nFaraday measurement in transmission [23], can evaluate the magnetization change induced only \nby the enhanced magnetic field around the sample  surface since the penetration depth of 400 nm \nprobe light for typical orthoferrites is on the orde r of tens of nm [35]. (The optical refractive \nindices of rare-earth orthoferrites in th e near ultraviolet region including HoFeO 3 are similar to \neach other, regardless of the rare-earth ion speci es, because it is mostly determined by the strong \noptical absorption due to charge transfer and orbital promotion transitions inside the FeO 6 \ntetragonal cluster [35].) All experiments in this  study were performed at room temperature. \n \n3. Results and discussions   \nFigure 2(a) (upper panel) shows the calculated  temporal magnetic waveform together with \nthe incident magnetic field. The maximum peak am plitude is four times that of the incident THz \npulse in the time domain and reaches 0.91 T. Th e magnetic field continues to ring until around \n25 ps after the incident pulse has decayed away. The spectrum of the pulse shown in figure 2(c) \nhas a peak at the LC resonance frequency ( LC=0.56 THz) of the SRR, which is designed to \ncoincide with the resonance frequency of the AF-mode ( νAF0=0.575 THz). Figure 2(a) (lower \npanel) shows the time development of the MOKE signal  for the highest THz excitation \nintensity (pump fluence I of 292 µJ/cm2 and maximum peak magnetic field Bmax of 0.91 T). The \ntemporal evolution of  is similar to that of the Faraday rotation measured in the previous \nstudy and the magnetization oscillates harmonically with a period of ~2 ps [23], implying that \nthe THz magnetic field coherently drives the AF-mode motion.  \n As shown in figure 2(b), as th e incident pump pulse intensity increases, the oscillation period \nbecomes longer. The Fourier transform spectra  of the MOKE signals for different pump \nintensities are plotted in figure 2(c). As the ex citation intensity increases, the spectrum becomes \nasymmetrically broadened on the lower freque ncy side and its peak frequency becomes \nredshifted. Figure 2(d) plots the center-of-mass fre quency (open circles) and the integral (closed \ncircles) of the power spectrum P(\n) as a function of incident pulse fluence. The center  6frequency monotonically redshifts and P() begins to saturate. As shown in figure 2(c), the \nMOKE spectra obtained at the center of the SRR (indicated in the inset of figure 1) does not \nshow a redshift even for the highest intensity excitation, suggesting that the observed redshift \noriginates from the nonlinearity of the precessional spin motion rather than that of the SRR \nresponse. We took the analytic signal approach  (ASA) to obtain the time development of the \ninstantaneous frequency (t) (figure 3(c)) and the envelope amplitude 0(t) (figure 3(d)) from \nthe measured magnetization change (t)=Mz(t)/|MS| (figure 3(b)) (see Appendix B for the \ndetails of the analytic signal approach). As is  described in the Appendix C, the MOKE signal \n6\n4\n2\n0\nIntegral of P( ) \n(arb. units)\n300 200 100 0\nFluence (µJ/cm2)0.575\n0.570\n0.565Frequency (THz)1.0\n0.8\n0.6\n0.4\n0.2\n0.0Intensity P( ) (arb. units)\n0.60 0.58 0.56 0.54\nFrequency (THz)  50%\n   100%\n 100% (x 3.7)\n (center)\n \n  10%\n |Bnr|2\n \nz \nm2 m1 \nM \nx Bnr \ny (a) (c) \n(d) (b) -0.020.000.02∆(degrees)\n40 30 20 10 0\nTime (ps)1.0\n0.5\n0.0\n-0.5B (T)  Bnr\n  Bin (x 3)\n0.08\n0.06\n0.04\n0.02\n0.00∆(degrees)\n24 20 16 12 8\nTime (ps)10%100%\nFigure 2. (a) Upper panel: Incident magnetic field of the THz pump pulse Bin estimated by\nelectro-optic sampling (dashed line) and the THz magnetic near-field Bnr calculated by the\nFDTD method (solid line). The illustration s hows the magnetization motion for the AF-mode.\nLower panel: The MOKE signal for a pump fluence of 292 µJ/cm2 (100%). (b) Comparison o f\ntwo MOKE signals for different pump fluences, vertically offset for clarity. (c) The FFT powe r\nspectrum of the magnetic near-field Bnr (black solid line). The spectra P() of the MOKE\nsignals for a series of pump fluences obtained at th e corner (solid lines) and at the center (blue\ndashed circle in the inset of figure 1) for a pump fluence of 100% (dashed line). Each spectru m\nof the MOKE signal is normalized by the peak amplitude at the corner for a pump fluence o f\n100%. (d) Intensity dependence of the center-of-m ass frequency (open circles) and the integral\n(closed circles) of the P().  7(t) is calibrated to the magnetization change (t) by using a linear relation, i.e., (t)=g(t), \nwhere g (=17.8 degrees−1) is a conversion coefficient. The tim e resolved experiment enables us \nto separate the contributions of the applied magnetic field and magnetization change to the \nfrequency shift in the time domain. A comparison of the temporal profiles between the driving \nmagnetic field (figure 3(a)) and the frequency e volution (figure 3(c)) shows that for the low \npump fluence (10%, closed blue circles), the frequency is redshifted only when the magnetic \nfield persists ( t < 25 ps), and after that, it recovers  to the constant AF mode frequency \n(νAF0=0.575 THz). This result indicates that the signals below t = 25 ps are affected by the \npersisting driving field and the redshift may orig inate from the forced oscillation. As long as the \n0.575\n0.570\n0.565\n0.560\n0.555Frequency (THz)\n50 40 30 20 10 0\nTime (ps)-0.4-0.20.00.20.4Magnetization \nchange 0.5\n0.0\n-0.5Bnr (T)\n0.4\n0.3\n0.2\n0.1\n0.0Amplitude 0  Experiment\n100%\n  10%\n  Experiment\n100%\nSimulation\n100% (1=0)\n100% \n  10% Simulation(a) \n(b) \n(c) \n(d) \nFigure 3. (a) FDTD calculated magnetic field Bnr for pump fluence of 100%. (b) Temporal\nevolution of the magnetization change obtained from the experimental data (gray circles) an d\nthe LLG model (red line). (c) Instantaneous frequencies and (d) envelope amplitudes fo r\npump fluences of 100% and 10% obtained by the analytic signals calculated from the\nexperimental data (circles) and the LLG simulation with nonlinear damping paramete r\n(1=1×10−3, solid lines) and without one ( 1=0, dashed line).  8magnetic response is under the linear regime, the instantaneous frequency is independent on the \npump fluence. However, for the high pump fluenc e (100%) a redshift (a maximum redshift of \n~15 GHz relative to the constant frequency νAF0) appears in the delay time ( t < 25 ps) and even \nafter the driving field decays away ( t > 25 ps) the frequency continues to be redshifted as long \nas the amplitude of the magnetization change is large. These results suggest that the frequency \nredshift in the high intensity case depends on the magnitude of the magnetization change, \nimplying that its origin is a nonlinear precessional spin motion with a large amplitude. \n \nThe temperature increase due to  the THz absorption (for HoFeO 3 T=1.7×10−3 K, for gold \nSRR T=1 K) is very small (see Appendix D). In ad dition, the thermal relaxation of the spin \nsystem, which takes more than a nanosecond [36], is much longer than the frequency \nmodulation decay (~50 ps) in figure 3(c). Therefore, laser heating can be ignored as the origin of the redshift.  \n \nFigure 4 shows a parametric plot of the instantaneous frequency \n(t) and envelope amplitude \n0(t) for the high pump fluence (100%). The instantaneous frequency shift for t > 25 ps has a \nsquare dependence on the amplitude, i.e., νAF=νAF0(1െCζ02). To quantify the relationship \nbetween the redshift and magnetization change, it  would be helpful to have an analytical \nexpression of the AF mode frequency AF as a function of the magnetization change, which is \nderived from the LLG equation based on the two- lattice model [32,33]. The dynamics of the \nsublattice magnetizations mi (i=1,2), as shown in the inset of figure 2(a), are described by \n \n dRi\ndt=െγ\n(1+α2)ቀRi×[B(t)+Beff,i]െαRi×൫Ri×[B(t)+Beff,i]൯ቁ,  (1) \n \nwhere Ri=mi/m0 (m0=|mi|) is the unit directional vector of the sublattice magnetizations, \n=1.76×1011 s−1T−1 is the gyromagnetic ratio, V(Ri) is the free energy of the iron spin system \nnormalized with m0, and Beff,i is the effective magnetic field given by െ∂V/∂Ri (i=1,2) (see \nAppendix E). The second term represents the ma gnetization damping with the Gilbert damping \nconstant \u001f \n \nSince Beff,i depends on the sublattice magnetizations mi and the product of these quantities \nappears on the right side of Eq. (1), the LLG e quation is intrinsically nonlinear. If the angle of  9the sublattice magnetization precession is sufficien tly small, Eq. (1) can be linearized and the \ntwo fixed AF- and F-modes for the weak excitation can be derived. However, as shown in figure \n3(b), the deduced maximum magnetization change  reaches ~0.4, corresponding to precession \nangles of 0.25° in the xz-plane and 15° in th e xy-plane. Thus, the magnetization change might \nbe too large to use the linear approximation. For such a large magnetization motion, assuming \nthe amplitude of the F-mode is zero and =0 in Eq. (1), the AF mode frequency AF in the \nnonlinear regime can be deduced as  \n \n νAF =νAF0ට1ିζ02tan2β0\nK(D),       ( 2 )  \n D =ඨ\t\t\t\t\t\tζ02(rAF2ି1) tan2β0\n1ିζ02tan2β0,      ( 3 )  \n \n0.575\n0.570\n0.565Frequency (THz)\n0.4 0.3 0.2 0.1 0.0\nAmplitude 0Experiment\n t > 25 ps\n t < 25 ps\n \n Analytic Solution\n 2nd order expansion\n \nFigure 4.  Relation between instantaneous frequency  and envelope amplitude 0 obtaine d\nfrom the magnetization change; for t < 25 ps (open circles) and for t > 25 ps (closed circles),\nthe analytic solution (blue line) and second orde r expansion of the analytic solution (gree n\ndashed line). Errors are estim ated from the spatial inhomogeneity of the driving magnetic\nfield (see Appendix H).  10where K(D) is the complete elliptic integral of the first kind, rAF(≈60) is the ellipticity of the \nsublattice magnetization precession trajectory of the AF-mode (see Appendix F), and 0 is the \namplitude of the (t). As shown in figure 4, the analytic solution can be approximated by the \nsecond order expansion νAF≈νAF0(1െtan2β0(rAF2െ1)ζ024⁄) and matches the observed redshift \nfor t > 25 ps, showing that the frequency appr oximately decreases with the square of (t). The \ndiscrepancy of the experimental data from the theoretical curve ( t < 25 ps) may be due to the \nforced oscillation of the AF-mode caused by the driving field.  \n \nTo elaborate the nonlinear damping effects, we compared the measured (t) with that \ncalculated from the LLG equation with the damping term. As shown in figures 3(c) and 3(d), the \nexperiment for the high intensity excitation devi ates from the simulation with a constant Gilbert \ndamping (dashed lines) even in the t > 25 ps time region, suggesting nonlinear damping \nbecomes significant in the large amplitude region. To describe the nonlinear damping \nphenomenologically, we modified the LLG equa tion so as to make the Gilbert damping \nparameter depend on the displacement of th e sublattice magnetization from its equilibrium \nposition, (Ri)=0+1Ri. As shown in figures 3(b)-(d), the magnetization change (t) derived \nwith Eq. (1) (solid line) with the damping parameters ( 0=2.27×10−4 and 1=1×10−3) nicely \nreproduces the experiments for both the high (100%) and low (10%) excitations.1 These results \nsuggest that the nonlinear damping plays a signifi cant role in the large amplitude magnetization \ndynamics. Most plausible mechanism for the nonlinear damping is four magnon scattering \nprocess, which has been introdu ced to quantitatively evaluate the magnon mode instability of \nferromagnet in the nonlinear response regime [37]. \n \n4. Conclusions  \nIn conclusion, we studied the nonlinear magnetization dynamics of a HoFeO 3 crystal excited \nby a THz magnetic field and measured by MOKE microscopy. The intense THz field can induce \nthe large magnetization change (~40%), and the ma gnetization change can be kept large enough \n                                            \n1 The damping parameter 0 (=2.27×10−4) and conversion coefficient  g (=17.8 degrees−1) are \ndetermined from the least-squares fit of the calculated result without the nonlinear damping \nparameter 1 to the experimental MOKE signal for the low pump fluence of 29.2 µJ/cm2. The \nnonlinear damping parameter 1 (=1×10−3) is obtained by fitting the experimental result for the \nhigh intensity case ( I=292 µJ/cm2) with the values of 0 and g obtained for the low excitation \nexperiment. The estimated value of g is consistent with the stat ic MOKE measurement; the Kerr \nellipticity induced by the spontaneous magnetization MS is ~0.05 degrees ( g~20 degrees−1). See \nAppendix G for details on the static Kerr measurement.  11to induce the redshift even after the field has gone , enabling us to separate the contributions of \nthe applied magnetic field and ma gnetization change to the frequency shift in the time domain. \nThe resonance frequency decreases in proportion to  the square of the magnetization change. A \nmodified LLG equation with a phenomenologi cal nonlinear damping term quantitatively \nreproduced the nonlinear dynamics. This suggest s that a nonlinear spin relaxation process \nshould take place in a strongly driven regime. Th is study opens the way to the study of the \npractical limits of the speed and efficiency of magnetization reversal, which is of vital \nimportance for magnetic recording and information processing technologies. \n   12Acknowledgments \nWe are grateful to Shintaro Takayoshi, Masah iro Sato, and Takashi Oka for their discussions \nwith us. This study was supported by a J SPS grants (KAKENHI 26286061 and 26247052) and \nIndustry-Academia Collaborative R&D grant fro m the Japan Science and Technology Agency \n(JST). \n   13Appendix A. Detection sche me of MOKE measurement \nWe show the details of the detection scheme  of the MOKE measurement. A probe pulse for \nthe MOKE measurement propagates along the z direction. By using the Jones vector [38], an electric field E\n0 of the probe pulse polarized linearly along the x-axis is described as  \n \n E0 =ቀ1\n0ቁ.       ( A . 1 )  \n \nThe probe pulse E1 reflected from the HoFeO 3 surface becomes elliptically polarized with a \npolarization rotation angle and a ellipticity angle . It can be written as \n \n E1 =R(െ߶)MR(θ)E0ൌ൬cos θ cos ߶െ\t݅ sin θ sin ߶\ncos θ sin ߶\t݅ sin θ cos ߶൰,  (A.2) \n  \nwhere  M is the Jones matrix describing \u001f\u001f phase retardation of the y component with \nrespect to the x component \n \nM=ቀ10\n0െiቁ,       ( A . 3 )  \n \nand R(ψ) is the rotation matrix \n \nR(ψ)=൬cosψ sinψ\nെsinψcosψ൰.      (A.4)  \n \nThe reflected light passes through the quarter wave plate, which is arranged such that its fast \naxis is tilted by an angle of 45° to the x-axis. The Jones matrix of the wave plate is given by \n \nRቀെπ\n4ቁMRቀπ\n4ቁ.        ( A . 5 )  \n Thus, the probe light E\n2 after the quarter wave plate is described as follows, \n  14E2 = ൬E2,x\nE2,y൰=Rቀെπ\n4ቁMRቀπ\n4ቁE1  \n=1\n2൬cosሺθ߶ሻെsin (θെ߶+)i(െcosሺθെ߶ሻsin (θ߶))\ncosሺθെ߶ሻsin (θϕ)+i(c o sሺθ߶ሻsin (θെ߶))൰.  (A.6)  \n \nThe Wollaston prism after the quarter wave plat e splits the x and y-polarization components of \nthe probe light E2. The spatially separated two pulses are incident to the balanced detector and \nthe detected probe pulse intensity ratio of the di fferential signal to the total corresponds to the \nKerr ellipticity angle as follows,    \n〈หாమ,ೣหమ〉ି〈หாమ,หమ〉\n〈หாమ,ೣหమ〉ା〈หாమ,หమ〉ൌെsin2θ.      ( A . 7 )  \n \nIn the main text, we show the Kerr ellipticity change =w−wo, where the ellipticity angles \n(w and wo) are respectively obtained with and without the THz pump excitation. \n \nAppendix B. Analytic signal approach and short time Fourier transform \nThe Analytic signal approach (ASA) allows the extraction of the time evolution of the \nfrequency and amplitude by a simple procedure and assumes that the signal contains a single \noscillator component. In our study, we measure only the MOKE signal originating from the \nAF-mode and it can be expected that the single oscillator assumption is valid. In the ASA, the \ntime profile of the magnetization change (t) is converted into an analytic signal (t), which is a \ncomplex function defined by using the Hilbert transform [39]; \n \nψ(t)=ζ0(t)exp( i߶(t))=ζ(t)+i ζ෨(t),     (B.1) \nζ෨(t)ൌ1\nπ pζ(t)\ntିτ∞\n-∞ dτ.       ( B . 2 )  \n \nwhere the p is the Cauthy principal value. The real part of (t) corresponds to (t). The real \nfunction 0(t) and (t) represent the envelope amplitude  and instantaneous phase of the \nmagnetization change. The instantaneous frequency (t)(=2(t)) is given by (t)=d(t)/dt. In \nthe analysis, we averaged 0(t) and (t) over a ten picosecond time range. \n \nTo confirm whether the ASA gives appropriate results, as shown in figure B.1 we compare  15them with those obtained by the short time Fourie r transform (STFT). As shown in figure B.1(a), \nthe time-frequency plot shows only one oscillato ry component of the AF-mode. As shown in \nfigures B.1(b) and (c), the instantaneous freque ncies and amplitudes obtained by the ASA and \nthe STFT are very similar. Because the ASA provides us the instantaneous amplitude with a \nsimple procedure, we showed the time evolu tions of frequency and amplitude derived by the \nASA in the main text. \n \nAppendix C. Determination of conversion coefficient g and linear damping parameter 0 \nThe conversion coefficient  g and the linear damping parameter 0(=) in Eq. (1) are \ndetermined by fitting the experimental MOKE signal (t) for the low pump fluence of 29.2 \nµJ/cm2 with the LLG calculation of the magnetization change (t). Figure C.1 shows the MOKE \nsignal (t) (circle) and the calculated magnetization change (t) (solid line). From the \nleast-squares fit of the calculated result to th e experiment by using a linear relation, i.e., \n(t)=g(t), we obtained the parameters g(=17.8 degrees−1) and 0(=2.27×10−4). 0.575\n0.570\n0.565\n0.560\n0.555\n0.550Frequency (THz)\n50 40 30 20 100\nTime (ps)ASA\n 100%\n   10%\nSTFT\n 100%\n   10%\n 1.0\n0.8\n0.6\n0.4\n0.2\n0.0\nFourier am plitude (arb. units)\n50 40 30 20 100\nTime (ps)0.4\n0.3\n0.2\n0.1\n0.0Amplitude 0ASA\n   100%\n     10% \nSTFT\n   100%\n     10%\n (a) (b) (c) \n1.2\n1.0\n0.8\n0.6\n0.4\n0.2\n0.0Frequency (THz)\n5040302010\nTime (ps)(arb. units)\n1.0 0.0\nFigure B.1.  (a) Time-dependence of the power spectrum of the magnetization \noscillation for the highest THz excitation ( I=292 µJ/cm2) obtained by the STFT. \nComparison of (b) instantaneous frequencies and (c) amplitudes obtained by the ASA \nand STFT with a time window with FWHM of 10 ps.   16 \n \nAppendix D. Laser heating effect \nThe details of the calculation of the temperature change are as follows: \n \nFor HoFeO 3:  \nThe absorption coefficient abs of HoFeO 3 at 0.5 THz is ~4.4 cm−1 [40]; the fluence IHFO \nabsorbed by HoFeO 3 can be calculated as IHFO=I(1−exp(−absd)), where d (=145 µm) is the \nsample thickness and I is the THz pump fluence. For the highest pump fluence, I=292 µJ/cm2, \nIHFO is 18.1 µJ/cm2. Since the sample thickness is much smaller than the penetration depth, \nd≪abs−1, we assume that the heating of the sample due to the THz absorption is homogeneous. \nBy using the heat capacity Cp of 100 J mol−1 K−1 [27], and the molar volume v of ~1.4×102 \ncm3/mol [27], the temperature change T can be estimated as\u001f T=IHFOv/Cpd ~1.7×10−3 K. \n \nFor gold resonator (SRR): \nThe split ring resonator has an absorption band (center frequency ~0.56 THz, band width ~50 \nGHz) originated from the LC resonance (figure 2( c)). Assuming the SRR absorbs all incident \nTHz light in this frequency band, the absorbed energy accounts for 3 % of the total pulse energy. \nHence, for the highest THz pump fluence, I=292 µJ/cm2, the fluence absorbed by the SRR is \nIgold=8.76 µJ/cm2. By using the heat capacity Cp of 0.13 J g−1 K−1 [41], the number of the SRRs \nper unit area N of 4×104 cm−2, and the mass of the SRR m of 1.6×10−9 g, the temperature change -10x10-3-50510degrees)\n50 40 30 20 10 0\nTime (ps)-0.10.00.1\nMagnetization \nchange  Experiment\n Simulation\nFigure C.1.  Experimentally observed MOKE signal \u001f(circle) and LLG simulatio n\nresult of the magnetization change \u001f(solid line) for the pump fluence of 29.2\nµJ/cm2.  17T can be estimated as\u001f T=Igold/CpNm ~ 1 K \n \nAppendix E. Free energy of HoFeO 3 \nThe free energy F of the iron spin (Fe3+) system based on the two-lattice model is a function \nof two different iron sublattice magnetizations mi, and composed of the exchange energy and \none-site anisotropy energy [32,33]. The free en ergy normalized by the sublattice magnetization \nmagnitude, V=F/m0 (m0=|mi|), can be expanded as a power series in the unit directional vector of \nthe sublattice magnetizations, Ri=mi/m0=(Xi,Yi,Zi). In the magnetic phase 4 (T > 58K), the \nnormalized free energy is given as follows [32,33]: \n \nV=ER1·R2+D(X1Z2െX2Z1)െAxx(X12+X22)െAzz(Z12+Z22),  (E.1) \n \nwhere E(=6.4×102 T) and D(=1.5×10 T) for HoFeO 3 are respectively the symmetric and \nantisymmetric exchange field [42]. Axx and Azz are the anisotropy constants. As mentioned in \nAppendix F, the temperature dependent values of the anisotropy constants can be determined \nfrom the antiferromagnetic resonance frequencies. The canting angle of Ri to the x-axis β0 \nunder no magnetic field is given by \n \ntan 2β0=D\nE+AxxିAzz.        ( E . 2 )  \n \nAppendix F. Linearized resonance modes and anisotropy constants ( Axx and Azz) \nThe nonlinear LLG equation of Eq. (1) can be  linearized and the two derived eigenmodes \ncorrespond to the AF and F-mode. The sublatti ce magnetization motion for each mode is given \nby the harmonic oscillation of mode coordinates; for the AF-mode ( QAF, \nPAF)=((X1−X2)s i nβ0+(Z1+Z2)c o sβ0, Y1−Y2), and for the F-mode ( QF, \nPF)=((X1+X2)sinβ0−(Z1−Z2)cosβ0, Y1+Y2), \n \nQAF=AAFcosωAFt,       ( F . 1 )  \nPAF=AAFrAFsinωAFt,      ( F . 2 )  \n \nQF=AFcosωFt,       ( F . 3 )  \nPF=AFrAFsinωFt,       ( F . 4 )   18 \nwhere AAF,F represents the amplitude of each mode. AF,F, and rAF,F are the resonance frequencies \nand ellipticities, which are given by    \n \nωAF=γට(b+a)(d-c),        ( F . 5 )  \nωF=γට(b-a)(d+c),        ( F . 6 )  \n rAF=γටሺௗିሻ\n(b+a),        ( F . 7 )  \n rF=γටሺௗାሻ\n(b-a),        ( F . 8 )  \n \nwhere =1.76×1011 s−1T−1 is the gyromagnetic ratio, and \n \n a=െ2Axxcos2β0െ2Azzsin2β0െEcos 2β0െDsin 2β0,  (F.9) \n b=E,         ( F . 1 0 )  \n c=2Axxcos2β0െ2Azzcos2β0+Ecos 2β0+Dsin 2β0,    (F.11) \n d=െEcos 2β0െDsin 2β0.       ( F . 1 2 )  \n \nSubstituting the literature values of the exchange fields ( E=6.4×102 T and D=1.5×10 T [42]) and \nthe resonance frequencies at room temperature ( AF/2=0.575 THz and F/2=0.37 THz) to \nEqs. (F.5) and (F.6), Axx and Azz can be determined to 8.8×10−2 T and 1.9×10−2 T. \n \nAppendix G. MOKE measurement for the spontaneous magnetization \nFigure G.1 shows time-development of the MOKE signals for the different initial condition \nwith oppositely directed magnetization. We applied the static magnetic field (~0.3 T) to saturate \nthe magnetization along the z-axis before the TH z excitation. The spontaneous magnetization of \nsingle crystal HoFeO a can be reversed by the much smaller magnetic field (~0.01 T) because of \nthe domain wall motion [27]. Then, we separately measured the static Kerr ellipticity angle \n\u001f\u001f\u001f\u001f\u001f\u001f  and THz induced ellipticity change  for different initial magnetization Mz=±Ms \nwithout the static magnetic field \u001f In figure G.1 we plot the summation of the time resolved \nMOKE signal \u001fand the static Kerr ellipticity \u001f\u001f\u001f\u001f\u001f\u001f  The sings of the ellipticity offset angle  19\u001f\u001f\u001f\u001f\u001f\u001f  for the different spontaneous magnetization (±M S) are different and their magnitudes \nare ~0.05 degrees. The conversion coefficient g(=1/~\u001f/0.05 degrees) is estimated to be ~20 \ndegrees−1, which is similar to the value dete rmined by the LLG fitting (~17.8 degrees−1). In the \ncase of the AF-mode excitation, the phases of the magnetization oscillations are in-phase \nregardless of the direction of the spontaneous magnetization M=±Ms, whereas they are \nout-of-phase in the case of the F-mode excitation. We can explain this claim as follows: In the \ncase of AF-mode excitation, the external THz magne tic field is directed along the z-direction as \nshown in the inset of figure 2(a), the signs of the torques acting on the sublattice magnetization \nmi (i=1,2) depends on the direction of mi, however, the resultant oscillation of the macroscopic \nmagnetization M= m1+m2 along the z-direction has same phase for the different initial condition \nM=±Ms. In the case of the F-mode excitation with the external THz magnetic field along the x \nor y-direction, the direction of the torques acting on the magnetization M depends on the initial \ndirection and the phase of the F-mode osc illation changes depending on the sign of the \nspontaneous magnetization ±Ms. \n \nAppendix H. Influence of the spatial distri bution of magnetic field on magnetization \nchange \nAs shown in the inset of figure 1, the pump magnetic field  strongly localizes near the metallic \narm of the SRR and the magnetic field strength significantly depends on the spatial position r \nwithin the probe pulse spot area. The intensity distribution of the probe pulse Iprobe(r) has an 0.05\n0.00\n-0.05Kerr ellipticity  (degrees)\n25 20 15 10 5\nTime (ps) +MS\n -MS\n \nFigure G.1. The MOKE signals, the temporal change of the Kerr ellipticity , measured \nfor different initial conditions with oppositely directed magnetizations.       20elongated Gaussian distribution with spatial widt hs of 1.1 µm along the x-axis and 1.4 µm along \nthe y-axis [full width at half maximum (FWHM)  intensity]. The maximum magnetic field is 1.2 \ntimes larger than the minimum one in the spot  diameter, causing the different magnetization \nchange dynamics at different positions. To take into account this spatial inhomogeneity to the \nsimulation, the spatially weighted average of magnetization change ζ̅(t) has to be calculated as \nfollows:  \n \n ζ̅(t)=ζ(r,t)Iprobe(r)dr\nIprobe(r)ௗr ,       ( H . 1 )  \n \nwhere (r,t) is a magnetization change at a position r and time t.  \n \nFigure H.1(a) shows the simulation result of the spatially averaged magnetization change ζ̅(t) \nand the non-averaged (r0,t) without the nonlinear damping term ( 1=0), where r0 denotes the \npeak position of Iprobe(r). For the low excitation intensity (10%), ζ̅(t) is almost the same as \n(r0,t) as shown in figure H.1(a). On the other hand, for the high excitation intensity, the spatial \ninhomogeneity of magnetization change dyna mics induces a discrepancy between the ζ̅(t) and (a) (b) (c) \n-0.100.000.10Magnetization change\n50 40 30 20 100\nTime (ps)-0.6-0.4-0.20.00.20.4 Averaged\n Non-averaged\n Averaged\n Non-averaged100%  10%\n0.575\n0.570\n0.565\n0.560\n0.55550 40 30 20 100\nTime (ps)0.575\n0.570\n0.565\n0.560\n0.555Frequency (THz) Averaged\n Non-averaged\n Experiment\n Averaged\n Non-averaged\nExperiment\n100%  10%\n0.4\n0.3\n0.2\n0.1\n0.0\n50 40 3020 100\nTime (ps)0.12\n0.08\n0.04\n0.00Amplitude   Averaged\n Non-averaged\n Experiment\n Averaged\n Non-averaged\n Experiment100%  10%\nFigure H.1.  Comparison of the spatially averag ed and non-averaged magnetization \nchange for the different pump fluences of 10% and 100%. (a) Temporal evolutions of \nthe magnetization change, (b) instantaneous  frequencies and (c) normalized envelope \namplitudes. Open circles show the experimental results.      21(r0,t). This discrepancy is caused by the quasi-i nterference effect between the magnetization \ndynamics with different frequencies and amplit udes at different positions. Figures H.1(b) and \n(c) show the instantaneous freque ncy and envelope amplitude obt ained from the data shown in \nfigure H.1(a) by using analytic signal approach with the experimental result. For the averaged \nmagnetization change, the frequency redshift is more emphasized (figure H.1(b)) and the decay \ntime becomes shorter (figure H.1(c)). Nonetheless,  neither spatially averaged nor non-averaged \nsimulation reproduces the experimental result of  the instantaneous frequency (figure H.1(b)) \nwithout nonlinear damping term.  \n \n   22References \n[1] Kirilyujk A, Kimel A V and Rasing T 2010 Rev. Mod. Phys.  82 2731 \n[2] Beaurepaire E, Merle J-C, Daunois A and Bigot J-Y 1996 Phys. Rev. Lett.  76 4250 \n[3] Barman A, Wang S, Maas J D, Hawkins A R, Kwon S, Liddle A, Bokor J and Schmidt H \n2006 Nano Lett.  6 2939 \n[4] Bovensiepen U 2007 J. Phys.: Condens. Matter  19 083201 \n[5] Wang J, Cotoros I, Dani K M, Liu X, Furdyna J K and Chemla D S 2007 Phys. Rev. Lett.  \n98 217401 \n[6] Fiebig M, Duong N P, Satoh T, Van Aken B B, Miyano K, Tomioka Y and Tokura Y 2008 \nJ. Phys. D: Appl. Phys.  41 164005 \n[7] Au Y , Dvornik M, Davison T, Ahmad E, Kea tley P S, Vansteenkiste A, Van Waeyenberge \nB and Kruglyak V V 2013 Phys. Rev. Lett.  110 097201 \n[8] Bigot J-Y and V omir M 2013 Ann. Phys. (Berlin)  525 2 \n[9] Koshihara S, Oiwa A, Hirasawa M, Katsumoto S, Iye Y , Urano C, Takagi H and Munekata \nH 1997 Phys. Rev. Lett.  78 4617 \n[10] Vahaplar K, Kalashnikova A, Kimel A, Hinzke D, Nowak U, Chantrell R, Tsukamoto A, \nItoh A, Kirilyuk A and Rasing T 2009 Phys. Rev. Lett.  103 117201 \n[11] Ostler T A, Barker J, Evans R and Chantrell R W 2012 Nat. Commun.  3 666 \n[12] Kimel A V , Kirilyuk A, Usachev P A, Pisarev R V , Balbashov A M and Rasing T 2005 \nNature  435 655 \n[13] Kim J-W, V omir M and Bigot J-Y 2012 Phys. Rev. Lett.  109 166601 \n[14] Kim J-W, V omir M and Bigot J-Y 2015 Sci. Rep.  5 8511 \n[15] Satoh T, Cho S-J, Iida R, Shimura T, Kuroda K, Ueda H, Ueda Y , Ivanov B A, Nori F and \nFiebig M 2010 Phys. Rev. Lett.  105 077402 \n[16] Le Guyader L, Kleibert A, Nolting F, Joly L, Derlet P, Pisarev R, Kirilyuk A, Rasing T and \nKimel A 2013 Phys. Rev. B  87 054437 \n[17] Yamaguchi K, Nakajima M and Suemoto T 2010 Phys. Rev. Lett.  105 237201 \nYamaguchi K, Kurihara T, Minami Y , Nakajima M and Suemoto T 2013 Phys. Rev. Lett.  \n110 137204 \n[18] Kampfrath T, Sell A, Klatt G, Pashkin A and Mährlein S 2011 Nature Photon. 5 31 \n[19] Wienholdt S, Hinzke D and Nowak U 2012 Phys. Rev. Lett.  108 247207 \n[20] Back C H, Weller D, Heidmann J, Mauri D, Guarisco D, Garwin E L and Siegmann H C \n1998 Phys. Rev. Lett.  81 3251  23[21] Tudosa I, Stamm C, Kashuba A B, King F, Siegmann H C, Stöhr J, Ju G, Lu B and Weller \nD 2004 Nature  428 831 \n[22] Afanasiev D, Razdolski I, Skibinsky K M,  Bolotin D, Yagupov S V , Strugatsky M B, \nKirilyuk A, Rasing T and Kimel A V 2014 Phys. Rev. Lett.  112 147403 \n[23] Mukai Y , Hirori H, Yamamoto T, Kageyama H and Tanaka K 2014 Appl. Phys. Lett.  105 \n022410 \n[24] Yeh K-L, Hoffmann M C, Hebling J and Nelson K A 2007 Appl. Phys. Lett.  90 171121 \n[25] Hebling J, Yeh K-L, Hoffmann M C, Bartal B and Nelson K A 2008 J. Opt. Soc. Am. B  25 \nB6 \n[26] Hirori H, Blanchard F and Tanaka K 2011 Appl. Phys. Lett.  98 091106 \n[27] Shao M, Cao S, Wang Y , Yuan S, Ka ng B, Zhang J, Wu A and XU J 2011 J. Cryst. Growth  \n318 947 \n[28] Padilla W J, Taylor A J, Highstrete C, Lee M and Averitt R D 2006 Phys. Rev. Lett.  96 \n107401 \n[29] Kumar N, Strikwerda A C, Fan K, Zhang X, Averitt R D, Planken P and Adam A J 2012 \nOpt. Express  20 11277 \n[30] V orob'ev G P, Kadomtseva A M, Krynetkii I B and Mukhin A A 1989 Sov. Phys. JETP 68 \n604 \n[31] Balbashov A, Kozlov G and Lebedev S 1989 Sov. Phys. JETP 68 629 \n[32] Herrmann G F 1963 J. Phys. Chem. Solids 24 597 \n[33] Perroni C A and Liebsch A 2006 Phys. Rev. B  74 134430 \n[34] Iida R, Satoh T, Shimura T, Kuroda K, Ivanov B, Tokunaga Y and Tokura Y 2011 Phys. \nRev. B  84 064402 \n[35] Kahn F J, Pershan P S and Remeika J P 1969 Phys. Rev.  186 891 \n[36] Kimel A, Ivanov B, Pisarev R, Usachev P, Kirilyuk A and Rasing T 2009 Nat. Phys.  5 727 \n[37] Suhl H 1957 Phys. Chem. Solids  1 209 \n[38] Gerrard M and Burch J 1975 Introduction to Matrix Methods in Optics  (Wiley, London) \n[39] Cohen L 1989 Proc. IEEE  77 941 \n[40] Kozlov G V , Lebedev S P, Mukhin A A, Prokhorov A S, Fedorov I V , Balbashov A M and \nParsegov I Y 1993 IEEE Trans. Magn. 29 3443 \n[41] Takahashi Y and Akiyama H 1986 Thermochimica Acta 109 105 \n[42] Koshizuka N and Hayashi K 1988 J. Phys. Soc. Jpn.  57 4418 "    },    {        "title": "1602.06201v2.A_systematic_study_of_magnetodynamic_properties_at_finite_temperatures_in_doped_permalloy_from_first_principles_calculations.pdf",        "content": "arXiv:1602.06201v2  [cond-mat.mtrl-sci]  23 Jun 2016A systematic study of magnetodynamic properties at finite te mperatures in doped\npermalloy from first principles calculations\nFan Pan,1,2,∗Jonathan Chico,3Johan Hellsvik,1Anna Delin,1,2,3Anders Bergman,3and Lars Bergqvist1,2\n1Department of Materials and Nano Physics, School of Informa tion and Communication Technology,\nKTH Royal Institute of Technology, Electrum 229, SE-16440 K ista, Sweden\n2Swedish e-Science Research Center (SeRC), KTH Royal Instit ute of Technology, SE-10044 Stockholm, Sweden\n3Department of Physics and Astronomy, Materials Theory Divi sion,\nUppsala University, Box 516, SE-75120 Uppsala, Sweden\n(Dated: June 24, 2018)\nBy means of first principles calculations, we have systemati cally investigated how the magnetody-\nnamicproperties Gilbert damping, magnetization andexcha ngestiffness areaffected whenpermalloy\n(Py) (Fe 0.19Ni0.81) is doped with 4d or 5d transition metal impurities. We find th at the trends in\nthe Gilbert damping can be understood from relatively few ba sic parameters such as the density of\nstates at the Fermi level, the spin-orbit coupling and the im purity concentration. The temperature\ndependence of the Gilbert damping is found to be very weak whi ch we relate to the lack of intraband\ntransitions in alloys. Doping with 4 delements has no major impact on the studied Gilbert damping,\napart from diluting the host. However, the 5 delements have a profound effect on the damping and\nallows it to be tuned over a large interval while maintaining the magnetization and exchange stiff-\nness. As regards spin stiffness, doping with early transitio n metals results in considerable softening,\nwhereas late transition metals have a minor impact. Our resu lt agree well with earlier calculations\nwhere available. In comparison to experiments, the compute d Gilbert damping appears slightly\nunderestimated while the spin stiffness show good general ag reement.\nI. INTRODUCTION\nSpintronics and magnonic applications have attracted\na large degree of attention due to the potential of cre-\nating devices with reduced energy consumption and im-\nprovedperformancecomparedtotraditionalsemiconduc-\ntor devices1–3. An important ingredient for understand-\ning and improving the performance of these devices is\na good knowledge of the magnetic properties. In this\nstudy, we focus on the saturation magnetization Ms, the\nexchange stiffness Aand the Gilbert damping α4. The\nlatter is related to the energy dissipation rate of which a\nmagnetic system returns to its equilibrium state from an\nexcited state, e.g. after the system has been subjected to\nanexternalstimuliisuchasanelectricalcurrentwhichal-\ntersits magneticstate. The threeparameters, Ms,Aand\nαdescribe the magnetodynamical properties of the sys-\ntem of interest. Ultimately one would like to have com-\nplete independent control and tunability of these proper-\nties. In this study, the magnetodynamical properties of\nPermalloy (Py) doped with transition metal impurities\nare systematically investigated within the same compu-\ntational framework.\nThe capability of tuning the damping for a material\nwith such a technological importance as Py is important\nfor the development of possible new devices in spintron-\nics and magnonics. The understanding of how transi-\ntion metals or rare earth dopants can affect the prop-\nerties of Py has been the focus of a number of recent\nexperimental studies5–9. Typically in these studies, the\nferromagnetic resonance10(FMR) technique is employed\nandαandMsare extracted from the linewidth of the\nuniform precession mode while Ais extracted from the\nfirst perpendicularstandingspin-wavemode11,12. On thetheory side, calculations of Gilbert damping from first-\nprinciples density functional theory methods have only\nrecently become possible due to the complexity of such\ncalculations. Two main approaches have emerged, the\nbreathing Fermi surface model13,14and the torque corre-\nlation models15,16. Common to both approaches is that\nspin-orbit coupling along with the density of states at\nthe Fermi level are the main driving forces behind the\ndamping. The breathing Fermi surface model only takes\nonly into account intraband transitions while torque cor-\nrelation model also includes interband transitions. The\ntorque correlation model in its original form contains a\nfree parameter, namely the scattering relaxation time.\nBrataaset. al17later lifted this restriction by employing\nscattering theory and linear response theory. The result-\ning formalism provides a firm foundation of calculating\nαquantitative from first principles methods and allows\nfurther investigations of the source of damping. Gilbert\ndamping in pure Py as well as doping with selected el-\nements have been calculated in the past9,18–20, however\nno systematic study of the magnetodynamic properties\nwithin the same computational framework has been con-\nducted which the present paper aims to address.\nThe paper is outlined as follows: In Section II we\npresent the formalism and details of the calculations, in\nSection III we present the results of our study and in\nSection IV we summarize our findings and provide an\noutlook.2\nII. THEORY\nA. Crystal structure of permalloy and treatment of\ndisorder in the first-principles calculations\nPure Permalloy (Py), an alloy consisting of iron (Fe)\nandnickel(Ni) withcompositionFe 0.19Ni0.81, crystallizes\nin the face centered cubic (fcc) crystal structure, where\nFe and Ni atoms are randomly distributed. Additional\ndoping with 4 dand 5dimpurities (M) substitutes Fe (or\nNi) sothat it becomes a threecomponent alloywith com-\nposition Py 1−xMx, wherexis the concentration of the\ndopant.\nAll first principles calculations in this study were\nperformed using the spin polarized relativistic (SPR)\nKorringa-Kohn-Rostoker (KKR)21Green’s function\n(GF) approach as implemented in the SPR-KKR\nsoftware22. The generalized gradient approximation\n(GGA)23wasusedintheparametrizationoftheexchange\ncorrelation potential and both the core and valence elec-\ntrons were solved using the fully relativistic Dirac equa-\ntion. The broken symmetry associated with the chemical\nsubstitution in the system was treated using the coherent\npotential approximation (CPA)24,25.\nB. Calculation of magnetodynamical properties of\nalloys: Gilbert damping within linear response\ntheory and spin stiffness\nOne of the merits with the KKR-CPA method is that\nit has a natural way of incorporating calculations of re-\nsponse properties using linear response formalism17,19,20.\nThe formalism for calculating Gilbert damping in the\npresent first principles method has been derived in Refs.\n[17] and [20], here we only give a brief outline of the\nmost important points. The damping can be related as\nthe dissipation rate of the magnetic energy which in turn\ncan be associated to the Landau-Lifshitz-Gilbert (LLG)\nequation4, leading to the expression\n˙E=Heff·dM\ndτ=1\nγ2˙ˆm[˜G(m)˙ˆm], (1)\nwhereˆm=M/Msdenotesthe normalizedmagnetization\nvector,Msthe saturation magnetization, γthe gyromag-\nnetic ratio and ˜G(m) the Gilbert relaxation rate tensor.\nPerturbing a magnetic moment from its equilibrium\nstate by a small deviation, ˆm(τ) =ˆm0+u(τ), gives an\nalternative expression of the dissipation rate by employ-\ning linear response theory\n˙Edis=π/planckover2pi1/summationdisplay\nij/summationdisplay\nµν˙uµ˙uν∝angbracketleftψi|∂ˆH\n∂uµ|ψj∝angbracketright∝angbracketleftψj|∂ˆH\n∂uν|ψi∝angbracketright×\nδ(EF−Ei)δ(EF−Ej),(2)where the δ-functions restrict the summation over\neigenstates to the Fermi level which can be rewrit-\nten in terms of Green’s function as Im G+(EF) =\n−π/summationtext\ni|ψi∝angbracketright∝angbracketleftψi|δ(EF−Ei). By comparing Eqs. (1) and\n(2), the Gilbert damping parameter αis obtained, which\nis dimensionless and is related to the Gilbert relaxation\ntensorα=˜G/(γMs). This can be expressed as a trans-\nport Kubo-Greenwood-like equation26,27in terms of the\nretarded single-particle Green’s functions\nαµν=−/planckover2pi1γ\nπMsTrace/angbracketleftBig∂ˆH\n∂uµImG+(EF)∂ˆH\n∂uνImG+(EF)/angbracketrightBig\nc,\n(3)\nwhere∝angbracketleft...∝angbracketrightcdenotes a configurational average. For the\ncubic systems treated in this study, the tensorial form of\nthe damping can with no loss of generality be replaced\nwith a scalar damping parameter. Thermal effects from\natomicdisplacementsandspinfluctuationswereincluded\nusing the alloy-analogy model28within CPA.\nThe spin-wave stiffness Dis defined as the curvature\nof the spin wave dispersion spectrum at small wave vec-\ntors (ω(q)≈Dq2).Din turn is directly related to the\nexchange interactions in the Heisenberg model which are\nobtained using the LKAG formalism29,30such that\nD=2\n3/summationdisplay\nijJijR2\nij√mimj, (4)\nwhereJijis the interatomic exchangeparameterbetween\nthei-thandj-thmagneticmoment, Rijthe distancecon-\nnecting the atomic sites with index iandjandmi(mj)\nthe magneticmoment atsite i(j). It is worthnotingthat\nEq. (4) only holds for cubic systems as treated here, for\nlower symmetries the relation needs modifications. The\nexchange couplings in metallic systems are typically long\nranged and could have oscillations of ferromagnetic and\nantiferromagnetic character, such as present in RKKY\ntype interactions. Due to the oscillations in exchange in-\nteractions, care is needed to reach numerical convergence\nof the series in Eq. (4) and it is achieved following the\nmethodology as outlined in Refs. 31 and 32.\nC. Calculation of finite temperature magnetic\nproperties\nOnce the exchange interactions within the Heisen-\nberg model have been calculated, we obtained finite\ntemperature properties from Metropolis33Monte Carlo\nsimulations as implemented in the UppASD software\npackage34,35. In particular, the temperature dependent\nmagnetization was obtained, and enters the expression\nfor micromagnetic exchange stiffness A, defined as36–39\nA(T) =DM(T)\n2gµB, (5)3\nwhereµBis the Bohr magneton, gis the Land´ e g-factor\nandM(T) the magnetization at temperature T.\nD. Details of the calculations\nForeachconcentrationofthedifferentimpuritiesinPy,\nthe lattice parameter was optimized by varying the vol-\nume and finding the energy minimum. The k-point mesh\nfor the self consistent calculations and exchange interac-\ntions was set to 223giving around 800 k-points in the ir-\nreducible wedge of the Brillouin zone (IBZ). The Gilbert\ndamping calculation requires a very fine mesh to resolve\nallthe Fermisurfacefeaturesandthereforeasignificantly\ndenser k-point mesh of 2283(∼1.0×106k-points in\nIBZ) was employed in these calculations to ensure nu-\nmericalconvergence. Moreover,vertexcorrections40were\nincluded in the damping calculationssince it has been re-\nvealedtobeimportantinpreviousstudies20forobtaining\nquantitative results.\nIII. RESULTS\nA. Equilibrium volumes and induced magnetic\nmoments\n6.656.706.756.806.856.906.957.00\nNbMoTcRuRhPdAgTaWReOsIrPtAuLattice constant (Bohr)   \n10%M\n15%M\nPy\nFIG. 1. Calculated equilibrium volumes of Py-M, where M\nstands for a 4 d(left) or 5 d(right) transition metal. Values\nfor 10% and 15% doping concentrations are shown. Reference\nvalue of pure Py is diplayed with a dashed line.\nFigure 1 shows the calculated equilibrium volume of\ndoped Pyfor twodifferent concentrations(10%and 15%)\nofimpuritiesfrom the 4 dand5dseriesofthe PeriodicTa-\nble. Firstofall, itis notedthat thevolumeincreaseswith\nthe concentration, and the volume within a series (4 dor\n5d) has a parabolicshape with minimum in the middle of\nthe series. This is expected since bonding states are con-secutivelyfilledand maximizedinthe middle ofthe series\nandthusthebondingstrengthreachesamaximum. Mov-\ning further through the series, anti-bonding states start\nto fill, giving rise to weaker bonding and larger equilib-\nriumvolumes. Thisisconsistentwiththeatomicvolumes\nwithin the two series41.\n0.500.600.700.800.901.001.10Total moment ( µΒ)\nM 5%\n10%\n15%\n20%\nPy\n-0.4-0.20.00.20.40.6\nNbMoTcRuRhPdAgTaWReOsIrPtAuLocal moment ( µΒ)M\nFIG.2. (Upper)Totalmagneticmoment(spinandorbital)for\ndifferent impurities and concentrations. Reference value f or\npure Py marked with a dashed line. (Lower) Local impurity\nmagnetic moment for Py 0.95M0.05\nThe local moments of the host atoms are only weakly\ndependent on the type of impurity atom present. More-\nover, the magnetic moments are dominated by the spin\nmomentµSwhile the orbital moments µLare much\nsmaller. As an example, in pure Py without additional\ndoping, the spin (orbital) moments of Fe is calculated to\n≈2.64 (0.05) µBand for Ni ≈0.64 (0.05) µB, respec-\ntively. This adds up to an average spin (orbital) moment\nof≈1.04(0.05)µBby taking into account the concentra-\ntion of Fe and Ni in Py. The total moment is analyzed\nin more detail in Fig. 2 (upper panel). As mentioned\nabove, one would like to achieve tunable and indepen-\ndent control of the saturation magnetization. Reducing\nthe magnetization reduces the radiative extrinsic damp-\ning but could at the same time affect the other properties\nin an unwanted manner. In many situations, one strive\nfor keeping the value of the total moment (saturation\nmagnetization) at least similar to pure Py, even for the\ndoped systems. It is immediately clear from Fig. 2 that\ndoping elements late in the series are the most preferable\nin that respect, for instance Rh and Pd in the 4 dseries\nand Ir, Pt and Au in the 5 dseries.\nIn Fig. 2 (lower panel) we show the local impurity\nmagnetic moments for 5% impurities in Py. In the be-\nginning of the 4 d(5d) series, the impurity atoms have an\nantiferromagnetic coupling, reflected in the negative mo-\nments compared to the host (Fe and Ni) atoms while lat-4\nter in the series couples ferromagnetically (positive mo-\nments). The antiferromagnetic coupling may not be pre-\nferred since it will tend to soften the magnetic properties\nand maybe even cause more complicated non-collinear\nmagnetic configurations to occur.\nB. Band structure\nSince Py and doped-Py are random alloys, they lack\ntranslational symmetry and calculations using normal\nband structure methods are more challenging due to the\nneed for large supercells. However, employing CPA re-\nstores the translational symmetry and more importantly,\nthe band structure of disordered systems can be ana-\nlyzedthroughtheBlochspectralfunction (BSF) A(E,k),\nwhich can be seen as a wave vector k-dependent density\nof states (DOS) function. For ordered systems the BSF\nis aδ-like function at energy E( k) while for disordered\nsystems each peak has an associated broadening with a\nlinewidth proportional to the amount of disorder scatter-\ning. In the upper panel of Fig. 3 the calculated BSF for\npure Py is displayed. Despite being a disordered system,\nthe electron bands are rather sharp below the Fermi level\nwhile in the vicinity ofthe Fermi level the bands becomes\nmuch more diffuse indicating that most of the disorder\nscattering takes place around these energies.\nIf Py is doped with 20% Pt impurities, the positions\nof the electron bands do not change much as shown by\nthe BSF in the lower panel of Fig. 3. The most strik-\ning change is the large increase of the disorder scatter-\ning compared to than Py causing diffuse electron bands\nthroughout the Brillouin zone and energies. However,\nexactly at the Fermi level the differences between the\ndoped and undoped system is not very pronounced and\nthesestatesarethe mostimportantforthe determination\nof the Gilbert damping, as seen from Eq. 3.\nC. Gilbert damping: effect of doping\nThe calculated Gilbert damping of the doped Py sys-\ntems fordifferent concentrationsofimpurities isshownin\nFig. 4 (upper panel). The 4 dimpurities only marginally\ninfluence the damping while the 5 dimpurities dramati-\ncally change the damping. The first observation is that\nweobtainverygoodagreementasinthe previousstudy19\nfor the 5dseries with 10% impurities, howevernot so sur-\nprising since we use same methodology. Secondly, the\nmost dramatic effect on damping upon doping is for the\ncase of Py doped with 20% Os impurities in which the\ndamping increases with approximately 800% compared\nto pure Py, as previously reported in Ref. [20]. How-\never, in the present study we have systematically var-\nied the impurity elements and concentrations and tried\nto identify trends over a large interval. Compared to\nexperiments5, the calculated values of the Gilbert damp-\ning are consistently underestimated. However it is worth\nFIG. 3. The Bloch spectral function A(E,k) of Py (upper\npanel) and Py doped with 20% Pt impurities (lower panel).\nThe Fermi level is indicated with a horizontal black line at\nzero energy.\nremembering that calculations only shows the intrinsic\npart of the damping while experiments may still have\nsome additional portion of extrinsic damping left such\nas Eddy current damping and radiation damping, since\nit is difficult to fully separate the different contributions.\nMoreover,incalculationsacompleterandomdistribution\nof atoms is assumed while there may be sample inhomo-\ngeneities such as clustering in the real samples.\nFrom most theoretical models, the two main material\nproperties that determine the damping are the density\nof states (DOS) at the Fermi level and the strength of\nthe spin-orbit coupling. In the following, we first inves-\ntigate separately how these properties affect the damp-\ning and later the combination of the two. In the lower\npanel of Fig. 4, the total DOS and the impurity-DOS are\ndisplayed for 10% impurity concentration of 4 dand 5d\nseries transition metals. In the both 4 dand 5dseries the\nimpurity-DOS exhibits a maximum in the middle of the\nseries. However, the value of the DOS are similar for the\n4dand 5dseries and therefore cannot solely explain the\nlarge difference in damping found between the two series.5\nFor the 4dseries, the calculated damping is not directly\nproportional to the DOS while there is a significant cor-\nrelation of the DOS and damping in the 5 dseries.\n0.000.010.020.030.04Gilbert damping αM 5%\n10%\n15%\n20%\nPy\nExp.5%\n0.200.400.600.801.001.201.40\nNbMoTcRuRhPdAgTaWReOsIrPtAun(EF) (sts./eV)\nM\nPy-M\nFIG. 4. (Upper) Calculated Gilbert damping parameter for\nPy+M in different concentrations of 4 dand 5dtransition\nmetal M at low temperatures ( T= 10K). Experimental re-\nsults from Ref. [5] measured at room temperature are dis-\nplayed by solid squares and dashed line indicate reference\nvalue for pure Py. (Lower) Total (blue) and impurity (black)\ndensity of states at the Fermi level EFfor 10% impurities in\nPy.\nIn order to analyze the separate influence of spin-orbit\ncoupling on the damping, we show in upper panel of\nFig. 5 the spin-orbit parameter ξ∝1\nrdV(r)\ndr, where V(r)\nis the radial potential, of the impurity d-states. The\ncalculations include all relativistic effects by solving the\nDirac equation but here we have specifically extracted\nthe main contribution from the spin-orbit coupling. As\nexpected, the spin-orbit parameter increases with atomic\nnumberZ, and is therefore considerably larger in the\n5dseries compared to the 4 dseries. This is the most\nlikely explanation why the damping is found to be larger\nin the 5dseries than the 4 dseries. However, within a\nsingle element in either the 4d or 5d series, the damp-\ning is quadratically dependent on the relatve strength of\nthe spin-orbit strength20. The calculated values of the\nspin-orbit parameter are in good agreement with previ-\nous calculations42,43and reaches large values of 0.6-0.9\neV for the late 5 delements Ir,Pt and Au while all val-ues are below 0.3 eV for the 4 dseries. If the damping\nacross elements would only be proportional to the spin-\norbit coupling, then the damping would monotonously\nincrease with atomic number and since this is not what\nhappens, we conclude that there is a delicate balance be-\ntween spin-orbit coupling and DOS that determines the\ndamping which is further highlighted through a qualita-\ntive analysis of the involved scattering processes.\n0.000.300.600.90ξ (eV)Spin-orbit coupling\n0.000.010.02\nNbMoTcRuRhPdAgTaWReOsIrPtAuα (Norm.)TC model\nCalculation\nFIG. 5. Upper: the spin-orbit parameter of d-electrons of\nthe impurity atoms. Lower: qualitative comparison between\ncalculations and torque correlation (TC) model for damping\nwith 10% impurity concentration.\nIn the torque correlation model, the dominant con-\ntribution to damping is through the scattering44,45and\ntakes the following form\nα=1\nγMs(γ\n2)2n(EF)ξ2(g−2)2/τ, (6)\nwhereτis the relaxation time between scattering events,\nandgthe Lande g-factor, for small orbital contributions,\ncan be related as46g= 2(1 +µL\nµS). We assume that τ\nis the same for all impurities, which is clearly an ap-\nproximation but calculating τis beyond the scope of the\npresent study. By normalizing the damping from Eq. (6)\nsuch that the value for Os (10% concentration) coincides\nwith the first principles calculations, we obtain a quali-\ntatively comparison between the model and calculations,\nas illustrated in lower panel of Fig. 5. It confirms the6\ntrend in which 5 dseries lead to a larger damping than\nthe4dseriesandcapturesqualitativelythemainfeatures.\nHowever, the peak value of the damping within the 5 d\nseries in the TC model occurs for Ir while calculations\ngive Os as in experiment. Another model developed for\nlow dimensional magnetic systems such as adatoms and\nclusters suggests that the damping is proportional to the\nproduct of majority and minority density of states at the\nFermi level47. It produces a parabolic trend but with\nmaximum at incorrect position and fails to capture the\nincreased damping of the 5 delements.\nTo further analyse the role of impurity atoms on the\ndamping we also performed calculations where instead of\nimpurities we added vacancies in the system, i.e. void\natoms. The results are shown in Fig. 6 where damping\nas a function of concentrationofAg (4 d), Os (5d) and va-\ncancies are compared to each other along with Os results\nfrom experimental5and previous calculations. Surpris-\ningly, vacancies have more or less the same effect as Ag\nwith the damping practically constant when increasing\nconcentration. Since Ag has a zero moment, small spin-\norbit coupling and small density of states at the Fermi\nlevel, the net effect of Ag from a damping (or scatter-\ning) point of view is mainly diluting the host similar to\nadding vacancies. In contrast, in the Os case, being a\n5dmetal, there is a strong dependence on the concentra-\ntion that was previously analyzed in terms of density of\nstates and Os having a strong spin-orbit coupling. Our\nresults from Os is slightly lower than the previous re-\nported values19,20, despite using same software. How-\never, the most likely reason for the small discrepancy is\nthe use of different exchange-correlationpotentials in the\ntwo cases.\n0.000.010.020.030.040.050.060.070.08\n 0  5  10  15  20Gilbert damping α\n(%) of MPyOs\nPyAg\nPyVac\n10%Os ref(theo.1)\n15%Os ref(theo.2)\nPyOs ref(expt.)\nFIG. 6. Calculated Gilbert damping as a function of Os,\nAg and vacancy (Vac) concentration in Py. Open red circle:\ncalculation from Ref. [19], solid red circle: calculation f rom\nRef. [20] and red solid square: experimental data from Ref. [ 5]D. Gilbert damping: effect of temperature\nIntheprevioussectionwestudiedhowthedampingde-\npends on the electronic structure and spin-orbit coupling\nat low temperatures. However, with increasing tempera-\nture additional scattering mechanisms contribute to the\ndamping, most importantly phonon and magnon scatter-\ning. The phonon scattering is indirectly taken into ac-\ncount by including a number of independent atomic dis-\nplacementsbringingtheatomsoutfromtheirequilibrium\npositions and magnon scattering is indirectly included by\nreducing the magnetic moment for a few configurations\nand then average over all atomic and magnetic configu-\nrations within CPA. It should be noted that the present\nmethodology using the alloy-analogymodel28has limita-\ntions for pure systems at very low temperatures where\nthe damping diverges, but we are far from that situation\nin this study since all systems have intrinsic chemical\ndisorder. However, the limitations for pure systems can\nbe lifted using a more advanced treatment using explicit\ncalculation of the dynamical susceptibility48.\n0.000.010.02α(T)\n1.001.25\n 0 50 100 150 200 250 300 350 400α(T)pho+mag/α(T)pho\nTemperature (K) Py+20% Mo\nPy+20% Rh\nPy+20% W\nPy+20% Pt\nFIG. 7. Gilbert damping parameter including temperature\neffects from both atomic displacements and spin fluctuations\n(upper panel). The effect of spin fluctuations on the Gilbert\ndamping (lower panel), see text.\nThe temperature dependence of damping for a few se-\nlected systems is displayed in Fig. 7 where both atomic\ndisplacements and spin fluctuations are taken into ac-\ncount. From the 4 d(5d) series, we choose to show results7\nfor Mo and Rh (W and Pt), where Mo (W) has a small\nantiferromagnetic moment and Rh (Pt) a sizeable ferro-\nmagnetic moment, from Fig. 2. All systems display an\noverall weak temperature dependence on damping which\nonly marginally increases with temperature, as shown\nin upper panel of Fig. 7. However, in order to sepa-\nrate the temperature contributions from atomic displace-\nments and spin fluctuations, we show the ratio between\nthe total damping and damping where only atomic dis-\nplacements are taken into account in the lower panel of\nFig. 7. The two systems with sizable moments (Rh and\nPt), clearly have a dominant contribution from spin fluc-\ntuations when the moments are reduced upon increased\nscattering due to temperature. In contrast, the two sys-\ntems with (small) antiferromagnetic moments (Mo and\nW), the effect of the spin fluctuations on the damping is\nnegligible and atomic displacements are solely responsi-\nble. The weak temperature dependence found in these\ndoped Py systems is somewhat surprising since in pure\nmetals like Fe and Ni, a strong temperature dependence\nhas been both measured and calculated20, however data\nfor other random alloy systems is scarce.\nThe temperature dependence of damping from the\nband structure is often attributed to interband and\nintraband transitions which arises from the torque-\ncorrelation model. Intraband transitions has conduc-\ntivity like dependence on temperature while interband\nshows resistivty-like dependence. The weak overall de-\npendence found in the systems in Fig. 7 suggests lack of\nintraband transitions but a more detailed analysis of the\nband structure and thermal disorder are left for a future\nstudy.\nE. Spin-wave stiffness and exchange stiffness\nIn the previous sections, we investigated saturation\nmagnetization and damping and we are therefore left\nwith the exchange stiffness. The calculated spin-wave\nstiffnessDatT= 0 K, from Eq. 4, is displayed in the up-\nperpanelofFig.8. Dcanbedirectlymeasuredfromneu-\ntron scattering experiments but as far as we are aware,\nno such data exist. For the late elements in the 4 dand\n5dseries, the spin wave stiffness is maximized and have\nvalues rather similar to pure Py, however with a reduc-\ntion of approximately 20%. In micromagnetic modelling,\nit is common to use the exchange stiffness Ainstead of\nD.Ais proportional to D, from Eq. 5, and the sole\ntemperature dependence of Atherefore comes from the\nmagnetization. In the lower panel of Fig. 8, we show the\ncalculated room temperature ( T= 300 K) values of A,\ntogether with values for pure Py and available experi-\nmental data. In the beginning of the 4 d(5d) series, the\nexchangestiffnessbecomes smalluponincreasingconcen-\ntration of impurities and the systems are magnetically\nvery soft. It follows from the fact that magnetization\nis small because the systems are close to their ordering\ntemperature. Contrary, for the late elements in the 4 d100200300400500Spin−wave stiffness\nD (meV Å2)5% M\n10% M15% M\nPy\n  5 10 15\nNbMoTcRuRhPdAgTaWReOsIrPtAuExchange stiffness\nA (pJ/M)\nPy(expt.) 15%(expt.)\nFIG. 8. Spin-wave stiffness Dof Py-M in the ground state\n(top) and exchange stiffness constant Aat room temperature\nT= 300K (bottom) as a function of doping concentration.\nThe strict dashed lines show the reference value of pure Py\nfrom calculation and experiments. The scattered dots indi-\ncate the experimental data for Py+15%M (Ag/Pt/Au) from\nRef.9\n(5d)series,themagnetizationhasalargefinitevalueeven\nat room temperature and thereforethe exchangestiffness\nalso has a large value, howeverreduced by approximately\n15% compared to pure Py.\nIV. SUMMARY AND CONCLUSIONS\nAsystematic study ofthe intrinsicmagnetic properties\nof transition metal doped Py has been presented. It is\nfound that the Gilbert damping is strongly dependent on\nthe spin-orbit coupling of the impurity atoms and more\nweakly dependent on the density of states that deter-\nmines disorder scattering. The strong influence of the\nspin-orbit coupling makes the 5 delements much more\neffective to change the Gilbert damping and more sen-\nsible to the concentration. As a result, the damping\ncan be increased by an order of magnitude compared to\nundoped Py. Overall, the damping features are quali-\ntatively rather well explained by the torque correlation\nmodel, yet it misses some quantitative predictive power\nthatonlyfirstprinciplesresultscanprovide. Moreover,it8\nisfoundthatthedampingoverallhasaweaktemperature\ndependence, howeverit is slightly enhanced with temper-\nature due to increased scattering caused by atomic dis-\nplacements and spin fluctuations. Elements in the begin-\nning of the 4 dor 5dseries are found to strongly influence\nthe magnetization and exchange stiffness due to antifer-\nromagnetic coupling between impurity and host atoms.\nIncontrast,elementsinthe endofthe4 dor5dserieskeep\nthe magnetization and exchange stiffness rather similar\nto undoped Py. More specifically, doping of the 5 dele-\nments Os, Ir and Pt are found to be excellent candidates\nfor influencing the magnetodynamical properties of Py.\nRecently, there have been an increasing attention for\nfinding metallic materials with small intrinsic damping,\nfor instance half metallic Heusler materials and FeCoalloys32,49. Controlling and varying the magnetodynam-\nical properties in these systems through doping or by\nother means, like defects, are very relevant and left for a\nfuture study.\nACKNOWLEDGMENTS\nThe work was financed through VR (the Swedish Re-\nsearch Council) and GGS (G¨ oran Gustafssons Founda-\ntion). A.B. acknowledge support from eSSENCE. The\ncomputations were performed on resources provided by\nSNIC (Swedish National Infrastructure for Computing)\nat NSC (National Supercomputer Centre) in Link¨ oping.\n∗fanpan@kth.se\n1I.ˇZuti´ c, J. Fabian, and S. Das Sarma,\nRev. Mod. Phys. 76, 323 (2004).\n2A. Brataas, A. D. Kent, and H. Ohno, Nat Mater 11, 372\n(2012).\n3E. V. Gomonay and V. M. Loktev,\nLow Temperature Physics 40, 17 (2014).\n4T. Gilbert, Magnetics, IEEE Transactions on 40, 3443\n(2004).\n5J. O. Rantschler, R. D. McMichael, A. Castillo, A. J.\nShapiro, W. F. Egelhoff, B. B. Maranville, D. Pulugurtha,\nA.P.Chen, andL.M.Connors,JournalofAppliedPhysics\n(2007).\n6S. Mizukami, T. Kubota, X. Zhang, H. Na-\nganuma, M. Oogane, Y. Ando, and T. Miyazaki,\nJapanese Journal of Applied Physics 50, 103003 (2011).\n7G. Woltersdorf, M. Kiessling, G. Meyer, J.-U. Thiele, and\nC. H. Back, Phys. Rev. Lett. 102, 257602 (2009).\n8Y. Endo, Y. Mitsuzuka, Y. Shimada, and M. Yamaguchi,\nMagnetics, IEEE Transactions on 48, 3390 (2012).\n9Y. Yin, F. Pan, M. Ahlberg, M. Ranjbar, P. D¨ urrenfeld,\nA. Houshang, M. Haidar, L. Bergqvist, Y. Zhai,\nR. K. Dumas, A. Delin, and J. ˚Akerman,\nPhys. Rev. B 92, 024427 (2015).\n10J. H. E. Griffiths, Nature 158, 670 (1946).\n11M. H. Seavey and P. E. Tannenwald,\nPhys. Rev. Lett. 1, 168 (1958).\n12A. M. Portis, Applied Physics Letters 2, 69 (1963).\n13V. Kambersk´ y, Canadian Journal of Physics 48, 2906\n(1970).\n14M. F¨ ahnle and D. Steiauf,\nPhys. Rev. B 73, 184427 (2006).\n15V.Kambersk´ y,Czechoslovak Journal of Physics B 26, 1366 (1976).\n16K. Gilmore, Y. U. Idzerda, and M. D. Stiles,\nPhys. Rev. Lett. 99, 027204 (2007).\n17A. Brataas, Y. Tserkovnyak, and G. E. W. Bauer, Phys.\nRev. Lett. 101, 037207 (2008).\n18A. A. Starikov, P. J. Kelly, A. Brataas, Y. Tserkovnyak,\nand G. E. W. Bauer, Phys. Rev. Lett. 105, 236601 (2010).\n19H. Ebert, S. Mankovsky, D. K¨ odderitzsch, and P. J. Kelly,\nPhys. Rev. Lett. 107, 066603 (2011).\n20S. Mankovsky, D. K¨ odderitzsch, G. Woltersdorf, and\nH. Ebert, Phys. Rev. B 87, 014430 (2013).21H. Ebert, D. K¨ odderitzsch, and J. Min´ ar, Reports on\nProgress in Physics 74, 096501 (2011).\n22H. Ebert, http://ebert.cup.uni-muenchen.de/SPRKKR .\n23K. J.P.Perfew and M.Ernzerhof, Phys. Rev. 77, 3865\n(1996).\n24P. Soven, Phys. Rev. 156, 809 (1967).\n25G. M. Stocks, W. M. Temmerman, and B. L. Gyorffy,\nPhys. Rev. 41, 339 (1978).\n26R. Kubo, Canadian Journal of Physics 34, 1274 (1956).\n27D. A. Greenwood, Proceedings of the Physical Society 71,\n585 (1958).\n28H. Ebert, S. Mankovsky, K. Chadova, S. Polesya, J. Min´ ar,\nand D. K¨ odderitzsch, Phys. Rev. B 91, 165132 (2015).\n29A. Liechtenstein, M. Katsnelson, V. Antropov, and\nV. Gubanov, Journal of Magnetism and Magnetic Mate-\nrials67, 65 (1987).\n30H. Ebert and S. Mankovsky, Phys. Rev. B 79, 045209\n(2009).\n31M. Pajda, J. Kudrnovsk´ y, I. Turek, V. Drchal, and\nP. Bruno, Phys. Rev. B 64, 174402 (2001).\n32P. D¨ urrenfeld, F. Gerhard, J. Chico, R. K. Dumas,\nM. Ranjbar, A. Bergman, L. Bergqvist, A. Delin,\nC. Gould, L. W. Molenkamp, and J. ˚Akerman,\nPhys. Rev. B 92, 214424 (2015).\n33N. Metropolis, A. W. Rosenbluth, M. N. Rosenbluth, A. H.\nTeller, and E. Teller, The Journal of Chemical Physics 21,\n1087 (1953).\n34B. Skubic, J. Hellsvik, L. Nordstr¨ om, and O. Eriksson,\nJournal of Physics: Condensed Matter 20, 315203 (2008).\n35http://physics.uu.se/uppasd .\n36C. A. F. Vaz, J. A. C. Bland, and G. Lauhoff, Reports on\nProgress in Physics 71, 056501 (2008).\n37J. Hamrle, O. Gaier, S.-G. Min, B. Hille-\nbrands, Y. Sakuraba, and Y. Ando,\nJournal of Physics D: Applied Physics 42, 084005 (2009).\n38A. Aharoni, Introduction to the theory of ferromagnetism ,\nInternational series of monographs on physics (Clarendon\nPress, 1996).\n39J. M. D. Coey, Magnetism and Magnetic Materials (Cam-\nbridge University Press, 2010).\n40W. H. Butler, Phys. Rev. B 31, 3260 (1985).\n41D. G. Pettifor, Journal of Physics F: Metal Physics 7, 1009\n(1977).9\n42V. Popescu, H. Ebert, B. Nonas, and P. H. Dederichs,\nPhys. Rev. B 64, 184407 (2001).\n43N. E. Christensen, Int. J. Quantum. Chem. XXV, 233\n(1984).\n44V. Kambersk´ y, Phys. Rev. B 76, 134416 (2007).\n45M. Farle, T. Silva, and G. Woltersdorf, Magnetic Nanos-\ntructures: Spin dynamics and spin transport (Springer,\n2013).46A. J. P. Meyer and G. Asch, Journal of Applied Physics\n32(1961).\n47S. Lounis, M. dos Santos Dias, and B. Schweflinghaus,\nPhys. Rev. B 91, 104420 (2015).\n48A. T. Costa and R. B. Muniz,\nPhys. Rev. B 92, 014419 (2015).\n49M. A. W. Schoen, D. Thonig, M. L. Schneider, T. J. Silva,\nH. T. Nembach, O. Eriksson, O. Karis, and J. M. Shaw,\narXiv:1512:3610 1512, 3610 (2015)."    },    {        "title": "1602.06673v3.Effects_of_Landau_Lifshitz_Gilbert_damping_on_domain_growth.pdf",        "content": "arXiv:1602.06673v3  [cond-mat.stat-mech]  1 Dec 2016Effects of Landau-Lifshitz-Gilbert damping on domain growt h\nKazue Kudo\nDepartment of Computer Science, Ochanomizu University,\n2-1-1 Ohtsuka, Bunkyo-ku, Tokyo 112-8610, Japan\n(Dated: May 25, 2021)\nDomain patterns are simulated by the Landau-Lifshitz-Gilb ert (LLG) equation with an easy-axis\nanisotropy. If the Gilbert damping is removed from the LLG eq uation, it merely describes the\nprecession of magnetization with a ferromagnetic interact ion. However, even without the damping,\ndomains that look similar to those of scalar fields are formed , and they grow with time. It is demon-\nstrated that the damping has no significant effects on domain g rowth laws and large-scale domain\nstructure. In contrast, small-scale domain structure is aff ected by the damping. The difference in\nsmall-scale structure arises from energy dissipation due t o the damping.\nPACS numbers: 89.75.Kd,89.75.Da,75.10.Hk\nI. INTRODUCTION\nCoarseningorphase-orderingdynamicsisobservedina\nwidevarietyofsystems. Whenasystemisquenchedfrom\na disordered phase to an ordered phase, many small do-\nmainsareformed, andtheygrowwithtime. Forexample,\nin the case of an Ising ferromagnet, up-spin and down-\nspin domains are formed, and the characteristic length\nscale increases with time. The Ising spins can be inter-\npreted as two different kinds of atoms in the case of a\nbinary alloy. At the late stage of domain growth in these\nsystems, characteristic length L(t) follows a power-law\ngrowth law,\nL(t)∼tn, (1)\nwherenis the growth exponent. The growth laws in\nscalarfieldshavebeenderivedbyseveralgroups: n= 1/2\nfornon-conservedscalarfields, and n= 1/3forconserved\nscalar fields [1–8].\nSimilar coarsening dynamics and domain growth have\nbeen observed alsoin Bose-Einstein condensates (BECs).\nThe characteristic length grows as L(t)∼t2/3in two-\ndimensional (2D) binary BECs and ferromagnetic BECs\nwith an easy-axis anisotropy [9–11]. The same growth\nexponent n= 2/3 is found in classical binary fluids in\nthe inertial hydrodynamic regime [1, 12]. It is remark-\nable that the same growth law is found in both quan-\ntum and classical systems. It should be also noted that\ndomain formation and coarsening in BECs occur even\nwithout energy dissipation. The dynamics in a ferro-\nmagnetic BEC can be described not only by the so-\ncalled Gross-Pitaevskii equation, which is a nonlinear\nSchr¨ odinger equation, but also approximately by a mod-\nified Landau-Lifshitz equation in which the interaction\nbetween superfluid flow and local magnetization is incor-\nporated[13–15]. Ifenergydissipationexists, theequation\nchanges to an extended Landau-Lifshitz-Gilbert (LLG)\nequation [9, 15, 16]. The normal LLG equation is usu-\nally used to describe spin dynamics in a ferromagnet.\nThe LLG equation includes a damping term which is\ncalled the Gilbert damping. When the system has an\neasy-axis anisotropy, the damping has the effect to directa spin to the easy-axis direction. The Gilbert damping\nin the LLG equation corresponds to energy dissipation\nin a BEC. In other words, domain formation without en-\nergy dissipation in a BEC implies that domains can be\nformed without the damping in a ferromagnet. However,\nthe LLG equation without the damping describes merely\nthe precession of magnetization with a ferromagnetic in-\nteraction.\nInthispaper, wefocusonwhateffectsthedampinghas\nondomainformationanddomaingrowth. UsingtheLLG\nequation (without flow terms), we investigate the mag-\nnetic domain growth in a 2D system with an easy-axis\nanisotropy. Since our system is simpler than a BEC, we\ncan also give simpler discussions on what causes domain\nformation. When the easy axis is perpendicular to the\nx-yplane, the system is an Ising-like ferromagnetic film,\nand domains in which the zcomponent of each spin has\nalmostthesamevalueareformed. Inordertoobservedo-\nmain formation both in damping and no-damping cases,\nwe limit the initial condition to almost uniform in-plane\nspins. Actually, without the damping, domain formation\ndoes not occur from an initial configuration of spins with\ntotally random directions. Without the damping, the z\ncomponent is conserved. The damping breaks the con-\nservation of the zcomponent as well as energy. Here,\nwe should note that the growth laws for conserved and\nnonconserved scalar fields cannot simply be applied to\nthe no-damping and damping cases, respectively, in our\nsystem. Although the zcomponent corresponds to the\norderparameterofascalarfield, oursystemhastheother\ntwo components. It is uncertain whether the difference\nin the number of degrees of freedom can be neglected in\ndomain formation.\nThe restofthe paperis organizedas follows. In Sec. II,\nwe describe the model and numerical procedures. Ener-\ngies and the characteristic length scale are also intro-\nduced in this section. Results of numerical simulations\nare shown in Sec. III. Domain patterns at different times\nand the time evolution of energies and the average do-\nmain size are demonstrated. Scaling behavior is con-\nfirmed in correlation functions and structure factors at\nlate times. In Sec. IV, we discuss why domain formation2\ncan occur even in the no-damping case, focusing on an\nalmost uniform initial condition. Finally, conclusions are\ngiven in Sec. V.\nII. MODEL AND METHOD\nThe model we use in numerical simulations is the LLG\nequation, which is widely used to describe the spin dy-\nnamics in ferromagnets. The dimensionless normalized\nform of the LLG equation is written as\n∂m\n∂t=−m×heff+αm×∂m\n∂t, (2)\nwheremis the unit vector of spin, αis the dimensionless\nGilbert damping parameter. We here consider the 2D\nsystemlyinginthe x-yplane,andassumethatthesystem\nhas a uniaxial anisotropy in the zdirection and that no\nlong-range interaction exists. Then, the dimensionless\neffective field is given by\nheff=∇2m+Canimzˆz, (3)\nwhereCaniis the anisotropy parameter, and ˆzis the unit\nvector in the zdirection.\nEquation (2) is mathematically equivalent to\n∂m\n∂t=−1\n1+α2m×heff+α\n1+α2m×(m×heff).\n(4)\nIn numerical simulations, we use a Crank-Nicolson\nmethod to solve Eq. (4). The initial condition is given as\nspins that are aligned in the xdirection with a little ran-\ndom noises: mx≃1 andmy≃mz≃0. Simulations are\nperformed in the 512 ×512 lattice with periodic bound-\nary conditions. Averages are taken over 20 independent\nruns.\nThe energy in this system is written as\nE=Eint+Eani\n=1\n2/integraldisplay\ndr(∇m(r))2−1\n2Cani/integraldisplay\ndrmz(r)2,(5)\nwhich gives the effective field as heff=−δE/δm. The\nfirst and second terms are the interfacial and anisotropy\nenergies, respectively. When Cani>0, thezcomponent\nbecomes dominant since a large m2\nzlowers the energy.\nWe take Cani= 0.2 in the simulations. The damping\nparameter αexpresses the rate of energy dissipation. If\nα= 0, the spatial average of mzas well as the energy E\nis conserved.\nConsidering mzas the order parameter of this system,\nwe here define the characteristic length scale Lof a do-\nmain pattern from the correlation function\nG(r) =1\nA/integraldisplay\nd2x/angb∇acketleftmz(x+r)mz(x)/angb∇acket∇ight,(6)\nwhereAis the area of the system and /angb∇acketleft···/angb∇acket∇ightdenotes an\nensemble average. The average domain size Lis defined\nby the distance where G(r), i.e., the azimuth average of\nG(r), first drops to zero, and thus, G(L) = 0.\nFIG. 1. (Color online) Snapshots of z-component mzat time\nt= 102((a) and (b)), 103((c) and (d)), and 104((e) and\n(f)). Snapshots (g) and (h) are enlarged parts of (e) and (f),\nrespectively. Profiles (i) and (j) of mzare taken along the\nbottom lines of snapshots (g) and (h), respectively. Left an d\nright columns are for the no-damping ( α= 0) and damping\n(α= 0.03) cases, respectively.\nIII. SIMULATIONS\nDomain patterns appear, regardless of the damping\nparameter α. The snapshots of the no-damping ( α= 0)\nand damping ( α= 0.03) cases are demonstrated in the\nleft and right columns of Fig. 1, respectively. Domain\npatterns at early times have no remarkable difference be-\ntweenthe twocases. Thecharacteristiclengthscalelooks\nalmostthesamealsoatlatertimes. However,asshownin\nthe enlarged snapshots at late times, difference appears\nespecially around domain walls. Domain walls, where\nmz≃0, are smooth in the damping case. However, in\nthe no-damping case, they look fuzzy. The difference ap-\npears more clearly in profiles of mz(Figs. 1(i) and 1(j)).\nWhile the profile in the damping case is smooth, that\nin the no-damping case is not smooth. Such an uneven\nprofile makes domain walls look fuzzy.\nThe difference in domain structure is closely connected\nwith energydissipation, which is shownin Fig. 2. The in-\nterfacial energy, which is the first term of Eq. (5), decays\nforα= 0.03 but increases for α= 0 in Fig. 2 (a). In con-3\n0 2000 4000 6000 8000 10000t00.020.040.060.080.1Eint α = 0\nα = 0.03(a)\n0 2000 4000 6000 8000 10000t-0.1-0.08-0.06-0.04-0.020Eaniα = 0\nα = 0.03(b)\nFIG. 2. (Color online) Time dependence of (a) the inter-\nfacial energy Eintand (b) the anisotropy energy Eani. The\ninterfacial energy increases with time in the no-damping ca se\n(α= 0) and decreases in the damping case ( α= 0.03). The\nanisotropy energy decreases with time in both cases.\ntrast, the anisotropy energy, which comes from the total\nofm2\nz, decreases with time for both α= 0 and α= 0.03.\nIn other words, the energy dissipation relating to the in-\nterfacial energy mainly causes the difference between the\ndamping and no-damping cases. In the damping case,\nthe interfacial energy decreases with time after a shot-\ntime increase as domain-wall structure becomes smooth.\nHowever, in the no-damping case, the interfacial energy\nincreases with time to conserve the total energy that is\ngiven by Eq. (5). This corresponds to the result that\nthe domain structure does not become smooth in the no-\ndamping case.\nBeforediscussinggrowthlaws, we shouldexaminescal-\ning laws. Scaled correlation functions of mzat different\ntimes are shown in Fig. 3. The functions look pretty\nsimilar in both damping and no-damping cases, which\nreflects the fact that the characteristic length scales in\nboth cases looks almost the same in snapshots. At late\ntimes, the correlation functions that are rescaled by the\naverage domain size L(t) collapse to a single function.\nHowever, the scaled correlation functions at early times\n(t= 100 and 1000) do not agree with the scaling func-\ntion especially in the short range. The disagreement at\nearly times is related with the unsaturation of mz. How\nmzsaturates is reflected in the time dependence of the0 0.5 1 1.5 2\nr/L(t)-0.200.20.40.60.8G(r)t =   100\nt =  1000\nt =  6000\nt =  8000\nt = 10000(a)\n0 0.5 1 1.5 2\nr/L(t)-0.200.20.40.60.8G(r)t =   100\nt =  1000\nt =  6000\nt =  8000\nt = 10000(b)\nFIG.3. (Color online) Scaledcorrelation functions atdiffe rent\ntimes in (a) no-damping ( α= 0) and (b) damping ( α= 0.03)\ncases. The correlation functions at late times collapse to a\nsingle function, however, the ones at early times do not.\nanisotropy energy which is shown in Fig. 2(b). At early\ntimes (t/lessorsimilar1000),Eanidecays rapidly. This implies that\nmzis not saturated enough in this time regime. The de-\ncreasein theanisotropyenergyslowsatlatetimes. In the\nlate-time regime, mzis sufficiently saturated except for\ndomain walls, and the decrease in the anisotropy energy\nis purely caused by domain growth. This corresponds to\nthe scaling behavior at late times.\nIn Fig. 4, the average domain size Lis plotted for\nthe damping and no-damping cases. In both cases,\nthe average domain size grows as L(t)∼t1/2at late\ntimes, although growth exponents at early times look\nliken= 1/3. Since scaling behavior is confirmed only\nat late times, the domain growth law is considered to be\nL(t)∼t1/2rather than t1/3in this system. In our pre-\nvious work, we saw domain growth as L(t)∼t1/3in a\nBEC without superfluid flow [9], which was essentially\nthe same system as the present one. However, the time\nregion shown in Ref. [9] corresponds to the early stage\n(t/lessorsimilar1830) in the present system.\nAlthough the growth exponent is supposed to be n=\n1/3 for conserved scalar fields, the average domain size\ngrows as L(t)∼t1/2, in our system, at late times even\nin the no-damping case. This implies that our system\nwithout damping cannot be categorized as a model of a4\n100 1000 10000t10100 Lα = 0\nα = 0.03\nt1/2\nt1/3\nFIG. 4. (Color online) Time dependence of the average do-\nmain size Lforα= 0 and 0 .03. In both damping and no-\ndamping cases, domain size grows as L(t)∼t1/2at late\ntimes. Before the scaling regime, early-time behavior look s\nas ifL(t)∼t1/3.\n1 10 100\nkL(t)10-1010-910-810-710-610-510-410-3S(k)/L(t)2\nt =  6000\nt =  8000\nt = 10000\n1 10 100\nkL(t)10-1010-910-810-710-610-510-410-3\nt =  6000\nt =  8000\nt = 10000(a) (b)\nk-3k-3\nFIG. 5. (Color online) Scaling plots of the structure factor\nscaled with L(t) at different times in (a) no-damping ( α= 0)\nand (b) damping ( α= 0.03) cases. In both cases, S(k)∼k−3\nin the high- kregime. However, they gave different tails in the\nultrahigh- kregime.\nconserved scalar field. Although we consider mzas the\norder parameter to define the characteristic length scale,\nthe LLG equation is described in terms of a vector field\nm.\nScaling behavior also appears in the structure factor\nS(k,t), which is given by the Fourier transformation of\nthe correlation function G(r). According to the Porod\nlaw, the structure factor has a power-law tail,\nS(k,t)∼1\nL(t)kd+1, (7)\nin the high- kregime [1]. Here, dis the dimension of\nthe system. Since d= 2 in our system, Eq. (7) leads\ntoS(k,t)/L(t)2∼[kL(t)]−3. In Fig. 5, S(k,t)/L(t)2is\nplotted as a function of kL(t). The data at different late\ntimes collapse to one curve, and they show S(k)∼k−3in the high- kregime (kL∼10) in both the damping and\nno-damping cases. In the ultrahigh- kregime (kL∼100),\ntails are different between the two cases, which reflects\nthe difference in domain structure. Since domain walls\nare fuzzy in the no-damping case, S(k) remains finite.\nHowever, in the damping case, S(k) decays faster in the\nultrahigh- kregime, which is related with smooth domain\nwalls.\nIV. DISCUSSION\nWe here have a naive question: Why does domain\npattern formation occur even in the no-damping case?\nWhenα= 0, Eq. (2) is just the equation of the pre-\ncession of spin, and the energy Eas well as mzis con-\nserved. We here discuss why similar domain patterns are\nformed from our initial condition in both damping and\nno-damping cases.\nUsing the stereographic projection of the unit sphere\nof spin onto a complex plane [17], we rewrite Eq. (4) as\n∂ω\n∂t=−i+α\n1+α2/bracketleftbigg\n∇2ω−2ω∗(∇ω)2\n1+ωω∗−Caniω(1−ωω∗)\n1+ωω∗/bracketrightbigg\n,\n(8)\nwhereωis a complex variable defined by\nω=mx+imy\n1+mz. (9)\nEquation (8) implies that the effect of the Gilbert damp-\ning is just a rescaling of time by a complex constant [17].\nThe fixed points of Eq. (8) are |ω|2= 1 and ω= 0.\nThelinearstabilityanalysisaboutthesefixedpointsgives\nsome clues about domain formation.\nAt the fixed point ω= 1,mx= 1 and my=mz= 0,\nwhich corresponds to the initial condition of the numer-\nical simulation. Substituting ω= 1 +δωinto Eq. (8),\nwe obtain linearized equations of δωandδω∗. Perform-\ning Fourier expansions δω=/summationtext\nkδ˜ωkeik·randδω∗=/summationtext\nkδ˜ω∗\n−keik·r, we have\nd\ndt/parenleftbiggδ˜ωk\nδ˜ω∗\n−k/parenrightbigg\n=/parenleftbigg\n˜α1(Cani−k2) ˜α1Cani\n˜α2Cani˜α2(Cani−k2)/parenrightbigg/parenleftbiggδ˜ωk\nδ˜ω∗\n−k/parenrightbigg\n,\n(10)\nwhere ˜α1=1\n2(−i+α)/(1+α2), ˜α2=1\n2(i+α)/(1+α2),\nk= (kx,ky), andk=|k|. The eigenvalues of the 2 ×2\nmatrix of Eq. (10) are\nλ(k) =α\n2(1+α2)(Cani−2k2)±/radicalbig\n4k2(Cani−k2)+α2C2\nani\n2(1+α2).\n(11)\nEven when α= 0,λ(k) has a positive real part for\nk <√Cani. Thus, the uniform pattern with mx= 1\nis unstable, and inhomogeneous patterns can appear.\nThe positive real parts of Eq. (11) for α= 0 and\nα= 0.03 have close values, as shown in Fig. 6. This cor-\nresponds to the result that domain formation in the early5\n0 0.1 0.2 0.3 0.4 0.5\nk00.020.040.060.080.1λ(k)α = 0\nα = 0.03\nFIG. 6. (Color online) Positive real parts of λ(k) that is given\nby Eq. (11), which has a positive real value for k <√Cani.\nThe difference between α= 0 and α= 0.03 is small.\nstage has no remarkable difference between the damping\n(α= 0.03) and no-damping ( α= 0) cases (See Fig. 1).\nFrom the view point of energy, the anisotropy energy\ndoes not necessarily keep decaying when α= 0. For con-\nservation of energy, it should be also possible that both\nanisotropy and interfacial energies change only a little.\nBecause of the instability of the initial state, mzgrows,\nand thus, the anisotropy energy decreases.\nThe initial condition, which is given as spins aligned in\nonedirection with somenoisesin the x-yplane, is the key\nto observe domain pattern formation in the no-damping\ncase. Actually, if spins have totally random directions,\nno large domains are formed in the no-damping case,\nalthough domains are formed in damping cases ( α >0)\nfrom such an initial state.\nWhenω= 0,mx=my= 0 and mz= 1, which is also\none of the fixed points. Substituting ω= 0 +δωinto\nEq. (8) and performing Fourier expansions, we have the\nlinearized equation of δ˜ωk,\nd\ndtδ˜ωk=i−α\n1+α2(k2+Cani)δ˜ωk. (12)\nThis implies that the fixed point is stable for α >0 andneutrally stable for α= 0. Although mz=−1 corre-\nsponds to ω→ ∞, the same stability is expected for\nmz=−1 by symmetry.\nSincetheinitialconditionisunstable, the z-component\nof spin grows. Moreover, linear instability is similar for\nα= 0 and α= 0.03. Since mz=±1 are not unstable,\nmzcan keep its value at around mz=±1. This is why\nsimilar domain patters are formed in both damping and\nno-damping cases. The main difference between the two\ncases is that mz=±1 are attracting for α >0 and neu-\ntrally stable for α= 0. Since mz=±1 are stable and at-\ntractingin the dampingcase, homogeneousdomainswith\nmz=±1 are preferable, which leads to a smooth profile\nofmzsuch as Fig. 1(j). In the damping case, mz=±1\nare neutrally stable (not attracting) fixed points, which\ndoes not necessarily make domains smooth.\nV. CONCLUSIONS\nWe have investigated the domain formation in 2D vec-\ntor fields with an easy-axis anisotropy, using the LLG\nequation. When the initial configuration is given as al-\nmost uniform spins aligned in an in-plane direction, sim-\nilar domain patterns appear in the damping ( α/negationslash= 0) and\nno-damping ( α= 0) cases. The average domain size\ngrows as L(t)∼t1/2in late times which are in a scal-\ning regime. The damping gives no remarkable effects\non domain growth and large-scale properties of domain\npattern. In contrast, small-scale structures are different\nbetween the two cases, which is shown quantitatively in\nthe structure factor. This difference is induced by the re-\nduction of the interfacial energy due to the damping. It\nshould be noted that the result and analysis especially\nin the no-damping case are valid for a limited initial\ncondition. Although domains grow in a damping case\neven from spins with totally random directions, domain\ngrowth cannot occur from such a random configuration\nin the no-damping case.\nACKNOWLEDGMENTS\nThis work was supported by MEXT KAKENHI\n(No. 26103514, “Fluctuation & Structure”).\n[1] A. Bray, Adv. Phys. 43, 357 (1994)\n[2] I. M. Lifshitz and V. V. Slyozov, J. Phys. Chem. Solids\n19, 35 (1961)\n[3] C. Wagner, Z. Elektrochem 65, 581 (1961)\n[4] T. Ohta, D. Jasnow, and K. Kawasaki, Phys. Rev. Lett.\n49, 1223 (1982)\n[5] D. A. Huse, Phys. Rev. B 34, 7845 (1986)\n[6] A. J. Bray, Phys. Rev. Lett. 62, 2841 (1989)\n[7] A. J. Bray, Phys. Rev. B 41, 6724 (1990)\n[8] A. J. Bray and A. D. Rutenberg, Phys. Rev. E 49, R27\n(1994)[9] K. Kudo and Y. Kawaguchi, Phys. Rev. A 88, 013630\n(2013)\n[10] J. Hofmann, S. S. Natu, and S. Das Sarma, Phys. Rev.\nLett.113, 095702 (2014)\n[11] L. A. Williamson and P. B. Blakie, Phys. Rev. Lett. 116,\n025301 (2016)\n[12] H. Furukawa, Phys. Rev. A 31, 1103 (1985)\n[13] A. Lamacraft, Phys. Rev. A 77, 063622 (2008)\n[14] D. M. Stamper-Kurn and M. Ueda, Rev. Mod. Phys. 85,\n1191 (2013)\n[15] Y. Kawaguchi and M. Ueda, Phys. Rep. 520, 253 (2012)6\n[16] K. Kudo and Y. Kawaguchi, Phys. Rev. A 84, 043607\n(2011)\n[17] M. Lakshmanan and K. Nakamura, Phys. Rev. Lett. 53,\n2497 (1984)"    },    {        "title": "1602.07317v2.Relaxation_of_a_classical_spin_coupled_to_a_strongly_correlated_electron_system.pdf",        "content": "Relaxation of a classical spin coupled to a strongly correlated electron system\nMohammad Sayad, Roman Rausch and Michael Pottho\u000b\nI. Institute for Theoretical Physics, University of Hamburg, Jungiusstra\u0019e 9, D-20355 Hamburg, Germany\nA classical spin which is antiferromagnetically coupled to a system of strongly correlated con-\nduction electrons is shown to exhibit unconventional real-time dynamics which cannot be described\nby Gilbert damping. Depending on the strength of the local Coulomb interaction U, the two main\nelectronic dissipation channels, transport of excitations via correlated hopping and via excitations of\ncorrelation-induced magnetic moments, become active on largely di\u000berent time scales. We demon-\nstrate that correlations can lead to a strongly suppressed relaxation which so far has been observed\nin purely electronic systems only and which is governed here by proximity to the divergent magnetic\ntime scale in the in\fnite- Ulimit.\nPACS numbers: 75.78.Jp, 71.10.Fd, 75.10.Hk, 05.70.Ln\nMotivation. A classical spin in an external magnetic\n\feld shows a precessional motion but when exchange-\ncoupled to a conduction-electron system the spin addi-\ntionally relaxes and \fnally aligns to the \feld direction.\nThis is successfully described on a phenomenological level\nby the Landau-Lifschitz-Gilbert (LLG) equation [1] and\nextensions of this concept [2, 3]. The Gilbert damping\nconstant\u000bis often taken as a phenomenological parame-\nter but can also be computed ab initio for real materials\n[4{6] within a framework of e\u000bectively independent elec-\ntrons using band theory [7] and then serves as an impor-\ntant input for atomistic spin-dynamics calculations [8].\nElectron correlations are expected to have an impor-\ntant e\u000bect on the spin dynamics. This has been demon-\nstrated in a few pioneering studies [9{11] { within di\u000ber-\nent models and using various approximations { but only\nindirectly by computing the e\u000bect of the Coulomb inter-\naction on the Gilbert damping. One hallmark of strong\ncorrelations, however, is the emergence and the separa-\ntion of energy (and time) scales { with the correlation-\ninduced Mott insulator [12] as a paradigmatic example.\nWith the present study we address correlation e\u000bects\nbeyond an LLG-type approach and keep the full tempo-\nral memory e\u000bect. It is demonstrated that correlation-\ninduced time-scale separation has profound and qualita-\ntively new consequences for the spin dynamics. These\nare important, e.g., for the microscopic understanding\nof the emerging relaxation time scales in modern nano-\nspintronics devices involving various transition metals\nand compounds [13{15].\nConcretely, we consider a generic model with a clas-\nsical spinSthat is antiferromagnetically exchange cou-\npled (J > 0) to a Hubbard system and study the spin\ndynamics as a function of the Hubbard- U. To tackle\nthis quantum-classical hybrid problem, we develop a\nnovel combination of linear-response theory [6, 16, 17] for\nthe spin dynamics with time-dependent density-matrix\nrenormalization group (t-DMRG) [18{20] for the corre-\nlated electron system. For technical reasons we consider\na Hubbard chain but concentrate on generic e\u000bects which\nare not bound to the one-dimensionality of the model.In the metallic phase at quarter \flling, a complex phe-\nnomenology is found where two di\u000berent channels for en-\nergy and spin dissipation, namely dissipation via corre-\nlated hopping and via excitations of local magnetic mo-\nments, become active on characteristic time scales, de-\npending on U. While magnetic excitations give the by\nfar dominating contribution to the Gilbert damping in\nthe strong-coupling limit, they contribute to the spin dy-\nnamics to a much lesser extent and on later and later\ntime scales when Uis increased.\nIt is demonstrated that electron correlations can have\nextreme consequences: At half-\flling and strong U, the\nspin relaxation is incomplete on intermediate time scales.\nThis represents a novel e\u000bect in a quantum-classical hy-\nbrid model which is reminiscent of prethermalization [21{\n24] or metastability of excitations due to lack of phase\nspace for decay [25{28], i.e., physics which so far has been\nobserved in purely electronic quantum systems only.\nGilbert damping. We consider the Hubbard model for\nNelectrons on an open chain of length Las a prototypical\nmodel of correlated conduction electrons:\nHe=\u0000Tn:n:X\ni<jX\n\u001b(cy\ni\u001bcj\u001b+ H.c.) +ULX\ni=1ni\"ni#:(1)\nThe nearest-neighbor (n.n.) hopping T= 1 sets the en-\nergy and time scale ( ~= 1). Using standard arguments\n[16, 17], the Gilbert damping parameter \u000bcan be com-\nputed as\n\u000b=\u0000J2Z1\n0dtt\u001f loc(t) (2)\nand depends on the Hubbard interaction Uvia the local\n(diagonal and isotropic) retarded spin susceptibility\n\u001floc(t) =\u0000i\u0002(t)h0j[si0z(t);si0z(0)]j0i (3)\nat the site i0where the classical spin is coupled to. J\nis the strength of the exchange interaction [see Eq. (4)\nbelow]. Furthermore, j0iis the ground state of He,\nsiz(t) =eiHetsize\u0000iHet, andsizis thez-component of thearXiv:1602.07317v2  [cond-mat.str-el]  16 Sep 20162\nlocal conduction-electron spin si=P\n\u001b\u001b0cy\ni\u001b\u001c\u001b\u001b0ci\u001b0=2,\nwith the vector of Pauli matrices \u001cand with\u001b;\u001b0=\";#.\nWe choose i0= 1 for two reasons: (i) As compared to\nthe symmetric choice i0=L=2, this allows us to double\nthe accessible time scale (before \fnite-size e\u000bects set in).\n(ii) The time-integral in (2) is sensitive to the long-time\nbehavior of \u001floc(t) which, at least for U= 0, is related\nto the strength of the van Hove singularities in the local\ndensity of states [17]. At the edge of the open chain,\nthose are weak and characteristic for a three-dimensional\nsystem.\nLocal spin correlations at quarter \flling. To compute\n\u001floc(t), we apply t-DMRG and the framework of matrix-\nproduct states [18] for systems with L= 80{120 sites.\nConcretely, we use the two-site version of the algorithm\nsuggested in Ref. [19, 20] which is based on the time-\ndependent variational principle (TDVP).\nElectron correlations are expected to speed up the re-\nlaxation of the classical spin since electron scattering fa-\ncilitates the transport of energy and spin density from\ni0to the bulk of the system. An increasingly e\u000ecient\ndissipation implies an increase of \u000bwithU. This can be\nnicely seen in \u001floc(t), which determines \u000bvia Eq. (2) and\nwhich is shown in Fig. 1 (upper panel) for quarter \flling\nn\u0011N=L = 0:5 where we have a (correlated) metal in the\nentireUrange. In fact, the absolute value of the integral\nweightR\ndt\u001floc(t) grows with increasing U.\nNote that via the \ructuation-dissipation theorem, the\ntotal weightR\ndt\u001floc(t) is given by the negative local\nstatic spin susceptibility. This explains that \u001floc(t) is\nmainly negative.\nSeparation of time scales. A central observation is\nthat\u001floc(t) develops a pronounced two-peak structure for\nstrongU. ForU= 0 and in the weak-coupling regime,\nthere is essentially a single (negative) peak around t\u00181\nonly. This corresponds to fast correlated-hopping pro-\ncesses on a scale set by the inverse hopping 1 =T. As is\nseen in the \fgure, the contribution of these processes to\nthe Gilbert damping grows with increasing U.\nThe second (negative) peak is clearly present for U&\n8. The almost linear shift of its position with Uhints\ntowards a time scale set by an e\u000bective magnetic interac-\ntionJH\u00181=Ubetween local magnetic moments formed\nby strong correlations in the conduction-electron system.\nEven forU!1 , however, local-moment formation is\nnot perfect at quarter \flling: We have hs2\nii=3\n4n=3\n8<\ns(s+ 1) withs= 1=2 for the size of the correlated lo-\ncal moment [12]. This explains the residual contributions\nfrom correlated hopping processes (\frst peak).\nFor strong Uthe Gilbert damping is dominated by\nmagnetic processes: Because of the extra factor tunder\nthe integral in Eq. (2), the contribution of the second\npeak in\u001floc(t) by far exceeds the hopping contribution\n(note the logarithmic scale in Fig. 1). Clearly, \u000bstrongly\nincreases with Uthough a precise value cannot be given\ndue to limitations of the t-DMRG in accessing the long-\n-0.10-0.050.00\u0000loc(t)\n10\u00001100101102timet0.00.10.20.30.40.5S(t)-0.10-0.050.00\u0000loc(t)\n10\u00001100101102timet0.00.10.20.30.40.5S(t)U=011Sx(t)Sz(t)U=22U=01214U=01“hopping”“magnetic”256128643216842-0.10-0.050.00\u0000loc(t)\n10\u00001100101102timet0.00.10.20.30.40.5S(t)-0.10-0.050.00\u0000loc(t)\n10\u00001100101102timet0.00.10.20.30.40.5S(t)U=0141Sx(t)Sz(t)U=22U=01214U=01“hopping”“magnetic”256128643216842-0.10-0.050.00\u0000loc(t)\n10\u00001100101102timet0.00.10.20.30.40.5S(t)-0.10-0.050.00\u0000loc(t)\n10\u00001100101102timet0.00.10.20.30.40.5S(t)U=0141Sx(t)Sz(t)U=22U=01214U=01“hopping”“magnetic”256128643216842-0.10-0.050.00\u0000loc(t)\n10\u00001100101102timet0.00.10.20.30.40.5S(t)-0.10-0.050.00\u0000loc(t)\n10\u00001100101102timet0.00.10.20.30.40.5S(t)U=0141Sx(t)Sz(t)U=22U=01214U=01“hopping”“magnetic”256128643216842-0.10-0.050.00\u0000loc(t)\n10\u00001100101102timet0.00.10.20.30.40.5S(t)-0.10-0.050.00\u0000loc(t)\n10\u00001100101102timet0.00.10.20.30.40.5S(t)U=0141Sx(t)Sz(t)U=22U=01214U=01“hopping”“magnetic”256128643216842-0.10-0.050.00\u0000loc(t)\n10\u00001100101102timet0.00.10.20.30.40.5S(t)-0.10-0.050.00\u0000loc(t)\n10\u00001100101102timet0.00.10.20.30.40.5S(t)U=0141Sx(t)Sz(t)U=22U=01214U=01“hopping”“magnetic”256128643216842\n1\nn=0.584100101102\n6080100U!0U!1⌧(U)UFIG. 1: (Color online) Upper panel: Local spin correlation\n\u001floc(t) ati0= 1 for an open Hubbard chain with L= 80 sites\nas obtained by t-DMRG for quarter \flling and di\u000berent Uas\nindicated. ( U\u001532:t\u0000Jmodel with three-site terms [29],\nL= 100 sites; U=1:L= 120). Energy and time scales are\n\fxed by the n.n. hopping T= 1. Lower panel: Resulting real-\ntime dynamics of a classical spin S(t) (withjS(t)j=1\n2, only\nSzis shown) coupled at i0to the local conduction-electron\nspin as obtained from Eq. (5) for J= 1 and di\u000berent U. The\nspin dynamics is initiated by switching the local magnetic\n\feld in Eq. (5) at time t= 0 fromx- toz-direction ( B\fn=\n1).Inset:U-dependence of the relaxation time \u001c, de\fned as\nSz(\u001c) = 0:98jSj.\ntime limit.\nSpin-dynamics model. The simple LLG equation for a\nclassical spin, i.e., _S=S\u0002B\u0000\u000bS\u0002_S, can only provide\nan overall picture of the spin dynamics and in fact ig-\nnores the electronic time-scale separation. We therefore\napply a re\fned approach which explicitly accounts for\nthe conduction-electron degrees of freedom in a model\nHwhich, besides He, Eq. (1), includes the local and\nisotropic coupling between si0and the classical spin S\n(jSj= 1=2):\nH=He+He\u0000spin=He+Jsi0S\u0000BS: (4)\nWe have also added a local magnetic \feld Bwhich, at\ntimet= 0, is suddenly switched from B=Bini^x, forcing\nthe spin to point in xdirection, toB=B\fn^zwithB\fn=\n1 to initiate the spin dynamics. Complete relaxation is\nachieved ifS(t)!1\n2^zfort!1 .\nThe Hamiltonian (4) represents a semiclassical Kondo-\nimpurity model with \fnite Hubbard- U. For a strong\nlocal \feld B\fn, the dynamical Kondo e\u000bect [30] that\nwould show up in case of a quantum-spin S=1\n2is sup-\npressed { this justi\fes the classical-spin approximation.\nThe quantum-classical hybrid (4) also results in the limit3\nof large spin quantum numbers S[31] of a correlated\nquantum-spin Kondo impurity model [32].\nS(t) satis\fes the classical equation of motion _S(t) =\nS(t)\u0002B\u0000JS(t)\u0002hsi0it[33]. Applying lowest-order per-\nturbation theory in J, the Kubo formula yields hsi0it=\nJRt\n0dt0\u001floc(t\u0000t0)S(t0) where the retarded local spin cor-\nrelation\u001floc(t) now plays the role of the linear-response\nfunction. As has been demonstrated in Ref. [17] for\nU= 0, this approach is perfectly reliable even for fairly\nstrong couplings Jand up to the time scale necessary\nfor complete spin relaxation. Here, we choose J= 1\nto generate relaxation times accessible to the t-DMRG\napproach.\nThe resulting e\u000bective integro-di\u000berential equation of\nmotion [6, 16] (see also Refs. [34, 35]),\n_S(t) =S(t)\u0002B\u0000J2S(t)\u0002Zt\n0dt0\u001floc(t\u0000t0)S(t0);(5)\nis numerically solved using a high-order Runge-Kutta [36]\nand quadrature techniques.\nCorrelation e\u000bects in the spin dynamics. The result-\ning spin dynamics (Fig. 1, lower panel) is characterized\nby precessional motion ( Sx(t);Sy(t) not shown) with Lar-\nmor frequency !L/Baround the ^ zaxis, and by relax-\nation driven by dissipation of energy and spin into the\nbulk of the electronic system. In the \fnal state there is\ncomplete alignment, S(t)\"\"B.\nComparing the results for the di\u000berent U, we \fnd that\n(i) signi\fcant relaxation starts at times t\u00181=T, i.e., on\nthe time scale for dissipation through correlated-hopping\nprocesses. (ii) Correlation e\u000bects lead to a considerably\nshorter relaxation time, e.g., by about a factor two when\ncomparing the results for U= 0 andU= 16 (see inset).\n(iii) To some extent this is due to an additional damping\nmechanism, namely via excitations of correlation-induced\nmagnetic moments { at least for moderate U. (iv) For\nstrongU, however, the relaxation time increases again.\nThis is counterintuitive but easily explained: Since the\nsecond, \\magnetic\" peak in \u001floc(t) shifts with increasing\nUto later and later times, relaxation is already com-\npleted before dissipation through spin-\rip processes can\nbecome active. This is most obvious for U!1 where\nmagnetic damping is never activated, and where a renor-\nmalized band picture may apply. (v) At intermediate U,\nhowever, the picture is di\u000berent. Here, spin-\rip processes\ndo contribute to the relaxation but more than an order\nof magnitude later ( t&U=T2) than the hopping time\nscale. Despite their dominating contribution to \u000b, their\ne\u000bect is weaker as compared to the correlated-hopping\nprocesses. Still, spin-\rips leave a clear characteristics in\nSz(t): their additional torque produces oscillations (with\na period largely independent of U) which superimpose\nthe monotonic relaxation dynamics. The onset of these\n\\magnetic\" oscillations linearly grows with U. Note that\nthe interpretation of these (and the following) \fndings\ndoes not rely on the one-dimensionality of the model.Gilbert damping of a Mott insulator. Dissipation\nthrough correlated hopping is impeded or even sup-\npressed at half-\flling where the system is a Mott insula-\ntor for allU > 0. Fig. 2 (upper panel) shows the t-DMRG\ndata for\u001floc(t) atn= 1 and di\u000berent U. Its time depen-\ndence is dominated by a single (negative) structure which\ngrows with increasing Uup to, say,U\u00198. In the weak-\ncoupling regime, U.4, the local magnetic moments are\nnot yet well-formed since the charge gap \u0001 \u0018e\u00001=U(as\nobtained from the Bethe ansatz [37] for U!0) is small\nas compared to T. Hence, residual hopping processes still\ncontribute signi\fcantly.\nIn the strong-coupling limit, on the other hand, spin-\n\rip processes dominate. Here, we observe scaling behav-\nior,\u001floc(t) =F(4tT2=U) with a universal function F(x).\nIndeed, due to the suppression of charge \ructuations, the\nlong-time, low-energy dynamics is captured by a Heisen-\nberg chainHHeis:=JHP\nisisi+1with antiferromagnetic\ninteraction JH= 4T2=Ubetween rigid s= 1=2-spins. As\nJHis the only energy scale remaining, F(tJH) is the re-\ntarded local susceptibility of the Heisenberg chain. With\nF(x) obtained numerically by means of t-DMRG applied\ntoHHeis:atJH= 1, the t-DMRG data for strong Uare\n\ftted perfectly (see Fig. 2). Signi\fcant deviations from\nthe scaling behavior can be seen in Fig. 2 for U= 8 and\nt\u00193, for instance.\nScaling can be exploited to determine the U-\ndependence of the Gilbert damping for a Mott insulator.\nFrom Eq. (2) we get\n\u000b=J2\nJ2\nHZ1\n0dxxF (x) =J2\nJ2\nH\u000b0=J2U2\n16T4\u000b0;(6)\nand thus, for \fxed J;T, we have\u000b/U2. For the univer-\nsal dimensionless Gilbert damping constant \u000b0we \fnd\n\u000b0\u00194:8: (7)\nFor a correlated Mott insulator, Eqs. (6) and (7) com-\npletely describe the U-dependence of the classical-spin\ndynamics in the weak- J, weak-Blimit where the t-\ndependence of S(t) is so slow, as compared to the typical\nmemory time \u001cmemcharacterizing \u001floc(t), that the Taylor\nexpansionS(t0)\u0019S(t) +_S(t)(t0\u0000t) can be cut at the\nlinear order under the t0-integral in Eq. (5), such that the\nLLG equation is obtained as a Red\feld equation [38].\nIncomplete spin relaxation. As demonstated with\nFig. 2 (lower panel), there is an anomalous U-dependence\nof the spin dynamics at n= 1. Only in the weak-\ncoupling regime, U.2, do damping e\u000bects increase and\nlead to a decrease of the relaxation time with increasing\nU. ForU= 4, however, the relaxation time increases\nagain. This behavior is clearly beyond the LLG the-\nory and is attributed to the fact that the memory time,\n\u001cmem/1=JH/Ufor strongU, becomes comparable to\nand \fnally exceeds the precession time scale \u001cB= 2\u0019=B\n(see the Supplementary Material [40]).4\n-0.4-0.3-0.2-0.10.0\u0000loc(t)\n10\u00001100101102timet0.00.10.20.30.40.5S(t)56Sx(t)Sz(t)U=84U=01201U=3216876542-0.10-0.050.00\u0000loc(t)\n10\u00001100101102timet0.00.10.20.30.40.5S(t)-0.10-0.050.00\u0000loc(t)\n10\u00001100101102timet0.00.10.20.30.40.5S(t)U=0141Sx(t)Sz(t)U=22U=01214U=01“hopping”“magnetic”256128643216842-0.10-0.050.00\u0000loc(t)\n10\u00001100101102timet0.00.10.20.30.40.5S(t)-0.10-0.050.00\u0000loc(t)\n10\u00001100101102timet0.00.10.20.30.40.5S(t)U=0141Sx(t)Sz(t)U=22U=01214U=01“hopping”“magnetic”256128643216842-0.10-0.050.00\u0000loc(t)\n10\u00001100101102timet0.00.10.20.30.40.5S(t)-0.10-0.050.00\u0000loc(t)\n10\u00001100101102timet0.00.10.20.30.40.5S(t)U=0141Sx(t)Sz(t)U=22U=01214U=01“hopping”“magnetic”256128643216842-0.10-0.050.00\u0000loc(t)\n10\u00001100101102timet0.00.10.20.30.40.5S(t)-0.10-0.050.00\u0000loc(t)\n10\u00001100101102timet0.00.10.20.30.40.5S(t)U=0141Sx(t)Sz(t)U=22U=01214U=01“hopping”“magnetic”256128643216842\nU=27U=81632HeisenbergmodelHeisenbergmodel\nn=1.0\nFIG. 2: (Color online) The same as Fig. 1 but for n= 1\n(L= 60). Thin black lines: Heisenberg model with JH=4T2\nU\n(L= 400) and, for improved accuracy at U= 8, with n.n.\nand n.n.n. couplings JH=4T2\nU\u000016T4\nU3andJ0\nH=4T4\nU3[39]\n(L= 300).\nIn addition, as for n= 0:5, we note a non-monotonic\nbehavior of Sz(t) with superimposed oscillations (see U=\n6, for example). With increasing Uthese oscillations die\nout, and at a \\critical\" interaction Uc\u00188 and for all\nU > U cthe relaxation time seems to diverge. Namely,\nthez-component of S(t) approaches a nearly constant\nvalue which decreases with increasing UwhileSx(and\nSy) still precess around B(see inset). Hence, on the\naccessible time scale, Ucmarks a transition or crossover\nto an incompletely relaxed but \\stationary\" state.\nThe same type of dynamical transition is also seen for\na classical spin coupled to a Heisenberg chain for which\nmuch larger system sizes ( L= 400) and thus about an\norder of magnitude longer time scales are accessible to t-\nDMRG. Here, the crossover coupling is JH;c\u00180:5. How-\never, these calculations as well as analytical arguments\nclearly indicate that a state with Sz= const.6= 1=2 is\nunstable and that \fnally, for t!1 , the fully relaxed\nstate withS(t)\"\"Bis reached (see [40] for details).\nThe \\stationary state\" on an intermediate time scale\noriginates when the bandwidth of magnetic excitations\ngets smaller than the \feld { as can be studied in detail\nalready for U= 0 (and very strong B). On the time\naxis, the missing relaxation results from a strong memory\ne\u000bect which, in the strong- Ulimit, shows up for JH.B.\nHere,\u001cmem&\u001cBwhich implies that the z-component of\nthe spin torque on S(t) averages to zero [40].\nThe incomplete spin relaxation can also be understood\nas a transient \\phase\" similar to the concept of a prether-\nmalized state. The latter is known for purely electronicsystems [21{24] which, in close parametric distance to\nintegrability, do not thermalize directly but are trapped\nfor some time in a prethermalized state. Here, for the\nquantum-classical hybrid, the analogue of an \\integrable\"\npoint is given by the U!1 limit where, for every \fnite\nt, the integral kernel \u001floc(t)\u00110, and Eq. (5) reduces to\nthe simple (linear) Landau-Lifschitz equation [1].\nThe situation is also reminiscent of quantum excita-\ntions which are metastable on an exponentially long time\nscale due to a small phase space for decay. An example\nis given by doublons in the Hubbard model which, for U\nmuch larger than the bandwidth and due to energy con-\nservation, can only decay in a high-order scattering pro-\ncess [25{28]. The relaxation time diverges in the U!1\nlimit where the doublon number is conserved. Here, for\naclassical spin, one would expect that relaxation via dis-\nsipation of (arbitrarily) small amounts of energy is still\npossible. Our results show, however, that this would hap-\npen on a longer time scale not accessible to the linear-\nresponse approach while the \\stationary state\" on the\nintermediate time scale is well captured [40].\nOutlook. Correlation-induced time-scale separation\nand incomplete relaxation represent phenomena with fur-\nther general implications. While slow correlation-induced\nmagnetic scales dominate the Gilbert damping \u000b, their\nactivation has been found to depend on microscopic\ndetails. This calls for novel correlated spin-dynamics\napproaches. The combination of t-DMRG with non-\nMarkovian classical spin dynamics is an example how to\nlink the \felds of strongly correlated electron systems and\nspin dynamics, but further work is necessary. Combi-\nnation with dynamical mean-\feld theory [42] is another\npromising option. Also spin dynamics based on LLG-\ntype approaches combined with ab initio band theory\ncould be successful in the case of very strong Uwhere\ndue to the absence of magnetic damping a renormalized\nband picture may be adequate. Further progress is even\nneeded for the very theory of a consistent hybrid-system\ndynamics [33, 43]. Generally, hybrid systems are not\nwell understood and call for a merger of known quantum\nand classical concepts, such as eigenstate thermalization,\nprethermalization, (non-)integrability etc [44]. However,\nalso concrete practical studies with classical spins cou-\npled to conduction electrons [17] are needed, as those\nhold the key for the microscopic understanding of nano-\nspintronics devices [15] or skyrmion dynamics [45, 46].\nAcknowledgements. We thank F. Hofmann for in-\nstructive discussions. Support of this work by the DFG\nwithin the SFB 668 (project B3) and within the SFB 925\n(project B5) is gratefully acknowledged.\n[1] L. D. Landau and E. M. Lifshitz, Physik. Zeits. Sowje-\ntunion 8,153 (1935); T. Gilbert, Phys. Rev. 100, 12435\n(1955); T. Gilbert, Magnetics, IEEE Transactions on 40,\n3443 (2004).\n[2] G. Tatara, H. Kohno, and J. Shibata, Phys. Rep. 468,\n213 (2008).\n[3] G. Bertotti, I. D. Mayergoyz, and C. Serpico, Nonlin-\near Magnetization Dynamics in Nanosystemes (Elsevier,\nAmsterdam, 2009).\n[4] K. Gilmore, Y. U. Idzerda, and M. D. Stiles, Phys. Rev.\nLett. 99, 027204 (2007).\n[5] H. Ebert, S. Mankovsky, D. K odderitzsch, and P. J.\nKelly, Phys. Rev. Lett. 107, 066603 (2011).\n[6] A. Sakuma, J. Phys. Soc. Jpn. 81, 084701 (2012).\n[7] M. F ahnle and C. Illg, J. Phys.: Condens. Matter 23,\n493201 (2011).\n[8] B. Skubic, J. Hellsvik, L. Nordstr om, and O. Eriksson,\nJ. Phys.: Condens. Matter 20, 315203 (2008).\n[9] E. M. Hankiewicz, G. Vignale, and Y. Tserkovnyak,\nPhys. Rev. B 78, 020404(R) (2008).\n[10] I. Garate and A. MacDonald, Phys. Rev. B 79, 064404\n(2009).\n[11] K. M. D. Hals, K. Flensberg, and M. S. Rudner, Phys.\nRev. B 92, 094403 (2015).\n[12] F. Gebhard, The Mott Metal-Insulator Transition\n(Springer, Berlin, 1997).\n[13] M. Morgenstern, Science 329, 1609 (2010).\n[14] S. Loth, M. Etzkorn, C. P. Lutz, D. M. Eigler, and A. J.\nHeinrich, Science 329, 1628 (2010).\n[15] A. A. Khajetoorians, J. Wiebe, B. Chilian, and\nR. Wiesendanger, Science 332, 1062 (2011).\n[16] S. Bhattacharjee, L. Nordstr om, and J. Fransson, Phys.\nRev. Lett. 108, 057204 (2012).\n[17] M. Sayad and M. Pottho\u000b, New J. Phys. 17, 113058\n(2015).\n[18] U. Schollw ock, Ann. Phys. (N.Y.) 326, 96 (2011).\n[19] J. Haegeman, J. I. Cirac, T. J. Osborne, I. Pi\u0014 zorn, H. Ver-\nschelde, and F. Verstraete, Phys. Rev. Lett. 107, 070601\n(2011).\n[20] J. Haegeman, C. Lubich, I. Oseledets, B. Vandereycken,\nand F. Verstraete, arXiv:1408.5056.\n[21] M. Moeckel and S. Kehrein, Phys. Rev. Lett. 100, 175702\n(2008).\n[22] M. Moeckel and S. Kehrein, New J. Phys. 12, 055016\n(2010).\n[23] M. Kollar, F. A. Wolf, and M. Eckstein, Phys. Rev. B\n84, 054304 (2011).\n[24] M. Marcuzzi, J. Marino, A. Gambassi, and A. Silva,\nPhys. Rev. Lett. 111, 197203 (2013).\n[25] A. Rosch, D. Rasch, B. Binz, and M. Vojta, Phys. Rev.\nLett. 101, 265301 (2008).\n[26] N. Strohmaier, D. Greif, R. J ordens, L. Tarruell, H.\nMoritz, T. Esslinger, R. Sensarma, D. Pekker, E. Altman,\nand E. Demler, Phys. Rev. Lett. 104, 080401 (2010).\n[27] F. Hofmann and M. Pottho\u000b, Phys. Rev. B 85, 205127\n(2012).\n[28] R. Rausch and M. Pottho\u000b, New J. Phys. 18, 023033\n(2016).\n[29] B. Ammon, M. Troyer, and H. Tsunetsugu, Phys. Rev.\nB52, 629 (1995).\n[30] M. Nuss, M. Ganahl, E. Arrigoni, W. von der Linden,\nand H. G. Evertz, Phys. Rev. B 91, 085127 (2015).\n[31] D. A. Garanin, Phys. Rev. B 78, 144413 (2008).\n[32] M. Sayad, R. Rausch, and M. Pottho\u000b, in preparation.\n[33] H. Elze, Phys. Rev. A 85, 052109 (2012).\n[34] T. Bose and S. Trimper, Phys. Rev. B 83, 134434 (2011).[35] D. Thonig, J. Henk, and O. Eriksson, Phys. Rev. B 92,\n104403 (2015).\n[36] J. H. Verner, Numerical Algorithms 53, 383 (2009).\n[37] A. A. Ovchinnikov, Zh. Eksp. Teor. Fiz. 57, 2137 (1969).\n[38] H. P. Breuer and F. Petruccione, The Theory of Open\nQuantum Systems (Oxford University Press, New York,\n2002).\n[39] L. N. Buleavski \u0014i, Soviet Journal of Experimental and\nTheoretical Physics 24, 154 (1967).\n[40] See Supplemental Material at URL .\n[41] F. H. L. Essler, H. Frahm, F. G ohmann, A. Kl umper,\nand V. Korepin, The One-Dimensional Hubbard Model\n(Cambridge University Press, Cambridge, 2005).\n[42] A. Georges, G. Kotliar, W. Krauth and M. J. Rozenberg,\nRev. Mod. Phys. 68, 13 (1996).\n[43] L. L. Salcedo, Phys. Rev. A 85, 022127 (2012).\n[44] M. Rigol, V. Dunjko, and M. Olshanii, Nature 452, 854\n(2008).\n[45] A. Fert, V. C. J., and Sampaio, Nat. Nanotechnol. 8, 152\n(2013).\n[46] J. Iwasaki, M. Mochizuki, and N. Nagaosa, Nat. Nan-\notechnol. 8, 742 (2013).6\nSupplementary material\nValidity of linear-response theory.\nThe reliability of the linear-response approach (see Eq.\n(5) of the main text) can be tested by comparing with the\nresults of the full (non-perturbative) quantum-classical\nhybrid dynamics for the model given by Eq. (4) of the\nmain text. This is easily accessible for the case U= 0 (see\nRef. [17] for details). Fig. 3 displays the time dependence\nof thez-component of the classical spin for J= 1, for a\nhalf-\flled system ( n= 1) of non-interacting conduction\nelectrons (U= 0) and for di\u000berent strengths of the \feld\nBafter switching from x- toz-direction.\nWhile there are some discrepancies visible, as ex-\npected, the \fgure demonstrates that the agreement on\na qualitative level is in fact excellent for weak as well\nas for strong \felds. Both approaches also predict a\ncrossover from complete to incomplete spin relaxation\natB=Bc\u00194. We conclude that the linear-response\napproach provides reliable results for the classical spin\ndynamics.\nThis can be explained by the observation that jhsi0itjis\nsmall and that the classical spin S(t) and the conduction-\nelectron momenthsi0itare nearly collinear at any instant\nof time (see Ref. [17] for a detailed and systematic discus-\nsion). Hence, even for moderately strong couplings J, the\nlinear-response contribution J2S(t)\u0002hsi0itto the equa-\ntion of motion for S(t) is small (and the quadratic and\nhigher-order corrections are expected to be even smaller).\n10\u00001100101102timet0.00.10.20.30.40.5SzJ=1B=1B=2B=3B=3.5B=4B=5\nFIG. 3: Time dependence of the z-component of the classical\nspin forJ= 1,n= 1,U= 0 and di\u000berent values of the \feld\nBas indicated. Calculations based on the linear-response\napproach (fat solid lines) are compared to the results of the\nfull quantum-classical hybrid theory (thin solid lines) for L=\n500.Mechanism for incomplete relaxation.\nFig. 3 shows that the relaxation of the classical spin\nbecomes incomplete for strong B. On the basis of the\nlinear-response theory this can be explained as follows:\nThexandycomponents of the linear response\nhsi0it=Zt\n0d\u001c\u001f loc(\u001c)S(t\u0000\u001c); (8)\ntend to zero if the characteristic memory time \u001cmem of\nthe kernel\u001floc(\u001c) is much larger than the precession time\nscale\u001cB= 2\u0019=B since the integral produces a vanishing\naverage in this case. This means that the corresponding\ntorque,\u0000J2S(t)\u0002hsi0it, is perpendicular to the \feld\ndirection and hence there is no relaxation of the spin.\nThe same argument can also be formulated after trans-\nformation to frequency space: After some transient ef-\nfect, we havehsi0i!=\u001floc(!)S(!), and thus the x;y-\ncomponents of the linear response will vanish if \u001floc(!=\nB) = 0, i.e., if Bis stronger than the bandwidth of the\nmagnetic excitations (here: Bc\u00194). Note that this\nrequires an unrealistically strong \feld in case of non-\ninteracting conduction electrons.\nIn the case of correlated conduction electrons, jS(t)\u0002\nhsi0itjremains small (of the order of 0.1 or smaller), for\nweak and for strong B, as has been checked numerically.\nWe therefore expect the linear-response approach to pro-\nvide qualitatively correct results for U > 0 as well.\nAt half-\flling and for strong U, the memory time\n\u001cmem/J\u00001\nH/U, i.e.,\u001cmem can easily become large as\ncompared to \u001cB, and thus incomplete spin relaxation can\noccur at comparatively weak and physically meaningful\n\feld strengths. For example, from the Bethe ansatz [41]\nwe have\nWspinon = 2Z1\n0dx\nxJ1(x)\ncosh(Ux=4)!\u0019\n2JHforU!1\n(9)\nfor the spinon bandwidth Wspinon whereJ1(x) is the \frst\nBessel function. Hence, for strong Hubbard interaction,\nBc\u00192Wspinon =\u0019JH.\nClassical spin coupled to a Heisenberg model.\nAt half-\flling and in the limit U!1 the low-energy\nphysics of the Hubbard model is captured by an antifer-\nromagnetic Heisenberg model,\nHs=X\ni(JHsisi+1+J0\nHsisi+2); (10)\nwhere up to order O(T2=U) the nearest-neighbor and\nthe next-nearest-neighbor couplings [39] are JH= 4T2=U\nandJ0\nH= 0, and up to order O(T4=U3),\nJH=4T2\nU\u000016T4\nU3; J0\nH=4T4\nU3: (11)7\n10\u00001100101102timet-0.4-0.3-0.2-0.10.0\u0000loc(t)\nJH=1\nFIG. 4: Local susceptibility at the edge of an open Heisen-\nberg chain ( JH= 1).\nAnalytically, by perturbation theory in x= 4tT2=U=\ntJH, one veri\fes the linear short-time behavior\n\u001floc(t) = \u0002(t)t2\n3(JHhsi0si0+1i+J0\nHhsi0si0+2i) +O(x2);\n(12)\nvalid to leading order for both, the Hubbard and the\ne\u000bective Heisenberg model. However, as can be seen in\nFig. 2 of the main text, the Heisenberg dynamics also\napplies to intermediate times; the e\u000bective model with\ncoupling constants (11) almost perfectly reproduces the\nresults of the Hubbard model for U\u00158.\nHere, we treat the Heisenberg model with n.n. cou-\npling as an independent system. Fig. 4 shows the cor-\nresponding local spin susceptibility for JH= 1 as ob-\ntained by t-DMRG calculations with L= 400 Heisen-\nberg spins. Since JHis the only energy scale, we have\n\u001floc(t) =F(tJH) for arbitrary JHwhereF(x) is a func-\ntion independent of JH. This implies that the dominant\n(negative) peak of \u001floc(t) shifts to later and later times\nasJHdecreases.\nFig. 5 displays the spin dynamics resulting from the\nfull model\nH=Hs+Hs\u0000spin=Hs+Jsi0S\u0000BS; (13)\nas obtained by the linear-response approach. One clearly\nnotes that for J.Jc\u00180:5 (corresponding to Uc\u00188)\nthe time dependence of Szdevelops a prethermalization-\nlike plateau on an intermediate time scale t\u0018100 (in\nunits of 1=JH).\nFor the Heisenberg model, using the scaling property\nof\u001floc(t), it is easily possible to perform calculations up\ntot= 1000. On this longer time scale, it is clearly vis-\nible (see Fig. 5) that Szdoes notapproach a constant\nvalue asymptotically. For JH= 0:4 andJH= 0:2 thez-\ncomponent of S(t) is even found to decrease and appears\nto approach the trivial solution Sz(t)\u00110.\nHowever, it is straightforwardly seen that a \\stationary\nstate\" of the form\nS(t) =Sz^z+S?cos(!t+')^x+S?sin(!t+')^y(14)\n10\u00001100101102timet-0.5-0.4-0.3-0.2-0.10.00.10.20.30.40.5S(t)Sx(t)Sz(t)JH=1.0JH=0.80.60.50.40.2FIG. 5: Classical spin dynamics for J= 1 and di\u000berent JH.\nwith arbitrary parameters S?;!;' and with constant\n(time-independent) Szdoes not solve the integro-\ndi\u000berential equation (5) of the main text for t! 1 .\nThere is one exception only, namely the trivial case where\n\u001floc(t)\u00110 which can be realized, up to arbitrarily long\ntimes, in the limit JH!0.\nFor small but \fnite JH>0, we therefore expect that\nthe classical spin develops a dynamics on an extremely\nlong time scale t\u001d103, the onset of which is already seen\nin Fig. 5, which \fnally terminates in the fully relaxed\nstate withS(t)!S0\"\"B.\nIt is in fact easy to see from the integro-di\u000berential\nequation that, if there is spin relaxation to a time-\nindependent constant, S(t)!S0fort!1 , the relaxed\nstate hasS0= 0:5^z. This implies that if there is com-\nplete relaxation at all, the spin relaxes to the equilibrium\ndirection.\nOscillations at short times.\nAs can be seen in Fig. 5 for weak JH, thez-component\nof the spin develops oscillations at short times, which can\nalso be seen for the case of the Hubbard model (cf. Fig. 2\nof the main text). These oscillations can be understood\nin the following way: Inserting the expression (12) with\nJ0\nH= 0 for the behavior of \u001floc(t) at short times into Eq.\n(5) of the main text,\n_S(t) =S(t)\u0002B\u00002\n3J2JHhsi0si0+1iS(t)\n\u0002Zt\n0dt0(t\u0000t0)S(t0) +O(t3J4); (15)\nand approximating S(t) by theJ= 0 resultS0(t) =\nS(cos!t;sin!t;0) (withB=B^z,!=B,S= 1=2) in\nthe second term on the right-hand side, a straightforward\ncalculations yields:\nSz(t) =2\n3J2JHhsi0si0+1iS2!tsin!t+ 2 cos!t\u00002\n!3\n+O(t4J2J2\nH): (16)8\nThis is found to perfectly describe the short-time oscil-\nlations for weak JHin Fig. 5 and for strong Uin Fig. 2in the main text. For longer times the oscillations are\ndamped and eventually die out."    },    {        "title": "1602.07325v1.Experimental_Investigation_of_Temperature_Dependent_Gilbert_Damping_in_Permalloy_Thin_Films.pdf",        "content": "1    Experimental Investigation of Temperature-Dependent Gilbert \nDamping in Permalloy Thin Films  \nYuelei Zhao1,2†, Qi Song1,2†, See-Hun Yang3, Tang Su1,2, Wei Yuan1,2, Stuart S. P. Parkin3,4, Jing \nShi5*, and Wei Han1,2*   \n1International Center for Quantum Materials,  Peking University, Beijing, 100871, P. R. China \n2Collaborative Innovation Center of Quantum Matter, Beijing 100871, P. R. China \n3IBM Almaden Research Center, San Jose, California 95120, USA \n4Max Planck Institute for Microstructu re Physics, 06120 Halle (Saale), Germany \n5Department of Physics and Astronomy, Univers ity of California, Riverside, California 92521, \nUSA \n†These authors contributed equally to the work \n*Correspondence to be addressed to: jing.shi @ucr.edu (J.S.) and weihan@pku.edu.cn (W.H.) \n \n \nAbstract \nThe Gilbert damping of ferromagnetic materials is arguably the most important but least \nunderstood phenomenological parameter that dictates real-time magnetization dynamics. \nUnderstanding the physical origin of  the Gilbert damping is highly relevant to developing future \nfast switching spintronics devices such as magnetic sensors and magnetic random access memory. Here, we report an experimental stud y of temperature-dependent Gilbert damping in \npermalloy (Py) thin films of varying thicknesses  by ferromagnetic resonance. From the thickness \ndependence, two independent cont ributions to the Gilbert damping are identified, namely bulk \ndamping and surface damping. Of particular inte rest, bulk damping decreases monotonically as \nthe temperature decreases, while surface da mping shows an enhancement peak at the 2 temperature of ~50 K. These results provide an important insight to the physical origin of the \nGilbert damping in ultr athin magnetic films.  \n \nIntroduction \nIt is well known that the magnetization dynamics  is described by the Landau-Lifshitz-Gilbert \nequation with a phenomenological parameter called the Gilbert damping ( α),1,2:   \n eff\nSdM dMMH Mdt M dtαγ=− × + × \n   (1) \nwhere M\nis the magnetization vector,  γis the gyromagne tic ratio, and SM M=\n is the saturation \nmagnetization. Despite intense theore tical and experimental efforts3-15, the microscopic origin of \nthe damping in ferromagnetic (FM) metallic ma terials is still not well understood. Using FM \nmetals as an example, vanadium doping decreases the Gilbert damping of Fe3 while many other \nrare-earth metals doping increase s the damping of permalloy (Py)4-6,16. Theoretically, several \nmodels have been developed to explain some  key characteristics. For example, spin-orbit \ncoupling is proposed to be the intrinsic or igin for homogenous time-varying magnetization9. The \ns-d exchange scattering model assumes that damp ing results from scattering of the conducting \nspin polarized electrons with the magnetization10. Besides, there is the Fermi surface breathing \nmodel taking account of the spin scattering with  the lattice defects ba sed on the Fermi golden \nrule11,12. Furthermore, other damping mechanisms in clude electron-electron scattering, electron-\nimpurity scattering13 and spin pumping into the adjacent nonmagnetic layers14, as well as the two \nmagnon scattering model, which refers to that pa irs of magnon are scatte red by defects, and the \nferromagnetic resonance (FMR) mode  moves into short wavelength spin waves, leading to a 3 dephasing contribution to the linewidth15. In magnetic nanostructu res, the magnetization \ndynamics is dictated by the Gilbert damping of the FM materials which can be simulated by \nmicromagnetics given the boundaries and dimens ions of the nanostructures. Therefore, \nunderstanding the Gilbert damping in FM materials is particularly important for characterizing \nand controlling ultrafast responses in magnetic  nanostructures that ar e highly relevant to \nspintronic applications such as  magne tic sensors and magnetic random access memory17.  \nIn this letter, we report an expe rimental investigation of the G ilbert damping in Py thin films \nvia variable temperature FMR in a modified  multi-functional insert  of physical property \nmeasurement system with a coplanar waveguide (see methods for details). We choose Py thin \nfilms since it is an interesting FM metallic material for spintronics due to its high permeability, nearly zero magnetostriction, low coercivity, a nd very large anisotropi c magnetoresistance. In \nour study, Py thin films are gr own on top of ~25 nm SiO\n2/Si substrates with a thickness ( d) range \nof 3-50 nm by magnetron sputtering (see methods  for details). A capping layer of TaN or Al 2O3 \nis used to prevent oxidation of the Py during m easurement. Interestingly, we observe that the \nGilbert damping of the thin Py films ( d <= 10 nm) shows an enhanced peak at ~ 50 K, while \nthicker films ( d >= 20 nm) decreases monotonically as the temperature decreases. The distinct \nlow-temperature behavior in the Gilbert dampi ng in different thickness regimes indicates a \npronounced surface contribution in the thin limit. In  fact, from the linear  relationship of the \nGilbert damping as a function of the 1/ d, we identify two contribu tions, namely bulk damping \nand surface damping. Interestingl y, these two contributions show  very different temperature \ndependent behaviors, in whic h the bulk damping decreases m onotonically as the temperature \ndecreases, while the surface damping indicates an  enhancement peak at ~ 50 K. We also notice \nthat the effective magnetization sh ows an increase at the same temperature of ~50 K for 3 and 5 4 nm Py films.  These observations could be all related to the magnetization reorientation on the \nPy surface at a certain temperatur e. Our results are important for theoretical investigation of the \nphysical origins of Gilbert damping and also us eful for the purpose of designing fast switching \nspintronics devices.  \nResults and Discussion \nFigure 1a shows five representative curves  of the forward amplitude of the complex \ntransmission coefficients (S 21) vs. in plane magnetic field meas ured on the 30 nm Py film with \nTaN capping at the frequencies of 4, 6, 8, 10 an d 12 GHz and at 300 K after renormalization by \nsubtracting a constant background. These experiment al results could be fitted using the Lorentz \nequation18: \n 2\n21 0 22()\n() ( )resHSSHH HΔ∝Δ+ −  (2) \nwhere S0 is the constant describing the coefficient for the transmitted microwave power, H is the \nexternal magnetic field, Hres is the magnetic field under the resonance condition, and ΔH is the \nhalf linewidth. The extracted ΔH vs. the excitation frequency ( f) is summarized in Figures 1b and \n1c for the temperature of 300 K and 5 K respect ively. The Gilbert damp ing could be obtained \nfrom the linearly fitted curves (red lin es), based on the following equation:  \n 02()H fHπαγΔ= + Δ   (3) \nin which γ is the geomagnetic ratio and ΔH0 is related to the inhom ogeneous properties of the \nPy films. The Gilbert damping at 300 K and 5 K is calculated to be 0.0064 ± 0.0001 and 0.0055 \n± 0.0001 respectively.  5 The temperature dependence of the Gilbert damp ing for 3-50 nm Py films with TaN capping \nlayer is summarized in Figure 2a. As d decreases, the Gilbert damping increases, indicative of \nthe increasing importance of the film surfaces. Interestingly, fo r thicker Py films (e.g. 30 nm), \nthe damping decreases monotonically as the temper ature decreases, which is expected for bulk \nmaterials due to suppressed sca ttering at low temperature. As d decreases down to 10 nm, an \nenhanced peak of the damping is obser ved at the temperature of ~ 50 K. As d decreases further, \nthe peak of the damping becomes more pronounce d. For the 3 nm Py film, the damping shows a \nslight decrease first from 0.0126 ± 0.0001 at 3 00 K to 0.0121 ± 0.0001 at 175 K, and a giant \nenhancement up to 0.0142 ± 0.0001 at 50 K, and then a sharp decrease back down to 0.0114 ± \n0.0003 at 5 K.  \nThe Gilbert damping as a function of the Py th icknesses at each temperature is also studied. \nFigure 2b shows the thickness dependence of the Py damping at 300 K. As d increases, the \nGilbert damping decreases, which indicates a surface/interface enhanced damping for thin Py \nfilms19. To separate the damping due to the bul k and the surface/interface contribution, the \ndamping is plotted as a function of 1/ d, as shown in Figure 2c, and it follows this equation as \nsuggested by theories19-21. \n 1()BSdαα α=+   (4) \nin which the Bα and Sα represent the bulk and surface da mping, respectively. From these \nlinearly fitted curves, we are able to separate  the bulk damping term and the surface damping \nterm out. In Figure 2b, the best fitted parameters for Bα and Sα are 0.0055 ± 0.0003 and 0.020 ± \n0.002 nm. To be noted, there are two insulating mate rials adjacent to the Py films in our studies. 6 This is very different from previous studies on Py/Pt bilayer systems, where the spin pumping \ninto Pt leads to an enhanced magnetic dampi ng in Py. Hence, the enhanced damping in our \nstudies is very unlikely resulti ng from spin pumping into SiO 2 or TaN. To our knowledge, this \nsurface damping could be related to  interfacial spin f lip scattering at the interface between Py \nand the insulating layers, which ha s been included in a generalized  spin-pumping theory reported \nrecently21. \nThe temperature dependence of the bulk damp ing and the surface damping are summarized \nin Figures 3a and 3b. The bulk damping of Py is  ~0.0055 at 300 K. As the temperature decreases, \nit shows a monotonic decrea se and is down to ~0.0049 at 5 K. Th ese values are consistent with \ntheoretical first principle calculations21-23 and the experimental valu es (0.004-0.008) reported for \nPy films with d ≥ 30 nm24-27. The temperature dependence of the bulk damping could be \nattributed to the magnetization rela xation due to the spin-lattice scattering in the Py films, which \ndecreases as the temperature decreases. \nOf particular interest, the surface damping sh ows a completely different characteristic, \nindicating a totally different mechanism from th e bulk damping. A strong enhancement peak is \nobserved at ~ 50 K for the surface damping. Could this enhancement of this surface/interface \ndamping be due to the strong spin-orbit coupli ng in atomic Ta of Ta N capping layer? To \ninvestigate this, we measure the damping of the 5 nm and 30 nm Py films with Al 2O3 capping \nlayer, which is expected to exhibit much lo wer spin-orbit coupling compared to TaN. The \ntemperature dependence of the Py damping is su mmarized in Figures 4a and 4b. Interestingly, \nthe similar enhancement of the damping at ~ 50 K is observed for 5 nm Py film with either Al 2O3 \ncapping layer or TaN layer, whic h excludes that the origin of the feature of the enhanced 7 damping at ~50 K results from th e strong spin-orbit coupling in TaN layer. These results also \nindicate that the mechanism of this feature is most likely related to the common properties of Py \nwith TaN and Al 2O3 capping layers, such as the crysta lline grain boundary and roughness of the \nPy films, etc. \nOne possible mechanism for the observed peak of  the damping at ~50 K could be related to a \nthermally induced spin reorientation transition  on the Py surface at that temperature. For \nexample, it has been show n that the spin reorientation of Py  in magnetic tunnel junction structure \nhappens due to the competition of different magne tic anisotropies, which c ould give rise to the \npeak of the FMR linewidth around the temperature of ~60 K28. Furthermore, we measure the \neffective magnetization ( Meff) as a function of temperature. Meff is obtained from the resonance \nfrequencies ( fres) vs. the external magnetic field via the Kittel formula29:  \n 12() [ ( 4 ) ]2res res res efffH H Mγππ=+   (4) \nin which Hres is the magnetic field at the resonance condition, and Meff is the effective \nmagnetization which contains the saturation ma gnetization and other anisotropy contributions. \nAs shown in Figures 5a and 5b, the 4π*M eff for 30 nm Py films w ith TaN capping layer are \nobtained to be ~10.4 and ~10.9 kG at 300 K and 5 K respectively. The temperature dependences \nof the 4π*M eff for 3nm, 5 nm, and 30 nm Py films are s hown in Figures 6a-6c. Around ~50 K, an \nanomaly in the effective magnetization for thin Py films (3 and 5 nm) is observed. Since we do \nnot expect any steep change in Py’s saturation magnetization at this temperature, the anomaly in \n4π*M eff should be caused by an anisot ropy change which coul d be related to a sp in reorientation. \nHowever, to fully understand the underlying mechan isms of the peak of the surface damping at ~ \n50 K, further theoretical and e xperimental studies are needed. 8 Conclusion  \nIn summary, the thickness and temperature dependences of the Gilbert damping in Py thin \nfilms are investigated, from which the contributio n due to the bulk damping and surface damping \nare clearly identified. Of particular interest, the bulk damping decreases monotonically as the \ntemperature decreases, while the surface damping develops an enhancement peak at ~ 50 K, \nwhich could be related to a thermally induced spin reorientation for the surface magnetization of \nthe Py thin films. This model is also consistent  with the observation of an enhancement of the \neffective magnetization below ~50 K. Our expe rimental results will contribute to the \nunderstanding of the intrinsic and ex trinsic mechanisms of the Gilber t damping in FM thin films.  \n \nMethods   \nMaterials growth. The Py thin films are deposited on ~25 nm SiO 2/Si substrates at room \ntemperature in 3×10- 3 Torr argon in a magnetron sputtering sy stem with a base pressure of ~ \n1×10-8 Torr. The growth rate of the Py is ~ 1  Å/s. To prevent ex situ  oxidation of the Py film \nduring the measurement,  a ~ 20 Å TaN or Al 2O3 capping layer is grown in situ environment. The \nTaN layer is grown by reactive sputtering of a Ta target in an argon-nitrogen gas mixture (ratio: \n90/10). For Al 2O3 capping layer, a thin Al (3 Å) layer is deposited first, and the Al 2O3 is \ndeposited by reactive spu ttering of an Al target in an ar gon-oxygen gas mixture (ratio: 93/7).  \nFMR measurement.  The FMR is measured using the vector network analyzer (VNA, Agilent \nE5071C) connected with a coplanar wave guide30 in the variable temperature insert of a \nQuantum Design Physical Properties Measuremen t System (PPMS) in the temperature range \nfrom 300 to 2 K. The Py sample is cut to be 1 × 0.4 cm and attached to the coplanar wave guide 9 with insulating silicon paste. For each temper ature from 300 K to 2 K, the forward complex \ntransmission coefficients (S 21) for the frequencies between 1 - 15 GHz are recorded as a function \nof the magnetic field sweeping from ~2500 Oe to 0 Oe.  \n \nContributions  \nJ.S. and W.H. proposed and supervised the studies. Y.Z. and Q.S. performed the FMR \nmeasurement and analyzed the data. T.S. and W.Y.  helped the measurement. S.H.Y. and S.S.P.P. \ngrew the films. Y.Z., J.S. and W.H. wrote the manuscript. All authors commented on the \nmanuscript and contributed to its final version. \n \nAcknowledgements   \nWe acknowledge the fruitful discussions with  Ryuichi Shindou, Ke Xia, Ziqiang Qiu, Qian \nNiu, Xincheng Xie and Ji Feng and the support of National Basic Research Programs of China \n(973 Grants 2013CB921903, 2014CB920902 and 2015 CB921104). Wei Han also acknowledges \nthe support by the 1000 Talents Program for Young Scientists of China. \n \nCompeting financial interests \nThe authors declare no compe ting financial interests. \n \n \nReferences: \n \n1 Landau, L. & Lifshitz, E. On the theory of the dispersion of magnetic permeability in \nferromagnetic bodies. Phys. Z. Sowjetunion  8, 153 (1935). \n2 Gilbert, T. L. A phenomenological theory  of damping in ferromagnetic materials. \nMagnetics, IEEE Transactions on  40, 3443-3449, doi:10.1109/TMAG.2004.836740 \n(2004). 10 3 Scheck, C., Cheng, L., Barsukov, I., Frait, Z.  & Bailey, W. E. Low Relaxation Rate in \nEpitaxial Vanadium-Doped Ultrathin Iron Films. Phys. Rev. Lett.  98, 117601 (2007). \n4 Woltersdorf, G., Kiessling, M., Meyer, G., Th iele, J. U. & Back, C. H. Damping by Slow \nRelaxing Rare Earth Impurities in Ni 80Fe20 Phys. Rev. Lett.  102, 257602 (2009). \n5 Radu, I., Woltersdorf, G., Kiessling, M., Meln ikov, A., Bovensiepen, U., Thiele, J. U. & \nBack, C. H. Laser-Induced Magnetization Dynamics of Lanthanide-Doped Permalloy \nThin Films. Phys. Rev. Lett.  102, 117201 (2009). \n6 Yin, Y., Pan, F., Ahlberg, M., Ranjbar, M ., Dürrenfeld, P., Houshang, A., Haidar, M., \nBergqvist, L., Zhai, Y., Dumas, R. K., Deli n, A. & Åkerman, J. Tunable permalloy-based \nfilms for magnonic devices. Phys. Rev. B  92, 024427 (2015). \n7 Rantschler, J. O., McMichae l, R. D., Castillo, A., Shapiro, A. J., Egelhoff, W. F., \nMaranville, B. B., Pulugurtha, D., Chen, A. P. & Connors, L. M. Effect of 3d, 4d, and 5d \ntransition metal doping on damping in permalloy thin films. J. Appl. Phys.  101, 033911 \n(2007). \n8 Ingvarsson, S., Ritchie, L., Liu, X. Y., Xia o, G., Slonczewski, J. C., Trouilloud, P. L. & \nKoch, R. H. Role of electron scattering in the magnetization relaxation of thin Ni 81Fe19 \nfilms. Phys. Rev. B  66, 214416 (2002). \n9 Hickey, M. C. & Moodera, J. S. Origin of Intrinsic Gilbert Damping. Phys. Rev. Lett.  \n102, 137601 (2009). \n10 Zhang, S. & Li, Z. Roles of Nonequilibri um Conduction Electrons  on the Magnetization \nDynamics of Ferromagnets. Phys. Rev. Lett.  93, 127204 (2004). \n11 Kuneš, J. & Kamberský, V. First-principl es investigation of the damping of fast \nmagnetization precession in ferromagnetic 3d metals. Phys. Rev. B  65, 212411 (2002). \n12 Kamberský, V. Spin-orbital Gilbert damping in common magnetic metals. Phys. Rev. B  \n76, 134416 (2007). \n13 Hankiewicz, E. M., Vignale, G. & Ts erkovnyak, Y. Inhomogeneous Gilbert damping \nfrom impurities and electron-electron interactions. Phys. Rev. B  78, 020404 (2008). \n14 Brataas, A., Tserkovnyak, Y. & Bauer, G. E. W. Scattering Theory of Gilbert Damping. \nPhys. Rev. Lett.  101, 037207 (2008). \n15 Arias, R. & Mills, D. L. Extrinsic contri butions to the ferromagne tic resonance response \nof ultrathin films. Phys. Rev. B  60, 7395-7409 (1999). \n16 Walowski, J., Müller, G., Djordjevic, M., M ünzenberg, M., Kläui, M., Vaz, C. A. F. & \nBland, J. A. C. Energy Equilibration Pro cesses of Electrons, Magnons, and Phonons at \nthe Femtosecond Time Scale. Phys. Rev. Lett.  101, 237401 (2008). \n17 Stiles, M. D. & Miltat, J. in Spin Dynamics in Confined Magnetic Structures III  Vol. 101 \nTopics in Applied Physics  (eds Burkard Hillebrands & André Thiaville) Ch. 7, 225-308 \n(Springer Berlin Heidelberg, 2006). \n18 Celinski, Z., Urquhart, K. B. & Heinrich, B. Using ferromagnetic resonance to measure \nthe magnetic moments of ultrathin films. J. Magn. Magn. Mater.  166, 6-26 (1997). \n19 Barati, E., Cinal, M., Edwards, D. M. & Umerski, A. Gilbert damping in magnetic \nlayered systems. Phys. Rev. B  90, 014420 (2014). \n20 Tserkovnyak, Y., Brataas, A., Bauer, G. E. W.  & Halperin, B. I. Nonlocal magnetization \ndynamics in ferromagnetic heterostructures. Rev. Mod. Phys.  77, 1375-1421 (2005). \n21 Liu, Y., Yuan, Z., Wesselink, R. J. H., St arikov, A. A. & Kelly, P. J. Interface \nEnhancement of Gilbert Damping from First Principles. Phys. Rev. Lett.  113, 207202 \n(2014). 11 22 Starikov, A. A., Kelly, P. J ., Brataas, A., Tserkovnyak, Y. & Bauer, G. E. W. Unified \nFirst-Principles Study of Gilbert Dampi ng, Spin-Flip Diffusion, and Resistivity in \nTransition Metal Alloys. Phys. Rev. Lett.  105, 236601 (2010). \n23 Mankovsky, S., Ködderitzsch, D., Woltersdo rf, G. & Ebert, H. First-principles \ncalculation of the Gilbert damping paramete r via the linear response formalism with \napplication to magnetic transition metals and alloys. Physical Review B  87, 014430 \n(2013). \n24 Bailey, W., Kabos, P., Mancoff, F. & Russe k, S. Control of magnetization dynamics in \nNi81Fe19 thin films through the us e of rare-earth dopants. Magnetics, IEEE Transactions \non 37, 1749-1754 (2001). \n25 Rantschler, J. O., Maranville, B. B., Malle tt, J. J., Chen, P., McMichael, R. D. & \nEgelhoff, W. F. Damping at no rmal metal/permalloy interfaces. Magnetics, IEEE \nTransactions on  41, 3523-3525 (2005). \n26 Luo, C., Feng, Z., Fu, Y., Zha ng, W., Wong, P. K. J., Kou, Z. X., Zhai, Y., Ding, H. F., \nFarle, M., Du, J. & Zhai, H. R. Enhancem ent of magnetization damping coefficient of \npermalloy thin films with dilute Nd dopants. Phys. Rev. B  89, 184412 (2014). \n27 Ghosh, A., Sierra, J. F., Auffret, S., Ebels,  U. & Bailey, W. E. Dependence of nonlocal \nGilbert damping on the ferromagnetic layer t ype in ferromagnet/Cu/Pt heterostructures. \nAppl. Phys. Lett.  98, 052508 (2011). \n28 Sierra, J. F., Pryadun, V. V., Russek, S. E ., García-Hernández, M., Mompean, F., Rozada, \nR., Chubykalo-Fesenko, O., Snoeck, E., Miao, G. X., Moodera, J. S. & Aliev, F. G. \nInterface and Temperature Dependent Magnetic  Properties in Permalloy Thin Films and \nTunnel Junction Structures. Journal of Nanoscience and Nanotechnology  11, 7653-7664 \n(2011). \n29 Kittel, C. On the Theory of Ferromagnetic Resonance Absorption. Phys. Rev.  73, 155 \n(1948). \n30 Kalarickal, S. S., Krivosik, P., Wu, M., Patt on, C. E., Schneider, M. L., Kabos, P., Silva, \nT. J. & Nibarger, J. P. Ferromagnetic reso nance linewidth in metallic thin films: \nComparison of measurement methods. J. Appl. Phys.  99, 093909 (2006). \n 12  \nFigure Captions  \n \nFigure 1.  Measurement of Gilbert damping in Py thin films via ferromagnetic resonance \n(Py thickness = 30 nm).   a, Ferromagnetic resonance spectra of the absorption for 30 nm Py thin \nfilms with TaN capping layer at gigahertz frequencies of 4, 6, 8, 10 and 12 GHz at 300 K after \nnormalization by background subtraction. b, c, The half linewidths as a function of the resonance \nfrequencies at 300 K and 5 K respectively. The red solid lines indicate the fitted lines based on \nequation (3), where the Gilbert damp ing constants could be obtained.  \n \nFigure 2. Temperature dependence of the Gilber t damping of Py thin films with TaN \ncapping.  a, The temperature dependence of the Gilbert damping fo r 3, 5, 10, 15, 20, 30, and 50 \nnm Py films. b, The Gilbert damping as a function of the Py thickness, d, measured at 300 K. c, \nThe Gilbert damping as a function of 1/ d measured at 300 K. The linear fitting corresponds to \nequation (4), in which the slope and the intercep t are related to the surf ace contribution and bulk \ncontribution to the total Gilber t damping. Error bars correspond to one standard deviation. \n Figure 3. Bulk and surface damping of Py  thin films with TaN capping layer.   a, b,  The \ntemperature dependence of the bulk damping an d surface damping, respectively. The inset table \nsummarizes the experimental values reported in early studies. Error bars correspond to one \nstandard deviation.  \nFigure 4. Comparison of the Gilbert damping of Py films with different capping layers.  a, \nb, Temperature dependence of the Gilbert dampi ng of Py thin films with TaN capping layer 13 (blue) and Al 2O3 capping layer (green) for 5 nm Py a nd 30 nm Py, respectively. Error bars \ncorrespond to one standard deviation.  \nFigure 5. Measurement of effective magnetizat ion in Py thin films via ferromagnetic \nresonance (Py thickness = 30 nm).   a, b, The resonance frequencies vs. the resonance magnetic \nfield at 300 K and 5 K, respectively. The fitted li nes (red curves) are obtained using the Kittel \nformula.   \nFigure 6. Effective magnetization of Py fi lms as a function of the temperature.  a, b, c, \nTemperature dependence of the effective magnetizati on of Py thin films of a thickness of 3 nm,  \n5 nm and 30 nm Py respectively. In b, c, the blue/green symbols correspond to the Py with \nTaN/Al\n2O3 capping layer.  \n \n 0\n500\n1000\n1500\n2000\n-0.3\n-0.2\n-0.1\n0.0\n0.1\n   4 \n   6 \n   8\n 10 \n 12 \n \nS\n21\n (dB) \n \nH (Oe)\nT=300 K\nf\n (GHz)\n0\n4\n8\n12\n16\n0\n10\n20\n30\n \n\nH (Oe)\n \nf (GHz)\nT=300 K\n0\n4\n8\n12\n16\n0\n10\n20\n30\n \n\nH (Oe)\n \nf (GHz)\nT=5 K\nb\nc\na\nFigure 10\n50\n100\n150\n200\n250\n300\n0.006\n0.008\n0.010\n0.012\n0.014\nd\n (nm)\n 3      \n 15        \n 5      \n 20\n 10    \n 30    \n               \n 50 \n \na\n \nTemperature (K)\n0.0\n0.1\n0.2\n0.3\n0.004\n0.006\n0.008\n0.010\n0.012\n0.014\n \na\n \n \n1/\nd\n (nm\n-1\n)\n0\n10\n20\n30\n0.006\n0.008\n0.010\n0.012\n0.014\n \n \nd\n (nm)\n \na\na\nb\nc\nFigure \n20\n50\n100\n150\n200\n250\n300\n0.0040\n0.0045\n0.0050\n0.0055\n0.0060\nTheory\n Ref. 21, 22\n Ref. 23\n Temperature (K)\n \na\nB\n \na\na\nExp.\n0.006\nRef. 24\n0.004\n-\n0.008\nRef.\n25\n0.007\nRef.\n26\n0.0067\nRef. 27\n0\n50\n100\n150\n200\n250\n300\n0.016\n0.018\n0.020\n0.022\n0.024\n0.026\n0.028\n0.030\n Temperature (K)\na\nS\n (nm)\n \nb\nFigure \n30\n50\n100\n150\n200\n250\n300\n0.004\n0.006\n0.008\n0.010\n 5 nm Py/TaN\n 5 nm Py/Al\n2\nO\n3\n Temperature (K)\na\n \n \n0\n50\n100\n150\n200\n250\n300\n0.004\n0.005\n0.006\n0.007\n 30 nm Py/TaN\n 30 nm Py/Al\n2\nO\n3\n Temperature (K)\na\n \n \na\nb\nFigure \n4a\nb\n0\n500\n1000\n1500\n2000\n0\n4\n8\n12\n16\n \nf\n (GHz)\n \nH\n (Oe)\nT=300 K\n0\n500\n1000\n1500\n2000\n0\n4\n8\n12\n16\n \nf\n (GHz)\n \nH (Oe)\nT=5 K\nFigure \n58.6\n8.8\n9.0\n9.2\n9.4\n9.6\n4\n\nM\neff\n (kG) \n 5 nm Py/TaN\n 5 nm Py/Al\n2\nO\n3\n \n6.2\n6.3\n6.4\n6.5\n6.6\n6.7\n6.8\n6.9\n4\n\nM\neff\n (kG) \n 3 nm Py/TaN\n \n0\n50\n100\n150\n10.6\n10.7\n10.8\n10.9\n11.0\n 30 nm Py/TaN\n 30 nm Py/Al\n2\nO\n3\n4\n\nM\neff\n (kG) \n \na\nb\nc\nTemperature (K) \nFigure \n6"    },    {        "title": "1603.07977v1.Large_spin_pumping_effect_in_antisymmetric_precession_of_Ni___79__Fe___21___Ru_Ni___79__Fe___21__.pdf",        "content": "Large spin pumping e\u000bect in antisymmetric precession of\nNi79Fe21/Ru/Ni 79Fe21\nH. Yang,1Y. Li,1and W.E. Bailey1,a)\nMaterials Science and Engineering, Dept. of Applied Physics and Applied Mathematics, Columbia University,\nNew York NY 10027\n(Dated: 16 September 2021)\nIn magnetic trilayer structures, a contribution to the Gilbert damping of ferromagnetic resonance arises from\nspin currents pumped from one layer to another. This contribution has been demonstrated for layers with\nweakly coupled, separated resonances, where magnetization dynamics are excited predominantly in one layer\nand the other layer acts as a spin sink. Here we show that trilayer structures in which magnetizations are\nexcited simultaneously, antisymmetrically, show a spin-pumping e\u000bect roughly twice as large. The antisym-\nmetric (optical) mode of antiferromagnetically coupled Ni 79Fe21(8nm)/Ru/Ni 79Fe21(8nm) trilayers shows a\nGilbert damping constant greater than that of the symmetric (acoustic) mode by an amount as large as\nthe intrinsic damping of Py (\u0001 \u000b'0.006). The e\u000bect is shown equally in \feld-normal and \feld-parallel to\n\flm plane geometries over 3-25 GHz. The results con\frm a prediction of the spin pumping model and have\nimplications for the use of synthetic antiferromagnets (SAF)-structures in GHz devices.\nPumped spin currents1,2are widely understood to in-\n\ruence the magnetization dynamics of ultrathin \flms\nand heterostructures. These spin currents increase the\nGilbert damping or decrease the relaxation time for thin\nferromagnets at GHz frequencies. The size of the e\u000bect\nhas been parametrized through the e\u000bective spin mixing\nconductance g\"#\nr, which relates the spin current pumped\nout of the ferromagnet, transverse to its static (time-\naveraged) magnetization, to its precessional amplitude\nand frequency. The spin mixing conductance is inter-\nesting also because it determines the transport of pure\nspin current across interfaces in quasistatic spin trans-\nport, manifested in e.g. the spin Hall e\u000bect.\nIn the spin pumping e\u000bect, spin current is pumped\naway from a ferromagnet / normal metal (F 1/N) in-\nterface, through precession of F1, and is absorbed else-\nwhere in the structure, causing angular momentum loss\nand damping of F1. The spin current can be absorbed\nthrough di\u000berent processes in di\u000berent materials. When\ninjected into paramagnetic metals (Pt, Pd, Ru, and oth-\ners), the spin current relaxes exponentially with para-\nmagnetic layer thickness3{5. The relaxation process has\nbeen likened to spin-\rip scattering as measured in CPP-\nGMR, where spin-\rip events are localized to heavy-metal\nimpurities6and the measurement reveals the spin di\u000bu-\nsion length \u0015SD. When injected into other ferromagnets\n(F2in F 1/N/F 2), the spin current is absorbed through\nits torque on magnetization5,7. A similar process appears\nto be relevant for antiferromagnets as well8.\nIn F 1/N/F 2structures, only half of the total possible\nspin pumping e\u000bect has been detected up until now. For\nwell-separated resonances of F1andF2, only one layer\nwill precess with large amplitude at a given frequency\n!, and spin current is pumped from a precessing F1into\na staticF2. If both layers precess symmetrically, with\na)Electronic mail: Contact author. web54@columbia.eduthe same amplitude and phase, equal and opposite spin\ncurrents are pumped into and out of each layer, causing\nno net e\u000bect on damping. The di\u000berence between the\nsymmetric mode and the uncoupled mode, increased by\na spin pumping damping \u000bspwas detected \frst in epi-\ntaxial Fe/Au/Fe structures9. However, if the magnetiza-\ntions can be excited with antisymmetric precession, the\ncoupled mode should be damped by twice that amount,\n2\u000bsp. Takahashi10has published an explicit prediction of\nthis \"giant spin pumping e\u000bect\" very recently, including\nan estimate of a fourfold enhanced spin accumulation in\nthe central layer.\nIn this paper, we show that a very large spin pump-\ning e\u000bect can be realized in antisymmetric precession of\nPy(8 nm)/Ru(0.70-1.25 nm)/Py(8 nm) synthetic antifer-\nromagnets (SAF, Py=Ni 79Fe21). The e\u000bect is roughly\ntwice that measured in Py trilayers with uncoupled\nprecession. Variable-frequency ferromagnetic resonance\n(FMR) measurements show, for structures with magne-\ntization saturated in the \flm plane or normal to the \flm\nplane, that symmetric (acoustic mode) precession of the\ntrilayer has almost no additional damping, but the an-\ntisymmetric (optical mode) precession has an additional\nGilbert damping of \u00180.006, compared with an uncou-\npled Py(8nm) layer in a F 1/N/F 2structure of\u00180.003.\nThe interaction stabilizes the antiparallel magnetization\nstate of SAF structures, used widely in di\u000berent elements\nof high-speed magnetic information storage, at GHz fre-\nquencies.\nMethod: Ta(5 nm)/Cu(3 nm)/Ni 79Fe21(8\nnm)/Ru(tRu)/Ni 79Fe21(8 nm)/Cu(3 nm)/SiO 2(5 nm),\ntRu= 0.7 - 1.2 nm heterostructures were deposited by\nultrahigh vacuum (UHV) sputtering at a base pressure\nof 5\u000210\u00009Torr on thermally oxidized Si substrates.\nThe Ru thckness range was centered about the second\nantiferromagnetic maximum of interlayer exchange\ncoupling (IEC) for Py/Ru/Py superlattices, 8-12 \u0017A,\nestablished \frst by Brillouin light scattering (BLS)\nmeasurement11. Oscillatory IEC in this system, as aarXiv:1603.07977v1  [cond-mat.mtrl-sci]  25 Mar 20162\nfunction of tRu, is identical to that in the more widely\nstudied Co/Ru( tRu)/Co superlattices12, 11.5 \u0017A, but is\nroughly antiphase to it. An in-plane magnetic \feld bias\nof 200 G, rotating in phase with the sample, was applied\nduring deposition as described in13.\nThe \flms were characterized using variable fre-\nquency, swept-\feld, magnetic-\feld modulated ferromag-\nnetic resonance (FMR). Transmission measurements\nwere recorded through a coplanar waveguide (CPW) with\ncenter conductor width of 300 \u0016m, with the \flms placed\ndirectly over the center conductor, using a microwave\ndiode signal locked in to magnetic \feld bias modulation.\nFMR measurements were recorded for magnetic \feld bias\nHBapplied both in the \flm plane (parallel condition, pc)\nand perpendicular to the plane (normal condition, nc.)\nAn azimuthal alignment step was important for the nc\nmeasurements. For these, the sample was rotated on twoaxes to maximize the \feld for resonance at 3 GHz.\nFor all FMR measurements, the sample magnetization\nwas saturated along the applied \feld direction, simplify-\ning extraction of Gilbert damping \u000b. The measurements\ndi\u000ber in this sense from low-frequency measurements of\nsimilar Py/Ru/Py trilayer structures by Belmenguenai et\nal14, or broadband measurements of (sti\u000ber) [Co/Cu] \u000210\nmultilayers by Tanaka et al15. In these studies, e\u000bects\non\u000bcould not be distinguished from those on inhomo-\ngeneous broadening.\nModel: In the measurements, we compare the mag-\nnitude of the damping, estimated by variable-frequency\nFMR linewidth through \u0001 H1=2= \u0001H0+ 2\u000b!=j\rj, and\nthe interlayer exchange coupling (IEC) measured through\nthe splitting of the resonances. Coupling terms between\nlayersiandjare introduced into the Landau-Lifshitz-\nGilbert equations for magnetization dynamics through\n_mi=\u0000mi\u0002(\riHe\u000b+!ex;imj) +\u000b0mi\u0002_mi+\u000bsp;i(mi\u0002_ mi\u0000mj\u0002_ mj) (1)\nincgsunits, where we de\fned magnetization unit vec-\ntors as m1=M1=Ms;1,m2=M2=Ms;2withMs;ithe\nsaturation moments of layer i. The coupling constants\nare, for the IEC term, !ex;i\u0011\riAex=(Ms;iti), where\nthe energy per unit area of the system can be written\nuA=\u0000Aexmi\u0001mj, andtiis the thickness of layer i. Pos-\nitive values of Aexcorrespond to ferromagnetic coupling,\nnegative values to antiferromagnetic coupling. The spin\npumping damping term is \u000bsp;i\u0011\r\u0016h~gFNF\n\"#=(4\u0019Ms;iti),\nwhere ~gFNF\n\"# is the spin mixing conductance of the tri-\nlayer in units of nm\u00002.\u000b0is the bulk damping for the\nlayer.\nThe collective modes of 1 ;2 are found from small-\namplitude solutions of Equations 1 for i= 1;2. General\nsolutions for resonance frequencies with arbitrary mag-\nnetization alignment, not cognizant of any spin pump-\ning damping or dynamic coupling, were developed by\nZhang et al12. In our experiment, to the extent possi-\nble, layers 1 ;2 are identical in deposited thickness, mag-\nnetization, and interface anisotropy (each with Cu the\nopposite side from Ru). Therefore if !irepresents the\nFMR frequency (dependent on bias \feld HB) of each\nlayeri, the two layers have !1=!2=!0. In this\nlimit, there are two collective modes: a perfectly sym-\nmetric mode Sand a perfectly antisymmetric mode A\nwith complex frequencies f!S= (1\u0000i\u000b0)!0andf!A=\n(1\u0000i\u000b0\u00002i\u000bsp) (!0+ 2!ex). The Gilbert damping for\nthe modes, \u000bk=\u0000Im(f!k)=Re(!k\nf), wherek= (S;A),\nand the resonance \felds Hk\nBsatisfy\nHA\nB=HS\nB+ 2HexHex=\u0000Aex=(MstF) (2)\n\u000bA=\u000bS+ 2\u000bsp\u000bsp=\r\u0016h~gFNF\n\"#=(4\u0019Ms;iti) (3)\nand!ex=\rHex. Note that there is no relationshipin this limit between the strength of the exchange cou-\nplingAexand the spin-pumping damping 2 \u000bspexpressed\nin the antisymmetric mode. The spin pumping damping\nand the interlayer exchange coupling can be read sim-\nply from the di\u000berences in the the Gilbert damping \u000b\nand resonance \felds between the antisymmetric and sym-\nmetric modes. The asymmetric mode will have a higher\ndamping by 2 \u000bspfor anyAexand a higher resonance\n\feld forAex<0, i.e. for antiferromagnetic IEC: because\nthe ground state of the magnetization is antiparallel at\nzero applied \feld, antisymmetric excitation rotates mag-\nnetizations towards the ground state and is lower in fre-\nquency than symmetric excitation.\nResults: Samplepc\u0000andnc\u0000FMR data are shown in\nFigure 1. Raw data traces (lock-in voltage) as a func-\ntion of applied bias \feld HBat 10 GHz are shown in the\ninset. We observe an intense resonance at low \feld and\nresonance weaker by a factor of 20-100 at higher \feld. On\nthe basis of the intensities, as well as supporting MOKE\nmeasurements, we assign the lower-\feld resonance to the\nsymmetric, or \"acoustic\" mode and the higher-\feld res-\nonance to the antisymmetric, or \"optical\" mode. Similar\nbehavior is seen in the nc- andpc\u0000FMR measurements.\nIn Figure 1a) and c), which summarizes the \felds-\nfor-resonance !(HB), there is a rigid shift of the\nantisymmetric-mode resonances to higher bias \felds HB,\nas predicted by the theory. The lines show \fts to\nthe Kittel resonance, !pc=\rr\nHeff\u0010\nHeff+ 4\u0019Meff\ns\u0011\n,\n!nc=\r\u0000\nHeff\u00004\u0019Meff\ns\u0001\nwith an additional e\u000bective\n\feld along the magnetization direction for the antisym-\nmetric mode: Heff;S =HB, andHeff;A =HB\u0000\n8\u0019Aex=(4\u0019MstF).\nIn Figure 2, we show coupling parameters, as a func-3\nHBHeffm(t)\nHBHeff m(t)(a)\n(c)(b)\n(d)10 GHz10 GHz\n1/21/2\nFIG. 1. FMR measurement of Ni 79Fe21(8\nnm)/Ru(tRu)/Ni 79Fe21(8 nm) trilayers; example shown\nfortRu= 1.2 nm. Inset : lock-in signal, transmitted power\nat 10 GHz, as a function of bias \feld HB, for a) pc-FMR\nand c) nc-FMR. A strong resonance is observed at lower HB\nand a weaker one at higher HB, attributed to the symmetric\n(S) and antisymmetric (A) modes, respectively. a), c): Field\nfor resonance !(HB) for the two resonances. Lines are \fts\nto the Kittel resonance expression, assuming an additional,\nconstant, positive \feld shift for !A,Hex=\u00002Aex=(MstF)\ndue to antiferromagnetic interlayer coupling Aex<0. b)\npc-FMR and d) nc-FMR linewidth as a function of frequency\n\u0001Hpp(!) for \fts to Gilbert damping \u000b.\ntion of Ru thickness, extracted from the FMR measure-\nments illustrated in Figure 1. Coupling \felds are mea-\nsured directly from the di\u000berence between the symmet-\nric and antisymmetric mode positions and plotted in\nFigure 2a. We convert the \feld shift to antiferromag-\nnetic IEC constant Aex<0 through Equation 2, us-\ning the thickness tF= 8 nm and bulk magnetization\n4\u0019Ms= 10.7 kG4. The extracted exchange coupling\nstrength in pc-FMR has a maximum antiferromagnetic\nvalue ofAex=\u00000.2 erg/cm2, which agrees to 5% with\nthat measured by Fassbender et al11for [Py/Ru] Nsu-\nperlattices.\nThe central result of the paper is shown in Figure 2 b).\nWe compare the damping \u000bof the symmetric ( S) and an-\ntisymmetric ( A) modes, measured both through pc-FMR\nandnc-FMR. The values measured in the two FMR ge-\nometries agree closely for the symmetric modes, for which\nsignals are larger and resolution is higher. They agree\nroughly within experimental error for the antisymmetric\nmodes, with no systematic di\u000berence. The antisymmetric\nmodes clearly have a higher damping than the symmetric\nmodes. Averaged over all thickness points, the enhanced\ndamping is roughly \u000bA\u0000\u000bS= 0.006.\nDiscussion: The damping enhancement of the anti-\nsymmetric ( A\u0000) mode over the symmetric ( S\u0000) mode,\nshown in Figure 2b), is a large e\u000bect. The value is close\nto the intrinsic bulk damping \u000b0\u00180.007 for Ni 79Fe21.\n0.7 0.8 0.9 1.0 1.1 1.2 1.301002003004005006002Hex (Oe)a)\nnc,Hexpc,HexMOKE,Hex\n0.7 0.8 0.9 1.0 1.1 1.2 1.3\ntRu (nm)0.0060.0080.0100.0120.0140.0160.018α\nα0α0+αspα0+2αspb)pc, S\nnc, Spc, A\nnc, A0.000.050.100.150.20\n−Aex (erg/cm2)FIG. 2. Coupling parameters for Py/Ru/Py trilayers. a):\nInterlayer (static) coupling from resonance \feld shift of an-\ntisymmetric mode; see Fig 1 a),c). The antiferromagnetic\nexchange parameter Aexis extracted through Eq 2, in agree-\nment with values found in Ref11. The line is a guide to the\neye. b) Spin pumping (dynamic) coupling from damping of\nthe symmetric (S) and antisymmetric (A) modes; see Fig 1\nb), d). The spin pumping damping for uncoupled layer pre-\ncession in Py/Ru/Py, \u000bspis shown for comparsion. Dotted\nlines show the possible e\u000bect of \u0018100 Oe detuning for the two\nPy layers. See text for details.\nWe compare the value with the value 2 \u000bspexpected from\ntheory for the antisymmetric mode and written in Eq 3.\nThe interfacial spin mixing conductance for Ni 79Fe21/Ru,\nwas found in Ref.16to be ~gFN\n\"#= 24 nm\u00002. For a F/N/F\nstructure, in the limit of ballistic transport with no spin\nrelaxation through N, the e\u000bective spin mixing conduc-\ntance is ~gFN\n\"#=2: spin current must cross two interfaces\nto relax in the opposite Flayer, and the conductance re-\n\rects two series resistances17. This yields \u000bsp= 0:0027.\nThe observed enhancement matches well with, and per-\nhaps exceeds slightly, the \"giant\" spin pumping e\u000bect of\n2\u000bsp, as shown.\nLittle dependence of the Gilbert damping enhancement\n\u000bA\u0000\u000bSon the resonance \feld shift HA\u0000HScan be ob-\nserved in Figure 2 a,b. We believe that this independence\nre\rects close tuning of the resonance frequencies for Py\nlayers 1 and 2, as designed in the depositions. For \f-\nnite detuning \u0001 !de\fned through !2=!0+ \u0001!and\n!1=!0\u0000\u0001!, the modes change. Symmetric and anti-\nsymmetric modes become hybridized as S0andA0, and4\nthe di\u000berence in damping is reduced. De\fning g\u0001!2=\n(1\u0000i\u000b0\n) (1\u0000i\u000b0\n\u00002i\u000bsp\n) \u0001!2, it is straightfor-\nward to show that for the nc-case, the mode frequen-\ncies are!S0;A0= (f!S+f!A)=2\u0006q\n(f!S\u0000f!A)2=4 +g\u0001!2.\nThe relevant parameter is the frequency detuning nor-\nmalized to the exchange (coupling) frequency, z\u0011\n\u0001!=(2!ex); ifz\u001d1, the layers have well-separated\nmodes, and each recovers the uncoupled damping en-\nhancement of \u000bsp,\u000bS0;A0=\u000b0+\u000bspidenti\fed in Refs5,9.\nThe possibility of \fnite detuning, assuming ( !2\u0000\n!1)=\r= 100 Oe, is shown in Figure 2b), with the dot-\nted lines. The small \u0000zlimit for detuning \fnds sym-\nmetric e\u000bects on damping of the S0andA0modes, with\n\u000bS0=\u000b0+ 2\u000bspz2and\u000bA0=\u000b0+ 2\u000bsp(1\u0000z2), respec-\ntively, recovering perfect symmetric and antisymmetric\nmode values for z= 0. We assume that the \feld split-\nting shown in Figure 2 a) gives an accurate measure of\n2!ex=\r, as supported by the MOKE results. This value\ngoes into the denominator of z. We \fnd a reasonable \ft\nto the dependence of SandAdamping on Ru thickness,\nimplicit in the coupling. For the highest coupling pionts,\nthe damping values closely reach the low- zlimit, and we\nbelieve that the \"giant\" spin pumping result of 2 \u000bspis\nevident here.\nWe would like to point out next that it was not a-\npriori obvious that the Py/Ru/Py SAF would exhibit\nthe observed damping. Ru could behave in two limits in\nthe context of spin pumping: either as a passive spin-\nsink layer, or as a ballistic transmission layer supporting\ntransverse spin-current transmission from one Py layer to\nthe other. Our results show that Ru behaves as the latter\nin this thickness range. The symmetric-mode damping of\nthe SAF structure, extrapolated back to zero Ru thick-\nness, is identical within experimental resolution ( \u001810\u00004)\nto that of a single Py \flm 16 nm thick measured in nc-\nFMR (\"\u000b0\" line in Fig 2b.) If Ru, or the Py/Ru interface,\ndepolarized pumped spin current very strongly over this\nthickness range as has been proposed for Pt18, we would\nexpect an immediate increase in damping of the acous-\ntic mode by the amount of \u0018\u000bsp. Instead, the volume-\ndependent Ru depolarization in spin pumping has an (ex-\nponential) characteristic length of \u0015SD\u001810 nm5, and\nattenuation over the range explored of \u00181 nm is negli-\ngible.\nPerspectives: Finally, we would like to highlight some\nimplications of the study. First, as the study con\frms\nthe prediction of a \"giant\" spin pumping e\u000bect as pro-\nposed by Takahashi10, it is plausible that the greatly en-\nhanced values of spin accumulation predicted there may\nbe supported by Ru in Py/Ru/Py synthetic antiferro-\nmagnets (SAFs). These spin accumulations would di\u000ber\nstrongly in the excitation of symmetric and antisymmet-\nric modes, and may then provide a clear signature intime-resolved x-ray magnetooptical techniques19, similar\nto the observation of static spin accumulation in Cu re-\nported recently20.\nSecond, in most device applications of synthetic an-\ntiferromagnets, it is not desirable to excite the antisym-\nmetric (optical) mode. SAFs are used in the pinned layer\nof MTJ/spin valve structures to increase exchange bias\nand in the free layer to decrease (magnetostatic) stray\n\felds. Both of these functions are degraded if the opti-\ncal, or asymmetric mode of the SAF is excited. Accord-\ning to our results, at GHz frequencies near FMR, the\nsusceptibility of the antisymmetric mode is reduced sub-\nstantially, here by a factor of two (from 1/ \u000b) for nc-FMR ,\ndue to spin pumping. This reduction of \u001fon resonance\nwill scale inversely with layer thickness. The damping,\nand susceptibility, of the desired symmetric (acoustic)\nmode is unchanged, on the other hand, implying that\nspin pumping favors the excitation the symmetric mode\nfor thin Ru, the typical operating point.\nWe acknowledge NSF-DMR-1411160 for support.\n1Y. Tserkovnyak, A. Brataas, and G. E. W. Bauer, Phys. Rev.\nLett. 88, 117601 (2002).\n2Y. Tserkovnyak, A. Brataas, G. Bauer, and B. Halperin, Reviews\nin Modern Physics 77, 1375 (2005).\n3S. Mizukami, Y. Ando, and T. Miyazaki, Journal of Magnetism\nand Magnetic Materials 239, 42 (2002).\n4A. Ghosh, J. F. Sierra, S. Au\u000bret, U. Ebels, and W. E. Bailey,\nApplied Physics Letters 98, (2011).\n5A. Ghosh, S. Au\u000bret, U. Ebels, and W. E. Bailey, Phys. Rev.\nLett. 109, 127202 (2012).\n6J. Bass and W. Pratt, Journal of Physics: Condensed Matter 19,\n41 pp. (2007).\n7G. Woltersdorf, O. Mosendz, B. Heinrich, and C. H. Back, Phys-\nical Review Letters 99, 246603 (2007).\n8P. Merodio, A. Ghosh, C. Lemonias, E. Gautier, U. Ebels,\nV. Baltz, and W. Bailey, Applied Physics Letters 104(2014).\n9B. Heinrich, Y. Tserkovnyak, G. Woltersdorf, A. Brataas, R. Ur-\nban, and G. E. W. Bauer, Phys. Rev. Lett. 90, 187601 (2003).\n10S. Takahashi, Applied Physics Letters 104(2014).\n11J. Fassbender, F. Nortemann, R. Stamps, R. Camley, B. Hille-\nbrands, G. Guntherodt, and S. Parkin, Journal of Magnetism\nand Magnetic Materials 121, 270 (1993).\n12Z. Zhang, L. Zhou, P. E. Wigen, and K. Ounadjela, Phys. Rev.\nB50, 6094 (1994).\n13C. Cheng, N. Sturcken, K. Shepard, and W. Bailey, Review of\nScienti\fc Instruments 83, 063903 (2012).\n14M. Belmeguenai, T. Martin, G. Woltersdorf, M. Maier, and\nG. Bayreuther, Physical Review B 76(2007).\n15K. Tanaka, T. Moriyama, M. Nagata, T. Seki, K. Takanashi,\nS. Takahashi, and T. Ono, Applied Physics Express 7(2014).\n16N. Behera, M. S. Singh, S. Chaudhary, D. K. Pandya, and P. K.\nMuduli, Journal of Applied Physics 117(2015).\n17See eqs. 31, 74, 81 in Ref2.\n18J.-C. Rojas-S\u0013 anchez, N. Reyren, P. Laczkowski, W. Savero, J.-\nP. Attan\u0013 e, C. Deranlot, M. Jamet, J.-M. George, L. Vila, and\nH. Ja\u000br\u0012 es, Phys. Rev. Lett. 112, 106602 (2014).\n19W. Bailey, C. Cheng, R. Knut, O. Karis, S. Au\u000bret, S. Zohar,\nD. Keavney, P. Warnicke, J.-S. Lee, and D. Arena, Nature Com-\nmunications 4, 2025 (2013).\n20R. Kukreja, S. Bonetti, Z. Chen, D. Backes, Y. Acremann, J. A.\nKatine, A. D. Kent, H. A. D urr, H. Ohldag, and J. St ohr, Phys.\nRev. Lett. 115, 096601 (2015)."    },    {        "title": "1604.02998v1.All_Optical_Study_of_Tunable_Ultrafast_Spin_Dynamics_in__Co_Pd__NiFe_Systems__The_Role_of_Spin_Twist_Structure_on_Gilbert_Damping.pdf",        "content": "All-Optical Study of Tunable Ultrafast Spin Dynamics in [Co/Pd]-NiFe Systems: The\nRole of Spin-Twist Structure on Gilbert Damping\nChandrima Banerjee,1Semanti Pal,1Martina Ahlberg,2T. N.\nAnh Nguyen,3, 4Johan \u0017Akerman,2, 4and Anjan Barman1,\u0003\n1Department of Condensed Matter Physics and Material Sciences,\nS. N. Bose National Centre for Basic Sciences, Block JD, Sec. III, Salt Lake, Kolkata 700 098, India\n2Department of Physics, University of Gothenburg, 412 96, Gothenburg, Sweden\n3Laboratory of Magnetism and Superconductivity,\nInstitute of Materials Science, Vietnam Academy of Science and Technology,\n18 Hoang Quoc Viet, Cau Giay, Hanoi, Vietnam.\n4Department of Materials and Nano Physics, School of Information and Communication Technology,\nKTH Royal Institute of Technology, Electrum 229, SE-16440 Kista, Sweden\n(Dated: April 12, 2016)\nWe investigate optically induced ultrafast magnetization dynamics in [Co(0.5 nm)/Pd(1\nnm)] 5/NiFe( t) exchange-spring samples with tilted perpendicular magnetic anisotropy using a time-\nresolved magneto-optical Kerr e\u000bect magnetometer. The competition between the out-of-plane\nanisotropy of the hard layer, the in-plane anisotropy of the soft layer and the applied bias \feld reor-\nganizes the spins in the soft layer, which are modi\fed further with the variation in t. The spin-wave\nspectrum, the ultrafast demagnetization time, and the extracted damping coe\u000ecient all depend on\nthe spin distribution in the soft layer, while the latter two also depend on the spin-orbit coupling\nbetween the Co and Pd layers. The spin-wave spectra change from multimode to single-mode as t\nincreases. At the maximum \feld reached in this study, H=2.5 kOe, the damping shows a nonmono-\ntonic dependence on twith a minimum at t= 7.5 nm. For t<7.5 nm, intrinsic e\u000bects dominate,\nwhereas for t>7.5 nm, extrinsic e\u000bects govern the damping mechanisms.\nI. INTRODUCTION\nNonuniform magnetic structures, including exchange\nbias (ferromagnet/antiferromagnet)3,24and exchange-\nspring (ferromagnet/ferromagnet)5{8systems, have\nrecently been explored extensively on account of their\nintrinsic advantages for applications in both permanent\nmagnets and recording media. Exchange-spring (ES)\nmagnets are systems of exchanged-coupled hard and soft\nmagnetic layers that behave as a single magnet. Here,\nthe high saturation magnetization ( Ms) of the soft phase\nand the high anisotropy ( Hk) of the hard phase result in\na large increase in the maximum energy product. This\nmakes them useful as permanent magnets in energy ap-\nplications such as engines or generators in miniaturized\ndevices. On the other hand, for spintronic applications,\nthe soft phase is used to improve the writability of\nthe magnetic media, which in turn is stabilized by the\nmagnetic con\fguration of the hard layer. Consequently,\na wealth of research has been devoted to investigating\nthe static and dynamic magnetic properties, including\nthe switching behavior and exchange coupling strength,\nin ES systems.\nIn case of ES systems with tilted anisotropy, the hard\nand soft phases consist of materials with out-of-plane\n(OOP) and in-plane (IP) anisotropies, respectively. This\ncombination results in a canting of the magnetization\nof the soft layer with a wide and tunable range of tilt\nangles. The advantage of such a hybrid anisotropy sys-\ntem is that it is neither plagued by the poor writability\nand thermal instability of systems with IP anisotropy,\nnor does it lead to very high switching \felds, as in OOPsystems. As a result, these materials provide additional\ndegrees of freedom to control the magnetization dynam-\nics in magnetic nanostructures, and hint at potential\napplications in novel spintronic devices utilizing the\nspin-transfer torque (STT) e\u000bect|such as spin-torque\noscillators (STOs)25,26and STT-MRAMs.\nSo far, numerous studies have been performed on\nsuch systems where the exchange coupling between\nthe hard and soft layers has been tailored by varying\nthe layer thickness,12,13layer composition,19number\nof repeats,15and interfacial anisotropy.13The litera-\nture describes investigations of domain structure and\nother static magnetic properties for [Co/Pd]/Co,14\n[Co/Pd]/NiFe,12,14,19,21[Co/Pd]/CoFeB,14,15,20\n[Co/Pd]-Co-Pd-NiFe,13[Co/Ni]/NiFe,4and CoCrPt-\nNi11|these systems being studied with static mag-\nnetometry, magnetic force microscopy (MFM), and\nmicromagnetic simulations. The magnetization dy-\nnamics in such systems have also been measured using\nBrillouin light scattering (BLS)19,20and ferromagnetic\nresonance (FMR)21experiments, where the spin-wave\n(SW) modes have been investigated by varying the thick-\nness of the soft layer and changing the con\fguration of\nthe hard layer. In any process involving magnetization\ndynamics, the Gilbert damping constant ( \u000b) plays a key\nrole in optimizing writing speeds and controlling power\nconsumption. For example, in case of STT-MRAM\nand magnonic devices, low \u000bfacilitates a lower writing\ncurrent and the longer propagation of SWs, whereas a\nhigher\u000bis desirable for increasing the reversal rates and\nthe coherent reversal of magnetic elements, which are\nrequired for data storage devices.arXiv:1604.02998v1  [cond-mat.mtrl-sci]  11 Apr 20162\n46810121416350400450500\n  )sf( emit noitazitengameDt (nm)(d)(a) \n-202 600 1200 1800-2-10\n  Kerr rotation (a rb. unit)\nTime (ps)(b)0 10 20 30\n  Power (arb. unit)\nFrequency (GHz)(c)\n(b)\n-202 60012001800-2-10Kerrrotation(arb.unit)\nTime(ps)\nFigure 1. (color online) (a) Schematic of the two-color pump-\nprobe measurement of the time-resolved magnetization dy-\nnamics of exchange-spring systems. The bias \feld is applied\nwith a small angle to the normal of the sample plane. (b)\nTypical time-resolved Kerr rotation data revealing ultrafast\ndemagnetization, fast and slow relaxations, and precession\nof magnetization for the exchange-spring system with t=\n7.5 nm at H= 2.5 kOe. (c) FFT spectrum of the background-\nsubtracted time-resolved Kerr rotation. (d) Variation of de-\nmagnetization time with t.\nIn this paper, we present all-optical excitation and de-\ntection of magnetization dynamics in [Co(0.5 nm)/Pd(1\nnm)] 5/NiFe( t) tilted anisotropy ES systems, with varying\nsoft layer thickness ( t), using a time-resolved magneto-\noptical Kerr e\u000bect (TR-MOKE) magnetometer. The dy-\nnamical magnetic behavior of similar systems has previ-\nously been studied using BLS19and FMR21measure-\nments. However, a detailed study of the precessional\nmagnetization dynamics and relaxation processes in such\ncomposite hard/soft systems is yet to be carried out.\nThe advantage of implementing TR-MOKE is that here\nthe magnetization dynamics can be measured on di\u000ber-\nent time scales and the damping is measured directly\nin the time domain, and is therefore more reliable. We\ninvestigate the ultrafast magnetization dynamics over pi-\ncosecond and picosecond time scales. The ultrafast de-\nmagnetization is examined and found to change due to\nthe modi\fed spin structure in the soft layer for di\u000berent\ntvalues. The extracted SW spectra are strongly depen-\ndent on t. An extensive study of the damping coe\u000ecient\nreveals that the extrinsic contribution to the damping\nis more dominant in the higher thickness regime, while\nintrinsic mechanisms govern the behavior at lower thick-\nnesses.II. EXPERIMENTAL DETAILS\nA. Sample fabrication\nThe samples were fabricated using dc mag-\nnetron sputtering and have the following structure:\nTa(5nm)/Pd(3nm)/[Co(0.5nm)/Pd(1nm)] \u00025=Ni80Fe20(t)\n/Ta(5nm), where t= 4{20 nm. The chamber base pres-\nsure was below 3 \u000210\u00008Torr, while the Ar work\npressure was 2 and 5 mTorr for the Ta, NiFe and Co,\nPd layers, respectively. The samples were deposited\nat room temperature on naturally oxidized Si(100)\nsubstrates. The 5 nm Ta seed layer was used to induce\nfcc-(111) orientation in the Pd layer, which improves\nthe perpendicular magnetic anisotropy of the Co/Pd\nmultilayers; a Ta cap layer was used to avoid oxidation,\nwhich has been reported in previous studies.12{14The\nlayer thicknesses are determined from the deposition\ntime and calibrated deposition rates.\nB. Measurement technique\nTo investigate the precessional frequency and damp-\ning of these samples, the magnetization dynamics were\nmeasured by using an all-optical time-resolved magneto-\noptical Kerr e\u000bect (TR-MOKE) magnetometer2based on\na two-color optical pump-probe experiment. The mea-\nsurement geometry is shown in Fig. 1(a). The magne-\ntization dynamics were excited by laser pulses of wave-\nlength (\u0015) 400 nm (pulse width = 100 fs, repetition rate\n= 80 MHz) of about 16 mJ/cm2\ruence and probed by\nlaser pulses with \u0015= 800 nm (pulse width = 88 fs, rep-\netition rate = 80 MHz) of about 2 mJ/cm2\ruence. The\npump and probe beams are focused using the same micro-\nscope objective with N.A. of 0.65 in a collinear geometry.\nThe probe beam is tightly focused to a spot of about\n800 nm on the sample surface and, as a result, the pump\nbecomes slightly defocused in the same plane to a spot\nof about 1 \u0016m. The probe beam is carefully aligned at\nthe centre of the pump beam with slightly larger spot\nsize. Hence, the dynamic response is probed from a ho-\nmogeneously excited volume. The bias \feld was tilted\nat around 15\u000eto the sample normal (and its projection\nalong the sample normal is referred to as Hin this ar-\nticle) in order to have a \fnite demagnetizing \feld along\nthe direction of the pump beam. This \feld is eventually\nmodi\fed by the pump pulse which induces precessional\nmagnetization dynamics in the samples. The Kerr rota-\ntion of the probe beam, back-re\rected from the sample\nsurface, is measured by an optical bridge detector us-\ning phase sensitive detection techniques, as a function of\nthe time-delay between the pump and probe beams. Fig-\nure 1(b) presents typical time-resolved Kerr rotation data\nfrom the ES sample with t= 7.5 nm at a bias \feld H=\n2.5 kOe. The data shows a fast demagnetization within\n500 fs and a fast remagnetization within 8 ps, followed by\na slow remagnetization within 1800 ps. The precessional3\n(b) \n010 20 30 0 1 2\nPower (arb. unit)  Kerr Rotation(arb. unit)\n  \n  \n    \n  \n    \n  \nFrequency (GHz)4.5 nm\n5.5 nm\n7.5 nm\n8 nm\n15 nm\n  \nTime (ns)20 nm \n  \n  B\nA \n   \nNiFe  (t = 20 nm)  \nCo/Pd \n1 -1 Normalized Mz Co/Pd NiFe  (t = 6 nm)  \nCo/Pd NiFe ( t = 10 nm) (a) \nFigure 2. (color online) (a) Background-subtracted time-\nresolved Kerr rotation and the corresponding FFT spectra\nfor samples with di\u000berent tvalues at H= 2.5 kOe. The\nblack lines show the \ft according to Eq. 1. (b) Simulated\nstatic magnetic con\fgurations for samples with t= 20, 10,\nand 6 nm with a bias \feld H= 2.5 kOe in the experimental\ncon\fguration. The simulated samples are not to scale. The\ncolor map is shown at the bottom of the \fgure.\ndynamics appear as an oscillatory signal above the slowly\ndecaying part of the time-resolved Kerr rotation data.\nThis part was further analyzed and a fast Fourier trans-\nform (FFT) was performed to extract the corresponding\nSW modes, as presented in Fig. 1(c).III. RESULTS AND DISCUSSIONS\nIn order to closely observe the ultrafast demagnetiza-\ntion and fast remagnetization, we recorded the transient\nMOKE signals for delay times up to 30 ps at a resolution\nof 50 fs. In Fig. 1(d), the demagnetization times are plot-\nted as a function of t. We observe that the demagnetiza-\ntion is fastest in the thinnest NiFe layer ( t= 4 nm) and\nincreases sharply with the increase in t, becoming con-\nstant at 500 fs at t= 5 nm. At t= 10 nm, it decreases\ndrastically to 400 fs and remains constant for further in-\ncreases in t. For t<5 nm, the laser beam penetrates\nto the Co/Pd layer. In this regime, the large spin-orbit\ncoupling of Pd enhances the spin-\rip rate, resulting in a\nfaster demagnetization process. As tincreases, the top\nNiFe layer is primarily probed. Here, the spin con\fgura-\ntion across the NiFe layer, which is further a\u000bected by the\ncompetition between the in-plane and the out-of-plane\nanisotropies of the NiFe and [Co/Pd] layers, governs the\ndemagnetization process. Qualitatively, ultrafast demag-\nnetization can be understood by direct transfer of spin\nangular momentum between neighboring domains10,23.\nwhich may be explained as follows: For t>8 nm, the\nmagnetization orientation in the NiFe layer varies over a\nwide range of angles across the \flm thickness, where the\nmagnetization gradually rotates from nearly perpendicu-\nlar at the Co/Pd and NiFe interface to nearly parallel to\nthe surface plane in the topmost NiFe layer. Such a spin\nstructure across the NiFe layer thickness can be seen as a\nnetwork of several magnetic sublayers, where the spin ori-\nentation in each sublayer deviates from that of the neigh-\nboring sublayer. This canted spin structure accelerates\nthe spin-\rip scattering between the neighboring sublay-\ners and thus results in a shorter demagnetization time,\nsimilar to the work reported by Vodungbo et al.23On the\nother hand, for 5 nm <t<8 nm, the strong out-of-plane\nanisotropy of the Co/Pd layer forces the magnetization\nin the NiFe \flm towards the direction perpendicular to\nthe surface plane, giving rise to a uniform spin structure.\nThe strong coupling reduces the transfer rate of spin an-\ngular momentum and causes the demagnetization time\nto increase.\nTo investigate the variation of the precessional dynamics\nwith t, we further recorded the time-resolved data for a\nmaximum duration of 2 ns at a resolution of 10 ps. Fig-\nure 2(a) shows the background-subtracted time-resolved\nKerr rotation data for di\u000berent values of tatH=\n2.5 kOe and the corresponding fast Fourier transform\n(FFT) power spectra. Four distinct peaks are observed\nin the power spectrum of t= 20 nm, which reduces to\ntwo for t= 15 nm. This is probably due to the rela-\ntive decrease in the nonuniformity of the magnetization\nacross the NiFe thickness, which agrees with the varia-\ntion in the demagnetization time, as described earlier. To\ncon\frm this, we simulated the static magnetic con\fgura-\ntions of these samples in a \feld of H= 2:5 kOe using the\nLLG micromagnetic simulator.18Simulations were per-\nformed by discretizing the samples in arrays of cuboidal4\ncells with two-dimensional periodic boundary conditions\napplied within the sample plane. The simulations assume\nthe Co/Pd multilayer as an e\u000bective medium16with sat-\nuration magnetization Ms= 690 emu/cc, exchange sti\u000b-\nness constant A= 1.3\u0016erg/cm, and anisotropy constant\nKu1= 5.8 Merg/cc along the (001) direction, while the\nmaterial parameters used for the NiFe layer were Ms=\n800 emu/cc, A= 1.3\u0016erg/cm, and Ku1= 0.17The in-\nterlayer exchange between Co/Pd and NiFe is set to 1.3\n\u0016erg/cm and the gyromagnetic ratio \r= 18.1 MHz Oe\u00001\nis used for both layers. The Co/Pd layer was discretized\ninto cells of dimension 5 \u00025\u00022 nm3and the NiFe layer\nwas discretized into cells of dimension 5 \u00025\u00021 nm3.\nThe results are presented in Fig. 2(b) for t= 20, 10, and\n6 nm samples. The nonuniform spin structure is promi-\nnent in the NiFe layer of the t= 20 nm sample, which\nmodi\fes the SW spectrum of this sample, giving rise to\nthe new modes.17With the reduction of t, the spin struc-\nture in the NiFe layer gradually becomes more uniform,\nwhile at t= 7.5 nm it is completely uniform over the\nwhole thickness pro\fle. Hence, for low values of t, the\npower spectra shows a single peak due to the collective\nprecession of the whole stack. The variation in precession\nfrequency with tis plotted in Fig. 3(a). The frequency\nof the most intense mode shows a slow decrease down to\nt= 7.5 nm, below which it increases sharply down to the\nlowest thickness t= 4.5 nm, exhibiting precessional dy-\nnamics. This mode is basically the uniform mode of the\nsystem and follows Kittel's equation.9The variation in\nfrequency depicts the evolution of the e\u000bective anisotropy\nfrom OOP to IP with increasing t, which is in agreement\nwith previously reported results.21For lower t, the sys-\ntem displays an OOP easy axis, owing to the strong OOP\nanisotropy of the Co/Pd multilayer. This is manifested\nas a sharp increase in the frequency with decreasing t\nbelow 7.5 nm. For greater thicknesses, the e\u000bect of the\nperpendicular anisotropy of the Co/Pd multilayer gradu-\nally decreases and the e\u000bect of the in-plane NiFe becomes\nmore prominent. These two anisotropies cancel near t=\n7.5 nm, resulting in a minimum frequency, as shown in\nFig. 3(a).\nTo extract the damping coe\u000ecient, the time domain\ndata was \ftted with an exponentially damped harmonic\nfunction given by Eq. 1.\nM(t) =M(0)e\u0000t\n\u001csin(2\u0019ft\u0000\u001e) (1)\nwhere the relaxation time \u001cis related to the Gilbert\ndamping coe\u000ecient \u000bby the relation \u001c= 1/(2\u0019f\u000b).\nHere, fis the experimentally obtained precession fre-\nquency and \u001eis the initial phase of the oscillation. The\n\ftted data for various values of tis shown by the solid\nblack lines in Fig. 2. We did not extract a value of \u000bfort\n= 15 and 20 nm due to the occurrence of multimode os-\ncillations, which may lead to an erroneous estimate of the\ndamping. The extracted \u000bvalues are plotted against tin\nFig. 3(b) for two di\u000berent \feld values of 2.5 and 1.3 kOe.\nThe evolution of \u000bas a function of tdepends signi\fcantly\n5 6 7 8 9 100.0160.0200.0240.0280.032\n5 10 15 2041216\nt(nm) t(nm)(a) (b)Frequency(GHz)Figure 3. (color online) Evolution of (a) spin-wave frequency\nand (b) Gilbert damping constant as a function of tat 1.3 kOe\n(green circles) and 2.5 kOe (violet circles).\nonH, as can be seen from the \fgure. This is because of\nthe di\u000berent mechanisms responsible for determining the\ndamping in di\u000berent samples, as will be discussed later.\nAn interesting trend in the \u000bvs.tplot is observed\nforH= 2.5 kOe. For 10 nm \u0014t\u00147.5 nm,\u000bdecreases\nwith decreasing tand reaches a minimum value of about\n0.014 for t= 7.5 nm. Below this thickness, \u000bincreases\nmonotonically and reaches a value of about 0.022 for the\nlowest thickness. This variation of \u000bis somewhat cor-\nrelated with the variation of precession frequency with\nthickness. In the thinner regime, we probe both the\nNiFe layer and a fraction of the Co/Pd multilayer and\nthe relative contribution from the latter increases as t\ndecreases. The occurrence of a single mode oscillation\npoints towards a collective precession of the stack, which\nmay be considered a medium with e\u000bective magnetic pa-\nrameters consisting of both NiFe and Co/Pd layers. The\nvariation in damping may be related to the variation in\nthe anisotropy of the material. The competing IP and\nOOP anisotropies of the NiFe and Co/Pd layers lead\nto the appearance of a minimum in the damping. The\ndamping in this system may have multiple contributions,\nnamely (a) dephasing of the uniform mode in the spin-\ntwist structure1(b) interfacial d-dhybridization at the\nCo/Pd interface16, and (c) spin pumping into the Pd\nlayer.22The \frst is an extrinsic mechanism and is dom-\ninant in samples with higher NiFe thicknesses, while the\nother two mechanisms are intrinsic damping mechanisms.\nFort>7.5 nm, due to the nonuniformity of the spin\ndistribution, the dominant mode undergoes dynamic de-\nphasing and the damping thus increasescompared to the\nmagnetically uniform samples. With the increase in NiFe\nthickness, the nonuniformity of spin distribution and the\nconsequent mode dephasing across its thickness increases,\nleading to an increase in the damping value. Hence, in\nsamples with higher tvalues, dephasing is the dominant\nmechanism, while at lower tvalues|i.e., when the con-\ntribution from the Co/Pd multilayer is dominant|the\nspin-orbit coupling and spin pumping e\u000bects dominate.\nAt intermediate tvalues, the extrinsic and intrinsic ef-\nfects compete with each other, leading to a minimum\nin the damping. However, the damping increases mono-\ntonically with tin a lower \feld of H=1.3 kOe. For a\ndeeper understanding of this e\u000bect, we have measured \u000b5\n24681012140.0120.0160.0200.0240.0280.0324\n56789100.0140.0210.0280.0350.042(b) \n  \n5nm \n5.5nm \n6.5nm \n7nm/s61537F\nrequency (GHz)(a) \n  \n10nm \n8.5nm \n8nm \n7.5nm \n7nm/s61537F\nrequency (GHz)\nFigure 4. (color online) Dependence of Gilbert damping co-\ne\u000ecient on soft layer thickness ( t) for (a) 7{10 nm and (b)\n5{7 nm, respectively.\nas a function of precession frequency f. Figures 4(a){(b)\nshow the variation of \u000bwith f. Two di\u000berent regimes in\nthe thickness are presented in (a) and (b) to show the\nrate of variation more clearly. For 10 nm \u0014t\u00147 nm,\u000b\ndecreases strongly with the decrease in fand the rate of\nvariation remains nearly constant with t. This is the sig-\nnature of extrinsic damping generated by the nonuniform\nspin distribution. However, for t= 6.5 nm, the rate falls\ndrastically and for t\u00145.5 nm,\u000bbecomes nearly indepen-\ndent of t, which indicates that purely intrinsic damping is\noperating in this regime. This con\frms the competition\nbetween two di\u000berent types of damping mechanisms in\nthese samples.\nThe study demonstrates that various aspects of ul-\ntrafast magnetization dynamics|namely demagnetiza-\ntion time, precession frequency, number of modes, and\ndamping|are in\ruenced by the spin distribution in the\nsoft magnetic layer, as well as by the properties of the\nhard layer. By changing the thickness of the soft layer,\nthe relative contributions of these factors can be tuned\ne\u000bectively. This enables e\u000ecient control of the damp-\ning and other magnetic properties over a broad range,\nand will hence be very useful for potential applications\nin spintronic and magnonic devices.IV. CONCLUSION\nIn summary, we have employed the time-resolved\nMOKE technique to measure the evolution of ul-\ntrafast magnetization dynamics in exchange-coupled\n[Co/Pd] 5/NiFe( t) multilayers, with varying NiFe layer\nthicknesses, by applying an out-of-plane bias magnetic\n\feld. The coupling of a high-anisotropy multilayer with\na soft layer allows broad control over the spin struc-\nture, and consequently other dynamic magnetic prop-\nerties which are strongly dependent on t. The ultra-\nfast demagnetization displayed a strong variation with\nt. The reason for this was ascribed to the chiral-spin-\nstructure-dependent spin-\rip scattering in the top NiFe\nlayer, as well as to interfacial 3 d-4dhybridization of\nCo/Pd layer. The precessional dynamics showed mul-\ntiple spin-wave modes for t= 20 nm and 15 nm, whereas\na single spin-wave mode is observed for thinner NiFe lay-\ners following the change in the magnetization pro\fle with\ndecreasing t. The precession frequency and the damp-\ning show strong variation with the thickness of the NiFe\nlayer. The changes in frequency are understood in terms\nof the modi\fcation of the anisotropy of the system, while\nthe variation in damping originates from the competition\nbetween intrinsic and extrinsic mechanisms, which are\nsomewhat related to the anisotropy. The observed dy-\nnamics will be important for understanding the utiliza-\ntion of tilted anisotropy materials in devices such as spin-\ntransfer torque MRAM and spin-torque nano-oscillators.\nV. ACKNOWLEDGEMENTS\nWe acknowledge \fnancial support from the G oran\nGustafsson Foundation, the Swedish Research Coun-\ncil (VR), the Knut and Alice Wallenberg Foundation\n(KAW), and the Swedish Foundation for Strategic Re-\nsearch (SSF). This work was also supported by the Euro-\npean Research Council (ERC) under the European Com-\nmunity's Seventh Framework Programme (FP/2007{\n2013)/ERC Grant 307144 \"MUSTANG\". AB acknowl-\nedges the \fnancial support from the Department of Sci-\nence and Technology, Government of India (Grant no.\nSR/NM/NS-09/2011(G)) and S. N. Bose National Centre\nfor Basic Sciences, India (Grant no. SNB/AB/12-13/96).\nC.B. thanks CSIR for the senior research fellowship.\n\u0003abarman@bose.res.in\n1A. Barman and S. Barman. Dynamic dephasing of mag-\nnetization precession in arrays of thin magnetic elements.\nPhys. Rev. B , 79:144415, 2009.\n2A. Barman and A. Haldar. Chapter One - Time-Domain\nStudy of Magnetization Dynamics in Magnetic Thin Films\nand Micro- and Nanostructures. volume 65 of Solid State\nPhys. , pages 1{108. Academic Press, 2014.3R. E. Camley, B. V. McGrath, R. J. Astalos, R. L. Stamps,\nJ.-V. Kim, and L. Wee. Magnetization dynamics: A study\nof the ferromagnet/antiferromagnet interface and exchange\nbiasing. J. Vac. Sci. Technol. A , 17:1335, 1999.\n4S. Chung, S. M. Mohseni, V. Fallahi, T. N. A. Nguyen,\nN. Benatmane, R. K. Dumas, and J. \u0017Akerman. Tunable\nspin con\fguration in [Co/Pd]{NiFe spring magnets. J.\nPhys. D: Appl. Phys. , 46:125004, 2013.6\n5D. C. Crew and R. L. Stamps. Ferromagnetic resonance in\nexchange spring thin \flms. J. Appl. Phys. , 93:6483, 2003.\n6T. J. Fal, K. L. Livesey, and R. E. Camley. Domain wall\nand microwave assisted switching in an exchange spring\nbilayer. J. Appl. Phys. , 109:093911, 2011.\n7E. E. Fullerton, J. S. Jiang, M. Grimsditch, C. H. Sowers,\nand S. D. Bader. Exchange-spring behavior in epitaxial\nhard/soft magnetic bilayers. Phys. Rev. B , 58:12193, 1998.\n8A. Haldar, C. Banerjee, P. Laha, and A. Barman. Brillouin\nlight scattering study of spin waves in NiFe/Co exchange\nspring bilayer \flms. J. Appl. Phys. , 115:133901, 2014.\n9C Kittel. On the theory of ferromagnetic reso-\nnance absorption. Phys. Rev. , 73:155, 1948. doi:\n10.1103/PhysRev.73.155. URL http://link.aps.org/\ndoi/10.1103/PhysRev.73.155 .\n10B. Koopmans, G. Malinowski, F. Dalla Longa, D. Steiauf,\nM. Fahnle, T. Roth, M. Cinchetti, and M. Aeschlimann.\nExplaining the paradoxical diversity of ultrafast laser-\ninduced demagnetization. Nat. Mater. , 9:259, 2010.\n11D. Navas, J. Torrejon, F. B\u0013 oeron, C. Redondo, F. Batal-\nlan, B. P. Toperverg, A. Devishvili, B. Sierra, F. Casta~ no,\nK. R. Pirota, and C. A. Ross. Magnetization reversal and\nexchange bias e\u000bects in hard/soft ferromagnetic bilayers\nwith orthogonal anisotropies. New J. Phys. , 14:113001,\n2012.\n12T. N. A. Nguyen, Y. Fang, V. Fallahi, N. Benatmane, S. M.\nMohseni, R. K. Dumas, and J. \u0017Akerman. [Co/Pd]{NiFe ex-\nchange springs with tunable magnetization tilt angle. Appl.\nPhys. Lett. , 98:172502, 2011.\n13T. N. A. Nguyen, N. Benatmane, V. Fallahi, Y. Fang, S. M.\nMohseni, R. K. Dumas, and J. \u0017Akerman. [Co/Pd]Co{Pd{\nNiFe spring magnets with highly tunable and uniform mag-\nnetization tilt angles. J. Magn. Magn. Mater. , 324:3929,\n2012.\n14T. N. A. Nguyen, V. Fallahi, Q. T. Le, S. Chung, S. M.\nMohseni, R. K. Dumas, C. W. Miller, and J. \u0017Akerman.\nInvestigation of the Tunability of the Spin Con\fguration\nInside Exchange Coupled Springs of Hard/Soft Magnets.\nIEEE Trans. Magn. , 50:2004906, 2014.\n15T. N. A. Nguyen, R. Knut, V. Fallahi, S. Chung, Q. T. Le,\nS. M. Mohseni, O. Karis, S. Peredkov, R. K. Dumas, C. W.\nMiller, and J. \u0017Akerman. Depth-Dependent Magnetization\nPro\fles of Hybrid Exchange Springs. Phys. Rev. Appl. , 2:\n044014, 2014.\n16S. Pal, B. Rana, O. Hellwig, T. Thomson, and A. Barman.\nTunable magnonic frequency and damping in [Co/Pd] 8multilayers with variable co layer thickness. Appl. Phys.\nLett., 98:082501, 2011.\n17S. Pal, S. Barman, O. Hellwig, and A. Barman. E\u000bect\nof the spin-twist structure on the spin-wave dynamics in\nFe55Pt45/Ni 80Fe20exchange coupled bi-layers with vary-\ning Ni 80Fe20thickness. J. Appl. Phys. , 115:17D105, 2014.\n18M. R. Scheinfein. LLG Micromagnetics Simulator. [On-\nline]. Available: http://llgmicro.home.mindspring.com/ .\n19S. Tacchi, T. N. A. Nguyen, G. Carlotti, G. Gubbiotti,\nM. Madami, R. K. Dumas, J. W. Lau, J. \u0017Akerman, A. Ret-\ntori, and M. G. Pini. Spin wave excitations in exchange-\ncoupled [Co/Pd]-NiFe \flms with tunable tilting of the\nmagnetization. Phys. Rev. B , 87:144426, 2013.\n20S. Tacchi, T. N. A. Nguyen, G. Gubbiotti, M. Madami,\nG. Carlotti, M. G. Pini, A. Rettori, V. Fallahi, R. K. Du-\nmas, and J. \u0017Akerman. [Co/Pd]{CoFeB exchange spring\nmagnets with tunable gap of spin wave excitations. J.\nPhys. D: Appl. Phys. , 47:495004, 2014.\n21L. Tryputen, F. Guo, F. Liu, T. N. A. Nguyen, M. S.\nMohseni, S. Chung, Y. Fang, J. \u0017Akerman, R. D.\nMcMichael, and C. A. Ross. Magnetic structure and\nanisotropy of [Co/Pd] 5{NiFe multilayers. Phys. Rev. B ,\n91:014407, 2015.\n22Y. Tserkovnyak, A. Brataas, and G. E. W. Bauer.\nEnhanced Gilbert Damping in Thin Ferromagnetic\nFilms. Phys. Rev. Lett. , 88:117601, Feb 2002. doi:\n10.1103/PhysRevLett.88.117601. URL http://link.aps.\norg/doi/10.1103/PhysRevLett.88.117601 .\n23B. Vodungbo, J. Gautier, G. Lambert, A. B. Sardinha,\nM. Lozano, S. Sebban, M. Ducousso, W. Boutu, K. Li,\nB. Tudu, M. Tortarolo, R. Hawaldar, R. Delaunay,\nV. L\u0013 opez-Flores, J. Arabski, C. Boeglin, H. Merdji,\nP. Zeitoun, and J. L uning. Laser-induced ultrafast demag-\nnetization in the presence of a nanoscale magnetic domain\nnetwork. Nat. Commun. , 3:999, 2012.\n24M. C. Weber, H. Nembach, S. Blomeier, B. Hillebrands,\nR. Kaltofen, J. Schumann, M. J. Carey, and J. Fassbender.\nAll-optical probe of magnetization dynamics in exchange\nbiased bilayers on the picosecond timescale. Eur. Phys. J.\nB, 45:243, 2005.\n25Y. Zhou, C. L. Zha, S. Bonetti, J. Persson, and\nJ.\u0017Akerman. Spin-torque oscillator with tilted \fxed layer\nmagnetization. Appl. Phys. Lett. , 92:262508, 2008.\n26Y. Zhou, C. L. Zha, S. Bonetti, J. Persson, and\nJ.\u0017Akerman. Microwave generation of tilted-polarizer spin\ntorque oscillator. J. Appl. Phys. , 105:07D116, 2009."    },    {        "title": "1604.04688v1.A_broadband_Ferromagnetic_Resonance_dipper_probe_for_magnetic_damping_measurements_from_4_2_K_to_300_K.pdf",        "content": "arXiv:1604.04688v1  [cond-mat.mtrl-sci]  16 Apr 2016A broadband Ferromagnetic Resonance dipper probe for magne tic damping\nmeasurements from 4.2 K to 300 K\nShikun Hea)and Christos Panagopoulosb)\nDivision of Physics and Applied Physics, School of Physical\nand Mathematical Sciences, Nanyang Technological Univers ity,\nSingapore 637371\nAdipper probefor broadband FerromagneticResonance (FMR)op erating from4.2K\nto room temperature is described. The apparatus is based on a 2-p ort transmitted\nmicrowave signal measurement with a grounded coplanar waveguide . The waveguide\ngenerates a microwave field and records the sample response. A 3- stagedipper design\nis adopted for fast and stable temperature control. The tempera ture variation due\nto FMR is in the milli-Kelvin range at liquid helium temperature. We also desig ned\na novel FMR probe head with a spring-loaded sample holder. Improve d signal-to-\nnoise ratio and stability compared to a common FMR head are achieved . Using a\nsuperconducting vector magnet we demonstrate Gilbert damping m easurements on\ntwo thin film samples using a vector network analyzer with frequency up to 26GHz:\n1) A Permalloy film of 5 nm thickness and 2) a CoFeB film of 1.5nm thicknes s. Ex-\nperiments were performed with the applied magnetic field parallel and perpendicular\nto the film plane.\na)Electronic mail: skhe@ntu.edu.sg\nb)Electronic mail: christos@ntu.edu.sg\n1I. INTRODUCTION\nIn recent years, the switching of a nanomagnet by spin transfer t orque (STT) using a spin\npolarized current has been realized and intensively studied.1–3This provides avenues to new\ntypes of magnetic memory and devices, reviving the interest on mag netization dynamics in\nultrathinfilms.4,5Highfrequencytechniques playanimportantroleinthisresearchdir ection.\nAmongthem, FerromagneticResonance(FMR)isapowerfultool. M ostFMRmeasurements\nhave been performed using commercially available systems, such as e lectron paramagnetic\nresonance (EPR) or electron spin resonance (ESR).6These techniques take advantage of\nthe high Q-factor of a microwave cavity, where the field modulation a pproach allows for the\nutilization of a lock-in amplifier.7The high signal-to-noise ratio enables the measurement of\nevensub-nanometerthickmagneticfilms. However, theoperating frequencyofametalcavity\nis defined by its geometry and thus is fixed. To determine the damping of magnetization\nprecession, whichisinprincipleanisotropic, several cavitiesarereq uiredtostudytherelation\nbetween the linewidth and microwave frequency at a given magnetiza tion direction.8–10The\napparent disadvantage isthat changing cavities canbetedious and prolongthe measurement\ntime.\nRecently, an alternative FMR spectrometer has attracted consid erable attention.11–17\nThe technique is based on a state of the art vector network analyz er (VNA) and a coplanar\nwaveguide (CPW). Both VNA and CPW can operate in a wide frequency range hence this\ntechnique is also referred to as broadband FMR or VNA-FMR. The br oadband FMR tech-\nnique offers several advantages. First, it is rather straightforw ard to measure FMR over a\nwide frequency range. Second, one may fix the applied magnetic field and acquire spectra\nwith sweeping frequency in a matter of minutes.17Furthermore, a CPW fabricated on a chip\nusing standard photolithography enables FMR measurements on pa tterned films as well as\non a single device.18In brief, it is a versatile tool suitable for the characterization of ma g-\nnetic anisotropy, investigation of magnetization dynamics and the s tudy of high frequency\nresponse of materials requiring a fixed field essential to avoid any ph ase changes caused by\nsweeping the applied field.\nAlthough homebuilt VNA-FMRs are designed mainly for room temperat ure measure-\nments, a setup with variable temperature capability is of great inter est both for fundamental\nstudies and applications. Denysenkov et al. designed a probe with va riable sample temper-\nature, namely, 4-420K,19however, the spectrometer only operates in reflection mode. In\na more recent effort, Harward et al. developed a system operating at frequencies up to\n70GHz.12However the lower bound temperature of the apparatus is limited to 27K. Here\nwe present a 2-port broadband FMR apparatus based on a superc onducting magnet. A\n3-stage dipper probe has been developed which allows us to work in th e temperature range\n4.2- 300K. Taking advantage of a superconducting vector magnet , measurements can be\nperformed with the magnetic moment saturated either parallel or p erpendicular to the film\nplane. Wealsodesigned aspring-loadedsample holderforfastandre liablesample mounting,\nquicktemperatureresponseandimprovedstability. Thissetupallow sforswiftchangesofthe\nFMR probe heads and requires little effort for the measurement of d evices. To demonstrate\nthe capability of this FMR apparatus we measured the temperature dependence of magne-\n2FIG. 1. View of the FMR dipper probe. Top panel: The schematic of the entire design with a\nstraight type FMR head. All RF connectors are 2.4mm. The vacu um cap mounted on the 4K\nstage, using In seal, and the radiation shield mounted on the second stage are not shown for clarity.\nBottom panel: photograph of the components inside the vacuu m cap.\ntization dynamics of thin film samples of Permalloy (Py) and CoFeB in diffe rent applied\nmagnetic field configurations.\nII. APPARATUS\nA. Cryostat and superconducting magnet system\nOur customized cryogenic system was developed by Janis Research Company Inc. and\nincludes a superconducting vector magnet manufactured by Cryo magnetics Inc. A vertical\nfield up to 9T is generated by a superconducting solenoid. The field ho mogeneity is ±0.1%\nover a 10mm region. A horizontal split pair superconducting magnet provides a field up to\n4T with uniformity ±0.5% over a 10mm region. The vector magnet is controlled by a\nModel 4G-Dual power supply. Although the power supply gives field r eadings according to\nthe initial calibration, to avoid the influence of remnant field we employ an additional Hall\nsensor. The cryostat has a 50mm vertical bore to accommodate v ariable temperature\ninserts and dipper probes. Our dipper probe described below is confi gured for this\ncryostat, however, the principle can be applied also to other comme rcially available\nsuperconducting magnets and cryostats.\n3B. Dipper probe\nFig. 1shows a schematic of our dipper probe assembly and a photograph o f the com-\nponents inside the vacuum cap. The dipper probe is 1.2m long and is mou nted to the\ncryogenic system via a KF50 flange. The sliding seal allows a slow insert ion of the dipper\nprobe directly into the liquid helium space. Supporting arms (not show n) lock the probe\nand minimize vibration, with the sample aligned to the field center. The c onnector box on\ntop has vacuum tight Lemo and Amphenol connectors for 18 DC sign al feedthrough. Two\n2.4mm RF connection ports allow for frequencies up to 50GHz . A vacu um pump port can\nbe shut by a Swagelok valve. We adopted a three stage design as sho wn in the photograph\nofFig. 1. The 4K stage and the vacuum cap immersed in the He bath provide co oling power\nfor the probe. The intermediate second stage acts as an isolator o f heat flow and as thermal\nsink for the RF cables, providing improved temperature control. Fu rthermore, it allows one\nto change probe heads conveniently as we discuss later. A separat e temperature sensor on\nthe second stage is used for monitoring purpose. The third stage, namely, the FMR probe\nhead with the spring loaded sample holder, is attached to the lower en d of the intermediate\nstage using stainless steel rods.\nApairof0.086”stainlesssteelSemi-RigidRFcablesrunfromthetopo ftheconnectorbox\nto the non-magnetic bulkhead connector (KEYCOM Corp.) mounted on the second stage.\nBeCu non-magnetic Semi-Rigid cables (GGB Industries, Inc.) are use d for the connection\nbetween the second stage and the probe head. The cables are car efully bent to minimize\nlosses. The rods connecting the stages are locked by set screws. Loosening the set screw\nallows the rod length to be adjusted to match the length of the RF ca bles. Reflection\ncoefficient (S 11) and transmission coefficient (S 21) can be recorded simultaneously with this\n2-port design. The leads for the temperature sensors, heater, Hall sensor and for optional\ntransport measurements are wrapped around Cu heat-sinks at t he 4K stage before being\nsoldered to the connection pins.\nC. Probe head with spring-loaded sample holder\nThe key part of the dipper probe, namely, the FMR probe head is sch ematically shown in\nFig. 2. Theassemblyisplacedinaradiationshieldtubewithaninnerdiametero f32mm. To\nmaximize thermal conduction between parts, homebuilt component s are machined from Au\nplated Cu. The 1” long customized grounded coplanar waveguide (GC PW) has a nominal\nimpedanceof50Ohm. Thestraight-lineshapeGCPWwasmadeonduro idR/circlecopyrtR6010(Rogers)\nboard, with a thickness of 254 µm and dielectric constant 10.2. The width of the center\nconductor is 117 µm and the gap between the latter and the ground planes is 76 µm. For\nthe connection, first the GCPW is soldered to the probe head, and s ubsequently the center\npin of the flange connector (Southwest Microwave) is soldered to t he center conductor of the\nGCPW. The response of the dipper with the straight-line shape GCPW installed is shown\ninFig. 3. The relatively large insertion loss (-16.9dB at 26GHZ) is due to a tota l cable\nlength of more than 3m and multiple connectors. The high frequency current flowing in the\nCPW generates a magnetic field of the same frequency. This RF field d rives the precession\n4FIG. 2. Schematic of the spring-loaded FMR probe head with st raight shape grounded coplanar\nwaveguide (GCPW). 1 Au plated Cu housing; 2 straight shape GC PW; 3 flange connector; 4 strain\ngauge thin film heater; 5 CernoxTMtemperature sensor; 6 Hall-sensor housing; 7 housing for 4- pin\nDip socket or pingo pin; 8 sample; 9 sample holder; 10 Cu sprin g; 11 spring housing; 12 sample\nholder handle nut.\nof the magnetic material placed on top of the signal line, and gives ris e to a change in the\nsystem’s impedance, which in turn alters the transmitted and reflec ted signals.\nA spring-loaded sample holder depicted in Fig. 2by items 9 to 12 is designed to mount\nthe sample. The procedure for loading a sample is as follows: 1) Pull up the handle nut\nand apply a thin layer of grease (Apiezon N type) to the sample holder ; 2) Place the sample\nat the center of the sample holder; 3) Mount the spring-loaded sam ple holder to the FMR\nhead; 4) Release the handle nut gradually so that the spring pushes the sample towards\nthe waveguide. The mounting-hole of the spring-housing is slightly lar ger than the outer\ndiameterofthespring. Thisallowsthesampleholdertomatchthesur faceoftheGCPWself-\nadaptively. With the spring-loaded FMR head design, the sample moun ting is simple and\nleaves no residue from the commonly used tapes. It maximizes the sig nal by minimizing the\ngap between waveguide and sample, and enhances the stability. Fur thermore, it is suitable\nfor variable temperature measurements due to the enhanced the rmal coupling between the\nsample, cold head and sensors ( items 9 to 12 in Fig. 2.).\nThe temperature sensor is mounted at the backside of the probe h ead. Due to limited\nspace, the heater consists of three parallel connected strain ga uges with a resistance of 120\nOhm. TheHallsensor canbemountedaccordingtotherequiredmea surement configuration.\nThe position of the Hall sensor shown in Fig. 2is an example for measurements in the\npresence of a magnetic field applied parallel to the sample surface.\nD. Probe-head using end-launch connector\nAlthough the probe head with straight-line CPW works well in our expe riment, the\nnecessary replacement of CPW due to unavoidable performance fa tigue over time, or for\ntesting new CPW designs can be time consuming. In response, end-la unch connectors\n(ELC) utilizing a clamping mechanism allow for a smooth transition from R F cables to\nCPW. Soldering the launch pin to the center conductor of CPW is optio nal and reduces\nthe effort for modifications. In Fig. 4, we show our design of a FMR probe-head using ELC\n50 5 10 15 20 25-30-20-100S (dB)\nf (GHz) S11\n S21\nFIG. 3. The reflection (S 11) and transmission (S 21) coefficients of the dipper probe with the\nstraight-line shape GCPW mounted. The measurement was perf ormed at room temperature.\nform Southwest Microwave, Inc. and a homebuilt U-shape GCPW. Sim ilar to the design of\nFig. 2, a Au plated Cu housing is used to mount the GCPW, ELC and the tempe rature and\nHall sensors. There are two locations for sample mounting. In posit ion A, the vertical field\nis used for measurements with the magnetic field applied parallel to th e surface of the thin\nfilm sample whereas the horizontal field is used for measurements wit h field perpendicular\nto the sample surface. On the other hand, measurements for bot h configurations can be\naccomplishedonlybyusingthehorizontalfieldifthesampleisplacedinp ositionB.Asshown\ninFig. 4(b) and (c), to change between configurations simply requires rot ating the dipper\nprobeby90degrees. Nevertheless, weprefertoplacethesample inpositionAfortheparallel\nconfigurationsincethesolenoidfieldismoreuniform. However, weno tethatthesamedesign\nwith the sample placed in position B is suitable also for an electromagnet . Furthermore,\nadding a rotary stage to the probe enables angular dependent FMR measurements.\nIII. EXPERIMENTAL TEST\nIn this section, we present data to assess the performance of th e FMR probe head and\ndiscuss two sets of magnetic damping measurements, demonstrat ing the capabilities and\nperformance of the appratus.\nA. Spring-loaded sample holder\nWe tested our setup using a Keysight PNA N5222A vector network a nalyzer with maxi-\nmum frequency 26.5GHz. The output power of the VNA is always 0dBm in our test. Note\nthat with 2.4mm connectors and customized GCPW, our design can in p rinciple operate\nup to 50GHz. The performance of the spring-loaded sample holder is first studied at room\ntemperature with a 2nm thick Co 40Fe40B20film. For direct comparison, the FMR spectra\nare recorded with two sample loading methods: One with a spring-load ed sample holder\n6FIG. 4. FMR probe-head with u-shape GCPW and end-launch conn ector. (a) Photograph of the\nprobe-head using end-launch connector and U-shape GCPW. Se nsors are mounted at the backside\nand at the bottom of the Cu housing. Simplified sketch of the co nfiguration for measuring with an\nexternal field generated by the split coils (b) parallel and ( c) perpendicular to the sample plane.\nRotating the dipper probe in the horizontal plane changes fr om one configuration to the other.\n(Fig. 2) and the other using the common method12which only requires Kapton tape. The\nmagnetic field is applied parallel to the plane of the thin film sample. Six se ts of data were\nobtained by reloading the sample for each measurement. In Fig. 5, we show the amplitude\nof the power transmission coefficient from Port 1 to Port 2 (S 21) at a frequency of 10GHz\nand a temperature of 300K. The open circles represent a spectru m for a spring-loaded sam-\nple mounting whereas the open squares is the spectrum showing larg est signal for the six\nflip-sample loadings. The averaged spectra for all six spectra are s hown by solid line and\ndotted line, for spring and flip-sample loading, respectively. Two obs ervations are evident:\nFirst, the best signal we obtained using the flip sample method is appr oximately 20 percent\nlower compared to the spring-loaded method. Thus the spring-load ed method gives a better\nsignal to noise ratio and sensitivity. Second, for the spring-loaded method, the difference\nbetween the averaged spectrum and single spectra is negligibly small. On the other hand the\nvariation between measurements for the flip-sample method can be as large as 20 percent.\nHence the spring-loaded method has better stability and is reprodu cible.\nB. Temperature response\nAs detailed in the previous section, the probe head is made of Au plate d Cu blocks with\nhighinternalthermalconductionandgoodthermalcontact. Con sequently, theresponsetime\nof the temperature control will be small as the characteristic the rmal relaxation time of a\nsystem is C/k, whereCis the heat capacity and kis the overall thermal conduction. Also,\nthe temperature difference between sample and sensor is minimized e ven with the heater\nturned on. Shown in Fig. 6are the FMR spectra and temperature variation for a CoFeB\nfilm of 3nm thickness measured at 4.4K. The external field was swept at a rate of about -\n10Oe/s. Forfields close to which FMRpeaks areobserved, we detec ted a temperaturerise of\n7FIG. 5. Comparison between S 21signals obtained using spring-loaded sample holder mounti ng and\nflip-sample mounting at 300K. The sample has a stack of MgO(3n m)—CoFeB(2nm)—MgO(3nm)\ndeposited on silicon substrate. (Numbers in parenthesis of the sample composition represent the\nthickness of the respective layer.) The frequency is 10 GHz a nd the FMR center field is at 1520\nOe.\na few mK. In fact, the field values corresponding to maximum temper atures are about 20Oe\nlower than the fields satisfying FMR condition, showing that the char acteristic relaxation\ntime between the sample and cold head is approximately 2 seconds. Th e temperature rise\nof the probe head due to FMR indicates that the magnetic system ab sorbs energy from\nthe microwave and dissipates into the thermal bath. Specifically, at the field satisfying\nthe FMR condition, the damping torque is balanced by the torque gen erated by the RF\nfield. However, the dissipation power of such process is propotiona l to the thickness of\nthe magnetic film hence is very small. The successful detection of a t emperature rise adds\ncredence to the high thermal conduction within the probe head and relative low thermal\nconduction between different stages. This demonstration shows t hat the probe head is\ncapable of measuring samples with phase transitions in a narrow temp erature range, such\nas a superconducting/ferromagnetic bilayer system.20\nC. Magnetic damping measurements\nAlthough the FMR probe can be used to determine the energy anisot ropy of magnetic\nmaterials, our primary purpose is to study magnetic damping parame ter. In the following,\ntwo examples of such measurements are briefly described. Shown in Fig. 7is FMR response\nof a Py film of 5nm thickness deposited on a silicon substrate, measur ed at 4.4K. The\nsweeping external magnetic field is parallel to the sample surface. R eal and Imaginary parts\nof the spectra obtained at selected frequencies are plotted with o pen circles in Fig. 7(a)\nand (b), respectively. In FMR measurements, the change in the tr ansmittance, S21, is a\ndirect measure of the field-dependent susceptibility of the magnet ic layer. According to the\n8FIG. 6. Sample temperature variation due to FMR at selected f requencies. (upper panel) Ampli-\ntude of S 21and (lower panel) temperature variation of MgO(3nm)—CoFeB (3nm)—MgO(3nm) at\n4.4K measured with external field parallel to the film plane.\nLandauLifshitzGilbert formalism, the dynamic susceptibility of the ma gnetic material in the\nconfiguration where the field is applied parallel to the plane of the thin film be described\nas:21\nχIP=4πMs(H0+Huni+4πMeff+i∆H/2)\n(H0+Huni)(H0+Huni+4πMeff)−H2\nf+i(∆H/2)·[2(H0+Huni)+4πMeff](1)\nHere, 4πMsis the saturation magnetization, Huniis the in-plane uniaxial anisotropy,\n4πMeffis the effective magnetization, Hf= 2πf/γ, and ∆His the linewidth of the spectrum\n– the last term is of key importance to determine the damping parame ter. As shown by\nsolid lines in Fig. 7(a) and (b), the spectra can be fitted very well by adding a backgr ound,\na drift proportional to time, and a phase factor.11,22The field linewidth as a function of\nfrequency – ∆ H(f) is plotted in Fig. 7(c). The data points fall on a straight line. The\ndamping parameter αGL= 0.012±0.001 is therefore determined by the slope through9,23:\n∆H=4π\nγαGLf+∆H0 (2)\nThe error bar here is calculated from the confidence interval of th e fit.\nWe have also tested the setup with a magnetic field applied perpendicu lar to the sample\nplane. The results for a MgO(3nm)—Co 40Fe40B20(1.5nm)—MgO(3nm) stack deposited on\nsilicon substrate are shown in Fig. 8. Comparing the spectra obtained at different tempera-\ntures and fixed frequency, two observations are evident. First, the FMR peak position shifts\nto higher field as the temperature is lowered due to changes in the eff ective magnetization.\n9FIG. 7. FMR data of a Py thin film of thickness 5nm measured at 4. 4K with magnetic field\napplied parallel to the sample plane. (a) Real and (b) Imagin ary parts of transmitted signal S 21\nat selected frequencies. The data are normalized and the rel ative strength between the spectra at\ndifferent frequencies are kept. (c) FMR linewidth as a functio n of frequency. The damping was\ncalculated to be 0.012 ±0.001, using a linear fit.\nSecond, the FMR linewidth increases with decreasing temperature. Although the interfacial\nanisotropy can be determined by fitting the FMR peak positions to th e Kittel formula,24\nhere, we are more interested in the damping parameter as a functio n of temperature. The\ndynamic susceptibility in this configuration is25:\nχOP=4πMs(H−4πMeff−i∆H/2)\n(H−4πMeff)2−H2\nf+i∆H·(H−4πMeff)(3)\nFollowing the same procedure as for Py, the real and imaginary part of the spectra are\nfitted simultaneously to obtain the linewidth. In Fig. 8(b), we plot the linewidth as a\nfunction of frequency at the two boundaries of our measured tem peratures. Although the\nmeasured linewidth at lower temperature is larger, the slope of the t wo curves is in good\nagreement. The additional linewidth at 6K is primarily due to zero freq uency broadening,\nwhich quantifies the magnitude of dispersion of the effective magnet ization. The results\nare summarized in Fig. 8(c). Gilbert damping is essentially independent of temperature\nalthough there is a minimum at 40 K. The room temperature value obta ined is in agreement\nwith the value for a thicker CoFeB.21,26On the other hand, the inhomogeneous broaden-\ning increases with lowering temperatue. The value at 6K is more than d ouble compared\nto room temperatue. Notably, neglecting the zero-frequency off set ∆H(0), arising due to\ninhomogeneity, would give rise to an enhanced effective damping comp ared to the intrinsic\ncontribution. Cavity based, angulardependent FMRmayalso disting uish theGilbertdamp-\ning from inhomogeneity effects. A shortcoming however, is the need to take into account\nthe possible contribution of two magnon scattering, which causes in creased complications\nin the analysis of the data.27,28On the other hand, broadband FMR using a dipper probe\nwith the applied magnetic field in the perpendicular configuration, rule s out two magnon\nscattering making this technique relatively straightforward to imple ment.29\nThedipperprobediscussedhereisnotlimitedtomeasurementsofth edampingcoefficient.\nThe broadband design is also useful for time-domain measurements .30Furthermore, a spin\n10FIG. 8. Temperature dependent FMR measurement for a CoFeB th in film of thickness 1.5nm\nwith the magnetic field applied perpendicular to the film plan e. (a) Transmitted FMR signal at\n20GHz obtained at different temperatures. (b) FMR linewidth a s a function of frequency at 6K\nand 280K. (c) Damping constant and inhomogeneous broadenin g as a function of temperature.\nThe solid lines are the guides for the eye.\ntransfer torque ferromagnetic resonance31measurement on a single device can be performed\nwith variable temperature using bias Tee and a separate sample holde r.\nIV. CONCLUSION\nWe have developed a variable temperature FMR to measure the magn etic damping pa-\nrameter in ultra thin films. The 3-stage dipper and FMR head with a spr ing-loaded sample\nholder design have a temperature stability of milli Kelvin during the FMR measurements.\nThis apparatus demonstrates improved signal stability compared t o traditional flip-sample\nmounting. The results for Py and CoFeB thin films show that the FMR d ipper can measure\nthe damping parameter of ultra thin films with: Field parallel and perpe ndicular to the film\nplane in the temperature range 4.2-300K and frequency up to at lea st 26 GHz.\nACKNOWLEDGMENTS\nTheauthorsaregratefultoSzeTerLimatDataStorageInstitut eforpreparingtheCoFeB\nsamples. We acknowledge Singapore Ministry of Education (MOE), Ac ademic Research\nFund Tier 2 (Reference No: MOE2014-T2-1-050) and National Res earch Foundation (NRF)\nof Singapore, NRF-Investigatorship (Reference No: NRF-NRFI2 015-04) for the funding of\nthis research.\nREFERENCES\n1J. C. Slonczewski, Journal of Magnetism and Magnetic Materials 159, L1 (1996) .\n2J. A. Katine, F. J. Albert, R. A. Buhrman, E. B. Myers, and D. C. Ra lph,\nPhysical Review Letters 84, 3149 (2000) .\n113S. Mangin, D. Ravelosona, J. A. Katine, M. J. Carey, B. D. Terris, a nd E. E.\nFullerton, Nat Mater 5, 210 (2006) .\n4N. Locatelli, V. Cros, and J. Grollier, Nat Mater 13, 11 (2014) .\n5A. Brataas, A. D. Kent, and H. Ohno, Nat Mater 11, 372 (2012) .\n6M. Farle, Reports on Progress in Physics 61, 755 (1998) .\n7B. Heinrich, Measurement Techniques and Novel Magnetic Properties (Springer\nBerlin Heidelberg, 2006).\n8B. Heinrich, G. Woltersdorf, R. Urban, and E. Simanek,\nJournal of Applied Physics 93, 7545 (2003) .\n9K. Lenz, H. Wende, W. Kuch, K. Baberschke, K. Nagy, and A. J´ an ossy,\nPhysical Review B 73, 144424 (2006) .\n10B. Heinrich, C. Burrowes, E. Montoya, B. Kardasz, E. Girt, Y.-Y. S ong, Y. Sun,\nand M. Wu, Physical Review Letters 107, 066604 (2011) .\n11S. S. Kalarickal, P. Krivosik, M. Wu, C. E. Patton, M. L. Schneider, P . Kabos, T. J.\nSilva, and J. P. Nibarger, Journal of Applied Physics 99, 093909 (2006) .\n12I. Harward, T. O’Keevan, A. Hutchison, V. Zagorodnii, and Z. Celins ki,\nReview of Scientific Instruments 82, 095115 (2011) .\n13J. Wei, J. Wang, Q. Liu, X. Li, D. Cao, and X. Sun,\nReview of Scientific Instruments 85, 054705 (2014) .\n14E. Montoya, T. McKinnon, A. Zamani, E. Girt, and B. Heinrich,\nJournal of Magnetism and Magnetic Materials 356, 12 (2014) .\n15H. Gowiski, M. Schmidt, I. Gociaska, J.-P. Ansermet, and J. Dubowik ,\nJournal of Applied Physics 116, 053901 (2014) .\n16M. Bailleul, Applied Physics Letters 103, 192405 (2013) .\n17C. Bilzer, T. Devolder, P. Crozat, C. Chappert, S. Cardoso, and P . P. Freitas,\nJournal of Applied Physics 101, 074505 (2007) .\n18I. Neudecker, K. Perzlmaier, F. Hoffmann, G. Woltersdorf, M. Bue ss, D. Weiss, and\nC. H. Back, Physical Review B 73, 134426 (2006) .\n19V.P.DenysenkovandA.M.Grishin, Review of Scientific Instruments 74, 3400 (2003) .\n20C. Bell, S. Milikisyants, M. Huber, and J. Aarts,\nPhysical Review Letters 100, 047002 (2008) .\n21C. Bilzer, T. Devolder, J.-V. Kim, G. Counil, C. Chappert, S. Cardoso , and P. P.\nFreitas,Journal of Applied Physics 100, 053903 (2006) .\n22H. T. Nembach, T. J. Silva, J. M. Shaw, M. L. Schneider, M. J. Carey , S. Maat, and\nJ. R. Childress, Physical Review B 84, 054424 (2011) .\n23Z. Celinski and B. Heinrich, Journal of Applied Physics 70, 5935 (1991) .\n24M. Sparks, R. Loudon, and C. Kittel, Physical Review 122, 791 (1961) .\n25T. Devolder, P.-H. Ducrot, J.-P. Adam, I. Barisic, N. Vernier, J.-V. Kim, B. Ockert,\nand D. Ravelosona, Applied Physics Letters 102, 022407 (2013) .\n26S. Ikeda, K. Miura, H. Yamamoto, K. Mizunuma, H. D. Gan, M. Endo, S. Kanai,\nJ. Hayakawa, F. Matsukura, and H. Ohno, Nat Mater 9, 721 (2010) .\n27S. Mizukami, Y. Ando, and T. Miyazaki, Physical Review B 66, 104413 (2002) .\n1228J.Lindner, I.Barsukov, C.Raeder,C.Hassel, O.Posth, R.Meck enstock, P.Landeros,\nand D. L. Mills, Physical Review B 80, 224421 (2009) .\n29R. Arias and D. L. Mills, Physical Review B 60, 7395 (1999) .\n30Y.-T. Cui, G. Finocchio, C. Wang, J. A. Katine, R. A. Buhrman, and D. C. Ralph,\nPhys. Rev. Lett. 104, 097201 (2010) .\n31L. Liu, T. Moriyama, D. C. Ralph, and R. A. Buhrman,\nPhys. Rev. Lett. 106, 036601 (2011) .\n13"    },    {        "title": "1604.05167v1.Parameter_Estimation_of_Gaussian_Damped_Sinusoids_from_a_Geometric_Perspective.pdf",        "content": "Parameter Estimation of Gaussian-Damped Sinusoids from a Geometric Perspective Thomas A. Pelaia II - ORCID: 0000-0002-5879-9340 Oak Ridge National Lab, Oak Ridge, TN 37831, USA ABSTRACT The five parameter gaussian damped sinusoid equation is a reasonable model for betatron motion with chromatic decoherence of the proton bunch centroid signal in the ring at the Spallation Neutron Source. A geometric method for efficiently fitting this equation to the turn by turn signals to extract the betatron tune and damping constant will be presented. This method separates the parameters into global and local parameters and allows the use of vector arithmetic to eliminate the local parameters from the parameter search space. Furthermore, this method is easily generalized to reduce the parameter search space for a larger class of problems. I. INTRODUCTION The five parameter gaussian damped sinusoid equation is a reasonable model for the transverse motion of a single proton bunch injected into the accumulator ring at the Spallation Neutron Source (SNS) [ ‑] at Oak Ridge National Lab and stored for several dozen turns. Each 1beam position monitor (BPM) in the ring can capture this turn by turn signal for each transverse plane resulting in a waveform (one per plane) with one element per turn. A waveform depends on both local beam parameters (orbit distortion, amplitude and phase) at the BPM and global beam parameters (betatron tune and damping constant) independent of BPM. Our goal is to measure the global beam parameters, and more specifically we are most interested in efficiently and accurately measuring the betatron tune with concurrently captured waveforms across all BPMs. In this paper we present a method based on a geometric perspective to fit the global parameters to one or more waveforms eliminating the local parameters from the parameter search space. With just two global parameters to fit, the optimization complexity is significantly reduced and especially ideal when fitting to multiple waveforms across all BPMs. By eliminating the unknown local parameters from the parameter search space, it is expected the fits should !1This manuscript has been authored by UT-Battelle, LLC, under Contract No. DE-AC05-00OR22725 with the U.S. Department of Energy. The United States Government retains and the publisher, by accepting the article for publication, acknowledges that the United States Government retains a non-exclusive, paid-up, irrevocable, world-wide license to publish or reproduce the published form of this manuscript, or allow others to do so, for United States Government purposes.consistently converge to the optimal global parameters faster. It will be shown that this technique naturally extends to a wider class of problems. II. BETATRON MOTION WITH CHROMATIC DECOHERENCE A charged particle at the nominal energy in the ring will exhibit simple betatron motion in the absence of nonlinearities. As is well known, if a particle has an energy that differs from the  nominal, the tune will shift accordingly as the product of the relative energy shift and the chromaticity of the ring optics. For a bunch with a distribution of energies about the nominal energy, the motion of its constituent particles will decohere over time due to the corresponding tune spread. The equation of motion has previously been derived [ ‑] for general synchrotron 2motion by Meller et. al; however, for our case a simpler model can be used. For times much smaller than the synchrotron period, it is straight forward to show that the bunch centroid signal (such as measured by a BPM) will be a gaussian-damped sinusoid. At SNS, this is indeed the region of relevance as the bunch signal is typically analyzed for several dozen stored turns, and the synchrotron period is greater than 1400 turns [ ‑]. That the observed signal damping for 3nominal chromaticity is due to chromatic decoherence is consistent with measurements showing that the damping time increases by several dozen turns as the chromaticity is reduced to zero. By definition, the betatron tune is shifted from the nominal tune by the product of the chromaticity and the relative energy shift. The orbit is also distorted by an amount equal to the product of the dispersion by the relative energy shift. For times much shorter than the synchrotron period, the relative energy shift is effectively constant, so we can approximate the position of a single particle as a function of turn by equation 1 where b is the orbit distortion, A is the amplitude, φ is the phase, µ is 2π times the tune, µ0 is 2π times the nominal tune, t is the turn index, η is the dispersion and ξ is the chromaticity. Given a bunch of particles, equation 2 shows the resulting centroid position of the bunch as a function of time in terms of the distribution function over µ. If the energy spread follows a gaussian distribution, then the distribution over µ can be written according to equation 3 where σµ is the standard deviation of µ. (1)!Gaussian-Damped Sinusoid EquationsParticle position as a function of timeq=b+Asin(μt+φ)+η2πξ(μ−μ0)Centroid motion integral<q>=b+∞∫-∞dμρ(μ)⎛⎝⎜Asin(μt+φ)+η2πξ(μ−μ0)⎞⎠⎟Distribution over angular frequencyρ(μ)=1σμ(2π)1/2e−(μ−μ0)2/2σμ2Distribution over angular frequencyρ(μ)=1σμ(2π)1/2e−12σμ2(μ−μ0)2Gaussian-damped Sinusoid solution<q>=b+Ae−12(σμt)2sin(μ0t+φ)Gaussian-damped Sinusoid standard formqn=Ae−γn2sin(μn+φ)+bGaussian-damped Sinusoid with linear coefficientsqn=e−γn2(r1sinμn+r2cosμn)+r3Waveform Vector Decomposition→q=r1→Sμ,γ+r2→Cμ,γ+r3→ZGaussian-Damped Sinusoid Fit Error→Ew=→qw−3∑j=1→Uj(→qw⋅→Uj)(2)!Gaussian-Damped Sinusoid EquationsParticle position as a function of timeq=b+Asin(μt+φ)+η2πξ(μ−μ0)Centroid motion integral<q>=b+∞∫-∞dμρ(μ)⎛⎝⎜Asin(μt+φ)+η2πξ(μ−μ0)⎞⎠⎟Distribution over angular frequencyρ(μ)=1σμ(2π)1/2e−(μ−μ0)2/2σμ2Distribution over angular frequencyρ(μ)=1σμ(2π)1/2e−12σμ2(μ−μ0)2Gaussian-damped Sinusoid solution<q>=b+Ae−12(σμt)2sin(μ0t+φ)Gaussian-damped Sinusoid standard formqn=Ae−γn2sin(μn+φ)+bGaussian-damped Sinusoid with linear coefficientsqn=e−γn2(r1sinμn+r2cosμn)+r3Waveform Vector Decomposition→q=r1→Sμ,γ+r2→Cμ,γ+r3→ZGaussian-Damped Sinusoid Fit Error→Ew=→qw−3∑j=1→Uj(→qw⋅→Uj)(3)!Gaussian-Damped Sinusoid EquationsParticle position as a function of time!=#+%sin)*++()+.2π1)−)3()Centroid motion integral!<>=#+6-889):)()%sin)*++()+.2π1)−)3()()Distribution over angular frequency:)()=1<=2π√e−@=−=ABC/EFGCDistribution over angular frequency:)()=1<=(2π)H/Ee−ICJGC@=−=ABCGaussian-damped Sinusoid solution!<>=#+%e−IC(FGK)Csin)3*++()Gaussian-damped Sinusoid standard form!L=%e−MLCsin()N++)+#Gaussian-damped Sinusoid with linear coefficients!L=e−MLC(OHsin)N+OEcos)N)+ORWaveform Vector Decomposition!→=OHT=,MVW⎯⎯+OEY=,MVW⎯⎯+ORZ→Gaussian-Damped Sinusoid Fit ErrorGaussian-Damped Sinusoid Equationsfile:///Users/t6p/Documents/Papers/2016/Parameter Estimation ...\n1 of 33/17/16, 9:48 AM!2Assuming this distribution and performing the integral, the centroid of motion is indeed determined to be in the form of a gaussian damped sinusoid as shown by equation 4. Indeed, BPM waveforms seem to be consistent with the gaussian-damped sinusoid over the first several dozen turns. Figure 1 shows a representative ring BPM waveform of a single injected bunch at nominal chromaticity which fits well to a gaussian-damped sinusoid. \nFigure 1. Representative SNS ring BPM horizontal waveform with both exponentially damped sinusoid and gaussian-damped sinusoid fits. The gaussian-damped sinusoid fit is best in the first sixty-five turns after which it damps away into the noise and other physics appears to dominate. III. GAUSSIAN-DAMPED SINUSOID PARAMETER ESTIMATION Equation 5 shows the general form of the gaussian-damped sinusoid with amplitude A, damping constant ɣ, angular frequency µ, phase φ and orbit distortion (offset) b. The waveform signal qn is evaluated at the zero based turn index n. (4)!Gaussian-Damped Sinusoid EquationsParticle position as a function of timeq=b+Asin(μt+φ)+η2πξ(μ−μ0)Centroid motion integral<q>=b+∞∫-∞dμρ(μ)⎛⎝⎜Asin(μt+φ)+η2πξ(μ−μ0)⎞⎠⎟Distribution over angular frequencyρ(μ)=1σμ2πe−(μ−μ0)2/2σμ2Distribution over angular frequencyρ(μ)=1σμ(2π)1/2e−12σμ2(μ−μ0)2Gaussian-damped Sinusoid solution<q>=b+Ae−(σμt)2/2sin(μ0t+φ)Gaussian-damped Sinusoid standard formqn=Ae−γn2sin(μn+φ)+bGaussian-damped Sinusoid with linear coefficientsqn=e−γn2(r1sinμn+r2cosμn)+r3Waveform Vector Decomposition→q=r1→Sμ,γ+r2→Cμ,γ+r3→ZGaussian-Damped Sinusoid Fit Error→Ew=→qw−3∑j=1→Uj(→qw⋅→Uj)\n(5)!Gaussian-Damped Sinusoid EquationsParticle position as a function of timeq=b+Asin⎛⎝⎜μt+φ⎞⎠⎟+η2πξ⎛⎝⎜μ−μ0⎞⎠⎟Centroid motion integral<q>=b+∞∫-∞dμρ⎛⎝⎜μ⎞⎠⎟⎛⎝⎜Asin⎛⎝⎜μt+φ⎞⎠⎟+η2πξ⎛⎝⎜μ−μ0⎞⎠⎟⎞⎠⎟Distribution over angular frequencyρ(μ)=1σμ(2π)1/2e−12σμ2(μ−μ0)2Gaussian-damped Sinusoid solution<q>=b+Ae−12(σμt)2sin⎛⎝⎜⎜μ0t+φ⎞⎠⎟⎟Gaussian-damped Sinusoid standard formqn=Ae−γn2sin(μn+φ)+bGaussian-damped Sinusoid with linear coefficientsqn=e−γn2(r1sinμn+r2cosμn)+r3Waveform Vector Decomposition→q=r1→Sμ,γ+r2→Cμ,γ+r3→ZGaussian-Damped Sinusoid Fit Error→Ew=→qw−3∑j=1→Uj⎛⎝⎜→qw⋅→Uj⎞⎠⎟Generalized waveform fitJ→⎛⎝⎜⎞⎠⎟!3\nBy inspection there are five parameters to fit and at least that many turns within the waveform will be needed to fit the data. Two of the parameters (damping constant and frequency) are global parameters which are shared in common throughout the ring and the remaining three (amplitude, phase and orbit distortion) are local parameters which vary throughout the ring. Since we are most interested in using the BPM turn by turn signals to measure tune, we are only interested in estimating the global parameters, and this is especially true when fitting over many concurrent waveforms corresponding to the BPMs distributed throughout the ring.  Conventional parameter estimation methods based on least squares fitting to a signal typically require a search over all five parameters. Even worse, if multiple waveforms are to be fit as we wish, the search becomes unnecessarily expensive as there will be two global parameters plus three local parameters per waveform to find. The search typically also requires good initial estimates of the parameters, and good phase and amplitude estimates can be challenging to compute. However, a duality exists between least squares estimation and geometric shortest distance problems [ ‑] that leads to the desired algorithm. The geometric 4viewpoint most naturally provides the mechanics for effectively eliminating the local parameters and reducing the parameter search space to just that of the global parameters.  To help visualize the geometric approach, consider the simpler problem presented in figure 2 in which three turn waveforms are fitted to simple sinusoids with (local) unknown amplitudes and phases and a common (global) unknown frequency to determine. In this case, for a specified frequency, the set of all possible amplitudes and phases form a plane passing through the origin, and the best frequency is the one for which its plane is closest to the measured waveforms represented by points in the three dimensional space. \nFigure 2. Three element (turn) waveforms form a three dimensional space with each measured waveform represented by a point (red dot in figure). Consider the simple model in which the waveforms are represented by sinusoids sharing a common (global) unknown frequency, but each with (local) unknown amplitude and phase. For fixed frequency, the set of amplitudes and phases form a plane passing through the origin. Varying phase forms !4\nellipses and varying amplitude changes the size of these ellipses all lying on a plane. The best fitting frequency is the one for which the corresponding plane passes closest to the measured waveforms (red dots). For the five parameter gaussian-damped sinusoid problem, geometric manipulation can reduce the parameter search space to just the two global parameters (frequency and damping constant). In the geometric perspective, any waveform can be mapped one to one to a single point in a space of dimension equal to the length of the waveform where each element corresponds to a coordinate in the space. Within this space, all perfectly gaussian-damped sinusoid waveforms form a five dimensional subspace (one degree of freedom per parameter). If the two global parameters (µ and ɣ) are fixed, there is a smaller corresponding subspace that is three dimensional and covers all possible values of the local parameters (amplitude, phase and offset). This three dimensional subspace is in fact a three dimensional vector subspace which allows the local parameters to be eliminated from the optimization. To see that this subspace is in fact a vector subspace, we can use the simple trigonometric identity where a sine with a phase can be expressed as the sum of a sine and cosine with the same frequency and appropriate coefficients. So the gaussian-damped sinusoid can be rewritten as follows where the amplitude and phase are absorbed into the coefficients on the sine and cosine terms and the offset, b, is simply renamed to r3 for consistency in the new form. In this form, this equation can be rewritten as a vector equation with coefficients on three constant (for fixed µ and ɣ) basis vectors where the vector elements correspond to the turns indexed by n. S⃗µ,ɣ and C⃗µ,ɣ are vectors with sine like and cosine like gaussian-damped elements that depend only on µ, ɣ and the turn index n and Z⃗ is a constant vector of just ones. For µ strictly in the open interval from zero to π, these three basis vectors are linearly independent. Linear independence can be verified through inspection as follows. The first element of S⃗µ,ɣ is identically zero and that of C⃗µ,ɣ and Z⃗ are identically one so there can be no nonzero factor from S⃗µ,ɣ to C⃗µ,ɣ or Z⃗. Since Z⃗ is a vector of all ones, there can be no common factor from C⃗µ,ɣ to Z⃗. Also, S⃗µ,ɣ and C⃗µ,ɣ cannot be linearly combined to make Z⃗ because summing sine like and cosine like vectors results in another sinusoid vector with a phase shift and there can be no common factor from this resulting sinusoid vector to a constant vector. Gaussian damping cannot compensate the sinusoid terms to make them constant because it provides monotonic damping over the elements. It is expected that a measured BPM turn by turn waveform will only be approximately gaussian-damped sinusoid due to noise and other contributions such as nonlinear effects and coupling not accounted for by our model. Such a waveform will be outside of the five dimensional solution space (and hence the three dimension µ-ɣ subspace for the optimal global (6)!Gaussian-Damped Sinusoid EquationsParticle position as a function of timeq=b+Asin⎛⎝⎜μt+φ⎞⎠⎟+η2πξ⎛⎝⎜μ−μ0⎞⎠⎟Centroid motion integral<q>=b+∞∫-∞dμρ⎛⎝⎜μ⎞⎠⎟⎛⎝⎜Asin⎛⎝⎜μt+φ⎞⎠⎟+η2πξ⎛⎝⎜μ−μ0⎞⎠⎟⎞⎠⎟Distribution over angular frequencyρ(μ)=1σμ(2π)1/2e−12σμ2(μ−μ0)2Gaussian-damped Sinusoid solution<q>=b+Ae−12(σμt)2sin⎛⎝⎜⎜μ0t+φ⎞⎠⎟⎟Gaussian-damped Sinusoid standard formqn=Ae−γn2sin(μn+φ)+bGaussian-damped Sinusoid with linear coefficientsqn=e−γn2(r1sinμn+r2cosμn)+r3Waveform Vector Decomposition→q=r1→Sμ,γ+r2→Cμ,γ+r3→ZGaussian-Damped Sinusoid Fit Error→Ew=→qw−3∑j=1→Uj⎛⎝⎜→qw⋅→Uj⎞⎠⎟Generalized waveform fitJ→⎛⎝⎜⎞⎠⎟(7)!Gaussian-Damped Sinusoid EquationsParticle position as a function of timeq=b+Asin(μt+φ)+η2πξ(μ−μ0)Centroid motion integral<q>=b+∞∫-∞dμρ(μ)⎛⎝⎜Asin(μt+φ)+η2πξ(μ−μ0)⎞⎠⎟Distribution over angular frequencyρ(μ)=1σμ(2π)1/2e−(μ−μ0)2/2σμ2Distribution over angular frequencyρ(μ)=1σμ(2π)1/2e−12σμ2(μ−μ0)2Gaussian-damped Sinusoid solution<q>=b+Ae−12(σμt)2sin(μ0t+φ)Gaussian-damped Sinusoid standard formqn=Ae−γn2sin(μn+φ)+bGaussian-damped Sinusoid with linear coefficientsqn=e−γn2(r1sinμn+r2cosμn)+r3Waveform Vector Decomposition→q=r1→Sμ,γ+r2→Cμ,γ+r3→ZGaussian-Damped Sinusoid Fit Error→Ew=→qw−3∑j=1→Uj(→qw⋅→Uj)\n!5parameters). However, such a waveform should be near the optimal µ-ɣ subspace, and so we identify the best fit by finding the two global parameters (µ and ɣ) that minimize the shortest distance from the waveform to the µ-ɣ subspace for these global parameters. We seek to develop an algorithm to directly compute this distance from a waveform to a µ-ɣ subspace without searching. Such an algorithm would effectively reduce the parameter search space from five down to just the two global parameters. It would allow for efficient global parameter estimation over several concurrent waveforms sharing the same global parameters (e.g. BPM turn by turn waveforms throughout the accumulator ring) by simply computing the root mean square (RMS) of distances over the waveforms to the µ-ɣ subspace and reducing the overall parameter search space by three local parameters per waveform. It will be shown that this measure of error is both convenient using simple vector arithmetic and identical to the usual RMS signal error validating it as the ideal measure for goodness of fit. Geometrically, the shortest distance from a measured waveform vector, q⃗w (with waveform index w), to the µ-ɣ subspace is the magnitude of the component of q⃗w orthogonal to the µ-ɣ subspace. The first step is to compute S⃗µ,ɣ and C⃗µ,ɣ for µ and ɣ recalling that Z⃗ is just a constant vector of ones. We have already established that S⃗µ,ɣ and C⃗µ,ɣ and Z⃗ are linearly independent, but they are not generally orthogonal or normalized. The second step is to form an orthonormal bases, U⃗i (i=1,2,3), from these vectors. Orthogonalization and normalization can be done using standard procedures such as the Modified Gram-Schmidt algorithm involving basic vector arithmetic. The shortest distance is given by the magnitude of the waveform component orthogonal to these bases vectors, |E⃗w| where the error vector is given by the following equation. By construction, it is clear that E⃗w is orthogonal to each of the basis vectors and hence to the µ-ɣ subspace and is the difference between the measured waveform vector and its projection onto this subspace. Thus it must be shortest distance from the measured waveform point to the µ-ɣ subspace. By inspection, it is also clear that E⃗w is identical to the signal error and thus minimizing the magnitude of E⃗w over µ and ɣ is equivalent to minimizing the RMS signal error over all five parameters. We can apply this method to multiple concurrent waveforms using the same bases for each µ and ɣ pair and compute the RMS distance over the waveforms as the appropriate error to minimize. Thus we have achieved our goal of reducing the parameter search space to just the global parameters and eliminating the search over the local parameters we don’t need.  (8)!Gaussian-Damped Sinusoid EquationsParticle position as a function of timeq=b+Asin(μt+φ)+η2πξ(μ−μ0)Centroid motion integral<q>=b+∞∫-∞dμρ(μ)⎛⎝⎜Asin(μt+φ)+η2πξ(μ−μ0)⎞⎠⎟Distribution over angular frequencyρ(μ)=1σμ(2π)1/2e−(μ−μ0)2/2σμ2Distribution over angular frequencyρ(μ)=1σμ(2π)1/2e−12σμ2(μ−μ0)2Gaussian-damped Sinusoid solution<q>=b+Ae−12(σμt)2sin(μ0t+φ)Gaussian-damped Sinusoid standard formqn=Ae−γn2sin(μn+φ)+bGaussian-damped Sinusoid with linear coefficientsqn=e−γn2(r1sinμn+r2cosμn)+r3Waveform Vector Decomposition→q=r1→Sμ,γ+r2→Cμ,γ+r3→ZGaussian-Damped Sinusoid Fit Error→Ew=→qw−3∑j=1→Uj(→qw⋅→Uj)\n!6IV . PERFORMANCE SIMULATIONS Simulations were performed to both verify the algorithm and measure its performance using parameters relevant to the operation of SNS. The simulations were performed using the Open XAL [‑] solver which is a black box optimizer. 5The plot in Figure 3 shows the performance of the geometric parameter estimation comparing single waveform fits and multiple waveform fits based on 40 sources (since SNS has just over 40 BPMs in the ring). Using multiple concurrent waveforms to the fit the tune offers a clear advantage over fitting to a single waveform as expected. Based on statistical noise, one would expect roughly a reduction in error by a factor of the square root of the number of waveforms used in the fitting. \nFigure 3. The log plot shows the relative tune error versus tune for both multiple waveform fits and single waveform fits. The waveforms were generated using a gaussian damping constant of 0.0005, random phases from 0 to 2π, random amplitudes from 5 mm to 15 mm, random offsets ranging over ±10 mm and gaussian signal noise with standard deviation of 1 mm. The solid lines are just trend lines. Figure 4 shows the performance of the geometric fitting algorithm compared with conventional direct least squares parameter estimation. 40 waveforms were used in the fits. In the conventional method, each waveform was fit to the gaussian damped sinusoid using the black box solver to find the five parameters that best fit to the waveform, and then the tune was determined by averaging over the tunes from all the waveforms. The total optimization time was !7Tune Fit Error - Single vs. 40 WaveformsRelative Tune Error1E-071E-061E-051E-041E-031E-021E-01\nTune00.10.20.30.40.5\nSingle Waveform40 Waveformslimited to 1 second wall time for both methods per run with one run per trial tune. The geometric method consistently outperformed the conventional least squares method by roughly two orders of magnitude in the relative error for the same total amount of wall clock time.  \nFigure 4. The log plot shows the relative tune error versus tune for the geometric and the conventional methods. For each point, the two methods were each given 1 second total wall clock optimization time using the black box solver. The waveforms were generated using a gaussian damping constant of 0.0005, random phases from 0 to 2π, random amplitudes from 5 mm to 15 mm, random offsets ranging over ±10 mm and gaussian signal noise with standard deviation of 1 mm. The solid lines are just trend lines.  V . GENERALIZATION OF GEOMETRIC PARAMETER ESTIMATION The algorithm just presented for reducing the parameter search space using geometric parameter estimation can be generalized to fitting equations in a broader form beyond the gaussian-damped sinusoid. Consider K sources of waveforms q⃗k of length N which can be modeled as the linear sum of J linearly independent vector functions F⃗j of M unknown global parameters λm and with coefficients rk,j which are unknown parameters local to the sources. !8Tune Fit Error - Geometric vs. Conventional MethodsRelative Tune Error1E-081E-061E-041E-021E+00\nTune00.10.20.30.40.5\nGeometric MethodConventional MethodThe goal is to find the global parameters which best fit this equation to the real waveform data from all the sources. The geometric perspective allows this problem to be reduced to a search for just the global parameters. For each set of trial global parameters there exists a subspace covering all possible values for the local parameters, and the measure of fit error is taken to be the shortest distance from the measured waveforms to this subspace. Because the local parameters appear just as coefficients on linearly independent vector functions, this subspace is a vector subspace which allows the shortest distance to be computed directly with vector arithmetic. The optimization procedure is a search over the space of global parameters, λm. For each set of trial global parameters, the vector functions F⃗j are to be computed. From these evaluated vector functions, an orthonormal bases is formed using vector arithmetic (e.g. using Modified Gram-Schmidt). The fit error for a specific waveform fit is computed to be the magnitude of the difference between the waveform vector and the projection onto the orthonormal bases. When fitting multiple concurrent waveforms with the same global parameters, compute the overall fit error as the RMS of the individual waveform fit errors. Vary the global parameters λm to minimize the overall fit error. If signal noise variance among the waveform sources is known, then variance weighting can be applied as usual when computing the overall fit error. This solution is equivalent to standard matrix based least squares estimation, but the geometric perspective provides a geometrically intuitive procedure involving vector operations to eliminate the unknown coefficients without the need for explicit matrix inversion. V . CONCLUSIONS Ring BPM turn by turn waveforms can be modeled as five parameter gaussian-damped sinusoids for charged particle betatron motion with chromatic decoherence over turns much less than the synchrotron period. Two of these parameters are global (independent of BPM) and three are local (BPM dependent). A geometric perspective in which a waveform is viewed as a vector in a space of dimension equal to the length of the waveform provides important insight into efficiently solving this problem. Because the equation can be written in a form where the local parameters only appear as linear coefficients, vector arithmetic was used to eliminate the local parameters and reduce the parameter search space to be just the two global parameters.  (9)!→Ew=→qw−3∑j=1→Uj(→qw⋅→Uj)Generalized waveform fit→qk=J∑j=1rkj→Fj(λ1,…,λM)Shortest Distance (generalized waveform error)Ek=‖‖‖‖→qk−J∑j=1→Uj(→qk⋅→Uj)‖‖‖‖\n(10)!Generalized waveform fit→qk=J∑j=1rkj→Fj(λ1,…,λM)Shortest Distance (generalized waveform error)Ek=‖‖‖‖→qk−J∑j=1→Uj(→qk⋅→Uj)‖‖‖‖\n!9The geometric approach was then generalized for parameter reduction of a class of problems in which the parameters to eliminate appear only as linear coefficients on linearly independent vector functions of other parameters to fit. REFERENCES [!] M.A. Plum, “Commissioning Of The Spallation Neutron Source Accelerator Systems,” 1Proceedings of PAC07, Albuquerque, New Mexico, 2007.[!] R.E. Meller et. al., “Decoherence of Kicked Beams,” SSC-N-360, 1987.2[!] The SNS Synchrotron period estimate was provided by internal communication with Michael 3Plum.[!] Gilbert Strang, “Introduction to Applied Mathematics,” Wellesley-Cambridge Press, 1986,  4pp. 34-39, 385-389.[!] T Pelaia II et al., Open XAL Status Report 2015, http://accelconf.web.cern.ch/AccelConf/5IPAC2015/papers/mopwi050.pdf, Proceedings of IPAC 2015, Richmond, V A (2015)\n!10"    },    {        "title": "1604.07053v3.Coupled_Spin_Light_dynamics_in_Cavity_Optomagnonics.pdf",        "content": "Coupled Spin-Light dynamics in Cavity Optomagnonics\nSilvia Viola Kusminskiy,1Hong X. Tang,2and Florian Marquardt1, 3\n1Institute for Theoretical Physics, University Erlangen-Nürnberg, Staudtstraße 7, 91058 Erlangen, Germany\n2Department of Electrical Engineering, Yale University, New Haven, Connecticut 06511, USA\n3Max Planck Institute for the Science of Light, Günther-Scharowsky-Straße 1, 91058 Erlangen, Germany\nExperiments during the past two years have shown strong resonant photon-magnon coupling in\nmicrowave cavities, while coupling in the optical regime was demonstrated very recently for the first\ntime. Unlike with microwaves, the coupling in optical cavities is parametric, akin to optomechanical\nsystems. This line of research promises to evolve into a new field of optomagnonics, aimed at\nthe coherent manipulation of elementary magnetic excitations by optical means. In this work we\nderive the microscopic optomagnonic Hamiltonian. In the linear regime the system reduces to the\nwell-known optomechanical case, with remarkably large coupling. Going beyond that, we study the\noptically induced nonlinear classical dynamics of a macrospin. In the fast cavity regime we obtain\nan effective equation of motion for the spin and show that the light field induces a dissipative term\nreminiscent of Gilbert damping. The induced dissipation coefficient however can change sign on\nthe Bloch sphere, giving rise to self-sustained oscillations. When the full dynamics of the system is\nconsidered, the system can enter a chaotic regime by successive period doubling of the oscillations.\nI. INTRODUCTION\nThe ability to manipulate magnetism has played his-\ntorically an important role in the development of infor-\nmation technologies, using the magnetization of materi-\nals to encode information. Today’s research focuses on\ncontrolling individual spins and spin currents, as well as\nspin ensembles, with the aim of incorporating these sys-\ntems as part of quantum information processing devices.\n[1–4]. In particular the control of elementary excitations\nof magnetically ordered systems –denominated magnons\nor spin waves, is highly desirable since their frequency is\nbroadly tunable (ranging from MHz to THz) [2, 5] while\ntheycanhaveverylonglifetimes, especiallyforinsulating\nmaterials like the ferrimagnet yttrium iron garnet (YIG)\n[6]. The collective character of the magnetic excitations\nmoreoverrendertheserobustagainstlocalperturbations.\nWhereas the good magnetic properties of YIG have\nbeenknownsincethe60s, itisonlyrecentlythatcoupling\nandcontrollingspinwaveswithelectromagneticradiation\nin solid-state systems has started to be explored. Pump-\nprobe experiments have shown ultrafast magnetization\nswitching with light [7–9], and strong photon-magnon\ncoupling has been demonstrated in microwave cavity ex-\nperiments [10–18] –including the photon-mediated cou-\npling between a superconducting qubit and a magnon\nmode [19]. Going beyond microwaves, this points to the\ntantalizing possibility of realizing optomagnonics : the\ncoupled dynamics of magnons and photons in the op-\ntical regime, which can lead to coherent manipulation\nof magnons with light. The coupling between magnons\nand photons in the optical regime differs from that of\nthe microwave regime, where resonant matching of fre-\nquencies allows for a linear coupling: one magnon can be\nconvertedintoaphoton, andviceversa[20–22]. Intheop-\ntical case instead, the coupling is a three-particle process.\nThis accounts for the frequency mismatch and is gener-\nz\nyzx\noptical mode\noptical shift\u0000GSx0magnonmodeopticalmodeGˆ~SˆaabcFigure 1. (Color online) Schematic configuration of the model\nconsidered. (a)Optomagnoniccavitywithhomogeneousmag-\nnetization along the z-axis and a localized optical mode with\ncircular polarization in the y-z-plane. (b) The homogeneous\nmagnon mode couples to the optical mode with strength G.\n(c) Representation of the magnon mode as a macroscopic spin\non the Bloch sphere, whose dynamics is controlled by the cou-\npling to the driven optical mode.\nally called parametric coupling. The mechanism behind\nthe optomagnonic coupling is the Faraday effect, where\nthe angle of polarization of the light changes as it prop-\nagates through a magnetic material. Very recent first\nexperiments in this regime show that this is a promising\nroute, by demonstrating coupling between optical modes\nand magnons, and advances in this field are expected to\ndevelop rapidly [23–27].\nIn this work we derive and analyze the basic op-\ntomagnonic Hamiltonian that allows for the study of\nsolid-state cavity optomagnonics. The parametric op-\ntomagnonic coupling is reminiscent of optomechanicalarXiv:1604.07053v3  [cond-mat.mes-hall]  19 Sep 20162\nmodels. In the magnetic case however, the relevant oper-\natorthatcouplestotheopticalfieldisthespin, insteadof\nthe usual bosonic field representing a mechanical degree\nof freedom. Whereas at small magnon numbers the spin\ncan be replaced by a harmonic oscillator and the ideas of\noptomechanics [28] carry over directly, for general trajec-\ntories of the spin this is not possible. This gives rise to\nrich non-linear dynamics which is the focus of the present\nwork. Parametric spin-photon coupling has been studied\npreviously in atomic ensembles [29, 30]. In this work we\nfocus on solid-state systems with magnetic order and de-\nrive the corresponding optomagnonic Hamiltonian. After\nobtaining the general Hamiltonian, we consider a simple\nmodel which consists of one optical mode coupled to a\nhomogeneous Kittel magnon mode [31]. We study the\nclassical dynamics of the magnetic degrees of freedom\nand find magnetization switching, self-sustained oscilla-\ntions, and chaos, tunable by the light field intensity.\nThe manuscript is ordered as follows. In Sec. (II) we\npresent the model and the optomagnonic Hamiltonian\nwhich is the basis of our work. In Sec. (IIA) we discuss\nbriefly the connection of the optomagnonic Hamiltonian\nderived in this work and the one used in optomechanic\nsystems. In Sec. (IIB) we derive the optomagnonic\nHamiltonian from microscopics, and give an expression\nfor the optomagnonic coupling constant in term of ma-\nterial constants. In Sec. (III) we derive the classical\ncoupled equations of motion of spin and light for a ho-\nmogeneous magnon mode, in which the spin degrees of\nfreedomcanbetreatedasamacrospin. InSec. (IIIA)we\nobtain the effective equation of motion for the macrospin\nin the fast-cavity limit, and show the system presents\nmagnetization switching and self oscillations. We treat\nthe full (beyond the fast-cavity limit) optically induced\nnonlinear dynamics of the macrospin in Sec. (IIIB), and\nfollow the route to chaotic dynamics. In Sec. (IV) we\nsketch a qualitative phase diagram of the system as a\nfunction of coupling and light intensity, and discuss the\nexperimental feasibility of the different regimes. An out-\nlook and conclusions are found in Sec. (V). In the Ap-\npendix we give details of some of the calculations in the\nmain text, present more examples of nonlinear dynamics\nasafunctionofdifferenttuningparameters, andcompare\noptomagnonic vs.optomechanic attractors.\nII. MODEL\nFurther below, we derive the optomagnonic Hamilto-\nnian which forms the basis of our work:\nH=\u0000~\u0001^ay^a\u0000~\n^Sz+~G^Sx^ay^a; (1)\nwhere ^ay(^a) is the creation (annihilation) operator for a\ncavity mode photon. We work in a frame rotating at the\nlaser frequency !las, and \u0001 =!las\u0000!cavis the detuning\nwith respect to the optical cavity frequency !cav. Eq. (1)assumes a magnetically ordered system with (dimension-\nless) macrospin S= (Sx;Sy;Sz)with magnetization axis\nalong ^ z, and a precession frequency \nwhich can be con-\ntrolled by an external magnetic field [32]. The coupling\nbetween the optical field and the spin is given by the\nlast term in Eq. (1), where we assumed (see below) that\nlight couples only to the x\u0000component of the spin as\nshown in Fig. (1). The coefficient Gdenotes the para-\nmetric optomagnonic coupling. We will derive it in terms\nof the Faraday rotation, which is a material-dependent\nconstant.\nA. Relation to optomechanics\nClose to the ground state, for deviations such that\n\u000eS\u001cS(withS=jSj), we can treat the spin in the\nusual way as a harmonic oscillator, ^Sx\u0019p\nS=2(^b+^by),\nwithh\n^b;^byi\n= 1. Then the optomagnonic interaction\n~G^Sx^ay^a\u0019~Gp\nS=2^ay^a(^b+^by)becomes formally equiv-\nalent to the well-known opto mechanical interaction [28],\nwith bare coupling constant g0=Gp\nS=2. All the phe-\nnomena of optomechanics apply, including the “optical\nspring” (here: light-induced changes of the magnon pre-\ncession frequency) and optomagnonic cooling at a rate\n\u0000opt, and the formulas (as reviewed in Ref. [28]) can be\ntaken over directly. All these effects depend on the light-\nenhanced coupling g=g0\u000b, where\u000b=pnphotis the\ncavity light amplitude. For example, in the sideband-\nresolved regime ( \u0014\u001c\n, where\u0014is the optical cavity\ndecay rate) one would have \u0000opt= 4g2=\u0014. Ifg > \u0014,\none enters the strong-coupling regime, where the magnon\nmode and the optical mode hybridize and where coher-\nent state transfer is possible. A Hamiltonian of the form\nof Eq. (1) is also encountered for light-matter interaction\nin atomic ensembles [29], and its explicit connection to\noptomechanics in this case was discussed previously in\nRef. [30]. In contrast to such non-interacting spin en-\nsembles, the confined magnon mode assumed here can\nbe frequency-separated from other magnon modes.\nB. Microscopic magneto-optical coupling G\nIn this section we derive the Hamiltonian presented in\nEq. (1) starting from the microscopic magneto-optical\neffect in Faraday-active materials. The Faraday effect is\ncaptured by an effective permittivity tensor that depends\non the magnetization Min the sample. We restrict our\nanalysis to non-dispersive isotropic media and linear re-\nsponseinthemagnetization, andrelegatemagneticlinear\nbirefringence effects which are quadratic in M(denomi-\nnated the Cotton-Mouton or Voigt effect) for future work\n[5, 33]. In this case, the permittivity tensor acquires an\nantisymmetric imaginary component and can be written3\nas\"ij(M)=\"0(\"\u000eij\u0000ifP\nk\u000fijkMk), where\"0(\") is the\nvacuum (relative) permittivity, \u000fijkthe Levi-Civita ten-\nsorandfamaterial-dependentconstant[33](hereandin\nwhat follows, Latin indices indicate spatial components).\nThe Faraday rotation per unit length\n\u0012F=!fMs\n2cp\"; (2)\ndepends on the frequency !, the vacuum speed of light\nc, and the saturation magnetization Ms. The magneto-\noptical coupling is derived from the time-averaged energy\n\u0016U=1\n4\u0001\ndrP\nijE\u0003\ni(r;t)\"ijEj(r;t), using the complex\nrepresentation of the electric field, (E+E\u0003)=2. Note\nthat \u0016Uis real since \"ijis hermitean [5, 33]. The magneto-\noptical contribution is\n\u0016UMO=\u0000i\n4\"0f\u0002\ndr M(r)\u0001[E\u0003(r)\u0002E(r)]:(3)\nThis couples the magnetization to the spin angular mo-\nmentum density of the light field. Quantization of this\nexpression leads to the optomagnonic coupling Hamilto-\nnian. A similar Hamiltonian is obtained in atomic en-\nsemble systems when considering the electric dipolar in-\nteraction between the light field and multilevel atoms,\nwhere the spin degree of freedom (associated with M(r)\nin our case) is represented by the atomic hyperfine struc-\nture [29]. The exact form of the optomagnonic Hamil-\ntonian will depend on the magnon and optical modes.\nIn photonic crystals, it has been demonstrated that opti-\ncal modes can be engineered by nanostructure patterning\n[34], and magnonic-crystals design is a matter of intense\ncurrent research [3]. The electric field is easily quantized,\n^E(+)(r;t) =P\n\fE\f(r)^a\f(t), where E\f(r)indicates the\n\ftheigenmode of the electric field (eigenmodes are indi-\ncated with Greek letters in what follows). The magne-\ntization requires more careful consideration, since M(r)\ndependsonthelocalspinoperatorwhich, ingeneral, can-\nnot be written as a linear combination of bosonic modes.\nThere are however two simple cases: (i) small deviations\nof the spins, for which the Holstein-Primakoff representa-\ntion is linear in the bosonic magnon operators, and (ii) a\nhomogeneous Kittel mode M(r) =Mwith macrospin S.\nIn the following we treat the homogeneous case, to cap-\nture nonlinear dynamics. From Eq. (3) we then obtain\nthe coupling Hamiltonian ^HMO =~P\nj\f\r^SjGj\n\f\r^ay\n\f^a\r\nwith\nGj\n\f\r=\u0000i\"0fMs\n4~SX\nmn\u000fjmn\u0002\ndrE\u0003\n\fm(r)E\rn(r);(4)\nwhere we replaced Mj=Ms=^Sj=S, withSthe extensive\ntotal spin (scaling like the mode volume). One can diago-\nnalize the hermitean matrices Gj, though generically not\nsimultaneously. In the present work, we treat the con-\nceptually simplest case of a strictly diagonal coupling tosome optical eigenmodes ( Gj\n\f\f6= 0butGj\n\u000b\f= 0). This is\nprecludedonlyiftheopticalmodesarebothtime-reversal\ninvariant ( E\freal-valued) and non-degenerate. In all the\nother cases, a basis can be found in which this is valid.\nFor example, a strong static Faraday effect will turn op-\ntical circular polarization modes into eigenmodes. Al-\nternatively, degeneracy between linearly polarized modes\nimplies we can choose a circular basis.\nConsider circular polarization (R/L) in the y\u0000z-plane,\nsuch thatGxis diagonal while Gy=Gz= 0. Then we\nfind\nGx\nLL=\u0000Gx\nRR=G=1\nSc\u0012F\n4p\"\u0018; (5)\nwhere we used Eq. (2) to express the coupling via the\nFaraday rotation \u0012F, and where \u0018is a dimensionless over-\nlap factor that reduces to 1if we are dealing with plane\nwaves (see App. A). Thus, we obtain the coupling Hamil-\ntonianHMO=~G^Sx(^ay\nL^aL\u0000^ay\nR^aR). This reduces to\nEq. (1) if the incoming laser drives only one of the two\ncircular polarizations.\nThe coupling Ggives the magnon precession frequency\nshift perphoton. It decreases for larger magnon mode\nvolume, in contrast to GS, which describes the overall\nopticalshift for saturated spin ( Sx=S). For YIG,\nwith\"\u00195and\u0012F\u0019200ocm\u00001[5, 35], we obtain\nGS\u00191010Hz(taking\u0018= 1), which can easily become\ncomparable to the precession frequency \n. The ultimate\nlimit for the magnon mode volume is set by the optical\nwavelength,\u0018(1\u0016m)3, which yields S\u00181010. There-\nforeG\u00191Hz, whereas the coupling to a single magnon\nwould be remarkably large: g0=Gp\nS=2\u00190:1MHz.\nThis provides a strong incentive for designing small mag-\nnetic structures, by analogy to the scaling of piezoelectri-\ncal resonators [36]. Conversely, for a macroscopic volume\nof(1mm)3, withS\u00181019, this reduces to G\u001910\u00009Hz\nandg0\u001910Hz.\nIII. SPIN DYNAMICS\nThe coupled Heisenberg equations of motion are ob-\ntainedfromtheHamiltonianinEq. (1)byusing\u0002\n^a;^ay\u0003\n=\n1,h\n^Si;^Sji\n=i\u000fijk^Sk. Wenextfocusontheclassicallimit,\nwhere we replace the operators by their expectation val-\nues:\n_a=\u0000i(GSx\u0000\u0001)a\u0000\u0014\n2(a\u0000\u000bmax)\n_S= (Ga\u0003aex\u0000\nez)\u0002S+\u0011G\nS(_S\u0002S):(6)\nHere we introduced the laser amplitude \u000bmaxand the in-\ntrinsic spin Gilbert-damping [37], characterized by \u0011G,\ndue to phonons and defects ( \u0011G\u001910\u00004for YIG [38]).\nAfter rescaling the fields (see App.. B), we see that the4\nclassical dynamics is controlled by only five dimension-\nless parameters:GS\n\n;G\u000b2\nmax\n\n;\u0001\n\n;\u0014\n\n; \u0011G. These are inde-\npendent of ~as expected for classical dynamics.\nIn the following we study the nonlinear classical dy-\nnamics of the spin, and in particular we treat cases where\nthe spin can take values on the whole Bloch sphere and\ntherefore differs significantly from a harmonic oscilla-\ntor, deviating from the optomechanics paradigm valid\nfor\u000eS\u001cS. The optically induced tilt of the spin\ncan be estimated from Eq. (6) as \u000eS=S =Gjaj2=\n\u0018\nG\u000b2\nmax=\n =B\u000bmax=\n, whereB\u000bmax=G\u000b2\nmaxis an op-\ntically induced effective magnetic field. We would ex-\npect therefore unique optomagnonic behavior (beyond\noptomechanics) for large enough light intensities, such\nthatB\u000bmaxis of the order of or larger than the preces-\nsion frequency \n. We will show however that, in the case\nof blue detuning, even small light intensity can destabi-\nlize the original magnetic equilibrium of the uncoupled\nsystem, provided the intrinsic Gilbert damping is small.\nA. Fast cavity regime\nAs a first step we study a spin which is slow compared\nto the cavity, where G_Sx\u001c\u00142. In that case we can\nabyzx-0.10-0.0500.050.10\n-0.2-0.100.10.2\nFigure 2. (Color online) Spin dynamics (fast cavity limit)\nat blue detuning \u0001 = \n and fixedGS=\n = 2,\u0014=\n = 5,\n\u0011G= 0. The left column depicts the trajectory (green full\nline) of a spin (initially pointing near the north pole) on the\nBloch sphere. The color scale indicates the optical damping\n\u0011opt. The right column shows a stereographic projection of\nthe spin’s trajectory (red full line). The black dotted line\nindicates the equator (invariant under the mapping), while\nthe north pole is mapped to infinity. The stream lines of the\nspin flow are also depicted (blue arrows). (a) Magnetization\nswitching behavior for light intensity G\u000b2\nmax=\n = 0:36. (b)\nLimit cycle attractor for larger light intensity G\u000b2\nmax=\n =\n0:64.expand the field a(t)in powers of _Sxand obtain an ef-\nfective equation of motion for the spin by integrating out\nthe light field. We write a(t) =a0(t) +a1(t) +:::, where\nthe subscript indicates the order in _Sx. From the equa-\ntion fora(t), we find that a0fulfills the instantaneous\nequilibrium condition\na0(t) =\u0014\n2\u000bmax1\n\u0014\n2\u0000i(\u0001\u0000GSx(t));(7)\nfrom which we obtain the correction a1:\na1(t) =\u00001\n\u0014\n2\u0000i(\u0001\u0000GSx)@a0\n@Sx_Sx:(8)\nTo derive the effective equation of motion for the spin,\nwe replacejaj2\u0019ja0j2+a\u0003\n1a0+a\u0003\n0a1in Eq. (6) which\nleads to\n_S=Be\u000b\u0002S+\u0011opt\nS(_Sxex\u0002S) +\u0011G\nS(_S\u0002S):(9)\nHereBe\u000b=\u0000\nez+Bopt, where Bopt(Sx) =Gja0j2ex\nacts as an optically induced magnetic field. The second\nterm is reminiscent of Gilbert damping, but with spin-\nvelocity component only along ex. Both the induced field\nBoptand dissipation coefficient \u0011optdepend explicitly on\nthe instantaneous value of Sx(t):\nBopt=G\n[(\u0014\n2)2+ (\u0001\u0000GSx)2]\u0010\u0014\n2\u000bmax\u00112\nex(10)\n\u0011opt=\u00002G\u0014SjBoptj(\u0001\u0000GSx)\n[(\u0014\n2)2+ (\u0001\u0000GSx)2]2:(11)\nThis completes the microscopic derivation of the optical\nLandau-Lifshitz-Gilbert equation for the spin, an impor-\ntant tool to analyze effective spin dynamics in different\ncontexts [39]. We consider the nonlinear adiabatic dy-\nnamics of the spin governed by Eq. (9) below. Two\ndistinct solutions can be found: generation of new sta-\nble fixed points (magnetic switching) and optomagnonic\nlimit cycles (self oscillations).\nGiven our Hamiltonian (Eq. (1)), the north pole is sta-\nble in the absence of optomagnonic coupling – the se-\nlection of this state is ensured by the intrinsic damping\n\u0011G>0. By driving the system this can change due to\nthe energy pumped to (or absorbed from) the spin, and\nthe new equilibrium is determined by Be\u000band\u0011opt, when\n\u0011optdominates over \u0011G. Magnetic switching refers to the\nrotation of the macroscopic magnetization by \u0018\u0019, to a\nnew fixed point near the south pole in our model. This\ncan be obtained for blue detuning \u0001>0, in which case\n\u0011optis negative either on the whole Bloch sphere (when\n\u0001> GS) or on a certain region, as shown in Fig. (2)a.\nSimilar results were obtained in the case of spin opto-\ndynamics for cold atoms systems [30]. The possibility of\nswitching the magnetization direction in a controlled way\nis of great interest for information processing with mag-\nnetic memory devices, in which magnetic domains serve5\nas information bits [7–9]. Remarkably, we find that for\nblue detuning, magnetic switching can be achieved for\narbitrary small light intensities in the case of \u0011G= 0.\nThis is due to runaway solutions near the north pole for\n\u0001>0, as discussed in detail in App. C. In physical sys-\ntems, the threshold of light intensity for magnetization\nswitching will be determined by the extrinsic dissipation\nchannels.\nFor higher intensities of the light field, limit cycle at-\ntractors can be found for j\u0001j<GS, where the optically\ninduced dissipation \u0011optcan change sign on the Bloch\nsphere (Fig. (2)b). The combination of strong nonlinear-\nity and a dissipative term which changes sign, leads the\nsystem into self sustained oscillations. The crossover be-\ntween fixed point solutions and limit cycle attractors is\ndetermined by a balance between the detuning and the\nlight intensity, as discussed in App. C. Limit cycle at-\ntractors require B\u000bmax=\n>j\u0001j=GS(note that from (11)\nBopt\u0018B\u000bmaxif\u0014\u001d(\u0001\u0000GS)).\nWe note that for both examples shown in Fig. (2), for\nthe chosen parameters we have \u0011opt\u001d\u0011Gin the case of\nYIG, and hence taking \u0011G= 0is a very good approx-\nimation. More generally, from Eqs. (10) we estimate\n\u0011opt\u0018GSB opt=\u00143and therefore we can safely neglect\n\u0011Gfor(\u000bmaxG)2S\u001d\u0011G\u00143. The qualitative results (limit\ncycle, switching) survive up to \u0011opt&\u0011G, although quan-\ntitatively modified as \u0011Gis increased: for example, the\nsize of the limit cycle would change, and there would be\na threshold intensity for switching.\nB. Full nonlinear dynamics\nThe nonlinear system of Eq. (6) presents even richer\nbehavior when we leave the fast cavity regime. For limit\ncycles near the north pole, when \u000eS\u001cS, the spin is\nwell approximated by a harmonic oscillator, and the dy-\nnamics is governed by the attractor diagram established\nfor optomechanics [40]. In contrast, larger limit cycles\nwill display novel features unique to optomagnonics, on\nwhich we focus here.\nBeyond the fast cavity limit, we can no longer give\nanalytical expressions for the optically induced magnetic\nfield and dissipation. Moreover, we can not define a coef-\nficient\u0011optsince an expansion in _Sxis not justified. We\ntherefore resort to numerical analysis of the dynamics.\nFig. (3) shows a route to chaos by successive period dou-\nbling, upon decreasingthe cavity decay \u0014. This route can\nbe followed in detail as a function of any selected param-\neter by plotting the respective bifurcation diagram. This\nis depicted in Fig. (4). The plot shows the evolution\nof the attractors of the system as the light intensity is\nincreased. The figure shows the creation and expansion\nof a limit cycle from a fixed point near the south pole,\nfollowed by successive period doubling events and finally\nentering into a chaotic region. At high intensities, a limit\nt⌦\nGSz⌦\nyzxabc\ndeIncreasing period of the limit cycle\nChaos2⇡\u00003\u00000.5\n\u000013\u00002.52\n\u00003\u00002\u00001Figure 3. (Color Online) Full non-linear spin dynamics and\nroute to chaos for GS=\n = 3andG\u000b2\nmax=\n = 1(\u0011G= 0).\nThe system is blue detuned by \u0001 = \n and the dynamics,\nafter a transient, takes place in the southern hemisphere. The\nsolid red curves represent the spin trajectory after the initial\ntransient, on the Bloch sphere for (a) \u0014=\n = 3, (b)\u0014=\n = 2,\n(c)\u0014=\n = 0:9, (d)\u0014=\n = 0:5. (f)Szprojection as a function\nof time for the chaotic case \u0014=\n = 0:5.\nspin projectionGSz/⌦chaoslimit cycleperioddoublingcoexistence\n1.0laser amplitudepG|↵max|2/⌦210-1-21.5\nFigure 4. Bifurcation density plot for GS=\n = 3and\u0014=\n = 1\nat\u0001 = \n(\u0011G= 0), as a function of light intensity. We plot\ntheSzvalues attained at the turning points ( _Sz= 0). For\nother possible choices ( eg. _Sx= 0) the overall shape of\nthe bifurcation diagram is changed, but the bifurcations and\nchaotic regimes remain at the same light intensities. For the\nplot, 30 different random initial conditions were taken.\ncycle can coexist with a chaotic attractor. For even big-\nger light intensities, the chaotic attractor disappears and\nthesystemprecessesaroundthe exaxis, asaconsequence\nof the strong optically induced magnetic field. Similar bi-\nfurcation diagrams are obtained by varying either GS=\nor the detuning \u0001=\n(see App. D).6\n11\n2\nxy-plane limit cycles\"optomechanics\"chaosoptomagnonic limit cyclesswitching\nchaos yz plane limit cycles⌦GS\nxy-plane limit cyclesB↵max⌦\nFigure 5. Phase diagram for blue detuning with \u0001 = \n, as a\nfunction of the inverse coupling strength \n=GSand the op-\ntically induced field B\u000bmax=\n =G\u000b2\nmax=\n. Boundaries are\nqualitative. Switching, in white, refers to a fixed point solu-\ntion with the spin pointing near the south pole. Limit cycles\nin thexyplane are shaded in blue, and they follow the op-\ntomechanical attractor diagram discussed in Ref. [40]. For\nhigherB\u000bmax, chaos can ensue. Orange denotes the param-\neter space in which limit cycles deviate markedly from op-\ntomechanical predictions. These are not in the xyplane and\nalso undergo period doubling leading to chaos. In red is de-\npicted the area where pockets of chaos can be found. For\nlargeB\u000bmax=\n, the limit cycles are in the yzplane. In the\ncase of red detuning \u0001 =\u0000\n, the phase diagram remains as\nis, except that instead of switching there is a fixed point near\nthe north pole.\nIV. DISCUSSION\nWe can now construct a qualitative phase diagram for\nour system. Specifically, we have explored the qualitative\nbehavior (fixed points, limit cycles, chaos etc.) as a func-\ntion of optomagnonic coupling and light intensity. These\nparameters can be conveniently rescaled to make them\ndimensionless. We chose to consider the ratio of magnon\nprecessionfrequencytocoupling, intheform \n=GS. Fur-\nthermore, we express the light intensity via the maxi-\nmal optically induced magnetic field B\u000bmax=G\u000b2\nmax.\nThe dimensionless coupling strength, once the material\nof choice is fixed, can be tuned viaan external magnetic\nfieldwhichcontrolstheprecessionfrequency \n. Thelight\nintensity can be controlled by the laser.\nWe start by considering blue detuning, this is shown\nin Fig. (5). The “phase diagram” is drawn for \u0001 = \n,\nand we set \u0014= \nand\u0011G= 0. We note that some of the\ntransitions are rather crossovers (“optomechanical limit\ncycles” vs.“optomagnonic limit cycles”). In addition, the\nother “phase boundaries” are only approximate, obtained\nfrom direct inspection of numerical simulations. These\nare not intended to be exact, and are qualitatively validfor departures of the set parameters, if not too drastic\n– for example, increasing \u0014will lead eventually to the\ndisappearance of the chaotic region.\nAs the diagram shows, there is a large range of pa-\nrameters that lead to magnetic switching, depicted in\nwhite. This area is approximately bounded by the con-\nditionB\u000bmax=\n.\u0001=GS, which in Fig. (5) corresponds\nto the diagonal since we took \u0001 = \n. This condition\nis approximate since it was derived in the fast cavity\nregime, see App. C. As discussed in Sec. III, magnetic\nswitching should be observable in experiments even for\nsmall light intensity in the case of blue detuning, pro-\nvided that all non-optical dissipation channels are small.\nThe caveat of low intensity is a slow switching time. For\nB\u000bmax=\n&\u0001=GS, the system can go into self oscilla-\ntions and even chaos. For optically induced fields much\nsmaller than the external magnetic field, B\u000bmax\u001c\nwe\nexpect trajectories of the spin in the xyplane, precessing\naround the external magnetic field along ezand therefore\nthe spin dynamics (after a transient) is effectively two-\ndimensional. This is depicted by the blue-shaded area\nin Fig.(5). These limit cycles are governed by the op-\ntomechanical attractor diagram presented in Ref. [40],\nas we show in App. E. There is large parameter region\nin which the optomagnonic limit cycles deviate from the\noptomechanical attractors. This is marked by orange in\nFig.(5). As the light intensity is increased, for \n=GS\u001c1\nthe limit cycles remain approximately confined to the xy\nplane but exhibit deviations from optomechanics. This\napproximate confinement of the trajectories to the xy\nplane at large B\u000bmax=\n(B\u000bmax=\n&0:5for\u0001 = \n)\ncan be understood qualitatively by looking at the ex-\npression of the induced magnetic field Boptdeduced in\nthe fast cavity limit, Eq. (10). Since we consider \u0001 = \n,\n\n=GS\u001c1impliesGS\u001d\u0001. In this limit, Bopt=\ncan\nbecome very small and the spin precession is around the\nezaxis. For moderate B\u000bmax=\nand\n=GS, the limit cy-\ncles are tilted and precessing around an axis determined\nby the effective magnetic field, a combination of the opti-\ncalinducedfieldandtheexternalmagneticfield. Bluede-\ntuning causes these limit cycles to occur in the southern\nhemisphere. Period doubling leads eventually to chaos.\nThe region where pockets of chaos can be found is rep-\nresented by red in the phase diagram. For large light\nintensity, such that B\u000bmax\u001d\n, the optical field domi-\nnates and the effective magnetic field is essentially along\ntheexaxis. The limit cycle is a precession of the spin\naround this axis.\nAccording to our results optomagnonic chaos is at-\ntained for values of the dimensionless coupling GS=\n\u0018\n1\u000010and light intensities G\u000b2\nmax=\n\u00180:1\u00001. This\nimplies a number of circulating photons similar to the\nnumber of locked spins in the material, which scales with\nthe cavity volume. This therefore imposes a condition\non the minimum circulating photon density in the cavity.\nFor YIG with characteristic frequencies \n\u00181\u000010GHz,7\ntheconditiononthecouplingiseasilyfulfilled(remember\nGS= 10GHz as calculated above). However the condi-\ntion on the light intensity implies a circulating photon\ndensity of\u0018108\u0000109photons/\u0016m3which is outside\nof the current experimental capabilities, limited by the\npower a typical microcavity can support (around \u0018105\nphotons/\u0016m3). On the other hand, magnetic switching\nand self-sustained oscillations of the optomechanical type\n(but taking place in the southern hemisphere) can be at-\ntained for low powers, assuming all external dissipation\nchannels are kept small. While self-sustained oscillations\nand switching can be reached in the fast-cavity regime,\nmorecomplexnonlinearbehaviorsuchasperioddoubling\nand chaos requires approaching sideband resolution. For\nYIG the examples in Figs. 3, 4 correspond to a preces-\nsion frequency \n\u00193\u0001109Hz(App. D), whereas \u0014can be\nestimated to be\u00181010Hz, taking into account the light\nabsorption factor for YIG ( \u00180:3cm\u00001) [35].\nFor red detuning \u0001<0, the regions in the phase dia-\ngram remain the same, except that instead of magnetic\nswitching, the solutions in this parameter range are fixed\npoints near the north pole. This can be seen by the sym-\nmetry of the problem: exchanging \u0001! \u0000 \u0001together\nwithex!\u0000exandez!\u0000ezleaves the problem un-\nchanged. The limit cycles and trajectories follow also\nthis symmetry, and in particular the limit cycles in the\nxyplane remain invariant.\nV. OUTLOOK\nThe observation of the spin dynamics predicted here\nwill be a sensitive probe of the basic cavity optomagnonic\nmodel, beyond the linear regime. Our analysis of the op-\ntomagnonic nonlinear Gilbert damping could be general-\nized to more advanced settings, leading to optomagnonic\nreservoir engineering (e.g. two optical modes connected\nby a magnon transition). Although the nonlinear dy-\nnamics presented here requires light intensities outside of\nthe current experimental capabilities for YIG, it should\nbe kept in mind that our model is the simplest case for\nwhich highly non-linear phenomena is present. Increas-\ning the model complexity, for example by allowing for\nmultiple-mode coupling, could result in a decreased light\nintensity requirement. Materials with a higher Faraday\nconstantwouldbealsobeneficial. Inthisworkwefocused\non the homogeneous Kittel mode. It will be an interest-\ning challenge to study the coupling to magnon modes at\nfinite wavevector, responsible for magnon-induced dissi-\npation and nonlinearities under specific conditions [41–\n43]. The limit cycle oscillations can be seen as “opto-\nmagnonic lasing”, analogous to the functioning principle\nof a laser where energy is pumped and the system set-\ntles in a steady state with a characteristic frequency, and\nalso discussed in the context of mechanics (“cantilaser”\n[44]). These oscillations could serve as a novel sourceof traveling spin waves in suitable geometries, and the\nsynchronization of such oscillators might be employed to\nimprove their frequency stability. We may see the de-\nsign of optomagnonic crystals and investigation of opto-\nmagnonic polaritons in arrays. In addition, future cav-\nity optomagnonics experiments will allow to address the\ncompletely novel regime of cavity-assisted coherent op-\ntical manipulation of nonlinear magnetic textures, like\ndomain walls, vortices or skyrmions, or even nonlinear\nspatiotemporal light-magnon patterns. In the quantum\nregime, prime future opportunities will be the conversion\nof magnons to photons or phonons, the entanglement be-\ntween these subsystems, and their applications to quan-\ntum communication and sensitive measurements.\nWe note that different aspects of optomagnonic sys-\ntems have been investigated in a related work done\nsimultaneously [45]. Our work was supported by an\nERC-StG OPTOMECH and ITN cQOM. H.T. ac-\nknowledges support by the Defense Advanced Research\nProjectsAgency(DARPA)MicrosystemsTechnologyOf-\nfice/Mesodynamic Architectures program (N66001-11-1-\n4114) and an Air Force Office of Scientific Research\n(AFOSR) Multidisciplinary University Research Initia-\ntive grant (FA9550-15-1-0029).\n[1] Y. Tserkovnyak, A. Brataas, G. E. W. Bauer, and B. I.\nHalperin, Reviews of Modern Physics 77, 1375 (2005).\n[2] A. V. Chumak, V. I. Vasyuchka, A. A. Serga, and\nB. Hillebrands, Nature Physics 11, 453 (2015).\n[3] M. Krawczyk and D. Grundler, Journal of physics. Con-\ndensed matter : an Institute of Physics journal 26,\n123202 (2014).\n[4] G. Kurizki, P. Bertet, Y. Kubo, K. Mølmer, D. Pet-\nrosyan, P. Rabl, and J. Schmiedmayer, Proceedings of\nthe National Academy of Sciences 112, 3866 (2015).\n[5] D. D. Stancil and A. Prabhakar, Spin Waves (Springer,\n2009).\n[6] A. A. Serga, A. V. Chumak, and B. Hillebrands, Journal\nof Physics D: Applied Physics 43, 264002 (2010).\n[7] C. D. Stanciu, F. Hansteen, A. V. Kimel, A. Kirilyuk,\nA. Tsukamoto, A. Itoh, and T. Rasing, Physical Review\nLetters99, 047601 (2007).\n[8] A. Kirilyuk, A. V. Kimel, and T. Rasing, Reviews of\nModern Physics 82, 2731 (2010).\n[9] C.-H. Lambert, S. Mangin, B. S. D. C. S. Varaprasad,\nY. K. Takahashi, M. Hehn, M. Cinchetti, G. Malinowski,\nK.Hono, Y.Fainman, M.Aeschlimann, andE.E.Fuller-\nton, Science 345, 1337 (2015).\n[10] H.Huebl, C.W.Zollitsch, J.Lotze, F.Hocke, M.Greifen-\nstein, A. Marx, R. Gross, and S. T. B. Goennenwein,\nPhysical Review Letters 111, 127003 (2013).\n[11] X. Zhang, C.-l. Zou, L. Jiang, and H. X. Tang, Physical\nReview Letters 113, 156401 (2014).\n[12] Y. Tabuchi, S. Ishino, T. Ishikawa, R. Yamazaki, K. Us-\nami, and Y. Nakamura, Physical Review Letters 113,\n083603 (2014).\n[13] M. Goryachev, W. G. Farr, D. L. Creedon, Y. Fan,8\nM. Kostylev, and M. E. Tobar, Physical Review Applied\n2, 054002 (2014).\n[14] J.Bourhill, N.Kostylev, M.Goryachev, D.Creedon, and\nM. Tobar, (2015).\n[15] J. A. Haigh, N. J. Lambert, A. C. Doherty, and A. J.\nFerguson, Physical Review B 91, 104410 (2015).\n[16] L. Bai, M. Harder, Y. P. Chen, X. Fan, J. Q. Xiao, and\nC.-M. Hu, Physical Review Letters 114, 227201 (2015).\n[17] D. Zhang, X.-M. Wang, T.-F. Li, X.-Q. Luo, W. Wu,\nF. Nori, and J. You, npj Quantum Information 1, 15014\n(2015).\n[18] N. J. Lambert, J. A. Haigh, and A. J. Ferguson, Journal\nof Applied Physics 117, 41 (2015).\n[19] Y. Tabuchi, S. Ishino, A. Noguchi, T. Ishikawa, R. Ya-\nmazaki, K. Usami, and Y. Nakamura, Science 349, 405\n(2015), arXiv:1410.3781.\n[20] O. Ì. O. Soykal and M. E. Flatté, Physical Review Letters\n104, 077202 (2010).\n[21] Y. Cao, P. Yan, H. Huebl, S. T. B. Goennenwein, and\nG. E. W. Bauer, Physical Review B 91, 094423 (2015).\n[22] B. Zare Rameshti, Y. Cao, and G. E. W. Bauer, Physical\nReview B 91, 214430 (2015).\n[23] X. Zhang, N. Zhu, C.-L. Zou, and H. X. Tang, (2015),\narXiv:1510.03545.\n[24] J. A. Haigh, S. Langenfeld, N. J. Lambert, J. J. Baum-\nberg, A. J. Ramsay, A. Nunnenkamp, and A. J. Fergu-\nson, Physical Review A 92, 063845 (2015).\n[25] X. Zhang, C.-L. Zou, L. Jiang, and H. X. Tang, Science\nAdvances 2, 1501286 (2016).\n[26] R. Hisatomi, A. Osada, Y. Tabuchi, T. Ishikawa,\nA. Noguchi, R. Yamazaki, K. Usami, and Y. Nakamura,\nPhysical Review B 93, 174427 (2016).\n[27] A. Osada, R. Hisatomi, A. Noguchi, Y. Tabuchi, R. Ya-\nmazaki, K. Usami, M. Sadgrove, R. Yalla, M. Nomura,\nand Y. Nakamura, Physical Review Letters 116, 223601\n(2016).\n[28] M. Aspelmeyer, T. J. Kippenberg, and F. Marquardt,\nReviews of Modern Physics 86, 1391 (2014).[29] K.Hammerer, A.S.Soerensen, andE.S.Polzik,Reviews\nof Modern Physics 82, 1041 (2010).\n[30] N. Brahms and D. M. Stamper-Kurn, Physical Review A\n82, 041804 (2010).\n[31] C. Kittel, Physical Review 73, 155 (1948).\n[32] Note that this frequency however depends on the mag-\nnetic field insidethe sample, and hence it depends on its\ngeometry and the corresponding demagnetization fields.\n[33] L. D. Landau and E. M. Lifshitz, Electrodynamics of con-\ntinuous media , second edi ed., edited by E. M. Lifshitz\nand L. P. Pitaevskii (Pergamon Press, 1984).\n[34] J. D. Joannopoulos, S. G. Johnson, J. N. Winn, and\nR. D. Meade, Photonic Crystals , 2nd ed. (Princeton Uni-\nversity Press, 2008).\n[35] M. J. Weber, CRC Handbook of Laser Science and Tech-\nnology Supplement 2: Optical Materials (CRC Press,\n1994).\n[36] L. Fan, K. Y. Fong, M. Poot, and H. X. Tang, Nature\ncommunications 6, 5850 (2015).\n[37] T. Gilbert, IEEE Transactions on Magnetics 40, 3443\n(2004).\n[38] In the magnetic literature, \u0011Gis denoted as \u000b[5].\n[39] Y. Tserkovnyak, A. Brataas, and G. E. W. Bauer, Phys-\nical review letters 88, 117601 (2002), 0110247.\n[40] F.Marquardt,J.G.E.Harris, andS.M.Girvin,Physical\nReview Letters 96, 103901 (2006).\n[41] A. M. Clogston, H. Suhl, L. R. Walker, and P. W. An-\nderson, J. Phys. Chem. Solids 1, 129 (1956).\n[42] H. Suhl, Journal of Physics and Chemistry of Solids 1,\n209 (1957).\n[43] G. Gibson and C. Jeffries, Physical Review A 29, 811\n(1984).\n[44] I. Bargatin and M. L. Roukes, Physical review letters 91,\n138302 (2003).\n[45] T. Liu, X. Zhang, H. X. Tang, and M. E. Flatté, (2016),\narXiv:1604.07052.\nAppendix A: Optomagnonic coupling Gfor plane waves\nIn this section we calculate explicitly the optomagnonic coupling presented in Eq.. (5) for the case of plane\nwaves mode functions for the electric field. We choose for definiteness the magnetization axis along the ^ zaxis, and\nconsider the case Gx\f\r6= 0. The Hamiltonian HMOis then diagonal in the the basis of circularly polarized waves,\neR=L=1p\n2(ey\u0007iez). The rationale behind choosing the coupling direction perpendicular to the magnetization axis,\nis to maximize the coupling to the magnon mode, that is to the deviations of the magnetization with respect to the\nmagnetization axis. The relevant spin operator is therefore ^Sx, which represents the flipping of a spin. In the case of\nplane waves, we quantize the electric field according to ^E+(\u0000)(r;t) = +(\u0000)iP\njejq\n~!j\n2\"0\"V^a(y)\nj(t)e+(\u0000)ikj\u0001r;whereV\nis the volume of the cavity, kjthe wave vector of mode jand we have identified the positive and negative frequency\ncomponents of the field as E!^E+,E\u0003!^E\u0000. The factor of \"0\"in the denominator ensures the normalization\n~!j=\"0\"hjj\u0001\nd3rjE(r)j2jji\u0000\"0\"h0j\u0001\nd3rjE(r)j2j0i, which corresponds to the energy of a photon in state jjiabove\nthe vacuumj0i. For two degenerate (R/L) modes at frequency !, using Eq. (2) we see that the frequency dependence\ncancels out and we obtain the simple form for the optomagnonic Hamiltonian HMO=~G^Sx(^ay\nL^aL\u0000^ay\nR^aR)with\nG=1\nSc\u0012F\n4p\". Therefore the overlap factor \u0018= 1in this case.9\nAppendix B: Rescaled fields and linearized dynamics\nTo analyze Eq. (6) it is convenient to re-scale the fields such that a=\u000bmaxa0,S=SS0and measure all times and\nfrequencies in \n. We obtain the rescaled equations of motion (time-derivatives are now with respect to t0= \nt)\n_a0=\u0000i(GS\n\nS0\nx\u0000\u0001\n\n)a0\u0000\u0014\n2\n(a0\u00001) (B1)\n_S0=\u0012G\u000b2\nmax\n\nja0j2ex\u0000ez\u0013\n\u0002S0+\u0011G\nS\u0010\n_S0\u0002S0\u0011\n(B2)\nIf we linearize the spin-dynamics (around the north-pole, e.g.), we should recover the optomechanics behavior. In\nthis section we ignore the intrinsic Gilbert damping term. We set approximately S0\u0019(S0\nx;S0\ny;1)Tand from Eq. (B1)\nwe obtain\n_S0\nx=S0\ny (B3)\n_S0\ny=\u0000G\u000b2\nmax\n\nja0j2\u0000S0\nx (B4)\nWe can now choose to rescale further, via S0\nx=\u0010\n\u000bmax=p\nS\u0011\nS00\nxand likewise for S0\ny. We obtain the following\nspin-linearized equations of motion:\n_S00\nx=S00\ny (B5)\n_S00\ny=\u0000Gp\nS\u000bmax\n\nja0j2\u0000S00\nx (B6)\n_a0=\u0000i(Gp\nS\u000bmax\n\nS00\nx\u0000\u0001\n\n)a0\u0000\u0014\n2\n(a0\u00001) (B7)\nThis means that the number of dimensionless parameters has been reduced by one, since the two parameters initially\ninvolving G, S, and\u000bmaxhave all been combined into\nGp\nS\u000bmax\n\n(B8)\nIn other words, for S0\nx;y=Sx;y=S\u001c1, the dynamics should only depend on this combination, consistent with the\noptomechanicalanalogyvalidinthisregimeasdiscussedinthemaintext(wherewearguedbasedontheHamiltonian).\nAppendix C: Switching in the fast cavity limit\nFrom Eq. (9) in the weak dissipation limit ( \u0011G\u001c1) we obtain\n_Sx=\nSy\n_Sy=\u0000SzBopt\u0000\nSx\u0000\u0011opt\nS_SxSz;\nfrom where we obtain an equation of motion for Sx. We are interested in studying the stability of the north pole once\nthe driving is turned on. Hence we set Sz=S,\nSx=\u0000\nSBopt\u0000\n2Sx\u0000\u0011opt\n_Sx;\nand we consider small deviations \u000eSxofSxfrom the equilibrium position that satisfies S0\nx=\u0000SBopt=\n, whereBopt\nis evaluated at S0\nx. To linear order we obtain\n\u000eSx=\u0000\n\u0012\n\n +S@Bopt\n@Sx\u0013\n\u000eSx+ 2GS\u0014\nBopt(\u0001 +GSB opt=\n)\nh\n(\u0014=\n)2+ (\u0001 +GSB opt=\n)2i2_\u000eSx:\nWe see that the dissipation coefficient for blue detuning ( \u0001>0) is always negative, giving rise to runaway solutions.\nTherefore the solutions near the north pole are always unstable under blue detuning, independent of the light intensity.10\nThese trajectories run to a fixed point near the south pole, which accepts stable solutions for \u0001>0(switching) or to\na limit cycle. Near the south pole, Sz=\u0000S,S0\nx=SBopt=\nand\n\u000eSx=\u0000\n\u0012\n\n\u0000S@Bopt\n@Sx\u0013\n\u000eSx\u00002GS\u0014\nBopt(\u0001\u0000GSB opt=\n)\nh\n(\u0014=\n)2+ (\u0001\u0000GSB opt=\n)2i2_\u000eSx:\nTherefore for \u0001> GSB opt=\nthere are stable fixed points, while in the opposite case there are also runaway\nsolutions that are caught in a limit cycle. For red detuning, \u0001!\u0000 \u0001and the roles of south and north pole are\ninterchanged.\nAppendix D: Nonlinear dynamics\nIn this section we give more details on the full nonlinear dynamics described in the main text. In Figs. 3 and (4) of\nthe main text we chose a relative coupling GS=\n = 3, around which a chaotic attractor is found. With our estimated\nGS\u00191010Hzfor YIG, this implies a precession frequency \n\u00193\u0001109Hz. In Fig. (3) the chaotic regime is reached at\n\u0014\u0019\n=2withG\u000b2\nmax=\n = 1, which implies \u000b2\nmax\u0019S=3, that is, a number of photons circulating in the (unperturbed)\n2.53\n-0.5\n-1Spin projectionSz/S\nGS/⌦Normalized coupling\nFigure 6. (Color online) Bifurcation density plot for G\u000b2\nmax=\n = 1and\u0014=\n = 1at\u0001 = \n(\u0011G= 0), as a function of the\nrelative coupling strength GS=\n. The dotted blue line indicates GS=\n = 3, for comparison with Fig. (4). As in the main\ntext, the points (obtained after the transient) are given by plotting the values of Szattained whenever the trajectory fulfills\nthe turning point condition _Sz= 0, for 20 different random initial conditions.11\n1.01.50-1-2Spin projectionGSz/⌦\n\u0000/⌦Detuning\nFigure 7. (Color online) Bifurcation density plot for GS=\n = 3,G\u000b2\nmax=\n = 1and\u0014=\n = 1(\u0011G= 0), as a function of the\ndetuning \u0001=\n. The dotted blue line indicates \u0001=\n = 1, for comparison with Fig. (4).\ncavity of the order of the number of locked spins and hence scaling with the cavity volume. Bigger values of the cavity\ndecay rate are allowed for attaining chaos at the same frequency, at the expense of more photons in the cavity, as can\nbe deduced from Fig. (4) where we took \u0014= \n. On the other hand we can think of varying the precession frequency\n\nby an applied external magnetic field and explore the nonlinearities by tuning GS=\nin this way (note that GSis\na material constant). This is done in Fig. (6). Alternatively, the nonlinear behavior can be controlled by varying the\ndetuning \u0001, as shown in Fig. (7).\nAppendix E: Relation to the optomechanical attractors\nIn this appendix we show that the optomagnonic system includes the higher order nonlinear attractors found in\noptomechanics as a subset in parameter space.\nIn optomechanics, the high order nonlinear attractors are self sustained oscillations with amplitudes Asuch that\nthe optomechanical frequency shift GAis a multiple of the mechanical frequency \n. Translating to our case, this\nmeansG\u000eS\u0018n\n. Since\u000eS=S\u0018Gj\u000bmaxj2=\n =B\u000bmax=\nwe obtain the condition\nGS\n\nB\u000bmax\n\n\u0018n (E1)\nfor observing these attractors. We can vary B\u000bmaxaccording to Eq. (E1). For \n=GS\u001c1we are in the limit of small\nB\u000bmax=\nand we expect limit cycles precessing along ezas discussed in Sec. (IV). In Fig. 8 the attractor diagram12\n5101520\n2015105GS/⌦GSx/⌦\n10302020301040GS/⌦GSx/⌦\nFigure 8. Attractor diagram for \u0001 = 1:5\nand\u0014=\n = 1with condition G2Sj\u000bmaxj2=n\n2. Top:n= 1, bottomn= 10. We\nplot theSxvalues attained at the turning points ( _Sx= 0) forSx>0. The diagram is symmetric for Sx<0as expected for\na limit cycle on the Bloch sphere. The diagram at the left coincides to a high degree of approximation with the predictions\nobtained for optomechanical systems (i.e. replacing the spin by a harmonic oscillator). In contrast, this is no longer the case\nfor the diagram on the right, which involves higher light intensities.13\nobtained by imposing condition (E1) is plotted. Since the trajectories are in the xyplane, we plot the inflection point\nof the coordinate Sx. We expect GSx=\nevaluated at the inflection point, which gives the amplitude of the limit\ncycle, to coincide with the optomechanic attractors for small B\u000bmax=\nand hence flat lines at the expected amplitudes\n(as calculated in Ref. [40]) as GS=\nincreases. Relative evenly spaced limit cycles increasing in number as larger\nvalues ofGS=\nare considered are observed, in agreement with Ref. [40]. Remarkable, these limit cycles attractors\nare found on the whole Bloch sphere, and not only near the north pole where the harmonic approximation is strictly\nvalid. These attractors are reached by allowing initial conditions on the whole Bloch sphere. For n= 1, (Fig. 8, top),\nswitching is observed up to GS=\n\u00184and then perfect optomechanic behavior. For higher values of n, deviations\nfrom the optomechanical behavior are observed for small GS=\n(implying large B\u000bmax=\naccording to Eq. (E1)) and\nlarge amplitude limit cycles, as compared to the size of the Bloch sphere. An example is shown in Fig. 8, bottom,\nforn= 10."    },    {        "title": "1604.07552v1.First_principles_studies_of_the_Gilbert_damping_and_exchange_interactions_for_half_metallic_Heuslers_alloys.pdf",        "content": "arXiv:1604.07552v1  [cond-mat.mtrl-sci]  26 Apr 2016First principles studies of the Gilbert damping and exchang e interactions for\nhalf-metallic Heuslers alloys\nJonathan Chico,1,∗Samara Keshavarz,1Yaroslav Kvashnin,1Manuel Pereiro,1Igor\nDi Marco,1Corina Etz,2Olle Eriksson,1Anders Bergman,1and Lars Bergqvist3,4\n1Department of Physics and Astronomy, Materials Theory Divi sion,\nUppsala University, Box 516, SE-75120 Uppsala, Sweden\n2Department of Engineering Sciences and Mathematics,\nMaterials Science Division, Lule˚ a University of Technolo gy, Lule˚ a, Sweden\n3Department of Materials and Nano Physics, School of Informa tion and Communication Technology,\nKTH Royal Institute of Technology, Electrum 229, SE-16440 K ista, Sweden\n4SeRC (Swedish e-Science Research Center), KTH Royal Instit ute of Technology, SE-10044 Stockholm, Sweden\n(Dated: September 28, 2018)\nHeusler alloys havebeen intensivelystudied dueto thewide varietyof properties thatthey exhibit.\nOne of these properties is of particular interest for techno logical applications, i.e. the fact that some\nHeusler alloys are half-metallic. In the following, a syste matic study of the magnetic properties\nof three different Heusler families Co 2MnZ, Co 2FeZ and Mn 2VZ with Z = (Al, Si, Ga, Ge) is per-\nformed. A key aspect is the determination of the Gilbert damp ing from first principles calculations,\nwith special focus on the role played by different approximat ions, the effect that substitutional\ndisorder and temperature effects. Heisenberg exchange inte ractions and critical temperature for\nthe alloys are also calculated as well as magnon dispersion r elations for representative systems,\nthe ferromagnetic Co 2FeSi and the ferrimagnetic Mn 2VAl. Correlations effects beyond standard\ndensity-functional theory are treated using both the local spin density approximation including the\nHubbard Uand the local spin density approximation plus dynamical mea n field theory approx-\nimation, which allows to determine if dynamical self-energ y corrections can remedy some of the\ninconsistencies which were previously reported for these a lloys.\nI. INTRODUCTION\nThe limitations presented by traditional electronic de-\nvices, such as Joule heating, which leads to higher en-\nergyconsumption, leakagecurrentsandpoorscalingwith\nsize amongothers1, havesparkedprofoundinterest in the\nfields of spintronics and magnonics. Spintronics applica-\ntions rely in the transmission of information in both spin\nand charge degrees of freedom of the electron, whilst in\nmagnonics information is transmitted via magnetic exci-\ntations, spin waves or magnons. Half-metallic materials\nwith a large Curie temperature are of great interest for\nthese applications. Due to the fact that they are con-\nductors in only one of the spin channels makes them\nideal candidates for possible devices2. Half-metals also\nhave certain advantages for magnonic applications, due\nto the fact that they are insulators in a spin channel and\nthus can have a smaller total density of states at the\nFermi energy than metals. This can result into a small\nGilbert damping, which is an instrumental prerequisite\nfor magnonic applications3.\nThe name “full Heusler alloys”refer to a set of com-\npounds with formula X 2YZ with X and Y typically being\ntransition metals4. The interest in them stems from the\nfactthattheirpropertiescanbecompletelydifferentfrom\nthose of their constituents. Heusler compounds can be\nsuperconducting5(Pd2YSn), semiconductors6(TiCoSb),\nhalf-metallic7(Co2MnSi), and can show a wide array of\nmagnetic configurations: ferromagnetic7(Co2FeSi), fer-\nrimagnetic8(Mn2VAl) or antiferromagnetic9(CrMnSb).\nDue to such a wide variety of behaviours, full Heusleralloys have been studied in great detail since their dis-\ncovery in 1903, leading to the discovery of new Heusler\nfamilies such as the half-Heuslers, with formula XYZ,\nand the inverse Heuslers, with formula X 2YZ. The lat-\nter tend to exhibit a different crystal structure and have\nbeen predicted to show quite remarkable properties10.\nMany Heusler alloys have also been predicted to be\nhalf-metallic, in particular Co 2MnSi has been the focus\nofmany theoreticaland experimental works7,11,12, due to\nits large Curie temperature of 985 K13, half-metallicity\nand low damping parameter, which makes it an ideal\ncandidate for possible spintronic applications. Despite\nthe large amount of research devoted to the half-metallic\nHeusleralloys,suchasCo 2MnSi, onlyrecentlytheoretical\npredictions of the Gilbert damping parameter have been\nmade for some Heusler alloys14,15.\nIn the present work first principle calculations of the\nfull Heusler families Co 2MnZ, Co 2FeZ and Mn 2VZ with\nZ = (Al, Si, Ga, Ge) are performed, with special empha-\nsis on the determination of the Gilbert damping and the\ninteratomic exchange interactions. A study treatment of\nthesystemswithdifferentexchangecorrelationpotentials\nis also performed.\nThe paper is organized as follows, in section II the\ncomputational methods used are presented. Then, in\nsection III, magnetic moments and spectral properties\nare discussed. In section IV the results for the exchange\nstiffness parameter, the critical temperature obtained via\nMonteCarlosimulationsandmagnondispersionrelations\nare presented. Finally in section V, the calculated damp-\ning parameter for the different Heusler is presented and2\ndiscussed.\nII. COMPUTATIONAL METHODS\nThe full Heusler alloys(X 2YZ) havea crystalstructure\ngiven by the space group Fm-3m with X occupying the\nWyckoffposition 8c (1\n4,1\n4,1\n4), while Ysits in the 4a(0,0,0)\nand Z in the 4b (1\n2,1\n2,1\n2).\nTo determine the properties of the systems first prin-\nciples electronic structure calculations were performed.\nThey were mainly done by means of the Korringa-Kohn-\nRostocker Green’s function formalism as implemented in\nthe SPR-KKRpackage16. The shape ofthe potential was\nconsidered by using both the Atomic Sphere Approxi-\nmation (ASA) and a full potential (FP) scheme. The\ncalculations of exchange interactions were performed in\nscalar relativistic approximation while the full relativis-\ntic Dirac equation was used in the damping calculations.\nThe exchange correlation functional was treated using\nboth the Local Spin Density Approximation (LSDA), as\nconsidered by Vosko, Wilk, and Nusair (VWN)17, and\nthe Generalized Gradient Approximation (GGA), as de-\nvised by Perdew, Burke and Ernzerhof (PBE)18. For\ncases in which substitutional disorder is considered, the\nCoherent Potential Approximation (CPA) is used19,20.\nStatic correlation effects beyond LSDA or GGA are\ntaken into account by using the LSDA+ Uapproach,\nwherethe Kohn-ShamHamiltonianissupplemented with\nan additional term describing local Hubbard interac-\ntions21, for thed-states of Co, Mn and Fe. The U-matrix\ndescribing this on-site interactions was parametrized\nthrough the Hubbard parameter Uand the Hund ex-\nchangeJ, using values UCo=UMn=UFe= 3 eV and\nJCo=JMn=JFe= 0.8 eV, which are in the range of the\nvalues considered in previous theoretical studies13,22–24.\nThis approach is used for the Heusler alloys families\nCo2MnZ and Co 2FeZ, as previous studies have shown\nthat for systems such as Co 2FeSi it might be necessary to\nreproduce several experimental observations, although,\nthis topic is still up for debate23. Since part of correla-\ntioneffectsofthe3 dorbitalsisalreadyincludedinLSDA,\ntheir contribution has to be subtracted before adding the\n+Uself-energy. This contribution to be removed is usu-\nally called “double-counting”(DC) correction and there\nis no unique way of defining it (see e.g. Ref. 25). We\nhave used two of the most widely used schemes for the\nDC, namely the Atomic Limit (AL), also known as Fully\nLocalized Limit (FLL)26, and the Around Mean Field\n(AMF)27. The dependence of the results on this choice\nwill be extensively discussed in the following sections.\nIn order to shine some light on the importance of\nthe dynamical correlations for the magnetic properties\nof the selected Heusler alloys, a series of calculations\nwere performed in the framework of DFT plus Dynami-\ncal Mean Field Theory (DMFT)28,29, as implemented in\nthe full-potential linear muffin-tin orbital (FP-LMTO)\ncode RSPt30. As for LSDA+ U, the DMFT calculationsare performed for a selected set of metal 3 dorbitals on\ntop of the LSDA solution in a fully charge self-consistent\nmanner.31,32Theeffectiveimpurityproblem, whichisthe\ncore of the DMFT, is solved through the spin-polarized\nT-matrix fluctuation-exchange (SPTF) solver33. This\ntype of solver is perturbative and is appropriate for the\nsystems with moderate correlationeffects, where U/W <\n1 (Wdenotes the bandwidth).34Contrary to the prior\nDMFT studies35,36, we have performed the perturba-\ntion expansion of the Hartree-Fock-renormalizedGreen’s\nfunction ( GHF) and not of the bare one. Concerning the\nDC correction, we here use both the FLL approach, de-\nscribed above, as well as the so-called “Σ(0)”correction.\nIn the latter case, the orbitally-averaged static part of\nthe DMFT self-energy is removed, which is often a good\nchoice for metals29,37. Finally, in order to extract infor-\nmationaboutthemagneticexcitationsin thesystems, we\nhave performed a mapping onto an effective Heisenberg\nHamiltonian\nˆH=−/summationdisplay\ni/negationslash=jJij/vector ei/vector ej, (1)\nwhereJijis anexchangeinteractionbetweenthe spinslo-\ncated at site iandj, while the /vector ei(/vector ej) representsthe unity\nvectoralongthe magnetizationdirectionatsite i (j). The\nexchange parameters then are computed by making use\nof the well established LKAG (Liechtenstein, Katsnel-\nson, Antropov, and Gubanov) formalism, which is based\non the magnetic force theorem38–40. More specific de-\ntails about the implementation of the LKAG formalism\nin RSPt can be found in Ref. 41. We also note that the\nperformance of the RSPt method was recently published\nin Ref.42and it was found that the accuracy was similar\nto that of augmented plane wave methods.\nFrom the exchange interactions between magnetic\natoms, it is possible to obtain the spin wave stiffness,\nD, which, for cubic systems is written as43\nD=2\n3/summationdisplay\ni,jJij√mimj|rij|2exp/parenleftbigg\n−ηrij\nalat/parenrightbigg\n,(2)\nwhere the mi’s are the magnetic moments of a given\natom,rijisthedistancebetweenthetwoconsideredmag-\nneticmoments, alatisthelatticeparameter, ηisaconver-\ngence parameter used to ensure the convergence of Eq. 2,\nthe value of Dis taken under the limit η→0. To ensure\nthe convergence of the summation, it is also important\nto take into consideration long range interactions. Hence\nthe exchange interactions are considered up to 6 lattice\nconstants from the central atom.\nThe obtained exchange interactions were then used to\ncalculate the critical temperature by making use of the\nBindercumulant, obtainedfromMonteCarlosimulations\nas implemented in the UppASD package44. This was\ncalculated for three different number of cell repetitions\n(10x10x10, 15x15x15 and 20x20x20), with the intersec-\ntion point determining the critical temperature of the\nsystem45.3\nThe Gilbert damping, α, is calculated via linear re-\nsponse theory46. Temperature effects in the scattering\nprocess of electrons are taken into account by consider-\ning an alloy analogy model within CPA with respect to\nthe atomic displacements and thermal fluctuations of the\nspin moments47. Vertex corrections are also considered\nhere, because they provide the “scattering in”term of the\nBoltzmann equation and it corrects significant error in\nthe damping, whenever there is an appreciable s-p or s-d\nscattering in the system16,48.\nFrom the calculated exchange interactions, the adia-\nbatic magnon spectra (AMS) can be determined by cal-\nculating the Fourier transform of the interatomic ex-\nchange interactions49. This is determined for selected\ncases and is compared with the magnon dispersion re-\nlation obtained from the dynamical structure factor,\nSk(q,ω), resulting fromspin dynamics calculations. The\nSk(q,ω) is obtained from the Fourier transform of the\ntime and spatially displaced spin-spin correlation func-\ntion,Ck(r−r′,t)50\nSk(q,ω) =1√\n2πN/summationdisplay\nr,r′eiq·(r−r′)/integraldisplay∞\n−∞eiωtCk(r−r′,t)dt.\n(3)\nThe advantage of using the dynamical structure factor\nover the adiabatic magnon spectra is the capability of\nstudying temperature effects as well as the influence of\nthe damping parameter determined from first principles\ncalculations or from experimental measurements.\nIII. ELECTRONIC STRUCTURE\nThe calculated spin magnetic moments for the selected\nsystems are reported in Table I. These values are ob-\ntained from SPR-KKR with various approximations of\nthe exchange correlation potential and for different geo-\nmetrical shapes of the potential itself. For the Co 2MnZ\nfamily, when Z = (Si ,Ge), the obtained spin mag-\nnetic moments do not seem to be heavily influenced by\nthe choice of exchange correlation potential or potential\nshape. However, for Z = (Al ,Ga) a large variation is\nobserved in the spin moment when one includes the Hub-\nbard parameter U.\nFor the Co 2FeZ systems, a pronounced difference can\nbe observed in the magnetic moments between the LSDA\nand the experimental values for Z = (Si ,Ge). Previ-\nous theoretical works13,22,24suggested that the inclusion\nof a +Uterm is necessary to obtain the expected spin\nmagnetic moments, but such a conclusion has been re-\ncently questioned23. To estimate which double counting\nschemewould be most suitableto treatcorrelationeffects\nin this class of systems, an interpolation scheme between\nthe FLL and AMF treatments was tested, as described\nin Ref. 59 and implemented in the FP-LAPW package\nElk60. It was found that both Co 2MnSi and Co 2FeSi\nare better described with the AMF scheme, as indicatedby their small αUparameter of ∼0.1 for both materials\n(αU= 0denotes completeAMF and αU= 1FLL), which\nis in agreement with the recent work by Tsirogiannis and\nGalanakis61.\nTo test whether a more sophisticated way to treat cor-\nrelation effects improves the description of these mate-\nrials, electronic structure calculations for Co 2MnSi and\nCo2FeSi using the DMFT scheme were performed. The\nLSDA+DMFT[Σ(0)] calculations yielded total spin mo-\nments of 5.00 µBand 5.34 µBfor respectively Co 2MnSi\nand Co 2FeSi. These values are almost equal to those ob-\ntained in LSDA, which is also the case in elemental tran-\nsition metals32. As mentioned above for LSDA+ U, the\nchoice of the DC is crucial for these systems. The main\nreason why no significant differences are found between\nDMFT and LSDA values is that the employed “Σ(0)”DC\nalmost entirely preserves the static part of the exchange\nsplitting obtained in LSDA62. For instance, by using\nFLL DC, we obtained a total magnetization of 5.00 µB\nand 5.61 µBin Co2MnSi and Co 2FeSi, respectively. We\nnote that the spin moment of Co 2FeSi still does not reach\nthe value expected from the Slater-Pauling rule, but the\nDMFT modifies it in a right direction, if albeit to a\nsmaller degree that the LSDA+ Uschemes.\nAnother important aspect of the presently studied sys-\ntems is the fact that they are predicted to be half-\nmetallic. In Fig. 1, the density of states (DOS) for\nboth Co 2MnSi and Co 2FeSi is presented using LSDA and\nLSDA+U. For Co 2MnSi, the DOS at the Fermi energy\nis observed to exhibit a very clear gap in one of the spin\nchannels, in agreement with previous theoretical works7.\nFor Co 2FeSi, instead a small pseudo-gap region is ob-\nserved in one of the spin channels, but the Fermi level\nis located just at the edge of the boundary as shown in\nprevious works24. Panels a) and b) of Fig. 1 also show\nthat some small differences arise depending on the ASA\nor FP treatment. In particular, the gap in the minority\nspin channel is slightly reduced in ASA.\nWhen correlation effects are considered within the\nLSDA+Umethod, the observed band gap for Co 2MnSi\nbecomes larger, while the Fermi level is shifted and still\nremainsin the gap. When applyingLSDA+ Uto Co2FeSi\nin the FLL scheme, EFis shifted farther away from the\nedgeofthe gap, whichexplainswhythemoment becomes\nalmostanintegerasexpected fromtheSlater-Paulingbe-\nhaviour7,24,63. Moreover,onecanseethatinASAthegap\nin the spin down channel is much smaller in comparison\nto the results obtained in FP.\nWhen the dynamical correlation effects are considered\nvia DMFT, the overall shape of DOS remains to be quite\nsimilartothatofbareLSDA,especiallyclosetotheFermi\nlevel, as seen in Fig. A.1 in the Appendix A. This is re-\nlated to the fact that we use a perturbative treatment\nof the many-body effects, which favours Fermi-liquid be-\nhaviour. Similarly to LSDA+ U, the LSDA+DMFT cal-\nculations result in the increased spin-down gaps, but the\nproducedshiftofthebandsisnotaslargeasinLSDA+ U.\nThis is quite natural, since the inclusion ofthe dynamical4\nTABLE I. Summary of the spin magnetic moments obtained using different approximations as obtained from SPR-KKR for the\nCo2MnZ and Co 2FeZ families with Z = (Al ,Si,Ga,Ge). Different exchange correlation potential approximati ons and shapes of\nthe potential have been used. The symbol†signifies that the Fermi energy is located at a gap in one of the spin channels.\nQuantity Co2MnAl Co 2MnGa Co 2MnSi Co 2MnGe Co 2FeAl Co 2FeGa Co 2FeSi Co 2FeGe\nalat[˚A] 5.75515.77515.65525.743535.730515.737515.640235.75054\nmASA\nLDA[µB] 4.04†4.09†4.99†4.94†4.86†4.93†5.09 5.29\nmASA\nGGA[µB] 4.09†4.15†4.99†4.96†4.93†5.00†5.37 5.53\nmASA\nLDA+UAMF [µB] 4.02†4.08 4.98†4.98†4.94†4.99†5.19 5.30\nmASA\nLDA+UFLL [µB] 4.77 4.90 5.02†5.11 5.22 5.36 5.86†5.94†\nmFP\nLDA[µB] 4.02†4.08†4.98†4.98†4.91†4.97†5.28 5.42\nmFP\nGGA[µB] 4.03†4.11 4.98†4.99†4.98†5.01†5.55 5.70\nmFP\nLDA+UAMF [µB] 4.59 4.99 4.98†5.13 5.12 5.40 5.98†5.98†\nmFP\nLDA+UFLL [µB] 4.03†4.17 4.99†4.99†4.99†5.09 5.86†5.98†\nmexp[µB] 4.04554.09564.96574.84574.96555.15576.00245.7458\n0369n↑tot[sts./eV]\n0\n3\n6\n9\n-6 -3 0 3n↓tot[sts./eV]\nE-EF[eV]ASA\nFPa)\n0369n↑tot[sts./eV]\n0\n3\n6\n9\n-6 -3 0 3n↓tot[sts./eV]\nE-EF[eV]ASA\nFPb )\n0369n↑tot[sts./eV]\n0\n3\n6\n9\n-6 -3 0 3n↓tot[sts./eV]\nE-EF[eV]FP [FLL]\nFP [AMF]c)\n0369n↑tot[sts./eV]\n0\n3\n6\n9\n-6 -3 0 3n↓tot[sts./eV]\nE-EFASA [AMF]\nFP [FLL]\nFP [AMF]d )\n[eV]\nFIG. 1. (Color online) Total density of states for different e xchange correlation potentials with the dashed line indica ting the\nFermi energy, sub-figures a) and b) when LSDA is used for Co 2MnSi and Co 2FeSi respectively. Sub-figures c) and d) show the\nDOS when the systems (Co 2MnSi and Co 2FeSi respectively) are treated with LSDA+ U. It can be seen that the half metalicity\nof the materials can be affected by the shape of the potential a nd the choice of exchange correlation potential chosen.\ncorrelations usually tends to screen the static contribu-\ntions coming from LSDA+ U.\nAccording to Ref. 35 taking into account dynami-\ncal correlations in Co 2MnSi results in the emergence of\nthe non-quasiparticle states (NQS’s) inside the minority-\nspin gap, which at finite temperature tend to decrease\nthe spin polarisation at the Fermi level. These NQS’s\nwere first predicted theoretically for model systems64and stem from the electron-magnon interactions, which\nare accounted in DMFT (for review, see Ref. 2). Our\nLSDA+DMFT results for Co 2MnSi indeed show the ap-\npearance of the NQS’s, as evident from the pronounced\nimaginary part of the self-energy at the bottom of the\nconduction minority-spin band (see Appendix B). An\nanalysis of the orbital decomposition of the self-energy\nreveals that the largest contribution to the NQS’s comes5\nfrom the Mn- TEgstates. However, in our calculations,\nwhere the temperature was set to 300K, the NQS’s ap-\npeared above Fermi level and did not contribute to the\nsystem’s depolarization, in agreementwith the recent ex-\nperimental study12.\nWe note that a half-metallic state with a magnetic\nmoment of around 6 µBfor Co 2FeSi was reported in a\nprevious LSDA+DMFT[FLL] study by Chadov et al.36.\nIn their calculations, both LSDA+ Uand LSDA+DMFT\ncalculations resulted in practically the same positions of\nthe unoccupied spin-down bands, shifted to the higher\nenergies as compared to LSDA. This is due to techni-\ncal differences in the treatment of the Hartree-Fock con-\ntributions to the SPTF self-energy, which in Ref. 36 is\ndone separately from the dynamical contributions, while\nin this study a unified approach is used. Overall, the\nimprovements in computational accuracy with respect to\npreviousimplementationscouldberesponsiblefortheob-\ntained qualitative disagreement with respect to Refs. 35\nand 36. Moreover, given that the results qualitatively\ndepend on the choice of the DC term, the description of\nthe electronic structure of Co 2FeSi is not conclusive.\nThe discrepancies in the magnetic moments presented\nin Table I with respect to the experimental values can in\npart be traced back to details of the density of the states\naround the Fermi energy. The studied Heusler alloys are\nthought to be half-metallic, which in turn lead to inte-\nger moments following the Slater-Pauling rule7. There-\nfore, any approximation that destroys half-metallicity\nwill have a profound effect on their magnetic properties7.\nFor example, for Co 2FeAl when the potential is treated\nin LSDA+ U[FLL] with ASA the Fermi energy is located\nat a sharp peak close to the edge of the band gap, de-\nstroyingthehalf-metallicstate(Seesupplementarymate-\nrial Fig.1). A similar situation occurs in LSDA+ U[AMF]\nwith a full potential scheme. It is also worth mention-\ning that despite the fact that the Fermi energy for many\nof these alloys is located inside the pseudo-gap in one of\nthe spin channels, this does not ensure a full spin po-\nlarization, which is instead observed in systems as e.g.\nCo2MnSi. Another important factor is the fact that EF\ncan be close to the edge of the gap as in Co 2MnGa when\nthe shape of the potential is considered to be given by\nASA and the exchange correlation potential is dictated\nby LSDA, hence the half-metallicity of these alloys could\nbe destroyed due to temperature effects.\nThe other Heusler family investigated here is the ferri-\nmagnetic Mn 2VZ with Z = (Al ,Si,Ga,Ge). The lattice\nconstants used in the simulations correspond to either\nexperimental or previous theoretical works. These data\nare reported in Table II together with appropriate ref-\nerences. Table II also illustrates the magnetic moments\ncalculated using different exchange correlation potentials\nand shapes of the potential. It can be seen that in gen-\neral there is a good agreement with previous works, re-\nsulting in spin moments which obey the Slater-Pauling\nbehaviour.\nFor these systems, the Mn atoms align themselves inTABLE II. Lattice constants used for the electronic struc-\nture calculations and summary of the magnetic properties fo r\nMn2VZ with Z = (Al ,Si,Ga,Ge). As for the ferromagnetic\nfamilies, different shapes of the potential and exchange cor -\nrelations potential functionals were used. The magnetic mo -\nments follow quite well the Slater-Pauling behavior with al l\nthe studied exchange correlation potentials. The symbol†\nsignifies that the Fermi energy is located at a gap in one of\nthe spin channels.\nQuantity Mn2VAl Mn 2VGa Mn 2VSi Mn 2VGe\nalat[˚A] 5.687655.905666.06656.09567\nmASA\nLDA[µB] 1.87 1.97†1.00†0.99†\nmASA\nGGA[µB] 1.99†2.04†1.01†1.00†\nmFP\nLDA[µB] 1.92 1.95†0.99†0.99\nmFP\nGGA[µB] 1.98†2.02†0.99†0.99†\nmexp[µB] — 1.8666— —\nan anti-parallel orientation with respect to the V mo-\nments, resulting in a ferrimagnetic ground state. As for\nthe ferromagnetic compounds, the DOS shows a pseu-\ndogap in one of the spin channels (see supplementary\nmaterial Fig.8-9) indicating that at T= 0 K these com-\npoundscouldbehalf-metallic. An importantfactoristhe\nfact that the spin polarization for these systems is usu-\nally considered to be in the opposite spin channel than\nfor the ferromagneticalloys presently studied, henceforth\nthe total magnetic moment is usually assigned to a neg-\native sign such that it complies with the Slater-Pauling\nrule7,65.\nIV. EXCHANGE INTERACTIONS AND\nMAGNONS\nIn this section, the effects that different exchange cor-\nrelation potentials and geometrical shapes of the poten-\ntial haveoverthe exchangeinteractionswill be discussed.\nA. Ferromagnetic Co 2MnZ and Co 2FeZ with\nZ= (Al,Si,Ga,Ge)\nIn Table III the calculated spin wave stiffness, D, is\nshown. In general there is a good agreement between\nthe calculated values for the Co 2MnZ family, with the\nobtained values using LSDA or GGA being somewhat\nlarger than the experimental measurements. This is in\nagreement with the observations in the previous section,\nin which the same exchange correlation potentials were\nfound to be able to reproducethe magnetic moments and\nhalf-metallicbehaviourfortheCo 2MnZfamily. Inpartic-\nular, for Co 2MnSi the ASA calculations are in agreement\nwith experiments68,69and previous theoretical calcula-\ntions70. It is important to notice that the experimen-\ntal measurements are performed at room temperature,\nwhich can lead to softening of the magnon spectra, lead-\ning to a reduced spin wave stiffness.6\nHowever, for the Co 2FeZ family neither LSDA or GGA\ncan consistently predict the spin wave stiffness, with\nZ=(Al, Ga) resulting in an overestimated value of D,\nwhile for Co 2FeSi the obtained value is severely underes-\ntimated. However, for some materials in this family, e.g.\nCo2FeGathespinwavestiffnessagreeswith previousthe-\noretical results70. These data reflect the influence that\ncertain approximations have on the location of the Fermi\nlevel, which previously has been shown to have profound\neffects on the magnitude of the exchange interactions71.\nThis can be observed in the half-metallic Co 2MnSi; when\nit is treated with LSDA+ U[FLL] in ASA the Fermi level\nis located at the edge of the gap (see Fig. 1c). Result-\ning in a severely underestimated spin wave stiffness with\nrespect to both the LSDA value and the experimental\nmeasurements (see Table III). The great importance of\nthe location of the Fermi energy on the magnetic proper-\nties can be seen in the cases of Co 2MnAl and Co 2MnGa.\nIn LSDA+ U[FLL], these systems show non integer mo-\nments which are overestimated with respect to the ex-\nperimental measurements (see Table I), but also results\nin the exchange interactions of the system preferring a\nferrimagnetic alignment. Even more the exchange inter-\nactions can be severely suppressed when the Hubbard U\nisused. Forexample, forCo 2MnGe inASAthe dominant\ninteraction is between the Co-Mn moments, in LSDA the\nobtainedvalueis0.79mRy, while inLSDA+ U[FLL]isre-\nduced to 0.34 mRy, also, the nearest neighbour Co 1-Co2\nexchange interaction changes from ferromagnetic to anti-\nferromagnetic when going from LSDA to LSDA+ U[FLL]\nwhich lead the low values obtainedfor the spin wavestiff-\nness. As will be discussed below also for the low Tcfor\nsome of these systems.\nIt is important to notice, that the systems that exhibit\nthe largest deviation from the experimental values, are\nusually those that under a certain exchange correlation\npotential and potential geometry loosetheir half-metallic\ncharacter. Such effect are specially noticeable when one\ncompares LSDA+ U[FLL] results in ASA and FP, where\nhalf-metallicity is more easily lost in ASA due to the\nfact that the pseudogap is much smaller under this ap-\nproximation than under FP (see Fig. 1). In general, it\nis important to notice that under ASA the geometry of\nthe potential is imposed, that is non-spherical contribu-\ntions to the potential are neglected. While this has been\nshown to be very successful to describe many properties,\nit does introduce an additional approximation which can\nlead to anill treatment ofthe properties ofsome systems.\nHence, care must be placed when one is considering an\nASA treatment for the potential geometry, since it can\nlead to large variations of the exchange interactions and\nthus is one of the causes of the large spread on the values\nobserved in Table III for the exchange stiffness and in\nTable IV for the Curie temperature.\nOne of the key factors behind the small values of the\nspin stiffness for Co 2FeSi and Co 2FeGe, in comparison\nwith the rest of the Co 2FeZ family, lies in the fact that\nin LSDA and GGA an antiferromagnetic long-range Fe-Fe interaction is present (see Fig. C.2 in Appendix C).\nAs the magnitude of the Fe-Fe interaction decreases the\nexchange stiffness increases, e.g. as in LSDA+ U[AMF]\nwith afull potential scheme. Theseexchangeinteractions\nare one of the factors behind the reduced value of the\nstiffness, this is evident when comparing with Co 2FeAl,\nwhich while having similar nearest neighbour Co-Fe ex-\nchange interactions, overall displays a much larger spin\nwave stiffness for most of the studied exchange correla-\ntion potentials.\nUsing LSDA+DMFT[Σ(0)] for Co 2MnSi and Co 2FeSi,\nthe obtained stiffness is 580 meV ˚A2and 280 meV ˚A2re-\nspectively, whilst in LSDA+DMFT[FLL] for Co 2MnSi\nthestiffnessis630meV ˚A2andforCo 2FeSiis282meV ˚A2.\nAs can be seen for Co 2MnSi there is a good agree-\nment between the KKR LSDA+ U[FLL], the FP-LMTO\nLSDA+DMFT[FLL] and the experimental values.\nThe agreement with experiments is particularly good\nwhen correlation effects are considered as in the\nLSDA+DMFT[Σ(0)] approach. On the other hand, for\nCo2FeSi the spin wave stiffness is severely underesti-\nmated which is once again consistent with what is shown\nin Table III.\nUsing the calculated exchange interactions, the criti-\ncal temperature, Tc, for each system can be calculated.\nUsing the ASA, the Tcof both the Co 2MnZ and Co 2FeZ\nsystems is consistently underestimated with respect to\nexperimental results, as shown in Table IV. The same\nunderestimation has been observed in previous theo-\nretical studies78, for systems such as Co 2Fe(Al,Si) and\nCo2Mn(Al,Si). However, using a full potential scheme\ninstead leads to Curie temperatures in better agreement\nwith the experimental values, specially when the ex-\nchange correlation potential is considered to be given by\nthe GGA (see Table IV). Such observation is consistent\nwith what was previouslymentioned, regardingthe effect\nofthe ASA treatmentonthe spin wavestiffness andmag-\nnetic moments, where in certain cases, ASA was found to\nnot be the best treatment to reproduce the experimen-\ntal measurements. As mentioned above, this is strongly\nrelated to the fact that in general ASA yields a smaller\npseudogapin the half-metallic materials, leading to mod-\nification of the exchange interactions. Thus, in general, a\nfullpotentialapproachseemstobeabletobetterdescribe\nthe magnetic properties in the present systems, since the\npseudogaparoundthe Fermienergyisbetter describedin\na FP approach for a given choice of exchange correlation\npotential.\nThe inclusionofcorrelationeffects forthe Co 2FeZfam-\nily, lead to an increase of the Curie temperature, as for\nthe spin stiffness. This is related to the enhancement of\nthe interatomic exchange interactions as exemplified in\nthe case of Co 2FeSi. However, the choice of DC once\nmore is shown to greatly influence the magnetic proper-\nties. For the Co 2FeZ family, AMF results in much larger\nTcthan the FLL scheme, whilst for Co 2MnZ the dif-\nferences are smaller, with the exception of Z=Al. All\nthese results showcase how important a proper descrip-7\nTABLE III. Summary of the spin wave stiffness, Dfor Co 2MnZ and Co 2FeZ with Z = (Al ,Si,Ga,Ge). For the Co 2MnZ family\nboth LSDA and GGA exchange correlation potentials yield val ues close to the experimental measurements. However, for th e\nCo2FeZ family a larger data spread is observed. The symbol∗implies that the ground state for these systems was found to b e\nFerri-magnetic from Monte-Carlo techniques and the critic al temperature presented here is calculated from the ferri- magnetic\nground state.\nQuantity Co2MnAl Co 2MnGa Co 2MnSi Co 2MnGe Co 2FeAl Co 2FeGa Co 2FeSi Co 2FeGe\nDASA\nLDA[meV˚A2] 282 291 516 500 644 616 251 206\nDASA\nGGA[meV˚A2] 269 268 538 515 675 415 267 257\nDASA\nLDA+UFLL [meV ˚A2] 29∗487∗205 94 289 289 314 173\nDASA\nLDA+UAMF [meV ˚A2] 259 318 443 417 553 588 235 214\nDFP\nLDA[meV˚A2] 433 405 613 624 692 623 223 275\nDFP\nGGA[meV˚A2] 483 452 691 694 740 730 323 344\nDFP\nLDA+UFLL [meV ˚A2] 447 400 632 577 652 611 461 436\nDFP\nLDA+UAMF [meV ˚A2] 216 348 583 579 771 690 557 563\nDexp[meV˚A2] 190722647357568-5346941374370754967671577—\ntion of the pseudogap region is in determining the mag-\nnetic properties of the system.\nAnother observation, is the fact that even if a given\ncombination of exchange correlation potential and geo-\nmetrical treatment of the potential can yield a value of\nTcin agreementwith experiments, it does not necessarily\nmeans that the spin wave stiffness is correctly predicted\n(see Table III and Table IV).\nWhen considering the LSDA+DMFT[Σ(0)] scheme,\ncritical temperatures of 688 K and 663 K are ob-\ntained for Co 2MnSi and Co 2FeSi, respectively. Thus,\nthe values of the Tcare underestimated in compari-\nson with the LSDA+ Uor LSDA results. The reason\nfor such behaviour becomes clear when one looks di-\nrectly on the Jij’s, computed with the different schemes,\nwhich are shown in Appendix C. These results sug-\ngest that taking into account the dynamical correlations\n(LSDA+DMFT[Σ(0)]) slightly suppresses most of the\nJij’s as compared to the LSDA outcome. This is an\nexpected result, since the employed choice of DC correc-\ntion preserves the exchange splitting obtained in LSDA,\nwhile the dynamical self-energy, entering the Green’s\nfunction, tends to lower its magnitude. Since these two\nquantities are the key ingredients defining the strength\nof the exchange couplings, the Jij’s obtained in DMFT\nare very similar to those of LSDA (see e.g. Refs. 41\nand 81). The situation is a bit different if one employs\nFLL DC, since an additional static correction enhances\nthe local exchange splitting.82For instance, in case of\nCo2MnSi the LSDA+DMFT[FLL] scheme provided a Tc\nof 764 K, which is closer to the experiment. The con-\nsistently better agreement of the LSDA+ U[FLL] and\nLSDA+DMFT[FLL] estimates of the Tcwith experimen-\ntal values might indicate that explicit account for static\nlocal correlations is important for the all considered sys-\ntems.\nUsing the calculated exchange interactions, it is also\npossible to determine the adiabatic magnon spectra\n(AMS). In Fig. 2 is shown the effect that different ex-\nchange correlation potentials have overthe description ofthe magnon dispersion relation of Co 2FeSi is shown. The\nmost noticeable effect between different treatments of\nthe exchange correlation potential is shifting the magnon\nspectra, while its overall shape seems to be conserved.\nThis is a direct result from the enhancement of nearest\nneighbour interactions (see Fig. C.2).\nWhen comparing the AMS treatment with the dy-\nnamical structure factor, S(q,ω), atT= 300 K and\ndamping parameter αLSDA= 0.004, obtained from first\nprinciples calculations (details explained in section V),\na good agreement at the long wavelength limit is found.\nHowever, a slight softening can be observed compared\nto the AMS. Such differences can be explained due to\ntemperature effects included in the spin dynamics sim-\nulations. Due to the fact that the critical temperature\nof the system is much larger than T= 300 K (see Ta-\nble IV), temperature effects are quite small. The high\nenergy optical branches are also softened and in general\nare much less visible. This is expected since the correla-\ntion was studied using only vectors in the first Brillouin\nzone and as has been shown in previous works50, a phase\nshift is sometimes necessary to properly reproduce the\noptical branches, implying the need of vectors outside\nthe first Brillouin zone. Also, Stoner excitations dealing\nwith electron-holeexcitations arenot included in this ap-\nproach,whichresultintheLandaudampingwhichaffects\nthe intensity of the optical branches. Such effects are not\ncaptured by the present approach, but can be studied\nby other methods such as time dependent DFT83. The\nshape of the dispersion relationalong the path Γ −Xalso\ncorresponds quite well with previous theoretical calcula-\ntions performed by K¨ ubler84.\nB. Ferrimagnetic Mn 2VZ with Z = (Al,Si,Ga,Ge)\nAsmentionedabove,theMnbasedMn 2VZfullHeusler\nfamily has a ferrimagnetic ground state, with the Mn\natoms orienting parallel to each other and anti-parallel\nwith respect to the V moments. For all the studied sys-8\nTABLE IV. Summary of the critical temperature for Co 2MnZ and Co 2FeZ with Z = (Al ,Si,Ga,Ge), with different exchange\ncorrelation potentials and shape of the potentials. The sym bol∗implies that the ground state for these systems was found to b e\nFerri-magnetic from Monte-Carlo techniques and the critic al temperature presented here is calculated from the ferri- magnetic\nground state.\nQuantity Co2MnAl Co 2MnGa Co 2MnSi Co 2MnGe Co 2FeAl Co 2FeGa Co 2FeSi Co 2FeGe\nTLDA\ncASA [K] 360 350 750 700 913 917 655 650\nTGGA\ncASA [K] 350 300 763 700 975 973 800 750\nTLDA+U\ncASAFLL[K] 50∗625∗125 225 575 550 994 475\nTLDA+U\ncASAAMF[K] 325 425 650 600 950 950 650 625\nTLDA\ncFP [K] 525 475 875 825 1050 975 750 750\nTGGA\ncFP [K] 600 525 1000 925 1150 1100 900 875\nTLDA+U\ncFPFLL[K] 525 475 950 875 1050 975 1050 1075\nTLDA+U\ncFPAMF[K] 450 450 1000 875 1275 1225 1450 1350\nTexp\nc[K] 69778694 98513905 10007910938011002498158\nTABLE V. Summary of the spin wave stiffness, D, and the\ncritical temperature for Mn 2VZ with Z = (Al ,Si,Ga,Ge) for\ndifferent shapes of the potential and exchange correlation p o-\ntentials.\nQuantity Mn2VAl Mn 2VGa Mn 2VSi Mn 2VGe\nDASA\nLDA[meV˚A2] 314 114 147\nDASA\nGGA[meV˚A2] 324 73 149\nDFP\nLDA[meV˚A2] 421 206 191\nDFP\nGGA[meV˚A2] 415 91 162\nDexp[meV˚A2] 53485— — —\nTLDA\ncASA [K] 275 350 150 147\nTGGA\ncASA [K] 425 425 250 250\nTLDA\ncFP [K] 425 450 200 200\nTGGA\ncFP [K] 600 500 350 350\nTexp\nc[K] 7688578366— —\ntemstheMn-Mnnearestneighbourexchangeinteractions\ndominates. In Table V the obtained spin wave stiffness,\nD, and critical temperature Tcare shown. For Mn 2VAl,\nit can be seen that the spin wave stiffness is trend when\ncompared to the experimental value. The same under-\nestimation can be observed in the critical temperature.\nFor Mn 2VAl, one may notice that the best agreement\nwith experiments is obtained for GGA in FP. An inter-\nesting aspect of the high Tcobserved in these materials\nis the fact that the magnetic order is stabilized due to\nthe anti-ferromagnetic interaction between the Mn and\nV sublattices, since the Mn-Mn interaction is in general\nmuch smaller than the Co-Co, Co-Mn and Co-Fe inter-\nactions present in the previously studied ferromagnetic\nmaterials.\nFor these systems it can be seen that in general the FP\ndescriptionyields Tc’swhichareinbetter agreementwith\nexperiment, albeit if the values are still underestimated.\nAs for the Co based systems the full potential technique\nimproves the description of the pseudogap, it is impor-\ntant to notice that for most systems both in ASA and\nFP the half-metallic characteris preserved. However, the\ndensity of states at the Fermi level changes which could\nlead to changes in the exchange interactions.As for the ferromagnetic systems one can calculate the\nmagnon dispersion relation and it is reported in Fig. 3\nfor Mn 2VAl. A comparison with Fig. 2 illustrates some\nof the differences between the dispersion relation of a fer-\nromagnet and of a ferrimagnetic material. In Fig. 3 some\noverlap between the acoustic and optical branches is ob-\nserved, as well as a quite flat dispersion relation for one\nof the optical branches. Such an effect is not observed in\nthe studied ferromagnetic cases. In general the different\nexchange correlation potentials only tend to shift the en-\nergy of the magnetic excitations, while the overall shape\nof the dispersion does not change noticeably, which is\nconsistent with what was seen in the ferromagnetic case.\nThe observed differences between the LSDA and GGA\nresults in the small qlimit, corresponds quite well with\nwhatisobservedinTableV, wherethe spinwavestiffness\nfor GGA with the potential given by ASA is somewhat\nlargerthan the LSDA case. This is directly related to the\nobservation that the nearest neighbour Mn-Mn and Mn-\nV interactions are large in GGA than in LSDA. Again,\nsuch observation is tied to the DOS at the Fermi level,\nsince Mn 2VAl is not half-metallic in LSDA, on the other\nhand in GGA the half-metallic state is obtained (see Ta-\nble. II.\nV. GILBERT DAMPING\nThe Gilbert damping is calculated for all the previ-\nously studied systems using ASA and a fully relativistic\ntreatment. In Fig. 4, the temperature dependence of the\nGilbert damping for Co 2MnSi is reported for different\nexchange-correlationpotentials. Whencorrelationeffects\nare neglected or included via the LSDA+ U[AMF], the\ndampingincreaseswith temperature. Onthe otherhand,\nin the LSDA+ U[FLL] scheme, the damping decreases as\na function of temperature, and its overall magnitude is\nmuch larger. Such observation can be explained from the\nfact that in this approximation a small amount of states\nexists at the Fermi energyin the pseudogapregion, hence\nresulting in a larger damping than in the half-metallic9\n0100200300400500\nΓ X W L ΓEnergy [meV]FP-LSDA\nDMFT[Σ(0)]a)\nFIG. 2. (Color online) a) Adiabatic magnon spectra for\nCo2FeSi for different exchange correlation potentials. In the\ncase of FP-LSDA and LSDA+DMFT[Σ(0)] the larger devia-\ntionsareobservedinthecase ofhighenergies, withtheDMFT\ncurve having a lower maximum than the LSDA results. In b)\na comparison of the adiabatic magnon spectra (solid lines)\nwith the dynamical structure factor S(q,ω) atT= 300 K,\nwhen the shape of the potential is considered to be given\nby the atomic sphere approximation and the exchange cor-\nrelation potential to be given by LSDA, some softening can\nbe observed due to temperature effects specially observed at\nhigher q-points.\ncases(see Fig. 1c).\nIn general the magnitude of the damping, αLSDA=\n7.4×10−4, is underestimated with respect to older ex-\nperimental measurements at room temperature, which\nyielded values of α= [0.003−0.006]86andα∼0.025\nfor polycrystalline samples87, whilst it agrees with previ-\nously performed theoretical calculations14. Such discrep-\nancy between the experimental and theoretical results\ncould stem from the fact that in the theoretical calcula-\ntions only the intrinsic damping is calculated, while in\nexperimental measurements in addition extrinsic effects\nsuch as eddy currents and magnon-magnon scattering\ncan affect the obtained values. It is also known that sam-FIG. 3. (Color online) Adiabatic magnon dispersion relatio n\nfor Mn 2VAl when different exchange correlation potentials\nare considered. In general only a shift in energy is observed\nwhen considering LSDA or GGA with the overall shape being\nconserved.\n00.511.522.533.54\n50 100 150 200 250 300 350 400 450 500Gilbert damping (10-3)\nTemperature [K]LSDA\nGGA\nLSDA+U [FLL]\nLSDA+U [AMF]\nFIG.4. (Color online)TemperaturedependenceoftheGilber t\ndamping for Co 2MnSi for different exchange correlation po-\ntentials. For LSDA, GGA and LSDA+ U[AMF] exchange cor-\nrelation potentials the damping increases with temperatur e,\nwhilst for LSDA+ U[FLL]thedampingdecreases as afunction\nof temperature.\nple capping or sample termination, can have profound ef-\nfects over the half-metallicity of Co 2MnSi88. Recent ex-\nperiments showed that ultra-low damping, α= 7×10−4,\nfor Co 1.9Mn1.1Si can be measured when the capping\nis chosen such that the half-metallicity is preserved89,\nwhich is in very good agreement with the present theo-\nretical calculations.\nIn Fig. 5, the Gilbert damping at T= 300 K for the\ndifferent Heusler alloys as a function of the density of\nstates at the Fermi level is presented. As expected, the\nincreased density of states at the Fermi energy results in10\nFIG. 5. (Color online) Gilbert damping for different Heusler\nalloys at T= 300 K as a function of density of states at the\nFermi energy for LSDA exchange correlation potential. In\ngeneral the damping increases as the density of states at the\nFermi Energy increases (the dotted line is to guide the eyes) .\nan increased damping. Also it can be seen that in gen-\neral, alloys belonging to a given family have quite similar\ndamping parameter, except for Co 2FeSi and Co 2FeGe.\nTheir anomalous behaviour, stems from the fact that\nin the LSDA approach both Co 2FeSi and Co 2FeGe are\nnot half-metals. Such clear dependence on the density of\nstates is expected, since the spin orbit coupling is small\nfor these materials, meaning that the dominating con-\ntribution to the damping comes from the details of the\ndensity of states around the Fermi energy90,91.\n1. Effects of substitutional disorder\nIn order to investigate the possibility to influ-\nence the damping, we performed calculations for the\nchemically disordered Heusler alloys Co 2Mn1−xFexSi,\nCo2MeAl1−xSixand Co 2MeGa 1−xGexwhere Me =\n(Mn,Fe).\nDue to the small difference between the lattice param-\neters of Co 2MnSi and Co 2FeSi, the lattice constant is\nunchanged when varying the concentration of Fe. This\nis expected to play a minor role on the following results.\nWhen one considers only atomic displacement contribu-\ntions to the damping (see Fig. 6a), the obtained values\nare clearlyunderestimated in comparisonwith the exper-\nimental measurements at room temperature92. Under\nthe LSDA, GGA and LSDA+ U[AMF] treatments, the\ndamping is shown to increase with increasing concentra-\ntion of Fe. On the other hand, in LSDA+ U[FLL] the\ndamping at low concentrations of Fe is much larger than\ninthe othercases, andit decreaseswith Feconcentration,\nuntil a minima is found at Fe concentration of x∼0.8.\nThis increase can be related to the DOS at the Fermi\nenergy, which is reported in Fig. 1c for Co 2MnSi. One00.511.522.533.544.55\n0 0.2 0.4 0.6 0.8 1Gilbert damping (10-3)\nFe concentrationLSDA\nGGA\nLSDA+ U[FLL]\nLSDA+ U[AMF]a)\n00.511.522.533.544.5\n0 0.2 0.4 0.6 0.8 1Gilbert damping (10-3)\nFe concentrationLSDA\nGGA\nLSDA+ U[FLL]\nLSDA+ U[AMF]b)\nFIG. 6. (Color online) Gilbert damping for the random alloy\nCo2Mn1−xFexSi as a function of the Fe concentration at T=\n300 K when a) only atomic deisplacements are considered and\nb) when both atomic displacements and spin fluctuations are\nconsidered.\ncan observe a small amount of states at EF, which could\nlead to increased values of the damping in comparison\nwith the ones obtained in traditional LSDA. As for the\npure alloys, a general trend relating the variation of the\nDOS at the Fermi level and the damping with respect to\nthe variation of Fe concentration can be obtained, anal-\nogous to the results shown in Fig. 5.\nWhen spin fluctuations are considered in addition to\nthe atomic displacements contribution, the magnitude of\nthe damping increases considerably, as shown in Fig. 6b.\nThis is specially noticeable at low concentrations of Fe.\nMn rich alloys have a Tclower than the Fe rich ones,\nthus resulting in larger spin fluctuations at T= 300 K.\nThe overall trend for LSDA and GGA is modified at low\nconcentrations of Fe when spin fluctuations are consid-\nered, whilst for LSDA+ U[FLL] the changes in the trends\noccur mostly at concentrations between x= [0.3−0.8].\nAn important aspect is the overall good agreement of11\nLSDA, GGA and LSDA+ U[AMF]. Instead results ob-\ntained in LSDA+ U[FLL] stand out as different from the\nrest. This is is expected since as was previously men-\ntioned the FLL DC is not the most appropriate scheme\nto treat these systems. An example of such inadequacy\ncan clearly be seen in Fig. 6b for Mn rich concentrations,\nwhere the damping is much larger with respect to the\nother curves. As mentioned above, this could result from\nthe appearance of states at the Fermi level.\nOverall the magnitude of the intrinsic damping pre-\nsented here is smaller than the values reported in experi-\nments92, whichreportvaluesforthedampingofCo 2MnSi\nofα∼0.005 and α∼0.020 for Co 2FeSi, in comparison\nwith the calculated values of αLSDA= 7.4×10−4and\nαLSDA= 4.1×10−3for Co 2MnSi and Co 2FeSi, respec-\ntively. In experiments also a minimum at the concentra-\ntion of Fe of x∼0.4 is present, while such minima is not\nseen in the present calculations. However, similar trends\nas those reported here (for LSDA and GGA) are seen in\nthe work by Oogane and Mizukami15. A possible reason\nbehind the discrepancy between theory and experiment,\ncould stem from the fact that as the Fe concentration\nincreases, correlation effects also increase in relative im-\nportance. Such a situation cannot be easily described\nthrough the computational techniques used in this work,\nandwill affectthe detailsofthe DOSatthe Fermienergy,\nwhich in turn could modify the damping. Another im-\nportant factor influencing the agreement between theory\nand experiments arise form the difficulties in separating\nextrinsic and intrinsic damping in experiments93. This,\ncombined with the large spread in the values reported in\nvarious experimental studies87,94,95, points towards the\nneed of improving both theoretical and experimental ap-\nproaches,ifoneintendstodeterminetheminimumdamp-\ning attainable for these alloys with sufficient accuracy.\nUp until now in the present work, disorder effects\nhave been considered at the Y site of the Heusler struc-\nture. In the following chemical disorder will be consid-\nered on the Z site instead. Hence, the chemical structure\nchanges to the type Co 2MeZA\n1−xZB\nx(Me=Fe,Mn). The\nalloys Co 2MeAlxSi1−xand Co 2MeGa xGe1−xare consid-\nered. The lattice constant for the off stoichiometric com-\npositions is treated using Vegard’s law96, interpolating\nbetween the values given in Table I.\nIn Fig. 7 the dependence of the damping on the con-\ncentration of defects is reported, as obtained in LSDA.\nFor Co 2FeGaxGe1−xas the concentration of defects in-\ncreases the damping decreases. Such a behaviour can\nbe understood by inspecting the density of states at the\nFermi level which follows the same trend, it is important\nto notice that Co 2FeGa is a half-metallic system, while\nCo2FeGe is not (see table I). On the other hand, for\nCo2FeAlxSi1−x, the damping increases slightly with Al\nconcentration, however, for the stoichiometric Co 2FeAl\nis reached the damping decreases suddenly, as in the pre-\nvious case. This is a direct consequence of the fact that\nCo2FeAl is a half metal and Co 2FeSi is not, hence when\nthe half-metallic state is reached a sudden decrease ofthe damping is observed. For the Mn based systems, as\nthe concentration of defects increases the damping in-\ncreases, this stark difference with the Fe based systems.\nFor Co 2MnAlxSi1−xthis is related to the fact that both\nCo2MnAl and Co 2MnSi are half-metals in LSDA, hence,\nthe increase is only related to the fact that the damp-\ning for Co 2MnAl is larger than the one of Co 2MnSi, it\nis also relevant to mention, that the trend obtained here\ncorresponds quite well with what is observed in both ex-\nperimental and theoretical results in Ref.86. A similar\nexplanation can be used for the Co 2MnGa xGe1−xalloys,\nas both are half-metallic in LSDA. As expected, the half\nmetallic Heuslers have a lower Gilbert damping than the\nother ones, as shown in Fig. 7.\n00.511.522.533.544.5\n0 0.2 0.4 0.6 0.8 1Gilbert damping (10-3)\nConcentration of defectsCo2FeAlxSi1-xCo2FeGaxGe1-xCo2MnAlxSi1-xCo2MnGaxGe1-x\nFIG. 7. (color online) Dependence of the Gilbert damping\nfor the alloys Co 2MeAlxSi1−xand Co 2MeGa xGe1−xwith Me\ndenoting Mn or Fe under the LSDA exchange correlation po-\ntential.\nVI. CONCLUSIONS\nThe treatment of several families of half-metallic\nHeusler alloys has been systematically investigated us-\ning several approximations for the exchange correlation\npotential, as well as for the shape of the potential. Spe-\ncial care has been paid to the calculation of their mag-\nnetic properties, such as the Heisenberg exchange inter-\nactions and the Gilbert damping. Profound differences\nhave been found in the description of the systems de-\npending on the choice of exchange correlation potentials,\nspeciallyforsystems in whichcorrelationeffects might be\nnecessarytoproperlydescribethepresumedhalf-metallic\nnature of the studied alloy.\nIn general, no single combination of exchange correla-\ntion potential and potential geometry was found to be\nable to reproduce all the experimentally measured mag-\nnetic properties of a given system simultaneously. Two\nof the key contributing factors are the exchange correla-12\ntion potential and the double counting scheme used to\ntreat correlation effects. The destruction of the half-\nmetallicity of any alloy within the study has profound\neffects on the critical temperature and spin wave stiff-\nness. A clear indication of this fact is that even if the\nFLL double counting scheme may result in a correct de-\nscription of the magnetic moments of the system, the\nexchange interactions may be severely suppressed. For\nthe systems studied with DMFT techniques either mi-\nnor improvement or results similar to the ones obtained\nfrom LSDA is observed. This is consistent with the in-\nclusion of local d−dscreening, which effectively dimin-\nishes the strength of the effective Coulomb interaction\nwith respect to LSDA+ U(for the same Hubbard param-\neterU). In general, as expected, the more sophisticated\ntreatment forthe geometricalshape ofthe potential, that\nis a full potential scheme, yields results closer to experi-\nments, which in these systems, is intrinsically related to\nthe description of the pseudogap region.\nFinally, the Gilbert damping is underestimated with\nrespect to experimental measurements, but in good\nagreement with previous theoretical calculations. One of\nthe possible reasons being the difficulty from the experi-\nmental point of view of separating intrinsic and extrinsic\ncontributions to the damping, as well as the strong de-\npendence of the damping on the crystalline structure.\nA clear correlation between the density of states at the\nFermi level and the damping is also observed, which is\nrelated to the presence of a small spin orbit coupling\nin these systems. This highlights the importance that\nhalf-metallic materials, and their alloys, have in possible\nspintronic and magnonic applications due to their low in-\ntrinsic damping, and tunable magnetodynamic variables.\nThese results could spark interest from the experimental\ncommunity due to the possibility of obtaining ultra-low\ndamping in half-metallic Heusler alloys.\nVII. ACKNOWLEDGEMENTS\nThe authors acknowledge valuable discussions with\nM.I. Katsnelsson and A.I. Lichtenstein. The work was\nfinanced through the VR (Swedish Research Council)\nand GGS (G¨ oran Gustafssons Foundation). O.E. ac-\nknowledges support form the KAW foundation (grants\n2013.0020 and 2012.0031). O.E. and A.B acknowledge\neSSENCE. L.B acknowledge support from the Swedish\ne-Science Research Centre (SeRC). The computer sim-\nulations were performed on resources provided by the\nSwedish National Infrastructure for Computing (SNIC)\nat the National Supercomputer Centre (NSC) and High\nPerformance Computing Center North (HPC2N).\nAppendix A: DOS from LSDA+DMFT\nHere we show the DOS in Co 2MnSi and Co 2FeSi ob-\ntained from LSDA and LSDA+DMFT calculations. The0369n↑tot[sts./eV]\n0\n3\n6\n9-6 -3 0 3n↓tot[sts./eV]\nE-EF[eV]LSDA\n0369n↑tot[sts./eV]\n0\n3\n6\n9-6 -3 0 3n↓tot[sts./eV]\nE-EF[eV]LSDA\nDMFT[Σ(0)]\nDMFT[FLL]\nFIG. A.1. (color online) DOS in Co 2FeSi (top panel) and\nCo2MnSi (bottom panel) obtained in different computational\nsetups.\nresults shown in Fig. A.1 indicate that the DMFT in-\ncreases the spin-down (pseudo-)gap in both Co 2FeSi and\nCo2MnSi. In the latter casethe shift ofthe bands is more\npronounced. InCo 2FeSiitmanifestsitselfinanenhanced\nvalue of the total magnetization. For both studied sys-\ntems, the FLL DC results in relatively larger values of\nthe gaps as compared with the “Σ(0)”estimates. How-\never, for the same choice of the DC this gap appears to\nbe smaller in LSDA+DMFT than in LSDA+ U. Present\nconclusion is valid for both Co 2FeSi and Co 2MnSi (see\nFig. 1 for comparison.)\nAppendix B: NQS in Co 2MnSi\nHere we show the calculated spectral functions in\nCo2MnSi obtained with LSDA+DMFT[Σ(0)] approach.\nAs discussed in the main text, the overall shape of DOS\nis reminiscent of that obtained in LSDA. However, a cer-\ntain amount of the spectral weight appears above the\nminority-spin gap. An inspection of the imaginary part\nof the self-energy in minority-spin channel, shown in the\nbottom panel of Fig. B.1, suggests a strong increase of\nMn spin-down contribution at the corresponding ener-\ngies, thus confirming the non-quasiparticle nature of the\nobtained states. We note that the use of FLL DC formu-\nlationresultsinanenhancedspin-downgapwhichpushes\nthe NQS to appear at even higher energies above EF(see\nAppendix A).13\n-30030PDOS [sts./Ry]\n-0.2-0.10Im [ Σ↑]\nCo Eg\nCo T2g\n-0.2 -0.1 0 0.1 0.2\nE-EF [Ry]-0.2-0.10Im [ Σ↓]\nMn Eg\nMn T2g\nFIG. B.1. (color online) Top panel: DOS in Co 2MnSi pro-\njected onto Mn and Co 3 dstates of different symmetry. Mid-\ndle and bottom panels: Orbital-resolved spin-up and spin-\ndown imaginary parts of the self-energy. The results are\nshown for the “Σ(0)”DC.\nAppendix C: Impact of correlation effects on the\nJij’s in Co 2MnSi and Co 2FeSi\nIn this section we present a comparison of the ex-\nchange parameters calculated in the framework of the\nLSDA+DMFT using different DC terms. The calculated\nJij’s between different magnetic atoms within the first\nfew coordination spheres are shown in Fig. C.1. One can\nsee that the leading interactions which stabilize the fer-\nromagnetism in these systems are the nearest-neighbour\nintra-sublattice couplings between Co and Fe(Mn) atoms\nand, to a lower extend, the interaction between two Co\natoms belonging to the different sublattices. This qual-\nitative behaviour is obtained independently of the em-\nployed method for treating correlation effects and is in\ngood agreement with prior DFT studies. As explained\nin the main text, the LSDA and LSDA+DMFT[Σ(0)] re-\nsults are more similar to each other, whereas most of the\nJij’s extracted from LSDA+DMFT[FLL] are relatively\nenhanced due to inclusion of an additional static contri-\nbution to the exchange splitting. This is also reflected in\nboth values of the spin stiffness and the Tc.\nIn order to have a further insight into the details of\nthe magnetic interactions in the system, we report here\nthe orbital-resolved Jij’s between the nearest-neighbours\nobtained with LSDA. The results, shown in Table. C.1,\nreveal few interesting observations. First of all, all the0.6 1.2 1.8 2.4-0.0500.050.1Jij[mRy]\n0.6 1.2 1.8 2.400.10.2\n0.6 1.2 1.8 2.4\nRij/aalat00.10.20.30.4Jij[mRy]\n0.6 1.2 1.8 2.4\nRij/aalat00.511.5\nLSDA\nLSDA+DMFT [ Σ0]\nLSDA+DMFT [FLL]Co1-Co1Mn-Mn\nCo1-Co2Co-Mn\nFIG. C.1. (color online) The calculated exchange parameter s\nin Co 2MnSi within LSDA and LSDA+DMFT for different\nchoice of DC.\nTABLEC.1. Orbital-resolved Jij’sbetweenthenearestneigh-\nbours in Co 2MnSi in mRy. In the case of Co 1-Co1, the second\nnearest neighbour value is given, due to smallness of the firs t\none. The results were obtained with LSDA.\nTotalEg−EgT2g−T2gEg−T2gT2g−Eg\nCo1-Co10.070 0.077 -0.003 -0.002 -0.002\nCo1-Co20.295 0.357 -0.058 -0.002 -0.002\nCo-Mn 1.237 0.422 -0.079 0.700 0.194\nMn-Mn 0.124 -0.082 0.118 0.044 0.044\nT2g-derived contributions are negligible for all the inter-\nactions involving Co atoms. This has to do with the\nfact that these orbitals are practicallyfilled and therefore\ncan not participate in the exchange interactions. As to\nthe most dominant Co-Mn interaction, the Eg−Egand\nEg−T2gcontributions are both strong and contribute\nto the total ferromagnetic coupling. This is related to\nstrong spin polarisation of the Mn- Egstates.\n00.050.1\n0.6 1.2 1.8 2.4Jij[mRy]\n-0.15-0.1-0.0500.050.1\n0.61.21.82.4\n00.20.40.6\n0.61.21.82.4Jij[mRy]\nRij/alat00.511.522.53\n0.61.21.82.4\nRij/alatLSDA\nLSDA+U[FLL]\nLDA+U[AMF]Co1-Co1 Fe-Fe\nCo1-Co2\nCo-Fe\nFIG. C.2. Exchange interactions for Co 2FeSi within LSDA\nand LSDA+ Uschemes and a full potential approach for dif-\nferent DC choices.14\nCorrelationeffectsalsohaveprofoundeffectsontheex-\nchange interactions of Co 2FeSi. In particular, the Fe-Fe\ninteractions can be dramatically changed when consid-\nering static correlation effects. It is specially noticeable\nhow the anti-ferromagneticexchangeinteractions can de-\ncreasesignificantlywhichcanaffecttheexchangestiffness\nand the critical temperature as described in the maintext. Also the long-range nature of the Fe-Fe interac-\ntions is on display, indicating that to be able to predict\nmacroscopic variables from the present approach, atten-\ntion must be paid to the cut-off range. As in Co 2MnSi,\ncorrelation effect do not greatly affect the Co-Co ex-\nchange interactions.\n∗jonathan.chico@physics.uu.se\n1D. D. Awschalom and M. E. Flatt´ e, Nat. Phys. 3, 153\n(2007).\n2M. I. Katsnelson, V. Y. Irkhin, L. Chioncel, A. I. Licht-\nenstein, and R. A. de Groot, Rev. Mod. Phys. 80, 315\n(2008).\n3V. V. Kruglyak, S. O. Demokritov, and D. Grundler, J.\nPhys. D: Appl. Phys. 43, 264001 (2010).\n4T. Graf, C. Felser, and S. S. Parkin, Prog. Solid State Ch.\n39, 1 (2011), ISSN 0079-6786.\n5S. K. Malik, A. M. Umarji, and G. K. Shenoy, Phys. Rev.\nB31, 6971 (1985).\n6Y. Xia, V. Ponnambalam, S. Bhattacharya, A. L. Pope,\nS. J. Poon, and T. M. Tritt, J. Phys. Condens. Matter.\n13, 77 (2001).\n7I. Galanakis, P. H. Dederichs, and N. Papanikolaou, Phys.\nRev. B66, 174429 (2002).\n8I. Galanakis, K. ¨Ozdo˘ gan, E. S ¸a¸ sıo˘ glu, and B. Akta¸ s, Phys.\nRev. B75, 092407 (2007).\n9R. A. de Groot, Physica B 172, 45 (1991), ISSN 0921-\n4526.\n10S. Skaftouros, K. ¨Ozdo˘ gan, E. S ¸a¸ sıo˘ glu, and I. Galanakis,\nPhys. Rev. B 87, 024420 (2013).\n11S. Trudel, O. Gaier, J. Hamrle, and B. Hillebrands, J.\nPhys. D: Appl. Phys. 43, 193001 (2010).\n12M. Jourdan, J. Min´ ar, J. Braun, A. Kronenberg,\nS. Chadov, B. Balke, A. Gloskovskii, M. Kolbe, H. Elmers,\nG. Sch¨ onhense, et al., Nat. Commun. 5(2014).\n13B. Balke, G. H. Fecher, H. C. Kandpal, C. Felser,\nK. Kobayashi, E. Ikenaga, J.-J. Kim, and S. Ueda, Phys.\nRev. B74, 104405 (2006).\n14A. Sakuma, J. Phys. D: Appl. Phys. 48, 164011 (2015).\n15M. Oogane and S. Mizukami, Philos. T. R. Soc. A 369,\n3037 (2011), ISSN 1364-503X.\n16H. Ebert, D. K¨ odderitzsch, and J. Min´ ar, Rep. Prog. Phys.\n74, 096501 (2011).\n17S. H. Vosko, L. Wilk, and M. Nusair, Can. J. Phys. 58,\n1200 (1980).\n18J. P. Perdew, K. Burke, and M. Ernzerhof, Phys. Rev.\nLett.77, 3865 (1996).\n19P. Soven, Phys. Rev. 156, 809 (1967).\n20G. M. Stocks, W. M. Temmerman, andB.L. Gyorffy, Phys.\nRev. Lett. 41, 339 (1978).\n21J. Hubbard, P. R. Soc. Lond. A Mat. 276, 238 (1963),\nISSN 0080-4630.\n22G. H. Fecher and C. Felser, J. Phys. D: Appl. Phys. 40,\n1582 (2007).\n23L. Makinistian, M. M. Faiz, R. P. Panguluri, B. Balke,\nS. Wurmehl, C. Felser, E. A. Albanesi, A. G. Petukhov,\nand B. Nadgorny, Phys. Rev. B 87, 220402 (2013).\n24S. Wurmehl, G. H. Fecher, H. C. Kandpal, V. Ksenofontov,\nC. Felser, H.-J. Lin, and J. Morais, Phys. Rev. B 72,184434 (2005).\n25M. Karolak, G. Ulm, T. Wehling, V. Mazurenko,\nA. Poteryaev, and A. Lichtenstein, J. Electron Spectrosc.\n181, 11 (2010), ISSN 0368-2048, proceedings of Inter-\nnational Workshop on Strong Correlations and Angle-\nResolved Photoemission Spectroscopy 2009.\n26A. I. Liechtenstein, V. I. Anisimov, and J. Zaanen, Phys.\nRev. B52, R5467 (1995).\n27M. T. Czy˙ zyk and G. A. Sawatzky, Phys. Rev. B 49, 14211\n(1994).\n28A. Georges, G. Kotliar, W. Krauth, and M. J. Rozenberg,\nRev. Mod. Phys. 68, 13 (1996).\n29G. Kotliar, S. Y. Savrasov, K. Haule, V. S. Oudovenko,\nO. Parcollet, and C. A. Marianetti, Rev. Mod. Phys. 78,\n865 (2006).\n30J. M. Wills, O. Eriksson, M. Alouani, and D. L. Price,\ninElectronic Structure and Physical Properies of Solids ,\nedited by H. Dreysse (Springer Berlin Heidelberg, 2000),\nvol. 535 of Lecture Notes in Physics , pp. 148–167, ISBN\n978-3-540-67238-8.\n31O. Gr˚ an¨ as, I. D. Marco, P. Thunstr¨ om, L. Nordstr¨ om,\nO. Eriksson, T. Bj¨ orkman, and J. M. Wills, Comp. Mater.\nSci.55, 295 (2012), ISSN 0927-0256.\n32A. Grechnev, I. Di Marco, M. I. Katsnelson, A. I. Lichten-\nstein, J. Wills, and O. Eriksson, Phys. Rev. B 76, 035107\n(2007).\n33Katsnelson, M. I. and Lichtenstein, A. I., Eur. Phys. J. B\n30, 9 (2002).\n34A. I. Lichtenstein and M. I. Katsnelson, Phys. Rev. B 57,\n6884 (1998).\n35L. Chioncel, Y. Sakuraba, E. Arrigoni, M. I. Katsnelson,\nM. Oogane, Y. Ando, T. Miyazaki, E. Burzo, and A. I.\nLichtenstein, Phys. Rev. Lett. 100, 086402 (2008).\n36S. Chadov, G. H. Fecher, C. Felser, J. Min´ ar, J. Braun,\nand H. Ebert, J. Phys. D: Appl. Phys. 42, 084002 (2009).\n37A. I. Lichtenstein, M. I. Katsnelson, and G. Kotliar, Phys.\nRev. Lett. 87, 067205 (2001).\n38A. Liechtenstein, M. Katsnelson, V. Antropov, and\nV. Gubanov, J. Magn. Magn. Mater. 67, 65 (1987), ISSN\n0304-8853.\n39A. I. Liechtenstein, M. I. Katsnelson, and V. A. Gubanov,\nJ. Phys. F: Met. Phys. 14, L125 (1984).\n40M. I. Katsnelson and A. I. Lichtenstein, Phys. Rev. B 61,\n8906 (2000).\n41Y. O. Kvashnin, O. Gr˚ an¨ as, I. Di Marco, M. I. Katsnel-\nson, A. I. Lichtenstein, and O. Eriksson, Phys. Rev. B 91,\n125133 (2015).\n42K. Lejaeghere, G. Bihlmayer, T. Bj¨ orkman, P. Blaha,\nS. Bl¨ ugel, V. Blum, D. Caliste, I. E. Castelli, S. J. Clark,\nA. Dal Corso, et al., Science 351(2016), ISSN 0036-8075.\n43M. Pajda, J. Kudrnovsk´ y, I. Turek, V. Drchal, and\nP. Bruno, Phys. Rev. B 64, 174402 (2001).15\n44B. Skubic, J. Hellsvik, L. Nordstr¨ om, and O. Eriksson, J.\nPhys. Condens. Matter. 20, 315203 (2008).\n45K. Binder, Rep. Prog. Phys. 60, 487 (1997).\n46H. Ebert, S. Mankovsky, D. K¨ odderitzsch, and P. J. Kelly,\nPhys. Rev. Lett. 107, 066603 (2011).\n47H. Ebert, S. Mankovsky, K. Chadova, S. Polesya, J. Min´ ar,\nand D. K¨ odderitzsch, Phys. Rev. B 91, 165132 (2015).\n48W. H. Butler, Phys. Rev. B 31, 3260 (1985).\n49S. V. Halilov, H. Eschrig, A. Y. Perlov, and P. M. Oppe-\nneer, Phys. Rev. B 58, 293 (1998).\n50C. Etz, L. Bergqvist, A. Bergman, A. Taroni, and O.Eriks-\nson, J. Phys. Condens. Matter. 27, 243202 (2015).\n51K. Buschow and P. van Engen, J. Magn. Magn. Mater. 25,\n90 (1981), ISSN 0304-8853.\n52P. Webster, J. Phys. Chem. Solid 32, 1221 (1971), ISSN\n0022-3697.\n53T. Ambrose, J. J. Krebs, and G. A. Prinz, J. Appl. Phys.\n87, 5463 (2000).\n54H. C. Kandpal, G. H. Fecher, and C. Felser, J. Phys. D:\nAppl. Phys. 40, 1507 (2007).\n55K. Buschow, P. van Engen, and R. Jongebreur, J. Magn.\nMagn. Mater. 38, 1 (1983), ISSN 0304-8853.\n56K.¨Ozdo˘ gan, E. S ¸a¸ sıo˘ glu, B. Akta¸ s, andI. Galanakis, Phys .\nRev. B74, 172412 (2006).\n57P. J. Brown, K. U. Neumann, P. J. Webster, and K. R. A.\nZiebeck, J. Phys. Condens. Matter. 12, 1827 (2000).\n58K. Ramesh Kumar, K. Bharathi, J. Chelvane,\nS. Venkatesh, G. Markandeyulu, and N. Harishkumar,\nIEEE Trans. Magn. 45, 3997 (2009), ISSN 0018-9464.\n59A. G. Petukhov, I. I. Mazin, L. Chioncel, and A. I. Licht-\nenstein, Phys. Rev. B 67, 153106 (2003).\n60The elk fp-lapw code ,http://elk.sourceforge.net/ , ac-\ncessed: 2016-03-17.\n61C. Tsirogiannis and I. Galanakis, J. Magn. Magn. Mater.\n393, 297 (2015), ISSN 0304-8853.\n62Note1, if one utilizes the “Σ(0)”DC for SPTF performed\nfor the bare Green’s function, then the static terms are\nstrictly preserved. In our case there is some renormaliza-\ntion due to the Hartree-Fock renormalization.\n63E. S ¸a¸ sıo˘ glu, L. M. Sandratskii, P. Bruno, and I. Galanaki s,\nPhys. Rev. B 72, 184415 (2005).\n64V. Y. Irkhin, M. I. Katsnelson, and A. I. Lichtenstein, J.\nPhys. Condens. Matter. 19, 315201 (2007).\n65K.¨Ozdogan, I. Galanakis, E. S ¸a¸ sioglu, and B. Akta¸ s, J.\nPhys. Condens. Matter. 18, 2905 (2006).\n66K. R. Kumar, N. H. Kumar, G. Markandeyulu, J. A. Chel-\nvane, V. Neu, and P. Babu, J. Magn. Magn. Mater. 320,\n2737 (2008), ISSN 0304-8853.\n67E. S ¸a¸ sioglu, L. M. Sandratskii, and P. Bruno, J. Phys.\nCondens. Matter. 17, 995 (2005).\n68J. Hamrle, O. Gaier, S.-G. Min, B. Hillebrands,\nY. Sakuraba, and Y. Ando, J. Phys. D: Appl. Phys. 42,\n084005 (2009).\n69H. Pandey, P. C. Joshi, R. P. Pant, R. Prasad, S. Auluck,\nand R. C. Budhani, J. Appl. Phys. 111, 023912 (2012).\n70J. Thoene, S. Chadov, G. Fecher, C. Felser, and J. K¨ ubler,\nJ. Phys. D: Appl. Phys. 42, 084013 (2009).\n71B. Belhadji, L. Bergqvist, R. Zeller, P. H. Dederichs,\nK. Sato, and H. Katayama-Yoshida, J. Phys. Condens.\nMatter.19, 436227 (2007).72T. Kubota, J. Hamrle, Y. Sakuraba, O. Gaier, M. Oogane,\nA. Sakuma, B. Hillebrands, K. Takanashi, and Y. Ando,\nJ. Appl. Phys. 106(2009).\n73R. Umetsu, A. Okubo, A. Fujita, T. Kanomata, K. Ishida,\nand R. Kainuma, IEEE Trans. Magn. 47, 2451 (2011),\nISSN 0018-9464.\n74S. Trudel, J. Hamrle, B. Hillebrands, T. Taira, and M. Ya-\nmamoto, J. Appl. Phys. 107(2010).\n75O.Gaier, J.Hamrle, S.Trudel, A.C. Parra, B.Hillebrands,\nE. Arbelo, C. Herbort, and M. Jourdan, J. Phys. D: Appl.\nPhys.42, 084004 (2009).\n76M. Zhang, E. Brck, F. R. de Boer, Z. Li, and G. Wu, J.\nPhys. D: Appl. Phys. 37, 2049 (2004).\n77O. Gaier, J. Hamrle, S. Trudel, B. Hillebrands, H. Schnei-\nder, and G. Jakob, J. Phys. D: Appl. Phys. 42, 232001\n(2009).\n78D. Comtesse, B. Geisler, P. Entel, P. Kratzer, and L. Szun-\nyogh, Phys. Rev. B 89, 094410 (2014).\n79M. Belmeguenai, H. Tuzcuoglu, M. S. Gabor, T. Petrisor,\nC. Tiusan, F. Zighem, S. M. Chrif, and P. Moch, J. Appl.\nPhys.115(2014).\n80X. B. Liu and Z. Altounian, J. Appl. Phys. 109(2011).\n81S. Keshavarz, Y. O. Kvashnin, I. Di Marco, A. Delin, M. I.\nKatsnelson, A.I.Lichtenstein, andO.Eriksson, Phys.Rev.\nB92, 165129 (2015).\n82Note2, this manifests itself in the larger magnetic moment\nvalues, as was discussed above.\n83P. Buczek, A. Ernst, P. Bruno, and L. M. Sandratskii,\nPhys. Rev. Lett. 102, 247206 (2009).\n84J. K¨ ubler, Theory of itinerant electron magnetism , vol. 106\n(Oxford University Press, 2000).\n85R. Y. Umetsu and T. Kanomata, Physics Procedia 75, 890\n(2015), ISSN 1875-3892, 20th International Conference on\nMagnetism, ICM 2015.\n86M. Oogane, T. Kubota, Y. Kota, S. Mizukami, H. Na-\nganuma, A. Sakuma, and Y. Ando, Appl. Phys. Lett. 96\n(2010).\n87R. Yilgin, M. Oogane, Y. Ando, and T. Miyazaki, J. Magn.\nMagn. Mater. 310, 2322 (2007), ISSN 0304-8853, pro-\nceedings of the 17th International Conference on Mag-\nnetismThe International Conference on Magnetism.\n88S. J. Hashemifar, P. Kratzer, and M. Scheffler, Phys. Rev.\nLett.94, 096402 (2005).\n89S. Andrieu, A. Neggache, T. Hauet, T. Devolder, A. Hallal,\nM. Chshiev, A. M. Bataille, P. Le F` evre, and F. Bertran,\nPhys. Rev. B 93, 094417 (2016).\n90S. Mankovsky, D. K¨ odderitzsch, G. Woltersdorf, and\nH. Ebert, Phys. Rev. B 87, 014430 (2013).\n91S. Lounis, M. dosSantosDias, and B. Schweflinghaus, Phys.\nRev. B91, 104420 (2015).\n92T. Kubota, S. Tsunegi, M. Oogane, S. Mizukami,\nT. Miyazaki, H. Naganuma, and Y. Ando, Appl. Phys.\nLett.94(2009).\n93R. D. McMichael, D. J. Twisselmann, and A. Kunz, Phys.\nRev. Lett. 90, 227601 (2003).\n94R.Yilgin, Y.Sakuraba, M. Oogane, S.Mizukami, Y.Ando,\nand T. Miyazaki, Jpn. J. Appl. Phys. 46, L205 (2007).\n95S. Qiao, J. Zhang, R. Hao, H. Zhaong, Y. Kang, and\nS. Kang, in Magnetics Conference (INTERMAG), 2015\nIEEE(2015), pp. 1–1.\n96L. Vegard, Z. Phys. 5, 17 (1921), ISSN 0044-3328."    },    {        "title": "1605.01694v2.Theory_of_magnon_motive_force_in_chiral_ferromagnets.pdf",        "content": "Theory of magnon motive force in chiral ferromagnets\nUtkan G ung ord u\u0003and Alexey A. Kovalev\nDepartment of Physics and Astronomy and Nebraska Center for Materials and Nanoscience,\nUniversity of Nebraska, Lincoln, Nebraska 68588, USA\nWe predict that magnon motive force can lead to temperature dependent, nonlinear chiral damping\nin both conducting and insulating ferromagnets. We estimate that this damping can signi\fcantly\nin\ruence the motion of skyrmions and domain walls at \fnite temperatures. We also \fnd that in\nsystems with low Gilbert damping moving chiral magnetic textures and resulting magnon motive\nforces can induce large spin and energy currents in the transverse direction.\nPACS numbers: 85.75.-d, 72.20.Pa, 75.30.Ds, 75.78.-n\nEmergent electromagnetism in the context of spintron-\nics [1] brings about interpretations of the spin-transfer\ntorque [2, 3] and spin-motive force (SMF) [4{10] in terms\nof \fctitious electromagnetic \felds. In addition to pro-\nviding beautiful interpretations, these concepts are also\nvery useful in developing the fundamental understanding\nof magnetization dynamics. A time-dependent magnetic\ntexture is known to induce an emergent gauge \feld on\nelectrons [5]. As it turns out, the spin current gener-\nated by the resulting \fctitious Lorentz force (which can\nalso be interpreted as dynamics of Berry-phase leading\nto SMF) in\ruences the magnetization dynamics in a dis-\nsipative way [5, 6, 11{16], a\u000becting the phenomenologi-\ncal Gilbert damping term in the Landau-Lifshitz-Gilbert\n(LLG) [17] equation. Inadequacy of the simple Gilbert\ndamping term has recently been seen experimentally in\ndomain wall creep motion [18]. Potential applications\nof such studies include control of magnetic solitons such\nas domain walls and skyrmions [19{31], which may lead\nto faster magnetic memory and data storage devices\nwith lower power requirements [32{34]. Recently, phe-\nnomena related to spin currents and magnetization dy-\nnamics have also been studied in the context of energy\nharvesting and cooling applications within the \feld of\nspincaloritronics [35{39].\nMagnons, the quantized spin-waves in a magnet, are\npresent in both conducting magnets and insulating mag-\nnets. Treatment of spin-waves with short wavelengths as\nquasiparticles allows us to draw analogies from systems\nwith charge carriers. For instance, the \row of thermal\nmagnons generates a spin transfer torque (STT) [40{42]\nand a time-dependent magnetic texture exerts a magnon\nmotive force. According to the Schr odinger-like equa-\ntion which governs the dynamics of magnons in the adia-\nbatic limit [41], the emergent \\electric\" \feld induced by\nthe time-dependent background magnetic texture exerts\na \\Lorentz force\" on magnons, which in turn generates\na current by \\Ohm's law\" (see Fig. 1). Despite the sim-\nilarities, however, the strength of this feedback current\nhas important di\u000berences from its electronic analog: it is\ninversely proportional to the Gilbert damping and grows\n\u0003ugungordu@unl.eduwith temperature.\nIn this paper, we formulate a theory of magnon feed-\nback damping induced by the magnon motive force. We\n\fnd that this additional damping strongly a\u000bects the dy-\nnamics of magnetic solitons, such as domain walls and\nskyrmions, in systems with strong Dzyaloshinskii-Moriya\ninteractions (DMI). We also \fnd that the magnon mo-\ntive force can lead to magnon accumulation (see Fig. 1),\nnon-vanishing magnon chemical potential, and large spin\nand energy currents in systems with low Gilbert damp-\ning. To demonstrate this, we assume di\u000busive transport\nof magnons in which the magnon non-conserving relax-\nation time, \u001c\u000b, is larger compared to the magnon conserv-\ning one,\u001cm(\u001c\u000b\u001d\u001cm). For the four-magnon thermal-\nization,\u001cm=~=(kBT)(Tc=T)3, and for LLG damping,\n\u001c\u000b=~=\u000bkBT, this leads to the constraint \u000b(Tc=T)3\u001c1\n[43, 44].\nEmergent electromagnetism for magnons. We initially\nassume that the magnon chemical potential is zero. The\nvalidity of this assumption is con\frmed in the last sec-\ntion. E\u000bects related to emergent electromagnetism for\nmagnons can be captured by considering a ferromagnet\nwell below the Curie temperature. We use the stochastic\nLLG equation:\ns(1 +\u000bn\u0002)_n=n\u0002(He\u000b+h); (1)\nwheresis the spin density along n,He\u000b=\u0000\u000enF[n]\nis the e\u000bective magnetic \feld, F[n] =R\nd3rF(n) is the\nfree energy and his the random Langevin \feld. It is\nconvenient to consider the free energy density F(n) =\nJ(@in)2=2 +^Dei\u0001(n\u0002@in) +H\u0001n+Kun2\nzwhereJis\nthe exchange coupling, ^Dis a tensor which describes the\nDMI [47],H=Ms\u00160Haezdescribes the magnetic \feld,\nKudenotes the strength of uniaxial anisotropy, Msis\nthe saturation magnetization, Hais the applied magnetic\n\feld, and summation over repeated indices is implied. At\nsu\u000eciently high temperatures, the form of anisotropies is\nunimportant for the discussion of thermal magnons and\ncan include additional magnetostatic and magnetocrys-\ntalline contributions.\nLinearized dynamics of magnons can be captured by\nthe following equation [48]:\ns(i@t+ns\u0001At) =\u0002\nJ(@i=i\u0000ns\u0001[Ai\u0000Di=J])2+'\u0003\n ;\n(2)arXiv:1605.01694v2  [cond-mat.mes-hall]  13 Jul 20162\nμ/μ0\n-0.3-0.2-0.100.10.20.3\nμ/μ0\n-0.75-0.50-0.2500.250.500.75\nFIG. 1. (Color online) A moving magnetic texture, such as\na domain wall (top) or an isolated skyrmion (bottom), gen-\nerates an emergent electric \feld and accumulates a cloud of\nmagnons around it. In-plane component of nsand electric\n\feldEare represented by small colored arrows and large black\narrows, respectively. Magnon chemical potential \u0016is mea-\nsured in\u00160=\u0018~vD=J \u0001 for domain wall and \u00160=\u0018~vD=JR\nfor skyrmion, where \u0018is the magnon di\u000busion length. Soli-\ntons are moving along + xaxis with velocity v,nz=\u00061 at\nx=\u00071 for domain wall and nz= 1 at the center for the\nisolated hedgehog skyrmion. Material parameters for Co/Pt\n(J= 16pJ/m, D= 4mJ/m2,Ms= 1:1MA/m,\u000b= 0:03,\nat room temperature [45]) were used for domain wall lead-\ning to \u0001\u00197nm, and Cu 2OSeO 3parameters ( J= 1:4pJ/m,\nD= 0:17mJ/m2,s= 0:5~=a3,a= 0:5nm with\u000b= 0:01, at\nT\u001850K [46]) for skyrmion leading to R\u001950nm. System\nsize is taken to be 6\u0001 \u00022\u0001 for domain wall and 6 R\u00026Rfor\nskyrmion.\nwhere'absorbs e\u000bect of anisotropies, DMI and the\nmagnetic \feld,  =nf\u0001(e0\nx+ie0\ny) describes \ructu-\nationsnf=n\u0000nsq\n1\u0000n2\nfaround slow component\nns(jnj=jnsj= 1,ns?nf) in a rotated frame in\nwhiche0\nz=ns,Di=^Dei, andA\u0016\u0002\u0011 ^R@\u0016^RTcorre-\nsponds to the gauge potential with \u0016=x;y;z;t . Note\nthat in the rotated frame, we have n!n0=^Rnand\n@\u0016!(@\u0016\u0000A0\n\u0016\u0002) withA0\n\u0016\u0002= (@\u0016^R)^RT. In deriv-\ning Eq. 2, we assumed that the exchange interaction is\nthe dominant contribution and neglected the coupling\nbetween the circular components of  and ydue to\nanisotropies [41, 49].\nThe gauge potential in Eq. (2) leads to a reactive\ntorque in the LLG equation for the slow dynamics [50].\nAlternatively, one can simply average Eq. (1) over the\nfast oscillations arriving at the LLG equation with the\nmagnon torque term [40]:\ns(1 +\u000bns\u0002)_ns\u0000ns\u0002Hs\ne\u000b=~(j\u0001D)ns;(3)\nwhereHs\ne\u000b=\u0000\u000ensF[ns] is the e\u000bective \feld for theslow magnetization calculated at zero temperature [51],\nji= (J=~)hns\u0001(nf\u0002@inf)iis the magnon current and\nDi=@i+ (^Dei=J)\u0002is the chiral derivative [48, 52, 53].\nMagnon feedback damping. The magnon current jis\ninduced in response to the emergent electromagnetic po-\ntential, and can be related to the driving electric \feld\nEi=~ns\u0001(@tns\u0002Dins) by local Ohm's law j=\u001bE\nwhere\u001bis magnon conductivity. The induced elec-\ntric \feld Ecan be interpreted as a magnon generaliza-\ntion of the spin motive force [6]. The magnon feed-\nback torque\u001c=~\u001b(E\u0001D)nshas dissipative e\u000bect on\nmagnetization dynamics and leads to a damping tensor\n^\u000bemf=\u0011(ns\u0002Dins)\n(ns\u0002Dins) in the LLG equation\nwith\u0011=~2\u001b=s[54].\nA general form of the feedback damping should also\ninclude the contribution from the dissipative torque\n[40]. Here we introduce such \f-terms phenomenologically\nwhich leads to the LLG equation:\ns(1 +ns\u0002[^\u000b+^\u0000])_ns\u0000ns\u0002Hs\ne\u000b=\u001c; (4)\nwhere\u001cis the magnon torque term and we separated the\ndissipative ^ \u000band reactive ^\u0000 contributions:\n^\u000b=\u000b+^\u000bemf\u0000\u0011\f2Dins\nDins; (5)\n^\u0000 =\u0011\f[(ns\u0002Dins)\nDins\u0000Dins\n(ns\u0002Dins)];\nwhere in general the form of chiral derivatives in the \f-\nterms can be di\u000berent. Given that \fand\u000bare typically\nsmall for magnon systems, the term ^ \u000bemfwill dominate\nthe feedback damping tensor. An unusual feature of the\nchiral part of the damping is that it will be present even\nfor a uniform texture. While the DMI prefers twisted\nmagnetic structures, this can be relevant in the presence\nof an external magnetic \feld strong enough to drive the\nsystem into the ferromagnetic phase.\nIn conducting ferromagnets, charge currents also lead\nto a damping tensor of the same form where the strength\nof the damping is characterized by \u0011e=~2\u001be=4e2swith\n\u001beas the electronic conductivity [11, 13, 15], which\nshould be compared to \u0011in conducting ferromagnets\nwhere both e\u000bects are present. Since the magnon feed-\nback damping \u0011grows as/1=\u000b, the overall strength of\nmagnon contribution can quickly become dominant con-\ntribution in ferromagnets with small Gilbert damping.\nUnder the assumption that magnon scattering is domi-\nnated by the Gilbert damping such that the relaxation\ntime is given by \u001c\u000b= 1=2\u000b!, magnon conductivity is\ngiven by\u001b3D\u00181=6\u00192\u0015~\u000bin three-dimensions and \u001b2D\u0018\n1=4\u0019~\u000bin two-dimensions [40] where \u0015=p\n~J=skBTis\nthe wavelength of the thermal magnons. For Cu 2OSeO 3\nin ferromagnetic phase, we \fnd \u0011\u00192nm2. Similarly,\nfor a Pt/Co/AlO xthin \flm of thickness t= 0:6nm yield\n\u0011\u00191nm2at room temperature. This shows that the\nmagnon feedback damping can become signi\fcant in fer-\nromagnets with sharp textures and strong DMI.\nDomain wall dynamics. We describe the domain wall\npro\fle in a ferromagnet with DMI by Walker ansatz3\ntan(\u0012(x;t)=2) = exp(\u0006[x\u0000X(t)]=\u0001) whereX(t) and\n\u001e(t) denote the center position and tilting angle of the\ndomain wall [55], \u0001 =p\nJ=K 0is the domain wall width,\nK0=Ku\u0000\u00160M2\ns=2 includes the contributions from\nuniaxial anisotropy as well as the demagnetizing \feld\nand ^D=\u0000D(sin\r11 + cos\rez\u0002) contains DMI due to\nbulk and structure inversion asymmetries whose relative\nstrength is determined by \r. After integrating the LLG\nequation, we obtain the equations of motion for a domain\nwall driven by external perpendicular \feld [56]:\n\u0000XX_X=\u0001 + _\u001e=FX;\u0000\u001e\u001e_\u001e\u0000_X=\u0001 =F\u001e;(6)\nwhere \u0000XX=\u000b+\u0011(D=J)2sin2(\r+\u001e)=3 and \u0000\u001e\u001e=\n\u000b+\u0011[2=3\u00012+ (\u0019D=2J\u0001) cos(\r+\u001e) + (D=J)2cos2(\r+\n\u001e)] are dimensionless angle-dependent drag coe\u000ecients,\nFX=H=s andF\u001e= [Ksin 2\u001e+ sin(\r+\u001e)D\u0019=2\u0001]=sare\ngeneralized \\forces\" associated with the collective coordi-\nnatesXand\u001e,Kis the strength of an added anisotropy\ncorresponding, e.g., to magnetostatic anisotropy K=\nNx\u00160M2\ns=2 whereNxis the demagnetization coe\u000ecient.\nIn deriving these equations, we have neglected higher or-\nder terms in \u000band\f[57].\nTime-averaged domain wall velocity obtained from\nnumerical integration of the equations of motion for\na Co/Pt interface with Rashba-like DMI is shown in\nFig. 2. Thermal magnon wavelength at room tempera-\nture (\u00190:3nm) is much shorter than the domain wall size\n\u0001 =p\nJ=K 0\u00197nm, so the quasiparticle treatment of\nmagnons is justi\fed. We observe that damping reduces\nthe speed at \fxed magnetic \feld, and this e\u000bect is en-\nhanced with increasing DMI strength Dand diminishing\nthe Gilbert damping \u000b(see Fig. 2).\nAnother important observation is that in the presence\nof the feedback damping, the relation between applied\n\feld and average domain wall velocity becomes nonlin-\near. This is readily seen from steady state solution of the\nequations of motion before the Walker breakdown with\n\u001e=\u001e0which solves sin( \r+\u001e0)D\u0019=2s\u0001 =\u0000[H=s][\u000b+\n\u0011(D=J)2sin2(\r+\u001e0)=3]\u00001(noting that D=\u0001\u001dK,\nimplying a N\u0013 eel domain wall [56, 58]) and X=vt,\nleading to the cubic velocity-\feld relation ( sv=\u0001)(\u000b+\n\u0011[2sv=J\u0019 ]2=3) =Hfor \feld-driven domain wall motion.\nThe angle\u001e0also determines the tilting of Eas seen in\nFig. 1.\nSkyrmion dynamics. Under the assumption that the\nskyrmion retains its internal structure as it moves, we\ntreat it as a magnetic texture ns=ns(r\u0000q(t)) with\nq(t) being the time-dependent position (collective coor-\ndinate [61]) of the skyrmion. We consider the motion\nof a skyrmion under the temperature gradient r\u001f=\n\u0000rT=T, which exerts a magnon torque:\n\u001c= (1 +\fTns\u0002)(Lr\u001f\u0001D)ns; (7)\nwhereLis the spin Seebeck coe\u000ecient and \fTis the \\\f-\ntype\" correction. These are given by L3D\u0018kBT=6\u00192\u0015\u000b\nand\fT\u00193\u000b=2 in three-dimensions and L2D\u0018kBT=4\u0019\u000b\nand\fT\u0019\u000bin two-dimensions within the relaxation time\nD=4mJ/m2\nD=2mJ/m2\nD=1mJ/m2\n0 10 20 30 4002004006008001000\nμ0Ha(mT)〈X〉(m/s)Co/Pt\nD=1.5mJ/m2D=1mJ/m2D=0.5mJ/m2\n0 1 2 3 402004006008001000\nμ0Ha(mT)〈X〉(m/s)Pt/CoFeB/MgOFIG. 2. (Color online) Domain wall velocity as a func-\ntion of the magnetic \feld and varying strength of DMI for\nCo/Pt and Pt/CoFeB/MgO \flms. Solid (dashed) lines cor-\nrespond to dynamics at zero (room) temperature. We used\nmaterial parameters Ms= 1:1MA/m,J= 16pJ/m, K0=\n0:34MJ/m3,\u000b= 0:03 [45] for Co/Pt, and Ms= 0:43MA/m,\nJ= 31pJ/m, K0= 0:38MJ/m3,\u000b= 4\u000210\u00003[31, 59, 60] for\nPt/CoFeB/MgO.\napproximation [41, 42]. Multiplying the LLG equation\nEq. (4) withR\nd2r@qjns\u0001ns\u0002and substituting _ns=\n\u0000_qi@qins, we obtain the equation of motion for v=_q:\ns(W\u0000Qz\u0002)v+ (\fT\u0011D\u0000Qz\u0002)Lr\u001f=F:(8)\nAbove,W=\u00110\u000b+\u0011\u000b0can be interpreted as the con-\ntribution of the renormalized Gilbert damping, Q=R\nd2rns\u0001(@xns\u0002@yns)=4\u0019is the topological charge of the\nskyrmion,\u00110is the dyadic tensor, \u0011Dis the chiral dyadic\ntensor which is\u0018\u00110for isolated skyrmions and vanishes\nfor skyrmions in SkX lattice [48] (detailed de\fnitions of\nthese coe\u000ecients are given in the Supplemental Material\n[62]). The \\force\" term F=\u0000rU(q) due to the e\u000bec-\ntive skyrmion potential U(q) is relevant for systems with\nspatially-dependent anisotropies [63], DMI [64], or mag-\nnetic \felds. In deriving this equation, we only considered\nthe dominant feedback damping contribution ^ \u000bemfwhich\nis justi\fed for small \u000band\f. For temperature gradients\nand forces along the x-axis we obtain velocities:\nvx=\u0000L@x\u001f(Q2+W\fT\u0011D) +FxW\ns(Q2+W2);\nvy=\u0000L@x\u001fQ(\fT\u0011D\u0000W) +FxQ\ns(Q2+W2): (9)\nThe Hall angle de\fned as tan \u0012H=vy=vxis strongly af-\nfected by the renormalization of Wsince tan\u0012H=Q=W\nfor a \\force\" driven skyrmion and tan \u0012H\u0019(\fT\u00110\u0000\nW)=Qfor a temperature gradient driven skyrmion. Sim-\nilar to the domain wall velocity in Fig. 2, the Hall e\u000bect\nwill depend on the overall temperature of the system.\nWe \fnd that for a skyrmion driven by @x\u001f, the Hall an-\ngle\u0012Hmay \rip the sign in magnets with strong DMI\nas the temperature increases. We estimate this should\nhappen in Cu 2OSeO 3atT\u001850K using a typical radial\npro\fle for a rotationally symmetric skyrmion given by\nUsov ansatz cos( \u0012=2) = (R2\u0000r2)=(R2+r2) forr\u0014R\nandR\u00192\u0019J=D\u001952nm.\nMagnon pumping and accumulation. The motion of\nskyrmions induces a transverse magnon current across4\nthe sample. This e\u000bect can be quanti\fed by the av-\nerage magnon current due to magnon motive force per\nskyrmion:\nj=\u001bZ\nd2rE=\u0019R2= (v\u0002ez)4\u001b~2Q=R2: (10)\nThe current can only propagate over the magnon di\u000bu-\nsion length; thus, it can be observed in materials with\nlarge magnon di\u000busion length or small Gilbert damping.\nμ/μ0\n-1.5-1.0-0.500.51.01.5\nFIG. 3. (Color online) An array of moving skyrmions (only 3\nshown in the \fgure) induces a transverse current and accumu-\nlation of magnons along the edges . \u0016is obtained by numer-\nically solving the di\u000busion equation using material parame-\nters for Pt/CoFeB/MgO given in the caption of Fig. 2 with\nR= 35nm. System height and distance between skyrmion\ncenters are taken to be 3 R.\nSo far, we have assumed a highly compressible limit in\nwhich we disregard any build up of the magnon chem-\nical potential \u0016. In a more realistic situation the build\nup of the chemical potential will lead to magnon di\u000bu-\nsion. To illustrate the essential physics, we consider a\nsituation in which the temperature is uniform. For slow\nmagnetization dynamics in which magnons quickly estab-\nlish a stationary state (i.e. R=v\u001d\u001c\u000bfor skyrmions and\n\u0001=_X\u001d\u001c\u000bfor domain walls, which is satis\fed at high\nenough temperatures) we write a stationary magnon dif-\nfusion equation:\nr2\u0016=\u0016\n\u00182+r\u0001E; (11)\nwhere\u0018=\u0015=2\u0019\u000bis the magnon di\u000busion length and\nwe used the local Ohm's law \u0000r\u0016=j=\u001b\u0000E. Renor-\nmalization of magnon current in Eq. (3) then follows\nfrom solution of the screened Poisson equation j=\u001bE+(\u001b=4\u0019)rR\nd3r0(r0\u0001E)e\u0000jr\u0000r0j=\u0018=jr\u0000r0jin three dimen-\nsions andj=\u001bE+ (\u001b=2\u0019)rR\nd2r0(r0\u0001E)K0(jr\u0000r0j=\u0018)\nin two dimensions where K0is the modi\fed Bessel func-\ntion for an in\fnitely large system [65]. By analyzing\nthe magnon current due to magnon accumulation ana-\nlytically and numerically, we \fnd that renormalization\nbecomes important when the length associated with the\nmagnetic texture is much smaller than the magnon dif-\nfusion length.\nFinally, we numerically solve Eq. (11) for isolated soli-\ntons (see Fig. 1) and for an array of moving skyrmions\n(see Fig. 3). Given that the width of the strip in Fig. 3\nis comparable to the magnon di\u000busion length one can\nhave substantial accumulation of magnons close to the\nboundary. Spin currents comparable to the estimate in\nEq. (10) can be generated in this setup and further de-\ntected by the inverse spin Hall e\u000bect [66]. From Eq. (10),\nfor a skyrmion with R= 35nm moving at 10m/s in\nPt/CoFeB/MgO with D= 1:5mJ/m2[31], we obtain an\nestimate for spin current js=j~\u001810\u00007J/m2which\nroughly agrees with the numerical results. This spin cur-\nrent will also carry energy and as a result will lead to a\ntemperature drop between the edges.\nConclusion. We have developed a theory of magnon\nmotive force in chiral conducting and insulating ferro-\nmagnets. The magnon motive force leads to tempera-\nture dependent, chiral feedback damping. The e\u000bect of\nthis damping can be seen in the non-linear, temperature\ndependent behavior of the domain wall velocity. In ad-\ndition, observation of the temperature dependent Hall\nangle of skyrmion motion can also reveal this additional\ndamping contribution. We have numerically con\frmed\nthe presence of the magnon feedback damping in \fnite-\ntemperature micromagnetic simulations of Eq. (1) using\nMuMaX3 [67].\nMagnon pumping and accumulation will also result\nfrom the magnon motive force. Substantial spin and en-\nergy currents can be pumped by a moving chiral tex-\nture in systems in which the size of magnetic textures\nis smaller or comparable to the magnon di\u000busion length.\nFurther studies could concentrate on magnetic systems\nwith low Gilbert damping, such as yttrium iron garnet\n(YIG), in which topologically non-trivial bubbles can be\nrealized.\nThis work was supported primarily by the DOE Early\nCareer Award de-sc0014189, and in part by the NSF un-\nder Grants Nos. Phy-1415600, and DMR-1420645 (UG).\n[1] I. \u0014Zuti\u0013 c, J. Fabian, and S. Das Sarma, Rev. Mod. Phys.\n76, 323 (2004).\n[2] J. Slonczewski, J. Magn. Magn. Mater. 159, L1 (1996).\n[3] L. Berger, Phys. Rev. B 54, 9353 (1996).\n[4] L. Berger, Phys. Rev. B 33, 1572 (1986).\n[5] G. E. Volovik, J. Phys. C Solid State Phys. 20, L83\n(1987).\n[6] S. E. Barnes and S. Maekawa, Phys. Rev. Lett. 98,246601 (2007).\n[7] S. A. Yang, G. S. D. Beach, C. Knutson, D. Xiao, Q. Niu,\nM. Tsoi, and J. L. Erskine, Phys. Rev. Lett. 102, 067201\n(2009).\n[8] J.-i. Ohe, S. E. Barnes, H.-W. Lee, and S. Maekawa,\nApplied Physics Letters 95, 123110 (2009).\n[9] Y. Yamane, K. Sasage, T. An, K. Harii, J. Ohe, J. Ieda,\nS. E. Barnes, E. Saitoh, and S. Maekawa, Phys. Rev.5\nLett. 107, 236602 (2011).\n[10] Y. Yamane, S. Hemmatiyan, J. Ieda, S. Maekawa, and\nJ. Sinova, Sci. Rep. 4, 6901 (2014).\n[11] S. Zhang and S. S.-L. Zhang, Phys. Rev. Lett. 102,\n086601 (2009).\n[12] Y. Tserkovnyak and C. H. Wong, Phys. Rev. B 79,\n014402 (2009).\n[13] C. H. Wong and Y. Tserkovnyak, Phys. Rev. B 81,\n060404 (2010).\n[14] M. F ahnle and C. Illg, J. Phys. Condens. Matter 23,\n493201 (2011).\n[15] K.-W. Kim, J.-H. Moon, K.-J. Lee, and H.-W. Lee, Phys.\nRev. Lett. 108, 217202 (2012).\n[16] J.-V. Kim, Phys. Rev. B 92, 014418 (2015).\n[17] T. Gilbert, IEEE Trans. Magn. 40, 3443 (2004).\n[18] E. Ju\u0013 e, C. K. Safeer, M. Drouard, A. Lopez, P. Balint,\nL. Buda-Prejbeanu, O. Boulle, S. Au\u000bret, A. Schuhl,\nA. Manchon, et al., Nat. Mater. 15, 272 (2015).\n[19] A. Bogdanov and A. Hubert, J. Magn. Magn. Mater.\n138, 255 (1994).\n[20] U. K. R o\u0019ler, A. N. Bogdanov, and C. P\reiderer, Nature\n442, 797 (2006).\n[21] S. Muhlbauer, B. Binz, F. Jonietz, C. P\reiderer,\nA. Rosch, A. Neubauer, R. Georgii, and P. Boni, Science\n323, 915 (2009).\n[22] X. Z. Yu, Y. Onose, N. Kanazawa, J. H. Park, J. H. Han,\nY. Matsui, N. Nagaosa, and Y. Tokura, Nature 465, 901\n(2010).\n[23] F. Jonietz, S. Muhlbauer, C. P\reiderer, A. Neubauer,\nW. Munzer, A. Bauer, T. Adams, R. Georgii, P. Boni,\nR. A. Duine, et al., Science 330, 1648 (2010).\n[24] S. Heinze, K. von Bergmann, M. Menzel, J. Brede, A. Ku-\nbetzka, R. Wiesendanger, G. Bihlmayer, and S. Bl ugel,\nNat. Phys. 7, 713 (2011).\n[25] N. S. Kiselev, A. N. Bogdanov, R. Sch afer, U. K. R. Ler,\nand U. K. R o\u0019ler, J. Phys. D. Appl. Phys. 44, 392001\n(2011).\n[26] T. Schulz, R. Ritz, A. Bauer, M. Halder, M. Wagner,\nC. Franz, C. P\reiderer, K. Everschor, M. Garst, and\nA. Rosch, Nat. Phys. 8, 301 (2012).\n[27] J. Iwasaki, M. Mochizuki, and N. Nagaosa, Nat. Com-\nmun. 4, 1463 (2013).\n[28] A. Fert, V. Cros, and J. Sampaio, Nat. Nanotechnol. 8,\n152 (2013).\n[29] F. B uttner, C. Mouta\fs, M. Schneider, B. Kr uger, C. M.\nG unther, J. Geilhufe, C. v. K. Schmising, J. Mohanty,\nB. Pfau, S. Scha\u000bert, et al., Nat. Phys. 11, 225 (2015).\n[30] Y. Tokunaga, X. Z. Yu, J. S. White, H. M. R\u001cnnow,\nD. Morikawa, Y. Taguchi, and Y. Tokura, Nat. Commun.\n6, 7638 (2015).\n[31] S. Woo, K. Litzius, B. Kr uger, M.-y. Im, L. Caretta,\nK. Richter, M. Mann, A. Krone, R. M. Reeve,\nM. Weigand, et al., Nat. Mater. 15, 501 (2016).\n[32] S. S. P. Parkin, M. Hayashi, and L. Thomas, Science 320,\n190 (2008).\n[33] J. Sampaio, V. Cros, S. Rohart, A. Thiaville, and A. Fert,\nNat. Nanotechnol. 8, 839844 (2013).\n[34] A. Brataas, A. D. Kent, and H. Ohno, Nat. Mater. 11,\n372 (2012).\n[35] M. Hatami, G. E. W. Bauer, Q. Zhang, and P. J. Kelly,\nPhys. Rev. Lett. 99, 066603 (2007).\n[36] G. E. W. Bauer, S. Bretzel, A. Brataas, and\nY. Tserkovnyak, Phys. Rev. B 81, 024427 (2010).\n[37] A. A. Kovalev and Y. Tserkovnyak, Phys. Rev. B 80,100408 (2009).\n[38] A. B. Cahaya, O. A. Tretiakov, and G. E. W. Bauer,\nAppl. Phys. Lett. 104, 042402 (2014).\n[39] A. A. Kovalev and Y. Tserkovnyak, Solid State Commun.\n150, 500 (2010).\n[40] A. A. Kovalev and Y. Tserkovnyak, EPL (Europhysics\nLett. 97, 67002 (2012).\n[41] A. A. Kovalev, Phys. Rev. B 89, 241101 (2014).\n[42] S. K. Kim and Y. Tserkovnyak, Phys. Rev. B 92, 020410\n(2015).\n[43] F. J. Dyson, Phys. Rev. 102, 1217 (1956).\n[44] S. A. Bender, R. A. Duine, A. Brataas, and\nY. Tserkovnyak, Phys. Rev. B 90, 094409 (2014).\n[45] H. Yang, A. Thiaville, S. Rohart, A. Fert, and\nM. Chshiev, Phys. Rev. Lett. 115, 267210 (2015).\n[46] A. A. Kovalev and U. G ung ord u, Europhys. Lett. 109,\n67008 (2015).\n[47] Note1, the DM tensor ^Drepresents a general form\nof DMI. In particular, bulk inversion asymmetry con-\ntributes to ^DasD011whereas structure inversion asym-\nmetry contributes as DRez\u0002.\n[48] U. G ung ord u, R. Nepal, O. A. Tretiakov, K. Be-\nlashchenko, and A. A. Kovalev, Phys. Rev. B 93, 064428\n(2016).\n[49] V. K. Dugaev, P. Bruno, B. Canals, and C. Lacroix, Phys.\nRev. B 72, 024456 (2005).\n[50] G. Tatara, Phys. Rev. B 92, 064405 (2015).\n[51] Note2, we disregard O(hn2\nfi) corrections assuming tem-\nperatures well below the Curie temperature. These cor-\nrections can be readily reintroduced.\n[52] K.-W. Kim, H.-W. Lee, K.-J. Lee, and M. D. Stiles, Phys.\nRev. Lett. 111, 216601 (2013).\n[53] Y. Tserkovnyak and S. A. Bender, Phys. Rev. B 90,\n014428 (2014).\n[54] Note3, in a quasi two-dimensional or two-dimensional\nsystem, surface spin density should be used which can\nbe obtained from the bulk spin density as s2D=s3Dt\nwheretis the layer thickness.\n[55] Note4, while a rigorous analysis of domain wall dynamics\nin the presence of a strong DMI should take domain wall\ntilting into account in general, in the particular case of a\ndomain wall driven by a perpendicular \feld, the ( X;\u001e)\nmodel remains moderately accurate for studying the ef-\nfects of feedback damping [16, 68].\n[56] A. Thiaville, S. Rohart, \u0013E. Ju\u0013 e, V. Cros, and A. Fert,\nEurophys. Lett. 100, 57002 (2012).\n[57] Note5, this equation is similar to the equation obtained\nfor electronic feedback damping in [16] but for magnons\n\u0000XXand \u0000\u001e\u001eare determined by D=J rather than ~ \u000bR\n(a parameter which is taken to be independent from D),\nwhich leads to di\u000berent conclusions. In [16], the chiral\nderivative associated with SMF is parameterized by ~ \u000bR\nasDi=@i+ (~\u000bRez\u0002)ei\u0002. For magnons, D=J corre-\nsponds to ~\u000bR.\n[58] F. J. Buijnsters, Y. Ferreiros, A. Fasolino, and M. I. Kat-\nsnelson, Phys. Rev. Lett. 116, 147204 (2016).\n[59] M. Yamanouchi, A. Jander, P. Dhagat, S. Ikeda, F. Mat-\nsukura, and H. Ohno, IEEE Magn. Lett. 2, 3000304\n(2011).\n[60] X. Liu, W. Zhang, M. J. Carter, and G. Xiao, J. Appl.\nPhys. 110, 033910 (2011).\n[61] A. A. Thiele, Phys. Rev. Lett. 30, 230 (1973).\n[62] Note6, see Supplemental Material at the end.[63] G. Yu, P. Upadhyaya, X. Li, W. Li, S. K. Kim, Y. Fan,\nK. L. Wong, Y. Tserkovnyak, P. K. Amiri, and K. L.\nWang, Nano Lett. 16, 1981 (2016).\n[64] S. A. D\u0013 \u0010az and R. E. Troncoso, arXiv:1511.04584 (2015).\n[65] Note7, as such, dynamics of magnetic solitons depend on\ntemperature through \u001band\u0018.\n[66] M. Weiler, M. Althammer, M. Schreier, J. Lotze,\nM. Pernpeintner, S. Meyer, H. Huebl, R. Gross,\nA. Kamra, J. Xiao, et al., Phys. Rev. Lett. 111, 176601\n(2013).\n[67] A. Vansteenkiste, J. Leliaert, M. Dvornik, M. Helsen,\nF. Garcia-Sanchez, and B. Van Waeyenberge, AIP Adv.\n4, 107133 (2014).\n[68] O. Boulle, S. Rohart, L. D. Buda-Prejbeanu, E. Ju\u0013 e, I. M.\nMiron, S. Pizzini, J. Vogel, G. Gaudin, and A. Thiaville,\nPhys. Rev. Lett. 111, 217203 (2013).Supplemental Material for\nTheory of Magnon Motive Force in Chiral Ferromagnets\nUtkan G¨ ung¨ ord¨ u and Alexey A. Kovalev\nDepartment of Physics and Astronomy and Nebraska Center for Materials and Nanoscience,\nUniversity of Nebraska, Lincoln, Nebraska 68588, USA\nTHIELE’S EQUATION OF MOTION FOR SKYRMION\nWe consider the motion of a rotationally symmetric skyrmion under the influence of applied temperature gradient.\nAssuming that skyrmion drifts without any changes to its internal structure, ns(r,t) =ns(r−q(t)) where qis the\nposition of the skyrmion, we multiply the LLG equation with the operator/integraltext\nd2r∂qjns·ns×and integrate over the\nregion containing the skyrmion and obtain the following equation motion:\ns(W−Q/primez×)v+ (βTηD−Qz×)L∂χ=F. (1)\nwhere v=˙qis the skyrmion velocity, W=η0α+η(α0−β2α2),Lis the spin Seebeck coefficient, χ= 1/T,F=−∇U,\nQis the topological charge defined in the main text and Q/prime=Q−ηβα 1. In terms of the polar coordinates ( θ,φ) of\nns, the dyadic tensor η0and the chiral dyadic tensor ηDare given by\nη0=π/integraldisplayR\n0dr/parenleftbigg(r∂rθ)2+ sin2θ\nr/parenrightbigg\n, ηD=η0+D\nJπ/integraldisplayR\n0dr(sinθcosθ+r∂rθ), (2)\nThe damping terms αican be expanded in powers of D/J asαi=/summationtext\njαβi,Dj(D/J)j, whereαβi,Djis given by\nαβ0,D2=π/integraldisplayR\n0dr(r∂rθ)2cos2θ+ sin2θ\nr\nαβ0,D=π/integraldisplayR\n0dr(r∂rθ)r∂rθtanθcos2θ+ sin2θ\nr2\nαβ0,D0=π/integraldisplayR\n0dr(r∂rθ)22 sin2θ\nr3(3)\nαβ,D2=2π/integraldisplayR\n0dr(∂rθ) sinθ(cos2θ+ 1)\nαβ,D=4π/integraldisplayR\n0dr(∂rθ) sinθcos2θtanθ+r∂rθ\nr\nαβ,D0=2π/integraldisplayR\n0dr(∂rθ) sinθcos2θtan2θ+ (r∂rθ)2\nr2(4)\nαβ2,D2=π/integraldisplayR\n0dr/parenleftbiggcos2θsin2θ+ (r∂rθ)2\nr/parenrightbigg\nαβ2,D=2π/integraldisplayR\n0dr/parenleftbiggcos2θsin2θtanθ+ (r∂rθ)3\nr2/parenrightbigg\nαβ2,D0=π/integraldisplayR\n0dr/parenleftbiggcos2θsin2θtan2θ+ (r∂rθ)4\nr3/parenrightbigg\n(5)\nThe integrals can be evaluated by using an approximate radial profile for θ(r). Using Usov ansatz yields the values\nenumerated in Table I for αβi,Dj. Only the dominant terms are kept in the main text.2\n1 D/J (D/J)2\n1496π\n15R252π\n5R16π\n5\nβC\nR2472π\n15R16π\n3\nβ2 5056\n105R22804π\n105R464π\n105\nTABLE I. List of feedback damping coefficients αβi,Djfor a rotationally symmetric skyrmion using Usov ansatz cos( θ/2) =\n(R2−r2)/(R2+r2) forr≤Rand 0 forr > R . Rows correspond to α0,α1andα2, expanded in powers of D/J. Above,\nC≈449. Remaining parameters are given as η0= 16π/3,ηD=η0+ (4πR/3)(D/J).\nTRANSPORT COEFFICIENTS\nTexture-independent part of the transport coefficients can be obtained using the Boltzmann equation within the\nrelaxation-time approximation in terms of the integral [1, 2]\nJij\nn=1\n(2π)3/planckover2pi1/integraldisplay\nd/epsilon1τ(/epsilon1)(/epsilon1−µ)n(−∂/epsilon1f0)/integraldisplay\ndS/epsilon1vivj\n|v|(6)\nasσ=J0and Π =−J1/J0. Above,τ(/epsilon1) is the relaxation time, /epsilon1(k) =/planckover2pi1ωk,vi=∂ωk/∂ki,dS/epsilon1is the area d2k\ncorresponding to a constant energy surface with /epsilon1(k) =/epsilon1,f0is the Bose-Einstein equilibrium distribution. Under the\nassumption that the scattering processes are dominated by Gilbert damping, we set τ(/epsilon1)≈1/2αω. By evaluating the\nintegral after these substitutions, we obtain σ2D≈F−1/6π2λ/planckover2pi1αin three dimensions ( d= 3), where λ=/radicalbig\n/planckover2pi1J/skBT\nis the wavelength of the thermal magnons, F−1=/integraltext∞\n0d/epsilon1/epsilon1d/2e/epsilon1+x/(/epsilon1+x)(e/epsilon1+x−1)2∼1 evaluated at the magnon gap\nx=/planckover2pi1ω0/kBT. Similarly for d= 2, we obtain σ2D≈F−1/4π/planckover2pi1α.\nThe spin Seebeck coefficient Lis given by−/planckover2pi1σΠ = /planckover2pi1J1, for which we obtain L3D≈F0kBT/6π2λαin 3D and\nL2D≈F0kBT/4παin 2D, where F0=/integraltext∞\n0d/epsilon1/epsilon1d/2/(e/epsilon1+x−1)2∼1. Ford >2 and small x, the numerical factor F0\ncan be expressed in terms of Riemann zeta function and Euler gamma function as ζ(d/2)Γ(d/2 + 1) [3]. In the main\ntext, the numerical factors F−1andF0are omitted.\n[1] N. Ashcroft and N. Mermin, Solid State Physics (Saunders College, Philadelphia, 1976).\n[2] A. A. Kovalev and Y. Tserkovnyak, EPL (Europhysics Lett. 97, 67002 (2012).\n[3] R. K. Pathria, Statistical Mechanics (Butterworth-Heinemann, 1996), 2nd ed."    },    {        "title": "1605.03996v1.Classical_limit_of_Rabi_nutations_in_spins_of_ferromagnets.pdf",        "content": "1 \n Title:  \n \nClassical limit  of Rabi nutation s in spins of  ferromagnet s \n \nAuthors:  \nAmir Capua1, Charles Rettner1, See-Hun Yang1, Stuart S. P. Parkin1,2 \n \nAffiliations:  \n1 IBM Research Division, Almaden Research Center, 650 Harry Rd., San Jose, California \n95120, USA  \n2 Max Planck Institute for Microstructure Physics, Halle (Saale), D -06120, Germany  \n \n \nAbstract:  \n \nRabi oscillations  describe the interaction of a two-level system  with a rotating \nelectromagnetic field. As such, they  serve as the principle method for manipulating \nquantum bits. By using a combination of femtosecond laser pulses and microwave \nexcitations, we have observed the classical form  of Rabi nutations in a ferr omagnetic \nsystem  whose equations of motion mirror the case of a precessing quantum two -level \nsystem . Key to our experiments is the selection of a subset of spins that is in resonance \nwith the microwave excitation and whose coherence time is thereby extende d. Taking \nadvantage of Gilbert damping, the relaxation times are further increased  such that mode -\nlocking takes place . The observation of such Rabi nutations  is the first step towards \npotential applications based on phase -coherent spin manipulation in ferr omagnets.  \n \n \n  2 \n Main Text:  \nA practical gateway to the quantum world is provided by m acroscopic quantum systems \nthat are large cooperative ensembles (1). Superfluids (2, 3), superconductors (4, 5) and \nultracold dilute atomic vapors (6-8), are examples of such systems. Another example are \nmagnon gases in ordered ensembles of magnetic moments that form a macroscopic state \nwhere the quantum nature is unveiled  even at room temperature  (9). In ferromagnets, t he \nmacroscopic  quantum behavior asserts itself at low  temperatures (mK)  and/or small enough  \nlength scales  (nanometer)  (10-12). In that limit the angular momentum observables obey \nthe classical equations of motion. Hence , a great deal of insight  into the quantum word is \ngained from  studies  of classical analog s (13). \nAlthough isolated electron or nuclear s pin states are ideal candidates for quantum  \ninformation processing (14-18), the abundant spin states in ferromagnetic systems are not \ncurrently considered suitable for such applica tions . Their s pin states lack prot ection due to \nspin-spin and spin -lattice interactions. While these  may be overcome  in the quantum \nregime  (10), their coherent manipulation r emains unexplored, even in the  classical limit.  \nThe initialization, manipulation, and readout of spin ensembles in ferromagnetic \nsystems  requires operation in the non -adiabatic regime. This regime pertains whenever an \noscillatory field and the state repres enting the ensemble are not in equilibrium . The \nadiabatic interaction  has been primarily explored  using  ferromag netic resonance (FMR) \nmethod s where  steady state  spin precessions are driven  continuously  by an oscillatory \nmicrowave field . Similar studies  of the magnetic order  have  also been conducted by 3 \n studying the impulse  responses in the absence of the rotating  field using the  time-resolved \nmagneto -optical Kerr effect (TR -MOKE) (19-23). In this technique the free induction \ndecay response  of a ferromagnet  is triggered by an intense optical pulse which  disturbs the \nmagnetic order  and drives the system away  from the equilibrium  state.  Despite  the \nextensive studies  of spin dynamics  in ferromagnetic metals , little attention has been given  \nto non -adiabatic transitions. This mode of operation can be accessed whenever the driven \nspin precessions are disrupted and is achieved by either modifying the state of the \noscillatory magnetic field, or that of the magnetization. While the former method is more \ncommonly used, for instance by applying a \n -pulse  (24), the latter is adopted here.  \nUsing ultra short  optical pulse s to perturb a microwave dri ven ferromagnetic  system s \n(19), we study the non-adiabatic regime  and show that Rabi nutations  in their classical form  \ncan be  revealed  in a ferromagnet . We observe  a chirping of the precession frequency , and \nstudy the  ability  to manipulate the spin states in the presence of  significant inhomogeneous \nbroadening.  In agreement with Gilbert’s damping theory (25), the intrinsic relaxation \ntimes, which represent the loss of spin angular momentum to the environment , can be \nextended  by tuning the external magnetic field (26). In such cases, consecutive optical  \npulses act to synchronize the phase s of the precessing spins (27). Consequently, spin -mode -\nlocking is initiated having the form of  intense pulsations of the magnetization . Our \nexperiments reveal that the microwave signal induces  coherence in the ensemble by \nselecting a subset of spins that are driven resonantly with the field. Hence , the ensemble 4 \n dephasing is suppressed and  relaxation times that represent more closely th ose of  \nindividual spin s result. \nThe sample studied  was a Co36Fe44B20 film with a  thickness of 11 Å that was \nperpendicularly magnetized and grown by magnetron sputtering . The effective anisotropy \nfield, \n0 KeffH , was measured to be ~ 140 mT, with \n0  being the magnetic permeability . \nFrom TR-MOKE measurements of the free induction decay responses  as a function of \napplied field  (Fig. 1A) a Gilbert damping  constant , \n, of 0.023 and a distribution of the \neffective anisotropy field,  \n0 KeffH , of 17 .5 mT were determined  (28, 29). \n represents  \nthe losses of spin angular momentum without  the ensemble dephasing while \nKeffH  allows \nto determine the inhomogeneous broadening of the resonance  linewidth  (28).  \nThe basic concept of  our experiment is presented in Fig. 1 B. A microwave field is \nused to drive spin precessions in the film which are then perturbed by a femtosecond optical \npulse  while being phase locked with the microwave oscillator (30). The temporal recovery \nis recorded by a weak optical pulse probe as a func tion of a pump -probe delay time . The \nexternal magnetic field was applied in the sample plane causing precession s to occur about \nthe x-axis (see figure) while the out -of-plane component of the magnetization, mz, was \ndetected in a polar -MOKE configuration  (28).  \nA measurement of the non -adiabatic interaction is presented in Fig. 1C for three  \nvalues of externally applied field , H0, and a microwave frequency of 10 GHz. T he \nresonance field, Hres, corresponding to this frequency is \n0 resH  ~ 450 mT (Fig. S1) . It is 5 \n seen that a distinct envelope modulates the 10 GHz oscillations  of the precessional motion . \nFurther more,  this envelope exhibits a systematic behavior; the time of its minim um, as \nindicated by the arrows in the figure, increases  as H0 approaches Hres. \nThe responses for a complete  set of applied  fields, are illustrated in Fig. 1 D. At H0 \n< Hres, the shift in time of the minima  is seen clearly and forms a “valley”. At H0 > Hres, a \nmaxim um is initially formed instead , making the response asymmetric . Given that the \nconditions for the non-adiabatic  interaction prevail , the related time s of these signatures , \nfor example the time of the minima , T, should be describable by  the inverse of the \ngeneralized  Rabi  frequency . In the absence of the magnetocrystalline and demagnetization \nfields,  T is given by :  \n22\n0 0 02\nrf rfT\nHh\n  \n\n    (1) \nHere, \n , hrf, and \nrf  are the gyromagnetic ratio, microwave amplitude,  and microwave \nangular frequency,  respectively.  The times  obtained by  the Rabi formula are overlaid on \nthe measured responses  of Fig. 1D and are seen to agree  well with the observed signature s, \ndemonstrating Rabi nutations in a ferromagnet in the classical limit . Also the second  \nnutation is readily seen. It is instructive  to notice that the microwave amplit ude is only \n0rfh\n ~ 0.8 mT in these  measurement s. Hence , the main contribution to the minima time, \nT, stems from t he off-resonance term, \n00 rf H  .  6 \n  The field dependent phase responses also reveal intricate detail s of the dynamic s \nand are analyzed by plotting the dataset  of Fig. 1D  as a two -dimensional contour plot  (Fig. \n2A). Before the pump pulse arrives, a net phase shift of ~ 0.75  π is measured across the \nresonance (enlarged in Fig. 2B). This phase shift is smaller than the expected theoretical \nvalue of π and is related to the relatively large Gilbert  damping.  Surprisingly , at times  \n(pump -probe delays)  well after the  perturbation , a phase shift of  ~ 2.75 π is observed  as H0 \nis varied  (Fig. 2C). To understand  the origin of  this behavior we analyze the instantaneous \nfrequency profiles  (Fig. 2 D). Apart from the sharp  transient  at t = 0, we extract negative, \nzero, and positive chirp profiles corresponding to H0 < Hres, H0 = Hres, and H0 > Hres, \nrespectively . At long delays, the instantaneous frequency recovers to the driving frequency , \nindependent  of H0. This behavior  is explained  qualitatively  by recalling that Rabi \noscillations  can be  regarded as a beating of the  natural  transient response of the system  at \nthe angular  frequency  of \n00H with the steady state  response at  \nrf (31). Hence,  when\n00H\n < \nrf, a negative chirp initially takes place which  recovers to \nrf . The same \nexplanation holds  also for other  H0 values . This reasoning  also account s for the asymmetry \nseen in the responses of Fig. 1D.  As H0 varies, the pump pulse perturbs the magnetization \nat different points along  the precession trajectory  owing to the p hase shif t associated with \nthe resonance . The resultant  beating response then changes from a destructive nature to a \nconstructive interaction. Therefore, variation of H0 provides  a means  of controlling the \neffective  “area” (the time -integrated Rabi frequency)  of the microwave radiation.  A 7 \n theoretical description of the interaction using the Landau -Lifshitz -Gilbe rt equation is \nfurther discussed  in the supplementary  materials section.  \n An important aspect  of Rabi nutations  is the dependence of their frequency on the \nmagnitude of the microwave field. This dependence  is most readily  seen under resonance \nconditions  in which case  the angular Rabi frequency simplifies to  \n0r rf h . The \nmeasured results are shown in Fig.  3A. In contrast  to our expectation s, no dependence  of \nthe envelope  on the amplitude of the microwave  is revealed . This apparent discrepancy is \nresolved by c onsidering  the contributions to Eq. (1). The maximal applied  microwave field \namplitude was  \n0rfh  ~ 7.5 mT  while the inhomogeneous linewidth broadening  at 10 GHz , \nas derived from the value of  \nKeffH , is 10.5 mT  (28). Hence, the detuning term in Eq. (1) \nis still significant so that the Rabi frequency is mainly determined by the off-resonant  \ncontribution rather than by the microwave power . In order to observe the dependence on \nthe microwave power , the contribution of the inhomogeneous broadening must first  be \nsuppressed . This was achieved  by repeating the measurements on a  single crystal sample, \nin the form of a 4 nm thick  epitaxially grown Fe film. In contrast to the sputter deposited \nfilm, the envelope exhibits a clear dependence on the microwave amplitud e (Fig. 3 B). The  \nexpected  increase in the associated time scales describing the envelope  is seen  for \nincreasing a mplitude s as predicted by Rabi’s formula.  \n Next, we turn to s how that the train of optical  pulses can synchronize the phases of \nthe spin precessions and induce pulsation s of the magnetization, namely , spin mode -8 \n locking. This mode of operation can be reached if the responses generated by subsequent \npump pulses of the pulse train interfere. Therefore, this regime requires that the transient  \npart of the responses persist for a duration longer than  the laser  repetition time,  TR (Fig. \n4A).  \nAs follows from Gilbert’s theory for damping, the rate of transfer of spin angular \nmomentum to the lattice can be controlled by the magnitude of H0. This process is \nquantified using  the intrinsic relaxation time, \nint , and is given by  \n1\n00H   in the limit \nwhere only the externally applied field is present . Accordingly, interference effects are \nexpected  at low H0 values . The nature of the interference will then depend on the arrival \ntime of the optical pump pulse within th e microwave cycle. This time is represented by  \n \nwhich is the relative phase between the microwave signal and the pump pulse (Fig. 4A).   \nThe measured response s as a function of \n  are presented in Figs. 4B & 4C.  At high \nmagnetic field  (\n00H  = 450 mT ) and short \nint  (~ 1.1 ns) (28) compared to  the laser  \nrepetition time, TR, of 12.5 ns, the interaction of each pump pulse within the train of pulses \ncan be regarded as an isolated e vent (Fig. 4B). V ariation of \n  has no effect on the \nenvelope; the carrier wave simply  shifts within the same envelope.  In contrast, for low \nexternal magnetic fields (\n00H  = 90 mT) and correspondingly long \nint  (~ 5 ns), \ninterference occur s and the moment at which the optical pulse is sent becomes  critical  (Fig. \n4C). For \n  90\n , constru ctive interference results in a sharp pulsation of the \nmagnetization. Likewise, for additional \n180\n , at \n  270\n , pulsations of opposite polarity 9 \n are generated. However, when the phase is tuned to \n  0\n  and \n  180\n , destructive \ninterference takes place and no pulsations are observed. The existence of the pulsations \nindicates that  the spins have become synchron ized, i.e., mode -locking takes place  (27). \nIn addition to the intrinsic relaxation,  the decay of the transient response is governed \nalso by the dephasing of the inhomogeneously broadened ensemble. This process is \nrepresented by the ensemble dephasing time, \nIH , so that the effective decay time of the \nmagnetization of the entire ensemble , \neff , is given by : \nint 1/ 1/ 1/eff IH    (28, 29, 32).  \nInterestingly,  while a fundamentally different dependence on \n  is observed in the \ntwo regimes of Figs. 4B and 4C, the inhomogeneous broadening causes \neff  to be very \nsimilar  in both cases  and correspond s to ~ 0.51 ns and ~ 0.49 ns for \n00H  = 450 mT  and \n00H\n = 90 mT , respectively . This fact shows that the  long intrinsic  relaxation time , \nint , in \nthe case of low H0 (Fig. 4C) can be  sensed  despite the significant ensemble dephasing. By \nuse of the relations \nint int2/  and \n2/IH IH  for the intrinsic  resonance linewidth  \nand inhomogeneou s broaden ing, respectively,  \nint1.75 rad GHz     and \n2.15 radIH GHz   \n were  extracted for \n00H  = 450 mT , while \nint0.43 rad GHz     and \n3.66 radIH GHz   \n were  found for \n00H  = 90 mT . The se linewidths are illustrated in \nthe lower schematic  of Figs . 4B & 4C . In contrast to the high H0 case, at low H0 only a \nsubset of  spins which  exhibit long \nint  are interacting, namely, the microwave induces \ncoherence in the ensemble. The action of the oscillatory field is to s timulate  the subset of 10 \n spins that are driven resonantly , while suppress ing the off -resonance subsets. Hence, the \ninhomogeneity is overcome and \neff  extends towards its upper limit of \nint . \nThe action of  “filtering” by the microwave signal  is further emphasized by \nexamining the free induction decay responses. Similar long intrinsic relaxation times  that \nare responsible for the mode -locking with \n00H  = 90 mT, also dominate the corresponding \nTR-MOKE m easurement  at \n00H  = 100 mT (Fig. 1A ), for example . In contrast to the \nmicrowave driven measurement, t his respons e show s that the ensemble dephases within  TR \nand exhibits  no signs of coherent interference , as apparent from times around t = 0 (27). \nFurthe rmore,  a closer inspection of  the measurements at the high H0 of Fig. 1D also turn \nout to reveal slight signatures of extended coherence that last for the duration of TR, and \nare clearly not found in the corresponding TR -MOKE responses. These  signatures  are \ndiscussed in the supplementary materials section  and once more demonstrate the ability  of \nour technique to observe details that are obscured by the ensemble dephasing . \nLastly, in the high m agnetic field limit of Fig. 4B , variation of \n  was shown to \ncause  no effect on the envelope . This  observation  seemingly contradicts  the explanation \nbehind the  appearance of the asymmetric signature  of Fig. 1D  which was attributed to the \nphase shift associated with the resonanc e. However, a fundamental difference  exists  \nbetween the two measurements;  in the former  case,  the phase -shift stems from delaying the \nmicrowave  signal  with respect to the optical pulse s while  in the latter  it stems from the \nphase  response of  the resonance.  11 \n In summary, w e have demonstrated a technique that has revealed  the classical form  \nof Rabi nutations in a ferromagnetic system . Our experiments show that a “purified” sub-\nensemble  is generated  and whose dephasing is largely reduced . Extension s of the prese nt \nwork  include  studies of the quantum regime in mesoscopic ferromagnetic structures , as \nwell as  studies of more complex dynamics such as the spin -Hall effect or the spin transfer \ntorque in the non -adiabatic regime . Connecting these effects and other spint ronic effects \nwith quantum processing may prove to be useful in overcoming problems that hinder  even \nsome mature quantum information technologies (33). \n \nAcknowledgments :  \nWe thank  Dr. Dan Ruga r, Dr. Chris Lutz  and Dr. John Mamin for fruitful discussions and \nChris Lada for expert technical assistance. A.C. thanks the Viterbi Foundation and the \nFeder Family  Foundation for supporting this research . \n \n 12 \n References and Note s: \n \n1. A. J. Leggett  et al. , Dynamics of the dissipative two -state system. Reviews of \nModern Physics  59, 1 (1987).  \n2. D. J. Bishop, J. D. Reppy, Study of the Superfluid Transition in Two -Dimensional \n4^He Films. Physical Review Letters  40, 1727 (1978).  \n3. R. Desbuquois  et al. , Superfluid behaviour of a two -dimensional Bose gas. Nat \nPhys  8, 645 (2012).  \n4. J. Clarke, A. N. C leland, M. H. Devoret, D. Esteve, J. M. Martinis, Quantum \nMechanics of a Macroscopic Variable: The Phase Difference of a Josephson \nJunction. Science  239, 992 (1988).  \n5. A. J. Berkley  et al. , Entangled Macroscopic Quantum States in Two \nSuperconducting Qubit s. Science  300, 1548 (2003).  \n6. M. H. Anderson, J. R. Ensher, M. R. Matthews, C. E. Wieman, E. A. Cornell, \nObservation of Bose -Einstein Condensation in a Dilute Atomic Vapor. Science  \n269, 198 (1995).  \n7. I. Bloch, T. W. Hansch, T. Esslinger, Measurement of the spatial coherence of a \ntrapped Bose gas at the phase transition. Nature  403, 166 (2000).  \n8. C. I. Hancox, S. C. Doret, M. T. Hummon, L. Luo, J. M. Doyle, Magnetic trapping \nof rare -earth atoms at millikelvin temperatures. Nature  431, 281 (2004).  \n9. S. O. Demokritov  et al. , Bose -Einstein condensation of quasi -equilibrium \nmagnons at room temperature under pumping. Nature  443, 430 (2006).  \n10. D. D. Awschalom, D. P. Divincenzo, J. F. Smyth, Macroscopic quantum effects in \nnanometer -scale magnets. Science  258, 414 (Oct 16, 1992).  \n11. S. Kleff, J. v. Delft, M. M. Deshmukh, D. C. Ralph, Model for ferromagnetic \nnanograins with discrete electronic states. Physical Review B  64, 220401 (2001).  \n12. M. M. Deshmukh  et al. , Magnetic Anisotropy Variations and Nonequilibri um \nTunneling in a Cobalt Nanoparticle. Physical Review Letters  87, 226801 (2001).  \n13. A. Abragam, The Principles of Nuclear Magnetism .  (Clarendon Press, Oxford \n1961).  \n14. G. D. Fuchs, V. V. Dobrovitski, D. M. Toyli, F. J. Heremans, D. D. Awschalom, \nGigahe rtz Dynamics of a Strongly Driven Single Quantum Spin. Science  326, \n1520 (2009).  \n15. P. Neumann  et al. , Quantum register based on coupled electron spins in a room -\ntemperature solid. Nat Phys  6, 249 (2010).  \n16. E. A. Chekhovich  et al. , Nuclear spin effects in semiconductor quantum dots. Nat \nMater  12, 494 (2013).  \n17. D. J. Christle  et al. , Isolated electron spins in silicon carbide with millisecond \ncoherence times. Nat Mater  14, 160 (2015).  \n18. C. G. Yale  et al. , Optical manipulation of  the Berry phase in a solid -state spin \nqubit. Nat Photon  10, 184 (2016).  13 \n 19. E. Beaurepaire, J. C. Merle, A. Daunois, J. Y. Bigot, Ultrafast Spin Dynamics in \nFerromagnetic Nickel. Physical Review Letters  76, 4250 (1996).  \n20. B. Koopmans, M. van Kampen, J. T. Kohlhepp, W. J. M. de Jonge, Ultrafast \nMagneto -Optics in Nickel: Magnetism or Optics? Physical Review Letters  85, 844 \n(2000).  \n21. J.-Y. Bigot, M. Vomir, E. Beaurepaire, Coherent ultrafast magnetism induced by \nfemtosecond laser pulses. Nat Phys  5, 515 (2 009).  \n22. B. Koopmans  et al. , Explaining the paradoxical diversity of ultrafast laser -induced \ndemagnetization. Nat Mater  9, 259 (2010).  \n23. D. Rudolf  et al. , Ultrafast magnetization enhancement in metallic multilayers \ndriven by superdiffusive spin current.  Nat Commun  3, 1037 (2012).  \n24. D. J. Griffiths, Introduction to Quantum Mechanics .  (Pearson Prentice Hall, \nUpper Saddle River, NJ, ed. 2nd ed., 2005).  \n25. T. L. Gilbert, A phenomenological theory of damping in ferromagnetic materials. \nMagnetics, IEEE Tra nsactions on  40, 3443 (2004).  \n26. R. Hanson, V. V. Dobrovitski, A. E. Feiguin, O. Gywat, D. D. Awschalom, \nCoherent Dynamics of a Single Spin Interacting with an Adjustable Spin Bath. \nScience  320, 352 (2008 -04-18 00:00:00, 2008).  \n27. A. Greilich  et al. , Mod e locking of electron spin coherences in singly charged \nquantum dots. Science  313, 341 (Jul 21, 2006).  \n28. Materials and methods are available as supplementary materials on Science \nOnline.  \n29. A. Capua, S. -H. Yang, T. Phung, S. S. P. Parkin, Determination of intrinsic \ndamping of perpendicularly magnetized ultrathin films from time -resolved \nprecessional magnetization measurements. Physical Review B  92, 224402 (2015).  \n30. I. Neudecker  et al. , Moda l spectrum of permalloy disks excited by in -plane \nmagnetic fields. Physical Review B  73, 134426 (2006).  \n31. H. C. Torrey, Transient Nutations in Nuclear Magnetic Resonance. Physical \nReview  76, 1059 (1949).  \n32. S. Iihama  et al. , Gilbert damping constants of  Ta/CoFeB/MgO(Ta) thin films \nmeasured by optical detection of precessional magnetization dynamics. Physical \nReview B  89, 174416 (2014).  \n33. L. M. K. Vandersypen  et al. , Experimental realization of Shor's quantum factoring \nalgorithm using nuclear magnetic r esonance. Nature  414, 883 (2001).  \n \n 14 \n Fig. 1   \n \n \n15 \n Fig. 1. Measurement in the non -adiabatic regime.  (A) Free induction decay responses \nof a TR-MOKE experiment. Traces are shifted along  the y-axis for clarity.  (B) Schematic \nof the experimental setup. The m agnetic film was patterned into a  square island  while t he \nmicrowave signal was transmitted by a shorted Au microwire. The m easured signal is \nproportional to the angle of polarization rotation , \n, of the optical beam . Lower schemati c \nshows the signals in time. (C) Temporal responses of the pu mp-probe ferromagnetic \nresonance measurement for three values of H0 at 10 GHz  and microwave field amplitude \nof ~ 0.8 mT . Each trace is normalized to the peak value.  Arrows indicate the position of \nthe minimum.  The p ump pulse arrives at t = 0 ps.  (D) Measured temporal responses at 10 \nGHz for a complete range of applied fields. Each trace is normalized  individually  to the \npeak value. The solid red lines were plotted us ing the Rabi formula.  Second oscillation is \nindicated by the guiding black dashed line. Inset illustrates the motion of the magnetization  \nvector .  \n  16 \n Fig. 2  \n \n17 \n Fig. 2. Phase response . (A) Dependence of  phase responses  on the applied field  at 10 GHz . \nData set of Fig. 1D is presented  in a two-dimensional contour  plot to represent the phase  \ninformation. Each temporal response was normalized individually. Blue curved  guiding  \nline indicates the location of the “valley” in Fig. 1D.  (B) Phase r esponse prior to the \nperturbation . The figure presents a close -up of the b lack dashed area of panel (A) for times \nbetween -300 ps and -100 ps.  An overall phase shift of ~ 0.75  π is measured across the \nresonance.  Data is not normalized . (C) Phase r esponse at long delays corresponding to \nblack dashed area in panel (A) which starts at 2200 ps . Data is presented in normalized \nunits. The measured net phase shift across  the resonance is ~ 2.75 π. (D) Instantaneous \nfrequency  profiles at \n00H  values  of 424 mT (blue), 444 mT (red), and 468 mT (yellow) \ncorresponding the blue , red, and yellow dashed lines of panel (A), respectively . Inset \nillustrates a schematic of the beating of the steady state  response  of the system at \nrf  with \nthe na tural response at the angular frequency of \n00H . \n 18 \n Fig. 3  \n \n \n19 \n Fig. 3. Microwave power dependence of  the nutations . (A) Temporal responses  at \nvarious microwave field amplitude s for the CoFeB sample . Responses are presented for a \nfrequency of 10 GHz and \n00H  = 446 mT. (B) Temporal responses of the MBE grown Fe \nsample at 12 GHz and \n00H  = 143 mT.  The measurements i n (A) and (B) were carried out \nat the resonance conditions . Envelopes  of the  responses  are indicated by the guiding  dashed \nlines.  20 \n Fig. 4  \n \n \n \n21 \n Fig. 4. Phase dependent t emporal responses for long and short intrinsic relaxation \ntimes.  (A) Schematic arrangement of signals  in time . TR represents the laser  repetition time . \n(B) Dependence of the temporal response on \n  for short \nint . Data presented for a \nfrequency of 10 GHz and \n00H  = 450 mT.  The relative phase , \n, does not have  a \nsignificant effect on the envelope. S imilar behavior is recorded  also at other bias fields (not \nshown) . (C) Dependence of temporal response  on \n for long \nint . Data presented for a \nfrequency of 1  GHz and \n00H  = 90 mT . Data is shown for the CoFeB sample.  Lower \nschematic is panels  (B) and (C) illustrate  the inhomogeneous broadening. Blue solid line \nindicates the total effective  resonance  linewidth  which  includes contributions of the \ninhomogeneous broadening. Shaded resonance linewidths indicate the subgroups that are \nselected by the microwave. Red solid lines indicate the subgroups that are not interacting \nwith the microwave  signal . \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n 22 \n  \n \n \n \nSupplementary Materials for  \n \nClassical limit  of Rabi nutations in  spins of  ferromagnet s \n \nAmir Capua, Charles Rettner, See -Hun Yang, Stuart S. P. Parkin  \n \n \n \n \n 23 \n Materials and Methods  \n \nMaterials and Fabrication  \nThe CoFeB film in this study was prepared on thermally oxidized Si(100) substrate and \nconsisted of the following structure , starting from the substrate side : SiO 2 (250)/Ta (100)/CoFeB \n(11)/MgO (11)/Ta (30) (numbers are in nominal thicknesses in angstroms). The MgO layer was \ndeposited by RF sputtering . The sample was annealed at a temperature of  \n275\nC  for 30 minutes \nwhile applying a 1 T field in  the out -of-plane direction. In -plane and out -of-plane magnetization \nloops are shown in Fig. S1. The single crystalline 40 Å thick Fe film was grown on MgO(100) \nusing molecular beam epitaxy.  \nFor the pump -probe ferromagnetic resonance measurement the samples were patterned to a \nmagnetic island of  \n220 20 m  using electron -beam lithography. A shorted Au microwire serving \nas an RF -transmission line was patterned at a distance of \n1 m  away from the island by lift -off. \n \nFerromagnetic resonance pump -probe measurement  \nA Ti:Sapphire oscillato r emitting ~ 70 fs pulses  at 800 nm  having energy of ~ 5 nJ per pulse  \nwas used for the optical measurements. The beam was focused to a spot size of approximately 10 \nm\n. The probe pulses were attenuated by 20 dB relative to the pump.  The time jitter between the \noptical pump and the microwave signal was measured to be smaller than 1 ps. All measurements \nwere carried out at room temperature.  The maximum microwave power applied was 1 W and \ncorresponded to an amplitude of ~ 7.5 mT.  \nIn the  pump -probe ferromagnetic resonance measurements a double lock -in detection scheme \nwas used by modulating the microwave signal at 50 KHz and the optical probe at 1 KHz.  In order \nto exert sufficient torque by the optical pump, the external magnetic field wa s applied at an angle \nof \n4\n  away from the sample plane. The same arrangement was applied also in the TR -MOKE \nmeasurements.  \n \nNumerical simulation  \nCalculation of the non -adiabatic interaction was carried out by numerically integrating  the \nLandau -Lifshitz -Gilbert equation.  Since the calculation does not account for the inhomogeneous \nbroadening, it describes the experiment in a qualitative manner.  In the calculation, the steady \nprecessional state was first obtained before applying the pe rturbation.  Two sources for the \nperturbation were introduced that gave the best results: quenching of the magnetization and \nintroduction of a momentary  phenomenological  magnetic field. The latter was required in order to \nreproduce the phase response at pos itive times near t = 0, namely the curvature in the vertical \ncontours of Fig. S2 appearing at times that immediately follow the pump. In Fig. S2 , the wave \nfronts shift to later times as the field increases to a value of ~ 440 mT. When the field is further \nincreased , the wave fronts shift to earlier  times . Since the magnetization acquires a phase shift that \nis associated with the resonance at a field of about 450 mT, presenting an additional \nphenomenological magnetic field causes the magnetization to alter i ts motion. This additional \ntorque was applied in the form of a 3 ps pulsed magnetic field of 60 mT which lied in the film \nplane orthogonal to the axis of precession.  The recovery profile of the magnetization after 24 \n quenching consisted of two time constants of 50 ps and 500 ps while the modulation depth was \n5%.  \nThe simulation result is shown in Fig. S3. Imprints of nutations on the amplitude of the \nprecessions are readily seen. The formation of the “valley” as in Fig. 1D is also observed. This \nvalley however  appears also at field values which are larger than the resonance field in contrast to \nthe measurement. At magnetic fields near resonance and immediately after zero time, similar \ncontours to the ones shown in Fig. S2 are seen.  \n \nExtraction of decay times f rom TR -MOKE measurements  \nExtraction of the ensemble dephasing times and the intrinsic spin relaxation times from TR -\nMOKE measurements was based on the analysis presented in  Ref. (29). Accordingly, \n  and \nKeffH\n were obtained by fitting the measured effective linewidths, \neff , with the equation:  \n \n0\n0 0 0 02\n00\n0\n0 0 0 022\n002         for        \n2\n2     for         eff Keff Keff keff\nKeff\nKeff Keff\neff Keff Keff\nKeffKeffHH H H H H\nH H H\nHH HH H H HHH HH  \n       \n\n\n       \n. \n          \n \nHere \n2/eff eff , and \neff  is the overall decay time of the precessional motion as measured in \nthe TR -MOKE experiment. This analysis is valid for H0 much larger or much smaller than HKeff. \nThe first terms of the equation represent the intrinsic linewidth \nint  while the second terms \nrepresent the inhomogeneous broadening \nIH . For H0 ~ HKeff , as for the case where \n00 90 mT H\n, \nint and \nIH  were extracted using a numerical method (32).  \n \nSignatures of extended coherence at high applied field and 10 GHz  \nSignatures of the extended coherences were even found in the phase responses at 10 GHz \n(Fig. 2A) where \neff and \nint  are shorter. At negative times, before the pump pulse arrives, the phase \nin Fig. 2B shifts slightly to negative values as H0 increases to \n00H  ~ 450 mT.  This response \ndiffers from the responses measured when the optical pump is completely turned off (Fig. S 4), \nindicating the slight remnant coherence from the previous cycle. Moreover, the negative phase \nshift resembles the behavior of the phases seen immed iately after t = 0 (Fig. S 2) implying a link \nbetween the responses manifested by long lasting coherence despite the short \neff  and \nint  (27). A \nnon-coherent process, such as a thermal process would have had an equal effect for all H0 and \nwould not have affected the phase in the manner observed. Once more, the corresponding TR -\nMOKE traces show no sign of the coherent interaction after TR, demonstrating the ability to \nobserve details that were obscured by the ensemble dephasing.  \n 25 \n  Figures  \n \n \n  \nFig. S1.  \nCoFeB film characterization. (A) In-plane and out -of-plane magnetization loops.  (B) Frequency \nvs. applied field as measured in a TR -MOKE experiment. The magnetic field was applied at an \nangle of \n4\n  away from the sample plane. B lack dashed line indicates HKeff. \n \n \n26 \n  \nFig. S 2 \nClose -up of Fig. 2A for times between for times between 100 ps and 300 ps. A negative phase \nshift is seen as the field increases to a value of 440 mT  after which a positive phase shift occurs \nwhen the field is further increased.  \n  \n27 \n  \n \nFig. S 3 \nCalculation of the out -of-plane component of the magnetization, mz. The r esponse at each bias \nfield was normalized independently to reach a maximum value of unity.  \n \n  \n28 \n  \nFig. S 4 \nResonance response without the optical pump for the CoFeB sample. Measurement shows the \nout-of-plane component of the magnetization, mz, at 10 GHz. The phase increases monotonically \nwith the field in contrast to Fig. 2B of  the main text. Data  is not normalized.  \n \n \n \n"    },    {        "title": "1605.04543v1.Propagation_of_Thermally_Induced_Magnonic_Spin_Currents.pdf",        "content": "arXiv:1605.04543v1  [cond-mat.mtrl-sci]  15 May 2016Propagation of Thermally Induced Magnonic Spin Currents\nUlrike Ritzmann, Denise Hinzke, and Ulrich Nowak\nFachbereich Physik, Universit¨ at Konstanz, D-78457 Konst anz, Germany\n(Dated: 19.12.2013)\nThe propagation of magnons in temperature gradients is inve stigated within the framework of an\natomistic spin model with the stochastic Landau-Lifshitz- Gilbert equation as underlying equation\nof motion. We analyze the magnon accumulation, the magnon te mperature profile as well as the\npropagation length of the excited magnons. The frequency di stribution of the generated magnons\nis investigated in order to derive an expression for the influ ence of the anisotropy and the damping\nparameter on the magnon propagation length. For soft ferrom agnetic insulators with low damping\na propagation length in the range of some µm can be expected for exchange driven magnons.\nPACS numbers: 75.30.Ds, 75.30.Sg, 75.76.+j\nI. INTRODUCTION\nSpin caloritronics is a new, emerging field in mag-\nnetism describing the interplay between heat, charge and\nspin transport1,2. A stimulation for this field was the dis-\ncovery of the spin Seebeck effect in Permalloy by Uchida\net al.3. Analog to the Seebeck effect, where in an elec-\ntric conductor an electrical voltage is created by apply-\ning a temperature gradient, in a ferromagnet a temper-\nature gradient excites a spin current leading to a spin\naccumulation. The generated spin accumulation was de-\ntected by measuring the spin current locally injected into\na Platinum-contact using the inverse spin Hall effect3,4.\nA first explanation of these effect was based on a spin-\ndependent Seebeckeffect, wherethe conductionelectrons\npropagate in two different channels and, due to a spin\ndependent mobility, create a spin accumulation in the\nsystem5.\nInterestingly, it was shown later on that this effect also\nappears in ferromagnetic insulators6. This shows that in\naddition to conduction-electron spin-currents, chargeless\nspin-currents exist as well, where the angular momentum\nis transported by the magnetic excitations of the system,\nso-called magnons. A first theoretical description of such\na magnonic spin Seebeck effect was developed by Xiao et\nal.7. With a two temperature model including the local\nmagnon (m) and phonon (p) temperatures the measured\nspin Seebeck voltage is calculated to be linearly depen-\ndent on the local difference between magnon and phonon\ntemperature, ∆ Tmp=Tm−Tp. This temperature dif-\nference decays with the characteristic lengthscale λ. For\nthe ferromagnetic material YIG they estimate the length\nscale in the range of several millimeters.\nThe contribution of exchange dominated magnons to\nthe spin Seebeck effect was investigated in recent experi-\nments by Agrawalet al.8. Using Brillouin lightscattering\nthe difference between the magnon and the phonon tem-\nperature in a system with a linear temperature gradient\nwas determined. They found no detectable temperature\ndifference and estimate a maximal characteristic length\nscale of the temperature difference of 470 µm. One pos-\nsible conclusion from this results might be be that in-\nstead of exchangemagnons, magnetostatic modes mainlycontribute to the spin Seebeck effect and are responsible\nfor the long-range character of this effect. Alternatively,\nphononsmightcontributetothemagnonaccumulationas\nwell viaspin-phonon drag9,10. A complete understanding\nof these different contributions to the spin Seebeck effect\nis still missing.\nIn this paper thermally excited magnonic spin currents\nand their length scale of propagation are investigated.\nUsingatomisticspinmodelsimulationwhichdescribethe\nthermodynamics of the magnetic system in the classical\nlimit including the whole frequency spectra of excited\nmagnons,wedescribespincurrentsbyexchangemagnons\nin the vicinity of a temperature step. After introducing\nour model, methods and basic definitions in Section II\nwe determine the magnon accumulation as well as the\ncorresponding magnon temperature and investigate the\ncharacteristic lengthscale of the decay of the magnon ac-\ncumulation in Section III. In Section IV we introduce\nan analytical description which is supported by our sim-\nulations shown in Section V and gives insight into the\nmaterial properties dependence of magnon propagation.\nII. MAGNETIZATION PROFILE AND\nMAGNON TEMPERATURE\nFor the investigationofmagnonic spin currentsin tem-\nperature gradients we use an atomistic spin model with\nlocalized spins Si=µi/µsrepresenting the normalized\nmagnetic moment µsof a unit cell. The magnitude of\nthe magnetic moment is assumed to be temperature in-\ndependent. Wesimulateathree-dimensionalsystemwith\nsimple cubic lattice structure and lattice constant a. The\ndynamics of the spin system are described in the classical\nlimit by solving the stochastic Landau-Lifshitz-Gilbert\n(LLG) equation,\n∂Si\n∂t=−γ\nµs(1+α2)Si×(Hi+α(Si×Hi)), (1)\nnumerically with the Heun method11withγbeing the\ngyromagnetic ratio. This equation describes a preces-\nsion of each spin iaround its effective field Hiand the2\ncoupling with the lattice by a phenomenological damp-\ning term with damping constant α. The effective field\nHiconsists of the derivative of the Hamiltonian and an\nadditional white-noise term ζi(t),\nHi=−∂H\n∂Si+ζi(t) . (2)\nThe Hamiltonian Hin our simulation includes exchange\ninteraction of nearest neighbors with isotropic exchange\nconstant Jand an uniaxial anisotropy with an easy axis\ninz-direction and anisotropy constant dz,\nH=−J\n2/summationdisplay\n<i,j>SiSj−dz/summationdisplay\niS2\ni,z. (3)\nThe additional noise term ζi(t) of the effective field Hi\nincludes the influence of the temperature and has the\nfollowing properties:\n/angbracketleftζ(t)/angbracketright= 0 (4)\n/angbracketleftBig\nζi\nη(0)ζj\nθ(t)/angbracketrightBig\n=2kBTpαµs\nγδijδηθδ(t) . (5)\nHerei,jdenote lattice sites and ηandθCartesian com-\nponents of the spin.\nWe simulate a model with a given phonon temperature\nTpwhich is space dependent and includes a temperature\nstep inz-direction in the middle of the system at z= 0\nfromatemperature T1\npinthehotterareato T2\np= 0K(see\nFig. 1). We assume, that this temperature profile stays\nconstant during the simulation and that the magnetic\nexcitationshavenoinfluence onthe phonontemperature.\nThe system size is 8 ×8×512, large enough to minimize\nfinite-size effects.\nAll spins are initialized parallel to the easy-axis in z-\ndirection. Due tothetemperaturestepanon-equilibrium\nin the magnonic density of states is created. Magnons\npropagate in every direction of the system, but more\nmagnons exist in the hotter than in the colder part of\nthe system. This leads to a constant net magnon current\nfrom the hotter towards the colder area of the system.\nDue to the damping of the magnons the net current ap-\npears around the temperature step with a finite length\nscale.After an initial relaxation time the system reaches\na steadystate. In this steady state the averagedspin cur-\nrent from the hotter towardsthe colderregionis constant\nand so the local magnetization is time independent. We\ncan now calculate the local magnetization m(z) depend-\ning on the space coordinate zas the time average over\nall spins in the plane perpendicular to the z-direction.\nWe use the phonon temperature T1\np= 0.1J/kBin the\nheated area, the anisotropy constant dz= 0.1Jand vary\nthe damping parameter α. The resulting magnetization\nversus the space coordinate zfor different damping pa-\nrameters in a section around the temperature step is\nshown in Fig. 1. For comparison the particular equi-\nlibrium magnetization m0of the two regions is also cal-\nculated and shown in the figure.m0α= 1α= 0.1α= 0.06Tp\nspace coordinate z/a\nphonon temperature kBTp/Jmagnetization m0.1\n0\n403020100-10-20-30-401\n0.995\n0.99\n0.985\n0.98\n0.975\nFIG. 1. Steady state magnetization mand equilibrium mag-\nnetization m0over space coordinate zfor a given phonon\ntemperature profile and for different damping parameters α\nin a small section around the temperature step.\nFar away from the temperature step on both sides the\namplitudes of the local magnetization m(z) converge to\nthe equilibrium values, only in the vicinity of the tem-\nperature step deviations appear. These deviations de-\nscribe the magnon accumulation, induced by a surplus\nof magnons from the hotter region propagating towards\nthe colder one. This leads to a less thermal excitation\nin the hotter area and the value of the local magneti-\nzation increases. In the colder area the surplus of in-\ncoming magnons decrease the value of the local magneti-\nzation. For smaller values of αthe magnons can propa-\ngate overlargerdistances before they are finally damped.\nThis leads to a damping-dependent magnon accumula-\ntion which increases with decreasing damping constant\nα.\nForafurtheranalysisinthecontextofthespin-Seebeck\neffect we define a local magnon temperature Tm(z) via\nthe magnetization profile m(z). For that the equilib-\nrium magnetization m0(T) is calculated for the same\nmodel but homogeneous phonon temperature Tp. In\nequilibrium magnon temperature Tmand the phonon\ntemperature Tpare the same and we can determine the\nfitequilibrium data\nmagnon temperature kBTm/Jmagnetization m0\n10.90.80.70.60.50.40.30.20.101\n0.95\n0.9\n0.85\n0.8\n0.75\n0.7\n0.65\n0.6\nFIG. 2. Equilibrium magnetization m0over the magnon\ntemperature Tm. Red points show the simulated equilibrium\nmagnetization and the black line shows a fit of the data.3\nT0\nmα= 1α= 0.1α= 0.06Tp\nspace coordinate z/a\nphonon temperature kBTp/Jmagnon temperature kBTm/J 0.1\n0\n403020100-10-20-30-400.1\n0.08\n0.06\n0.04\n0.02\n0\nFIG. 3. Magnon temperature Tmover the space coordinate\nzfor different damping parameters αcorresponding to the\nresults in Fig. 1.\n(magnon) temperature dependence of the equilibrium\nmagnetization m0(Tm) of the system. The magnetiza-\ntion of the equilibrium system decreases for increasing\nmagnontemperatureasshownin Fig. 2 andthe behavior\ncan be described phenomenologically with a function12\nm0(T) = (1−Tm/Tc)βwhereTcistheCurietemperature.\nFitting ourdatawefind Tc= (1.3326±0.00015)J/kBand\nfor the exponent we get β= 0.32984±0.00065. This fit\nof the data is also shown in Fig. 2 and it is a good\napproximation over the whole temperature range. The\ninverse function is used in the following to determine the\nmagnon temperature for a given local magnetization and\nwith that a magnon temperature profile Tm(z).\nThe resulting magnon temperature profiles are shown\nin Fig. 3. Far away from the temperature step the\nmagnon temperature Tm(z) coincides with the given\nphonon temperature Tp, and deviations — dependent on\nthe damping constant α— appear only around the tem-\nperature step. These deviations correspond to those of\nthe local magnetization discussed in connection with Fig.\n1.\nIII. MAGNON PROPAGATION LENGTH\nTo describe the characteristic lengthscale of the\nmagnon propagationaround the temperature step we de-\nfine the magnon accumulation ∆ m(z) as the difference\nbetween the relative equilibrium magnetization m0(z) at\nthe given phonon temperature Tp(z) and the calculated\nlocal magnetization m(z):\n∆m(z) =m0(z)−m(z) . (6)\nWeinvestigatethemagnonpropagationinthecolderpart\nof the system, where Tp(z) = 0. For a small magnon\ntemperature, the temperature dependence of the magne-\ntization can be approximated as\nm(Tm)≈1−β\nTcTm. (7)α= 1.00α= 0.50α= 0.10α= 0.08α= 0.06\nspace coordinate z/amagnon accumulation ∆m\n2502001501005001\n10−2\n10−4\n10−6\n10−8\n10−10\nFIG. 4. Magnon accumulation ∆ mover space coordinate z\nin the colder region of the system at Tp= 0K for different\ndamping constants αshows exponential decay with magnon\npropagation length ξ. The points show the data from our\nsimulation and the lines the results from an exponential fit.\nThese linear equation is in agreement with an analytical\nsolution for low temperatures presented by Watson et\nal.12. For low phonon temperatures one obtains for the\ndifference between phonon and magnon temperature\n∆T=Tm−Tp=β\nTc∆m. (8)\nNote, that the proportionality between magnon accumu-\nlation and temperature difference holds for higher tem-\nperatures as well as long as magnon and phonon temper-\nature are sufficiently close so that a linear approximation\napplies,thoughtheproportionalityfactorincreases. Note\nalso, that this proportionalitywas determined in theoret-\nical descriptions of a magnonic spin Seebeck effect7. Our\nresults for the magnon accumulation should hence be rel-\nevantfortheunderstandingofthemagnonicspinSeebeck\nwhere the temperature difference between the magnons\nin the ferromagnet and the electrons in the non-magnet\nplays a key role.\nWe further investigate our model as before with a tem-\nperature in the heated area of T1\np= 0.1J/kB, anisotropy\nconstant dz= 0.1Jand different damping parameters.\nThe magnon accumulation ∆ mversus the space coordi-\nnatezin the colder region of the system at Tp= 0K is\nshown in Fig. 4. Apart from a sudden decay close to the\ntemperature step the magnon accumulation ∆ m(z) then\ndecays exponentially on a length scale that depends on\nthe damping constant α. To describe this decay we fit\nthe data with the function\n∆m(z) = ∆m(0)·e−z\nξ. (9)\nWe define the fitting parameter ξas the propagation\nlength of the magnons. Here, the deviations from the\nexponential decay at the beginning of the system are ne-\nglected. The fits for the data are shown in Fig. 4 as\ncontinuous lines.\nThe propagation length dependence on the damping\nparameter αis shown in Fig. 5. The values of the prop-\nagation length from our simulations, shown as points,4\ndz= 0.01Jdz= 0.05Jdz= 0.10Jdz= 0.50J\ndamping constant αpropagation length ξ/a\n1 0.1100\n10\n1\nFIG. 5. Magnon propagation length ξover the damping con-\nstantαfor different anisotropy constant dz. Numerical data\nis shown as points and the solid lines are from Eq. (19).\nare inversely proportional to the damping constant α\nand, furthermore, show also a strong dependence on the\nanisotropy constant dz. This behavior will be discussed\nin the next two sections with an analytical analysisof the\nmagnon propagation and an investigation of the frequen-\ncies of the propagating magnons. A simple approxima-\ntion for the propagation length leads to Eq. (19) which\nis also shown as solid lines in Fig. 5.\nIV. ANALYTICAL DESCRIPTION WITH\nLINEAR SPIN-WAVE THEORY\nFor the theoretical description of the magnon accu-\nmulation, excited by a temperature step in the system,\nwe solve the LLG equation (Eq. (1)), analytically in\nthe area with Tp= 0K. We consider a cubical system\nwith lattice constant awhere all spins are magnetized in\nz-direction parallel to the easy-axis of the system. As-\nsuming only small fluctuations in the x- andy-direction\nwe have Sz\ni≈1 andSx\ni,Sy\ni≪1. In that case we can\nlinearize the LLG-equation and the solution of the re-\nsulting equation consists of a sum over spin waves with\nwavevectors qand the related frequency ωqwhich decay\nexponentially in time dependent on their frequency and\nthe damping constant αof the system,\nS±\ni(t) =1√\nN/summationdisplay\nqS±\nq(0)e∓iqri±iωqt·e−αωqt. (10)\nThe frequency ωqof the magnons is described by the\nusual dispersion relation\n¯hωq=1\n(1+α2)/parenleftBig\n2dz+2J/summationdisplay\nθ(1−cos(qθaθ))/parenrightBig\n. (11)\nThe dispersion relation includes a frequency gap due to\nthe anisotropy constant and a second wavevector depen-\ndent term with a sum over the Cartesian components13.\nConsidering now the temperature step, magnons from\nthe hotter area propagate towards the colder one. Weinvestigate the damping process during that propagation\ninordertodescribethe propagatingfrequenciesaswellas\nto calculate the propagation length ξof the magnons for\ncomparisonwiththeresultsfromsectionIII.Themagnon\naccumulation will depend on the distance to the temper-\nature step and — for small fluctuations of the SxandSy\ncomponents — can be expressed as\n∆m(z) = 1−/angbracketleftSz(z)/angbracketright ≈1\n2/angbracketleftbig\nSx(z)2+Sy(z)2/angbracketrightbig\n, (12)\nwhere the brackets denote a time average. We assume\nthat the local fluctuations of the SxandSycomponents\ncan be described with a sum over spin waves with differ-\nent frequencies and damped amplitudes aq(z),\nSx(z) =/summationdisplay\nqaq(z)cos(ωqt−qr) , (13)\nSy(z) =/summationdisplay\nqaq(z)sin(ωqt−qr) . (14)\nIn that case for the transverse component of the magne-\ntization one obtains\n/angbracketleftbig\nSx(z)2+Sy(z)2/angbracketrightbig\n=/angbracketleftBigg/summationdisplay\nqaq(z)2/angbracketrightBigg\n, (15)\nwhere mixed terms vanish upon time averaging. The\nmagnon accumulation can be written as:\n∆m(z) =1\n2/angbracketleftBigg/summationdisplay\nqaq(z)2/angbracketrightBigg\n. (16)\nThe amplitude aq(z) of a magnon decays exponentially\nas seen in Eq. (10) dependent on the damping constant\nand the frequency of the magnons. In the next step we\ndescribe the damping process during the propagation of\nthe magnons. In the one-dimensional limit magnons only\npropagate in z-direction with velocity vq=∂ωq\n∂q. Then\nthe propagation time can be rewritten as t=z/vqand\nwe can describe the decay of the amplitude with aq(z) =\naq(0)·f(z) with a damping function\nf(z) = exp/parenleftBig\n−αωqz\n∂ωq\n∂qz/parenrightBig\n. (17)\nThe amplitudes are damped exponentially during the\npropagation which defines a frequency dependent propa-\ngation length\nξωq=/radicalbigg\nJ2−/parenleftBig\n1\n2(1+α2)(¯hωq−2dz)−J/parenrightBig2\nα(1+α2)¯hωq,(18)\nwhere we used γ=µs/¯h. In the low anisotropy limit\nthis reduces to ξωq=λ/παwhereλ= 2π/qis the wave\nlength of the magnons.5\nThe total propagation length is then the weighted av-\nerage over all the excited frequencies. The minimal fre-\nquency is defined by the dispersion relation with a fre-\nquency gap of ωmin\nq= 1/(¯h(1 +α2))2dz. For small fre-\nquencies above that minimum the velocity is small, so\nthe magnons are damped within short distances. Due to\nthe fact that the damping process is also frequency de-\npendent higher frequencies will also be damped quickly.\nIn the long wave length limit the minimal damping is\nat the frequency ωmax\nq≈4dz/(¯h(1 +α2)) which can be\ndetermined by minimizing Eq. (17) .\nIn a three-dimensional system, besides the z-\ncomponent of the wavevector, also transverse compo-\nnents of the wavevector have to be included. The\ndamping of magnons with transverse components of\nthe wavevector is higher than described in the one-\ndimensionalcase,becausethe additionaltransverseprop-\nagationincreasethepropagationtime. Inoursimulations\nthe cross-section is very small, so that transverse compo-\nnents of the wave-vectors are very high and get damped\nquickly. Thisfact andthe highdamping forhighfrequen-\ncies described in Eq. (17) can explain the very strong\ndamping at the beginning of the propagation shown in\nFig. 4.\nV. FREQUENCIES AND DAMPING OF\nPROPAGATING MAGNONS\nIn this section we investigate the frequency distribu-\ntionofthemagnonicspincurrentwhilepropagatingaway\nfrom the temperature step. First we determine the fre-\nquencies of the propagating magnons in our simulations\nwith Fourier transformation in time to verify our as-\nsumptions from the last section. As before a system of\n8×8×512spins with a temperature step in the center of\nthe system is simulated with an anisotropy of dz= 0.1J.\nThe temperature of the heated area is T1\np= 0.1J/kBand\nthe damping constant is α= 0.1. After an initial relax-\nation to a steady-state the frequency distribution of the\npropagating magnons in the colder area is determined by\nFourier transformation in time of S±(i) =Sx(i)±iSy(i).\nThe frequency spectra are averaged over four points in\nthex-y-plane and analyzed depending on the distance z\nof the plane to the temperature step.\nThe results for small values of zare shown in Fig. 6(a)\nand for higher values of z, far away from the temper-\nature step, for the regime of the exponential decay, in\nFig. 6(b). For small values of z, near the temperature\nstep, the frequency range of the propagating magnons is\nvery broad. The minimum frequency is given by ωmin\nq=\n2dz/(¯h(1+α2)) and far away from the temperature step\nthe maximum peak is around ωmax\nq= 4dz/(¯h(1 +α2)).\nThese characteristic frequencies are in agreement with\nour findings in section IV.\nFurthermore, a stronger damping for higher frequen-\ncies can be observed. This effect corresponds to the\nstrong damping of magnons with wavevector compo-10864204·10−3\n3·10−3\n2·10−3\n1·10−3\n0z= 20az= 10az= 1a(a)\nfrequency ¯hωq/Jamplitude |S+(ωq)|\n1.41.210.80.60.40.208·10−5\n6·10−5\n4·10−5\n2·10−5\n0ωminz= 100az= 90az= 80a(b)\nfrequency ¯hωq/Jamplitude |S+(ωq)|\nFIG.6. Amplitude |S+(ωq)|versusthefrequency ωqfor asys-\ntem with 8 ×8×512 spins. (a):after propagation over short\ndistances form 1 to 20 lattice constants. (b): after propaga -\ntion over longer distances from 80 to 100 lattice constants.\nnents transverse to the z-direction and it explains the\nhigher initial damping, which was seen in the magnon\naccumulation in Fig. 4. A much narrower distribu-\ntion propagates over longer distances and reaches the\narea shown in Fig. 6(b). In that area the damping\ncan be described by one-dimensional propagation of the\nmagnons in z-direction with a narrow frequency distri-\nbution around the frequency with the lowest damping\nωmax\nq= 4dz/(¯h(1+α2)).The wavelength and the belong-\ning group velocity of the magnons depending on their\nfrequency in the one-dimensional analytical model are\nshown in Fig.7(a). In the simulated system magnons\nwith the longest propagation length have a wavelength\nofλ= 14a. Depending on the ratio dz/Jthe wave-\nlength increases for systems with lower anisotropy. As\ndiscussed in the last chapter, magnons with smaller fre-\nquencies are less damped in the time domain, but due to\ntheir smaller velocity the magnons very close to the min-\nimum frequency also have a smaller propagation length.\nTo investigate the frequency-dependent damping-\nprocess during the propagation of the magnons we calcu-\nlatetheratiooftheamplitudeofthemagnons |S+(ωq,x)|\nforz= 80aandz= 80a+∆ with ∆ = 10 a,20a,50aand\nnormalize it to a damping per propagation of one spin.\nThe resulting ratios ( |S+(ωq,x)|/|S+(ωq,x−∆)|)1/∆are\nshown in Fig. 7 in comparison with the frequency-6\n2\n1.5\n1\n0.5\n0\n43.532.521.510.50100\n80\n60\n40\n20\n0ωmin\nqvqλ(a)\nfrequency ¯hωq/J\nvelocityvq¯h/(Ja)wave length λ/a\ndamping function∆ = 10a∆ = 20a∆ = 50a(b)\nfrequency ω[J/¯h]damping ratio\n10.90.80.70.60.50.40.30.21\n0.98\n0.96\n0.94\n0.92\n0.9\nFIG. 7. (a): Wavelength λand group velocity vqof the\nmagnons in a one-dimensional model dependent on the fre-\nquencyωq. (b):Damping ratio as explained in the text versus\nthe frequency ωqfor different distances ∆ and compared to\nthe damping function (Eq. (17)).\ndependent damping-function (Eq. (17)). The figure\nshows a good agreement between simulation and our an-\nalytical calculations.\nThese results explain the dependence of the magnon\npropagation length on the model parameters. The fre-\nquency with the maximal amplitude is determined by\nthe anisotropy constant. Under the assumption that the\nfrequency with the lowest damping is dominant and the\ncontribution of other frequencies can be neglected the\npropagation length can be calculated as\nξ=a\n2α/radicalbigg\nJ\n2dz, (19)\nwhere the square-root term is the domain wall width of\nthe model. This formula is also plotted in Fig. 5.\nThe comparison with our simulations shows good\nagreement though the equation above gives only the\npropagation of those magnons with the smallest damp-\ningduringthe propagation.Inthe consideredsystemwith\nα= 0.1 anddz= 0.1Jwe get a propagation length of\naboutξ= 11aat a wavelength of the magnons λ= 14a.\nFor smaller values of the anisotropy and smaller damp-\ning parameters the frequency distribution of the thermal\nmagnons is broader and Eq. (19) is an overestimation\nof the real propagation length since the magnon accu-mulation is no longer exponentially decaying due to the\nbroader spectrum of propagating frequencies. However\nwe would expect for soft ferromagnetic insulators with a\nsmalldamping constantof10−4−10−3and ananisotropy\nconstant in the range of 10−3J−10−2Ja propagation\nlength of 103a−105awhich would be in the micrometer-\nrange.\nVI. SUMMARY AND DISCUSSION\nUsing the frameworkof an atomistic spin model we de-\nscribe thermally induced magnon propagationin a model\ncontaining a temperature step. The results give an im-\npression of the relevant length scale of the propagation\nof thermally induced exchange magnons and its depen-\ndence on system parameters as the anisotropy, the ex-\nchange and the damping constant. In the heated area\nmagnons with a broad frequency distribution are gener-\natedandbecauseoftheverystrongdampingformagnons\nwith high frequency, especially those with wave-vector\ncomponents transverse to the propagation direction in\nz-direction, most of the induced magnons are damped\non shorter length scales. Behind this region of strong\ndamping near the temperature step, the propagation of\nmagnons is unidirectional and the magnon accumula-\ntion decays exponentially with the characteristic prop-\nagation length ξ. This propagation length depends on\nthe damping parameter but also on system properties as\nthe anisotropy of the system, because of the dependence\non the induced frequencies.\nIn contrast to long range magnetostatic spin waves,\nwhich can propagate over distances of some mm14,15, we\nfind that for exchange magnons the propagation length\nis considerably shorter and expect from our findings for\nsoft ferromagnetic insulators with a low damping con-\nstant a propagation length in the range of some µm for\nthose magnons close to the frequency gap and the lowest\ndamping. These findings will contribute to the under-\nstanding of length scale dependent investigations of the\nspin Seebeck effect8,16–18.\nRecent experiments investigate the longitudinal spin\nSeebeck effect, where the generated spin current\nlongitudinal to the applied temperature gradient is\nmeasured19–22. In this configuration Kehlberger et al.\nshow that the measured spin current is dependent on the\nthickness of the YIG layer and they observe a saturation\nofthespincurrentonalengthscaleof100 nm16. Thissat-\nuration can be explained by the lengthscale of the prop-\nagation of the thermally excited magnons. Only those\nmagnons reaching the YIG/Pt interface of the sample\ncontribute to the measured spin current and — as shown\nhere — exchange magnons thermally excited at larger\ndistances are damped before they can reach the inter-\nface. In this paper, we focus on the propagation length\nof those magnons with the lowest damping, however the\nlengthscale of the magnon accumulation at the end of\na temperature gradient is dominated by a broad range7\nof magnons with higher frequencies which are therefore\ndamped on shorter length scales.\nACKNOWLEDGMENTS\nThe authors would like to thank the Deutsche\nForschungsgemeinschaft (DFG) for financial support viaSPP 1538 “Spin Caloric Transport” and the SFB 767\n“Controlled Nanosystem: Interaction and Interfacing to\nthe Macroscale”.\n1G. E. W. Bauer, E. Saitoh, and B. J. van Wees,\nNature Mater. 11, 391 (2012).\n2G. E. W. Bauer, A. H. MacDonald, and S. Maekawa,\nSolid State Commun. 150, 459 (2010).\n3K. Uchida, S. Takahashi, K. Harii, J. Ieda,\nW. Koshibae, K. Ando, S. Maekawa, and E. Saitoh,\nNature455, 778 (2008).\n4E. Saitoh, M. Ueda, H. Miyajima, and G. Tatara,\nAppl. Phys. Lett. 88, 182509 (2006).\n5K. Uchida, S. Takahashi, J. Ieda, K. Harii,\nK. Ikeda, W. Koshibae, S. Maekawa, and E. Saitoh,\nJ. Appl. Phys. 105, 07C908 (2009).\n6K. Uchida, J. Xiao, H. Adachi, J. Ohe, S. Taka-\nhashi, J. Ieda, T. Ota, Y. Kajiwara, H. Umezawa,\nH. Kawai, G. E. W. Bauer, S. Maekawa, and E. Saitoh,\nNature Mater. 9, 894 (2010).\n7J. Xiao, G. E. W. Bauer, K.-c. Uchida, E. Saitoh, and\nS. Maekawa, Phys. Rev. B 81, 214418 (2010).\n8M. Agrawal, V. I. Vasyuchka, A. A. Serga, A. D.\nKarenowska, G. A. Melkov, and B. Hillebrands,\nPhys. Rev. Lett. 111, 107204 (2013).\n9H. Adachi, K. ichi Uchida, E. Saitoh, J. ichiro\nOhe, S. Takahashi, and S. Maekawa,\nAppl.Phys. Lett. 97, 252506 (2010).\n10H. Adachi, J.-i. Ohe, S. Takahashi, and S. Maekawa,\nPhys. Rev. B 83, 094410 (2011).\n11U. Nowak, “Handbook of magnetism and advanced mag-\nnetic materials,” (John Wiley & Sons, 2007) Chap. Clas-\nsical Spin-Models.\n12R. E. Watson, M. Blume, and G. H. Vineyard,\nPhys. Rev. 181, 811 (1969).\n13U. Atxitia, D. Hinzke, O. Chubykalo-Fesenko, U. Nowak,H. Kachkachi, O. N. Mryasov, R. F. Evans, and R. W.\nChantrell, Phys. Rev. B 82, 134440 (2010).\n14Y. Kajiwara, K. Harii, S. Takahashi, J. Ohe,\nK. Uchida, M. Mizuguchi, H. Umezawa, H. Kawai,\nK. Ando, K. Takanashi, S. Maekawa, and E. Saitoh,\nNature464, 262 (2010).\n15T. An, V. I. Vasyuchka, K. Uchida, A. V. Chumak, K. Ya-\nmaguchi, K. Harii, J. Ohe, M. B. Jungfleisch, Y. Kajiwara,\nH. Adachi, B. Hillebrands, S. Maekawa, and E. Saitoh,\nNature Mater. 12, 549 (2013).\n16A. Kehlberger, R. R¨ oser, G. Jacob, U. Ritzmann,\nD. Hinzke, U. Nowak, M. Ombasli, D. H. Kim, C. A. Ross,\nM. B. Jungfleisch, B. Hillebrands, and M. Kl¨ aui, arXiv:\n1306.0784.\n17S. Hoffman, K. Sato, and Y. Tserkovnyak,\nPhys. Rev. B 88, 064408 (2013).\n18M. Agrawal, V. I. Vasyuchka, A. A. Serga, A. Kirihara,\nP. Pirro, T. Langner, M. B. Jungfleisch, A. V. Chumak,\nE. T. Papaioannou, and B. Hillebrands, arXiv:1309.2164.\n19K. Uchida, H. Adachi, T. Ota, H. Nakayama, S. Maekawa,\nand E. Saitoh, Appl. Phy. Lett. 97, 172505 (2010).\n20M. Weiler, M. Althammer, F. D. Czeschka, H. Huebl,\nM. S. Wagner, M. Opel, I.-M. Imort, G. Reiss,\nA. Thomas, R. Gross, and S. T. B. Goennenwein,\nPhys. Rev. Lett. 108, 106602 (2012).\n21D. Qu, S. Y. Huang, J. Hu, R. Wu, and C. L. Chien,\nPhys. Rev. Lett. 110, 067206 (2013).\n22T. Kikkawa, K. Uchida, Y. Shiomi, Z. Qiu, D. Hou,\nD. Tian, H. Nakayama, X.-F. Jin, and E. Saitoh,\nPhys. Rev. Lett. 110, 067207 (2013)."    },    {        "title": "1605.06578v1.Landau_Lifshitz_theory_of_the_magnon_drag_thermopower.pdf",        "content": "Landau-Lifshitz theory of the magnon-drag thermopower\nBenedetta Flebus,1, 2Rembert A. Duine,1, 3and Yaroslav Tserkovnyak2\n1Institute for Theoretical Physics and Center for Extreme Matter and Emergent Phenomena,\nUtrecht University, Leuvenlaan 4, 3584 CE Utrecht, The Netherlands\n2Department of Physics and Astronomy, University of California, Los Angeles, California 90095, USA\n3Department of Applied Physics, Eindhoven University of Technology,\nPO Box 513, 5600 MB, Eindhoven, The Netherlands\nMetallic ferromagnets subjected to a temperature gradient exhibit a magnonic drag of the electric\ncurrent. We address this problem by solving a stochastic Landau-Lifshitz equation to calculate the\nmagnon-drag thermopower. The long-wavelength magnetic dynamics result in two contributions to\nthe electromotive force acting on electrons: (1) An adiabatic Berry-phase force related to the solid\nangle subtended by the magnetic precession and (2) a dissipative correction thereof, which is rooted\nmicroscopically in the spin-dephasing scattering. The \frst contribution results in a net force pushing\nthe electrons towards the hot side, while the second contribution drags electrons towards the cold\nside, i.e., in the direction of the magnonic drift. The ratio between the two forces is proportional\nto the ratio between the Gilbert damping coe\u000ecient \u000band the coe\u000ecient \fparametrizing the\ndissipative contribution to the electromotive force.\nThe interest in thermoelectric phenomena in ferromag-\nnetic heterostructures has been recently revived by the\ndiscovery of the spin Seebeck e\u000bect [1]. This e\u000bect is now\nunderstood to stem from the interplay of the thermally-\ndriven magnonic spin current in the ferromagnet and the\n(inverse) spin Hall voltage generation in an adjacent nor-\nmal metal [2]. Lucassen et al. [3] subsequently proposed\nthat the thermally-induced magnon \row in a metallic\nferromagnet can also produce a detectable (longitudinal)\nvoltage in the bulk itself, due to the spin-transfer mecha-\nnism of magnon drag. Speci\fcally, smooth magnetization\ntexture dynamics induce an electromotive force [4], whose\nnet average over thermal \ructuations is proportional to\nthe temperature gradient. In this Letter, we develop\na Landau-Lifshitz theory for this magnon drag, which\ngeneralizes Ref. [3] to include a heretofore disregarded\nBerry-phase contribution. This additional magnon drag\ncan reverse the sign of the thermopower, which can have\npotential utility for designing scalable thermopiles based\non metallic ferromagnets.\nElectrons propagating through a smooth dynamic tex-\nture of the directional order parameter n(r;t) [such that\njn(r;t)j\u00111, with the self-consistent spin density given\nbys=sn] experience the geometric electromotive force\nof [4]\nFi=~\n2(n\u0001@tn\u0002@in\u0000\f@tn\u0001@in) (1)\nfor spins up along nand\u0000Fifor spins down. The resul-\ntant electric current density is given by\nji=\u001b\"\u0000\u001b#\nehFii=~P\u001b\n2ehn\u0001@tn\u0002@in\u0000\f@tn\u0001@ini;(2)\nwhere\u001b=\u001b\"+\u001b#is the total electrical conductivity, P=\n(\u001b\"\u0000\u001b#)=\u001bis the conducting spin polarization, and eis\nthe carrier charge (negative for electrons). The averaging\nh:::iin Eq. (2) is understood to be taken over the steady-\nstate stochastic \ructuations of the magnetic orientation.The latter obeys the stochastic Landau-Lifshitz-Gilbert\nequation [5]\ns(1 +\u000bn\u0002)@tn+n\u0002(Hz+h) +X\ni@iji= 0;(3)\nwhere\u000bis the dimensionless Gilbert parameter [6], H\nparametrizes a magnetic \feld (and/or axial anisotropy)\nalong thezaxis, and ji=\u0000An\u0002@inis the magnetic spin-\ncurrent density, which is proportional to the exchange\nsti\u000bnessA. ForH > 0, the equilibrium orientation is\nn!\u0000 z, which we will suppose in the following. The\nLangevin \feld stemming from the (local) Gilbert damp-\ning is described by the correlator [7]\nhhi(r;!)h\u0003\nj(r0;!0)i=2\u0019\u000bs~!\u000eij\u000e(r\u0000r0)\u000e(!\u0000!0)\ntanh~!\n2kBT(r);\n(4)\nupon Fourier transforming in time: h(!) =R\ndtei!th(t).\nAt temperatures much less than the Curie tempera-\nture,Tc, it su\u000eces to linearize the magnetic dynamics\nwith respect to small-angle \ructuations. To that end, we\nswitch to the complex variable, n\u0011nx\u0000iny, parametriz-\ning the transverse spin dynamics. Orienting a uniform\nthermal gradient along the xaxis,T(x) =T+x@xT,\nwe Fourier transform the Langevin \feld (4) also in real\nspace, with respect to the yandzaxes. Linearizing\nEq. (3) for small-angle dynamics results in the Helmholtz\nequation:\nA(@2\nx\u0000\u00142)n(x;q;!) =h(x;q;!); (5)\nwhere\u00142\u0011q2+[H\u0000(1+i\u000b)s!]=A,h\u0011hx\u0000ihy, and qis\nthe two-dimensional wave vector in the yzplane. Solving\nEq. (5) using the Green's function method, we substitute\nthe resultant ninto the expression for the charge current\ndensity (2), which can be appropriately rewritten in thearXiv:1605.06578v1  [cond-mat.mes-hall]  21 May 20162\nfollowing form (for the nonzero xcomponent):\njx=~P\u001b\n2eZd2qd!\n(2\u0019)3!\n\u0002Re(1 +i\f)hn(x;q;!)@xn\u0003(x;q0;!0)i\n(2\u0019)3\u000e(q\u0000q0)\u000e(!\u0000!0): (6)\nTedious but straightforward manipulations, using the\ncorrelator (4), \fnally give the following thermoelectric\ncurrent density:\njx=\u000bsP\u001b@xT\n4eA2kBT2Zd2qd!\n(2\u0019)3(~!)3\nsinh2~!\n2kBTRe [(1 +i\f)I];\n(7)\nwhereI(\u0014)\u0011\u0014=j\u0014j2(Re\u0014)2, having made the convention\nthat Re\u0014>0.\nTo recast expression (7) in terms of magnon modes, we\nincorporate the integration over qxby noticing that, in\nthe limit of low damping, \u000b!0,\nI=2\n\u0019Z\ndqx1 +iq2\nx=\u000b~!\n(~!\u0000q2x\u0000q2\u0000\u0018\u00002)2+ (\u000b~!)2:(8)\nHere, we have introduced the magnetic exchange length\n\u0018\u0011p\nA=H and de\fned ~ !\u0011s!=A . After approximating\nthe Lorentzian in Eq. (8) with the delta function when\n\u000b\u001c1, Eq. (7) can \fnally be expressed in terms of a\ndimensionless integral\nJ(a)\u0011Z1\na=p\n2dxx5p\n2x2\u0000a2\nsinh2x2; (9)\nas\nj=\u0012\n1\u0000\f\n3\u000b\u0013\nJ\u0012\u0015\n\u0018\u0013kBP\u001b\n\u00192e\u0012T\nTc\u00133=2\nrT: (10)\nHere,Tis the ambient temperature, kBTc\u0011A(~=s)1=3\nestimates the Curie temperature, and \u0015\u0011p\n~A=skBT\nis the thermal de Broglie wavelength in the absence of\nan applied \feld. We note that \u000b;\f\u001c1 while\u000b\u0018\f, in\ntypical transition-metal ferromagnets [8].\nFor temperatures much larger than the magnon gap\n(typically of the order of 1 K in metallic ferromagnets),\n\u0015\u001c\u0018and we can approximate J(\u0015=\u0018)\u0019J(0)\u0018\n1. This limit e\u000bectively corresponds to the gapless\nmagnon dispersion of \u000fq\u0011~!q\u0019~Aq2=s. Within\nthe Boltzmann phenomenology, the magnonic heat cur-\nrent induced by a uniform thermal gradient is given by\njQ=\u0000rTR\n[d3q=(2\u0019)3](@qx!q)2\u001c(!q)\u000fq@TnBE, where\n\u001c\u00001(!q) = 2\u000b!qis the Gilbert-damping decay rate of\nmagnons (to remain within the consistent LLG phe-\nnomenology) and nBE= [exp(\u000fq=kBT)\u00001]\u00001is the Bose-\nEinstein distribution function. By noticing that\n\u000fq@TnBE=kB\u0014~!q=2kBT\nsinh( ~!q=2kBT)\u00152\n; (11)\nrThydrodynamic\nr⌦<0geometricˆxˆzˆy\n⌦\ne\u0000e\u0000e\u0000e\u0000e\u0000e\u0000e\u0000e\u0000\nFIG. 1. Schematics for the two contributions to the electron-\nmagnon drag. In the absence of decay (i.e., \u000b!0), magnons\ndrifting from the hot (left) side to the cold (right) side drag\nthe charge carriers viscously in the same direction, inducing\na thermopower /\f. The (geometric) Berry-phase drag gov-\nerned by the magnon decay is proportional to \u000band acts in\nthe opposite direction. It is illustrated for a spin wave that is\nthermally emitted from the left. As the spin wave propagates\nto the right, the solid angle \n subtended by the spin preces-\nsion shrinks, inducing a force oriented to the left for spins\nparallel to n.\nit is easy to recast the second, /\fcontribution to\nEq. (10) in the form\nj(\f)=\f~P\u001b\n2eAjQ; (12)\nwhich reproduces the main result of Ref. [3].\nThe magnon-drag thermopower (Seebeck coe\u000ecient),\nS=\u0000@xV\n@xT\f\f\f\f\njx=0; (13)\ncorresponds to the voltage gradient @xVinduced under\nthe open-circuit condition. We thus get from Eq. (10):\nS=\u0012\f\n3\u000b\u00001\u0013\nJkBP\n\u00192e\u0012T\nTc\u00133=2\n= (\f\u00003\u000b)~P\u0014m\n2eA;\n(14)\nwhere\u0014m= (2=3\u00192)JkBA(T=Tc)3=2=\u000b~is the magnonic\ncontribution to the heat conductivity. Such magnon-drag\nthermopower has recently been observed in Fe and Co\n[9], with scaling/T3=2over a broad temperature range\nand opposite sign in the two metals. Note that the sign\ndepends on \f=\u000b and the e\u000bective carrier charge e.\nEquations (10) and (14) constitute the main results of\nthis paper. In the absence of Gilbert damping, \u000b!0,\nthe magnon-drag thermopower Sis proportional to the\nheat conductivity. This contribution was studied in\nRef. [3] and is understood as a viscous hydrodynamic\ndrag. In simple model calculations [8], \fP > 0 and this\nhydrodynamic thermopower thus has the sign of the ef-\nfective carrier charge e. WhenP > 0, so that the ma-\njority band is polarized along the spin order parameter3\nn, the/\u000bcontribution to the thermopower is opposite\nto the/\fcontribution. (Note that \u000bis always>0,\nin order to yield the positive dissipation.) The underly-\ning geometric meaning of this result is sketched in Fig. 1.\nNamely, the spin waves that are generated at the hot end\nand are propagating towards the cold end are associated\nwith a decreasing solid angle, @x\n<0. The \frst term\nin Eq. (1), which is rooted in the geometric Berry con-\nnection [10], is proportional to the gradient of this solid\nangle times the precession frequency, /!@i\n, resulting\nin a net force towards the hot side acting on the spins\ncollinear with n.\nNote that we have neglected the Onsager-reciprocal\nbackaction of the spin-polarized electron drift on the\nmagnetic dynamics. This is justi\fed as including the\ncorresponding spin-transfer torque in the LLG equation\nwould yield higher-order e\u000bects that are beyond our\ntreatment [11]. The di\u000busive contribution to the See-\nbeck e\u000bect,/T=EF, whereEFis a characteristic Fermi\nenergy, which has been omitted from our analysis, is ex-\npected to dominate only at very low temperatures [9].\nThe conventional phonon-drag e\u000bects have likewise been\ndisregarded. A systematic study of the relative impor-\ntance of the magnon and phonon drags is called upon in\nmagnetic metals and semiconductors.\nThis work is supported by the ARO under Contract\nNo. 911NF-14-1-0016, FAME (an SRC STARnet center\nsponsored by MARCO and DARPA), the Stichting voor\nFundamenteel Onderzoek der Materie (FOM), and the\nD-ITP consortium, a program of the Netherlands Orga-\nnization for Scienti\fc Research (NWO) that is funded by\nthe Dutch Ministry of Education, Culture, and Science\n(OCW).[1] K. Uchida, S. Takahashi, K. Harii, J. Ieda, W. Koshibae,\nK. Ando, S. Maekawa, and E. Saitoh, Nature 455,\n778 (2008); K. Uchida, J. Xiao, H. Adachi, J. Ohe,\nS. Takahashi, J. Ieda, T. Ota, Y. Kajiwara, H. Umezawa,\nH. Kawai, G. E. W. Bauer, S. Maekawa, and E. Saitoh,\nNat. Mater. 9, 894 (2010).\n[2] J. Xiao, G. E. W. Bauer, K. Uchida, E. Saitoh, and\nS. Maekawa, Phys. Rev. B 81, 214418 (2010).\n[3] M. E. Lucassen, C. H. Wong, R. A. Duine, and\nY. Tserkovnyak, Appl. Phys. Lett. 99, 262506 (2011).\n[4] R. A. Duine, Phys. Rev. B 77, 014409 (2008);\nY. Tserkovnyak and M. Mecklenburg, ibid.77, 134407\n(2008).\n[5] S. Ho\u000bman, K. Sato, and Y. Tserkovnyak, Phys. Rev. B\n88, 064408 (2013).\n[6] T. L. Gilbert, IEEE Trans. Magn. 40, 3443 (2004).\n[7] W. F. Brown, Phys. Rev. 130, 1677 (1963).\n[8] Y. Tserkovnyak, A. Brataas, and G. E. W. Bauer, J.\nMagn. Magn. Mater. 320, 1282 (2008).\n[9] S. J. Watzman, R. A. Duine, Y. Tserkovnyak, H. Jin,\nA. Prakash, Y. Zheng, and J. P. Heremans, \\Magnon-\ndrag thermopower and Nernst coe\u000ecient in Fe and Co,\"\narXiv:1603.03736.\n[10] M. V. Berry, Proc. R. Soc. London A 392, 45 (1984);\nG. E. Volovik, J. Phys. C: Sol. State Phys. 20, L83\n(1987); S. E. Barnes and S. Maekawa, Phys. Rev. Lett.\n98, 246601 (2007); Y. Tserkovnyak and C. H. Wong,\nPhys. Rev. B 79, 014402 (2009).\n[11] The backaction by the spin-transfer torque would be ab-\nsent when the longitudinal spin current, ji=\u001b(PE i+\nFi=e)n, vanishes, where Eiis the electric \feld and Fi\nis the spin-motive force (1). Understanding Eq. (10) as\npertaining to the limit of the vanishing spin current ji\nrather than electric current ji=\u001b(Ei+PFi=e)nwould,\nhowever, result in higher-order (in T=T c) corrections to\nthe Seebeck coe\u000ecient (13). These are beyond the level\nof our approximations."    },    {        "title": "1605.06797v1.Low_Gilbert_damping_in_Co2FeSi_and_Fe2CoSi_films.pdf",        "content": "arXiv:1605.06797v1  [cond-mat.mtrl-sci]  22 May 2016Low Gilbert damping in Co 2FeSi and Fe 2CoSi films\nChristian Sterwerf,1,∗Soumalya Paul,2Behrouz Khodadadi,2Markus Meinert,1\nJan-Michael Schmalhorst,1Mathias Buchmeier,2Claudia K. A. Mewes,2Tim Mewes,2and G¨ unter Reiss1\n1Center for Spinelectronic Materials and Devices,\nPhysics Department, Bielefeld University, Germany\n2Department of Physics and Astronomy/MINT Center,\nThe University of Alabama, Tuscaloosa, AL 35487, USA\n(Dated: August 15, 2018)\nThin highly textured Fe 1+xCo2−xSi (0≤x≤1) films were prepared on MgO (001) substrates\nby magnetron co-sputtering. The magneto-optic Kerr effect ( MOKE) and ferromagnetic resonance\n(FMR) measurements were used to investigate the compositio n dependence of the magnetization,\nthe magnetic anisotropy, the gyromagnetic ratio and the rel axation of the films. The effective mag-\nnetization for the thin Fe 1+xCo2−xSi films, determined by FMR measurements, are consistent wit h\nthe Slater Pauling prediction. Both MOKE and FMR measuremen ts reveal a pronounced fourfold\nanisotropy distribution for all films. In addition we found a strong influence of the stoichiometry\non the anisotropy as the cubic anisotropy strongly increase s with increasing Fe concentration. The\ngyromagnetic ratio is only weakly dependent on the composit ion. We find low Gilbert damping pa-\nrameters for all films with values down to 0 .0012±0.00012 for Fe 1.75Co1.25Si. The effective damping\nparameter for Co 2FeSi is found to be 0 .0018±0.0004. We also find a pronounced anisotropic relax-\nation, which indicates significant contributions of two-ma gnon scattering processes that is strongest\nalong the easy axes of the films. This makes thin Fe 1+xCo2−xSi films ideal materials for the appli-\ncation in STT-MRAM devices.\nI. INTRODUCTION\nHalf-metallic ferromagnets have attracted great inter-\nest during the past few years because they promise to\nboost the performance of spintronic devices. High spin\npolarization at the Fermi level can generate high tun-\nnel magnetoresistance (TMR) ratios. A TMR effect can\nbe measured in a magnetic tunnel junction (MTJ) that\nconsists of two ferromagnetic films separated by a thin\ninsulator. The same structures can also be utilized to\nspin transfer torque induced magnetization switching [1],\nhoweverin this casea lowswitching currentdensity is de-\nsirable. Thus, low magnetic damping and a high spin po-\nlarization are frequently required for spin transfer torque\nbased devices [2]. A high spin polarization can be found\nin half-metals where one spin band structure is semicon-\nducting while the other spin band structure is metallic.\nCo- and Fe-based Heusler compounds are good candi-\ndates for materials with high Curie temperatures and\nhalf-metallic behavior.\nFull Heusler compounds have the formula X 2YZ, where\nX and Y are transition metals and Z is a main group\nelement. There are two different ordered structures: the\nL21structure and the X astructure with a different occu-\npation sequence. Both structures consist of a four-atom\nbasis and an fcc lattice. The prototype of the L2 1struc-\nture is Cu 2MnAl (space group Fm ¯3m) with the occupa-\ntion sequence X-Y-X-Z [3]. The prototypes for the X a\nstructure are Hg 2CuTi and Li 2AgSb with an occupation\nsequence Y-X-X-Z, with the two X-atoms at inequivalent\npositions in the lattice [4, 5]. In this work, we investigate\n∗csterwerf@physik.uni-bielefeld.dethe magnetic properties of a stoichiometric series rang-\ning from Co 2FeSi to Fe 2CoSi, where Co 2FeSi crystalizes\nin the L2 1structure and Fe 2CoSi in the X astructure, re-\nspectively. Both compounds should have a (pseudo-)gap\nin the minority states as predicted by first principle cal-\nculations. By substituting Co and Fe atoms the number\nof electrons varies and the Fermi level is expected to be\nshifted to lower energies when the Fe concentration is\nincreased. As we reported previously, magnetic tunnel\njunctions based on the Fe 1+xCo2−xSi films exhibit very\nhigh TMR ratios for all stoichiometries [6]. At 15K a\nmaximum TMR ratio of 262% was found for the inter-\nmediate stoichiometry Fe 1.75Co1.25Si, while the Co 2FeSi\nand Fe 2CoSi based MTJs showed a TMR ratio of 167%\nand 227%, respectively. One possible explanation for the\nhigh TMR ratio is that for Fe 1.75Co1.25Si the Fermi en-\nergy is shifted inside the pseudo-gap. In this work we\npresent results of the magnetic properties for the mag-\nnetization dynamics in particular including anisotropy\nand the Gilbert damping parameter of the Fe 1+xCo2−xSi\nfilms, as the intrinsic relaxation is are expected to be low\nfor half-metals [7].\nII. PREPARATION AND\nCHARACTERIZATION TECHNIQUES\nThin Fe 1+xCo2−xSi (x=0, 0 .25, 0.5, 0.75, 1) films were\nfabricated using co-sputtering in an UHV sputtering sys-\ntemwithabasepressureof1 ·10−9mbar. TheArpressure\nduringsputteringwas2 ·10−3mbar. Thefilmsweregrown\nby dc- and rf-magnetron sputtering from elemental tar-\ngets ontoMgO (001) substrates. Additional MgOand Cr\nseed layers were used to accommodate small lattice mis-2\nmatches and to promote coherent and epitaxial growth,\nas the Cr seed layer grows in 45◦direction on the MgO\nlayer, which has a lattice parameter of 4 .212˚A. The lat-\ntice mismatch between two unit cells of Cr (2 ×2.885˚A\nat 20◦C [8]) and one unit cell of Co 2FeSi (5.64˚A [9])\nor Fe2CoSi (5.645˚A [10]) is about 2%. The 5nm thick\nMgO and Cr films were in-situ annealed at 700◦C to ob-\ntain smooth surfaces. Fe 1+xCo2−xSi films with a thick-\nness of 20nm were deposited at room temperature and\nex-situ vacuum annealed at 500◦C. A 2nm thick MgO\ncapping layer was used to prevent oxidation of the films.\nTo determine the stoichiometry and to adjust the sput-\ntering powers, x-rayfluorescencemeasurementswere car-\nried out. To obtain information about the magnetization\ndynamics, in-plane ferromagneticresonance(FMR) mea-\nsurements were performed using a broadband coplanar\nwaveguide setup up to a maximum frequency of 40GHz.\nLeast square fits of the raw data using a first derivative\nof a Lorenzian line shape were done to precisely deter-\nmine the resonance field and the peak-to-peak linewidth\n∆H[11, 12]. For the FMR in-plane angle dependent\nmeasurements the samples were mounted on a rotating\nstage and the resonance spectra were measured at a fre-\nquencyof30GHz whilethe in-planeanglewaschangedin\n5◦steps. In addition quasistatic magnetization reversal\nmeasurements were carried out using the magneto-optic\nKerr effect (MOKE) in a vector MOKE setup with an s-\npolarized laser with a wavelength of 488nm. Anisotropy\nmeasurements were carried out using a rotating sample\nholder. The magnetic field was applied in the plane of\nthe films.\nIII. CRYSTALLOGRAPHIC PROPERTIES\nX-ray diffraction measurements were used to investi-\ngate the crystallographic properties of the Fe 1+xCo2−xSi\nfilms. Ordering parameters, determined from x-ray\ndiffraction, were already discussed in our previous work\n[6] and found to be high for Co 2FeSi and decrease when\ngoing to Fe 2CoSi. In order to test the films for crystal-\nlographic symmetry ϕscans are performed on the (220)\nplanes of the Fe 1+xCo2−xSi films. Figure 1 shows the\nresults together with the (220) plane of the MgO (001)\nsubstrate. The result shows that the (100) Heusler plane\nis rotated by 45◦with respect to the MgO (100) plane.\nThe fourfold symmetry of the ϕ-scans clearly verifies the\nhighly textured growth of all Fe 1+xCo2−xSi films of this\nstudy.\nIV. MAGNETIZATION DYNAMICS\nIn this section we present in-plane broadband FMR\nmeasurements for the Fe 1+xCo2−xSi samples to obtain\ninformation about the magnetic properties of the films.\nThe Landau-Lifshitz-Gilbert equation describes the dy-\nnamics of the magnetization vector /vectorMin the presencesqrt intensity (a.u.) \n360 315 270 225 180 135 90 45 0\n (°)x=0.75 x=0.5 x=0.25 x=0   MgO \nsubstrate \nx=1 \nFIG. 1. ϕ-scans of the (220) Fe 1+xCo2−xSi peak and (220)\nMgO substrate peak showing the fourfold symmetry of the\nfilms.\n40 \n30 \n20 \n10 \n0f (GHz) \n8 6 4 2 0\nH (kOe)  Fe2CoSi [100]\n Fe2CoSi [110]\nFIG. 2. Resonance frequency versus magnetic field (Kittel\nplot) along the in-plane magnetic hard [110] and the magneti c\neasy [100] axis for Fe 2CoSi. The experimental data are fitted\nusing a combined fit (equations (3 and 4)) to determine Meff\nandγ′.\nof an effective field /vectorHeff, which contains both dc and ac\nfields.\nIt is given by [13]:\nd/vectorM\ndt=−γ/vectorM×/vectorHeff+α\nM/parenleftBigg\n/vectorM×d/vectorM\ndt/parenrightBigg\n,(1)\nwhereγis the gyromagnetic ratio and the quantity pa-\nrameter αis the Gilbert damping parameter. Accord-\ning to the Landau-Lifshitz-Gilbert equation (1), the res-\nonance condition can be expressed in terms of the sec-\nond derivatives of the free-energy density Eby the Smit-\nBeljers formula [14]:\n/parenleftbiggf\nγ′/parenrightbigg2\n=1\n(Msinθ)2/bracketleftBigg\n∂2E\n∂θ2∂2E\n∂ϕ2−/parenleftbigg∂2E\n∂θ∂ϕ/parenrightbigg2/bracketrightBigg/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle\nθ0,ϕ0,(2)\nwhereγ′=γ/2π,θandϕare the polar and azimuthal\nangles of the magnetization /vectorMandθ0andϕ0the corre-3\nMeff ( B / f.u.) \n0 0.5 1 0.25 0.75\nx7\n6\n5\n4\n3\n2\n1\n07\n6\n5\n4\n3\n2\n1\n0\nMS ( B / f.u.) \nFIG. 3. Dependence of the fitted effective magnetic moment\nper formula unit for Fe 1+xCo2−xSi films with x=0, 0 .25, 0.5,\n0.75, 1shownontheleftaxis. Thedashedlineshows theinter-\npolated expected magnetic moments according to the Slater-\nPauling rule (right axis).\nsponding equilibrium values. Measurements of the mag-\nnetic field dependent resonance frequency were carried\nout in two different orientations of the sample: in [100]\nand [110] direction of the Fe 1+xCo2−xSi Heusler alloy, as\nthe [100] direction is the magnetic easy axis and the [110]\ndirection the magnetic hard axis, respectively. Figure 2\nshowsthe exemplaryKittel plots along[100]and [110]di-\nrections for the Fe 2CoSi sample. The experimental data\nwere fitted simultaneously using the Kittel equation for\nboth easy and hard configurations [15]:\nf=γ′/radicalbigg\n(Hres−ha−H4)(Hres−ha+H4\n2+4πMeff) (3)\nf=γ′/radicalbig\n(Hres−ea+H4)(Hres−ea+H4+4πMeff) (4)\nwhereMeff,γ′andH4are shared fit parameters. H4de-\nscribes the magnitude of the in-plane fourfold anisotropy\nfield.Hres−haandHres−eadenote the resonance field\nalong the magnetic hard and the magnetic easy axis,\nrespectively. The resulting fit parameters for the gyro-\nmagnetic ratio γ′are presented in Fig. 6 a) for all xin\nFe1+xCo2−xSi. Within the errorbarsitisnearlyconstant\nfor x≥0.25 and slightly smaller for Co 2FeSi. The fitted\neffective magnetization, which includes any perpendicu-\nlar anisotropy present in the films, is shown in Fig. 3\nfor the Fe 1+xCo2−xSi samples. The error bars originate\nfrom fitting of the Kittel equations and the determina-\ntion of the volume of the unit cell. For bulk Co 2FeSi\nand Fe 2CoSi the experimentally determined magnetiza-\ntionsare5 .95µB/f.u.[9]and4 .99µB/f.u.[10], respectively,\nwhich match the expected magnetizations according to\nthe Slater-Pauling rule (visualized by the dashed line\nin Fig. 3 on the right axis). The deviation from the\nexpected values might be attributed to residual atomic\ndisorder in the films or the presence of a perpendicular\nanisotropy caused by a small tetragonal distortion in the\n[001] direction.100\n80 \n60 \n40 \n20 \n0H (Oe)\n40 30 20 10 0\nf (GHz) Co2FeSi\n Fe1.25Co 1.75Si \n Fe1.5Co 1.5Si \n Fe1.75 Co1.25Si \n Fe2CoSi\nFIG. 4. Frequency dependent FMR linewidth for all sam-\nples measured along the magnetic hard axis [110] of the\nFe1+xCo2−xSi films.\nThe frequency dependence of the linewidth of the fer-\nromagnetic resonance absorption provides direct infor-\nmation about the magnetic relaxation. The frequency\ndependence of the linewidth [16, 17] can under certain\nconditions be characterized by an inhomogeneous resid-\nual linewidth at zero field ∆ H0and an intrinsic contri-\nbution [18]:\n∆H= ∆H0+2√\n3αeff\nγ′f. (5)\nFor correct determination of the effective damping pa-\nrameter it is necessary to measure the linewidth over a\nwide frequency range to determine the slope. It is not\nsufficient to measure ∆ Hat a fixed frequency, because\na non-zero extrinsic linewidth ∆ H0results in an over-\nestimated damping parameter αeff. Figure 4 shows the\npeak-to-peak linewidth ∆ Hfor all frequencies and all\nx. The measurements were performed in the direction\nof the magnetic hard axis of the Heusler films. The ex-\nperimental data were fitted by equation (5) to determine\nthe effective damping parameters. The slope at higher\nfrequencies was used to determine the damping parame-\nters. The inhomogeneous residual linewidth at zero field\n∆H0is presented in Fig. 6 b) for all stoichiometries. The\nerror margins result from the different slopes in the ∆ H\nvs.fcurves. The residual linewidth decreases as the\nFe concentration increases and reaches its lowest value of\n∆H0= 12Oe for Fe 2CoSi. McMichael et al.[19] found\nthat small grain size distributions can lead to low inho-\nmogeneous line broadening.\nThe effective Gilbert damping parameter αeffis shown\nin Fig. 6 c). All damping parameters have the same\norder of magnitude and vary between 0 .0012±0.00012\nto 0.0019±0.00013. The very upper limit of the er-\nror margins was calculated assuming that the linewidth\nmeasured at 40GHz is caused solely by Gilbert type\ndamping. Co 2FeSi exhibits a damping parameter of\n0.0018±0.0004, while Fe 2CoSi shows a slightly larger\nvalue of 0 .0019±0.00013. Kasatani et al.found damping4\n60 \n50 \n40 \n30 \n20 \n10 \n0) e O (  H\n40 30 20 10 0\nf (GHz)[100]\n[110]Fe 2CoSi\nmagnetic hard axis magnetic easy axis \nFIG. 5. FMR linewidth for Fe 2CoSi measured along both the\nmagnetic hard [110] and magnetic easy [100] axis.\nparameters from 0 .0023 to 0 .0061 for Co 2FeSi films and\n0.002for Fe 2CoSi [20]. In general, the Gilbert damping is\nexpected to be low in half-metallic materials, where spin-\nflip processes are suppressed [7, 21]. The small damping\nparameters of the metallic films show that a pseudo-gap\naspresentinthe Fe 1+xCo2−xSisystemissufficienttogive\nrise to a low Gilbert damping.\nFigure 5 shows the frequency dependent linewidth\nalong easy and hard axes for the Fe 2CoSi. The linewidth\nexhibits almost linearbehavior(the Gilbert model) along\nthe hard axis. We observed non-linear behavior in the\nlinewidth vs. frequency responsealongthe magneticeasy\naxis. ThisnonlineardependenceoftheFMRlinewidthon\nfrequencyisatypicalobservationwhentwomagnonscat-\ntering contributes significantly to the relaxation [22, 23].\nTwo-magnon scattering is an extrinsic relaxation mecha-\nnism and can be induced by means of different scattering\ncenters such as voids or pores [24], surface roughness [22]\nand grain size [25] or by network of misfit dislocations\nwhich causes scattering of the FMR mode (k=0) into\npropagating spin waves (k /negationslash=0).\nA. FMR in-plane rotation measurements\nTo obtain further information about the magnetic\nanisotropies and magnetic relaxation additional FMR\nmeasurements were carried out as a function of the\nin-plane angle of the applied field with respect to the\nFe1+xCo2−xSi[110]axis. Theoperatingfrequencyforthe\nrotation measurements was 30GHz. At this frequency\nthe resonancefields are high enough to saturate the mag-\nnetization along the easy and hard axes. All measure-\nments were performed at room temperature.\nA fourfold symmetry is observed in the in-plane angle\ndependence of the ferromagnetic resonance field for all\nsamples. Figure7a)exemplarilyshowstheferromagnetic\nresonancefield Hresversus the in-plane rotation angle for\nFe2CoSi. The dependence of the resonance field on the\nin-plane angle was simulated numerically using equationH0) e O (  2.94\n2.92\n2.90\n2.88\n2.86' (MHz/Oe) \n60 \n40 \n20 \n0K4e k (  rmc/g3)\n0 0.5 1 0.25 0.75\nxa) \nb) \nc) \nd) eff 60 \n50 \n40 \n30 \n20 \n10 \n0\n0.005\n0.004\n0.003\n0.002\n0.001\n0.000\nFIG. 6. a) Gyromagnetic ratio γ′, b)Extrinsic contribution to\nthe linewidth ∆ H0of the FMR spectra, c) effective Gilbert\ndamping parameter and d) cubic magnetic anisotropy con-\nstantK4for Fe 1+xCo2−xSi films with x= 0, 0.25, 0.5, 0.75\n,1.\n(2), assuming a cubic magnetic anisotropy contribution\nto the Gibbs free energy [26, 27]:\nEcubic=−1\n2K4/parenleftbig\nα4\n1+α4\n2+α4\n3/parenrightbig\n, (6)\nwhereK4is the cubic magnetic anisotropy constant and\nα1,α2,α3are the directional cosines with respect to\nthe cubic principal axes. The experimentally determined\nin-plane angle dependent Hresdata were fitted with the\nnumerical solution (red line in Fig. 7 a)) to determine\nthe cubic anisotropy constant. Figure 7 b) shows the\ncorresponding linewidth data, which also shows a clear\nfourfoldsymmetry. The linewidth exhibitsmaximaalong\nthe easy axes and minima along the hard axes of the\ncubic magnetic anisotropy. Randomly distributed crys-\ntalline defects oriented along the in-plane principal crys-\ntallographic axis [28] or a fourfold distribution in misfit\ndislocations [29] which induce the same symmetry on the\nstrength of two magnon scattering can explain the ob-\nserved anisotropic relaxation.\nThe magnetic fourfold symmetry matches the crystal-\nlographic symmetry of the highly textured Fe 1+xCo2−xSi\nfilms mentioned before. A polar plot of the MOKE\nsquareness versus the rotational angle for Fe 2CoSi is pre-\nsented in Fig. 8. This measurement confirms the cubic5\n5100520053005400\n5200\n5300\n5400045 90 \n135\n180\n225\n270315\n020 40 60 \n20 \n40 \n60 045 90 \n135\n180\n225\n270315Hres  (O e) \n[110] [110] \n[100][100] a) \nb) \nH (Oe) \nFIG. 7. Polar plots of a) the resonance fields H resand b)\nthe linewidth ∆ Has a function of the in-plane angle of the\napplied field with respect to the [110] axis of a 20nm thick\nFe2CoSi film measured at a microwave frequency of 30GHz.0.60.81\n0.8\n1045 90 \n135\n180\n225\n270315[110] [100]\nMR/M S\nFIG.8. Polar plotsofthesquarenessMR\nMSforFe 2CoSiobtained\nby MOKE measurements.\nanisotropy present in the films as seen in the FMR mea-\nsurement. The magnetic easy axis is located along the\n[100] crystallographic axis and the magnetic hard axis\nis located along the [110] crystallographic axis. A cubic\nanisotropy with the easy magnetic axis in the Heusler\n[100] direction is found for all samples. The cubic mag-\nnetic anisotropy constant K4obtained from the FMR\nmeasurements changes significantly in this series from\n55.8kerg\ncm3for Fe 2CoSi to 16 .6kerg\ncm3for Co 2FeSi, respec-\ntively. The cubic anisotropy constants for all stoichiome-\ntries are presented in Fig. 6 d). Hashimoto et al.found a\nsimilarcubicanisotropyconstantof18kerg\ncm3forcrystalline\nCo2FeSi with a film thickness of 18 .5nm [30]. Some films\nshowanadditionaluniaxialanisotropycomponent,which\ncan originate from miscut substrates.\nV. CONCLUSION\nIn summary we found very small damping parame-\ntersforthehalf-metallicFe 1+xCo2−xSifilmsvaryingfrom\n0.0012±0.00012 to 0 .0019±0.00013. Co 2FeSi exhibits\na damping parameter of 0 .0018±0.0004. Thus, the\nfilms are suitable for the use in STT-MRAMs. FMR\nandMOKEmeasurementsrevealafourfoldmagnetocrys-\ntalline anisotropy for all films in accordance with the\nfourfold crystalline symmetry in the highly textured\nfilms. The need for frequency dependent FMR measure-\nments was exemplified by the finding that the residual\nlinewidth changes both with composition and with the\nmeasurement direction.\nACKNOWLEDGMENTS\nThe authors gratefully acknowledge financial support\nfrom Bundesministerium f¨ ur Bildung und Forschung\n(BMBF) and Deutsche Forschungsgemeinschaft (DFG,\ncontract no. RE 1052/32-1) as well as support through\nthe MINT Center summer program. S. Paul, B. Kho-\ndadadi and T. Mewes would like to acknowledge sup-\nport by the NSF-CAREER Award No. 0952929, C.K.A.\nMewes would like to acknowledge support by the NSF-\nCAREER Award No. 1452670.6\n[1] L. Berger, Physical Review B 54, 9353 (1996).\n[2] J. C. Slonczewski, Journal of Magnetism and Magnetic\nMaterials 159, L1 (1996).\n[3] A. J. Bradley and J. W. Rodgers, Proceedings of the\nRoyal Society of London Series A 144, 340 (1934).\n[4] M. Puselj and Z. Ban, Croat. Chem. Acta 41, 79 (1969).\n[5] H. Pauly, A. Weiss, and H. Witte, Z. Metallkunde 59,\n47 (1968).\n[6] C. Sterwerf, M. Meinert, J.-M. Schmalhorst, and\nG. Reiss, IEEE Transactions on Magnetics 49, 4386\n(2013).\n[7] C. Liu, C. Mewes, M. Chshiev, and T. Mewes, Applied\nPhysics Letters 95, 022509 (2009).\n[8] M. E. Straumanis and C. C. Weng,\nActa Crystallographica 8, 367 (1955).\n[9] S. Wurmehl, G. Fecher, H. Kandpal, V. Ksenofontov,\nC. Felser, H.-J. Lin, and J. Morais, Physical Review B\n72, 184434 (2005).\n[10] H. Luo, Z. Zhu, L. Ma, S. Xu, H. Liu, J. Qu, Y. Li, and\nG. Wu, Journal of Physics D: Applied Physics 40, 7121\n(2007).\n[11] M. E. Schabes, H. Zhou, and H. N. Bertram, Journal of\nApplied Physics 87, 5666 (2000).\n[12] N. Pachauri, B. Khodadadi, and M. Althammer, Journal\nof Applied Physics 117, 233907 (2015).\n[13] B. Heinrich and J. F. Cochran, Advances in Physics 42,\n523 (1993).\n[14] H. Suhl, Physical Review 97, 555 (1955).\n[15] X. Liu, Y. Sasaki, and J. K. Furdyna, Physical Review\nB67, 205204 (2003).[16] B. Heinrich, J. F. Cochran, and R. Hasegawa, Journal\nof Applied Physics 57, 3690 (1985).\n[17] H. Lee, L. Wen, M. Pathak, and P. Janssen, Journal of\nPhysics D: Applied Physics 41, 215001 (2008).\n[18] C. Mewes and T. Mewes, Relaxation in Magnetic Ma-\nterials for Spintronics, in: Handbook of Nanomagnetism\n(Pan Stanford, 2015) p. 74.\n[19] R. D. McMichael, D. J. Twisselmann, and A. Kunz,\nPhys. Rev. Lett. 90, 227601 (2003).\n[20] Y. Kasatani, S. Yamada, H. Itoh, M. Miyao, K. Hamaya,\nand Y. Nozaki, Applied Physics Express 7, 123001\n(2014).\n[21] G. M. M¨ uller, J. Walowski, M. Djordjevic, and G. X.\nMiao, Nature Materials 8, 56 (2009).\n[22] H. Lee, Y. Wang, C. Mewes, and W. H. Butler, Applied\nPhysics Letters 95, 082502 (2009).\n[23] P. Landeros, R. E. Arias, and D. L. Mills, Physical Re-\nview B77, 214405 (2008).\n[24] M. J. Hurben and C. E. Patton, Journal of Applied\nPhysics83, 4344 (1998).\n[25] R. D. McMichael, M. D. Stiles, and P. J. Chen, Journal\nof Applied Physics 83, 7037 (1998).\n[26] M. Farle, Reports on Progress in Physics 61, 755 (1998).\n[27] B. Heinrich and J. Bland, eds., Radio Frequency Tech-\nniques, in: Ultrathin Magnetic Structures II (Springer,\n1994) p. 195.\n[28] I. Barsukov, P. Landeros, R. Meckenstock, J. Lindner,\nD. Spoddig, Z.-A. Li, B. Krumme, H. Wende, D. L. Mills,\nand M. Farle, Phys. Rev. B 85, 014420 (2012).\n[29] G. Woltersdorf and B. Heinrich, Physical Review B 69,\n184417 (2004).\n[30] M. Hashimoto, J. Herfort, H. P. Schonherr, and K. H.\nPloog, Applied Physics Letters 87, 102506 (2005)."    },    {        "title": "1605.07232v1.Large_time_behaivor_of_global_solutions_to_nonlinear_wave_equations_with_frictional_and_viscoelastic_damping_terms.pdf",        "content": "arXiv:1605.07232v1  [math.AP]  23 May 2016Large time behaivor of global solutions\nto nonlinear wave equations with frictional\nand viscoelastic damping terms\nRyo Ikehata,\nDepartment of Mathematics,\nGraduate School of Education, Hiroshima University\nHigashi-Hiroshima 739-8524, Japan\nHiroshi Takeda,\nDepartment of Intelligent Mechanical Engineering,\nFaculty of Engineering, Fukuoka Institute of Technology,\n3-30-1 Wajiro-higashi, Higashi-ku, Fukuoka, 811-0295 JAPAN\nAbstract\nIn this paper, we study the Cauchy problem for a nonlinear wave equ ation with frictional\nand viscoelastic damping terms. As is pointed out by [8], in this combinat ion, the frictional\ndamping term is dominant for the viscoelastic one for the global dyna mics of the linear\nequation. In this note we observe that if the initial data is small, the f rictional damping\nterm is again dominant even in the nonlinear equation case. In other w ords, our main result\nis diffusion phenomena: the solution is approximated by the heat kern el with a suitable\nconstant. Our proof is based on several estimates for the corre sponding linear equations.\nKeywords: critical exponent, nonlinear wave equation, damping terms , asymptotic profile, the\nCauchy problem\n11 Introduction\nIn this paper we are concerned with the following Cauchy prob lem for a wave equation with two\ntypes of damping terms\n/braceleftBigg\n∂2\ntu−∆u+∂tu−∆∂tu=f(u), t >0, x∈Rn,\nu(0,x) =u0(x), ∂tu(0,x) =u1(x), x∈Rn,(1.1)\nwhereu0(x) andu1(x) are given initial data, and about the nonlinearity f(u) we shall consider\nonly the typical case such as\nf(r) :=|r|p,(p >1),\nwithout loss of generality.\nIn the Cauchy problem case of the following equation with a fr ictional damping term\n/braceleftBigg\n∂2\ntu−∆u+∂tu=f(u), t >0, x∈Rn,\nu(0,x) =u0(x), ∂tu(0,x) =u1(x), x∈Rn,(1.2)\nnowadays one knows an important result called as the critica l exponent problem such as: there\nexists an exponent p∗>1 such that if the power pof nonlinearity f(u) satisfies p∗< p, then\nthe corresponding problem (1.2) has a small data global in ti me solution, while in the case\nwhen 1< p≤p∗the problem (1.2) does not admit any nontrivial global solut ions. In this\nfrictional damping case, one has p∗=pF:= 1+2\nn, which is called as the Fujita exponent in the\nsemi-linear heat equation case. About these contributions , one can cite so many research pa-\nperswritten by [4], [5], [7], [10], [12], [13], [14], [15], [ 16], [20], [21], [22] and thereferences therein.\nQuite recently, Ikehata-Takeda [9] has treated the origina l problem (1.1) motivated by a\nprevious result concerning the linear equation due to Ikeha ta-Sawada [8], and solved the Fujita\ncritical exponent one. They have discovered the value p∗= 1+2\nnagain only in the low dimen-\nsional case (i.e., n= 1,2). So, the problem is still open for n≥3. Anyway, this result due\nto [9] implies an important recognition that the dominant te rm is still the frictional damping\n∂tualthough the equation (1.1) has two types of damping terms. N ote that in the viscoelastic\ndamping case:/braceleftBigg\n∂2\ntu−∆u−∆∂tu=f(u), t >0, x∈Rn,\nu(0,x) =u0(x), ∂tu(0,x) =u1(x), x∈Rn,(1.3)\nwe still do not know the “exact” critical exponent p∗. Several interesting results about this\ncritical exponent problem including optimal linear estima tes for (1.3) can be observed in the\nliterature due to D’Abbicco-Reissig [2, see Theorem 2, and S ection 4]. But, it seems a little far\nfrom complete solution on the critical exponent problem of ( 1.3). In fact, in [2] they studied\nmore general form of equations such as\n∂2\ntu−∆u+(−∆)σ∂tu=µf(u)\nwithσ∈[0,1] andµ≥0. Pioneering and/or important contributions for the case σ= 1 (i.e.,\nstrong damping one) can be seen in some papers due to [6], [11] ( both in abstract theory), [17],\n[19] and the references therein.\nFrom observations above one naturally encounters an import ant problem such as:\neven in the higher dimensional case for n≥3, can one also solve the critical exponent problem\n2of (1.1)?\nOur first purpose is to prove the following global existence r esult of the solution together with\nsuitable decay properties to problem (1.1).\nTheorem 1.1. Letn= 1,2,3,ε >0andp >1 +2\nn. Assume that (u0,u1)∈(Wn\n2+ε,1∩\nWn\n2+ε,∞)×(L1∩L∞)with sufficiently small norms. Then, there exists a unique glob al solution\nu∈C([0,∞);L1∩L∞)to problem (1.1)satisfying\n/ba∇dblu(t,·)/ba∇dblLq(Rn)≤C(1+t)−n\n2(1−1\nq)(1.4)\nforq∈[1,∞].\nOur second aim is to study the large time behavior of the globa l solution given in Theorem\n1.1.\nTheorem 1.2. Under the same assumptions as in Theorem 1.1, the corresponding global solution\nu(t,x)satisfies\nlim\nt→∞tn\n2(1−1\nq)/ba∇dblu(t,·)−MGt/ba∇dblLq(Rn)= 0, (1.5)\nfor1≤q≤ ∞, whereM:=/integraldisplay\nRn(u0(y)+u1(y))dy+/integraldisplay∞\n0/integraldisplay\nRnf(u(s,y))dyds.\nRemark 1.3.By combining the blowup result given in [9, Theorem 1.3] and T heorems 1.1 and\n1.2 with n= 3, one can make sure that even in the n= 3 dimensional case the critical exponent\np∗of the nonlinearity f(u) is given by the Fujita number p∗=pF. Such sharpness has already\nbeen announced in the low dimensional cases (i.e., n= 1,2) by [9, Theorems 1.1 and 1.3]. So,\nthe result for n= 3 is essentially new. This is one of our main contributions t o problem (1.1) in\nthis paper. It is still open to show the global existence part for alln≥4.\nBefore closing this section, wesummarizenotation, whichw ill beusedthroughoutthis paper.\nLetˆfdenote the Fourier transform of fdefined by\nˆf(ξ) :=cn/integraldisplay\nRne−ix·ξf(x)dx\nwithcn= (2π)−n\n2. Also, let F−1[f] orˇfdenote the inverse Fourier transform.\nWe introduce smooth cut-off functions to localize the freque ncy region as follows:\nχL,χMandχH∈C∞(R) are defined by\nχL(ξ) =/braceleftBigg\n1,|ξ| ≤1\n2,\n0,|ξ| ≥3\n4,χH(ξ) =/braceleftBigg\n1,|ξ| ≥3,\n0,|ξ| ≤2,\nχM(ξ) = 1−χL(ξ)−χH(ξ).\nFork≥0 and 1≤p≤ ∞, letWk,p(Rn) be the usual Sobolev spaces\nWk,p(Rn) :=/braceleftBig\nf:Rn→R;/ba∇dblf/ba∇dblWk,p(Rn):=/ba∇dblf/ba∇dblLp(Rn)+/ba∇dbl|∇x|kf/ba∇dblLp(Rn)<∞/bracerightBig\n,\nwhereLp(Rn) is the Lebesguespace for 1 ≤p≤ ∞as usual. When p= 2, we denote Wk,2(Rn) =\nHk(Rn). For the notation of the function spaces, the domain Rnis often abbreviated. We\nfrequently use the notation /ba∇dblf/ba∇dblp=/ba∇dblf/ba∇dblLp(Rn)without confusion. Furthermore, in the following\nCdenotes a positive constant, which may change from line to li ne.\nThe paper is organized as follows. Section 2 presents some pr eliminaries. In Section 3, we\nshow the point-wise estimates of the propagators for the cor responding linear equation in the\nFourier space. Section 4 is devoted to the proof of linear est imates, which play crucial roles to\nget main results. In sections 5 and 6, we give the proof of our m ain results.\n32 Preliminaries\nIn this section, we collect several basic facts on the Fourie r multiplier theory, the decay estimates\nof the solution for the heat equation and elementary inequal ities to obtain the decay property\nof the solutions.\n2.1 Fourier multiplier\nForf∈L2∩Lp, 1≤p≤ ∞, letm(ξ) be the Fourier multiplier defined by\nF−1[mˆf](x) =cn/integraldisplay\nRne−ix·ξm(ξ)ˆf(ξ)dξ.\nWe define Mpas the class of the Fourier multiplier with 1 ≤p≤ ∞:\nMp:=/braceleftbigg\nm:Rn→R|There exists a constant Ap>0 such that /ba∇dblF−1[mˆf]/ba∇dblp≤Ap/ba∇dblf/ba∇dblp/bracerightbigg\n.\nForm∈Mp, we let\nMp(m) := sup\nf/ne}ationslash=0/ba∇dblF−1[mˆf]/ba∇dblp\n/ba∇dblf/ba∇dblp.\nThe following lemma describes the inclusion among the class of multipliers.\nLemma 2.1. Let1\np+1\np′= 1with1≤p≤p′≤ ∞. ThenMp=Mp′and form∈C∞(Rn), it\nholds that\nMp(m) =Mp′(m).\nMoreover, if m∈Mp, thenm∈Mqfor allq∈[p,p′]and\nMq(m)≤Mp(m) =Mp′(m). (2.1)\nWe use the Carleson-Beurling inequality, which is applied t o show the Lpboundedness of the\nFourier multipliers.\nLemma 2.2 (Carleson-Beurling’s inequality) .Ifm∈Hswiths >n\n2, thenm∈Mrfor all\n1≤r≤ ∞.Moreover, there exists a constant C >0such that\nM∞(m)≤C/ba∇dblm/ba∇dbl1−n\n2s\n2/ba∇dblm/ba∇dbln\n2s\n˙Hs. (2.2)\nFor the proof of Lemmas 2.1 and 2.2, see [1].\n2.2 Decay property of the solution of heat equations\nThe following Lemma is also well-known as the decay property and approximation formula of\nthe solution of the heat equation. For the proof, see e.g. [3] .\nLemma 2.3. Letn≥1,ℓ≥0,k≥˜k≥0and1≤r≤q≤ ∞. Then there exists C >0such\nthat\n/ba∇dbl∂ℓ\nt∇k\nxet∆g/ba∇dblq≤Ct−n\n2(1\nr−1\nq)−ℓ−k−˜k\n2/ba∇dbl∇k−˜k\nxg/ba∇dblr. (2.3)\nMoreover, if g∈L1∩Lq, then it holds that\nlim\nt→∞tn\n2(1−1\nq)+k\n2/ba∇dbl∇k\nx(et∆g−mGt)/ba∇dblq= 0, (2.4)\nwherem=/integraldisplay\nRng(y)dyfor1≤q≤ ∞.\n42.3 Useful formula\nIn this subsection, we recall useful estimates to show resul ts in this paper. The following well-\nknown estimate is frequently used to obtain time decay estim ates.\nLemma 2.4. Letn≥1,k≥0and1≤r≤2. Then there exists a constant C >0such that\n/ba∇dbl|ξ|ke−(1+t)|ξ|2/ba∇dblr≤C(1+t)−n\n2r−k\n2. (2.5)\nThe next lemma is also useful to compute the decay order of the nonlinear term in the\nintegral equation.\nLemma 2.5. (i)Leta >0andb >0withmax{a,b}>1. There exists a constant Cdepending\nonly onaandbsuch that for t≥0it is true that\n/integraldisplayt\n0(1+t−s)−a(1+s)−bds≤C(1+t)−min{a,b}. (2.6)\n(ii)Let1> a≥0,b >0andc >0. There exists a constant C, which is independent of tsuch\nthat fort≥0it holds that\n/integraldisplayt\n0e−c(t−s)(t−s)−a(1+s)−bds≤C(1+t)−b. (2.7)\nThe proof of Lemma 2.5 is well-known (see e.g. [18]).\n3 Point-wise estimates in the Fourier space\nIn this section, we show point-wise estimates of the Fourier multipliers, which are important to\nobtain linear estimates in the next section. Now, we recall t he Fourier multiplier expression of\nthe evolution operator to the linear problem. According to t he notation of [8] and [9] we define\nthe Fourier multipliers K0(t,ξ) andK1(t,ξ) as\nK0(t,ξ) :=−λ−eλ+t+λ+eλ−t\nλ+−λ−=e−t|ξ|2−|ξ|2e−t\n1−|ξ|2,\nK1(t,ξ) :=−eλ−t+eλ+t\nλ+−λ−=e−t|ξ|2−e−t\n1−|ξ|2,\nand the evolution operators K0(t)gandK1(t)gto problem (1.1) by\nKj(t)g:=F−1[Kj(t,ξ)ˆg] (3.1)\nforj= 0,1, where λ±are the characteristic roots computed through the correspo nding algebraic\nequations (see Section 3 of [9])\nλ2+(1+|ξ|2)λ+|ξ|2= 0.\nMoreover, using the cut-off functions χk(k=L,M,H), we introduce the “localized” evolution\noperators by\nKjk(t)g:=F−1[Kjk(t,ξ)ˆg], (3.2)\nwhereKjk(t,ξ) :=Kj(t,ξ)χk, forj= 0,1,k=L,M,H.\n53.1 Estimates for the low frequency parts\nWe begin with the following point-wise estimates on small |ξ|region in the Fourier space.\nLemma 3.1. Letn≥1be an integer and |ξ| ≤1/2. Then there exists a constant C >0such\nthat\n|e−t|ξ|2−e−t|ξ|2| ≤Ce−(1+t)|ξ|2, (3.3)\n|∇ξ(e−t|ξ|2−e−t|ξ|2)| ≤Ce−(1+t)|ξ|2(1+t)|ξ|, (3.4)\n|∇2\nξ(e−t|ξ|2−e−t|ξ|2)| ≤Ce−(1+t)|ξ|2(1+t+t2|ξ|2). (3.5)\nProof.The proof is straightforward. Noting |ξ| ≤1\n2, we easily see that\n|e−t|ξ|2−e−t|ξ|2| ≤C(e−t|ξ|2+e−t)≤Ce−(1+t)|ξ|2,\nand\n|∇ξ(e−t|ξ|2−e−t|ξ|2)| ≤Ce−t|ξ|2t|ξ|+Ce−t|ξ| ≤Ce−(1+t)|ξ|2(1+t)|ξ|,\nwhich prove the estimates (3.3) and (3.4), respectively. Fi nally we show the estimate (3.5).\nTaking the second derivative and using |ξ| ≤1\n2again, we have\n|∇2\nξ(e−t|ξ|2−e−t|ξ|2)|= 2|∇ξ(e−t|ξ|2tξ−e−tξ)|\n≤C(e−t|ξ|2(|tξ|2+t)+e−t)\n≤Ce−(1+t)|ξ|2(1+t+t2|ξ|2),\nwhich is the desired estimate (3.5), and the proof is complet e.\nThe following estimate is useful to obtain the decay propert y and the large time behavior of\nthe evolution operator K1(t)g.\nLemma 3.2. Letn≥1be an integer and |ξ| ≤1/2. Then there exists constant C >0such that\n|e−t|ξ|2−e−t| ≤Ce−(1+t)|ξ|2, (3.6)\n|∇ξ(e−t|ξ|2−e−t)| ≤Ce−(1+t)|ξ|2t|ξ|, (3.7)\n|∇2\nξ(e−t|ξ|2−e−t)| ≤Ce−(1+t)|ξ|2(t+t2|ξ|2). (3.8)\nProof.The proof is standard. We have (3.6) by similar arguments to ( 3.3). When k >0, by\napplying ∇k\nξ(e−t|ξ|2−e−t) =∇k\nξe−t|ξ|2, (3.7) and (3.8) can be derived.\nAs an easy consequence of Lemmas 3.1 and 3.2, we arrive at the p oint-wise estimates for the\nFourier multipliers with small |ξ|.\nCorollary 3.3. Under the assumptions as in Lemmas 3.1and Lemma 3.2, it holds that\n|KjL(t,ξ)| ≤Ce−(1+t)|ξ|2χL, (3.9)\n|∇ξKjL(t,ξ)| ≤Ce−(1+t)|ξ|2(1+t)|ξ|χL+Ce−t\n4χ′\nL, (3.10)\n|∇2\nξKjL(t,ξ)| ≤Ce−(1+t)|ξ|2(1+t+t2|ξ|2)χL+Ce−t\n4(χ′\nL+χ′′\nL) (3.11)\nforj= 0,1.\n6Proof.The estimates (3.9), (3.10) and (3.11) for j= 1 are shown by the same argument. Here\nwe only show (3.11) with j= 0. We first note that\n|∇k\nξ(1−|ξ|2)−1| ≤/braceleftBigg\nC|ξ|,fork= 1,\nC,for integers k≥0.(3.12)\nIn addition, it is easy to see that\n|∇k\nξK0L(t,ξ)| ≤Ce−t\n4 (3.13)\non suppχ′\nL∪suppχ′′\nLby (3.3) - (3.5) and (3.12) with k= 0,1. Thus, a direct calculation, (3.12),\n(3.13) and Lemma 3.1 show that\n|∇2\nξK0L(t,ξ)| ≤C/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle∇2\nξ/parenleftBigg\ne−t|ξ|2−e−t|ξ|2\n1−|ξ|2χL/parenrightBigg/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle\n≤CχL|∇2\nξ(e−t|ξ|2−e−t|ξ|2)|+CχL|ξ||∇(e−t|ξ|2−e−t|ξ|2)|\n+CχL|e−t|ξ|2−e−t|ξ|2|+Ce−t\n4(χ′\nL+χ′′\nL)\n≤CχL(1+t+t2|ξ|2)e−(1+t)|ξ|2+CχL|ξ|2e−(1+t)|ξ|2\n+CχLe−(1+t)|ξ|2+Ce−t\n4(χ′\nL+χ′′\nL)\n≤Ce−(1+t)|ξ|2(1+t+t2|ξ|2)χL+Ce−t\n4(χ′\nL+χ′′\nL),\nwhich is the desired estimate (3.11) with j= 0. The proof of Corollary 3.3 is now complete.\nThe following result plays an important role to obtain asymp totic profiles of the evolution\noperators K0(t)gandK1(t)g.\nCorollary 3.4. Under the same assumption as in Lemmas 3.1and Lemma 3.2, it holds that\n|KjL(t,ξ)−e−t|ξ|2χL| ≤C|ξ|2e−(1+t)|ξ|2χL, (3.14)\n|∇ξ(KjL(t,ξ)−e−t|ξ|2χL)| ≤Ce−(1+t)|ξ|2|ξ|(1+t|ξ|2)χL+Ce−t\n4χ′\nL, (3.15)\n|∇2\nξ(KjL(t,ξ)−e−t|ξ|2χL)| ≤Ce−(1+t)|ξ|2(1+t|ξ|2+t2|ξ|4)χL+Ce−t\n4(χ′\nL+χ′′\nL) (3.16)\nforj= 0,1.\nProof.We first consider the case j= 0. Combining the estimate (3.9) with j= 1 and the fact\nthat\nK0L(t,ξ)−e−t|ξ|2χL=|ξ|2K1L(t,ξ), (3.17)\none can get (3.14) with j= 0. In order to show (3.15) and (3.16), by using (3.17) again w e see\nthat\n|∇k\nξ(K0L(t,ξ)−e−t|ξ|2χL)|\n≤/braceleftBigg\nC(|ξ||K1L(t,ξ)|+|ξ|2|∇K1L(t,ξ)|) fork= 1,\nC(|K1L(t,ξ)|+|ξ||∇K1L(t,ξ)|+|ξ|2|∇2K1L(t,ξ)|) fork= 2.(3.18)\nCombining (3.18) and (3.10) with j= 1 yields the estimate (3.15) with j= 0. We now apply\nthis argument again to (3.10) with j= 1 replaced by (3.11) with j= 1, to obtain the estimate\n(3.16) with j= 0. Finally we prove (3.14) - (3.16) with j= 1. Noting that\nK1L(t,ξ)−e−t|ξ|2χL=e−t|ξ|2|ξ|2−e−t\n1−|ξ|2, (3.19)\n7andapplyingasimilarargumentto(3.6),onegets(3.14)wit hj= 1. Moreover, using ∇k\nξ(e−t|ξ|2|ξ|2−\ne−t) =∇k\nξe−t|ξ|2|ξ|2fork >0, we can deduce that\n|∇ξ(e−t|ξ|2|ξ|2−e−t)| ≤C|ξ|(1+t|ξ|2)e−t|ξ|2, (3.20)\n|∇2\nξ(e−t|ξ|2|ξ|2−e−t)| ≤C(1+t|ξ|2+t2|ξ|4)e−t|ξ|2. (3.21)\nTherefore, by (3.14) with j= 1 and (3.20), we obtain (3.15) with j= 1. Likewise, we use (3.14)\nand (3.15) with j= 1 and (3.21) to meet (3.16) with j= 1, and the Corollary follows.\n3.2 Estimates for the middle and high frequency parts\nThe following lemma states that the middle part for |ξ|has a sufficient regularity and decays\nfast.\nLemma 3.5. Letn≥1andk≥0. Then there exists a constant C >0such that\n|∇k\nξKjM(t,ξ)χM| ≤Ce−t\n4χM (3.22)\nforj= 0,1.\nProof.The support of the middle part ∇k\nξKjM(t,ξ)χMis compact and does not contain a neigh-\nborhood of the origin ξ= 0. Therefore, we can estimate the polynomial of |ξ|by a constant.\nThis implies the desired estimate (3.22), and the proof is no w complete.\nThe rest part of this subsection is devoted to the point-wise estimates for the high frequency\npartsKjH(t)gforj= 0,1.\nLemma 3.6. Letn= 1,2,3,ε >0andα∈ {2,n\n2+ε}. Then it holds that\n∇k\nξ/parenleftbigg|ξ|2−α\n1−|ξ|2χH/parenrightbigg\n∈L2(Rn) (3.23)\nfork= 0,1,2.\nProof.It is easy to see that\n∇k\nξ/parenleftbigg|ξ|2−α\n1−|ξ|2/parenrightbigg\n=O(|ξ|−α−k) (3.24)\nas|ξ| → ∞ and 2(−α−k)<−n. Moreover the support of|ξ|2−α\n1−|ξ|2χHdoes not have a\nneighborhood of |ξ|= 1. Summing up these facts, we can assert (3.23), and the proo f is\ncomplete.\nLemma 3.7. Letn≥1and|ξ| ≥3. Then there exists a constant C >0such that\n|∇ξ(e−t|ξ|2−e−t)| ≤Ce−t|ξ|2t|ξ|, (3.25)\n|∇2\nξ(e−t|ξ|2−e−t)| ≤Ce−t|ξ|2(t+t2|ξ|2). (3.26)\nProof.Applying ∇k\nξ(e−t|ξ|2−e−t) =∇k\nξe−t|ξ|2again, we easily have Lemma 3.7.\nCorollary 3.8. Under the same assumptions as in Lemma 3.6, there exists a constant C >0\nsuch that\n|K1H(t,ξ)| ≤Ce−t|ξ|−2χH, (3.27)\n|∇ξK1H(t,ξ)| ≤Ce−t(te−t|ξ|2+|ξ|−3)χH+Ce−tχ′\nH, (3.28)\n/vextendsingle/vextendsingle∇2\nξK1H(t,ξ)/vextendsingle/vextendsingle≤Ce−t(t+t2)e−t|ξ|2χH+e−t\n2|ξ|−2+Ce−t\n2(χ′\nH+χ′′\nH).(3.29)\n8Proof.Since (3.27) - (3.29) are shown by the similar way, we only che ck the validity of (3.29).\nWe first note that\n/vextendsingle/vextendsingleKj(t,ξ)χ′\nH/vextendsingle/vextendsingle+/vextendsingle/vextendsingle∇ξKj(t,ξ)χ′\nH/vextendsingle/vextendsingle+/vextendsingle/vextendsingleKj(t,ξ)χ′′\nH/vextendsingle/vextendsingle≤Ce−t\n2(χ′\nH+χ′′\nH) (3.30)\nforj= 0,1. Indeed, the support of χ′\nHandχ′′\nHis compact and does not include a neighborhood\nofξ= 0. So, the direct calculation and (3.24) - (3.26) show\n|∇2\nξK1(t,ξ)|\n≤C|∇2\nξ(e−t|ξ|2)||ξ|−2+C|∇ξe−t|ξ|2||∇ξ(1−|ξ|2)−1|+Ce−t|ξ|2|∇2\nξ(1−|ξ|2)−1|\n≤Ce−te−t|ξ|2/braceleftbig\n(t+t2|ξ|2)|ξ|−2+|ξ|−3t|ξ|+|ξ|−4)/bracerightbig\n≤Ce−t{(t+t2)e−t|ξ|2+|ξ|−2t+|ξ|−4}\n≤Ce−t(t+t2)e−t|ξ|2+e−t\n2|ξ|−2(3.31)\nfor|ξ| ≥3. Thus combining (3.30) and (3.31), we see\n|∇2\nξK1H(t,ξ)| ≤CχH|∇2K1(t,ξ)|+Ce−t\n2(χ′\nH+χ′′\nH)\n≤Ce−t(t+t2)e−t|ξ|2χH+e−t\n2|ξ|−2χH+Ce−t\n2(χ′\nH+χ′′\nH),\nwhich is the desired conclusion.\nThe following estimates are useful for the estimates for K0H(t)g.\nCorollary 3.9. Under the same assumptions as in Lemma 3.6, there exists a constant C >0\nsuch that/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/parenleftBigg\ne−t|ξ|2−e−t|ξ|2−(n\n2+ε)\n1−|ξ|2χH/parenrightBigg/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle≤Ce−t|ξ|−2χH+Ce−t\n2|ξ|−n\n2−ε−kχH,(3.32)\n/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle∇ξ/parenleftBigg\ne−t|ξ|2−e−t|ξ|2−(n\n2+ε)\n1−|ξ|2χH/parenrightBigg/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle\n≤Ce−t(te−t|ξ|2+|ξ|−3)χH+Ce−tχ′\nH+Ce−t\n2|ξ|−n\n2−ε−kχH,(3.33)\n/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle∇2\nξ/parenleftBigg\ne−t|ξ|2−e−t|ξ|2−(n\n2+ε)\n1−|ξ|2χH/parenrightBigg/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle\n≤Ce−t(t+t2)e−t|ξ|2χH+e−t\n2|ξ|−2χH+Ce−t\n2(χ′\nH+χ′′\nH)+Ce−t\n2|ξ|−n\n2−ε−kχH.(3.34)\nProof.Observing the fact that\n/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle∇k\nξ/parenleftBigg\ne−t|ξ|2−e−t|ξ|2−(n\n2+ε)\n1−|ξ|2χH/parenrightBigg/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle\n≤/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle∇k\nξ/parenleftBigg\ne−t|ξ|2\n1−|ξ|2χH/parenrightBigg/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle+e−t/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle∇k\nξ/parenleftBigg\n|ξ|2−(n\n2+ε)\n1−|ξ|2χH/parenrightBigg/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle,(3.35)\nwe see that the first factor in the right hand side of (3.35) sat isfy the following estimates\n/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsinglee−t|ξ|2\n1−|ξ|2χH/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle≤Ce−t|ξ|−2χH,\n/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle∇ξ/parenleftBigg\ne−t|ξ|2\n1−|ξ|2χH/parenrightBigg/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle≤Ce−t(te−t|ξ|2+|ξ|−3)χH+Ce−tχ′\nH,\n/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle∇2\nξ/parenleftBigg\ne−t|ξ|2\n1−|ξ|2χH/parenrightBigg/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle≤Ce−t(t+t2)e−t|ξ|2χH+e−t\n2|ξ|−2χH+Ce−t\n2(χ′\nH+χ′′\nH),(3.36)\n9as in Corollary 3.8. Furthermore, by using (3.24) with α=n\n2+ε, and (3.31) with j= 1, the\nsecond factor in the right hand side of (3.35) is estimated as follows\ne−t/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle∇k\nξ/parenleftBigg\n|ξ|2−(n\n2+ε)\n1−|ξ|2χH/parenrightBigg/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle≤Ce−t\n2|ξ|−n\n2−ε−kχH. (3.37)\nSumming up these estimates (3.35) - (3.37), one can conclude (3.32) - (3.34).\n4 Linear estimates\nIn this section, we shall study an important decay property o f the solution u(t,x) to the corre-\nsponding linear equation in order to handle with the origina l semi-linear problem (1.1)\n/braceleftBigg\n∂2\ntu−∆u+∂tu−∆∂tu= 0, t >0, x∈Rn,\nu(0,x) =u0(x), ∂tu(0,x) =u1(x), x∈Rn.(4.1)\nOur purpose is to show the following proposition, which sugg ests large time behaviors of the\nsolution to the linear problem above in L1∩L∞framework.\nProposition 4.1. Letn= 1,2,3andε >0. Assume that (u0,u1)∈(Wn\n2+ε,1∩Wn\n2+ε,∞)×\n(L1∩L∞). Then, there exists a unique solution u∈C([0,∞);L1∩L∞)to problem (4.1)such\nthat\n/ba∇dblu(t,·)/ba∇dblLq(Rn)≤C(1+t)−n\n2(1−1\nq), (4.2)\n/ba∇dblu(t,·)−˜MGt/ba∇dblLq(Rn)=o(t−n\n2(1−1\nq)) (t→ ∞) (4.3)\nforq∈[1,∞], where ˜M=/integraldisplay\nRn(u0(y)+u1(y))dy.\n4.1 Decay estimates for “localized” evolution operators\nIn this subsection, we prepare several decay properties of t he evolution operators.\nLemma 4.2. Letn= 1,2,3,1≤r≤q≤ ∞. Then there exists a constant C >0such that\n/ba∇dblKjL(t)g/ba∇dblq≤C(1+t)−n\n2(1\nr−1\nq)/ba∇dblg/ba∇dblr (4.4)\nforj= 0,1.\nLemma 4.3. Letn= 1,2,3,ε >0and1≤r≤q≤ ∞. Then there exists a constant C >0\nsuch that\n/ba∇dblK0H(t)g/ba∇dblq≤/braceleftBigg\nCe−t\n2/ba∇dbl|∇x|n\n2+εg/ba∇dblrforn= 1,\nCe−t\n2/ba∇dbl|∇x|n\n2+εg/ba∇dblqforn= 2,3,(4.5)\n/ba∇dblK1H(t)g/ba∇dblq≤/braceleftBigg\nCe−t\n2/ba∇dblg/ba∇dblrforn= 1,\nCe−t\n2/ba∇dblg/ba∇dblqforn= 2,3,(4.6)\nand\n/ba∇dblKjM(t)g/ba∇dblq≤Ce−t\n2/ba∇dblg/ba∇dblrforj= 0,1. (4.7)\n10Proof of Lemma 4.2. To show (4.4), it is sufficient to show that\n/ba∇dblKjL(t)g/ba∇dbl∞≤C(1+t)−n\n2/ba∇dblg/ba∇dbl1, (4.8)\n/ba∇dblKjL(t)g/ba∇dblq≤C/ba∇dblg/ba∇dblq, (4.9)\nfor 1≤q≤ ∞. Indeed, once we have (4.8) and (4.9), the Riesz-Thorin comp lex interpolation\ntheorem yields (4.4). So, we first show (4.8). By the Hausdorff- Young inequality and (2.5), we\nsee that\n/ba∇dblKjL(t)g/ba∇dbl∞≤C/ba∇dblKjL(t,ξ)ˆg/ba∇dbl1≤ /ba∇dblKjL(t)/ba∇dbl1/ba∇dblˆg/ba∇dbl∞\n≤ /ba∇dble−(1+t)|ξ|2/ba∇dbl1/ba∇dblg/ba∇dbl1=C(1+t)−n\n2/ba∇dblg/ba∇dbl1,\nwhich show the desired estimate (4.8). Next, we prove (4.9) b y applying (2.2). Then by using\n(3.9) - (3.11) and (2.5), we can assert the upper bounds of /ba∇dbl∇k\nξKjL(t)/ba∇dbl2fork= 0,1,2 as follows:\n/ba∇dbl∇k\nξKjL(t)/ba∇dbl2≤C(1+t)−n\n4+k\n2. (4.10)\nTherefore for n= 1, we apply (4.10) with k= 0,1 and (2.2) with s= 1 to have\nM∞(KjL(t))≤C/ba∇dblKjL(t)/ba∇dbl1−1\n2\n2/ba∇dblKjL(t)/ba∇dbl1\n2\n˙H1\n≤C/ba∇dblKjL(t)/ba∇dbl1−1\n2\n2/ba∇dbl∇ξKjL(t)/ba∇dbl1\n2\n2\n≤C(1+t)−1\n4(1+t)−1\n4+1\n2≤C.(4.11)\nOn the other hand, for n= 2,3, we use (4.10) with k= 0,2 and (2.2) with s= 2 to see\nM∞(KjL(t))≤C/ba∇dblKjL(t)/ba∇dbl1−n\n4\n2/ba∇dblKjL(t)/ba∇dbln\n4\n˙H2\n≤C/ba∇dblKjL(t)/ba∇dbl1−n\n4\n2/ba∇dbl∇2\nξKjL(t)/ba∇dbln\n4\n2\n≤C(1+t)−n\n4(1−n\n4)(1+t)n\n4(−n\n4+1)≤C.(4.12)\nBy combining (4.11), (4.12) and (2.1) one can obtain\nMq(KjL(t))≤M∞(KjL(t))≤C\nfor 1≤q≤ ∞, which proves the desired estimate (4.9) by definition of Mq.\nProof of Lemma 4.3. Firstly, we remark that (4.5) and (4.6) can be derived by the s ame idea.\nHence we only check (4.6). As in the proof of Lemma 4.2, we only need to show\n/ba∇dblK1H(t)g/ba∇dbl∞≤Ce−t/ba∇dblg/ba∇dbl1, (4.13)\nforn= 1 and\n/ba∇dblK1H(t)g/ba∇dblq≤Ce−t\n2/ba∇dblg/ba∇dblq, (4.14)\nfor 1≤q≤ ∞andn= 1,2,3. Forn= 1, the Hausdorff-Young inequality and (3.27) yield\n/ba∇dblK1H(t)g/ba∇dbl∞≤ /ba∇dblK1H(t,ξ)ˆg/ba∇dbl1≤Ce−t/ba∇dbl|ξ|−2χH/ba∇dbl1/ba∇dblˆg/ba∇dbl∞≤Ce−t/ba∇dblg/ba∇dbl1,\nsince|ξ|−2χH∈L1(R), which is the desired estimate (4.13). In order to show (4.1 4), we again\napply the same argument as (4.9). Indeed, by (3.27) - (3.29), we see\n/ba∇dbl∇k\nξK1H(t,ξ)/ba∇dbl2≤Ce−t\n2 (4.15)\n11fork= 0,1,2. Here we have just used the fact that\ne−tt/ba∇dble−t|ξ|2/ba∇dbl2≤Ce−tt1−n\n4≤Ce−t\n2,\nsince 1−n\n4>0 forn= 1,2,3. Therefore, we apply (4.15) with k= 0,1, (2.1) and (2.2) with\ns= 1 to have\nMq(K1H(t))≤M∞(K1H(t))≤C/ba∇dblK1H(t)/ba∇dbl1−1\n2\n2/ba∇dblK1H(t)/ba∇dbl1\n2\n˙H1\n≤C/ba∇dblK1H(t)/ba∇dbl1−1\n2\n2/ba∇dbl∇ξK1H(t)/ba∇dbl1\n2\n2\n≤Ce−t\n2,(4.16)\nfor the case n= 1, and in the case when n= 2,3, by (4.15) with k= 0,2, (2.1) and (2.2) with\ns= 2 one can find that\nMq(K1H(t))≤M∞(K1H(t))≤C/ba∇dblK1H(t)/ba∇dbl1−n\n4\n2/ba∇dblK1H(t)/ba∇dbln\n4\n˙H2\n≤C/ba∇dblK1H(t)/ba∇dbl1−n\n4\n2/ba∇dbl∇2\nξK1H(t)/ba∇dbln\n4\n2\n≤Ce−t\n2.(4.17)\nBy definition of Mq, with the help of (4.16) and (4.17), we obtain the desired est imate (4.14)\nforn= 1,2,3.\nFinally, we check (4.7). The proof of (4.7) is immediate. Ind eed, we now apply the argument\nfor (4.4), with (4.10) replaced by (3.22) to obtain (4.7), an d the proof of Lemma 4.3 is now\ncomplete.\n4.2 Asymptotic behavior of the low frequency part\nInthissubsection,westatethattheevolutionoperators KjL(t)gforj= 0,1arewell-approximated\nby the solution of the heat equation in the small |ξ|region.\nLemma 4.4. Letn= 1,2,3,1≤r≤q≤ ∞. Then there exists a constant C >0such that\n/ba∇dblKjL(t)g−et∆(ˇχL∗g)/ba∇dblq≤C(1+t)−n\n2(1\nr−1\nq)−1/ba∇dblg/ba∇dblr (4.18)\nforj= 0,1.\nProof.For the proof, we again apply the similar argument to the proo f of Lemma 4.2. Namely,\nwe claim that\n/ba∇dblKjL(t)g−et∆(ˇχL∗g)/ba∇dbl∞≤C(1+t)−n\n2−1/ba∇dblg/ba∇dbl1, (4.19)\n/ba∇dblKjL(t)g−et∆(ˇχL∗g)/ba∇dblq≤C(1+t)−1/ba∇dblg/ba∇dblq, (4.20)\nfor 1≤q≤ ∞. Here we recall that (4.19), (4.20) and the Riesz-Thorin int erpolation theorem\nshow (4.18). Therefore it suffices to prove (4.19) and (4.20) i n order to get (4.18).\nWe first show (4.19). The Hausdorff - Young inequality, (3.14) and (2.5) with k= 2 and\nr= 1 show\n/ba∇dblKjL(t)g−et∆(ˇχL∗g)/ba∇dbl∞≤C/ba∇dbl(KjL(t)−e−t|ξ|2χL)ˆg/ba∇dbl1\n≤C/ba∇dblKjL(t)−e−t|ξ|2χL/ba∇dbl1/ba∇dblˆg/ba∇dbl∞\n≤C/ba∇dbl|ξ|2e−(1+t)|ξ|2χL/ba∇dbl1/ba∇dblg/ba∇dbl1≤C(1+t)−n\n2−1/ba∇dblg/ba∇dbl1,\n12which is the desired estimate (4.19).\nNext, we prove (4.20). By observing (3.14) - (3.16) and (2.5) , we get\n/ba∇dbl∇k\nξ(KjL(t)−e−t|ξ|2χL)/ba∇dbl2≤C(1+t)−n\n4−1+k\n2 (4.21)\nfork= 0,1,2.\nIn order to check (4.20) for the case n= 1, we apply (2.2) with s= 1 and (4.21) with k= 0,1\nto get\nM∞(KjL(t)−e−t|ξ|2χL)≤C/ba∇dblKjL(t)−e−t|ξ|2χL/ba∇dbl1−1\n2\n2/ba∇dblKjL(t)−e−t|ξ|2χL/ba∇dbl1\n2\n˙H1\n≤C/ba∇dblKjL(t)−e−t|ξ|2χL/ba∇dbl1−1\n2\n2/ba∇dbl∇ξ(KjL(t)−e−t|ξ|2χL)/ba∇dbl1\n2\n2\n≤C(1+t)1\n2(−1\n4−1)(1+t)1\n2(−1\n4−1\n2)≤C(1+t)−1.(4.22)\nNamely, we have arrived at (4.20) with n= 1 since combining (2.1) and (4.22) gives (4.20).\nIn the case when n= 2,3, we use (4.21) with k= 0,2 and (2.2) with s= 2 to obtain\nM∞(KjL(t)−e−t|ξ|2χL)≤C/ba∇dblKjL(t)−e−t|ξ|2χL/ba∇dbl1−n\n4\n2/ba∇dblKjL(t)−e−t|ξ|2χL/ba∇dbln\n4\n˙H2\n≤C/ba∇dblKjL(t)−e−t|ξ|2χL/ba∇dbl1−n\n4\n2/ba∇dbl∇2\nξ(KjL(t)−e−t|ξ|2χL)/ba∇dbln\n4\n2\n≤C(1+t)(−n\n4−1)(1−n\n4)(1+t)−n\n4n\n4=C(1+t)−1.\nThat is, Mq(KjL(t)−e−t|ξ|2χL)≤M∞(KjL(t)−e−t|ξ|2χL)≤C(1+t)−1for 1≤q≤ ∞by (2.1)\nagain. This shows (4.10) with n= 2,3, which proves Lemma 4.4.\n4.3 Proof of Proposition 4.1\nIn this subsection, we shall prove Proposition 4.1.\nWe start with the observation that the results obtained in pr evious subsections guarantee\nthe decay property and large time behavior of the evolution o perators K0(t) andK1(t).\nCorollary 4.5. Letn= 1,2,3,ε >0and1≤r≤q≤ ∞. Then there exists a constant C >0\nsuch that\n/ba∇dblK0(t)g/ba∇dblq≤C(1+t)−n\n2(1\nr−1\nq)/ba∇dblg/ba∇dblr+Ce−t\n2/ba∇dbl|∇x|n\n2+εg/ba∇dblq, (4.23)\n/ba∇dblK1(t)g/ba∇dblq≤C(1+t)−n\n2(1\nr−1\nq)/ba∇dblg/ba∇dblr+Ce−t\n2/ba∇dblg/ba∇dblq, (4.24)\n/ba∇dbl(K0(t)−et∆)g/ba∇dblq≤C(1+t)−n\n2(1\nr−1\nq)−1/ba∇dblg/ba∇dblr+Ce−t\n2/ba∇dbl|∇x|n\n2+εg/ba∇dblq, (4.25)\n/ba∇dbl(K1(t)−et∆)g/ba∇dblq≤C(1+t)−n\n2(1\nr−1\nq)−1/ba∇dblg/ba∇dblr+Ce−t\n2/ba∇dblg/ba∇dblq. (4.26)\nRemark 4.6.We note that under the statement above for n= 1, we see that\n/ba∇dblK1(t)g/ba∇dblq≤C(1+t)−1\n2(1\nr−1\nq)/ba∇dblg/ba∇dblr,\n/ba∇dbl(K1(t)−et∆)g/ba∇dblq≤C(1+t)−1\n2(1\nr−1\nq)−1/ba∇dblg/ba∇dblr,\nsinceCe−t\n2/ba∇dblg/ba∇dblris estimated by C(1+t)−1\n2(1\nr−1\nq)−1/ba∇dblg/ba∇dblr. The same reasoning can be applied to\nthe case q=r, namely,\n/ba∇dblK1(t)g/ba∇dblq≤ /ba∇dblg/ba∇dblq, (4.27)\n/ba∇dbl(K1(t)−et∆)g/ba∇dblq≤C(1+t)−1/ba∇dblg/ba∇dblq. (4.28)\n13Proof.The proof of the estimates (4.23) - (4.26) is similar. Here we only show the proof of\n(4.23). Combining (4.4) with j= 0, (4.5) and (4.7) with j= 0, and the definition of the\nlocalized operators, we see that\n/ba∇dblK0(t)g/ba∇dblq≤/summationdisplay\nk=L,M,H/ba∇dblK0K(t)g/ba∇dblq\n≤C(1+t)−n\n2(1\nr−1\nq)/ba∇dblg/ba∇dblr+Ce−t\n2/ba∇dblg/ba∇dblr+Ce−t\n2/ba∇dbl|∇x|n\n2+εg/ba∇dblq\n≤C(1+t)−n\n2(1\nr−1\nq)/ba∇dblg/ba∇dblr+Ce−t\n2/ba∇dbl|∇x|n\n2+εg/ba∇dblq,\nwhich show the desired estimate (4.23). This completes the p roof of Corollary 4.5.\nBy combining (4.25), (4.26) and (2.4), we can assert the appr oximation formula of the\nevolution operators K0(t) andK1(t) in terms of the heat kernel for large t.\nCorollary 4.7. Letn= 1,2,3,ε >0and(g0,g1)∈(Wn\n2+ε,1∩Wn\n2+ε,q)×(L1∩Lq). Then it is\ntrue that\n/ba∇dblKj(t)gj−mjGt/ba∇dblq=o(t−n\n2(1−1\nq)), (4.29)\nast→ ∞forj= 0,1, wheremj=/integraldisplay\nRngj(y)dy.\nProof.Forj= 0, we apply (4.25) and (2.4) to get\ntn\n2(1−1\nq)/ba∇dblK0(t)g0−m0Gt/ba∇dblq\n≤tn\n2(1−1\nq)/ba∇dbl(K0(t)−et∆)g0/ba∇dblq+tn\n2(1−1\nq)/ba∇dblet∆g0−m0Gt/ba∇dblq\n≤C(1+t)−1/ba∇dblg0/ba∇dbl1+Ce−t\n2/ba∇dbl|∇x|n\n2+εg/ba∇dblq+tn\n2(1−1\nq)/ba∇dblet∆g0−m0Gt/ba∇dblq\n→0\nast→ ∞, which is the desired estimate (4.29) with j= 0. We now apply this argument with\n(4.25) replaced by (4.26), to obtain the estimate (4.29) wit hj= 1, and Corollary 4.7 now\nfollows.\nNow, we are in a position to prove Proposition 4.1 by combinin g Corollaries 4.5 and 4.7.\nProof of Proposition 4.1. We recall that the solution to (4.1) is expressed as u(t,·) =K0(t)u0+\nK1(t)u1.Then it follows from (4.23) and (4.24) with r= 1,\n/ba∇dblu(t)/ba∇dblq≤ /ba∇dblK0(t)u0/ba∇dblq+/ba∇dblK1(t)u1/ba∇dblq≤C(1+t)−n\n2(1−1\nq),\nwhich is the desired estimate (4.2). Also we see at once (4.3) . Indeed, (4.25), (4.26) with r= 1\nand (4.29) give\n/ba∇dbl(u(t,·)−˜MGt)/ba∇dblq≤ /ba∇dbl(K0(t)−et∆)u0/ba∇dblq+/ba∇dbl(K1(t)−et∆)u1/ba∇dblq\n+/ba∇dbl(et∆(u0+u1)−˜MGt/ba∇dblq\n≤C(1+t)−n\n2(1−1\nq)−1+o(t−n\n2(1−1\nq))\nast→ ∞, which is the desired estimate (4.3). This proves Propositi on 4.1.\n145 Existence of global solutions\nThis section is devoted to the proof of Theorem 1.1. Here we pr epare some notation, which will\nbe used soon. We define the closed subspace of C([0,∞);L1∩L∞) as\nX:={u∈C([0,∞);L1∩L∞);/ba∇dblu/ba∇dblX≤M},\nwhere\n/ba∇dblu/ba∇dblX:= sup\nt≥0{/ba∇dblu(t)/ba∇dbl1+(1+t)n\n2/ba∇dblu(t)/ba∇dbl∞}\nandM >0 will be determined later. We also introduce the mapping Φ on Xby\nΦ[u](t) :=K0(t)u0+K1(t)u1+/integraldisplayt\n0K1(t−τ)f(u)(τ)dτ. (5.1)\nFor simplicity of notation, we denote the integral term of (5 .1) byI[u](t):\nI[u](t) :=/integraldisplayt\n0K1(t−τ)f(u)(τ)dτ. (5.2)\nIn this situation, we claim that\n/ba∇dblΦ[u]/ba∇dblX≤M (5.3)\nfor allu∈Xand\n/ba∇dblΦ[u]−Φ[v]/ba∇dblX≤1\n2/ba∇dblu−v/ba∇dblX (5.4)\nfor allu,v∈X. For the proof of Theorem 1.1, it suffices to show (5.3) and (5.4 ). Indeed, once\nwe have (5.3) and (5.4), we see that Φ is a contraction mapping onX. Therefore it is immediate\nfrom the Banach fixed point theorem that Φ has a unique fixed poi nt inX. Namely, there exists\na unique global solution u= Φ[u] inXand Theorem 1.1 can be proved. We remark that the\nlinear solution K0(t)u0+K1(t)u1is estimated suitably by linear estimates stated in Proposi tion\n4.1. In what follows, we concentrate on estimates for I[u](t) defined by (5.2). Firstly we prepare\nseveral estimates of the norms for f(u) andf(u)−f(v), which will be used below.\nBy using the mean value theorem, we can see that there exists θ∈[0,1] such that\nf(u)−f(v) =f′(θu+(1−θ)v)(u−v).\nTherefore, by noting the definition of /ba∇dbl·/ba∇dblX, we arrive at the estimate\n/ba∇dblf(u)−f(v)/ba∇dbl1≤ /ba∇dblf′(θu+(1−θ)v)/ba∇dbl∞/ba∇dblu−v/ba∇dbl1\n≤C/ba∇dbl(θu+(1−θ)v/ba∇dblp−1\n∞/ba∇dblu−v/ba∇dbl1\n≤C(/ba∇dblu/ba∇dblp−1\n∞+/ba∇dblv/ba∇dblp−1\n∞)/ba∇dblu−v/ba∇dbl1\n≤C(1+τ)−n\n2(p−1)(/ba∇dblu/ba∇dblp−1\nX+/ba∇dblv/ba∇dblp−1\nX)/ba∇dblu−v/ba∇dblX\n≤C(1+τ)−n\n2(p−1)Mp−1/ba∇dblu−v/ba∇dblX(5.5)\nforu,v∈X. By the similar way, we have\n/ba∇dblf(u)−f(v)/ba∇dbl∞≤C(/ba∇dblu/ba∇dblp−1\n∞+/ba∇dblv/ba∇dblp−1\n∞)/ba∇dblu−v/ba∇dbl∞\n≤C(1+τ)−np\n2Mp−1/ba∇dblu−v/ba∇dblX(5.6)\nforu,v∈X. If we take v= 0 in (5.5) and (5.6), and if we recall /ba∇dblu/ba∇dblX≤M, we easily see that\n/ba∇dblf(u)/ba∇dbl1≤C(1+τ)−n\n2(p−1)Mp,\n/ba∇dblf(u)/ba∇dbl∞≤C(1+τ)−np\n2Mp(5.7)\n15foru∈X.\nNow, by using the above estimates in (5.7), let us derive the e stimate of /ba∇dblI[u](t)/ba∇dbl1for\nn= 1,2,3.\nTo begin with, we apply (4.27) with q= 1, (5.8), (2.4) and (2.5) to have\n/ba∇dblI[u](t)/ba∇dbl1≤/integraldisplayt\n0/ba∇dblK1(t−τ)f(u)/ba∇dbl1dτ≤C/integraldisplayt\n0/ba∇dblf(u)/ba∇dbl1dτ\n≤C/ba∇dblu/ba∇dblp\nX/integraldisplayt\n0(1+τ)−n\n2(p−1)dτ≤CMp,(5.8)\nsince−n\n2(p−1)<−1 forp >1+2\nn.\nSecondly by the similar way to (5.8), we calculate /ba∇dblI[u](t)−I[v](t)/ba∇dbl1as follows:\n/ba∇dblI[u](t)−I[v](t)/ba∇dbl1≤/integraldisplayt\n0/ba∇dblK1(t−τ)(f(u)−f(v))/ba∇dbl1dτ\n≤C/integraldisplayt\n0/ba∇dblf(u)−f(v)/ba∇dbl1dτ\n≤CMp−1/ba∇dblu−v/ba∇dblX/integraldisplayt\n0(1+τ)−n\n2(p−1)dτ\n≤CMp−1/ba∇dblu−v/ba∇dblX,(5.9)\nforu,v∈X, where we have just used (5.5) and (5.6).\nFor the proof of Theorem 1.1, it still remains to get the estim ates for /ba∇dblΦ[u](t)/ba∇dbl∞and\n/ba∇dblΦ[u](t)−Φ[v](t)/ba∇dbl∞.\nNow, in order to obtain the estimate for /ba∇dblΦ[u](t)/ba∇dbl∞, we split the nonlinear term into two\nparts:\n/ba∇dblI[u](t)/ba∇dbl∞≤/integraldisplayt\n2\n0/ba∇dblK1(t−τ)f(u)/ba∇dbl∞dτ+/integraldisplayt\nt\n2/ba∇dblK1(t−τ)f(u)/ba∇dbl∞dτ\n=:J1(t)+J2(t).(5.10)\nTo obtain the estimate of J1(t), we apply (4.24) with q=∞andr= 1 and (5.7) to have\nJ1(t)≤C/integraldisplayt\n2\n0(1+t−τ)−n\n2/ba∇dblf(u)/ba∇dbl1dτ+C/integraldisplayt\n2\n0e−t−τ\n2/ba∇dblf(u)/ba∇dbl∞dτ\n≤C(1+t)−n\n2/integraldisplayt\n2\n0(1+τ)−n\n2(p−1)dτMp+Ce−1\n2t/integraldisplayt\n2\n0(1+τ)−np\n2dτMp\n≤C(1+t)−n\n2Mp,(5.11)\nwhere we have used the fact that −n\n2(p−1)<−1.\nFor the term J2(t), by using (4.27) with q=∞and (5.7) we obtain\nJ2(t)≤C/integraldisplayt\nt\n2/ba∇dblf(u)/ba∇dbl∞dτ≤C/integraldisplayt\nt\n2(1+τ)−np\n2dτMp≤C(1+t)−np\n2+1Mp,(5.12)\nwhere we remark that the power in the right hand side −np\n2+ 1 is strictly smaller than −n\n2\nsince−np\n2+ 1 =−n\n2(p−1) + 1−n\n2and−n\n2(p−1)<−1. By combining (5.10) - (5.12), we\narrive at\n/ba∇dblI[u](t)/ba∇dbl∞≤J1(t)+J2(t)≤C(1+t)−n\n2Mp. (5.13)\n16Next, we estimate /ba∇dblΦ[u](t)−Φ[v](t)/ba∇dbl∞. Again, we divide /ba∇dblI[u](t)−I[v](t)/ba∇dbl∞into two parts:\n/ba∇dblI[u](t)−I[v](t)/ba∇dbl∞≤/integraldisplayt\n2\n0/ba∇dblK1(t−τ)(f(u)−f(v))/ba∇dbl∞dτ\n+/integraldisplayt\nt\n2/ba∇dblK1(t−τ)(f(u)−f(v))/ba∇dbl∞dτ\n=:J3(t)+J4(t).(5.14)\nAs in the proof of (5.11), we can deduce that\nJ3(t)≤C/integraldisplayt\n2\n0(1+t−τ)−n\n2/ba∇dblf(u)−f(v)/ba∇dbl1dτ\n+C/integraldisplayt\n2\n0e−t−τ\n2/ba∇dblf(u)−f(v)/ba∇dbl∞dτ\n≤C(1+t)−n\n2/integraldisplayt\n2\n0(1+τ)−n\n2(p−1)dτMp−1/ba∇dblu−v/ba∇dblX\n+Ce−1\n2t/integraldisplayt\n2\n0(1+τ)−np\n2dτMp−1/ba∇dblu−v/ba∇dblX\n≤C(1+t)−n\n2Mp−1/ba∇dblu−v/ba∇dblX,(5.15)\nwhere we have used the fact that −np\n2+1<−n\n2again. In the same manner as (5.12), we can\nget\nJ4(t)≤C/integraldisplayt\nt\n2/ba∇dblf(u)−f(v)/ba∇dbl∞dτ\n≤C/integraldisplayt\nt\n2(1+τ)−np\n2dτMp−1/ba∇dblu−v/ba∇dblX\n≤C(1+t)−np\n2+1Mp−1/ba∇dblu−v/ba∇dblX.(5.16)\nThus, (5.14) - (5.16) yield\n/ba∇dblI[u](t)−I[v](t)/ba∇dbl∞≤J3(t)+J4(t)≤C(1+t)−n\n2Mp−1/ba∇dblu−v/ba∇dblX. (5.17)\nBy (4.2), (5.9) and (5.13), we deduce that\n/ba∇dblΦ[u]/ba∇dblX≤ /ba∇dblK0(t)u0+K1(t)u1/ba∇dblX+/ba∇dblI[u]/ba∇dblX\n≤C0(/ba∇dblu0/ba∇dblWn\n2+ε,1∩Wn\n2+ε,∞+/ba∇dblu1/ba∇dblL1∩L∞)+Cp\n1(5.18)\nfor some C0>0 andC1>0.\nSimilar arguments can be applied to /ba∇dblΦ[u]−Φ[v]/ba∇dblXby using (5.9) and (5.17), and then one\ncan assert that\n/ba∇dblΦ[u]−Φ[v]/ba∇dblX≤ /ba∇dblI[u]−I[v]/ba∇dblX≤C2Mp−1/ba∇dblu−v/ba∇dblX (5.19)\nfor some C2>0. By choosing /ba∇dblu0/ba∇dblWn\n2+ε,1∩Wn\n2+ε,∞+/ba∇dblu1/ba∇dblL1∩L∞sufficiently small, we can make\nsure the validity of the inequality such as\nC1Mp<1\n2M, C 2Mp−1<1\n2, (5.20)\nbecauseof therelation M= 2C0(/ba∇dblu0/ba∇dblWn\n2+ε,1∩Wn\n2+ε,∞+/ba∇dblu1/ba∇dblL1∩L∞). By combining (5.18), (5.19)\nand (5.20) one has the desired estimates (5.3) and (5.4), and the proof is now complete.\n176 Asymptotic behavior of the solution\nIn this section, we show the proof of Theorem 1.2. For the proo f of Theorem 1.2, we pre-\npare slightly general setting. Here, we introduce the funct ionF=F(t,x)∈L1(0,∞;L1(Rn))\nsatisfying\n/ba∇dblF(t)/ba∇dblq≤C(1+t)−n\n2(p−1)−n\n2(1−1\nq), (6.1)\nfor 1≤q≤ ∞andp >1+2\nn. We can now formulate our main statement in this section.\nProposition 6.1. Letn≥1andp >1+2\nn, and assume (6.1). Then it holds that\n/vextenddouble/vextenddouble/vextenddouble/vextenddouble/parenleftbigg/integraldisplayt\n0K1(t−τ)F(τ)dτ−/integraldisplay∞\n0/integraldisplay\nRnF(τ,y)dydτ·Gt(x)/parenrightbigg/vextenddouble/vextenddouble/vextenddouble/vextenddouble\nq=o(t−n\n2(1−1\nq)) (6.2)\nast→ ∞.\nAs a first step of the proof of Proposition 6.1, we split the non linear terms into five parts.\nNamely, we see that\n/integraldisplayt\n0K1(t−τ)F(τ)dτ−/integraldisplay∞\n0/integraldisplay\nRnF(τ,y)dydτ·Gt(x)\n=/integraldisplayt\n2\n0(K1(t−τ)−e(t−τ)∆)F(τ)dτ+/integraldisplayt\nt\n2K1(t−τ)F(τ)dτ\n+/integraldisplayt\n2\n0(e(t−τ)∆−et∆)F(τ)dτ+/integraldisplayt\n2\n0/parenleftbigg\net∆F(τ)−/integraldisplay\nRnF(τ,y)dy·Gt(x)/parenrightbigg\ndτ\n−/integraldisplay∞\nt\n2/integraldisplay\nRnF(τ,y)dydτ·Gt(x),\nand here we set each terms as follows:\nA1(t) :=/integraldisplayt\n2\n0(K1(t−τ)−e(t−τ)∆)F(τ)dτ,\nA2(t) :=/integraldisplayt\nt\n2K1(t−τ)F(τ)dτ, A3(t) :=/integraldisplayt\n2\n0(e(t−τ)∆−et∆)F(τ)dτ,\nA4(t) :=/integraldisplayt\n2\n0/parenleftbigg\net∆F(τ)−/integraldisplay\nRnF(τ,y)dy·Gt(x)/parenrightbigg\ndτ\nA5(t) :=−/integraldisplay∞\nt\n2/integraldisplay\nRnF(τ,y)dydτ·Gt(x).\nIn what follows, we estimate each Aj(t) forj= 1,···,5, respectively.\nLemma 6.2. Under the same assumptions as in Proposition 6.1, there exists a constant C >0\nsuch that\n/ba∇dblA1(t)/ba∇dblq≤C(1+t)−n\n2(1−1\nq)−1, (6.3)\n/ba∇dblAj(t)/ba∇dblq≤Ct−n\n2(1−1\nq)−n\n2(p−1)+1(j= 2,5), (6.4)\n/ba∇dblA3(t)/ba∇dblq≤/braceleftBigg\nCt−n\n2(1−1\nq)−1log(2+t), p≥1+4\nn,\nCt−n\n2(1−1\nq)−n\n2(p−1)+1,1+2\nn< p <1+4\nn,(6.5)\n/ba∇dblA4(t)/ba∇dblq=o(t−n\n2(1−1\nq)), (6.6)\nast→ ∞for1≤q≤ ∞.\n18Proof.First, we show (6.3). By (4.26) with r= 1 and (6.1) we see that\n/ba∇dblA1(t)/ba∇dblq≤/integraldisplayt\n2\n0/ba∇dbl(K1(t−τ)−e(t−τ)∆)F(τ)/ba∇dblqdτ\n≤C/integraldisplayt\n2\n0(1+t−τ)−n\n2(1−1\nq)−1/ba∇dblF(τ)/ba∇dbl1dτ+C/integraldisplayt\n2\n0e−t−τ\n2/ba∇dblF(τ)/ba∇dblqdτ\n≤C(1+t)−n\n2(1−1\nq)−1/integraldisplayt\n2\n0(1+τ)−n\n2(p−1)dτ\n+Ce−t\n2/integraldisplayt\n2\n0(1+τ)−n\n2(p−1)−n\n2(1−1\nq)dτ\n≤C(1+t)−n\n2(1−1\nq)−1,\nwhich is the desired estimate (6.3). Next, we show (6.4) wit j= 2. By (4.27) and (6.1), we see\nthat\n/ba∇dblA2(t)/ba∇dblq≤/integraldisplayt\nt\n2/ba∇dblK1(t−τ)F(τ)/ba∇dblqdτ≤C/integraldisplayt\nt\n2/ba∇dblF(τ)/ba∇dblqdτ\n≤C/integraldisplayt\nt\n2(1+τ)−n\n2(1−1\nq)−n\n2(p−1)dτ\n≤C(1+t)−n\n2(1−1\nq)−n\n2(p−1)+1,\nwhich is the desired estimate (6.4) with j= 2.\nThirdly, we show (6.4) with j= 5. By the combination of (6.1) and the direct computation,\nwe get\n/ba∇dblA5(t)/ba∇dblq≤/integraldisplay∞\nt\n2/ba∇dblF(τ)/ba∇dbl1dτ/ba∇dblGt/ba∇dblq\n≤/integraldisplay∞\nt\n2(1+τ)−n\n2(p−1)dτ/ba∇dblGt/ba∇dblq≤Ct−n\n2(1−1\nq)−n\n2(p−1)+1,\nwhich is the desired estimate (6.4) with j= 5.\nLet us prove (6.5). To begin with, observe that there exists θ∈[0,1] such that\nGt−τ(x−y)−Gt(x−y) = (−τ)∂tGt−θτ(x−y),\nbecause of the mean value theorem on t. Then, we can apply (2.6) with ˜k= 0,ℓ= 1 and r= 1\nto have\n/ba∇dblA3(t)/ba∇dblq≤/integraldisplayt\n2\n0/ba∇dbl(e(t−τ)∆−et∆)F(τ)/ba∇dblqdτ\n=/integraldisplayt\n2\n0τ/ba∇dbl∂te(t−θτ)∆F(τ)/ba∇dblqdτ\n≤C/integraldisplayt\n2\n0τ(t−τ)−n\n2(1−1\nq)−1/ba∇dblF(τ)/ba∇dbl1dτ\n≤Ct−n\n2(1−1\nq)−1/integraldisplayt\n2\n0τ(1+τ)−n\n2(p−1)dτ\n≤/braceleftBigg\nCt−n\n2(1−1\nq)−1log(2+t), p≥1+4\nn,\nCt−n\n2(1−1\nq)−n\n2(p−1)+1,1+2\nn< p <1+4\nn,\n19which implies (6.5).\nFinally, we prove (6.6). To show the estimate for A4(t), we first divide the integrand into\ntwo parts:\n/integraldisplayt\n2\n0/parenleftbigg\net∆F(τ,x)−/integraldisplay\nRnF(τ,y)dy·Gt(x)/parenrightbigg\ndτ\n=/integraldisplayt\n2\n0/integraldisplay\n|y|≤t1\n4+/integraldisplayt\n2\n0/integraldisplay\n|y|≥t1\n4(Gt(x−y)−Gt(x))F(τ,y)dydτ=:A41(t)+A42(t).(6.7)\nIn what follows, we estimate A41(t) andA42(t), respectively. For the estimate of A41(t), we\napply the mean value theorem again on xto have\nGt(x−y)−Gt(x) = (−y)·∇xGt(x−˜θy)\nwith some ˜θ∈[0,1], where ·denotes the standard Euclid inner product. Then we arrive at the\nestimate\n/ba∇dblA41(t)/ba∇dblq≤/integraldisplayt\n2\n0/integraldisplay\n|y|≤t1\n4/ba∇dblGt(x−y)−Gt(x)/ba∇dblLq\nx|F(τ,y)|dydτ\n=/integraldisplayt\n2\n0/integraldisplay\n|y|≤t1\n4/vextenddouble/vextenddouble/vextenddouble(−y)·∇xGt(x−˜θy)/vextenddouble/vextenddouble/vextenddouble\nLq\nx|F(τ,y)|dydτ\n≤Ct−n\n2(1−1\nq)−1\n2+1\n4/integraldisplayt\n2\n0/ba∇dblF(τ)/ba∇dbl1dτ\n≤Ct−n\n2(1−1\nq)−1\n4/integraldisplayt\n2\n0(1+τ)−n\n2(p−1)dτ≤Ct−n\n2(1−1\nq)−1\n4,(6.8)\nby direct calculations. On the other hand, for the term A42(t), we recall the fact that/integraldisplay∞\n0/integraldisplay\nRn|F(τ,y)|dydτ <∞implies\nlim\nt→∞/integraldisplay∞\n0/integraldisplay\n|y|≥t1\n4|F(τ,y)|dydτ= 0.\nThus we see that\n/ba∇dblA42(t)/ba∇dblq≤/integraldisplayt\n2\n0/integraldisplay\n|y|≥t1\n4(/ba∇dblGt(x−y)/ba∇dblLq\nx+/ba∇dblGt(x)/ba∇dblLq\nx)|F(τ,y)|dydτ\n≤Ct−n\n2(1−1\nq)/integraldisplay∞\n0/integraldisplay\n|y|≥t1\n4|F(τ,y)|dydτ,\nso that\ntn\n2(1−1\nq)/ba∇dblA42(t)/ba∇dblq→0 (6.9)\nast→ ∞. Therefore, by combining (6.7), (6.8) and (6.9) one has\n/ba∇dblA4(t)/ba∇dblq≤ /ba∇dblA41(t)/ba∇dblq+/ba∇dblA42(t)/ba∇dblq=o(t−n\n2(1−1\nq))\nast→ ∞, which is the desired estimate (6.6). We complete the proof o f Lemma 6.2.\n20Proof of Proposition 6.1. For 1≤q≤ ∞, Lemma 6.2 immediately yields (6.4). Indeed, from\n(6.5) - (6.8) it follows that\ntn\n2(1−1\nq)/vextenddouble/vextenddouble/vextenddouble/vextenddouble/parenleftbigg/integraldisplayt\n0K1(t−τ)F(τ)dτ−/integraldisplay∞\n0/integraldisplay\nRnF(τ,y)dydτ·Gt(x)/parenrightbigg/vextenddouble/vextenddouble/vextenddouble/vextenddouble\nq\n≤Ctn\n2(1−1\nq)5/summationdisplay\nj=1/ba∇dblAj(t)/ba∇dblq→0\nast→ ∞, which is the desired conclusion.\nNow we are in a position to prove Theorem 1.2.\nProof of Theorem 1.2. From the proof of Theorem 1.1, we see that the nonlinear term f(u)\nsatisfies the condition (6.1). Then we can apply Proposition 6.1 toF(τ,y) =f(u(τ,y)), and the\nproof is now complete.\nAcknowledgments. The work of the first author (R. IKEHATA) was supported in part by\nGrant-in-Aid for Scientific Research (C)15K04958 of JSPS. T he work of the second author (H.\nTAKEDA) was supported in part by Grant-in-Aid for Young Scie ntists (B)15K17581 of JSPS.\nReferences\n[1] Brenner, P., Thom´ ee, V. and Wahlbin, L., Besov spaces and applications to difference\nmethods for initial value problems , LectureNotes inMathematics, Vol. 434. Springer-Verlag,\nBerlin-New York, 1975.\n[2] D’Abbicco, M. and Reissig, M., Semilinear structural da mped waves, Math. Methods Appl.\nSci. 37(11)(2014), 1570-1592.\n[3] Giga, M., Giga, Y. and Saal, J., Nonlinear partial differen tial equations. Asymptotic be-\nhavior of solutions and self-similar solutions. Progress i n Nonlinear Differential Equations\nand their Applications, 79. Birkh¨ auser Boston, Inc., Bost on, MA, 2010.\n[4] Hayashi, N., Kaikina, E.I. and Naumkin, P. I., Damped wav e equation with super critical\nnonlinearities, Diff. Int. Eqns 17 (2004), 637-652.\n[5] Hosono, T. and Ogawa, T., Large time behavior and Lp-Lqestimate of 2-dimensional\nnonlinear damped wave equations, J. Diff. Eqns 203 (2004), 82- 118.\n[6] Ikehata, R., Asymptotic profiles for wave equations with strong damping, J. Diff. Eqns 257\n(2014), 2159-2177.\n[7] Ikehata, R., Miyaoka, Y. andNakatake, T., Decay estimat es of solutionsfordissipative wave\nequations in RNwith lower power nonlinearities, J.Math.Soc.Japan 56 (200 4), 365-373.\n[8] Ikehata, R. and Sawada, A., Asymptotic profiles of soluti ons for wave equations\nwith frictional and viscoelastic damping terms, Asymptoti c Anal. 98 (2016), 59-77.\nDOI10.3233/ASY-161361\n[9] Ikehata, R. and Takeda, H., Critical exponent for nonlin ear wave equations with frictional\nand viscoelastic damping terms, arXiv:1604.08265v1 [math .AP] 27 Apr 2016.\n21[10] Ikehata, R. and Tanizawa, K., Global existence of solut ions for semilinear damped wave\nequations in RNwith noncompactly supported initial data, Nonlinear Anal. 61 (2005),\n1189-1208.\n[11] Ikehata, R., Todorova, G.andYordanov, B., Wave equati onswithstrongdampinginHilbert\nspaces, J. Diff. Eqns 254 (2013), 3352-3368.\n[12] Karch, G., Selfsimilar profiles in large time asymptoti cs of solutions to damped wave equa-\ntions, Studia Math. 143 (2000), 175-197.\n[13] Kawakami, T. and Ueda, Y., Asymptotic profiles to the sol utions for a nonlinear damped\nwave equation, Diff. Int. Eqns 26 (2013), 781-814.\n[14] Marcati, P. and Nishihara, K., The Lp-Lqestimates of solutions to one-dimensional damped\nwave equations and their application to compressible flow th rough porous media, J. Diff.\nEqns 191 (2003), 445-469.\n[15] Narazaki, T., Lp-Lqestimates for damped wave equations and their applications to semi-\nlinear problem, J. Math. Soc. Japan 56 (2004), 585-626.\n[16] Nishihara, K., Lp-Lqestimates to the damped wave equation in 3-dimensional spac e and\ntheir application, Math. Z. 244 (2003), 631-649.\n[17] Ponce, G., Global existence of small solutions to a clas s of nonlinear evolution equations,\nNonlinear Anal. 9(5) (19), 399-418.\n[18] Segal, I., Dispersion for non-linear relativistic equ ations. II, Ann. Sci. ´Ecole Norm. Sup. 1\n(1968) 459-497.\n[19] Shibata, Y., On the rate of decay of solutions to linear v iscoelastic equation, Math. Meth.\nAppl. Sci. 23 (2000), 203-226.\n[20] Takeda, H., Higher-order expansion of solutions for a d amped wave equation, Asymptotic\nAnal. 94 (2015), 1-31. DOI: 10.3233/ASY-151295\n[21] Todorova, G. and Yordanov, B., Critical exponent for a n onlinear wave equation with\ndamping, J. Diff. Eqns 174 (2001), 464-489.\n[22] Zhang, Qi S., A blow-up result for a nonlinear wave equat ion with damping: the critical\ncase, C. R. Acad. Sci. Paris S´ er. I Math. 333(2001), 109-114.\n22"    },    {        "title": "1605.08698v1.A_reduced_model_for_precessional_switching_of_thin_film_nanomagnets_under_the_influence_of_spin_torque.pdf",        "content": "A reduced model for precessional switching of thin-\flm nanomagnets under the\nin\ruence of spin-torque\nRoss G. Lund1, Gabriel D. Chaves-O'Flynn2, Andrew D. Kent2, Cyrill B. Muratov1\n1Department of Mathematical Sciences, New Jersey Institute of Technology , University Heights, Newark, NJ 07102, USA\n2Department of Physics, New York University, 4 Washington Place, New York, NY 10003, USA\n(Dated: May 25, 2022)\nWe study the magnetization dynamics of thin-\flm magnetic elements with in-plane magnetization\nsubject to a spin-current \rowing perpendicular to the \flm plane. We derive a reduced partial\ndi\u000berential equation for the in-plane magnetization angle in a weakly damped regime. We then apply\nthis model to study the experimentally relevant problem of switching of an elliptical element when\nthe spin-polarization has a component perpendicular to the \flm plane, restricting the reduced model\nto a macrospin approximation. The macrospin ordinary di\u000berential equation is treated analytically\nas a weakly damped Hamiltonian system, and an orbit-averaging method is used to understand\ntransitions in solution behaviors in terms of a discrete dynamical system. The predictions of our\nreduced model are compared to those of the full Landau{Lifshitz{Gilbert{Slonczewski equation for\na macrospin.\nI. INTRODUCTION\nMagnetization dynamics in the presence of spin-\ntransfer torques is a very active area of research with ap-\nplications to magnetic memory devices and oscillators1{3.\nSome basic questions relate to the types of magnetiza-\ntion dynamics that can be excited and the time scales on\nwhich the dynamics occurs. Many of the experimental\nstudies of spin-transfer torques are on thin \flm magnetic\nelements patterned into asymmetric shapes (e.g. an el-\nlipse) in which the demagnetizing \feld strongly con\fnes\nthe magnetization to the \flm plane. Analytic models\nthat capture the resulting nearly in-plane magnetization\ndynamics (see e.g.4{8) can lead to new insights and guide\nexperimental studies and device design. A macrospin\nmodel that treats the entire magnetization of the ele-\nment as a single vector of \fxed length is a starting point\nfor most analyses.\nThe focus of this paper is on a thin-\flm magnetic el-\nement excited by a spin-polarized current that has an\nout-of-plane component. This out-of-plane component\nof spin-polarization can lead to magnetization precession\nabout the \flm normal or magnetization reversal. The for-\nmer dynamics would be desired for a spin-transfer torque\noscillator, while the latter dynamics would be essential in\na magnetic memory device. A device in which a perpen-\ndicular component of spin-polarization is applied to an\nin-plane magnetized element was proposed in Ref. [9] and\nhas been studied experimentally10{12. There have also\nbeen a number of models that have considered the in\ru-\nence of thermal noise on the resulting dynamics, e.g., on\nthe rate of switching and the dephasing of the oscillator\nmotion13{15.\nHere we consider a weakly damped asymptotic regime\nof the Landau{Lifshitz{Gilbert{Slonczewski (LLGS)\nequation for a thin-\flm ferromagnet, in which the oscil-\nlatory nature of the in-plane dynamics is highlighted. In\nthis regime, we derive a reduced partial di\u000berential equa-\ntion (PDE) for the in-plane magnetization dynamics un-\nder applied spin-torque, which is a generalization of theunderdamped wave-like model due to Capella, Melcher\nand Otto8. We then analyze the solutions of this equa-\ntion under the macrospin (spatially uniform) approxima-\ntion, and discuss the predictions of such a model in the\ncontext of previous numerical studies of the full LLGS\nequation16.\nThe rest of this article is organized as follows. In Sec.\nII, we perform an asymptotic derivation of the reduced\nunderdamped equation for the in-plane magnetization\ndynamics in a thin-\flm element of arbitrary cross sec-\ntion, by \frst making a thin-\flm approximation to the\nLLGS equation, then a weak-damping approximation. In\nSec. III, we then further reduce to a macrospin ordinary\ndi\u000berential equation (ODE) by spatial averaging of the\nunderdamped PDE, and restrict to the particular case of\na soft elliptical element. A brief parametric study of the\nODE solutions is then presented, varying the spin-current\nparameters. In Sec. IV, we make an analytical study of\nthe macrospin equation using an orbit-averaging method\nto reduce to a discrete dynamical system, and compare\nits predictions to the full ODE solutions. In Sec. V, we\nseek to understand transitions between the di\u000berent so-\nlution trajectories (and thus predict current-parameter\nvalues when the system will either switch or precess) by\nstudying the discrete dynamical system derived in Sec.\nIV. Finally, we summarize our \fndings in Sec. VI.\nII. REDUCED MODEL\nWe consider a domain \n \u001aR3occupied by a ferromag-\nnetic \flm with cross-section D\u001aR2and thickness d, i.e.,\n\n =D\u0002(0;d). Under the in\ruence of a spin-polarized\nelectric current applied perpendicular to the \flm plane,\nthe magnetization vector m=m(r;t), withjmj= 1 in \nand 0 outside, satis\fes the LLGS equation (in SI units)\n@m\n@t=\u0000\r\u00160m\u0002He\u000b+\u000bm\u0002@m\n@t+\u001cSTT (1)arXiv:1605.08698v1  [cond-mat.mes-hall]  27 May 20162\nin \n, with@m=@n= (n\u0001r)m= 0 on@\n, where nis the\noutward unit normal to @\n. In the above, \u000b > 0 is the\nGilbert damping parameter, \ris the gyromagnetic ratio,\n\u00160is the permeability of free space, He\u000b=\u00001\n\u00160Ms\u000eE\n\u000emis\nthe e\u000bective magnetic \feld,\nE(m) =Z\n\n\u0010\nAjrmj2+K\b(m)\u0000\u00160MsHext\u0001m)\u0011\nd3r\n+\u00160M2\nsZ\nR3Z\nR3r\u0001m(r)r\u0001m(r0)\n8\u0019jr\u0000r0jd3rd3r0(2)\nis the micromagnetic energy with exchange constant A,\nanisotropy constant K, crystalline anisotropy function\n\b, external magnetic \feld Hext, and saturation magne-\ntizationMs. Additionally, the Slonczewski spin-transfer\ntorque\u001cSTTis given by\n\u001cSTT=\u0000\u0011\r~j\n2deMsm\u0002m\u0002p; (3)\nwherejis the density of current passing perpendicularly\nthrough the \flm, eis the elementary charge (positive),\npis the spin-polarization direction, and \u00112(0;1] is the\nspin-polarization e\u000eciency.\nWe now seek to nondimensionalize the above system.\nLet\n`=s\n2A\n\u00160M2s; Q =2K\n\u00160M2s;hext=Hext\nMs:(4)\nWe then rescale space and time as\nr!`r; t!t\n\r\u00160Ms; (5)\nobtaining the nondimensional form\n@m\n@t=\u0000m\u0002he\u000b+\u000bm\u0002@m\n@t\u0000\fm\u0002m\u0002p;(6)\nwhere he\u000b=He\u000b=Ms, and\n\f=\u0011~j\n2de\u00160M2s(7)\nis the dimensionless spin-torque strength.\nSince we are interested in thin \flms, we now assume\nthatmis independent of the \flm thickness. Then, after\nrescaling\nE!\u00160M2\nsd`2E; (8)\nwe have he\u000b'\u0000\u000eE\n\u000em, whereEis given by a local energy\nfunctional de\fned on the (rescaled) two-dimensional do-\nmainD(see, e.g., Ref. [17]):\nE(m)'1\n2Z\nD\u0000\njrmj2+Q\b(m)\u00002hext\u0001m\u0001\nd2r\n+1\n2Z\nDm2\n?d2r+1\n4\u0019\u000ejln\u0015jZ\n@D(m\u0001n)2ds;(9)in which now m:D!S2,m?is its out-of-plane com-\nponent,\u000e=d=`is the dimensionless \flm thickness, and\n\u0015=d=L\u001c1 (whereLis the lateral size of the \flm) is the\n\flm's aspect ratio. The e\u000bective \feld is given explicitly\nby\nhe\u000b= \u0001m\u0000Q\n2rm\b(m)\u0000m?ez+hext; (10)\nandmsatis\fes equation (6) in Dwith the boundary\ncondition\n@m\n@n=\u00001\n2\u0019\u000ejln\u0015j(m\u0001n)(n\u0000(m\u0001n)m) (11)\non@D.\nWe now parametrize min terms of spherical angles as\nm= (\u0000sin\u0012cos\u001e;cos\u0012cos\u001e;sin\u001e); (12)\nand the current polarization direction pin terms of an\nin-plane angle  and its out-of-plane component p?as\np=1p\n1 +p2\n?(\u0000sin ;cos ;p?): (13)\nWriting\f\u0003=\f=p\n1 +p2\n?, after some algebra, one may\nthen write equation (6) as the system\n@\u001e\n@t=\u00001\ncos\u001ehe\u000b\u0001m\u0012+\u000bcos\u001e@\u0012\n@t\n+\f\u0003(p?cos\u001e\u0000sin\u001ecos(\u0012\u0000 ));(14)\n\u0000cos\u001e@\u0012\n@t=\u0000he\u000b\u0001m\u001e+\u000b@\u001e\n@t+\f\u0003sin(\u0012\u0000 );(15)\nwhere m\u0012=@m=@\u0012andm\u001e=@m=@\u001eformgiven by\n(12). Again, since we are working in a soft thin \flm, we\nassume\u001e\u001c1 and that the out-of-plane component of\nthe e\u000bective \feld in equation (10) is dominated by the\ntermhe\u000b\u0001ez'\u0000m?=\u0000sin\u001e. Note that this assumes\nthat the crystalline anisotropy and external \feld terms\nin the out-of-plane directions are relatively small, so we\nassume the external \feld is only in plane, though it is still\npossible to include a perpendicular anisotropy simply by\nrenormalizing the constant in front of the m?term in\nhe\u000b. We then linearize the above system in \u001e, yielding\n@\u001e\n@t=\u000eE\n\u000e\u0012+\u000b@\u0012\n@t+\f\u0003(p?\u0000\u001ecos(\u0012\u0000 )); (16)\n\u0000@\u0012\n@t=\u001e+\f\u0003sin(\u0012\u0000 )\n+\u001e(\u0000hxsin\u0012+hycos\u0012) +\u000b@\u001e\n@t:(17)\nwherehx=he\u000b\u0001exandhy=he\u000b\u0001ey, andE(\u0012) isE(m)\nevaluated at \u001e= 0.3\nWe now note that the last two terms in (17) are neg-\nligible relative to \u001ewheneverjhxj;jhyjand\u000bare small,\nwhich is true of typical clean thin-\flm samples of su\u000e-\nciently large lateral extent. Neglecting these terms, one\nhas\n@\u001e\n@t=\u000eE\n\u000e\u0012+\u000b@\u0012\n@t+\f\u0003(p?\u0000\u001ecos(\u0012\u0000 )); (18)\n\u0000@\u0012\n@t=\f\u0003sin(\u0012\u0000 ) +\u001e: (19)\nThen, di\u000berentiating (19) with respect to tand using the\nresult along with (19) to eliminate \u001eand@\u001e\n@tfrom (18),\nwe \fnd a second-order in time equation for \u0012:\n0 =@2\u0012\n@t2+@\u0012\n@t(\u000b+ 2\f\u0003cos(\u0012\u0000 )) +\u000eE\n\u000e\u0012\n+\f\u0003p?+\f2\n\u0003sin(\u0012\u0000 ) cos(\u0012\u0000 );(20)\nwhere, explicitly, one has\n\u000eE\n\u000e\u0012=\u0000\u0001\u0012+Q\n2~\b0(\u0012) +hext\u0001(cos\u0012;sin\u0012); (21)\nand~\b(\u0012) = \b( m(\u0012)). In turn, from the boundary condi-\ntion on min (11), we can derive the boundary condition\nfor\u0012as\nn\u0001r\u0012=1\n2\u0019\u000ejln\u0015jsin(\u0012\u0000') cos(\u0012\u0000'); (22)\nwhere'is the angle parametrizing the normal nto@D\nvian= (\u0000sin';cos').\nThe model comprised of (20){(22) is a damped-driven\nwave-like PDE for \u0012, which coincides with the reduced\nmodel of Ref. [8] for vanishing spin-current density in\nan in\fnite sample. This constitutes our reduced PDE\nmodel for magnetization dynamics in thin-\flm elements\nunder the in\ruence of out-of-plane spin currents. It is\neasy to see that all of the terms in (20) balance when the\nparameters are chosen so as to satisfy\n\f\u0003\u0018p?\u0018\u000b\u0018Q1=2\u0018jhextj1=2\u0018`\nL\u0018\u000ejln\u0015j:(23)\nThis shows that it should be possible to rigorously obtain\nthe reduced model in (20){(22) in the asymptotic limit\nofL!1 and\u000b;\f\u0003;p?;Q;jhextj;\u000e!0 jointly, so that\n(23) holds.\nIII. MACROSPIN SWITCHING\nIn this section we study the behavior of the reduced\nmodel (20){(22) in the approximation that the magneti-\nzation is spatially uniform on an elliptical domain, and\ncompare the solution phenomenology to that found by\nsimulating the LLGS equation in the same physical situ-\nation, as studied in Ref. [16].A. Derivation of macrospin model\nIntegrating equation (20) over the domain Dand using\nthe boundary condition (22), we have\nZ\nD\u0012@2\u0012\n@t2+@\u0012\n@t(\u000b+ 2\f\u0003cos(\u0012\u0000 ))\n+\f\u0003p?+\f2\n\u0003sin(\u0012\u0000 ) cos(\u0012\u0000 )\n+Q\n2~\b0(\u0012) +hext\u0001(cos\u0012;sin\u0012)\u0013\nd2r\n=1\n2\u0019\u000ejln\u0015jZ\n@Dsin(\u0012\u0000') cos(\u0012\u0000') ds:(24)\nAssume now that \u0012does not vary appreciably across the\ndomainD, which makes sense in magnetic elements that\nare not too large. This allows us to replace \u0012(r;t) by\nits spatial average \u0016\u0012(t) =1\njDjR\nD\u0012(r;t) d2r, wherejDj\nstands for the area of Din the units of `2. Denoting\ntime derivatives by overdots, and omitting the bar on \u0016\u0012\nfor notational simplicity, this spatial averaging leads to\nthe following ODE for \u0012(t):\n\u0012+_\u0012(\u000b+ 2\f\u0003cos(\u0012\u0000 )) +\f2\n\u0003sin(\u0012\u0000 ) cos(\u0012\u0000 )\n+\f\u0003p?+Q\n2~\b0(\u0012) +hext\u0001(cos\u0012;sin\u0012)\n=\u000ejln\u0015j\n4\u0019jDjsin 2\u0012Z\n@Dcos(2') ds\n\u0000\u000ejln\u0015j\n4\u0019jDjcos 2\u0012Z\n@Dsin(2') ds:(25)\nNext, we consider a particular physical situation in\nwhich to study the macrospin equation, motivated by\nprevious work10,11. As in Refs. [14{16], we consider an\nelliptical thin-\flm element (recall that lengths are now\nmeasured in the units of `):\nD=\u001a\n(x;y) :x2\na2+y2\nb2<1\u001b\n; (26)\nwith no in-plane crystalline anisotropy, Q= 0, and no\nexternal \feld, hext= 0. We take the long axis of the\nellipse to be aligned with the ey-direction, i.e. b > a ,\nwith the in-plane component of current polarization also\naligned along this direction, i.e., taking  = 0. One can\nthen compute the integral over the boundary in equation\n(25) explicitly, leading to the equation\n\u0012+_\u0012(\u000b+\f\u0003cos\u0012) + \u0003 sin\u0012cos\u0012\n+\f2\n\u0003sin\u0012cos\u0012+\f\u0003p?= 0;(27)\nwhere we introduced the geometric parameter 0 <\u0003\u001c1\nobtained by an explicit integration:\n\u0003 =\u000ejln\u0015j\n2\u00192abZ2\u0019\n0b2cos2\u001c\u0000a2sin2\u001cp\nb2cos2\u001c+a2sin2\u001cd\u001c: (28)4\n(d)\n(c)\n(a)\n(b)\nFIG. 1: Solutions of macrospin equation (30) for \u000b= 0:01, \u0003 = 0:1. In (a),p?= 0:2,\u001b= 0:03: decaying solution; in (b),\np?= 0:2,\u001b= 0:06: limit cycle solution (the initial conditions in (a) and (b) are \u0012(0) = 3:5, to better visualize the behavior).\nIn (c),p?= 0:3,\u001b= 0:08: switching solution; in (d), p?= 0:6,\u001b= 0:1: precessing solution.\nThis may be computed in terms of elliptic integrals,\nthough the expression is cumbersome so we omit it here.\nImportantly, up to a factor depending only on the eccen-\ntricity the value of \u0003 is given by\n\u0003\u0018d\nLlnL\nd: (29)\nFor example, for an elliptical nanomagnet with dimen-\nsions 100\u000230\u00022:5 nm (similar to those considered in\nRef. [16]), this yields \u0003 '0:1.\nIt is convenient to rescale time byp\n\u0003 and divide\nthrough by \u0003, yielding\n\u0012+1p\n\u0003_\u0012(\u000b+ 2\u001b\u0003 cos\u0012) + sin\u0012cos\u0012\n+\u001bp?+\u001b2\u0003 sin\u0012cos\u0012= 0;(30)\nwhere we introduced \u001b=\f\u0003=\u0003. We then apply this\nODE to model the problem of switching of the thin-\flm\nelements, taking the initial in-plane magnetization direc-tion to be static and aligned along the easy axis, an-\ntiparallel to the in-plane component of the spin-current\npolarization. Thus, we take\n\u0012(0) =\u0019; _\u0012(0) = 0; (31)\nand study the resulting initial value problem.\nB. Solution phenomenology\nLet us brie\ry investigate the solution phenomenology\nas the dimensionless spin-current parameters \u001bandp?\nare varied, with the material parameters, \u000band \u0003, \fxed.\nWe take all parameters to be constant in time for simplic-\nity. We \fnd, by numerical integration, 4 types of solution\nto the initial value problem de\fned above. The sample\nsolution curves are displayed in Fig. 1 below. The \frst\n(panel (a)) occurs for small values of \u001b, and consists sim-\nply of oscillations of \u0012around a \fxed point close to the\nlong axis of the ellipse, which decay in amplitude towards\nthe \fxed point, without switching.5\nSecondly (panel (b)), still below the switching thresh-\nold, the same oscillations about the \fxed point can reach\na \fnite \fxed amplitude and persist without switching.\nThis behavior corresponds to the onset of relatively small\namplitude limit-cycle oscillations around the \fxed point.\nThirdly (panel (c)), increasing either \u001b;p?or both, we\nobtain switching solutions. These have initial oscillations\nin\u0012about the \fxed point near \u0019, which increase in ampli-\ntude, and eventually cross the short axis of the ellipse at\n\u0012=\u0019=2. Then\u0012oscillates about the \fxed point near 0,\nand the oscillations decay in amplitude toward the \fxed\npoint.\nFinally (panel (d)), further increasing \u001bandp?we\nobtain precessing solutions. Here, the initial oscillations\nabout the \fxed point near \u0019quickly grow to cross \u0019=2,\nafter which \u0012continues to decrease for all t, the magne-\ntization making full precessions around the out-of-plane\naxis.\nIV. HALF-PERIOD ORBIT-AVERAGING\nAPPROACH\nWe now seek to gain some analytical insight into the\ntransitions between the solution types discussed above.\nWe do this by averaging over half-periods of the oscil-\nlations observed in the solutions to generate a discrete\ndynamical system which describes the evolution of the\nenergy of a solution \u0012(t) on half-period time intervals.\nFirstly, we observe that in the relevant parameter\nregimes the reduced equation (30) can be seen as a weakly\nperturbed Hamiltonian system. We consider both \u000band\n\u0003 small, with \u000b.p\n\u0003, and assume \u001b\u0018\u000b=\u0003 and\n\u001bp?.1. The arguments below can be rigorously jus-\nti\fed by considering, for example, the limit \u0003 !0 while\nassuming that \u000b=O(\u0003) and that the values of \u001band\np?are \fxed. This limit may be achieved in the origi-\nnal model by sending jointly d!0 andL!1 , while\nkeeping17\nLd\n`2lnL\nd.1: (32)\nThe last condition ensures the consistency of the assump-\ntion that\u0012does not vary appreciably throughout D.\nIntroducing !(t) =_\u0012(t), (30) can be written to leading\norder as\n_\u0012=@H\n@!;_!=\u0000@H\n@\u0012; (33)\nwhere we introduced\nH=1\n2!2+V(\u0012); V (\u0012) =1\n2sin2\u0012+\u001bp?\u0012: (34)\nAt the next order, the e\u000bects of \fnite \u000band \u0003 appear\nin the \frst-derivative term in (30), while the other forc-\ning term is still higher order. The behavior of (30) is\ntherefore that of a weakly damped Hamiltonian system\nwith Hamiltonian H, with the e\u000bects of \u000band\u001bservingto slowly change the value of Has the system evolves.\nThus, we now employ the technique of orbit-averaging to\nreduce the problem further to the discrete dynamics of\nH(t), where the discrete time-steps are equal (to the lead-\ning order) to half-periods of the underlying Hamiltonian\ndynamics (which thus vary with H).\nLet us \frst compute the continuous-in-time dynamics\nofH. From (34),\n_H=!( _!+V0(\u0012)); (35)\nwhich vanishes to leading order. At the next order, from\n(30), one has\n_H=\u0000!2\np\n\u0003(\u000b+ 2\u001b\u0003 cos\u0012): (36)\nWe now seek to average this dynamics over the Hamil-\ntonian orbits. The general nature of the Hamiltonian\norbits is either oscillations around a local minimum of\nV(\u0012) (limit cycles) or persistent precessions. If the lo-\ncal minimum of Vis close to an even multiple of \u0019,H\ncannot increase, while if it is close to an odd multiple\nthenHcan increase if \u001bis large enough. The switching\nprocess involves moving from the oscillatory orbits close\nto one of these odd minima, up the energy landscape,\nthen jumping to oscillatory orbits around the neighbor-\ning even minimum, and decreasing in energy towards the\nnew local \fxed point.\nWe focus \frst on the oscillatory orbits. We may de\fne\ntheir half-periods as\nT(H) =Z\u0012\u0003\n+\n\u0012\u0003\n\u0000d\u0012\n_\u0012; (37)\nwhere\u0012\u0003\n\u0000and\u0012\u0003\n+are the roots of the equation V(\u0012) =\nHto the left and right of the local minimum of V(\u0012)\nabout which \u0012(t) oscillates. To compute this integral, we\nassume that \u0012(t) follows the Hamiltonian trajectory:\n_\u0012=\u0006p\n2(H\u0000V(\u0012)): (38)\nWe then de\fne the half-period average of a function\nf(\u0012(t)) as\nhfi=1\nT(H)Z\u0012\u0003\n+\n\u0012\u0003\n\u0000f(\u0012) d\u0012p\n2(H\u0000V(\u0012)); (39)\nwhich agrees with the time average over half-period to\nthe leading order. Note that this formula applies irre-\nspectively of whether the trajectory connects \u0012\u0003\n\u0000to\u0012\u0003\n+\nor\u0012\u0003\n+to\u0012\u0003\n\u0000. Applying this averaging to _H, we then have\nD\n_HE\n=\u00001\nT(H)Z\u0012\u0003\n+\n\u0012\u0003\n\u0000\u001f(\u0012;H) d\u0012; (40)\nwhere we de\fned\n\u001f(\u0012;H) =(\u000b+ 2\u001b\u0003 cos\u0012)p\n2(H\u0000V(\u0012))p\n\u0003: (41)6\nIf the value ofHis such that either of the roots \u0012\u0003\n\u0006no\nlonger exist, this indicates that the system is now on a\nprecessional trajectory. In order to account for this, we\ncan de\fne the period on a precessional trajectory instead\nas\nT(H) =Z\u0012C\n\u0012C\u0000\u0019d\u0012\n_\u0012; (42)\nwhere\u0012Cis a local maximum of V(\u0012). On the preces-\nsional trajectories, we then have\nD\n_HE\n=\u00001\nT(H)Z\u0012C\n\u0012C\u0000\u0019\u001f(\u0012;H) d\u0012: (43)\nIn order to approximate the ODE solutions, we now\ndecompose the dynamics of Hinto half-period time in-\ntervals. We thus take, at the n'th timestep,Hn=H(tn),\ntn+1=tn+T(Hn) and\nHn+1=Hn\u0000Z\u0012\u0003\n+(Hn)\n\u0012\u0003\n\u0000(Hn)\u001f(\u0012;Hn) d\u0012; (44)\nifHncorresponds to a limit cycle trajectory. The same\ndiscrete map applies to precessional trajectories, but with\nthe integration limits replaced with \u0012C\u0000\u0019and\u0012C, re-\nspectively.\nA. Modelling switching with discrete map\nIn order to model switching starting from inside a well\nofV(\u0012), we can iterate the discrete map above, starting\nfrom an initial energy H0. We chooseH0by choosing a\nstatic initial condition \u0012(0) =\u00120close to an odd multiple\nof\u0019(let us assume without loss of generality that we are\nclose to\u0019), and computing H0=V(\u00120).\nOn the oscillatory trajectories, the discrete map then\npredicts the maximum amplitudes of oscillation ( \u0012\u0003\n\u0006(Hn))\nat each timestep, by locally solving Hn=V(\u0012) for each\nn. After some number of iterations, the trajectory will\nescape the local potential well, and one or both roots of\nHn=V(\u0012) will not exist. Due to the positive average\nslope ofV(\u0012) the most likely direction for a trajectory to\nescape the potential well is _\u0012<0 (`downhill'). Assuming\nthis to be the case, at some timestep tN, it will occur that\nthe equationHN=V(\u0012) has only one root \u0012=\u0012\u0003\n+>\u0019,\nimplying that the trajectory has escaped the potential\nwell, and will proceed on a precessional trajectory in a\nnegative direction past \u0012=\u0019=2 towards\u0012= 0.\nTo distinguish whether a trajectory results in switching\nor precession, we then perform a single half-period step\non the precessional orbit from \u0012Cto\u0012C\u0000\u0019, and check\nwhetherH< V (\u0012C\u0000\u0019): if this is the case, the tra-\njectory moves back to the oscillatory orbits around the\nwell close to \u0012= 0, and decreases in energy towards the\n\fxed point near \u0012= 0, representing switching. If how-\neverH> V(\u0012C\u0000\u0019) after the precessional half-period,\nthe solution will continue to precess.In Fig. 2 below, we display the result of such an iter-\nated application of the discrete map, for the same param-\neters as the switching solution given in Fig. 1(c). In Fig.\n2(a), the continuous curve represents the solution to (30),\nand the points are the predicted peaks of the oscillations,\nfrom the discrete map (44). Fig. 2(b) shows the energy\nof the same solution as a function of \u0012. Again the blue\ncurve givesH(t) for the ODE solution, the green points\nare the prediction of the iterated discrete map, and the\nred curve is V(\u0012). The discrete map predicts the switch-\ning behavior quite well, only su\u000bering some error near\nthe switching event, when the change of His signi\fcant\non a single period.\nB. Modelling precession\nHere we apply the discrete map to a precessional\nsolution|one in which the trajectory, once it escapes\nthe potential well near \u0019, does not get trapped in the\nnext well, and continues to rotate. Fig. 3(a) below dis-\nplays such a solution \u0012(t) and its discrete approximation,\nand Fig. 3(b) displays the energy of the same solution.\nAgain, the prediction of the discrete map is excellent.\nV. TRANSITIONS IN TRAJECTORIES\nIn this section we seek to understand the transi-\ntions between the trapping, switching, and precessional\nregimes as the current parameters \u001bandp?are varied.\nA. Escape Transition\nFirstly, let us consider the transition from states which\nare trapped in a single potential well, such as those in\nFigs. 1(a,b), to states which can escape and either switch\nor precess. E\u000bectively, the absolute threshold for this\ntransition is for the value of Hto be able to increase for\nsome value \u0012close to the minimum of V(\u0012) near\u0019. Thus,\nwe consider the equation of motion (36) for H, and wish\nto \fnd parameter values such that _H>0 for some\u0012near\n\u0019. This requires that\n!2\np\n\u0003(\u000b+ 2\u001b\u0003 cos\u0012)<0: (45)\nAssuming that !6= 0, we can see that the optimal value\nof\u0012to hope to satisfy this condition is \u0012=\u0019, yield-\ning a theoretical minimum \u001b=\u001bsfor the dimensionless\ncurrent density for motion to be possible, with\n\u001bs=\u000b\n2\u0003: (46)\nThis is similar to the critical switching currents derived\nin previous work14. We then require \u001b>\u001bsfor the possi-\nbility of switching or precession. Note that this estimate\nis independent of the value of p?.7\n(b) (a)\nFIG. 2: Switching solution (blue line) and its discrete approximation (green circles). Parameters: \u000b= 0:01, \u0003 = 0:1,p?= 0:3,\n\u001b= 0:08. Panel (a) shows the solution \u0012(t), and panel (b) shows the trajectory for this solution in the H\u0000\u0012plane. The red\nline in (b) shows V(\u0012).\nB. Switching{Precessing Transition\nWe now consider the transition from switching to pre-\ncessional states. This is rather sensitive and there is not\nin general a sharp transition from switching to precession.\nIt is due to the fact that for certain parameters, the path\nthat the trajectory takes once it escapes the potential\nwell depends on how much energy it has as it does so. In\nfact, for a \fxed \u000b;\u0003, and values of \u001b > \u001bswe can sep-\narate the (\u001b;p?)-parameter space into three regions: (i)\nafter escaping the initial well, the trajectory always falls\ninto the next well, and thus switches; (ii) after escaping,\nthe trajectory may either switch or precess depending on\nits energy as it does so (and thus depending on its initial\ncondition); (iii) after escaping, the trajectory completely\npasses the next well, and thus begins to precess.\nWe can determine in which region of the parameter\nspace a given point ( \u001b;p?) lies by studying the discrete\nmap (44) close to the peaks of V(\u0012). Assume that the\ntrajectory begins at \u0012(0) =\u0019, and is thus initially in\nthe potential well spanning the interval \u0019=2\u0014\u0012\u00143\u0019=2.\nDenote by\u0012Cthe point close to \u0012=\u0019=2 at which V(\u0012)\nhas a local maximum. It is simple to compute\n\u0012C=\u0019\n2+1\n2sin\u00001(2\u001bp?): (47)\nMoreover, it is easy to see that all other local maxima of\nV(\u0012) are given by \u0012=\u0012C+k\u0019, fork2Z.\nWe now consider trajectories which escape the initial\nwell by crossing \u0012C. These trajectories have, for some\nvalue of the timestep nwhile still con\fned in the initial\nwell, an energy value Hnin the range\nHtrap<Hn<V(\u0012C+\u0019); (48)\nwhere we de\fneHtrapto be the value of Hnsuch that\nthe discrete map (44) gives Hn+1=V(\u0012C). We thushaveHn+1> V (\u0012C). In order to check whether the\ntrajectory switches or precesses, we then compute Hn+2\nand compare it to V(\u0012C\u0000\u0019). We may then classify the\ntrajectories as switching if Hn+2\u0000V(\u0012C\u0000\u0019)<0, and\nprecessional ifHn+2\u0000V(\u0012C\u0000\u0019)>0.\nFigure 4 displays a plot of Hn\u0000V(\u0012C+\u0019) against\nHn+2\u0000V(\u0012C\u0000\u0019). The blue line shows the result of\napplying the discrete map, while the red line is the iden-\ntity line. Values of Hn\u0000V(\u0012C+\u0019) which are inside the\nrange speci\fed in (48) are thus on the negative x-axis\nhere. We can classify switching trajectories as those for\nwhich the blue line lies below the x-axis, and precessing\ntrajectories as those which lie above. In Fig. 4, the pa-\nrameters are such that both of these trajectory types are\npossible, depending on the initial value of Hn, and thus\nthis set of parameters are in region (ii) of the parameter\nspace. We note that, since the curve of blue points and\nthe identity line intersect for some large enough value of\nH, this \fgure implies that if the trajectory has enough\nenergy to begin precessing, then after several precessions\nthe trajectory will converge to one which conserves en-\nergy on average over a precessional period (indicated by\nthe arrows). In region (i) of the parameter space, the\nportion of the blue line for Hn\u0000V(\u0012C+\u0019)<0 would\nhaveHn+2\u0000V(\u0012C\u0000\u0019)<0, while in region (iii), they\nwould all haveHn+2\u0000V(\u0012C\u0000\u0019)>0.\nWe can classify the parameter regimes for which\nswitching in the opposite direction (i.e. \u0012switches from\n\u0019to 2\u0019) is possible in a similar way. It is not possible\nto have a precessional trajectory moving in this direction\n(_\u0012>0), though.\nWe may then predict, for a given point ( \u001b;p?) in pa-\nrameter space, by computing relations similar to that in\nFig. 4, which region that point is in, and thus generate\na theoretical phase diagram.\nIn Fig. 5 below, we display the phase diagram in the8\n(a) (b)\nFIG. 3: Precessing solution (blue line) and its discrete approximation (green circles). Parameters: \u000b= 0:01, \u0003 = 0:1,p?= 0:6,\n\u001b= 0:1. Panel (a) shows the solution \u0012(t), and panel (b) shows the trajectory for this solution in the H\u0000\u0012plane. The red\nline in (b) shows V(\u0012).\n(\u001b;p?)-parameter space, showing the end results of solv-\ning the ODE (30) as a background color, together with\npredictions of the bounding curves of the three regions\nof the space, made using the procedure described above.\nThe predictions of the discrete map, while not perfect,\nare quite good, and provide useful estimates on the dif-\nferent regions of parameter space. In particular, we note\nthat the region where downhill switching reliably occurs\n(the portion of region (i) above the dashed black line) is\nestimated quite well. We would also note that we would\nexpect the predictions of the discrete map to improve if\nthe values of \u0003 and \u000bwere decreased.\nVI. DISCUSSION\nWe have derived an underdamped PDE model for mag-\nnetization dynamics in thin \flms subject to perpendic-\nular applied spin-polarized currents, valid in the asymp-\ntotic regime of small \u000band \u0003, corresponding to weak\ndamping and strong penalty for out-of-plane magnetiza-\ntions. We have examined the predictions of this model\napplied to the case of an elliptical \flm under a macrospin\napproximation by using an orbit-averaging approach. We\nfound that they qualitatively agree quite well with pre-\nvious simulations using full LLGS dynamics16.\nThe bene\fts of our reduced model are that they should\nfaithfully reproduce the oscillatory nature of the in-\nplane magnetization dynamics, reducing computational\nexpense compared to full micromagnetic simulations. In\nparticular, in su\u000eciently small and thin magnetic ele-\nments the problem further reduces to a single second-\norder scalar equation.\nThe orbit-averaging approach taken here enables the\ninvestigation of the transition from switching to preces-\nsion via a simple discrete dynamical system. The regionsin parameter space where either switching or precession\nare predicted, as well as an intermediate region where\nthe end result depends sensitively on initial conditions.\nIt may be possible to further probe this region by includ-\ning either spatial variations in the magnetization (which,\nin an earlier study16were observed to simply `slow down'\nthe dynamics and increase the size of the switching re-\ngion), or by including thermal noise, which could result\nin instead a phase diagram predicting switching proba-\nbilities at a given temperature, or both.\n−0.05 −0.04 −0.03 −0.02 −0.01 0 0.01 0.02 0.03 0.04 0.05−0.05−0.04−0.03−0.02−0.0100.010.020.030.040.05\nHn−V(θC+π)Hn+2−V(θC−π)Switch Precess\nFIG. 4: Precession vs switching prediction from the discrete\nmap. Parameters: \u000b= 0:01, \u0003 = 0:1,p?= 0:35,\u001b= 0:08.\nValues ofHn\u0000V(\u0012C+\u0019) to the left of the dashed line switch\nafter the next period, the trajectory becoming trapped in the\nwell around \u0012= 0. Values to the right begin to precess, and\nconverge to a precessional \fxed point of the discrete map.9\nσp⊥\n0 0.05 0.1 0.15 0.2 0.25 0.300.10.20.30.40.50.60.70.80.91\n(i)(ii)(iii)\nFIG. 5: Macrospin solution phase diagram: \u000b= 0:01;\u0003 = 0:1.\nThe background color indicates the result of solving the ODE\n(30) with initial condition (31): the dark region to the left of\nthe \fgure indicates solutions which do not escape their initial\npotential well, and the vertical dashed white line shows the\ncomputed value of the minimum current required to escape,\n\u001bs=\u000b=(2\u0003). The black band represents solutions which\ndecay, like in Fig. 1(a), while the dark grey band represents\nsolutions like in Fig. 1(b). In the rest of the \fgure, the\ngreen points indicate switching in the negative direction like\nin Fig. 1(c), grey indicate switching in the positive direction,\nand white indicates precession like in Fig. 1(d). The solid\nblack curves are the predictions of boundaries of the regions\n(as indicated in the \fgure) by using the discrete map, and\nthe dashed line is the prediction of the boundary below which\nswitching in the positive direction is possible.ACKNOWLEDGMENTS\nResearch at NJIT was supported in part by NSF via\nGrant No. DMS-1313687. Research at NYU was sup-\nported in part by NSF via Grant No. DMR-1309202.\n1S. D. Bader and S. S. P. Parkin, Annu. Rev. Condens.\nMatter Phys. 1, 71 (2010).\n2A. Brataas and A. D. Kent and H. Ohno, Nature Mat. 11,\n372 (2012).\n3A. D. Kent and D. C. Worledge, Nature Nanotechnol. 10,\n187 (2015).\n4C. J. Garc\u0013 \u0010a-Cervera and W. E, J. Appl. Phys. 90, 370\n(2001).\n5A. DeSimone, R. V. Kohn, S. M uller and F. Otto Comm.\nPure Appl. Math. 55, 1408 (2002).\n6R. V. Kohn and V. V. Slastikov, Proc. R. Soc. Lond. Ser.\nA461, 143 (2005).\n7C. B. Muratov and V. V. Osipov, J. Comput. Phys. 216,\n637 (2006).\n8A. Capella, C. Melcher and F. Otto, Nonlinearity 20, 2519\n(2007).\n9A. D. Kent, B. Ozyilmaz and E. del Barco Appl. Phys.\nLett.84, 3897 (2004).10H. Liu, D. Bedau, D. Backes, J. A. Katine, J. Langer and\nA. D. Kent Appl. Phys. Lett. 97, 242510 (2010).\n11H. Liu, D. Bedau, D. Backes, J. A. Katine, and A. D. Kent,\nAppl. Phys. Lett. 101, 032403 (2012).\n12L. Ye, G. Wolf, D. Pinna, G. D. Chaves-O'Flynn and A.\nD. Kent, J. Appl. Phys. 117, 193902 (2015).\n13K. Newhall and E. Vanden-Eijnden, J. Appl. Phys. 113,\n184105 (2013).\n14D. Pinna, A. D. Kent and D. L. Stein Phys. Rev. B 88,\n104405 (2013).\n15D. Pinna, D. L. Stein and A. D. Kent Phys. Rev. B 90,\n174405 (2014).\n16G. D. Chaves-O'Flynn, G. Wolf, D. Pinna and A. D. Kent,\nJ. Appl. Phys. 117, 17D705 (2015).\n17R. V. Kohn and V. V. Slastikov, Arch. Rat. Mech. Anal.\n178, 227 (2005)."    },    {        "title": "1606.00086v1.Existence_of_arbitrarily_smooth_solutions_of_the_LLG_equation_in_3D_with_natural_boundary_conditions.pdf",        "content": "arXiv:1606.00086v1  [math.AP]  1 Jun 2016EXISTENCE OF ARBITRARILY SMOOTH SOLUTIONS OF THE LLG\nEQUATION IN 3D WITH NATURAL BOUNDARY CONDITIONS\nMICHAEL FEISCHL AND THANH TRAN\nAbstract. We prove that the Landau-Lifshitz-Gilbert equation in three space dimen-\nsions with homogeneous Neumann boundary conditions admits arbitr arily smooth solu-\ntions, given that the initial data is sufficiently close to a constant fun ction.\n1.Introduction\nThe Landau-Lifshitz-Gilbert (LLG) equation is widely considered as a valid model\nof micromagnetic phenomena occurring in, e.g., magnetic sensors, r ecording heads, and\nmagneto-resistive storage device [12, 14, 20]. It describes the pr ecessional motion of\nmagnetization in ferromagnets. The main difficulty of the LLG equatio n is its strongly\nnon-linear character.\nClassical results concerning existence and non-uniqueness of solu tions can be found\nin [5, 22]. The existence of weak solutions is proved for 2D and 3D in [2]. I t is known that\nweak solutions are in general not unique but exist globally. Througho ut the literature,\nthere are various works on weakly-convergent numerical approx imation methods for the\nLLG (coupled to the Maxwell-equations) equations [2, 4, 6, 7, 9, 15, 16] (the list is not\nexhausted) even without an artificial projection step [1, 11].\nThis paper considers the question of existence of arbitrarily smoot h strong solutions of\nthis equation. For the case of the 2D torus, the book [20] gives an e xhaustive overview on\nresults concerning the existence and regularity of strong solution s. A brief summary of\nthe state of the art for 2D domains with periodic boundary condition s could be phrased\nas follows: There exist arbitrarily smooth solutions provided that th e initial data is\nsufficiently close to a constant function. Moreover, there exist ar bitrarily smooth local-\nin-time solutions for initial data of finite energy (see, e.g., [13]). For t he 3D case, much\nless is known in terms of strong solvability. For the 3D torus (with per iodic boundary\nconditions) [8] proves H2-regularity local in time for the coupled system of LLG and\nMaxwell-equations. The work [3] proves global existence of stron g solutions for small ini-\ntial energies on small ellipsoids. The survey article [21] summarizes re sults in the context\nof the evolution of harmonic maps (which however does not cover th e LLG equation). A\nrecent paper [19] studies the existence, uniqueness and asympt otic behavior of solutions\nin the whole spatial space R3.\nTo the authors best knowledge, this work is the first which proves e xistence of arbi-\ntrarily smooth (non-trivial) solutions on bounded 3D domains. It also gives a first result\non existence of arbitrarily smooth strong solutions with natural bo undary conditions (in\n2D and 3D). It is worth mentioning that the proof is constructive in t he sense that a\nconvergent sequence of approximate solutions is designed algorith mically. The limit of\nthis sequence turns out to be a smooth strong solution of the LLG e quation.\nThe main motivation to prove existence of smooth strong solutions f or the LLG equa-\ntion originated in the recent work [11] by the authors. There, we pr oved a priori error\n1estimates for a time integrator for the LLG equations (as well as th e coupled LLG-\nMaxwell system) which imply strong convergence of the numerical m ethod in case of\nsmooth strong solutions. Thus, the present work justifies the as sumptions in [11].\n2.The Landau-Lifshitz-Gilbert equation\nConsider a bounded smooth domain D⊂R3with connected boundary Γ having the\noutward normal vector n. Note that all the results in this paper also hold true for\nD⊂Rn,n≥2. For brevity of presentation, however, we only consider the phy sically\nmost relevant case n= 3. We define DT:= (0,T)×Dand Γ T:= (0,T)×Γ forT >0.\nWe start with the LLG equation which reads as\nmt−αm×mt=−Cem×∆minDT (1)\nfor some constant Ce>0. Here the parameter αis a positive constant. It follows\nfrom eq. (1) that |m|is constant. We follow the usual practice to normalize |m|. The\nfollowing conditions are imposed on the solution of eq. (1):\n∂nm= 0 on Γ T, (2a)\n|m|= 1 in DT, (2b)\nm(0,·) =m0inD, (2c)\nwhere∂ndenotes the normal derivative.\nThe initial data m0satisfies|m0|= 1 inD. The condition eq. (2b) together with basic\nproperties of the cross product leads to the following equivalent fo rmulation of eq. (1):\nαmt+m×mt=Ce∆m−Ce(m·∆m)minDT. (3)\nBefore stating the main result of the article, we set some notations . Bold letters\n(e.g.v) will be used for vector functions. However, as there is no confus ion, we still\nuseL2(DT) to denote the Lebesgue space of vector functions taking values inR3, i.e., we\nwill write v∈L2(DT) instead of v∈L2(DT)3. The same rule applies to other function\nspaces.\nThe following function spaces will be frequently used. For any non-n egative integer k∈\nN0={0,1,2,...}, we define\nHk,2k(DT) :=/braceleftbig\nv∈L2(DT) :/ba∇⌈blv/ba∇⌈blHk,2k(DT)<∞/bracerightbig\nwhere the norm is defined by\n/ba∇⌈blv/ba∇⌈blHk,2k(DT):=k/summationdisplay\nℓ=0/ba∇⌈bl∂ℓ\ntv/ba∇⌈blL2(0,T;H2k−2ℓ(D)).\nThe corresponding seminorm is\n|v|Hk,2k(DT):=/parenleftBig2k/summationdisplay\nℓ=1/ba∇⌈blDℓv/ba∇⌈bl2\nL2(DT)/parenrightBig1/2\n+k/summationdisplay\nℓ=1/ba∇⌈bl∂ℓ\ntv/ba∇⌈blL2(0,T;H2k−2ℓ(D)),\nwhereDℓdenotesℓth-order partial derivatives with respect to the spatial variables.\nFinally, we define\nH1\n⋆(D) :=/braceleftbig\nv∈H1(D) : ∆v∈L2(D) and ∂nv= 0 on Γ/bracerightbig\n. (4)\nWe are now ready to state the main result of the paper.\nTheorem 1. Assume that the initial data m0satisfies |m0|= 1inDand, for some\nintegerk≥3,\n2(i)m0∈H2k(D)∩H1\n⋆(D);\n(ii)Djm0∈H1\n⋆(D)for allj/2≤k−1;\n(iii)|m0|H2k(D)is sufficiently small.\nThen the problem eq. (1)–eq.(2)has a smooth strong solution m∈Hk,2k(DT)which\nsatisfies\n/ba∇⌈blm/ba∇⌈blHk,2k(DT)≤Csmooth/ba∇⌈blm0/ba∇⌈blH2k(D), (5)\nwhereCsmooth>0depends only on α,Ce,T, andk.\n3.Auxiliary Results\nFor the reader’s convenience, we state in the following lemma some we ll-known results\nregarding Sobolev embeddings and traces.\nLemma 2.\n(i)The embeddings H1(D)֒→L6(D)as well as H1,2(DT)֒→L2(0,T;L∞(D))∩\nL∞(0,T;L2(D))are continuous.\n(ii)The embedding Hk+2,2k+4(DT)֒→Wk,∞(DT)is continuous for all k∈N0.\n(iii)Ifw∈Hk,2k(DT)fork≥1then∂i\ntDjw(0)∈H1(D)for alli+j/2≤k−1.\nProof.We first prove (i). The embedding H1(D)֒→L6(D) follows from the standard\nSobolev inequality. By definition of H1,2(DT), there holds\nH1,2(DT) :=H1(0,T;L2(D))∩L2(0,T;H2(D)).\nThewell-known embeddings H1(0,T;L2(D))֒→L∞(0,T;L2(D))andL2(0,T;H2(D))֒→\nL2(0,T;L∞(D)) (since D⊂R3) conclude (i).\nSecond, we prove (ii). Since D⊂R3, it is well-known that the embeddings\nH1(0,T;Hℓ+2(D))֒→H1(0,T;Wℓ,∞(D))֒→L∞(0,T;Wℓ,∞(D))\nare continuous for any ℓ≥0; see e.g. [17]. On the other hand, we can write\nWk,∞(DT) =/braceleftbig\nv:∂i\ntv∈L∞(0,T;Wk−i,∞(D)), i= 0,...,k/bracerightbig\n.\nHence the embedding\n/braceleftbig\nv:∂i\ntv∈H1(0,T;Hk−i+2(D)), i= 0,...,k/bracerightbig\n֒→Wk,∞(DT)\nis continuous. Consequently, the embedding\nk/intersectiondisplay\ni=0Hi+1(0,T;Hk−i+2(D))֒→Wk,∞(DT)\nis continuous. Since Hk+2,2k+4(DT)⊂/intersectiontextk\ni=0Hi+1(0,T;Hk−i+2(D)), part (ii) is proved.\nStatement (iii) can be derived from [10, Theorem 4, Section 5.9.2, p. 288] as follows:\n/ba∇⌈bl∂i\ntDjw(0)/ba∇⌈blH1(D)/lessorsimilar/ba∇⌈bl∂i\ntDjw/ba∇⌈blL2(0,T;H2(D))+/ba∇⌈bl∂i+1\ntDjw/ba∇⌈blL2(0,T;L2(D))\n/lessorsimilar/ba∇⌈bl∂i\ntw/ba∇⌈blL2(0,T;Hj+2(D))+/ba∇⌈bl∂i+1\ntw/ba∇⌈blL2(0,T;Hj(D))\n/lessorsimilar/ba∇⌈blw/ba∇⌈blHk,2k(DT)\nifi+j/2≤k−1 andk≥1. The lemma is proved. /square\nThe following lemma states some useful inequalities involving the norm a nd seminorm\nofHk,2k(DT).\nLemma 3. Letv,w,v, andwbe scalar and vector functions in Hk,2k(DT)fork≥2.\n3(i)Ifi,j∈N0satisfy0< m=⌈i+j/2⌉ ≤kthen∂i\ntDjv∈Hk−m,2k−2m(DT)and\n/ba∇⌈bl∂i\ntDjv/ba∇⌈blHk−m,2k−2m(DT)≤C|v|Hk,2k(DT). (6)\n(ii)Furthermore, vw,vw,v×w,v·w, and|v|2−|w|2belong to the corresponding\nspaceHk,2k(DT)and satisfy\n/ba∇⌈blvw/ba∇⌈blHk,2k(DT)≤C/ba∇⌈blv/ba∇⌈blHk,2k(DT)/ba∇⌈blw/ba∇⌈blHk,2k(DT), (7a)\n/ba∇⌈blvw/ba∇⌈blHk,2k(DT)≤C/ba∇⌈blv/ba∇⌈blHk,2k(DT)/ba∇⌈blw/ba∇⌈blHk,2k(DT), (7b)\n/ba∇⌈blv×w/ba∇⌈blHk,2k(DT)≤C/ba∇⌈blv/ba∇⌈blHk,2k(DT)/ba∇⌈blw/ba∇⌈blHk,2k(DT), (7c)\n/ba∇⌈blv·w/ba∇⌈blHk,2k(DT)≤C/ba∇⌈blv/ba∇⌈blHk,2k(DT)/ba∇⌈blw/ba∇⌈blHk,2k(DT), (7d)\n/ba∇⌈bl|v|2−|w|2/ba∇⌈blHk,2k(DT)≤C(/ba∇⌈blv/ba∇⌈blHk,2k(DT)+/ba∇⌈blw/ba∇⌈blHk,2k(DT))/ba∇⌈blv−w/ba∇⌈blHk,2k(DT).(7e)\nThe constant C >0depends only on an upper bound of kand onDT.\nProof.To see eq. (6), we use the definition of the Hk,2k(DT)-norm and write\n/ba∇⌈bl∂i\ntDjv/ba∇⌈blHk−m,2k−2m(DT)=k−m/summationdisplay\nℓ=0/ba∇⌈bl∂ℓ+i\ntDjv/ba∇⌈blL2(0,T;H2k−2m−2ℓ(D))\n=k−m+i/summationdisplay\nℓ=i/ba∇⌈bl∂ℓ\ntDjv/ba∇⌈blL2(0,T;H2k−2m−2ℓ+2i(D)).\nSincem=⌈i+j/2⌉, we have\nk−m+i≤kand 2k−2m+2i+j≤2k. (8)\nHence, if i >0 then\n/ba∇⌈bl∂i\ntDjv/ba∇⌈blHk−m,2k−2m(DT)≤k/summationdisplay\nℓ=1/ba∇⌈bl∂ℓ\ntv/ba∇⌈blL2(0,T;H2k−2m−2ℓ+2i+j(D))\n≤k/summationdisplay\nℓ=1/ba∇⌈bl∂ℓ\ntv/ba∇⌈blL2(0,T;H2k−2ℓ(D))≤ |v|Hk,2k(DT).\nIfi= 0 then 1 ≤j≤2k(as 0< m≤k) and thus\n/ba∇⌈bl∂i\ntDjv/ba∇⌈blHk−m,2k−2m(DT)\n=k−m/summationdisplay\nℓ=0/ba∇⌈bl∂ℓ\ntDjv/ba∇⌈blL2(0,T;H2k−2m−2ℓ(D))\n≤ /ba∇⌈blDjv/ba∇⌈blL2(0,T;H2k−2m(D))+k/summationdisplay\nℓ=1/ba∇⌈bl∂ℓ\ntv/ba∇⌈blL2(0,T;H2k−2m−2ℓ+j(D))\n=/parenleftBig2k−2m/summationdisplay\nj′=0/ba∇⌈blDj+j′v/ba∇⌈bl2\nL2(DT)/parenrightBig1/2\n+k/summationdisplay\nℓ=1/ba∇⌈bl∂ℓ\ntv/ba∇⌈blL2(0,T;H2k−2m−2ℓ+j(D))\n=/parenleftBig2k−2m+j/summationdisplay\nj′=j/ba∇⌈blDj′v/ba∇⌈bl2\nL2(DT)/parenrightBig1/2\n+k/summationdisplay\nℓ=1/ba∇⌈bl∂ℓ\ntv/ba∇⌈blL2(0,T;H2k−2m−2ℓ+j(D))\n≤ |v|Hk,2k(DT),\nwhere in the last step we used eq. (8) and the definition of the semino rm.\n4We next show eq. (7a). The product rule implies\n/ba∇⌈blvw/ba∇⌈blHk,2k(DT)=k/summationdisplay\nℓ=0/ba∇⌈bl∂ℓ\nt(vw)/ba∇⌈blL2(0,T;H2k−2ℓ(D))\n/lessorsimilark/summationdisplay\nℓ=0/summationdisplay\nj1+j2=ℓ/parenleftBig/integraldisplayT\n0/ba∇⌈bl(∂j1\ntv)(∂j2\ntw)/ba∇⌈bl2\nH2k−2ℓ(D))dt/parenrightBig1/2\n/lessorsimilark/summationdisplay\nℓ=0/summationdisplay\nj1+j2=ℓ2k−2ℓ/summationdisplay\nn=0/summationdisplay\ni1+i2=n/parenleftBig/integraldisplayT\n0/integraldisplay\nD|Di1∂j1\ntv|2|Di2∂j2\ntw|2dxdt/parenrightBig1/2\n.\nNote that\ni1\n2+j1+i2\n2+j2=n\n2+ℓ≤k−ℓ+ℓ=k.\nHence, putting I:=/braceleftbig\n(i1,i2,j1,j2)∈N0:i1/2+j1+i2/2+j2≤k/bracerightbig\nwe obtain\n/ba∇⌈blvw/ba∇⌈blHk,2k(DT)/lessorsimilar/summationdisplay\n(i1,i2,j1,j2)∈I/parenleftBig/integraldisplayT\n0/integraldisplay\nD|Di1∂j1\ntv|2|Di2∂j2\ntw|2dxdt/parenrightBig1/2\n≤S1+S2+S3,\nwhere\nSν:=/summationdisplay\n(i1,i2,j1,j2)∈Iν/parenleftBig/integraldisplayT\n0/integraldisplay\nD|Di1∂j1\ntv|2|Di2∂j2\ntw|2dxdt/parenrightBig1/2\n, ν= 1,2,3,\nwith\nI1:=/braceleftbig\n(i1,i2,j1,j2)∈ I:i1/2+j1≥1 andi2/2+j2≥1/bracerightbig\n,\nI2:=/braceleftbig\n(i1,i2,j1,j2)∈ I:i1/2+j1= 0 ori2/2+j2= 0/bracerightbig\n,\nI3:=/braceleftbig\n(i1,i2,j1,j2)∈ I:i1/2+j1= 1/2 ori2/2+j2= 1/2/bracerightbig\n.\nEach term in S1is estimated by using the H¨ older inequality separately in space and tim e\nas\nS1≤/summationdisplay\n(i1,i2,j1,j2)∈I1/parenleftBig/integraldisplayT\n0/ba∇⌈blDi1∂j1\ntv(t)/ba∇⌈bl2\nL∞(D)/ba∇⌈blDi2∂j2\ntw(t)/ba∇⌈bl2\nL2(D)dt/parenrightBig1/2\n≤/summationdisplay\n(i1,i2,j1,j2)∈I1/ba∇⌈blDi1∂j1\ntv/ba∇⌈blL2(0,T;L∞(D))/ba∇⌈blDi2∂j2\ntw/ba∇⌈blL∞(0,T;L2(D))\n≤/summationdisplay\n(i1,i2,j1,j2)∈I1/ba∇⌈blDi1∂j1\ntv/ba∇⌈blH1,2(DT)/ba∇⌈blDi2∂j2\ntw/ba∇⌈blH1,2(DT),\nwhere in the last step we used lemma 2 (i). Note that in this index set I1there\nhold⌈i1/2+j1⌉ ≤k−1 and⌈i2/2+j2⌉ ≤k−1. Hence, estimate eq. (6) gives\n/ba∇⌈blDi1∂j1\ntv/ba∇⌈blH1,2(DT)/lessorsimilar/ba∇⌈blv/ba∇⌈blHk,2k(DT)and/ba∇⌈blDi2∂j2\ntw/ba∇⌈blH1,2(DT)/lessorsimilar/ba∇⌈blw/ba∇⌈blHk,2k(DT),\nimplying S1/lessorsimilar/ba∇⌈blv/ba∇⌈blHk,2k(DT)/ba∇⌈blw/ba∇⌈blHk,2k(DT).\n5The sum S2is estimated with the help of lemma 2 (ii) by\nS2≤k/summationdisplay\nℓ=02k−2ℓ/summationdisplay\ni=0/parenleftBig/integraldisplayT\n0/integraldisplay\nD|v|2|Di∂ℓ\ntw|2dxdt/parenrightBig1/2\n+k/summationdisplay\nℓ=02k−2ℓ/summationdisplay\ni=0/parenleftBig/integraldisplayT\n0/integraldisplay\nD|Di∂ℓ\ntv|2|w|2dxdt/parenrightBig1/2\n/lessorsimilar/ba∇⌈blv/ba∇⌈blL∞(DT)k/summationdisplay\nℓ=02k−2ℓ/summationdisplay\ni=0/ba∇⌈bl∂ℓ\ntw/ba∇⌈blL2(0,T;Hi(D))\n+/ba∇⌈blw/ba∇⌈blL∞(DT)k/summationdisplay\nℓ=02k−2ℓ/summationdisplay\ni=0/ba∇⌈bl∂ℓ\ntv/ba∇⌈blL2(0,T;Hi(D))\n/lessorsimilar/ba∇⌈blv/ba∇⌈blHk,2k(DT)/ba∇⌈blw/ba∇⌈blHk,2k(DT).\nFinally, for S3, since the problem is symmetric, we just consider the case when i1= 1\nandj1= 0. Since H1(D)⊆L6(D)⊆L4(D) (see lemma 2 (i)) we have\n/parenleftBig/integraldisplayT\n0/integraldisplay\nD|D1v|2|Di2∂j2\ntw|2dxdt/parenrightBig1/2\n≤/parenleftBig/integraldisplayT\n0/ba∇⌈blD1v(t)/ba∇⌈bl2\nL4(D)/ba∇⌈blDi2∂j2\ntw(t)/ba∇⌈bl2\nL4(D)dt/parenrightBig1/2\n≤ /ba∇⌈blD1v/ba∇⌈blL∞(0,T;L4(D))/ba∇⌈blDi2∂j2\ntw/ba∇⌈blL2(0,T;L4(D))\n/lessorsimilar/ba∇⌈blD1v/ba∇⌈blH1(0,T;H1(D))/ba∇⌈blDi2∂j2\ntw/ba∇⌈blL2(0,T;H1(D))\n≤ /ba∇⌈blv/ba∇⌈blH2,4(DT)/ba∇⌈blw/ba∇⌈blHk,2k(DT)\n≤ /ba∇⌈blv/ba∇⌈blHk,2k(DT)/ba∇⌈blw/ba∇⌈blHk,2k(DT),\nwhere in the penultimate step we used eq. (6), noting that i2/2+j2< k. This and the\nanalogous result for i2= 1 and j2= 0 prove\nS3/lessorsimilar/ba∇⌈blv/ba∇⌈blHk,2k(DT)/ba∇⌈blw/ba∇⌈blHk,2k(DT).\nAltogether, we obtain eq. (7a).\nThe remaining multiplicative estimates eq. (7b)–eq. (7d) follow from e q. (7a) by the\nfact that all of them can be expressed as (sums of) products of s calar functions.\nFinally, we show eq. (7e) by using the identity |v|2−|w|2= (v+w)·(v−w) and the\nalready proved estimate eq. (7d). This concludes the proof. /square\nThe following lemma is a slight generalization to the vector case of a well- known result\non the existence of solutions of the heat equation.\nLemma 4. LetL:R3→R3denote a linear operator which satisfies\nLa·a≥cL|a|2for alla∈R3, (9)\nfor some cL>0. For a given r∈L2(DT), the vector-valued heat equation\nL∂tw−∆w=rinDT,\nw= 0in{0}×D,\n∂nw= 0onΓT(10)\nhas a weak solution which satisfies\n/ba∇⌈blw/ba∇⌈blH1,2(DT)≤Cheat/ba∇⌈blr/ba∇⌈bl2\nL2(DT). (11)\nThe constant Cheat>0depends only on T,L, andD.\n6Proof.Note that eq. (9) implies the existence of L−1which satisfies\nL−1b·b≥cL\n/ba∇⌈blL/ba∇⌈bl2|b|2for allb∈R3. (12)\nThus, we can reformulate eq. (10) into\n∂tw−L−1∆w=L−1rinDT,\nw= 0 in{0}×D,\n∂nw= 0 on Γ T.(13)\nWe want to use the result [17, Theorem 3.2]. To that end, and in the no tation of [17], we\ndefineA:=−L−1∆ and\nD(A) :=/braceleftbig\nv∈H2(D) :∂nv= 0 on Γ/bracerightbig\n⊆L2(D).\nDefine the graph norm /ba∇⌈bl·/ba∇⌈bl2\nD(A):=/ba∇⌈bl·/ba∇⌈bl2\nL2(D)+/ba∇⌈blA(·)/ba∇⌈bl2\nL2(D). Then, there holds for all p∈C\nsatisfying Re( p)≥p0>0 and for all v∈D(A)\n/ba∇⌈bl(A+p)v/ba∇⌈blL2(D)≤(1+|p|)/ba∇⌈blv/ba∇⌈blD(A)\nas well as\n/ba∇⌈bl(A+p)v/ba∇⌈bl2\nL2(D)=/ba∇⌈blAv/ba∇⌈bl2\nL2(D)−2Re(p/an}b∇ack⌉tl⌉{tAv,v/an}b∇ack⌉t∇i}htD)+|p|2/ba∇⌈blv/ba∇⌈bl2\nL2(D)\n=/ba∇⌈blAv/ba∇⌈bl2\nL2(D)+2Re(p/an}b∇ack⌉tl⌉{tL−1∇v,∇v/an}b∇ack⌉t∇i}htD)+|p|2/ba∇⌈blv/ba∇⌈bl2\nL2(D).\nIt follows from eq. (12) that\n2Re(p/an}b∇ack⌉tl⌉{tL−1∇v,∇v/an}b∇ack⌉t∇i}htD)≥2p0cL\n/ba∇⌈blL/ba∇⌈bl2/ba∇⌈bl∇v/ba∇⌈bl2\nL2(D)≥0,\nso that\n/ba∇⌈bl(A+p)v/ba∇⌈bl2\nL2(D)≥ /ba∇⌈blAv/ba∇⌈bl2\nL2(D)+|p|2/ba∇⌈blv/ba∇⌈bl2\nL2(D)≥min{1,p2\n0}/ba∇⌈blv/ba∇⌈bl2\nD(A).\nStandardellipticregularitytheory(seee.g.[18, Theorem4.18])show sthatA+p:D(A)→\nL2(D) is surjective. Hence, A+p:D(A)→L2(D) is a bijective isomorphism. Moreover,\nwe have for v∈L2(D)\n/ba∇⌈bl(A+p)−1v/ba∇⌈blL2(D)/lessorsimilar1\n1+|p|/ba∇⌈blv/ba∇⌈blL2(D).\nfor allp∈Csatisfying Re( p)> p0. Thus, the requirements of [17, Theorem 3.2] are\nsatisfied which yields the existence of w∈L2(0,T;D(A)) satisfying eq. (13) and hence\nalso eq. (10).\nStandard elliptic regularity theory (see e.g. [18, Theorem 4.18]) gives\n/ba∇⌈blw/ba∇⌈blH2(D)/lessorsimilar/ba∇⌈bl∆w/ba∇⌈blL2(D)+/ba∇⌈blw/ba∇⌈blH1(D)for allw∈D(A).\nSince/ba∇⌈blw/ba∇⌈blH1(D)/lessorsimilar/ba∇⌈bl∆w/ba∇⌈blL2(D)+/ba∇⌈blw/ba∇⌈blL2(D)for all functions satisfying ∂nw= 0, we deduce\nthatw∈L2(0,T;H2(D)). The proof of [17, Theorem 3.2] also reveals\n/ba∇⌈blw/ba∇⌈blL2(0,T;H2(D))/lessorsimilar/ba∇⌈blw/ba∇⌈blL2(0,T;D(A))/lessorsimilar/ba∇⌈blL−1r/ba∇⌈blL2(DT)≃ /ba∇⌈blr/ba∇⌈blL2(DT).\nThis estimate and eq. (13) yield /ba∇⌈bl∂tw/ba∇⌈blL2(0,T;L2(D))/lessorsimilar/ba∇⌈blr/ba∇⌈blL2(DT), completing the proof of\nthe lemma. /square\nThe next lemma is a result on higher regularity for solutions to eq. (10 ).\n7Lemma 5. Under the assumption of lemma 4, if r∈Hk−1,2k−2(DT)fork≥2satisfies\n∂i\ntDjr(0)∈H1\n⋆(D)for alli+j/2≤k−2,\nthen the solution wof the vector-valued heat equation eq. (10)satisfies\n/ba∇⌈blw/ba∇⌈blHk,2k(DT)≤Cr/ba∇⌈blr/ba∇⌈blHk−1,2k−2(DT) (14)\nand\n∂i\ntDjw(0)∈H1\n⋆(D)for alli+j/2≤k−1. (15)\nProof.Wefirstrecallthatif r∈Hk−1,2k−2(DT)then∂i\ntDjr(0)∈H1(D)fori+j/2≤k−2;\nsee lemma 2. The proof is an induction on k∈N, where lemma 4 confirms the case k= 1.\nLetk >1 and assume that eq. (14) and eq. (15) hold for k−1. Then, differentiation\nreveals that v:=∂tw−L−1r(0) is the unique solution of\nL∂tv−∆v=∂tr+L−1∆r(0) inDT,\nv= 0 in{0}×D,\n∂nv= 0 on Γ T.\nTheright-handside /tildewider:=∂tr+L−1∆r(0)satisfies ∂i\ntDj/tildewider(0)∈H1\n⋆(D)foralli+j/2≤k−3.\nThe induction hypothesis and lemma 3 show that\n/ba∇⌈blv/ba∇⌈blHk−1,2k−2(DT)/lessorsimilar/ba∇⌈bl∂tr+L−1∆r(0)/ba∇⌈blHk−2,2k−4(DT)/lessorsimilar/ba∇⌈blr/ba∇⌈blHk−1,2k−2(DT)(16)\nas well as\n∂n\ntDmv(0)∈H1\n⋆(D) for all n+m/2≤k−2. (17)\nThe definition of vand estimate eq. (16) imply\nk/summationdisplay\nj=1/ba∇⌈bl∂j\ntw/ba∇⌈blL2(0,T;H2k−2j(D))/lessorsimilar/ba∇⌈blv/ba∇⌈blHk−1,2k−2(DT)/lessorsimilar/ba∇⌈blr/ba∇⌈blHk−1,2k−2(DT). (18)\nAssumeforthemomentthat wandraresmooth. Then, wehavewithellipticregularity\n(see, e.g., [18, Theorem 4.18]) and −∆w=r−L∂twthat all 0 ≤t≤Tsatisfy\n/ba∇⌈blw(t)/ba∇⌈blH2k(D)/lessorsimilar/ba∇⌈blr(t)/ba∇⌈blH2k−2(D)+/ba∇⌈bl∂tw(t)/ba∇⌈blH2k−2(D).\nIntegration over t∈(0,T) reveals for smooth wandr\n/ba∇⌈blw/ba∇⌈blL2(0,T;H2k(D))/lessorsimilar/ba∇⌈blr/ba∇⌈blHk−1,2k−2(DT)+/ba∇⌈bl∂tw/ba∇⌈blL2(0,T;H2k−2(D)). (19)\nA density argument now proves w∈L2(0,T;H2k(D)) with eq. (19) even for non-smooth\nw. The combination of eq. (18) and eq. (19) shows\n/ba∇⌈blw/ba∇⌈blHk,2k(DT)/lessorsimilar/ba∇⌈blr/ba∇⌈blHk−1,2k−2(DT).\nTo see∂i\ntDjw(0)∈H1\n⋆(D) for alli+j/2≤k−1, we distinguish three cases: First, for\ni≥2, since∂i\ntDjw=∂i−1\ntDjv, property eq. (17) gives with n=i−1 andm=j≤\n2k−2−2ithat\n∂i\ntDjw(0) =∂n\ntDmv(0)∈H1\n⋆(D).\nSecond, for i= 1, eq. (17) shows with n= 0 and m=j≤2k−4 that\nDmv(0) =∂tDjw(0)−L−1Djr(0)∈H1\n⋆(D).\nSinceDjr(0)∈H1\n⋆(D) for allj/2≤k−2 by definition, we obtain ∂tDjw(0)∈H1\n⋆(D)\nfor allj/2≤k−2. Finally, for i= 0, we have for Djw(0) = 0∈H1\n⋆(D) forj/2≤k−1\nby definition. Altogether, this proves ∂i\nt∆jw(0)∈H1\n⋆(D) for alli+j/2≤k−1 and thus\nconcludes the proof. /square\n8The next technical result will be used to prove that the solution of s ome nonlinear\nparabolic problem satisfies condition eq. (2b) for all t >0 if it satisfies that condition\natt= 0.\nLemma 6. Letu∈H1(0,T;L2(D))such that u(t)∈W2,∞(D)for all0≤t≤Twith\nu|{0}×D= 1be a strong solution of\nβ∂tu−u∆u= 0inDT,\n∂nu= 0onΓT\nfor some constant β >0. Then, there holds u= 1inDT.\nProof.Definee:=u−1. There holds\nβ∂te−e∆e−∆e= 0 and ∂ne= 0 on Γ T.\nMultiplication by eand integration by parts over Dshows\nβ\n2∂t/ba∇⌈ble(t)/ba∇⌈bl2\nL2(D)+/ba∇⌈bl∇e/ba∇⌈bl2\nL2(D)≤ /ba∇⌈bl∆e(t)/ba∇⌈blL∞(D)/ba∇⌈ble(t)/ba∇⌈bl2\nL2(D)/lessorsimilar/ba∇⌈ble(t)/ba∇⌈bl2\nL2(D),\nby use of the regularity assumptions for the last inequality. Thus, w e have\n∂t/ba∇⌈ble(t)/ba∇⌈bl2\nL2(D)/lessorsimilar/ba∇⌈ble(t)/ba∇⌈bl2\nL2(D)for all 0≤t≤T.\nGronwall’s inequality proves /ba∇⌈ble(t)/ba∇⌈blL2(D)/lessorsimilar/ba∇⌈ble(0)/ba∇⌈blL2(D)= 0, which concludes the proof.\n/square\nWe next define a residual operator which will be used to generate a s equence {mℓ}\nconverging to a solution mof eq. (1)–eq. (2).\nDefinition 7. Letx0be an arbitrary point in Dandm0be the initial data given\nin eq.(2c). For any v∈Hk,2k(DT)for some k >0, we define the residual\nR(v) :=αvt+v×vt−Ce|v|2∆v−Ce|∇v|2v. (20)\nWe also define a linear operator L:R3→R3by\nLa:=Lm0(x0)a:=αa+m0(x0)×a,a∈R3. (21)\nIt is easy to see that Lsatisfies eq. (9) with cL=αand that\nR(v) =αvt+v×vt−Ce∆v+Ce(1−|v|2)∆v−Ce|∇v|2v\n=αvt+m0(x0)×vt+(v−m0(x0))×vt−Ce∆v\n+Ce(1−|v|2)∆v−Ce|∇v|2v\n=Lvt+(v−m0(x0))×vt−Ce∆v+Ce(1−|v|2)∆v−Ce|∇v|2v,(22)\nwhereLis applied pointwise in time and space.\nThe following lemma gives some mapping properties of the operator R. (We recall the\ndefinition of H1\n⋆(D) in eq. (4).)\nLemma 8.\n(i)The residual operator Rdefined in definition 7 is continuous from Hk,2k(DT)into\nHk−1,2k−2(DT)fork≥3. More precisely, there holds\n/ba∇⌈blR(v)−R(w)/ba∇⌈blHk−1,2k−2(DT)\n≤CR(1+/ba∇⌈blv/ba∇⌈bl2\nHk,2k(DT)+/ba∇⌈blw/ba∇⌈bl2\nHk,2k(DT))/ba∇⌈blv−w/ba∇⌈blHk,2k(DT).\n9(ii)Fork≥3, ifw∈Hk,2k(DT)satisfies\n∂i\ntDjw(0)∈H1\n⋆(D)for alli+j/2≤k−1,\nthen\n∂i\ntDjR(w)(0)∈H1\n⋆(D)for alli+j/2≤k−2.\nProof.Statement (i) is proved by using lemma 3 (which is applicable because k≥3) as\nfollows:\n/ba∇⌈blR(v)−R(w)/ba∇⌈blHk−1,2k−2(DT)\n/lessorsimilar/ba∇⌈bl∂t(v−w)/ba∇⌈blHk−1,2k−2(DT)\n+/ba∇⌈bl(v−w)×∂tv/ba∇⌈blHk−1,2k−2(DT)+/ba∇⌈blw×∂t(v−w)/ba∇⌈blHk−1,2k−2(DT)\n+/ba∇⌈bl(|v|2−|w|2)∆v/ba∇⌈blHk−1,2k−2(DT)+/ba∇⌈bl|w|2∆(v−w)/ba∇⌈blHk−1,2k−2(DT)\n+/ba∇⌈bl(∇(v−w)·∇v)v/ba∇⌈blHk−1,2k−2(DT)+/ba∇⌈bl(∇w·∇(v−w))v/ba∇⌈blHk−1,2k−2(DT)\n+/ba∇⌈bl(∇w·∇w)(v−w)/ba∇⌈blHk−1,2k−2(DT)\n/lessorsimilar/parenleftbig\n1+/ba∇⌈blv/ba∇⌈blHk,2k(DT)+/ba∇⌈blw/ba∇⌈blHk,2k(DT)\n+/ba∇⌈blv/ba∇⌈bl2\nHk,2k(DT)+/ba∇⌈blw/ba∇⌈bl2\nHk,2k(DT)/parenrightbig\n/ba∇⌈blv−w/ba∇⌈blHk,2k(DT)\n/lessorsimilar(1+/ba∇⌈blv/ba∇⌈bl2\nHk,2k(DT)+/ba∇⌈blw/ba∇⌈bl2\nHk,2k(DT))/ba∇⌈blv−w/ba∇⌈blHk,2k(DT).\nTo prove (ii) we note that since R(w)∈Hk−1,2k−2(DT) fork≥3, lemma 2 gives\n∂i\ntDjR(w)(0)∈H1(D) for alli+j/2≤k−2. It remains to show that ∆ ∂i\ntDjR(w)(0)∈\nL2(D) and that the normal derivative of ∂i\ntDjR(w)(0) is zero. It is easy to see from the\ndefinition eq. (20) of Rand the product rule that for i+j/2≤k−2, the derivative\n∂i\ntDjR(w) is a sum of terms of the form\n(∂n1\ntDm1v1)⊙1/parenleftBig\n(∂n2\ntDm2v2)⊙2(∂n3\ntDm3v3)/parenrightBig\n(23)\nwithn1+n2+n3+(m1+m2+m3)/2≤k−1 andvs∈ {w,1},s= 1,2,3, where ⊙1\nand⊙2denote either the scalar, dot, or crossproduct. Thus at least 2 e lements in the set\n{(n1,m2),(n2,m2),(n3,m3)}satisfyni+mi/2≤(k−1)/2. Without loss of generality we\nassumei= 2,3. lemma 3 gives\n∂ni\ntDmiw∈Hk−⌈(k−1)/2⌉,2(k−⌈(k−1)/2⌉)(DT)⊆H2,4(DT)\n(because k≥3). lemma 2 (iii)&(i) imply\nD2(∂ni\ntDmiw(0)) =∂ni\ntDmi+2w(0)∈H1(D)⊂L6(D)⊂L4(D),\nand thus\n∂ni\ntDmiw(0),∂ni\ntDmi+1w(0)∈H2(D)⊆L∞(D), i= 1,2.\nThe product rule shows (with the definition ∆1/2:=D1) that ∆∂i\ntDjR(w)(0) is a sum\nof terms of the form\n(∂n1\nt∆r1Dm1v1(0))⊙1/parenleftBig\n(∂n2\nt∆r2Dm2v2(0))⊙2(∂n3\nt∆r3Dm3v3(0))/parenrightBig\n(24)\n10withrs∈ {0,1/2,1},s= 1,2,3, satisfying r1+r2+r3= 1. This and the considerations\nabove together with the assumption ∂n\ntDmv1(0)∈H1\n⋆(D) show\n/parenleftbig\n∂n1\nt∆r1Dm1v1(0),∂n2\nt∆r2Dm2v2(0),∂n3\nt∆r3Dm3v3(0)/parenrightbig\n∈\n\nL2(D)×L∞(D)×L∞(D) forr1∈ {1/2,1}, r2,r3∈ {0,1/2},\nH1(D)×L4(D)×L∞(D) forr1= 0,r2= 1,r3= 0,\nH1(D)×L∞(D)×L4(D) forr1= 0,r2= 0,r3= 1,\nH1(D)×L∞(D)×L∞(D) forr1= 0,r2=r3= 1/2,\n⊆\n\nL2(D)×L∞(D)×L∞(D) forr1∈ {1/2,1}, r2,r3∈ {0,1/2},\nL4(D)×L4(D)×L∞(D) forr1= 0,r2= 1,r3= 0,\nL4(D)×L∞(D)×L4(D) forr1= 0,r2= 0,r3= 1,\nL2(D)×L∞(D)×L∞(D) forr1= 0,r2=r3= 1/2.\nHence the product eq. (24) is in L2(D). This implies that ∆ ∂i\ntDjR(w)(0)∈L2(D).\nMoreover, the normal derivatives of each factor of eq. (23) are zero by definition, and\nthus the product rule implies that also ∂i\ntDjR(w)(0) = 0, completing the proof of the\nlemma. /square\nThefollowinglemmagivessufficientconditionsforagivenfunction msatisfying R(m) =\n0 to be a solution to eq. (1)–eq. (2).\nLemma 9. Ifm∈Hk,2k(DT)fork≥3satisfies\nR(m) = 0onDT,\n∂nm= 0onΓT,\n|m|= 1on{0}×D,\nm(0,·) =m0inD,(25)\nthenmis a strong solution to eq. (1)–eq.(2).\nProof.It suffices to show that msatisfies eq. (2b) andeq. (3). The first property is shown\nby invoking lemma 6. To this end, let u:=|m|2. lemma 2 (ii) shows m∈W1,∞(DT) and\nhenceu∈W1,∞(DT). lemma 2 (iii) proves m(t)∈H5(D)⊆W2,∞(D), which implies\nu(t)∈W2,∞(D) for all 0 ≤t≤T. Moreover, by using\n∆|m|2= 2∆m·m+2|∇m|2, (26)\ntogether with eq. (20) and eq. (25) we obtain\nα\n2∂tu−Ce\n2u∆u=α\n2∂t|m|2−Ce\n2|m|2∆|m|2\n=αmt·m−Ce|m|2∆m·m−Ce|∇m|2|m|2\n=R(m)·m= 0 inDT.\nAssumption eq. (25) also implies ∂nu=∂n|m|2= 2∂nm·m= 0 on Γ T. Hence, lemma 6\nyieldsu= 1 inDT, i.e. eq. (2b) holds, which in turn together with R(m) = 0 implies\nαmt+m×mt=Ce∆m+Ce|∇m|2m.\nIt follows from eq. (26) that |∇m|2=−∆m·mso thatmsatisfies eq. (3), completing\nthe proof of the lemma.\n/square\nFinally, since Ris not linear, we need the following lemma to estimate R(v−w).\n11Lemma 10. Letv,w∈Hk,2k(DT)fork≥3. Then, there holds\n/ba∇⌈blR(v−w)/ba∇⌈blHk−1,2k−2(DT)/lessorsimilar/ba∇⌈blR(v)−(L∂t−Ce∆)w/ba∇⌈blHk−1,2k−2(DT)\n+/ba∇⌈blv−m0(x0)/ba∇⌈blHk−1,2k−2(DT)/ba∇⌈blw/ba∇⌈blHk,2k(DT)\n+/ba∇⌈bl(1−|v|2)∆w/ba∇⌈bl2\nHk−1,2k−2(DT)\n+/ba∇⌈blw/ba∇⌈blHk,2k(DT)|v|Hk,2k(DT)/parenleftbig\n1+/ba∇⌈blv/ba∇⌈blHk,2k(DT)/parenrightbig\n+/ba∇⌈blw/ba∇⌈bl2\nHk,2k(DT)/parenleftbig\n1+/ba∇⌈blv/ba∇⌈blHk,2k(DT)/parenrightbig\n+/ba∇⌈blw/ba∇⌈bl3\nHk,2k(DT).(27)\nThe hidden constant depends only on Ceand on the constants from lemma 3.\nProof.It can be easily derived from eq. (22) that\nR(v−w)−R(v) =−(L∂t−Ce∆)w+(v−m0(x0))×wt+w×(vt−wt)\n−Ce/parenleftbig\n1−|v−w|2/parenrightbig\n∆w−Ce/parenleftbig\n|w|2−2(v·w)/parenrightbig\n∆v\n+Ce|∇v−∇w|2w−Ce/parenleftbig\n|∇w|2−2∇v·∇w/parenrightbig\nv,\nso that\nR(v−w) =R(v)−(L∂t−Ce∆)w+(v−m0(x0))×wt+w×(vt−wt)\n−Ce/parenleftbig\n1−|v|2+2v·w−|w|2/parenrightbig\n∆w−Ce/parenleftbig\n|w|2−2(v·w)/parenrightbig\n∆v\n+Ce/parenleftbig\n|∇v|2−2∇v·∇w+|∇w|2/parenrightbig\nw−Ce/parenleftbig\n|∇w|2−2∇v·∇w/parenrightbig\nv\n=R(v)−(L∂t−Ce∆)w+T1+···+T6.\nHence\n/ba∇⌈blR(v−w)/ba∇⌈blHk−1,2k−2(DT)\n/lessorsimilar/ba∇⌈blR(v)−(L∂t−Ce∆)w/ba∇⌈blHk−1,2k−2(DT)+6/summationdisplay\ni=1/ba∇⌈blTi/ba∇⌈blHk−1,2k−2(DT).\nDenoting Ti=/ba∇⌈blTi/ba∇⌈blHk−1,2k−2(DT), lemma 3 yields\nT1/lessorsimilar/ba∇⌈blv−m0(x0)/ba∇⌈blHk−1,2k−2(DT)|w|Hk,2k(DT)\n≤ /ba∇⌈blv−m0(x0)/ba∇⌈blHk−1,2k−2(DT)/ba∇⌈blw/ba∇⌈blHk,2k(DT),\nT2/lessorsimilar/ba∇⌈blw/ba∇⌈blHk−1,2k−2(DT)/parenleftbig\n|v|Hk,2k(DT)+|w|Hk,2k(DT)/parenrightbig\n≤ |v|Hk,2k(DT)/ba∇⌈blw/ba∇⌈blHk,2k(DT)+/ba∇⌈blw/ba∇⌈bl2\nHk,2k(DT),\nT3/lessorsimilar/ba∇⌈bl(1−|v|2)∆w/ba∇⌈blHk−1,2k−2(DT)+/ba∇⌈blv/ba∇⌈blHk−1,2k−2(DT)/ba∇⌈blw/ba∇⌈blHk−1,2k−2(DT)|w|Hk,2k(DT)\n+/ba∇⌈blw/ba∇⌈bl2\nHk−1,2k−2(DT)|w|Hk,2k(DT)\n≤ /ba∇⌈bl(1−|v|2)∆w/ba∇⌈blHk−1,2k−2(DT)+/ba∇⌈blv/ba∇⌈blHk,2k(DT)/ba∇⌈blw/ba∇⌈bl2\nHk,2k(DT)+/ba∇⌈blw/ba∇⌈bl3\nHk,2k(DT),\nT4/lessorsimilar/ba∇⌈blw/ba∇⌈bl2\nHk−1,2k−2(DT)|v|Hk,2k(DT)+/ba∇⌈blv/ba∇⌈blHk−1,2k−2(DT)/ba∇⌈blw/ba∇⌈blHk−1,2k−2(DT)|v|Hk,2k(DT)\n≤ /ba∇⌈blv/ba∇⌈blHk,2k(DT)/ba∇⌈blw/ba∇⌈bl2\nHk,2k(DT)+|v|Hk,2k(DT)/ba∇⌈blv/ba∇⌈blHk,2k(DT)/ba∇⌈blw/ba∇⌈blHk,2k(DT)\nT5/lessorsimilar|v|2\nHk,2k(DT)/ba∇⌈blw/ba∇⌈blHk−1,2k−2(DT)+|v|Hk,2k(DT)|w|Hk,2k(DT)/ba∇⌈blw/ba∇⌈blHk−1,2k−2(DT)\n+|w|2\nHk,2k(DT)/ba∇⌈blw/ba∇⌈blHk−1,2k−2(DT)\n≤ |v|2\nHk,2k(DT)/ba∇⌈blw/ba∇⌈blHk,2k(DT)+/ba∇⌈blv/ba∇⌈blHk,2k(DT)/ba∇⌈blw/ba∇⌈bl2\nHk,2k(DT)+/ba∇⌈blw/ba∇⌈bl3\nHk,2k(DT),\nT6/lessorsimilar|w|2\nHk,2k(DT)/ba∇⌈blv/ba∇⌈blHk−1,2k−2(DT)+|v|Hk,2k(DT)|w|Hk,2k(DT)/ba∇⌈blv/ba∇⌈blHk−1,2k−2(DT)\n≤ /ba∇⌈blv/ba∇⌈blHk,2k(DT)/ba∇⌈blw/ba∇⌈bl2\nHk,2k(DT)+|v|Hk,2k(DT)/ba∇⌈blv/ba∇⌈blHk,2k(DT)/ba∇⌈blw/ba∇⌈blHk,2k(DT).\n12Collecting all the terms we obtain the desired estimates, completing t he proof. /square\n4.Proof of the Main Result\nThis is a constructive proof. Starting with the initial guess m0(t,x) :=m0(x) for all\n(t,x)∈DT, we define a sequence ( mℓ)ℓ∈N0as follows. Having defined mℓ,ℓ= 0,1,2,...,\nthe construction involves the following tasks:\n•Definerℓ:=R(mℓ),\n•Solve\nL∂tRℓ−Ce∆Rℓ=rℓinDT,\n∂nRℓ= 0 on Γ T,\nRℓ= 0 on {0}×D.\n•Definemℓ+1:=mℓ−Rℓ.\nFirst we note that the above iteration is well-defined. Indeed, the a ssumptions on\nthe initial data m0imply that the initial guess m0belongs to Hk,2k(DT) and satis-\nfies∂i\ntDjm0(0)∈H1\n⋆(D) for alli+j/2≤k−1. Lemmas 5 and 8 then imply that R0also\nhas the same smoothness properties, and so does m1. By repeating the same argument,\nall functions mℓhave the same smoothness properties as m0, and the sequence {mℓ}is\nwell constructed. Next we note that, mℓ|{0}×D=m0|{0}×D=m0and∂nmℓ=∂nm0= 0\nfor allℓ∈N. Note also that due to lemma 5\n/ba∇⌈blRℓ/ba∇⌈blHk,2k(DT)≤Cr/ba∇⌈blrℓ/ba∇⌈blHk−1,2k−2(DT), ℓ= 0,1,2,.... (28)\nWe will show that the sequence ( mℓ)ℓ∈N0converges to a function m. lemma 8 then\nyields the convergence of R(mℓ) toR(m) asℓ→ ∞. lemma 9 will then be used to\nconclude that mis a strong solution of eq. (1)–eq. (2).\nTo show that {mℓ}is a Cauchy sequence we note that for 0 ≤ℓ′≤ℓ\n/ba∇⌈blmℓ−mℓ′/ba∇⌈blHk,2k(DT)≤ℓ−2/summationdisplay\nj=ℓ′−1/ba∇⌈blRj+1/ba∇⌈blHk,2k(DT). (29)\nDenoting\nRj:=/ba∇⌈blRj/ba∇⌈blHk,2k(DT), mj:=/ba∇⌈blmj/ba∇⌈blHk,2k(DT), mj,0:=|mj|Hk,2k(DT),\ninorder toestimate each termin thesum onthe right hand side ofeq. (29) we use eq. (28)\nand invoke lemma 10 with v=mjandw=Rj, noting that\nR(mj) =rj= (L∂t−Ce∆)Rj,\nto obtain\nRj+1/lessorsimilar/ba∇⌈blrj+1/ba∇⌈blHk−1,2k−2(DT)\n=/ba∇⌈blR(mj+1)/ba∇⌈blHk−1,2k−2(DT)=/ba∇⌈blR(mj−Rj)/ba∇⌈blHk−1,2k−2(DT)\n/lessorsimilarRj/ba∇⌈blmj−m0(x0)/ba∇⌈blHk−1,2k−2(DT)+/ba∇⌈bl(1−|mj|2)∆Rj/ba∇⌈blHk−1,2k−2(DT)\n+Rjmj,0(1+mj)+R2\nj(1+mj)+R3\nj.(30)\nFor the first term on the right hand side of eq. (30) we note that mℓ(0,x0)−m0(x0) = 0,\nand hence lemma 2 (i) yields (since k≥3)\n|mj(t,x)−m0(x0)| ≤(diam(D)2+T2)1/2/ba∇⌈bl(∂t,∇)mj/ba∇⌈blL∞(DT)\n/lessorsimilar/ba∇⌈bl(∂t,∇)mj/ba∇⌈blHk−1,2k−2(DT)/lessorsimilar|mj|Hk,2k(DT).\n13This implies\n/ba∇⌈blmj−m0(x0)/ba∇⌈blHk−1,2k−2(DT)≤ /ba∇⌈blmj−m0(x0)/ba∇⌈blL2(DT)+|mj|Hk−1,2k−2(DT)/lessorsimilarmj,0.(31)\nFor the second term on the right hand side of eq. (30), we first obs erve that since\nmj=m0−j−1/summationdisplay\ni=0Riand|m0(t,·)|=|m0|= 1, (32)\nthere holds\n|mj|2=|m0|2−2m0·j−1/summationdisplay\ni=0Ri+/vextendsingle/vextendsinglej−1/summationdisplay\ni=0Ri/vextendsingle/vextendsingle2\nso that\n1−|mj|2= 2m0·j−1/summationdisplay\ni=0Ri−/vextendsingle/vextendsinglej−1/summationdisplay\ni=0Ri/vextendsingle/vextendsingle2.\nThus, with the help of lemma 3, we obtain\n/ba∇⌈bl(1−|mj|2)∆Rj/ba∇⌈blHk−1,2k−2(DT)/lessorsimilarRj/parenleftBig\nm0j−1/summationdisplay\ni=0Ri+|j−1/summationdisplay\ni=0Ri|2/parenrightBig\n. (33)\nAltogether, eq. (30)–eq. (33) imply\nRj+1≤/tildewideCRj/parenleftBig\n(mj,0+Rj)(1+mj)+R2\nj+m0j−1/summationdisplay\ni=0Ri+/vextendsingle/vextendsinglej−1/summationdisplay\ni=0Ri/vextendsingle/vextendsingle2/parenrightBig\n=:/tildewideCQjRj, (34)\nfor some constant /tildewideC >0, where Qjis the sum of all the terms in the brackets. We will\nshow that for all q∈(0,1) there exists ε >0 such that |m0|H2k(D)≤εimplies\n/tildewideCQj≤qfor allj∈N0. (35)\nGivenq∈(0,1) (and with the constants CRfrom lemma 8 (i), and Crfrom eq. (28)), we\ndefineCrR:=CrCR(3+2|D|+|D|1/2) and choose 0 < ε <1 sufficiently small such that\nǫ/parenleftBig\n1+CrR+CrR\n1−q/parenrightBig/parenleftBig\n1+ǫ+|D|1/2+CrR\n1−q/parenrightBig\n+(CrRǫ)2\n+/parenleftBig\nǫ+|D|1/2/parenrightBigCrRǫ\n1−q+/parenleftBigCrRǫ\n1−q/parenrightBig2\n≤/tildewideC−1q. (36)\nThis allows us to prove eq. (35) by induction. By assumption, |m0|H2k(D)is sufficiently\nsmall such that\nm0,0=|m0|H2k(D)≤ǫ\nand\n/ba∇⌈blm0−c/ba∇⌈blHk,2k(D)=/ba∇⌈blm0−c/ba∇⌈blH2k(D)≤Cpc|m0|H2k(D)≤ǫ\nwithc:=|D|−1/integraltext\nDm0∈R3where the Poincar´ e constant Cpc>0 depends only on D.\nBy definition, we have |c|= 1 and hence\nm0≤ /ba∇⌈blm0−c/ba∇⌈blHk,2k(D)+/ba∇⌈blc/ba∇⌈blHk,2k(D)≤ǫ+|D|1/2.\n14Moreover, since R(c) = 0 we have, noting eq. (28),\nR0≤Cr/ba∇⌈blR(m0)/ba∇⌈blHk−1,2k−2(DT)=Cr/ba∇⌈blR(m0)−R(c)/ba∇⌈blHk−1,2k−2(DT)\n≤CrCR(1+m2\n0+/ba∇⌈blc/ba∇⌈bl2\nHk,2k(DT))/ba∇⌈blm0−c/ba∇⌈blHk,2k(DT)\n≤CrCR(3+2|D|+|D|1/2)ǫ=CrRε.\nHence\nQ0= (m0,0+R0)(1+m0)+R2\n0≤ǫ(1+CrR)(1+|D|1/2+ǫ)+(CrRǫ)2.\nOur choice of εguarantees /tildewideCQ0≤q. To conclude the induction, assume that /tildewideCQi≤q\nfor alli= 0,...,j−1. Then the induction assumption and eq. (34) give\nRj≤qRj−1≤ ··· ≤qjR0≤qjCrRǫ, (37)\nwhich implies\nj−1/summationdisplay\ni=0Ri≤j−1/summationdisplay\ni=0qiR0≤CrRǫ\n1−q.\nHence eq. (32) proves\nmj,0≤m0,0+j−1/summationdisplay\ni=0Ri≤ǫ/parenleftbigg\n1+CrR\n1−q/parenrightbigg\nas well as\nmj≤m0+j−1/summationdisplay\ni=0Ri≤ǫ+|D|1/2+CrRǫ\n1−q. (38)\nIt then follows from the definition of Qjandε >0 that eq. (35) holds for all j. This\nconcludes the induction and proves eq. (35) for all j∈N0.\nWe now prove that {mℓ}is a Cauchy sequence. It follows from eq. (29), eq. (37) that\n/ba∇⌈blmℓ−mℓ′/ba∇⌈blHk,2k(DT)≤ℓ−2/summationdisplay\nj=ℓ′−1qj+1R0≤CrRǫ\n1−qqℓ′→0 asℓ′→ ∞.\nTherefore, {mℓ}converges to some m∈Hk,2k(DT) which satisfies, by passing to the\nlimit in the first inequality in eq. (38),\n/ba∇⌈blm/ba∇⌈blHk,2k(DT)≤ /ba∇⌈blm0/ba∇⌈blHk,2k(DT)+∞/summationdisplay\nj=0/ba∇⌈blRj/ba∇⌈blHk,2k(DT)/lessorsimilar/ba∇⌈blm0/ba∇⌈blHk,2k(DT)+R0\n1−q.(39)\nIt remains to prove that R(m) = 0, which can easily be seen from the continuity of R\n(see lemma 8) and the definition of Rℓ:\n/ba∇⌈blR(m)/ba∇⌈blHk−1,2k−2(DT)= lim\nℓ→∞/ba∇⌈blR(mℓ)/ba∇⌈blHk−1,2k−2(DT)\n= lim\nℓ→∞/ba∇⌈blL∂tRℓ−Ce∆Rℓ/ba∇⌈blHk−1,2k−2(DT)/lessorsimilarlim\nℓ→∞/ba∇⌈blRℓ/ba∇⌈blHk,2k(DT)\n/lessorsimilarlim\nℓ→∞qℓ= 0.\nAs argued at the beginning of this proof, this shows that m|DTis a strong solution\nof eq. (1).\nFinally, to show eq. (5) we note that eq. (28), the continuity of R, and the fact that\nR(0) = 0 yield\nR0/lessorsimilar/ba∇⌈blr0/ba∇⌈blHk−1,2k−2(DT)=/ba∇⌈blR(m0)−R(0)/ba∇⌈blHk−1,2k−2(DT)/lessorsimilar/ba∇⌈blm0/ba∇⌈blHk,2k(DT).\n15Hence eq. (5) follows from eq. (39), completing the proof of the th eorem.\nReferences\n[1] Claas Abert, Gino Hrkac, Marcus Page, Dirk Praetorius, Michele R uggeri, and Dieter Suess. Spin-\npolarized transport in ferromagnetic multilayers: an unconditionally convergent FEM integrator.\nComput. Math. Appl. , 68(6):639–654, 2014.\n[2] Fran¸ cois Alouges. A new finite element scheme for Landau-Lifch itz equations. Discrete Contin. Dyn.\nSyst. Ser. S , 1(2):187–196, 2008.\n[3] Fran¸ cois Alouges and Karine Beauchard. Magnetization switchin g on small ferromagnetic ellipsoidal\nsamples. ESAIM Control Optim. Calc. Var. , 15(3):676–711, 2009.\n[4] Francois Alouges, Evaggelos Kritsikis, Jutta Steiner, and Jean- Christophe Toussaint. A convergent\nand precise finite element scheme for Landau-Lifschitz-Gilbert equ ation.Numer. Math. , 128(3):407–\n430, 2014.\n[5] Fran¸ cois Alouges and Alain Soyeur. On global weak solutions for L andau-Lifshitz equations: exis-\ntence and nonuniqueness. Nonlinear Anal. , 18(11):1071–1084, 1992.\n[6] S¨ oren Bartels, Joy Ko, and Andreas Prohl. Numerical analysis o f an explicit approximation scheme\nfor the Landau-Lifshitz-Gilbert equation. Math. Comp. , 77(262):773–788, 2008.\n[7] S¨ oren Bartels and Andreas Prohl. Convergence of an implicit finit e element method for the Landau-\nLifshitz-Gilbert equation. SIAM J. Numer. Anal. , 44(4):1405–1419 (electronic), 2006.\n[8] I. Cimr´ ak. Existence, regularity and local uniqueness of the so lutions to the Maxwell–Landau–\nLifshitz system in three dimensions. J. Math. Anal. Appl. , 329:1080–1093, 2007.\n[9] Ivan Cimr´ ak. A survey on the numerics and computations for th e Landau-Lifshitz equation of\nmicromagnetism. Arch. Comput. Methods Eng. , 15(3):277–309, 2008.\n[10] Lawrence C. Evans. Partial differential equations , volume 19 of Graduate Studies in Mathematics .\nAmerican Mathematical Society, Providence, RI, second edition, 2 010.\n[11] M. Feischl and T. Tran. The eddy current–LLG equations: FEM -BEM coupling and a priori error\nestimates. submitted to SIAM J. Numer. Anal. , 2016.\n[12] T. Gilbert. A Lagrangian formulation of the gyromagnetic equat ion of the magnetic field. Phys Rev ,\n100:1243–1255, 1955.\n[13] Bo Ling Guo and Min Chun Hong. The Landau-Lifshitz equation of t he ferromagnetic spin chain\nand harmonic maps. Calc. Var. Partial Differential Equations , 1(3):311–334, 1993.\n[14] L.LandauandE.Lifschitz.Onthetheoryofthedispersionofm agneticpermeabilityinferromagnetic\nbodies.Phys Z Sowjetunion , 8:153–168, 1935.\n[15] Kim-Ngan Le, Marcus Page, Dirk Praetorius, and Thanh Tran. O n a decoupled linear FEM inte-\ngrator for eddy-current-LLG. Appl. Anal. , 94(5):1051–1067, 2015.\n[16] Kim-Ngan Le and Thanh Tran. A convergent finite element appro ximation for the quasi-static\nMaxwell-Landau-Lifshitz-Gilbert equations. Comput. Math. Appl. , 66(8):1389–1402, 2013.\n[17] J.-L. Lions and E. Magenes. Non-homogeneous boundary value problems and applications . Vol.\nII. Springer-Verlag, New York-Heidelberg, 1972. Translated from t he French by P. Kenneth, Die\nGrundlehren der mathematischen Wissenschaften, Band 182.\n[18] William McLean. Strongly elliptic systems and boundary integral equations . Cambridge University\nPress, Cambridge, 2000.\n[19] ChristofMelcher. Global solvability of the Cauchy problem for th e Landau-Lifshitz-Gilbert equation\nin higher dimensions. Indiana Univ. Math. J. , 61(3):1175–1200, 2012.\n[20] Andreas Prohl. Computational micromagnetism . Advances in Numerical Mathematics. B. G. Teub-\nner, Stuttgart, 2001.\n[21] Michael Struwe. Geometric evolution problems. In Nonlinear partial differential equations in dif-\nferential geometry (Park City, UT, 1992) , volume 2 of IAS/Park City Math. Ser. , pages 257–339.\nAmer. Math. Soc., Providence, RI, 1996.\n[22] A. Visintin. On Landau-Lifshitz’ equations for ferromagnetism .Japan J. Appl. Math. , 2(1):69–84,\n1985.\n16"    },    {        "title": "1606.02072v1.The_temperature_dependence_of_FeRh_s_transport_properties.pdf",        "content": "The temperature dependence of FeRh's transport properties\nS. Mankovsky, S. Polesya, K. Chadova and H. Ebert\nDepartment Chemie, Ludwig-Maximilians-Universit at M unchen, 81377 M unchen, Germany\nJ. B. Staunton\nDepartment of Physics, University of Warwick, Coventry, UK\nT. Gruenbaum, M. A. W. Schoen and C. H. Back\nDepartment of Physics, Regensburg University, Regensburg, Germany\nX. Z. Chen, C. Song\nKey Laboratory of Advanced Materials (MOE), School of Materials\nScience and Engineering, Tsinghua University, Beijing 100084, China.\n(Dated: April 3, 2018)\nThe \fnite-temperature transport properties of FeRh compounds are investigated by \frst-principles\nDensity Functional Theory-based calculations. The focus is on the behavior of the longitudinal re-\nsistivity with rising temperature, which exhibits an abrupt decrease at the metamagnetic transition\npoint,T=Tmbetween ferro- and antiferromagnetic phases. A detailed electronic structure inves-\ntigation for T\u00150 K explains this feature and demonstrates the important role of (i) the di\u000berence\nof the electronic structure at the Fermi level between the two magnetically ordered states and (ii)\nthe di\u000berent degree of thermally induced magnetic disorder in the vicinity of Tm, giving di\u000berent\ncontributions to the resistivity. To support these conclusions, we also describe the temperature\ndependence of the spin-orbit induced anomalous Hall resistivity and Gilbert damping parameter.\nFor the various response quantities considered the impact of thermal lattice vibrations and spin \ruc-\ntuations on their temperature dependence is investigated in detail. Comparison with corresponding\nexperimental data \fnds in general a very good agreement.\nPACS numbers: Valid PACS appear here\nFor a long time the ordered equiatomic FeRh alloy has\nattracted much attention owing to its intriguing temper-\nature dependent magnetic and magnetotransport prop-\nerties. The crux of these features of this CsCl-structured\nmaterial is the \frst order transition from an antiferro-\nmagnetic (AFM) to ferromagnetic (FM) state when the\ntemperature is increased above Tm= 320 K [1, 2]. In\nthis context the drop of the electrical resistivity that is\nobserved across the metamagnetic transition is of central\ninterest. Furthermore, if the AFM to FM transition is\ninduced by an applied magnetic \feld, a pronounced mag-\nnetoresistance (MR) e\u000bect is found experimentally with\na measured MR ratio \u001850% at room temperature [2{\n4]. The temperature of the metamagnetic transition as\nwell as the MR ratio can be tuned by addition of small\namounts of impurities [2, 5{8]. These properties make\nFeRh-based materials very attractive for future applica-\ntions in data storage devices. The origin of the large MR\ne\u000bect in FeRh, however, is still under debate. Suzuki et\nal. [9] suggest that, for deposited thin FeRh \flms, the\nmain mechanism stems from the spin-dependent scatter-\ning of conducting electrons on localized magnetic mo-\nments associated with partially occupied electronic d-\nstates [10] at grain boundaries. Kobayashi et al. [11]\nhave also discussed the MR e\u000bect in the bulk ordered\nFeRh system attributing its origin to the modi\fcation of\nthe Fermi surface across the metamagnetic transition. Sofar only one theoretical investigation of the MR e\u000bect in\nFeRh has been carried out on an ab-initio level [12].\nThe present study is based on spin-polarized, electronic\nstructure calculations using the fully relativistic multiple\nscattering KKR (Korringa-Kohn-Rostoker) Green func-\ntion method [13{15]. This approach allowed to calcu-\nlate the transport properties of FeRh at \fnite tempera-\ntures on the basis of the linear response formalism using\nthe Kubo-St\u0014 reda expression for the conductivity tensor\n[16, 17]\n\u001b\u0016\u0017=~\n4\u0019N\nTrace\n^j\u0016(G+(EF)\u0000G\u0000(EF))^j\u0017G\u0000(EF)\n\u0000^j\u0016G+(EF)^j\u0017(G+(EF)\u0000G\u0000(EF))\u000b\nc;(1)\nwhere \n is the volume of the unit cell, Nis the num-\nber of sites, ^j\u0016is the relativistic current operator and\nG\u0006(EF) are the electronic retarded and advanced Green\nfunctions, respectively, calculated at the Fermi energy\nEF. In Eq. (1) the orbital current term has been omit-\nted as it only provides small corrections to the prevailing\ncontribution arising from the \frst term in the case of a\ncubic metallic system [18{20].\nHere we focus on the \fnite temperature transport\nproperties of FeRh. In order to take into account\nelectron-phonon and electron-magnon scattering e\u000bects\nin the calculations, the so-called alloy analogy modelarXiv:1606.02072v1  [cond-mat.mtrl-sci]  7 Jun 20162\n[21, 22] is used. Within this approach the tempera-\nture induced spin (local moment) and lattice excitations\nare treated as localized, slowly varying degrees of free-\ndom with temperature dependent amplitudes. Using the\nadiabatic approximation in the calculations of transport\nproperties, and accounting for the random character of\nthe motions, the evaluation of the thermal average over\nthe spin and lattice excitations in Eq. (1) is reduced to\na calculation of the con\fgurational average over the lo-\ncal lattice distortions and magnetic moment orientations,\nh:::ic, using the recently reported approach [21, 22] which\nis based on the coherent potential approximation (CPA)\nalloy theory [23{25].\nTo account for the e\u000bect of spin \ructuations, which\nwe describe in a similar way as is done within the dis-\nordered local moment (DLM) theory [26], the angular\ndistribution of thermal spin moment \ructuations is cal-\nculated using the results of Monte Carlo (MC) simula-\ntions. These are based on ab-initio exchange coupling\nparameters and reproduce the \fnite temperature mag-\nnetic properties for the AFM and FM state in both the\nlow- (T < Tm) and high-temperature ( T > Tm) regions\nvery well [27]. Figure 1(a), inset, shows the temperature\ndependent magnetization, M(T), for one of the two Fe\nsublattices aligned antiparallel/parallel to each other in\nthe AFM/FM state, calculated across the temperature\nregion covering both AFM and FM states of the system.\nThe di\u000bering behavior of the magnetic order M(T) in the\ntwo phases has important consequences for the transport\nproperties as discussed below.\nFigure 1(a) shows the calculated electrical resistiv-\nity as a function of temperature, \u001axx(T), accounting\nfor the e\u000bects of electron scattering from thermal spin\nand lattice excitations, and compares it with experi-\nmental data. There is clearly a rather good theory-\nexperiment agreement especially concerning the di\u000ber-\nence\u001aAFM\nxx (Tm)\u0000\u001aFM\nxx(Tm) at the AFM/FM transition,\nTm= 320K. The AFM state's resistivity increases more\nsteeply with temperature when compared to that of the\nFM state, that has also been calculated for temperatures\nbelow the metamagnetic transition temperature (dotted\nline). Note that the experimental measurements have\nbeen performed for a sample with 1% intermixing be-\ntween the Rh and Fe sublattices leading to a \fnite resid-\nual resistivity at T!0 K, and as a consequence there is\na shift of the experimental \u001axx(T) curve with respect to\nthe theoretical one [28].\nWe can separate out the contributions of spin \ructua-\ntions and lattice vibrations to the electrical resistivities,\n\u001afluc\nxx(T) and\u001avib\nxx(T), respectively. These two compo-\nnents have been calculated for \fnite temperatures keep-\ning the atomic positions undistorted to \fnd \u001afluc\nxx(T) and\n\fxed collinear orientations of all magnetic moments to\n\fnd\u001avib\nxx(T), respectively. The results for the AFM and\nFM states are shown in Fig. 1(b), where again the FM\n(AFM) state has also been considered below (above) the\n(a)\n(b)\nFIG. 1. (a) Calculated longitudinal resistivity (closed cir-\ncles - AFM state, open circles - FM state) in comparison\nwith experiment [2]. The dashed line represents the results\nfor Fe 0:49Rh0:51, while the dash-dotted line gives results for\n(Fe-Ni) 0:49Rh0:51with the Ni concentration x= 0:05 to sta-\nbilize the FM state at low temperature). The inset represents\nthe relative magnetization of a Fe sub-lattice as a function\nof temperature obtained from MC simulations. (b) electrical\nresistivity calculated for the AFM (closed symbols) and FM\n(open symbols) states accounting for all thermal scattering ef-\nfects (circles) as well as e\u000bects of lattice vibrations (diamond)\nand spin \ructuations (squares) separately. The inset shows\nthe temperature dependent longitudinal conductivity for the\nAFM and FM states due to lattice vibrations only.\ntransition temperature Tm. For both magnetic states the\nlocal moment \ructuations have a dominant impact on\nthe resistivity. One can also see that both components,\n\u001afluc\nxx(T) and\u001avib\nxx(T), in the AFM state have a steeper\nincrease with temperature than those of the FM state.\nThe origin of this behavior can be clari\fed by refer-\nring to Mott's model [29] with its distinction between\ndelocalized sp-electrons, which primarily determine the\ntransport properties owing to their high mobility, and\nthe more localized d-electrons. Accordingly, the conduc-\ntivity should depend essentially on (see. e.g. [30]): (i)\nthe carrier (essentially sp-character) concentration nand\n(ii) the relaxation time \u001c\u0018[V2\nscattn(EF)]\u00001, whereVscatt\nis the average scattering potential and n(EF) the total\ndensity of states at the Fermi level. This model has been\nused, in particular, for qualitative discussions of the ori-\ngin of the GMR e\u000bect in heterostructures consisting of3\nmagnetic layers separated by non-magnetic spacers. In\nthis case the GMR e\u000bect can be attributed to the spin\ndependent scattering of conduction electrons which leads\nto a dependence of the resistivities on the relative ori-\nentation of magnetic layers, parallel or antiparallel, as-\nsuming the electronic structure of non-magnetic spacer\nto be unchanged. These arguments, however, cannot be\nstraightforwardly applied to CsCl-structured FeRh, even\nthough it can be pictured as a layered system with one\natom thick layers, since the electronic structure of FeRh\nshows strong modi\fcations across the AFM-FM transi-\ntion as discussed, for example, by Kobayashi et al. [11]\nto explain the large MR e\u000bect in FeRh.\n(a) (b)\n(c) (d)\nFIG. 2. Comparison of the temperature dependent densities\nof states (DOS) for the FM and AFM states of FeRh for\nT= 40\u0000 \u0000400 K : (a) Fe s-DOS, (b) Fe p-DOS, (c) Rh\ns-DOS, and (d) Rh p-DOS.\nWe use the calculated density of states at the Fermi\nlevel as a measure of the concentration of the conducting\nelectrons. The change of the carriers concentration at the\nAFM-FM transition can therefore be seen from the mod-\ni\fcation of the sp-DOS at the Fermi level. The element-\nprojected spin-resolved sp-DOS (nsp(E)), calculated for\nboth FM and AFM states at di\u000berent temperatures, is\nshown in Fig. 2. At low temperature, for both Fe and\nRh sublattices, the sp-DOS atEFis higher in the FM\nthan in the AFM state, nFM\nsp(EF)> nAFM\nsp (EF). This\ngives a \frst hint concerning the origin of the large dif-\nference between the FM- and AFM-conductivities in the\nlow temperature limit (see inset for \u001bvib\nxxin Fig. 1(b)).\nIn this case the relaxation time \u001cis still long owing to\nthe low level of both lattice vibrations and spin \ructu-\nations which determines the scattering potential Vscatt.\nFor both magnetic states the decrease of the conductiv-ity with rising temperature is caused by the increase of\nscattering processes and consequent decrease of the re-\nlaxation time. At the same time, the conductivity di\u000ber-\nence, \u0001\u001b(T) =\u001bvib;FM\nxx (T)\u0000\u001bvib;AFM\nxx (T), reduces with\nincrease in temperature. This e\u000bect can partially be at-\ntributed to the temperature dependent changes of the\nelectronic structure (disorder smearing of the electronic\nstates) re\rected by changes in the density of states at\nthe Fermi level [28] (see Fig. 2). Despite this, up to the\ntransition temperature, T=Tm, the di\u000berence \u0001 \u001b(T) is\nrather pronounced leading to a signi\fcant change of the\nresistivity at T=Tm.\n0.0 0.1 0.2 0.3 0.4 0.50.00.10.20.30.40.5\n(a)\n0.0 0.1 0.2 0.3 0.4 0.50.00.10.20.30.40.5\n(b)\n0.0 0.1 0.2 0.3 0.4 0.50.00.10.20.30.40.5\n(c)\nFIG. 3. (a) Bloch spectral function of FeRh calculated for\nthe AFM state at T= 300 K (a) and for the FM state re-\nsolved into majority spin (b) and minority spin (c) electron\ncomponents, calculated for T= 320 K. The \fnite width of\nthis features determine the electronic mean free paths.4\nOne has to stress that in calculating the contribution\nof spin moment \ructuations to the resistivity, the dif-\nferent temperature dependent behavior of the magnetic\norder in the FM and AFM states must be taken into ac-\ncount. This means, that at the critical point, T=Tm,\nthe smaller sublattice magnetization in the AFM state\ndescribes a more pronounced magnetic disorder when\ncompared to the FM state which leads to both a smaller\nrelaxation time and shorter mean free path. The result\nis a higher resistivity in the AFM state.\nThe di\u000berent mean free path lengths in the FM and\nAFM states at a given temperature can be analyzed using\nthe Bloch spectral function (BSF), AB(~k;E) [15], calcu-\nlated forE=EF, since the electronic states at the Fermi\nlevel give the contribution to the electrical conductivity.\nFor a system with thermally induced spin \ructuations\nand lattice displacements the BSF has features with \f-\nnite width from which the mean free path length of the\nelectrons can be inferred. Fig. 3 shows an intensity con-\ntour plot for the BSF of FeRh averaged over local moment\ncon\fgurations appropriate for the FM and AFM states\njust above and just below the FM-AFM transition respec-\ntively. Fig. 3(a) shows the AFM Bloch spectral function\nwhereas Figs. 3(b) and (c) show the sharper features of\nthe spin-polarized BSF of the FM state especially for\nthe minority spin states. This implies a longer electronic\nmean free path in the FM state in comparison to that\nin the AFM state which is consistent with the drop in\nresistivity.\nIn particular concerning technical applications of\nFeRh, it is interesting to study further temperature de-\npendent response properties. In Fig. 4(a) we show our\ncalculations of the total anomalous Hall resistivity for\nFeRh in the FM state, represented by the o\u000b-diagonal\nterm\u001axyof the resistivity tensor and compare it with\nexperimental data [11]. As the FM state is unstable in\npure FeRh at low temperatures, the measurements were\nperformed for (Fe 0:965Ni0:035)Rh, for which the FM state\nhas been stabilized by Ni doping. The calculations have\nbeen performed both, for the pure FeRh compound as\nwell as for FeRh with 5% Ni doping, (Fe 0:95Ni0:05)Rh,\nwhich theory \fnds to be ferromagnetically ordered down\ntoT=0 K. As can be seen the magnitude of \u001axy(T)\nincreases in a more pronounced way for the undoped\nsystem. Nevertheless, both results are in a rather good\nagreement with experiment.\nIn addition to the temperature dependent transport\nproperties the inclusion of relativistic e\u000bects into the ab-\ninitio theory enables us to present results for the Gilbert\ndamping, which plays a crucial role for spin dynamics.\nWe have calculated this quantity taking into account all\ntemperature induced e\u000bects, i.e. spin \ructuations and\nlattice vibrations [32, 33]. As one can see in Fig. 4(b), the\ncalculated results are in rather good agreement with the\nexperimental value (shown by diamond) \u000b= 0:0012 ob-\ntained for a thick \flm at T= 420 K [31] as well as new ex-\n(a)\n(b)\nFIG. 4. (a) The temperature dependence of the anomalous\nHall resistivity for the FM state of (Fe 0:95Ni0:05)Rh in com-\nparison with experimental data [11]; (b) Gilbert damping pa-\nrameter as a function of temperature: theory accounting for\nall thermal contributions (squares) in comparison with the\nexperimental results for thick-\flm system (50 nm) [31] (open\ndiamond) and for FeRh thin \flm deposited on MgO(001) sur-\nface (up- and down-triangles). Up- and down-triangles rep-\nresent data for a heating and cooling cycles, respectively (for\ndetails see supplementary materials). The inset represents the\nresults for the individual sources for the Gilbert damping, i.e.,\nlattice vibrations (circles) and spin \ructuations (diamonds).\nThe total\u000bvalues calculated for FeRh crystal without (c)\nand with tetragonal (t) distortions ( c=a= 1:016) are shown\nby open and closed squares, respectively.\nperimental data for thin \flms [15]. The separate contri-\nbutions to the Gilbert damping due to spin \ructuations\nand lattice vibrations are presented in the inset to Fig.\n4(b) for a given temperature window again arti\fcially ex-\ntended to low temperatures. These results allow to iden-\ntify the leading role of lattice vibrations (circles in the\ninset to Fig. 4(b)) at high temperature region where the\nelectron spin-\rip interband transitions are most respon-\nsible for dissipation due to the magnetization dynamics.\nIn the low-temperature region, where the T-dependence\nof\u000bis determined by intraband spin-conserving scatter-\ning events, it stems dominantly from electron scattering\ndue to thermally induced spin-\ructuations (diamonds in\nthe inset to Fig. 4(b)).\nThe experimental data shown in Fig. 4(b)) by trian-\ngles represent results for rather thin \flms ( d= 25 nm)5\ndeposited on top of a MgO(001) substrate [15]. The FeRh\nunit cell with a lattice constantp\n2 times smaller than\nthat of MgO, is rotated around zaxis by 45owith respect\nto the MgO cell. Because of this, a compressive strain in\nthe FeRh \flm occurs. As it follows from the experimen-\ntal data [34], this implies a tetragonal distortion of the\nFM FeRh unit cell with c=a= 1:016. Results of corre-\nsponding calculations for \u000bare given in the inset of Fig.\n4(b) by full squares, demonstrating a rather weak e\u000bect\nof this distortion. The smaller value of \u000bcompared to\nexperiment, has therfore to be attributed to the use of\nbulk geometry instead of the experimental \flm geometry\nwith a corresponding impact on the damping parameter.\nIn summary, we have presented ab-initio calculations\nfor the \fnite temperature transport properties of the\nFeRh compound. A steep increase of the electric resis-\ntivity has been obtained for the AFM state leading to a\npronounced drop of resistivity at the AFM to FM transi-\ntion temperature. This e\u000bect can be attributed partially\nto the di\u000berence of the electronic structure of FeRh in the\nFM and AFM states, as well as to a faster increase of the\namplitude of spin \ructuations caused by temperature in\nthe AFM state. Further calculated temperature depen-\ndent response properties such as the AHE resistivity and\nthe Gilbert damping parameter for the FM system show\nalso good agreement with experimental data. This gives\nadditional con\fdence in the model used to account for\nthermal lattice vibrations and spin \ructuations.\nACKNOWLEDGEMENTS\nFinancial support by the DFG via SFB 689\n(Spinph anomene in reduzierten Dimensionen) and from\nthe EPSRC (UK) (Grant No. EP/J006750/1) is grate-\nfully acknowledged.\n[1] J. S. Kouvel and C. C. Hartelius, J. Appl. Phys. 33\n(1962).\n[2] N. Baranov and E. Barabanova, Journal of Alloys and\nCompounds 219, 139 (1995), eleventh international con-\nference on solid compounds of transition elements.\n[3] P. A. Algarabel, M. R. Ibarra, C. Marquina, A. del Moral,\nJ. Galibert, M. Iqbal, and S. Askenazy, Applied Physics\nLetters 66, 3061 (1995).\n[4] M. A. de Vries, M. Loving, A. P. Mihai, L. H. Lewis,\nD. Heiman, and C. H. Marrows, New Journal of Physics\n15, 013008 (2013).\n[5] P. H. L. Walter, Journal of Applied Physics 35, 938\n(1964).\n[6] J. S. Kouvel, Journal of Applied Physics 37, 1257 (1966).\n[7] R. Barua, F. Jimnez-Villacorta, and L. H. Lewis, Ap-\nplied Physics Letters 103, 102407 (2013).[8] W. Lu, N. T. Nam, and T. Suzuki, Journal of Applied\nPhysics 105, 07A904 (2009).\n[9] I. Suzuki, T. Naito, M. Itoh, T. Sato, and T. Taniyama,\nJournal of Applied Physics 109, 07C717 (2011).\n[10] M. Kn ulle, D. Ahlers, and G. Sch utz, Solid State Com-\nmun.94, 267 (1995).\n[11] Y. Kobayashi, K. Muta, and K. Asai, Journal of Physics:\nCondensed Matter 13, 3335 (2001).\n[12] J. Kudrnovsk\u0013 y, V. Drchal, and I. Turek, Phys. Rev. B\n91, 014435 (2015).\n[13] H. Ebert et al., The Munich SPR-KKR package , version\n6.3,\nH. Ebert et al.\nhttp://olymp.cup.uni-muenchen.de/ak/ebert/SPRKKR\n(2012).\n[14] H. Ebert, D. K odderitzsch, and J. Min\u0013 ar, Rep. Prog.\nPhys. 74, 096501 (2011).\n[15] see supplementary materials, .\n[16] P. St\u0014 reda, J. Phys. C: Solid State Phys. 15, L717 (1982).\n[17] S. Lowitzer, D. K odderitzsch, and H. Ebert, Phys. Rev.\nB82, 140402(R) (2010).\n[18] T. Naito, D. S. Hirashima, and H. Kontani, Phys. Rev.\nB81, 195111 (2010).\n[19] S. Lowitzer, M. Gradhand, D. K odderitzsch, D. V. Fe-\ndorov, I. Mertig, and H. Ebert, Phys. Rev. Lett. 106,\n056601 (2011).\n[20] I. Turek, J. Kudrnovsk\u0013 y, and V. Drchal, Phys. Rev. B\n86, 014405 (2012).\n[21] H. Ebert, S. Mankovsky, D. K odderitzsch, and\nP. J. Kelly, Phys. Rev. Lett. 107, 066603 (2011),\nhttp://arxiv.org/abs/1102.4551v1.\n[22] H. Ebert, S. Mankovsky, K. Chadova, S. Polesya,\nJ. Min\u0013 ar, and D. K odderitzsch, Phys. Rev. B 91, 165132\n(2015).\n[23] B. Velick\u0013 y, Phys. Rev. 184, 614 (1969).\n[24] W. H. Butler, Phys. Rev. B 31, 3260 (1985).\n[25] I. Turek, J. Kudrnovsk\u0013 y, V. Drchal, L. Szunyogh, and\nP. Weinberger, Phys. Rev. B 65, 125101 (2002).\n[26] B. L. Gyor\u000by, A. J. Pindor, J. Staunton, G. M. Stocks,\nand H. Winter, J. Phys. F: Met. Phys. 15, 1337 (1985).\n[27] S. Polesya, S. Mankovsky, D. K odderitzsch, J. Min\u0013 ar,\nand H. Ebert, Phys. Rev. B 93, 024423 (2016).\n[28] J. B. Staunton, M. Banerjee, dos Santos Dias, A. Deak,\nand L. Szunyogh, Phys. Rev. B 89, 054427 (2014).\n[29] N. F. Mott, Adv. Phys. 13, 325 (1964).\n[30] E. Y. Tsymbal, D. G. Pettifor, and S. Maekawa, \\Gi-\nant magnetoresistance: Theory,\" in Handbook of Spin\nTransport and Magnetism , edited by E. Y. Tsymbal and\nI. Zuti\u0013 c (Taylor and Francis Group, New York, 2012).\n[31] E. Mancini, F. Pressacco, M. Haertinger, E. E. Fullerton,\nT. Suzuki, G. Woltersdorf, and C. H. Back, Journal of\nPhysics D: Applied Physics 46, 245302 (2013).\n[32] S. Mankovsky, D. K odderitzsch, G. Woltersdorf, and\nH. Ebert, Phys. Rev. B 87, 014430 (2013).\n[33] H. Ebert, S. Mankovsky, K. Chadova, S. Polesya,\nJ. Min\u0013 ar, and D. K odderitzsch, Phys. Rev. B 91, 165132\n(2015).\n[34] C. Bordel, J. Juraszek, D. W. Cooke, C. Baldasseroni,\nS. Mankovsky, J. Min\u0013 ar, H. Ebert, S. Moyerman, E. E.\nFullerton, and F. Hellman, Phys. Rev. Lett. 109, 117201\n(2012)."    },    {        "title": "1606.06610v1.Torsion_Effects_and_LLG_Equation.pdf",        "content": "arXiv:1606.06610v1  [hep-th]  21 Jun 2016Torsion Effects and LLG Equation.\nCristine N. Ferreira†a, Cresus F. L. Godinho‡b, J. A. Helay¨ el Neto∗c\n†N´ ucleo de Estudos em F´ ısica, Instituto Federal de Educa¸ c ˜ ao, Ciˆ encia e Tecnologia\nFluminense, Rua Dr. Siqueira 273, Campos dos Goytacazes, 28 030-130 RJ, Brazil\n‡Grupo de F´ ısica Te´ orica, Departamento de F´ ısica, Univer sidade Federal Rural do Rio de\nJaneiro, BR 465-07, 23890-971, Serop´ edica, RJ, Brazil\n∗Centro Brasileiro de Pesquisas F´ ısicas (CBPF), Rua Dr. Xav ier Sigaud 150, Urca,\n22290-180, Rio de Janeiro, Brazil\nAbstract\nBased on the non-relativistic regime of the Dirac equation coupled to a torsion\npseudo-vector, we study the dynamics of magnetization and how it is affected by the\npresence of torsion. We consider that torsion interacting terms in Dirac equation\nappear in two ways one of these is thhrough the covariant derivativ e considering the\nspin connection and gauge magnetic field and the other is through a n on-minimal\nspin torsion coupling. We show within this framework, that it is possible to obtain\nthe most general Landau, Lifshitz and Gilbert (LLG) equation includ ing the torsion\neffects, where we refer to torsion as a geometric field playing an impo rtant role in the\nspin coupling process. We show that the torsion terms can give us tw o important\nlandscapes in the magnetization dynamics: one of them related with d amping and the\nother related with the screw dislocation that give us a global effect lik e a helix damping\nsharped. These terms are responsible for changes in the magnetiz ation precession\ndynamics.\nacrisnfer@iff.edu.br ,bcrgodinho@ufrrj.br ,chelayel@cbpf.br\n11 Introduction\nThe discovery of the graphene-like systems and topological insulators systems introduced a\nnew dynamic in the applications of the framework of the high e nergy physics in low energy\nsystems in special condensed matter systems. The fact that t hese systems can be described\nby Dirac equations give us new possibilities for theoretica l and experimental applications.\nIn this direction there are some effects in condensed matter sy stems still without a full\ndescription as the magnetizable systems that we described i n this work. In this sense the\nconstructionofthetheoreticalframeworksthatcan,insom elimit, beobtainedinlowenergy\nsystems is the crucial importance to understand the invaria nces and interactions in certain\nlimits. One of the important effects that we can study is relate d with the spin systems.\nSo, in this work we deal with the new framework to study the spi n systems considering\nthe Dirac equation in non-relativistic limit with torsion i nteraction[1]. Spin systems are\ngenerally connected with magnetic systems. It is well known that spin angular momentum\nisanintrinsicpropertyofquantumsystems. Whenamagnetic fieldisapplied, eachmaterial\npresents some level of magnetization, and Quantum Mechanic s says that magnetization is\nrelated to the expectation value of the spin angular momentu m operator. In the case\nof ferromagnetic materials, they can have a large magnetiza tion even under the action\nof a small magnetic field and the magnetization process is alw ays followed by hysteresis,\nand the magnetization is uniform and lined up with the magnet ic field, usually these\nmaterials exhibit a strong ordering process that results in a parallel line up the spins[2]. In\nmaterials graphene type we also can generate magnetic momen t. In this form it is possible\nto study the transport phenomena [3]. Overlapping between e lectronic wave functions are\ninteractions well understood, again thanks to Quantum Mech anics, however thereare other\nkinds of interactions occurring such as magnetocrystallin e anisotropy, connected with the\ntemperature dependence[4] and demagnetization fields [5], acting in low range. In such\nsystems if we only consider the precession we will not reach t he right limit. Certainly, the\nprecession equation has to include a damping term providing the magnetization alignment\nwith the magnetic field after a finite time [6]. In order to simu late these phenomena,\nseveral physical models have been presented. However, the L andau-Lifshitz model is still\nthe one widely used in the description of the dynamics of ferr omagnetic media. In their\npioneering work [7] in 1935, Landau and Lifshitz proposed a n ew theory based on the\nfollowing dynamical equation:\n∂tM=/vectorHeff×/vectorM+α\nM2s/vectorM×(/vectorM×/vectorHeff), (1)\nwhere/vectorHeffdenote an effective magnetic field, with the gyromagnetic rati o absorbed, inter-\nacting with the magnetization M=|/vectorM|. The first term is the precession of the magnetiza-\ntion vector around the direction of the effective magnetic fiel d and the second one describes\na damping of the dynamics. With this theory we are able to comp ute the thickness of walls\nbetween magnetic domains, and also understand the domain fo rmation in ferromagnetic\n2materials. This theory, which now goes under the name of micr omagnetics, has been in-\nstrumental in the understanding and development of magneti c memories. Landau and\nLifshitz considered the Gibbs energy G of a magnetic materia l to be composed of three\nterms: exchange, anisotropy and Zeeman energies (due to the external magnetic field), and\npostulated that the observed magnetization per unit volume M field would correspond to\na local minimum of the Gibbs energy. Later researchers added other terms to G such as\nmagnetoelastic energy and demagnetization energy. They al so derived the Landau Lifshitz\n(LL) equation using only physical arguments and not using th e calculus of variations. In\nsubsequent work, Gilbert [8] realized a more convincing for m for the damping term, based\non a variational approach, and the new combined form was then called Landau Lifshitz\nGilbert (LLG) equation, today it is a fundamental dynamic sy stem in applied magnetism.\nNowadays, the scientific and technological advances provid e a wide spectrum of ma-\nnipulations to the spin degrees of freedom. The complete for mulation for magnetization\ndynamics also include the excitation of magnons and their in teraction with other degrees\nof freedom, that remains as a challenge for modern theory of m agnetism [9]. These amaz-\ning and reliable kinds of procedures are propelling spintro nics as a consolidated sub-area\nof Condensed Matter Physics [10]. Since the experimental ad vances are increasingly pro-\nviding high-precision data, many theoretical works are bei ng presented [11, 14, 12, 13]\nand including strange materials, as the topological insula tors, connected with the mag-\nnetocondutivity [15] and graphene like structures [16] for a deeper understanding of the\nphenomenon including the spin polarization super currents for spintronics [17], holographic\nunderstanding of spin transport phenomena [18] and non-rel ativistic background [19].\nThe torsion field appears as one of the most natural extension s of General Relativity\nalong with the metric tensor, which couples to the energy-mo mentum distribution, inspects\nthe details of the spin density tensor. Actually, in General Relativity, fermions naturally\ncouple to torsion by means of their spin.\nIn this work we consider that the torsion interacts with the m atter in two types one of\nthese is present in covariant derivative that contains the s pin connection related with the\nChristoffell symbol given by the metric of the curve space time and the contorsion given\nby the torsion that have two antisymmetric index. The other c ontribution is given by the\nnon minimal spin torsion coupling that is important to the co nsistence of the theory. It is\npossible to study the non relativistic approach to the torsi on in connection with the spin\nparticles [20], in this work we only consider the torsion con tribution considering the plane\nspace-time.\nOur work is organized as follows, in Section 2, we analyse the torsion coupling in\nrelativistic limit, in Section 3 the modified version of Paul i equation is presented. Our\napproach starts with a field theoretical action where a Dirac fermion is non-minimally\ncoupled in the presence of a torsion term, a low relativistic approximation is considered\nand the equivalent Pauli equation is then obtained. We deriv e a very similar expression to\nthe Landau-Lifshitz-Gilbert (LLG) equation from our Pauli equation with torsion. Under\nspecific conditions for the magnetic moment, we show that the LLG equation can be\n3established with damping and dislocations terms.\n2 TheRelativistic andNon- Relativistic Discussions for Sp in\nCoupling with Torsion\nIn this Section, let us understand the way to describe the spi n interaction by taking into\naccount the torsion coupling. In our framework there are two terms in Dirac action, one\nof these is connected with spin current effect from the spin con nection, the other is a non-\nminimal spin torsion coupling whose effects are the subject of this work. Dirac’s equation\nis relativistic and we should justify why we use it in our mode l. Dispersion relations in\nCondensed Matter Physics (CMP) linear in the velocity appea r in a wide class of models\nand one adopts the framework of Dirac’s equation to approach them. However, the speed\nof light, c, is suitably replaced by the Fermi velocity, vF. Here, this is not what we are\ndoing. We actually start off from the Dirac’s equation and we t ake, to match with effects\nof CMP, the non-relativistic regime, for the electron moves with velocities v≤c\n300. So,\ncontrary to an analogue model where we describe the phenomen a by a sort of relativity\nwith c replaced by vF, we here consider that the non-relativistic electrons of ou r system is\na remnant of a more fundamental relativistic world. The non- relativistic limit is also more\ncomplete because it brings effects that do not directly appear in Galilean Physics. This is\nwhy we have taken the viewpoint of associating our physics to the Dirac’s equation.\n2.1 The Dirac Model for Torsion and the Spin Current Interpre tation\nIn this sub-section, we consider the microscopic discussio n that gives the explicit form of\nthe spin current in function of the gauge potential and torsi on coupling. The scenario we\nare setting up is justified by the following chain of argument s: (i) We are interested in\nspin effects. We assume that there is a space-time structures ( torsion) whose coupling with\nthe matter spin becomes relevant. But, we are actually inter ested in the possible non-\nrelativistic effects stemming from this coupling, which is mi nimal and taken into account\nin the covariant derivative though the spin connection. (ii ) The other point we consider\nis that, amongst the three irreducible torsion components, its pseudo-vector piece is the\nonly one that couples to the charged leptons. Then, with this results in mind, we realize\nthat the electron spin density may non-minimally couple, in a Pauli- like interaction, to\nthe field-strength of the torsion pseudo-vector degree of fr eedom.\nSo, our scenario is based on the relevant role space-time tor sion, here modeled by a\npseudo-vector, may place in the non-relativistic electron s of spin systems in CMP. The spin\ncurrent that we talk about is the spin magnetic moment and in g eneral is not conserved\nalone. Thequantity that is conserved is the total magnetic m oment that is the composition\nbetween both /vectorJ=/vectorJS+/vectorJL. In form that ∂µJµ= 0 where the spin current can be defined\nin related to the three component spin current as Jµ\nS=ǫµab\nρJρ\naband/vectorJLis the spin orbit\n4coupling . In analogy with the charge current, defined by the d erivative of the action in\nrelation to the gauge field Aµwe used the definition where the spin current is the derivativ e\nof the action in relation with the spin connection. We consid er here the spin current as\nthe derivative of the action in relation to the spin connecti onωab\nµthen the spin current is\ngiven by\nJµ\nab=δS\nδωabµ, (2)\nwhereωab\nµis\nωab\nµ=ea\nν∇µeνb+ea\nλΓλ\nµνeνb+ea\nλKλ\nµνeνb, (3)\nandKλ\nµνis the contorsion given by\nKλ\nµν=−1\n2(Tλ\nµ ν+Tλ\nν µ−Tλ\nµν), (4)\nwhereTλ\nµνis the torsion. In this work we consider two terms for torsion , on of these is the\ntotally anti symmetric tensor, that respecting the duality relation given by Tµνλ=ǫµνλρSρ\nwhereSρis the pseudo-vector part of torsion.1The other term that we consider is the 2-\nform tensor TµνwhereTµν=∂µSν−∂νSµthis term is analog to the field strength of the\nelectromagnetic gauge potential Aµthat in our case is changed to pseudo-vector Sµ. The\ninvariant fermionic action that contained these contribut ions for torsion is given by\nS=/integraldisplay\nd4xi¯ψ(γµDµ+λTµνΣµν+m)ψ, (5)\nwhere the covariant derivative is Dµ=Dµ−iηωab\nµΣabthat contain the covariant gauge\nderivativeDµ=∂µ−ieAµand the spin connection covariant derivative.\nWe consider the flat space-time where the only contribution f or the spin connection is\nthe contortion. In this form we have a spin current given by\nJµ\nab=1\n2¯ψγµΣabψ. (6)\nWe consider the ansatz where ωab\nµcontain the total antisymmetric part of the contorsion\ngiven by\nωκλ\nµ=µǫκλρ\nµSρ. (7)\nWe used the splitting γµΣκλ=ǫµ\nκλργργ5+δµ\nκγλ−δµ\nλγκThat give us the current in the\nform\nJµ\nαλ=δS\nδΓabµ=1\n2ǫµ\nαλρ¯ψγργ5ψ=Ji\njk+J0\nij (8)\n1Considering space times with torsion Tα\nβγ, the afine connection is not symmetric, Tα\nβγ= Γα\nβγ−Γα\nγβ, and\nwe can split it into three irreducible components, where one of them is the pseudo-trace Sκ=1\n6ǫαβγκTαβγ.\n5where the currents are\nJi\njk=1\n2ǫi\njk¯ψγ0γ5ψ; (9)\nJ0\nij=1\n2ǫijk¯ψγkγ5ψ, (10)\nthen the current part of the action coming from the covariant derivative is given by\nScurr=−η/integraldisplay\nd4xγµγ5Sµ=η/integraldisplay\nd4x/vectorS·/vectorJ. (11)\nThe action (5) considers the temporal component of the torsi on pseudo-vector S0= 0,\nwe have can be written as\nS=/integraldisplay\nd4xi¯ψ/parenleftBig\nγµ∂µ+igγµAµ+iηγµγ5Sµ+λTµνΣµν+m/parenrightBig\nψ, (12)\nOur sort of gravity background does not exhibit metric fluctu ations. The space-time is\ntaken to be flat, and we propose a scenario such that the type of gravitational background\nis parametrized by the torsion pseudo-trace Sµ, whose origin may be traced back to one\ngeometrical defect.\n2.2 Dirac equation in presence of torsion\nNow, we discuss the Dirac equation given by eq. (12). From the action above, taking the\nvariation with respect to ( δS/δ¯ψ), a modified Dirac’s equation reads as below:\n[iγµ∂µ−ηγµγ5Sµ−eγµAµ+λΣµν∂µSν+m]ψ= 0. (13)\nFor a vanishing λ−parameter, the equation (13) has been carefully studied in [ 21, 22, 23,\n24, 26].\nThe generation, manipulation, and detection of a spin curre nt, as well as the flow of\nelectron spins, are the main challenges in the field of spintr onics, which involves the study\nof active control and manipulation of the spin degree of free dom in solid-state systems,\n[27, 10, 28]. A spin current interacts with magnetization by exchanging the spin-angular\nmomentum, enabling the direct manipulation of magnetizati on without using magnetic\nfields [29, 30]. The interaction between spin currents and ma gnetization provides also\na method for spin current generation from magnetization pre cession, which is the spin\npumping [31, 32]. We showed in last Sections that there are tw o type of deformations\ninduced by the torsion. Both of these can be generating a spin current.\nAfter a suitable separation of components ( µ= 0,1,2,3), the equation of motion can\nbe written as,\ni∂tψ=iαi∂iψ+ηγ5S0ψ−ηαiγ5Siψ+eA0ψ+\n−eαiAiψ−iλ\n4ǫijkβγ5αk∂iSjψ+βmψ. (14)\n6Definingagauge-invariant momentum, πj=i∂j−eAj, andusingthatΣ ij=−i\n4ǫijkγ5αk\nwithγ0=β, the effective Hamiltonian takes the form,\nH=αkπk+ηγ5S0−ηαkγ5Sk+eA0+\n−iλ\n4ǫijkβγ5αk∂iSj+βm. (15)\nwhere we have the matricial definitions:\nαi=/parenleftbigg0σi\nσi0/parenrightbigg\n,γ5=/parenleftbigg0 1\n1 0/parenrightbigg\n,β=/parenleftbigg1 0\n0−1/parenrightbigg\n, (16)\nIn the Heisenberg picture, the position, /vector x, and momentum, /vector π, operators obey two\ndifferent kinds of relations; we consider the torsion as a func tion of position only, S=S(/vector x),\nso that\n˙/vector x=/vector α\n˙/vector π=e(/vector α×/vectorB)+e/vectorE+ηγ5∂\n∂x(/vector α·/vectorS)ˆx. (17)\nOne reproduces the usual relation for ˙/vector x, while the equation for ˙/vector πpresents a new term\napparently giving some tiny correction to the Lorentz force .\nHowever, if we consider the torsion in a broader context, now as a momentum- and\nposition-dependent background field, S=S(/vector x,/vectork), we have to deal with the following\npicture,\n˙/vector x=/vector α−ηγ5∂\n∂k(/vector α·/vectorS)ˆk\n˙/vector π=e(/vector α×/vectorB)+e/vectorE+ηγ5∂\n∂x(/vector α·/vectorS)ˆx+\n+eηγ5∂A\n∂x∂\n∂k(/vector α·/vectorS)ˆk. (18)\nThe two sets of dynamical equations above are clearly showin g us the small corrections\ninducedbythetorsionterm. Nevertheless, wearestill here intherelativistic domain, andit\nis necessary change this framework for a better understandi ng of the SHE phenomenology.\nFor this reason, in next Section we are going to approach the s ystem by going over into its\nnon-relativistic regimen\n2.3 Non-Relativistic Approach with torsion\nIn this sub-section, we consider the Dirac equation in its no n-relativistic limit. One im-\nportant requirement for the Dirac equation is that it reprod uces what we know from non-\nrelativistic quantum mechanics. We can show that, in the non -relativistic limit, two com-\nponents of the Dirac spinor are large and two are quite small. To make contact with the\n7non-relativistic description , we go back to the equations w ritten in terms of ϕandχof\nthe four component spinor ψ=eimt√\n2m/parenleftbiggϕ\nχ/parenrightbigg\n, just prior to the introduction of the γma-\ntrices. we obtain two equation on of these for ϕand the other for χ. We can solved in\nχand substiuted in the Dirac equation given by (12) and take th e non-relativistic regi-\nmen (|/vector p|<< m). So, in this physical landscape, from now and hereafter, ou r goal is to\nconsider a low-relativistic approximation based on an exte nded Pauli equation version by\nincluding torsion as presented before. Employing the Hamil tonian (15), we could carry\nout our calculations in the framework of the Fouldy-Wouthuy sen transformations; however\nfor the sake of our approximation at lowest order in v/c, we take that SHE is adequately\nwell described by the low-relativistic Pauli equation. We a re considering that the electron\nvelocities are in the range of Fermi’s velocity. In this case , we arrive at the version given\nbelow for the Pauli’s equation:\ni∂ϕ\n∂t=/bracketleftBig(/vector p−e/vectorA)2\n2m−e\n2m(/vector σeff·/vectorB)+eA0−(σ·/vectorSeff)+\n−λ\n8m(/vector∇×/vectorS)·(/vector σ×/vector p)+iλ\n8m(/vector∇×/vectorS)·/vector p+\n−eλ\n8m(/vector∇×/vectorS)·(/vector σ×/vectorA)+ieλ\n8m(/vector∇×/vectorS)·/vectorA/bracketrightBig\nϕ. (19)\nThe equation above displays the usual Pauli terms, but corre cted by new terms due to\nthe torsion coupling. The second and the fourth contributio ns in the RHS of eq.(19) can\nbe thought of as effective terms for /vector σand/vectorS, respectively given by\nσeff=/vector σ+η\n2m/vectorS+iη\n2m(/vector σ×/vectorS) (20)\n/vectorSeff=η/vectorS+iλ\n4(/vector∇×/vectorS). (21)\nThe fifth contribution in the RHS is proportional to the Rashb a SO coupling term; this\nterm yields an important effect on the behavior of spin.\n3 From the Modified Pauli Equation to Unfold in LLG\nIn this Section, we consider the magnetization equation der ivation given by Dirac non-\nrelativistic limit take into account the presence of torsio n. Let us start by considering the\nmodified Pauli equation eq.(19) and find the magnetization eq uation. By using the Landau\ngauge/vectorA=H/vector xand taking that /vector x·/vector σ= 0 (the spins are aligned orthogonally to the plane\nof motion), give us the Hamiltonian of the full system in the n on-relativistic limit as:\n8H=(/vector p−e/vectorA)2\n2m−e\n2m(/vector σeff·/vectorB)+eA0−(σ·/vectorSeff)−λ\n8m(/vector∇×/vectorS)·(/vector σ×/vector p)+iλ\n8m(/vector∇×/vectorS)·/vector p,\n(22)\nwith/vectorB(t) =µ0/vectorH(t), whereµ0is the gyromagnetic ratio. The Pauli equation associated\nwith (22) reads as follows below\ni∂ϕ\n∂t=/bracketleftBig(/vector p−eA)2\n2m−e\n2m(/vector σeff·/vectorH)+eA0−(/vector σ·/vectorSeff)+\n−λ\n8m(/vector∇×/vectorS)·(/vector σ×/vector p))+iλ\n8m(/vector∇×/vectorS)·p/bracketrightBig\nϕ. (23)\nLet us consider the magnetization vector equation related w in the spin magnetic mo-\nment/vector µ=e\n2m/vector σ. In our approach we consider the magnetization is defined by /vectorM=\n(/vector µϕ)†ϕ−ϕ†(/vector µϕ) whereϕgiven by Pauli equation (23) and ϕ†ϕ= 1 and/vectorˆSϕ=/vectorSϕ,\nwith the notation/vectorˆSis a torsion operator and /vectorSis the torsion autovalue. We have, by\nthe manipulation of Pauli equation the magnetization equat ion associated with a fermionic\nstate when we applied a external magnetic field /vectorHconsidering the Pauli product algebra\nas1\n2(σiσj−σjσi) =iǫijkσkand1\n2(σiσj+σjσi) =δij. The magnetization equation that\narrive that is\n∂/vectorM\n∂t=/vectorM×/vectorH+η/vectorM×/vectorS+β(/vectorM×/vectorL) +\n+ηe\n2m2(/vectorS×/vectorH) +eλ\n2m(/vector∇×/vectorS), (24)\nwith the magnetic moment given by /vectorL=/vector r×/vector pand/vectorSas the torsion pseudo-vector. We can\nobserved that there are two terms that arrived by the covaria nt derivative Dµdefined by\nthe coupling constant ηand the other is the parameter that arrived by non-minimal sp in\ntorsion coupling with the coupling constant λ. Where the effect of the new terms given\nwhenη/ne}ationslash= 0 andλ/ne}ationslash= 0.\nWe consider the scalar product of the magnetization /vectorM, the magnetic field /vectorHand the\ntorsion pseudo-vector /vectorSwith the equation (28) and we obtain2\n∂t/bracketleftBig1\n2(/vectorM·/vectorM)/bracketrightBig\n=ηe\n2m2/bracketleftBig\n/vectorM·(/vectorS×/vectorH)+λm\nη/vectorM·(/vector∇×/vectorS)/bracketrightBig\n; (25)\n2We used the vectorial relating given by A·(B×C) =B·(C×A) =C·(A×B) the other is\nA×(B×C) = (A·C)B−(A·B)Cand∇·(A×B) =B·(∇×A)−A·(∇×B).\n9∂t/bracketleftBig1\n2(/vectorH·/vectorM)/bracketrightBig\n=/bracketleftBig\nη/vectorM·(/vectorS×/vectorH)+β/vectorM·(/vectorL×/vectorH) +\n+e\n2mλ/vectorH·(/vector∇×/vectorS)/bracketrightBig\n; (26)\n∂t/bracketleftBig1\n2(/vectorS·/vectorM)/bracketrightBig\n=/vectorM·(/vectorS×/vectorH)+β/vectorM·(/vectorL×/vectorH). (27)\n∂t/bracketleftBig1\n2(/vectorL·/vectorM)/bracketrightBig\n=/vectorM·(/vectorL×/vectorH)+η/vectorM·(/vectorL×/vectorS) +\n+ηe\n2m2/vectorS·(/vectorL×/vectorH)+eλ\n2m/vectorL·(/vector∇×/vectorS),. (28)\nWith the equations dysplayed in (25)-(27), it is possible to inspect the general behavior of\nthe magnitude of the magnetization, /vectorM, that precesses around the magnetic field, /vectorH. In\nour framework, the magnetization also precesses around the torsion vector /vectorM·/vectorS. Without\ntorsion, we have∂/vectorM\n∂t=/vectorM×/vectorH, so that/vectorM·/vectorM= constant and /vectorM·/vectorH= constant as in the\nusual case of the electron under the action of a time-depende nt external magnetic field,\nwith the Zeeman term given by the Hamiltonian HM=/vectorM·/vectorH.\n3.1 Planar torsion analysis with damping\nHere, we intend to analyze some possibilities of solutions t o the magnetization that respect\nthe conditions given by (25)-( 27). The magnitude of the magn etization is not constant in\ngeneral, as we can see in equation (25), but, if this quantity is constant, there comes out a\nconstraint given by\n/vectorM·(/vectorS×/vectorH) =−λm\nη/vectorM·(/vector∇×/vectorS). (29)\nIf we considerd(/vectorL·/vectorM)\ndt= 0, we have\n/vectorS·(/vectorL×/vectorH) =−mλ\nη/vectorL·(/vector∇×/vectorS). (30)\nThis expression describes us the case where /vectorM·/vectorH/ne}ationslash= 0 and/vectorS·/vectorM/ne}ationslash= 0; then, there is a the\ndamping angle in both directions given by the precession aro und the magnetic field /vectorHand\naround the torsion pseudo-vector /vectorS:\n∂t/bracketleftBig1\n2(/vectorH·/vectorM)/bracketrightBig\n=λm(e\n2m2/vectorH−/vectorM)·(/vector∇×/vectorS); (31)\n∂t/bracketleftBig1\n2(/vectorS·/vectorM)/bracketrightBig\n=−λm\nη/vectorM·(/vector∇×/vectorS). (32)\n10Let us consider the first proposal in a very particular and ver y simple case for a planar\ntorsion field, /vectorS=1\n2χ(xˆy−yˆx); this choice allows us to realize the curl of torsion as an\neffective magnetic field, /vector∇×/vectorS=/vectorBeff=χˆz.\nIf we pick up the configuration of Fig. 1, we find the relation of the angle between the\nmagnetic field /vectorHand the magnetization /vectorM.\nFigure 1: Magnetization vector rotating around the magneti c field/vectorHwith damping given\nby the dynamics of the angle ζ. The system {M,H}rotates around the vector /vectorSin the xy-\nplane also with damping given by the angle φ. We consider φ=ωφtwithωφ=ωθ=2λmχ\nηS;\nwith this configuration ζ=ωζt=λχm\nηHM/parenleftBig\nH+ηM/parenrightBig\nt.\nWe started off by discussing the case where the torsion is plan ar with the magnitude\nof the magnetization being constant, /vectorM·/vectorM= 0, and\n∂t/bracketleftBig1\n2(/vectorS·/vectorM)/bracketrightBig\n=−λχm\nη/vectorM·/vector z,. (33)\nthis gives us the magnetic momentum precession around the to rsion. For consistency, we\nshow that this result is compatible with the equation\n∂t/bracketleftBig1\n2(/vectorH·/vectorM)/bracketrightBig\n=λmχ(e\n2m2/vectorH−/vectorM)·/vector z. (34)\nIn the case of the Fig. 1, the magnetization precesses around the magnetic field and around\nthe planar torsion vector both with damping.\n113.2 Helix-Damping Sharped Effect in a Planar Torsion Configur ation\nNow, let us consider the most general case, where the magnitu de of the magnetization is\nnot constant, but with ( /vectorL·/vectorM) = 0. The configuration is considered in Fig. 2. With the\nFigure 2: In this picture we show the effect of the torsion in mag netization dynamics. The\ngreen vector is the magnetization vector and the blue vector is the external magnetic field.\nexpressions (25)-(27), we can readily write the magnitude o f the magnetization\n∂t/bracketleftBig1\n2(/vectorM·/vectorM)/bracketrightBig\n=eη\n2m2/bracketleftBig\n∂t/bracketleftBig1\n2(/vectorS·/vectorM)/bracketrightBig\n+λm\nη/vectorM·(/vector∇×/vectorS)/bracketrightBig\n. (35)\nBy using of the equation (35) and considering ∂t/bracketleftBig\n1\n2(/vectorS·/vectorM)/bracketrightBig\n/ne}ationslash= 0, we can see, the example\nof the Fig. 3, that ∂t/bracketleftBig\n1\n2(/vectorM·/vectorM)/bracketrightBig\n/ne}ationslash= 0. This possibility gives us that the magnitude of\nmagnetization is not constant, as in the usual LLG. This effect is the effect of torsion that\ngives us that the rotational lines do not return around thems elves.\nEquation (35 ) does not involve the explicit dependence of th e magnetic field. We\nchoose to work out the equation\n∂t/bracketleftBig1\n2(/vectorHeff·/vectorM)/bracketrightBig\n=e\n2mλχ/vectorH·/vector z, (36)\nwhere/vectorHeff=/vectorH−η/vectorSgives us the explicit form of the magnetic field interaction w ith the\n12Figure 3: In this draw we show the effect of the torsion in magnet ization dynamics. The\ngreen vector is the magnetization vector and the blue vector is the external magnetic field.\nIn this representation we used |/vectorM|=M(t),|/vectorS|= constant and |/vectorH|= constant with\nωθ=2mλχ\nηSthenM(t) =λeχ\nωθmsinωθt.\nmagnetization.\nWe notice that this quantity is different from zero, then the an gle between the magnetic\nfield and the magnetization is not constant; this yields us th e damping precessing effect of\nthe magnetization vector around the magnetic field. The comp osition between these two\neffects, dislocation and damping, is what we refer to as the hel ix-sharped with damping,\neffect where the damping effect can be see in Fig. 4. We can show tha t there are two\nmagnetization effects: the damping given by the longitudinal magnetization function m(t)l\nand dislocations given by the longitudinal magnetization f unctionm(t)tas we can see in\nFig. 2. The trajectory of the magnetization is the conical in creasing spiral, where the\nmodulus of magnetization increases with the time. In the tor sion plan the behavior is\ngiven by Fig3.\n13Figure 4: Damping behavior in torsion plane. Show the behavi or of theφ=ωφtdynamic.\n4 Concluding Remarks\nIn this work, we have considered that the magnetization equa tion is a non-relativistic\nremnant of the non-relativistic limit of the Dirac equation with torsion couplings. We\nhave considered two types of couplings: one of these related with the spin current in Dirac\nequation, defined by the spin connection. When we derived the action in relation with\nthe spin connection we obtain the spin current, this descrip tion is analog to the charged\ncurrent when we have the derivation of the action in relation with the gauge field.\nWerefertotheothertermasthenon-minimaltorsiontermand Itgives ustherotational\nof the torsion. We have analyzed this term in the general cont ext and observed that it is\npossible to recover the Landau Lifshitz in the case were the t orsion is zero. Then, we can\npoint out that the non-relativistic limit of the Dirac equat ion reproduces the usual case\nwhere the magnetization vector precesses around the magnet ic field. When we introduce\nthe torsion terms we analyze, in the general regime the magni tude of magnetization /vectorM·/vectorM,\nthe precession of the magnetization around the magnetic fiel d/vectorH·/vectorM, and the precession\nof the magnetization around the torsion pseudo-vector /vectorS·/vectorMis not constant.\nWhen the magnitude of the magnetization is constant, in the c ase where the torsion is\nplanar, there are two possible magnetization precessions o ne around the magnetic field and\nother around the planar torsion pseudo-vector. In both dyna mics, there occurs damping.\nAn interesting example has been analyzed in Fig. 1, where we s how that it is possible\nto realized an apparatus in some experimental device. In thi s sense, our framework can\nreproduce the LLG equation. The most general approach shoul d consider that the mag-\nnitude of the magnetization is not constant. In this case, as we can see from Fig. 2, the\nloop drawn by the magnetization damping but the is not remain in the same plane. This\n14effect is typically a torsion effect, were the lines are not close d. This effect seems to be\nlike a dislocation in the material that presents topologica l defects like solitons and vortices.\nBoth dislocation and damping give us what we refer to as the he lix-damping sharp effect,\nwich is a new feature of the models with torsion[33].\nWe can observethat this resultis thenewfeatureintroduced bytheplanartorsion, ifwe\nconsider the comparation with dampingand dislocations ter ms presented in LLG equation.\nThis may help in the task of setting up new apparatuses and may be experimental purposes\nto explore such characteristics in this phenomenon. We have found that /vectorM·/vectorM= constant,\nits consequence is the dislocation effect. The damping effect is the usual one, where the\nangle dynamic can crease and decrease with the time. In this w ork we does not study the\npolarization of the spins that is subject of next work when we will consider these systems in\nterms or the spinup and spin down dynamic. In the literature, this effect is named pumped\nspin current [34], and we shall study the possibility of this current when the system is in\na helix-sharped configuration[35].\nReferences\n[1] A. Dyrdal, J. Barnas, Phys. Rev. B 92, 165404 (2015)\n[2] Z. Wang, C. Tang, R. Sachs, Y. Barlas, and J. Shi, Phys. Rev . Lett.114, 016603\n(2015).\n[3] K. M. McCreary, A. G. Swartz, W. Han, J. Fabian, R.K. Kawak ami Phys. Rev. Lett.\n109, 186604 (2012)\n[4] I. A. Zhuravlev, V. P. Antropov andK. D. Belashchenko, Ph ys.Rev. Lett. 115, 217201\n(2015).\n[5] V. Flovik, F. Maci, J. M. Hernndez, R. Brucas, M. Hanson an d E. Wahlstrm, Phys.\nRev. B92, 104406 (2015).\n[6] I. Turek, J. Kudrnovsky and V. Drchal, Physical Review B 92, 214407 (2015).\n[7] L.D. Landau, E.M. Lifshitz, ”On the theory of the dispers ion of magnetic permeability\nin ferromagnetic bodies”, Phys. Z. Soviet Union 8, 153 (1935); M. Lakshmanan, ”The\nfascinating world of the Landau Lifshitz Gilbert equation: an overview”, Phil. Trans.\nR. Soc. A 369, 1280 (2011).\n[8] Gilbert, T. L. ”A phenomenological theory of damping in f erromagnetic materials”,\nIEEE Trans. Magn. 40, 34433449 (2004).\n[9] S. Iihama, S. Mizukami, H. Naganuma, M. Oogane, Y. Ando, a nd T. Miyazaki, Phys.\nRev. B89, 174416 (2014).\n15[10] S. A. Wolf, D. D. Awschalom, R. A. Buhrman, J. M. Daughton ,S. von Moln´ ar,M. L.\nRoukes, A. Y. Chtchelkanova,D. M. Treger , Science 294,1488 (2001).\n[11] D. Culcer et al., Phys. Rev. Lett. 93, 046602 (2004).\n[12] J. Yao and Z. Q. Yang, Phys. Rev. B 73, 033314 (2006).\n[13] E. G. Mishchenko, A.V. Shytov and B. I. Halperin Phys. Re v. Lett.93, 226602 (2004).\n[14] Sh. Murakami, N. Nagaosa, S.-C. Zhang, Science 301, 1348 (2003).\n[15] P. Adroguer, E. L. Weizhe, D. Culcer and E. M. Hankiewicz , Phys. Rev. B 92, 241402\n(2015).\n[16] Xin-Zhong Yan and C. S. Ting, Physical Review B 92, 165404 (2015) .\n[17] M. Eschrig, Rept. Prog. Phys. 78, no. 10, 104501 (2015).\n[18] K. Hashimoto, N. Iizuka and T. Kimura, Phys. Rev. D 91, no. 8, 086003 (2015).\n[19] K. B. Fadafan and F. Saiedi, Eur. Phys. J. C 75612 (2015).\n[20] K. Bakke, C. Furtado and J. R. Nascimento, Eur. Phys. J. C 60, 501 (2009) : erratum\nEur. Phys. J. C 64169 (2009)\n[21] I.L. Buchbinder and I.L. Shapiro, Phys. Lett. B 151263 (1985)\n[22] V.G. Bagrov, I.L. Buchbinder and I.L. Shapiro, Sov. J. P hys.353 (1992),\nhep-th/9406122\n[23] R. T. Hammond, Phys. Rev. D 5212 (1995).\n[24] I.L. Shapiro, Phys. Rep. 357113 (2002)\n[25] R. T. Hammond Rep. Prog. Phys. 65599 (2002).\n[26] L.H. Ryder and I.L. Shapiro, Phys. Lett. A 24721 (1998)\n[27] Zutic I, Fabian J and Sarma S D, Rev. Mod. Phys. 76, 323 (2004).\n[28] Maekawa S., Concepts in Spin Electronics (Oxford: Oxfo rd University Press), (2006).\n[29] Grollier J, Cros V, Hamzic A, George J M, Jaffr‘es H, Fert A, Faini G, Youssef J B\nand Legall H., Appl.Phys. Lett. 78, 3663 (2001).\n[30] Ando K, Takahashi S, Harii K, Sasage K, Ieda J, Maekawa S. and Saitoh E., Phys.\nRev. Lett. 101, 036601 (2008).\n16[31] Tserkovnyak Y, Brataas A. and Bauer G. E. W. , Phys. Rev. L ett.88, 117601 (2002).\n[32] Mizukami S, Ando Y and Miyazaki T., Phys. Rev. B 66104413 (2002).\n[33] S. Azevedo, J. Phys. A 34, 6081 (2001).\n[34] P. A. Andreev, Phys. Rev. E 91, 033111 (2015)\n[35] TaikiYoda, TakehitoYokoyama, ShuichiMurakami, Scie ntificReports5, 12024(2015).\n17"    },    {        "title": "1606.09326v2.Skyrmion_dynamics_in_a_chiral_magnet_driven_by_periodically_varying_spin_currents.pdf",        "content": "arXiv:1606.09326v2  [cond-mat.mes-hall]  6 Dec 2016Skyrmion dynamics in a chiral magnet driven by periodically\nvarying spin currents\nRui Zhu*and Yin-Yan Zhang\nDepartment of Physics, South China University of Technolog y,\nGuangzhou 510641, People’s Republic of China\nAbstract\nIn this work, we investigated the spin dynamics in a slab of ch iral magnets induced by an\nalternating (ac) spin current. Periodic trajectories of th e skyrmion in real space are discovered\nunder the ac current as a result of the Magnus and viscous forc es, which originate from the Gilbert\ndamping, the spin transfer torque, and the β-nonadiabatic torque effects. The results are obtained\nby numerically solving the Landau-Lifshitz-Gilbert equat ion and can be explained by the Thiele\nequation characterizing the skyrmion core motion.\nPACS numbers: 75.78.-n, 72.25.-b, 71.70.-d\n*Corresponding author. Electronic address: rzhu@scut.edu.cn\n1I. INTRODUCTION\nThe skyrmion spin texture is a kind of topologically-nontrivial magnet ic vortex formed\nmost typically in the bulk chiral magnets (CMs) and magnetic thin films1–3. In CMs it is\nbelieved that the spin-orbit coupling induced Dzyaloshinskii-Moriya int eraction (DMI) gov-\nerns the spin twisting1. Recently the magnetic skyrmion structure attracts intensive fo cus,\nboth in the fundamental theoretic aspect and in its potential applic ation in the information\ntechnology4–7. In the magnetic skyrmion state the emergent electrodynamic effe ct originates\nfrom its nontrivial spin topology and gives rise to the topological Hall effect and a remark-\nable current-driven spin transfer torque effect1,8–14. The so-called skyrmionics makes use\nof the skyrmion as a memory unit favored by its topologically protect ed long lifetime and\nultralow driving current, which is five or six orders smaller than that f or driving a magnetic\ndomain wall1,15.\nAlthough the current-driven spin dynamics in the CMs with DMI has be en intensively\nstudied recently, less work of an alternating current (ac) driven s kyrmion dynamics was\nreported. The skyrmion-motion-induced ac current generation h as been predicted, which\nshares the reversed effect of our consideration16. In this work, we investigated the ac-\nspin-current driven skyrmion dynamics with the DMI, Gilbert damping17, adiabatic and\nnonadiabatic spin torques, and different current profiles taken int o account. Our proposition\nis inspired by the following several aspects. Firstly, it is both theore tically and technically\ninteresting to know the behavior of a skyrmion when an external ac current is applied.\nSecondly, ahigh-speedlow-powermodulationofaskyrmionisfavora bleforpotentialmemory\nprocessing. Lastly but not least, we noticed the mathematical sign ificance of the solution of\nthe Landau-Lifshitz-Gilbert (LLG) equation of a collinear magnet wit h periodically varying\nspin-currents applied, in which chaos is observed18,19.\nThe topological property of a spin texture can be described by the surface integral of the\nsolid angle of the unitary spin-field vector n(r). The skyrmion number is so defined as S=\n1\n4π/integraltext\nn·/parenleftBig\n∂n\n∂x×∂n\n∂y/parenrightBig\nd2rcounting how many times the spin field wraps the unit sphere. More\nspecific topological properties of a skyrmion can be considered by a nalyzing its radial and\nwhirling symmetric pattern1,20–22. In the continuum field theory, as a result of topological\nprotection, the skyrmion cannot be generated from a topologically trivial magnetic state\nsuch as a ferromagnet or a helimagnet by variation without a topolog ically nontrivial force\n2such as a spatially nontrivial spin current11,12, geometrical constriction13, domain wall pair\nsource5,6, the edge spin configuration22, etc., and vise versa. It is predicted by simulation\nthat the skyrmion can be generated from a quasi-ferromagnetic a nd helimagnetic state by\nexternal Lorentzian and radial spin current12and that transformation is possible between\ndifferent topologically-nontrivial states such as that between the domain-wall pair and the\nskyrmion5,6. The local current flowing from the scanning tunneling microscope t o generate\nthe skyrmion in experiment can be approximated by a radial spin curr ent, which imbues\nnontrivial topology into the helimagnet11. Also an artificial magnetic skyrmion can be\ntailored by an external magnetic field with nontrivial geometric distr ibution23. When the\nboundary geometry of the material is tailored such as by a notch in a long plate, a skyrmion\ncan be generated by a collinear spin current13. In this case, the nontrivial constriction\ntopology contributes to the formation of the skyrmion. The unifor m current can move and\nrotate a skyrmion without changing its topology15,24. In this work we will show that these\ntopological behaviors of the skyrmion are retained in the spin dynam ics driving by an ac\nspin current.\nAlmost all kinds of ferromagnetic and vortex spin dynamics can be de scribed by the LLG\nequation. The behavior of the LLG equation is of importance in both t he physical and\nmathematical sciences18. It has been shown by previous works that the spin torque effect\ndriven by a periodic varying spin current can be described as well by t he LLG equation\nwith the original time-independent current replaced by the time-de pendent current in the\nspin torque term18,19. Although chaotic behaviors are predicted in the spatially-uniform a c\nspin-current driven collinear ferromagnetic spin structure18,19, which is well described by the\nsingle-spin LLG equation, no similar phenomenon is reported in a spatia lly-nonuniform spin\nlattice, the latter of which can be attributed to the relaxation proc esses of the inter-site\nscattering. Even if some sort of chaotic behavior occurs after a lo ng time of evolution, it\nis workable to restore the original state by applying a magnetic field a fter some time. The\ninfluence of it on the skyrmionics exploitation is not large. In this work , we use a matrix-\nbased fourth-order Runge-Kutta method to solve the LLG equat ion with both the adiabatic\nand nonadiabatic spin torques taken into account. Analytical solut ion of the generalized\nThiele equation1,13,24reproduces our numerical results.\n3II. THEORETIC FORMALISM\nWe consider a thin slab of CM modulated by a constant magnetic field an d an ac spin\ncurrent. The strong DMI makes the material a skyrmion-host. In the continuum approxi-\nmation, the Hamiltonian of the localized magnetic spin in a CM can be desc ribed as1,12,13\nH=−J/summationtext\nrMr·/parenleftbig\nMr+ex+Mr+ey/parenrightbig\n−D/summationtext\nr/parenleftbig\nMr×Mr+ex·ex+Mr×Mr+ey·ey/parenrightbig\n−B·/summationtext\nrMr,(1)\nwithJandDthe ferromagnetic and Dzyaloshinskii-Moriya (DM) exchange energ ies, re-\nspectively. The dimensionless local magnetic moments Mrare defined as Mr≡ −Sr//planckover2pi1,\nwhereSris the local spin at rand/planckover2pi1is the plank constant divided by 2 π. We assume that\nthe length of the vector |Mr|=Mis fixed, therefore Mr=Mn(r) withn(r) the unitary\nspin field vector. The unit-cell dimension is taken to be unity. An exte rnal magnetic field\nBis applied perpendicular to the slab plane to stabilize the skyrmion confi guration. The\nBohr magneton µBis absorbed into Bto have it in the unit of energy. The typical DMI\nD= 0.18Jis used throughout this work13. This DM exchange strength corresponds to the\ncritical magnetic fields Bc1= 0.0075Jbetween the helical and skyrmion-crystal phases and\nBc2= 0.0252Jbetween the skyrmion-crystal and ferromagnetic phases, resp ectively. We\nadoptB= (0,0,0.01J) in our numerical considerations with J= 1 meV.\nTheextendedformoftheLLGequationthattakesintoaccountth eDMIandtheadiabatic\nand nonadiabatic spin torque effects can be expressed in the followin g formula1,12,13,25\ndMr\ndt=−γMr×Beff\nr+α\nMMr×dMr\ndt+pa3\n2eM[j(r,t)·∇]Mr\n−pa3β\n2eM2{Mr×[j(r,t)·∇]Mr}.(2)\nBy assuming that the energy of a magnet with the local magnetizatio nMrin a spatially\nvarying magnetic field Beff\nrhas the form of H=−γ/planckover2pi1/summationtext\nrMr·Beff\nr, we have\nBeff\nr=−1\n/planckover2pi1γ∂H\n∂Mr, (3)\nand therefore the first term in the right hand side of Eq. (2) is\n−γMr×Beff\nr=−J\n/planckover2pi1Mr×/parenleftbig\nMr+ex+Mr+ey+Mr−ex+Mr−ey/parenrightbig\n−1\n/planckover2pi1(Mr×B)\n−D\n/planckover2pi1Mr×\n/parenleftbig\nMr−ey,z−Mr+ey,z/parenrightbig\nex+(Mr+ex,z−Mr−ex,z)ey\n+/parenleftbig\nMr+ey,x−Mr+ex,y−Mr−ey,x+Mr−ex,y/parenrightbig\nez\n.(4)\n4The second to the last terms of Eq. (2) sequentially correspond to the effect of the Gilbert\ndamping, the time-dependent spin current j(t) =jesin(ωt)-induced adiabatic and nonadia-\nbatic spin torques, respectively. pmeasures the polarization of the conduction electrons, eis\nthe positive electron charge, and ais the average in-plane lattice constant of the CM. In our\nconsiderations the frequency of the ac spin current ωis small enough in comparison of the\nmagnetization evolution rate. Therefore, the spin torques can be satisfactorily described by\nusing the time-dependent current in thestandard torque expres sion, which has been justified\nby previous studies18,19. Here, the unit of time is set to be t0=/planckover2pi1/J≈6.6×10−13s. A\nphenomenologically expectedvalueof α= 0.1isusedinafterwardsnumerical considerations.\nBy looking deep into Eqs. (2) and (4), we can make some predictions o f the behavior of\nthe local magnetization. We know that the effect of the magnetic fie ld together with the\nGilbert term is to precess the magnetic spin into the direction of the e xternal field. The first\nterm in the right hand side of Eq. (4) is that the effective magnetic fie ld is in the direction\nof neighboring spins. Therefore the evolution tends to form a ferr omagnet. This contributes\nto the centripetal force of the magnetization in the direction of Mr×Mr′, which results in\nthe precession of one around the other. The effective field in the DM term in Eq. (4) is\n−∇×Mrwith unitary lattice constants. The integral counterpart of the c url is/contintegraltext\nMr′·dl.\nWhentheneighboringspinsformaring, theenergyisthelowest, hen cegeneratingaspiraling\nforce to the CM. It helps our understanding if we analogize all the ot her terms in the right\nhand side of Eq. (2) to the effect of a magnetic field. The local “magn etic field” of the\nphenomenological Gilbert damping force is proportional to −dMr/dtin the standard linear-\nresponse damping form, proportional to the velocity and pointing o ppositely to it. While\nthe local spin is precessing, the direction of Mr×dMr/dtpoints to the precession axis of\nMradding a force swaying to that axis. The last two terms are the effec t of the current-\ninduced spin torques. For convenience of interpretation, we discu ss the case that j(r,t) is\nspatially-uniform and along the x-direction. Then [ j(r,t)·∇]Mr=jx(t)∂Mr/∂x. In the\ncase of the adiabatic torque, this term adds a velocity to Mrmaking it sway to the direction\nofMr+ex, andMr+extoMr+2ex, and etc. if jx(t) is positive. Therefore, the complete spin\ntexture moves along the direction of the external spin current like a relay race no matter\nit is a skyrmion or a domain wall. In a periodic magnetic structure such a s a ferromagnet\nand a helimagnet the “relay race” goes back to itself and hence no sp in structure movement\noccurs. Following this physical picture, the local “magnetic field” of the nonadiabatic spin\n5torque is along the direction of jx(t)∂Mr/∂x. It exerts a velocity perpendicular to that\noriginates from the adiabatic spin torque. Its result is the motion of the spin texture in the\ndirection perpendicular to the spin current. We have already analyz ed the mechanisms of\ntheLLGequation termby term. However, they affectsthe system collaboratively. While the\nadiabatic spin torque moves the skyrmion along the spin current, th e Gilbert damping force\ncontributes a velocity in the direction of Mr×dMr/dtandtherefore the effect is a transverse\nmotion of the skyrmion, which is the so-called Hall-like motion12. Also it is noticeable that\nthe transverse velocity resulting from the Gilbert damping and the n onadiabatic spin torque\nis opposite to each other. In real situations, both αandβare much less than 1. The\nadiabatic spin torque makes the main contribution to the motion of th e skyrmion. And\nwhen the two transverse force is equal, the motion of the skyrmion is straightly along the\ndirection of the spin current. Therefore, periodic trajectories o f the skyrmion in real space\ncan be predicted under the influence of a spatially uniform ac spin cur rent.\nThe previous discussions are well expressed in the Thiele equation de scribing the motion\nof the center of mass of a skyrmion as1,13,24,26.\nG×[−j(t)−vd]+κ[−βj(t)−αvd]−∇U(r) =0, (5)\nwherevd=dR/dt=/parenleftBig\n˙X,˙Y/parenrightBig\nwithR= (X,Y) the center of mass coordinates, κis a\ndimensionless constant of the order of unity, and G= 2πSezis the gyrovector with ezin the\ndirection perpendicular to the CM plane. The minus sign before j(t) is because of that the\ndirection of the motion of conduction electrons is opposite to that o f the current. The Thiele\nequation (5) describes5coaction of the Magnus force Fg=G×[−j(t)−vd], the viscous\nforceFv=κ[−βj(t)−αvd], and the confining force Fp=−∇U(r). In our considerations,\nperiodic boundary conditions are used to justify an infinite two-dime nsional model. The\napplied magnetic field is spatially uniform and the impurity effect is neglec ted. Therefore\n∇U≈0. The analytical result of Eq. (5) assuming S=−1 andj(t) =jesin(ωt)excan be\nobtained as \n\nX=αβκ2+4π2\n(α2κ2+4π2)ωjecos(ωt),\nY=2πκ(β−α)\n(α2κ2+4π2)ωjecos(ωt).(6)\nSince the spin current is time dependent, FgandFvinstantaneously change their direc-\ntion with the motion of the skyrmion core and simultaneously react on the motion of the\nskyrmion, giving rise to the trigonometric trajectory of the skyrm ion shown in Eq. (6),\n6which agrees with the simulation results. Because the skyrmion vort ex moves in the relay\nfashion under the effect of the spin torque shown by the LLG equat ion, there is a π/2 phase\nlag between its core motion and the sinusoidally varying spin current.\nIII. NUMERICAL RESULTS AND INTERPRETATIONS\nBy multiplying ˜ α−1with\n˜α= 1−α\n0−(Mr)z(Mr)y\n(Mr)z0−(Mr)x\n−(Mr)y(Mr)x0\n, (7)\nfrom the left to Eq. (2), the matrix-based Runge-Kutta method is developed. In Figs. 1 to\n3, numerical results of our simulations are given. We set M= 1,p= 0.5, anda= 4˚A. The\nintegral step h= 0.1t0is used and its convergence is justified by comparison with the result s\nofh= 0.01t0. WithD= 0.18J, the natural helimagnet wavevector Q= 2π/λ=D/J\nwith the diameter of the skyrmion λ=D/J≈35 in the unit of a. A 30×30 square\nlattice is considered which approximately sustains a single skyrmion. P eriodic boundary\ncondition is used to allow the considered patch to fit into an infinite plan e. While part of the\nskyrmion moves out of the slab, complementary part enters from t he outside as the natural\nground state of a CM is the skyrmion crystal. We use the theoretica lly perfect skyrmion\nprofilen(r) = [cosΦ( ϕ)sinΘ(r),sinΦ(ϕ)sinΘ(r),cosΘ(r)] with Θ( r) =π(1−r/λ) and\nΦ(ϕ) =ϕin the polar coordinates as the initial state and it would change into a n atural\nskyrmion in less than one current period. The skyrmion number for t his state S=−1. The\nspatially-uniform ac spin current is applied in the x-direction as j(t) =jesin(ωt)exwithin\nthe CM plane.\nVariationofthe skyrmion number intimedriven bythe acspincurrent is shown inFig. 1.\nItcanbeseen thatcosinusoidal variationof Soriginatesfromthesinusoidal j(t)withexactly\nthe same period. Fig. 2 shows the snapshots of the spin profile at th e bottoms and peeks of\nthecosinusoidal variationof SandFig. 3shows thetrajectoriesof thecenter oftheskyrmion\n(see Ref. 27 for Supplementary Movie). The skyrmion number is a de monstration of the\nmotion pattern of the skyrmion. While the skyrmion moves to one side of the CM slab, only\npart of a skyrmion is within the view and hence the skyrmion number is r educed. Previous\nauthors have found that the velocity of the skyrmion increases line arly with the increase of\n7the current amplitude and that thedynamical threshold current t o move a skyrmion isin the\nsameorderofthatneededforadomainwall1. Herewehavereobtainedthetwo points. Itcan\nbeseeninFig. 1(a)thatthepeakhight oftheskyrmionnumber incre ases withtheamplitude\nof the current density and it becomes almost invisible when jeis as small as 1010Am−2. In\nFig. 1(b), the evolutions of Sfor different ac periods are shown. The frequency of the ac\ncurrent is in the order of GHz, which is sufficiently adiabatic as the rat e of the spin dynamics\nis in the order of 10−12s. We can see that the periodic pattern of Sis better kept with larger\namplitudes for smaller ac frequencies. It shows that the phenomen on is a good adiabatic\none. Within our numerical capacity, it can be predicted that strong cosinusoidal variation\ncan occur at MHz or smaller ac frequencies, which promises experime ntal realization.\nThe variation of Sis the result of the motion of the skyrmion. The periodic translation o f\nskyrmion is the result of the coaction of the instantaneous Magnus and viscous forces. The\nspin current gives rise to the drift velocity of the spin texture. As a combined result of the\nGilbert damping, the DMI, the adiabatic and nonadiabatic spin torque s, the skyrmion Hall\neffect, namely, the transverse motion of the skyrmion perpendicu lar to the spin current, is\nobserved in topologically-nontrivial spin textures. As shown in Fig. 2 , in spite of its motion,\nthe topological properties of the skyrmion are conserved becaus e the initial skyrmion state\nand the natural skyrmion ground state share similar topology and n o topology-breaking\nsource such as an in-plane magnetic field is present. When part of th e skyrmion moves out\nof the CM slab, only the remaining part contributes to the skyrmion n umber and hence Sis\ndecreased. The cosinusoidal variation of Sdirectly reflects the oscillating trajectory of the\nskyrmion Shown in Figs. 2 and 3. We can see that the skyrmion change s from the initial\nartificial skyrmion state into the natural skyrmion state with S=−1 conserved, as shown\nin Fig. 2(a) and (b). At the times of integer periods the skyrmion is at the center of the CM\nslab and at the times of half-integer periods the skyrmion moves to t he left side as shown\nin Fig. 2 (c) to (f).\nAs predicted by the Thiele equation, the trajectory of the skyrmio n follows a cosinusoidal\npattern expressed in Eq. (6). It is interesting that the trajecto ry of the skyrmion results\nfrom the competition between the drift motion of the skyrmion and t he skyrmion Hall effect\nunder the influence of the adiabatic and nonadiabatic spin torque eff ects. The adiabatic spin\ntorque effect exerts a force to align the spin at each site to its + x-direction neighbor while\nj(t) is in the exdirection, which results in the motion of the spin pattern to the + xdirection\n8in a relay fashion. The Gilbert damping effect and the nonadiabatic spin torque add a\ntransverse velocity to the moving skyrmion perpendicular to its orig inal velocity. These two\nforces are in opposite directions when αandβare both positive. Therefore the transverse\nmotion is determined by the sign and relative strength of these two e ffects. From Eq. (6)\nwe can see that when β−α >0 the skyrmion’s y-direction motion is in a cosinusoidal form\nand when β−α <0 it is in a negative cosinusoidal form. For the x-direction motion of the\nskyrmion, the direction is the same in the two cases and the magnitud e is slightly smaller for\nthe latter because |4π2| ≫ |αβκ2|holds for all physical parameter settings. And physically\nit is because the x-direction motion of the skyrmion is mainly determined by the adiabatic\nspin torque, which is the prerequisite for any motion of the skyrmion .\nOur simulation results of the skyrmion trajectories for β= 0.5α,α, and 2αwith fixed\nα= 0.1 are shown in Fig. 3. Good agreement with the prediction by the Thiele equation\nis obtained. In the three cases, Xevolves cosinusoidally with the initial position ( X,Y) =\n(15,15) at the center of the CM slab. For β= 0.5α,Yevolves minus-cosinusoidally; for\nβ=α,Yis constant at 15; for β= 2α,Yevolves cosinusoidally. As the difference between\nβandαis small in Fig. 3 (a) and (c), the cosinusoidal pattern shrinks into a s tep jump.\nBesides the oscillation, a tiny linear velocity of the skyrmion can be see n in Fig. 3 (a) and\n(c). And the directions of this velocity are different in the two cases . We attribute this\nlinear velocity to the whirling of the skyrmion from the artificial initial p rofile to the natural\nprofile sustained by the real CM. Because at this whirling step, the G ilbert damping and\nthe adiabatic and nonadiabatic torques are already in effect, the init ial linear velocities are\ndifferent in the two cases.\nIV. CONCLUSIONS\nIn this work, we have investigated the dynamics of the skyrmion in a C M driven by\nperiodically varying spin currents by replacing the static current in t he LLG equation by\nan adiabatic time-dependent current. Oscillating trajectories of t he skyrmion are found\nby numerical simulations, which are in good agreement with the analyt ical solution of the\nThiele equation. In the paper, physical behaviors of the general L LG equation with the\nGilbert damping and the adiabatic and nonadiabatic spin torques coex istent are elucidated.\nEspecially, the effect of the nonadiabatic spin torque is interpreted both physically and\n9numerically.\nV. AUTHOR CONTRIBUTION STATEMENT\nR.Z. wrote the program and the paper. Y.Y.Z. made the simulation.\nVI. ACKNOWLEDGEMENTS\nR.Z. would like to thank Pak Ming Hui for stimulation and encouragemen t of the work.\nThis project was supported by the National Natural Science Foun dation of China (No.\n11004063) and the Fundamental Research Funds for the Centra l Universities, SCUT (No.\n2014ZG0044).\n101N. Nagaosa and Y. Tokura, Nat. Nanotechnol. 8, 899 (2013).\n2S. M¨ uhlbauer, B. Binz, F. Jonietz, C. Pfleiderer, A. Rosch, A . Neubauer, R. Georgii, and P.\nB¨ oni, Science 323, 915 (2009).\n3S.Heinze, K. V. Bergmann, M. Menzel, J. Brede, A. Kubetzka, R . Wiesendanger, G. Bihlmayer,\nand S. Bl¨ ugel, Nat. Phys. 7, 713 (2011).\n4X. Zhang, M. Ezawa, and Y. Zhou, Sci. Rep. 5, 9400 (2015).\n5X. Xing, P. W. T. Pong, and Y. Zhou, Phys. Rev. B 94, 054408 (2016).\n6Y. Zhou and M. Ezawa, Nat. Commun. 5, 4652 (2014).\n7J. Sampaio, V. Cros, S. Rohart, A. Thiaville, and A. Fert, Nat . Nanotechnol. 8, 839 (2013).\n8K. Hamamoto, M. Ezawa, and N. Nagaosa, Phys. Rev. B 92, 115417 (2015).\n9A. Neubauer, C. Pfleiderer, B. Binz, A. Rosch, R. Ritz, P. G. Ni klowitz, and P. B¨ oni, Phys.\nRev. Lett. 102, 186602 (2009).\n10T. Schulz, R. Ritz, A. Bauer, M. Halder, M. Wagner, C. Franz, C . Pfleiderer, K. Everschor, M.\nGarst, and A. Rosch, Nat. Phys. 8, 301 (2012).\n11N. Romming, C. Hanneken, M. Menzel, J. E. Bickel, B. Wolter, K . V. Bergmann, A. Kubetzka,\nand R. Wiesendanger, Science 341, 636 (2013).\n12Y. Tchoe and J. H. Han, Phys. Rev. B 85, 174416 (2012).\n13J. Iwasaki, M. Mochizuki, and N. Nagaosa, Nat. Nanotechnol. 8, 742 (2013).\n14D.C. Ralph and M.D. Stiles, J. Magn. Magn. Mater. 320, 1190 (2008).\n15F. Jonietz, S. M¨ uhlbauer, C. Pfleiderer, A. Neubauer, W. M¨ u nzer, A. Bauer, T. Adams, R.\nGeorgii, P. B¨ oni, R. A. Duine, K. Everschor, M. Garst, and A. Rosch, Science 330, 1648 (2010).\n16S.-Z. Lin, C. D. Batista, C. Reichhardt, and A. Saxena, Phys. Rev. Lett. 112, 187203 (2014).\n17K. M. D. Hals and A. Brataas, Phys. Rev. B 89, 064426 (2014).\n18M. Lakshmanan, Phil. Trans. R. Soc. A 369, 1280 (2011).\n19Z. Yang, S. Zhang, and Y. C. Li, Phys. Rev. Lett. 99, 134101 (2007).\n20X. Zhang, Y. Zhou, and M. Ezawa, Phys. Rev. B 93, 024415 (2016).\n21A. Bogdanov and A. Hubert, J. Magn. Magn. Mater. 195, 182 (1999).\n22Y.-Y. Zhang and R. Zhu, Phys. Lett. A 380, 3756 (2016).\n23J. Li, A. Tan, K.W. Moon, A. Doran, M.A. Marcus, A.T. Young, E. Arenholz, S. Ma, R.F.\n11Yang, C. Hwang, and Z.Q. Qiu, Nat. Commun. 5, 4704 (2014).\n24K. Everschor, M. Garst, B. Binz, F. Jonietz, S. M¨ uhlbauer, C . Pfleiderer, and A. Rosch, Phys.\nRev. B86, 054432 (2012).\n25I. M. Miron, G. Gaudin, S. Auffret, B. Rodmacq, A. Schuhl, S. Piz zini, J. Vogel, and P.\nGambardella, Nat. Mater. 9, 230 (2010).\n26A. A. Thiele, Phys. Rev. Lett. 30, 230 (1972).\n27Supplementary Movie.\n12/s48 /s53/s48/s48 /s49/s48/s48/s48 /s49/s53/s48/s48 /s50/s48/s48/s48/s45/s49/s46/s48/s48/s45/s48/s46/s57/s53/s45/s48/s46/s57/s48/s45/s48/s46/s56/s53/s45/s49/s46/s48/s48/s45/s48/s46/s57/s53/s45/s48/s46/s57/s48/s45/s48/s46/s56/s53/s45/s48/s46/s56/s48/s45/s48/s46/s55/s53\n/s116/s83/s32/s106\n/s101/s61/s48\n/s32/s84/s61/s52/s48/s48\n/s32/s84/s61/s53/s48/s48\n/s32/s84/s61/s54/s48/s48/s40/s98/s41/s83/s32/s106\n/s101/s61/s49/s48/s49/s48\n/s65/s109/s45/s50\n/s32/s106\n/s101/s61/s49/s48/s49/s49\n/s65/s109/s45/s50\n/s32/s106\n/s101/s61/s50 /s49/s48/s49/s49\n/s65/s109/s45/s50/s40/s97/s41/s32\nFIG. 1: Variation of the skyrmion number S in time (a) for differ ent current amplitudes and (b)\nfor different ac current frequencies. The time tand ac spin current period Tare in the unit of\nt0≈6.6×10−13s.β= 0.05. In panel (a), T= 500. In panel (b), je= 1011Am−2.\n13(0,−0.91763)\n  \n(a)\n−1−0.500.51(750,−0.77054)\n(b)(1000,−0.97276)\n(c)\n(1250,−0.76819)\n(d)(1500,−0.97019)\n(e)(1750,−0.76716)\n(f)\nFIG. 2: Snapshots of the dynamical spin configurations at the bottoms and peaks of the skyrmion\nnumber shown in Fig. 1. The in-plane components of the magnet ic moments are represented by\narrows and their z-components are represented by the color plot. The paramete rs areje= 2×1011\nAm−2,T= 500, and β= 0.05. On the top of each panel are the ( t,S) values.\n14/s57/s49/s48/s49/s49/s49/s50/s49/s51/s49/s52/s49/s53/s49/s54\n/s57/s49/s48/s49/s49/s49/s50/s49/s51/s49/s52/s49/s53\n/s48 /s53/s48/s48 /s49/s48/s48/s48 /s49/s53/s48/s48 /s50/s48/s48/s48/s57/s49/s48/s49/s49/s49/s50/s49/s51/s49/s52/s49/s53\n/s32/s88\n/s32/s89/s40/s97/s41/s32 /s61/s48/s46/s48/s53\n/s40/s98/s41/s32 /s61/s48/s46/s49\n/s116/s40/s99/s41/s32 /s61/s48/s46/s50\nFIG. 3: Variation of the skyrmion center coordinates ( X,Y) in time (a) for β= 0.05, (b) for\nβ= 0.1, and (c) for β= 0.2. Other parameters are the same as Fig. 2.\n15"    },    {        "title": "1607.01307v1.Magnetic_moment_of_inertia_within_the_breathing_model.pdf",        "content": "Magnetic moment of inertia within the breathing model\nDanny Thonig,\u0003Manuel Pereiro, and Olle Eriksson\nDepartment of Physics and Astronomy, Material Theory, University Uppsala, S-75120 Uppsala, Sweden\n(Dated: June 20, 2021)\nAn essential property of magnetic devices is the relaxation rate in magnetic switching which\nstrongly depends on the energy dissipation and magnetic inertia of the magnetization dynamics.\nBoth parameters are commonly taken as a phenomenological entities. However very recently, a large\ne\u000bort has been dedicated to obtain Gilbert damping from \frst principles. In contrast, there is no\nab initio study that so far has reproduced measured data of magnetic inertia in magnetic materials.\nIn this letter, we present and elaborate on a theoretical model for calculating the magnetic moment\nof inertia based on the torque-torque correlation model. Particularly, the method has been applied\nto bulk bcc Fe, fcc Co and fcc Ni in the framework of the tight-binding approximation and the\nnumerical values are comparable with recent experimental measurements. The theoretical results\nelucidate the physical origin of the moment of inertia based on the electronic structure. Even though\nthe moment of inertia and damping are produced by the spin-orbit coupling, our analysis shows that\nthey are caused by undergo di\u000berent electronic structure mechanisms.\nPACS numbers: 75.10.-b,75.30.-m,75.40.Mg,75.78.-n,75.40.Gb\nThe research on magnetic materials with particular fo-\ncus on spintronics or magnonic applications became more\nand more intensi\fed, over the last decades [1, 2]. For\nthis purpose, \\good\" candidates are materials exhibiting\nthermally stable magnetic properties [3], energy e\u000ecient\nmagnetization dynamics [4, 5], as well as fast and stable\nmagnetic switching [6, 7]. Especially the latter can be\ninduced by i)an external magnetic \feld, ii)spin polar-\nized currents [8], iii)laser induced all-optical switching\n[9], or iv)electric \felds [10]. The aforementioned mag-\nnetic excitation methods allow switching of the magnetic\nmoment on sub-ps timescales.\nThe classical atomistic Landau-Lifshitz-Gilbert (LLG)\nequation [11, 12] provides a proper description of mag-\nnetic moment switching [13], but is derived within the\nadiabatic limit [14, 15]. This limit characterises the\nblurry boundary where the time scales of electrons and\natomic magnetic moments are separable [16] | usually\nbetween 10\u0000100 fs. In this time-scale, the applicabil-\nity of the atomistic LLG equation must be scrutinized\nin great detail. In particular, in its common formula-\ntion, it does not account for creation of magnetic inertia\n[17], compared to its classical mechanical counterpart of\na gyroscope. At short times, the rotation axis of the\ngyroscope do not coincide with the angular momentum\naxis due to a \\fast\" external force. This results in a\nsuperimposed precession around the angular-momentum\nand the gravity \feld axis; the gyroscope nutates. It is\nexpected for magnetisation dynamics that atomic mag-\nnetic moments behave in an analogous way on ultrafast\ntimescales [17, 18] (Fig. 1).\nConceptional thoughts in terms of \\magnetic mass\"\nof domain walls were already introduced theoretically by\nD oring [19] in the late 50's and evidence was found ex-\nperimentally by De Leeuw and Robertson [20]. More\nrecently, nutation was discovered on a single-atom mag-\nnetic moment trajectory in a Josephson junction [21{23]\nB\nprecession conenutation cone\nmFIG. 1. (Color online) Schematic \fgure of nutation in the\natomistic magnetic moment evolution. The magnetic moment\nm(red arrow) evolves around an e\u000bective magnetic \feld B\n(gray arrow) by a superposition of the precession around the\n\feld (bright blue line) and around the angular momentum\naxis (dark blue line). The resulting trajectory (gray line)\nshows an elongated cycloid.\ndue to angular momentum transfer caused by an elec-\ntron spin \rip. From micromagnetic Boltzman theory,\nCiornei et al. [18, 24] derived a term in the extended\nLLG equation that addresses \\magnetic mass\" scaled by\nthe moment of inertia tensor \u0013. This macroscopic model\nwas transferred to atomistic magnetization dynamics and\napplied to nanostructures by the authors of Ref. 17, and\nanalyzed analytically in Ref. 25 and Ref. 26. Even in the\ndynamics of Skyrmions, magnetic inertia was observed\nexperimentally [27].\nLike the Gilbert damping \u000b, the moment of inertia ten-\nsor\u0013have been considered as a parameter in theoretical\ninvestigations and postulated to be material speci\fc. Re-\ncently, the latter was experimentally examined by Li et\nal. [28] who measured the moment of inertia for Ni 79Fe21\nand Co \flms near room temperature with ferromagnetic\nresonance (FMR) in the high-frequency regime (aroundarXiv:1607.01307v1  [cond-mat.mtrl-sci]  5 Jul 20162\n200 GHz). At these high frequencies, an additional sti\u000b-\nening was observed that was quadratic in the probing fre-\nquency!and, consequently, proportional to the moment\nof inertia\u0013=\u0006\u000b\u0001\u001c. Here, the lifetime of the nutation \u001c\nwas determined to be in the range of \u001c= 0:12\u00000:47 ps,\ndepending not only on the selected material but also on\nits thickness. This result calls for a proper theoretical\ndescription and calculations based on ab-initio electronic\nstructure footings.\nA \frst model was already provided by Bhattacharjee\net al. [29], where the moment of inertia \u0013was derived\nin terms of Green's functions in the framework of the\nlinear response theory. However, neither \frst-principles\nelectronic structure-based numerical values nor a detailed\nphysical picture of the origin of the inertia and a poten-\ntial coupling to the electronic structure was reported in\nthis study. In this Letter, we derive a model for the\nmoment of inertia tensor based on the torque-torque cor-\nrelation formalism [30, 31]. We reveal the basic electron\nmechanisms for observing magnetic inertia by calculat-\ning numerical values for bulk itinerant magnets Fe, Co,\nand Ni with both the torque-torque correlation model\nand the linear response Green's function model [29]. In-\nterestingly, our study elucidate also the misconception\nabout the sign convention of the moment of inertia [32].\nThe moment of inertia \u0013is de\fned in a similar way\nas the Gilbert damping \u000bwithin the e\u000bective dissipation\n\feldBdiss[30, 33]. This ad hoc introduced \feld is ex-\npanded in terms of viscous damping \u000b@m=@tand magnetic\ninertia\u0013@2m=@t2in the relaxation time approach [32, 34]\n(see Supplementary Material). The o\u000b-equilibrium mag-\nnetic state induces excited states in the electronic struc-\nture due to spin-orbit coupling. Within the adiabatic\nlimit, the electrons equilibrate into the ground state at\ncertain time scales due to band transitions [35]. If this\nrelaxation time \u001cis close to the adiabatic limit, it will\nhave two implications for magnetism: i)magnetic mo-\nments respond in a inert fashion, due to formation of\nmagnetism, ii)the kinetic energy is proportional to mu2=2\nwith the velocity u=@m=@tand the \\mass\" m of mag-\nnetic moments, following equations of motion of classical\nNewtonian mechanics. The inertia forces the magnetic\nmoment to remain in their present state, represented in\nthe Kambersky model by \u000b=\u0000\u0013\u0001\u001c(Ref. 32 and 34);\ntheraison d'etre of inertia is to behave opposite to the\nGilbert damping.\nIn experiments, the Gilbert damping and the moment\nof inertia are measurable from the diagonal elements of\nthe magnetic response function \u001fvia ferromagnetic res-\nonance [31] (see Supplementary Material)\n\u000b=!2\n0\n!Mlim\n!!0=\u001f?\n!(1)\n\u0013=1\n2!2\n0\n!Mlim\n!!0@!<\u001f?\n!\u00001\n!0; (2)\nwhere!M=\rBand!0=\rB0are the frequencies re-lated to the internal e\u000bective and the external magnetic\n\feld, respectively. Thus, the moment of inertia \u0013is equal\nto the change of the FMR peak position, say the \frst\nderivative of the real part of \u001fwith respect to the prob-\ning frequency [29, 36]. Alternatively, rapid external \feld\nchanges induced by spin-polarized currents lead also to\nnutation of the macrospin [37].\nSetting\u001fonab-initio footings, we use the torque-\ntorque correlation model, as applied for the Gilbert\ndamping in Ref. 30 and 35. We obtain (see Supplemen-\ntary Material)\n\u000b\u0016\u0017=g\u0019\nmsX\nnmZ\nT\u0016\nnm(k)T\u0017\nnm(k)Wnmdk (3)\n\u0013\u0016\u0017=\u0000g~\nmsX\nnmZ\nT\u0016\nnm(k)T\u0017\nnm(k)Vnmdk; (4)\nwhere\u0016;\u0017 =x;y;z andmsis the size of the mag-\nnetic moment. The spin-orbit-torque matrix elements\nTnm=hn;kj[\u001b;Hsoc]jm;ki| related to the commuta-\ntor of the Pauli matrices \u001band the spin-orbit Hamilto-\nnian | create transitions between electron states jn;ki\nandjm;kiin bandsnandm. This mechanism is equal\nfor both, Gilbert damping and moment of inertia. Note\nthat the wave vector kis conserved, since we neglect non-\nuniform magnon creation with non-zero wave vector. The\ndi\u000berence between moment of inertia and damping comes\nfrom di\u000berent weighting mechanism Wnm;Vnm: for the\ndampingWnm=R\n\u0011(\")Ank(\")Amk(\")d\"where the elec-\ntron spectral functions are represented by Lorentzian's\nAnk(\") centred around the band energies \"nkand broad-\nened by interactions with the lattice, electron-electron\ninteractions or alloying. The width of the spectral func-\ntion \u0000 provides a phenomenological account for angular\nmomentum transfer to other reservoirs. For inertia, how-\never,Vnm=R\nf(\") (Ank(\")Bmk(\") +Bnk(\")Amk(\")) d\"\nwhereBmk(\") = 2(\"\u0000\"mk)((\"\u0000\"mk)2\u00003\u00002)=((\"\u0000\"mk)2+\u00002)3\n(see Supplementary Material). Here, f(\") and\u0011(\") are\nthe Fermi-Dirac distribution and the \frst derivative of it\nwith respect to \". Knowing the explicit form of Bmk, we\ncan reveal particular properties of the moment of inertia:\ni)for \u0000!0 (\u001c!1 ),Vnm=2=(\"nk\u0000\"mk)3. Sincen=m\nis not excluded, \u0013!\u00001 ; the perturbed electron system\nwill not relax back into the equilibrium. ii)In the limit\n\u0000!1 (\u001c!0), the electron system equilibrates imme-\ndiately into the ground state and, consequently, \u0013= 0.\nThese limiting properties are consistent with the expres-\nsion\u0013=\u0000\u000b\u0001\u001c. Eq. (4) also indicates that the time scale\nis dictated by ~and, consequently, on a femto-second\ntime scale.\nTo study these properties, we performed \frst-\nprinciples tight binding (TB) calculations [38] of the\ntorque-correlation model as described by Eq. (4) as well\nas for the Green's function model reported in Ref. 29.\nThe materials investigated in this letter are bcc Fe, fcc\nCo, and fcc Ni. Since our magnetic moment is \fxed3\n-1·10−3-5·10−405·10−41·10−3−ι(fs)\n10−110+0\nΓ (eV)Fe\nCo\nNiTorque\nGreen\n10−21α10−410−21Γ (eV)\nFIG. 2. (Color) Moment of inertia \u0013as a function of the band\nwidth \u0000 for bcc Fe (green dotes and lines), fcc Co (red dotes\nand lines), and fcc Ni (blue dotes and lines) and with two\ndi\u000berent methods: i)the torque-correlation method (\flled\ntriangles) and the ii)Greens function method [29](\flled cir-\ncles). The dotted gray lines indicating the zero level. The\ninsets show the calculated Gilbert damping \u000bas a function of\n\u0000. Lines are added to guide the eye. Notice the negative sign\nof the moment of inertia.\nin thezdirection, variations occur primarily in xory\nand, consequently, the e\u000bective torque matrix element is\nT\u0000=hn;kj[\u001b\u0000;Hsoc]jm;ki, where\u001b\u0000=\u001bx\u0000i\u001by. The\ncubic symmetry of the selected materials allows only di-\nagonal elements in both damping and moment of inertia\ntensor. The numerical calculations, as shown in Fig. 2,\ngive results that are consistent with the torque-torque\ncorrelation model predictions in both limits, \u0000 !0 and\n\u0000!1 . Note that the latter is only true if we assume\nthe validity of the adiabatic limit up to \u001c= 0. It should\nalso be noted that Eq. (4) is only valid in the adiabatic\nlimit (>10 fs). The strong dependency on \u0000 indicates,\nhowever, that the current model is not a parameter-free\napproach. Fortunately, the relevant parameters can be\nextracted from ab-initio methods: e.g., \u0000 is related ei-\nther to the electron-phonon self energy [39] or to electron\ncorrelations [40].\nThe approximation \u0013=\u0000\u000b\u0001\u001cderived by F ahnle et\nal. [32] from the Kambersk\u0013 y model is not valid for all\n\u0000. It holds for \u0000 <10 meV, where intraband transi-\ntions dominate for both damping and moment of inertia;\nbands with di\u000berent energies narrowly overlap. Here, the\nmoment of inertia decreases proportional to 1=\u00004up to a\ncertain minimum. Above the minimum and with an ap-\npropriate large band width \u0000, interband transitions hap-\npen so that the moment of inertia approaches zero for\nhigh values of \u0000. In this range, the relation \u0013=\u000b\u0001\u001c\nused by Ciornei et al [18] holds and softens the FMR res-\nonance frequency. Comparing qualitative the di\u000berence\n10−410−310−210−1−ι(fs)/α\n510+02510+12510+22\nτ(fs)\n5·10−310−22·10−23·10−2\nΓ (eV)−ι\nαFIG. 3. (Color online) Gilbert damping \u000b(red dashed line),\nmoment of inertia \u0013(blue dashed line), and the resulting nu-\ntation lifetime \u001c=\u0013=\u000b(black line) as a function of \u0000 in the\nintraband region for Fe bulk. Arrows indicating the ordinate\nbelonging of the data lines. Notice the negative sign of the\nmoment of inertia.\nbetween the itinerant magnets Fe, Co and Ni, we obtain\nsimilar features in \u0013and\u000bvs. \u0000, but the position of the\nminimum and the slope in the intraband region varies\nwith the elements: \u0013min= 5:9\u000110\u00003fs\u00001at \u0000 = 60 meV\nfor bcc Fe, \u0013min= 6:5\u000110\u00003fs\u00001at \u0000 = 50 meV for fcc\nCo, and\u0013min= 6:1\u000110\u00003fs\u00001at \u0000 = 80 meV for fcc Ni.\nThe crossing point of intra- and interband transitions for\nthe damping was already reported by Gilmore et al. [35]\nand Thonig et al. [41]. The same trends are also repro-\nduced by applying the Green's function formalism from\nBhattacharjee et al. [29] (see Fig. 2). Consequently, both\nmethods | torque-torque correlation and the linear re-\nsponse Green's function method | are equivalent as it\ncan also be demonstrated not only for the moment of\ninertia but also for the Gilbert damping \u000b(see Supple-\nmentary Material)[41]. In the torque-torque correlation\nmodel (4), the coupling \u0000 de\fnes the width of the en-\nergy window in which transitions Tnmtake place. The\nGreen function approach, however, provides a more ac-\ncurate description with respect to the ab initio results\nthan the torque-torque correlation approach. This may\nbe understood from the fact that a \fnite \u0000 broadens and\nslightly shifts maxima in the spectral function. In par-\nticular, shifted electronic states at energies around the\nFermi level causes di\u000berences in the minimum of \u0013in both\nmodels. Furthermore, the moment of inertia can be re-\nsolved by an orbital decomposition and, like the Gilbert\ndamping\u000b, scales quadratically with the spin-orbit cou-\npling\u0010, caused by the torque operator ^Tin Eq. (4). Thus,\none criteria for \fnding large moments of inertia is by hav-\ning materials with strong spin-orbit coupling.\nIn order to show the region of \u0000 where the approxi-\nmation\u0013=\u0000\u000b\u0001\u001cholds, we show in Fig. 3 calculated\nvalues of\u0013,\u000b, and the resulting nutation lifetime \u001cfor a\nselection of \u0000 that are below \u0013min. According to the data\nreported in Ref. 28, this is a suitable regime accessible4\nfor experiments. To achieve the room temperature mea-\nsured experimental values of \u001c= 0:12\u00000:47 ps, we have\nfurthermore to guarantee that \u0013 >> \u000b . An appropriate\nexperimental range is \u0000 \u00195\u000010 meV, which is realistic\nand caused, e.g., by the electron-phonon coupling. A nu-\ntation lifetime of \u001c\u00190:25\u00000:1 ps is revealed for these\nvalues of \u0000 (see Fig. 3), a value similar to that found in ex-\nperiment. The aforementioned electron-phonon coupling,\nhowever, is underestimated compared to the electron-\nphonon coupling from a Debye model (\u0000 \u001950 meV) [42].\nIn addition, e\u000bects on spin disorder and electron corre-\nlation are neglected, that could lead to uncertainties in\n\u0000 and hence discrepancies to the experiment. On the\nother hand, it is not excluded that other second order\nenergy dissipation terms, Bdiss, proportional to ( @e=@t)2\nwill also contribute [32] (see Supplementary material).\nThe derivation of the moment of inertia tensor from the\nKambersk\u0013 y model and our numerics corroborates that\nrecently observed properties of the Gilbert damping will\nbe also valid for the moment of inertia: i)the moment\nof inertia is temperature dependent [41, 43] and decays\nwith increasing phonon temperature, where the later usu-\nally increase the electron-phonon coupling \u0000 in certain\ntemperature intervals [42]; ii)the moment of inertia is\na tensor, however, o\u000b-diagonal elements for bulk mate-\nrials are negligible small; iii)it is non-local [36, 41, 44]\nand depends on the magnetic moment [45{47]. Note that\nthe sign change of the moment of inertia also e\u000bects the\ndynamics of the magnetic moments (see Supplementary\nMaterial).\nThe physical mechanism of magnetic moment of inertia\nbecomes understandable from an inspection of the elec-\ntron band structure (see Fig. 4 for fcc Co, as an example).\nThe model proposed here allows to reveal the inertia k-\nand band-index nresolved contributions (integrand of\nEq. (4)). Note that we analyse for simplicity and clarity\nonly one contribution, AnBm, in the expression for Vnm.\nAs Fig. 4 shows the contribution to Vnmis signi\fcant only\nfor speci\fc energy levels and speci\fc k-points. The \fg-\nure also shows a considerable anisotropy, in the sense that\nmagnetisations aligned along the z- or y-directions give\nsigni\fcantly di\u000berent contributions. Also, a closer in-\nspection shows that degenerate or even close energy levels\nnandm, which overlap due to the broadening of energy\nlevels, e.g. as caused by electron-phonon coupling, \u0000, ac-\ncelerate the relaxation of the electron-hole pairs caused\nby magnetic moment rotation combined with the spin\norbit coupling. This acceleration decrease the moment\nof inertia, since inertia is the tendency of staying in a\nconstant magnetic state. Our analysis also shows that\nthe moment of inertia is linked to the spin-polarization\nof the bands. Since, as mentioned, the inertia preserves\nthe angular momentum, it has largest contributions in\nthe electronic structure, where multiple electron bands\nwith the same spin-polarization are close to each other\n(cf. Fig. 4 c). However, some aspects of the inertia,\n-4-3-2-10E−EF(eV)\n-4-3-2-10E−EF(eV)\nι<0\nι>0\n-4-3-2-10E−EF(eV)\nΓ H N\nk(a−1\n0)(a)\n(b)\n(c)y\nz\nFIG. 4. (Color online) Moment of inertia in the electron band\nstructure for bulk fcc Co with the magnetic moment a) in y\ndirection and b) in zdirection. The color and the intensity\nindicates the sign and value of the inertia contribution (blue\n-\u0013 <0; red -\u0013 >0; yellow - \u0013\u00190). The dotted gray line\nis the Fermi energy and \u0000 is 0 :1 eV. c) Spin polarization of\nthe electronic band structure (blue - spin down; red - spin up;\nyellow - mixed states).\ne.g. being caused by band overlaps, is similar to the\nGilbert damping [48], although the moment of inertia is\na property that spans over the whole band structure and\nnot only over the Fermi-surface. Inertia is relevant in\nthe equation of motion [17, 35] only for \u001c&0:1 ps and\nparticularly for low dimensional systems. Nevertheless,\nin the literature there are measurements, as reported in\nRef. 37, where the inertia e\u000bects are present.\nIn summary, we have derived a theoretical model for\nthe magnetic moment of inertia based on the torque-\ntorque correlation model and provided \frst-principle\nproperties of the moment of inertia that are compared\nto the Gilbert damping. The Gilbert damping and the\nmoment of inertia are both proportional to the spin-\norbit coupling, however, the basic electron band struc-5\nture mechanisms for having inertia are shown to be dif-\nferent than those for the damping. We analyze details\nof the dispersion of electron energy states, and the fea-\ntures of a band structure that are important for having\na sizable magnetic inertia. We also demonstrate that\nthe torque correlation model provides identical results\nto those obtained from a Greens functions formulation.\nFurthermore, we provide numerical values of the moment\nof inertia that are comparable with recent experimen-\ntal measurements[28]. The calculated moment of inertia\nparameter can be included in atomistic spin-dynamics\ncodes, giving a large step forward in describing ultrafast,\nsub-ps processes.\nAcknowledgements The authors thank Jonas Frans-\nson and Yi Li for fruitful discussions. The support of\nthe Swedish Research Council (VR), eSSENCE and the\nKAW foundation (projects 2013.0020 and 2012.0031) are\nacknowledged. The computations were performed on re-\nsources provided by the Swedish National Infrastructure\nfor Computing (SNIC).\n\u0003danny.thonig@physics.uu.se\n[1] S. S. P. Parkin, J. X., C. Kaiser, A. Panchula, K. Roche,\nand M. Samant, Proceedings of the IEEE 91, 661 (2003).\n[2] Y. Xu and S. Thompson, eds., Spintronic Materials and\nTechnology , Series in Materials Science and Engineering\n(CRC Press, 2006).\n[3] T. Miyamachi, T. Schuh, T. M arkl, C. Bresch, T. Bal-\nashov, A. St ohr, C. Karlewski, S. Andr\u0013 e, M. Marthaler,\nM. Ho\u000bmann, et al., Nature 503, 242 (2013).\n[4] K.-W. Kim and H.-W. Lee, Nature Mat. 14, 253 (2016).\n[5] A. V. Chumak, V. I. Vasyuchka, A. A. Serga, and\nB. Hillebrands, Nature Phys. 11, 453 (2015).\n[6] I. Tudosa, C. Stamm, A. B. Kashuba, F. King, H. C.\nSiegmann, J. St ohr, G. Ju, B. Lu, and D. Weller, Nature\n428, 831 (2004).\n[7] A. Chudnovskiy, C. H ubner, B. Baxevanis, and\nD. Pfannkuche, Physica Status Solidi (B) 251, 1764\n(2014).\n[8] S. Maekawa, S. O. Valenzuela, E. Saitoh, and T. Kimura,\neds., Spin Current (Oxford University Press, 2012).\n[9] A. V. Kimel, Nature Mat. 13, 225 (2014).\n[10] J. St ohr, H. C. Siegmann, A. Kashuba, and S. J. Gamble,\nAppl. Phys. Lett. 94, 072504 (2009).\n[11] V. P. Antropov, M. I. Katsnelson, B. N. Harmon, M. van\nSchilfgaarde, and D. Kusnezov, Phys. Rev. B 54, 1019\n(1996).\n[12] B. Skubic, J. Hellsvik, L. Nordstr om, and O. Eriksson,\nJ. Phys.: Condens. Matt. 20, 315203 (2008).\n[13] R. Chimata, L. Isaeva, K. K\u0013 adas, A. Bergman, B. Sanyal,\nJ. H. Mentink, M. I. Katsnelson, T. Rasing, A. Kirilyuk,\nA. Kimel, et al., Phys. Rev. B 92, 094411 (2015).\n[14] M. Born and V. Fock, Z. Phys. 51, 165 (1928).\n[15] T. Kato, J. Phys. Soc. Jpn. 5, 435 (1950).\n[16] T. Moriya, Spin Fluctuations in Itinerant Electron Mag-\nnetism (Springer-Verlag Berlin Heidelberg, 1985).\n[17] D. B ottcher and J. Henk, Phys. Rev. B 86, 020404(R)\n(2012).[18] M.-C. Ciornei, Ph.D. thesis, Ecole Polytechnique, Uni-\nversidad de Barcelona (2010).\n[19] W. D oring, Zeit. Naturforsch 3a, 373 (1948).\n[20] F. De Leeuw and J. M. Robertson, J. Appl. Phys. 46,\n3182 (1975).\n[21] W.-M. Zhu and Z.-G. Ye, Appl. Phys. Lett. 89, 232904\n(pages 3) (2006).\n[22] J. Fransson, J. Phys.: Condens. Matt. 19, 285714 (2008).\n[23] J. Fransson and J.-X. Zhu, New J. Phys. 10, 013017\n(2008).\n[24] M.-C. Ciornei, J. Rubi, and J.-E. Wegrowe, Phys. Rev.\nB83, 020410 (2011).\n[25] E. Olive, Y. Lansac, M. Meyer, M. Hayoun, and J. E.\nWegrowe, Journal of Applied Physics 117, 213904 (2015).\n[26] T. Kikuchi and G. Tatara, Phys. Rev. B 92, 184410\n(2015).\n[27] F. B uttner, C. Mouta\fs, M. Schneider, B. Kr uger, C. M.\nG unther, and J. Geilhufe, Nature Physics 11, 225 (2015).\n[28] Y. Li, A. L. Barra, S. Au\u000bret, U. Ebels, and W. E. Bailey,\nPhysical Review B 92, 140413 (2015).\n[29] S. Bhattacharjee, L. Nordstr om, and J. Fransson, Phys.\nRev. Lett. 108, 057204 (2012).\n[30] V. Kambersk\u0013 y, Czechoslovak Journal of Physics, Section\nB34, 1111 (1984).\n[31] K. Gilmore, Ph.D. thesis, MONTANA STATE UNIVER-\nSITY (2007).\n[32] M. F ahnle, D. Steiauf, and C. Illg, Phys. Rev. B 88,\n219905 (2013).\n[33] V. Kambersk\u0013 y, Can. J. Phys. 48, 2906 (1970).\n[34] M. F ahnle, D. Steiauf, and C. Illg, Phys. Rev. B 84,\n172403 (2011).\n[35] K. Gilmore, Y. U. Idzerda, and M. D. Stiles, Phys. Rev.\nLett.99, 027204 (2007).\n[36] A. Brataas, Y. Tserkovnyak, and G. E. W. Bauer, Phys.\nRev. Lett. 101, 037207 (2008).\n[37] Y. Zhou, V. Tiberkevich, G. Consolo, E. Iacocca,\nB. Azzerboni, A. Slavin, and J. Akerman, Phys. Rev.\nB82, 012408 (2010).\n[38]CAHMD , 2013. A computer program package for\natomistic magnetisation dynamics simulations.\nAvailable from the authors; electronic address:\ndanny.thonig@physics.uu.se.\n[39] E. Pavarini, E. Koch, and U. Schollw ock, eds., Emer-\ngent Phenomena in Correlated Matter (Forschungszen-\ntrum J ulich GmbH, Institute for Advanced Simulation,\n2013).\n[40] M. Sayad, R. Rausch, and M. Pottho\u000b,\narXiv:1602.07317v1 [cond-mat.str-el] (2016).\n[41] D. Thonig and J. Henk, New J. Phys. 16, 013032 (2014).\n[42] S. H ufner, ed., Very High Resolution Photoelectron Spec-\ntroscopy (Springer Berlin, 2007).\n[43] H. Ebert, S. Mankovsky, D. K odderitzsch, and J. P.\nKelly, Phys. Rev. Lett. 107, 066603 (2011).\n[44] K. Gilmore and M. D. Stiles, Phys. Rev. B 79, 132407\n(2009).\n[45] D. Steiauf and M. F ahnle, Phys. Rev. B 72, 064450\n(2005).\n[46] M. F ahnle and D. Steiauf, Phys. Rev. B 73, 184427\n(2006).\n[47] Z. Yuan, K. M. D. Hals, Y. Liu, A. A. Starikov,\nA. Brataas, and P. J. Kelly, Phys. Rev. Lett. 113, 266603\n(2014).\n[48] K. Gilmore, Y. U. Idzerda, and M. D. Stiles, J. Appl.\nPhys. 103, 07D303 (2008)."    },    {        "title": "1607.03633v2.Optimal_decay_rate_for_the_wave_equation_on_a_square_with_constant_damping_on_a_strip.pdf",        "content": "arXiv:1607.03633v2  [math-ph]  2 Mar 2017OPTIMAL DECAY RATE FOR THE WAVE EQUATION ON A\nSQUARE WITH CONSTANT DAMPING ON A STRIP\nREINHARD STAHN\nAbstract. We consider the damped wave equation with Dirichlet boundar y\nconditions on the unit square parametrized by Cartesian coo rdinates xandy.\nWe assume the damping ato be strictly positive and constant for x < σand\nzero for x > σ. We prove the exact t−4/3-decay rate for the energy of classical\nsolutions. Our main result (Theorem 1) answers question (1) of [1, Section\n2C.].\n1.Introduction\n1.1.The main result. Let/Box= (0,1)2be the unit square. We parametrize it by\nCartesian coordinates xandy. Leta- the damping - be a function on /Boxwhich\ndepends only on xsuch thata(x) =a0>0 forx < σanda(x) = 0 forx > σ\nwhereσis some fixed number from the interval (0 ,1). We consider the damped\nwave equation:\n\n\nutt(t,x,y)−∆u(t,x,y)+2a(x)ut(t,x,y) = 0 (t∈(0,∞),(x,y)∈/Box),\nu(t,x,y) = 0 ( t∈(0,∞),(x,y)∈∂/Box),\nu(0,x,y) =u0(x,y), ut(0,x,y) =u1(x,y) ((x,y)∈/Box).\nWe are interested in the energy\nE(t,U0) =1\n2/integraldisplay /integraldisplay\n|∇u(t,x,y)|2+|ut(t,x,y)|2dxdy\nof a wave at time twith initial data U0= (u0,u1). LetD= (H2∩H1\n0)×H1\n0(/Box)\ndenote the set of classical initial data. The purpose of this paper is to prove\nTheorem 1. Let/Box,aandE(t,U0)be as above. Then supE(t,U0)1/2≈t−2/3\nwhere the supremum is taken over initial data /ba∇dblU0/ba∇dblD= 1.\nThe exact meaning of ‘ ≈‘ and other symbols is explained in Section 2. In Section\n4 we show that this theorem is equivalent to Theorem 3 below. Section 3 is devoted\nto the proof of Theorem 3.\nRemark 2. The proof of Theorem 1 shows that a higher dimensional analog ue is\nalso true. That is, one can replace y∈Rbyy∈Rd−1for any natural number\nd≥2. The exact decay rate remains the same for all d.\nMSC2010: Primary 35B40, 47D06. Secondary 35L05, 35P20.\nKeywords and phrases: damped wave equation, piecewise cons tant damping, energy, resolvent\nestimates, polynomial decay, C0-semigroups.\n12 REINHARD STAHN\n1.2.The semigroup approach. If we setU= (u,ut) andU0= (u0,u1) we may\nformulate the damped wave equation as an abstract Cauchy proble m\n˙U(t)+AU(t) = 0,U(0) =U0whereA=/parenleftbigg\n0−1\n−∆ 2a(x)/parenrightbigg\non the Hilbert space H=H1\n0×L2(/Box). The domain of AisD(A) = (H2∩H1\n0)×\nH1\n0(/Box). Since −Ais a dissipative (we equip H1\n0(/Box) with the gradient norm) and\ninvertible operator on a Hilbert space it generates a C0-semigroup of contractions\nby the Lumer-Phillips theorem. Note that the inclusion D(A)֒→ His compact by\nthe Rellich-Kondrachovtheorem. Thus the spectrum of Acontains only eigenvalues\nof finite multiplicity.\n1.3.Classification of the main result. Our situation is a very particular in-\nstance of the so called partially rectangular situation. A bounded domain Ω is\ncalledpartially rectangular if its boundary ∂Ω is piecewise C∞and if Ω contains\nan open rectangle Rsuch that two opposite sides of Rare contained in ∂Ω. We call\nthese two opposite sides horizontal . One can decompose Ω =R∪W, whereWis an\nopen set which is disjoint to R. In our particular situation we can Wchoose to be\nempty. Furthermore it is assumed, that a>0 onWanda= 0 onS, whereS⊆R\nis an open rectangle with two sides contained in the horizontal sides o fR. To avoid\nthe discussion of null-sets we assume for simplicity that either ais continuous up\nto the boundary or it is as in subsection 1.1.\nUnder these constraints one can show that the energy of classica l solutions can\nnever decay uniformly faster than 1 /t2, i.e.\n(1) sup\nU0∈D(A)E(t,U0)1\n2/greaterorsimilar1\nt.\nThis result seems to be well-known. Unfortunately we do not know an original\nreference to this bound on the energy. A short modern proof usin g [2, Proposition\n1.3] can be found in [1]. But there is also a geometric optics proof using quantified\nversions of the techniques of [8]. Unfortunately the latter appro ach seems to be\nnever published anywhere.\nOn the other hand: If we assume that the damping does not vanish c ompletely\ninR(this is an additional assumption only if Wis empty), then\n(2) ∀U0∈D(A) :E(t,U0)1\n2/lessorsimilar1\nt1\n2.\nThis is a corollary of one of the main results in [1]. There the authors sh owed\nthatstability at rate t−1/2for anabstract damped wave equation is equivalent to\nan observability condition for a related Schr¨ odinger equation. Ear lier contributions\ntowards (2) were given by [5] and [7].\nHaving the two bounds (1) and (2) at hand a natural question arise s: Are these\nboundssharp? Concerningthe fastdecayratesrelatedto(1) th is is partlyanswered\nby[5]and[1]. Essentiallytheauthorsshowedthatifthedampingfunc tionissmooth\nenough than one can get a decay rate as close to t−1as we wish. Unfortunately\nthey could not characterize theexactdecay rate in terms of properties of a. A\nbreakthrough into this direction was achieved in [6] in a slightly differen t situation\n(thereSdegenerates to a line).\nTo the best of our knowledge it is completely unknown if the slowest po ssible\nratet−1/2is attained. To us the only known result towards this direction is due t oOPTIMAL DECAY RATE FOR THE DAMPED WAVE EQUATION 3\nNonnenmacher: If we are in the very particular situation described in subsection\n1.1 then\nsup\nU0∈D(A)E(t,U0)1\n2/greaterorsimilar1\nt2\n3.\nSee [1, Appendix B]. So this situation is a candidate for the slow decay r ate. In this\npaper we show that Nonnenmacher’s bound is actually equal to the e xact decay\nrate.\nThis of course raises a new question: Is it possible to find a non-vanis hing\nbounded dampingin apartiallyrectangulardomain, satisfyingthe con straintsspec-\nified above, but discarding the continuity assumptions, such that t he exact decay\nrate forE(t,U0)1\n2is strictly slower than t−2/3? We think this is an interesting\nquestion for future research.\n1.4.From waves to stationary waves. Letf∈L2(/Box). Now we consider the\nstationary damped wave equation with Dirichlet boundary conditions\n(3)/braceleftbiggP(s)u(x,y) = (−∆−s2+2isa(x))u(x,y) =f(x,y) in /Box\nu(x,y) = 0 on∂/Box\nAs already said above, to prove Theorem 1 is essentially to show\nTheorem 3. The operator P(s) :H2∩H1\n0(/Box)→L2(/Box)from (3) is invertible for\neverys∈R. Moreover\n/vextenddouble/vextenddoubleP(s)−1/vextenddouble/vextenddouble\nL2→L2≈1+|s|1\n2.\nActually we only prove a /lessorsimilar-inequality since the reverse inequality is a conse-\nquence of Nonnenmacher’s appendix to [1] together with Propositio n 2.4 in the\nsame paper (see Section 4 for more details). Since it is well-known we a lso do\nnot prove the invertability of P(s). The (simple) standard proof is based on test-\ning the homogeneous stationary wave equation with u. From considering real and\nimaginary part of the resulting expression one easily checks u= 0 by a unique\ncontinuation principle .\nAcknowledgments. This paper was inspired and motivated by [1, Appendix B\n(by S. Nonnenmacher)] and [3]. I am grateful to Ralph Chill for read ing and\ncorrecting the very first version of this paper.\n2.Notations and conventions\nConvention . Because of the symmetry of (3) we have/vextenddouble/vextenddoubleP(−s)−1/vextenddouble/vextenddouble\nL2→L2=/vextenddouble/vextenddoubleP(s)−1/vextenddouble/vextenddouble\nL2→L2. Therefore in the following we always assume sto bepositive.\nConstants . We use two special constants c>0 andC >0. Special means, that\nthey may change their value from line to line. The difference between t hese two\nconstants is, that their usage implicitly means that we could always re placecby a\nsmaller constant and Cby a larger constant - if this is necessary . So one should\nkeep in mind that cis a small number and Ca large number.\nLandau notation . For this subsection let us denote by φ,φ1,φ2andψcomplex\nvalued functions defined on R\\K, whereKis a compact interval. Furthermore we4 REINHARD STAHN\nalways assume φ,φ1andφ2to be real valued and (not necessary strictly) positive.\nWe define\nφ1(s)/lessorsimilarφ2(s) :⇔ ∃s0>0,C >0∀|s| ≥s0:φ1(s)≤Cφ2(s),\nφ1(s)≈φ2(s) :⇔φ1(s)/lessorsimilarφ2(s) andφ2(s)/lessorsimilarφ1(s).\nFurthermore we define the following classes (sets) of functions:\nO(φ(s)) :={ψ;|ψ(s)|/lessorsimilarφ(s)},\no(φ(s)) :={ψ;∀ε>0∃sε>0∀|s| ≥sε:|ψ(s)| ≤εφ(s)}.\nBy abuse of notation we write for example ψ(s) =O(φ(s)) instead of ψ∈O(φ(s))\norφ(s) =φ1(s)+O(φ2(s)) instead of |φ(s)−φ1(s)|/lessorsimilarφ2(s). ByO(s−∞) we denote\nthe intersection of all O(s−N) forN∈N.\nFunction spaces. As usual, by L2(Ω) we mean the space of square-integrable\nfunctions on some open subset Ω of Rnfor somen∈N. Forka natural number\nHk(Ω) denotes the space of functions from L2(Ω) whose distributional derivatives\nup to order kare square integrable, too. Finally the space H1\n0(Ω) denotes the\nclosure of the set of compactly supported smooth functions in H1(Ω). We equip\nH1\n0(Ω) with the norm (/integraltext\nΩ|∇u|2dx)1/2which is equivalent to the usual norm.\n3.Proof of Theorem 3\nHere is the plan for the proof: First we separate the y-dependence of the sta-\ntionary wave equation from the problem. As a result we are dealing wit h a family\nof one dimensional problems which are parametrized by the vertical wave number\nn∈N. Then we derive explicit solution formulasfor the separated problem s. These\nformulasallow us to estimate the solutions of the separatedproblem s by their right-\nhand side with a constant essentially depending explicitly onsandn. In the final\nstep we introduce appropriate regimes for srelative to nwhich allow us to drop\nthen-dependence of the constant by a (short) case study.\n3.1.Separation of variables. First recall that the functions sn(y) =√\n2sin(nπy)\nforn∈ {1,2,...}formacompleteorthonormalsystemof L2(0,1). Thusconsidering\nuandfsatisfying (3) we may write\nu(x,y) =∞/summationdisplay\nn=1un(x)sn(y) andf(x,y) =∞/summationdisplay\nn=1fn(x)sn(y). (4)\nIn terms of this separation of variables the stationary wave equat ion is equivalent\nto the one dimensional problem Pn(s)un=fnwhere\nPn(s) =−∂2\nx−k2\nn+2isa(x),and (5)\nk2\nn=s2−(nπ)2.\nNote thatknmight be an imaginary number. In a few lines we see that only the\nreal case is important. In that case we choose kn≥0. But first we prove the\nfollowing simple\nLemma 4. Letφ:R→(0,∞). Then the estimate/vextenddouble/vextenddoublePn(s)−1/vextenddouble/vextenddouble\nL2→L2/lessorsimilarφ(s)\nuniformly in nis equivalent to the estimate/vextenddouble/vextenddoubleP(s)−1/vextenddouble/vextenddouble\nL2→L2/lessorsimilarφ(s).OPTIMAL DECAY RATE FOR THE DAMPED WAVE EQUATION 5\nProof.LetP(s)u=fand expand uandfas in (4). Then the implication from the\nleft to the right is aconsequence ofthe followingchain ofequations a nd inequalities:\n/ba∇dblu/ba∇dbl2\nL2=∞/summationdisplay\nn=1/ba∇dblun/ba∇dbl2\nL2/lessorsimilarφ(s)2∞/summationdisplay\nn=1/ba∇dblfn/ba∇dbl2\nL2=φ(s)2/ba∇dblf/ba∇dbl2\nL2.\nThe reverse implication follows from looking at f(x,y) =fn(x)sn(y) andu(x,y) =\nun(x)sn(y). /square\nSo below we are concerned with the separated stationary wave equ ation\n(6)/braceleftbiggPn(s)un(x) =fn(x) forx∈(0,1)\nun(0) =un(1) = 0\nwherePn(s) is defined in (5). In view of Lemma 4 we are left to show /ba∇dblun/ba∇dblL2/lessorsimilar\ns1/2/ba∇dblfn/ba∇dblL2uniformly in nin order to prove Theorem 3. It turns out that such an\nestimate is easy to prove if knis imaginary. More precisely:\nLemma 5. There exists a constant c >0such that/vextenddouble/vextenddoublePn(s)−1/vextenddouble/vextenddouble\nL2→H1\n0/lessorsimilar1holds\nuniformly in nwhenevers2≤(nπ)2+c.\nNote thatPn(s)−1is considered as an operator mapping to H1\n0(0,1). But it does\nnot really matter since we will only use this estimate after replacing H1\n0byL2.\nProof.Testing equation (6) by unand taking the real part leads to\n/integraldisplay1\n0|u′\nn|2−c/integraldisplay1\n0|un|2≤/integraldisplay1\n0|fnun|.\nRecall that /ba∇dblv′/ba∇dbl2\nL2≥π2/ba∇dblv/ba∇dbl2\nL2for allv∈H1\n0(0,1) sinceπ2is the lowest eigenvalue\nof the Dirichlet-Laplacian on the unit interval. Thus the conclusion of the Lemma\nholds for all c<π2. /square\nThis lemma allows us to assume\n(7) kn=/radicalbig\ns2−(nπ)2>c\nfor some universal constant c>0 not depending on neither snorn.\n3.2.Explicit formula for Pn(s)−1.From now on we consider (6) under the con-\nstraint (7). To avoid cumbersome notation we drop the subscript nfromkn,\ni.e. we write kinstead from now on. Next let v=un|[0,σ],g=fn|(0,σ)and\nw=un|[σ,1],h=fn|(σ,1). We may write (6) as a coupled system consisting of a\nwave equation with constant damping and an undamped wave equatio n:\n(8)\n\n(−∂2\nx−k2+2isa0)v(x) =g(x) forx∈(0,σ),\n(−∂2\nx−k2)w(x) =h(x) forx∈(σ,1),\nv(0) =w(1) = 0,\nv(σ) =w(σ),v′(σ) =w′(σ).\n3.2.1.Solution of the homogeneous equation. The following ansatz satisfies the first\nthree lines of (8) with g,h= 0:\nv0(x) =1\nk′sin(k′x), w0(x) =1\nksin(k(1−x)), (9)\nwherek′is the solution of k′2=k2−2isa0which has negative imaginary part.6 REINHARD STAHN\n3.2.2.Solution of the inhomogeneous equation. The following ansatz satisfies the\nfirst three lines of (8):\nvg(x) =−1\nk′/integraldisplayx\n0sin(k′(x−y))g(y)dy, wh(x) =−1\nk/integraldisplay1\nxsin(k(y−x))h(y)dy. (10)\nThis is simply the variation of constants (or Duhamel’s) formula. It is u seful to\nknow the derivatives of these particular solutions:\nv′\ng(x) =−/integraldisplayx\n0cos(k′(x−y))g(y)dy, w′\nh(x) = +/integraldisplay1\nxcos(k(y−x))h(y)dy. (11)\n3.2.3.General solution. The general solution of the first three lines of (6) has the\nform\nv=av0+vg, w=bw0+wh. (12)\nOur task is to find the coefficients a=a(s,n) andb=b(s,n). Therefore we have to\nanalyze the coupling condition in line four of (8). A short calculation sh ows that\nit is equivalent to\n/parenleftbiggv0−w0\nv′\n0−w′\n0/parenrightbigg/vextendsingle/vextendsingle/vextendsingle/vextendsingle\nx=σ/bracehtipupleft/bracehtipdownright/bracehtipdownleft/bracehtipupright\n=:M(s,n)/parenleftbigga\nb/parenrightbigg\n=/parenleftbiggwh−vg\nw′\nh−v′\ng/parenrightbigg/vextendsingle/vextendsingle/vextendsingle/vextendsingle\nx=σ.\nFrom the preceding equation we easily deduce\na=1\ndetM/bracketleftbig\nw′\n0(vg−wh)−w0(v′\ng−wh)/bracketrightbig\nx=σ, (13)\nb=1\ndetM/bracketleftbig\nv′\n0(vg−wh)−v0(v′\ng−wh)/bracketrightbig\nx=σ. (14)\nMoreover\n(15) det M=1\nk′sin(k′σ)cos(k(1−σ)+1\nkcos(k′σ)sin(k(1−σ))).\n3.3.Proving a general estimate /ba∇dblun/ba∇dblL2≤C(k,k′,M)/ba∇dblfn/ba∇dblL2.Forthis inequal-\nity we will derive an explicitformula for Cin terms of k,k′andM. In the next\nsubsection we identify the qualitatively different regimes in which scan live. By\nregimewe mean a relation which says how big s- the full momentum - is compared\ntonπ- the momentum in y-direction. For each of these regimes we then easily\ntranslate the explicitk,k′,Mdependence of Cto a anexplicitdependence on s.\n3.3.1.Elementary estimates for w0andwh.Directly from the definition of w0(see\n(9)) we deduce\n(16) /ba∇dblw0/ba∇dbl∞≤1\nk,/ba∇dblw′\n0/ba∇dbl∞≤1 and/ba∇dblw0/ba∇dbl2≤√1−σ\nk.\nIn the same manner for whfrom (10) and (11) we deduce:\n(17)/ba∇dblwh/ba∇dbl∞≤√1−σ\nk/ba∇dblh/ba∇dbl2,/ba∇dblw′\nh/ba∇dbl∞≤√\n1−σ/ba∇dblh/ba∇dbl2and/ba∇dblwh/ba∇dbl2≤1−σ\nk/ba∇dblh/ba∇dbl2.OPTIMAL DECAY RATE FOR THE DAMPED WAVE EQUATION 7\n3.3.2.Estimating w.Recall from (12) that w=bw0+wh. Recall the formula (14)\nforb. Note that\n(v′\n0vg−v0v′\ng)(σ) =1\nk′/integraldisplayσ\n0sin(k′y)g(y)dy.\nThus it seems to be natural to decompose\nb=1\ndetM/bracketleftbig\n(v0w′\nh−v′\n0wh)+(v′\n0vg−v0v′\ng)/bracketrightbig\nx=σ\n=:b1+b2.\nThis leads to the decomposition of w=b1w0+b2w0+whinto three parts. With\nthe help of (16) and (17) each part can easily be estimated as follows :\n(18)/ba∇dblb1w0/ba∇dbl2/lessorsimilare|ℑk′|σ\n|k′detM|/parenleftbigg1\nk+|k′|\nk2/parenrightbigg\n/ba∇dblh/ba∇dbl2,\n/ba∇dblb2w0/ba∇dbl2/lessorsimilare|ℑk′|σ\n|k′detM|1\nk/ba∇dblg/ba∇dbl2,/ba∇dblwh/ba∇dbl2/lessorsimilar1\nk/ba∇dblh/ba∇dbl2.\nWe could now add all three single estimates to get the desired estimat e onwbut\nwe wait until we have done the same thing for v.\n3.3.3.Estimating v. Recall from (12) that v=av0+vh. Recall the formula (13)\nfora. Note that\n(w0w′\nh−w′\n0wh)(σ) =1\nk/integraldisplay1\nσsin(k(1−y))h(y)dyand\nvg=(−w′\n0v0+w0v′\n0)(σ)\ndetMvg=:vg,2+vg,3.\nThus it seems to be natural to decompose\na=1\ndetM/bracketleftbig\n(w0w′\nh−w′\n0wh)+w′\n0vg−w0v′\ng/bracketrightbig\nx=σ\n=:a1+a2+a3.\nThis in turn leads to a decomposition of v=a1v0+ (a2v0+vg,2) + (a3v0+vg,3)\ninto three parts. Essentially it leaves to find a good representation of the second\nand the third part of v. First let us write\na2v0+vg,2=w′\n0(σ)\nk′detM(vg(σ)sin(k′x)−k′v0(σ)vg(x))/bracehtipupleft /bracehtipdownright/bracehtipdownleft /bracehtipupright\n=:I(x),\na3v0+vg,3=w0(σ)\nk′detM/parenleftbig\n−v′\ng(σ)sin(k′x)+k′v′\n0(σ)vg(x)/parenrightbig\n/bracehtipupleft /bracehtipdownright/bracehtipdownleft /bracehtipupright\n=:II(x).\nSimple calculations yield\n−2I(x) =/integraldisplayσ\n0cos(k′(σ−x−y))g(y)dy−/integraldisplayx\n0cos(k′(σ−x+y))g(y)dy\n−/integraldisplayσ\nxcos(k′(σ+x−y))g(y)dy,8 REINHARD STAHN\nand\n2II(x) =/integraldisplayσ\nxsin(k′(σ+x−y))g(y)dy−/integraldisplayx\n0sin(k′(σ−x+y))g(y)dy\n−/integraldisplayσ\n0sin(k′(σ−x+y))g(y)dy.\nUsing this and again the elementary estimates (16) and (17) for w0andwhwe\ndeduce\n(19)/ba∇dbla3v0+vg,3/ba∇dbl2/lessorsimilare|ℑk′|σ\n|k′detM|1\nk/ba∇dblg/ba∇dbl2,\n/ba∇dbla2v0+vg,2/ba∇dbl2/lessorsimilare|ℑk′|σ\n|k′detM|/ba∇dblg/ba∇dbl2,/ba∇dbla1v0/ba∇dbl2/lessorsimilare|ℑk′|σ\n|k′detM|1\nk/ba∇dblh/ba∇dbl2.\n3.3.4.Conclusion. Putting (18) and (19) together we get the desired inequality\n(20) /ba∇dblun/ba∇dblL2/lessorsimilar/bracketleftBigg\ne|ℑk′|σ\n|k′detM|/parenleftbigg\n1+|k′|\nk2/parenrightbigg\n+1\nk/bracketrightBigg\n/ba∇dblfn/ba∇dblL2.\n3.4.Regimes where scan live. Keeping (20) in mind, our task is now to find\nasymptotic dependencies of kandk′onsand a lower bound for |k′detM|. A\npriori there is no unique asymptotic behavior of k=/radicalbig\ns2−(nπ)2asstends to\ninfinity because of k’s dependence on n. To overcome this difficulty we introduce\nthe following four regimes:\n(i)c≤k≤cs1\n2,(ii)cs1\n2≤k≤Cs1\n2,(iii)Cs1\n2≤k≤cs,(iv)cs≤k<s.\nRecall from Section 2 that c(resp.C) means a small (resp. big) number. Both\nconstants may be different in each regime. But by the convention ma de in section\n2 we may assume that consecutive regimes overlap.\nSince we want to investigate the asymptotics s→ ∞we always may assume\ns>s0for some sufficiently large number s0>0.\n3.4.1.Regime (i): c≤k≤cs1\n2.For sufficiently small cthe first order Taylor\nexpansion of the square root at 1 gives a good approximation of\nk′=√\n2a0s1\n2e−iπ\n4/parenleftbigg\n1+ik2\na0s+O(k4s−2)/parenrightbigg\n.\nIn particular ℑk′=−√a0s1\n2(1+O(k2s−1)) tends with a polynomial rate to minus\ninfinity asstends to infinity. Therefore cot( k′σ) =i+O(s−∞). Together with (15)\nthis gives us the following useful formula for\n(21) det M=sin(k′σ)\nk′/bracketleftbigg\ncos(k(1−σ))+k′\nk(i+O(s−∞))sin(k(1−σ))/bracketrightbigg\n.\nIt is not difficult to see that the term within the brackets is bounded a way from\nzero. Thus |k′detM|/greaterorsimilarexp(|ℑk′|σ). From (20) now follows (recall also (7))\n/ba∇dblun/ba∇dblL2/lessorsimilar/parenleftbigg\n1+|k′|\nk2/parenrightbigg\n/ba∇dblfn/ba∇dblL2/lessorsimilars1\n2/ba∇dblfn/ba∇dblL2uniformly in n.OPTIMAL DECAY RATE FOR THE DAMPED WAVE EQUATION 9\n3.4.2.Regime (ii): cs1\n2≤k≤Cs1\n2.Because of k′2=k2−2isa0we see that both\nℜk′and−ℑk′are of order s1\n2. Therefore (21) is valid also in this regime. Again the\ntermwithinthe bracketsisboundedawayfromzero. Thus |k′detM|/greaterorsimilarexp(|ℑk′|σ)\nand (20) imply\n/ba∇dblun/ba∇dblL2/lessorsimilar/ba∇dblfn/ba∇dblL2uniformly in n.\n3.4.3.Regime (iii): Cs1\n2≤k≤cs.Using first order Taylor expansion for the\nsquare root at 1 gives\nk′=k/parenleftbig\n1−ia0sk−2+O(s2k−4)/parenrightbig\n.\nIn particular: If we choose Cbig enough we can assume the ratio k′/kto be as\nclose to 1 as we wish. Similarly: If we choose csmall enough we may assume −ℑk′\nto be as large as we want. Therefore we may assume cot( k′σ) to be as close to ias\nwe wish. This means that the following variant of (21) is true for this r egime\ndetM=sin(k′σ)\nk′[cos(k(1−σ))+(i+ε)sin(k(1−σ))],\nwhereε∈Cis some error term with a magnitude as small as we wish. If we choose\ncandCsuch that |ε| ≤1/2 we see that the term within the brackets is bounded\naway from zero. Thus |k′detM|/greaterorsimilarexp(|ℑk′|σ) and (20) imply\n/ba∇dblun/ba∇dblL2/lessorsimilar/ba∇dblfn/ba∇dblL2uniformly in n.\n3.4.4.Regime (iv): cs≤k<s.As in the previous regime\nk′=k/parenleftbig\n1−ia0sk−2+O(s−2)/parenrightbig\n.\nIn particular k′/k= 1 +O(s−1)→1 andℑk′=−a0sk−1+O(s−1) is bounded\naway from 0 ,+∞and−∞. Thus\ndetM=1\nk′[sin(k′σ)cos(k(1−σ)+cos(k′σ)sin(k(1−σ)))]+O(s−2)\n=sin(k+(k′−k)σ)\nk′+O(s−2).\nThis implies that |k′detM| ≈1. Thus from (20) we deduce\n/ba∇dblun/ba∇dblL2/lessorsimilar/ba∇dblfn/ba∇dblL2uniformly in n.\n3.5.Conclusion. LetunsolvePn(s)un(x) =fn(x), wherePn(s) is defined in (5).\nSection 3.4 together with Lemma 5 shows that the estimate /ba∇dblun/ba∇dblL2/lessorsimilars1/2/ba∇dblfn/ba∇dblL2\nholds uniformly for any n. Therefore, Lemma 4 implies Theorem 3.\n4.Exact decay rate for the damped wave equation\nNow we want to proveTheorem 1. Therefore recallthe definition of the energyE\nand the damped wave operator Afrom Section 1. Then [4, Theorem 2.4] together\nwith [2, Proposition 1.3] restricted to our situation says in particular that for any\nα>0\n(22) sup\n/bardblU0/bardblD(A)=1E(t,U0)1\n2≈t−1\nα⇔/vextenddouble/vextenddouble(is+A)−1/vextenddouble/vextenddouble≈sα.\nIn [1, Proposition 2.4] it was shown in particular that\n(23)/vextenddouble/vextenddouble(is+A)−1/vextenddouble/vextenddouble≈sα⇔/vextenddouble/vextenddoubleP(s)−1/vextenddouble/vextenddouble\nL2→L2≈sα−1.10 REINHARD STAHN\nActually this equivalence is stated there with ‘ ≈‘ replaced by ‘ /lessorsimilar‘. But the ‘ /greaterorsimilar‘-\nversion is included in [1, Lemma 4.6]. In the appendix of [1] St´ ephane No nnen-\nmacher proved\nProposition 6 (Nonnenmacher, 2014) .The spectrum of Acontains an infinite\nsequence (zj)withℑzj→ ∞such that 0<ℜzj/lessorsimilar(ℑzj)−3/2.\nActually he proved this theorem under periodic boundary conditions , but the\nproof applies also to Dirichlet or Neumann boundary conditions. Note that Propo-\nsition 6 together with (23) establishes the ‘ /greaterorsimilar‘-inequality of Theorem 3.\nUsing (22) and (23) together with Theorem 3 yields Theorem 1.\nReferences\n[1] Nalini Anantharaman and Matthieu L´ eautaud. Sharp polynomial decay rates\nfor the damped wave equation on the torus. Anal. PDE , 7(1):159–214, 2014.\nISSN 2157-5045; 1948-206X/e. doi: 10.2140/apde.2014.7.159.\n[2] CharlesBattyandThomasDuyckaerts. Non-uniformstabilityfo rboundedsemi-\ngroups on Banach spaces. J. Evol. Equ. , 8(4):765–780, 2008. ISSN 1424-3199;\n1424-3202/e. doi: 10.1007/s00028-008-0424-1.\n[3] Charles Batty, Lassi Paunonen, and David Seifert. Optimal ene rgy decay in\na one-dimensional coupled wave–heat system. Journal of Evolution Equations ,\npages 1–16, 2016. ISSN 1424-3202. doi: 10.1007/s00028-015-0 316-0. URL\nhttp://dx.doi.org/10.1007/s00028-015-0316-0 .\n[4] Alexander Borichev and Yuri Tomilov. Optimal polynomial decay of functions\nand operator semigroups. Math. Ann. , 347(2):455–478, 2010. ISSN 0025-5831;\n1432-1807/e. doi: 10.1007/s00208-009-0439-0.\n[5] Nicolas Burq and Michael Hitrik. Energy decay for damped wave eq uations\non partially rectangular domains. Math. Res. Lett. , 14(1):35–47, 2007. ISSN\n1073-2780; 1945-001X/e. doi: 10.4310/MRL.2007.v14.n1.a3.\n[6] Matthieu L´ eautaud and Nicolas Lerner. Energy decay for a loca lly undamped\nwave equation. arXiv:1411.7271v1, 2014.\n[7] Zhuangyi Liu and Bopeng Rao. Characterization of polynomial de cay rate for\nthe solutionoflinearevolutionequation. Z. Angew. Math. Phys. , 56(4):630–644,\n2005. ISSN 0044-2275; 1420-9039/e. doi: 10.1007/s00033-004 -3073-4.\n[8] J.V. Ralston. Solutions of the wave equation with localized energy. Commun.\nPure Appl. Math. , 22:807–823, 1969. ISSN 0010-3640; 1097-0312/e. doi: 10.\n1002/cpa.3160220605.\nFachrichtungMathematik, Institutf¨ urAnalysis,TechnischeUniv ersit¨ atDresden,\n01062, Dresden, Germany. Email: Reinhard.Stahn@tu-dresden.de"    },    {        "title": "1607.04983v3.Magnetic_Skyrmion_Transport_in_a_Nanotrack_With_Spatially_Varying_Damping_and_Non_adiabatic_Torque.pdf",        "content": "1\nMagnetic Skyrmion Transport in a Nanotrack With Spatially\nVarying Damping and Non-adiabatic Torque\nXichao Zhang1,2, Jing Xia1, G. P. Zhao3, Xiaoxi Liu4, and Yan Zhou1\n1School of Science and Engineering, The Chinese University of Hong Kong, Shenzhen 518172, China\n2School of Electronic Science and Engineering, Nanjing University, Nanjing 210093, China\n3College of Physics and Electronic Engineering, Sichuan Normal University, Chengdu 610068, China\n4Department of Information Engineering, Shinshu University, Wakasato 4-17-1, Nagano 380-8553, Japan\nReliable transport of magnetic skyrmions is required for any future skyrmion-based information processing devices. Here we\npresent a micromagnetic study of the in-plane current-driven motion of a skyrmion in a ferromagnetic nanotrack with spatially\nsinusoidally varying Gilbert damping and/or non-adiabatic spin-transfer torque coefficients. It is found that the skyrmion moves in\na sinusoidal pattern as a result of the spatially varying Gilbert damping and/or non-adiabatic spin-transfer torque in the nanotrack,\nwhich could prevent the destruction of the skyrmion caused by the skyrmion Hall effect. The results provide a guide for designing\nand developing the skyrmion transport channel in skyrmion-based spintronic applications.\nIndex Terms —magnetic skyrmions, racetrack memories, micromagnetics, spintronics.\nI. I NTRODUCTION\nMagnetic skyrmions are quasiparticle-like domain-wall\nstructures with typical sizes in the sub-micrometer regime [1]–\n[7]. They are theoretically predicted to exist in magnetic metals\nhaving antisymmetric exchange interactions [8], and confirmed\nby experiments [9], [10] just after the turn of the twenty-\nfirst century. Isolated skyrmions are expected to be used to\nencode information into bits [11], which might lead to the\ndevelopment of novel spintronic applications, such as the\nracetrack memories [12]–[19], storage devices [20]–[22], and\nlogic computing devices [23].\nThe write-in and read-out processes of skyrmions in thin\nfilms are realizable and controllable at low temperatures [24]–\n[26]. A recent experiment has realized the current-induced\ncreation and motion of skyrmions in Ta/CoFeB/TaO trilayers\nat room temperature [27]. Experimental investigations have\nalso demonstrated the increased stability of skyrmions in mul-\ntilayers [28]–[30], which makes skyrmions more applicable to\npractical room-temperature applications.\nHowever, the skyrmion experiences the skyrmion Hall effect\n(SkHE) [31], [32], which drives it away from the longitudinal\ndirection when it moves in a narrow nanotrack. As a con-\nsequence, in the high-speed operation, the transverse motion\nof a skyrmion may result in its destruction at the nanotrack\nedges [18], [33]–[36]. Theoretical and numerical works have\nproposed several intriguing methods to reduce or eliminate the\ndetrimental transverse motion caused by the SkHE. For ex-\nample, one could straightforwardly enhance the perpendicular\nmagnetic anisotropy near the nanotrack edges to better confine\nthe skyrmion motion [33]. An alternative solution is to trans-\nport skyrmions on periodic substrates [37]–[40], where the\nskyrmion trajectory can be effectively controlled. Moreover, by\nconstructing antiferromagnetic skyrmions [34], [35] and anti-\nThe first two authors contributed equally to this work. Corre-\nsponding authors: X. Liu (email: liu@cs.shinshu-u.ac.jp) and Y . Zhou\n(email: zhouyan@cuhk.edu.cn).ferromagnetically exchange-coupled bilayer skyrmions [18],\n[36], the SkHE can be completely suppressed. Recently, it is\nalso found that the skyrmionium can perfectly move along\nthe driving force direction due to its spin texture with a zero\nskyrmion number [41], [42].\nIn this paper, we propose and demonstrate that a skyrmion\nguide with spatially sinusoidally varying Gilbert damping\nand/or non-adiabatic spin-transfer torque (STT) coefficients\ncan be designed for transporting skyrmions in a sinusoidal\nmanner, which is inspired by a recent study on the magnetic\nvortex guide [43], where the vortex core motion is controlled\nvia spatially varying Gilbert damping coefficient. The results\nprovide a guide for designing and developing the skyrmion\ntransport channel in future spintronic devices based on the\nmanipulation of skyrmions.\nII. M ETHODS\nOur simulation model is an ultra-thin ferromagnetic nan-\notrack with the length land the width w, where the thick-\nness is fixed at 1nm. We perform the simulation using\nthe standard micromagnetic simulator, i.e., the 1.2 alpha 5\nrelease of the Object Oriented MicroMagnetic Framework\n(OOMMF) [44]. The simulation is accomplished by a set of\nbuilt-in OOMMF extensible solver (OXS) objects. We employ\nthe OXS extension module for modeling the interface-induced\nantisymmetric exchange interaction, i.e., the Dzyaloshinskii-\nMoriya interaction (DMI) [45]. In addition, we use the updated\nOXS extension module for simulating the in-plane current-\ninduced STTs [46]. The in-plane current-driven magnetization\ndynamics is governed by the Landau-Lifshitz-Gilbert (LLG)\nequation augmented with the adiabatic and non-adiabatic\nSTTs [44], [47]\ndM\ndt=\u0000\r0M\u0002Heff+\u000b\nMS(M\u0002dM\ndt) (1)\n+u\nM2\nS(M\u0002@M\n@x\u0002M)\u0000\fu\nMS(M\u0002@M\n@x);arXiv:1607.04983v3  [cond-mat.mes-hall]  15 Dec 20162\nFig. 1. (a) The magnetic damping coefficient \u000b(x)and non-adiabatic STT\ncoefficient\f(x)as functions of xin the nanotrack. (b) Trajectories of current-\ndriven skyrmions with \f=\u000b=2 = 0:15,\f=\u000b= 0:3, and\f= 2\u000b= 0:6.\nDot denotes the skyrmion center. Red cross indicates the skyrmion destruction.\n(c) Skyrmion Hall angle \bas a function of xfor skyrmion motion with \f=\n\u000b=2 = 0:15,\f=\u000b= 0:3, and\f= 2\u000b= 0:6. The dashed lines indicate\n\b =\u000614\u000e. (e) Real-space top-views of skyrmion motion with \f=\u000b=2 =\n0:15,\f=\u000b= 0:3, and\f= 2\u000b= 0:6.wandvdenote the nanotrack\nwidth and velocity direction, respectively. The dashed line indicates the central\nline of the nanotrack. The skyrmion is destroyed at t= 870 ps when\f=\n2\u000b= 0:6. The out-of-plane magnetization component is represented by the\nred (\u0000z)-white ( 0)-green ( +z) color scale.\nwhere Mis the magnetization, MSis the saturation magne-\ntization,tis the time, \r0is the Gilbert gyromagnetic ratio,\n\u000bis the Gilbert damping coefficient, and \fis the strength of\nthe non-adiabatic STT. The adiabatic STT coefficient is given\nbyu, i.e., the conduction electron velocity. The effective field\nHeffis expressed as\nHeff=\u0000\u0016\u00001\n0@E\n@M; (2)\nwhere\u00160is the vacuum permeability constant. The average\nenergy density Econtains the exchange, anisotropy, demag-\nnetization, and DMI energies, which is given as\nE=A[r(M\nMS)]2\u0000K(n\u0001M)2\nM2\nS\u0000\u00160\n2M\u0001Hd(M) (3)\n+D\nM2\nS(Mz@Mx\n@x+Mz@My\n@y\u0000Mx@Mz\n@x\u0000My@Mz\n@y);\nwhereA,K, andDare the exchange, anisotropy, and DMI\nenergy constants, respectively. nis the unit surface normal\nvector, and Hd(M)is the demagnetization field. Mx,My\nandMzare the three Cartesian components of M.\nThe model is discretized into tetragonal volume elements\nwith the size of 2nm\u00022nm\u00021nm, which ensures a\ngood compromise between the computational accuracy and ef-\nficiency. The magnetic parameters are adopted from Refs. [14],\nFig. 2. (a)vx, (b)vy, and (c) \bas functions of \u000band\fgiven by Eq. (11)\nand Eq. (12), respectively. vxandvyare reduced by u.\n[23]:\r0= 2:211\u0002105m/(A\u0001s),A= 15 pJ/m,D= 3mJ/m2,\nK= 0:8MJ/m3,MS= 580 kA/m. In all simulations, we\nassumeu= 100 m/s andw= 50 nm. The skyrmion is initially\nlocated at the position of x= 100 nm,y= 25 nm.\nThe Gilbert damping coefficient \u000bis defined as a function\nof the longitudinal coordinate xas follows [Fig. 1(a)]\n\u000b(x) =\u000bamp\u0001f1 + sin [2\u0019(x=\u0015\u000b)]g+\u000bmin; (4)\nwhere\u000bamp= (\u000bmax\u0000\u000bmin)=2is the amplitude of the \u000b\nfunction.\u000bmaxand\u000bminstand for the maximum and mini-\nmum values of the \u000bfunction, respectively. \u0015\u000bdenotes the\nwavelength of the \u000bfunction. It is worth mentioning that the\nspatially varying \u000bcan be achieved by gradient doping of\nlanthanides impurities in ferromagnets [43], [48], [49]. Exper-\niments have found that \u000bis dependent on the interface [50].\nThus it is also realistic to construct the varying \u000bby techniques\nsuch as interface engineering. Indeed, as shown in Ref. [51],\nlocal control of \u000bin a ferromagnetic/non-magnetic thin-film\nbilayer has been experimentally demonstrated by interfacial\nintermixing induced by focused ion-beam irradiation.\nIn a similar way, the non-adiabatic STT coefficient \fis\nalso defined as a function of the longitudinal coordinate xas\nfollows [Fig. 1(a)]\n\f(x) =\famp\u0001f1 + sin [2\u0019(x=\u0015\f)\u0000']g+\fmin;(5)\nwhere\famp= (\fmax\u0000\fmin)=2is the amplitude of the \f\nfunction.\fmaxand\fminstand for the maximum and minimum\nvalues of the \ffunction, respectively. \u0015\fand'denote the\nwavelength and phase of the \ffunction, respectively. Since\nthe value of \fdepends on the material properties [52], it is\nexpected to realize the spatial varying \fby constructing a\nsuperlattice nanotrack using different materials, similar to the\nmodel given in Ref. [43]. Note that the effect of varying \f\nhas also been studied in spin torque oscillators [53].\nIII. R ESULTS\nA. Nanotrack with spatially uniform \u000band\f\nWe first recapitulate the in-plane current-driven skyrmion\nmotion in a nanotrack with spatially uniform \u000band\f. As\nshown in Fig. 1(b), the skyrmion moves along the central line\nof the nanotrack when \f=\u000b= 0:3. However, due to the\nSkHE, it shows a transverse shift toward the upper and lower\nedges when \f= 2\u000b= 0:6and\f=\u000b=2 = 0:15, respectively.3\nThe skyrmion is destroyed by touching the upper edge when\n\f= 2\u000b= 0:6att= 870 ps.\nThe skyrmion Hall angle \b, which characterizes the trans-\nverse motion of the skyrmion caused by the SkHE, is defined\nas\n\b = tan\u00001(vy=vx): (6)\nFigure 1(c) shows \bas a function of xfor the skyrmion motion\nwith\f=\u000b=2 = 0:15,\f=\u000b= 0:3, and\f= 2\u000b= 0:6. It\ncan be seen that \b = 0\u000ewhen\f=\u000b= 0:3, indicating\nthe moving skyrmion has no transverse motion [Fig. 1(d)].\nWhen\f=\u000b=2 = 0:15,\bincreases from\u000015\u000eto0\u000e,\nindicating the moving skyrmion has a transverse shift toward\nthe lower edge which is balanced by the transverse force due to\nthe SkHE and the edge-skyrmion repulsive force [Fig. 1(d)].\nWhen\f= 2\u000b= 0:6,\bdecreases from 15\u000eto3\u000ewithin\n870 ps, indicating the moving skyrmion shows a transverse\nmotion toward the upper edge. At t= 870 ps, the skyrmion\nis destroyed as it touches the upper edge of the nanotrack\n[Fig. 1(d)]. It should be noted that the skyrmion profile is\nrigid before it touches the nanotrack edge. In order to better\nunderstand the transverse motion caused by the SkHE, we also\nanalyze the in-plane current-driven skyrmion motion using the\nThiele equation [54]–[57] by assuming the skyrmion moves in\nan infinite film, which is expressed as\nG\u0002(v\u0000u) +D(\fu\u0000\u000bv) =0; (7)\nwhere G= (0;0;\u00004\u0019Q)is the gyromagnetic coupling vector\nwith the skyrmion number\nQ=1\n4\u0019Z\nm\u0001\u0012@m\n@x\u0002@m\n@y\u0013\ndxdy: (8)\nm=M=M Sis the reduced magnetization and Dis the\ndissipative tensor\nD= 4\u0019\u0012DxxDxy\nDyxDyy\u0013\n: (9)\nu= (u;0)is the conduction electron velocity, and vis the\nskyrmion velocity. For the nanoscale skyrmion studied here,\nwe have\nQ=\u00001;Dxx=Dyy= 1;Dxy=Dyx= 0: (10)\nHence, the skyrmion velocity is given as\nvx=u(\u000b\f+ 1)\n\u000b2+ 1; vy=u(\f\u0000\u000b)\n\u000b2+ 1: (11)\nThe skyrmion Hall angle \bis thus given as\n\b = tan\u00001(vy=vx) = tan\u00001\u0012\f\u0000\u000b\n\u000b\f+ 1\u0013\n: (12)\nBy calculating Eq. (11), we show vxas functions of \u000band\f\nin Fig. 2(a). vxranges between 0:5uand1:21u, indicating the\nskyrmion always moves in the +xdirection. When \u000b= 0:42\nand\f= 1,vxcan reach the maximum value of vx= 1:21u.\nSimilarly, we show vyas functions of \u000band\fin Fig. 2(b).\nvyranges between\u00000:5uandu, indicating the skyrmion can\nmove in both the \u0006ydirections. When \u000b < \f ,vy>0, the\nskyrmion shows a positive transverse motion, while when \u000b>\n\f,vy<0, the skyrmion shows a negative transverse motion.\nFig. 3. (a) Trajectories of current-driven skyrmions with \u000bamp =\n0:315;0:225;0:215.\u0015\u000b= 2wand\f= 0:3. (b) \bas a function of x\nfor skyrmion motion with \u000bamp= 0:315;0:225;0:215.\u0015\u000b= 2wand\n\f= 0:3. (c) Trajectories of current-driven skyrmions with \u0015\u000b=w;2w;4w.\n\u000bamp= 0:225 and\f= 0:3. (d)\bas a function of xfor skyrmion motion\nwith\u0015\u000b=w;2w;4w.\u000bamp= 0:225 and\f= 0:3.\nFig. 4. (a) Trajectories of current-driven skyrmions with \famp =\n0:315;0:225;0:215.\u0015\f= 2w,'= 0, and\u000b= 0:3. (b) \bas a function\nofxfor skyrmion motion with \famp= 0:315;0:225;0:215.\u0015\f= 2w,\n'= 0 , and\u000b= 0:3. (c) Trajectories of current-driven skyrmions with\n\u0015\f=w;2w;4w.\famp= 0:225,'= 0, and\u000b= 0:3. (d)\bas a function\nofxfor skyrmion motion with \u0015\f=w;2w;4w.\famp= 0:225,'= 0, and\n\u000b= 0:3.\nBy calculating Eq. (12), we also show \bas functions of \u000b\nand\fin Fig. 2(c), where \bvaries between \b = 45\u000eand\n\b =\u000045\u000e. Obviously, one has \b = 0\u000e,\b<0\u000e, and \b>0\u000e\nfor\u000b=\f,\u000b>\f , and\u000b<\f , respectively, which agree with\nthe simulation results for the nanotrack when the edge effect\nis not significant, i.e., when the skyrmion moves in the interior\nof the nanotrack. For example, using Eq. (12), the skyrmion\nhas\b = 14\u000eand\b =\u000014\u000efor\f= 2\u000b= 0:6and\f=\n\u000b=2 = 0:15, respectively, which match the simulation results\natt\u00180ps where the edge effect is negligible [Fig. 1(c)].\nB. Nanotrack with spatially varying \u000bor\f\nWe first demonstrate the in-plane current-driven skyrmion\nmotion in a nanotrack with spatially varying \u000band spatially\nuniform\f, i.e.,\u000bis a function of x, as in Eq. (4), and \f=\n0:3. Figure 3(a) shows the trajectories of the current-driven\nskyrmions with different \u000b(x)functions where \u0015\u000b= 2wand\n\f= 0:3. For\u000bmax= 0:75,\u000bmin= 0:12, i.e.,\u000bamp= 0:315,\nthe skyrmion moves in the rightward direction in a sinusoidal\npattern. For \u000bmax= 0:6,\u000bmin= 0:15, i.e.,\u000bamp= 0:225, the\nmaximum transverse shift of skyrmion is reduced in compared\nto that of\u000bamp= 0:315. For\u000bmax= 0:45,\u000bmin= 0:2, i.e.,\n\u000bamp= 0:125, the amplitude of the skyrmion trajectory further4\nFig. 5. Trajectories of current-driven skyrmions with '= 0\u00182\u0019.\u000bamp=\n\famp= 0:225 and\u0015\u000b=\u0015\f= 2w.\ndecreases. \bas a function of xcorresponding to Fig. 3(a) for\ndifferent\u000b(x)functions are given in Fig. 3(b). Figure 3(c)\nshows the trajectories of the current-driven skyrmions with\ndifferent\u0015\u000bwhere\u000bamp= 0:225and\f= 0:3.\bas a function\nofxcorresponding to Fig. 3(c) for different \u0015\u000bare given in\nFig. 3(d).\nWe then investigate the in-plane current-driven skyrmion\nmotion in a nanotrack with spatially uniform \u000band spatially\nvarying\f, i.e.,\fis a function of x, as in Eq. (5), and \u000b=\n0:3. Figure 4(a) shows the trajectories of the current-driven\nskyrmions with different \f(x)functions where \u0015\f= 2w,'=\n0and\u000b= 0:3. The results are similar to the case with spatially\nvarying\u000b. For\fmax= 0:75,\fmin= 0:12, i.e.,\famp= 0:315,\nthe skyrmion moves in the rightward direction in a sinusoidal\npattern. For \fmax= 0:6,\fmin= 0:15, i.e.,\famp= 0:225, the\nmaximum transverse shift of skyrmion is reduced in compared\nto that of\famp= 0:315. For\fmax= 0:45,\fmin= 0:2, i.e.,\n\famp= 0:125, the amplitude of the skyrmion trajectory further\ndecreases. \bas a function of xcorresponding to Fig. 4(a) for\ndifferent\f(x)functions are given in Fig. 4(b). Figure 4(c)\nshows the trajectories of the current-driven skyrmions with\ndifferent\u0015\fwhere\famp= 0:225and\u000b= 0:3.\bas a function\nofxcorresponding to Fig. 4(c) for different \u0015\fare given in\nFig. 4(d).\nFrom the skyrmion motion with spatially varying \u000bor\nspatially varying \f, it can be seen that the amplitude of\ntrajectory is proportional to \u000bampor\famp. The wavelength of\ntrajectory is equal to \u0015\u000b;\f, while the amplitude of trajectory is\nproportional to \u0015\u000b;\f.\balso varies with xin a quasi-sinusoidal\nmanner, where the peak value of \b(x)is proportional to \u000bamp,\n\famp, and\u0015\u000b;\f. As shown in Fig. 2(c), when \fis fixed at a\nvalue between \u000bmaxand\u000bmin, larger\u000bampwill lead to larger\npeak value of \b(x). On the other hand, a larger \u0015\u000b;\fallows\na longer time for the skyrmion transverse motion toward a\ncertain direction, which will result in a larger amplitude of\ntrajectory as well as a larger peak value of \b(x).\nFig. 6. \bas a function of xfor skyrmion motion with '= 0\u00182\u0019.\n\u000bamp=\famp= 0:225 and\u0015\u000b=\u0015\f= 2w.\nC. Nanotrack with spatially varying \u000band\f\nWe also demonstrate the in-plane current-driven skyrmion\nmotion in a nanotrack with both spatially varying \u000band\f,\ni.e., both\u000band\fare functions of x, as given in Eq. (4) and\nEq. (5), respectively.\nFigure 5 shows the trajectories of the current-driven\nskyrmions with spatially varying \u000band\fwhere\u000bamp=\n\famp= 0:225and\u0015\u000b=\u0015\f= 2w. Here, we focus on the effect\nof the phase difference between the \u000b(x)and\f(x)functions.\nFor'= 0 and'= 2\u0019, as the\u000b(x)function is identical to\nthe\f(x)function, the skyrmion moves along the central line\nof the nanotrack. For 0<'< 2\u0019, as\u000b(x)could be different\nfrom\f(x)at a certainx, it is shown that the skyrmion moves\ntoward the right direction in a sinusoidal pattern, where the\nphase of trajectory is subject to '. Figure 6 shows \bas a\nfunction of xcorresponding to Fig. 5 for '= 0\u00182\u0019where\n\u000bamp=\famp= 0:225 and\u0015\u000b=\u0015\f= 2w. It shows that\n\b = 0\u000ewhen'= 0 and'= 2\u0019, while it varies with xin\na quasi-sinusoidal manner when 0<'< 2\u0019. The amplitude\nof trajectory as well as the peak value of \b(x)reach their\nmaximum values when '=\u0019.\nIV. C ONCLUSION\nIn conclusion, we have shown the in-plane current-driven\nmotion of a skyrmion in a nanotrack with spatially uniform\n\u000band\f, where \bis determined by \u000band\f, which can vary\nbetween \b = 45\u000eand\b =\u000045\u000ein principle. Then, we\nhave investigated the in-plane current-driven skyrmion motion\nin a nanotrack with spatially sinusoidally varying \u000bor\f.\nThe skyrmion moves on a sinusoidal trajectory, where the\namplitude and wavelength of trajectory can be controlled by\nthe spatial profiles of \u000band\f. The peak value of \b(x)is\nproportional to the amplitudes and wavelengths of \u000b(x)and\n\f(x). In addition, we have demonstrated the in-plane current-\ndriven skyrmion motion in a nanotrack having both spatially\nsinusoidally varying \u000band\fwith the same amplitude and\nwavelength. The skyrmion moves straight along the central5\nline of the nanotrack when \u000b(x)and\f(x)have no phase\ndifference, i.e., '= 0. When'6= 0, the skyrmion moves\nin a sinusoidal pattern, where the peak value of \b(x)reaches\nits maximum value when '=\u0019. This work points out the\npossibility to guide and control skyrmion motion in a nan-\notrack by constructing spatially varying parameters, where the\ndestruction of skyrmion caused by the SkHE can be prevented,\nwhich enables reliable skyrmion transport in skyrmion-based\ninformation processing devices.\nACKNOWLEDGMENT\nX.Z. was supported by JSPS RONPAKU (Dissertation\nPh.D.) Program. G.P.Z. was supported by the National Natural\nScience Foundation of China (Grants No. 11074179 and No.\n10747007), and the Construction Plan for Scientific Research\nInnovation Teams of Universities in Sichuan (No. 12TD008).\nY .Z. was supported by the Shenzhen Fundamental Research\nFund under Grant No. JCYJ20160331164412545.\nREFERENCES\n[1] H.-B. Braun, Adv. Phys. 61, 1 (2012).\n[2] N. Nagaosa and Y . Tokura, Nat. Nanotechnol. 8, 899 (2013).\n[3] Y .-H. Liu and Y .-Q. Li, Chin. Phys. B 24, 17506 (2015).\n[4] R. Wiesendanger, Nat. Rev. Mat. 1, 16044 (2016).\n[5] W. Kang, Y . Huang, X. Zhang, Y . Zhou, and W. Zhao, Proc. IEEE 104,\n2040 (2016).\n[6] G. Finocchio, F. Büttner, R. Tomasello, M. Carpentieri, and M. Kläui,\nJ. Phys. D: Appl. Phys. 49, 423001 (2016).\n[7] S. Seki and M. Mochizuki, Skyrmions in Magnetic Materials (Springer,\nSwitzerland, 2016).\n[8] U. K. Roszler, A. N. Bogdanov, and C. Pfleiderer, Nature 442, 797\n(2006).\n[9] S. Mühlbauer, B. Binz, F. Jonietz, C. Pfleiderer, A. Rosch, A. Neubauer,\nR. Georgii, and P. Böni, Science 323, 915 (2009).\n[10] X. Z. Yu, Y . Onose, N. Kanazawa, J. H. Park, J. H. Han, Y . Matsui, N.\nNagaosa, and Y . Tokura, Nature 465, 901 (2010).\n[11] A. Fert, V . Cros, and J. Sampaio, Nat. Nanotechnol. 8, 152 (2013).\n[12] S. S. P. Parkin, M. Hayashi, and L. Thomas, Science 320, 190 (2008).\n[13] J. Iwasaki, M. Mochizuki, and N. Nagaosa, Nat. Nanotechnol. 8, 742\n(2013).\n[14] J. Sampaio, V . Cros, S. Rohart, A. Thiaville, and A. Fert, Nat. Nan-\notechnol. 8, 839 (2013).\n[15] R. Tomasello, E. Martinez, R. Zivieri, L. Torres, M. Carpentieri, and G.\nFinocchio, Sci. Rep. 4, 6784 (2014).\n[16] X. Zhang, G. P. Zhao, H. Fangohr, J. P. Liu, W. X. Xia, J. Xia, and F.\nJ. Morvan, Sci. Rep. 5, 7643 (2015).\n[17] H. Du, R. Che, L. Kong, X. Zhao, C. Jin, C. Wang, J. Yang, W. Ning,\nR. Li, C. Jin, X. Chen, J. Zang, Y . Zhan, and M. Tian, Nat. Commun.\n6, 8504 (2015).\n[18] X. Zhang, Y . Zhou, and M. Ezawa, Nat. Commun. 7, 10293 (2016).\n[19] J. Müller, A. Rosch, and M. Garst, New J. Phys. 18, 065006 (2016).\n[20] M. Beg, R. Carey, W. Wang, D. Cortés-Ortuño, M. V ousden, M.-A.\nBisotti, M. Albert, D. Chernyshenko, O. Hovorka, R. L. Stamps, and H.\nFangohr, Sci. Rep. 5, 17137 (2015).\n[21] D. Bazeia, J.G.G.S. Ramos, and E.I.B. Rodrigues, J. Magn. Magn.\nMater. 423, 411 (2017).\n[22] H. Y . Yuan and X. R. Wang, Sci. Rep. 6, 22638 (2016).\n[23] X. Zhang, M. Ezawa, and Y . Zhou, Sci. Rep. 5, 9400 (2015).\n[24] N. Romming, C. Hanneken, M. Menzel, J. E. Bickel, B. Wolter, K.\nBergmannvon, A. Kubetzka, and R. Wiesendanger, Science 341, 636\n(2013).\n[25] C. Hanneken, F. Otte, A. Kubetzka, B. Dupé, N. Romming, K. Bergman-\nnvon, R. Wiesendanger, and S. Heinze, Nat. Nanotechnol. 10, 1039\n(2015).\n[26] D. M. Crum, M. Bouhassoune, J. Bouaziz, B. Schweflinghaus, S. Blüel,\nand S. Lounis, Nat. Commun. 6, 8541 (2015).\n[27] W. Jiang, P. Upadhyaya, W. Zhang, G. Yu, M. B. Jungfleisch, F. Y .\nFradin, J. E. Pearson, Y . Tserkovnyak, K. L. Wang, O. Heinonen, S. G.\nE. te Velthuis, and A. Hoffmann, Science 349, 283 (2015).[28] S. Woo, K. Litzius, B. Kruger, M.-Y . Im, L. Caretta, K. Richter, M.\nMann, A. Krone, R. M. Reeve, M. Weigand, P. Agrawal, I. Lemesh,\nM.-A. Mawass, P. Fischer, M. Klaui, and G. S. D. Beach, Nat. Mater.\n15, 501 (2016).\n[29] C. Moreau-Luchaire, C. Moutafis, N. Reyren, J. Sampaio, C. A. F. Vaz,\nN. Van Horne, K. Bouzehouane, K. Garcia, C. Deranlot, P. Warnicke, P.\nWohlhüter, J.-M. George, M. Weigand, J. Raabe, V . Cros, and A. Fert,\nNat. Nanotechnol. 11, 444 (2016).\n[30] O. Boulle, J. V ogel, H. Yang, S. Pizzini, D. de Souza Chavesde, A.\nLocatelli, T. O. Mente¸ s, A. Sala, L. D. Buda-Prejbeanu, O. Klein, M.\nBelmeguenai, Y . Roussigné, A. Stashkevich, S. M. Chérif, L. Aballe,\nM. Foerster, M. Chshiev, S. Auffret, I. M. Miron, and G. Gaudin, Nat.\nNanotechnol. 11, 449 (2016).\n[31] J. Zang, M. Mostovoy, J. H. Han, and N. Nagaosa, Phys. Rev. Lett. 107,\n136804 (2011).\n[32] W. Jiang, X. Zhang, G. Yu, W. Zhang, X. Wang, M. B. Jungfleisch, J. E.\nPearson, X. Cheng, O. Heinonen, K. L. Wang, Y . Zhou, A. Hoffmann,\nand S. G. E. te Velthuis, Nat. Phys. advance online publication , 19\nSeptember 2016 (doi:10.1038/nphys3883).\n[33] I. Purnama, W. L. Gan, D. W. Wong, and W. S. Lew, Sci. Rep. 5, 10620\n(2015).\n[34] X. Zhang, Y . Zhou, and M. Ezawa, Sci. Rep. 6, 24795 (2016).\n[35] J. Barker and O. A. Tretiakov, Phys. Rev. Lett. 116, 147203 (2016).\n[36] X. Zhang, M. Ezawa, and Y . Zhou, Phys. Rev. B 94, 064406 (2016).\n[37] C. Reichhardt and C. J. Olson Reichhardt, Phys. Rev. B 94, 094413\n(2016).\n[38] C. Reichhardt, D. Ray, and C. J. Olson Reichhardt, Phys. Rev. B 91,\n104426 (2015).\n[39] C. Reichhardt and C. J. Olson Reichhardt, Phys. Rev. B 92, 224432\n(2015).\n[40] C. Reichhardt, D. Ray, and C. J. Olson Reichhardt, New J. Phys. 17,\n073034 (2015).\n[41] X. Zhang, J. Xia, Y . Zhou, D. Wang, X. Liu, W. Zhao, and M. Ezawa,\nPhys. Rev. B 94, 094420 (2016).\n[42] S. Komineas and N. Papanicolaou, Phys. Rev. B 92, 174405 (2015).\n[43] H. Y . Yuan and X. R. Wang, AIP Adv. 5, 117104 (2015).\n[44] M. J. Donahue and D. G. Porter, OOMMF User’s Guide, Version 1.0,\nInteragency Report NO. NISTIR 6376, National Institute of Standards\nand Technology, Gaithersburg, MD (1999) http://math.nist.gov/oommf.\n[45] S. Rohart and A. Thiaville, Phys. Rev. B 88, 184422 (2013).\n[46] The updated OXS extension module can be downloaded at\nhttps://sites.google.com/site/xichaozhang/micromagnetics/oommf-\noxs-extensions.\n[47] A. Thiaville, Y . Nakatani, J. Miltat, and Y . Suzuki, Europhys. Lett. 69,\n990 (2005).\n[48] S. G. Reidy, L. Cheng, and W. E. Bailey, Appl. Phys. Lett. 82, 1254\n(2003).\n[49] J. He and S. Zhang, Appl. Phys. Lett. 90, 142508 (2007).\n[50] R. Urban, G. Woltersdorf, and B. Heinrich, Phys. Rev. Lett. 87, 217204\n(2001).\n[51] J. A. King, A. Ganguly, D. M. Burn, S. Pal, E. A. Sallabank, T. P. A.\nHase, A. T. Hindmarch, A. Barman, and D. Atkinson, Appl. Phys. Lett.\n104, 242410 (2014).\n[52] K. Gilmore, I. Garate, A. H. MacDonald, and M. D. Stiles, Phys. Rev.\nB84, 224412 (2011).\n[53] Y . Zhou and J. Åkerman, Appl. Phys. Lett. 94, 112503 (2009).\n[54] A. A. Thiele, Phys. Rev. Lett. 30, 230 (1973).\n[55] T. Schulz, R. Ritz, A. Bauer, M. Halder, M. Wagner, C. Franz, C.\nPfleiderer, K. Everschor, M. Garst, and A. Rosch, Nat. Phys. 8, 301\n(2012).\n[56] K. Everschor, M. Garst, R. A. Duine, and A. Rosch, Phys. Rev. B 84,\n064401 (2011).\n[57] J. Iwasaki, M. Mochizuki, and N. Nagaosa, Nat. Commun. 4, 1463\n(2013)."    },    {        "title": "1608.00984v2.Ferromagnetic_Damping_Anti_damping_in_a_Periodic_2D_Helical_surface__A_Non_Equilibrium_Keldysh_Green_Function_Approach.pdf",        "content": "arXiv:1608.00984v2  [cond-mat.mes-hall]  13 Aug 2016Ferromagnetic Damping/Anti-damping in a Periodic 2D Helic al surface; A\nNon-Equilibrium Keldysh Green Function Approach\nFarzad Mahfouzi1,∗and Nicholas Kioussis1\n1Department of Physics, California State University, North ridge, California 91330-8268, USA\nIn this paper, we investigate theoretically the spin-orbit torque as well as the Gilbert damping for\na two band model of a 2D helical surface state with a Ferromagn etic (FM) exchange coupling. We\ndecompose the density matrix into the Fermi sea and Fermi sur face components and obtain their\ncontributions to the electronic transport as well as the spi n-orbit torque (SOT). Furthermore, we\nobtain the expression for the Gilbert damping due to the surf ace state of a 3D Topological Insulator\n(TI) and predicted its dependence on the direction of the mag netization precession axis.\nPACS numbers: 72.25.Dc, 75.70.Tj, 85.75.-d, 72.10.Bg\nI. INTRODUCTION\nThe spin-transfer torque (STT) is a phenomenon in\nwhich spin current of large enough density injected into\na ferromagnetic layer switches its magnetization from\none static configuration to another [1]. The origin of\nSTT is absorption of itinerant flow of angular momen-\ntum components normal to the magnetization direc-\ntion. It represents one of the central phenomena of the\nsecond-generation spintronics, focused on manipulation\nof coherent spin states, since reduction of current den-\nsities (currently of the order 106-108A/cm2) required\nfor STT-based magnetization switching is expected to\nbring commercially viable magnetic random access mem-\nory (MRAM) [2]. The rich nonequilibrium physics [3]\narising in the interplay of spin currents carried by fast\nconduction electrons and collective magnetization dy-\nnamics, viewed as the slow classical degree of freedom,\nis of great fundamental interest.\nVery recent experiments [4, 5] and theoretical stud-\nies[6] havesoughtSTT innontraditionalsetupswhich do\nnot involvetheusual two(spin-polarizingandfree) F lay-\ners with noncollinear magnetizations [3], but rely instead\non the spin-orbit coupling (SOC) effects in structures\nlacking inversion symmetry. Such “SO torques” [7] have\nbeen detected [4] in Pt/Co/AlO xlateral devices where\ncurrent flows in the plane of Co layer. Concurrently, the\nrecent discovery [8] of three-dimensional (3D) topologi-\ncal insulators (TIs), which possess a usual band gap in\nthe bulk while hosting metallic surfaces whose massless\nDirac electrons have spins locked with their momenta\ndue to the strong Rashba-type SOC, has led to theoreti-\ncal proposals to employ these exotic states of matter for\nspintronics [9] and STT in particular [10]. For example,\nmagnetizationofa ferromagneticfilm with perpendicular\nanisotropy deposited on the TI surface could be switched\nby interfacial quantum Hall current [10].\nIn this paper, we investigate the dynamical properties\nof a FM/3DTI heterostructure, where the F overlayer\n∗farzad.mahfouzi@gmail.comcovers a TI surface and the device is periodic along in-\nplanex−ydirections. The effect of the F overlayer is a\nproximityinduced exchangefield −∆surf/vector m·/vectorσ/2superim-\nposed on the Dirac cone dispersion. For a partially cov-\nered FM/TI heterostructure, the spin-momentum-locked\nDirac electrons flip their spin upon entering into the in-\nterface region, thereby inducing a large antidamping-like\nSOT on the FM [15–17]. The antidamping-like SOT\ndriven by this mechanism which is unique to the sur-\nface of TIs has been predicted in Ref. [17], where a time-\ndependent nonequilibrium Green function [18] (NEGF)-\nbased framework was developed. The formalism made it\npossible to separate different torque components in the\npresenceofarbitraryspin-flipprocesseswithinthedevice.\nSimilaranti-dampingtorqueshasalsobeen predicted[19]\nto exist due to the Berry phase in periodic structures\nwhere the device is considered infinite in in-plane direc-\ntions and a Kubo formula was used to describe the SOT\nas a linear response to homoginiuos electric field at the\ninterface. However, the connection between the two ap-\nproaches is not clear and one of the goals of the current\npaper is to address the similarities and the differences\nbetween the two. In the following we present the theo-\nretical formalism of the SOT and damping in the regime\nofslowlyvaryingparametersofaperiodicsysteminspace\nand time.\nGenerally, in a quantum system with slowly varying\nparameters in space and/or time, the system stays close\ntoits equilibrium state ( i.e.adiabaticregime)and the ef-\nfects of the nonadiabaticity is taken into account pertur-\nbatively using adiabatic expansion. Conventionally, this\nexpansion is performed using Wigner representation [20]\nafter the separation of the fast and slow variations in\nspace and/or time. [21] The slow variation implies that\nthe NEGFs vary slowly with the central space ( time ),\n/vector xc= (/vector x+/vector x′)/2(tc= (t+t′)/2 ), while they changefast\nwith the relative space (time), /vector xr=/vector x−/vector x′(tr=t−t′).\nHere we use an alternative approach, where we consider\n(x,t) and (/vector xr,tr) as the natural variables to describe the\nclose to adiabatic apace-time evolution of NEGFs and\nthen perform the following Fourier transform\nˇG(/vector xt;/vector x′t′) =/integraldisplaydE\n2πd/vectork\nΩkeiE(t−t′)+i/vectork·(/vector x−/vector x′)ˇG/vectorkE(/vector xt).(1)2\nwhere, Ω kis the volume of the phase space that the\n/vectork-integration is being performed. The standard Dyson\nequation of motion for ˇG(/vector xt;/vector x′t′) is cumbersome to ma-\nnipulate[22,23]orsolvenumerically,[24]sotheyareusu-\nally transformed to some other representation.[11] Gen-\neralizing the equation to take into account slowly varying\ntime and spatial dependence of the Hamiltonian we ob-\ntain,\nˇG=/parenleftbigg\nGrG<\n0Ga/parenrightbigg\n, (2)\n=/parenleftbigg\nGr,−1\nad−iDxtΣ<\n0 Ga,−1\nad−iDxt/parenrightbigg−1\n,\nwhere,\nGr,−1\nad= (E−iη)1−H(/vectork,t)−µ(/vector x),(3a)\nΣ<=−2iηf(E−i∂\n∂t−µ(/vector x)), (3b)\nDxt=∂\n∂t+∂H\n∂/vectork·/vector∇, (3c)\nand,η=/planckover2pi1/2τis the phenomenological broadening pa-\nrameter, where τis the relaxation time. It is worth\nmentioning that for a finite ηthe number of particles\nis not conserved, and a more accurate interpretation of\nthe introduced broadening might be to consider it as an\nenergy-independent scape rate of electrons to fictitious\nreservoirs attached to the positions /vector x. Consequently, a\nfinite broadening could be interpreted as the existence\nof an interface in the model between each atom in the\nsystem and the reservoir that is spread homogeneously\nalong the infinite periodic system.\nEq. (2) shows that the effect of the space/time varia-\ntion is to replace E→E−i∂/∂tand/vectork→/vectork−i/vector∇in the\nequation of motion for the GFs in stationary state. To\nthe lowest order with respect to the derivatives we can\nwrite,\nˇG=ˇGad−i∂ˇGad\n∂E∂ˇG−1\nad\n∂tˇGad−i∂ˇGad\n∂/vectork·/vector∇ˇG−1\nadˇGad,\n(4)\nwhere,\nˇG−1\nad=/parenleftbigg\nGr,−1\nad−2iηf(E−µ(/vector x))\n0 Ga,−1\nad/parenrightbigg\n.(5)\nFor the density matrix of the system, ρ(t) =1\niG<(t,t),\nwe obtain,\nρneq\n/vectork,t≈ −/integraldisplaydE\n2πℜ/parenleftbigg\n[D(Gr\nad),Gr\nad]f+2iηD(Gr\nad)Ga\nad∂f\n∂E/parenrightbigg\n(6)\nwhereD=∂\n∂t−/vector∇µ·∂\n∂/vectorkis the differential operator act-\ning on the slowly varying parameters in space and time.The details of the derivation is presented in Appendix.A.\nThe density matrix in Eq. (6) is the central formula of\nthe paper and consists of two terms; the first term con-\ntains the equilibrium Fermi distribution function from\nthe electrons bellow the Fermi surface occupying a slowly\n(linearly) varying single particle states that has only in-\nterband contributions and can as well be formulated in\nterms of the Berry phase as we will show the following\nsections, and; the second term corresponds to the elec-\ntrons with Fermi energy (at zero temperature we have,\n∂f/∂E=δ(E−EF)) which are the only electrons al-\nlowed to get excited in the presence of the slowly varying\nperturbations. The fact that the first term originates\nfrom the assumption that the electric field is constant\ninside the metallic FM suggests that this term might dis-\nappearoncethescreeningeffect isincluded. Onthe other\nhand, duetothe factthat thesecondtermcorrespondsto\nthe nonequilibrium electrons injected from the fictitious\nreservoirs attached to the device through the scape rate\nη, it might capture the possible physical processes that\noccur at the contact region and makes it more suitable\nfor the calculation of the relevant physical observables in\nsuch systems.\nUsing the expression for the nonequilibrium density\nmatrix the local spin density can be obtained from,\n/vectorSneq(t) =/angb∇acketleft/vector σ/angb∇acket∇ightneq=1\n4π2/integraldisplay\nd2/vectorkTr[ρneq\n/vectork,t/vector σ],(7)\nwhere/angb∇acketleft.../angb∇acket∇ightneqrefers to the ensemble average over many-\nbody states out of equilibrium demonstrated by the\nnonequilibrium density matrix of the electrons and, Tr\nrefers to the trace. In this case the time derivative in the\ndifferential operator Dleads to the damping of the dy-\nnamics of the ferromagnet while the momentum deriva-\ntive leads to either damping or anti-damping of the FM\ndynamics depending on the direction of the applied elec-\ntric field. In the followingsection we apply the formalism\nto a two band helical surface state model attached to a\nFM.\nII. SOT AND DAMPING OF A HELICAL 2D\nSURFACE\nA two band Hamiltonian model for the system can be\ngenerally written as,\nH(/vectork,t) =ε0(/vectork)1+/vectorh(/vectork,t)·/vectorσ (8)\nwhere,/vectorh=/vectorhso(/vectork)+∆xc(/vectork)\n2/vector m(t), with/vectorhso(/vectork) =−/vectorhso(−/vectork)\nand ∆ xc(/vectork) = ∆ xc(−/vectork) being spin-orbit and magnetic\nexchange coupling terms respectively. In particular in\nthe case of Rashba type helical states we have /vectorhso=\nαsoˆez×/vectork. In this case for the adiabatic single particle\nGF we have,\nGr\nad(E,t) =(E−ε0−iη)1+/vectorh·/vectorσ\n(E−ε0−iη)2−|/vectorh|2(9)3\nFrom Eq. (7) for the local spin density, we obtain (See\nAppendix B for details),\n/vectorSneq(t) =/integraldisplayd2/vectork\n4π2/parenleftBigg/vectorh×D/vectorh\n2|/vectorh|3(f1−f2)−(/vector∇µ·/vector v0)/vectorh\n2η|/vectorh|(f′\n1−f′\n2)\n+(/vectorh×D/vectorh\n2|/vectorh|2+ηD/vectorh−1\nη(/vectorh·D/vectorh)/vectorh\n2|/vectorh|2)(f′\n1+f′\n2)/parenrightBigg\n(10)\nwhere,f1,2=f(ε0± |/vectorh|) and/vector v0=∂ε0/∂/vectorkis the group\nvelocity of electrons in the absence of the SOI. Here, we\nassumeη≪ |/vectorh|which corresponds to a system close to\nthe ballistic regime. In this expression we kept the ηD/vectorh\nbecause of its unique vector orientation characteristics.\nAs it becomes clear in the following, the first term in\nEq. (10) is a topological quantity which in the presence\nof an electric field becomes dissipative and leads to an\nanti-damping torque. The second term in this expression\nleads to the Rashba-Edelstein field-like torque which is a\nnondissipative observable. The third term has the exact\nformasthe firstterm with the difference that it is strictly\na Fermi surface quantity. The fourth term, also leads to\na field like torque that as we will see in the following has\nsimilar features as the Rashba-Edelstein effect. It is im-\nportant to pay attention that unlike the first term, the\nrest of the terms in Eq. (10) are solely due to the flow\nof the non-equilibrium electrons on the Fermi surface.\nFurthermore, we notice that the terms that lead to dissi-\npation in the presence of an electric field ( D ≡/vector∇µ·∂\n∂/vectork)\nbecome nondissipative when we consider D ≡∂/∂tand\nvice versa.\nA. Surface State of a 3D-TI\nIn the case of the surface state of a 3D-TI, as an ap-\nproximation we can ignore ε0(/vectork) and consider the helical\nterm as the only kinetic term of the Hamiltonian. In this\ncasethelocalchargecurrentandthenonequilibriumlocal\nspin density share a similar expression, /vectorI=/angb∇acketleft∂(/vectorh·/vector σ)/∂/vectork/angb∇acket∇ight.\nFor the conductivity, analogous to Eq. (10), we obtain,\nσij=e/integraldisplayd2/vectork\n4π2\n/vectorh·∂/vectorh\n∂ki×∂/vectorh\n∂kj\n2|/vectorh|2/parenleftBigg\nf1−f2\n|/vectorh|+f′\n1+f′\n2/parenrightBigg\nδi/negationslash=j\n+−η|∂/vectorh\n∂ki|2+1\nη(∂|/vectorh|2\n∂ki)2\n2|/vectorh|2(f′\n1+f′\n2)δij\n(11)\nThis shows that the Fermi sea component of the density\nmatrixcontributesonlytotheanomalousHallconductiv-\nity which is in terms of a winding number. On the other\nhand, the second term is finite only for the longitudinal\ncomponents of the conductivity and can be rewritten in\nterms of the group velocity of the electrons in the system\nwhich leads to the Drude-like formula.Should the linear dispersion approximation for the ki-\nnetic term in the Hamiltonian be valid in the range of\nthe energy scale corresponding to the magnetic exchange\ncoupling ∆ xc(i.e. when vF≫∆xc), the effect of the in-\nplane component of the magnetic exchange coupling is to\nshift the Dirac point (i.e. center of the k-space integra-\ntion) which does not affect the result ofthe k-integration.\nIn this case after performing the partial time-momentum\nderivatives, ( D(/vectorh) =∆xc\n2∂/vector m\n∂t−vFˆez×/vector∇µ), we use\n/vectorh(/vectork,t) =vFˆez×/vectork+∆xc\n2mz(t)ˆez, to obtain,\n/vectorSneq(t) =/integraldisplaykdk\n4π|/vectorh|2/parenleftBigg\n/vectorS1f1−f2\n|/vectorh|+(/vectorS1+/vectorS2)(f′\n1+f′\n2)/parenrightBigg\n,\n(12)\nwhere,\n/vectorS1(/vectork,t) =∆2\nxc\n4mz(t)ˆez×∂/vector m\n∂t+∆xcvF\n2mz(t)/vector∇µ(13)\n/vectorS2(/vectork,t) =∆xc\n4η(2η2−v2\nF|k|2)(∂mx\n∂tˆex+∂my\n∂tˆey)\n+∆xc\n4η(2η2−∆2\nxcm2\nz\n2)∂mz\n∂tˆez\n−vF\nη(η2−v2\nF|k|2\n2)ˆez×/vector∇µ (14)\nThe dynamics of the FM obeys the LLG equation where\nthe conductions electrons insert torque on the FM mo-\nments through the magnetic exchange coupling,\n∂/vector m\n∂t=/vector m×\nγ/vectorBext+∆xc\n2/vectorSneq(t)−/summationdisplay\nijαij\n0∂mi\n∂tˆej\n\n(15)\nwhere,αij\n0=αji\n0, withi,j=x,y,z, is the intrinsic\nGilbert damping tensor of the FM in the absence of\nthe TI surface state and /vectorBextis the total magnetic field\napplied on the FM aside from the contribution of the\nnonequilibrium electrons.\nWhile the terms that consist of /vector∇µare called SOT,\nthe ones that contain∂/vector m\n∂tare generally responsible for\nthe damping of the FM dynamics. However, we no-\ntice that ˆ ez×∂/vector m\n∂tterm in Eq. (13) which arises from\nthe Berry curvature, becomes mz∂/vector m\n∂tin the LLG equa-\ntion that does not contribute to the damping and only\nrenormalizes the coefficient of the left hand side of the\nEq. (15). The second term in the Eq. (13), is the\nanti-damping SOT pointing along ( ez×/vector∇µ)-axis. The\ncone angle dependence of the anti-damping term can\nbe checked by assuming an electric field along the x-\naxis when the FM precesses around the y-axis, (i.e.\n/vector m(t) = cos(θ)ˆey+sin(θ)cos(ωt)ˆex+sin(θ)sin(ωt)ˆez). In\nthis case the average of the SOT along the y-axis in one\nperiod of the precession leads to the average of the an-\ntidamping SOT that shows a sin2(θ) dependence, which\nis typical for the damping-like torques. Keeping in mind4\nthat in this section we consider vF≫∆xc, the first and\nsecond terms in Eq. (14) show that the Gilbert damp-\ning increases as the precession axis goes from in-plane ( x\nory) to out of plane ( z) direction. Furthermore, when\nthe precession axis is in-plane (e.g. along y-axis), the\ndamping rate due to the oscillation of the out of plane\ncomponent of the magnetization ( ∂mz/∂t) has a sin4(θ)\ndependence that can be ignored for low power measure-\nment of the Gilbert damping θ≪1. This leaves us with\nthe contribution from the in-plane magnetization oscilla-\ntion (∂mx∂t) only. Therefore, the Gilbert damping for\nin-plane magnetization becomes half of the case when\nmagnetization is out-of-plane. The anisotropic depen-\ndence of the Gilbert damping can be used to verify the\nexistence of the surface state of the 3DTI as well as the\nproximity induce magnetization at the interface between\na FM and a 3DTI. Finally, the third term in Eq. (14)\ndemonstrates a field like SOT with the same vector field\ncharacteristics as the Rashba-Edelstein effect.\nIII. CONCLUSION\nIn conclusion, we have developed a linear response\nNEGF framework which provides unified treatment of\nboth spin torque and damping due to SOC at interfaces.\nWe obtained the expressions for both damping and anti-\ndamping torques in the presence of a linear gradiance of\nthe electric field and adiabatic time dependence of the\nmagnetization dynamics for a helical state correspond-\ning to the surface state of a 3D topological insulator.\nWe present the exact expressions for the damping/anti-\ndamping SOT as well as the field like torques and showed\nthat, (i); Both Fermi surface and Fermi sea contribute\nsimilarly to the anti-damping SOT as well as the Hall\nconductivity and, ( ii); The Gilbert damping due to the\nsurface state of a 3D TI when the magnetization is in-\nplane is less than the Gilbert damping when it is in the\nout-of-plane direction. This dependence can be used as\na unique signature of the helicity of the surface states\nof the 3DTIs and the presence of the proximity induced\nmagnetic exchange from the FM overlayer.\nACKNOWLEDGMENTS\nWe thank Branislav K. Nikoli´ c for the fruitful discus-\nsions. F. M. and N. K. were supported by NSF PREM\nGrant No. 1205734.Appendix A: Derivation of the Density Matrix\nUsing Eqs. (2) and . (4) it is straightforwardto obtain,\nG<=(Gr\nad−Ga\nad)f−2ηf′∇µ·∂Gr\nad\n∂kGa\nad\n+i∂G<\nad\n∂E∂H\n∂tGa\nad+i∂Gr\nad\n∂E∂H\n∂tG<\nad\n+i∂G<\nad\n∂k·∇HGa\nad+i∂Gr\nad\n∂k·∇HG<\nad(A1)\nWe plug in the expression for the adiabatic lesser GF in\nequilibrium, G<\nad= 2iηfGr\nadGa\nad= (Gr\nad−Ga\nad)f, and\nobtain,\nG<= (Gr\nad−Ga\nad)f−2ηf′∇µ·∂Gr\nad\n∂kGa\nad\n+if∂(Gr\nad−Ga\nad)\n∂E∂H\n∂tGa\nad+if∂Gr\nad\n∂E∂H\n∂t(Gr\nad−Ga\nad)\n+if′(Gr\nad−Ga\nad)∂H\n∂tGa\nad+if∂(Gr\nad−Ga\nad)\n∂k·∇HGa\nad\n+if∂Gr\nad\n∂k·∇H(Gr\nad−Ga\nad). (A2)\nExpanding the terms, leads to,\nG<=/parenleftbigg\nGr\nad−Ga\nad+iGa\nad∂H(t)\n∂t∂Ga\nad\n∂E\n−iGr\nad∂H\n∂k·∇µ(x)∂Gr\nad\n∂E−i∂Ga\nad\n∂E∂H(t)\n∂tGa\nad\n+i∂Gr\nad\n∂E∂H\n∂k·∇µ(x)Gr\nad/parenrightbigg\nf\n+if′Gr\nad∇µ·∂H\n∂k(Gr\nad−Ga\nad)\n+if′(Gr\nad−Ga\nad)∂H\n∂tGa\nad, (A3)\nwhere,forthefirstandthirdlineswehaveusedtheiranti-\nHermitian forms instead. Since to calculate the density\nmatrix we integrate G<over energy, we can use integra-\ntion by parts and obtain,\nG<=/parenleftbigg\nGr\nad−Ga\nad+2iGa\nad∂H(t)\n∂t∂Ga\nad\n∂E\n−2iGr\nad∂H\n∂k·∇µ(x)∂Gr\nad\n∂E/parenrightbigg\nf\n−if′/parenleftbigg\nGr\nad∇µ·∂H\n∂kGa\nad−Gr\nad∂H\n∂tGa\nad/parenrightbigg\n= (Gr\nad−Ga\nad)f\n+i/parenleftbigg\n2Gr\nadD(H)∂Gr\nad\n∂Ef+Gr\nadD(H)Ga\nadf′/parenrightbigg\n.(A4)\nwhere we define, D=∂\n∂t− ∇µ·∂\n∂k. Finally, using the\nidentity,\n2Gr\nadD(H)∂Gr\nad\n∂E=Gr\nadD(H)∂Gr\nad\n∂E−∂Gr\nad\n∂ED(H)Gr\nad\n+∂(Gr\nadD(H)Gr\nad)\n∂E, (A5)5\nand performing the differential over the energy, we arrive\nat Eq. (6).\nAppendix B: Derivation of the Local Spin Density\nFrom Eqs. (6) and (7), the local spin density can be written as, /vectorSneq=/vectorS(1)\nneq+/vectorS(2)\nneq, where /vectorS(1)\nneqis due to the\nnonequilibrium electrons at the Fermi surface and for a two band mo del can be calculated as the following,\n/vectorS(1)\nneq≈−2ηℑ/integraldisplaydE\n2πTr\n/parenleftBig\nD(/vectorh·/vectorσ)+/vector∇µ·/vector v01/parenrightBig/parenleftBig\n(E−ε0+iη)1+/vectorh·/vectorσ/parenrightBig\n((E−ε0−iη)2−|/vectorh|2)((E−ε0+iη)2−|/vectorh|2)/vectorσ\n+D(1\n(E−ε0−iη)2−|/vectorh|2)((E−ε0+iη)1+/vectorh·/vectorσ)/vectorσ((E−ε0−iη)1+/vectorh·/vectorσ)\n(E−ε0+iη)2−|/vectorh|2/bracketrightBigg\nf′(E) (B1)\nPreforming the trace over the Pauli matrix, we obtain,\n/vectorS(1)\nneq≈ −4η/integraldisplaydE\n2π/parenleftBigg\nηD/vectorh+D(/vectorh)×/vectorh\n((E−ε0−iη)2−|/vectorh|2)(E−ε0+iη)2−|/vectorh|2)\n+4ℑ(/parenleftBig\n/vectorh·D(/vectorh)+(E−ε0−iη)(/vector∇µ·/vector v0)/parenrightBig\n(E−ε0)/vectorh\n((E−ε0−iη)2−|/vectorh|2)2((E−ε0+iη)2−|/vectorh|2))\nf′(E) (B2)\nIn the limit of small broadening, η≪ |/vectorh|, we obtain,\n/vectorS(1)\nneq≈−ηD/vectorh−/vectorh×D/vectorh+1\nη/vectorh·D(/vectorh)/vectorh\n2|/vectorh|2/parenleftBig\nf′(ε0+|/vectorh|)+f′(ε0−|/vectorh|)/parenrightBig\n−1\nη(/vector∇µ·/vector v0)/vectorh\n2|/vectorh|/parenleftBig\nf′(ε0+|/vectorh|)−f′(ε0−|/vectorh|)/parenrightBig\n(B3)\nFor the Fermi sea contribution to the nonequilibrium local spin densit y we have,\n/vectorS(2)\nneq≈ℜ/integraldisplaydE\n2πTr/bracketleftBigg\n∂\n∂E/parenleftBigg\n(E−ε0−iη)1+/vectorh·/vectorσ\n(E−ε0−iη)2−|/vectorh|2/parenrightBigg/parenleftBig\nD(/vectorh·/vectorσ)+/vector∇µ·/vector v01/parenrightBig(E−ε0−iη)1+/vectorh·/vectorσ\n(E−ε0−iη)2−|/vectorh|2/vectorσ\n−(E−ε0−iη)1+/vectorh·/vectorσ\n(E−ε0−iη)2−|/vectorh|2/parenleftBig\nD(/vectorh·/vectorσ)+/vector∇µ·/vector v01/parenrightBig∂\n∂E/parenleftBigg\n(E−ε0−iη)1+/vectorh·/vectorσ\n(E−ε0−iη)2−|/vectorh|2/parenrightBigg\n/vectorσ/bracketrightBigg\nf(E). (B4)\nSimilarly, we obtain,\n/vectorS(2)\nneq≈ℜ/integraldisplaydE\n2πTr\n/bracketleftBig\nD(/vectorh·/vectorσ)+/vector∇µ·/vector v01,(E−ε0−iη)1+/vectorh·/vectorσ/bracketrightBig\n((E−ε0−iη)2−|/vectorh|2)2/vectorσ\nf(E)\n≈2ℑ/integraldisplaydE\nπD(/vectorh)×/vectorh\n((E−ε0−iη)2−|/vectorh|2)2f(E)\n≈−1\n2D(/vectorh)×/vectorh\n|/vectorh|3/parenleftBig\nf(ε0+|/vectorh|)−f(ε0−|/vectorh|)/parenrightBig\n(B5)\n[1] D. Ralph and M. Stiles, Journal of Magnetism and Mag-\nnetic Materials 320, 1190 (2008).[2] J. Katine and E. E. Fullerton, Journal of Magnetism and\nMagnetic Materials 320, 1217 (2008).6\n[3] C. Wang et al., Nature Physics 7, 496 (2011).\n[4] I. M. Miron et al., Nature Materials 9, 230 (2010).\n[5] L. Liu et al., Phys. Rev. Lett. 109, 096602 (2012).\n[6] A.Manchon andS. Zhang, Physical ReviewB 78, 212405\n(2008).\n[7] P. Gambardella and I. M. Miron, Philos. Transact. A\nMath. Phys. Eng. Sci. 369, 3175 (2011).\n[8] M. Z. Hasan and C. L. Kane, Rev. Mod. Phys. 82, 3045\n(2010).\n[9] D. Pesin and A. H. MacDonald, Nature Mater. 11, 409\n(2012).\n[10] I. Garate and M. Franz, Phys. Rev. Lett. 104, 146802\n(2010).\n[11] F. Mahfouzi, J. Fabian, N. Nagaosa, and B. K. Nikoli´ c,\nPhys. Rev. B 85, 054406 (2012).\n[12] A. Manchon, Phys. Rev. B 83, 172403 (2011).[13] E. Zhao, C. Zhang, and M. Lababidi, Phys. Rev. B 82,\n205331 (2010).\n[14] N. P. Butch et al., Phys. Rev. B 81, 241301 (2010).\n[15] X.-L. Qi, T. L. Hughes, and S.-C. Zhang, Nat Phys 4,\n273 (2008).\n[16] F. Mahfouzi, B. K.Nikoli´ c, S.-H.Chen, andC.-R.Chang ,\nPhys. Rev. B 82, 195440 (2010).\n[17] F. Mahfouzi, B. K. Nikoli´ c, and N. Kioussis, Phys. Rev.\nB93, 115419 (2016).\n[18] L. Keldysh, Sov. Phys. JETP 20, 1018 (1965).\n[19] H. K. et al, Nature Nanotechnology 9, 211 (2014).\n[20] H. Haug and A.-P. Jauho, Quantum kinetics in transport\nand optics of semiconductors (Springer-Verlag, Berlin,\nADDRESS, 2008).\n[21] N. Bode et al., Phys. Rev. B 85, 115440 (2012).\n[22] L. Arrachea, Phys. Rev. B 75, 035319 (2007).\n[23] A.-P. Jauho, N. S. Wingreen, and Y. Meir, Phys. Rev. B\n50, 5528 (1994).\n[24] B. Gaury et al., Physics Reports 534, 1 (2014)."    },    {        "title": "1609.01094v1.Coarsening_dynamics_of_topological_defects_in_thin_Permalloy_films.pdf",        "content": "Coarsening dynamics of topological defects in thin Permalloy films\nIlari Rissanen∗and Lasse Laurson\nCOMP Centre of Excellence and Helsinki Institute of Physics,\nDepartment of Applied Physics, Aalto University,\nP.O.Box 11100, FI-00076 Aalto, Espoo, Finland.\nWestudythedynamicsoftopologicaldefectsinthemagnetictextureofrectangularPermalloythin\nfilm elements during relaxation from random magnetization initial states. Our full micromagnetic\nsimulations reveal complex defect dynamics during relaxation towards the stable Landau closure\ndomain pattern, manifested as temporal power-law decay, with a system-size dependent cut-off time,\nofvariousquantities. Theseincludetheenergydensityofthesystem, andthenumberdensitiesofthe\ndifferent kinds of topological defects present in the system. The related power-law exponents assume\nnon-trivial values, and are found to be different for the different defect types. The exponents are\nrobust against a moderate increase in the Gilbert damping constant and introduction of quenched\nstructural disorder. We discuss details of the processes allowed by conservation of the winding\nnumber of the defects, underlying their complex coarsening dynamics.\nPACS numbers: 75.78.-n,76.60.Es,75.70.Kw\nI. INTRODUCTION\nThe topic of coarsening dynamics of topological de-\nfects in diverse systems ranging from liquid crystals1–4\nto biosystems5and cosmology6has attracted consider-\nable interest as it is related to properties of symmetry-\nbreaking phase transitions. As the system is quenched\nfrom a high-temperature disordered phase to a low-\ntemperature ordered phase, the symmetry of the disor-\ndered phase is broken and topological defects are gener-\nated, subsequentlyexhibitingslowcoarseningdynamics7.\nSuch phase-ordering kinetics is often characterized by a\npower-law temporal decay ρ(t)∼t−ηof the density ρof\nthe defects, with the value of the exponent ηdepending\non the characteristics of the system, and/or the defects8.\nIn ferromagnetic thin films dominated by shape\nanisotropy, elementary topological defects within the\nmagnetic texture, i.e. vortices, antivortices and edge\ndefects, may occur9–11. For instance, magnetic domain\nwalls can be envisaged as composite objects consisting of\ntwo or more such elementary defects, each characterized\nbytheirintegerorfractionalwindingnumbers9. Also,the\npresence of vortices is intimately related to magnetic flux\nclosure patterns minimizing the stray field energy of mi-\ncron or submicron magnetic particles12–14. While coars-\nening of e.g. the domain structures in Ising and Potts\ntype of models15,16as well as that of the defect struc-\nture in the XY model1,17have been extensively studied,\nless is known about the details of coarsening dynamics\nin soft (low-anisotropy) ferromagnetic thin films, involv-\ning the collective dynamics of vortices, antivortices and\nedge defects, when all the relevant effective field terms\n(exchange, and demagnetizing energies) are included in\nthe description.\nHere we study, by performing an extensive set of full\nmicromagnetic simulations of the magnetization dynam-\nics in Permalloy thin films, the relaxation process of\nsuch magnetic topological defects, starting from ran-\ndom magnetization initial states, mimicking the high-temperature disordered paramagnetic phase. Our zero-\ntemperature simulations, resembling a rapid quench to\nthe low-temperature ferromagnetic phase, show how de-\nfects emerge from the disordered initial states, and sub-\nsequently exhibit coarsening dynamics. The focus of our\nstudy is on the time period after a short initial transient\nsuch that the length of the magnetization vectors is ap-\nproximately constant, and the system can be modeled by\nthe Landau-Lifshitz-Gilbert equation. In the coarsening\nprocess, the densities of the various defect types, as well\nas the energy density of the system18, follow power-law\ntemporal decay towards the ground state with a flux-\nclosure Landau pattern. These power laws are charac-\nterized by non-trivial exponent values which are differ-\nent for the different defect types, and exhibit a cut-off\ntime scale growing with the system size. We also ad-\ndress the question of how the values of these exponents\nare affected by changes in the Gilbert damping constant\nαand introduction of random structural disorder within\nthe film, and discuss the role of the conservation of the\nwinding number on the possible annihilation reactions,\nunderlying the complex coarsening dynamics of the var-\nious defect populations.\nThe paper is organized as follows: In the following sec-\ntion (Section II), properties of the elementary topological\ndefects in soft ferromagnetic thin films are reviewed, and\ndetails of the micromagnetic simulations and data anal-\nysis are presented in Section III. In Section IV, we show\nour results on the defect coarsening dynamics and ana-\nlyze the possible annihilation reactions underlying such\ndynamics. Finally, Section V finishes the paper with dis-\ncussion and conclusions.\nII. TOPOLOGICAL DEFECTS IN\nMAGNETICALLY SOFT THIN FILMS\nIn the absence of an external magnetic field, the ori-\nentation of the spins in thin films of magnetically softarXiv:1609.01094v1  [cond-mat.mes-hall]  5 Sep 20162\nmaterial such as Permalloy is determined by the compe-\ntition of shape anisotropy and exchange interaction. For\nsmall films or nanodots (up to few tens of nanometers\ndepending on the film/dot thickness19), the exchange in-\nteraction energy dominates and the ground state is a sin-\ngle magnetic domain20. In larger films, up to a couple of\ntens of microns21, the ground state configuration consists\nof one or more elementary topological defects, depending\non the geometry of the film9. Square thin films can con-\ntain three types of stable elementary magnetic defects:\nvortices, antivortices and edge defects. Other structures,\nsuch as domain walls, are composed of these elementary\ndefects.\nMagnetic vortex is a point defect with core radius ap-\nproximately equal to the magnetic exchange length of\nthe material (between 5 and 6 nm in Permalloy22). The\ncore magnetization, often referred to as the polarization\nof the vortex, points out of the thin film plane, while\nthe surrounding magnetization rotates around the core\n(Fig. 1 a). Vortices are thus characterized by two quan-\ntities: the core polarization, and the rotation direction\nof the surrounding magnetization (clockwise or counter-\nclockwise).\nAn antivortex is another type of point defect with out-\nof-plane polarization, with a core radius similar to that\nof vortices. Unlike vortices, however, the magnetization\naround an antivortex does not rotate around the core of\nthe defect. Instead, the magnetization points into the\ncore from two opposite directions and out of the core\nfrom two perpendicular directions, with the rest of the\nsurrounding magnetization assuming orientation in be-\ntween these main directions (Fig. 1 b)10.\nVortices and antivortices are both bulk defects, i.e.\nthey form in the bulk of the film and tend to stay away\nfrom the edges unless driven there by the relaxation or\nan outside influence such as an external magnetic field.\nThe third type of defects, edge defects, are confined to\nthe edge and cannot move to the bulk9. The edge defects\nare also different from vortices and antivortices in that\nthe core magnetization of the defect does not necessarily\npoint out of plane. Edge defects can further be divided\ninto two types, henceforth referred to as the positive edge\ndefect (Fig. 1 c) and negative edge defect (Fig. 1 d), ac-\ncording to their winding numbers.\nThe topological defects can be characterized by the\nwinding number W, defined as a normalized line integral\nof the magnetization vector angle θover a closed loop\naround the defect10,W=1\n2π/contintegraltext\nSθ(φ)ds, whereφis the\nangle of the vector from the defect core to the point on\nthelinebeingintegratedover, and Stheintegrationpath.\nThe winding number (or topological charge) is +1for\nvortices and−1for antivortices. Though edge defects\ncannot be similarly circled around, they can be shown to\nhave fractional winding numbers of ±1/2.9\nThe total winding number of a thin film is a conserved\nquantity. In a film with nholes, the total winding num-\nber isW=/summationtextk\ni=1Wi= 1−n, whereWiis the winding\nnumber of defect iandkis the number of defects9. The\na) b)\nc) d)Figure 1. The elementary topological defects: a)vortex,b)\nantivortex, c)positive edge defect, d)negative edge defect.\nThe film is viewed from the direction of the z-axis, with the\narrows showing the magnetization direction in the xy-plane.\nThe vortex/antivortex cores pointing out of plane are denoted\nwith⊗. In the case of edge defects, the cores (not necessarily\nout of plane) are marked with #and the black bar represents\nthe edge of the sample.\ntotal number of defects can be quite high in large mag-\nnetically unrelaxed films, but the number will eventually\ndecay due to the collision-induced annihilations of the\ndefects. In a film with no holes, such as the ones simu-\nlated in this paper, the total winding number is equal to\n1. This corresponds to a few possible configurations, out\nof which the single vortex state ( flux-closure orLandau\npattern) is energetically most favorable13.\nIII. MICROMAGNETIC SIMULATIONS\nA. Simulation details\nDuring the relaxation of the magnetization from a ran-\ndomized initial state, the time evolution of the mag-\nnetic moments m=M/Msis described by the Landau-\nLifshitz-Gilbert (LLG) equation,\n∂m\n∂t=γHeff×m+αm×∂m\n∂t, (1)\nwhereγis the gyromagnetic ratio, Heffthe effective\nmagnetic field, Msthe saturation magnetization, and\nαthe Gilbert damping constant. Hefftakes into ac-\ncount four energy contributions, which are the afore-\nmentioned exchange energy, energy due to magnetocrys-\ntalline anisotropy, Zeeman energy (energy of an external\nfield) and the demagnetizing field energy. In the con-\ntext of this work, the Zeeman and anisotropy contribu-3\ntionsarenegligible, asnoexternalfieldsarebeingapplied\nand the magnetocrystalline anisotropy of Permalloy is in-\nsignificant.\nSimulations were performed with a GPU-based micro-\nmagnetic code Mumax3, using the adaptive Dormand-\nPrince method and finite differences for temporal and\nspatial discretization, respectively23. Simulations were\nrun for square samples of thickness 20 nm, with four\ndifferent linear film sizes Lof 512 nm, 1024 nm, 2048\nnm and 4096 nm. The dimensions of a single simula-\ntion cell were chosen to be 4 nm ×4 nm×20 nm, so\nthat the smallest film corresponds to 128 ×128 cells; the\nnumber of out-of-plane z-direction cells is 1. Here, typ-\nical parameter values of Permalloy13,18,24are used, i.e.,\nMs= 860·103A/m, and A= 13·10−12J/m. Un-\nless stated otherwise, we consider α= 0.02, which corre-\nsponds to slightly Nd-doped25or Pt-doped26Permalloy.\nWe also investigate the effect of αon the relaxation pro-\ncess, using values ranging from the typical 0.01for pure\nPermalloyupto 0.1representinghighlydopedPermalloy,\nand a couple of very high values, α= 0.5andα= 0.9.\nWhile the latter two α-values are clearly too high to re-\nalistically describe magnetization dynamics of Permal-\nloy, they allow to address the more general question of\nthe effect of damping on the defect coarsening dynam-\nics. Moreover, we also check the stability of our results\nwith respect to adding quenched structural disorder to\nthe system27–31, by performing a Voronoi tessellation to\ndividethefilmsintograins, mimickingthepolycrystalline\nstructure of the material30,31; we consider average grain\nsizes of 10nm,20nm and 40nm. Disorder is then imple-\nmented by either setting a random saturation magnetiza-\ntion in each grain (from a normal distribution with mean\nMsand standard deviation of 0.1Msor0.2Ms)27,30, or\ndecreasing the exchange coupling across the grain bound-\naries by 10 %,30 %or50 %30,31.\nAt the start of the simulation, the magnetization is\nrandomized in each cell, after which the system is let re-\nlax at zero temperature without external magnetic fields.\nFour example snapshots of the relaxation process are\nshown in Fig. 2. The relaxation process consists of ap-\nproximately three phases. In the beginning of the relax-\nation (usually from 0 to approx. 1 ns, though less with\nstronger damping), the system experiences large fluctua-\ntions in magnetization without well-defined magnetic de-\nfects or domains. After these fluctuations have settled,\nthe dynamics consist of defects moving and annihilating\nwith each other; the properties of this defect coarsen-\ning dynamics are the main focus of the paper. The final\nrelaxation stage consists of a single vortex experiencing\ndamped gyrational motion towards the center of the film.\nThe configuration of the resulting ground state displays\nthe Landau pattern: four large domains separated by di-\nagonal domain walls starting from the corners of the film\nand meeting at a 90◦angle in a vortex at the center12–14.\nOne should note that during the very early stages of\nthe relaxation starting from the disordered paramagnetic\nstate, the LLG equation does not fully describe the mag-\nFigure 2. The relaxation process in a square thin film with\nlateral size L= 1024 nm. The color wheel in the center\nshows the direction of the magnetization corresponding to\neach color. In this case, the final vortex (black dot at the\ncenterofthebottomrightpicture)isa −z-polarizedclockwise\nvortex.\nnetization dynamics since it assumes that the lengths of\nthe magnetic moments are conserved; the latter is not\nstrictlyspeakingthecaseduringthefirststageofquench-\ning the system across the phase boundary from the high-\ntemperature paramagnetic to the ferromagnetic phase.\nHowever, multiple studies32–34concerning the longitudi-\nnal relaxation of the magnetic moments point out that\nin low temperatures, the longitudinal relaxation is or-\nders of magnitude faster than the transverse relaxation,\nand takes place in the femto- and picosecond timescale.\nThus, afterthefirstfewpicoseconds, andespeciallyinthe\nannihilation-dominatedrelaxationregimewhichisthefo-\ncus of the present study, longitudinal relaxation is nonex-\nistent, and the LLG equation suffices to fully describe the\nmagnetization dynamics.\nB. Locating and characterizing defects\nHere, we describe the algorithm used to find the de-\nfects from the magnetization data. Finding bulk de-\nfects (vortices and antivortices) is relatively simple, as\nthey have strong z-directional magnetization at the core.\nThe cores of the defects are determined by comparing\nthez-magnetization with the nearest neighbors and find-\ning the local maximum or minimum, and comparing it\nto a threshold value of 0.5Ms. The type of the defect\nis then determined by performing a discretized version4\nX12 3\n7654 8\nX12 3\n7654 8\n1234\n5 1234\n5a) b)\nc) d)\nFigure 3. A schematic view of the different defects and the\n(ideal) surrounding magnetization in the nearest neighbors\nin thexy-plane, similarly as in Fig. 1. The numbers in the\ncorners of the cells show the cell traversal order when deter-\nmining the defect type.\nof the winding number integration: The nearest neigh-\nbors are looped through in a circle, and the rotation di-\nrection of the magnetization vector is monitored. Do-\ning a counterclockwise loop, the vector in two consec-\nutive cells would turn left in the case of a vortex and\nright in the case of an antivortex (Fig. 3 a,b). Ideally,\nthe angle between two consecutive neighbors in the xy-\nplane would be 45◦. Since each magnetization vector\nis normalized to Ms, the length of the cross product of\ntwo consecutive magnetization vectors in the vortex case\n(choosing counterclockwise turn as positive) would yield\n||mi×mi+1||=M2\nssin 45◦=M2\ns√\n2. The corresponding re-\nsult for an antivortex would be −M2\ns/√\n2. Summing the\nresults for each neighbor pair, this method would ideally\ngive 2√\n2M2\nsfor vortices and−2√\n2M2\nsfor antivortices.\nDue to nonidealities in rotation and the fact that the\nspins in the cells neighboring the core also tend to have\nnonzeroz-components, the sums can be smaller. Thus,\nthresholdvaluesforrecognizingdefectsaresetto M2\nsand\n-M2\nsfor vortices and antivortices, respectively.\nIn addition to the winding number, the vortices can\nhave clockwise or counterclockwise rotation. The rota-\ntion is determined as above, but considering the center-\nto-neighbor vector and the magnetization vector for each\nneighbor. This approach would ideally yield 4Ms(1+√\n2)\nfor counterclockwise rotation and −4Ms(1 +√\n2)for\nclockwise rotation, since the angle between the vectors\nwould be 90◦. Due to nonidealities, threshold values were\nset to 5Msand−5Ms, respectively.\nEdge defects are somewhat harder to detect, as they\nusually do not have polarized cores. Thus the defects are\n2.0\n1.6\n1.2\n0.8\n0.4\n00            0.4           0.8          1.2           1.6          2.0y [µm]\nx [µm]\nFigure 4. The various kinds elementary defects present in the\nrelaxation of the largest film: clockwise vortices (squares),\ncounterclockwise vortices (circles), antivortices ( +-signs) and\nedge defects (triangles). The color of a bulk defect shows its\npolarization (black for −zand white for +z). For the edge\ndefects, the color indicates whether the winding number of\nthe defect is positive (black) or negative (white).\n10-1110-1010-910-8\nt [s]-2×104-1×10401×104WtotL = 512 nm\nL = 1024 nm\nL = 2048 nm\nL = 4096 nm\n10-910-8\nt [s]02468Wtot\nFigure 5. The total average winding numbers for the four film\nsizes used. The inset shows that the winding number for the\nlargest sizes takes more time to converge into 1.\ndetermined only by performing a loop through the near-\nest neighbors as in the case of bulk defects (Fig. 3 c,d).\nThough the method performed quite well in finding the\nedge defects, the difficulty of singling out the core some-\ntimes resulted in multiple detections in the same region.\nThis problem was somewhat mitigated by introducing\nan area around the edge defects in which similar defects\nwould be ignored. Fig. 4 shows a snapshot of the relax-\nation with defects pinpointed by the detection algorithm.\nIn the simulations, the initial random fluctuations\ncause many false defect detections. This can be seen\nas the total winding number fluctuating in the beginning5\na)                               b)\nFigure 6. The two most commonly encountered metastable\nstates, shown for the system with L= 1024nm.a)A two\nedge defect, two vortex state which also displays significant\nbending of domain walls. b)A more complex state, showing\nan isolated vortex and an arc of other defects.\nbefore converging close to 1 (Fig. 5). The tendency of\nthe total winding number to be below 0 in the beginning\nis due to the initial fluctuations being more easily cate-\ngorized as antivortices, since they do not have a rotation\ndirectionthreshold. Thewindingnumbercanalsochange\nmomentarilyduringannihilationsduetothedetectional-\ngorithm having difficulties determining defects that are\nvery close to one another. Moreover, sometimes a dis-\nturbance (such as a spin wave from an annihilation) near\na defect could cause the defect to become momentarily\nunrecognizable to the algorithm. To lessen the fluctua-\ntion in defect amounts, a persistence time of 20 ps for\nthe already detected defects was introduced. During the\npersistence time, the algorithm considers a defect to exist\nin the location it was last detected even if it can’t find\nit at the present time. The persistence time reduces the\nnoise in the number of defects.\nIV. RESULTS\nDepending on the system size, the relaxation from ran-\ndommagnetizationtothesinglevortexgroundstatetook\nusually approximately from 5 ns to 40 ns. Hence the sim-\nulations were run for 20 ns for the two smallest film sizes,\n30 ns for the L= 2048nm film and 50 ns for the largest\nfilm. Usually the ground state was reached relatively\nquickly compared to the simulation time. Only with the\nlargest film size there were a couple of instances in which\nthe system had not relaxed to the single-vortex state or\na metastable state before the simulation time ran out,\nthough in these cases the system was still close to the\nrelaxed state with only 2−4defects left.\nMetastable states were encountered in 10 of the total\n80 simulations. With the smallest film, only one simula-\ntion ended up in a metastable state, whereas each larger\nsize had three simulations finished in a metastable state.\nOf these states, two different kinds were common: a sim-\nple one with two negative edge defects on two opposite\nsides and two vortices close to the center (Fig. 6 a) and a\n10-1110-1010-910-8\nt [s]10-510-410-310-210-1100(E(t) - E(tR))/ E(0)L= 512 nm\nL= 1024 nm\nL= 2048 nm\nL= 4096 nm\nt-1.22 Defect\nannihilationInitial\nfluctuation\n0 10 20 30 4050\nρd[1/µm2]0.00.51.01.5ρE[J/µm2]L= 512 nm\nL= 1024 nm\nL= 2048 nm\nL= 4096 nm×10-14Figure 7. The time evolution of the total energy of the films,\nconsisting of an “initial fluctuation” phase independent of the\nsystem size, and a defect annihilation/coarsening phase dis-\nplaying power law relaxation terminated by a cut-off time\nincreasing with the film size. The linear dependence of en-\nergy density and defect density during the coarsening phase\nis shown in the inset.\nmore complicated state with an antivortex, three vortices\nand two edge defects (Fig. 6 b), in which one vortex is\nisolated and the other defects are in an arc close to one\nof the edges.\nA. Time evolution of energy and defect densities\nIn Fig. 7, the time evolution of the energy towards the\nvalueE(tR)is shown for all four system sizes, averaged\nover 20 simulations for each size. Here tRis the time\nafter which the number of defects in the system does not\ndecrease, which corresponds to either the single vortex\nstate, a metastable state with more than one defect, or\nthe state the system had time to reach before the simu-\nlation time ran out.\nThe total energy of the system drops very little in the\nfirst 0.1 ns, and then starts decreasing in a slightly oscil-\nlating fashion in the initial fluctuation regime (0 – 1 ns).\nStable defects start to form at roughly 0.5 ns, but the\nenergy evolution is dominated by the global fluctuations\nin the system. As can be seen from the figure, during the\ninitial phase the time evolution of the energy, normalized\nby the initial value, is independent of the system size.\nThis results from the fact that during the initial fluctua-\ntions the magnetization is largely random, and thus the\nenergy contributions from the exchange interaction and\nthe stray fields are proportional to the system size.\nIn the ”defect annihilation” or coarsening phase, the\ntime evolution of the energy resembles a power law\nE(t)−E(tR)∝t−ηE, with an exponent ηE= 1.22±0.08.\nIn this phase, the total energy of the system consists\nmostly of the energy contained in the domain walls con-\nnecting the elementary defects and the stray fields cre-\nated by the (anti)vortex cores in which magnetization6\n10-1110-1010-910-810-7\nt[s]10-310-210-1100101102103ρd(t) -ρd(tR) [1/µm2]L= 512 nm\nL= 1024 nm\nL= 2048 nm\nL= 4096 nm\nt-1.42\n10-1110-1010-910-810-7\nt[s]10-310-210-1100101102103ρav(t) -ρav(tR) [1/µm2]L= 512 nm\nL= 1024 nm\nL= 2048 nm\nL= 4096 nm\nt-1.62\n10-1110-1010-910-810-7\nt[s]10-210-1100101102ρed(t) -ρed(tR) [1/µm]L= 512 nm\nL= 1024 nm\nL= 2048 nm\nL= 4096 nm\nt-0.8210-1110-1010-910-810-7\nt[s]10-310-210-1100101102103ρv(t) -ρv(tR) [1/µm2]L= 512 nm\nL= 1024 nm\nL= 2048 nm\nL= 4096 nm\nt-1.51a) b)\nc) d)\nFigure 8. The time evolution of the densities of a)all defects, b)vortices,c)antivortices and d)edge defects, with solid lines\ncorresponding to power law fits as guides to the eye. The power laws can be seen most clearly in the largest film sizes. In the\ncase of edge defects, the positive edge defects are noted with dashed symbols.\npoints out of plane. The largest system sizes are the\nslowest to reach the energy minimum and thus show the\nmost clear-cut power-law behavior. The inset of Fig. 7\nshows that during the coarsening phase the energy den-\nsityρE=E/L2of the system is linearly proportional\nto the density of the defects ρd=Nd/L2, whereNdis\nthe total number of all defects. This implies that the de-\nfects are the main contributors to the energy at this stage\nof the relaxation, a result previously obtained for the\nXY-model by Qian et al.17. The slope changes slightly\nwithsystemsize, likelyduetothefactthatwhiletheedge\ndefects also contribute to the total energy, their number\nscales with the lateral film size as 4Linstead of the L2-\nscaling of bulk defects. Thus the smaller the film, the\ngreater the relative amount of edge defects at the begin-\nning of the simulation. In smaller films the reduction\nof total number of defects involves more annihilations\nof edge defects, compared to vortex-antivortex annihila-\ntions that dominate in larger films. If the edge defects\nare energetically less expensive than bulk defects (which\nwould seem reasonable, considering that they don’t usu-\nally have large out-of-plane components and the overallchange in magnetization direction around the defect is\nless than that of bulk defects), these annihilations re-\nsult in smaller decrease in energy than vortex-antivortex\nannihilations. Hence, for smaller systems, the energy de-\ncreases more slowly as a function of the total number of\ndefects.\nIn the coarsening phase, also the densities of the dif-\nferent defect types decay as power laws, with different\nexponents for each type of defect. These exponent are\ndetermined only by the topological charge of the defect,\ngiven that considering separately the chirality and/or\ncore polarization (in the case of vortices and antivortices)\ndid not have an effect on the exponent. The total num-\nber density of all defects (counting vortices, antivortices\nand both types of edge defects) decays as a power law\nρd(t)−ρd(tR)∝t−ηdwith the exponent ηd= 1.42±0.06\n(Fig. 8 a). Here tRis again the time after which no anni-\nhilations take place. The exponent value is considerably\nhigher than the asymptotic value ( ηd= 1) found in sim-\nulations of the XY-model with linear damping and local\ninteractions1,17,35.\nFor the total amounts of vortices (summing both7\na) b)0.4 µm\n0.4 µm\nΔt = 0 ps Δt = 50 ps\nΔt = 100 ps Δt = 150 psΔt = 0 ps Δt = 150 ps\nΔt = 300 ps Δt = 450 ps\nFigure 9. The annihilation of a vortex and an antivortex with a)antiparallel b)parallel polarizations.\nclockwise and counterclockwise rotations and both\ncore polarizations) the time evolution is well-described\nby a power law ρv(t)−ρv(tR)∝t−ηv, with\nηv= 1.51±0.05(Fig. 8 b). The exponent of the time\nevolution of the antivortex density ρavis consis-\ntently found to be somewhat larger: We find a\npower law ρav(t)−ρav(tR)∝t−ηavwith an exponent\nηav= 1.62±0.09(Fig. 8 c). Since the typical relaxed\nstate achieved in the simulations is a single-vortex state,\nρav(tR)is usually 0, though few unrelaxed/metastable\nend states have ρav(tR)between 1-2.\nThedensityofnegativeedgedefectsishigherthanthat\nof the positive ones in all the simulations. Positive edge\ndefects were observed to be short-lived byproducts of an-\nnihilations of vortices and negative edge defects. This\nsupports the notion in Ref.9that negative edge defects\nare energetically preferable over positive edge defects for\nfilms withLt>L2\nex, whereLandtare the lateral length\nand thickness of the film, respectively, and Lexis the\nexchange length. Like vortices and antivortices, the den-\nsity of negative edge defects appears to show power-law\nbehaviorρned(t)−ρned(tR)∝t−ηned, with an exponent\nηned= 0.82±0.09(Fig. 8 d). One should note here that\nfor edge defects, ρned=Nned/Lis a line density instead\nof an area density. The number density of positive edge\ndefects decays close to zero soon after the initial fluctua-\ntions and there’s no visible power-law behavior.\nB. Defect dynamics during relaxation\nExamining the motion of defects during the relax-\nation/coarsening process reveals complex dynamical de-\nfect behavior, including various kinds of annihilations,\nvortex and antivortex emissions and core switching. Allof these events are restricted by the conservation of the\ntotal winding number.\nThe possible annihilation events are limited to four\ntypes: positive and negative edge defect annihilation,\nvortex-antivortex annihilation, vortex and 2×negative\nedge defect annihilation and antivortex and 2×positive\nedge defect annihilation. Out of these four annihilation\nprocesses, only two were primarily encountered in the\nsimulations: vortex-antivortex annihilation, and vortex\nand2×edge defect annihilation. In the former case, the\nparallelityorantiparallelityofthepolarizationsofthean-\nnihilating vortex/antivortex pair affects the nature of the\nannihilation process. This is related to the conservation\nof another topological quantity, the skyrmion number36.\nWhen the polarizations of the annihilating vortex and\nantivortex are parallel, the skyrmion number is con-\nserved, resulting in a continuous and relatively slow an-\nnihilation process. The vortex and antivortex approach\neach other until they’re indistinguishable and start ac-\ncelerating in a direction perpendicular to a line connect-\ning them. During the acceleration, the combined vortex-\nantivortex defect widens and diffuses continuously into\nthe surrounding magnetization. This process is depicted\nin Fig. 9 a. By contrast, if the polarizations are antipar-\nallel, the skyrmion number is not conserved, and a more\nabrupt annihilation (referred to as ”exchange explosion”\nby some authors10) takes place: the vortex and antivor-\ntex circle around one another in decaying orbits until\nmeeting at the center and explosively releasing circular\nspins waves (Fig. 9 b).\nThe steps of the annihilation process where a vortex\nannihilates with two negative edge defects are harder to\npinpoint. In a typical vortex-edge defect annihilation,\none of the edge defects changes sign and emits an an-\ntivortex, which annihilates with the approaching vortex.\nThe remaining edge defects, now having opposite signs,8\nΔt = 100 ps\nΔt = 0 ps\nΔt = 200 ps Δt = 300 ps0.4 µm\nFigure 10. Though somewhat difficult to see, during this edge\ndefect-vortex annihilation, the lower edge defect emits an an-\ntivortex with which the vortex actually annihilates.\nthenannihilatewitheachother. Thiskindofannihilation\nalso causes an emission of spin waves (Fig. 10). An edge\ndefect could also absorb or emit a vortex or an antivor-\ntex and change sign without a vortex/antivortex close by\nto annihilate with, since a +1/2edge defect emitting a\nvortex or absorbing an antivortex and changing into a\n−1/2defect conserves the winding number. Such emis-\nsions and absorptions were observed in the simulations,\nthough in most cases the emitted vortex/antivortex was\nshortly absorbed again accompanied with an emission of\nspin waves.\nThevelocitiesofthedefectsdonottypicallyexceedthe\ncore switching velocity of Permalloy ( 340±20m/s)37.\nHowever, sometimes an exception occurs in antiparal-\nlel vortex-antivortex annihilations. In this case the in-\ncreasing velocity of the vortex and/or antivortex causes\nthe formation of a dip particle, an antiparallelly polar-\nizedmagnetizationregionclosetothefast-movingcore38.\nJust before annihilation, the vortex/antivortex exceeds\nthe core switching velocity and the dip particle separates\ninto a vortex-antivortex pair. The consecutive annihi-\nlations of the two pairs then take place (Fig. 11). In\naddition to velocity, the environment of a vortex also af-\nfects the possibility of a core switch. Some core switches\nwere observed to happen even for relatively stationary\nvortices, usually after being excited by a spin wave orig-\ninating from a nearby annihilation.\nAnother core switching behavior was sometimes found\nat the corners of the film: a vortex could ”bounce”\n(shortly get absorbed and then again emitted by the edge\ndefect) between two edge defects on different edges of\nthe film while reversing polarization with each bounce\n(Fig. 12) and emitting spin waves. This kind of bouncing\nΔt = 150 ps\nΔt = 0 ps\nΔt = 300 ps Δt = 450 ps0.4 µmDip particle\nforming\nAVO-VG annihilation  \nAVG-VO annihilation  Figure 11. In this antiparallel annihilation, the negatively\npolarizedantivortex(blackdot)generatesadipparticlewhich\nthen splits into a positively polarized vortex-antivortex pair.\nThus two annihilations occur: an antiparallel annihilation of\nthe original antivortex and the generated vortex (AV O-VG),\nand a parallel annihilation of the generated antivortex and\nthe original vortex (AV G-VO).\nΔt = 150 ps\nΔt = 0 ps\nΔt = 300 ps Δt = 450 ps0.4 µm\nFigure 12. The core switching of a vortex due to a momen-\ntary absorption into an edge defect. Usually before and after\nthe absorption and emission of the bouncing defect, the edge\ndefect cores gain short-lived out-of-plane magnetization com-\nponents.\nalways ended up in both the vortex and the edge defects\nannihilating at the corner. Typically there were two or\nthree such core switches before the final annihilation.9\nC. Effects of damping and quenched disorder\nHere, we discuss briefly how the above results are af-\nfected by changes in the damping constant α, and when\nintroducing quenched disorder to the system. Fig. 13\nshows the time evolution of the total defect density ρd(t)\ninapuresystemfordifferentvaluesof αintherangefrom\n0.01 to 0.9; notice that while the higher values of αcon-\nsideredareclearlyunphysicalforPermalloy, theyallowto\naddress the question of how the defect coarsening process\nis modified when the overdamped limit (as often consid-\nered in coarse-grained models of defect coarsening, such\nas the XY-model) is approached. As indicated by the in-\nsetofFig. 13,thepowerlawexponent ηdevolvesfromthe\nlow-αvalueofηd≈1.4toalowervalueof ηd= 1.07±0.05\nfor the highest α-value considered. We note that ηdob-\ntainedhereinthelimitoflarge αisclosetothatobtained\nfor XY-model in earlier works1,17,35. The correspond-\ning exponents for the different defect types also exhibit\nsimilar evolution with α, with the values obtained for\nα= 0.9found to be ηv= 1.09±0.06,ηav= 1.13±0.06,\nηned= 0.72±0.09for vortices, antivortices and edge\ndefects, respectively (not shown). Qualitatively, with in-\ncreasingαfrom 0.01 towards 0.1, the initial fluctuations\ntend to settle down somewhat faster, and core switching\nevents are found to be less abundant. For the highest\nα-values considered (0.5 and 0.9), the system forms well-\ndefined defects almost instantaneously, with their subse-\nquent motionbeingquite sluggish. Also, no core switches\nnor ”bounces” of vortices from edge defects are observed.\nAsaresult,thedurationofthecoarseningphaseincreases\nsignificantly, with the largest system taking more than 70\nns to fully relax in some simulation runs.\nFinally, introducing random structural disorder due\nto the polycrystalline nature of Permalloy to the films\nwithα= 0.02has the effect that some of the simula-\ntion runs finish with more than one defect pinned by the\ndisorder. However, for the parameter values used in our\nsimulations for the grain size, exchange coupling reduc-\ntions across the grain boundary, and saturation magne-\ntization variations in different grains (see Section III),\nthe exponents of the power law relaxations remain the\nsame as in the corresponding pure system (not shown).\nWhen the exchange coupling between grains is weakened,\nthe defects prefer to move along the grain boundaries.\nAdditionally, core switches were observed to occasion-\nally occur when a vortex/antivortex crosses over a grain\nboundary. The probability for such core switches ap-\npears to increase with weaker inter-grain exchange cou-\npling strength. Varying the saturation magnetization in\nthe grains makes the movement of the defects somewhat\nchoppy, and increases the chance of defect pinning, but\notherwise the dynamics of the relaxation process remains\nsimilartothatinthenon-disorderedPermalloyfilmscon-\nsidered above.\n10-1010-910-810-7\nt [s]10-210-1100101102103ρd(t) - ρd(tR) [1/µm2]α= 0.01\nα= 0.1\nα= 0.5\nα= 0.9\n0.0 0.2 0.4 0.6 0.8 1.0α 1.01.11.21.31.41.5ηdFigure13. Mainfigureshowstheaveragetimeevolutionofthe\ntotal number density of defects ρdforL= 4096nm with four\ndifferentα-values. Forlarger α, thepowerlawcharacterofthe\nrelaxation (black lines indicate the the power-law fits used)\nstartsearlierduetothestronglydampedinitialmagnetization\nfluctuations. The inset shows the resulting ηd-exponent as a\nfunction of α.\nV. CONCLUSIONS\nIn this paper, we have investigated the magnetic relax-\nation starting from disordered initial states of Permalloy\nthin films of various sizes by extensive micromagnetic\nsimulations. We conclude that the resulting coarsen-\ning dynamics involve complex processes and display a\nmultitude of phenomena, such as defect annihilations,\ncore switching and vortex absorption/emission, many of\nwhichhavepreviouslybeenindividuallystudiedindetail.\nTogether these phenomena result in highly nontrivial dy-\nnamics for single defects which then give rise to interest-\ning time evolution of system-wide quantities such as the\ntotal energy density and the defect densities.\nIn the defect coarsening/annihilation phase, this com-\nplexityismanifestedinparticularasslowpower-lawtem-\nporal decay characterized by non-trivial exponents of\nquantities such as the energy density of the system, of\nthe form of ρ(t)−ρ(tR)∝t−ηE, withηE= 1.22±0.08for\nthe energy density time evolution. For the defect den-\nsities, different values of ηwere observed depending on\nthe defect type: For vortices, antivortices and negative\nedge defects we find ηv= 1.51±0.05,ηav= 1.62±0.09\nandηned= 0.82±0.09, respectively. The temporal de-\ncay of the total density of defects is characterized by the\nexponentηd= 1.42±0.06. These exponents show lit-\ntle change (within error bars) when using the Gilbert\ndamping constant αwithin the range of 0.01 - 0.1, and\nare found to be robust against adding quenched disorder\nof moderate strength. When αis increased further, the\nrelaxation exponents approach the asymptotic value for\nthe XY-model with local interactions ( ηd= 1)1. This\nshould be due to the large damping practically eliminat-\ning the precessional motion of the magnetic moments so10\nthat they align with the local effective field almost im-\nmediately; thus, the dynamics of the magnetic moments\nstarts to resemble that of the XY-model in the no-inertia\n(overdamped) limit. Our results thus suggest that the\nrelatively low damping of Permalloy has a key role in the\nemergence of the non-trivial values of the relaxation ex-\nponents, and that quenched disorder, present in any real\nsamples, is irrelevant for the relaxation exponent values.\nDue to the relatively small size of the films and as\na consequence the number of defects (about 500 in the\nlargest films at the initial stages of the coarsening phase),\nthe power-law relaxation phase of energy and defect den-\nsities was limited in time to roughly one or two or-\nders of magnitude. Thus, simulations and experiments\nwith larger films and, consequently, longer relaxation\ntimes, would be useful. For experimental investigation,\ntime-resolvedX-rayimagingtechniquesshouldhavegood\nenough spatial and temporal resolutions (25 - 30 nm and\n70 - 100 ps, respectively)39–41to observe the defects andtheir dynamics. Even though these resolutions are still\nlimited when compared to our simulations, the longer\nrelaxation times and larger inter-defect separations ex-\npected during the later stages of the relaxation process\nin larger films (e.g., with linear sizes in the range of\ntens of microns) should make it possible to experimen-\ntally observe the approximate time evolution of the vor-\ntex/antivortex number densities.\nACKNOWLEDGMENTS\nWe thank Mikko Alava for useful comments. We ac-\nknowledge the support of the Academy of Finland via an\nAcademy Research Fellowship (LL, projects no. 268302\nand 273474), and the Centres of Excellence Programme\n(2012-2017, project no. 251748). We acknowledge the\ncomputational resources provided by the Aalto Univer-\nsity School of Science Science-IT project.\n∗ilari.rissanen@aalto.fi\n1B. Yurke, A. N. Pargellis, T. Kovacs, and D. A. Huse,\nPhys. Rev. E 47, 1525 (1993).\n2T. Nagaya, H. Hotta, and H. Oriharaand Yoshi-\nhiro Ishibashi, J. Phys. Soc. Jpn. 61, 3511 (1992).\n3I. Chuang, B. Yurke, A. N. Pargellis, and N. Turok,\nPhys. Rev. E 47, 3343 (1993).\n4C.Harrison, D.Angelescu, M.Trawick, Z.Cheng, D.Huse,\nP. Chaikin, D. Vega, J. Sebastian, R. Register, and\nD. Adamson, Europhys. Lett. 67, 800 (2004).\n5E. M. Kramer and J. V. Groves, Phys. Rev. E 67, 041914\n(2003).\n6I. Chuang, R. Durrer, N. Turok, and B. Yurke, Science\n251, 1336 (1991).\n7R. D. MacPherson and D. J. Srolovitz, Nature 446, 1053\n(2007).\n8A. J. Bray, Adv. Phys. 51, 481 (2002).\n9O. Tchernyshyov and G.-W. Chern, Phys. Rev. Lett. 95,\n197204 (2005).\n10R.HertelandC.M.Schneider,Phys.Rev.Lett. 97,177202\n(2006).\n11G.-W. Chern, H. Youk, and O. Tchernyshyov,\nJ. Appl. Phys. 99(2006).\n12L. D. Landau and E. Lifshitz, Phys. Z. Sowjetunion 8, 101\n(1935).\n13W. Rave and A. Hubert, IEEE Trans. Magn. 36, 3886\n(2000).\n14J. Raabe, C. Quitmann, C. H. Back, F. Nolting, S. John-\nson, and C. Buehler, Phys. Rev. Lett. 94, 217204 (2005).\n15C. Sire and S. N. Majumdar, Phys. Rev. E 52, 244 (1995).\n16C. Sire and S. N. Majumdar, Phys. Rev. Lett. 74, 4321\n(1995).\n17H. Qian and G. F. Mazenko, Phys. Rev. E 68, 021109\n(2003).\n18V. Estévez and L. Laurson, Phys. Rev. B 91, 054407\n(2015).\n19K. L. Metlov and K. Y. Guslienko, J. Magn. Magn. Mater.\n242, 1015 (2002).\n20A. Hubert and R. Schäfer, Magnetic Domains: The Anal-ysis of Magnetic Microstructures (Springer Berlin Heidel-\nberg, 2008).\n21K. Y. Guslienko, J. Nanosci. Nanotechnol. 8, 2745 (2008).\n22G. S. Abo, Y.-K. Hong, J. Park, J. Lee, W. Lee, and B.-C.\nChoi, IEEE Trans. Magn. 49, 4937 (2013).\n23A. Vansteenkiste, J. Leliaert, M. Dvornik, M. Helsen,\nF. Garcia-Sanchez, and B. Van Waeyenberge, AIP Adv.\n4, 107133 (2014).\n24K.-S. Lee and S.-K. Kim, Appl. Phys. Lett. 91, 132511\n(2007).\n25C. Luo, Z. Feng, Y. Fu, W. Zhang, P. K. J. Wong, Z. X.\nKou, Y. Zhai, H. F. Ding, M. Farle, J. Du, and H. R. Zhai,\nPhys. Rev. B 89, 184412 (2014).\n26S. Mizukami, T. Kubota, X. Zhang, H. Naganuma,\nM. Oogane, Y. Ando, and T. Miyazaki, Jpn. J. Appl.\nPhys.50, 103003.\n27H. Min, R. D. McMichael, M. J. Donahue, J. Miltat, and\nM. D. Stiles, Phys. Rev. Lett. 104, 217201 (2010).\n28J. Leliaert, B. Van de Wiele, A. Vansteenkiste, L. Laurson,\nG. Durin, L. Dupré, and B. Van Waeyenberge, J. Appl.\nPhys.115, 17D102 (2014).\n29J. Leliaert, B. Van de Wiele, A. Vansteenkiste, L. Laurson,\nG. Durin, L. Dupré, and B. Van Waeyenberge, Phys. Rev.\nB89, 064419 (2014).\n30J. Leliaert, B. Van de Wiele, A. Vansteenkiste, L. Laurson,\nG. Durin, L. Dupré, and B. Van Waeyenberge, J. Appl.\nPhys.115, 233903 (2014).\n31J. Leliaert, B. Van de Wiele, A. Vansteenkiste, L. Laurson,\nG. Durin, L. Dupré, and B. Van Waeyenberge, Sci. Rep.\n6(2016).\n32L. Xu and S. Zhang, J. Appl. Phys. 113, 163911 (2013),\nhttp://dx.doi.org/10.1063/1.4803150.\n33V. Baryakhtar, Zh. Eksp. Teor. Fiz. 87, 1501 (1984).\n34O. Chubykalo-Fesenko, U. Nowak, R. W. Chantrell, and\nD. Garanin, Phys. Rev. B 74, 094436 (2006).\n35H. Toyoki, Phys. Rev. A 42, 911 (1990).\n36O. A. Tretiakov and O. Tchernyshyov, Phys. Rev. B 75,\n012408 (2007).\n37S.-K. Kim, Y.-S. Choi, K.-S. Lee, K. Y. Guslienko, and11\nD.-E. Jeong, Appl. Phys. Lett. 91, 082506 (2007).\n38R. Hertel, S. Gliga, M. Fähnle, and C. M. Schneider,\nPhys. Rev. Lett. 98, 117201 (2007).\n39H. Stoll, M. Noske, M.Weigand, K. Richter, B. Krüger,\nR. M. Reeve, M. Hänze, C. F. Adolff, F. Stein, G. Meier,\nM. Kläui, and G. Schütz, Front. Phys. 3(2015).40P. Fischer, M.-Y. Im, S. Kasai, K. Yamada, T. Ono, and\nA. Thiaville, Phys. Rev. B 83, 212402 (2011).\n41A. Vansteenkiste, K. W. Chou, M. Weigand, M. Curcic,\nV. Sackmann, H. Stoll, T. Tyliszczak, G. Woltersdorf,\nC. H. Back, G. Schutz, and B. Van Waeyenberge, Nat.\nPhys.5, 332 (2009)."    },    {        "title": "1609.06286v1.Global_existence_and_asymptotic_behavior_of_solutions_to_the_Euler_equations_with_time_dependent_damping.pdf",        "content": "arXiv:1609.06286v1  [math.AP]  20 Sep 2016Global existence andasymptotic behavior of solutions\ntotheEuler equations with time-dependent damping\nXinghongPan∗\n(Department of Mathematics and IMS,Nanjing University, Na njing 210093, China.)\nAbstract\nWe study the isentropic Euler equations with time-dependen t damping, given byµ\n(1+t)λρu.\nHere,λ,µaretwonon-negativeconstantstodescribethedecay rateof dampingwithrespectto\ntime. We will investigatethe global existence and asymptot ic behavior of small data solutions\nto the Euler equations when 0< λ <1,0< µin multi-dimensions n≥1. The asymptotic\nbehavior will coincide with the one that obtained by many aut hors in the case λ= 0. We will\nalso show that the solutioncan onlydecay polynomiallyin ti mewhilein the three dimensions,\nthevorticitywilldecay exponentiallyfast.\nKeywords: Eulerequations,time-dependentdamping,globalexistenc e,asymptoticbehav-\nior, vorticity,exponentialdecay.\nMathematical Subject Classification2010: 35L70,35L65,76N15.\n1 Introduction\nThis paper deals with the isentropic Euler equations with ti me-dependent damping in multi-\ndimensions: \n\n∂tρ+∇·(ρu) = 0,\n∂t(ρu)+∇·(ρu⊗u)+∇p=−µ\n(1+t)λρu,\nρ|t=0= 1+ρ0(x),u|t=0=u0(x),(1.1)\nwhereρ0(x)∈R,u0(x)∈Rn,supportedin {x∈Rn||x| ≤R}. Hereρ(t,x),u(t,x)andp(t,x)\nrepresentthedensity,fluidvelocityandpressurerespecti velyandλ,µaretwopositiveconstants\ntodescribethedecayrateofthedampingintime. Weassumeth efluidisapolytropicgaswhich\nmeanswe can assume p(ρ) =1\nγργ,γ >1.\n∗E-mail:math.scrat@gmail.com.\n12 XINGHONG PAN\nAsiswellknown,whenthedampingvanishes,shockwillform. Forthemathematicalanal-\nysis of finite-time formation of singularities, readers can see Alinhac[1], Chemin[2], Courant-\nFriedrichs[3],Christodoulou[5],Makino-Ukai-Kawashim a[16],Rammaha[22]aswellasChen-\nLiu[4], Yin[29], Sideris[23,24]and references therein fo rmoredetail.\nTheEulerequationswithnontime-decayed dampingare\n/braceleftBigg\n∂tρ+∇·(ρu) = 0,\n∂t(ρu)+∇·(ρu⊗u)+∇p=−κρu,(1.2)\nwhereκisthedampingconstantand 1/κcanberegardedastherelaxationtimeofsomephysical\nfluid. Manyauthorshaveproventheglobalexistenceanduniq uenessofsmoothsolutionstosys-\ntem(1.2)withsmalldata. Alsotheasymptoticbehaviorofth esmoothsolutionwasstudied. For\nthe 1d Euler equations, see Dafermos[6], Hsiao-Liu[7], Hua ng-Marcati-Pan[10],Nishida[17],\nNishihara-Wang-Yang[19]andtheirreferences. Forthemul ti-dimensionalcase,Wang-Yang[28]\ngive the pointwise estimates of the solution by using some en ergy methods and estimating the\nGreenfunctiontothelinearizedsystem. Sideris-Thomases -Wang[25]provesasimilarresultby\nusingasimplerapproach. Theybothprovedthatthesmoothso lutiondecaysinmaximumnorm\nto the background state at a rate of (1 +t)−3\n2in 3 dimensions. Sideris-Thomases-Wang also\nshow that the smooth solution has a polynomially decayed low er bound in time while the vor-\nticity will decay exponentially. Tan-Wang[26], Jiu-Zheng [11] study this problem in the frame\nof Besov space and obtain similar asymptotic behavior of the solution. Also see Kong-Wang\n[14]forextension.\nItisnaturaltoaskwhethertheglobalsolutionexistswhent hedampingisdecayedandwhat\nisthecriticaldecayratetoseparatetheglobalexistencea ndthefinite-timeblowupofsolutions\nwith small data. The papers [8, 9],[20, 21] have done some ins pection on this topic where the\nauthorssystematicallystudythecasethatthedampingdeca yswithtimeasµ\n(1+t)λ. Theybelieve\nthat there is a pair of non-negative critical exponent (λc(n),µc(n)), depending on the space\ndimensionn, such that\nwhen0≤λ < λc(n),0< µorλ=λc(n),µ > µc(n),(1.1)have global existence of\nsmall-data solutions; while when λ=λc(n),µ≤µc(n)andλ > λc(n),0≤µ, the smooth\nsolutionsof (1.1)willblowupin finitetime.\nIn two and three dimensions, Hou-Witt-Yin [9], Hou-Yin [8] h ave shown that the critical\nexponent is (λc(3),µc(3))= (1,0)and(λc(2),µc(2))= (1,1); while in one dimension, the\ncritical exponentis (λc(1),µc(1)) = (1,2),partlypresented in[20, 21].\nThis paper deals with the case 0< λ < λ c(n) = 1,0< µin multi-dimensions n≥1. We\nwill obtain the global existence and asymptotic behavior of the solution to system (1.1). The\nmethod used here will be completely different from that in [8 , 9]. Also we will show that the\nconvergence rate of the solution to the background constant state(1,0)will coincide with that\noneobtainedbyWang-Yang[28]andSideris-Thomases-Wang[ 25]whenλ= 0whichindicates\nthat the time-asymptotic behavior of the solution is the dif fusion wave of the corresponding\nlinearsystem.\nTheproofoftheglobalexistenceofthesolutionisbasedont hemethodofweightedenergy\nestimates for symmetric hyperbolic system by introducing t he sound speed as a new variableEULEREQUATIONSWITH TIME-DEPENDENTDAMPING 3\nrather than the density. We will establish some weighted a pr ior estimates to the solution.\nThe choice of the weight is inspired by the corresponding lin ear wave equation with effective\ndampingsatisfiedbythesoundspeed. See(2.3). Theweightca nbefoundinNishihara[18]. The\nlocal-existence result stated in Kato[12] or Majda[15] and the continuity argument can assure\ntheglobalexistenceofthesolution.\nThe estimates of convergence rate of the solution to the back ground state come from the\ninvestigation of the fundamental solutions to the phase fun ction of the corresponding linear\nwave equation which can be found in Wirth[27]. We will show th atL2andL∞norms of the\nsolutiontosystem(1.1)willpresentadecayestimatesimil artothatofthecorrespondinglinear\ndissipativewaveequationwiththesamedamping.\nLetHl(Rn)betheusualSobolevspacewithitsnorm\n/ba∇dblf/ba∇dblHl/definesl/summationdisplay\nk=0/summationdisplay\n|α|=k/ba∇dbl∂α\nxf/ba∇dblL2.\nwhere∂α\nx=∂α1\n1...∂αnn,α= (α1,...,αn). Later for convenience,we willuse ∂k\nx=/summationtext\n|α|=k∂α\nx, use\n/ba∇dbl·/ba∇dblpto denote /ba∇dbl·/ba∇dblLpand/ba∇dbl·/ba∇dbl=/ba∇dbl·/ba∇dbl2.\nWe statetheglobalexistenceresultas follows.\nTheorem 1.1 DenoteB=(1+λ)n\n2−δ, whereδ∈(0,(1+λ)n\n2]can be arbitrarilysmall. Suppose\nthatn≥1,0< λ <1,0< µand(ρ0,u0)∈Hs+m(Rn), supported in {x∈Rn||x| ≤\nR}, wheres= [n/2] + 1andm≥2. Then there exists a ε0=ε0(δ,λ,µ,R)such that for\nany0≤ε≤ε0, when/ba∇dbl(ρ0,u0)/ba∇dblHs+m≤ε, there exists a unique global classical solution\n(ρ(t,x),u(t,x))of(1.1)satisfying\n(1+t)B+1+λ/parenleftBig\n/ba∇dbl∂tρ(t)/ba∇dbl2\nHs+m−1+/ba∇dbl∂xρ(t)/ba∇dbl2\nHs+m−1+/ba∇dbl∂xu(t)/ba∇dbl2\nHs+m−1/parenrightBig\n+(1+t)B+2λ/parenleftBig\n/ba∇dbl∂tu(t)/ba∇dbl2\nHs+m−1/parenrightBig\n+(1+t)B/parenleftBig\n/ba∇dbl(ρ(t)−1)/ba∇dbl2+/ba∇dblu(t)/ba∇dbl2/parenrightBig\n≤Cλ,µ,δ,R/parenleftBig\n/ba∇dblρ0/ba∇dbl2\nHs+m+/ba∇dblu0/ba∇dbl2\nHs+m/parenrightBig\n. (1.3)\nThroughoutthispaperwewilldenoteagenericconstantby Cwhichmay bedifferentfrom\nlineto line.\nRemark1.1 From(1.3), usingtheSobolevembedding,for k= 0,1,...,m−1,we have\n(1+t)B+1+λ\n2/parenleftBig\n/ba∇dbl∂k\nx∂tρ(t)/ba∇dbl∞+/ba∇dbl∂k+1\nxρ(t)/ba∇dbl∞+/ba∇dbl∂k+1\nxu(t)/ba∇dbl∞/parenrightBig\n+(1+t)B\n2+λ/parenleftBig\n/ba∇dbl∂k\nx∂tu(t)/ba∇dbl∞/parenrightBig\n+(1+t)B\n2/parenleftBig\n/ba∇dbl(ρ(t)−1)/ba∇dbl∞+/ba∇dblu(t)/ba∇dbl∞/parenrightBig\n≤Cε. (1.4)4 XINGHONG PAN\nFrom (1.4), we see that /ba∇dbl(ρ(t)−1)/ba∇dbl∞+/ba∇dblu(t)/ba∇dbl∞≤Cε(1 +t)−1+λ\n4n+δ\n2. Whenλ= 0, it\ndecays slower than what we expect for (1 +t)−n\n2as shown in [28] and [25]. So next, based\non theinvestigationoftheproperties tothecorresponding linearwaveequation(3.1)ofsystem\n(1.1), wehavethefollowingfurtherasymptoticbehaviorof thesolution.\nTheorem 1.2 Definekc/defines1+λ\n1−λ(n+1)−n−2δ\n1−λandm≥kc+2. Thenundertheassumption\nof Theorem1.1, we have the following asymptoticbehavior of the solution (ρ,u)inL2andL∞\nnorms.\nForρ:\n/ba∇dbl∂k\nx(ρ−1)/ba∇dbl∞≤Cε/braceleftBigg\n(1+t)−(1−λ)n+k\n2 0≤k≤kc;\n(1+t)−(1+λ)n+1\n2+δkc≤k≤m−2.(1.5)\n/ba∇dbl∂k\nx(ρ−1)/ba∇dbl ≤Cε\n\n(1+t)−(1−λ)(n\n4+k\n2)0≤k≤kc+n\n2;\n(1+t)−(1+λ)n+1\n2+δkc+n\n2≤k≤s+m−2.(1.6)\nWhileforu,duetothedamping,itwill decayslower than ρbya factor (1+t)λ. Thatis\n/ba∇dbl∂k\nxu/ba∇dbl∞≤Cε/braceleftBigg\n(1+t)−(1−λ)n+k+1\n2+λ0≤k≤kc−1;\n(1+t)−(1+λ)n+1\n2+λ+δkc−1≤k≤m−3.(1.7)\n/ba∇dbl∂k\nxu/ba∇dbl ≤Cε\n\n(1+t)−(1−λ)(n\n4+k+1\n2)+λ0≤k≤kc+n\n2−1;\n(1+t)−(1+λ)n+1\n2+λ+δkc+n\n2−1≤k≤s+m−3.(1.8)\nRemark1.2 Notingδ >0can be arbitrarily small, then lim\nλ→0kc= 1+. From(1.5)and(1.7),\nwe havewhen λ→0\n/ba∇dbl(ρ−1)(t)/ba∇dbl∞≤C(1+t)−n\n2,/ba∇dblu(t)/ba∇dbl∞≤C(1+t)−n+1\n2.\nThis coincideswiththedecayratethatobtainedinthenon-d ecayed dampingcase.\nIn 3-d Euler equations, denoting ω=∇×u, we will derive the exponential decay of ωin\ntime inL2norm under a positive integration assumption on ρ0. While the solution itself can\nnot decay so fast to itsbackground state. It has a polynomial lydecayed lowerbound. They are\npresented inthefollowingTheorem.\nTheorem 1.3 Supposeq0/defines/integraltext\nRnρ0dx >0. Under the assumption of Theorem1.1, there exists\nat0>0,dependingon λ,µ,Rsuchthatwhen t≥t0,\n/ba∇dbl(ρ−1)(t)/ba∇dbl ≥Cq0(1+t)−n\n2,/ba∇dblu(t)/ba∇dbl ≥Cq0(1+t)−n+2\n2. (1.9)\nOntheotherhand,inthreedimensions,the L2normofthevorticity ωdecays exponentially\nintime, satisfyingthefollowingestimate\n/ba∇dblω(·,t)/ba∇dbl ≤Ce−C(1+t)1−λ. (1.10)EULEREQUATIONSWITH TIME-DEPENDENTDAMPING 5\nRemark1.3 TheideaofprovingTheorem1.3comesfrom[25],wheretheaut horsdealwiththe\ncasen= 3andthedampingisnon-decayed intime.\nThe paper is organized as follows. In Section 2, we reformula te the Euler equations into a\nsymmetric hyperbolic system. Then based on a fundamental we ighted energy inequality in\nLemma2.1 and using detailed weighted energy estimates, we p rove the global existence of\nsmooth solutions with small data. In Section 3, by investiga ting the structure to the linear\nsystemof(1.1),wegivetheasymptoticbehaviorofthesolut ion. InSection4,weshowthatthe\nvorticitydecayexponentiallywhilethesolutionitselfha sapolynomiallydecayedlowerbound.\nIn theAppendix,wegivetheproofofthefundamentalweighte denergy inequality.\n2 GlobalExistence\nIn this Section, First we reformulate system (1.1) to a symme tric system. Remember c=/radicalbig\nP′(ρ) =ργ−1\n2. First wetransform(1.1)intothefollowingsystem\n\n\n2\nγ−1∂tc+c∇·u+2\nγ−1u∇·c= 0,\n∂tu+u·∇u+2\nγ−1c∇c+µ\n(1+t)λu= 0,\nc|t=0= 1+c0(x),u|t=0=u0(x),(2.1)\nwherec0(x)∈R, supportedin {x∈Rn||x| ≤R}.\nLetv=2\nγ−1(c−1), then(v,u)satisfies\n\n\n∂tv+∇·u=−u·∇v−γ−1\n2v∇·u,\n∂tu+∇v+µ\n(1+t)λu=−u·∇u−γ−1\n2v∇v,\nv|t=0=v0(x),u|t=0=u0(x),(2.2)\nwherev0(x) =2\nγ−1c0(x).\n2.1 Afundamental weighted energy inequality\nFrom (2.2),we have\n∂ttv−∆v+µ\n(1+t)λ∂tv=Q(v,u), (2.3)\nwhere\nQ(v,u) =µ\n(1+t)λ(−u·∇v−γ−1\n2v∇·u)6 XINGHONG PAN\n−∂t(u·∇v−γ−1\n2v∇·u)+∇·(u·∇u+γ−1\n2v∇v).\nInthefollowing,wewillobtainafundamentalweightedener gyinequalityabout(2.3). This\ntechniquecomes fromNishihara[18]. Introducetheweight\ne2ψ, ψ(t,x) =a|x|2\n(1+t)1+λ,\nwherea=(1+λ)µ\n8/parenleftBig\n1−δ\n(1+λ)n/parenrightBig\nandδisdescribedinTheorem1.1. Forsimplificationofnotation,\nwedenote\nJ(t;g) =/integraldisplay\nRne2ψg2(t,x)dx, Jψ(t;g) =/integraldisplay\nRne2ψ(−ψt)g2(t,x)dx.\nLemma 2.1 DenoteB=(1+λ)n\n2−δ, whereδ∈(0,(1+λ)n\n2]can be arbitrarilysmall. Then the\nequation (2.3)has thefollowingweightedenergy inequality.\n(1+t)B+1+λ/bracketleftBig\nJ(t;vt)+J(t;|∂xv|)/bracketrightBig\n+(1+t)BJ(t;v)\n+/integraldisplayt\n0(1+τ)B+1+λ/bracketleftBig\nJψ(τ;vτ)+Jψ(τ;|∇v|)/bracketrightBig\ndτ\n+/integraldisplayt\n0/bracketleftBig\n(1+τ)B+1J(τ,vτ)+(1+τ)B+λJ(τ,|∇v|)/bracketrightBig\ndτ\n+/integraldisplayt\n0/bracketleftBig\n(1+τ)BJψ(τ,v)+(1+τ)B−1J(τ,v)/bracketrightBig\ndτ\n≤C/ba∇dbl(v0,u0)/ba∇dbl2\nH1+CG(t),\nwhere\nG(t)/defines/integraldisplayt\n0/integraldisplay\nRne2ψ/braceleftBig/bracketleftbig\n(1+τ)B+1+λvτ+(1+τ)B+λv/bracketrightbig\nQ(v,u)/bracerightBig\ndxdτ,\nandCdepends on λ,µ,δ,R.\nProof.WegivetheproofofLemma2.1in theAppendix. /square\n2.2 Some apriori weighted energy estimates\nDefine theweightedSobolevspace Hs+m\nψ(Rn)as\nHs+m\nψ(Rn) ={f|eψ∂k\nxf∈L2(Rn),0≤k≤s+m,k∈N}\nwithitsnorm\n/ba∇dblf/ba∇dblHs+m\nψ=s+m/summationdisplay\nk=0/parenleftBig/integraldisplay\nRn(∂k\nxf)2e2ψdx/parenrightBig1\n2.EULEREQUATIONSWITH TIME-DEPENDENTDAMPING 7\nWedefine theweightedenergy asfollows\nEψ\ns+m(T) = : sup\n0<t<T/braceleftBig\n(1+t)B+1+λ/parenleftbig\n/ba∇dblvt(t)/ba∇dbl2\nHs+m−1\nψ+/ba∇dbl∂xv(t)/ba∇dbl2\nHs+m−1\nψ+/ba∇dbl∂xu(t)/ba∇dbl2\nHs+m−1\nψ/parenrightbig\n+(1+t)B/parenleftbig\n/ba∇dblv(t)/ba∇dbl2\nL2\nψ+/ba∇dblu(t)/ba∇dbl2\nL2\nψ/parenrightbig/bracerightBig1\n2.\nBecause theinitialdataiscompactlysupported,wehave\nEψ\ns+m(0)≤C/parenleftBig\n/ba∇dbl(vt(0),∂xv(0),∂xu(0))/ba∇dblHs+m−1+/ba∇dbl(v(0),u(0))/ba∇dbl/parenrightBig\n≤C/ba∇dbl(v0,u0)/ba∇dblHs+m≤Cε.\nIn thefollowing,wewillestimate (v,u)undertheaprioriassumption\nEψ\ns+m(t)≤Mε, (2.4)\nwhereM, independent of ε, will be determined later. By choosing Mlarge andεsufficient\nsmall,wewillprove\nEψ\ns+m(t)≤1\n2Mε. (2.5)\nWe will first obtain some a prior estimates for the 1-order der ivatives. Then the higher\nderivativeswillbehandledin asimilarway. From (2.4), by S obolevembedding,wehave\n/braceleftBig\n(1+t)B\n2/ba∇dbl(v(t),u(t))/ba∇dblL∞+(1+t)B+1+λ\n2/ba∇dbl(vt(t),∂xv(t),∂xu(t))/ba∇dblL∞/bracerightBig\n≤CEψ\n[n/2]+2(t)≤CMε. (2.6)\nFrom (2.2)2, wesee\n(1+t)B/2+λ/ba∇dblut(t)/ba∇dblL∞\n≤(1+t)B/2+λ/braceleftBig\n/ba∇dbl∇v(t)/ba∇dblL∞+(1+t)−λ/ba∇dblu(t)/ba∇dblL∞+/ba∇dblu·∇u/ba∇dblL∞+/ba∇dblv∇v/ba∇dblL∞/bracerightBig\n≤CMε. (2.7)\nNowforsimplificationofnotationin ourlatercomputation, we introduce\nP0(t)\n/definesP(t;v,u,vt,ut,∂xv,∂xu)\n/defines(1+t)B+1+λ/bracketleftBig\nJψ(t;vt)+Jψ(t;|∂xv|)+Jψ(t;|∂xu|)/bracketrightBig\n+(1+t)B+1/bracketleftbig\nJ(t;vt)+J(t;|∂xu|)/bracketrightbig\n+(1+t)B+λ/bracketleftbig\nJ(t;|ut|)+J(t;|∂xv|)/bracketrightbig\n+(1+t)B/bracketleftbig\nJψ(t;v)+Jψ(t;|u|)/bracketrightbig\n+(1+t)B−λJ(t;|u|)+(1+t)B−1J(t;v). (2.8)\nIn thefollowing,wewillgivethreeestimatesforthefirstor derderivativesundertheaprior\nassumption,whichcan reflect ourideaofprovingthe(2.5).\nEstimate18 XINGHONG PAN\nLemma 2.2 Underthea priorassumption (2.4), wehavethefollowingestimate\nG(t)≤CMε(1+t)B+1+λ/bracketleftbig\nJ(t;|vt|)+J(t;|∂xv|)/bracketrightbig\n+Cε/parenleftBig\nEψ\n1(0)/parenrightBig2\n+CMε/integraldisplayt\n0P0(τ)dτ, (2.9)\nwhereP0(τ)is definedin (2.8).\nProof.Remembertheformulationof Q(v,u),wehave\nG(t) =/integraldisplayt\n0/integraldisplay\nRne2ψ/parenleftBig\n(1+τ)B+1+λvτ+(1+τ)B+λv/parenrightBig\n×\n/braceleftBigµ\n(1+τ)λ/parenleftBig\n−u·∇v−γ−1\n2v∇·u/parenrightBig\n/bracehtipupleft /bracehtipdownright/bracehtipdownleft /bracehtipupright\nI1\n−/parenleftBig\n∂τ(u·∇v)+γ−1\n2∂τ(v∇·u)/parenrightBig\n/bracehtipupleft /bracehtipdownright/bracehtipdownleft /bracehtipupright\nI2\n+/parenleftBig\n∇·(u·∇u)+γ−1\n2∇·(v∇v)/parenrightBig\n/bracehtipupleft /bracehtipdownright/bracehtipdownleft /bracehtipupright\nI3/bracerightBig\ndxdτ.\nNowweestimate Ii(i= 1,2,3)term byterm undertheapriori estimate(2.4).\nUsing(2.6), (2.7)and Cauchy-Schwartz inequality,wehave\nI1=µ/integraldisplayt\n0/integraldisplay\nRne2ψ(1+τ)B+1vτ/parenleftBig\n−u·∇v−γ−1\n2v∇·u/parenrightBig\ndxdτ\n+µ/integraldisplayt\n0/integraldisplay\nRn(1+τ)Be2ψv/parenleftBig\n−u·∇v−γ−1\n2v∇·u/parenrightBig\ndxdτ\n≤C/ba∇dbl(1+τ)1+λ\n2∇v(τ)/ba∇dbl∞/integraldisplayt\n0/bracketleftBig\n(1+τ)B+1J(τ;vτ)+(1+τ)B−λJ(τ;|u|)/bracketrightBig\ndτ\n+C/ba∇dblv(τ)/ba∇dbl∞/integraldisplayt\n0/bracketleftBig\n(1+τ)B+1J(τ;vτ)+(1+τ)B+1J(τ;|∇u|)/bracketrightBig\ndτ\n+C/ba∇dblv(τ)/ba∇dbl∞/integraldisplayt\n0/bracketleftBig\n(1+τ)B−λJ(τ;|u|)+(1+τ)B+λJ(τ;|∇v|)/bracketrightBig\ndτ\n+C/ba∇dblv(τ)/ba∇dbl∞/integraldisplayt\n0/bracketleftBig\n(1+τ)B−1J(τ;v)+(1+τ)B−1J(τ;|∇u|)/bracketrightBig\ndτ\n≤CMε/integraldisplayt\n0P0(τ)dτ. (2.10)\nAnd\nI2=−/integraldisplayt\n0/integraldisplay\nRne2ψ/parenleftBig\n(1+τ)B+1+λvτ+(1+τ)B+λv/parenrightBig\n×EULEREQUATIONSWITH TIME-DEPENDENTDAMPING 9\n/braceleftBig/parenleftbig\nuτ·∇v+γ−1\n2vτ∇·u/parenrightbig\n+/parenleftBig\nu·∇vτ/parenrightBig\n/bracehtipupleft/bracehtipdownright/bracehtipdownleft/bracehtipupright\nI2,1+/parenleftBigγ−1\n2v∇·uτ/parenrightBig\n/bracehtipupleft/bracehtipdownright/bracehtipdownleft/bracehtipupright\nI2,2/bracerightBig\ndxdτ\n≤CMε/integraldisplayt\n0P0(τ)dτ+I2,1+I2,2. (2.11)\nNextweuseintegrationbyparts toestimate I2,1andI2,2.\nI2,1=−/integraldisplayt\n0/integraldisplay\nRne2ψ/parenleftBig\n(1+τ)B+1+λvτ+(1+τ)B+λv/parenrightBig/parenleftBig\nu·∇vτ/parenrightBig\ndxdτ\n=−1\n2/integraldisplayt\n0/integraldisplay\nRne2ψ(1+τ)B+1+λu·∇v2\nτdxdτ−/integraldisplayt\n0/integraldisplay\nRne2ψ(1+τ)B+λvu·∇vτdxdτ\n=1\n2/integraldisplayt\n0/integraldisplay\nRne2ψ(1+τ)B+1+λv2\nτ(∇·u+2u·∇ψ/bracehtipupleft/bracehtipdownright/bracehtipdownleft/bracehtipupright\nI1\n2,1)dxdτ\n+/integraldisplayt\n0/integraldisplay\nRne2ψ(1+τ)B+λvτ(v∇·u+u·∇v+2vu·∇ψ/bracehtipupleft/bracehtipdownright/bracehtipdownleft/bracehtipupright\nI2\n2,1)dxdτ\n≤CMε/integraldisplayt\n0P0(τ)dτ+I1\n2,1+I2\n2,1. (2.12)\nNoting(5.2),wehave\n|∇ψ| ≤C(1+t)−λ\n2(−ψt)1\n2. (2.13)\nUsingCauchy-Schwartz inequality,weobtain\n|I1\n2,1|+|I2\n2,1|\n≤C/ba∇dbl(1+τ)1+λ\n2vτ/ba∇dblL∞/integraldisplayt\n0/bracketleftbig\n(1+τ)B+1J(τ;vτ)+(1+τ)BJψ(τ;|u|)/bracketrightbig\ndxdτ\n≤C/ba∇dbl(1+τ)1+λ\n2vτ/ba∇dblL∞/integraldisplayt\n0/bracketleftbig\n(1+τ)B−1J(τ;v)+(1+τ)BJψ(τ;|u|)/bracketrightbig\ndxdτ\n≤CMε/integraldisplayt\n0P0(τ)dτ. (2.14)\nCombing(2.12)and (2.14), wehave\n|I2,1| ≤CMε/integraldisplayt\n0P0(τ)dτ. (2.15)\nForI2,2, wehave\nI2,2=/integraldisplayt\n0/integraldisplay\nRne2ψ/parenleftBig\n(1+τ)B+1+λvτ+(1+τ)B+λv/parenrightBig/parenleftBigγ−1\n2v∇·uτ/parenrightBig\ndxdτ10 XINGHONG PAN\n=C/integraldisplayt\n0/integraldisplay\nRne2ψ(1+τ)B+λv2∇·uτdxdτ\n/bracehtipupleft /bracehtipdownright/bracehtipdownleft /bracehtipupright\nI1\n2,2\n+C/integraldisplayt\n0/integraldisplay\nRne2ψ(1+τ)B+1+λvτv∇·uτdxdτ\n/bracehtipupleft /bracehtipdownright/bracehtipdownleft /bracehtipupright\nI2\n2,2. (2.16)\nUsingintegrationby parts,(2.13)andCauchy-Schwartz ine quality,wehave\nI1\n2,2=−C/integraldisplayt\n0/integraldisplay\nRne2ψ(1+τ)B+λuτ·(2v∇·v+2v2∇ψ)dxdτ\n≤CMε/integraldisplayt\n0P0(τ)dτ. (2.17)\nFortheestimateof I2\n2,2,we use(2.2)and integrationbyparts. From (2.2)1, wehave\n∇·u=−vt+u·∇v\n1+γ−1\n2v, (2.18)\n∇·ut=−vtt+ut·∇v+u·∇vt\n1+γ−1\n2v+γ−1\n2vt(vt+u·∇v)\n(1+γ−1\n2v)2. (2.19)\nFrom (2.6),we have\n1\n1+γ−1\n2v≤1\n1−CMε≤2. (2.20)\nInserting(2.19)and(2.20)into I2\n2,2, wehave\nI2\n2,2=/integraldisplayt\n0/integraldisplay\nRne2ψ(1+τ)B+1+λvτv∇·uτdxdτ\n=/integraldisplayt\n0/integraldisplay\nRne2ψ(1+τ)B+1+λvτv/braceleftBig\n−vττ+uτ·∇v+u·∇vτ\n1+γ−1\n2v\n+γ−1\n2vτ(vτ+u·∇v)\n(1+γ−1\n2v)2/bracerightBig\ndxdτ\n≤ −/integraldisplayt\n0/integraldisplay\nRne2ψ(1+τ)B+1+λvvτvττ\n1+γ−1\n2vdxdτ\n/bracehtipupleft /bracehtipdownright/bracehtipdownleft /bracehtipupright\nI2,1\n2,2\n−/integraldisplayt\n0/integraldisplay\nRne2ψ(1+τ)B+1+λvvτu·∇vτ\n1+γ−1\n2vdxdτ\n/bracehtipupleft /bracehtipdownright/bracehtipdownleft /bracehtipupright\nI2,2\n2,2\n+CMε/integraldisplayt\n0P0(τ)dτ. (2.21)EULEREQUATIONSWITH TIME-DEPENDENTDAMPING 11\nUsingintegrationby partsin timeand(2.20), wehave\nI2,1\n2,2=−/integraldisplayt\n0/integraldisplay\nRne2ψ(1+τ)B+1+λv(v2\nτ)τ\n2/parenleftbig\n1+γ−1\n2v/parenrightbigdxdτ\n=−/integraldisplay\nRne2ψ(1+τ)B+1+λvv2\nτ\n2/parenleftbig\n1+γ−1\n2v/parenrightbig/vextendsingle/vextendsingle/vextendsingle/vextendsingleτ=t\nτ=0dx\n+/integraldisplayt\n0/integraldisplay\nRne2ψv2\nτ/braceleftbigg\n(1+τ)B+1+λ∂τ/parenleftbiggv\n2/parenleftbig\n1+γ−1\n2v/parenrightbig/parenrightbigg\n+(B+1+λ)(1+τ)B+λv\n2/parenleftbig\n1+γ−1\n2v/parenrightbig\n+2ψt(1+τ)B+1+λv\n2/parenleftbig\n1+γ−1\n2v/parenrightbig/bracerightbigg\ndxdτ\n≤CMε(1+t)B+1+λJ(t;vt)+Cε/parenleftBig\nEψ\n1(0)/parenrightBig2\n+CMε/integraldisplayt\n0P0(τ)dτ.(2.22)\nAnd\nI2,2\n2,2=−/integraldisplayt\n0/integraldisplay\nRne2ψ(1+τ)B+1+λv\n2/parenleftbig\n1+γ−1\n2v/parenrightbigu·∇v2\nτdxdτ\n=/integraldisplayt\n0/integraldisplay\nRne2ψ(1+τ)B+1+λv2\nτ/braceleftbigg\nu·∇/parenleftBigv\n2/parenleftbig\n1+γ−1\n2v/parenrightbig/parenrightBig\n+v\n2/parenleftbig\n1+γ−1\n2v/parenrightbig∇·u\n+v\n1+γ−1\n2vu∇ψ/bracerightbigg\ndxdτ\n≤CMε/integraldisplayt\n0P0(τ)dτ. (2.23)\nFrom (2.21), (2.22)and (2.23), wehave\nI2\n2,2≤CMε(1+t)B+1+λJ(t;vt)+Cε/parenleftBig\nEψ\n1(0)/parenrightBig2\n+CMε/integraldisplayt\n0P0τ)dτ.\nSummingall theestimatesabout I2, wecan get\nI2≤CMε(1+t)B+1+λJ(t;vt)+Cε/parenleftBig\nEψ\n1(0)/parenrightBig2\n+CMε/integraldisplayt\n0P0(τ)dτ. (2.24)\nTheestimateof I3willbeessentiallythesamewith I2, actuallywecan get\nI3≤CMε(1+t)B+1+λJ(t;|∂xv|)+Cε/parenleftBig\nEψ\n1(0)/parenrightBig2\n+CMε/integraldisplayt\n0P0(τ)dτ. (2.25)12 XINGHONG PAN\nCombingtheestiamtes(2.10), (2.24)and (2.25), wegetthee stimate\nG(t)≤CMε(1+t)B+1+λ/bracketleftbig\nJ(t;|vt|)+J(t;|∂xv|)/bracketrightbig\n+Cε/parenleftBig\nEψ\n1(0)/parenrightBig2\n+CMε/integraldisplayt\n0P0(τ)dτ,\nwhichfinishes theproofoftheLemma. /square\nCombingLemma2.1and Lemma2.2,wehave\n(1+t)B+1+λ/bracketleftBig\nJ(t;vt)+J(t;|∇v|)/bracketrightBig\n+(1+t)BJ(t;v)\n+/integraldisplayt\n0(1+τ)B+1+λ/bracketleftBig\nJψ(τ;vτ)+Jψ(τ;|∇v|)/bracketrightBig\ndτ\n+/integraldisplayt\n0/bracketleftBig\n(1+τ)B+1J(τ,vτ)+(1+τ)B+λJ(τ,|∇v|)/bracketrightBig\ndτ\n+/integraldisplayt\n0/bracketleftBig\n(1+τ)BJψ(τ,v)+(1+τ)B−1J(τ,v)/bracketrightBig\ndτ\n≤C/parenleftBig\nEψ\n1(0)/parenrightBig2\n+CMε/integraldisplayt\n0P0(τ)dτ. (2.26)\nEstimate2\nNextweconsidertheweighted L2norm of(v,u). WehavethefollowingLemma.\nLemma 2.3 Underthea priorassumption (2.4), wehavethefollowingestimate\n(1+t)B/bracketleftbig\nJ(t;v)+J(t;|u|)]\n+/integraldisplayt\n0(1+τ)BJψ(τ;|u|)dτ+/integraldisplayt\n0(1+τ)B−λJ(τ;|u|)dτ\n−C/integraldisplayt\n0/bracketleftbig\n(1+τ)BJψ(τ;v)+(1+τ)B−1J(τ;v)/bracketrightbig\ndτ\n≤C/parenleftbig\nJ(0;v)+J(0;|u|)/parenrightbig\n+CMε/integraldisplayt\n0P0(τ)dτ. (2.27)\nwhereP0(τ)is definedin (2.8).\nProof.Multiplying(2.2)2by(K+t)Be2ψuyields\n∂t/bracketleftBig\n(K+t)Be2ψ\n2|u|2/bracketrightBig\n+e2ψ(K+t)B(−ψt)|u|2\n/parenleftBigµ(K+t)B\n(1+t)λ−B(K+t)B−1\n2/parenrightBig\ne2ψ|u|2+(K+t)Be2ψu·∇v\n= (K+t)Be2ψu·(−u·∇u−γ−1\n2v∇v). (2.28)EULEREQUATIONSWITH TIME-DEPENDENTDAMPING 13\nIntegrating(2.28)on Rn×[0,t]and choosing Kbelarge, wecan get\n(K+t)BJ(t;|u|)+/integraldisplayt\n0(K+τ)BJψ(τ;|u|)dτ+/integraldisplayt\n0(K+τ)B−λJ(τ;|u|)dτ\n+C/integraldisplayt\n0/integraldisplay\nRne2ψ(K+τ)Bu·∇vdxdτ\n/bracehtipupleft /bracehtipdownright/bracehtipdownleft /bracehtipupright\nI1\n≤C/integraldisplayt\n0/integraldisplay\nRn(K+τ)Be2ψu·(−u·∇u−γ−1\n2v∇v)dxdτ\n/bracehtipupleft /bracehtipdownright/bracehtipdownleft /bracehtipupright\nI2+CJ(0;|u|).(2.29)\nWecometoestimate I1andI2.\nUsingintegrationby partsand (2.2)1, wehave\nI1=−/integraldisplayt\n0/integraldisplay\nRn(K+τ)Be2ψv(∇·u+2u·∇ψ)dxdτ\n=/integraldisplayt\n0/integraldisplay\nRn(K+τ)Be2ψv(vτ+u·∇v+γ−1\n2v∇·u−2u·∇ψ)dxdτ\n=/integraldisplayt\n0/integraldisplay\nRn/braceleftBig\n∂τ/bracketleftBig\n(K+τ)Be2ψv2\n2/bracketrightBig\n+(K+τ)Be2ψ(−ψτ)v2−B/2(K+τ)B−1e2ψv2/bracerightBig\ndxdτ\n/bracehtipupleft /bracehtipdownright/bracehtipdownleft /bracehtipupright\nI1,1\n+/integraldisplayt\n0/integraldisplay\nRn(K+τ)Be2ψv(u·∇v+γ−1\n2v∇·u)dxdτ\n/bracehtipupleft /bracehtipdownright/bracehtipdownleft /bracehtipupright\nI1,2\n−2/integraldisplayt\n0/integraldisplay\nRn(K+τ)Be2ψvu·∇ψdxdτ\n/bracehtipupleft /bracehtipdownright/bracehtipdownleft /bracehtipupright\nI1,3. (2.30)\nWeseethat\nI1,1= 1/2(K+t)BJ(t;v)−CJ(0;v)\n+/integraldisplayt\n0(K+τ)BJψ(τ;v)dτ−B/2/integraldisplayt\n0(K+τ)B−1J(τ;v)dτ. (2.31)\nUsing(2.6)and(2.7), wehave\n|I1,2| ≤ /ba∇dbl(K+t)(1+λ)/2∂xv(t)/ba∇dblL∞/integraldisplayt\n0/bracketleftbig\n(K+t)B−λJ(τ;u)+(K+τ)B−1J(τ;v)/bracketrightbig\ndτ\n+/ba∇dbl(K+t)(B+1)/2∂xu(t)/ba∇dblL∞/integraldisplayt\n0(K+t)B−1J(τ;v)dτ\n≤CMε/integraldisplayt\n0P0(τ)dτ. (2.32)14 XINGHONG PAN\nUsingCauchy-Schwartz inequalityand (5.2),we obtain\n|I1,3| ≤ν/integraldisplayt\n0(K+t)B−λJ(τ;|u|)dτ+Cν/integraldisplayt\n0(K+t)BJψ(τ;v)dτ, (2.33)\nwhereνisasmallnumber. Combing(2.30) −(2.33),wehave\nI1≥1/2(K+t)BJ(t;v)−CJ(0;v)\n+/integraldisplayt\n0(K+τ)BJψ(τ;v)dτ−B/2/integraldisplayt\n0(K+τ)B−1J(τ;v)dτ\n−ν/integraldisplayt\n0(K+t)B−λJ(τ;|u|)dτ−Cν/integraldisplayt\n0(K+t)BJψ(τ;v)dτ\n−CMε/integraldisplayt\n0P0(τ)dτ. (2.34)\nI2can bedonethesamewith I1,2, so,wehave\n|I2| ≤CMε/integraldisplayt\n0P0(τ)dτ. (2.35)\nCombing(2.29),(2.34)and (2.35), by choosing νsmall,we have\n(K+t)B/bracketleftbig\nJ(t;v)+J(t;|u|)]\n+/integraldisplayt\n0(K+τ)BJψ(τ;|u|)dτ+/integraldisplayt\n0(K+τ)B−λJ(τ;|u|)dτ\n−C/integraldisplayt\n0/bracketleftbig\n(K+t)BJψ(τ;v)+(K+t)B−1J(t;v)/bracketrightbig\ndτ\n≤C/parenleftbig\nJ(0;v)+J(0;|u|)/parenrightbig\n+CMε/integraldisplayt\n0P0(τ)dτ.\nThisfinishestheproofofLemma2.3. /square\nEstimate3\nNextweconsidertheweighted L2norm of(∂xv,∂xu). Wehavethefollowinglemma.\nLemma 2.4 Underthea priorassumption (2.4), wehavethefollowingestimate\n(1+t)B+1+λ/bracketleftbig\nJ(t;|∂xv|)+J(t;|∂xu|)]\n+/integraldisplayt\n0(1+τ)B+1+λJψ(τ;|∂xu|)dτ+/integraldisplayt\n0(1+τ)B+1J(τ;|∂xu|)dτ\n−C/integraldisplayt\n0/bracketleftbig\n(1+τ)B+1+λJψ(τ;|∂xv|)+(1+τ)B+λJ(τ;|∂xv|)/bracketrightbig\ndτ\n≤C/parenleftBig\nEψ\n1(0)/parenrightBig2\n+CMε/integraldisplayt\n0P0(τ)dτ, (2.36)\nwhereP0(τ)is definedin (2.8).EULEREQUATIONSWITH TIME-DEPENDENTDAMPING 15\nProof.Bydifferentiating(2.2)2withrespectto xandthenmultiplyingitwith (K+t)B+1+λe2ψ∂xu,\nweget\n∂t/bracketleftBig\n(K+t)B+1+λe2ψ|∂xu|2\n2/bracketrightBig\n+e2ψ(K+t)B+1+λ(−ψt)|∂xu|2\n+/bracketleftbigg/parenleftBig\nµ(K+t)λ\n(1+t)λ−B+1+λ\n2(K+t)λ−1/parenrightBig\n(K+t)B+1e2ψ|∂xu|2/bracketrightbigg\n+(K+t)B+1+λe2ψ∂xu·∇∂xv\n= (K+t)B+1+λe2ψ∂xu·∂x(−u·∇u−γ−1\n2v∇v). (2.37)\nBychoosing Klargeandintegrating(2.37)on Rn×[0,t],wecanobtaintheestimate(2.36)\nby usingthesametrickas thatin theproofofLemma2.3. Weomi tthedetail. /square\nEstimate4\nFrom (2.2)2, wecan easily have\n/integraldisplayt\n0(1+τ)B+λJ(τ;|uτ|)dτ\n≤/integraldisplayt\n0(1+τ)B+λJ(τ;|∂xv|)dτ+/integraldisplayt\n0(1+τ)B−λJ(τ;|u|)dτ+CMε/integraldisplayt\n0P0(τ)dτ.\n(2.38)\nAdding(2.26)to ν1·(2.27)+ν2·(2.36)+ν3·(2.38),where ν1,ν2≪1andν3≪ν1,wecanget\n/parenleftBig\nEψ\n1(t)/parenrightBig2\n+/integraldisplayt\n0P0(τ)dτ\n≤/parenleftBig\nEψ\n1(0)/parenrightBig2\n+CMε/integraldisplayt\n0P0(τ)dτ. (2.39)\n2.3 Proofof Theorem1.1\nProof.Actually the estimate (2.39) can be obtained for higher deri vatives. Define Pk(t)the\nsameasP0.\nPk(t)\n/definesP(t;∂k\nxv,∂k\nxu,∂k\nxvt,∂k\nxut,∂x∂k\nxv,∂x∂k\nxu)\n/defines(1+t)B+1+λ/bracketleftBig\nJψ(t;∂k\nxvt)+Jψ(t;|∂x∂k\nxv|)+Jψ(t;|∂x∂k\nxu|)/bracketrightBig\n+(1+t)B+1/bracketleftbig\nJ(t;∂k\nxvt)+J(t;|∂x∂k\nxu|)/bracketrightbig\n+(1+t)B+λ/bracketleftbig\nJ(t;|∂k\nxut|)+J(t;|∂x∂k\nxv|)/bracketrightbig\n+(1+t)B/bracketleftbig\nJψ(t;∂k\nxv)+Jψ(t;|∂k\nxu|)/bracketrightbig\n+(1+t)B−λJ(t;|∂k\nxu|)+(1+t)B−1J(t;∂k\nxv).16 XINGHONG PAN\nForanyl∈N, differentiating(2.3) ltimeswithrespect tospace variable x, wehave\n∂tt∂l\nxv−∆∂l\nxv+µ\n(1+t)λ∂t∂l\nxv=∂l\nxQ(v,u). (2.40)\nThen do the same estimate as shown in Estimate 1 . We can get the following similar\nestimateto(2.26)\n(1+t)B+1+λ/bracketleftbig\nJ(t;∂l\nxvt)+J(t;|∂x∂l\nxv|)/bracketrightbig\n+/integraldisplayt\n0(1+τ)B+1+λ/bracketleftbig\nJψ(τ;∂l\nxvτ)+Jψ(τ;(|∂x∂l\nxv|)/bracketrightbig\ndτ\n+/integraldisplayt\n0(1+τ)B+1J(τ;∂l\nxvτ)+(1+τ)B+λJ(τ;|∂x∂l\nxv|)dτ\n+/integraldisplayt\n0(1+τ)BJψ(τ,∂l\nxv)+(1+τ)B−1J(τ;∂l\nxv)dτ\n≤CEψ\nl+1(0)+CMεl/summationdisplay\nk=0/integraldisplayt\n0Pk(τ)dτ. (2.41)\nDifferentiating(2.2)2ltimeswithrespectto x,multiplyingitby (K+t)Be2ψ∂l\nxuanddoing\nthesameestimateas in Estimate2 ,wehave\n(1+t)B/bracketleftbig\nJ(t;∂l\nxv)+J(t;|∂l\nxu|)]\n+/integraldisplayt\n0(1+τ)BJψ(τ;|∂l\nxu|)dτ+/integraldisplayt\n0(1+τ)B−λJ(τ;|∂l\nxu|)dτ\n−C/integraldisplayt\n0/bracketleftbig\n(1+τ)BJψ(τ;∂l\nxv)+(1+τ)B−1J(τ;∂l\nxv)/bracketrightbig\ndτ\n≤CEψ\nl(0)+CMεl−1/summationdisplay\nk=0/integraldisplayt\n0Pk(τ)dτ. (2.42)\nDifferentiating(2.2)2l+1timeswithrespectto x,multiplyingitby (K+t)B+1+λe2ψ∂l+1\nxu\nand doingthesameestimateas in Estimate3 , wehave\n(1+t)B+1+λ/bracketleftbig\nJ(t;|∂x∂l\nxv|)+J(t;|∂x∂l\nxu|)]\n+/integraldisplayt\n0(1+τ)B+1+λJψ(τ;|∂x∂l\nxu|)dτ+/integraldisplayt\n0(1+τ)B+1J(τ;|∂x∂l\nxu|)dτ\n−C/integraldisplayt\n0/bracketleftbig\n(1+τ)B+1+λJψ(τ;|∂x∂l\nxv|)+(1+τ)B+λJ(τ;|∂x∂l\nxv|)/bracketrightbig\ndτ\n≤CEψ\nl+1(0)+CMεl/summationdisplay\nk=0/integraldisplayt\n0Pk(τ)dτ. (2.43)\nFrom (2.2)2, wealsohave\n/integraldisplayt\n0(1+τ)B+λJ(τ;|∂l\nxuτ|)dτEULEREQUATIONSWITH TIME-DEPENDENTDAMPING 17\n≤/integraldisplayt\n0(1+τ)B+λJ(τ;|∂x∂l\nxv|)dτ+/integraldisplayt\n0(1+τ)B−λJ(τ;|∂l\nxu|)dτ\n+CMεl/summationdisplay\nk=0/integraldisplayt\n0Pk(τ)dτ. (2.44)\nAdding (2.41) −(2.44) together the same as (2.39) for lfrom0tos+m−1together. Then\nthereexistsa C0, dependingon λ,µ,δandRsuchthat\n/parenleftBig\nEψ\ns+m(t)/parenrightBig2\n+s+m−1/summationdisplay\nk=0/integraldisplayt\n0Pk(τ)dτ\n≤C/parenleftBig\nEψ\ns+m(0)/parenrightBig2\n+CMεs+m−1/summationdisplay\nk=0/integraldisplayt\n0Pk(τ)dτ.\n≤C2\n0ε2+C0Mεs+m−1/summationdisplay\nk=0/integraldisplayt\n0Pk(τ)dτ.\nLetM= 2C0andC0Mε<1. Thenwe obtain\nEψ\ns+m(t)≤1\n2Mε.\nThe local existence of symmetrizable hyperbolic equations have been proved by using the\nfixed point theorem. In order to get the global existence of th e system, we only need a priori\nestimate. Basedontheaprioriestimate(2.4),weyield(2.5 ). ThisfinishestheproofofTheorem\n1.1. /square\n3 AsymptoticBehaviorof theSolution\nIn this Section, we restudy the decay rate of the smooth solut ion to system (1.1) and obtain\nthe asymptotic behavior of the solution to the background st ate(1,0)τ. For this purpose, we\nconsiderthelinearequationof(2.3)\nwtt−∆w+µ\n(1+t)λwt=f. (3.1)\nThis equation involvesin thestudy of theCauchy problem to t he wave equation with time-\ndependentdamping.\n\nwtt−∆w+µ\n(1+t)λwt=f,\nv(0,x) =w0(x), wt(0,x) =w1(x).(3.2)\nwhere0<λ<1,0<µ.18 XINGHONG PAN\n3.1 Decay rate ofsolutions tothe linear wave equation\nIf we apply the partial Fourier transform with respect to the space variables to (3.2), we can\nget\n\nˆwtt+|ξ|2ˆw+µ\n(1+t)λˆwt=ˆf,\nˆw(0,ξ) = ˆw0(ξ),ˆwt(0,ξ) = ˆw1(ξ),(3.3)\nwhereξis thefrequency parameter. The solutionof (3.3) can be repr esented by thesum of the\nsolutionofthefollowingtwo equations\n\n\nˆw1\ntt+|ξ|2ˆw1+µ\n(1+t)λˆw1\nt= 0,\nˆw(0,ξ) = ˆw0(ξ),ˆwt(0,ξ) = ˆw1(ξ),(3.4)\nand \n\nˆw2\ntt+|ξ|2ˆw2+µ\n(1+t)λˆw2\nt=ˆf,\nˆw(0,x) = 0,ˆwt(0,x) = 0.(3.5)\nThesolutionof(3.4)can berepresented intheform\nˆw1(t,ξ) = Φ1(t,ξ)ˆw0(ξ)+Φ2(t,ξ)ˆw1(ξ).\nAndThesolutionof(3.5)can berepresented in theform\nˆw2(t,ξ) =/integraldisplayt\n0Φ2(t−τ,ξ)ˆf(ξ)dξ.\nΦ1(t,ξ)andΦ2(t,ξ)arethefundamentalsolutionsof(3.4)1satisfying\nΦ1(0,ξ) = 1, ∂tΦ1(0,ξ) = 0,\nΦ2(0,ξ) = 0, ∂tΦ2(0,ξ) = 1.\nIfweset\nKi(t,x) =F−1\nξ→x[Φi(t,ξ)]i= 1,2.\nThen thesolutionof(3.3)can berepresented by\nw(t,x) = (K1(t,·)∗w0(·))(x)+(K2(t,·)∗w1(·))(x)\n+/integraldisplayt\n0(K2(t−τ,·)∗f(·))(x)dτ. (3.6)\nThe asymptoticbehaviorof Φi(t,ξ)(i= 1,2)with respect to timehas already been investi-\ngated carefully in Wirth [27]. Here wewill givethemostimpo rtantTheorem therefor ouruse.\nBefore that,weneed to introducesomenotations.EULEREQUATIONSWITH TIME-DEPENDENTDAMPING 19\nIn thespace (t,ξ)∈(R+×Rn),denoteZ1,Z2andZ3as follows.\nZ1=/braceleftBig\n(t,ξ) :|ξ| ≤1\n4µ\n(1+t)λ/bracerightBig\n,\nZ2=/braceleftBig\n(t,ξ) :|ξ| ≥1\n4µ\n(1+t)λ/bracerightBig\n∩/braceleftBig\n(t,ξ) :|ξ| ≤1/bracerightBig\n,\nZ3=/braceleftBig\n(t,ξ) :|ξ| ≥1\n4µ\n(1+t)λ/bracerightBig\n∩/braceleftBig\n(t,ξ) :|ξ| ≥1/bracerightBig\n.\nLemma 3.1 (Theorem17 of[27]) There exists a constant C0, depending on λ,µsuch that Φi\nhasthefollowingestimatesindifferent zones\nWhen(t,ξ)∈Z1,\n|Φi(t,ξ)| ≤C0e−C0|ξ|2(1+t)1−λ. (3.7)\nWhen(t,ξ)∈Z2,\n|Φi(t,ξ)| ≤C0e−C0(1+t)1−λeC0(1−|ξ|2)(1+tξ)1−λ, (3.8)\nwheretξistheupperboundarylineofthezone Z1, whichmeans\n|ξ|=1\n4µ\n(1+tξ)λ. (3.9)\nWhen(t,ξ)∈Z3,\n|Φi(t,ξ)| ≤C0e−C0(1+t)1−λ. (3.10)\nProof.Thisisadirect resultofTheorem17in Wirth[27]. Weomitthe detail. /square\nNextwegivetwointegralpropertiesof Φiinthephasespace ξ.\nLemma 3.2\n/ba∇dblξαΦi(t,ξ)/ba∇dblL1(Zj)≤C(1+t)−(1−λ)(n/2+|α|/2), (3.11)\n/ba∇dblξαΦi(t,ξ)/ba∇dblL2(Zj)≤C(1+t)−(1−λ)(n/4+|α|/2), (3.12)\nwherei,j= 1,2.\nProof.In thezoneZ1, using(3.7), wehave\n/ba∇dblξαΦi(t,ξ)/ba∇dblL1(Z1)\n≤/integraldisplay\nξ∈Z1|ξ|α|Φi(t,ξ)|dξ\n≤C/integraldisplay∞\n0|ξ|n−1+αe−C0|ξ|2(1+t)1−λd|ξ|\n≤C(1+t)−(1−λ)(n/2+|α|/2)/integraldisplay∞\n0sn−1+|α|e−C0s2ds20 XINGHONG PAN\n≤C(1+t)−(1−λ)(n/2+|α|/2).\nIntegrationin Z2willusetherepresentationof tξ. From(3.7),wehave 1+tξ=/parenleftbig4\nµ|ξ|/parenrightbig−1\nλ. Then\nusing(3.8),we have\n/ba∇dblξαΦi(t,ξ)/ba∇dblL1(Z2)\n≤C0e−C0(1+t)1−λ/integraldisplay\nξ∈Z2|ξ|αeC0(1−|ξ|2)/parenleftbig\n4\nµ|ξ|/parenrightbig−1−λ\nλ\ndξ\n≤Ce−C0(1+t)1−λ/integraldisplay1\nµ\n4(1+t)−λ|ξ|n−1+|α|eC0/parenleftbig\n4\nµ|ξ|/parenrightbig−1−λ\nλ\ne−C|ξ|2−1−λ\nλd|ξ|\nNotingthat\n|ξ|n−1+|α|e−C|ξ|2−1−λ\nλ\n=/parenleftBig\n|ξ|2−1−λ\nλ/parenrightBign−1+|α|\n2e−C|ξ|2−1−λ\nλ|ξ|1−λ\n2λ(n−1+|α|)\n≤C|ξ|1−λ\n2λ(n−1+|α|).\nThen weget\n/ba∇dblξαΦi(t,ξ)/ba∇dblL1(Z2)\n≤Ce−C0(1+t)1−λ/integraldisplay1\nµ\n4(1+t)−λ|ξ|1−λ\n2λ(n−1+|α|)eC0/parenleftbig\n4\nµ|ξ|/parenrightbig−1−λ\nλ\nd|ξ|\n=Ce−C0(1+t)1−λ/integraldisplay(1+t)1−λ\n/parenleftbig\n4\nµ/parenrightbig−1−λ\nλs−1\n2(n−1+|α|)−1\n1−λeC0sds/parenleftBig\ns=/parenleftbig4\nµ|ξ|/parenrightbig−1−λ\nλ/parenrightBig\n≤Ce−C0(1+t)1−λ/integraldisplay(1+t)1−λ\n/parenleftbig\n4\nµ/parenrightbig−1−λ\nλs−1\n2(n+|α|)eC0sds\n/bracehtipupleft /bracehtipdownright/bracehtipdownleft /bracehtipupright\nI. (3.13)\nIcan bedealtwith integrationbyparts\nI=1\nC0s−1\n2(n+|α|)eC0s/vextendsingle/vextendsingle/vextendsingle/vextendsingles=(1+t)1−λ\ns=/parenleftbig\n4\nµ/parenrightbig−1−λ\nλ\n−n+|α|\n2C0/integraldisplay(1+t)1−λ\n/parenleftbig\n4\nµ/parenrightbig−1−λ\nλs−1\n2(n+|α|)−1eC0sds\n≤C(1+t)−1−λ\n2(n+|α|)eC0(1+t)1−λ.\nInsertingthisinto(3.13),weobtainthat\n/ba∇dblξαΦi(t,ξ)/ba∇dblL1(Z2)≤C(1+t)−(1−λ)(n/2+|α|/2).EULEREQUATIONSWITH TIME-DEPENDENTDAMPING 21\nTheproofof(3.12)willbeessentiallythesameas (3.11). We omitthedetail.\nCombining the estimates of Φi(t,ξ)in Lemma3.1 and Lemma3.2, we have the following\nestimatestothesolutionof(3.2).\nLemma 3.3 Assumethatforafunction g(x)∈L1∩Hs+l(Rn),wheres= [n/2]+1andl≥0.\nThen we have\n/ba∇dbl∂k\nx(Ki(t,·)∗g(·))/ba∇dbl∞≤C(1+t)−(1−λ)n+k\n2/ba∇dblg/ba∇dbl1+Ce−C0(1+t)1−λ/ba∇dbl∂s+k\nxg/ba∇dbl,(3.14)\nwherek∈Nand0≤k≤l.\n/ba∇dbl∂k\nx(Ki(t,·)∗g(·))/ba∇dbl ≤C(1+t)−(1−λ)(n\n4+k\n2)/ba∇dblg/ba∇dbl1+Ce−C0(1+t)1−λ/ba∇dbl∂k\nxg/ba∇dbl,(3.15)\nwherek∈Nand0≤k≤s+l.\nProof.UsingLemma3.1,Lemma3.2and H¨ olderinequality,wehave\n/ba∇dbl∂k\nx(Ki(t,·)∗g(·))/ba∇dbl∞\n≤ /ba∇dbl|ξ|kˆKi(t,ξ)ˆg(ξ)/ba∇dbl1\n=/parenleftBig/integraldisplay\nξ∈Z1∪Z2+/integraldisplay\nξ∈Z3/parenrightBig\n|ξ|k|Φi(t,ξ)||ˆg|dξ\n≤ /ba∇dblˆg/ba∇dbl∞/integraldisplay\nξ∈Z1∪Z2|ξ|k|Φi(t,ξ)|dξ+/integraldisplay\nξ∈Z3/ba∇dblΦi(t,ξ)/ba∇dbl∞|ξ|k|ˆg|dξ\n≤ /ba∇dblˆg/ba∇dbl∞/integraldisplay\nξ∈Z1∪Z2|ξ|k|Φi(t,ξ)|dξ\n+Ce−C0(1+t)1−λ/parenleftBig/integraldisplay\n|ξ|≥1|ξ|−2([n/2]+1)dξ/parenrightBig1\n2/parenleftBig/integraldisplay\n|ξ|≥1|ξ|2([n/2]+1+k)|ˆg|2dξ/parenrightBig1\n2\n≤C(1+t)−(1−λ)n+k\n2/ba∇dblg/ba∇dbl1+Ce−C0(1+t)1−λ/ba∇dbl∂k+s\nxg/ba∇dbl.\nThisproves(3.14).\nUsingPlancherel equality,Lemma3.1and Lemma3.2,wehave\n/ba∇dbl∂k\nx(Ki(t,·)∗g(·))/ba∇dbl2\n=/ba∇dbl|ξ|kˆKi(t,ξ)ˆg(ξ)/ba∇dbl2\n=/parenleftBig/integraldisplay\nξ∈Z1∪Z2+/integraldisplay\nξ∈Z3/parenrightBig\n|ξ|2k|Φi(t,ξ)|2|ˆg|2dξ\n≤ /ba∇dblˆg/ba∇dbl2\n∞/integraldisplay\nξ∈Z1∪Z2|ξ|2k|Φi(t,ξ)|2dξ+/integraldisplay\nξ∈Z3/ba∇dblΦi(t,ξ)/ba∇dbl2\n∞|ξ|2k|ˆg|2dξ\n≤ /ba∇dblg/ba∇dbl2\n1/integraldisplay\nξ∈Z1∪Z2|ξ|2k|Φi(t,ξ)|2dξ+Ce−2C0(1+t)1−λ/integraldisplay\nξ∈Z3|ξ|2k|ˆg|2dξ\n≤C(1+t)−(1−λ)(n\n2+k)/ba∇dblg/ba∇dbl2\n1+Ce−2C0(1+t)1−λ/ba∇dbl∂k\nxg/ba∇dbl2.\nThisproves(3.15). /square22 XINGHONG PAN\n3.2 Asymptoticbehavior of v\nFrom (2.3)and(3.6), wehave\nv(t,x) = (K1(t,·)∗v0(·))(x)+(K2(t,·)∗v1(·))(x)\n+/integraldisplayt\n0(K2(t−τ,·)∗Q(v,u)(τ,·))(x)dτ, (3.16)\nwhere\nv1(x) =∂tv|t=0=−∇·u0−u0·∇v0−γ−1\n2v0∇·u0. (3.17)\nBefore we do estimateson v, we first estimatethe nonlinearterm Q(v,u)underthehelp of\nTheorem1.1. Rememberthat\nQ(v,u) =µ\n(1+t)λ(−u·∇v−γ−1\n2v∇·u)\n−∂t(u·∇v−γ−1\n2v∇·u)+∇·(u·∇u+γ−1\n2v∇v).(3.18)\nProposition3.1 Assumef,g∈Hl(Rn). Then for |α| ≤l, wehave\n/ba∇dbl∂α\nx(fg)/ba∇dbl ≤Cl/parenleftbig\n/ba∇dblf/ba∇dbl∞/ba∇dbl∂l\nxg/ba∇dbl+/ba∇dblg/ba∇dbl∞/ba∇dbl∂l\nxf/ba∇dbl/parenrightbig\n.\nProof ofthepropositioncan befoundin manyliteratures. Se e [13]for example.\nLemma 3.4 UndertheassumptionofTheorem1.1, wehavethefollowinges timatesforQ.\n/ba∇dblQ(v,u)(t,·)/ba∇dbl1≤Cε2(1+t)−B−1+λ\n2. (3.19)\nAnd\n/ba∇dbl∂k\nxQ(v,u)(t,·)/ba∇dbl ≤Cε2(1+t)−B−1+λ\n2, (3.20)\nwherek= 0,1,2,...,s+m−2.\nProof.Thisis a direct computationofthe combinationofLeibnizfo rmula, H¨ olderinequality,\nProposition3.1and (1.4)in Theorem1.1. /square\nLemma 3.5 Suppose that a >1anda≥b >0. Then there exists a constant Csuch that for\nallt≥0,/integraldisplayt\n0(1+t−τ)−a(1+τ)−bdτ≤C(1+t)−b. (3.21)\nProof.Thisisjustadirect computation\n/integraldisplayt\n0(1+t−τ)−a(1+τ)−bdτ\n=/parenleftBig/integraldisplayt\n2\n0+/integraldisplayt\nt\n2/parenrightBig\n(1+t−τ)−a(1+τ)−bdτEULEREQUATIONSWITH TIME-DEPENDENTDAMPING 23\n≤C(1+t)−a/integraldisplayt\n2\n0(1+τ)−bdτ+C(1+t)−b/integraldisplayt\nt\n2(1+t−τ)−adτ\n≤C(1+t)−a+1−b+C(1+t)−b\n≤C(1+t)−b.\n/square\nL∞estimate\nFrom (3.16), using(3.14), for k= 0,1,...,m−2\n/ba∇dbl∂k\nxv/ba∇dbl∞\n≤C(1+t)−(1−λ)n+k\n2/ba∇dbl(v0,v1)/ba∇dbl1+Ce−C0(1+t)1−λ/ba∇dbl∂s+k\nx(v0,v1)/ba∇dbl\n+C/integraldisplayt\n0(1+t−τ)−(1−λ)n+k\n2/ba∇dblQ(v,u)/ba∇dbl1dτ\n+C/integraldisplayt\n0e−C0(1+t−τ)1−λ/ba∇dbl∂s+k\nxQ(v,u)/ba∇dbldτ\n≤C(1+t)−(1−λ)n+k\n2/ba∇dbl(v0,u0)/ba∇dblHs+m−1\n+Cε2/integraldisplayt\n0(1+t−τ)−(1−λ)n+k\n2(1+τ)−B−1+λ\n2dτ.\nNotingB=(1+λ)n\n2−δ, when(1−λ)n+k\n2=B+1+λ\n2, wehave\nk=kc=1+λ\n1−λ(n+1)−n−2δ\n1−λ.\nCase1:0≤k≤kc\nUsingLemma3.5andtheinitialdataassumption,wehave\n/ba∇dbl∂k\nxv/ba∇dbl∞\n≤Cε(1+t)−(1−λ)n+k\n2+Cε2(1+t)−(1−λ)n+k\n2\n≤Cε(1+t)−(1−λ)n+k\n2. (3.22)\nCase2:kc≤k≤m−2\n/ba∇dbl∂k\nxv/ba∇dbl∞\n≤Cε(1+t)−(1−λ)n+k\n2+Cε2(1+t)−B−1+λ\n2\n≤C(1+t)−(1+λ)n+1\n2+δ. (3.23)\nL2estimate\nFrom (3.16), using(3.15), for k= 0,1,...,s+m−2\n/ba∇dbl∂k\nxv/ba∇dbl24 XINGHONG PAN\n≤C(1+t)−(1−λ)(n\n4+k\n2)/ba∇dbl(v0,v1)/ba∇dbl1+Ce−C0(1+t)1−λ/ba∇dbl∂k\nx(v0,v1)/ba∇dbl\n+C/integraldisplayt\n0(1+t−τ)−(1−λ)(n\n4+k\n2)/ba∇dblQ(v,u)/ba∇dbl1dτ\n+C/integraldisplayt\n0e−C0(1+t−τ)1−λ/ba∇dbl∂k\nxQ(v,u)/ba∇dbldτ\n≤C(1+t)−(1−λ)(n\n4+k\n2)/ba∇dbl(v0,u0)/ba∇dblHs+m−1\n+Cε2/integraldisplayt\n0(1+t−τ)−(1−λ)(n\n4+k\n2)(1+τ)−B−1+λ\n2dτ.\nWhen(1−λ)(n\n4+k\n2) =B+1+λ\n2, wehave\nk=kc+n\n2.\nCase1:0≤k≤kc+n\n2\nUsingLemma3.5,wehave\n/ba∇dbl∂k\nxv/ba∇dbl\n≤Cε(1+t)−(1−λ)(n\n4+k\n2)+Cε2(1+t)−(1−λ)(n\n4+k\n2)\n≤Cε(1+t)−(1−λ)(n\n4+k\n2). (3.24)\nCase2:kc+n\n2≤k≤m−2\n/ba∇dbl∂k\nxv/ba∇dbl\n≤Cε(1+t)−(1−λ)(n\n4+k\n2)+Cε2(1+t)−B−1+λ\n2\n≤Cε(1+t)−(1+λ)n+1\n2+δ. (3.25)\nCombiningtheestimates(3.22)-(3.25), weproved(1.5)and (1.6).\n3.3 Asymptoticbehavior ofu\nDenoteu= (u1,...,un). From (2.2)2, differentiatingit ktimeinx, we have\n∂t∂k\nxui+µ\n(1+t)λ∂k\nxui=−∂k\nx/parenleftbig\n∂iv+uj∂jui+γ−1\n2v∂iv/parenrightbig\n. (3.26)\nL∞estimate\nFrom (3.26), wehave\nd\ndt/bracketleftBig\neµ\n1−λ(1+t)1−λ∂k\nxui/bracketrightBig\n=−eµ\n1−λ(1+t)1−λ∂k\nx/parenleftbig\n∂iv+uj∂jui+γ−1\n2v∂iv/parenrightbig\n. (3.27)\nIntegrating(3.27)from 0to t, weobtain\neµ\n1−λ(1+t)1−λ∂k\nxui=eµ\n1−λ∂k\nxui\n0(x)EULEREQUATIONSWITH TIME-DEPENDENTDAMPING 25\n−/integraldisplayt\n0eµ\n1−λ(1+τ)1−λ∂k\nx/parenleftbig\n∂iv+uj∂jui+γ−1\n2v∂iv/parenrightbig\ndτ.(3.28)\nFrom theestimatesof v(1.5)and (1.6)and (1.4),we have\n|−∂k\nx/parenleftbig\n∂iv+uj∂xjui+γ−1\n2v∂iv/parenrightbig\n|\n≤/braceleftBigg\n(1+t)−(1−λ)n+k+1\n20≤k≤kc−1,\n(1+t)−1+λ\n2(n+1)+δkc−1<k≤m−3.(3.29)\nThen from(3.28)and (3.29), wehave\nCase1:0≤k≤kc−1\n/ba∇dbl∂k\nxui/ba∇dbl∞≤Cεe−µ\n1−λ(1+t)1−λ\n+e−µ\n1−λ(1+t)1−λ/integraldisplayt\n0(1+τ)−(1−λ)n+k+1\n2eµ\n1−λ(1+τ)1−λdτ\n/bracehtipupleft /bracehtipdownright/bracehtipdownleft /bracehtipupright\nI.(3.30)\nUsingintegrationby parts,itis easy tosee\nI=C/integraldisplayt\n0(1+τ)−(1−λ)n+k+1\n2+λdeµ\n1−λ(1+τ)1−λ\n≤C(1+t)−(1−λ)n+k+1\n2+λeµ\n1−λ(1+t)1−λ.\nSo, from (3.30), weget\n/ba∇dbl∂k\nxui/ba∇dbl∞≤Cε(1+t)−(1−λ)n+k+1\n2+λ. (3.31)\nCase2:kc−1≤k≤m−3\n/ba∇dbl∂k\nxui/ba∇dbl∞≤Cεe−µ\n1−λ(1+t)1−λ\n+e−µ\n1−λ(1+t)1−λ/integraldisplayt\n0(1+τ)−(1+λ)n+1\n2+δeµ\n1−λ(1+τ)1−λdτ\n≤Cε(1+t)−(1+λ)n+1\n2+λ+δ. (3.32)\nL2estimate\nMultiplying(3.26)by ∂k\nxui, then integratingiton Rnand usingH¨ olderinequality,weget\nd\ndt/ba∇dbl∂k\nxui/ba∇dbl2+µ\n(1+t)λ/ba∇dbl∂k\nxui/ba∇dbl2\n=−/integraldisplay\nRn∂k\nxui∂k\nx/parenleftbig\n∂iv+uj∂jui+γ−1\n2v∂iv/parenrightbig\ndx\n≤µ\n2(1+t)λ/ba∇dbl∂k\nxui/ba∇dbl2+C(1+t)λ/ba∇dbl∂k\nx∂iv/ba∇dbl2\n+C(1+t)λ/parenleftBig\n/ba∇dbl∂k\nx(uj∂jui)/ba∇dbl2+/ba∇dbl∂k\nx(v∂iv)/ba∇dbl2/parenrightBig26 XINGHONG PAN\n≤µ\n2(1+t)λ/ba∇dbl∂k\nxui/ba∇dbl2+C(1+t)λ/ba∇dbl∂k\nx∂iv/ba∇dbl2\n+Cε4(1+t)−2B−1. (3.33)\nForm(1.5)and (1.6), wehave\n/ba∇dbl∂k\nx∂iv/ba∇dbl\n≤\n\nCε(1+t)−(1−λ)(n\n4+k+1\n2)0≤k≤kc+n\n2−1,\nCε(1+t)−(1+λ)n+1\n2+δkc+n\n2−1<k≤s+m−3.\nCase1:0≤k≤kc+n\n2−1\nd\ndt/ba∇dbl∂k\nxui/ba∇dbl2+µ\n2(1+t)λ/ba∇dbl∂k\nxui/ba∇dbl2\n≤Cε2(1+t)−(1−λ)(n\n2+k+1)+λ.\nThen wehave\nd\ndt/parenleftBig\n/ba∇dbl∂k\nxui/ba∇dbl2eµ\n2(1−λ)(1+t)1−λ/parenrightBig\n≤Cε2/integraldisplayt\n0(1+τ)−(1−λ)(n\n2+k+1)+λeµ\n2(1−λ)(1+τ)1−λdτ (3.34)\nIntegrating(3.34)from 0tot, weobtain\n/ba∇dbl∂k\nxui/ba∇dbl ≤Cε(1+t)−(1−λ)(n\n4+k+1\n2)+λ. (3.35)\nCase2:whenkc+n\n2−1<k≤s+m−3\nd\ndt/ba∇dbl∂k\nxui/ba∇dbl2+µ\n2(1+t)λ/ba∇dbl∂k\nxui/ba∇dbl2\n≤Cε2(1+t)−(1+λ)(n+1)+λ+2δ.\nThesameas (3.34), wehave\n/ba∇dbl∂k\nxui/ba∇dbl ≤Cε(1+t)−(1+λ)n+1\n2+λ+δ. (3.36)\nCombining(3.31), (3.32),(3.35)and (3.36), weproved(1.7 )and(1.8).\n4 ProofofTheorem1.3\nInthisSection,wederivethatthesmoothsolutionoftheEul erequationshaveapolynomially\ndecayed lowerboundintimewhilein threedimensions,thevo rticitywilldecay exponentially.EULEREQUATIONSWITH TIME-DEPENDENTDAMPING 27\nLower-bound decay rateof (ρ,u).\nDefine\nF(t) =/integraldisplay\nRnx·(ρu)dx, M (t) =/integraldisplay\nRn(ρ−1)dx,\nB(t) ={x||x| ≤R+t}.\nDuetothefinitepropagationofthesolutionandthecompacts upportoftheinitialdata,wehave\nsupp(ρ(t)−1)⊆B(t),suppu⊆B(t).\nFrom (1.1)1, wehave\nd\ndtM(t) =/integraldisplay\nRnρtdx=−/integraldisplay\nRn∇·(ρu) = 0,\nwhichmeans M(t) =M(0). So usingCauchy-Schwartz inequality,wehave\nq0=M(0)≤/vextendsingle/vextendsingle/vextendsingle/integraldisplay\nB(t)(ρ−1)dx/vextendsingle/vextendsingle/vextendsingle≤C/ba∇dbl(ρ−1)/ba∇dbl(R+t)n\n2. (4.1)\nUsing(1.1)2and integrationby parts,weget\nF′(t)+µ\n(1+t)λF(t) =/integraldisplay\nRn/parenleftbig\nρ|u|2+n(p(ρ)−p(1))/parenrightbig\ndx. (4.2)\nWhilenotingtheconvexityof p(ρ) =1\nγργ,γ >1, so wehave\n/integraldisplay\nRn(p(ρ)−p(1))dx≥/integraldisplay\nRnp′(1)(ρ−1)dx=M(t) =M(0). (4.3)\nCombining(4.2)and (4.3), wehave\nF′(t)+µ\n(1+t)λF(t)≥nq0. (4.4)\nMultiplying(4.4)by eµ\n1−λ(1+t)1−λand integratingit on [0,t], weobtain\nF(t)≥eµ\n1−λF(0)e−µ\n1−λ(1+t)1−λ\n+nq0e−µ\n1−λ(1+t)1−λ/integraldisplayt\n0eµ\n1−λ(1+τ)1−λdτ\n≥eµ\n1−λF(0)e−µ\n1−λ(1+t)1−λ\n+nq0\nµe−µ\n1−λ(1+t)1−λ/integraldisplayt\n0µ\n(1+t)λeµ\n1−λ(1+τ)1−λdτ\n=eµ\n1−λF(0)e−µ\n1−λ(1+t)1−λ28 XINGHONG PAN\n+nq0\nµe−µ\n1−λ(1+t)1−λ/parenleftBig\neµ\n1−λ(1+t)1−λ−eµ\n1−λ/parenrightBig\n≥Cq0.\nwhent>t0, wheret0is asuitablylarge constantdependingon λ,µ.\nUsing Cauchy-Schwartz inequality,finite propagationspee d, and|ρ|is uniformly bounded,\nwehave\nCq0≤F(t)≤C/parenleftBig/integraldisplay\nB(t)ρ2|x|2dx/parenrightBig1\n2/parenleftBig/integraldisplay\n|u|2dx/parenrightBig1\n2≤C(R+t)n+2\n2/ba∇dblu/ba∇dbl.(4.5)\n(4.1)and (4.5)imply(1.9).\nExponential decay ofthe vorticity.\nIn threedimensions,thevorticity ωoftheequationssatisfies\n∂tω+µ\n(1+t)λω+u·∇ω−ω∇u= 0. (4.6)\nMultiplying(4.6)by ωandintegratingit on Rn, weget\n1\n2d\ndt/integraldisplay\n|w|2dx+µ\n(1+t)λ/integraldisplay\n|ω|2dx≤C/integraldisplay\n(|ω|2|∇u|+|ω·∇u·ω|dx)\n≤C/ba∇dbl∇u/ba∇dbl∞/integraldisplay\n|ω|2dx.\nFrom Theorem 1.1, we have /ba∇dbl∇u/ba∇dbl∞≤Cε(1+t)B+1+λ\n2. NotingB+1+λ\n2≥λandεis small, so\nweobtain1\n2d\ndt/integraldisplay\n|w|2dx+µ\n2(1+t)λ/integraldisplay\n|ω|2dx≤0.\nThisimpliestheexponentialdecay of ω(1.10).\n5 Appendix\nProofofLemma2.1\nRemember\nψ(t,x) =a|x|2\n(1+t)1+λ, a=(1+λ)µ\n8/parenleftBig\n1−δ\n(1+λ)n/parenrightBig\n.\nThen\nψt=−(1+λ)a|x|2\n(1+t)2+λ=−1+λ\n1+tψ,\n∇ψ=2ax\n(1+t)1+λ,∆ψ=2an\n(1+t)1+λ.(5.1)\nAnd\n|∇ψ|2\n−ψt=4a\n1+λ1\n(1+t)λ=1\n2/parenleftBig\n1−δ\n(1+λ)n/parenrightBigµ\n(1+t)λ, (5.2)EULEREQUATIONSWITH TIME-DEPENDENTDAMPING 29\n∆ψ=/bracketleftBig(1+λ)n\n4−δ\n4/bracketrightBigµ\n(1+t)1+λ. (5.3)\nMultiplying(2.3)by e2ψ∂tvande2ψv, wehave\n∂t/bracketleftBige2ψ\n2/parenleftbig\n(∂tv)2+|∇v|2/parenrightbig/bracketrightBig\n−∇·(e2ψ∂tv∇v)\n+e2ψ/parenleftBigµ\n(1+t)λ−|∇ψ|2\n−ψt−ψt/parenrightBig\n(∂tv)2+e2ψ\n−ψt|ψt∇v−∂tv∇ψ|2\n/bracehtipupleft /bracehtipdownright/bracehtipdownleft /bracehtipupright\nI1\n=e2ψ∂tvQ(v,u), (5.4)\nand\n∂t/bracketleftBig\ne2ψ/parenleftBig\nv∂tv+µ\n2(1+t)λv2/parenrightBig/bracketrightBig\n−∇·(e2ψv∇v)\n+e2ψ/braceleftBig\n|∇v|2+/parenleftBig\n−ψt+λ\n2(1+t)/parenrightBigµ\n(1+t)λv2+2v∇ψ·∇v/bracehtipupleft/bracehtipdownright/bracehtipdownleft/bracehtipupright\nI2−2ψtv∂v−(∂tv)2/bracerightBig\n=e2ψvQ(v,u). (5.5)\nWeestimate I1,I2as follows:\nI1≥e2ψ\n−ψt/parenleftBig\n(1−δ1)ψ2\nt|∇v|2−(1/δ1−1)v2\nt|∇ψ|2/parenrightBig\n=e2ψ/braceleftBig\n(1−δ1)(−ψt)|∇v|2−1\n2/parenleftBig\n1−δ\n(1+λ)n/parenrightBig\n(1/δ1−1)µ\n(1+t)λv2\nt/bracerightBig\n.(5.6)\nChoosingδ1closeto1 such that\n1\n2/parenleftBig\n1−δ\n(1+λ)n/parenrightBig\n(1/δ1−1)≤δ\n2(1+λ)n, (5.7)\nand\nI2= 4e2ψv∇v·∇ψ−e2ψ∇v2·∇ψ\n= 4e2ψv∇v·∇ψ−∇·(e2ψv2∇ψ)+2e2ψv2|∇ψ|2+e2ψ(∆ψ)v2.(5.8)\nThen Inserting(5.6)and(5.8)into(5.4)and (5.5), wehave\n∂t/bracketleftBige2ψ\n2/parenleftbig\n(∂tv)2+|∇v|2/parenrightbig/bracketrightBig\n−∇·(e2ψ∂tv∇v)\n+e2ψ/braceleftBig/parenleftBigµ\n2(1+t)λ−ψt/parenrightBig\n(vt)2+(1−δ1)(−ψt)|∇v|2/bracerightBig\n≤e2ψ∂tvQ(v,u), (5.9)30 XINGHONG PAN\nand\n∂t/bracketleftBig\ne2ψ/parenleftBig\nv∂tv+µ\n2(1+t)λv2/parenrightBig/bracketrightBig\n−∇·{(e2ψ(v∇v+v2∇ψ)}\n+e2ψ/braceleftBig\n|∇v|2+4v∇ψ·∇v+/parenleftBigµ\n(1+t)λ(−ψt)+2|∇ψ|2/parenrightBig\nv2\n/bracehtipupleft /bracehtipdownright/bracehtipdownleft /bracehtipupright\nI3\n+/parenleftBig\nλ+(1+λ)n\n2−2δ/parenrightBigµ\n2(1+t)1+λv2−2ψtvvt−v2\nt/bracerightBig\n=e2ψvQ(v,u). (5.10)\nBy (5.2), wehave\nI3=|∇v|2+4v∇ψ·∇v+41−δ//parenleftbig\n2(1+λ)n/parenrightbig\n1−δ//parenleftbig\n(1+λ)n/parenrightbigv2|∇ψ|2\n≥(1−δ2)|∇v|2+4/parenleftBig1−δ//parenleftbig\n2(1+λ)n/parenrightbig\n1−δ//parenleftbig\n(1+λ)n/parenrightbig−1/δ2/parenrightBig\n|∇ψ|2v2\nChoosingδ2closeto1 can assurethat forsome δ3,δ4>0, wehave\nI3≥δ3/parenleftBig\n|∇v|2+|∇ψ|2v2/parenrightBig\n≥δ4/parenleftBig\n|∇v|2+µ\n(1+t)λ(−ψt)v2/parenrightBig\n(5.11)\nInserting(5.11)into(5.10), weget\n∂t/bracketleftBig\ne2ψ/parenleftBig\nv∂tv+µ\n2(1+t)λv2/parenrightBig/bracketrightBig\n−∇·{(e2ψ(v∇v+v2∇ψ)}\n+e2ψ/braceleftBig\nδ4/parenleftBig\n|∇v|2+µ\n(1+t)λ(−ψt)v2/parenrightBig\n+/parenleftBig\nλ+(1+λ)n\n2−2δ/parenrightBigµ\n2(1+t)1+λv2−2ψtvvt−v2\nt/bracerightBig\n≤e2ψvQ(v,u). (5.12)\nIntegrating(5.12)on Rn,we have\nd\ndt/integraldisplay\nRne2ψ/parenleftBig\nv∂tv+µ\n2(1+t)λv2/parenrightBig\ndx\n+/integraldisplay\nRne2ψ/braceleftBig\nδ4/parenleftBig\n|∇v|2+µ\n(1+t)λ(−ψt)v2/parenrightBig\n+/parenleftBig\nλ+(1+λ)n\n2−2δ/parenrightBigµ\n2(1+t)1+λv2\n−2ψtvvt−v2\nt/bracerightBig\ndx\n≤/integraldisplay\nRne2ψvQ(v,u)dx. (5.13)EULEREQUATIONSWITH TIME-DEPENDENTDAMPING 31\nToabsorbthenegativeterm −v2\nt,weintegrate(5.9)in Rnandmultiplyitby (K+t)λ,where\nKisa sufficientlylarge constant. Then weget\nd\ndt/bracketleftBig\n(K+t)λ/integraldisplay\nRne2ψ\n2/parenleftbig\nv2\nt+|∇v|2/parenrightbig\ndx/bracketrightBig\n−λ\n(K+t)1−λ/integraldisplay\nRne2ψ\n2/parenleftbig\nv2\nt+|∇v|2/parenrightbig\ndx\n+/integraldisplay\nRne2ψ/braceleftBig/parenleftBigµ\n2−ψt(K+t)λ/parenrightBig\nv2\nt+(1−δ1)(−ψt)(K+t)λ|∇v|2/bracerightBig\n≤(K+t)λ/integraldisplay\nRne2ψ∂tvQ(v,u)dx. (5.14)\nNowadding ν·(5.13)to(5.14),where νisasufficientsmallconstant,weget\nd\ndt/integraldisplay\nRne2ψ/braceleftBig(K+t)λ\n2/parenleftbig\nv2\nt+|∇v|2/parenrightbig\n+νvvt/bracehtipupleft/bracehtipdownright/bracehtipdownleft/bracehtipupright\nI4+νµ\n2(1+t)λv2/bracerightBig\ndx\n+/integraldisplay\nRne2ψ/braceleftBig/parenleftBigµ\n2−ν+(−ψt)(K+t)λ−λ\n2(K+t)1−λ/parenrightBig\nv2\nt\n+/parenleftBig\n(1−δ1)(−ψt)(K+t)λ+δ4ν−λ\n2(K+t)1−λ/parenrightBig\n|∇v|2\n+δ4νµ\n(1+t)λ(−ψt)v2+/parenleftBig\nλ+(1+λ)n\n2−δ\n2/parenrightBigνµ\n2(1+t)1+λv2\n−2νψtvvt/bracehtipupleft/bracehtipdownright/bracehtipdownleft/bracehtipupright\nI5/bracerightBig\ndx\n≤(K+t)λ/integraldisplay\nRne2ψ∂tvQ(v,u)dx+ν/integraldisplay\nRne2ψvQ(v,u)dx. (5.15)\nThe termsI4andI5can be absorbed by the other positiveterms by applying the sm allness\nofνand largeness of Kand thefollowingCauchy-Schwartz inequality\n|I4| ≤(K+t)λ\n4v2\nt+ν2\n(K+t)λv2\n≤(K+t)λ\n4v2\nt+ν2\n(1+t)λv2, (5.16)\nand\n|I5| ≤νδ4\n2(−ψt)µ\n(1+t)λv2+2ν\nµδ4(−ψt)(1+t)λv2\nt\n≤νδ4\n2(−ψt)µ\n(1+t)λv2+2ν\nµδ4(−ψt)(K+t)λv2\nt. (5.17)\nDenote\nE(t) =/integraldisplay\nRne2ψ/braceleftBig(K+t)λ\n2/parenleftbig\nv2\nt+|∇v|2/parenrightbig\n+νvvt/bracehtipupleft/bracehtipdownright/bracehtipdownleft/bracehtipupright\nI4+νµ\n2(1+t)λv2/bracerightBig\ndx.32 XINGHONG PAN\nWhenνis sufficiently small and Kis large, using (5.16), there exists a small constant cδsuch\nthat\nd\ndtE(t)+/integraldisplay\nRn/parenleftBig\nλ+(1+λ)n\n2−δ\n2/parenrightBigνµ\n2(1+t)1+λv2dx\n+cδ/integraldisplay\nRne2ψ/braceleftBig/parenleftBig\n1+(−ψt)(K+t)λ/parenrightBig\nv2\nt\n+/parenleftBig\n1+(−ψt)(K+t)λ/parenrightBig\n|∇v|2\n+(1+t)−λ(−ψt)v2/bracerightBig\ndxdτ\n≤(K+t)λ/integraldisplay\nRne2ψ∂tvQ(v,u)dx+/integraldisplay\nRne2ψvQ(v,u)dx. (5.18)\nUsing(5.16),wesee that\n1\n4(K+t)λ/bracketleftBig\nJ(t;vt)+J(t;|∇v|)/bracketrightBig\n+(νµ\n2−ν2)(1+t)−λJ(t;v)\n≤E(t)≤\n3\n4(K+t)λ/bracketleftBig\nJ(t;vt)+J(t;|∇v|)/bracketrightBig\n+(νµ\n2+ν2)(1+t)−λJ(t;v).(5.19)\nMultiplying(5.18)by (K+t)B+λand using(5.19), wehave\nd\ndt/bracketleftBig\n(K+t)B+λE(t)/bracketrightBig\n+/parenleftBig\ncδ−3\n4(B+λ)(K+t)λ−1/parenrightBig\n(K+t)B+λ/bracketleftBig\nJ(t;vt)+J(t;|∇v|)/bracketrightBig\n+(K+t)B+λ\n2(1+t)1+λ/parenleftbigg/parenleftBig\nλ+(1+λ)n\n2−δ\n2/parenrightBig\nνµ−2(B+λ)(νµ\n2+ν2)/parenrightbigg\nJ(t;v)\n+cδ(K+t)B+2λ/bracketleftBig\nJψ(vt)+Jψ(t;|∇v|)/bracketrightBig\n+cδ(K+t)BJψ(t;v)\n≤/integraldisplay\nRne2ψ/bracketleftbigg/parenleftBig\n(K+t)B+2λvt+ν(K+t)B+λv/parenrightBig\nQ(v,u)/bracketrightbigg\ndx. (5.20)\nIntegrating(5.20)over [0,t],wecanshowthatbychoosingsmall νandlargeK,thereexists\naconstantC0dependingon λ,µ,δ,R suchthat\n(K+t)B+2λ/bracketleftBig\nJ(t;vt)+J(t;|∇v|)/bracketrightBig\n+(K+t)BJ(t;v)\n+/integraldisplayt\n0(K+τ)B+λ/bracketleftBig\nJ(τ;vτ)+J(τ;|∇v|)/bracketrightBig\ndτ\n+/integraldisplayt\n0(K+τ)B+2λ/bracketleftBig\nJψ(τ;vτ)+Jψ(τ;|∇v|)/bracketrightBig\ndτ\n+/integraldisplayt\n0/bracketleftBig\n(K+τ)BJψ(τ;v)+(K+τ)B−1J(τ;v)/bracketrightBig\ndτEULEREQUATIONSWITH TIME-DEPENDENTDAMPING 33\n≤C/ba∇dbl(v(0),vt(0),∂xv(0))/ba∇dbl\n+C/integraldisplayt\n0/integraldisplay\nRne2ψ/braceleftBig/parenleftBig\n(K+τ)B+2λvτ+(K+τ)B+λv/parenrightBig\nQ(v,u)/bracerightBig\ndxdτ. (5.21)\nConsideringthat/integraltextt\n0(K+τ)B+λ/bracketleftbig\nJ(τ;vτ)+J(τ;|∇v|)/bracketrightbig\ndτhasbeen estimated,wemultiply\n(5.14)by (K+t)B+1and integrateitover [0,t]to obtain\n(K+t)B+1+λ/parenleftBig\nJ(t;vt)+J(t;|∇v|)/parenrightBig\n−/integraldisplayt\n0(λ+B+1)(K+τ)B+λ/bracketleftbig\nJ(τ;vτ)+J(τ;|∇v|)/bracketrightbig\ndτ\n+/integraldisplayt\n0(K+τ)B+1+λ[Jψ(τ;vτ)+Jψ(τ;|∇v|)/bracketrightbig\ndτ\n+/integraldisplayt\n0(K+τ)B+1J(τ;vτ)dτ\n≤C/ba∇dbl(vt(0),∂xv(0))/ba∇dbl+C/integraldisplayt\n0/integraldisplay\nRn(K+τ)B+1+λe2ψ∂τvQ(v,u)dxdτ. (5.22)\nForsmallν, addingν·(5.22)to(5.21), wehave\n(K+t)B+1+λ/bracketleftBig\nJ(t;vτ)+J(t;|∇v|)/bracketrightBig\n+(K+t)BJ(t;v)\n+/integraldisplayt\n0(K+τ)B+1+λ[Jψ(τ;vτ)+Jψ(τ;|∇v|)/bracketrightbig\ndτ\n+/integraldisplayt\n0/bracketleftbig\n(K+τ)B+1J(τ,vτ)+(K+τ)B+λJ(τ,|∇v|)/bracketrightbig\ndτ\n+/integraldisplayt\n0/bracketleftbig\n(K+τ)BJψ(τ,v)+(K+τ)B−1J(τ,v)/bracketrightbig\ndτ\n≤C/ba∇dbl(v(0),vt(0),∂xv(0))/ba∇dbl\n+C/integraldisplayt\n0/integraldisplay\nRne2ψ/braceleftBig/parenleftBig\n(K+τ)B+1+λvτ+(K+τ)B+λv/parenrightBig\nQ(v,u)/bracerightBig\ndxdτ.\nThisprovesLemma2.1. /square\nAcknowledgement. I want to express my gratitude to my advisor Professor Huiche ng Yin\ninNanjingNormalUniversityforhisguidanceaboutthiswor k. ThisworkisproceededwhenI\nam visiting Department of Mathematics in Universityof Cali fornia, Riverside. So, I also want\ntoexpress mythankstomy co-advisor,ProfessorQi S. Zhang, in UCRforhis encouragement.\nReferences\n[1] Alinhac,S. Blowupfornonlinearhyperbolicequations. Birkh¨ auserBoston,1995.34 XINGHONG PAN\n[2] Chemin, Jean-Yves Remarques sur l’apparition de singul arit´ es dans les ´ ecoulements\neul´ erienscompressibles.Comm.Math.Phys.133(1990), no .2, 323-329.\n[3] Courant, R.; Friedrichs,K. O. SupersonicFlowand Shock Waves.New York,1948.\n[4] Chen,Gui-Qiang;Liu,HailiangFormationof δ-shocksandvacuumstatesinthevanishing\npressurelimitofsolutionstotheEulerequationsforisent ropicfluids.SIAMJ.Math.Anal.\n34(2003), no.4, 925-938\n[5] Christodoulou,D. Theformationof shocks in3-dimensio nalfluids. EMSMonographsin\nMathematics.European MathematicalSociety (EMS), Z¨ uric h,2007.\n[6] Dafermos,ConstantineM.Hyperbolicconservationlaws incontinuumphysics.Springer-\nVerlag, Berlin, 2000.\n[7] Hsiao, Ling; Liu, Tai-Ping Convergence to nonlinear dif fusion waves for solutions of a\nsystem of hyperbolic conservation laws with damping. Comm. Math. Phys. 143 (1992),\nno.3, 599-605.\n[8] Hou, Fei; Yin, Huicheng On the global existence and blowu p of smooth solutions\nto the multi-dimensional compressible Euler equations wit h time-depending damping\narXiv:1606.08935.\n[9] Hou, Fei; Witt, Ingo; Yin, Huicheng On the global existen ce and blowup of\nsmooth solutions of 3-D compressible Euler equations with t ime-depending damping.\narXiv:1510.04613\n[10] Huang, Feimin; Marcati, Pierangelo; Pan, Ronghua Conv ergence to the Barenblatt solu-\ntionforthecompressibleEulerequationswithdampingandv acuum.Arch.Ration.Mech.\nAnal.176 (2005),no. 1,1-24.\n[11] Jiu, Quansen; Zheng, Xiaoxin Global well-posedness of the compressible Euler with\ndampinginBesovspaces. Math.MethodsAppl.Sci. 35 (2012), no. 13,1570-1586.\n[12] Kato, Tosio The Cauchy problem for quasi-linear symmet ric hyperbolic systems. Arch.\nRationalMech. Anal. 58(1975), no.3, 181-205.\n[13] Klainerman, Sergiu; Majda, Andrew Singular limits of q uasilinear hyperbolic systems\nwith large parameters and the incompressible limit of compr essible fluids. Comm. Pure\nAppl.Math.34 (1981),no.4, 481-524.\n[14] Kong, De-Xing; Wang, Yu-Zhu Long-time behaviour of smo oth solutions to the com-\npressibleEulerequationswithdampinginseveral spacevar iables.IMAJ.Appl. Math.77\n(2012),no. 4,473-494.\n[15] Majda, A. Compressible fluid flow and systems of conserva tion laws in several space\nvariables.AppliedMathematicalSciences, 53.Springer-V erlag, New York, 1984.EULEREQUATIONSWITH TIME-DEPENDENTDAMPING 35\n[16] Makino, Tetu; Ukai, Seiji; Kawashima, Shuichi Sur la so lution ` a support compact de\nequationsd’Eulercompressible.Japan J.Appl.Math.3 (198 6), no.2, 249-257.\n[17] Nishida, Takaaki Nonlinear hyperbolic equations and r elated topics in fluid dynamics.\nPublicationsMath´ ematiquesd’Orsay,No.78-02.D´ eparte mentdeMath´ ematique,Univer-\nsit´ edeParis-Sud, Orsay,1978.\n[18] Nishihara, Kenji Asymptotic behavior of solutions to t he semilinear wave equation with\ntime-dependentdamping.TokyoJ.Math.34 (2011),no. 2,327 -343.\n[19] Nishihara, Kenji; Wang, Weike; Yang, Tong Lp-convergence rate to nonlinear diffusion\nwavesforp-systemwithdamping.J.DifferentialEquations 161(2000),no. 1, 191-218.\n[20] Pan, Xinghong Global existence of solutionsto 1-d Eule r equations with time-dependent\ndamping.NonlinearAnal.132 (2016),327-336.\n[21] Pan,XinghongBlowupofsolutionsto1-dEulerequation swithtime-dependentdamping.\nJ.Math. Anal.Appl.442(2016), 435-445.\n[22] Rammaha, M. A. Formation of singularities in compressi ble fluids in two-space dimen-\nsions.Proc. Amer.Math.Soc. 107(1989),no. 3,705-714.\n[23] Sideris, T. Delayed singularity formation in 2D compre ssible flow. Amer. J. Math. 119\n(1997),no. 2,371-422.\n[24] Sideris, T. Formation of singularities in three-dimen sional compressible fluids. Comm.\nMath.Phys.101(1985), no.4, 475-485.\n[25] Sideris, T; Thomases, Becca; Wang, Dehua Long time beha vior of solutions to the 3D\ncompressible Euler equations with damping. Comm. Partial D ifferential Equations 28\n(2003),no. 3-4,795-816.\n[26] Tan,Zhong;Wang,YongGlobalsolutionandlarge-timeb ehaviorofthe3Dcompressible\nEulerequationswithdamping.J.Differential Equations25 4(2013), no.4, 1686-1704.\n[27] Wirth, Jens Wave equations with time-dependent dissip ation. II. Effective dissipation. J.\nDifferentialEquations232 (2007),no. 1,74-103.\n[28] Wang, Weike; Yang, Tong The pointwise estimates of solu tions for Euler equations with\ndampinginmulti-dimensions.J. DifferentialEquations17 3 (2001),no. 2,410-450.\n[29] Yin, Huicheng Formation and construction of a shock wav e for 3-D compressible Euler\nequationswiththespherical initialdata. NagoyaMath.J. 1 75(2004),125-164."    },    {        "title": "1609.07901v1.Relativistic_theory_of_spin_relaxation_mechanisms_in_the_Landau_Lifshitz_Gilbert_equation_of_spin_dynamics.pdf",        "content": "arXiv:1609.07901v1  [cond-mat.mtrl-sci]  26 Sep 2016Relativistic theory of spin relaxation mechanisms in the La ndau-Lifshitz-Gilbert equation\nof spin dynamics\nRitwik Mondal,∗Marco Berritta, and Peter M. Oppeneer\nDepartment of Physics and Astronomy, Uppsala University, P .O. Box 516, Uppsala, SE-75120, Sweden\n(Dated: October 14, 2018)\nStarting from the Dirac-Kohn-Sham equation we derive the re lativistic equation of motion of\nspin angular momentum in a magnetic solid under an external e lectromagnetic field. This equation\nof motion can be rewritten in the form of the well-known Landa u-Lifshitz-Gilbert equation for a\nharmonic external magnetic field, and leads to a more general magnetization dynamics equation for\na general time-dependent magnetic field. In both cases with a n electronic spin-relaxation term which\nstems from the spin-orbit interaction. We thus rigorously d erive, from fundamental principles, a\ngeneral expression for the anisotropic damping tensor whic h is shown to contain an isotropic Gilbert\ncontribution as well as an anisotropic Ising-like and a chir al, Dzyaloshinskii-Moriya-like contribution.\nThe expression for the spin relaxation tensor comprises fur thermore both electronic interband and\nintraband transitions. We also show that when the externall y applied electromagnetic field possesses\nspin angular momentum, this will lead to an optical spin torq ue exerted on the spin moment.\nPACS numbers: 75.78.-n, 76.20.+q, 71.15.Rf\nI. INTRODUCTION\nIn their seminal 1935-paper, L. D. Landau and E. M.\nLifshitz proposed the equation of motion governing the dy-\nnamics of a continuum magnetization [1]. Eighty years after\nits original formulation, the Landau-Lifshitz (LL) equati on\ncontinues to play a fundamental role in the understanding\nof magnetization dynamics [2] and forms the cornerstone of\ncontemporary micromagnetic simulations (see, e.g., Refs.\n[3, 4]).\nOriginally, the Landau-Lifshitz equation was derived on\nthe basis of phenomenological considerations [1]. It define s\nthe time-evolution of a volume magnetization M(r,t)as\n∂M\n∂t=−γM×Heff−λM×[M×Heff],(1)\nwhereγis the gyromagnetic ratio, Heffis the effective mag-\nnetic field, and λis an isotropic damping parameter. The\nfirst term describes the precession of the local magnetiza-\ntionM(r,t)around the effective field Heff. The second\nterm describes the magnetization relaxation such that the\nmagnetization vector relaxes to the direction of the effecti ve\nfield. The damping term in the LL equation was reformu-\nlated by Gilbert [5, 6] to give the Landau-Lifshitz-Gilbert\n(LLG) equation,\n∂M\n∂t=−γM×Heff+αM×∂M\n∂t, (2)\nwhereαis the Gilbert damping constant. Note that both\ndamping parameters αandλare here scalars, which cor-\nresponds to the assumption of an isotropic medium. Both\nLL and LLG equations preserve the length of the magneti-\nzation during the dynamics and are mathematically equiv-\nalent (see, e.g. [7]).\nA number of explanations have been proposed for the\nmicroscopic origin of the spin relaxation in magnetic met-\nals [8–18]. Already in their original work Landau and Lif-\nshitz attributed the damping constant to relativistic effects\n∗Ritwik.Mondal@physics.uu.se[1]. More specific microscopic theories of spin relaxation\nin ferromagnetic metals have been developed in the last\ndecennia. Kamberský proposed the breathing Fermi sur-\nface model [8] and the related torque-correlation model\n[14, 19]. Brataas et al. proposed a scattering theory for-\nmulation [15] of the Gilbert damping which is equiva-\nlent to a Kubo linear-response formulation. A different\nform of the relaxation term caused by spatial dispersion of\nthe exchange interaction—this in contrast to the isotropic\nmedium assumption made in the LL equation—was pro-\nposed by Bar’yakhtar and co-workers [10, 20, 21].\nMore recently the debate on what the appropriate the-\nory to describe damping would be has focused on first-\nprinciples electronic structure calculations and, in how\nfar these could provide quantitative values of the Gilbert\ndamping [22–30]. Recent ab initio calculations of the\nGilbert damping constant for transition-metal alloys pre-\ndicted values that correspond to the experimental values\nwithin a range of a factor of two to three [22–24, 26–28],\nwith significant deviations however for the pure elemen-\ntal ferromagnets. This indicates that there is still a need\nto improve the fundamental understanding of the origin\nof spin-moment relaxation. Also, very recent publications\nhave questioned the existing understanding of the Gilbert\ndamping [31, 32].\nHere we develop a theoretical description of spin relax-\nation on the basis of the relativistic Density Functional\nTheory (DFT). To this end, we start from the relativis-\ntic Dirac-Kohn-Sham (DKS) equation that adequately de-\nscribes the electronic states in a magnetic solid. From thes e\nwe derive the general equation of motion for spin angular\nmomentum, which adopts the form of the LLG equation.\nWithin this framework we obtain explicit expressions for\nthe tensorial form of the Gilbert damping term, which we\nfind to contain an isotropic Gilbert-like contribution and\nanisotropic Ising-like and chiral Dzyaloshinskii-Moryia -like\ncontributions. Our derivation follows similar steps as a pr e-\nvious derivation by Hickey and Moodera [17], however, as\ndiscussed below, it includes previously missing terms and\nthus leads to different expressions for the spin relaxation.2\nII. THE RELATIVISTIC DIRAC HAMILTONIAN\nAs mentioned before, relativistic effects such as the spin-\norbit interaction are at the heart of spin angular momen-\ntum dissipation in solids. To examine how these fundamen-\ntal physical interactions lead to magnetization damping we\nchoose therefore to start from the most general relativisti c\nHamiltonian, the DKS Hamiltonian. This Hamiltonian de-\nscribes the one-electron quantum state in an effective spin-\npolarized field due to other electrons and nuclei in the solid ,\nin addition to externally applied fields. For spin-polarize d\nelectrons in a magnetic material the DKS Hamiltonian is\ngiven as [33–35]\nHD=cα·(p−eA)+/parenleftbig\nβ−1/parenrightbig\nmc2+V1+eΦ1\n−µBβΣ·Bxc. (3)\nHereVis the unpolarized Kohn-Sham selfconsistent po-\ntential,Bxcis the spin-polarized part of the exchange-\ncorrelation potential in the material, A=A(r,t)is the vec-\ntor potential of an externally applied electromagnetic fiel d,\neΦ(r,t)is the scalar potential of this field, p=−i/planckover2pi1∇, and\nµBise/planckover2pi1\n2m, the Bohr magneton. 1is the4×4identity matrix\nandα,β, andΣare the well-known Dirac matrices in Dirac\nbi-spinor space, which contain the Pauli spin matrices σ\nand the2×2identity matrix. At this point, it is important\nto observe that there are two fundamentally different fields\npresent in the DKS Hamiltonian. There are the Maxwellfields, that is, (implicitly) the external magnetic inducti on\nB(r,t) =∇×A(r,t)as well as the external electric field,\nE(r,t) =−∂A(r,t)\n∂t−∇Φ. The strongest field in a magnetic\nmaterial is however the exchange field, which stems from\nthe Pauli exclusion principle. The exchange field Bxcis\nfundamentally different from the standard magnetic induc-\ntion, as it obviously acts only on the spin degree of freedom\n(see, e.g., [34]) and does not couple to the orbital angular\nmomentum. Also, it doesn’t fulfill the Maxwell equations\nas the auxiliary electromagnetic field (e.g., ∇·B= 0) and\nit cannot be included as a vector potential Axcin the linear\nmomentum, i.e. p−eAxc, but instead needs to be treated\nas a separate term in Eq. (3).\nNext, we want to investigate the relativistic spin evo-\nlution of spin-polarized electrons in a magnetic solid. To\nachieve this we need the positive energy, that is, the elec-\ntron solutions that are given by the large component of\nthe Dirac bi-spinor. To arrive at an elucidating formula-\ntion in terms of the spin operator we employ the Foldy-\nWouthuysen transformation approach [35, 36] on the DKS\nequation for the case where an exchange field Bxcis explic-\nitly present (for details, see Ref. [37]). Doing so, one ob-\ntains a Hamiltonian for the electron solutions only, which\nwe expand in orders of 1/c2to select the largest relativistic\ncontributions. This leads to a semi-relativistic, extende d\nPauli Hamiltonian (see Ref. [37]),\nHEP=(p−eA)2\n2m+V−µBσ·B−µBσ·Bxc\neff+eΦ−(p−eA)4\n8m3c2−1\n8m2c2/parenleftbig\np2V/parenrightbig\n−e/planckover2pi12\n8m2c2∇·E\n+i\n4m2c2σ·(pV)×(p−eA)−e/planckover2pi1\n8m2c2σ·{E×(p−eA)−(p−eA)×E}\n+iµB\n4m2c2[(p×Bxc)·(p−eA)]. (4)\nExcept from the last term in Eq. (4), all the appearing relati vistic corrections involving the exchange interaction can be\nadded together giving an effective exchange field [38],\nBxc\neff=Bxc−1\n8m2c2/braceleftBig/bracketleftbig\np2Bxc/bracketrightbig\n+2(pBxc)·(p−eA)+2(p·Bxc)(p−eA)+4[Bxc·(p−eA)](p−eA)/bracerightBig\n≡Bxc+Bxc\ncorr. (5)\nThe Hamiltonian HEPexactly includes all spin-dependent\nrelativistic terms (of the order of 1/c2) and all the terms\ninvolving Bxcand the external electromagnetic fields. We\nemphasize that for our purpose of unveiling the relativisti c\nmechanisms of spin dissipation it is obviously not sufficient\nto work with the conventional Pauli Hamiltonian, which\nonly consists of the five first terms in the nonrelativistic\nlimit. The correct form of all relativistic terms can solely\nbe obtained when one starts from the DKS equation with\nexchange field. We remark that in a previous study Hickey\nand Moodera [17] used a Pauli Hamiltonian different from\nthe above one, without exchange field and without crystal\npotential and thus without the intrinsic spin-orbit intera c-\ntion [the first term in the second line of Eq. (4)].\nThe meaning of the terms in Hamiltonian (4) can bereadily understood, see Ref. [37] for details. The fourth\nterm on the right is a Zeeman-like term due the presence of\nthe relativistically corrected exchange field, which acts a s\nan effective mean field. The ninth term is the one, which\nin a central potential V, gives rise to the conventional form\nof the spin-orbit coupling. The tenth term is a kind of\nspin-orbit interaction but due to the external fields. The\nvery last term is a relativistic correction which depends on\ntheBxcfield but is independent of the spin. As we will\nsee in the following, the terms that are responsible for spin\nrelaxation are the relativistic terms that involve a direct\ncoupling of the spin operator with either the exchange field\nBxcor one of the externally applied fields ( EorA).3\nIII. SPIN EQUATION OF MOTION\nThe spin angular momentum operator is given by S=\n(/planckover2pi1/2)σ. To obtain an equation of motion for the spin op-\nerator we have to evaluate the commutator [S,HEP(t)]. It\nis obvious from the expression of HEPthat only the terms\nwhich are explicitly spin dependent will contribute as oth-\nerwise the commutator vanishes. We can thus extract from\nHEPthe spin Hamiltonian\nHS(t) =H0+Hint\nsoc+Hext\nsoc (6)\nwhere the Zeeman-like fields are added up to an effective\nmagnetic induction,\nH0=−e\nmS·(B+Bxc+Bxc\ncorr)≡ −e\nmS·Beff.(7)\nThe part H0contains the main nonrelativistic contribution,\nall other terms in the spin Hamiltonian HSare of relativis-\ntic origin. The intrinsic spin-orbit coupling is given by th e\nHamiltonian\nHint\nsoc=i\n2/planckover2pi1m2c2S·(pV)×(p−eA). (8)\nThe crystal potential stems from the nuclei-electron and\nelectron-electron interactions and thus should have trans -\nlational symmetry. Consequently, also the intrinsic spin-\norbit Hamiltonian has translational symmetry [39]. If the\nposition of any j-th nucleus is Rj, the electron position is\nr, and the electron position with respect to the nucleus is\nrepresented by rj, then the crystal potential can be rep-\nresented by a sum of atom-centered potentials. Making\nnow in addition the central potential approximation (no\nangular dependence) for each of the atom-centered poten-\ntials, the potential can be written as V(rj) =V(|r−Rj|).\nThe translational symmetry is realized by the fact that\nrj=r−Rj. With the definition of spin-orbit interac-\ntion strength ξ(rj) =1\n2m2c21\nrdV(rj)/dr, and the Coulomb\ngauge,∇·A= 0, for homogeneous magnetic fields, i.e.,\nA= (B×r)/2, this Hamiltonian can further be written as\nHint\nsoc=1\n2m2c21\nrdV\ndrS·L−er\n4m2c2dV\ndrS·B\n+e\n4m2c21\nrdV\ndr(S·r)(r·B)\n=/summationdisplay\njξ(rj)/bracketleftBig\nS·L−e\n2/parenleftBig\nr2S·B−(S·r)(r·B)/parenrightBig/bracketrightBig\n.(9)\nWe note, first, that the full spin-orbit Hamiltonian, Hint\nsoc+\nHext\nsoc, is gauge invariant [40], but for deriving expressions\nwe need to make a choice. The Coulomb gauge is a suit-\nable choice here, yet it can be used exactly only when a\nslowly varying and homogeneous magnetic field is present.\nThis gauge further implies that only the transversal parts\nofEand ofAare retained, the latter being gauge invari-\nant. Doing so, we have thus recovered the “usual” spin-orbit\ncoupling term and other ultra-relativistic terms.\nThe external spin-orbit coupling Hamiltonian is given by\nHext\nsoc=−e\n4m2c2S·{E×(p−eA)−(p−eA)×E},\nwhich has a similar form as Hint\nsoc[Eq. (8)], but contains the\nexternal Maxwell fields instead. Making use of Maxwell’sequation ∇×E=−∂B/∂t, this Hamiltonian can be\nrewritten as\nHext\nsoc=−e\n2m2c2S·(E×p)+ie/planckover2pi1\n4m2c2S·∂B\n∂t\n+e2\n2m2c2S·(E×A). (10)\nThe last term in the Hamiltonian Hext\nsocdescribes the in-\nteraction of the photon spin angular momentum density,\njs=ǫ0(E×A)[41], with the electron spins [40, 42]. A\nrelated interaction energy due to a coupling of the angu-\nlar momentum density of the electromagnetic field with the\nmagnetic moment was proposed recently on phenomenolog-\nical grounds [43]. The relativistic light-spin interactio n in\nthe Hamiltonian (10) adopts thus the form\nHext\nlight−spin=e2\n2m2c2ǫ0S·js. (11)\nThis term, being second order in the external fields can\nbecome important in the strong field regime. As we focus\nin first instance on the damping, we will not consider it in\nthe derivation of the spin damping, but we come back to it\nlater on.\nNow we have the necessary parts of the spin Hamiltonian\nand we are ready to calculate the spin dynamics equations.\nAccording to the definition of magnetization, this quantity\nis given by the expectation value of spin angular momentum\n[44]\nM=/summationdisplay\njgµB\nVTr/braceleftbig\nρSj/bracerightbig\n, (12)\nwhereVis a suitably chosen volume element. The sum-\nmation is taken over all the electrons jand the definition\nof the density matrix is ρ=/summationtext\nipi|ψi/an}b∇acket∇i}ht/an}b∇acketle{tψi|, where the set\nof wave functions |ψi/an}b∇acket∇i}htare in a mixed state and piare the\noccupation numbers. As is customary in spin dynamics\nmodels [12–20, 24, 26, 45] the contribution of the orbital\nangular momentum to the total magnetization has been\nneglected because it is quenched for the common transition\nmetals (e.g., Fe, Ni, Co etc.). The equation of motion of\nthe magnetization is obtained by taking the time deriva-\ntive on both sides of Eq. (12), and using that ∂ρ/∂t= 0\nfor quasiadiabatic processes [46], which gives\n∂M\n∂t=gµB\nV1\ni/planckover2pi1/summationdisplay\njTr/braceleftbig\nρ[Sj,HS(t)]/bracerightbig\n. (13)\nTo obtain the magnetization dynamics we substitute the\nspin Hamiltonian HS(t) =H0+Hint\nsoc+Hext\nsocin the right-\nhand side of Eq. (13) and work out the trace term-by-term.\nBefore presenting the result we consider briefly the ap-\nproximations made in the derivation. Notably, Eq. (13)\nis valid for local processes and will hence provide a local\ndamping mechanism. However, it is known that nonlo-\ncalcontributions to the damping exist (see, e.g., [47–49])\nthat can be caused by spin transport from one region to\nanother [50–52]. Such effects can be treated using the con-\ntinuity equation, ∂ρ/∂t+∇·J= 0, withJthe current\noperator, leading to an additional spin current term (see,\ne.g., [52, 53]). A further remark due at this point concerns4\nthe time dependence of the exchange field. In line with the\nabove, we adopt the adiabatic approximation that is valid\nfor systems not too far from the ground state [54].\nWorking out the commutator, we find that the first or-\nder dynamical equation of motion is given by the mostly\nnonrelativistic part in the spin Hamiltonian, H0. Using\nthe commutation relations for spin angular momentum,\n[Sj,Sk] =i/planckover2pi1ǫjklSl, the first order equation of motion be-\ncomes\n∂M\n∂t/vextendsingle/vextendsingle/vextendsingle0\n=−γM×Beff, (14)\nwhereγ=g|e|/2mis the gyromagnetic ratio and g≈2for\nspin degrees of freedom. Using B=µ0(H+M), the right-\nhand term can be rewritten in the conventional form as\n−γ0M×Heff, whereγ0=µ0γ. This equation provides the\ncommon understanding of the Larmor precessional motion\nof magnetization around an effective magnetic field, with\na distinction that there is a relativistic correction Bxc\ncorrto\nthis field that has not been noted before.\nNext we treat the relativistic spin-orbit effects in the\nmagnetization dynamics. As we will see, these are the ones\nthat lead to local damping, i.e., the spin relaxation mech-\nanisms in a magnetic solid are of relativistic origin [1, 9].\nFirst, we focus on the relativistic intrinsic spin-orbit co u-\npling Hamiltonian Hint\nsocin Eq. (9). Due to the quenching\nof the orbital angular momentum, the first term vanishes.\nThe dynamics due to the remaining two terms in the Hamil-\ntonian is calculated as\n∂M\n∂t/vextendsingle/vextendsingle/vextendsingleint\nsoc=e\n4m2c2/angbracketleftBig\nrdV\ndr/angbracketrightBig\nM×B\n−e\n4m2c2M×/angbracketleftBig\nr1\nrdV\ndr(r·B)/angbracketrightbig\n=e\n2/summationdisplay\nj/bracketleftBig\n/an}b∇acketle{tξ(rj)r2/an}b∇acket∇i}htM×B\n−M×/an}b∇acketle{tξ(rj)r(r·B)/an}b∇acket∇i}ht/bracketrightBig\n. (15)\nThe first term in the dynamics of Eq. (15) can be seen as a\nfurther relativistic correction to the magnetization prec es-\nsion. The second term has a form similar to the first term,\nbut with opposite sign. The terms can be combined, but\nthey do not contribute to any relaxation processes as they\ndo not contain a time variation of the magnetic induction.\nNext we consider the dynamics related to Hext\nsoc. We will\nsee below that it is mainly the relativistic extrinsic spin-\norbit coupling, i.e., the first two terms of Eq. (10), which\ngive rise to dominant local spin relaxation mechanisms in\nmagnetic solids. In addition, we observe here that these\ncorrespond to the transverse spin relaxation. We consider\nhere the long wavelength approximation, where the wave-\nlength of the field is much larger than the size of the system.\nIn other words the GHz/THz electromagnetic field inside\nthe ferromagnetic film is assumed uniform throughout the\nfilm as long as the film thickness is sufficiently small. We\ncan thus use the Coulomb gauge, i.e., A= (B×r)/2. This\ngauge allows us to obtain the explicit time dependence of\nthe Hamiltonian. The transverse electric field in the Hamil-\ntonian is then written as E=1\n2(r×∂B/∂t). Employing\nthe gauge, the first two terms in Eq. (10) can be re-writtenin an explicit, time-dependent form:\nHext\nsoc=ie/planckover2pi1\n4m2c2S·∂B\n∂t/parenleftbigg\n1−(r·p)\ni/planckover2pi1/parenrightbigg\n+e\n4m2c2(S·r)/parenleftbigg∂B\n∂t·p/parenrightbigg\n. (16)\nAt this point it is needed to inspect the hermiticity of\nthe Hamiltonian. It can be shown that the total spin-orbit\nHamiltonian in Eq. (16) is hermitian (see Appendix A),\nhowever, for the individual terms it is different. Writing\ndown the Hamiltonian in component form with the usual\nsummation convention, we obtain\nHext\nsoc=ie/planckover2pi1\n4m2c2Si∂Bi\n∂t/bracehtipupleft/bracehtipdownright/bracehtipdownleft/bracehtipupright\nanti−hermitian−e\n4m2c2/summationdisplay\ni/negationslash=jSi∂Bi\n∂trjpj\n/bracehtipupleft/bracehtipdownright/bracehtipdownleft/bracehtipupright\nnon−hermitian\n+e\n4m2c2/summationdisplay\ni/negationslash=jSiri∂Bj\n∂tpj\n/bracehtipupleft/bracehtipdownright/bracehtipdownleft/bracehtipupright\nhermitian. (17)\nPreviously, Hickey and Moodera considered the effect of the\nspin-orbit Hamiltonian on damping, but only obtained the\nfirst two terms in Eq. (10) [17]. They proposed then only\nthe anti-hermitian part of the Hamiltonian as an intrinsic\nsource of Gilbert damping [17]. Anti-hermitian Hamilto-\nnians understandably are always dissipative [55, 56]. Con-\nsequently, their choice of taking the anti-hermitian term\nonly was criticized, given that the full spin-orbit Hamilto -\nnian should be hermitian and that it therefore should not\nexhibit dissipation [55].\nIn our case the total spin-orbit Hamiltonian (16) is man-\nifestly hermitian, yet we will show below that it does give\nrise to spin moment damping. The point is, that even\nwhen the full Hamiltonian is hermitian, it only has this\nproperty when one considers the dynamics of the full sys-\ntem. It is however customary in spin moment dynamics\n[12–20, 24, 26, 45] to integrate out the orbital degree of\nfreedom and other magnetic degrees of freedom (as back-\nground fluctuations of the system) thus restricting the fo-\ncus on the single spin moment dynamics. In the thereby\nrestricted Hilbert space the hermiticity is lost and hence\nthe whole Hamiltonian can contribute to the damping.\nCalculating now the commutation relation [S,Hext\nsoc]and\ntaking the summation of the trace over all electrons, the\nspin moment dynamics adopts the form\n∂M\n∂t/vextendsingle/vextendsingle/vextendsingleext\nsoc=−ie/planckover2pi1\n4m2c2M×∂B\n∂t/parenleftBigg\n1−/angbracketleftbig\nr·p/angbracketrightbig\ni/planckover2pi1/parenrightBigg\n−e\n4m2c2M×/angbracketleftBig\nr/parenleftbigg∂B\n∂t·p/parenrightbigg/angbracketrightBig\n. (18)\nA rewriting of these terms is required to elucidate further\nthe spin relaxation.\nIV. THE DAMPING EQUATIONS\nTo obtain explicit expressions for the damping terms, we\nemploy the general relation between magnetic induction B,5\nmagnetization M, and magnetic field H, given as B=\nµ0(M+H). We take the time derivative on both sides,\n∂B\n∂t=µ0/bracketleftbigg∂M\n∂t+∂H\n∂t/bracketrightbigg\n. (19)\nThis relation is generally valid, also for the stationary ca se,\neven though the magnetization M(t)and magnetic field\nH(t)are time dependent. At this point it is instructive to\nconsider what kinds of magnetic fields H(t)can occur. The\nsimplest case is when at some time t0only a static fieldH0\nis present, then obviously only the first term in Eq. (19)\ncontributes. If the field H(t)is explicitly time dependent,\nwe can distinguish to cases: an ac driven, periodic magnetic\nfield, as is commonly used in measurements, or a more gen-\neral field, for example a magnetic field pulse. In the latter\ncase, one could proceed to derive the spin dynamics by\nkeeping explicitly the term∂H\n∂t. As a result, one obtains a\nLLG-like equation, where however the magnetic field cou-\nples into the damping term. The thus-obtained modified\nLLG equation is given and analyzed further below, in Sect.\nV.\nIn the former case, the effect of the magnetic response\nbecomes apparent when an ac magnetic field is applied.\nFor ferromagnetic materials, where there is a net magneti-\nzation present even in the absence of the applied field, the\nmagnetic susceptibility can be introduced by the definition :\nχ=∂M/∂H. Using a chain rule for the time derivative,\n∂H\n∂t=∂H\n∂M∂M\n∂t, Eq. (19) can be written as\n∂B\n∂t=µ0/parenleftbig\n1+χ−1/parenrightbig\n·∂M\n∂t, (20)\nwhere 1is the3×3identity matrix. This relation has been\nused in the ensuing magnetization dynamics.\nSubstituting Eq. (20) in the first term of Eq. (18), we\nobtain\n∂M(1)\n∂t/vextendsingle/vextendsingle/vextendsingleext\nsoc=−ie/planckover2pi1µ0\n4m2c2M×/bracketleftbigg\n(1+χ−1)·∂M\n∂t/bracketrightbigg/parenleftbigg\n1−/an}b∇acketle{tr·p/an}b∇acket∇i}ht\ni/planckover2pi1/parenrightbigg\n.\n(21)\nThis term can already be recognized to have the form of\nthe Gilbert damping, M×/bracketleftbig\nα·∂M\n∂t/bracketrightbig\n, yet with a tensorial\ndamping constant.\nFor the full damping we have to combine with the second\nterm in Eq. (18), which is rewritten as\n∂M(2)\n∂t/vextendsingle/vextendsingle/vextendsingleext\nsoc=−eµ0\n4m2c2M×/angbracketleftBig\nr/parenleftbigg/bracketleftBig\n(1+χ−1)·∂M\n∂t/bracketrightBig\n·p/parenrightbigg/angbracketrightBig\n.\n(22)\nTo join the terms we proceed with using vector components.\nEquation (21) becomes\n∂M(1)\n∂t/vextendsingle/vextendsingle/vextendsingleext\nsoc=−eµ0\n4m2c2/summationdisplay\nijklnMk/bracketleftbigg\n(1+χ−1)ij∂Mj\n∂t/bracketrightbigg\n×(i/planckover2pi1−/an}b∇acketle{trnpn/an}b∇acket∇i}ht)εkilˆel, (23)\nwithεijkthe Levi-Civita tensor and ˆea unit vector. This\nterm can be written as\n∂M(1)\n∂t/vextendsingle/vextendsingle/vextendsingleext\nsoc=/summationdisplay\nijklMk∂Mj\n∂tΩijεkilˆel, (24)withΩij=−eµ0\n4m2c2/summationtext\nn(i/planckover2pi1−/an}b∇acketle{trnpn/an}b∇acket∇i}ht)(1+χ−1)ij. The sec-\nond term (22) can be written in a similar form, but with\na tensor ∆ij=−eµ0\n4m2c2/summationtext\nn/an}b∇acketle{tripn/an}b∇acket∇i}ht(1+χ−1)nj. Combining\nthese two terms gives the total damping term,\n∂M\n∂t/vextendsingle/vextendsingle/vextendsingleext\nsoc=/summationdisplay\nijklMk/bracketleftBig\nΩij+∆ij/bracketrightBig∂Mj\n∂tεkilˆel,(25)\nwhere it is convenient to define A ij≡Ωij+∆ij,\nAij=−eµ0\n4m2c2/summationdisplay\nn/bracketleftBig\ni/planckover2pi1−/an}b∇acketle{trnpn/an}b∇acket∇i}ht+/an}b∇acketle{trnpi/an}b∇acket∇i}ht/bracketrightBig\n(1+χ−1)ij\n=−eµ0\n8m2c2/summationdisplay\nn,k/bracketleftBig\n/an}b∇acketle{tripk+pkri/an}b∇acket∇i}ht−/an}b∇acketle{trnpn+pnrn/an}b∇acket∇i}htδik/bracketrightBig\n×(1+χ−1)kj.(26)\nNote that a summation over iis not intended in the right-\nhand side expressions. In vector form the spin-orbit damp-\ning term becomes\n∂M\n∂t/vextendsingle/vextendsingle/vextendsingleext\nsoc=M×/bracketleftBig\nA·∂M\n∂t/bracketrightBig\n. (27)\nSummarizing our result, we observe that we have obtained\na damping parameter A ijof Gilbert type that is however\nin its general form not a scalar but a tensor. The tenso-\nrial character of the Gilbert damping was also concluded\nrecently in other investigations [16, 57]. In this form it\naccounts for transversal spin relaxation that conserves th e\nlength of the magnetization, i.e., ∂(M·M)/∂t= 0.\nEvery tensor can be decomposed in a symmetric and an\nanti-symmetric part. Hence, the damping tensor can be\ndecomposed into a scalar ( α) multiplied by the unit matrix,\na symmetric tensor ( I), and an anti-symmetric tensor ( A,\nwithAij=1\n2(Aij−Aji)). The latter tensor can in turn be\nexpressed as Aij=εijkDkwithDbeing a vector. Finally,\nthe damping dynamics can then be written as\n∂M\n∂t/vextendsingle/vextendsingle/vextendsingleext\nsoc=αM×∂M\n∂t+M×/bracketleftBig\nI·∂M\n∂t/bracketrightBig\n+M×/bracketleftBig\nD×∂M\n∂t/bracketrightBig\n. (28)\nThe first term is the conventional Gilbert damping. It orig-\ninates from the decomposition of the symmetric part of the\ntensor into an isotropic Heisenberg-like (α1)contribution\nas well as an anisotropic Ising-like ( I) contribution which\nleads to the second term. Along with that it is not surpris-\ning that the last term implies a Dzyaloshinskii-Moriya-lik e\ncontribution. The anisotropic nature of the Gilbert damp-\ning has been noted before [18, 57], but not the appearance\nof the Dzyaloshinskii-Moriya-like damping. This type of\ndamping could be related to the chiral damping of mag-\nnetic domain walls that was reported recently [58].\nFor the case of a constant, scalar Gilbert damping param-\neter it is straightforward to transform the LLG equation to\nobtain the LL equation with the phenomenological damp-\ning term proposed by Landau and Lifshitz [1]. However,\nthis is no longer the case for tensorial Gilbert damping,\nfor which the transformation is much more involved. The\nspin-dynamics equation in the Landau-Lifshitz form now6\nbecomes (see Appendix B)\n/parenleftbig\nΨ21+G/parenrightbig\n·∂M\n∂t=\n−γ0ΨM×Heff−γ0M×/bracketleftBig\n(α1+I)·(M×Heff)/bracketrightBig\n,(29)\nwhereΨ = 1+ M·Dand the tensor Gis defined through\nG=α2M21−/bracketleftBig\n(M·I·M)−tM2/bracketrightBig\n(α1+I)\n−/parenleftBig\ntM−M·I/parenrightBig\nM·I−M2I2+M/parenleftBig\nM·I2/parenrightBig\n,(30)\nwith the trace, t= Tr( I). In general the trace of such a\nmatrix Iis non-zero, however its value will depend on how\nthe symmetric tensor Asym\nij=1\n2(Aij+Aji) =Iij+αδij\nis decomposed. If the decomposition in Ising and Heisen-\nberg parts is such that the isotropic part is chosen as\nα=1\n3Tr(Asym\nij), then the trace of Iwill vanish, t= 0. Note\nthat the term (15) due to the intrinsic spin-orbit interacti on\nhas been left out, as it is expected to give only a small cor-\nrection to the effective magnetic field. The damping term\nthus adopts the form −γ0M×[Λ·(M×Heff)], similar to\nthe phenomenological damping considered by Landau and\nLifshitz [1], but with damping tensor Λ. A more general\nform of the LL damping as a tensor was already considered\nmuch earlier (see, e.g. [59]), and it is reflected also in our\nderivation. However, a distinction is that here the leading\n∂M/∂tterm on the left-hand side in Eq. (29) is, in its gen-\neral form, multiplied not with a scalar ( 1+α2M2) but with\na tensor which moreover depends on the direction of M.\nIt is worth noting that in the absence of the\nDzyaloshinskii-Moriya and anisotropic relaxation contri bu-\ntions, i.e., setting D=I= 0we retrieve the original LL and\nLLG equations with scalar damping parameters. The va-\nlidity range of our derived equations of spin motion is thus\nlarger than the originally proposed equations of motion. It\nshould also be emphasized that the Dzyaloshinskii-Moriya-\nlike contribution appears in the Gilbert damping, however,\nit does not appear in the damping term of the LL equation\n(29). Instead, it leads to the renormalization of the stan-\ndard dynamical terms in the LL equation as can be seen\nfrom the appearance of the quantity Ψin Eq. (29). We\nlastly note that the here obtained relaxation terms do not\nallow a variation with respect to the coordinates i.e., they\ndo not include effects of spatial dispersion.\nV. DISCUSSION\n1. Analysis of the damping expression\nEquation (26) for the Gilbert damping pertains to the\nrelaxation of spin motion in the presence of spin-orbit in-\nteraction. This damping is of relativistic origin as is ex-\nemplified by its 1/c2dependence. The expression for the\nGilbert tensor is different from that obtained previously\n[17], where only the constant term i/planckover2pi1in the square bracket\nwas found. The new parts /an}b∇acketle{tripj/an}b∇acket∇i}htrelate to how the elec-\ntronic band energies Eνkof Bloch states |νk/an}b∇acket∇i}htdisperse with\nk-space direction. It can be rewritten as (see Appendix C)\n/an}b∇acketle{tripj/an}b∇acket∇i}ht=−i/planckover2pi1\n2m/summationdisplay\nν,ν′,kf(Eνk)−f(Eν′k)\nEνk−Eν′kpi\nνν′pj\nν′ν,(31)wherepνν′≡ /an}b∇acketle{tνk|p|ν′k/an}b∇acket∇i}htandf(Eνk)is the Fermi function.\nThe sum contains interband and intraband contributions.\nThe intraband (Fermi surface) contribution (ν=ν′)can\nbe written as\n/an}b∇acketle{tripj/an}b∇acket∇i}ht=−im\n2/planckover2pi1/summationdisplay\nνk/parenleftbigg∂f\n∂E/parenrightbigg\nEνk/parenleftbigg∂Eνk\n∂ki/parenrightbigg/parenleftbigg∂Eνk\n∂kj/parenrightbigg\n.(32)\nThis expression has a similarity with other previously de-\nrived expressions, as e.g. the breathing Fermi surface mode l\n[8, 24] that has been applied to metallic ferromagnets. The\nexpression for the /an}b∇acketle{tripj/an}b∇acket∇i}htterms has furthermore a form sim-\nilar to that for the conductivity tensor in linear-response\ntheory [60]; it is in particular well-suited for ab initio cal-\nculations. We note further that the influence of electron\ninteraction with quasiparticles can be introduced by repla c-\ningEνk−Eν′kbyEνk−Eν′k+iδ, where the small δgives\na finite relaxation time to the electronic states.\nFor numerical evaluation of the damping tensor the sus-\nceptibility tensor χis furthermore needed, which is in gen-\neral wavevector and frequency dependent, χ(q,ω). Thus,\nalso the Gilbert damping tensor is here a frequency and\nq-dependent quantity, in accordance with recent measure-\nments [32]. Suitable expressions for χhave been considered\npreviously in the context of Gilbert damping [13, 16, 45].\nLinear-response formulations that express χas a spin-spin\ncorrelation function include the Pauli and Van Vleck sus-\nceptibility contributions [61], and expressions for the or -\nbital susceptibility have been derived as well [62]. These\nexpressions are fitting for ab initio calculations of χwithin\na DFT framework. The spin-orbit interaction will have an\nadditional influence on χ, however, unlike the main Gilbert\ndamping contribution which is proportional to the spin-\norbit coupling, this will only be a higher order effect.\nWe can consequently distinguish here two origins for the\ndamping: the first one is related to the terms /an}b∇acketle{tripj/an}b∇acket∇i}ht, which\nrepresent dissipation contributions into the orbital degr ees\nof freedom. The second nature is due to the magnetic sus-\nceptibilityχwhich represents losses through the magnetic\nstructure of the material. Both effects are simultaneously\npresent, and nonzero, for metallic ferromagnets as well as\ninsulators.\nIt is also important to mention that the damping ten-\nsor in the our derivation does not include spin-relaxation\neffects due to interaction of spin-polarized electrons with\nquasiparticles as magnon or phonons or scattering with de-\nfects. Longitudinal spin relaxation due to spin-flip pro-\ncesses caused by electron-phonon scattering have been re-\ncently calculated ab initio for the transition-metal ferro-\nmagnets [63–65], and magnon spin-flip scattering has been\nconsidered as well [66]. Spin angular momentum transfer\ndue to explicit coupling of the spins to the lattice has been\ntreated in several models [67–69]. As mentioned above,\nalthough the spin-lattice dissipation channel is not encom -\npassed in our derivation, an approximate way to include its\ninfluence has been introduced before, by a suitable spec-\ntral broadening of the Bloch electron energies (see, e.g.,\n[24, 70]).\nLastly, we remark that in the present derivation we ob-\ntain only first-order time-derivatives of M(r,t). Second-\norder time-derivatives of M(r,t)have recently been related\nto moment of inertia of the magnetization [71].7\n2. Exchange field and nonlocal contributions\nThus far we have not explicitly discussed the exchange\ninteraction. The influence of the exchange field can be\naccounted for in various levels of approximation, for ex-\nample, within the Heisenberg model or evaluated within\ntime-dependent DFT [72, 73]. In the former, a suitable\nsimplification of the exchange interaction in a magnetic\nsolid is to express it through the Heisenberg Hamiltonian\nHxc=−/summationtext\nα>βJαβSα·Sβ, where the Jαβare exchange\nconstants and Sαis the atomic spin on atom α. Using this\nHamiltonian to express the exchange field leads to Landau-\nLifshitz-Gilbert equations of motion for the dynamics of\natomic moments (see, e.g., [74–76]).\nMore general, the exchange field depends on the spatial\nposition which implies that there can exist an influence of\nspatial nonuniformity of the exchange field on the spin re-\nlaxation. An influence on the dynamics occurring due to\nmagnetization inhomogeneity ( ∇2M) appearing in the ef-\nfective field was already suggested by Landau and Lifshitz\n[1]. Such a term is in fact needed to properly describe\nspin wave dispersions [77]. A nonlocal damping mecha-\nnism due to spatial dispersion of the exchange field was\nproposed by Bar’yakhtar on the basis of phenomenological\nconsiderations such as symmetry arguments and Onsager’s\nrelations [10]. This leads to a modified expression for the\ndamping term in the Landau-Lifshitz-Bar’yakhtar equation\nwhich contains the derivative of the exchange field ∇2Bxc\n[10, 20]. The existence of such nonlocal damping term can\nbe related to the continuity equation connecting the spin\ndensity and spin current; it is important for obtaining the\ncorrect asymptotic behavior of spin wave damping at large\nwavevectors k[20] known for magnetic dielectrics, see [59].\nSuch nonlocal damping is important, too, for describing\nspin current flow in magnetic metallic heterostructures [78 ].\nThese nonlocal damping terms are furthermore related to\nthe earlier proposed magnetization damping effects due to\nspin diffusion [52, 79–81] that have been studied recently\n[82]. As a consequence of the spin current flow the local\nlength of the magnetization is not conserved. In the present\nwork such nonlocal terms are not included since we focus\non the local dissipation and have thus omitted the spin cur-\nrent contribution of the continuity equation. A future full\ntreatment that takes into account both local and nonlocal\nspin dissipation mechanisms would permit to describe mag-\nnetization dynamics and spin transport on an equal footing\nin a broader range of inhomogeneous systems.\n3. General time-dependent magnetic fields\nWhen the driving magnetic field is not an ac harmonic\nfield the dependence of M(r,t)onH(t)will induce a more\ncomplex dynamics. In this case it is possible to derive a\nclosed expression for the spin dynamics by explicitly keep-\ning the term∂H\n∂tin Eq. (19). A similar derivation as pre-\nsented in Sect. IV for the ac driving field leads then to the\nfollowing expression for the magnetization dynamics\n∂M\n∂t=−γ0M×Heff+M×/bracketleftBig\n¯A·/parenleftBig∂M\n∂t+∂H\n∂t/parenrightBig/bracketrightBig\n,(33)where the damping tensor ¯A is given by\n¯Aij=−eµ0\n8m2c2/summationdisplay\nn/bracketleftBig\n/an}b∇acketle{tripj+pjri/an}b∇acket∇i}ht−/an}b∇acketle{trnpn+pnrn/an}b∇acket∇i}htδij/bracketrightBig\n.(34)\nThe time-dependent magnetic field thus leads to a new,\nmodified spin dynamics equation which has, to our knowl-\nedge, not been derived before. The time-derivate of H(t)\nintroduces here an additional torque, M×∂H\n∂t. This field-\nderivative torque might offer new ways to achieve fast mag-\nnetization switching. Consider for example an initially\nsteep magnetic field pulse that thereafter relaxes slowly\nback to its initial value. The derivative of such field will ex -\nert a large but shortly lasting torque on the magnetization,\nwhich could initiate switching. Irradiation of magnetic th in\nfilms with a picosecond THz field pulse was recently shown\nto trigger ultrafast magnetization dynamics [83], and suit -\nable shaping of the THz magnetic field pulse could hence\noffer a route to achieve switching on a picosecond time scale.\n4. The optical spin torque\nThe interaction of the spin moment with the optical spin\nangular moment jsis given by the Hamiltonian Hext\nlight−spin.\nWe note that such relativisitic interaction is important fo r\nrecent attempts to manipulate the magnetization in a ma-\nterial using optical angular momentum, i.e., helicity of th e\nlaser field [40, 84, 85]. This interaction leads to spin dy-\nnamics of the form\n∂M\n∂t/vextendsingle/vextendsingle/vextendsingleext\nlight−spin=−e2\n2m2c2ǫ0M×js, (35)\nwhereM×jsis the optical spin torque exerted by the\noptical angular moment on the spin moment. This equa-\ntion expresses that the spin moment in a material can be\nmanipulated by acting on it with the optical spin angular\nmoment of an external electromagnetic field in the strong\nfield regime.\nVI. CONCLUSIONS\nOn the basis of the relativistic Dirac-Kohn-Sham equa-\ntion we have derived the spin Hamiltonian to describe ad-\nequately the dynamics of electron spins in a solid, tak-\ning into account all the possible spin-related relativisti c\neffects up to the order 1/c2and the exchange field and ex-\nternal electromagnetic fields. From this manifestly hermi-\ntian spin Hamiltonian we have calculated the spin equation\nof motion which adopts the form of the Landau-Lifshitz-\nGilbert equation for applied harmonic fields. For univer-\nsal time-dependent external magnetic fields we obtain a\nmore general dynamics equation which involves the field-\nderivative torque. Our derivation does notably not rely\non phenomenological assumptions but provides a rigorous\ntreatment on the basis of fundamental principles, specifi-\ncally, Dirac theory with all relevant fields included.\nWe have shown the existence of a relativistic correction\nto the precessional motion in the obtained LLG equation\nand have derived an expression for the spin relaxation\nterms of relativistic origin. One of the most prominent8\nresults of the presented article is the derived expression f or\nthe tensorial Gilbert damping, which has been shown to\ncontain an isotropic Gilbert contribution, an anisotropic\nIsing-like contribution, and a chiral, Dzyaloshinskii-\nMoriya-like contribution. Transforming the LLG equation\nto the Landau-Lifshitz equation of motion, we showed\nthat the LLG equation with anisotropic tensorial Gilbert\ndamping cannot trivially be written as a LL equation with\nan anisotropic LL damping term, but an additional matrix\nappears in front of the ∂M/∂tterm. The Dzyaloshinskii-\nMoriya-like contribution serves as a renormalization fact or\nto the common LL dynamical terms. The obtained\nexpression for the Gilbert damping tensor in the case of\na periodic driving field depends on the spin-spin suscep-\ntibility response function along with a term representing\nthe electronic spin damping due to dissipation into the\norbital degrees of freedom. As there exist an on-going\ndiscussion on what the fundamental origin of the Gilbert\ndamping is and how it can accurately be evaluated from\nfirst-principles calculations [28, 30–32], we point out tha t\nthe two components of the derived damping expression\n(spin-spin and current-current response functions) are\nsuitable for future ab initio calculations within the density\nfunctional formalism.\nACKNOWLEDGMENTS\nWe thank B. A. Ivanov, P. Maldonado, A. Aperis, K.\nCarva, and H. Nembach for helpful discussions. We alsothank the anonymous reviewers for valuable comments.\nThis work has been supported by the European Com-\nmunity’s Seventh Framework Programme (FP7/2007-2013)\nunder grant agreement No. 281043, FemtoSpin, the Swedish\nResearch Council (VR), the Knut and Alice Wallenberg\nFoundation (Contract No. 2015.0060), and the Swedish Na-\ntional Infrastructure for Computing (SNIC).\nAppendix A: Hermiticity of Hamiltonian Hext\nsoc\nThe extrinsic spin-orbit Hamiltonian Hext\nsoc, given in Eq.\n(16), can indeed be shown to be hermitian, however its in-\ndividual terms are not all hermitian. Adapting the Einstein\nsummation convention, this Hamiltonian can be written in\ncomponent form as\nHext\nsoc=e\n4m2c2/parenleftBig\ni/planckover2pi1Si∂tBi\n−Si∂tBirjpj+Siri∂tBjpj/parenrightBig\n,(A1)\nwith∂t≡∂/∂t. To demonstrate that it is hermitian, we\ntake the Hermitian conjugate, and rewrite it in a few steps.\n/bracketleftBig\nHext\nsoc/bracketrightBig†\n=e\n4m2c2/parenleftBig\n−i/planckover2pi1Si∂tBi−Si∂tBipjrj+Si∂tBjpjri/parenrightBig\n=e\n4m2c2/parenleftBig\n−i/planckover2pi1Si∂tBi−Si∂tBirjpj+Si∂tBjripj−Si∂tBi(pjrj)+Si∂tBj(pjri)/parenrightBig\n=e\n4m2c2/parenleftBig\n−i/planckover2pi1S·∂tB−(S·∂tB)(r·p)+(S·r)(∂tB·p)−(S·∂tB)(p·r)+S·{(∂tB·p)r}/parenrightBig\n=e\n4m2c2/parenleftBig\n−i/planckover2pi1S·∂tB−(S·∂tB)(r·p)+(S·r)(∂tB·p)+i/planckover2pi1(S·∂tB)(∇·r)−i/planckover2pi1S·{(∂tB·∇)r}/parenrightBig\n=e\n4m2c2/parenleftBig\n−i/planckover2pi1S·∂tB−(S·∂tB)(r·p)+(S·r)(∂tB·p)+3i/planckover2pi1S·∂tB−i/planckover2pi1S·∂tB/parenrightBig\n=e\n4m2c2/parenleftBig\ni/planckover2pi1S·∂tB−(S·∂tB)(r·p)+(S·r)(∂tB·p)/parenrightBig\n=Hext\nsoc. (A2)\nFor the individual terms of the Hamiltonian it is straightfo rward to show their hermitian or non-hermitian character:\nHext\nsoc= =ie/planckover2pi1\n4m2c2Si∂tBi\n/bracehtipupleft/bracehtipdownright/bracehtipdownleft/bracehtipupright\nanti−hermitian−e\n4m2c2/summationdisplay\ni/negationslash=jSi∂tBirjpj\n/bracehtipupleft/bracehtipdownright/bracehtipdownleft/bracehtipupright\nnon−hermitian+e\n4m2c2/summationdisplay\ni/negationslash=jSiri∂tBjpj\n/bracehtipupleft/bracehtipdownright/bracehtipdownleft/bracehtipupright\nhermitian. (A3)\nAs noted before all three terms of the hermitian Hamiltonian contribute to the spin relaxation process.\nAppendix B: From LLG to LL equations of motion\nWe found that the generalized LLG equation of spin dynamics c an be written in the form [see Eq. (27)]\n∂M\n∂t=−γM×Beff+M×/bracketleftBig\nA·∂M\n∂t/bracketrightBig\n. (B1)9\nAs discussed earlier, using the tensor decomposition, one c an also write\n∂M\n∂t=−γM×Beff+αM×∂M\n∂t+M×/bracketleftBig\nI·∂M\n∂t/bracketrightBig\n+M×/bracketleftBig\nD×∂M\n∂t/bracketrightBig\n. (B2)\nThe Dzyaloshinskii-Moriya-like damping terms can be expan ded, using a×(b×c) =b(a·c)−c(a·b), to give M×/bracketleftBig\nD×∂M\n∂t/bracketrightBig\n=−∂M\n∂t(M·D). Since the magnetization length is conserved we therefore h aveM·∂M/∂t= 0. Defining\n(1+M·D) = Ψ, the LLG equation of spin motion reduces to\nΨ∂M\n∂t=−γM×Beff+αM×∂M\n∂t+M×/bracketleftBig\nI·∂M\n∂t/bracketrightBig\n. (B3)\nNote that Ψis both a magnetization and Dzyaloshinskii-Moriya vector d ependent quantity. Next, we have to calculate\nthe second and third terms on the right-hand side of Eq. (B3). Taking a cross product with Mon both sides of the last\nequation gives\nΨM×∂M\n∂t=−γM×(M×Beff)+αM×/parenleftBig\nM×∂M\n∂t/parenrightBig\n+M×/parenleftBig\nM×/bracketleftBig\nI·∂M\n∂t/bracketrightBig/parenrightBig\n=−γM×(M×Beff)−αM2∂M\n∂t−M2/bracketleftBig\nI·∂M\n∂t/bracketrightBig\n+M/parenleftBig\nM·/bracketleftBig\nI·∂M\n∂t/bracketrightBig/parenrightBig\n. (B4)\nSimilarly, to evaluate the last term of Eq. (B3), we take the d ot product with the symmetric part of the tensor, followed\nby a cross product with the magnetization,\nΨM×/bracketleftBig\nI·∂M\n∂t/bracketrightBig\n=−γM×/bracketleftBig\nI·(M×Beff)/bracketrightBig\n+αM×/bracketleftBig\nI·/parenleftBig\nM×∂M\n∂t/parenrightBig/bracketrightBig\n+M×/parenleftBig\nI·/braceleftBig\nM×/bracketleftBig\nI·∂M\n∂t/bracketrightBig/bracerightBig/parenrightBig\n.(B5)\nAt this point we already observe that the first term on the righ t hand side has adopted a form of the LL damping but\nwith a tensor. The second and third terms are treated in the fo llowing. The second term can be written in component\nform as\nαM×/bracketleftBig\nI·/parenleftBig\nM×∂M\n∂t/parenrightBig/bracketrightBig\n=αMlImkMi∂Mj\n∂tεijkεlmnˆen. (B6)\nWe use the following relation for the product of two anti-sym metric Levi-Civita tensors\nεijkεlmn=δil(δjmδkn−δjnδkm)−δim(δjlδkn−δjnδkl)+δin(δjlδkm−δjmδkl), (B7)\nand, defining the trace of the symmetric tensor Tr(I) =t, a little bit of tensor algebra results in\nαM×/bracketleftBig\nI·/parenleftBig\nM×∂M\n∂t/parenrightBig/bracketrightBig\n=αM2/parenleftBig\nI·∂M\n∂t/parenrightBig\n−αtM2∂M\n∂t+α/parenleftBig\nM·I·M/parenrightBig∂M\n∂t−αM/bracketleftBig\nM·/parenleftBig\nI·∂M\n∂t/parenrightBig/bracketrightBig\n.(B8)\nNow we proceed to calculate the last part of Eq. (B5); the comp onents of this term are given by\nM×/parenleftBig\nI·/braceleftBig\nM×/bracketleftBig\nI·∂M\n∂t/bracketrightBig/bracerightBig/parenrightBig\n=MmInlMkIij∂Mj\n∂tεkilεmnoˆeo. (B9)\nUsing once again the relation in Eq. (B7) and expanding in diff erent components we find\nM×/parenleftBig\nI·/braceleftBig\nM×/bracketleftBig\nI·∂M\n∂t/bracketrightBig/bracerightBig/parenrightBig\n=/bracketleftBig\n(M·I·M)−tM2/bracketrightBig/parenleftBig\nI·∂M\n∂t/parenrightBig\n+/parenleftBig\ntM−M·I/parenrightBig/bracketleftBig\nM·/parenleftBig\nI·∂M\n∂t/parenrightBig/bracketrightBig\n+(M·M)/bracketleftBig\nI·/parenleftBig\nI·∂M\n∂t/parenrightBig/bracketrightBig\n−M/bracketleftBig\nM·/braceleftbigg\nI·/parenleftBig\nI·∂M\n∂t/parenrightBig/bracerightbigg/bracketrightBig\n. (B10)\nNow we have the necessary terms to formulate the LL equation o f motion. Taking these together, the LLG dynamics\nof Eq. (B3) can be written as\nΨ2∂M\n∂t=−γΨM×Beff−γM×/bracketleftBig\n(α1+I)·(M×Beff)/bracketrightBig\n−G·∂M\n∂t, (B11)\nwith the general tensorial form of Gwhich is given by\nG=α2M21−/bracketleftBig\n(M·I·M)−tM2/bracketrightBig\n(α1+I)−/parenleftBig\ntM−M·I/parenrightBig\nM·I−M2I2+M/parenleftBig\nM·I2/parenrightBig\n.\nUsingB=µ0(H+M), the transformation from the LLG to the LL equation results i n the form\n/parenleftBig\nΨ21+G/parenrightBig\n·∂M\n∂t=−γ0ΨM×Heff−γ0M×/bracketleftBig\n(α1+I)·(M×Heff)/bracketrightBig\n. (B12)\nAs mentioned before, in general the Landau-Lifshitz dampin g cannot be described by a scalar. We find that in the\ndamping term the effect of the anisotropic Ising-like dampin g is present, while the influence of the Dzyaloshinskii-Mori ya-\nlike damping is accounted for through the renormalizing qua ntityΨ.10\nAppendix C: Expressions for matrix elements\nWe provide here suitable expressions for ab initio calcu-\nlations of the matrix elements /an}b∇acketle{tripj/an}b∇acket∇i}ht. We consider thereto\nthe Bloch states |νk/an}b∇acket∇i}htin a crystal to calculate the expecta-\ntion value\n/an}b∇acketle{tripj/an}b∇acket∇i}ht=/summationdisplay\nν,ν′,k/an}b∇acketle{tνk|ri|ν′k/an}b∇acket∇i}ht/an}b∇acketle{tν′k|pj|νk/an}b∇acket∇i}htf(Eνk),(C1)\nwheref(Eνk)is the Fermi-Dirac function. The momentum\nand position operators are connected through the Ehrenfest\ntheorem, p=im\n/planckover2pi1[H,r], which we employ to obtain matrix\nelements of the position operator\n/an}b∇acketle{tν′k|r|νk/an}b∇acket∇i}ht=−i/planckover2pi1\nm/an}b∇acketle{tν′k|p|νk/an}b∇acket∇i}ht\n(Eν′k−Eνk). (C2)Substitution in equation (C1) gives\n/an}b∇acketle{tripj/an}b∇acket∇i}ht=−i/planckover2pi1\nm/summationdisplay\nν,ν′,kf(Eνk)pi\nνν′pj\nν′ν\nEνk−Eν′k\n=−i/planckover2pi1\n2m/summationdisplay\nν,ν′,kf(Eνk)−f(Eν′k)\nEνk−Eν′kpi\nνν′pj\nν′ν.(C3)\nThe double sum over quantum numbers can be further\nrewritten by separating according to interband matrix el-\nements (ν/ne}ationslash=ν′) and intraband matrix elements ( ν=ν′).\nThe latter part becomes\n/an}b∇acketle{tripj/an}b∇acket∇i}ht=−1\n2i/planckover2pi1\nm/summationdisplay\nν,k/parenleftbigg∂f\n∂E/parenrightbigg\nEνkpi\nννpj\nνν, (C4)\nwhich can be reformulated using pi\nνν=m\n/planckover2pi1(∂Eνk/∂k)ito\ngive expression (32). The expressions (C3) and (C4) are\nsimilar to Kubo linear-response expressions for elements o f\nthe conductivity tensor and are suitable for first-principl es\ncalculations within a DFT framework.\n[1]L. D. Landau and E. M. Lifshitz, Phys. Z. Sowjetunion 8,\n101 (1935).\n[2]V. G. Bar’yakhtar and B. A. Ivanov, Low Temp. Phys. 41,\n663 (2015).\n[3]W. F. Brown, Micromagnetics (Wiley, New York, 1963).\n[4]M. Kruzik and A. Prohl, SIAM Rev. 48, 439 (2006).\n[5]T. L. Gilbert, Ph.D. thesis, Illinois Institute of Technolo gy,\nChicago, 1956.\n[6]T. L. Gilbert, IEEE Trans. Magn. 40, 3443 (2004).\n[7]M. Lakshmanan, Phil. Trans. Roy. Soc. London A 369, 1280\n(2011).\n[8]V. Kamberský, Can. J. Phys. 48, 2906 (1970).\n[9]V. Korenman and R. E. Prange, Phys. Rev. B 6, 2769\n(1972).\n[10]V. G. Bar’yakhtar, Sov. Phys. JETP 60, 863 (1984).\n[11]D. A. Garanin, Phys. Rev. B 55, 3050 (1997).\n[12]E. Šimánek and B. Heinrich, Phys. Rev. B 67, 144418\n(2003).\n[13]Y. Tserkovnyak, G. A. Fiete, and B. I. Halperin, Appl.\nPhys. Lett. 84, 5234 (2004).\n[14]V. Kamberský, Phys. Rev. B 76, 134416 (2007).\n[15]A. Brataas, Y. Tserkovnyak, and G. E. W. Bauer, Phys.\nRev. Lett. 101, 037207 (2008).\n[16]I. Garate and A. MacDonald, Phys. Rev. B 79, 064403\n(2009).\n[17]M. C. Hickey and J. S. Moodera, Phys. Rev. Lett. 102,\n137601 (2009).\n[18]M. Fähnle and C. Illg, J. Phys.: Condens. Matter 23,\n493201 (2011).\n[19]V. Kamberský, Czech. J. Physics B 26, 1366 (1976).\n[20]V. G. Bar’yakhtar, V. I. Butrim, and B. A. Ivanov, JETP\nLett.98, 289 (2013).\n[21]V. G. Bar’yakhtar and A. G. Danilevich, Low Temp. Phys.\n39, 993 (2013).\n[22]J. Kuneš and V. Kamberský, Phys. Rev. B 65, 212411\n(2002).\n[23]D. Steiauf and M. Fähnle, Phys. Rev. B 72, 064450 (2005).\n[24]K. Gilmore, Y. U. Idzerda, and M. D. Stiles, Phys. Rev.\nLett.99, 027204 (2007).[25]A. A. Starikov, P. J. Kelly, A. Brataas, Y. Tserkovnyak,\nand G. E. W. Bauer, Phys. Rev. Lett. 105, 236601 (2010).\n[26]H. Ebert, S. Mankovsky, D. Ködderitzsch, and P. J. Kelly,\nPhys. Rev. Lett. 107, 066603 (2011).\n[27]A. Sakuma, J. Phys. Soc. Jpn. 81, 084701 (2012).\n[28]E. Barati, M. Cinal, D. M. Edwards, and A. Umerski, Phys.\nRev. B 90, 014420 (2014).\n[29]D. Thonig and J. Henk, New J. Phys. 16, 013032 (2014).\n[30]A. Sakuma, J. Appl. Phys. 117, 013912 (2015).\n[31]D. M. Edwards, J. Phys.: Condens. Matter 28, 086004\n(2016).\n[32]Y. Li and W. E. Bailey, Phys. Rev. Lett. 116, 117602\n(2016).\n[33]A. H. MacDonald and S. H. Vosko, J. Phys. C: Solid State\nPhys.12, 2977 (1979).\n[34]H. Eschrig and V. D. P. Servedio, J. Comput. Chem. 20,\n23 (1999).\n[35]W. Greiner, Relativistic quantum mechanics. Wave equa-\ntions (Springer, Berlin, 2000).\n[36]L. L. Foldy and S. A. Wouthuysen, Phys. Rev. 78, 29\n(1950).\n[37]R. Mondal, M. Berritta, K. Carva, and P. M. Oppeneer,\nPhys. Rev. B 91, 174415 (2015).\n[38]Note that when the momentum operator and a (vector)\nfunction are enclosed in round brackets the momentum op-\nerator acts only on this function.\n[39]M. Cinal and D. M. Edwards, Phys. Rev. B 55, 3636 (1997).\n[40]R. Mondal, M. Berritta, C. Paillard, S. Singh, B. Dkhil,\nP. M. Oppeneer, and L. Bellaiche, Phys. Rev. B 92, 100402\n(2015).\n[41]L. Allen, S. M. Barnett, and M. J. Padgett, Optical Angular\nMomentum (Institute of Physics, Bristol, 2003).\n[42]H. Bauke, S. Ahrens, C. H. Keitel, and R. Grobe, New J.\nPhys.16, 103028 (2014).\n[43]A. Raeliarijaona, S. Singh, H. Fu, and L. Bellaiche, Phys.\nRev. Lett. 110, 137205 (2013).\n[44]R. M. White, Quantum Theory of Magnetism: Magnetic\nProperties of Materials (Springer, Heidelberg, 2007).\n[45]E. M. Hankiewicz, G. Vignale, and Y. Tserkovnyak, Phys.\nRev. B 78, 020404 (2008).11\n[46]J. Ho, F. C. Khanna, and B. C. Choi, Phys. Rev. Lett. 92,\n097601 (2004).\n[47]J. Walowski, M. D. Kaufmann, B. Lenk, C. Hamann, J. Mc-\nCord, and M. Münzenberg, J. Phys. D: Appl. Phys. 41,\n164016 (2008).\n[48]H. T. Nembach, J. M. Shaw, C. T. Boone, and T. J. Silva,\nPhys. Rev. Lett. 110, 117201 (2013).\n[49]T. Weindler, H. G. Bauer, R. Islinger, B. Boehm, J.-Y.\nChauleau, and C. H. Back, Phys. Rev. Lett. 113, 237204\n(2014).\n[50]S. Mizukami, Y. Ando, and T. Miyazaki, Phys. Rev. B 66,\n104413 (2002).\n[51]Y. Tserkovnyak, A. Brataas, and G. E. W. Bauer, Phys.\nRev. Lett. 88, 117601 (2002).\n[52]S. Zhang and S. S.-L. Zhang, Phys. Rev. Lett. 102, 086601\n(2009).\n[53]S. Zhang and Z. Li, Phys. Rev. Lett. 93, 127204 (2004).\n[54]M. A. L. Marques and E. K. U. Gross, Ann. Rev. Phys.\nChem. 55, 427 (2004).\n[55]A. Widom, C. Vittoria, and S. D. Yoon, Phys. Rev. Lett.\n103, 239701 (2009).\n[56]A. Galda and V. M. Vinokur, ArXiv e-prints (2015),\narXiv:1512.05408 [cond-mat.other].\n[57]K. Gilmore, M. D. Stiles, J. Seib, D. Steiauf, and\nM. Fähnle, Phys. Rev. B 81, 174414 (2010).\n[58]E. Jué, C. K. Safeer, M. Drouard, A. Lopez, P. Balint,\nL. Buda-Prejbeanu, O. Boulle, S. Auffret, A. Schuhl,\nA. Manchon, I. M. Miron, and G. Gaudin, Nat. Mater.\n15, 272 (2016).\n[59]A. I. Akhiezer, V. G. Bar’yakhtar, and S. V. Peletminskii,\nSpin waves (North-Holland, Amsterdam, 1968).\n[60]P. M. Oppeneer, in Handbook of Magnetic Materials ,\nVol. 13, edited by K. H. J. Buschow (Elsevier, Amsterdam,\n2001) pp. 229 – 422.\n[61]D. B. Ghosh, S. K. De, P. M. Oppeneer, and M. S. S.\nBrooks, Phys. Rev. B 72, 115123 (2005).\n[62]M. Deng, H. Freyer, and H. Ebert, Solid State Commun.\n114, 365 (2000).\n[63]K. Carva, M. Battiato, D. Legut, and P. M. Oppeneer,\nPhys. Rev. B 87, 184425 (2013).\n[64]S. Essert and H. C. Schneider, Phys. Rev. B 84, 224405\n(2011).\n[65]C. Illg, M. Haag, and M. Fähnle, Phys. Rev. B 88, 214404\n(2013).\n[66]M. Haag, C. Illg, and M. Fähnle, Phys. Rev. B 90, 014417\n(2014).[67]C. Vittoria, S. D. Yoon, and A. Widom, Phys. Rev. B 81,\n014412 (2010).\n[68]A. Baral, S. Vollmar, and H. C. Schneider, Phys. Rev. B\n90, 014427 (2014).\n[69]D. A. Garanin and E. M. Chudnovsky, Phys. Rev. B 92,\n024421 (2015).\n[70]J. Sinova, T. Jungwirth, X. Liu, Y. Sasaki, J. K. Furdyna,\nW. A. Atkinson, and A. H. MacDonald, Phys. Rev. B 69,\n085209 (2004).\n[71]S. Bhattacharjee, L. Nordström, and J. Fransson, Phys.\nRev. Lett. 108, 057204 (2012).\n[72]K. Capelle, G. Vignale, and B. L. Györffy, Phys. Rev. Lett.\n87, 206403 (2001).\n[73]Z. Qian and G. Vignale, Phys. Rev. Lett. 88, 056404 (2002).\n[74]U. Nowak, O. N. Mryasov, R. Wieser, K. Guslienko, and\nR. W. Chantrell, Phys. Rev. B 72, 172410 (2005).\n[75]R. F. L. Evans, W. J. Fan, P. Chureemart, T. A. Ostler,\nM. O. A. Ellis, and R. W. Chantrell, J. Phys.: Condens.\nMatter 26, 103202 (2014).\n[76]M. O. A. Ellis, R. F. L. Evans, T. A. Ostler, J. Barker,\nU. Atxitia, O. Chubykalo-Fesenko, and R. W. Chantrell,\nLow Temp. Phys. 41, 705 (2015).\n[77]C. Herring and C. Kittel, Phys. Rev. 81, 869 (1951).\n[78]I. A. Yastremsky, P. M. Oppeneer, and B. A. Ivanov, Phys.\nRev. B 90, 024409 (2014).\n[79]V. G. Bar’yakhtar, B. A. Ivanov, and K. A. Safaryan, Solid\nState Commun. 72, 1117 (1989).\n[80]E. G. Galkina, B. A. Ivanov, and V. A. Stephanovich, J.\nMagn. Magn. Mater. 118, 373 (1993).\n[81]Y. Tserkovnyak, E. M. Hankiewicz, and G. Vignale, Phys.\nRev. B 79, 094415 (2009).\n[82]W. Wang, M. Dvornik, M.-A. Bisotti, D. Chernyshenko,\nM. Beg, M. Albert, A. Vansteenkiste, B. V. Waeyenberge,\nA. N. Kuchko, V. V. Kruglyak, and H. Fangohr, Phys. Rev.\nB92, 054430 (2015).\n[83]C. Vicario, C. Ruchert, F. Ardana-Lamas, P. M. Derlet,\nB. Tudu, J. Luning, and C. P. Hauri, Nat. Photon. 7, 720\n(2013).\n[84]N. Tesarová, P. Nemec, E. Rozkotová, J. Zemen, T. Janda,\nD. Butkovicová, F. Trojánek, K. Olejnik, V. Novák,\nP. Maly, and T. Jungwirth, Nat. Photon. 7, 492 (2013).\n[85]C.-H. Lambert, S. Mangin, B. S. D. C. S. Varaprasad,\nY. K. Takahashi, M. Hehn, M. Cinchetti, G. Malinowski,\nK. Hono, Y. Fainman, M. Aeschlimann, and E. E. Fuller-\nton, Science 345, 1403 (2014)."    },    {        "title": "1609.08250v1.Anomalous_Feedback_and_Negative_Domain_Wall_Resistance.pdf",        "content": "Anomalous Feedback and Negative Domain Wall Resistance\nRan Cheng,1Jian-Gang Zhu,2and Di Xiao1\n1Department of Physics, Carnegie Mellon University, Pittsburgh, PA 15213, USA\n2Department of Electrical and Computer Engineering,\nCarnegie Mellon University, Pittsburgh, PA 15213, USA\nMagnetic induction can be regarded as a negative feedback e\u000bect, where the motive-force opposes\nthe change of magnetic \rux that generates the motive-force. In arti\fcial electromagnetics emerging\nfrom spintronics, however, this is not necessarily the case. By studying the current-induced domain\nwall dynamics in a cylindrical nanowire, we show that the spin motive-force exerting on electrons\ncan either oppose or support the applied current that drives the domain wall. The switching into the\nanomalous feedback regime occurs when the strength of the dissipative torque \fis about twice the\nvalue of the Gilbert damping constant \u000b. The anomalous feedback manifests as a negative domain\nwall resistance, which has an analogy with the water turbine.\nI. INTRODUCTION\nMagnetization dynamics and electron transport are\ncoupled together in a reciprocal manner. Their interplay\nintroduces a variety of feedback phenomena [1{12]. For\nexample, when a background magnetization varies slowly\nover space and time, conduction electron spins will follow\nthe magnetization orientation. By doing so, the electron\nwave function acquires a geometric phase changing with\ntime, which behaves as a time-varying magnetic \rux and\nproduces a spin motive-force (SMF) according to Fara-\nday's e\u000bect [13, 14]. As a feedback, electrons driven by\nthe SMF react on the magnetization via the spin-transfer\ntorque (STT) [15{17]. This reaction leads to a modi\fed\nmagnetic damping, which hinders the magnetization dy-\nnamics that generates the SMF [8]. In parallel, when a\nmagnetic texture is driven into motion by a current, it in\nturn exerts SMFs on the electrons, resulting in a modi-\n\fed electrical resistivity that inhibits the growth of the\ndriving current [2, 3].\nSimilar feedback mechanisms also apply to magnetic\nheterostructures [11]. For example, spin current pumped\nfrom a precessing ferromagnet into an adjacent normal\nmetal experiences a back\row, which, in turn acts on the\nferromagnet through STT [18]. Because of the back\row-\ninduced STT, the e\u000bective spin-mixing conductance on\nthe interface is renormalized [19]. If the pumped spin\ncurrent is absorbed by a second ferromagnet instead of\n\rowing back, it will mediate a dynamical interlayer cou-\npling between the two ferromagnets [4, 10]. Recently, it\nhas also been shown that in the presence of the spin Hall\ne\u000bect, spin pumping and spin-back\row are connected\nthrough a feedback loop due to the combined e\u000bect of\nthe spin Hall and its reverse process [11, 12]. This novel\nfeedback mechanism, despite quadratic in the spin Hall\nangle, gives rise to a crucial nonlinear damping e\u000bect\nthat qualitatively changes the dynamical behavior of the\nmagnetization.\nIn electromagnetics, a negative feedback is ensured by\nthe Lenz law [20], which requires that the emf generated\nby Faraday's e\u000bect must oppose the change of magnetic\n\rux that causes the emf. For instance, an electric motorworks simultaneously as a dynamotor so that the induced\nemf counteracts the applied emf. As a result, the electric\ncurrent \rowing through its coil is attenuated and the\nresistance from I\u0000Vmeasurement is larger than the\nresistance of the coil. In the context of spintronics, the\ncurrent-induced magnetization dynamics plays the role\nof an electric motor, which in turn drives the current in\na similar fashion as a dynamotor. Regarding the Lenz\nlaw, one may expect an increased resistivity.\nIn this paper, however, we show that this naive expec-\ntation is not always correct. The feedback acting on the\ndriving current can also give rise to a reduced resistiv-\nity. As an example, we study the current-driven domain\nwall (DW) dynamics in a nanowire with cylindrical sym-\nmetry [21], and demonstrate that when the DW is set\ninto motion by an applied current, its reaction in the\nform of SMF can either propel or repel the electron mo-\ntion, creating either a negative or a positive DW resis-\ntance. The sign of the DW resistance re\rects the style\nof the feedback, which depends only on two phenomeno-\nlogical parameters|the Gilbert damping constant \u000band\nthe strength of the dissipative torque \f. To interpret\nsuch an anomalous feedback phenomenon, we make an\nanalogy to the working mechanism of a water turbine. It\nis observed that if a DW propels electrons along with its\nmotion, just like a rotating turbine wheel carriers water,\na negative DW resistance is produced.\nThe paper is organized as follows. In Sec. II, we es-\ntablish the general formalism. In Sec. III, we apply the\nformalism to a slowly-varying spin texture and derive the\nfeedback-induced change of dissipations. In Section IV,\nwe explore the current-driven DW dynamics in a cylindri-\ncally symmetric nanowire, and derive the DW resistance\nin terms of\u000band\f. In Section V, we provide an intuitive\ninterpretation of the anomalous feedback.\nII. DYNAMIC FEEDBACKS\nAs illustrated in Fig. 1, the interplay between local\nmagnetization and conduction electrons is resolved in\na dynamic feedback loop connecting energy dissipationarXiv:1609.08250v1  [cond-mat.mes-hall]  27 Sep 20162\nmagnetization\nelectrons\nfeedbackloopSpinmotive-forcespin-transfertorqueEJouleheatingGilbertdampingHeff\nFIG. 1. (Color online) The interplay between magnetization\nand conduction electrons generates a dynamic feedback loop\nthat connects the magnetic and electronic dissipations.\nchannels of each individual process. Under the adiabatic\nassumption [22], we regard the magnetic order parame-\nterm(r;t) as a slowly-varying vector in space and time\nso that conduction electron spins are able to adjust to\nthe magnetization direction. Given the magnetic free en-\nergyU[m(r;t)], we de\fne the e\u000bective magnetic \feld as\nHe\u000b=\u0000\u000eU=\u000em. In the di\u000busive region, nonlocal pro-\ncesses are suppressed, and the coupled dynamics of the\nsystem is described by\n(1\u0000^\u000bm\u0002)_m=\rHe\u000b\u0002m+\u001c(j); (1a)\nj=^G(m)E+\"(_m); (1b)\nwhere\ris the gyromagnetic ratio, ^ \u000bis the magnetic\ndamping tensor, ^G(m) is the conductivity tensor. The\nSTT\u001cand the motive force \"arelocal functions ofjand\n_m, respectively; they mix the dynamics of mwith that\nof electrons. Note that \u001cand\"may also depend on the\nspatial gradient of the magnetization rm. With proper\ninitial conditions, the evolution of mandjcan be solved\nby iterations of Eq. (1) on discretized spacetime grid. At\nany particular point ( r;t), one is allowed to eliminate j\n(or_m) by substituting Eq. (1b) into Eq. (1a) [or Eq. (1a)\ninto Eq. (1b)] if both \u001cand\"are local functions of the\nspace and time coordinates.\nSuch an elimination operation ful\flls the feedback loop\nillustrated in Fig. 1. For example, if E= 0, the current\njis only induced by the motion of mthrough\", which\nis simultaneously reacting on mby virtue of\u001c. In this\nregard, we can eliminate jby inserting Eq. (1b) into\nEq. (1a), which modi\fes the magnetic damping tensor\n^\u000b. In a parallel sense, if the magnetization dynamics is\nsolely driven by j(no magnetic \feld), it also generates a\nfeedback onjand renormalizes the conductivity tensor\n^G. The latter corresponds to the elimination of _mby\ninserting Eq. (1a) into Eq. (1b).\nThe dynamic feedback e\u000bects can be further elucidated\nby energy dissipations. Swapping the roles of the ther-\nmodynamic forces He\u000bandEwith the corresponding\ncurrents _mandj[23], we can rewrite Eq. (1) as\n\u0014\nHe\u000b\nE\u0015\n=\u0014\nL11L12\nL21L22\u0015\u0014\n_m\nj\u0015\n: (2)\nHere,L11is pertaining to the Gilbert damping, L12the\ncurrent-induced torque, L21the motive force, and L22the electrical resistivity. The Onsager's reciprocity rela-\ntion implies that LT\n12(m;He\u000b) =L21(\u0000m;\u0000He\u000b) [24].\nIf magnetization and current decouple, i.e.,L12= 0, the\nmagnetic free energy dissipates only through the Gilbert\ndamping _Um=\u0000He\u000b\u0001_m=\u0000L11_m2, while the elec-\ntron free energy dissipates only through the Joule heating\n_Ue=\u0000E\u0001j=\u0000L22j2. However, when the STT ( L12)\nand the motive force ( L21) are introduced, a feedback\nloop will connect the two channels of energy dissipation\nas shown in Fig. 1. For example, the magnetic dissipation\nis implemented by not only the Gilbert damping, but also\nthe Joule heating, since a magnetic precession inevitably\ndrives the electron motion that carries away the magnetic\nenergy and subsequently dissipates into heat. This mani-\nfests as a renormalization of the magnetic damping tensor\n^\u000b(thusL11). In a similar fashion, electron current can\nexcite magnetic precession, which takes away the elec-\ntron kinetic energy and damped into heat through the\nGilbert damping. As a result, the resistivity tensor L22\nis e\u000bectively modi\fed. The rates of free energy loss are\nthus\n_Um=\u0000_m\u0002\nL11\u0000L12L\u00001\n22L21\u0003_m\u0011\u0000L 11_m2;(3a)\n_Ue=\u0000j\u0002\nL22\u0000L21L\u00001\n11L12\u0003\nj\u0011\u0000L 22j2; (3b)\nwhereL11andL22are the response coe\u000ecients modi\fed\nby the dynamic feedback.\nIn general, if a system is driven by a set of Nthermody-\nnamic forces [or currents in the \\swapped\" convention,\nsee Eq. (2)] X1,X2,\u0001\u0001\u0001XN, there are Ncurrents (or\nforces)J1,J2,\u0001\u0001\u0001JNsatisfyingJa=LabXb, where the\nrepeated index is summed. By a straightforward deriva-\ntion elaborated in the Appendix, the renormalized energy\ndissipation rate through a particular channel kis\n_Uk=\u0000X2\nk\n[L\u00001]kk; (4)\nwhereL\u00001denotes the inverse of the response matrix.\nForN= 2, Eq. (4) reduces to Eq. (3). We mention\nthat Eq. (4) is quite general, where the thermodynamic\nforces (or currents) can be magnetic, electric, thermalic,\netc. However, to simplify the following discussions, we\ndo not include any thermoelectric e\u000bect, although they\nmay become important in many circumstances [9].\nIII. SPIN TEXTURE\nA. Damping\nAs mentioned earlier, a spacetime dependent magne-\ntizationm(r;t) drives local spin currents via the SMF.\nThe SMF that exerts on spin-up electrons is opposite\nto its counterpart that exerts on spin-down electrons:\n\"\"=\u0000\"#, where the spin direction is determined with\nrespect to the local and instantaneous m(r;t). Since the\nspin current is polarized along m, we only keep its \row3\ndirection in the subscript, so the i-component of the spin\ncurrent density is\njs\ni=\u0016B\ne(G\"\nik\"\"\nk\u0000G#\nik\"#\nk)\n=\u0016B\u0016hGc\nik\n2e2[(@tm\u0002@km)\u0001m+\f@tm\u0001@km];(5)\nwhereGc\nik=G\"\nik+G#\nikis theik-component of the conduc-\ntivity tensor, \u0016Bis the Bohr magneton, and the Land\u0013 e g-\nfactor of electrons is taken to be 2. The term proportional\nto\fis the dissipative SMF [25, 26], which is the recipro-\ncal e\u000bect of the dissipative STT; \fis a phenomenological\nconstant that characterizes the relative strengths of the\ndissipative contribution.\nAs a feedback e\u000bect, the locally pumped spin current\nacts on the magnetization through the STT. De\fne the\nelectron velocity \feld as u=js=Ms, whereMsis the\nsaturation magnetization. Then the STT consists of two\northogonal terms [17]\n\u001c= (ui@i)m\u0000\fm\u0002(ui@i)m: (6)\nInserting Eq. (5) into Eq. (6) yields a damping term that\nrenormalizes the original Gilbert damping. The Landau-\nLifshitz-Gilbert (LLG) equation becomes\n@tm=\rHe\u000b\u0002m+m\u0002(D\u0001@tm); (7)\nwhereDis the damping tensor that can be decomposed\ninto ^D=^D0+^D0, where ^D0=\u000b0^I is the original Gilbert\ndamping, and the feedback correction is\n^D0=\u0011[^S+^A] (8)\nwith\u0011=\u0016B\u0016h=(2e2Ms). In Eq. (8), the element of the\nsymmetric part is\nSab=Gc\nik[(m\u0002@im)a(m\u0002@km)b\n\u0000\f2(@im)a(@km)b\u0003\n; (9)\nand that of the antisymmetric part is\nAab=\fGc\nik[(@im)a(m\u0002@km)b\u0000(a*)b)];(10)\nwhere summations over repeated indices are assumed.\nIn matrix form, the feedback correction can be written\nas^D0=\u0011TSTT\nTSMF=\u0011Gc\nik[(m\u0002@im) +\f@im]\n[(m\u0002@km)\u0000\f@km]. This suggestive form interprets\nthe feedback loop as two combined processes: a dynamic\nmpumps a local spin current, which in turn acts on m\nitself, implementing the feedback e\u000bect. When \f!0,\nEq. (8) reduces to Eq. (11) in Ref. [8].\nHere is an important remark. Although equations (8){\n(10) are similar to the results derived in Ref. [6, 7], the un-\nderlying physics is fundamentally distinct. In Ref. [6, 7],\nthe damping renormalization is attributed to the current-\ninduced noise, and thermal \ructuation is the primary\nstimulus. Consequently, the coe\u000ecient of the damping\ntensor depends on temperature. By contrast, our results\nare valid even at zero temperature.B. Resistance\nWhen closing the feedback loop the other way around,\ni.e., currentSTT\u0000\u0000\u0000! LLGSMF\u0000\u0000\u0000! current, we will obtain the\nfeedback modi\fcation of the resistance. To perform this\ncalculation, we start with the LLG equation\n@tm=\rHe\u000b\u0002m+\u000bm\u0002@tm\n+ (ui@i)m\u0000\fm\u0002(ui@i)m;(11)\nthen combine all @tmterms so that\n@tm=\r\n1 +\u000b2[He\u000b\u0002m+\u000bm\u0002(He\u000b\u0002m)]\n+1 +\u000b\f\n1 +\u000b2(ui@i)m+\u000b\u0000\f\n1 +\u000b2m\u0002(ui@i)m;(12)\nwhereu=P\u0016Bjc=(eMs) withP= (nF\n\"\u0000nF\n#)=(nF\n\"+nF\n#)\nthe polarization of carrier density at the Fermi level. The\ncharge current density is now driven by both the SMF\nand an external electric \feld E,\njc\ni=jc(E)\ni+jc(smf)\ni =Gc\nikEk\n+Gs\nik\u0016h\n2e[(@tm\u0002@km)\u0001m+\f(@tm\u0001@km)];(13)\nwhereGs\nik=G\"\nik\u0000G#\nikis theik-component of the spin\nconductivity. It should not be confused that for the SMF-\ninduced electron \row, the spin current depends on the\ncharge conductivity [see Eq. (5)], whereas the charge cur-\nrent depends on the spin conductivity [8].\nWhen substituting the LLG equation into the SMF to\neliminate@tm, terms involving He\u000bresult in nonlinear\ndependence between jcandE, which in principle should\nbe solved numerically . Nevertheless, those terms can be\ndiscarded in many special cases. For instance, if the mag-\nnetic free energy is invariant under a particular motion\nofm, we haveHe\u000bkmat all times, thus those terms\nvanish identically. In such circumstances, Eis linear in\njc, and the feedback can be expressed analytically as a\nrenormalization of the resistivity tensor. We will restrict\nthe following discussion to this category.\nTo proceed, we insert Eq. (12) into Eq. (13) and make\nthe approximation that He\u000bkm. After some manipula-\ntions, we obtain\njc\ni+Gs\nikRk`jc\n`=Gc\nikEk; (14)\nwhere the element of the feedback matrix ^Ris\nRk`=P\u0016B\u0016h\n2e2Ms\u0014\u000b(1\u0000\f2)\u00002\f\n1 +\u000b2@km\u0001@`m\n+1 + 2\u000b\f\u0000\f2\n1 +\u000b2(@km\u0002@`m)\u0001m\u0015\n\u0011P\u0016B\u0016h\n2e2Ms[f(\u000b;\f)gk`+h(\u000b;\f)\nk`]: (15)\nThe symmetric part of ^Ris proportional to the quantum\nmetricgk`=@km\u0001@`m[27], while the antisymmetric4\npart is proportional to the Berry curvature \n k`= (@km\u0002\n@`m)\u0001m. To appreciate the physical meaning of ^R, we\nturn to the resistivity by multiplying [ ^Gc]\u00001on Eq. (14),\nwhich givesE= ^\u001ajc. The resistivity tensor is\n^\u001a= ^\u001a0(1 + ^Gs^R); (16)\nwhere ^\u001a0= [^Gc]\u00001is the bare resistivity tensor without\nfeedback, and ^Gs^Ris the feedback-induced renormaliza-\ntion. Depending on the spatial pattern of m(r;t) and\nthe relative ratio between \u000band\f, a particular element\nof^Rcan be either positive or negative.\nIV. DOMAIN WALL RESISTANCE\nTransverse DWs in thin cylindrical magnetic nanowires\nhave two salient features that arouse recent interest [21].\n(1) The inner structure of these DWs remain unchanged\nduring their propagation, thus our assumption He\u000bkm\nis respected at all times. (2) These DWs are massless and\nthe critical currents required to initiate their motions are\nzero. Because of the latter property, the DW resistance\npractically measurable from I-V curve solely stems from\nthe dynamic feedback e\u000bect, whereas the conventional\ntheory based on stationary DW con\fgurations [28, 29] is\nincomplete.\nSuch a DW is a one-dimensional soliton characterized\nby two spherical angles \u0012and\u001especifying the local ori-\nentation of the magnetization\n\u0012(x;t) = 2 arctan e[x\u0000xc(t)]=w; (17a)\n\u001e(x;t) =\u001e(t); (17b)\nwherexc(t) is the center of the DW, and wis the width\nof the DW (supposed to be much larger than the lat-\ntice spacing). In one dimensions, the antisymmetric part\nof Eq. (15) vanishes, ^\n = 0; ^Rhas only one compo-\nnent, andGs=PGc. In this case, Eq. (14) reduces\nto\u001ajc=E, where\u001a=\u0002\n\u001a0+P2\u0011f(\u000b;\f)j@xmj2\u0003\nwith\n\u0011=\u0016B\u0016h=(2e2Ms). The pro\fle function given by Eq. (17)\nyieldsj@xmj2= 1=[w2cosh2(x=w)]. By integrating \u001a\noverx2(\u00001;+1), we obtain the total resistance\nR=R0+\u000b(1\u0000\f2)\u00002\f\n1 +\u000b2\u0014P2\u0016B\u0016h\ne2Ms\u00151\nAw; (18)\nwhereAis the area of the cross section of the cylindrical\nnanowire. The second term of Eq. (18) is ascribed to the\ndynamic feedback e\u000bect, which scales inversely with w.\nSinceP2\u0016B\u0016h=(e2MsAw)>0, the sign of this correction\nis only determined by f(\u000b;\f) = [\u000b(1\u0000\f2)\u00002\f]=(1+\u000b2).\nConsider\u000b\u001c1 and\f\u001c1, thenf(\u000b;\f)\u0019\u000b\u00002\f. As\na result, the dynamical correction of the DW resistance\nis positive for \f <\u000b= 2, and negative for \f >\u000b= 2. Using\ntypical material parameters of permalloy, the feedback-\ninduced resistance of a 100nm wide DW with A\u001830nm2\nis in the range of 10\u00005to 10\u00004\n.A negative DW resistance indicates that the feedback\nexerting on the electrons by the DW is positive. To be\nspeci\fc, when the DW is set into motion by a current,\nit propels the electrons in their direction of motion, thus\nreducing the electrical resistance. In terms of the Lenz\nlaw, this means that the SMF induction enhances the\n\rux (geometric phase) change by making the electrons\nmore mobile, contrasting to the normal case where the\nSMF opposes the \rux change. It worths emphasizing that\nsuch an anomalous situation is unique to cylindrically\nsymmetric nanowires, while nanostrips are not applicable\nas the approximation He\u000bkmis invalid.\nV. DISCUSSION\nDi\u000berent from the static DW resistance [28, 29] that\nis absorbed by R0in our theory, the feedback-induced\nDW resistance is associated with the DW dynamics. The\npeculiarity of using a cylindrical nanowire is that the\nthreshold current to initiate the DW dynamics is tech-\nnically zero [21]. So, what we mean by DW resistance\nrefers to the di\u000berence in Rwhen comparing the results\nofI\u0000Vmeasurements between a freely moving DW and\na pinned DW on identical cylindrical nanowires under\nthe same voltage drop.\nThe key to understand why such di\u000berence is negative\nfor\f > 2\u000blies in the reaction SMF that propels the\nelectrons along the direction of the DW motion. It con-\ntradicts the case of an electric motor where the back emf\ninduction opposes the driving current and raises the sys-\ntem resistance. At the same time, we need to justify that\nsuch an anomalous feedback e\u000bect does not violate any\nfundamental physical law. To this end, we make a heuris-\ntic analogy between the current-induced DW dynamics in\ncylindrical nanowires and a water turbine with constant\npump, where the rotating wheel represents our moving\nDW. In fact, the linear velocity of the DW is proportional\nto its angular velocity, and their ratio is independent of\nthe current [21]. Therefore, it is equivalent to character-\nize the DW motion by its angular velocity, which is more\ncurrent without turbineI!terminal velocity\npumpI!!!\nFIG. 2. (Color online) Comparison between an electric motor\ndriven by a constant voltage and a water turbine driven by\na constant pump. The overall current Ias a function of the\nangular velocity !signals the nature of the feedback e\u000bect.5\ntransparent to compare with a turbine wheel. Drawing\nsuch an analogy is to show that a negative resistance is\nnot surprising, while the analogy itself is by no means\nexact.\nAs schematically illustrated in Fig. 2, the working\nmechanism of a water turbine is compared with an elec-\ntric motor. They have one thing in common: the steady-\nstate angular velocity !increases with decreasing load.\nSo by controlling the load, one can tune !in both cases.\nHowever, the feedback mechanisms in the two cases are\nremarkably di\u000berent. In an electric motor, if one raises !\nby reducing the load, the back emf induced by Faraday's\ne\u000bect will get larger, which counteracts the applied volt-\nage more strongly and reduces the overall current. Con-\nsequently, the resistance read o\u000b from the I\u0000Vcurve\nincreases. This realizes the usual negative feedback e\u000bect\nand respects the Lenz law since Idecreases when the mo-\ntor rotates faster. In sharp contrast, if one increases !of\na water turbine, the water \rows more easily in the pipe as\nthe turbine blades less block the water. As a result, the\n\\resistance\" of the entire turbine system appears to be\nsmaller. This feature marks an anomalous feedback: the\nwater current increases when the turbine rotates faster.\nIgnoring the mass and friction of the wheel, the max-\nimum achievable angular velocity (in the limit of zero\nload), hence the maximum water current, is set by the\nwater \row in the absence of the turbine. Now go back\nto our DW dynamics: reducing the DW pinning corre-\nsponds to reducing the load on a water turbine, which\nenhances the driving current in just a similar way as the\nenhancement of water \row.\nFinally, we comment on why the anomalous feedback\nis more likely to occur in one dimensions. Since \u000b;\f\u001c1,\nthe second term of Eq. (15) dominates the \frst term, andits coe\u000ecient is unlikely to \rip sign unless \fis greater\nthan unity. However, in higher dimensions, the second\nterm always exist, so the \frst term that could lead to the\nanomaly is suppressed. Although the second term only\nrefers to the transverse components of the transport, the\nboundary conditions on the edges can considerably com-\nplicate the e\u000bective value of the longitudinal component\nand obscure the observation.\nACKNOWLEDGMENTS\nThe authors are grateful to A. Brataas for insightful\ndiscussions. We also thank J. Xiao and M. W. Daniels\nfor useful comments. This study was supported by the\nU.S. Department of Energy, O\u000ece of BES, Division of\nMSE under Grant No. DE-SC0012509.\nAppendix: Derivation of Eq. (4)\nIf all channels are in open circuit conditions except for\na particular channel k, only the current Jkis nonzero\neven in the presence of all Nthermodynamic forces\nX1\u0001\u0001\u0001XN. The energy dissipation rate is then\n_Uk=\u0000JkXk=\u0000LkkX2\nk\u0000NX\ni6=kLkiXiXk; (A.1)\nwhere the \frst term is the usual dissipation term. We\nnow eliminate those cross terms XiXk(i6=k) in terms\nofX2\nk. Since all currents but Jkare zero, multiplying Xk\nonJi=LijXjwithi6=kgives:\n2\n6666666664L11L12\u0001\u0001\u0001L1;k\u00001L1;k+1\u0001\u0001\u0001L1N\n.....................\nLk\u00001;1Lk\u00001;2\u0001\u0001\u0001Lk\u00001;k\u00001Lk\u00001;k+1\u0001\u0001\u0001Lk\u00001;N\nLk+1;1Lk+1;2\u0001\u0001\u0001Lk+1;k\u00001Lk+1;k+1\u0001\u0001\u0001Lk+1;N\n.....................\nLN1LN2\u0001\u0001\u0001LN;k\u00001LN;k+1\u0001\u0001\u0001LNN3\n77777777752\n6666666664X1Xk\n...\nXk\u00001Xk\nXk+1Xk\n...\nXNXk3\n7777777775=\u0000X2\nk2\n6666666664L1k\n...\nLk\u00001;k\nLk+1;k\n...\nLNk3\n7777777775: (A.2)\nThe coe\u000ecient matrix consists of the remaining elements\nofLafter taking away the k-th row and the k-th column.\nRegarding the Cramer's rule, the cross term is solved as\nXiXk=X2\nkAki\nAkkfori6=k, whereAijis thei;j-th alge-\nbraic cofactor (minor) of L. Inserting this relation into\nEq. (A.1), and considering the identity of row expansiondet[L] =PN\ni=1LkiAki, we \fnally obtain\n_Uk=\u0000\u0014\nLkk+det[L]\u0000LkkAkk\nAkk\u0015\nX2\nk=\u0000X2\nk\n[L\u00001]kk;\nwhich proves Eq. (4).\n[1] G. E. Volovik, J. Phys. C 20, L83 (1987). [2] J. Zang, M. Mostovoy, J. H. Han, and N. Nagaosa, Phys.\nRev. Lett. 107, 136804 (2011).6\n[3] T. Schulz et al. , Nat. Phys. 8, 301 (2012).\n[4] B. Heinrich, Y. Tserkovnyak, G. Woltersdorf, A. Brataas,\nR. Urban, and G. E. W. Bauer, Phys. Rev. Lett. 90,\n187601 (2003).\n[5] J. Xiao, A. Zangwill, and M. D. Stiles, Phys. Rev. B 70,\n172405 (2004).\n[6] J\u001crn Foros, A. Brataas, Y. Tserkovnyak, and G. E. W.\nBauer, Phys. Rev. B 78, 140402(R) (2008).\n[7] C. H. Wong and Y. Tserkovnyak, Phys. Rev. B 80,\n184411 (2009).\n[8] S. Zhang and S. S.-L. Zhang, Phys. Rev. Lett. 102,\n086601 (2009).\n[9] G. E. W. Bauer, S. Bretzel, A. Brataas, and Y.\nTserkovnyak, Phys. Rev. B 81, 024427 (2010).\n[10] H. Skarsv\u0017 ag, A. Kapelrud, and A. Brataas, Phys. Rev. B\n90, 094418 (2014).\n[11] R. Cheng, J.-G. Zhu, and D. Xiao, Phys. Rev. Lett., 117,\n097202 (2016).\n[12] R. Cheng, D. Xiao, and A. Brataas, Phys. Rev. Lett.\n116, 207603 (2016).\n[13] S. E. Barnes and S. Maekawa, Phys. Rev. Lett. 98,\n246601 (2007).\n[14] S. A. Yang et al. , Phys. Rev. Lett. 102, 067201 (2009);\nS. A. Yang et al. , Phys. Rev. B 82, 054410 (2010).\n[15] J. Slonczewki, J. Magn. Magn. Mater. 159, L1 (1996);\nL. Berger, Phys. Rev. B 54, 9353 (1996).\n[16] Y. B. Bazaliy, B. A. Jones, and S. -C. Zhang, Phys. Rev.\nB57, R3213 (1998).\n[17] S. Zhang and Z. Li, Phys. Rev. Lett. 93, 127204 (2004).[18] Y. Tserkovnyak and A. Brataas, Phys. Rev. B 66, 224403\n(2002); X. Wang, G. E. W. Bauer, B. J. van Wees, A.\nBrataas, Y. Tserkovnyak, Phys. Rev. Lett 97, 216602\n(2006); H.-J. Jiao and G. E. W. Bauer, Phys. Rev. Lett.\n110, 217602 (2013).\n[19] Y. Zhou, H.-J. Jiao, Y.-T. Chen, G. E. W. Bauer, and J.\nXiao, Phys. Rev. B 88, 184403 (2013).\n[20] D. J. Gri\u000eths, Introduction to Electrodynamics , 3rd. Ed.,\nUpper Saddle River, NJ: prentice Hall, 1999.\n[21] M. Yan, A. K\u0013 akay, S. Gliga, and R. Hertel, Phys. Rev.\nLett. 104, 057201 (2010).\n[22] D. Xiao, M.-C. Chang, and Q. Niu, Rev. Mod. Phys. 82,\n1959 (2010).\n[23] The swapped convention is easy to describe current-\ninduced torques and SMFs. For voltage-induced torques,\nhowever, the standard convention is more suitable.\n[24] E. M. Lifshitz and L. P. Pitaevskii, Statistical Physics ,\nCourse of Theo. Phys. Vol. 9 (Pergamon, Oxford, 1980),\nPart 2.\n[25] R. A. Duine, Phys. Rev. B 77, 014409 (2008).\n[26] Y. Tserkovnyak and M .Mecklenburg, Phys. Rev. B 77,\n134407 (2008).\n[27] J. P. Provost and G. Vallee, Commun. Math. Phys. 76,\n289 (1980).\n[28] P. M. Levy and S. Zhang, Phys. Rev. Lett. 79, 5110\n(1997).\n[29] R. P. van Gorkom, A. Brataas, and G. E. W. Bauer,\nPhys. Rev. Lett. 83, 4401 (1999)."    },    {        "title": "1610.01072v2.Magnetomechanical_coupling_and_ferromagnetic_resonance_in_magnetic_nanoparticles.pdf",        "content": "Magnetomechanical coupling and ferromagnetic resonance in magnetic nanoparticles\nHedyeh Keshtgar,1Simon Streib,2Akashdeep Kamra,3Yaroslav M. Blanter,2and Gerrit E. W. Bauer2, 4\n1Institute for Advanced Studies in Basic Science, 45195 Zanjan, Iran\n2Kavli Institute of NanoScience, Delft University of Technology, Lorentzweg 1, 2628 CJ Delft, The Netherlands\n3Fachbereich Physik, Universität Konstanz, D-78457 Konstanz, Germany\n4Institute for Materials Research and WPI-AIMR, Tohoku University, Sendai 980-8577, Japan\n(Dated: March 20, 2017)\nWe address the theory of the coupled lattice and magnetization dynamics of freely suspended\nsingle-domain nanoparticles. Magnetic anisotropy generates low-frequency satellite peaks in the\nmicrowave absorption spectrum and a blueshift of the ferromagnetic resonance (FMR) frequency.\nThe low-frequency resonances are very sharp with maxima exceeding that of the FMR, because\ntheir magnetic and mechanical precessions are locked, thereby suppressing the effective Gilbert\ndamping. Magnetic nanoparticles can operate as nearly ideal motors that convert electromagnetic\ninto mechanical energy. The Barnett damping term is essential for obtaining physically meaningful\nresults.\nPACS numbers: 75.10.Hk, 75.80.+q , 75.75.Jn , 76.50.+g\nI. INTRODUCTION\nMagnetic nanoparticles (nanomagnets) are of funda-\nmental interest in physics by forming a link between the\natomic and macroscopic world. Their practical impor-\ntance stems from the tunability of their magnetic prop-\nerties [1], which is employed in patterned media for high\ndensity magnetic data storage applications [2] as well as\nin biomedicine and biotechnology [3–6]. Superparamag-\nnetic particles are used for diagnostics, stirring of liq-\nuids, and magnetic tweezers [7]. The heat generated by\nthe magnetization dynamics under resonance conditions\nis employed for hyperthermia cancer treatment [8–10].\nMolecular based magnets can cross the border from the\nclassical into the quantum regime [11, 12]. The magnetic\nproperties of individual atomic clusters can be studied by\nmolecular beam techniques [13–15].\nEinstein, de Haas, and Barnett [16, 17] established the\nequivalence of magnetic and mechanical angular momen-\ntum of electrons by demonstrating the coupling between\nmagnetization and global rotations. Spin and lattice are\nalso coupled by magnetic anisotropy, induced either by\ndipolar forces or crystalline fields. A quite different in-\nteraction channel is the magnetoelastic coupling between\nlattice waves (phonons) and spin waves (magnons) with\nfinite wave vectors. This magnetoelastic coupling be-\ntween the magnetic order and the underlying crystalline\nlattice has been explored half a century ago by Kittel [18]\nand Comstock [19, 20]. The coupling between spin and\nlattice causes spin relaxation including Gilbert damping\nof the magnetization dynamics [21, 22].\n“Spin mechanics” of thin films and nanostructures en-\ncompasses many phenomena such as the actuation of the\nmagnetization dynamics by ultrasound [23–25], the dy-\nnamics of ferromagnetic cantilevers [26–28], spin current-\ninduced mechanical torques [22, 29], and rotating mag-\nnetic nanostructures [30]. The Barnett effect by rotation\nhas been observed experimentally by nuclear magnetic\nresonance [31]. The coupled dynamics of small magneticspheres has been studied theoretically by Usov and Li-\nubimov [32] and Rusconi and Romero-Isart [33] in clas-\nsical and quantum mechanical regimes, respectively. A\nprecessing single-domain ferromagnetic needle is a sen-\nsitive magnetometer [34], while a diamagnetically levi-\ntated nanomagnet can serve as a sensitive force and in-\nertial sensor [35]. A stabilization of the quantum spin of\nmolecular magnets by coupling to a cantilever has been\npredicted [36, 37] and observed recently [38].\nHere we formulate the dynamics of rigid and single-\ndomain magnetic nanoparticles with emphasis on the\neffects of magnetic anisotropy and shape. We derive\nthe equations of motion of the macrospin and macro-\nlattice vectors that are coupled by magnetic anisotropy\nand Gilbert damping. We obtain the normal modes and\nmicrowave absorption spectra in terms of the linear re-\nsponse to ac magnetic fields. We demonstrate remark-\nable changes in the normal modes of motion that can be\nexcited by microwaves. We predict microwave-activated\nnearly undamped mechanical precession. Anisotropic\nmagnetic nanoparticles are therefore suitable for stud-\nies of non-linearities, chaos, and macroscopic quantum\neffects.\nIn Sec. II we introduce the model of the nanomag-\nnet and give an expression for its energy. In Sec. III we\ndiscuss Hamilton’s equation of motion for the magneti-\nzation of a freely rotating particle, which is identical to\nthe Landau-Lifshitz equation. We then derive the cou-\npled equations of motion of magnetization and lattice in\nSec. IV. Our results for the easy-axis and easy-plane con-\nfigurations are presented in Secs. V and VI. We discuss\nand summarize our results in Secs. VII and VIII. In the\nAppendices A to D we present additional technical de-\ntails and derivations.arXiv:1610.01072v2  [cond-mat.mes-hall]  28 Apr 20172\nz\ny\nxnyb\nxbzb\nθ(a)\nn\nm\nm\nn(b)\n(c)\nFigure 1. (a) Laboratory frame ( x,y,z) and (moving) body\nframe (xb,yb,zb) of a nanomagnet with principal axis nalong\nthezb-axis. The directions of nand magnetization mare\nshown for (b) oblate and (c) prolate spheroids with dipolar\nmagnetic anisotropy.\nII. MACROSPIN MODEL\nWe consider a small isolated nanomagnet that justi-\nfies the macrospin and macrolattice approximations, in\nwhich all internal motion is adiabatically decoupled from\nthe macroscopic degrees of freedom, rendering the mag-\nnetoelastic coupling irrelevant.\nWe focus on non-spherical nanoparticles with mass\ndensity\u001a(r)and tensor of inertia\nI=Z\nd3r\u001a(r)\u0002\n(r\u0001r)^1\u0000r\nr\u0003\n;(2.1)\nwhere ^1is the 3x3 unit matrix. The mechanical proper-\nties of an arbitrarily shaped rigid particle is identical to\nthat of an ellipsoid with a surface that in a coordinate\nsystem defined along the symmetry axes (in which Iis\ndiagonal) reads\n\u0010x\nc\u00112\n+\u0010y\nb\u00112\n+\u0010z\na\u00112\n= 1; (2.2)\nwherea;b;care the shape parameters (principal radii).\nThe volume is V= 4\u0019abc= 3, total mass Q=\u001aV,\nand principal moments of inertia I1=Q\u0000\na2+b2\u0001\n=5;\nI2=Q\u0000\na2+c2\u0001\n=5;I3=Q\u0000\nb2+c2\u0001\n=5. We focus in the\nfollowing on prolate (a > b =c)and oblate (a < b =c)\nspheroids, because this allows analytic solutions of the\ndynamics close to the minimum energy state.\nWe assume that the particle is smaller than the crit-\nical sizedcr\u001836pAKA=(\u00160M2\ns)for magnetic domain\nformation [39], where Ais the exchange constant, KA\nthe anisotropy constant, Msthe saturation magnetiza-\ntion, and\u00160= 4\u0019\u000210\u00007N A\u00002the vacuum permeability.\nFor strong ferromagnets these parameters are typically in\nthe rangeA2[5;30] pJ m\u00001,KA2[10;20000] kJ m\u00003,\nMs2[0:4;1:7] MA m\u00001, leading to dcr2[1;500] nm\n[39]. For a spherical particle of radius Rwith sound\nvelocityv, the lowest phonon mode frequency is approx-imately [40]\n!ph\n2\u0019\u0019v\n4R= 0:25\u0012v=(103m\ns)\nR=nm\u0013\nTHz;(2.3)\nwhile the lowest magnon mode (for bulk dispersion rela-\ntion ~!mag=Dk2)\n!mag\n2\u0019\u0019\u0019D\n8~R2= 0:6\u0012D=(meV nm2)\nR2=nm2\u0013\nTHz;(2.4)\nwhere the spin wave stiffness D= 2g\u0016BA=Msis typically\nof the order meV nm2[39], e.g.,D= 2:81 meV nm2for\niron [41]. We may disregard spin and lattice waves and\nthe effects of their thermal fluctuations when the first\nexcited modes are at sufficiently higher frequencies than\nthat of the total motion (the latter is typically in the\nGHz range) and therefore adiabatically decoupled [33,\n40], i.e. the macrospin and macrolattice model is valid.\nThermal fluctuations of the magnetization with respect\nto the lattice do not play an important role below the\nblocking temperature, TB\u0018KAV=(25kB)[42], where\nkBis the Boltzmann constant. For kBT\u001cVMs\u00160H0,\nthermal fluctuations of the magnetization with respect to\nthe static external magnetic field H0are suppressed.\nUnder the conditions stipulated above the classical dy-\nnamics (disregarding translations of the center of mass)\nis described in terms of the magnetization vector M=\nMsm(withjmj= 1) and the three Euler angles ( \u0012;\u001e; )\nof the crystal orientation direction in terms of the axis\nn(\u0012;\u001e)and a rotation angle  around it (see Appendix A\nfor details). The total energy can be split up into several\ncontributions,\nE=ET+EZ+ED+EK: (2.5)\nET=1\n2\nTI\nis the kinetic energy of the rotational mo-\ntion of the nanomagnet in terms of the angular frequency\nvector \n.EZ=\u0000\u00160VM\u0001Hextis the Zeeman energy\nin a magnetic field Hext.ED=1\n2\u00160VMTDMis the\nmagnetostatic self-energy with particle shape-dependent\ndemagnetization tensor D.EK=K1V(m\u0002n)2is the\n(uniaxial) magnetocrystalline anisotropy energy, assum-\ning that the easy axis is along n, andK1is the material-\ndependent anisotropy constant.\nWe consider an inertial lab frame with origin at the\ncenter of mass and a moving frame with axes fixed in the\nbody. The lab frame is spanned by basis vectors ex,ey,\nez, and the body frame by basis vectors exb,eyb,ezb(see\nFig. 1). The body axes are taken to be the principal axes\nthat diagonalize the tensor of inertia. For spheroids with\nb=cthe inertia and demagnetizing tensors in the body\nframe have the form\nIb=0\n@I?0 0\n0I?0\n0 0I31\nA;Db=0\n@D?0 0\n0D?0\n0 0D31\nA;(2.6)\nwithI?=Q\u0000\na2+b2\u0001\n=5andI3= 2Qb2=5; the elements\nD?andD3for magnetic spheroids are given in [43]. The3\nparticle shape enters the equations of motion via I?,I3,\nand the difference D3\u0000D?, the latter reduces to \u00001=2\nfor a thin needle and 1for a thin disk. When\nE?\u0000Ek=KAV=K1V\u00001\n2\u00160VM2\ns(D3\u0000D?)(2.7)\nis larger than zero, the configuration mknis sta-\nble (“easy axis”); otherwise m?n(“easy plane”).\nThe anisotropy constant KAincludes both magnetocrys-\ntalline and shape anisotropy.\nIII. LANDAU-LIFSHITZ EQUATION\nFor reference we rederive here the classical equation of\nmotion of the magnetization. The magnetization of the\nparticle at rest is related to the angular momentum S=\n\u0000VMsm=\r, where\r= 1:76\u00021011s\u00001T\u00001is (minus) the\ngyromagnetic ratio of the electron. The Poisson bracket\nrelations for angular momentum are\nfS\u000b;S\fg=\u000f\u000b\f\rS\r: (3.1)\nHamilton’s equation of motion reads\nd\ndtS=fS;Hg; (3.2)\nwhereH\u0011Eis the Hamiltonian. We consider a general\nmodel Hamiltonian of a single macrospin coupled to the\nmacrolattice,\nH=X\ni;j;k2N0aijk(n;L)Si\nxSj\nySk\nz; (3.3)\nwhere the coefficients aijk(n;L)may depend on the ori-\nentation nof the lattice and its mechanical angular\nmomentum L=I\n. Since lattice and magnetization\nare different degrees of freedom, the Poisson brackets\nfn;Sg=fL;Sg= 0and thereforefaijk(n;L);Sg= 0.\nWe derive in Appendix B\nfS;Hg=X\ni;j;k2N0aijk(n;L)0\n@iSi\u00001\nxSj\nySk\nz\njSi\nxSj\u00001\nySk\nz\nkSi\nxSj\nySk\u00001\nz1\nA\u0002S;(3.4)\nwhich is the Landau-Lifshitz equation [44],\nd\ndtS=rSHjn;L=const:\u0002S: (3.5)\nIn accordance with Eq. (3.4), the gradient in Eq. (3.5)\nhas to be evaluated for constant nandL.\nThe rotational kinetic energy ET=1\n2\nTI\ndoes\nnot contribute to this equation of motion directly since\nfS;ETg= 0. However, ETis crucial when considering\nthe energy of the nanomagnet under the constraint of\nconserved total angular momentum J=L+S. Minimiz-\ning the energy of the nanomagnet under the constraint\nof constant Jis equivalent to\n~He\u000b=\u00001\n\u00160VMsrmE\f\f\f\f\nJ=const:= 0;(3.6)where the rotational kinetic energy ETcontributes the\nBarnett field\nHB=\u00001\n\u00160VMsrmET\f\f\f\f\nJ=const:=\u0000\n\r\u00160;(3.7)\nwhich gives rise to the Barnett effect (magnetization by\nrotation)[17]. AlthoughtheBarnettfieldappearsherein\nthe effective field ~He\u000bwhen minimizing the energy, it is\nnot part of the effective field He\u000bof the Landau-Lifshitz\nequation,\nHe\u000b=\u00001\n\u00160VMsrmE\f\f\f\f\nn;L=const:;(3.8)\nwhere Lis kept constant instead of J. In the Landau-\nLifshitz-Gilbert equation in the laboratory frame the\nBarnett effect operates by modifying the Gilbert damp-\ning torque as shown below.\nIV. EQUATIONS OF MOTION\nWe now derive the coupled equations of motion of the\nmagnetization mand the Euler angles ( \u001e;\u0012; ). The\nmagnetization dynamics is described by the Landau-\nLifshitz-Gilbert equation [21, 44]\n_m=\u0000\r\u00160m\u0002He\u000b+\u001c(\u000b)\nm; (4.1)\nwhere the effective magnetic field Eq. (3.8) follows from\nthe energy Eq. (2.5),\nHe\u000b=Hext+HD+HK; (4.2)\nand\u001c(\u000b)\nmis the (Gilbert) damping torque. The external\nmagnetic field Hextis the only source of angular mo-\nmentum; all other torques acting on the total angular\nmomentum J=L\u0000VMsm=\rcancel. From\n_J=\u00160VMsm\u0002Hext; (4.3)\nwe obtain the mechanical torque as time-derivative of the\nmechanical angular momentum, which leads to Newton’s\nLaw\n_L=VMs\n\r_m+\u00160VMsm\u0002Hext:(4.4)\nThe dissipation parameterized by the Gilbert constant\n[21] damps the relative motion of magnetization and lat-\ntice. In the body frame of the lattice [30]\n\u001c(\u000b)\nm;b=\u000bmb\u0002_mb; (4.5)\nwherethesubscript bindicatesvectorsinthebodyframe.\nTransformed into the lab frame (see Appendix A)\n\u001c(\u000b)\nm=\u000b[m\u0002_m+m\u0002(m\u0002\n)]:(4.6)\nThis torque is an angular momentum current that flows\nfrom the magnet into lattice [22]. Angular momentum4\n2200 2300 2400 2500\nω/(2π) [MHz]01234567−ωImχxx[1013s−1]\nQf= 3900\nQf= 2900\n40 45 50 55\nω/(2π) [GHz]0.00.51.01.52.0−ωImχxx[1013s−1]\nQf= 50\nFigure 2. Low- and high-frequency resonances in the FMR\nspectrum of an Fe nanosphere of 2 nmdiameter in a static\nmagnetic field of 0:65 Twith Gilbert damping constant \u000b=\n0:01; quality factor Qf=!=(2\u0011).\nis conserved, but the generated heat is assumed to ulti-\nmately be radiated away. In vacuum there is no direct\ndissipation of the rigid mechanical dynamics.\nThe Barnett field \u00160HB=\u0000\n=\renters in the lab\nframe only in the damping term \u001c(\u000b)\nm. To leading order\nin\u000b\n_m\u0019\u0000\r\u00160m\u0002He\u000b\u0000\u000b\r\u0016 0m\u0002[m\u0002(He\u000b+HB)]+O(\u000b2):\n(4.7)\nThe contribution of HBin the damping term causes the\nBarnett effect [17]. We find that this Barnett damping is\nvery significant for the coupled dynamics even though no\nfast lattice rotation is enforced: without Barnett damp-\ning the FMR absorption of the low-frequency modes de-\nscribed below would become negative.\nV. EASY-AXIS CONFIGURATION\nWe first consider an easy-axis configuration ( mknk\nez) in the presence of an external magnetic field with\na large dc component H0along ezand a small trans-\nverse ac component, Hext=\u0000hx(t); hy(t); H 0\u0001T, with\nhx(t)/hy(t)/ei!t:Linearizing the equations of mo-\ntion in terms of small transverse amplitudes, we can solve\n(4.1) and (4.4) analytically to obtain the linear response\ntoh(see Appendix C for the derivation), i.e. the trans-\nverse magnetic susceptibility. Since we find _\nz= 0, we\ndisregard an initial net rotation by setting \nz= 0. Forsmall damping \u000b\u001c1, the normal modes are given by\nthe positive solutions of the equations\n!3\u0007!2!0\u0000!!c!A\u0006!c!A!H= 0;(5.1)\nwhere!H=\r\u00160H0,!A= 2\rKA=Ms,!0=!H+!A, and\n!c=MsV=(\rI?)is the natural mechanical frequency\ngoverned by the spin angular momentum. Note that the\nequivalent negative solutions of Eq. (5.1) have the same\nabsolute values as the positive solutions. We find that\nthe FMR mode !0is blueshifted to !k=!0+\u000e!kwith\n\u000e!k\u0019!2\nA!c\n!2\n0>0; (5.2)\nwhich is significant for small nanomagnets with large sat-\nuration magnetization and low mass density. It is a coun-\nterclockwise precession of mwithnnearly at rest.\nTwo additional low-frequency modes emerge. For !\u001c\n!0;!Awe may disregard the cubic terms in Eq. (5.1) and\nfind\n!l1;2\u0019s\u0012!c!A\n2!0\u00132\n+!H!c!A\n!0\u0006!c!A\n2!0:(5.3)\nAt low frequencies, the magnetization can follow the lat-\ntice nearly adiabatically, so these modes correspond to\nclockwise and counterclockwise precessions of nearly par-\nallel vectors mandn, but with a phase lag that gener-\nates the splitting. The frequency of the clockwise mode\n!l1> !l2(see Fig. 3). Since magnetization and mass\nprecess in unison, the effective Gilbert damping is ex-\npected to be strongly suppressed as observable in FMR\nabsorption spectra as shown below.\nThe absorbed FMR power is (see Appendix D)\nP=\u0000\u00160V\n2!Im\u0000\nh\u0003T\n?\u001fh?\u0001\n; (5.4)\nwhere h?is the ac field normal to the static magnetic\nfieldH0ezand\n\u001f\u000b\f=M\u000b\nh\f\f\f\f\f\nh?=0(5.5)\nis the transverse magnetic susceptibility tensor ( \u000b;\f =\nx;y). The diagonal ( \u001fxx=\u001fyy) and the off-diagonal\ncomponents( \u001fxy=\u0000\u001fyx)bothcontributetotheabsorp-\ntion spectrum near the resonance frequencies, jIm\u001fxxj\u0019\njRe\u001fxyj. For\u000b\u001c1, we find that the sum rule\nZ1\n0d!(\u0000!Im\u001fxx(!))\u0019\u0019\n2!0!M;(5.6)\nwhere!M=\r\u00160Ms, does not depend on !c, meaning\nthat the coupling does not generate oscillator strengths,\nonly redistributes it. Close to a resonance\n\u0000!Im\u001fxx(!)\u0018F\u00112\n(!\u0000!i)2+\u00112;(5.7)5\n10−310−210−1100101\nωH/ωA0.00.51.01.52.02.5angular frequency [1010s−1]\nωl1: clockwise mode\nωl2: countercl. mode\nFigure 3. Low-frequency magnetomechanical modes !l1and\n!l2of an Fe nanosphere of 2 nmdiameter.\nwith integral \u0019\u0011F. For the low-frequency modes the\nmaximumF\u00181\n2!M!2\nA=(\u000b!2\nH)with broadening \u0011\u0018\n1\n2\u000b!c!2\nH=(!A+!H)2; for the FMR mode F\u00181\n2!M=\u000b\nwith\u0011\u0018\u000b!0.\nLetusconsideranironspherewith 2 nmdiameter(a=\nb= 1 nm) under\u00160H0= 0:65 Tor!H=(2\u0019) = 18:2 GHz.\nIts magnetization !M=(2\u0019) = 60:33 GHz, crystalline\nanisotropy !A=(2\u0019) = 29:74 GHz [45], and the mag-\nnetomechanical coupling !c=(2\u0019) = 0:5(nm=a)2GHz.\nThe blocking temperature is TB\u001811(a=nm)3Kand\njEZj=(kBTB)\u001930, while the critical size for domain\nformationdcr\u001820 nm[46, 47]. We adopt a typi-\ncal Gilbert damping constant \u000b= 0:01. The calcu-\nlated FMR spectra close to the three resonances are\nshown in Fig. 2. Both low-frequency resonances are very\nsharp with a peak value up to 3.5 times larger than\nthat of the high-frequency resonance, although the in-\ntegrated intensity ratio is only 0.2 %. Long relaxation\ntimes of low-frequency modes that imply narrow reso-\nnances have been predicted for spherical nanomagnets\n[32]. The blueshift of the high-frequency resonance is\n\u000e!k=(2\u0019)\u00190:2(nm=a)2GHz. In Fig. 3 we plot the low-\nfrequency modes !l1and!l2as a function of !H=!A. For\n!H=!A!0,!l1\u0019!cand!l2!0. The low-frequency\nmodes become degenerate in the limit !H=!A!1.\nIn\"-Fe2O3[48] magnetization is reduced, resulting in\n!M=(2\u0019) = 2:73 GHz and!c=(2\u0019) = 35(nm=a)2MHz.\nFor the single-molecule magnet TbPc 2[38], we esti-\nmate!A=(2\u0019)\u00185 THz[49],!M=(2\u0019)\u001810 GHz,\n!c=(2\u0019)\u0018100 MHz [50], giving access to the strong-\nanisotropy regime with ultra-low effective damping.\nVI. EASY-PLANE CONFIGURATION\nAn easy-plane anisotropy aligns the equilibrium mag-\nnetization normal to the principal axis ( m?n), which is\ntypically caused by the shape anisotropy of pancake-like\noblate spheroids corresponding to !A<0. We choose an\nexternal magnetic field with a static component in the\nplaneH0eyand an ac field along xandz, while the equi-\n284.40 284.45 284.50 284.55 284.60\nω/(2π) [MHz]024681012141618FMR spectrum [1013s−1]\n−ωImχxx\n−ωImχzz\n−ωReχxz\n+ωReχzx\n11.011.512.012.513.013.514.0\nω/(2π) [GHz]0.00.51.01.52.02.53.0FMR spectrum [1013s−1]\n−ωImχxx\n−ωImχzz\n−ωReχxz\n+ωReχzxFigure 4. FMR spectrum of an Fe disk with 15 nmdiameter\nand2 nmthickness in a static magnetic field of 0:25 Twith\nGilbert damping constant \u000b= 0:01.\nlibrium npoints along ez(see Fig. 1(b)). For \u0012\u001c1,\nmy\u00191,nz\u00191, we again obtain analytic solutions for\nmandn(see Appendix C). We find two singularities in\nthe magnetic susceptibility tensor with frequencies (for\n\u000b\u001c1)\n!?\u0019!Hr\n1\u0000!A\n!H\u0000!c!A\n!2\nH; (6.1)\n!l\u0019s\n!2\nH!c!A\n!A!H\u0000!2\nH+!c!A: (6.2)\nSincenxdoes not depend on time there is only one\nlow-frequency mode !l, viz. an oscillation about the x-\naxis of the nanomagnet. Linearization results in _Ly\u0019\nVMs_my=\r\u00190and implies _Ly\u0019I?nx\u00190. The high-\nfrequency resonance !?is blueshifted by \u000e!?\u0018!c. As\nbefore, the lattice hardly moves in the high-frequency\nmode, while at low frequencies the magnetization is\nlocked to the lattice.\nIn Fig. 4 we plot the FMR spectrum of an Fe nan-\nodisk with shape parameters a= 1 nmandb= 7:5 nm\nunder\u00160H0= 0:25 Tor!H=(2\u0019) = 7 GHz . The\ncharacteristic frequencies are !c=(2\u0019) = 17:2 MHzand\n!A=(2\u0019) =\u000014:4 GHz. The blocking temperature with\njEZj=(kBTB)\u001924is now about 300 K:Again, the low-\nfrequency resonance is very sharp and relatively weak.\nThe contribution of Im\u001fxxto the low-frequency reso-\nnance is by a factor of 600 smaller than the dominant\nIm\u001fzzand therefore not visible in the plot.6\nVII. DISCUSSION\nThe examples discussed above safely fulfill all condi-\ntions for the validity of the theory either at reduced tem-\nperatures (T < 11 K, Fe sphere with 2 nmdiameter) or\neven up to room temperature ( 2 nm\u000215 nmFe disk).\nThe levitation of the particle can be achieved in cluster\nbeams [13, 15, 51], in aerosols [52], or by confinement to\na magnetic trap [33, 35, 53]. FMR experiments should\npreferably be carried out in a microwave cavity, e.g., a\ncoplanar wave guide that can also serve as a trap [54].\nMetal oxide nanoparticles, such as \"-Fe2O3[48], have\ncrystal anisotropies of the same order as that of pure\niron but smaller magnetization, which reduces the mag-\nnetomechanical coupling strength, leading to similar re-\nsults for somewhat smaller particles. The strongest\nanisotropies and couplings can be found in single-\nmolecule magnets, e.g., TbPc 2[49], but FMR experi-\nmentshavetobecarriedoutatlowtemperaturesinorder\nto suppress thermal fluctuations.\nOur theory holds for isolated particles at suffi-\nciently low temperatures and disregards quantum ef-\nfects. According to the fluctuation-dissipation theorem\na Gilbert damping is at finite temperatures associated\nwith stochastic fields [55]. A full statistical treatment of\nthe dynamics of magnetic nanoparticles at elevated tem-\nperatures, subject to microwaves, and weakly coupled to\ntheenvironmentisbeyondthescopeofthepresentpaper.\nWhen not suspended in vacuum but in, e.g., a liquid, the\nmechanical motion encounters viscous damping and ad-\nditional random torques acting on the lattice. Vice versa,\nthe liquid in proximity of the particle will be stirred by\nits motion. These effects can be included in principle\nby an additional torque term in Eq. (4.4). The external\ntorque will cause fluctuations in \nzand a temperature\ndependent broadening of the low-frequency resonances.\nMicrowave cavities loaded with thin films or spheres of\nthe high-quality ferrimagnet yttrium iron garnet have re-\nceived recent attention because of the relative ease with\nwhich the (ultra) strong coupling between magnons and\nphotons can be achieved (for references and evidence\nfor coherent magnon-phonon interaction, see [56]). The\nsharp low-frequency modes of free magnetic nanoparti-\nclescoupledtorfcavitymodesat10-100MHzcorrespond\nto co-operativities that are limited only by the quality\nfactor of the cavity. This appears to be a promising\nroute to access non-linear, chaotic, or quantum dynami-\ncal regimes. This technique would work also for magnets\nwith large damping and could break the monopoly of\nyttrium iron garnet for quantum cavity magnonics. Ma-\nterials with a large anisotropy are most attractive by the\nenhanced magnetization-lattice coupling.\nVIII. SUMMARY\nIn conclusion, we discussed the effect of the mag-\nnetomechanical coupling on the dynamics of levitatedsingle-domain spheroidal magnetic nanoparticles, e.g., in\nmolecular cluster beams and aerosols. We predict a blue\nshift of the high-frequency resonance and additional low-\nfrequency satellites in FMR spectra that reflect parti-\ncle shape and material parameters. In the low-frequency\nmodes the nanomagnet precesses together with the mag-\nnetization with strongly reduced effective damping and\nthereby spectral broadening.\nACKNOWLEDGMENTS\nThis work is part of the research program of the Sticht-\ning voor Fundamenteel Onderzoek der Materie (FOM),\nwhich is financially supported by the Nederlandse Or-\nganisatie voor Wetenschappelijk Onderzoek (NWO) as\nwellasJSPSKAKENHIGrantNos. 25247056,25220910,\n26103006. A. K. acknowledges financial support from the\nAlexander v. Humboldt foundation. H. K. would like to\nexpress her gratitude toward her late supervisor Malek\nZareyan for the opportunity to collaborate with the TU\nDelft researchers. S. S. is grateful to Alejandro O. León\nfor insightful discussions.\nAppendix A: Coordinate systems and\ntransformations\nWe derive the coordinate transformation from the lab\nwith basis vectors ex,ey,ezto the body frame exb,eyb,\nezb. The position of the particle is specified by the three\nEuler angles ( \u001e;\u0012; ). These three angles are defined by\nthetransformationmatrixfromthelabtothebodyframe\n(rb=Ar),\nA=0\n@cos sin 0\n\u0000sin cos 0\n0 0 11\nA0\n@1 0 0\n0 cos\u0012sin\u0012\n0\u0000sin\u0012cos\u00121\nA\n\u00020\n@cos\u001esin\u001e0\n\u0000sin\u001ecos\u001e0\n0 0 11\nA: (A1)\nThe main axis nof the particle is given by the local zb-\naxis in the body frame and can be directly obtained via\nthe inverse transformation AT,\nn=0\n@sin\u0012sin\u001e\n\u0000sin\u0012cos\u001e\ncos\u00121\nA: (A2)\nThe angular velocity vector of the rotating particle reads\nin the lab frame\n\n=_ AT0\n@0\n0\n11\nA+_\u00120\n@cos\u001e\u0000sin\u001e0\nsin\u001ecos\u001e0\n0 0 11\nA0\n@1\n0\n01\nA+_\u001e0\n@0\n0\n11\nA\n=0\n@_\u0012cos\u001e+_ sin\u0012sin\u001e\n_\u0012sin\u001e\u0000_ sin\u0012cos\u001e\n_\u001e+_ cos\u00121\nA; (A3)7\nand in the body frame,\n\nb=A\n=0\n@_\u001esin\u0012sin +_\u0012cos \n_\u001esin\u0012cos \u0000_\u0012sin \n_\u001ecos\u0012+_ 1\nA:(A4)\nThe mechanical angular momentum Land the principal\naxisnof the nanomagnet can be related by considering\nthe mechanical angular momentum in the body frame\nLb=Ib\nb: (A5)\nTransforming (A5) to the lab frame and expanding for\nsmall angles \u0012,\nLx\u0019I?d\ndt(\u0012cos\u001e)\u0019\u0000I?_ny;(A6a)\nLy\u0019I?d\ndt(\u0012sin\u001e)\u0019I?_nx; (A6b)\nLz\u0019I3(_\u001e+_ )\u0019I3\nz; (A6c)\nwhich is a valid approximation when \nz=O(\u0012):Fur-\nthermore,nz\u00191and _nz\u00190is consistent with \u0012\u001c1.\nThe Gilbert damping is defined for the relative motion\nof the magnetization with respect to the lattice, i.e. in\nthe rotating frame. The damping in the lab frame is\nobtained by the coordinate transformation\n\u001c(\u000b)\nm=AT\u001c(\u000b)\nm;b=AT(\u000bmb\u0002_mb);(A7)\nwhere mb=Am. Expanding the time derivative\n\u001c(\u000b)\nm=\u000bm\u0002_m+\u000bm\u0002\u0010\nAT_Am\u0011\n:(A8)\nThe angular frequency vector \nis defined by\n_r=\n\u0002r; (A9)\nwhere ris a point in the rotating body, i.e. _rb= 0, and\n_r=_ATrb=_ATAr: (A10)\nUsingd\ndt(ATA) =AT_A+_ATA= 0and comparing\nEqs. (A9) and (A10),\nAT_Ar=r\u0002\n; (A11)\nand therefore\n\u001c(\u000b)\nm=\u000bm\u0002_m+\u000bm\u0002(m\u0002\n):(A12)\nAppendix B: Poisson bracket in Hamilton’s equation\nIn the following, we show how to derive Hamilton’s\nequation of motion (3.4). Using the linearity of the Pois-\nson bracket together with the product rule\nfAB;Cg=AfB;Cg+fA;CgB; (B1)andfaijk(n;L);Sg= 0, we get\nfS;Hg=X\ni;j;k2N0aijk(n;L)\b\nS;Si\nxSj\nySk\nz\t\n:(B2)\nWe only consider the x-component, as the other compo-\nnents can be derived similarly. Using the product rule\n(B1), we may write\n\b\nSx;Si\nxSj\nySk\nz\t\n=Si\nx\b\nSx;Sj\nySk\nz\t\n=Si\nxSj\ny\b\nSx;Sk\nz\t\n+Si\nxSk\nz\b\nSx;Sj\ny\t\n:\n(B3)\nNext, we prove by induction that\n\b\nSx;Sk\nz\t\n=\u0000kSySk\u00001\nz; (B4)\nwhere the base case ( k= 0)\n\b\nSx;S0\nz\t\n= 0 (B5)\nand the inductive step ( k!k+ 1)\n\b\nSx;Sk+1\nz\t\n=Sz\b\nSx;Sk\nz\t\n+Sk\nzfSx;Szg\n=\u0000(k+ 1)SySk\nz (B6)\ncomplete the proof. Similarly, it follows\n\b\nSx;Sj\ny\t\n=jSj\u00001\nySz: (B7)\nSummarizing\n\b\nSx;Si\nxSj\nySk\nz\t\n=jSi\nxSj\u00001\nySk+1\nz\n\u0000kSi\nxSj+1\nySk\u00001\nz; (B8)\nwhich gives with Eq. (B2) the x-component of Eq. (3.4).\nAppendix C: Linearized equations of motion\n1. Easy-axis configuration\nIn the easy-axis case ( mknkez), the linearized equa-\ntions of motion of the magnetization mand mechanical\nangular momentum Lread\n_mx=\u0000!Hmy+!Mhy\nMs\u0000!A(my\u0000ny)\u0000\u000b( _my\u0000_ny);\n(C1a)\n_my=!Hmx\u0000!Mhx\nMs+!A(mx\u0000nx) +\u000b( _mx\u0000_nx);\n(C1b)\n_mz= 0; (C1c)\n_Lx=\u0000I?ny; (C2a)\n_Ly=I?nx; (C2b)\n_Lz=I3_\nz= 0; (C2c)8\n0.0 0.5 1.0 1.5 2.0\nωH/ωA050100150200\nReχxx(ωl1)\nReχxx(ωl2)\n0.0 0.5 1.0 1.5 2.0\nωH/ωA−50000−40000−30000−20000−100000\nImχxx(ωl1)\nImχxx(ωl2)\n0.0 0.5 1.0 1.5 2.0\nωH/ωA−20000−15000−10000−500005000100001500020000\nReχxy(ωl1)\nReχxy(ωl2)\n0.0 0.5 1.0 1.5 2.0\nωH/ωA−200−150−100−50050100150200\nImχxy(ωl1)\nImχxy(ωl2)\nFigure 5. Real and imaginary parts of the magnetic susceptibility tensor \u001f(!)of the low-frequency modes !l1and!l2for an\nFe nanosphere of 2 nmdiameter with Gilbert damping \u000b= 0:01.\nwith\nnx=!2\nN(mx\u0000nx) +\u000b!c( _mx\u0000_nx);(C3a)\nny=!2\nN(my\u0000ny) +\u000b!c( _my\u0000_ny);(C3b)\nnz= 0; (C3c)\nwhere!2\nN=!c!A. Since _\nz= 0and with initial condi-\ntion\nz= 0, there is no net rotation \nz. Introducing the\nchiral modes,\nm\u0006=mx\u0006imy; n\u0006=nx\u0006iny; h\u0006=hx\u0006ihy;(C4)\nwecanwritetheequationsofmotioninthecompactform\n_m\u0006=\u0006i\u0012\n!0m\u0006\u0000!Mh\u0006\nMs\u0000!An\u0006\u0013\n\u0006i\u000b\u0000\n_m\u0006\u0000_n\u0006\u0001\n;\n(C5)\nn\u0006=!2\nN\u0000\nm\u0006\u0000n\u0006\u0001\n+\u000b!c\u0000\n_m\u0006\u0000_n\u0006\u0001\n: (C6)\nFor ac magnetic fields\nh\u0006(t) =h\u0006\n0ei!t; (C7)\nwe solve the equations of motion by the ansatz\nm\u0006(t) =m\u0006\n0ei!t; n\u0006(t) =n\u0006\n0ei!t:(C8)\nThe observables correspond to the real part of the com-\nplexm,n;andh. The susceptibilities are defined\nm\u0006=\u001f\u0006h\u0006=Ms; n\u0006=\u001f\u0006\nnm\u0006;(C9)and read\n\u001f\u0006\nn(!) =!2\nN+i\u000b!!c\n\u0000!2+!2\nN+i\u000b!!c;(C10)\n\u001f\u0006(!) =\u0007!M(\u0000!2+!2\nN+i\u000b!!c)\n\u0002\u0002\n(!\u0007!0\u0007i\u000b!)(\u0000!2+!2\nN+i\u000b!!c)\n\u0006!c(!A+i\u000b!)2\u0003\u00001: (C11)\nClose to a resonance of \u001f\u0006at!ithe absorbed microwave\npower is determined by the contributions\n\u0000!\n2Im\u001f\u0006(!)\u0018F\u0006 (\u0011\u0006)2\n(!\u0000!i)2+ (\u0011\u0006)2;(C12)\nwith\n\u0011\u0006=\u0006\u000b!i\u0000\n!2\ni+!c(\u0006!i\u0000!H)\u0001\n3!2\ni\u00072!i!0\u0000!c!A; (C13)\nF\u0006=1\n2!M(!2\ni\u0000!c!A)\n\u000b(!2\ni+!c(\u0006!i\u0000!H)):(C14)\nNote that for each resonance of \u001f+at!ithere is a cor-\nresponding resonance of \u001f\u0000at\u0000!i.\nThe magnitudes of the x- andy-components of nare\nrelated to mvia the susceptibility \u001f\u0006\nngiven in Eq. (C10).9\nFor high frequencies !we find\u001f\u0006\nn\u00190and for low fre-\nquencies\u001f\u0006\nn\u00191. Therefore, the main axis nis nearly\nstatic for the high-frequency mode, while for the low-\nfrequency modes nstays approximately parallel to m.\nThesusceptibility \u001f\u0006giveninEq.(C11)canberelated\nto the usual magnetic susceptibilities (\u000b;\f=x;y),\n\u001f\u000b\f=M\u000b\nh\f\f\f\f\f\nh?=0: (C15)\nDefining the symmetric and antisymmetric parts of the\nsusceptibility \u001f\u0006,\n\u001f\u0006=\u001fs\u0006\u001fa: (C16)\nwe find the relations\n\u001fxx=\u001fyy=\u001fs; (C17a)\n\u001fxy=\u0000\u001fyx=i\u001fa: (C17b)\nThe magnetization dynamics in terms of the magnetic\nsusceptibility reads\nRe\u0012\nmx(t)\nmy(t)\u0013\n= Re\u0014\u0012\n\u001fxx\u001fxy\n\u0000\u001fxy\u001fxx\u0013\u0012\nhx(t)=Ms\nhy(t)=Ms\u0013\u0015\n;\n(C18)\nwhere\u001fyy=\u001fxxand\u001fyx=\u0000\u001fxy. For linear polariza-\ntionhx(t) =jhxjei!tandhy(t) = 0,\nRe\u0012\nmx(t)\nmy(t)\u0013\n=jhxj\nMs\u0012\nRe\u001fxxcos(!t)\u0000Im\u001fxxsin(!t)\n\u0000Re\u001fxycos(!t) + Im\u001fxysin(!t)\u0013\n:\n(C19)\nAccording to Fig. 5, jRe\u001fxxj;jIm\u001fxyj \u001c j Re\u001fxyj \u0019\njIm\u001fxxj, and Im\u001fxx<0for both low-frequency modes\n!l1and!l2. The direction of the precession depends now\non the sign of Re\u001fxy, which is negative for !l1and posi-\ntive for!l2. The mode !l1is a clockwise precession,\nRe\u0012\nmx(t)\nmy(t)\u0013\n/\u0012\nsin(!l1t)\ncos(!l1t)\u0013\n; (C20)\nwhereas the mode !l2precesses counterclockwise:\nRe\u0012\nmx(t)\nmy(t)\u0013\n/\u0012\nsin(!l2t)\n\u0000cos(!l2t)\u0013\n:(C21)\nNote that\u001f\u0000(!)has a low-frequency peak only at !l1\nand\u001f+(!)only at!l2(for!>0).\n2. Easy-plane configuration\nHere, we consider an equilibrium magnetization nor-\nmal to the principal axis ( m?n) due to the shape\nanisotropy of an oblate spheroid. Linearizing for small\ndeviations from the equilibrium ( \u0012\u001c1,my\u00191,nz\u00191),\nthe equations of motion for the magnetization and me-\nchanical angular momentum read_mx=!Hmz\u0000!Mhz\nMs\u0000!A(mz+ny) +\u000b( _mz+ _ny);\n(C22a)\n_my= 0; (C22b)\n_mz=\u0000!Hmx+!Mhx\nMs\u0000\u000b_mx\u0000\u000b\nz; (C22c)\n_Lx=\u0000I?ny; (C23a)\n_Ly=I?nx; (C23b)\n_Lz=I3_\nz=VMs\n\r(\u0000\u000b_mx\u0000\u000b\nz);(C23c)\nwith\nnx= 0; (C24a)\nny=!2\nN(mz+ny)\u0000\u000b!c( _mz+ _ny);(C24b)\nnz= 0: (C24c)\nIn the presence of ac magnetic fields\nhx(t) =hx;0ei!t; hz(t) =hz;0ei!t;(C25)\nwe use the ansatz\nmx(t) =mx;0ei!t; mz(t) =mz;0ei!t; ny(t) =ny;0ei!t:\n(C26)\nFrom Eq. (C23c)\n\nz=\u0000!I!\u000bmx\n!\u0000i\u000b!I\u0019\u0000\u000b!Imx;(C27)\nwhere!I=VMs=(\rI3)and provided \u000b!Iis sufficiently\nsmaller than all the other relevant frequencies. We ap-\nproximate\u000b\nz=O(\u000b2)\u00190in Eq. (C22c). Due to\nthe reduced symmetry for m?n, we cannot simplify\nthe equations of motion by introducing chiral modes, but\nhave to calculate the Cartesian components of the mag-\nnetic susceptibility tensor \u001fas\n\u001fxx=!M\u0002\n!2(!A\u0000!H)\u0000i\u000b(!3\u0000!!c!H)\u0000!H!2\nN\u0003\n=\u001fd;\n(C28a)\n\u001fzz=\u0000!M(!H+i\u000b!)(!2+!2\nN\u0000i\u000b!c!)=\u001fd;(C28b)\n\u001fxz=i!!M(!2+!2\nN\u0000i\u000b!c!)=\u001fd; (C28c)\n\u001fzx=\u0000\u001fxz; (C28d)\nwhere the denominator\n\u001fd=!4(1 +\u000b2) +i\u000b!3(!A\u0000!c\u00002!H)\n+!2(!A!H\u0000!2\nH+!2\nN\u0000\u000b2!c!H)\n+i\u000b!!H(!c!H\u0000!2\nN)\u0000!2\nH!2\nN:(C29)\nThe singularities in \u001fmark the two resonance frequen-\ncies. For small damping ( \u000b\u001c1)\n!2\n1;2=\u00001\n2(!A!H\u0000!2\nH+!2\nN)\n\u00061\n2q\n(!A!H\u0000!2\nH+!2\nN)2+ 4!2\nH!2\nN:(C30)10\nFrom Eq. (C24b), we obtain the following relation be-\ntween the magnetic and mechanical motion\nny=\u0000!2\nN+i\u000b!c!\n!2+!2\nN\u0000i\u000b!c!mz: (C31)\nFor high frequencies ny\u00190and for low frequencies ny\u0019\n\u0000mz. This implies that for the high frequency mode\n!?=!1we recover the bulk FMR, while in the low-\nfrequency mode !l=!2the magnetization is locked to\nthe lattice.\nAppendix D: FMR absorption\nFMRabsorptionspectraareproportionaltotheenergy\ndissipated in the magnet [25]. The energy density of the\nmagnetic field is given by\nw(t) =1\n2H(t)\u0001B(t); (D1)\nwhere B=\u00160\u001fH. The absorbed microwave power by a\nmagnet of volume Vis\nP(t) =V_w(t) =VH(t)\u0001_B(t): (D2)\nThe average over one cycle T= 2\u0019=!,\nP\u0011hP(t)i=1\nTZT\n0dtP(t); (D3)can be calculated using the identity\n\nRe(Aei!t)\u0001Re(Bei!t)\u000b\n=1\n2Re (A\u0003\u0001B):(D4)\nWhen a monochromatic ac component of the magnetic\nfieldh?is normal to its dc component, the power reads\nP=\u0000\u00160V\n2!Im (h\u0003\n?\u0001M?); (D5)\nwhere M?is the transverse magnetization. When the\nmagnetization and static magnetic field are parallel to\nthe principal axis of the particle, we can write\nP=\u0000\u00160V\n2!h\n(jhxj2+jhyj2)Im\u001fs(!)\n\u00002Im(h\u0003\nxhy)Im\u001fa(!)]; (D6)\nwhere the symmetric and antisymmetric parts of the\nsusceptibility \u001f\u0006Eq. (C11) as defined by Eq. (C16)\nobeythesymmetryrelations Im\u001fs(\u0000!) =\u0000Im\u001fs(!)and\nIm\u001fa(\u0000!) = Im\u001fa(!). The term proportional to Im\u001fa\ncan therefore be negative, depending on the signs of !\nandIm(h\u0003\nxhy), whereas the term involving Im\u001fs(as well\nas the total absorbed power) is always positive.\nWhen magnetization and static magnetic field are nor-\nmal to the principal axis, both real and imaginary parts\nof the off-diagonal components of \u001fcontribute to the ab-\nsorbed power\nP=\u0000\u00160V\n2!\u0002\njhxj2Im\u001fxx(!) +jhzj2Im\u001fzz(!)\n+ Im(\u001fxzh\u0003\nxhz+\u001fzxhxh\u0003\nz)]: (D7)\n1A. G. Kolhatkar, A. C. Jamison, D. Litvinov, R. C. Will-\nson, and T. R. Lee, Int. J. Mol. Sci. 14, 15977 (2013).\n2N. A. Frey and S. Sun, in Inorganic Nanoparticles: Synthe-\nsis, Applications, and Perspectives , edited by C. Altavilla\nand E. Ciliberto (CRC Press, Boca Raton, London, 2011)\npp. 33–68.\n3Q. A. Pankhurst, J. Connolly, S. K. Jones, and J. Dobson,\nJ. Phys. D 36, R167 (2003).\n4P. Tartaj, M. del Puerto Morales, S. Veintemillas-\nVerdaguer, T.González-Carreño, andC.J.Serna,J.Phys.\nD36, R182 (2003).\n5C. C. Berry and A. S. G. Curtis, J. Phys. D 36, R198\n(2003).\n6A. van Reenen, A. M. de Jong, J. M. J. den Toonder, and\nM. W. J. Prins, Lab Chip 14, 1966 (2014).\n7M. M. van Oene, L. E. Dickinson, F. Pedaci, M. Köber,\nD. Dulin, J. Lipfert, and N. H. Dekker, Phys. Rev. Lett.\n114, 218301 (2015).\n8I. Nándori and J. Rácz, Phys. Rev. E 86, 061404 (2012).\n9G. Vallejo-Fernandez, O. Whear, A. G. Roca, S. Hussain,\nJ. Timmis, V. Patel, and K. O’Grady, J. Phys. D 46,\n312001 (2013).10Z. Jánosfalvi, J. Hakl, and P. F. de Châtel, Adv. Cond.\nMat. Phys. 2014, 125454 (2014).\n11O. Kahn, Acc. Chem. Res. 33, 647 (2000).\n12E. M. Chudnovsky, in Molecular Magnets: Physics and\nApplications , edited by J. Bartolomé, F. Luis, and F. J.\nFernández (Springer, Berlin, Heidelberg, 2014) pp. 61–75.\n13J. P. Bucher, D. C. Douglass, and L. A. Bloomfield, Phys.\nRev. Lett. 66, 3052 (1991).\n14I. M. L. Billas, J. A. Becker, A. Châtelain, and W. A.\nde Heer, Phys. Rev. Lett. 71, 4067 (1993).\n15L. Ma, R. Moro, J. Bowlan, A. Kirilyuk, and W. A.\nde Heer, Phys. Rev. Lett. 113, 157203 (2014).\n16A. Einstein and W. J. de Haas, Proceedings KNAW 18 I,\n696 (1915).\n17S. J. Barnett, Phys. Rev. 6, 239 (1915).\n18C. Kittel, Phys. Rev. 110, 836 (1958).\n19R. L. Comstock and B. A. Auld, J. Appl. Phys. 34, 1461\n(1963).\n20R. L. Comstock, J. Appl. Phys. 34, 1465 (1963).\n21T. L. Gilbert, IEEE Trans. Magn. 40, 3443 (2004).\n22P. Mohanty, G. Zolfagharkhani, S. Kettemann, and\nP. Fulde, Phys. Rev. B 70, 195301 (2004).11\n23K. Uchida, H. Adachi, T. An, T. Ota, M. Toda, B. Hille-\nbrands, S. Maekawa, and E. Saitoh, Nat. Mater. 10, 737\n(2011).\n24M.Weiler,H.Huebl,F.S.Goerg,F.D.Czeschka,R.Gross,\nand S. T. B. Goennenwein, Phys. Rev. Lett. 108, 176601\n(2012).\n25D. Labanowski, A. Jung, and S. Salahuddin, Appl. Phys.\nLett.108, 022905 (2016).\n26A. A. Kovalev, G. E. W. Bauer, and A. Brataas, Appl.\nPhys. Lett. 83, 1584 (2003).\n27A. A. Kovalev, G. E. W. Bauer, and A. Brataas, Phys.\nRev. Lett. 94, 167201 (2005).\n28A. A. Kovalev, G. E. W. Bauer, and A. Brataas, Japan.\nJ. Appl. Phys. 45, 3878 (2006).\n29A. G. Mal’shukov, C. S. Tang, C. S. Chu, and K. A. Chao,\nPhys. Rev. Lett. 95, 107203 (2005).\n30S. Bretzel, G. E. W. Bauer, Y. Tserkovnyak, and\nA. Brataas, Appl. Phys. Lett. 95(2009).\n31H. Chudo, M. Ono, K. Harii, M. Matsuo, J. Ieda,\nR. Haruki, S. Okayasu, S. Maekawa, H. Yasuoka, and\nE. Saitoh, Appl. Phys. Express 7, 063004 (2014).\n32N. Usov and B. Y. Liubimov, J. Magn. Magn. Mater. 385,\n339 (2015).\n33C. C. Rusconi and O. Romero-Isart, Phys. Rev. B 93,\n054427 (2016).\n34D. F. Jackson Kimball, A. O. Sushkov, and D. Budker,\nPhys. Rev. Lett. 116, 190801 (2016).\n35J. Prat-Camps, C. Teo, C. C. Rusconi, W. Wieczorek, and\nO. Romero-Isart, arXiv:1703.00221 (2017).\n36A. A. Kovalev, L. X. Hayden, G. E. W. Bauer, and\nY. Tserkovnyak, Phys. Rev. Lett. 106, 147203 (2011).\n37D. A. Garanin and E. M. Chudnovsky, Phys. Rev. X 1,\n011005 (2011).\n38M. Ganzhorn, S. Klyatskaya, M. Ruben, and W. Werns-\ndorfer, Nat. Comm. 7, 11443 (2016).\n39J. Coey, Magnetism and Magnetic Materials (Cambridge\nUniversity Press, Cambridge, UK, 2009).40O. Romero-Isart, A. C. Pflanzer, M. L. Juan, R. Quidant,\nN. Kiesel, M. Aspelmeyer, and J. I. Cirac, Phys. Rev. A\n83, 013803 (2011).\n41G. Shirane, V. J. Minkiewicz, and R. Nathans, J. Appl.\nPhys.39, 383 (1968).\n42M. Knobel, W. C. Nunes, L. M. Socolovsky, E. De Biasi,\nJ. M. Vargas, and J. C. Denardin, J. Nanosci. Nanotech-\nnol.8, 2836 (2008).\n43J. A. Osborn, Phys. Rev. 67, 351 (1945).\n44L. D. Landau and E. Lifshitz, Phys. Z. Sowjet. 8, 153\n(1935).\n45S.-J. Park, S. Kim, S. Lee, Z. G. Khim, K. Char, and\nT. Hyeon, J. Am. Chem. Soc. 122, 8581 (2000).\n46R. F. Butler and S. K. Banerjee, J. Geophys. Res. 80, 252\n(1975).\n47A. Muxworthy and W. Williams, Geophys. J. Int. 202, 578\n(2015).\n48S. Ohkoshi, A. Namai, K. Imoto, M. Yoshikiyo, W. Tarora,\nK. Nakagawa, M. Komine, Y. Miyamoto, T. Nasu, S. Oka,\nand H. Tokoro, Sci. Rep. 5, 14414 (2015).\n49N.Ishikawa, M.Sugita, T.Okubo, N.Tanaka, T.Iino, and\nY. Kaizu, Inorg. Chem. 42, 2440 (2003).\n50M. Ganzhorn, S. Klyatskaya, M. Ruben, and W. Werns-\ndorfer, Nat. Nanotechnol. 8, 165 (2013).\n51I. M. Billas, A. Châtelain, and W. A. de Heer, Science\n265, 1682 (1994).\n52C. Wang, S. K. Friedlander, and L. Mädler, China Partic-\nuol.3, 243 (2005).\n53D. E. Pritchard, Phys. Rev. Lett. 51, 1336 (1983).\n54S. Bernon, H. Hattermann, D. Bothner, M. Knufinke,\nP. Weiss, F. Jessen, D. Cano, M. Kemmler, R. Kleiner,\nD. Koelle, and J. Fortágh, Nat. Comm. 4, 2380 (2013).\n55W. F. Brown, Phys. Rev. 130, 1677 (1963).\n56X. Zhang, C.-L. Zou, L. Jiang, and H. X. Tang, Sci. Adv.\n2, e1501286 (2016)."    },    {        "title": "1610.01622v1.Finite_dimensional_colored_fluctuation_dissipation_theorem_for_spin_systems.pdf",        "content": "arXiv:1610.01622v1  [cond-mat.stat-mech]  5 Oct 2016Finite-dimensional colored fluctuation-dissipation theo rem for spin systems\nStam Nicolis,1,a)Pascal Thibaudeau,2,b)and Julien Tranchida2,1,c)\n1)CNRS-Laboratoire de Math´ ematiques et Physique Th´ eoriqu e (UMR 7350), F´ ed´ eration de Recherche ”Denis\nPoisson” (FR2964), D´ epartement de Physique, Universit´ e de Tours, Parc de Grandmont, F-37200, Tours,\nFRANCE\n2)CEA DAM/Le Ripault, BP 16, F-37260, Monts, FRANCE\nWhen nano-magnetsare coupled to random external sources, th eir magnetization becomes a random variable,\nwhose properties are defined by an induced probability density, tha t can be reconstructed from its moments,\nusing the Langevin equation, for mapping the noise to the dynamical degrees of freedom. When the spin\ndynamics is discretized in time, a general fluctuation-dissipation the orem, valid for non-Markovian noise, can\nbe established, even when zero modes are present. We discuss the subtleties that arise, when Gilbert damping\nis present and the mapping between noise and spin degrees of freed om is non–linear.\nPACS numbers: 05.40.Ca, 05.10.-a, 75.78.-n\nI. INTRODUCTION\nFor any system, in equilibrium with a bath, the\nfluctuation-dissipation relation (FDR) plays an impor-\ntant role in defining consistently its closure, since it re-\nlates the fluctuations of the subsystem of the dynamical\ndegrees of freedom, that one is, by definition, interested\nin, with the fluctuations of the degrees of freedom that\nare defined as uninteresting and are lumped under the\nterm “dissipation”.\nThe essential reason behind this relation is that, for\nequilibrium situations, it is possible to define a proba-\nbility measure on the space of states, with respect to\nwhich the average values, that enter in the FDR, can\nbe unambiguously computed. So this can be modified,\nif the dynamical degrees of freedom are so affected by\nthe immersion in the bath, that they must be replaced\nby others–the interaction with the bath leads to a phase\ntransition and the equilibrium measure is not unitarily\nequivalent to the measure of the dynamical degrees of\nfreedom, in the absence of the bath.\nWhile it is possible to address these questions by nu-\nmerical simulations, and reconstruct the density that\nway, what has, really, changedin the lastyearsis that ex-\nperiments ofgreatprecision, that probe both issues, have\nbecome possible, particularly in magnetic systems1. It is\nin such a context that the FDR has become of topical\ninterest2–4.\nIn such systems, since the noise affects the magnetic\nfield, that makes the spin precess, it is not additive, but\nmultiplicative. While, already, for additive noise, the\nissue of the “backreaction” of the dynamical degrees of\nfreedom on the bath is quite delicate, for multiplicative\nnoise it becomes even more difficult to evade and must\nbe addressed.\nFurther complications arise when the fluctuations are\ncolored, namely posses finite intrinsic correlation time5,6.\na)Electronic mail: stam.nicolis@lmpt.univ-tours.fr\nb)Electronic mail: pascal.thibaudeau@cea.fr\nc)Electronic mail: julien.tranchida@cea.frIn such a situation, no FDR has been unequivocally ob-\ntained, that relates the intensity of the fluctuations to\nthe damping constant7.\nIn this note, we wish to study these issues in the con-\ntext of magnetic systems placed in random magnetic\nfields, whose distribution can have an auto–correlation\ntime comparable to the time scale defined by the pre-\ncession frequency. The aim of this communication is to\nsketch out a route for establishing a FDR in a quite gen-\neralsetting8, that will be shownto be consistentto previ-\nous results for magnetic systems, obtained in the limit of\nwhite-noise fluctuations, and can be readily adapted be-\nyond this context, especially for explicit calculations. A\nremaining challenge is to obtain the stochastic equation,\nthat defines the mapping between noise and the dynam-\nical degrees of freedom, that are identified with the spin\ncomponents of a nanomagnet, and whose solution does,\nindeed, describe a normalizable density for the spin con-\nfigurations.\nII. GAUSSIAN APPROXIMATION\nIn order to better grasp the issues at stake, we shall\nstart with a finite number of dynamical degrees of free-\ndom,sA\nn. The time index nruns from 0 to N−1 and\nwill be identified with the evolution time instant, in the\ncontinuum limit; the flavor index Aruns from 1 to Nf\nand labels “internal” degrees of freedom–it will label the\ncomponents of the spin. The summation convention on\nrepeated indices is assumed.\nWe assume that these dynamical degrees of freedom\nare immersed in a bath. The bath is described by vari-\nablesηA\nnand is defined by the partition function\nZ=/integraldisplayNf/productdisplay\nA=1N−1/productdisplay\nn=0dηA\nne−1\n2ηA\nnFABDnmηB\nm (1)\nThe matrix Facts on the flavor indices and the ma-\ntrixDon the “target space” indices–that describe the\ninstants in time. The white noise case corresponds to\ntakingDnm=δnm/σ2. The simplest colored noise case2\ncorresponds to taking Dnm=δnm/σ2\nn, with not all the\nσnequal. Furthermore, if it cannot be put in diagonal\nform at all, then it describes higher derivative effects.\nThe average of a functional Fof the variables ηA\nnis\nthen well defined as\n/angbracketleftF/angbracketright=1\nZ/integraldisplayNf/productdisplay\nA=1N−1/productdisplay\nn=0dηA\nnF[η]e−1\n2ηA\nnFABDnmηB\nm(2)\nFrom this expression we may deduce the moments of the\ndegrees of freedom of the bath:\n/angbracketleftbig\nηA\nn/angbracketrightbig\n= 0/angbracketleftbig\nηA\nnηB\nm/angbracketrightbig\n=/bracketleftbig\nF−1/bracketrightbigAB/bracketleftbig\nD−1/bracketrightbig\nnm(3)\nwith the others deduced from Wick’s theorem. What\nwe notice here is that, for non–diagonal matrices, Fand\nD, the degrees of freedom of the bath that have well–\ndefined properties, i.e. the degrees of freedom that are\neigenstates of these matrices, are linear combinations of\ntheηA\nn. So it makes sense to work in that basis. In this\ncontext, the white noise limit corresponds to the case in\nwhichDis the identity matrix–all components have the\nsame relaxation time. The colored noise case, then can\nbe identified as that, where Dis not the identity matrix.\nWhen we immerse a physical system in such a bath it\ncan happen that the eigenbases of the system and of the\nbath do not match.\nThe map between the degrees of freedom of the bath\nand the dynamical degrees of freedom is provided by\na stochastic equation. For instance, one consider the\nLandau-Lifshitz-Gilbert equation ˙s=ω×s+αs×˙s+\nE(s)η, where the vielbein Econtains both an antisym-\nmetric part ×sand at least an additional non-zero diag-\nonal element. Because this vielbein is invertible, we can\nexpressηas a function of s.\nTo illustrate the procedure, we start with the case of\nlinear equations:\nηA\nn=fA\nBCm\nnsB\nm (4)\nAssuming that the matrices are invertible, we obtain the\nchange of variables (we shall study presently what hap-\npens when the matrices have zero modes)\nsA\nn=/bracketleftbig\nf−1/bracketrightbigA\nB/bracketleftbig\nC−1/bracketrightbigm\nnηB\nm (5)\nThe Jacobian is a constant that can be absorbed in the\nnormalizationofthe partition function9, sowe obtain the\npartition function for the dynamical degrees of freedom,\nZ=/integraldisplayNf/productdisplay\nA=1N−1/productdisplay\nn=0dsA\nne−1\n2sA′\nn′fB\nB′FABfA\nA′Cn′\nnDnmCm′\nmsB′\nm′(6)\nthat defines the correlation functions–for the finite-\ndimensional case the moments–of the dynamical degrees\nof freedom. The 1–point function vanishes, /angbracketleftsA\nn/angbracketright= 0,\nwhile the 2–point function is given by the expression\n/angbracketleftbig\nsA\nnsB\nm/angbracketrightbig\n=/bracketleftbig\n[f−1Ff]−1/bracketrightbigAB/bracketleftbig\n[C−1DC]−1/bracketrightbig\nnm(7)This is the FDR for the present case, that relates the\nparameters, fA\nBandCm\nn, of the spin dynamics, with the\nparameters, FABandDnm, of the bath.\nIII. WHEN ZERO MODES ARE RELEVANT\nLet us now consider the case when the matrices fA\nB\nand/orCm\nnhave zero modes, a case that is relevant for\nthe physical system studied in this paper.\nThe zero modes imply, quite simply, that we cannot\nreplace all of the ηA\nnby thesA\nn, since we cannot invert\neq. (4); we can, only, replace the non–zero modes. The\nmatrices fand/orCare not of full rank–but they surely\nhave positive rank, otherwise the stochastic map does\nnot make sense. When we replace the non–zero modes,\nwe shall generate quadratic terms in the sA\nn–but, since\nwe do not replace all of the ηA\nn, on the one hand there\nwill be mixed terms, while there will remain the terms\nquadratic in the ηA\nn, that correspond to the zero modes.\nWhen we integrate over the zero modes, the ηA\nnthat we\ncould not express directly as linear combinations of the\nsA\nn, we shall encounter Gaussian integrals over them that\ncontaintermslinearin thezeromodesandthe sA\nnalready\nreplaced. The result of these Gaussian integrations will\nbequadraticcontributionstothealreadypresent sA\nn, that\nenter in the action with the opposite sign to their coeffi-\ncients. The system will be stable, if these contributions\ndo not completely cancel the existing ones and will lead\nto a modification of the FDR.\nLet us see this in action. We shall take Nf= 3 and\nfA\nB=εA\nBCωC, withωa fixed vector in flavor space. In\nthe magnetic case it will correspond to the fixed part of\nthe precession frequency. We immediately remark that\nfA\nBhas one zeromode, along the vector ω. Since this\nvector is fixed, without loss of generality, we may take it\nto lie along the z−axis:ω= (0,0,ω3).\nThe stochastic equation, eq. (4), takes the form\nη1\nn=ω3Cm\nns2\nm\nη2\nn=−ω3Cm\nns1\nm(8)\nWe may replace these in the partition function for the\nnoise; but we must integrate over η3\nnseparately. We re-\nmark that they do not involve s3\nn, the component of the\ndynamical degrees of freedom, parallel to the precession\nvector.\nIfFAB=δAB, i.e. the spherical symmetry is imposed,\nwe immediately deduce that the integration over η3\nnde-\ncouples from the rest and just gives a contribution to the\nnormalization. The partition function for the dynamical\ndegrees of freedom, s1\nnands2\nn, is given by the expression\nZ=/integraldisplay2/productdisplay\nA′=1N−1/productdisplay\nn=0dsA\nne−(ω3)2\n2sA′\nn′[CDC]n′m′sA′\nm′(9)\nThere’s a subtle point here: the motion of the A′=\n1,2 flavor components is a rotation, with precession3\nfrequency ω3, about the z−axis, so the combination,\n(s1\nm)2+(s2\nm)2should appear–and it does. Therefore we\ndeduce the FDR for this case, that corresponds to Lar-\nmor precession:\n/angbracketleftbig\nsA\nnsB\nm/angbracketrightbig\n=/parenleftbig\nω3/parenrightbig−2/bracketleftbig\nC−1DC/bracketrightbig−1\nnm(10)\nIf the spherical symmetry is not imposed, in flavor space,\ne.g.FAB=κδAB+λAB(1−δAB), we would have had\nterms linear in η3\nn, along with the quadratic terms and\nadditional contributions when we would have integrated\nover the η3\nn.\nIV. BEYOND THE GAUSSIAN APPROXIMATION\nNow let us address the issue of non–linear stochas-\ntic maps, also relevant for the Landau–Lifshitz–Gilbert\nequation. Let us replace eq. (4) by\nηA\nn=fA\n(1)BC(1)m\nnsB\nm+fA\n(2)BCC(2)ml\nnsB\nmsC\nl(11)In this case the Jacobian of the transformation, between\nthe degrees of freedom of the bath and the degrees of\nfreedom that describe the “interesting” dynamics, is not\na constant:\nJAB\nnk(s)≡δηA\nn\nδsB\nk=fA\n(1)BC(1)k\nn+\n+/bracketleftBig\nfA\n(2)BCC(2)kl\nn+fA\n(2)CBC(2)lk\nn/bracketrightBig\nsC\nl(12)\nThismeansthat, ifitispossibletoneglectthezeromodes\nand the concomitant fluctuations in the sign of the de-\nterminant, which is true in perturbation theory, the par-\ntition function for the spin degrees of freedom is given by\nthe expression\nZ=/integraldisplay/bracketleftbig\ndsA\nn/bracketrightbig\ndetJAB\nnk(s)e−1\n2ηA\nn(s)FABDnmηB\nm(s)(13)\nwhere the η(s) are defined by eq. (11). The expression\nin the exponent contains terms that are quadratic and\nquartic in the spin variables. The fluctuation–dissipation\nrelation can then be deduced from the Schwinger–Dyson\nequations9,\nZ−1/integraldisplay/bracketleftbig\ndsA\nn/bracketrightbig∂\n∂sL\nk/braceleftBig\nsA1n1···sAInIdetJAB\nnk(s)e−1\n2ηA\nn(s)FABDnmηB\nm(s)/bracerightBig\n= 0 (14)\nThese relations can be used to generalize eqs. (10) and\nexpress the fact that the spin degrees of freedom are in\nequilibriumwiththebath. Thedeterminantcanbeintro-\nduced into the exponent using anti-commuting variables,\nthat describe the dynamics of the bath9.\nIt should be stressed that, since the η(s) are polynomi-\nals in the spin degrees of freedom, once the determinant\nhas been expressed in terms of anti-commuting fields,\nthere is a finite number of parameters that define the\ndynamics and, thus, enter in the fluctuation–dissipation\nrelation. Indeed, if Dnmis not the identity matrix, which\nmeans that the dynamics is not ultra–local in time, tun-\nneling between configurations implies that the effects of\nthe determinant and it sign will be, inevitably and, thus,\nimplicitly, be generatedby the dynamics, thereforeit suf-\nfices to sample the correlation functions by the action\nof the spin degrees of freedom. The subtleties of the\ndynamics are encoded in the relation between the noise\nfields and the spins, so it is at that point that the zero\nmodes need to be taken into account. There are not any\nissues of principle, involved, however, precisely because\nthe system is consistently closed10. How to sample the\ncorrelation functions will be reported in detail in future\nwork.REFERENCES\n1Markus G. M¨ unzenberg. “Magnetization dynamics: Ferromag -\nnets stirred up”, Nat Mater 9(3):184–185 (2010).\n2Aditi Mitra and A. J. Millis. “Spin dynamics and violation of\nthe fluctuation dissipation theorem in a nonequilibrium ohm ic\nspin-boson model”, Phys. Rev. B72(12), 121102 (2005)\n3Vladimir L. Safonov and H. Neal Bertram, “Fluctuation-\ndissipation considerations and damping models for ferroma gnetic\nmaterials”, Phys. Rev. B71(22):224402 (2005).\n4William T. Coffey and Yuri P. Kalmykov. “Thermal fluctuations\nof magnetic nanoparticles: Fifty years after Brown”, Journal of\nApplied Physics 112(12):121301, (2012).\n5R. Kupferman, G. A. Pavliotis, and A. M. Stuart. “Itˆ o versus\nStratonovich white-noise limits for systems with inertia a nd col-\nored multiplicative noise”, Phys. Rev. E70(3):036120 (2004).\n6Masamichi Nishino and Seiji Miyashita, “Realization of the ther-\nmal equilibrium in inhomogeneous magnetic systems by the\nLandau-Lifshitz-Gilbert equation with stochastic noise, and its\ndynamical aspects”, Phys. Rev. ,91(13):134411 (2015).\n7Marco Baiesi, Christian Maes, and Bram Wynants. “Fluctua-\ntions and Response of Nonequilibrium States”, Phys. Rev. Lett. ,\n103(1):010602 (2009).\n8Camille Aron, Giulio Biroli, and Leticia F Cugliandolo. “Sy mme-\ntries of generating functionals of Langevin processes with colored\nmultiplicative noise”, Journal of Statistical Mechanics: Theory\nand Experiment , 2010(11):P11018 (2010).\n9Jean Zinn-Justin. Quantum Field Theory and Critical Phe-\nnomena. Number 113 in International series of monographs on\nphysics. Clarendon Press, Oxford, 4. ed., reprinted editio n, 2011.\nOCLC: 767915024.\n10S. Nicolis, “How quantum mechanics probes superspace”, con tri-\nbution to SQS15, Dubna. [arXiv:1606.08284 [hep-th]]."    },    {        "title": "1610.04598v2.Nambu_mechanics_for_stochastic_magnetization_dynamics.pdf",        "content": "arXiv:1610.04598v2  [cond-mat.mes-hall]  19 Jan 2017Nambu mechanics for stochastic magnetization\ndynamics\nPascal Thibaudeaua,∗, Thomas Nusslea,b, Stam Nicolisb\naCEA DAM/Le Ripault, BP 16, F-37260, Monts, FRANCE\nbCNRS-Laboratoire de Math´ ematiques et Physique Th´ eoriqu e (UMR 7350), F´ ed´ eration de\nRecherche ”Denis Poisson” (FR2964), D´ epartement de Physi que, Universit´ e de Tours, Parc\nde Grandmont, F-37200, Tours, FRANCE\nAbstract\nThe Landau-Lifshitz-Gilbert (LLG) equation describes the dynamic s of a damped\nmagnetization vector that can be understood as a generalization o f Larmor spin\nprecession. The LLG equation cannot be deduced from the Hamilton ian frame-\nwork, by introducing a coupling to a usual bath, but requires the int roduction of\nadditional constraints. It is shown that these constraints can be formulated ele-\ngantly and consistently in the framework of dissipative Nambu mecha nics. This\nhas many consequences for both the variational principle and for t opological as-\npects of hidden symmetries that control conserved quantities. W e particularly\nstudy how the damping terms of dissipative Nambu mechanics affect t he con-\nsistent interaction of magnetic systems with stochastic reservoir s and derive a\nmaster equation for the magnetization. The proposals are suppor ted by numer-\nical studies using symplectic integrators that preserve the topolo gical structure\nof Nambu equations. These results are compared to computations performed\nby direct sampling of the stochastic equations and by using closure a ssumptions\nfor the moment equations, deduced from the master equation.\nKeywords: Magnetization dynamics, Fokker-Planck equation, magnetic\nordering\n∗Corresponding author\nEmail addresses: pascal.thibaudeau@cea.fr (Pascal Thibaudeau),\nthomas.nussle@cea.fr (Thomas Nussle), stam.nicolis@lmpt.univ-tours.fr (Stam Nicolis)\nPreprint submitted to Elsevier September 18, 20181. Introduction\nIn micromagnetism, the transverse Landau-Lifshitz-Gilbert (LLG ) equation\n(1 +α2)∂si\n∂t=ǫijkωj(s)sk+α(ωi(s)sjsj−ωj(s)sjsi) (1)\ndescribes the dynamics of a magnetization vector s≡M/MswithMsthe sat-\nuration magnetization. This equation can be seen as a generalization of Larmor\nspin precession, for a collection of elementary classical magnets ev olving in an\neffective pulsation ω=−1\n¯hδH\nδs=γBand within a magnetic medium, charac-\nterized by a damping constant αand a gyromagnetic ratio γ[1].His here\nidentified as a scalar functional of the magnetization vector and ca n be consis-\ntently generalized to include spatial derivatives of the magnetizatio n vector [2]\nas well. Spin-transfer torques, that are, nowadays, of particula r practical rele-\nvance [3, 4] can be, also, taken into account in this formalism. In the following,\nwe shall work in units where ¯ h= 1, to simplify notation.\nIt is well known that this equation cannot be derived from a Hamiltonia n\nvariational principle, with the damping effects described by coupling t he magne-\ntization to a bath, by deforming the Poisson bracket of Hamiltonian m echanics,\neven though the Landau–Lifshitz equation itself is Hamiltonian. The r eason is\nthat the damping cannot be described by a “scalar” potential, but b y a “vector”\npotential.\nThis has been made manifest [5] first by an analysis of the quantum ve rsion\nof the Landau-Lifshitz equation for damped spin motion including arb itrary\nspin length, magnetic anisotropy and many interacting quantum spin s. In par-\nticular, this analysis has revealed that the damped spin equation of m otion is\nan example of metriplectic dynamical system [6], an approach which t ries to\nunite symplectic, nondissipative and metric, dissipative dynamics into one com-\nmon mathematical framework. This dissipative system has been see n afterwards\nnothing but a natural combination of semimetric dynamics for the dis sipative\npart and Poisson dynamics for the conservative ones [7]. As a conse quence, this\nprovided a canonical description for any constrained dissipative sy stems through\n2an extension of the concept of Dirac brackets developed originally f or conserva-\ntive constrained Hamiltonian dynamics. Then, this has culminated rec ently by\nobserving the underlying geometrical nature of these brackets a s certain n-ary\ngeneralizations of Lie algebras, commonly encountered in conserva tive Hamilto-\nnian dynamics [8]. However, despite the evident progresses obtaine d, no clear\ndirection emerges for the case of dissipative n-ary generalizations, and even\nno variational principle have been formulated, to date, that incorp orates such\nproperties.\nWhat we shall show in this paper is that it is, however, possible to de-\nscribe the Landau–Lifshitz–Gilbert equation by using the variationa l principle\nof Nambu mechanics and to describe the damping effects as the resu lt of in-\ntroducing dissipation by suitably deforming the Nambu–instead of th e Poisson–\nbracket. In this way we shall find, as a bonus, that it is possible to de duce\nthe relation between longitudinal and transverse damping of the ma gnetization,\nwhen writing the appropriate master equation for the probability de nsity. To\nachieve this in a Hamiltonian formalism requires additional assumptions , whose\nprovenance can, thus, be understood as the result of the prope rties of Nambu\nmechanics. We focus here on the essential points; a fuller account will be pro-\nvided in future work.\nNeglecting damping effects, if one sets H1≡ −ω·sandH2≡s·s/2, eq.(1)\ncan be recast in the form\n∂si\n∂t={si,H1,H2}, (2)\nwhere for any functions A,B,Cofs,\n{A,B,C} ≡ǫijk∂A\n∂si∂B\n∂sj∂C\n∂sk(3)\nis the Nambu-Poisson (NP) bracket, or Nambu bracket, or Nambu t riple bracket,\na skew-symmetric object, obeying both the Leibniz rule and the Fun damental\nIdentity [9, 10]. One can see immediately that both H1andH2are constants of\nmotion, because of the anti-symmetric property of the bracket. This provides the\ngeneralization of Hamiltonian mechanics to phase spaces of arbitrar y dimension;\n3in particular it does not need to be even. This is a way of taking into acc ount\nconstraints and provides a natural framework for describing the magnetization\ndynamics, since the magnetization vector has, in general, three co mponents.\nThe constraints–and the symmetries–can be made manifest, by no ting that\nit is possible to express vectors and vector fields in, at least, two wa ys, that can\nbe understood as special cases of Hodge decomposition.\nFor the three–dimensional case that is of interest here, this mean s that a\nvector field V(s) can be expressed in the “Helmholtz representation” [11] in the\nfollowing way\nVi≡ǫijk∂Ak\n∂sj+∂Φ\n∂si(4)\nwhereAis a vector potential and Φ a scalar potential.\nOn the other hand, this same vector field V(s) can be decomposed according\nto the “Monge representation” [12]\nVi≡∂C1\n∂si+C2∂C3\n∂si(5)\nwhich defines the “Clebsch-Monge potentials”, Ci.\nIf one identifies as the Clebsch–Monge potentials, C2≡H1,C3≡H2and\nC1≡D,\nVi=∂D\n∂si+H1∂H2\n∂si, (6)\nand the vector field V(s)≡˙s, then one immediately finds that eq. (2) takes the\nform\n∂s\n∂t={s,H1,H2}+∇sD (7)\nthat identifies the contribution of the dissipation in this context, as the expected\ngeneralization from usual Hamiltonian mechanics. In the absence of the Gilbert\nterm, dissipation is absent.\nMore generally, the evolution equation for any function, F(s) can be written\nas [13]\n∂F\n∂t={F,H1,H2}+∂D\n∂si∂F\n∂si(8)\nfor a dissipation function D(s).\n4The equivalence between the Helmholtz and the Monge representat ion im-\nplies the existence of freedom of redefinition for the potentials, CiandDand\nAiand Φ. This freedom expresses the symmetry under symplectic tra nsforma-\ntions, that can be interpreted as diffeomorphism transformations , that leave the\nvolume invariant. These have consequences for the equations of m otion.\nFor instance, the dissipation described by the Gilbert term in the Lan dau–\nLifshitz–Gilbert equation (1)\n∂D\n∂si≡α(˜ωi(s)sjsj−˜ωj(s)sjsi) (9)\ncannot be derived from a scalar potential, since the RHS of this expr ession is not\ncurl–free, so the function Don the LHS is not single valued; but it does conserve\nthe norm of the magnetization, i.e. H2. Because of the Gilbert expression,\nbothωandηare rescaled such as ˜ω≡ω/(1 +α2) andη→η/(1 +α2).\nSo there are two questions: (a) Whether it can lead to stochastic e ffects, that\ncan be described in terms of deterministic chaos and/or (b) Whethe r its effects\ncan be described by a bath of “vector potential” excitations. The fi rst case\nwas described, in outline in ref. [14], where the role of an external to rque was\nshown to be instrumental; the second will be discussed in detail in the following\nsections. While, in both cases, a stochastic description, in terms of a probability\ndensity on the space of states is the main tool, it is much easier to pre sent for\nthe case of a bath, than for the case of deterministic chaos, which is much more\nsubtle.\nTherefore, we shall now couple our magnetic moment to a bath of flu ctuating\ndegrees of freedom, that will be described by a stochastic proces s.\n2. Nambu dynamics in a macroscopic bath\nTo this end, one couples linearly the deterministic system such as (8) , to\na stochastic process, i.e. a noise vector, random in time, labelled ηi(t), whose\nlaw of probability is given. This leads to a system of stochastic differen tial\nequations, that can be written in the Langevin form\n∂si\n∂t={si,H1,H2}+∂D\n∂si+eij(s)ηj(t) (10)\n5whereeij(s) can be interpreted as the vielbein on the manifold, defined by the\ndynamical variables, s. It should be noted that it is the vector nature of the\ndynamical variables that implies that the vielbein, must, also, carry in dices.\nWe may note that the additional noise term can be used to “renorma lize”\nthe precession frequency and, thus, mix, non-trivially, with the Gilb ert term.\nThis means that, in the presence of either, the other cannot be ex cluded.\nWhen this vielbein is the identity matrix, eij(s) =δij, the stochastic cou-\npling to the noise is additive, whereas it is multiplicative otherwise. In th at\ncase, if the norm of the spin vector has to remain constant in time, t hen the\ngradient of H2must be orthogonal to the gradient of Dandeij(s)si= 0∀j.\nHowever, it is important to realize that, while the Gilbert dissipation te rm\nis not a gradient, the noise term, described by the vielbein is not so co nstrained.\nFor additive noise, indeed, it is a gradient, while for the case of multiplic ative\nnoise studied by Brown and successors there can be an interesting interference\nbetween the two terms, that is worth studying in more detail, within N ambu\nmechanics, to understand, better, what are the coordinate art ifacts and what\nare the intrinsic features thereof.\nBecause {s(t)}, defined by the eq.(10), becomes a stochastic process, we\ncan define an instantaneous conditional probability distribution Pη(s,t), that\ndepends, on the noise configuration and, also, on the magnetizatio ns0at the\ninitial time and which satisfies a continuity equation in configuration sp ace\n∂Pη(s,t)\n∂t+∂( ˙siPη(s,t)))\n∂si= 0. (11)\nAn equation for /an}b∇acketle{tPη/an}b∇acket∇i}htcan be formed, which becomes an average over all the\npossible realizations of the noise, namely\n∂/an}b∇acketle{tPη/an}b∇acket∇i}ht\n∂t+∂/an}b∇acketle{t˙siPη/an}b∇acket∇i}ht\n∂si= 0, (12)\nonce the distribution law of {η(t)}is provided. It is important to stress here\nthat this implies that the backreaction of the spin degrees of freed om on the\nbath can be neglected–which is by no means obvious. One way to chec k this is\nby showing that no “runaway solutions” appear. This, however, do es not ex-\n6haust all possibilities, that can be found by working with the Langevin equation\ndirectly. For non–trivial vielbeine, however, this is quite involved, so it is useful\nto have an approximate solution in hand.\nTo be specific, we consider a noise, described by the Ornstein-Uhlen beck\nprocess [15] of intensity ∆ and autocorrelation time τ,\n/an}b∇acketle{tηi(t)/an}b∇acket∇i}ht= 0\n/an}b∇acketle{tηi(t)ηj(t′)/an}b∇acket∇i}ht=∆\nτδije−|t−t′|\nτ\nwhere the higher point correlation functions are deduced from Wick ’s theorem\nand which can be shown to become a white noise process, when τ→0. We\nassume that the solution to eq.(12) converges, in the sense of ave rage over-the-\nnoise, to an equilibrium distribution, that is normalizable and, whose co rrelation\nfunctions, also, exist. While this is, of course, not at all obvious to p rove, evi-\ndence can be found by numerical studies, using stochastic integra tion methods\nthat preserve the symplectic structure of the Landau–Lifshitz e quation, even\nunder perturbations (cf. [16] for earlier work).\n2.1. Additive noise\nWalton [17] was one of the first to consider the introduction of an ad ditive\nnoise into an LLG equation and remarked that it may lead to a Fokker- Planck\nequation, without entering into details. To see this more thoroughly and to\nillustrate our strategy, we consider the case of additive noise, i.e. w heneij=\nδijin our framework. By including eq.(10) in (12) and in the limit of white\nnoise, expressions like /an}b∇acketle{tηiPη/an}b∇acket∇i}htmust be defined and can be evaluated by either an\nexpansion of the Shapiro-Loginov formulae of differentiation [18] an d taking the\nlimit ofτ→0, or, directly, by applying the Furutsu-Novikov-Donsker theore m\n[19, 20, 21]. This leads to\n/an}b∇acketle{tηiPη/an}b∇acket∇i}ht=−˜∆∂/an}b∇acketle{tPη/an}b∇acket∇i}ht\n∂si. (13)\n7where ˜∆≡∆/(1 +α2). Using the dampened current vector Ji≡ {si,H1,H2}+\n∂D\n∂si, the (averaged) probability density /an}b∇acketle{tPη/an}b∇acket∇i}htsatisfies the following equation\n∂/an}b∇acketle{tPη/an}b∇acket∇i}ht\n∂t+∂\n∂si(Ji/an}b∇acketle{tPη/an}b∇acket∇i}ht)−˜˜∆∂2/an}b∇acketle{tPη/an}b∇acket∇i}ht\n∂si∂si= 0 (14)\nwhere˜˜∆≡∆/(1 +α2)2and which is of the Fokker-Planck form [22]. This last\npartial differential equation can be solved directly by several nume rical methods,\nincluding a finite-element computer code or can lead to ordinary differ ential\nequations for the moments of s.\nFor example, for the average of the magnetization, one obtains th e evolution\nequation\nd/an}b∇acketle{tsi/an}b∇acket∇i}ht\ndt=−/integraldisplay\ndssi∂/an}b∇acketle{tPη(s,t)/an}b∇acket∇i}ht\n∂t=/an}b∇acketle{tJi/an}b∇acket∇i}ht. (15)\nFor the case of Landau-Lifshitz-Gilbert in a uniform precession field B, we\nobtain the following equations, for the first and second moments,\nd\ndt/an}b∇acketle{tsi/an}b∇acket∇i}ht=ǫijk˜ωj/an}b∇acketle{tsk/an}b∇acket∇i}ht+α[˜ωi/an}b∇acketle{tsjsj/an}b∇acket∇i}ht−˜ωj/an}b∇acketle{tsjsi/an}b∇acket∇i}ht] (16)\nd\ndt/an}b∇acketle{tsisj/an}b∇acket∇i}ht= ˜ωl(ǫilk/an}b∇acketle{tsksj/an}b∇acket∇i}ht+ǫjlk/an}b∇acketle{tsksi/an}b∇acket∇i}ht) +α[˜ωi/an}b∇acketle{tslslsj/an}b∇acket∇i}ht\n+ ˜ωj/an}b∇acketle{tslslsi/an}b∇acket∇i}ht−2˜ωl/an}b∇acketle{tslsisj/an}b∇acket∇i}ht] + 2˜˜∆δij (17)\nwhere ˜ω≡γB/(1 +α2). In order to close consistently these equations, one can\ntruncate the hierarchy of moments; either on the second /an}b∇acketle{t/an}b∇acketle{tsisj/an}b∇acket∇i}ht/an}b∇acket∇i}ht= 0 or third\ncumulants /an}b∇acketle{t/an}b∇acketle{tsisjsk/an}b∇acket∇i}ht/an}b∇acket∇i}ht= 0, i.e.\n/an}b∇acketle{tsisj/an}b∇acket∇i}ht=/an}b∇acketle{tsi/an}b∇acket∇i}ht/an}b∇acketle{tsj/an}b∇acket∇i}ht, (18)\n/an}b∇acketle{tsisjsk/an}b∇acket∇i}ht=/an}b∇acketle{tsisj/an}b∇acket∇i}ht/an}b∇acketle{tsk/an}b∇acket∇i}ht+/an}b∇acketle{tsisk/an}b∇acket∇i}ht/an}b∇acketle{tsj/an}b∇acket∇i}ht+/an}b∇acketle{tsjsk/an}b∇acket∇i}ht/an}b∇acketle{tsi/an}b∇acket∇i}ht\n−2/an}b∇acketle{tsi/an}b∇acket∇i}ht/an}b∇acketle{tsj/an}b∇acket∇i}ht/an}b∇acketle{tsk/an}b∇acket∇i}ht. (19)\nBecause the closure of the hierarchy is related to an expansion in po wers of\n∆, for practical purposes, the validity of eqs.(16,17) is limited to low v alues\nof the coupling to the bath (that describes the fluctuations). For example, if\none sets /an}b∇acketle{t/an}b∇acketle{tsisj/an}b∇acket∇i}ht/an}b∇acket∇i}ht= 0, eq.(16) produces an average spin motion independent of\nvalue that ∆ may take. This is in contradiction with the numerical expe riments\n8performed by the stochastic integration and noise average of eq.( 10) quoted in\nreference [23] and by experiments. This means that it is mandatory to keep\nat least eqs.(16) and (17) together in the numerical evaluation of t he thermal\nbehavior of the dynamics of the average thermal magnetization /an}b∇acketle{ts/an}b∇acket∇i}ht. This was\npreviously observed [24, 25] and circumvented by alternate secon d-order closure\nrelationships, but is not supported by direct numerical experiment s.\nThis can be illustrated by the following figure (1). For this given set of\nFigure 1: Magnetization dynamics of a paramagnetic spin in a constant magnetic field,\nconnected to an additive noise. The upper graphs (a) plot som e of the first–order moments of\nthe averaged magnetization vector over 102realizations of the noise, when the lower graphs\n(b) plot the associated model closed to the third-order cumu lant (eqs.(16)-(17), see text).\nParameters of the simulations : {∆ = 0.13 rad.GHz; α= 0.1;ω= (0,0,18) rad.GHz; timestep\n∆t= 10−4ns}. Initial conditions: s(0) = (1,0,0),/an}bracketle{tsi(0)sj(0)/an}bracketri}ht= 0 but /an}bracketle{ts1(0)s1(0)/an}bracketri}ht= 1.\nparameters, the agreement between the stochastic average an d the effective\nmodel is fairly decent. As expected, for a single noise realization, th e norm\nof the spin vector in an additive stochastic noise cannot be conserv ed during\nthe dynamics, but, by the average-over-the-noise accumulation process, this is\n9observed for very low values of ∆ and very short times. However, t his agreement\nwith the effective equations is lost, when the temperature increase s, because of\nthe perturbative nature of the equations (16-17). Agreement c an, however, be\nrestored by imposing this constraint in the effective equations, for a given order\nin perturbation of ∆, by appropriate modifications of the hierarchic al closing\nrelationships /an}b∇acketle{t/an}b∇acketle{tsisj/an}b∇acket∇i}ht/an}b∇acket∇i}ht=Bij(∆) or/an}b∇acketle{t/an}b∇acketle{tsisjsk/an}b∇acket∇i}ht/an}b∇acket∇i}ht=Cijk(∆).\nIt is of some interest to study the effects of the choice of initial con ditions. In\nparticular, how the relaxation to equilibrium is affected by choosing a c omponent\nof the initial magnetization along the precession axis in the effective m odel, e.g.\ns(0) = (1/√\n2,0,−1/√\n2) and by taking all the initial correlations,\n/an}b∇acketle{tsi(0)sj(0)/an}b∇acket∇i}ht=\n1\n20−1\n2\n0 0 0\n−1\n201\n2\n(20)\nThe results are shown in figure (2).\nBoth in figures (1) and (2), it is observed that the average norm of the spin\nvector increases over time. This can be understood with the above arguments.\nIn general, according to eq.(10) and because Jis a transverse vector,\n(1 +α2)sidsi\ndt=eij(s)siηj(t). (21)\nThis equation describes how the LHS depends on the noise realization ; so the\naverage over the noise can be found by computing the averages of the RHS. The\nsimplest case is that of the additive vielbein, eij(s) =δij. Assuming that the\naverage-over-the noise procedure and the time derivative commu te, we have\nd\ndt/angbracketleftbig\ns2/angbracketrightbig\n=2/an}b∇acketle{tsiηi/an}b∇acket∇i}ht\n1 +α2. (22)\nFor any Gaussian stochastic process, the Furutsu-Novikov-Don sker theorem\nstates that\n/an}b∇acketle{tsi(t)ηi(t)/an}b∇acket∇i}ht=/integraldisplay+∞\n−∞dt′/an}b∇acketle{tηi(t)ηj(t′)/an}b∇acket∇i}ht/angbracketleftbiggδsi(t)\nδηj(t′)/angbracketrightbigg\n. (23)\nIn the most general situation, the functional derivativesδsi(t)\nδηj(t′)can be calculated\n[26], and eq.(23) admits simplifications in the white noise limit. In this limit,\n10-2-1012\n0 1 2 3 4 5\nt (ns)-2-1012\nsxsy\nsz(a)\n(b)\nFigure 2: Magnetization dynamics of a paramagnetic spin in a constant magnetic field, con-\nnected to an additive noise. The upper graphs (a) plot some of the first–order moments of\nthe averaged magnetization vector over 103realizations of the noise, when the lower graphs\n(b) plot the associated model closed to the third-order cumu lant (eqs.(16)-(17), see text).\nParameters of the simulations : {∆ = 0.0655 rad.GHz; α= 0.1;ω= (0,0,18) rad.GHz;\ntimestep ∆ t= 10−4ns,s(0) =/an}bracketle{ts(0)/an}bracketri}ht= (1/√\n2,0,−1/√\n2),/an}bracketle{tsisj/an}bracketri}ht(0) = 0 except for (11)=1/2,\n(13)=(31)=-1/2, (33)=1/2 }.\n11the integration is straightforward and we have\n/angbracketleftbig\ns2(t)/angbracketrightbig\n=s2(0) + 6˜˜∆t, (24)\nwhich is a conventional diffusion regime. It is also worth noticing that w hen\ncomputing the trace of (17), the only term which remains is indeed\nd\ndt/an}b∇acketle{tsisi/an}b∇acket∇i}ht= 6˜˜∆ (25)\nwhich allows our effective model to reproduce exactly the diffusion re gime. Fig-\nure (3) compares the time evolution of the average of the square n orm spin\nvector. Numerical stochastic integration of eq.(10) is tested by in creasing the\n0 1 2 3 4 5\nt (ns)11,522,53\n<|s|2>mean over 103 runs\nmean over 104 runs\ndiffusion regime\nFigure 3: Mean square norm of the spin in the additive white no ise case for the following\nconditions: integration step of 10−4ns; ∆ = 0 .0655 rad.GHz; s(0) = (0 ,1,0);α= 0.1;\nω= (0,0,18) rad.GHz compared to the expected diffusion regime (see te xt).\nsize of the noise sampling and reveals a convergence to the predicte d linear\ndiffusion regime.\n122.2. Multiplicative noise\nBrown [27] was one of the first to propose a non–trivial vielbein, tha t takes\nthe form eij(s) =ǫijksk/(1 +α2) for the LLG equation. We notice, first of\nall, that it is present, even if α= 0, i.e. in the absence of the Gilbert term.\nAlso, that, since the determinant of this matrix [ e] is zero, this vielbein is not\ninvertible. Because of its natural transverse character, this vie lbein preserves the\nnorm of the spin for any realization of the noise, once a dissipation fu nctionD\nis chosen, that has this property. In the white-noise limit, the aver age over-the-\nnoise continuity equation (12) cannot be transformed strictly to a Fokker-Planck\nform. This time\n/an}b∇acketle{tηiPη/an}b∇acket∇i}ht=−˜∆∂\n∂sj(eji/an}b∇acketle{tPη/an}b∇acket∇i}ht), (26)\nwhich is a generalization of the additive situation shown in eq.(13). The conti-\nnuity equation thus becomes\n∂/an}b∇acketle{tPη/an}b∇acket∇i}ht\n∂t+∂\n∂si(Ji/an}b∇acketle{tPη/an}b∇acket∇i}ht)−˜˜∆∂\n∂si/parenleftbigg\neij∂\n∂sk(ekj/an}b∇acketle{tPη/an}b∇acket∇i}ht)/parenrightbigg\n= 0. (27)\nWhat deserves closer attention is, whether, in fact, this equation is invariant\nunder diffeomeorphisms of the manifold [28] defined by the vielbein, o r whether\nit breaks it to a subgroup thereof. This will be presented in future w ork. In the\ncontext of magnetic thermal fluctuations, this continuity equatio n was encoun-\ntered several times in the literature [22, 29], but obtaining it from fir st principles\nis more cumbersome than our latter derivation, a remark already qu oted [18].\nMoreover, our derivation presents the advantage of being easily g eneralizable\nto non-Markovian noise distributions [23, 30, 31], by simply keeping th e partial\nderivative equation on the noise with the continuity equation, and so lving them\ntogether.\nConsequently, the evolution equation for the average magnetizat ion is now\nsupplemented by a term provided by a non constant vielbein and one h as\nd/an}b∇acketle{tsi/an}b∇acket∇i}ht\ndt=/an}b∇acketle{tJi/an}b∇acket∇i}ht+˜˜∆/angbracketleftbigg∂eil\n∂skekl/angbracketrightbigg\n. (28)\nWith the vielbein proposed by Brown and assuming a constant extern al field,\n13one gets\nd/an}b∇acketle{tsi/an}b∇acket∇i}ht\ndt=ǫijk˜ωj/an}b∇acketle{tsk/an}b∇acket∇i}ht+α(˜ωi/an}b∇acketle{tsjsj/an}b∇acket∇i}ht−˜ωj/an}b∇acketle{tsjsi/an}b∇acket∇i}ht)\n−2∆\n(1 +α2)2/an}b∇acketle{tsi/an}b∇acket∇i}ht. (29)\nThis equation highlights both a transverse part, coming from the av erage over\nthe probability current Jand a longitudinal part, coming from the average\nover the extra vielbein term. By imposing, further, the second-or der cumulant\napproximation /an}b∇acketle{t/an}b∇acketle{tsisj/an}b∇acket∇i}ht/an}b∇acket∇i}ht= 0, i.e. “small” fluctuations to keep the distribution of\nsgaussian, a single equation can be obtained, in which a longitudinal rela xation\ntimeτL≡(1 +α2)2/2∆ may be identified.\nThis is illustrated by the content of figure (4). In that case, the ap proxima-\nFigure 4: Magnetization dynamics of a paramagnetic spin in a constant magnetic field, con-\nnected to a multiplicative noise. The upper graphs (a) plot s ome of the first–order moments\nof the averaged magnetization vector over 102realizations of the noise, when the lower graphs\n(b) plot the associated model closed to the third-order cumu lant (eq.(29), see text). Param-\neters of the simulations : {∆ = 0.65 rad.GHz; α= 0.1;ω= (0,0,18) rad.GHz; timestep\n∆t= 10−4ns}. Initial conditions: s(0) = (1,0,0),/an}bracketle{tsi(0)sj(0)/an}bracketri}ht= 0 but /an}bracketle{tsx(0)sx(0)/an}bracketri}ht= 1.\n14tion/an}b∇acketle{t/an}b∇acketle{tsisjsj/an}b∇acket∇i}ht/an}b∇acket∇i}ht= 0 has been retained in order to keep two sets of equations, three\nfor the average magnetization components and nine on the averag e second-order\nmoments, that have been solved simultaneously using an eight-orde r Runge-\nKutta algorithm with variable time-steps. This is the same numerical im ple-\nmentation that has been followed for the studies of the additive nois e, solving\neqs.(16) and (17) simultaneously. We have observed numerically tha t, as ex-\npected, the average second-order moments are symmetrical by an exchange of\ntheir component indices, both for the multiplicative and the additive n oise. In-\nterestingly, by keeping identical the number of random events tak en to evaluate\nthe average of the stochastic magnetization dynamics between th e additive and\nmultiplicative noise, we observe a greater variance in the multiplicative case.\nAs we have done in the additive noise case, we will also investigate briefl y the\nbehavior of this equation under different initial conditions, and in par ticular with\na non vanishing component along the z-axis. This is illustrated by the c ontent\nof figure (5). It is observed that for both figures (4) and (5), th e average spin\nconverges to the same final equilibrium state, which depends ultimat ely on the\nvalue of the noise amplitude, as shown by equation (27).\n3. Discussion\nMagnetic systems describe vector degrees of freedom, whose Ha miltonian\ndynamics implies constraints. These constraints can be naturally ta ken into\naccount within Nambu mechanics, that generalizes Hamiltonian mecha nics to\nphase spaces of odd number of dimensions. In this framework, diss ipation can\nbe described by gradients that are not single–valued and thus do no t define\nscalar baths, but vector baths, that, when coupled to external torques, can lead\nto chaotic dynamics. The vector baths can, also, describe non-tr ivial geometries\nand, in that case, as we have shown by direct numerical study, the stochastic\ndescription leads to a coupling between longitudinal and transverse relaxation.\nThis can be, intuitively, understood within Nambu mechanics, in the fo llowing\nway:\n15-1-0.500.51\n0 1 2 3 4 5\nt (ns)-1-0.500.51\nsxsy\nsz(a)\n(b)\nFigure 5: Magnetization dynamics of a paramagnetic spin in a constant magnetic field, con-\nnected to a multiplicative noise. The upper graphs (a) plot s ome of the first–order moments\nof the averaged magnetization vector over 104realizations of the noise, when the lower graphs\n(b) plot the associated model closed to the third-order cumu lant (eq.(29), see text). Param-\neters of the simulations : {∆ = 0.65 rad.GHz; α= 0.1;ω= (0,0,18) rad.GHz; timestep\n∆t= 10−4ns}. Initial conditions: s(0) =/parenleftbig\n1/√\n2,0,1/√\n2/parenrightbig\n,/an}bracketle{tsi(0)sj(0)/an}bracketri}ht= 0 except for\n/an}bracketle{ts1(0)s1(0)/an}bracketri}ht=/an}bracketle{ts1(0)s3(0)/an}bracketri}ht=/an}bracketle{ts3(0)s3(0)/an}bracketri}ht= 1/2.\n16The dynamics consists in rendering one of the Hamiltonians, H1≡ω·s,\nstochastic, since ωbecomes a stochastic process, as it is sensitive to the noise\nterms–whether these are described by Gilbert dissipation or couplin g to an\nexternal bath. Through the Nambu equations, this dependence is “transferred”\ntoH2≡ ||s||2/2. This is one way of realizing the insights the Nambu approach\nprovides.\nIn practice, we may summarize our numerical results as follows:\nWhen the amplitude of the noise is small, in the context of Langevin-\ndynamics formalism for linear systems and for the numerical modeling ofsmall\nthermal fluctuations in micromagnetic systems, as for a linearized s tochastic\nLLG equation, the rigorous method of Lyberatos, Berkov and Cha ntrell might\nbe thought to apply [32] and be expected to be equivalent to the app roach\npresented here. Because this method expresses the approach t o equilibrium of\nevery moment, separately, however, it is restricted to the limit of s mall fluctua-\ntions around an equilibrium state and, as expected, cannot captur e the transient\nregime of average magnetization dynamics, even for low temperatu re. This is a\nuseful check.\nWe have also investigated the behaviour of this system under differe nt sets of\ninitial conditions as it is well-known and has been thoroughly studied in [ 1] that\nin the multiplicative noise case (where the norm is constant) this syst em can\nshow strong sensitivity to initial conditions and it is possible, using ste reographic\ncoordinates to represent the dynamics of this system in 2D. In our additive noise\ncase however, as the norm of the spin is not conserved, it is not eas y to get long\nrun behavior of our system and in particular equilibrium solutions. Mor eover as\nwe no longer have only two independent components of spin, it is not p ossible\nto obtain a 2D representation of our system and makes it more comp licated to\nstudy maps displaying limit cycles, attractors and so on. Thus under standing\nthe dynamics under different initial conditions would require somethin g more\nand, as it is beyond the scope of this work, will be done elsewhere.\nTherefore, we have focused on studying the effects of the prese nce of an\ninitial longitudinal component and of additional, diagonal, correlation s. No\n17differences have been observed so far.\nAnother issue, that deserves further study, is how the probabilit y density\nof the initial conditions is affected by the stochastic evolution. In th e present\nstudy we have taken the initial probability density to be a δ−function; so it will\nbe of interest to study the evolution of other initial distributions in d etail, in\nparticular, whether the averaging procedures commute–or not. In general, we\nexpect that they won’t. This will be reported in future work.\nFinally, our study can be readily generalized since any vielbein can be ex -\npressed in terms of a diagonal, symmetrical and anti-symmetrical m atrices,\nwhose elements are functions of the dynamical variable s. Because ˙sis a pseu-\ndovector (and we do not consider that this additional property is a cquired by the\nnoise vector), this suggests that the anti-symmetric part of the vielbein should\nbe the “dominant” one. Interestingly, by numerical investigations , it appears\nthat there are no effects, that might depend on the choice of the n oise connection\nfor the stochastic vortex dynamics in two-dimensional easy-plane ferromagnets\n[33], even if it is known that for Hamiltonian dynamics, multiplicative and a d-\nditive noises usually modify the dynamics quite differently, a point that also\ndeserves further study.\nReferences\n[1] Giorgio Bertotti, Isaak D. Mayergoyz, and Claudio Serpico. Nonlinear\nMagnetization Dynamics in Nanosystems . Elsevier, April 2009. Google-\nBooks-ID: QH4ShV3mKmkC.\n[2] Amikam Aharoni. Introduction to the Theory of Ferromagnetism . Claren-\ndon Press, 2000.\n[3] Jacques Miltat, Gon¸ calo Albuquerque, Andr´ e Thiaville, and Caro le Vouille.\nSpin transfer into an inhomogeneous magnetization distribution. Journal\nof Applied Physics , 89(11):6982, 2001.\n[4] Dmitry V. Berkov and Jacques Miltat. Spin-torque driven magnet ization\n18dynamics: Micromagnetic modeling. Journal of Magnetism and Magnetic\nMaterials , 320(7):1238–1259, April 2008.\n[5] Janusz A. Holyst and Lukasz A. Turski. Dissipative dynamics of qu antum\nspin systems. Physical Review A , 45(9):6180–6184, May 1992.\n[6] /suppress Lukasz A. Turski. Dissipative quantum mechanics. Metriplectic d ynamics\nin action. In Zygmunt Petru, Jerzy Przystawa, and Krzysztof Ra pcewicz,\neditors,From Quantum Mechanics to Technology , number 477 in Lecture\nNotes in Physics, pages 347–357. Springer Berlin Heidelberg, 1996.\n[7] Sonnet Q. H. Nguyen and /suppress Lukasz A. Turski. On the Dirac approa ch to\nconstrained dissipative dynamics. Journal of Physics A: Mathematical and\nGeneral , 34(43):9281–9302, November 2001.\n[8] Josi A. de Azc´ arraga and Josi M. Izquierdo. N-ary algebras: A review\nwith applications. Journal of Physics A: Mathematical and Theoretical ,\n43(29):293001, July 2010.\n[9] Yoichiro Nambu. Generalized Hamiltonian Dynamics. Physical Review D ,\n7(8):2405–2412, April 1973.\n[10] Ra´ ul Ib´ a˜ nez, Manuel de Le´ on, Juan C. Marrero, and Dav id Martı´ n de\nDiego. Dynamics of generalized Poisson and Nambu–Poisson bracket s.\nJournal of Mathematical Physics , 38(5):2332, 1997.\n[11] Jerrold E. Marsden and Tudor S. Ratiu. Introduction to Mechanics and\nSymmetry , volume 17 of Texts in Applied Mathematics . Springer New York,\nNew York, NY, 1999.\n[12] Phillip Griffiths and Joseph Harris. Principles of Algebraic Geometry . John\nWiley & Sons, August 2014.\n[13] Minos Axenides and Emmanuel Floratos. Strange attractors in dissipative\nNambu mechanics: Classical and quantum aspects. Journal of High Energy\nPhysics , 2010(4), April 2010.\n19[14] Julien Tranchida, Pascal Thibaudeau, and Stam Nicolis. Quantum Mag-\nnets and Matrix Lorenz Systems. Journal of Physics: Conference Series ,\n574(1):012146, 2015.\n[15] George Eugene Uhlenbeck and Leonard S. Ornstein. On the The ory of the\nBrownian Motion. Physical Review , 36(5):823–841, September 1930.\n[16] Pascal Thibaudeau and David Beaujouan. Thermostatting the atomic spin\ndynamics from controlled demons. Physica A: Statistical Mechanics and\nits Applications , 391(5):1963–1971, March 2012.\n[17] Derek Walton. Rate of transition for single domain particles. Journal of\nMagnetism and Magnetic Materials , 62(2-3):392–396, December 1986.\n[18] V. E. Shapiro and V. M. Loginov. “Formulae of differentiation” an d their\nuse for solving stochastic equations. Physica A: Statistical Mechanics and\nits Applications , 91(3-4):563–574, May 1978.\n[19] Koichi Furutsu. On the statistical theory of electromagnetic waves in a\nfluctuating medium (I). Journal of Research of the National Bureau of\nStandards , 67D:303–323, May 1963.\n[20] Evgenii A. Novikov. Functionals and the Random-force Method in Tur-\nbulence Theory. Soviet Physics Journal of Experimental and Theoretical\nPhysics , 20(5):1290–1294, May 1964.\n[21] Valery I. Klyatskin. Stochastic Equations through the Eye of the Physicist:\nBasic Concepts, Exact Results and Asymptotic Approximatio ns. Elsevier,\nAmsterdam, 1 edition, 2005. OCLC: 255242261.\n[22] Hannes Risken. The Fokker-Planck Equation , volume 18 of Springer Series\nin Synergetics . Springer-Verlag, Berlin, Heidelberg, 1989.\n[23] Julien Tranchida, Pascal Thibaudeau, and Stam Nicolis. Closing th e hier-\narchy for non-Markovian magnetization dynamics. Physica B: Condensed\nMatter , 486:57–59, April 2016.\n20[24] Dmitry A. Garanin. Fokker-Planck and Landau-Lifshitz-Bloch e quations\nfor classical ferromagnets. Physical Review B , 55(5):3050–3057, February\n1997.\n[25] Pui-Wai Ma and Sergei L. Dudarev. Langevin spin dynamics. Physical\nReview B , 83:134418, April 2011.\n[26] Julien Tranchida, Pascal Thibaudeau, and Stam Nicolis. A functio nal\ncalculus for the magnetization dynamics. arXiv:1606.02137 [cond-mat,\nphysics:nlin, physics:physics] , June 2016.\n[27] William Fuller Brown. Thermal Fluctuations of a Single-Domain Partic le.\nPhysical Review , 130(5):1677–1686, June 1963.\n[28] Jean Zinn-Justin. QuantumField Theory and Critical Phenomena . Number\n113 in International series of monographs on physics. Clarendon P ress,\nOxford, 4. ed., reprinted edition, 2011. OCLC: 767915024.\n[29] Jos´ e Luis Garc´ ıa-Palacios and Francisco J. L´ azaro. Langev in-dynamics\nstudy of the dynamical properties of small magnetic particles. Physical\nReview B , 58(22):14937–14958, December 1998.\n[30] Pascal Thibaudeau, Julien Tranchida, and Stam Nicolis. Non-Mar kovian\nMagnetization Dynamics for Uniaxial Nanomagnets. IEEE Transactions\non Magnetics , 52(7):1–4, July 2016.\n[31] Julien Tranchida, Pascal Thibaudeau, and Stam Nicolis. Colored- noise\nmagnetization dynamics: From weakly to strongly correlated noise. IEEE\nTransactions on Magnetics , 52(7):1300504, 2016.\n[32] Andreas Lyberatos, Dmitry V. Berkov, and Roy W. Chantrell. A method\nfor the numerical simulation of the thermal magnetization fluctuat ions in\nmicromagnetics. Journal of Physics: Condensed Matter , 5(47):8911–8920,\nNovember 1993.\n21[33] Till Kamppeter, Franz G. Mertens, Esteban Moro, Angel S´ an chez, and\nA. R. Bishop. Stochastic vortex dynamics in two-dimensional easy- plane\nferromagnets: Multiplicative versus additive noise. Physical Review B ,\n59(17):11349–11357, May 1999.\n22"    },    {        "title": "1610.06661v1.Spin_transport_and_dynamics_in_all_oxide_perovskite_La___2_3__Sr___1_3__MnO__3__SrRuO__3__bilayers_probed_by_ferromagnetic_resonance.pdf",        "content": "Spin transport and dynamics in all-oxide perovskite La 2=3Sr1=3MnO 3/SrRuO 3bilayers\nprobed by ferromagnetic resonance\nSatoru Emori,1,\u0003Urusa S. Alaan,1, 2Matthew T. Gray,1, 2Volker\nSluka,3Yizhang Chen,3Andrew D. Kent,3and Yuri Suzuki1, 4\n1Geballe Laboratory for Advanced Materials, Stanford University, Stanford, CA 94305 USA\n2Department of Materials Science and Engineering, Stanford University, Stanford, CA 94305 USA\n3Department of Physics, New York University, New York, NY 10003, USA\n4Department of Applied Physics, Stanford University, Stanford, CA 94305 USA\n(Dated: November 11, 2021)\nThin \flms of perovskite oxides o\u000ber the possibility of combining emerging concepts of strongly\ncorrelated electron phenomena and spin current in magnetic devices. However, spin transport and\nmagnetization dynamics in these complex oxide materials are not well understood. Here, we ex-\nperimentally quantify spin transport parameters and magnetization damping in epitaxial perovskite\nferromagnet/paramagnet bilayers of La 2=3Sr1=3MnO 3/SrRuO 3(LSMO/SRO) by broadband ferro-\nmagnetic resonance spectroscopy. From the SRO thickness dependence of Gilbert damping, we\nestimate a short spin di\u000busion length of <\u00181 nm in SRO and an interfacial spin-mixing conductance\ncomparable to other ferromagnet/paramagnetic-metal bilayers. Moreover, we \fnd that anisotropic\nnon-Gilbert damping due to two-magnon scattering also increases with the addition of SRO. Our\nresults demonstrate LSMO/SRO as a spin-source/spin-sink system that may be a foundation for\nexamining spin-current transport in various perovskite heterostructures.\nI. INTRODUCTION\nManipulation and transmission of information by spin\ncurrent is a promising route toward energy-e\u000ecient mem-\nory and computation devices1. Such spintronic devices\nmay consist of ferromagnets interfaced with nonmagnetic\nconductors that exhibit spin-Hall and related spin-orbit\ne\u000bects2{4. The direct spin-Hall e\u000bect in the conductor\ncan convert a charge current to a spin current, which ex-\nerts torques on the adjacent magnetization and modi\fes\nthe state of the device5,6. Conversely, the inverse spin-\nHall e\u000bect in the conductor can convert a propagating\nspin current in the magnetic medium to an electric signal\nto read spin-based information packets7. For these device\nschemes, it is essential to understand the transmission of\nspin current between the ferromagnet and the conductor,\nwhich is parameterized by the spin-mixing conductance\nand spin di\u000busion length. These spin transport parame-\nters can be estimated by spin pumping at ferromagnetic\nresonance (FMR), in which a spin current is resonantly\ngenerated in the ferromagnet and absorbed in the adja-\ncent conductor8,9. Spin pumping has been demonstrated\nin various combinations of materials, where the magnetic\nlayer may be an alloy (e.g., permalloy) or insulator (e.g.,\nyttrium iron garnet) and the nonmagnetic conductor may\nbe a transition metal, semiconductor, conductive poly-\nmer, or topological insulator10{16.\nTransition metal oxides, particularly those with the\nperovskite structure, o\u000ber the intriguing prospect of\nintegrating a wide variety of strongly correlated elec-\ntron phenomena17,18with spintronic functionalities19,20.\nAmong these complex oxides, La 2=3Sr1=3MnO 3(LSMO)\nand SrRuO 3(SRO) are attractive materials for epitaxial,\nlattice-matched spin-source/spin-sink heterostructures.\nLSMO, a metallic ferromagnet known for its colossalmagnetoresistance and Curie temperature of >300 K,\ncan be an excellent resonantly-excited spin source be-\ncause of its low magnetization damping21{26. SRO, a\nroom-temperature metallic paramagnet with relatively\nhigh conductivity27, exhibits strong spin-orbit coupling28\nthat may be useful for emerging spintronic applications\nthat leverage spin-orbit e\u000bects2{4.\nA few recent studies have reported dc voltages at FMR\nin LSMO/SRO bilayers that are attributed to the in-\nverse spin-Hall e\u000bect in SRO generated by spin pump-\ning24{26. However, it is generally a challenge to separate\nthe inverse spin-Hall signal from the spin recti\fcation sig-\nnal, which is caused by an oscillating magnetoresistance\nmixing with a microwave current in the conductive mag-\nnetic layer29{31. Moreover, while the spin-mixing con-\nductance is typically estimated from the enhancement in\nthe Gilbert damping parameter \u000b, the quanti\fcation of \u000b\nis not necessarily straightforward in epitaxial thin \flms\nthat exhibit pronounced anisotropic non-Gilbert damp-\ning23,32{37. It has also been unclear how the Gilbert and\nnon-Gilbert components of damping in LSMO are each\nmodi\fed by an adjacent SRO layer. These points above\nhighlight the need for an alternative experimental ap-\nproach for characterizing spin transport and magnetiza-\ntion dynamics in LSMO/SRO.\nIn this work, we quantify spin transport parameters\nand magnetization damping in epitaxial LSMO/SRO bi-\nlayers by broadband FMR spectroscopy with out-of-plane\nandin-plane external magnetic \felds. Out-of-plane FMR\nenables straightforward extraction of Gilbert damping as\na function of SRO overlayer thickness, which is repro-\nduced by a simple \\spin circuit\" model based on di\u000busive\nspin transport38,39. We \fnd that the spin-mixing conduc-\ntance at the LSMO/SRO interface is comparable to other\nferromagnet/conductor interfaces and that spin current\nis absorbed within a short length scale of <\u00181 nm in thearXiv:1610.06661v1  [cond-mat.mtrl-sci]  21 Oct 20162\n42 44 46 48 50LSMO(10)\n  /SRO(18)LSMO(10)\n  log(intensity) (a.u.)\n2 (deg.)LSAT(002)  \nLSMO(002)  \nSRO(002)  \nFigure 1. 2 \u0012-!x-ray di\u000braction scans of a single-layer\nLSMO(10 nm) \flm and LSMO(10 nm)/SRO(18 nm) bilayer.\nconductive SRO layer. From in-plane FMR, we observe\npronounced non-Gilbert damping that is anisotropic and\nscales nonlinearly with excitation frequency, which is ac-\ncounted for by an existing model of two-magnon scat-\ntering40. This two-magnon scattering is also enhanced\nwith the addition of the SRO overlayer possibly due to\nspin pumping. Our \fndings reveal key features of spin\ndynamics and transport in the prototypical perovskite\nferromagnet/conductor bilayer of LSMO/SRO and pro-\nvide a foundation for future all-oxide spintronic devices.\nII. SAMPLE AND EXPERIMENTAL DETAILS\nEpitaxial \flms of LSMO(/SRO) were grown on\nas-received (001)-oriented single-crystal (LaAlO 3)0:3\n(Sr2AlTaO 6)0:7(LSAT) substrates using pulsed laser de-\nposition. LSAT exhibits a lower dielectric constant than\nthe commonly used SrTiO 3substrate and is therefore\nbetter suited for high-frequency FMR measurements.\nThe lattice parameter of LSAT (3.87 \u0017A) is also closely\nmatched to the pseudocubic lattice parameter of LSMO\n(\u00193.88 \u0017A). By using deposition parameters similar to\nthose in previous studies from our group41,42, all \flms\nwere deposited at a substrate temperature of 750\u000eC with\na target-to-substrate separation of 75 mm, laser \ruence\nof\u00191 J/cm2, and repetition rate of 1 Hz. LSMO was de-\nposited in 320 mTorr O 2, followed by SRO in 100 mTorr\nO2. After deposition, the samples were held at 600\u000eC\nfor 15 minutes in \u0019150 Torr O 2and then the substrate\nheater was switched o\u000b to cool to room temperature. The\ndeposition rates were calibrated by x-ray re\rectivity mea-\nsurements. The thickness of LSMO, tLSMO , in this study\nis \fxed at 10 nm, which is close to the minimum thickness\nat which the near-bulk saturation magnetization can be\nattained.\nX-ray di\u000braction results indicate that both the LSMO\n\flms and LSMO/SRO bilayers are highly crystalline\nand epitaxial with the LSAT(001) substrate, with high-\nresolution 2 \u0012-!scans showing distinct Laue fringes\naround the (002) Bragg re\rection (Fig. 1). In this study,\nthe maximum thickness of the LSMO and SRO layers\n770 780 790 800 810 main mode\n sec. mode\n  dIFMR/dH (a.u.)\n0H (mT) data\n fit(a) (b) \n770 780 790 800 810 data\n fit\n  dIFMR/dH (a.u.)\n0H (mT)Figure 2. Exemplary FMR spectra and \ftting curves: (a)\none mode of Lorentzian derivative; (b) superposition of a main\nmode and a small secondary mode due to slight sample inho-\nmogeneity.\ncombined is less than 30 nm and below the threshold\nthickness for the onset of structural relaxation by mis\ft\ndislocation formation41,42. The typical surface roughness\nof LSMO and SRO measured by atomic force microscopy\nis<\u00184\u0017A, comparable to the roughness of the LSAT sub-\nstrate surface.\nSQUID magnetometry con\frms that the Curie tem-\nperature of the LSMO layer is \u0019350 K and the room-\ntemperature saturation magnetization is Ms\u0019300 kA/m\nfor 10-nm thick LSMO \flms. The small LSMO thickness\nis desirable for maximizing the spin-pumping-induced en-\nhancement in damping, since spin pumping scales in-\nversely with the ferromagnetic layer thickness8,9. More-\nover, the thickness of 10 nm is within a factor of \u00192\nof the characteristic exchange lengthp\n2Aex=\u00160M2s\u00195\nnm, assuming an exchange constant of Aex\u00192 pJ/m in\nLSMO (Ref. 43), so standing spin-wave modes are not\nexpected.\nBroadband FMR measurements were performed at\nroom temperature. The \flm sample was placed face-\ndown on a coplanar waveguide with a center conductor\nwidth of 250 \u0016m. Each FMR spectrum was acquired at a\nconstant excitation frequency while sweeping the exter-\nnal magnetic \feld H. The \feld derivative of the FMR\nabsorption intensity (e.g., Fig. 2) was acquired using an\nrf diode combined with an ac (700 Hz) modulation \feld.\nEach FMR spectrum was \ftted with the derivative of the\nsum of the symmetric and antisymmetric Lorentzians, as\nshown in Fig. 2, from which the resonance \feld HFMR\nand half-width-at-half-maximum linewidth \u0001 Hwere ex-\ntracted. In some spectra (e.g., Fig. 2(b)), a small sec-\nondary mode in addition to the main FMR mode was\nobserved. We \ft such a spectrum to a superposition of\ntwo modes, each represented by a generalized Lorentzian\nderivative, and analyze only the HFMR and \u0001Hof the\nlarger-amplitude main FMR mode. The secondary mode\nis not a standing spin-wave mode because it appears\nabove or below the resonance \feld of the main mode\nHFMR with no systematic trend in \feld spacing. We\nattribute the secondary mode to regions in the \flm with3\n0.360.400.440.48\n  0Meff (T)\n0 5 10 15 201.952.002.05\ntSRO (nm)\n  gop\n0 5 10 15 200.00.20.40.60.81.01.2\nLSMO/SRO\nLSMO\n  0HFMR (T)\nf (GHz)(a) (b) \n(c) \nFigure 3. (a) Out-of-plane resonance \feld\nHFMR versus excitation frequency ffor\na single-layer LSMO(10 nm) \flm and a\nLSMO(10 nm)/SRO(3 nm) bilayer. The\nsolid lines indicate \fts to the data using\nEq. 1. (b,c) SRO-thickness dependence of\nthe out-of-plane Land\u0013 e g-factor (b) and ef-\nfective saturation magnetization Me\u000b(c).\nThe dashed lines indicate the values aver-\naged over all the data shown.\nslightly di\u000berent Msor magnetic anisotropy. More pro-\nnounced inhomogeneity-induced secondary FMR modes\nhave been observed in epitaxial magnetic \flms in prior\nreports22,44.\nIII. OUT-OF-PLANE FMR AND ESTIMATION\nOF SPIN TRANSPORT PARAMETERS\nOut-of-plane FMR allows for conceptually simpler ex-\ntraction of the static and dynamic magnetic properties\nof a thin-\flm sample. For \ftting the frequency depen-\ndence ofHFMR, the Land\u0013 e g-factor gopand e\u000bective sat-\nuration magnetization Me\u000bare the only adjustable pa-\nrameters in the out-of-plane Kittel equation. The fre-\nquency dependence of \u0001 Hfor out-of-plane FMR arises\nsolely from Gilbert damping, so that the conventional\nmodel of spin pumping8,9,38,39can be used to analyze the\ndata without complications from non-Gilbert damping.\nThis consideration is particularly important because the\nlinewidths of our LSMO(/SRO) \flms in in-plane FMR\nmeasurements are dominated by highly anisotropic non-\nGilbert damping (as shown in Sec. IV). Furthermore, a\nsimple one-dimensional, time-independent model of spin\npumping outlined by Boone et al.38is applicable in the\nout-of-plane con\fguration, since the precessional orbit of\nthe magnetization is circular to a good approximation.\nThis is in contrast with the in-plane con\fguration with\na highly elliptical orbit from a large shape anisotropy\n\feld. By taking advantage of the simplicity in out-of-\nplane FMR, we \fnd that the Gilbert damping parame-\nter in LSMO is approximately doubled with the addition\nof a su\u000eciently thick SRO overlayer due to spin pump-\ning. Our results indicate that spin-current transmission\nat the LSMO/SRO interface is comparable to previously\nreported ferromagnet/conductor bilayers and that spin\ndi\u000busion length in SRO is <\u00181 nm.\nWe \frst quantify the static magnetic properties of\nLSMO(/SRO) from the frequency dependence of HFMR.\nThe Kittel equation for FMR in the out-of-plane con\fg-\nuration takes a simple linear form,\nf=gop\u0016B\nh\u00160(HFMR\u0000Me\u000b); (1)\nwhere\u00160is the permeability of free space, \u0016Bis the Bohrmagneton, and his the Planck constant. As shown in\nFig. 3(a), we only \ft data points where \u00160HFMR is at\nleast 0.2 T above \u00160Me\u000bto ensure that the \flm is sat-\nurated out-of-plane. Figures 3(b) and (c) plot the ex-\ntractedMe\u000bandgop, respectively, each exhibiting no\nsigni\fcant dependence on SRO thickness tSROto within\nexperimental uncertainty. The SRO overlayer therefore\nevidently does not modify the bulk magnetic proper-\nties of LSMO, and signi\fcant interdi\u000busion across the\nSRO/LSMO interface can be ruled out. The averaged\nMe\u000bof 330\u000610 kA/m (\u00160Me\u000b= 0:42\u00060:01 T) is close\ntoMsobtained from static magnetometery and implies\nnegligible out-of-plane magnetic anisotropy; we thus as-\nsumeMs=Me\u000bin all subsequent analyses. The SRO-\nthickness independence of gop, averaging to 2 :01\u00060:01,\nimplies that the SRO overlayer does not generate a signif-\nicant orbital contribution to magnetism in LSMO. More-\nover, the absence of detectable change in gopwith in-\ncreasingtSRO may indicate that the imaginary compo-\nnent of the spin-mixing conductance8,9is negligible at\nthe LSMO/SRO interface.\nThe Gilbert damping parameter \u000bis extracted from\nthe frequency dependence of \u0001 H(e.g., Figure 4(a)) by\n\ftting the data with the standard linear relation,\n\u0001H= \u0001H0+h\ngop\u0016B\u000bf: (2)\nThe zero-frequency linewidth \u0001 H0is typically attributed\nto sample inhomogeneity. We observe sample-to-sample\nvariation of \u00160\u0001H0in the range\u00191\u00004 mT with no\nsystematic correlation with tSRO or the slope in Eq. 2.\nMoreover, similar to the analysis of HFMR, we only \ft\ndata obtained at \u00150.2 T above \u00160Me\u000bto minimize spu-\nrious broadening of \u0001 Hat low \felds. The linear slope of\n\u0001Hplotted against frequency up to 20 GHz is therefore\na reliable measure of \u000bdecoupled from \u0001 H0in Eq. 2.\nFigure 4(a) shows an LSMO single-layer \flm and an\nLSMO/SRO bilayer with similar \u0001 H0. The slope, which\nis proportional to \u000b, is approximately a factor of 2 greater\nfor LSMO/SRO. Figure 4(b) summarizes the dependence\nof\u000bon SRO-thickness, tSRO. For LSMO single-layer\n\flms we \fnd \u000b= (0:9\u00060:2)\u000210\u00003, which is on the same\norder as previous reports of LSMO thin \flms21{23,26.\nThis low damping is also comparable to the values re-4\n0 5 10 15 200123\n   (10-3)\ntSRO (nm)\n1/Gext 1/G↑↓m LSMO SRO\n0 5 10 15 201.01.52.02.53.0\nLSMO\n  0H (mT)\nf (GHz)LSMO/SRO(a) (b) (c) \nFigure 4. (a) Out-of-plane FMR linewidth \u0001 Hversus excitation frequency for LSMO(10 nm) and LSMO(10 nm)/SRO(3 nm).\nThe solid lines indicate \fts to the data using Eq. 2. (b) Gilbert damping parameter \u000bversus SRO thickness tSRO. The solid\ncurve shows a \ft to the di\u000busive spin pumping model (Eq. 5). (c) Schematic of out-of-plane spin pumping and the equivalent\n\\spin circuit.\"\nported in Heusler alloy thin \flms45,46and may arise from\nthe half-metal-like band structure of LSMO (Ref. 47).\nLSMO can thus be an e\u000ecient source of spin current\ngenerated resonantly by microwave excitation.\nWith a few-nanometer thick overlayer of SRO, \u000bin-\ncreases to\u00192\u000210\u00003(Fig. 4(b)). This enhanced damping\nwith the addition of SRO overlayer may arise from (1)\nspin scattering48,49at the LSMO/SRO interface or (2)\nspin pumping8,9where nonequilibrium spins from LSMO\nare absorbed in the bulk of the SRO layer. Here, we\nassume that interfacial spin scattering is negligible, since\n<\u00181 nm of SRO overlayer does not enhance \u000bsigni\fcantly\n(Fig. 4(b)). This is in contrast with the pronounced in-\nterfacial e\u000bect in ferromagnet/Pt bilayers48,49, in which\neven<1 nm of Pt can increase \u000bby as much as a fac-\ntor of\u00192 (Refs. 50{52). In the following analysis and\ndiscussion, we show that spin pumping alone is su\u000ecient\nfor explaining the enhanced damping in LSMO with an\nSRO overlayer.\nWe now analyze the data in Fig. 4(b) using a one-\ndimensional model of spin pumping based on di\u000busive\nspin transport38,39. The resonantly-excited magnetiza-\ntion precession in LSMO generates non-equilibrium spins\npolarized along ^ m\u0002d ^m=dt, which is transverse to the\nmagnetization unit vector ^ m. This non-equilibrium spin\naccumulation di\u000buses out to the adjacent SRO layer\nand depolarizes exponentially on the characteristic length\nscale\u0015s. The spin current density ~jsat the LSMO/SRO\ninterface can be written as38,53\n~jsjinterface =~2\n2e2^m\u0002d^m\ndt\u0010\n1\nG\"#+1\nGext\u0011; (3)\nwhere ~is the reduced Planck constant, G\"#is the inter-\nfacial spin-mixing conductance per unit area, and Gextis\nthe spin conductance per unit area in the bulk of SRO.\nIn Eq. 3, 1/ G\"#and 1/Gextconstitute spin resistors in\nseries such that the spin transport from LSMO to SRO\ncan be regarded analogously as a \\spin circuit,\" as il-\nlustrated in Fig. 4(c). In literature, these interfacialand bulk spin conductances are sometimes lumped to-\ngether as an \\e\u000bective spin-mixing conductance\" Ge\u000b\n\"#=\n(1=G\"#+ 1=Gext)\u00001(Refs. 10{13, 16, 20, 23, 26, 44). We\nalso note that the alternative form of the (e\u000bective) spin-\nmixing conductance g(e\u000b)\ne\u000b, with units of m\u00002, is related to\nG(e\u000b)\n\"#, with units of \n\u00001m\u00002, byg(e\u000b)\ne\u000b= (h=e2)G(e\u000b)\n\"#\u0019\n26 k\n\u0002G(e\u000b)\n\"#.\nThe functional form of Gextis obtained by solving the\nspin di\u000busion equation with appropriate boundary condi-\ntions38,39,53. In the case of a ferromagnet/nonmagnetic-\nmetal bilayer, we obtain\nGext=1\n2\u001aSRO\u0015stanh\u0012tSRO\n\u0015s\u0013\n; (4)\nwhere\u001aSROis the resistivity of SRO, tSROis the thick-\nness of the SRO layer, and \u0015sis the di\u000busion length of\npumped spins in SRO. Finally, the out\row of spin cur-\nrent (Eq. 3) is equivalent to an enhancement of Gilbert\ndamping9with respect to \u000b0of LSMO with tSRO = 0\nsuch that\n\u000b=\u000b0+gop\u0016B~\n2e2MstLSMO\u00141\nG\"#+ 2\u001aSRO\u0015scoth\u0012tSRO\n\u0015s\u0013\u0015\u00001\n:\n(5)\nThus, two essential parameters governing spin transport\nG\"#and\u0015scan be estimated by \ftting the SRO-thickness\ndependence of \u000b(Fig. 4(b)) with Eq. 5.\nIn carrying out the \ft, we \fx \u000b0= 0:9\u000210\u00003. We note\nthat\u001aSROincreases by an order of magnitude compared\nto the bulk value of \u00192\u000210\u00006\nm astSROis reduced to\na few nm; also, at thicknesses of 3 monolayers ( \u00191.2 nm)\nor below, SRO is known to be insulating54. We there-\nfore use the tSRO-dependent \u001aSRO shown in Appendix\nA while assuming \u0015sis constant. An alternative \ftting\nmodel that assumes a constant \u001aSRO, which is a common\napproach in literature, is discussed in Appendix A.\nThe curve in Fig. 4(b) is generated by Eq. 5 with G\"#=\n1:6\u00021014\n\u00001m\u00002and\u0015s= 0:5 nm. Given the scatter of5\n170175180185\n170\n175\n180\n185[010]\n[110]\n[100] \n  \n0HFMR (mT)\nLSMO\nLSMO/SRO\n0 5 10 15 200100200300400500\n  0HFMR (mT)\nf (GHz)H||[100]\nH||[110](a) (b) (c) \n0 5 10 15 201.952.002.05\ntSRO (nm)\n  gip\n-6-4-20\n  0H||,4 (mT)\n(d) \n14 15330360 \n \n  \nFigure 5. (a) Angular dependence of HFMR at 9 GHz for LSMO(10 nm) and LSMO(10 nm)/SRO(7 nm). The solid curves\nindicate \fts to the data using Eq. 6. (b) Frequency dependence of HFMR for LSMO(10 nm)/SRO(7 nm) with \feld applied in the\n\flm plane along the [100] and [110] directions. Inset: close-up of HFMR versus frequency around 14-15 GHz. In (a) and (b), the\nsolid curves show \fts to the Kittel equation (Eq. 6). (c,d) SRO-thickness dependence of the in-plane cubic magnetocrystalline\nanisotropy \feld (c) and in-plane Land\u0013 e g-factor (d). The dashed lines indicate the values averaged over all the data shown.\nthe experimental data, acceptable \fts are obtained with\nG\"#\u0019(1:2\u00002:5)\u00021014\n\u00001m\u00002and\u0015s\u00190:3\u00000:9\nnm. The estimated ranges of G\"#and\u0015salso depend\nstrongly on the assumptions behind the \ftting model.\nFor example, as shown in Appendix A, the constant- \u001aSRO\nmodel yields G\"#>\u00183\u00021014\n\u00001m\u00002and\u0015s\u00192:5 nm.\nNevertheless, we \fnd that the estimated G\"#\nis on the same order of magnitude as those\nof various ferromagnet/transition-metal heterostruc-\ntures39,55,56, signifying that the LSMO/SRO interface\nis reasonably transparent to spin current. More impor-\ntantly, the short \u0015simplies the presence of strong spin-\norbit coupling that causes rapid spin scattering within\nSRO. This \fnding is consistent with a previous study on\nSRO at low temperature in the ferromagnetic state show-\ning extremely fast spin relaxation with Gilbert damping\n\u000b\u00181 (Ref. 28). The short \u0015sindicates that SRO may be\nsuitable as a spin sink or detector in all-oxide spintronic\ndevices.\nIV. IN-PLANE FMR AND ANISOTROPIC\nTWO-MAGNON SCATTERING\nIn epitaxial thin \flms, the analysis of in-plane FMR\nis generally more complicated than that of out-of-plane\nFMR. High crystallinity of the \flm gives rise to a non-\nnegligible in-plane magnetocrystalline anisotropy \feld,\nwhich manifests in an in-plane angular dependence of\nHFMR and introduces another adjustable parameter in\nthe nonlinear Kittel equation for in-plane FMR. More-\nover, \u0001Hin in-plane FMR of epitaxial thin \flms often\ndepends strongly on the magnetization orientation and\nexhibits nonlinear scaling with respect to frequency due\nto two-magnon scattering, a non-Gilbert mechanism for\ndamping23,32{37. We indeed \fnd that damping of LSMO\nin the in-plane con\fguration is anisotropic and domi-\nnated by two-magnon scattering. We also observe ev-idence of enhanced two-magnon scattering with added\nSRO layers, which may be due to spin pumping from\nnonuniform magnetization precession.\nFigure 5(a) plots HFMR of a single-layer LSMO \flm\nand an LSMO/SRO bilayer as a function of applied \feld\nangle within the \flm plane. For both samples, we observe\nclear four-fold symmetry, which is as expected based on\nthe epitaxial growth of LSMO on the cubic LSAT(001)\nsubstrate. Similar to previous FMR studies of LSMO on\nSrTiO 3(001)57,58, the magnetic hard axes (corresponding\nto the axes of higher HFMR) are alongh100i. The in-\nplane Kittel equation for thin \flms with in-plane cubic\nmagnetic anisotropy is59,\nf=gip\u0016B\nh\u00160\u0002\nHFMR +Hjj;4cos(4\u001e)\u00031\n2\u0002\n\u0014\nHFMR +Me\u000b+1\n4Hjj;4(3 + cos(4\u001e))\u00151\n2\n;\n(6)\nwheregipis the Land\u0013 e g-factor that is obtained from in-\nplane FMR data, Hjj;4is the e\u000bective cubic anisotropy\n\feld, and\u001eis the in-plane \feld angle with respect to\nthe [100] direction. Given that LSMO is magnetically\nvery soft (coercivity on the order of 0.1 mT) at room\ntemperature, we assume that the magnetization is par-\nallel to the \feld direction, particularly with \u00160H\u001d10\nmT. In \ftting the angular dependence (e.g., Fig. 5(a))\nand frequency dependence (e.g., Fig. 5(b)) of HFMR to\nEq. 6, we \fx Me\u000bat the values obtained from out-of-\nplane FMR (Fig. 3(b)) so that Hjj;4andgipare the\nonly \ftting parameters. For the two samples shown in\nFig. 5(a), the \fts to the angular dependence and fre-\nquency dependence data yield consistent values of Hjj;4\nandgip. For the rest of the LSMO(/SRO) samples, we\nuse the frequency dependence data with Hjj[100] and\nHjj[110] to extract these parameters. Figures 5(c) and\n(d) show that Hjj;4andgip, respectively, exhibit no sys-\ntematic dependence on tSRO, similar to the \fndings from6\nout-of-plane FMR (Figs. 3(b),(c)). The in-plane cubic\nmagnetocrystalline anisotropy in LSMO(/SRO) is rela-\ntively small, with \u00160Hjj;4averaging to\u00192.5 mT.gipav-\nerages out to 1 :99\u00060:02, which is consistent with gop\nfound from out-of-plane FMR.\nWhile the magnetocrytalline anisotropy in\nLSMO(/SRO) is found to be modest and indepen-\ndent oftSRO, we observe much more pronounced\nin-plane anisotropy and tSRO dependence in linewidth\n\u0001H, as shown in Figs. 6(a) and (b). Figure 6(a)\nindicates that the in-plane dependence of \u0001 His four-\nfold symmetric for both LSMO(10 nm) and LSMO(10\nnm)/SRO(7 nm). \u0001 His approximately a factor of 2\nlarger when the sample is magnetized along h100icom-\npared to when it is magnetized along h110i. One might\nattribute this pronounced anisotropy to anisotropic\nGilbert damping60, such that the sample magnetized\nalong the hard axes h100imay lead to stronger damp-\ning. However, we \fnd no general correlation between\nmagnetocrystalline anisotropy and anisotropic \u0001 H: As\nwe show in Appendix B, LSMO grown on NdGaO 3(110)\nwith pronounced uniaxial magnetocrystalline anisotropy\nexhibits identical \u0001 Hwhen magnetized along the easy\nand hard axes. Moreover, whereas Gilbert damping\nshould lead to a linear frequency dependence of \u0001 H,\nfor LSMO(/SRO) the observed frequency dependence\nof \u0001His clearly nonlinear as evidenced in Fig. 6(b).\nThe pronounced anisotropy and nonlinear frequency\ndependence of \u0001 Htogether suggest the presence of a\ndi\u000berent damping mechanism.\nA well-known non-Gilbert damping mechanism in\nhighly crystalline ultrathin magnetic \flms is two-magnon\nscattering23,32{37,40,61,62, in which uniformly precessing\nmagnetic moments (a spin wave, or magnon mode, with\nwavevector k= 0) dephase to a k6= 0 magnon mode with\nadjacent moments precessing with a \fnite phase di\u000ber-\nence. By considering both exchange coupling (which re-\nsults in magnon energy proportional to k2) and dipolar\ncoupling (magnon energy proportional to \u0000jkj) among\nprecessing magnetic moments, the k= 0 andk6= 0\nmodes become degenerate in the magnon dispersion re-\nlation61as illustrated in Fig. 6(c).\nThe transition from k= 0 tok6= 0 is activated by\ndefects that break the translational symmetry of the\nmagnetic system by localized dipolar \felds40,61,62. In\nLSMO(/SRO), the activating defects may be faceted such\nthat two-magnon scattering is more pronounced when\nthe magnetization is oriented along h100i. One possibil-\nity is that LSMO thin \flms naturally form pits or islands\nfaceted alongh100iduring growth. However, we are un-\nable to consistently observe signs of such faceted defects\nin LSMO(/SRO) samples with an atomic force micro-\nscope (AFM). It is possible that these crystalline defects\nare smaller than the lateral resolution of our AFM setup\n(<\u001810 nm) or that these defects are not manifested in sur-\nface topography. Such defects may be point defects or\nnanoscale clusters of distinct phases that are known to\nexist intrinsically even in high-quality crystals of LSMO(Ref. 63).\nAlthough the de\fnitive identi\fcation of defects that\ndrive two-magnon scattering would require further in-\nvestigation, we can rule out (1) atomic step terraces\nand (2) mis\ft dislocations as sources of anisotropic two-\nmagnon scattering. (1) AFM shows that the orienta-\ntion and density of atomic step terraces di\u000ber randomly\nfrom sample to sample, whereas the anisotropy in \u0001 H\nis consistently cubic with larger \u0001 HforHjjh100ithan\nHjjh110i. This is in agreement with the recent study\nby Lee et al. , which shows anisotropic two-magnon scat-\ntering in LSMO to be independent of regularly-spaced\nparallel step terraces on a bu\u000bered-oxide etched SrTiO 3\nsubstrate23. (2) Although Woltersdorf and Heinrich have\nfound that mis\ft dislocations in Fe/Pd grown on GaAs\nare responsible for two-magnon scattering33, such dis-\nlocations are expected to be virtually nonexistent in\nfully strained LSMO(/SRO) \flms on the closely-latticed\nmatched LSAT substrates41,42.\nWe assume that the in-plane four-fold anisotropy and\nnonlinear frequency dependence of \u0001 Hare entirely due\nto two-magnon scattering. For a sample magnetized\nalong a given in-plane crystallographic axis hhk0i=h100i\norh110i, the two-magnon scattering contribution to \u0001 H\nis given by40\n\u0001Hhhk0i\n2m = \u0000hhk0i\n2m sin\u00001sp\nf2+ (fM=2)2\u0000fM=2p\nf2+ (fM=2)2+fM=2;(7)\nwherefM= (gip\u0016B=h)\u00160Msand \u0000hhk0i\n2m is the two-\nmagnon scattering parameter. The angular dependence\nof \u0001His \ftted with33\n\u0001H= \u0001H0+h\ngip\u0016B\u000bf\n+ \u0001Hh100i\n2m cos2(2\u001e) + \u0001Hh110i\n2m cos2(2[\u001e\u0000\u0019\n4]):(8)\nSimilarly, the frequency dependence of \u0001 Hwith the sam-\nple magnetized along [100] or [110], i.e., \u001e= 0 or\u0019=4,\nis well described by Eqs. 7 and 8. In principle, it should\nbe possible to \ft the linewidth data with \u0001 H0,\u000b, and\n\u00002mas adjustable parameters. In practice, the \ft car-\nried out this way is overspeci\fed such that wide ranges\nof these parameters appear to \ft the data. We there-\nfore impose a constraint on \u000bby assuming that Gilbert\ndamping for LSMO(/SRO) is isotropic: For each SRO\nthicknesstSRO,\u000bis \fxed to the value estimated from\nthe \ft curve in Fig. 4(c) showing out-of-plane FMR data.\n(This assumption is likely justi\fed, since the damping\nfor LSMO(10 nm) on NdGdO 3(110) with strong uniaxial\nmagnetic anisotropy is identical for the easy and hard\ndirections, as shown in Appendix B.) To account for the\nuncertainty in the Gilbert damping in Fig. 4(c), we vary \u000b\nby\u000625% for \ftting the frequency dependence of in-plane\n\u0001H. Examples of \fts using Eqs. 7 and 8 are shown in\nFig. 6(a),(b).\nFigure 6(d) shows that the SRO overlayer enhances\nthe two-magnon scattering parameter \u0000 2mby up to a7\n0 5 10 15 2004812\nLSMOLSMO/SRO\n  0H (mT)\nf (GHz)(a) (b) \n(c) (d) \n  \nFMR \nfreq. \nk f \nk=0 k≠0 \n0 5 10 15 200102030\n  02m (mT)\ntSRO (nm)H||[100]\nH||[110]\n0510\n0\n5\n10\nLSMO\nLSMO/SRO[110][010]\n[100]\n   0H (mT)\nFigure 6. (a) In-plane angular dependence of\nlinewidth \u0001H at 9 GHz for LSMO(10 nm) and\nLSMO(10 nm)/SRO(7 nm). The solid curves\nindicate \fts to Eq. 8. (b) Frequency depen-\ndence of \u0001H for LSMO(10 nm) and LSMO(10\nnm)/SRO(7 nm) with Happlied along the [100]\ndirection. The solid curves indicate \fts to\nEq. 7. The dashed and dotted curves indicate\nestimated two-magnon and Gilbert damping\ncontributions, respectively. (c) Schematic of a\nspin wave dispersion curve (when the magne-\ntization is in-plane and has a \fnite component\nparallel to the spin wave wavevector k) and two-\nmagnon scattering. (d) Two-magnon scattering\ncoe\u000ecient \u0000 2m, estimated for the cases with H\napplied along the [100] and [110] axes, plotted\nagainst SRO thickness tSRO. The dashed curve\nis the same as that in Fig. 4(c) scaled to serve\nas a guide for the eye for \u0000 2mwith H along\n[100].\nfactor of\u00192 forHjj[100]. By contrast, for Hjj[110], al-\nthough LSMO/SRO exhibits enhanced \u0001 Hcompared to\nLSMO, the enhancement in \u0000 2mis obscured by the un-\ncertainty in Gilbert damping. In Table I, we summa-\nrize the Gilbert and two-magnon contributions to \u0001 H\nfor LSMO single layers and LSMO/SRO (averaged val-\nues for samples with tSRO>4 nm) with Hjj[100] and\nHjj[110]. Comparing the e\u000bective spin relaxation rates,\n(gip\u0016B=h)\u00160Ms\u000band (gip\u0016B=h)\u00160\u00002m, reveals that two-\nmagnon scattering dominates over Gilbert damping.\nWe now speculate on the mechanisms behind the\nenhancement in \u0000 2min LSMO/SRO, particularly for\nHjj[100]. One possibility is that SRO interfaced with\nLSMO directly increases the rate of two-magnon scat-\ntering, perhaps due to formation of additional defects at\nthe surface of LSMO. If this were the case we might ex-\npect a signi\fcant increase and saturation of \u0000 2mat small\ntSRO. However, in reality, \u0000 2mincreases for tSRO>1 nm\n(Fig. 6(d)), which suggests spin scattering in the bulk\nof SRO. We thus speculate another mechanism, where\nk6= 0 magnons in LSMO are scattered by spin pump-\nTable I. Spin relaxation rates extracted from in-plane FMR\n(106s\u00001)\nLSMO LSMO/SRO*\nGilbert:gip\u0016B\nh\u00160Ms\u000b 11\u00062 23\u00064\ntwo-magnon:gip\u0016B\nh\u00160\u00002m(Hjj[100]) 290\u000650 550\u0006100\ntwo-magnon:gip\u0016B\nh\u00160\u00002m(Hjj[110]) 140\u000660 250\u000660\n* Averaged over samples with tSRO>4 nm.ing into SRO. As shown by the guide-for-the-eye curve\nin Fig. 6(d), the tSROdependence of \u0000 2m(forHjj[100])\nmay be qualitatively similar to the tSROdependence of\n\u000bmeasured from out-of-plane FMR (Fig. 4(c)); this cor-\nrespondence would imply that the same spin pumping\nmechanism, which is conventionally modeled to act on\nthek= 0 mode, is also operative in the degenerate k6= 0\nmagnon mode in epitaxial LSMO. Indeed, previous stud-\nies have electrically detected the presence of spin pump-\ning fromk6= 0 magnons by the inverse spin-Hall e\u000bect in\nY3Fe5O12/Pt bilayers64{66. However, we cannot conclu-\nsively attribute the observed FMR linewidth broadening\nin LSMO/SRO to such k6= 0 spin pumping, since it\nis unclear whether faster relaxation of k6= 0 magnons\nshould necessarily cause faster relaxation of the k= 0\nFMR mode. Regardless of its origin, the pronounced\nanisotropic two-magnon scattering introduces additional\ncomplexity to the analysis of damping in LSMO/SRO\nand possibly in other similar ultrathin epitaxial magnetic\nheterostructures.\nV. SUMMARY\nWe have demonstrated all-oxide perovskite bilayers\nof LSMO/SRO that form spin-source/spin-sink systems.\nFrom out-of-plane FMR, we deduce a low Gilbert damp-\ning parameter of \u00191\u000210\u00003for LSMO. The two-fold en-\nhancement in Gilbert damping with an SRO overlayer\nis adequately described by the standard model of spin\npumping based on di\u000busive spin transport. We ar-\nrive at an estimated spin-mixing conductance G\"#\u0019\n(1\u00002)\u00021014\n\u00001m\u00002and spin di\u000busion length \u0015s<\u00181\nnm, which indicate reasonable spin-current transparency\nat the LSMO/SRO interface and strong spin scattering8\nwithin SRO. From in-plane FMR, we reveal pronounced\nnon-Gilbert damping, attributed to two-magnon scatter-\ning, which results in a nonlinear frequency dependence\nand anisotropy in linewidth. The magnitude of two-\nmagnon scattering increases with the addition of an SRO\noverlayer, pointing to the presence of spin pumping from\nnonuniform spin wave modes. Our \fndings lay the foun-\ndation for understanding spin transport and magneti-\nzation dynamics in epitaxial complex oxide heterostruc-\ntures.\nACKNOWLEDGEMENTS\nWe thank Di Yi, Sam Crossley, Adrian Schwartz, Han-\nkyu Lee, and Igor Barsukov for helpful discussions, and\nTianxiang Nan and Nian Sun for the design of the copla-\nnar waveguide. This work was funded by the National\nSecurity Science and Engineering Faculty Fellowship of\nthe Department of Defense under Contract No. N00014-\n15-1-0045.\nAPPENDIX A: SPIN PUMPING AND SRO\nRESISTIVITY\nWhen \ftting the dependence of the Gilbert damping\nparameter\u000bon spin-sink thickness, a constant bulk re-\nsistivity for the spin sink layer is often assumed in lit-\nerature. By setting the resistivity of SRO to the bulk\nvalue\u001aSRO= 2\u000210\u00006\nm and \ftting the \u000b-versus-tSRO\ndata (Fig. 4(c) and reproduced in Fig. 7(a)) to Eq. 5,\nwe arrive at G\"#>\u00183\u00021014\n\u00001m\u00002and\u0015s\u00192:5 nm.\nThe \ft curve is insensitive to larger values of G\"#because\nthe bulk spin resistance 1/ Gext, with the relatively large\nresistivity of SRO, dominates over the interfacial spin re-\nsistance 1/G\"#(see Eqs. 4 and 5). As shown by the dot-\nted curve in Fig. 7, this simple constant- \u001aSROmodel ap-\npears to mostly capture the tSRO-dependence of \u000b. This\nmodel of course indicates \fnite spin pumping at even\nvery small SRO thickness <\u00181 nm, which is likely non-\nphysical since SRO should be insulating in this thickness\nregime54. Indeed,\u0015sestimated with this model should\nprobably be considered a phenomenological parameter:\nAs pointed out by recent studies, strictly speaking, a\nphysically meaningful estimation of \u0015sshould take into\naccount the thickness dependence of the resistivity of the\nspin sink layer39,56,67, especially for SRO whose thickness\ndependence of resistivity is quite pronounced.\nFigure 7(b) plots the SRO-thickness dependence of the\nresistivity of SRO \flms deposited on LSAT(001) mea-\nsured in the four-point van der Pauw geometry. The\ntrend can be described empirically by\n\u001aSRO=\u001ab+\u001as\ntSRO\u0000tth; (9)\nwhere\u001ab= 2\u000210\u00006\nm is the resistivity of SRO in the\nbulk limit,\u001as= 1:4\u000210\u000014\nm2is the surface resistivity\n0 5 10 15 200123\n   (10-3)\ntSRO (nm)\n0 10 20 3010-61x10-51x10-4\n  SRO (m)\ntSRO (nm)(a) (b) Figure 7. (a) Gilbert damping parameter \u000bversus SRO\nthicknesstSRO. The solid curve is a \ft taking into account\nthetSROdependence of SRO resistivity, whereas the dotted\ncurve is a \ft assuming a constant bulk-like SRO resistivity.\n(b) Resistivity of SrRuO 3\flms on LSAT(001) as a function\nof thickness.\n0 5 10 15 20012345\nhard\neasy\n  0H (mT)\nf (GHz)\n0 5 10 15 200123450H (mT)\n  \nf (GHz)hard\neasy(b) (a) \nFigure 8. Frequency dependence of in-plane FMR\nlinewidth \u0001 Hof LSMO(10 nm) on (a) LSAT(001) and (b)\nNdGaO 3(110), with the magnetization along the magnetic\neasy and hard axes. The solid curves are \fts to Eq. 7 with\nthe Gilbert damping parameter \u000b\fxed to 0:9\u000210\u00003.\ncoe\u000ecient, and tth= 1 nm is the threshold thickness\nbelow which the SRO layer is essentially insulating. The\nvalue oftthagrees with literature reporting that SRO\nis insulating at thickness of 3 monolayers ( \u00191.2 nm) or\nbelow54. Given the large deviation of \u001aSROfrom the bulk\nvalue, especially at small tSRO, the trend in Fig. 7(b)\nsuggests that taking into account the tSRO dependence\nof\u001aSROis a sensible approach.\nAPPENDIX B: IN-PLANE DAMPING OF LSMO\nON DIFFERENT SUBSTRATES\nIn Fig. 8, we compare the frequency dependence of \u0001 H\nfor 10-nm thick LSMO \flms deposited on di\u000berent sub-\nstrates: LSAT(001) and NdGaO 3(110). (NdGaO 3is an\northorhombic crystal and has ap\n2-pseudocubic param-\neter of\u00193.86 \u0017A, such that (001)-oriented LSMO grows\non the (110)-oriented surface of NdGaO 3.) As shown\nin Sec. IV, LSMO on LSAT(001) exhibits cubic mag-\nnetic anisotropy within the \flm plane with the h110iand\nh100ias the easy and hard axes, respectively. LSMO\non NdGaO 3(110) exhibits uniaxial magnetic anisotropy\nwithin the \flm plane with [1 \u001610] and [001] of NdGaO 3as9\nthe easy and hard axes, respectively.68Whereas LSMO\non LSAT(001) shows distinct magnitudes of damping\nwhen the \flm is magnetized along the easy and hard\naxes (Figs. 8(a) and 6(a)), in LSMO on NdGaO 3(110)damping is identical for both the easy and hard axes\n(Figs. 8(b)). These results demonstrate that higher\ndamping (wider linewidth) is in general not linked to the\nmagnetic hard axis of LSMO.\n\u0003satorue@stanford.edu\n1R. L. Stamps, S. Breitkreutz, J. \u0017Akerman, A. V. Chu-\nmak, Y. Otani, G. E. W. Bauer, J.-U. Thiele, M. Bowen,\nS. A. Majetich, M. Kl aui, I. L. Prejbeanu, B. Dieny, N. M.\nDempsey, and B. Hillebrands, J. Phys. D. Appl. Phys. 47,\n333001 (2014).\n2A. Ho\u000bmann, IEEE Trans. Magn. 49, 5172 (2013).\n3J. Sinova, S. O. Valenzuela, J. Wunderlich, C. H. Back,\nand T. Jungwirth, Rev. Mod. Phys. 87, 1213 (2015).\n4A. Ho\u000bmann and S. D. Bader, Phys. Rev. Appl. 4, 047001\n(2015).\n5A. Brataas, A. D. Kent, and H. Ohno, Nat. Mater. 11,\n372 (2012).\n6N. Locatelli, V. Cros, and J. Grollier, Nat. Mater. 13, 11\n(2014).\n7A. V. Chumak, V. I. Vasyuchka, A. A. Serga, and B. Hille-\nbrands, Nat. Phys. 11, 453 (2015).\n8Y. Tserkovnyak, A. Brataas, and G. Bauer, Phys. Rev.\nLett.88, 117601 (2002).\n9Y. Tserkovnyak, A. Brataas, and G. Bauer, Phys. Rev. B\n66, 224403 (2002).\n10F. D. Czeschka, L. Dreher, M. S. Brandt, M. Weiler, M. Al-\nthammer, I.-M. Imort, G. Reiss, A. Thomas, W. Schoch,\nW. Limmer, H. Huebl, R. Gross, and S. T. B. Goennen-\nwein, Phys. Rev. Lett. 107, 046601 (2011).\n11Y. Sun, H. Chang, M. Kabatek, Y.-Y. Song, Z. Wang,\nM. Jantz, W. Schneider, M. Wu, E. Montoya, B. Kar-\ndasz, B. Heinrich, S. G. E. te Velthuis, H. Schultheiss, and\nA. Ho\u000bmann, Phys. Rev. Lett. 111, 106601 (2013).\n12W. Zhang, M. B. Jung\reisch, W. Jiang, J. Sklenar, F. Y.\nFradin, J. E. Pearson, J. B. Ketterson, and A. Ho\u000bmann,\nJ. Appl. Phys. 117, 172610 (2015).\n13C. Du, H. Wang, P. C. Hammel, and F. Yang, J. Appl.\nPhys. 117, 172603 (2015).\n14K. Ando, S. Takahashi, J. Ieda, H. Kurebayashi, T. Tryp-\niniotis, C. H. W. Barnes, S. Maekawa, and E. Saitoh, Nat.\nMater. 10, 655 (2011).\n15K. Ando, S. Watanabe, S. Mooser, E. Saitoh, and H. Sir-\nringhaus, Nat. Mater. 12, 622 (2013).\n16M. Jamali, J. S. Lee, J. S. Jeong, F. Mahfouzi, Y. Lv,\nZ. Zhao, B. K. Nikoli\u0013 c, K. A. Mkhoyan, N. Samarth, and\nJ.-P. Wang, Nano Lett. 15, 7126 (2015).\n17P. Zubko, S. Gariglio, M. Gabay, P. Ghosez, and J.-M.\nTriscone, Annu. Rev. Condens. Matter Phys. 2, 141 (2011).\n18H. Y. Hwang, Y. Iwasa, M. Kawasaki, B. Keimer, N. Na-\ngaosa, and Y. Tokura, Nat. Mater. 11, 103 (2012).\n19S. Majumdar and S. van Dijken, J. Phys. D. Appl. Phys.\n47, 034010 (2014).\n20E. Lesne, Y. Fu, S. Oyarzun, J. C. Rojas-S\u0013 anchez, D. C.\nVaz, H. Naganuma, G. Sicoli, J.-P. Attan\u0013 e, M. Jamet,\nE. Jacquet, J.-M. George, A. Barth\u0013 el\u0013 emy, H. Ja\u000br\u0012 es,\nA. Fert, M. Bibes, and L. Vila, Nat. Mater. (2016).\n21G. Y. Luo, C. R. Chang, and J. G. Lin, IEEE Trans. Magn.\n49, 4371 (2013).22G. Y. Luo, M. Belmeguenai, Y. Roussign\u0013 e, C. R. Chang,\nJ. G. Lin, and S. M. Ch\u0013 erif, AIP Adv. 5, 097148 (2015).\n23H. K. Lee, I. Barsukov, A. G. Swartz, B. Kim, L. Yang,\nH. Y. Hwang, and I. N. Krivorotov, AIP Adv. 6, 055212\n(2016).\n24S. M. Haidar, Y. Shiomi, J. Lustikova, and E. Saitoh, Appl.\nPhys. Lett. 107, 152408 (2015).\n25V. A. Atsarkin, B. V. Sorokin, I. V. Borisenko, V. V. Demi-\ndov, and G. A. Ovsyannikov, J. Phys. D. Appl. Phys. 49,\n125003 (2016).\n26M. Wahler, N. Homonnay, T. Richter, A. M uller, C. Eisen-\nschmidt, B. Fuhrmann, and G. Schmidt, Sci. Rep. 6, 28727\n(2016).\n27G. Koster, L. Klein, W. Siemons, G. Rijnders, J. S. Dodge,\nC.-B. Eom, D. H. A. Blank, and M. R. Beasley, Rev. Mod.\nPhys. 84, 253 (2012).\n28M. C. Langner, C. L. S. Kantner, Y. H. Chu, L. M. Martin,\nP. Yu, J. Seidel, R. Ramesh, and J. Orenstein, Phys. Rev.\nLett.102, 177601 (2009).\n29A. Azevedo, L. H. Vilela-Le~ ao, R. L. Rodr\u0013 \u0010guez-Su\u0013 arez,\nA. F. Lacerda Santos, and S. M. Rezende, Phys. Rev. B\n83, 144402 (2011).\n30L. Bai, P. Hyde, Y. S. Gui, C.-M. Hu, V. Vlaminck, J. E.\nPearson, S. D. Bader, and A. Ho\u000bmann, Phys. Rev. Lett.\n111, 217602 (2013).\n31M. Obstbaum, M. H artinger, H. G. Bauer, T. Meier,\nF. Swientek, C. H. Back, and G. Woltersdorf, Phys. Rev.\nB89, 060407 (2014).\n32J. Lindner, K. Lenz, E. Kosubek, K. Baberschke, D. Spod-\ndig, R. Meckenstock, J. Pelzl, Z. Frait, and D. L. Mills,\nPhys. Rev. B 68, 060102 (2003).\n33G. Woltersdorf and B. Heinrich, Phys. Rev. B 69, 184417\n(2004).\n34K. Lenz, H. Wende, W. Kuch, K. Baberschke, K. Nagy,\nand A. J\u0013 anossy, Phys. Rev. B 73, 144424 (2006).\n35K. Zakeri, J. Lindner, I. Barsukov, R. Meckenstock,\nM. Farle, U. von H orsten, H. Wende, W. Keune, J. Rocker,\nS. S. Kalarickal, K. Lenz, W. Kuch, K. Baberschke, and\nZ. Frait, Phys. Rev. B 76, 104416 (2007).\n36I. Barsukov, F. M. R omer, R. Meckenstock, K. Lenz,\nJ. Lindner, S. Hemken to Krax, A. Banholzer, M. K orner,\nJ. Grebing, J. Fassbender, and M. Farle, Phys. Rev. B 84,\n140410 (2011).\n37H. Kurebayashi, T. D. Skinner, K. Khazen, K. Olejnik,\nD. Fang, C. Ciccarelli, R. P. Campion, B. L. Gallagher,\nL. Fleet, A. Hirohata, and A. J. Ferguson, Appl. Phys.\nLett.102, 062415 (2013).\n38C. T. Boone, H. T. Nembach, J. M. Shaw, and T. J. Silva,\nJ. Appl. Phys. 113, 153906 (2013).\n39C. T. Boone, J. M. Shaw, H. T. Nembach, and T. J. Silva,\nJ. Appl. Phys. 117, 223910 (2015).\n40R. Arias and D. L. Mills, Phys. Rev. B 60, 7395 (1999).\n41Y. Takamura, R. V. Chopdekar, E. Arenholz, and\nY. Suzuki, Appl. Phys. Lett. 92, 162504 (2008).10\n42A. Grutter, F. Wong, E. Arenholz, M. Liberati, and\nY. Suzuki, J. Appl. Phys. 107, 09E138 (2010).\n43M. Golosovsky, P. Monod, P. K. Muduli, and R. C. Bud-\nhani, Phys. Rev. B 76, 184413 (2007).\n44M. Haertinger, C. H. Back, J. Lotze, M. Weiler, S. Gepr ags,\nH. Huebl, S. T. B. Goennenwein, and G. Woltersdorf,\nPhys. Rev. B 92, 054437 (2015).\n45S. Mizukami, D. Watanabe, M. Oogane, Y. Ando,\nY. Miura, M. Shirai, and T. Miyazaki, J. Appl. Phys.\n105, 07D306 (2009).\n46P. D urrenfeld, F. Gerhard, J. Chico, R. K. Dumas, M. Ran-\njbar, A. Bergman, L. Bergqvist, A. Delin, C. Gould, L. W.\nMolenkamp, and J. \u0017Akerman, Phys. Rev. B 92, 214424\n(2015).\n47C. Liu, C. K. A. Mewes, M. Chshiev, T. Mewes, and W. H.\nButler, Appl. Phys. Lett. 95, 022509 (2009).\n48H. Nguyen, W. Pratt, and J. Bass, J. Magn. Magn. Mater.\n361, 30 (2014).\n49J.-C. Rojas-S\u0013 anchez, N. Reyren, P. Laczkowski, W. Savero,\nJ.-P. Attan\u0013 e, C. Deranlot, M. Jamet, J.-M. George, L. Vila,\nand H. Ja\u000br\u0012 es, Phys. Rev. Lett. 112, 106602 (2014).\n50S. Azzawi, A. Ganguly, M. Toka\u0018 c, R. M. Rowan-Robinson,\nJ. Sinha, A. T. Hindmarch, A. Barman, and D. Atkinson,\nPhys. Rev. B 93, 054402 (2016).\n51M. Caminale, Phys. Rev. B 94, 014414 (2016).\n52S. Emori, T. Nan, A. M. Belkessam, X. Wang, A. D.\nMatyushov, C. J. Babroski, Y. Gao, H. Lin, and N. X.\nSun, Phys. Rev. B 93, 180402 (2016).\n53M. Polianski and P. Brouwer, Phys. Rev. Lett. 92, 026602\n(2004).\n54J. Xia, W. Siemons, G. Koster, M. R. Beasley, and A. Ka-\npitulnik, Phys. Rev. B 79, 140407 (2009).\n55C.-F. Pai, Y. Ou, L. H. Vilela-Le~ ao, D. C. Ralph, and R. A.\nBuhrman, Phys. Rev. B 92, 064426 (2015).\n56E. Montoya, P. Omelchenko, C. Coutts, N. R. Lee-Hone,\nR. H ubner, D. Broun, B. Heinrich, and E. Girt, Phys. Rev.B94, 054416 (2016).\n57M. Belmeguenai, S. Mercone, C. Adamo, L. M\u0013 echin,\nC. Fur, P. Monod, P. Moch, and D. G. Schlom, Phys.\nRev. B 81, 054410 (2010).\n58\u0017A. Monsen, J. E. Boschker, F. Maci\u0012 a, J. W. Wells,\nP. Nordblad, A. D. Kent, R. Mathieu, T. Tybell, and\nE. Wahlstr om, J. Magn. Magn. Mater. 369, 197 (2014).\n59M. Farle, Reports Prog. Phys. 61, 755 (1998).\n60A. A. Baker, A. I. Figueroa, C. J. Love, S. A. Cavill,\nT. Hesjedal, and G. van der Laan, Phys. Rev. Lett. 116,\n047201 (2016).\n61D. Mills and S. M. Rezende, Spin Damping in Ultrathin\nMagnetic Films, in Spin Dyn. Con\fn. Magn. Struct. II ,\nedited by B. Hillebrands and K. Ounadjela, chap. 2, pp.\n27{58, Springer, 2002.\n62R. D. McMichael, M. D. Stiles, P. J. Chen, and W. F.\nEgelho\u000b, J. Appl. Phys. 83, 7037 (1998).\n63E. Dagotto, J. Burgy, and A. Moreo, Solid State Commun.\n126, 9 (2003).\n64C. W. Sandweg, Y. Kajiwara, A. V. Chumak, A. A. Serga,\nV. I. Vasyuchka, M. B. Jung\reisch, E. Saitoh, and B. Hille-\nbrands, Phys. Rev. Lett. 106, 216601 (2011).\n65G. L. da Silva, L. H. Vilela-Leao, S. M. Rezende, and\nA. Azevedo, Appl. Phys. Lett. 102, 012401 (2013).\n66S. A. Manuilov, C. H. Du, R. Adur, H. L. Wang, V. P.\nBhallamudi, F. Y. Yang, and P. C. Hammel, Appl. Phys.\nLett.107, 042405 (2015).\n67M.-H. Nguyen, D. C. Ralph, and R. A. Buhrman, Phys.\nRev. Lett. 116, 126601 (2016).\n68H. Boschker, M. Mathews, E. P. Houwman, H. Nishikawa,\nA. Vailionis, G. Koster, G. Rijnders, and D. H. A. Blank,\nPhys. Rev. B 79, 214425 (2009)."    },    {        "title": "1612.02360v1.Gilbert_damping_of_magnetostatic_modes_in_a_yttrium_iron_garnet_sphere.pdf",        "content": "Gilbert damping of magnetostatic modes in a yttrium iron garnet sphere\nS. Klingler,1, 2,a)H. Maier-Flaig,1, 2C. Dubs,3O. Surzhenko,3R. Gross,1, 2, 4H. Huebl,1, 2, 4\nS.T.B. Goennenwein,1, 5, 6and M. Weiler1, 2\n1)Walther-Mei\u0019ner-Institut, Bayerische Akademie der Wissenschaften, 85748 Garching,\nGermany\n2)Physik-Department, Technische Universit at M unchen, 85748 Garching, Germany\n3)INNOVENT e.V. Technologieentwicklung, 07745 Jena, Germany\n4)Nanosystems Initiative Munich, 80799 Munich, Germany\n5)Institut f ur Festk orperphysik, Technische Universit at Dresden, 01062 Dresden,\nGermany\n6)Center for Transport and Devices of Emergent Materials, Technische Universit at Dresden, 01062 Dresden,\nGermany\n(Dated: 8 December 2016)\nThe magnetostatic mode (MSM) spectrum of a 300 \u0016m diameter single crystalline sphere of yttrium iron\ngarnet is investigated using broadband ferromagnetic resonance (FMR). The individual MSMs are identi\fed\nvia their characteristic dispersion relations and the corresponding mode number tuples ( nmr) are assigned.\nTaking FMR data over a broad frequency and magnetic \feld range allows to analyze both the Gilbert\ndamping parameter \u000band the inhomogeneous line broadening contribution to the total linewidth of the\nMSMs separately. The linewidth analysis shows that all MSMs share the same Gilbert damping parameter\n\u000b= 2:7(5)\u000210\u00005irrespective of their mode index. In contrast, the inhomogeneous line broadening shows a\npronounced mode dependence. This observation is modeled in terms of two-magnon scattering processes of\nthe MSMs into the spin-wave manifold, mediated by surface and volume defects.\nThe ferrimagnetic insulator yttrium iron garnet (YIG)\nhas numerous applications in technology and funda-\nmental research due to its low intrinsic Gilbert damp-\ning and large spin-wave propagation length.1It is used\nas prototypical material in various experiments in spin\nelectronics2{4and spin caloritronics5,6and is indispens-\nable for microwave technology.\nRecently, YIG spheres attracted attention in the\n\feld of quantum information technology.7{15For exam-\nple, strong coupling between magnons and photons in\nYIG/cavity hybrid systems can be employed for the up-\nand down-conversion of quantum signals between mi-\ncrowave and optical frequencies, enabling a long-range\ntransmission of quantum information between microwave\nquantum circuits.14{16Here, the damping of the mag-\nnetic excitation plays a crucial role, since it limits the\ntime-scale in which energy and information is exchanged\nand stored in the magnon-photon hybrid system.\nOne type of magnetic excitations in YIG spheres17{19\nare magnetostatic modes (MSMs) which resemble stand-\ning spin-wave patterns within the sphere. Although the\nlinewidth of MSMs in YIG spheres has been studied at\n\fxed frequencies in the past,20{22the respective contri-\nbutions of intrinsic Gilbert damping and inhomogeneous\nline broadening23to the total linewidth have not yet been\ninvestigated. In particular, it is not evident from the\nliterature, whether di\u000berent MSMs feature the same or\ndi\u000berent Gilbert damping.24,25\nHere, we report on the study of dynamic properties of\nmultiple MSMs for a 300 \u0016m diameter YIG sphere us-\ning broadband ferromagnetic resonance. The frequency\na)Electronic mail: stefan.klingler@wmi.badw.deand magnetic \feld resolved FMR data allows to separate\nGilbert damping and inhomogeneous line broadening of\nthe MSMs. One and the same Gilbert damping parame-\nter\u000b= 2:7(5)\u000210\u00005is found for all MSMs, independent\nof their particular mode index. However, the inhomoge-\nneous line broadening markedly di\u000bers between the ob-\nserved MSMs. This \fnding is attributed to two-magnon\nscattering processes of the MSMs into the spin-wave man-\nifold, mediated by surface and volume defects.\nThe MSM pro\fles and eigenfrequencies of a magnetic\nsphere can be calculated in the magnetostatic approx-\nimationr\u0002H= 0,17{19using the Landau-Lifshitz-\nGilbert equation (LLG).26,27The resonance frequencies\n\n of the MSMs are obtained by solving the characteristic\nequation:17{19\nn+ 1 +\u00180dPm\nn(\u00180)=d\u00180\nPmn(\u00180)\u0006m\u0017= 0; (1)\nwhere\u00182\n0= 1 + 1=\u0014,\u0014= \n H=\u0000\n\n2\nH\u0000\n2\u0001\n,\u0017=\n\n=\u0000\n\n2\nH\u0000\n2\u0001\n, \nH=\u00160Hi=\u00160Msand \n =!=\r\u0016 0Ms.\nHere,\r=gJ\u0016B=~is the gyromagnetic ratio, gJis the\nLand\u0013 eg-factor,\u0016Bis the Bohr magneton, ~is the reduced\nPlanck constant, \u00160is the vacuum permeability and Ms\nis the saturation magnetization. The angular frequency\nof the applied microwave \feld is denoted as != 2\u0019f.\nThe internal \feld is given by Hi=H0+Hani+Hdemag ,\nwhereH0is the applied static magnet \feld, Haniis the\nanisotropy \feld, and Hdemag =\u0000Ms=3 is the demagneti-\nzation \feld of a sphere.\nThe mode pro\fles of the MSMs have the form of asso-\nciated Legendre polynomials Pm\nn, where the localization\nof the MSMs at the surface is related to the mode index\nn2N.21The indexjmj\u0014ncorresponds to an angular-\nmomentum quantum number of the MSM,28where thearXiv:1612.02360v1  [cond-mat.mtrl-sci]  7 Dec 20162\nbar above the mode index mis used for indices m < 0.\nThe index r\u00150 enumerates the solutions of the char-\nacteristic equation (1) for given nandmfor increasing\nfrequencies.18,29In total, each MSM is uniquely identi-\n\fed by the index tuple ( nmr). For more information and\nplots of the MSM mode patterns, the review of Ref. 19\nis recommended.\nThe Gilbert damping parameter phenomenologically\naccounts for the viscous (linearly frequency-dependent)\nrelaxation of magnetic excitations. Assuming a domi-\nnant Gilbert-type damping for all MSM modes, the full\nlinewidth at half maximum (FWHM) \u0001 f(nmr)of a MSM\nresonance line at frequency f(nmr)\nres is given by:30\n\u0001f(nmr)= 2\u000bf(nmr)\nres + \u0001f(nmr)\n0: (2)\nHere, \u0001f0denotes the inhomogeneous line broadening\ncontributions to the total linewidth. For a two-magnon\nscattering process mediated by volume and surface de-\nfects the latter can be written as:21\n\u0001f(nmr)\n0 = \u0001fm-mF(nmr)+ \u0001f0\n0: (3)\nHere, \u0001fm-maccounts for the two-magnon scattering pro-\ncess of the MSMs into the spin-wave manifold.21,22,31The\nfactorF(nmr)represents the ratio of the linewidth of a\nparticular MSM with respect to the uniform precessing\n(110)-mode.21,22,32,33It therefore accounts for the surface\nsensitivity of the speci\fc mode compared to the (110)-\nmode. The two-magnon scattering processes can be sup-\npressed if a perfectly polished YIG sphere is used, due to\nthe vanishing ability of the system to transfer linear and\nangular momentum from and to the lattice.21The term\n\u0001f0\n0represents a constant contribution to the linewidth\nin which all other frequency-independent broadening ef-\nfects are absorbed. The complete scattering theory used\nin this letter is presented in Ref. 21.\nFig. 1 (a) shows a sketch of the measurement setup.\nThe YIG sphere with a diameter of d= 300\u0016m is placed\nin a disk shaped Vespel sample holder (diameter 6 mm,\nnot shown), which has a centered hole with a diameter\nof 350\u0016m. The sphere in the sample holder is exposed\nto a static magnetic \feld in order to align the easy [111]-\ndirection of the YIG crystal parallel to the \feld direc-\ntion. The orientation of the sphere is subsequently \fxed\nusing photoresist and the alignment is con\frmed by Laue\ndi\u000braction.\nThe oriented YIG sphere is placed on a 50 \n impedance\nmatched coplanar waveguide (CPW) structure. The\nsphere is placed in the middle of the w= 300\u0016m wide\ncenter conductor, with the YIG [110]-axis aligned par-\nallel to the long axis of the center conductor of the\nCPW. Additionally, a pressed crumb of Diphenylpicryl-\nhydrazyl (DPPH) is glued on the center conductor, where\nthe distance between the YIG sphere and the DPPH is\nl\u00191 cm. DPPH is a spin marker with a g-factor34of\ngDPPH = 2:0036(3). The measurement of its resonance\nfrequency\nfDPPH =gDPPH\u0016B\n2\u0019~\u00160HDPPH\n0 (4)\nP1 P2 z, [111] y, [110] \nx, h xVNAelectro magnet top view (a)  side view \nYIG \nDPPH CPW \nH0\nIm ∆S 21   ,Re ∆S 21 (a.u.)  \n-10 0 10\nf-f res  (MHz)(b) (530)-mode  \nH0w\nlaa/2 P1 \nP2 hrf FIG. 1. (a) The CPW with the YIG sphere and the DPPH\nis positioned in the homogeneous \feld of an electromagnet.\nThe CPW is connected to port 1 (P1) and port 2 (P2) of a\nvector network analyzer (VNA). The YIG sphere is placed\non top of the center conductor of the CPW with its [111]-\naxis parallel to the applied magnetic \feld H0inz-direction.\n(b) Typical normalized transmission spectrum of the (530)-\nmode at\u00160H0= 0:8 T (symbols) including a \ft to Eq. (5)\n(lines).\nprovides an independent magnetic \feld reference at the\nsample position, in addition to Hall probe measurements.\nThe static magnetic \feld calculated from the DPPH\nresonance frequency is denoted as HDPPH\n0 . The stray\n\feld originating from the YIG sphere at the location of\nthe DPPH creates a systematic measurement error of\n\u000e\u00160Hstray\u001440\u0016T, as estimated using a dipole approxi-\nmation.\nFor the broadband FMR experiments, the CPW is po-\nsitioned between the pole shoes of an electromagnet with\na maximum \feld strength of j\u00160H0j\u00142:25 T. The pole\nshoe diameter is a= 6 cm, while the pole shoe sepa-\nration isa=2, to ensure a su\u000ecient homogeneity of the\napplied magnetic \felds. The measured radial \feld gra-\ndient creates a systematic \feld measurement error of\n\u000e\u00160Hdisp= 0:3 mT forl= 1 cm displacement from the\ncenter axis.\nThe CPW is connected to port 1 (P1) and port 2 (P2)\nof a vector network analyzer (VNA) and the complex\nscattering parameter S21is recorded as a function of H0\nandf\u001426:5 GHz. The applied microwave power is -\n20 dBm to avoid non-linear e\u000bects causing additional line\nbroadening. The microwave current \rowing along the\ncenter conductor generates a microwave magnetic \feld\npredominately in the x-direction at the location of the\nYIG sphere. This results in an oscillating torque on\nthe magnetization, which is aligned in parallel to the z-\ndirection by the external static \feld H0. Forf=f(nmr)\nres ,\nthe excited resonant precession of the magnetization re-\nsults in an absorption of microwave power.\nIn order to eliminate the e\u000bect of the frequency depen-\ndent background transmission of the CPW, the following\nmeasurement protocol is applied: First, S21is measured\nfor \fxedH0in a frequency range fDPPH\u00061 GHz. Second,\nS21is measured for the same frequency range at a slightly3\nFIG. 2. (a) Normalized transmission magnitude j\u0001S21j\nplotted versus applied magnetic \feld \u00160H0and microwave\nfrequencyfrelative to the DPPH resonance fDPPH . The\ncontrast between the dashed lines is stretched for better vis-\nibility. (b) Calculated and measured dispersions of various\nMSMs (lines and open circles, respectively).\nlarger magnetic \feld H0+ \u0001H0, with\u00160\u0001H0= 100 mT.\nSince for this \feld no YIG and DPPH resonances are\npresent in the observed frequency range, the latter mea-\nsurement contains the pure background transmission.\nThird, the normalized transmission spectra is obtained\nas \u0001S21=S21(H0)=S21(H0+ \u0001H0), which corrects the\nmagnitude and the phase of the signal. This procedure is\nrepeated for all applied magnetic \felds. The transmitted\nmagnitude around the resonance can be expressed as:30\n\u0001S21(f) =A+Bf+Z\n\u0010\nf(nmr)\nres\u00112\n\u0000if2\u0000if\u0001f(nmr):(5)\nHere,Ais a complex o\u000bset parameter, Bis a complex lin-\near background and Zis a complex scaling parameter.35\nFig. 1 (b) exemplary shows the real and imaginary part of\n\u0001S21for the (530)-mode at \u00160H0= 0:8 T. In addition, a\n\ft of Eq. 5 to the data is shown, which adequately models\nthe shape of the resonances.\nFig. 2 (a) shows the normalized transmitted magnitude\nj\u0001S21jas a function of H0andf\u0000fDPPH on a linear\ncolor-coded scale. The frequency axis is chosen relative\nto the DPPH resonance frequency, so that all modes with\na linear dispersion f(nmr)\nres/H0appear as straight lines,whereas modes with a non-linear dispersion are curved.\nNote, that the \feld values displayed on the y-axis repre-\nsent the magnetic \feld strength measured with the Hall\nprobe.\nThe di\u000berent modes appearing in the color plot in\nFig. 2 (a) can be identi\fed in a straightforward manner.\nAt \frst, all visible resonances are \ftted using Eq. (5)\nin order to extract f(nmr)\nres and \u0001f(nmr). Furthermore,\nthe DPPH resonance line is identi\fed as straight line at\nf\u0000fDPPH = 0 MHz and the resonance \felds HDPPH\n0 are\ncalculated using Eq. (4).\nSecond, the straight lines at about f\u0000fDPPH\u0019\n\u000060 MHz and f\u0000fDPPH\u0019\u0000740 MHz are identi\fed as\nthe (110)- and (210)-mode, respectively. A simultaneous\n\ft of the dispersion relations18\nf(110)\nres =gYIG\u0016B\n2\u0019~\u00160(H0+Hani) (6)\nand\nf(210)\nres =gYIG\u0016B\n2\u0019~\u00160\u0012\nH0+Hani\u00002\n15Ms\u0013\n(7)\nto the measured values of f(110)\nres,f(210)\nres and\u00160HDPPH\n0\nyieldsgYIG = 2:0054(3),\u00160Ms= 176:0(4) mT and\n\u00160Hani=\u00002:5(4) mT. The error of gYIGis given by the\nsystematic error introduced by the \feld normalization\nusinggDPPH . The errors in \u00160Haniand\u00160Msare given\nby\u000e\u00160Hdisp+\u000e\u00160Hstray. All values are in good agree-\nment with previously reported material parameters36{40\nfor YIG (gYIG = 2:005(2),\u00160Hani=\u00005:7 mT and\n\u00160Ms= 180 mT) and, hence, justify the (110)- and (210)-\nmode assignments.\nThird, the complete MSM manifold is computed using\nthe extracted material parameters. The mode numbers\nof the remaining modes are determined from the charac-\nteristic dispersions. Fig. 2 (b) shows the dispersions of\nthe identi\fed modes as function of f(nmr)\nres\u0000fDPPH and\nHDPPH\n0 , with very good agreement of theory (lines) and\nexperiment (circles). Slight deviations between model\npredictions and data might be attributed to a non-perfect\nspherical shape of the sample, which would change the\nboundary conditions for the magnetization dynamic in\nthe YIG spheroid, and thus the dispersion relations.\nIn Fig. 3 (a) the linewidth \u0001 f(nmr)of each MSM is\nplotted versus its resonance frequency f(nmr)\nres . The o\u000bset\n\u0001f(nmr)\n0 is magni\fed by a factor of 5 to emphasize the\ndi\u000berences in the inhomogeneous line broadening. Indi-\nvidual \fts of all \u0001 f(nmr)to Eq. (2) yield identical slopes\nfor all modes within a small scatter, which is also evident\nfrom the linewidth data in Fig. 3 (a). Hence, the Gilbert\ndamping parameter and inhomogeneous line broadening\nare obtained from a simultaneous \ft of Eq. (2) to the\nextracted data points. Here, \u000bis a shared \ft parameter\nfor all MSMs, but the inhomogeneous line broadening\n\u0001f(nmr)\n0 is \ftted separately for each mode. To avoid\n\ftting errors, the linewidths data are disregarded when\na mode anti-crossing is observed, since this results in a4\n5 10 15 20 25 ∆f (nmr) (MHz) \nfres       (GHz) (a) \n0(110)\n(440)\n(531)\n(530)\n(511)\n(631)\n(502)\n(nmr) 246810 Offset x5 \n(b) \n0.00.51.0 1.5 2.0 ∆f0(nmr) (MHz) \n-500 0 500 -250 250 ∆f00=0.3 MHz (110) \n(440) (531) (530) (511) \n(631) \n(502) Measurement \nTheory \n fres      - fDPPH  (MHz) (nmr) \nFIG. 3. (a) Linewidth vs. resonance frequency of the\nmeasured MSMs. The Gilbert damping of all MSMs is\n\u000b= 2:7(5)\u000210\u00005as evident from the same slope of all\ncurves. The inhomogeneous line broadening is di\u000berent for\neach MSM. Note that the data points are plotted with an o\u000b-\nset proportional to the inhomogeneous line broadening. (b)\nInhomogeneous line broadening as a function of f\u0000fDPPH .\npronounced increase in linewidth.41As evident from the\nsolid \ft curves in Fig. 3 (a) the evolution of the linewidth\nwith resonance frequency of all measured MSMs can be\nwell described with a shared Gilbert damping parameter\nof\u000b= 2:7(5)\u000210\u00005, independent of the mode num-\nber and the mode intensity. The latter strongly sug-\ngests a negligible e\u000bect of radiative damping on the mea-\nsured linewidths.42The error in \u000bis given by the scat-\nter of\u000bfrom the independent \fts. Other groups report\nGilbert damping parameters for YIG \flms43{49larger\nthan\u000b= 6:15\u000210\u00005, whereas for bulk YIG37,49,50values\nof\u000b= 4\u000210\u00005are found. Hence, the Gilbert damp-\ning parameter obtained here is the smallest experimen-\ntal value reported so far. The results are in agreement\nwith the notion, that the Gilbert damping parameter is a\nbulk property which only depends on intrinsic damping\ne\u000bects. However, the inhomogeneous line broadening is\nindeed di\u000berent for the various MSMs.\nFig. 3 (b) shows the extracted values for the inhomo-\ngeneous line broadening (\flled dots) as a function of\nf(nmr)\nres\u0000fDPPH . The error bars indicate the variation of\nthe inhomogeneous line broadening between global andindividual \fts. In order to show the approximate posi-\ntion of the modes in comparison to Fig. 2, the x-scale is\ncalculated for a magnetic \feld strength of \u00160H= 0:5 T.\nAdditionally, the linewidths \u0001 f(nmr)\n0 for all modes are\ncalculated using the two-magnon scattering theory, given\nin Eq. (4) of Ref. 21 (open circles). For the calculations of\nthe linewidths, a pit radius R= 350 nm and a constant\nlinewidth contribution of \u0001 f0\n0= 30 kHz was assumed.\nSince the calculated \u0001 fm-mare slightly frequency depen-\ndent, the average linewidth values for the measured \feld\nand frequency range are used and the standard deviation\nis indicated by the error bars of the open symbols. For\nmost MSMs the variation is smaller than 10 kHz. Never-\ntheless, the (440)-mode should show a prominent peak in\nthe linewidth measurement at about f(440)\nres = 10 GHz in\nFig. 3 (a),21which is however not observed in the experi-\nmental data. Additionally, the (110)-MSM shows a much\nlarger linewidth than expected from the calculations. In\na perfect sphere the (110)-mode is degenerate with the\n(430)-mode,18but in a real sphere this degeneracy might\nbe lifted. If the di\u000berence of the (110)- and (430)-mode\nfrequencies is smaller than the linewidth of the measured\nresonance, an additional inhomogeneous line broadening\nis expected. Indeed, a careful analysis of the (110)-MSM\nline shape reveals a second resonance line in very close\nvicinity to the (110)-mode, yielding an arti\fcial inhomo-\ngeneous line broadening of this mode. Besides these two\nMSMs, an excellent quantitative agreement between the\ntwo-magnon scattering model and experiment is found.\nIn conclusion, broadband ferromagnetic resonance ex-\nperiments on magnetostatic modes in a YIG sphere are\npresented and various magnetostatic modes are identi-\n\fed. The linewidth analysis of the data allows to distin-\nguish between the Gilbert damping and inhomogeneous\nline broadening. A very small Gilbert damping parame-\nter of\u000b= 2:7(5)\u000210\u00005is found for all MSMs, indepen-\ndent of their mode indices. Furthermore, the inhomoge-\nneous line broadening di\u000bers between the various magne-\ntostatic modes, in agreement with the expectations due\nto two-magnon scattering processes of the magnetostatic\nmodes into the spin-wave manifold.\nFinancial support from the DFG via SPP 1538 \"Spin\nCaloric Transport\\ (project GO 944/4) is gratefully ac-\nknowledged.\n1A. A. Serga, A. V. Chumak, and B. Hillebrands, Journal of\nPhysics D: Applied Physics 43, 264002 (2010).\n2A. V. Chumak, A. A. Serga, and B. Hillebrands, Nature Com-\nmunications 5, 4700 (2014).\n3S. Klingler, P. Pirro, T. Br acher, B. Leven, B. Hillebrands, and\nA. V. Chumak, Applied Physics Letters 106, 212406 (2015).\n4K. Ganzhorn, S. Klingler, T. Wimmer, S. Gepr ags, R. Gross,\nH. Huebl, and S. T. B. Goennenwein, Applied Physics Letters\n109, 022405 (2016).\n5K. Uchida, S. Takahashi, K. Harii, J. Ieda, W. Koshibae,\nK. Ando, S. Maekawa, and E. Saitoh, Nature 455, 778 (2008).\n6J. Xiao, G. E. W. Bauer, K.-C. Uchida, E. Saitoh, and\nS. Maekawa, Physical Review B 81, 214418 (2010).\n7H. Huebl, C. W. Zollitsch, J. Lotze, F. Hocke, M. Greifenstein,\nA. Marx, R. Gross, and S. T. B. Goennenwein, Physical Review\nLetters 111, 127003 (2013).5\n8O. O. Soykal and M. E. Flatt\u0013 e, Physical Review Letters 104,\n077202 (2010).\n9O. O. Soykal and M. E. Flatt\u0013 e, Physical Review B 82, 104413\n(2010).\n10A. Imamoglu, Physical Review Letters 102, 083602 (2009).\n11J. H. Wesenberg, A. Ardavan, G. A. D. Briggs, J. J. L. Morton,\nR. J. Schoelkopf, D. I. Schuster, and K. M\u001clmer, Physical Review\nLetters 103, 070502 (2009).\n12Y. Tabuchi, S. Ishino, A. Noguchi, T. Ishikawa, R. Yamazaki,\nK. Usami, and Y. Nakamura, Science 349, 405 (2015).\n13H. Maier-Flaig, M. Harder, R. Gross, H. Huebl, and S. T. B.\nGoennenwein, Physical Review B 94, 054433 (2016).\n14S. Klingler, H. Maier-Flaig, R. Gross, C.-M. Hu, H. Huebl,\nS. T. B. Goennenwein, and M. Weiler, Applied Physics Letters\n109, 072402 (2016).\n15R. Hisatomi, A. Osada, Y. Tabuchi, T. Ishikawa, A. Noguchi,\nR. Yamazaki, K. Usami, and Y. Nakamura, Physical Review B\n93, 174427 (2016).\n16X. Zhang, N. Zhu, C.-L. Zou, and H. X. Tang, Physical Review\nLetters 117, 123605 (2016).\n17L. R. Walker, Physical Review 105, 390 (1957).\n18P. C. Fletcher and R. O. Bell, Journal of Applied Physics 30,\n687 (1959).\n19P. R oschmann and H. D otsch, Physica Status Solidi (b) 82, 11\n(1977).\n20P. R oschmann, IEEE Transactions on Magnetics 11, 1247 (1975).\n21J. Nemarich, Physical Review 136, A1657 (1964).\n22Y. Lam, Solid-State Electronics 8, 923 (1965).\n23Here, inhomogeneous refers to all non-Gilbert like damping mech-\nanisms, while the lineshape remains Lorentzian.\n24D. H olzer, Physics Letters A 170, 45 (1992).\n25D. D. Stancil and A. Prabhakar, Spin Waves , 1st ed. (Springer\nUS, Boston, MA, 2009).\n26L. D. Landau and E. Lifshitz, Phys. Z. Sowjetunion 5, 153 (1935).\n27T. L. Gilbert, IEEE Transactions on Magnetics 40, 3443 (2004).\n28G. Wiese, Zeitschrift f ur Physik B - Condensed Matter 82, 453\n(1991).\n29P. Fletcher, I. H. Solt, and R. Bell, Physical Review 114, 739\n(1959).\n30S. S. Kalarickal, P. Krivosik, M. Wu, C. E. Patton, M. L. Schnei-\nder, P. Kabos, T. J. Silva, and J. P. Nibarger, Journal of Applied\nPhysics 99, 093909 (2006).\n31M. Sparks, R. Loudon, and C. Kittel, Physical Review 122, 791(1961).\n32R. L. White, Journal of Applied Physics 30, S182 (1959).\n33R. L. White, Journal of Applied Physics 31, S86 (1960).\n34C. P. Poole, Electron Spin Resonance , 2nd ed. (Dover Publica-\ntions, Mineola, New York, 1996).\n35AandBaccount for a small drift in S21, resulting in an imperfect\nbackground correction.\n36J. F. Dillon, Physical Review 105, 759 (1957).\n37P. R oschmann and W. Tolksdorf, Materials Research Bulletin 18,\n449 (1983).\n38P. Hansen, Journal of Applied Physics 45, 3638 (1974).\n39G. Winkler, Magnetic Garnets , 5th ed. (Vieweg, Braunschweig,\nWiesbaden, 1981).\n40P. Hansen, P. R oschmann, and W. Tolksdorf, Journal of Applied\nPhysics 45, 2728 (1974).\n41M. Sparks, Ferromagnetic-Relaxation Theory (McGraw-Hill,\n1964).\n42M. A. W. Schoen, J. M. Shaw, H. T. Nembach, M. Weiler, and\nT. J. Silva, Physical Review B 92, 184417 (2015).\n43M. Haertinger, C. H. Back, J. Lotze, M. Weiler, S. Gepr ags,\nH. Huebl, S. T. B. Goennenwein, and G. Woltersdorf, Physical\nReview B 92, 054437 (2015).\n44O. d'Allivy Kelly, A. Anane, R. Bernard, J. Ben Youssef,\nC. Hahn, A. H. Molpeceres, C. Carretero, E. Jacquet, C. Der-\nanlot, P. Bortolotti, R. Lebourgeois, J.-C. Mage, G. de Loubens,\nO. Klein, V. Cros, and A. Fert, Applied Physics Letters 103,\n082408 (2013).\n45C. Hauser, T. Richter, N. Homonnay, C. Eisenschmidt, M. Qaid,\nH. Deniz, D. Hesse, M. Sawicki, S. G. Ebbinghaus, and\nG. Schmidt, Scienti\fc Reports 6, 20827 (2016).\n46Y. Sun, Y.-Y. Song, H. Chang, M. Kabatek, M. Jantz, W. Schnei-\nder, M. Wu, H. Schultheiss, and A. Ho\u000bmann, Applied Physics\nLetters 101, 152405 (2012).\n47P. Pirro, T. Br acher, A. V. Chumak, B. L agel, C. Dubs,\nO. Surzhenko, P. G ornert, B. Leven, and B. Hillebrands, Ap-\nplied Physics Letters 104, 012402 (2014).\n48C. Hahn, V. V. Naletov, G. de Loubens, O. Klein, O. d'Allivy\nKelly, A. Anane, R. Bernard, E. Jacquet, P. Bortolotti, V. Cros,\nJ. L. Prieto, and M. Mu~ noz, Applied Physics Letters 104, 152410\n(2014).\n49C. Dubs, O. Surzhenko, R. Linke, A. Danilewsky, U. Br uckner,\nand J. Dellith, (2016), arXiv:1608.08043.\n50P. R oschmann, IEEE Transactions on Magnetics 17, 2973 (1981)."    },    {        "title": "1612.07020v2.Spin_Pumping__Dissipation__and_Direct_and_Alternating_Inverse_Spin_Hall_Effects_in_Magnetic_Insulator_Normal_Metal_Bilayers.pdf",        "content": "Spin Pumping, Dissipation, and Direct and Alternating Inverse Spin Hall E\u000bects in\nMagnetic Insulator-Normal Metal Bilayers\nAndr\u0013 e Kapelrud and Arne Brataas\nDepartment of Physics, Norwegian University of Science and Technology, NO-7491 Trondheim, Norway\nWe theoretically consider the spin-wave mode- and wavelength-dependent enhancement of the\nGilbert damping in magnetic insulatornormal metal bilayers due to spin pumping as well as the\nenhancement's relation to direct and alternating inverse spin Hall voltages in the normal metal. In\nthe long-wavelength limit, including long-range dipole interactions, the ratio of the enhancement\nfor transverse volume modes to that of the macrospin mode is equal to two. With an out-of-\nplane magnetization, this ratio decreases with both an increasing surface anisotropic energy and\nmode number. If the surface anisotropy induces a surface state, the enhancement can be an order of\nmagnitude larger than for to the macrospin. With an in-plane magnetization, the induced dissipation\nenhancement can be understood by mapping the anisotropy parameter to the out-of-plane case\nwith anisotropy. For shorter wavelengths, we compute the enhancement numerically and \fnd good\nagreement with the analytical results in the applicable limits. We also compute the induced direct-\nand alternating-current inverse spin Hall voltages and relate these to the magnetic energy stored\nin the ferromagnet. Because the magnitude of the direct spin Hall voltage is a measure of spin\ndissipation, it is directly proportional to the enhancement of Gilbert damping. The alternating spin\nHall voltage exhibits a similar in-plane wave-number dependence, and we demonstrate that it is\ngreatest for surface-localized modes.\nPACS numbers: 76.50.+g, 75.30.Ds, 75.70.-i, 75.76.+j, 75.78.-n\nI. INTRODUCTION\nIn magnonics, one goal is to utilize spin-based sys-\ntems for interconnects and logic circuits1. In previous\ndecades, the focus was to gain control over these systems\nby exploiting long-range dipole interactions in combina-\ntion with geometrical shaping. However, the complex\nnature of the nonlinear magnetization dynamics persis-\ntently represents a challenge in using geometrical shaping\nalone to realize a variety of desired properties1.\nIn magnonic systems, a unique class of materials con-\nsists of magnetic insulators. Magnetic insulators are elec-\ntrically insulating, but localized magnetic moments cou-\nple to form a long-range order. The prime example is\nYttrium Iron Garnet (YIG). YIG is a complex crystal2\nin the Garnet family, where the Fe2+and Fe3+ions at\ndi\u000berent sites in the unit cell contribute to an overall fer-\nrimagnetic ordering. What di\u000berentiates YIG from other\nferromagnetic (ferrimagnetic) systems is its extremely\nlow intrinsic damping. The Gilbert damping parame-\nter measured in YIG crystals is typically two orders of\nmagnitude smaller than that measured in conventional\nmetallic ferromagnets (Fe, Co, Ni, and alloys thereof).\nThe recent discovery that the spin waves in mag-\nnetic insulators strongly couple to spin currents in ad-\njacent normal metals has re-invigorated the \feld of\nmagnonics3{12. Although there are no mobile charge car-\nriers in magnetic insulators, spin currents \row via spin\nwaves and can be transferred to itinerant spin currents in\nnormal metals via spin transfer and spin pumping13,14.\nThese interfacial e\u000bects open new doors with respect to\nlocal excitation and detection of spin waves in magnonic\nstructures. Another key element is that we can transfer\nknowledge from conventional spintronics to magnonics,opening possibilities for novel physics and technologies.\nTraditionally, spin-wave excitation schemes have focused\non the phenomenon of resonance or the use of \u001frsted\n\felds from microstrip antennas.\nA cornerstone for utilizing these systems is to estab-\nlish a good understanding of how the itinerant elec-\ntrons in normal metals couple across interfaces with\nspin-wave dynamics in magnetic insulators. Good mod-\nels for adressing uniform (macrospin) magnetization\nthat agrees well with experiments have been previously\ndeveloped13{15. We recently demonstrated that for long-\nwavelength magnons the enhanced Gilbert damping for\nthe transverse volume modes is twice that of the uniform\nmode, and for surface modes, the enhancement can be\nmore than ten times stronger. These results are con-\nsistent with the theory of current-induced excitations\nof the magnetization dynamics16because spin pump-\ning and spin transfer are related by Onsager reciprocity\nrelations17. Moreover, mode- and wave-vector-dependent\nspin pumping and spin Hall voltages have been clearly\nobserved experimentally4.\nIn this paper, we extend our previous \fndings18in the\nfollowing four aspects. i) We compute the in\ruence of\nthe spin back\row on the enhanced spin dissipation. ii)\nWe also compute the induced direct and alternating in-\nverse spin Hall voltages. We then relate these voltages to\nthe enhanced Gilbert damping and the relevant energies\nfor the magnetization dynamics. The induced voltages\ngive additional information about the spin-pumping pro-\ncess, which can also be directly measured. iii) We also\nprovide additional information on the e\u000bects of interfa-\ncial pinning of di\u000berent types in various \feld geometries.\niv) Finally, we explain in more detail how the numerical\nanalysis is conducted for a greater number of in-planearXiv:1612.07020v2  [cond-mat.mes-hall]  6 Apr 20172\nwave numbers.\nIt was discovered19{23and later quantitatively\nexplained13,15,24,25that if a dynamic ferromagnetic mate-\nrial is put in contact with a normal metal, the magnetiza-\ntion dynamics will exert a torque on the spins of electrons\nin the immediate vicinity of the magnet. This e\u000bect is\nknown as spin pumping (SP)13,15,25. As the electrons are\ncarried away from the ferromagnet-normal metal inter-\nface, the electrons spin with respect to each other, caus-\ning an overall loss of angular momentum. The inverse\ne\u000bect, in which a spin-polarized current can a\u000bect the\nmagnetization of a ferromagnet, is called spin-transfer\ntorque (STT)26{28.\nThe discovery that a precessing magnetization in mag-\nnetic insulators3, such as YIG, also pumps spins into an\nadjacent metal layer was made possible by the fact that\nthe mixing conductance in YIG-normal metal systems is\nof such a size that the extra dissipation of the magneti-\nzation due to the spin pumping is of the same order of\nmagnitude as the intrinsic Gilbert damping. A conse-\nquence of this e\u000bect is that the dissipation of the magne-\ntization dynamics is enhanced relative to that of a system\nin which the normal metal contact is removed.\nThis paper is organized in the following manner. Sec-\ntion II presents the equation of motion for the magne-\ntization dynamics and the currents in the normal metal\nand the appropriate boundary conditions, both for gen-\neral nonlinear excitations and in the fully linear response\nregime. In Section III, we derive approximate solutions\nto the linearized problem, demonstrating how the mag-\nnetization dissipation is enhanced by the presence of an\nadjacent metal layer. Section IV presents our numerical\nmethod and results. Finally, we summarize our \fndings\nin Section V.\nII. EQUATIONS OF MOTION\nThe equation of motion for the magnetization is given\nby the Landau-Lifshitz-Gilbert equation29(presented\nhere in CGS units)\n@M\n@t=\u0000\rM\u0002He\u000b+\u000b\nMsM\u0002@M\n@t; (1)\nwhere\r=jg\u0016B=~jis the magnitude of the gyromagnetic\nratio;g\u00192 is the Land\u0013 e g-factor for the localized elec-\ntrons in the ferromagnetic insulator (FI); and \u000bis the di-\nmensionless Gilbert damping parameter. In equilibrium,\nthe magnitude of the magnetization is assumed to be\nclose to the saturation magnetization Ms. The magneti-\nzation is directed along the z-axis in equilibrium. Out of\nequilibrium, we assume that we have a small transverse\ndynamic magnetization component, such that\nM=M(r;t) =Ms+m(r;t) =Ms^z+m(r;t);(2)\nwherejmj\u001cMsandm\u0001^z= 0. Furthermore, we assume\nthat the dynamic magnetization can be described by a\nx\nhzx\nyz\nfq(a)\nd\nL2\n-L2NM\nFI\nSUBx (b)\nFIG. 1. a) The coordinate system. ^\u0018is the \flm normal\nand ^\u0010is the spin-wave propagation direction. \u0018\u0011\u0010form a\nright-handed coordinate system. The ^zaxis is the direc-\ntion of the magnetization in equilibrium, such that xyis the\nmagnetization-precession plane. b) The \flm stack is in the\nnormal direction.\nplane wave traveling along the in-plane \u0010-axis. In the\n(\u0018;\u0011;\u0010 ) coordinate system (see Figure 1), we have\nm(r;t) =m(\u0018;\u0010;t ) =mQ(\u0018)ei(!t\u0000Q\u0010); (3)\nwhere!is the harmonic angular frequency, Qis the in-\nplane wave number, and mQ(\u0018) =XQ(\u0018)^x+YQ(\u0018)^y,\nwhereXQandYQare complex functions. Note that m\nis independent of the \u0011coordinate due to translational\ninvariance.\nHe\u000bis the e\u000bective \feld, given as the functional deriva-\ntive of the free energy29,30\nHe\u000b(r;t) =\u0000\u000eU[M(r;t)]\n\u000eM(r;t)=Hi+2A\nM2sr2M(r;t)+\n+ 4\u0019ZL\n2\n\u0000L\n2d\u00180bGxy(\u0018\u0000\u00180)m(\u00180;\u0010;t);(4)\nwhere Hiis the internal \feld, which is composed of\nthe applied external \feld and the static demagnetization\n\feld. The direction of Hide\fnes the z-axis (see Fig-\nure 1). The second term of Eq. (4) is the \feld, Hex,\ninduced by to the exchange interaction (assuming cu-\nbic symmetry), where Ais the exchange sti\u000bness pa-\nrameter. The last term is the dynamic \feld, hd(r;t),\ninduced by dipole-dipole interactions, where bGxyis the\nupper 2\u00022 part of the dipole{dipole tensorial Green's\nfunctionbG\u0018\u0011\u0010in the magnetostatic approximation31ro-\ntated to the xyzcoordinate system (see Appendix A for\ncoordinate-transformation matrices).32\nThe e\u000bect of the dipolar interaction on the spin-wave\nspectrum depends on the orientation of the internal \feld\nwith respect to both the interface normals of the thin\n\flm, ^\u0018, and the in-plane spin-wave propagation direc-\ntion, ^\u0010. Traditionally, the three main con\fgurations are\nthe out-of-plane con\fguration ( \u0012= 0), in the forward\nvolume magnetostatic wave (FVMSW) geometry (see\nFig. 2a); the in-plane and parallel-to- ^\u0010con\fguration, in3\nthebackward volume magnetostatic wave (BVMSW) ge-\nometry (see Fig. 2b); and the in-plane and perpendicular-\nto-^\u0010con\fguration, in the magnetostatic surface wave\n(MSSW) geometry (see Fig. 2c).1,32{36Here, the term\n\\forward volume modes\" denotes modes that have posi-\ntive group velocities for all values of QL, whereas back-\nward volume modes can have negative group velocities\nin the range of QL, where both exchange and dipolar\ninteractions are signi\fcant. Volume modes are modes in\nwhich mQ(\u0018) is distributed across the thickness of the\nentire \flm, whereas the surface modes are localized more\nclosely near an interface.\nA. Spin-Pumping Torque\nWe consider a ferromagnetic insulator (FI) in contact\nwith a normal metal (NM) (see Figure 1). If the magneti-\nzation in the FI close to the interface is precessing around\nthe e\u000bective \feld, electron spins in the NM re\rected at\nthe interface will start to precess due to the local ex-\nchange coupling to the magnetization in the FI. The re-\n\rected electrons carry the angular momentum away from\nthe interface, where the spin information can get lost\nthrough dephasing of the spins within a typical spin di\u000bu-\nsion length lsf. This loss of angular momentum manifests\nitself as an increased local damping of the magnetization\ndynamics in the FI. The magnetization dissipation due\nto the spin-pumping e\u000bect can be taken into account by\nadding the local dissipation torque15\n\u001csp=\r~2g?\n2e2M2s\u000e(\u0018\u0000L\n2)M(r;t)\u0002@M(r;t)\n@t;(5)\nto the right-hand side (rhs) of Eq. (1). Here, g?is the\nreal part of the spin-mixing conductance per area, and e\nis the electron charge. We neglect the contribution from\nthe imaginary part of the mixing conductance, because\nthis has been shown to be signi\fcantly smaller than that\nof the real part, in addition to a\u000becting only the gyro-\nmagnetic ratio.15The spin-current density pumped from\nthe magnetization layer is thus given by\nj(s)\nsp=\u0000~2g?\n2e2M2s\u0014\nM(r;t)\u0002@M(r;t)\n@t\u0015\n\u0018=L=2;(6)\nin units of erg. Next, we will see how the spin pumping\na\u000bects the boundary conditions.\nB. Spin-Pumping Boundary Conditions\nFollowing the procedure of Rado and Weertman37, we\nintegrate Eq.(1) with the linear expansion of Eq. (2) over\na small pill-box volume straddling one of the interfaces\nof the FI. Upon letting the pill box thickness tend to\nzero, only the surface torques of the equation survive.\nAccounting for the direction of the outward normal ofthe lid on the di\u000berent top and bottom interfaces, we\narrive at the exchange-pumping boundary condition\n\u00142A\nM2sM\u0002@M\n@\u0018+~2\n2e2M2sg?M\u0002@m\n@t\u0015\n\u0018=\u0006L=2= 0:(7)\nThere is no spin current pumped at the interface to the\ninsulating substrate; thus, a similar derivation results in\na boundary condition that gives an unpinned magnetiza-\ntion,\n@M(r;t)\n@\u0018\f\f\f\f\n\u0018=\u0000L=2= 0: (8)\nIn the next section, we will generalize the bound-\nary conditions of Eqs. (7) by also considering possible\nsurface-anisotropy energies.\nIncluding surface anisotropy:\nIn the presence of surface anisotropy at an interface\nwith an easy-axis (EA) pointing along the direction ^n,\nthe surface free energy is\nUs[M(r;t)] =Z\ndV Ks\"\n1\u0000\u0012M(r;t)\u0001^n\nMs\u00132#\n\u000e(\u0018\u0000\u0018i);\n(9)\nwhereKsis the surface-anisotropy energy density at the\ninterface, which is assumed to be constant; ^nis the direc-\ntion of the anisotropy easy axis; and\u0018iis the transverse\ncoordinate of the interface. The contribution from the\nEA surface-anisotropy energy to the e\u000bective \feld is de-\ntermined by\nHs=\u0000\u000eUs[M(r;t)]\n\u000eM(r;t)=2Ks\nM2s(M\u0001^n)\u000e(\u0018\u0000\u0018j)^n:\nHowever, if we have an easy-plane (EP) surface\nanisotropy with, ^nbeing the direction of the hard axis,\nthe e\u000bective \feld is the same as that for the EA case,\nexcept for a change of sign of Ks. We unify both cases\nby de\fning Ks>0 to imply that we have an EA surface\nanisotropy with its easy axis along ^n, whereasKs<0\nimplies that we have an EP surface anisotropy with its\nhard axis along ^n.\nFollowing the approach from Section II B, the total\nboundary condition, including exchange, pumping and\nsurface anisotropy, becomes\n\u0014\n\u00062A\nM2sM\u0002@M\n@\u0018\u00002Ks\nM2s(M\u0001^n) (M\u0002^n) +\n+~2\n2e2M2sg?M\u0002@M\n@t\u0015\n\u0018=\u0006L=2= 0;(10)\nwhere the positive (negative) sign in front of the exchange\nterm indicates that the bulk FI is located below (above)\nthe interface coordinate.4\nx,z\nz\n(a)\nx\nz,zq (b)\nx\nz\nzfq (c)\nFIG. 2. Laboratory \feld con\fgurations, i.e., directions of ^z(green arrow) in relation to \flm normal ^\u0018and the spin-wave\npropagation direction ^\u0010, resulting in the di\u000berent geometries: a) FVMSW geometry; b) BVMSW geometry; c) MSSW geometry.\nC. Linearization\nWe linearize the equation of motion using Eq. (2) with\nrespect to the dynamic magnetization m. The linearized\nequation of motion for the bulk magnetization Eq. (1)\nbecomes32\n\u001a\ni!\n!M\u0012\n\u000b\u00001\n1\u000b\u0013\n+11\u0014!H\n!M+ 8\u0019\r2A\n!2\nM\u0012\nQ2\u0000d2\nd\u00182\u0013\u0015\u001b\n\u0001\n\u0001mQ(\u0018) =ZL\n2\n\u0000L\n2d\u00180bGxy(\u0018\u0000\u00180)mQ(\u00180);(11)\nwhere!H\u0011\rHi,!M\u00114\u0019\rMs, and 11 =\u00001 0\n0 1\u0001\n.\nNext, we linearize the boundary conditions of Eq. (10).\nWe choose the anisotropy axis to be perpendicular to the\n\flm plane, ^n=^\u0018, which in the xyzcoordinate system\nis given by ^\u0018xyz= (sin\u0012;0;cos\u0012), where\u0012is the angle\nbetween the z-axis and the \flm normal (see Fig. 1). The\n\fnite surface anisotropy forces the magnetization to be\neither perpendicular or coplanar with the \flm surface so\nthat\u0012= 0;\u0019=2;\u0019. Linearizing to 1st order in the dy-\nnamic magnetization, we arrive at the linearized bound-\nary conditions for the top interface\n\u0012\nL@\n@\u0018+i!\n!M\u001a+dcos(2\u0012)\u0013\nmQ;x(\u0018)\f\f\f\f\n\u0018=L\n2= 0;(12a)\n\u0012\nL@\n@\u0018+i!\n!M\u001a+dcos2(\u0012)\u0013\nmQ;y(\u0018)\f\f\f\f\n\u0018=L\n2= 0;(12b)\nwhered\u0011LKs=Ais the dimensionless surface-\npinning parameter that relates the exchange to the\nsurface anisotropy and the \flm thickness and \u001a\u0011\n!ML~2g?=4Ae2is a dimensionless constant relating the\nexchange sti\u000bness and the spin-mixing conductance.\nD. Spin Accumulation in NM and Spin Back\row\nThe pumped spin current induces a spin accumulation,\n\u0016(s)=\u0016(s)^s, in the normal metal. Here, ^sis the spin-\npolarization axis, and \u0016(s)= (\u0016\"\u0000\u0016#)=2 is half of thedi\u000berence between chemical potentials for spin-up and\nspin-down electrons in the NM.\nAs the spin accumulation is a direct consequence of\nthe spin dynamics in the FI (see Eq. (6)), the spin ac-\ncumulation cannot change faster than the magnetization\ndynamics at the interface. Thus, assuming that spin-\rip\nprocesses in the NM are must faster than the typical pre-\ncession frequency of the magnetization in the FI25, we can\nneglect precession of the spin accumulation around the\napplied \feld and any decay in the NM. With this assump-\ntion, the spin-di\u000busion equation@\u0016(s)\n@t=Dr2\u0016(s)\u0000\u0016(s)\n\u001csf,\nwhereDis the spin-di\u000busion constant, and \u001csfis the\nmaterial-speci\fc average spin-\rip relaxation time, be-\ncomes\n\u0016(s)\u0019l2\nsfr2\u0016(s); (13)\nwherelsf\u0011p\u001csfDis the average spin-\rip relaxation\nlength.\nThe spin accumulation results in a back\rowing spin-\ncurrent density, given by\nj(s)\nbf(L=2) =~g?\ne2M2sh\nM(r;t)\u0002\u0010\nM(r;t)\u0002\u0016(s)(r;t)\u0011i\n\u0018=L=2;\n(14)\nwhere the positive sign indicates \row from the NM into\nthe FI. This spin current creates an additional spin-\ntransfer torque on the magnetization at the interface\n\u001cbf=\u0000\r~g?\ne2M2s\u000e\u0010\n\u0018\u0000L\n2\u0011\nM(r;t)\u0002\u0010\nM(r;t)\u0002\u0016(s)\u0011\n:\n(15)\nBecause the spin accumulation is a direct result of the\npumped spin current, it must have the same orientation\nas the M(r;t)\u0002@tM(r;t) term in Eq. (5). That term is\ncomprised of two orthogonal components: the 1st-order\ntermMs^z\u0002_min thexyplane, and the 2nd-order term\nm\u0002_moriented along ^z. Because the magnetization is\na real quantity, care must be taken when evaluating the\n2nd-order term. Using Eq. (3), the 2nd-order pumped5\nspin current is proportional to\nRefmg\u0002@tRefmg\f\f\f\n\u0018=L=2=e\u00002Imf!gtRef!g\u0002\n\u0002^zh\nImXQReYQ\u0000ReXQImYQi\n;(16)\nwhich is a decaying direct-current (DC) term. This is in\ncontrast to the 1st-order term, which is an alternating-\ncurrent (AC) term. Thus, we write the spin accumula-\ntion as\n\u0016(s)=\u0016(s)\nAC(^z\u0002^mt) +\u0016(s)\nDC^z; (17)\nwhere we have used the shorthand notation mt=_m(\u0018=\nL=2), such that ^mt=mt=jmtj, which in general is not\nparallel to mbut guaranteed to lie in the xyplane. In-\nserting Eq. (17) into Eq. (13) gives one equation each for\nthe AC and DC components of the spin accumulation,\n@2\u0016(s)\nj\n@\u00182=l\u00002\nsf;j\u0016(s)\nj; (18)\nwherejdenotes either the AC or DC case and lsf,DC =lsf\nwhilelsf,AC =lsf(1+l2\nsfQ2)\u00001=2because mt/exp(i(!t\u0000\nQ\u0010)). Eq. (18) can be solved by demanding spin-current\nconservation at the NM boundaries: at the free surface of\nthe NM, there can be no crossing spin current; thus, the \u0018\ncomponent of the spin-current density must vanish there,\n@\u0018\u0016(s)\njj\u0018=L=2+d= 0. Similarly, by applying conservation\nof angular momentum at the FI-NM interface, the net\nspin-current density crossing the interface, due to spin\npumping and back\row, must equal the spin current in\nthe NM layer, giving\n\u0014\n\u0000~2g?\n2e2M2sM\u0002@M\n@t+~g?\ne2M2sM\u0002\u0010\nM\u0002\u0016(s)\u0011\u0015\n\u0018=L=2\n=\u0000~\u001b\n2e2@\u0018\u0016(s)j\u0018=L=2;(19)\nwhere\u001bis the conductivity of the NM. Using these\nboundary conditions, we recover the solutions (see,\ne.g.,25,38)\n\u0016(s)\nj=\u0016(s)\nj;0sinh\u0010\nl\u00001\nsf;j\u0002\n\u0018\u0000(L=2 +d)\u0003\u0011\nsinh\u0010\n\u0000d\nlsf;j\u0011; (20)\nwhere\u0016(s)\nj;0is time dependent, and depends on the \u0010co-\nordinate only in the AC case. We \fnd that the AC and\nDC spin accumulations \u0016(s)\nj;0are given by\n\u0016(s)\nAC;0=\u0000~\n2mt\nMs\u0014\n1 +\u001b\n2g?lsf;ACcoth\u0012d\nlsf;AC\u0013\u0015\u00001\n;\n(21)\n\u0016(s)\nDC;0=\u0000lsf~\n\u001bM2s~g?tanh\u0012d\nlsf\u0013\n^z\u0001[m\u0002_m]\u0018=L=2;\n(22)TABLE I. Typical values for the parameters used in the\ncalculations.6,7,11,39{41\nParameter Value Unit\nA 3:66\u000110\u00007erg cm\u00001\n\u000b 3\u000110\u00004{\nKs 0:05 erg cm\u00002\ng? 8:18\u00011022cm\u00001s\u00001\n\r 1:76\u0001107G\u00001s\u00001\n4\u0019M s 1750 G\n\u001b 8:45\u00011016s\u00001\nd 50 nm\nlsf 7.7 nm\n\u0002 0.1 {\nwhere ~g?is a renormalized mixing conductance, which is\ngiven by\n~g?=g?(\n1\u0000\u0014\n1 +\u001b\n2g?lsf;ACcoth\u0012d\nlsf;AC\u0013\u0015\u00001)\n:\n(23)\nThis scaling of g?occurring in the DC spin accumulation\noriginates from the second-order spin back\row due to the\nAC spin accumulation that is generated in the normal\nmetal.\nAdding both the spin-pumping and the back\row\ntorques to Eq. (1) and repeating the linearization pro-\ncedure from Sec. II C, we \fnd that the AC spin accumu-\nlation renormalizes the pure spin-mixing conductance.\nThus, the addition of the back\row torque can be ac-\ncounted for by replacing g?with ~g?in the boundary con-\nditions of Eqs. (12), making the boundary conditions Q-\ndependent in the process. Using the values from Table I,\nwhich are based on typical values for a YIG-Pt bilayer\nsystem, we obtain ~ g?=g?\u00180:4 forQL\u001c1, whereas\n~g?=g?!1 for large values of QL. Thus, AC back\row is\nsigni\fcant for long-wavelength modes and should be con-\nsidered when estimating g?from the linewidth broaden-\ning in ferromagnetic resonance (FMR) experiments.11\nInverse Spin Hall E\u000bect\nThe inverse spin Hall e\u000bect (ISHE) converts a spin\ncurrent in the NM to an electric potential through the\nspin-orbit coupling in the NM. For a spin current in\nthe^\u0018direction, the ISHE electric \feld in the NM layer\nisEISHE =\u0000e\u00001\u0002h(@\u0018\u0016(s))\u0002^\u0018i\u0018, where \u0002 is the di-\nmensionless spin-Hall angle, and h\u0001i\u0018is a spatial average\nacross the NM layer, i.e., for \u00182(L=2;L=2 +d). Using\nthe previously calculated spin accumulation, we \fnd that6\nthe AC electric \feld is\nEAC\nISHE =\u0000\u0002~\n2deMs\u0014\n1 +\u001b\n2g?lsf;ACcoth\u0012d\nlsf;AC\u0013\u0015\u00001\n\u0002\n\u0002\u0002\n\u0000^\u0011(\u0000mt;ycos\u0012cos\u001e+mt;xsin\u001e)+\n+^\u0010(\u0000mt;xcos\u001e\u0000mt;ycos\u0012sin\u001e)\u0003\n;(24)\nwhere\nmt;i=\u0000[Im!Remi+Re!Immi]\u0018=L=2; (25)\nandi=x;y. For BVMSW ( \u0012=\u0019=2;\u001e= 0) modes,\nthe AC \feld points along ^\u0010, whereas for MSSW ( \u0012=\n\u001e=\u0019=2) modes, it points along ^\u0011(i.e., in plane, but\ntransverse to \u0010; see Fig. 1). Notice that for both BVMSW\nand MSSW mode geometries, only the xcomponent of\nmtcontributes to the \feld. In contrast, for FVMSW\n(\u0012= 0) modes, the \feld points somewhere in the \u0011\u0010\nplane, depending on the ratio of mt;xtomt;y.\nSimilarly to the AC \feld, the DC ISHE electric \feld is\ngiven by\nEDC\nISHE = \u0002\u0016(s)\nDC;0\ndesin\u0012(^\u0011cos\u001e\u0000^\u0010sin\u001e);(26)\nwhich is perpendicular to the AC electric \feld and zero\nfor the FVMSW mode geometry.\nThe total time-averaged energy in the ferromagnet\nEtotal(see Morgenthaler42) is given by\nhEtotaliT=Z\nferriteRe\u0014\n\u0000i\u0019!\u0003\n!M(m\u0002m\u0003)^z\u0015\ndV; (27)\nwhere the integral is taken over the volume of the ferro-\nmagnet.\nBecause the DC ISHE \feld is in-plane, the voltage\nmeasured per unit distance along the \feld direction,\n^\u0003=^\u0011cos\u001e\u0000^\u0010sin\u001e, can be used to construct an esti-\nmate of the mode e\u000eciency. Taking the one-period time\naverage of Eq. (26) using Eq. (22) and normalizing it by\nEq. (27) divided by the in-plane surface area, A, we \fnd\nan amplitude-independent measure of the DC ISHE:\n\u000fDC=he^\u0003\u0001EDC\nISHEiT\nhEtotaliT=A=\u00002\r\u0002lsf~\nd\u001bMs~g?tanh\u0012d\nlsf\u0013\nsin\u0012\u0002\n\u0002Reh\n\u0000i!\u0003\n!M(m\u0002m\u0003)^zi\n\u0018=L=2RL=2\n\u0000L=2Reh\n\u0000i!\u0003\n!M(m\u0002m\u0003)^zi\nd\u0018;(28)\ngiven in units of cm, and where f\u0001g\u0003denotes complex\nconjugation.\nSimilarly, the AC ISHE electric \feld, being time-\nvarying, will contribute a power density that, when nor-\nmalized by the power density in the ferromagnet, be-comes\n\u000fAC=h\u001b\u0000\nEAC\nISHE\u00012iT\nRef!g\n2\u0019ALhEtotaliT=\u0019\u001b\nRef!g\u0012\u0002~\n2deMs\u00132\n\u0002\n\u0002\u0014\n1 +\u001b\n2g?lsf;ACcoth\u0012d\nlsf;AC\u0013\u0015\u00002\n\u0002\n\u0002jmt;xj2+ cos2\u0012jmt;yj2\n1\nLRL=2\n\u0000L=2Reh\n\u0000i!\u0003\n!M(m\u0002m\u0003)^zi\nd\u0018:(29)\nTo be able to calculate explicit realizations of the mode-\ndependent equations Eqs. (28) and (29), one will need to\n\frst calculate the dispersion relation and mode pro\fles\nin the ferromagnet.\nIII. SPIN-PUMPING THEORY FOR\nTRAVELLING SPIN WAVES\nBecause, the linearized boundary conditions (see\nEqs. (12)) explicitly depend on the eigenfrequency !, we\ncannot apply the method of expansion in the set of pure\nexchange spin waves, as was performed by Kalinikos and\nSlavin32. Instead, we analyze and solve the system di-\nrectly for small values of QL, whereas the dipole-dipole\nregime ofQL\u00181 is explored using numerical computa-\ntions in Sec. IV.\nA. Long-Wavelength Magnetostatic Modes\nWhenQL\u001c1 Eq. (11) is simpli\fed to\n( \nsin2\u00120\n0 0!\n+i!\n!M \n\u000b\u00001\n1\u000b!\n+\n+11\u0014!H\n!M\u00008\u0019\r2A\n!2\nMd2\nd\u00182\u0015\u001b\n\u0001mQ(\u0018) = 0;(30)\nwhere the 1st-order matrix term describe the dipole-\ninduced shape anisotropy and stems from bGxy(see32).\nWe make the ansatz that the magnetization vector in\nEq. (3) is composed of plane waves, e.g., mQ(\u0018)/eik\u0018.\nInserting this ansatz into Eq. (30) produces the disper-\nsion relation\n\u0010!\n!M\u00112\n=\u0010!H\n!M+\u00152\nexk2+i\u000b!\n!M\u0011\n\u0002\n\u0002\u0010!H\n!M+\u00152\nexk2+ sin2\u0012+i\u000b!\n!M\u0011\n;(31)\nwhere\u0015ex\u0011p\n8\u0019\r2A=!2\nMis the exchange length . Keep-\ning only terms to \frst order in the small parameter \u000b,\nwe arrive at\n!(k)\n!M=\u0006r\u0010!H\n!M+\u00152exk2\u0011\u0010!H\n!M+\u00152exk2+ sin2\u0012\u0011\n+\n+i\u000b\u0010!H\n!M+\u00152\nexk2+sin2\u0012\n2\u0011\n: (32)7\nThe boundary conditions in Eq. (12) depend explicitly on\n!andkand give another equation k=k(!) to be solved\nsimultaneously with Eq. (32). However, in the absence\nof spin pumping, i.e., when the spin-mixing conductance\nvanishesg?!0, it is su\u000ecient to insert the constant k\nsolutions from the boundary conditions into Eq. (32) to\n\fnd the eigenfrequencies.\nDi\u000berent wave vectors can give the same eigenfre-\nquency. It turns out that this is possible when !(k) =\n!(i\u0014), which has a non-trivial solution relating \u0014tok:\n\u00152\nex\u00142= sin2\u0012+\u00152\nexk2+ 2!H\n!M\u0006i2\u000b!(k)=!M:(33)\nWith these \fndings, a general form of the magnetiza-\ntion is\nmQ(\u0018) = \n1\nr(k)!h\nC1cos\u0000\nk(\u0018+L\n2)\u0001\n+C2sin\u0000\nk(\u0018+L\n2)\u0001i\n+\n+ \n1\nr(i\u0014)!h\nC3cosh\u0000\n\u0014(\u0018+L\n2)\u0001\n+C4sinh\u0000\n\u0014(\u0018+L\n2)\u0001i\n;\n(34)\nwherefCigare complex coe\u000ecients to be determined\nfrom the boundary conditions, and where \u0014=\u0014(k) is\ngiven by Eq. (33). The ratio between the transverse com-\nponents of the magnetization, r(k) =YQ=XQ, is deter-\nmined from the bulk equation of motion (see Eq. (30))\nand is in linearized form\nr(k) =\u0000\u000bsin2\u0012\u00062ir\u0010\n!H\n!M+\u00152exk2\u0011\u0010\n!H\n!M+\u00152exk2+ sin2\u0012\u0011\n2\u0010\n!H\n!M+\u00152exk2\u0011 ;\n(35)\nimplying elliptical polarization of mQwhen\u00126= 0.\nInserting Eq. (34) into Eq. (8) only leads to a solution\nwhenk= 0, such that C2=C4= 0 in the general case.\nBy solving Eq. (12b) for C3, we \fnd\nC3\nC1=\u0000!H\n!M+\u00152\nexk2+ sin2\u0012+i\u000b!\n!M\n!H\n!M\u0000\u00152ex\u00142+ sin2\u0012+i\u000b!\n!M\u0002\n\u0002(i!\n!M~\u001a+dcos2\u0012) cos(kL)\u0000kLsin(kL)\n(i!\n!M~\u001a+dcos2\u0012) cosh(\u0014L) +\u0014Lsinh(\u0014L);(36)\nwhere ~\u001a\u0011\u001ajg?!~g?is the pumping parameter altered by\nthe AC spin back\row from the NM (see Section II D). C1\nis chosen to be the free parameter that parameterizes the\ndynamic magnetization amplitude, which can be deter-\nmined given a particular excitation scheme. Lineariza-\ntion of Eq. (36) with respect to \u000bis straightforward, but\nthe expression is lengthy; we will therefore not show it\nhere.\nInserting the ansatz with C2=C4= 0 andC3given\nby Eq. (36) into Eq. (12a) gives the second equation for\nkand!(the \frst is Eq. (32)). In the general case, thenumber of terms in this equation is very large; thus, we\ndescribe it as\nf(k;!;\u000b; ~\u001a) = 0; (37)\ni.e., an equation that depends on the wave vector k, fre-\nquency!, Gilbert damping constant \u000band spin-pumping\nparameter ~\u001a.\nBecause both the bulk and interface-induced dissipa-\ntion are weak, \u000b\u001c1, ~\u001a\u001c1, the wavevector is only\nslightly perturbed with respect to a system without dis-\nsipation, i.e., k!k+\u000ekwhere\u0015ex\u000ek\u001c1. It is therefore\nsu\u000ecient to expand fup to 1storder in these small quan-\ntities:\nf(k;!; 0;0) + (~\u001a)@f\n@~\u001a\f\f\f\f\n0+\u000b@f\n@\u000b\f\f\f\f\n0+\n+ (\u0015ex\u000ek)@f\n@(\u0015ex\u000ek)\f\f\f\f\n0\u00190;(38)\nwhere the sub-index 0 means evaluation in a system with-\nout dissipation, i.e., when ( \u000b;~\u001a;\u000ek) = (0;0;0). By solv-\ning the system of equations in the absence of dissipation,\nf(k;!; 0;0) = 0, the dissipation-induced change in the\nwave vector \u000ekis given by\n\u000ek\u0019\u0000~\u001a@f\n@~\u001a\f\f\f\n0+\u000b@f\n@\u000b\f\f\f\n0\n\u0015ex@f\n@(\u0015ex\u000ek)\f\f\f\n0: (39)\nIn turn, this change in the wave vector should be in-\nserted into the dispersion relation of Eq. (31) to \fnd\nthe dissipation. Inspecting Eq. (31), we note that \u000ek-\ninduced additional terms proportional to !are of the\nform (k+\u000ek)2\u0000k2\u00192k\u000ekwhich renormalize the Gilbert-\ndamping term i\u000b!\n!M. Thus, in Eq. (39), there are terms\nproportional to the frequency in both terms in the numer-\nator. We extract these terms /i!\n!Mby di\u000berentiating\nwith respect to !and de\fne the renormalization of the\nGilbert damping, i.e., \u000b!\u000b+ \u0001\u000b, from spin pumping\nas\n\u0001\u000b=i2\u0015exk!M@!\u0000\n\u0015ex\u000ekj\u000b=0\u0001\ni2\u0015exk!M@!\u0000\n\u0015ex\u000ekj~\u001a=0\u0001\n\u00001; (40)\nwhere@!represents the derivative with respect to !and\nkis the solution to the 0th-order equation. Note that in\nperforming a further local analysis around some point k0\nin thek-space of Eq. (37), a series expansion of faround\nk0must be performed before evaluating Eqs. (39) and\n(40).\nEq. (40) is generally valid, except when d= 0 and\nkL!0, which we discuss below. In the following sec-\ntion, we will determine explicit solutions of the 0th-order\nequation for some key cases, and mapping out the spin-\nwave dispersion relations and dissipation in the process.8\nB. No Surface Anisotropy ( d= 0)\nLet us \frst investigate the case of a vanishing sur-\nface anisotropy. In this case, the 0th-order expansion\nof Eq. (37) has a simple form and is independent of the\nmagnetization angle \u0012. The equation to determine kis\ngiven by\nkLtan(kL) = 0; (41)\nwith solutions k=n\u0019=L , wheren2Z. Similarly, the\nexpression for \u000ekis greatly simpli\fed, \u000ekn=i!\n!M~\u001a\nn\u0019\u0015ex\nL,\nn6= 0, such that the mode-dependent Gilbert damping\nis\n\u0001\u000bn= 2~\u001a\u0012\u0015ex\nL\u00132\n; n6= 0: (42)\nFor the macrospin mode, when n= 0, the linear ex-\npansion in \u000ekbecomes insu\u000ecient. This is because\nkLtan(kL)\u0018(kL)2forkL!0; thus, we must expand\nthe function fto second order in the deviation \u000ekaround\nkL= 0. Ford= 0, we \fnd that the boundary condi-\ntion becomes \u000ek2L2=i!\n!M~\u001a\u00152\nex, and when inserted into\nEq. (31), it immediately gives\n\u0001\u000b0= ~\u001a\u0012\u0015ex\nL\u00132\n=1\n2\u0001\u000bn; (43)\nwhich is the macrospin renormalization factor found in\nRef. 15. Using a di\u000berent approach, our results in this\nsection reproduce our previous result that the renormal-\nization of the Gilbert damping for standing waves is\ntwice the renormalization of the Gilbert damping of the\nmacrospin.18Next, we will obtain analytical results be-\nyond the description in Ref. 18 for the enhancement of\nthe Gilbert damping in the presence of surface anisotropy.\nC. Including Surface Anisotropy ( d6= 0)\nIn the presence of surface anisotropy, the out-of-plane\nand in-plane \feld con\fgurations must be treated sepa-\nrately. This distinction is because the boundary condi-\ntion Eq. (37) has di\u000berent forms for the two con\fgura-\ntions in this scenario.\n1. Out-of-plane Magnetization\nWhen the magnetization is out of plane, i.e., \u0012= 0, the\nspin-wave excitations are circular and have a high degree\nof symmetry. A simpli\fcation in this geometry is that\nthe coe\u000ecient C3= 0. In the absence of dissipation,\nthe boundary condition Eq. (37) determining the wave\nvectors becomes\nkLtan(kL) =d: (44)\nLet us consider the e\u000bects of the two di\u000berent\nanisotropies in this geometry.\n0510152025300.00.51.01.52.02.5\nLKsADaEA,nDa0\nn=0n=5FIG. 3. The ratio of enhanced Gilbert damping \u0001 \u000bEA,n=\u0001\u000b0\nin a system with easy-axis surface anisotropy versus the en-\nhanced Gilbert damping of macrospin modes in systems with\nno surface anisotropy as a function of surface-anisotropy en-\nergy.nrefers to the mode number, where n= 0 is the\nuniform-like mode. The dashed line represents the ratio\n\u0001\u000bn=\u0001\u000b0in the case of no surface anisotropy (see Eq. (42)).\na. Easy-Axis Surface Anisotropy ( d > 0):When\nd\u00181 or larger, the solutions of Eq. (44) are displaced\nfrom the zeroes of tan( kL), i.e., the solutions we found in\nthe case of no surface anisotropy, and towards the upper\npoles located at kuL= (2n+1)\u0019=2, wheren= 0;1;2;:::.\nWe therefore expand fin Eq. (37) (and thus also in\nEq. (44)) into a Laurent series around the poles from\nthe \frst negative order up to the \frst positive order in\nkLto solve the boundary condition for kL, giving\nkL\u0019\u0015ex\nL3(1 +d) + 2(kuL)2\u0000p\n12(kuL)2+ 9(1 +d)2\n2kuL:\n(45)\nUsing this result and the Laurent-series expansion for\nfin Eq. (39) and Eq. (40), we \fnd the Gilbert-damping\nrenormalization term ( \u000b!\u000b+ \u0001\u000b(oop)\nEA,n) and the ratio\nbetween the modes\n\u0001\u000b(oop)\nEA,n\n\u0001\u000b0\u00193\u0000\n3(1 +d) + 2(kuL)2\u0000p\n12(kuL)2+ 9(1 +d)2\u0001\n\u0002\n\u0002\u0000p\n4(kuL)2+ 3(1 +d)2\u0000p\n3(1 +d)\u0001\n2(kuL)2p\n4(kuL)2+ 3(1 +d)2:\n(46)\nThis ratio is plotted in Figure 3 for n\u00145. We see that\nthe ratio vanishes for large values of d. For small values\nof the anisotropy energy d, the approximate ratio exceeds\nthe exact result of the ratio we found in the limiting case\nof no surface anisotropy (see Eq. (42)). For moderate\nvalues ofd\u00185, the expansion around the upper poles\nis su\u000ecient, but only for the \frst few modes. This im-\nplies that moderate-strength easy-axis surface anisotropy\nquenches spin pumping for the lowest excited modes but\ndoes not a\u000bect modes with higher transverse exchange\nenergy.9\n05101520253001234\nLÈKsÈADaEP,nDa0\nn=1n=5\nFIG. 4. Plot of \u0001 \u000b(oop)\nEP,n=\u0001\u000b0. The dashed line represents\nthe ratio \u0001 \u000bn=\u0001\u000b0in the case of no surface anisotropy (see\nEq. (42)).\nb. Easy-Plane Surface Anisotropy ( d < 0):Easy-\nplane surface anisotropy is represented by a negative sur-\nface anisotropy din Eq. (44). In this case, the boundary\ncondition must be treated separately for the uniform-\nlike (n= 0) mode and the higher excitations. When\njdj>1, we can obtain a solution by expanding along the\nimaginary axis of kL. This corresponds to expressing the\nboundary condition in the form \u0000ikLtanh(ikL) =\u0000jdj,\nwith the asymptotic behavior kL\u0019 \u0000ijdj. Using the\nasymptotic form of the boundary condition in Eqs. (39)\nand calculating the renormalization of the Gilbert damp-\ning using Eq. (40), we \fnd that the renormalization is\n\u000b!\u000b+ \u0001\u000b(oop)\nEP,0, where\n\u0001\u000b(oop)\nEP,0\n\u0001\u000b0= 2jdj: (47)\nThus, the Gilbert damping of the lowest mode is much\nenhanced by increasing surface anisotropy. The surface-\nanisotropy mode is localized at the surface because it\ndecays from the spin-active interface and into the \flm.\nBecause the e\u000bective volume of the mode is reduced,\nspin pumping more strongly causes dissipation out of the\nmode and into the normal metal.\nFor the higher modes ( n > 0), the negative term on\nthe rhs of Eq. (44) forces the kLsolutions closer to thenegative, lower poles of tan( kL), located at k(l)\nnL= (2n\u0000\n1)\u0019=2, wheren= 1;2;3;:::. We repeat the procedure\nused for the EA case by expanding finto a Laurent series\naround these lower poles, arriving at\nkL\u00193(1\u0000jdj) + 2(k(l)\nnL)2+q\n12(k(l)\nnL)2+ 9(1\u0000jdj)2\n2k(l)\nnL:\n(48)\nUsing this relation and the new lower-pole Laurent ex-\npansion for f, Eqs. (39) and (40) give us the renormal-\nization of the Gilbert damping ( \u000b!\u000b+ \u0001\u000b(oop)\nEP,n) and\nthe ratio\n\u0001\u000b(oop)\nEP,n\n\u0001\u000b0\u00193\u0000\n3(1\u0000jdj) + 2(kuL)2+p\n12(kuL)2+ 9(1\u0000jdj)2\u0001\n\u0002\n\u0002\u0000p\n4(kuL)2+ 3(1\u0000jdj)2+p\n3(1\u0000jdj)\u0001\n2(kuL)2p\n4(kuL)2+ 3(1\u0000jdj)2:\n(49)\nThis ratio is plotted in Figure 4 from n= 1 up ton= 5.\nWe see that the ratio vanishes for large values of jdj.\nSimilar to the case of EA surface anisotropy, the approx-\nimation breaks down for large nand/or small values of\njdj.\nWhereas the n= 0 mode exhibits a strong spin-\npumping enhanced dissipation in this \feld con\fguration,\nthe DC ISHE \feld vanishes when \u0012= 0 (see Eq. (26)).\nThis is one of the reasons why this con\fguration is sel-\ndom used in experiments. However, this con\fguration\ncan lead to a signi\fcant AC ISHE, and a similar AC sig-\nnal was recently detected12. Because of the strong dissi-\npation enhancement, the EP surface anisotropy induced\nlocalized mode in perpendicular magnetization geometry\ncould be important in future experimental work.\n2. In-plane Magnetization\nWe will now complete the discussion of the spin-\npumping enhanced Gilbert damping by treating the case\nin which the magnetization is in plane ( \u0012=\u0019=2). For\nsuch systems, the coe\u000ecient C36= 0, and the 0th-order\nexpansion of Eq. (37) becomes\nkLtankL=\u0000d\u0000\n(\u0015exk)2+!H\n!M\u0001q\n1 + (\u0015exk)2+ 2!H\n!Mq\n1 + (\u0015exk)2+ 2!H\n!M\u0000\n1 + 2(\u0015exk)2+ 2!H\n!M\u0001\n\u0000d\u0015ex\nL\u0000\n1 + (\u0015exk)2+!H\n!M\u0001\ncoth\u0010\nL\n\u0015exq\n1 + (\u0015exk)2+ 2!H\n!M\u0011:\n(50)\nFor typical \flm thicknesses, of some hundred nanome-\nters, we have L=\u0015 ex\u001d1 and (\u0015exk)2\u001c1 for the lowest\neigenmodes. Thus, we take the asymptotic coth \u00181 and\nneglect the ( \u0015exk)2terms, ridding the rhs of Eq. (50) ofanykdependence. Eq (50) now becomes similar to the\nout-of-plane case\nkLtan(kL) =de\u000b; (51)10\nwhere\nde\u000b=\u0000d!H\n!Mq\n1 + 2!H\n!M\n\u0000\n1 + 2!H\n!M\u00013=2\u0000d\u0015ex\nL\u0000\n1 +!H\n!M\u0001: (52)\nde\u000bis positive if d<0 and negative for d>0 up to a crit-\nical valued\u0015ex=L=\u0015exKs=A=\u0000\n1 + 2!H\n!M\u00013=2=\u0000\n1 +!H\n!M\u0001\n,\nwhere the denominator becomes zero. For negative d,\njde\u000bj<jdj, whereas for positive d,jde\u000bjis initially smaller\nthan that ofjdjbut quickly approaches the critical value.\nWith the value Ksfrom Tab. I, we have jde\u000bj<jdj, in-\ndependent of the sign of d.\nWith this relation, we can calculate an approximate\nGilbert damping renormalization in both the EA and\nEP cases using the EP and EA relations, respectively,\nobtained in the out-of-plane con\fguration. Thus,\n\u0001\u000bip\nEA,0\u0019\u0001\u000boop\nEP,0jd!deff= 2jde\u000bj; (53)\n\u0001\u000bip\nEA,n\u0019\u0001\u000boop\nEP,njd!deff; (54)\n\u0001\u000bip\nEP,n\u0019\u0001\u000boop\nEA,njd!deff: (55)\nTo summarize this section regarding the enhancement\nof Gilbert damping, we see that the enhancement can be\nvery strong for the surface modes because their e\u000bective\nsizes are smaller than the thickness of the \flm. For all\nother modes, the enhancement decreases with increasing\nmagnitude of the surface-anisotropy energy.\nIV. NUMERICAL CALCULATIONS\nThe \frst step in the numerical method is to approxi-\nmate the equation of motion of Eq. (11) into by \fnite-size\nmatrix eigenvalue problem. We discretize the transverse\ncoordinate\u0018on the interval [\u0000L=2;L=2] intoNpoints la-\nbeled byj= 1;2;:::;N , and characterize the transverse\ndiscrete solutions of the dynamic magnetization vectors\nmQby (mx;j;my;j) of size 2N.\nWe approximate the 2nd-order derivative arising from\nthe exchange interaction using a nth-order central dif-\nference method. For the n\u00002 discretized points next to\nthe boundaries, we also use nth-order methods, using for-\nward (backward) di\u000berence schemes for the lower (upper)\n\flm boundary. This strategy avoids the introduction of\n\\ghost\" points outside the interval [ \u0000L=2;L=2] to satisfy\nthe boundary conditions.\nThus, the total operator acting on the magnetiza-\ntion on the left-hand side of Eq. (11) becomes a sparse\n2N\u00022Nmatrix operator. On the right-hand side of\nEq. (11), we also represent the convolution integral as\na 2N\u00022Ndense matrix operator, where each row is\nweighted according to the extended integration formu-\nlas for closed integrals to nth order43. The four N\u0002N\nsub-blocks of this integration operator correspond to the\nfour tensor elements of bGxy. In the \fnal discrete form,\nwe obtained a 2 N\u00022N !-dependent matrix.Next, the 4 boundary conditions (at the left and right\nboundaries for the two components, mxandmy) are used\nto reduce the number of equations to 2 N\u00004. This is per-\nformed by algebraically solving the discretized boundary\nconditions with respect to the boundary points, i.e., by\ndetermining miwherei2f1;N;N + 1;2Ngin terms of\nthe magnetizations at the interior points.\nFinally, each (2 N\u00004)\u0002(2N\u00004) matrix is separated\ninto two parts: a term independent of the frequency !\nand a term proportional to !. The dipole interaction\ncauses the eigenvalue problem to be non-Hermitian and\ntherefore computationally more demanding than a gen-\neralized eigenvalue problem. We \fnd the dispersion rela-\ntion and magnetization vectors by solving this eigenvalue\nproblem. The resulting eigenvectors are used to \fnd the\nmagnetization at the boundary by back-substitution into\nthe equations for the boundary conditions.\nWe are interested in \fnding the mode and wave-vector\ndependence of the spin-pumping enhanced Gilbert damp-\ning. To obtain this information numerically, we perform\ntwo independent calculations of the (complex) eigenval-\nues. First, we calculate the complex eigenvalues !dwhen\nthere is no spin pumping, but dissipation occurs via the\nconventional bulk Gilbert damping. Second, we calcu-\nlate the complex eigenvalues !spwhen spin pumping is\nactive at the FI-NM interface but there is no bulk Gilbert\ndamping. A mode- and wave-vector-dependent measure\nof the e\u000bective enhanced Gilbert damping enhancement\nis then given by\n\u0001\u000b=\u000bIm!sp\nIm!d: (56)\nTo ensure that we treat the same modes in the two in-\ndependent calculations, we check the convergence of the\nrelative di\u000berence in the real part of the eigenvalues. Ta-\nble I lists the values for the di\u000berent system parameters\nthat are used throughout this section.\nLet us \frst discuss the renormalization of the Gilbert\ndamping when there is no surface anisotropy. We will\npresent the numerical results for the three main geome-\ntries described in Sec. I and compare the results to the\nanalytical results of Sec. III A.\nA. FVMSW ( \u0012= 0)\nFigure 5 shows the wave-vector dependent renormal-\nization of the Gilbert damping \u0001 \u000bdue to spin pumping\nat the FI-NM interface in the FVMSW geometry. In\nthis geometry, waves travelling along \u0006^\u0010have the same\nsymmetry; thus, each line is doubly degenerate and cor-\nresponds to two waves of \u0006!. The \\spikes\" in the \fgure\nare due to degeneracies, i.e., mode crossings, and upon\ninspection, these spikes can be observed in the dispersion\nrelation.11\n0.010.1110100QL0.050.100.150.200.250.30DaH10-2L\n0.11100.60.70.80.91.0Re8wwM<-L2xL2m®\nHxL¤\nFIG. 5. \u0001\u000bversus wave vector for the FVMSW geometry\nof the four smallest eigenvalues. Top inset: Magnitudes of\neigenvectors (in arbitrary units) across the \flm at QL= 10.\nBottom inset: dispersion relation in the dipole-dipole active\nregime.\n1. Easy-Axis Surface Anisotropy ( ^\u0018easy axis)\n0.010.1110100QL0.20.40.60.81.01.2DaH10-3L\n0.11100.60.70.80.91.0Re8wwM<\n-L2xL2m®\nHxL¤\nFIG. 6. \u0001\u000bEAversus wave vector for the FVMSW geometry\nshowing the four smallest eigenvalues. The horizontal dashed\nlines indicate solutions of Eq. (46). Left inset: Magnitudes of\neigenvectors (in arbitrary units) across the \flm at QL= 5.\nRight inset: Dispersion relation in the dipole-dipole active\nregime.\nFigure 6 shows \u0001 \u000bEAfor the FVMSW geometry with\nan EA surface anisotropy at the spin-active interface. As\npredicted in Sec. III C 1 a, all modes exhibit a decreased\n\u0001\u000bcompared with those in Eqs. (43) and (42). For small\nQLand the chosen value of Ks(see Tab. I), the 1stfour\nmodes match the analytical result of Eq. (46), which is\nconsistent with the plot in Figure 3. For even higher ex-\ncited modes, the e\u000bect of the EA surface anisotropy be-\ncomes weaker due to the increase in transverse exchange\nenergy. These modes (not shown in the \fgure) approach\nthe value of \u0001 \u000bn.2. Easy-Plane Surface Anisotropy ( ^\u0018hard axis)\naLDaH10-3L45\n0.010.11100.00.10.20.3\nQL0.11100.50.60.70.80.91.0Re8wwM<bL\n-L2xL2m®\nHxL¤cL\nFIG. 7. a) \u0001 \u000bEPversus wave vector for the FVMSW geom-\netry, showing the four smallest eigenvalues. The dashed lines\nrepresent the analytic solutions from Sec. III C 1 b. b) Disper-\nsion relation in the dipole-dipole active regime. c) Magnitude\nof eigenvectors (in arbitrary units) across the \flm at QL= 5.\nFigure 7 shows \u0001 \u000bEPfor the FVMSW geometry with\nan EP surface anisotropy. We see that the mode corre-\nsponding to n= 0 has been promoted to a surface mode\nwith a large \u0001 \u000b, which for small values of QLmatches\nEq. (47). For the higher excited modes, we observe a\ndecrease in \u0001 \u000bcompared to the case with no surface\nanisotropy.\nB. BVMSW ( \u0012=\u0019=2and\u001e= 0)\n0.010.1110100QL0.20.40.60.81.0DaH10-3L\n0.11100.70.80.91.01.11.2Re8wwM<\n-L2xL2m®\nHxL¤\nFIG. 8. \u0001\u000bversus wave vector for the BVMSW geometry\n(\u0012=\u0019=2 and\u001e= 0) withKs= 0, plotted for the four small-\nest eigenvalues. Left inset: magnitudes of normalized eigen-\nvectors across the \flm at QL= 5. Right inset: dispersion\nrelation in the dipole-dipole active regime.\nFigure 8 shows the QL-dependent renormalization of\nthe Gilbert damping due to spin pumping at the FI-\nNM interface in the BVMSW geometry. We see that\nthe enhancement \u0001 \u000bagrees with the analytic limits in12\nEqs. (43) and (42) for small values of QL. For large val-\nues ofQL, we are in the strong exchange regime, in which\nthe in-plane exchange energy becomes large compared to\nall other energy contributions. This in-plane exchange\nsti\u000bness e\u000bectively quenches the coupling to the normal\nmetal layer, causing \u0001 \u000b!0 for large values of QL.\nAlthough Figure 8 only appears to show the three \frst\neigenvalues and eigenvectors, it actually contains dou-\nble this amount. Because ^zis parallel to the wave-\npropagation direction ^\u0010in this geometry, there is no\nchange in dipolar energies, regardless of whether the wave\ntravels in the + ^\u0010direction or in the \u0000^\u0010direction; thus,\nthe Gilbert damping is enhanced equally in both wave\ndirections. A slight o\u000bset from this con\fguration, taking\neither\u0012 < \u0019= 2 or\u001e6= 0, would result in a splitting of\neach line in Figure 8 into two distinct lines.\nIncluding Surface Anisotropy\nFigure 9 shows both the EA and the EP surface-\nanisotropy calculations in the BVMSW geometry. In the\ncase of an EA surface anisotropy, the mode correspond-\ning ton= 0 gets promoted to a surface mode, similarly\nto the case in which there is EP surface anisotropy in the\nFVMSW geometry. The increase in \u0001 \u000bis much smaller\nfor the same magnitude of Ks, as explained in detail in\nSec. III C. The higher modes, corresponding to n > 0,\nexhibit increased quenching of the Gilbert damping en-\nhancement. In the case of EP surface anisotropy, all\nmodes exhibit quenched Gilbert damping enhancement.\nC. MSSW ( \u0012=\u001e=\u0019=2)\nFigure 10 shows the QL-dependent renormalization of\nthe Gilbert damping due to spin pumping at the FI-NM\ninterface in the MSSW geometry. The computed eigen-\nvalues agree with Eqs. (43) and (42) for small values of\nQL. We see in the inset of Figure 10 that in this geom-\netry, the macrospin-like mode behaves as predicted by\nDamon and Eshbach3433, cutting through the dispersion\nrelations of the higher excited modes for increasing val-\nues ofQLin the dipole-dipole regime. A prominent fea-\nture of this geometry is the manner in which the modes\nwith di\u000berent signs of Ref!gbehave di\u000berently due to\nthe dipole-dipole interaction. This is because the inter-\nnal \feld direction ( ^z) is not parallel to the direction of\ntravel ( ^\u0010) of the spin wave. Hence, changing the sign of\n!is equivalent to inverting the externally applied \feld,\nchanging the xyzcoordinate system in Figure 1 from a\nright-handed coordinate system to a left-handed system.\nIn the middle of the dipole regime, the lack of symme-\ntry with respect to propagation direction has di\u000berent\ne\u000bects on the eigenvectors; e.g., in the dipole-dipole ac-\ntive region the modes with positive or negative Ref!g\nexperience an increased or decreased magnitude of the\ndynamic magnetization, depending on the value of QL,as shown in Figure 10e & f. This magnitude di\u000berence\ncreates di\u000berent renormalizations of the Gilbert damp-\ning, as the plot of \u0001 \u000b(\u0006)in Figure 10b & c shows.\nIncluding Surface Anisotropy\nFigure 11 shows \u0001 \u000bcomputed for modes in the MSSW\ngeometry with EA and EP surface anisotropies. We can\nclearly see that for small QLan exponentially localized\nmode exists in the EA case, and as predicted in Sec. III C,\nall the lowest-energy modes have spin pumping quenched\nby EP surface anisotropy. This is similar to the corre-\nsponding case in the BVMSW geometry.\nD. AC and DC ISHE\nFigure 12 shows the DC and AC ISHE measures for\nthe BVMSW geometry corresponding to the data repre-\nsented in Figure 8. In this geometry, the angular term,\nsin\u0012, in Eq. (28) is to equal one, ensuring that the DC\nmeasure is nonzero. This is not the case for all geometries\nbecause the DC electric \feld vanishes in the FVMSW ge-\nometry. The mode-dependent DC ISHE measure exhibits\nthe sameQL-dependence as the spectrum of the Gilbert\ndamping enhancement in all geometries where sin \u00126= 0.\nWe have already presented the renormalization of the\nGilbert damping in the most general cases above. There-\nfore, we restrict ourselves to presenting the simple case of\nthe BVMSW geometry with no surface anisotropy here.\nThe AC ISHE measure plotted in Figure 12 exhibits\na similarQLdependence to the Gilbert damping renor-\nmalization (and hence the DC ISHE measure), but with a\nslight variation in the spectrum towards higher values of\nQL. Note that because Eq. (24) is non-zero for all values\nof\u0012, the AC e\u000bect should be detectable in the FVMSW\ngeometry. By comparing the computed renormalization\nof the Gilbert damping for the di\u000berent geometries in\nthe previous subsections, we see that the strong renor-\nmalization of the n= 0 induced surface mode that oc-\ncurs in the FVMSW geometry with easy-plane surface\nanisotropy (see Sec. IV A 2 and Fig. 7) can have a pro-\nportionally strong AC ISHE signal in the normal metal.\nV. CONCLUSION\nIn conclusion, we have presented analytical and numer-\nical results for the spin-pumping-induced Gilbert damp-\ning and direct- and alternating terms of the inverse spin-\nHall e\u000bect. In addition to the measures of the magnitudes\nof the DC and AC ISHE, the e\u000bective Gilbert damp-\ning constants strongly depend on the modes through the\nwave numbers of the excited eigenvectors.\nIn the long-wavelength limit with no substantial sur-\nface anisotropy, the spectrum is comprised of standing-\nwave volume modes and a uniform-like (macrospin)13\n0.010.1110100QL0.51.01.52.0aLDaH10-3L\n0.11100.70.80.91.01.11.2Re8wwM<\n-L2xL2m®\nHxL¤\n0.0010.010.1110100QL0.20.40.60.81.0bLDaH10-3L\n0.11100.70.80.91.01.11.2Re8wwM<\n-L2xL2m®\nHxL¤\nFIG. 9. a) Dispersion relation versus wave vector for the BVMSW geometry ( \u0012=\u0019=2,\u001e= 0) for the four lowest eigenvalues in\nthe case of EA surface anisotropy. b) Dispersion relation in the case of EP surface anisotropy. In both \fgures, the horizontal\ndashed lines mark the value of \u0001 \u000bnin the case of no surface anisotropy.\n0.010.11101000.00.20.40.60.8Da+H10-3LaL\n0.010.11101000.00.20.40.60.81.0\nQLDa-H10-3LbL\n0.010.11101000.91.1.1ÈReHwwMLÈcL\nQL\n-L2xL20.0.51.1.5m®\nHxL¤dL\n-L2xL20.0.51.m®\nHxL¤eL\nFIG. 10. Gilbert damping renormalization in the MSSW geometry. Subplots a) and b) show Gilbert damping renormalization\n\u0001\u000bfor modes with positive (negative) Ref!g. The horizontal dashed lines represent the analytical values \u0001 \u000b0and \u0001\u000bnfor\nsmallQL. c) Dispersion relation versus wave vector for the MSSW geometry ( \u0012=\u001e=\u0019=2) for the four smallest eigenvalues,\ncolored pairwise in \u0006!. Subplot d (e) shows the magnitude of normalized eigenvectors (in arbitrary units) at QL= 3 across\nthe \flm modes with positive (negative) Ref!g.\n0.010.11101000.0.51.1.5Da+H10-3LaL\n0.010.11101000.0.51.1.5\nQLDa-H10-3LbL\n0.010.11101000.00.20.40.60.8Da+H10-3LcL\n0.010.11101000.00.20.40.60.8\nQLDa-H10-3LdL\nFIG. 11. a) and b) Gilbert damping renormalization from spin pumping in the MSSW geometry ( \u0012=\u001e=\u0019=2) for modes with\npositive (negative) Ref!gin the case of EA surface anisotropy. The four smallest eigenvalues are colored pairwise in \u0006!across\nthe plots. c) and d) show the Gilbert damping renormalization in the case of EP surface anisotropy.\nmode. These results are consistent with our previous\n\fndings18: in the long-wavelength limit, the ratio be-\ntween the enhanced Gilbert damping for the higher vol-ume modes and that of the macrospin mode is equal\nto two. When there is signi\fcant surface anisotropy,\nthe uniform mode can be altered to become a pure lo-14\n0.010.11101000.00.10.20.30.40.50.6\nQLeACH10-4LaL\n0.010.11101000.00.51.01.5\nQLeDCH10-9cmLbL\nFIG. 12. ISHE as a function of in-plane wave vector in the\nBVMSW geometry with Ks= 0. a) AC ISHE measure of\nEq. (28); b) DC ISHE measure of Eq. (28).\ncalized surface mode (in the out-of-plane geometry and\nwith EP surface anisotropy), a blend between a uniform\nmode and a localized mode (in-plane geometries and EA\nsurface anisotropy), or quenched uniform modes (out-of-\nplane \feld con\fguration and EA surface anisotropy, or\nin-plane \feld con\fguration and EP surface anisotropy).\nThe e\u000bective Gilbert damping is strongly enhanced for\nthe surface modes but decreases with increasing surface-\nanisotropy energies for all the other modes.\nThe presented measures for both the AC and DC in-\nverse spin-Hall e\u000bects are strongly correlated with the\nspin-pumping renormalization of the Gilbert damping,\nwith the DC e\u000bect exhibiting the same QLdependency,\nwhereas the AC e\u000bect exhibits a slighthly di\u000berent vari-\nation for higher values of QL. Because the AC e\u000bect\nis nonzero in both in-plane and out-of-plane geometries\nand because both EP and EA surface anisotropies in-\nduce surface-localized waves at the spin-active interface,\nthe AC ISHE can be potentially large for these modes.\nACKNOWLEDGMENTS\nWe acknowledge support from EU-FET grant no.\n612759 (\\InSpin\"), ERC AdG grant no. 669442 (\\In-sulatronics\"), and the Research Council of Norway grant\nno. 239926.\nAppendix A: Coordinate transforms\nThe transformation for vectors from \u0018\u0011\u0010toxyzcoor-\ndinates (see Fig. 1) is given by an a\u000ene transformation\nmatrixT, so that\nf(xyz)=T\u0001f(\u0018\u0011\u0010);\nfor some arbitrary vector f. Tensor{vector products are\ntransformed by inserting a unity tensor I=T\u00001Tbe-\ntween the tensor and vector and by left multiplication by\nthe tensor T, such that the tensor transforms as TbGT\u00001\nfor some tensor bGwritten in the \u0018\u0011\u0010basis.\nTis given by the concatenated rotation matrices T=\nR2\u0001R1, whereR1is a rotation \u001earound the \u0018-axis,\nandR2is a rotation \u0012\u0000\u0019\n2around the new \u0011-axis/y-axis.\nHence,\nR1=0\nB@1 0 0\n0 cos\u001e\u0000sin\u001e\n0 sin\u001ecos\u001e1\nCA; (A1)\nR2=0\nB@sin\u00120\u0000cos\u0012\n0 1 0\ncos\u00120 sin\u00121\nCA; (A2)\nsuch that\nT=0\nB@sin\u0012\u0000cos\u0012sin\u001e\u0000cos\u0012cos\u001e\n0 cos\u001e\u0000sin\u001e\ncos\u0012sin\u0012sin\u001e sin\u0012cos\u001e1\nCA: (A3)\nThis transformation matrix consists of orthogonal trans-\nformations; thus, the inverse transformation, which\ntransforms xyz!\u0018\u0011\u0010, is just the transpose, T\u00001=TT.\n1A. Serga, A. Chumak, and B. Hillebrands, J. Phys. D 43,\n264002 (2010).\n2V. Cherepanov, I. Kolokolov, and V. L'vov, Physics Re-\nports 229, 81 (1993).\n3Y. Kajiwara, K. Harii, S. Takahashi, J. Ohe, K. Uchida,\nM. Mizuguchi, H. Umezawa, H. Kawai, K. Ando,\nK. Takanasahi, S. Maekawa, and E. Saitoh, Nature 464,\n262 (2010).\n4C. W. Sandweg, Y. Kajiwara, K. Ando, E. Saitoh, and\nB. Hillebrands, Appl. Phys. Lett. 97(2010).5C. W. Sandweg, Y. Kajiwara, A. V. Chumak, A. A. Serga,\nV. I. Vasyuchka, M. B. Jung\reisch, E. Saitoh, and B. Hille-\nbrands, Phys. Rev. Lett. 106, 216601 (2011).\n6B. Heinrich, C. Burrowes, E. Montoya, B. Kardasz, E. Girt,\nY.-Y. Song, Y. Sun, and M. Wu, Phys. Rev. Lett. 107,\n066604 (2011).\n7S. M. Rezende, R. L. Rodriguez-Suarez, M. M. Soares,\nL. H. Vilela-Leao, D. L. Dominguez, and A. Azevedo,\nAppl. Phys. Lett. 102, 012402 (2013).\n8L. H. Vilela-Leao, A. A. C. Salvador, and S. M. Rezende,15\nAppl. Phys. Lett. 99, 102505 (2011).\n9K. Ando and E. Saitoh, Phys. Rev. Lett. 109, 026602\n(2012).\n10C. Burrowes, B. Heinrich, B. Kardasz, E. A. Montoya,\nE. Girt, Y. Sun, Y.-Y. Song, and M. Wu, Appl. Phys.\nLett. 100, 092403 (2012).\n11M. B. Jung\reisch, V. Lauer, R. Neb, A. V. Chumak,\nand B. Hillebrands, Applied Physics Letters 103 (2013),\nhttp://dx.doi.org/10.1063/1.4813315.\n12C. Hahn, G. de Loubens, M. Viret, O. Klein, V. V. Naletov,\nand J. Ben Youssef, Phys. Rev. Lett. 111, 217204 (2013).\n13Y. Tserkovnyak, A. Brataas, G. E. W. Bauer, and B. I.\nHalperin, Rev. Mod. Phys. 77, 1375 (2005).\n14A. Brataas, A. D. Kent, and H. Ohno, Nat. Mater. 11,\n372 (2012).\n15Y. Tserkovnyak, A. Brataas, and G. E. W. Bauer, Phys.\nRev. Lett. 88, 117601 (2002).\n16J. Xiao and G. E. W. Bauer, Phys. Rev. Lett. 108, 217204\n(2012).\n17A. Brataas, Y. Tserkovnyak, G. E. W. Bauer, and P. J.\nKelly, in Spin Current , edited by S. Maekawa, S. O. Valen-\nzuela, E. Saitoh, and T. Kimura (Oxford University Press,\n2012).\n18A. Kapelrud and A. Brataas, Phys. Rev. Lett. 111, 097602\n(2013).\n19P. Monod, H. Hurdequint, A. Janossy, J. Obert, and\nJ. Chaumont, Phys. Rev. Lett. 29, 1327 (1972).\n20R. H. Silsbee, A. Janossy, and P. Monod, Phys. Rev. B\n19, 4382 (1979).\n21A. Janossy and P. Monod, Phys. Rev. Lett. 37, 612 (1976).\n22S. Mizukami, Y. Ando, and T. Miyazaki, Japanese journal\nof applied physics 40, 580 (2001).\n23R. Urban, G. Woltersdorf, and B. Heinrich, Phys. Rev.\nLett. 87, 217204 (2001).\n24A. Brataas, Y. Tserkovnyak, G. E. W. Bauer, and B. I.\nHalperin, Phys. Rev. B 66, 060404 (2002).25Y. Tserkovnyak, A. Brataas, and G. E. W. Bauer, Phys.\nRev. B 66, 224403 (2002).\n26L. Berger, Physical Review B 54, 9353 (1996).\n27J. Slonczewski, J. Magn. Magn. Mater. 159, L1 (1996).\n28A. Brataas, A. D. Kent, and H. Ohno, Nature materials\n11, 372 (2012).\n29T. Gilbert, Phys. Rev. 100, 1243 (1955).\n30T. Gilbert, IEEE Trans. Magn. 40, 3443 (2004).\n31B. A. Kalinikos, Sov. Phys. J. 24, 719 (1981).\n32B. A. Kalinikos and A. N. Slavin, J. Phys. C 19, 7013\n(1986).\n33J. R. Eshbach and R. W. Damon, Phys. Rev. 118, 1208\n(1960).\n34R. Damon and J. Eshbach, J. Phys. Chem. Solids 19, 308\n(1961).\n35H. Puszkarski, IEEE Trans. Magn. 9, 22 (1973).\n36R. E. D. Wames and T. Wolfram, J. Appl. Phys. 41, 987\n(1970).\n37G. Rado and J. Weertman, J. Phys. Chem. Solids 11, 315\n(1959).\n38H. Jiao and G. E. W. Bauer, Phys. Rev. Lett. 110, 217602\n(2013).\n39M. B. Jung\reisch, V. Lauer, R. Neb, A. V. Chumak, and\nB. Hillebrands, ArXiv e-prints (2013), arXiv:1302.6697\n[cond-mat.mes-hall].\n40Z. Qiu, K. Ando, K. Uchida, Y. Kajiwara, R. Takahashi,\nT. An, Y. Fujikawa, and E. Saitoh, ArXiv e-prints (2013),\narXiv:1302.7091 [cond-mat.mes-hall].\n41H. L. Wang, C. H. Du, Y. Pu, R. Adur, P. C. Hammel,\nand F. Y. Yang, Phys. Rev. Lett. 112, 197201 (2014).\n42F. R. Morgenthaler, IEEE Trans. Magn. 8, 130 (1972).\n43W. Press, S. Teukolsky, W. Vetterling, and B. Flannery,\nNumerical Recipies , 3rd ed. (Cambridge University Press,\n2007)."    },    {        "title": "1612.09551v2.Spectroscopic_evidence_of_Alfvén_wave_damping_in_the_off_limb_solar_corona.pdf",        "content": "arXiv:1612.09551v2  [astro-ph.SR]  6 Feb 2017Draftversion August12, 2021\nPreprint typesetusingL ATEX styleemulateapjv. 01 /23/15\nSPECTROSCOPIC EVIDENCEOF ALFV ´ENWAVEDAMPINGIN THEOFF-LIMBSOLARCORONA\nG.R.Gupta\nInter-University Centre for Astronomy and Astrophysics, P ostBag-4, Ganeshkhind, Pune411007, India\nDraft version August12, 2021\nABSTRACT\nWe investigate off-limb active region and quiet Sun corona using spectroscopi c data. Active region is clearly\nvisible in several spectral lines formed in the temperature range of 1.1–2.8 MK. We derive electron number\ndensityusinglineratio method,andnon-thermalvelocityi n theoff-limbregionupto the distanceof140Mm.\nWe compare density scale heights derived from several spect ral line pairs with expected scale heights as per\nhydrostatic equilibrium model. Using several isolated and unblended spectral line profiles, we estimate non-\nthermal velocities in active region and quiet Sun. Non-ther mal velocities obtained from warm lines in active\nregionfirstshowincreaseandlatershoweitherdecreaseora lmostconstantvaluewithheightinthefaro ff-limb\nregion, whereas hot lines show consistent decrease. Howeve r, in the quiet Sun region, non-thermal velocities\nobtainedfromvariousspectrallinesshoweithergradualde creaseorremainalmostconstantwithheight. Using\nthese obtained parameters, we further calculate Alfv´ en wa ve energy flux in the both active and quiet Sun\nregions. We findsignificantdecreaseinwave energyfluxeswit hheight,andhenceprovideevidenceofAlfv´ en\nwave damping. Furthermore,we derivedampinglengthsofAlf v´ enwaves in thebothregionsandfind themto\nbein therangeof 25-170Mm. Di fferentdampinglengthsobtainedat di fferenttemperaturesmaybeexplained\nas either possible temperature dependent damping or measur ements obtained in di fferent coronal structures\nformed at different temperatures along the line-of-sight. Temperature d ependent damping may suggest some\nroleofthermalconductioninthedampingofAlfv´ enwavesin thelowercorona.\nKeywords: Sun: Corona— Sun: UVradiation— Waves— Turbulence\n1.INTRODUCTION\nHeatingofsolaratmosphereandaccelerationofsolarwind\nremain two of the most puzzling problems in the solar and\nspace physics. Thereare several theoriesproposedto expla in\nthe phenomena, however, to identify any one dominant pro-\ncess is extremely di fficult to do. For details, see Parnell &\nDe Moortel (2012), andDe Moortel & Browning (2015)and\nreferences therein for current progress in the field. Most of\nthe modelsproposedso farare attributed to eitherdissipat ion\nof magnetohydrodynamics(MHD) waves or magneticrecon-\nnection. Amongthe severalproposedideas, role of wave tur-\nbulencein the heatingof solar coronaand accelerationof so -\nlar wind is one of the well studied model (see recent reviews\nbyArregui 2015 ;Cranmer et al. 2015 ).Alfv´ en(1942) first\nsuggested existence of electromagnetic-hydrodynamicwav es\nin the solar atmosphere and its importance in the heating of\nsolar corona ( Alfv´ en 1947 ). This led to the wave heating\nmodel of solar corona. In this model, convective motions at\nthe footpoints of magnetic flux tubes are assumed to gener-\natewave-likefluctuationsthatpropagateupintothe extend ed\ncorona (Cranmer & van Ballegooijen 2005 ;Suzuki & Inut-\nsuka2005 ). Thesefluctuationsareoftenproposedtopartially\nreflectbackdowntowardtheSun,developintostrongmagne-\ntohydrodynamics(MHD) turbulence, and dissipate graduall y\n(Cranmeretal.2007 ;Verdinietal.2010 ). Recently, VanBal-\nlegooijenetal. (2011)developedathree-dimensionalmagne-\ntohydrodynamics (MHD) Alfv´ en wave turbulence model to\nexplaintheheatingofbothsolar chromosphereandcoronain\nthecoronalloop. Anothermodelusedtoexplaincoronalheat -\ning is nanoflare heating model (see recent review by Klim-\nchuk2015 ). Inthismodel,randomphotosphericmotionsand\nflowsleadtotwistingandbraidingofcoronalfieldlines. Thi s\nresultsinbuildingupofmagneticstress,andthus,leadsto re-\ne-mail: girjesh@iucaa.inleaseofenergyintheformofimpulsiveheatingeventscalle d\nasnanoflares( Parker1988 ).\nIn order to understand wave heating mechanism of solar\natmosphere, observations of detection, propagation, and d is-\nsipation of waves are essential. Tomczyk et al. (2007) and\nMcIntosh et al. (2011) reported the ubiquitous presence of\noutwardpropagatingAlfv´ enic (transverse)waves in the so lar\ncorona. Propagating Alfv´ enic waves were also found in the\npolarregion( Guptaetal.2010 ;Mortonetal.2015 ). Compre-\nhensivereviewexistsonthedetectionofpropagatingwaves in\nthe solar atmosphere, e.g., Banerjee et al. (2011);De Moor-\ntel & Nakariakov (2012);Jess et al. (2015). In recent stud-\nies, evidence of damping of propagating waves are also re-\nported(Mortonetal.2014 ;Gupta2014 ;KrishnaPrasadetal.\n2014). SignaturesofAlfv´ enwavescanalsobefoundthrough\nthe study of broadening of spectral line profiles in the solar\ncorona(e.g., Banerjee et al. 2009 ;Jess et al. 2009 ). Alfv´ enic\nwave motions are transverse to the direction of propagation .\nIn case of field lines aligned in the plane of sky, plasma mo-\ntions due to Alfv´ enic waves will either be directed towards\nor away from the line-of-sight. In the o ff-limb corona, sev-\neral spatially unresolved structures may be present along t he\nline-of-sight with di fferent phases of oscillations. These un-\nresolvedwavemotionscanleadtonon-thermalbroadeningof\nspectrallineprofiles. Thus,observednon-thermalbroaden ing\nof spectral line profiles in the corona will be proportional t o\nAlfv´ enwaveamplitude,e.g., Moran(2001).\nThere are numerous studies devoted to measure o ff-limb\nnon-thermal broadening of spectral lines to search for any\nwave activity. Hassler et al. (1990)performedthe first obser-\nvations of high temperature line profiles in the solar o ff-limb\nregion using sounding rocket experiments. They found in-\ncreaseinlinewidthwithheightabovethelimbandinterpret ed\nasa signatureofpropagatinghydromagneticwavesin theso-\nlar corona. Later, more studies were carried out using space2 Gupta\nbased SUMER instrument on-board SOHO. Using SUMER,\nDoyle et al. (1998) andBanerjee et al. (1998)found increase\ninnon-thermalline widthwitho ff-limbheightandassociated\ndensity decrease. Their results were in excellent agreemen t\nwithpredictionsfromoutwardpropagatingundampedAlfv´ e n\nwaves.Harrison et al. (2002) performed similar analysis on\noff-limb part of quiet Sun corona using CDS instrument on-\nboardSOHO.Theyfoundnarrowingoflinewidthwithheight\nand interpreted as indication of wave dissipation in a close d\nloop system in the low corona. Banerjee et al. (2009) per-\nformed similar analysis on polar plume and interplume re-\ngion using Extreme-Ultraviolet Imaging Spectrometer (EIS ,\nCulhane et al. 2007 ) on-board Hinode ( Kosugi et al. 2007 ).\nTheyfoundsignaturesofoutwardpropagatingundampedlin-\near Alfv´ en waves within 1 .1R⊙. Recently, Bemporad &\nAbbo(2012) andHahn et al. (2012) measured non-thermal\nline width up to 1 .4R⊙in the open magnetic field of polar\nregions using EIS/Hinode. They found signature of damp-\ning of Alfv´ en waves beyond 1 .1−1.14R⊙.Lee et al. (2014)\ninvestigatedcoolloopanddarklaneoverao ff-limbactivere-\ngion and obtained basic plasma parameters as a function of\nheight above the limb. They found slight decrease in non-\nthermalvelocityalongthe coolloopwhereassharpfall alon g\nthedarklane. Theyattributedthesefindingstowavedamping .\nHahn&Savin (2014)alsomeasuredenergyanddissipationof\nAlfv´ enicwavesinthequietSunregion.\nRecently, Van Ballegooijen et al. (2011) developed a 3-D\nMHDmodelofAlfv´ enwaveturbulencetoexplaintheheating\nof solar chromosphere and corona in the coronal loop. This\nmodel has attracted lot of attention from the community to\nlook for such signatures (e.g. Asgari-Targhi et al. 2014 ). In\nthiswork,wefocusono ff-limbactiveregionloopsystemand\nquiet Sun corona to study propagation of Alfv´ en waves with\nheight and search for any signature of their damping over a\nwide rangeof temperature. As Alfv´ en wave energyflux den-\nsity isgivenby(e.g. Moran2001 ),\nED=ρξ2VA=/radicalbiggρ\n4πξ2B (1)\nwhereρismassdensity(ρ=mpNe,mpisprotonmass,and\nNeiselectronnumberdensity), ξisAlfv´ enwavevelocityam-\nplitude, and VAis Alfv´ en wave propagationvelocitygiven as\nB//radicalbig\n4πρ. Therefore,totalwaveenergyfluxcrossingasurface\nareaA willbe givenby,\nEF=1√\n4π/radicalbig\nmpNeξ2B A (2)\nHenceforth,totalAlfv´ enwaveenergyfluxdependsonelec-\ntronnumberdensity,waveamplitude,magneticfield,andare a\nof cross section. In this paper, our main focus is to estimate\ntotalwaveenergyfluxwithheightintheo ff-limbsolarcorona,\nandthus,tofindanysignaturesofwavedamping. Forthepur-\npose, we identified a unique set of good spectroscopic data\ncoveringtheoff-limbactiveregionandquietSunobservedby\nEIS/Hinode. Data coversvariousspectrallinesformedovera\nwiderangeoftemperature. Previoussuchstudiesweremainl y\ncarriedoutwithfewspectrallinesformedatverysimilarte m-\nperature e.g. Fexii, and Fexiii. Therefore, current study\nprovidesan uniqueopportunityto carry out such analysis fo r\ncoronal structures formed over a wide range of temperature.\nThismayalsoenableustofindanypossibleexistenceoftem-\nperature dependence. Related details of observations are d e-\nFigure1. Intensity map of o ff-limb active region and quiet Sun in Fe xii\n195.12 Å spectral line rastered by EIS /Hinode on 17 December 2007. Con-\ntinuous white line indicates location ofsolar limb.\nscribed in§2. We employ spectroscopic methods to obtain\nelectron number density, and non-thermalvelocity which ar e\ndescribed in§3.1, and§3.2respectively. In§3.3, we de-\nscribe calculation of Alfv´ en wave energyflux using obtaine d\nparameters. Obtained results are discussed in §4, and final\nsummaryandconclusionsareprovidedin §5.\n2.OBSERVATIONSAND DATAANALYSIS\nOff-limb active region AR 10978 was observed by\nEIS/Hinode on 17 December 2007. EIS observations were\ncarried out with 2′′slit and exposure time of 45 s. Obser-\nvations were performed over the wavelength range of 180–\n204 Å and 248–284 Å. Raster scan started at 10:42:20 UT\nand completed at 13:02:17 UT and covered a field of view\nof 360′′×512′′. This dataset was previously analyzed by\nO’Dwyer et al. (2011) to study electron density and temper-\nature structure of a limb active region. We followed stan-\ndard procedures for preparing the EIS data using IDL rou-\ntine EISPREP1available in the Solar Software (SSW;Free-\nland&Handy1998 ). Recently, Brooks&Warren (2016)and\nTestaetal. (2016)showedthatabsolutecalibrationofEISdata\nleadstoasystematicoverestimationofspectrallinewidth sfor\nmostofthepixelsalongslit. Thus,forthepurposeofmeasur -\ning line widths, we obtained EIS spectra in the data number\n(DN) unit by applying EIS PREP routine with /noabs key-\nword. Moreover,wealsoobtainedEISspectrainthephysical\nunits (erg cm−2s−1sr−1) to further perform electron number\ndensity diagnostics. Routine also provides errorbars on th e\n1ftp://sohoftp.nascom.nasa.gov /solarsoft/hinode/eis/doc/eisnotes/01EISPREP/eisswnote01.pdf3\nobtainedintensities. In addition,there also exists 22% un cer-\ntaintyintheobservedintensitybasedonpre-flightcalibra tion\nof EIS (Lang et al. 2006 ). All the EIS spectral line profiles\nwere fitted with Gaussian function using EIS AUTOFIT2.\nRoutine also provides one-sigma errorbars on the fitted pa-\nrameters. Comparison between both type of spectra recon-\nfirmsthe systematicoverestimationoflinewidthsfromabso -\nlutely calibrated data as recently reported by Brooks & War-\nren(2016) andTesta et al. (2016). However, magnitude of\nthissystematicoverestimationofline-widthswerefoundt obe\nvery small in the current dataset. As EIS sensitivity is evol v-\ning over time, absolutely calibrated data (in physical unit s)\nand related errors were further recalibrated using the meth od\nofWarren et al. (2014). There exists spatial o ffsets in the\nsolar-Xandsolar-Ydirectionsbetweenimagesobtainedfro m\ndifferent wavelengths. These o ffsets were corrected with re-\nspect to image obtained from Fe xii195.12 Å spectral line3.\nFigure1shows intensity map of observed o ff-limb active re-\ngion obtained from Fe xii195.12Å line. Observed active re-\ngion is very bright and has several saturated image pixels at\nfewlocations.\nFigure2. Contribution function of spectral lines selected for detai led analy-\nsisofoff-limb active andquietSunregions recorded byEIS /Hinode(seealso\nTable1).\nTo identify spectral line wavelengths and corresponding\npeak formation temperatures, all the atomic data used in thi s\nstudy are taken from CHIANTI atomic database ( Dere et al.\n1997;Del Zanna et al. 2015 ). To perform line width anal-\nysis, we identified several unblended and isolated spectral\nlineswith goodsignalstrengthas highlightedby Younget al.\n(2007) (see Table 1). Although there exist some blend in\nFexiv274 Å, and Fexv284 Å lines, their contribution can\nsafely be ignored in the active region conditions. Lines are\nchosen in such a way to get good coverage over temperature\nrange. Contribution function of selected spectral lines we re\ncalculated using the CHIANTI v.8 ( Del Zanna et al. 2015 )at\nconstant electron number density Ne=109cm−3. Obtained\ncontributionfunctioncurvesareplottedinFigure 2. Peakfor-\nmation temperatureof all the selected lines are also provid ed\nin Table1. We identify all the spectral lines formed below 2\nMK temperature as warm lines whereas those formed above\n2ftp://sohoftp.nascom.nasa.gov /solarsoft/hinode/eis/doc/eisnotes/16AUTOFIT/eisswnote16.pdf\n3ftp://sohoftp.nascom.nasa.gov /solarsoft/hinode/eis/doc/eisnotes/03GRATING DETECTOR TILT/eisswnote03.pdf2 MK as hot lines. We also identified several density sensi-\ntive lines and utilized them only for the purpose of deriving\nelectronnumberdensity.\nTable1\nListof emission lines used in the present study. Lines marke d with asterisks\n(*) are density sensitive lines and used only to calculate el ectron number\ndensities.\nIon Wavelength (Å)aTpeak(MK)a\nFex184.537 1.12\nFexi180.401, 182.167∗1.37\nSix258.374∗,261.056 1.41\nSx264.231 1.55\nFexii192.394, 196.640∗1.58\nFexiii196.525∗,202.044 1.78\nFexiv264.789∗,274.204b2.00\nFexv284.163c2.24\nFexvi262.976 2.82\naWavelengths andpeak formationtemperatures are taken from\nCHIANTIdatabase.\nbblended withSivii274.180 Å.\ncblended withAlix284.042 Å.\nIn Figure 3, we plot monochromatic intensity maps of\noff-limb active region obtained from di fferent emission lines\nformed over temperature range of 1.1 MK to 2.8 MK. Inten-\nsity mapsclearly show that structuresin the active regiona re\nnot well defined as discrete loop, instead emissions are more\nlikelydiffuseinnaturewithoutanysharpboundaries. Di ffuse\nemissions observed from active region in Fe x–Fexvilines\narereal. Suchdiffuseemissionswerealsohighlightedbysev-\neral authors in the past (e.g., O’Dwyer et al. 2011 ). Here,\nweshowdiffusenatureofactiveregionwithseveralemission\nlinesformedover1.1MK to2.8MK.\nTo study variation of several physical parameters with\nheight,we choseseveral structuresandstripes in the o ff-limb\nactiveregionandquietSun. Althoughactiveregionis di ffuse\ninnature,wecanstillidentifysomebrightloop-likestruc tures\nextendingfarintothecoronafromFe xii192.394Åintensity\nmap. We traced and analyzed several such structures. Here,\nwe present result from one such structure which was traced\nup to very far off-limb distance. Traced stripe is named as\nAR1. Furthermore, to get the average behavior of active re-\ngion, we binned over whole active region data in the solar-Y\ndirection. Similarly, we also binned over small quiet Sun re -\ngion in the solar-Y direction to study the quiet Sun. Boxes\nchosentoobtaintheaveragedataarelabeledasAR2 andQS.\nWe alsotracedanotherstripeparallelto AR1inthequietSun\nregiononlyforthebackgroundstudypurpose. Allthechosen\nstripesandboxesareshowninFigure 3.\nAs selected spectral lines are isolated and unblended, we\nfitted all the profiles with single Gaussian function. In Fig-\nure4, we show examples of spectral line profiles and fitted\nGaussian. Profiles were obtained at the o ff-limb distance of\n61 Mm along AR1, AR2, and QS. From the plots, it is clear\nthatalltheprofilesaresymmetricandcanbewellrepresente d\nbya singleGaussianfunction.\nOne of the factor which may a ffect our analysis would be\ncontaminationfrominstrumentalscatteredorstraylight. Min-\nimumstray light contributionabovethe darkcurrentis foun d\ntobearound2%ofthetotalon-diskcountsforthatrespectiv e4 Gupta\nFigure3. Monochromatic intensity mapsofo ff-limb active region and quiet Sunobtained indi fferent wavelengths usingEIS /Hinode (aslabeled). Dashed white\nlines on each panel indicate o ff-limb locations (active region AR1, AR2, and quiet Sun QS) ch osen for detailed analysis. Continuous white line on each pa nel\nindicates location of solar limb.\nline4. Wechosesufficientboxsizetocalculateaveragecounts\nfrom each spectral line in the on-disk part of Sun. As stray\nlightcontaminationintheo ff-limbcoronawillbesimply0.02\ntimes of average counts, we obtained fraction of stray light\ncontribution along all the stripes for all the lines. As inte n-\nsity drops-offwith height in the o ff-limb corona, stray light\ncontributionincreaseswith height. We foundstray light co n-\ntributions to be less than 8% up to the o ff-limb distance of\n≈140 Mm along AR1 and AR2 in all the spectral lines ex-\ncept for Fexvi263 Å. These contributionswere obtained af-\nter using 2% weightage of on-disk counts. For Fe xvi263 Å\nspectral line, stray light contributions were below 9% up to\nthe distance of 100 Mm. Beyond that distance, contribution\nincreasessharplytoabout20%and35%alongAR1andAR2\nrespectively. For the quiet Sun region QS, stray light contr i-\nbutions were less than 15% up to very far distance ( ≈125\nMm) in most of the spectral lines. Stray light contributions\nin Fex185 Å, Fexiv274 Å, and Fexv284 Å lines increase\nsharplytoabove40%fordistancesbeyond130Mm,whereas\nthat in Fexvi262.976 Å goes beyond 100% for most of the\ndistances along QS. This may indicate that signal in Fe xvi\n262.976Åline alongQS is mainlyfromscatteredlight. Near\nthe limb, stray light contributionsfor all the stripes are m uch\nsmaller (<3% for heights<80 Mm and<90 Mm along\nAR1 and AR2 respectively, whereas <5% for heights<50\nMm for QS). As noted by Hahnet al. (2012),stray light con-\n4ftp://sohoftp.nascom.nasa.gov /solarsoft/hinode/eis/doc/eisnotes/12STRAYLIGHT/eisswnote12.pdftaminationsstartaffectinglinewidthmeasurementsonlyifits\ncontributions are more than 45%. In the current analysis, as\nstraylightcontributionsarefoundtobeverysmall,thus,i tef-\nfectsarealmost insignificantin the line widthmeasurement s.\nResults obtained in the present analysis are mainly derived\nfrom heights where stray light contaminationsare very smal l\nfor most of the spectral lines ( <8% for distances up to ≈140\nMmalongAR1andAR2,and <15%fordistancesupto ≈125\nMmalongQS).\n3.RESULTS\nAsperEquation 2,Alfv´ enwaveenergyfluxisproportional\nto√Neξ2BA. Wedescribeestimationofelectronnumberden-\nsityandwavevelocityamplitudeinthefollowingsubsectio ns.\n3.1.Intensity,Density,andEmission Measure\nIn Figure 5, we plot variation of intensity obtained from\nselected spectral lines with height along active region AR1 ,\nAR2,andquietSunQS.Associated1 σerrorbarsarealsoplot-\ntedwiththedata-points. Fromtheplots,itisclearthatdat aset\nhas good signal strength in all the selected spectral lines i n\nthe off-limb region of solar corona. This makes it suitable to\nestimate electron number density for the calculation of tot al\nAlfv´ enwaveenergyfluxwith height.\nElectron number densities obtained from various spec-\ntral line pairs in the corona can be compared with hydro-\nstatic equilibrium model. This have been done in the past\nwithimagingobservationsusingTransitionRegionandCoro -\nnal Explorer (TRACE; e.g., Aschwanden et al. 1999 ), and5\nFigure4. Examplesofspectral lineprofiles and fitted Gaussian profile s obtained attheoff-limb height of61Mmalong AR1(bluelines), AR2 (black lines ), and\nQS(green lines).\nrecently with spectroscopic observations using EIS /Hinode\n(e.g.,Leeet al.2014 ;Guptaetal. 2015 ).\nElectron number density profile in hydrostatic equilibrium\nisgivenby,\nNe(h)=Ne(0)exp/parenleftBigg\n−h\nλ(Te)/parenrightBigg\n(3)\nwhereλisdensityscale heightgivenby,\nλ(Te)=2kbTe\nµmHg≈46/bracketleftbiggTe\n1MK/bracketrightbigg\n[Mm] (4)\nwherekbistheBoltzmannconstant, Teiselectrontempera-\nture,µismeanmolecularweight( ≈1.4forthesolar corona),\nmHismassofthehydrogenatom,and gisaccelerationdueto\ngravityatthesolarsurface(seee.g., Aschwandenetal.1999 ).\nMoreover, observationally measured quantity such as in-\ntensity of an optically thin emission line dependson electr on\nnumber density, i.e. I∝Nβ\newhere 1<β<3, and value of\nβdependsuponwhether the given line is allowed, forbidden,\nor inter-system ( Mason & Monsignori Fossi 1994 ). In this\nstudy,we havechosenonlyallowedlinesto dotheline width\nanalysis. Some density sensitive forbidden lines were also\nchosen to calculate electron numberdensity (see Table 1). In\nFigure5, we plot variation of intensity obtained from all the\nspectrallineswith heightalongAR1, AR2,andQSstripes.\nSincedatasetcoversactiveandquietSunregionsoverwide\nrange of wavelength, we identified several density sensitiv e\nline pairs formed over range of temperature. We selected\nFexiλ182.167/λ180.401, Sixλ258.374/λ261.056, Fexii\nλ196.640/λ192.394, Fexiiiλ196.525/λ202.044, and Fexiv\nλ264.789/λ274.204line pairs to obtain the electron numberdensity (Young et al. 2007 ). In order to perform density\nandtemperaturediagnosticsonthe activeregionloops,bac k-\nground subtraction plays an important role (e.g., Del Zanna\n& Mason 2003 ).O’Dwyer et al. (2011) analyzed the current\ndatasetbeforeandusedquietSunregiontostudybackground\nemission. Wefollowthesamestrategyanduseintensityalon g\nthequietSunstripestoperformbackgroundsubtractionalo ng\nactive regionstripes. Therefore,we use quiet Sun stripes B G\nand QS to subtract background emission from active region\nstripes AR1 and AR2 respectively (stripes are shown in Fig-\nure3). Allplasmadiagnosticsareperformedovertheseback-\ngroundsubtractedintensities.\nIn Figure 6, we plot variation of electron number density\nderived from selected spectral line pairs with height along\nAR1, AR2, and QS. Plots show that as height increases elec-\ntronnumberdensitydecreases,however,correspondingerr or-\nbar increases with height. Some of the lines in quiet Sun re-\ngion show estimates with larger errorbars. Near the active\nregion limb, electron number densities were estimated to be\nof the order of>109cm−3which drops to around 108cm−3\nin the far off-limbregion. Densities obtained from Fe xi, and\nFexiiline pairs show almost similar numbers, whereas those\nobtainedfromFexiii, andSixline pairsshowsimilar values.\nNear the limb region, densities from Fe xi, and Fexiipairs\nshow consistently larger values than that of Fe xiii, and Six\npairs. However, they all seem to converge towards similar\nvalues beyond the distance of 80 Mm ( <5×108cm−3) and\n95Mm(<4.5×108cm−3) alongAR1 andAR2 respectively.\nDensities estimated from Fe xivline pair are smaller in num-\nberas comparedto other pairsandalso falls-o ffmore rapidly\nwithheightin bothAR1 andAR2.\nIn the quiet Sun region, we found number densities to be6 Gupta\nFigure5. Intensity variation with height along active region AR1, AR 2, and quiet Sun QS obtained from various spectral lines as la beled. True errors will also\ninclude 22% uncertainty of theobserved intensity based on t he pre-flight calibration of EIS( Lang etal. 2006 ).\nlower than those in active region. In this case, densities es ti-\nmated from Fexii, and Fexiiiline pairs converge to similar\nnumbers beyond 45 Mm ( <1.6×108cm−3). However, near\nthe limb, densities obtained from Fe xiipair are higher than\nthat from Fexiiipair. Densities obtained from Si xpair are\nhigherthanthat estimatedfromFe xii, andFexiiipairs. Den-\nsities estimated from Fe xiand Fexivpairs near the limb are\ncomparativelyhigherthanthoseobtainedfromotherlinepa irs\nbutdrops-offveryrapidlywith height.\nWe fitted electron number density variation with height\nalong AR1, AR2, and QS with exponential function Ne=\nN0exp(−h/Hd)+cusing MPFIT routines( Markwardt2009 ).\nFitsprovidedensityscaleheights Hdatdifferenttemperatures\nobtained from different spectral line pairs (see Table 2). Ex-\npected electron density scale heights λ(Te) as per hydrostatic\nequilibrium model at at di fferent temperatures (see Equa-\ntion4) are also provided in the table. Comparison between\ntwo density scale heights indicate that both the active and\nquiet Sun regions are basically underdense with few excep-\ntionsfromquietSunregion.\nAs active region and quiet Sun stripes were observed over\narangeoftemperature,weemployedemissionmeasure(EM)\nloci technique to examine the thermal structure of di fferent\nstripesasa functionofo ff-limbheight. SeveralEMloci plots\nwere constructed at di fferent heights along the active region\nAR1, AR2,andquietSun QS stripes. InFigure 7,we present\nsample EM loci plots obtained at the height of 55 Mm above\nthe off-limb. From the plots, it is clear that plasma along the\nline-of-sight is not isothermal at that height. Based on EM\nloci plots, distribution of plasma along all the stripes wer e\nfoundto be multi-thermalat all the heights. Theseresults a re\ningoodagreementwiththefindingsof O’Dwyeretal. (2011)who analyzed the same dataset before. They found plasma\nin the active region to be multi-thermal at di fferent distances\nfrom the limb. Similarly, Warren et al. (2008) also studied\nisolated coronal loops from the same active region when ob-\nserved on-disk and found them to be not isothermal. Thus,\nbasedoncurrentandpreviousstudies,plasmaalongthedi ffer-\nent active region stripes can be considered as multi-therma l.\nThis may indicate that emission in di fferent lines are coming\nfromeithersinglecoronalstructureformedoverwiderange of\ntemperatureorthereexistmultiplestructuresatdi fferenttem-\nperaturesalongtheline-of-sight. Moreover,plasmaalong the\noff-limb quiet Sun region appears to be nearly isothermal if\nthecontributionsfromhotlinesareexcluded. Figure 5shows\nthat intensities obtained from hot lines along QS are increa s-\ningwithheightnearo ff-limbregions. Thismaysuggestsome\npossible contaminations in the hot lines from nearby active\nregion.\n3.2.Non-thermalVelocity\nNon-thermalvelocitiesare an importantingredient for cal -\nculation of Alfv´ en wave energy flux. These have been ex-\ntracted from the observed emission line profiles as follows.\nObserved full width half maximum (FWHM) of any coronal\nspectrallineisgivenby,\nFWHM=/bracketleftBigg\n4ln2/parenleftbiggλ\nc/parenrightbigg2/parenleftBigg2kbTi\nMi+ξ2/parenrightBigg\n+W2\ninst/bracketrightBigg1/2\n(5)\nwhereTiis ion temperature, Miion mass,ξis non-thermal\nvelocity, and Winstis the instrumental width. EIS /Hinode in-\nstrumental width is not constant, and is found to vary with7\nFigure6. Electron number density variation with height along active region AR1,AR2, and quiet Sun QSobtained fromdi fferent density sensitive spectral line\npairs as labeled. Over-plotted continuous lines represent fitted exponential decay profile to obtain density scale heig hts from various spectral line pairs (see also\nTable2).\nTable2\nElectron number density scale heights obtained from variou s spectral line pairs along active region AR1, AR2, and quiet Sun QS.\nIon Wavelength T peakHydrostatic Densityscale height (Mm)\n(Å) (MK) height(Mm) AR1 AR2 QS\nFexi182.167/180.401 1.37 63.02 24 .86±0.31 34.22±0.26 27.13±0.98\nSix258.374/261.056 1.41 64.98 39 .84±2.15 53.61±2.25 78.12±9.91\nFexii196.640/192.394 1.58 72.91 26 .98±0.36 28.54±0.20 38.70±0.66\nFexiii196.525/202.044 1.78 81.80 55 .23±1.44 63.44±0.22 90.61±5.48\nFexiv264.789/274.204 2.00 91.78 41 .37±0.56 43.53±0.32 14.00±0.96\nFigure7. EM loci plots obtained at o ff-limb height of 55 Mmalong AR1, AR2, and QS.8 Gupta\nFigure8. Variation ofnon-thermal velocity with height along active region AR1obtained fromvarious spectral lines aslabeled. Overplotted continuous lines in\nall the panels show smooth variation of data-points obtaine d using 20-point running average.\nFigure9. Same as Figure 8but for active region AR2.9\nCCD Y-pixelposition alongthe slit5. EIS instrumentalwidth\nfor2′′slitvariesbetween64–74mÅforadownloadedcentral\n512-pixels (starting from pixels 256 to 767). These widths\nwere calculated using IDL routine EIS SLITWIDTH pro-\nvidedbyEISteam. Instrumentalwidthswere thensubtracted\nfrom the FWHM of spectral lines accordingly. We further\ncalculated non-thermal components by subtracting thermal\ncomponents from each spectral lines. Thermal components\nwere calculated after assuming ion temperatures to be equal\nto the peak formation temperature of spectral lines as found\nfrom contribution functions (see Figure 2and Table 1). Af-\nter subtraction of instrumental width, line widths were pri -\nmarily dominated by non-thermal components. Errorbars on\nnon-thermalvelocitieswerecalculatedusingerrorsinthe pro-\nfile fitting, 3 mÅ error in instrumental width, and errors in\nassumed thermal temperatures which were taken to be half\nwidthhalf maxima(HWHM) of Gaussianfits appliedto con-\ntributionfunctionsofrespectivespectrallines.\nIn Figures 8,9, and10, we plot non-thermal velocities ( ξ)\nobtained from various spectral lines with height along acti ve\nregions AR1, AR2, and quiet Sun QS respectively. We also\nover-plot20-pointrunningaverageofdata-pointstovisua lize\nvariationson longerspatial scale. Non-thermalvelocitie sob-\ntainedfromwarmlinessuchasFe x185Å,Fexii192Å,and\nFexiii202Å show initial increasefrom ≈24kms−1nearthe\nlimbto≈33kms−1ataroundheightof80Mm,whereasthose\nobtainedfromSix261Å, andSx264Å show increasefrom\n≈34kms−1to≈39kms−1atthesimilarheightsalongAR1.\nBeyond these heights, non-thermal velocities either decre ase\norremainalmostconstantwithsomescattereddata-points.\nVariation of non-thermal velocities with height along AR2\nalso show similar pattern as in AR1 but their values are en-\nhanced by≈2−3 km s−1. This possible enhancementalong\nAR2couldbeduetointegrationtakenoverlargerspatialsca le\nto deduce the non-thermal velocities. On comparison with\npolar regions (e.g., Banerjee et al. 2009 ;Bemporad & Abbo\n2012),non-thermalvelocitiesobtainedfromFe xiiline in the\nactiveregionsareconsistentlysmallerinmagnitudebutsh ows\nsharp increase with height. However, recent findings of Lee\netal.(2014)showedconsistentdecreaseinnon-thermalveloc-\nities along the cool loop and dark lane in the o ff-limb active\nregion. Moreover, non-thermal velocities obtained from ho t\nlines such as Fexv284 Å, and Fexvi263 Å show gradual\ndecreasewith height. Velocitiesobtainedfromhot Fe xv284\nÅlineshowsdecreasefrom ≈45kms−1nearthelimbto≈36\nkms−1beyond100MmalongAR1,whereasFe xvi263Åline\nshowsdecreasefrom ≈38 kms−1to≈32 km s−1. Variations\nobtainedfromhot Fe xv284Å, andFexvi263Å lines along\nAR2 show pattern again similar to that in AR1 with veloci-\nties being again enhanced by ≈2−3 km s−1. Surprisingly,\nwarm Fexi180 Å line shows pattern similar to that of hot\nlines whereas Fexiv274 Å line shows intermediate behav-\nior of hot and warm lines. Singh et al. (2006)performedline\nwidth study along steady coronal structures using data from\nNorikuracoronagraph. TheyfounddecreaseinFWHMofhot\nFexiv5303Å line up to the distance of 300′′abovethe limb\nwhichbecameconstantthereafter. Theyalsofoundincrease in\nFWHM of warm Fe x6374Å line up to the distance of 250′′\nwhich remained unchanged further. FWHM of intermediate\nlines(Fexi7892ÅandFexiii10747Å)showedintermediate\n5ftp://sohoftp.nascom.nasa.gov /solarsoft/hinode/eis/doc/eisnotes/07LINEWIDTH/eisswnote07.pdfbehavior. Thus, results of line width variation with height do\nindicatesometemperaturedependence. Findingsinthisstu dy\nare almost similar to those of Singh et al. (2006) with some\nshift in temperaturedependence. This shift might be specifi c\nto the active regionsstudied. However,cause for exception al\nbehavior of warm Fe xi180 Å line in this study is unknown\nand can not be speculated at this stage. Recently, Brooks &\nWarren(2016)surveyed15non-flaringon-diskactiveregions\nusing EIS/Hinode. They measured non-thermal velocities at\nspecific locationsin the coresof solar active regionsovert he\ntemperaturerangeof1–4MK.However,theydidnotfindany\nsignificanttrendwithtemperature.\nIn the quiet Sun region, non-thermal velocities obtained\nfrom warm iron lines show consistent decrease with height.\nNon-thermalvelocitiesobtainedfromwarmSi xandSxlines\nshow almost constant value of ≈34 km s−1and≈36 km\ns−1respectively with height, however, they do show some\nlarge scatter around. Non-thermal velocities obtained fro m\nhot Fexivand Fexvlines also show decrease with height\nsimilartowarmlines. Novisiblepatterncanbeinferredfro m\nhot Fexviline as signal in this line in the quiet Sun region\nis mostly due to scattered light as mentioned earlier. These\nfinding are similar to those of Harrison et al. (2002) where\nthey studied spectral line profiles of warm Mg x625 Å line\nfrom quiet clean corona. They found narrowing of emission\nlines as a function of height similar to findings in the curren t\nstudy. They attributed narrowing of profiles with height to\ndissipationofwaveactivity.\n3.3.Alfv´ enwave energyflux\nAlfv´ en wave energy flux can be calculated by using Equa-\ntion2. Inafluxtubegeometry, B×Awillalwaysbeaconstant.\nBecause in a constant magnetic field model, cross-sectional\nareawill remainconstant,thus,productwill also remainco n-\nstant. However, in the case of expanding flux tube model, B\nwill decrease with height (let us assume inverse square field\ndependence),whereas Awillincreasewithsquaredradiusde-\npendence,thus,productof BandAwillagainbeconstant(see\nMoran 2001 ). Therefore, total Alfv´ en wave energy flux will\nalways be proportionalto√Neξ2in either case. Henceforth,\nif total Alfv´ en wave energy flux is conserved as waves prop-\nagates outward,√Neξ2will remain constant with height. In\nFigures11,12, and13, we plot variations of√Neξ2with\nheight obtained from selected spectral lines along active r e-\ngion AR1, AR2, and quiet Sun QS respectively. As electron\nnumber densities were estimated only from few spectral line\npairs, therefore for rest of the lines, we choose number den-\nsities obtainedfrom line pairs formedat nearest temperatu re.\nPlotsclearlyshow thatproduct√Neξ2decreaseswithheight\ninall spectrallinesin all the regions. Thisprovidesclear evi-\ndence of damping of Alfv´ en wave energy flux with height in\nthebothoff-limbactiveandquietSunregion. Alfv´ enwaveen-\nergyfluxesarefoundtobe ≈1.85×107ergcm−2s−1nearthe\nlimb which decreasesto ≈0.86×107ergcm−2s−1at around\nheight of 70 Mm as calculated from Fe xii192 Å spectral\nline. To calculate the Alfv´ en wave energy flux, we assumed\ncoronal magnetic field strength of 39 G as measured by Van\nDoorsselaereet al. (2008)using looposcillations. Calculated\nAlfv´ en wave energy fluxes are of similar order of magnitude\nwhichis requiredtomaintainthe active regioncorona( ≈107\nergcm−2s−1asestimatedby Withbroe&Noyes1977 ). More-\nover, coronal magnetic field strength can vary like 10 G and\n33 G as measured by Lin et al. (2000) in two active regions10 Gupta\nFigure10. Sameas Figure 8butfor quiet Sun QS.\nFigure11. Variation of proportional Alfv´ en wave energy flux (√Neξ2) with height along active region AR1 obtained from various s pectral lines as labeled.\nOver-plotted continuous lines arefitted exponential decay profiletoobtain wavedamping lengths fromvarious spectral lines formed atrangeoftemperature (see\nalso Table 3).11\nFigure12. Sameas Figure 11butfor active region AR2.\nFigure13. Sameas Figure 11butfor quiet Sun QS.12 Gupta\nTable3\nDamping lengths derived from various spectral lines along a ctive region AR1,AR2, and quiet Sun QS.\nIon Wavelength T peak Dampinglength Dl(Mm)\n(Å) (MK) AR1 AR2 QS\nFex184.537 1.12 28 .31±6.72 65.19±3.45 27.54±2.39\nFexi180.401 1.37 46 .52±7.62 52.93±0.82 33.29±2.44\nSix261.056 1.41 114 .87±77.31 161.69±11.01 169.95±181.20\nSx264.231 1.55 65 .02±12.47 56.32±5.72 84.47±23.77\nFexii192.394 1.58 78 .49±4.38 73.70±13.44 91.95±20.12\nFexiii202.044 1.78 144 .1±12.44 118.30±6.85 159.98±10.49\nFexiv274.204 2.00 75 .99±6.00 70.06±4.45 29.31±3.50\nFexv284.163 2.24 57 .68±2.97 57.99±2.66 26.59±2.79\nFexvi262.976 2.82 47 .05±2.65 53.85±2.90−−−−\nat distances of 0.12 and 0.15 R⊙using longitudinal Zeeman\neffect in Fexiii10747 Å spectral line. Therefore, if assumed\nmagnetic field strength is of the order of 10 G, then Alfv´ en\nwave energy fluxes will be slightly less than the energy flux\nrequired to maintain the corona. One thing to be noted here\nthatalthoughAlfv´ enwavesaregettinggraduallydampedwi th\nheight, non-thermal velocities obtained from warm spectra l\nlineswereinitiallyincreasingwithheightintheactivere gion.\nThis indicates that damping of Alfv´ en waves can only be in-\nferredfromcompletecalculationoftotalAlfv´ enwaveener gy\nflux with height. Only non-thermal velocity estimates with\nheightwill notservethepurpose.\nUpon finding the evidence of damping of Alfv´ en wave en-\nergy flux with height, we further obtain damping length in\nall the spectral lines covering range of temperature. E ffect\nof damping can be calculated by multiplying e−h/Dlto the\nproportional Alfv´ en wave energy flux√Neξ2, whereDlis\ntermedas‘dampinglength’fortotalAlfv´ enwaveenergyflux ,\nFwt∝/radicalbig\nNeξ2e−h/Dl(6)\nFwt≈A/radicalbig\nNeξ2e−h/Dl+B (7)\nwhere A and B are appropriate constants. Henceforth, we\nobtaineddampinglengthby fitting the Fwtvaluesin different\nspectrallinesasperEquation 7usingMPFITroutines( Mark-\nwardt2009 ). Deriveddampinglengths Dlfromvariousspec-\ntrallinesalongactiveregionAR1,AR2,andquietSunQSare\nin the range of 25-170 Mm and providedin the Table 3.Be-\nmporad & Abbo (2012) also reported decay of Alfv´ en wave\nenergyfluxwithheightinthepolarcoronalholeregion. How-\never, they performed linear fit to decay profile and estimated\ndecay rates to be−1.07×10−3erg cm−1below 0.03 R ⊙and\n−4.5×10−5ergcm−1between0.03-0.4R ⊙. Equivalentdamp-\ning length for the decay rate between 0.03-0.4 R ⊙is calcu-\nlated to be around 95 Mm. They performed measurements\nusing EIS Fexii195 Å spectral line. In this work, damping\nlengths obtained from Fe xii192 Å lines are in the range of\n75–90 Mm in both active and quiet Sun regions. This sug-\ngeststhatdampinglengthsobtainedfromboththestudiesar e\ncomparable.\n4.DISCUSSIONS\nInthiswork,wefoundclearevidenceofdampingofAlfv´ en\nwaves in the off-limb active and quiet Sun region. Damping\nlengths were found to be di fferent for different spectral lines\nformed at different temperatures (see Table 3). We further\nexplore existence of any temperature dependence on variousdecay lengths obtained in this study. Henceforth, we analyz e\ndensity scale heights and Alfv´ en wave damping lengths with\npeak formation temperature of their respective spectral li nes\n(seeTable 1). Weplotdensityscaleheightsobtainedfromthe\ndifferent line pairs with respect to their peak formation tem-\nperature (see top panels of Figure 14). Density scale heights\nfirst increase and later decrease with temperature. However ,\ndensity scale height obtained from Fe xiipair does not fol-\nlow this trend. Moreover, hydrostatic scale heights as ex-\npected from Equation 4are also provided in Table 2. As\nmentionedearlier, comparisonbetween the two density scal e\nheightsindicatethatbothactiveandquietSunregionsareb a-\nsically underdense, with few exceptions from quiet Sun re-\ngion. However, it appears that emissions coming from spec-\ntral lines formed near the temperature of 1.8 MK are more\ncloser to be in hydrostaticequilibriumthan those formeddi f-\nferent from 1.8 MK in the active region. This may be specu-\nlated as observed region to be filled with plasma of tempera-\nture nearly 1.8 MK and has poor supply for other cooler and\nhotter plasma. This result might be a characteristic of ob-\nserved active region, and di fferent active regions might have\ndifferenttemperaturedistribution.\nInthebottompanelsofFigure 14,weplotdampinglengths\nobtained from different spectral lines with respect to their\npeak formation temperature. Di fferent panels show that\ndamping lengths first increase and later decrease with tem-\nperature. Maximum damping length is attained at around\ntemperature of 1.78 MK (corresponds to Fe xiii202 Å) for\nall the active region and quiet Sun stripes. We would also\nlike to point out that several structures were traced and ana -\nlyzed as mentioned earlier. Although obtained decay length s\nwere not same, results followedsimilar pattern for all the a n-\nalyzed structures. Henceforth, obtained results indicate mea-\nsurement of different damping lengths for di fferent tempera-\ntures. These results can either be interpreted as temperatu re\ndependent damping of Alfv´ en waves or measurement of dif-\nferentdampinglengthsin di fferent coronalstructuresformed\noverwide rangeoftemperaturealongtheourline-of-sight.\nPossible temperature dependent damping length of Alfv´ en\nwaves may indicate that thermal conduction plays some im-\nportant role in the damping of these waves. However, role\nof thermal conductionin the dampingof Alfv´ en waves is not\nmuchexplored(e.g., VanBallegooijenet al.2011 ). Although\nit is very well studied for the case of slow magneto-acoustic\nwaves (e.g., De Moortel et al. 2002 ). Role of thermal con-\nductioninthedampingofslowmagneto-acousticwaveswere\nrecently observed by Gupta(2014) andKrishna Prasad et al.\n(2014)based on possible period(of waves) dependentdamp-\ning length. In this study, although we do not have any infor-13\nFigure14. Variation of density scale heights (top panels) and Alfv´ en waves damping lengths (bottom panels) obtained from di fferent spectral lines with respect\nto their peak formation temperature.\nmationonwaveperiod,we havecoverageoverwiderangeof\ntemperature. Workof DeMoortelet al. (2002)suggestedthat\nslightlyenhancedthermalconductivitymayexplainobserv ed\ndamping lengths of 40-50 Mm for slow waves. These en-\nhancementsinthermalconductivitywerelateralso suggest ed\nbyGupta(2014). Inthis study,observeddampinglengthsfor\nAlfv´ enwavesareintherangeof25-170Mmasobtainedfrom\ndifferenttemperaturelines. Henceforth,these resultsdemand\nfordetailed investigationof role of thermalconductionin the\ndampingofAlfv´ enwaves.\nSlowmagneto-acousticwavesinthesolarcoronapropagate\nalong the field lines with propagation speed of the order of\n100 km s−1and velocity amplitude of the order of 5-10 km\ns−1. Activeregionstudiedinthisworkislocatednearthelimb\nandderivedresultsaremainlyfocusedono ff-limbregions. In\nthe off-limb region, magnetic field lines are generally found\nto be oriented nearly perpendicular to the observers line-o f-\nsight. Therefore, contribution from observed Doppler velo c-\nities due to propagation of slow magneto-acoustic waves in\nthe measurement of non-thermal velocities will be minimal.\nSimilarly, studies on measurementof plasma flows in the ac-\ntiveregionloopsindicatetemperaturedependentflowspeed s.\nDel Zanna (2008) andTripathi et al. (2009) measured abso-\nlute flow speeds to be less than 30 km s−1along the active\nregionloopsusing similar spectral lines formedover tempe r-\naturerangeof0.6–2MK.Theyfounddecreaseinflowspeeds\nwith increase in temperature (redshift to blueshift). More -\nover,Brooks & Warren (2011) also measured on an aver-\nage Doppler velocity of −22 km s−1from the edges of ac-\ntive regions. Generally loops cross an active region in the\nEast-West direction, so flows along the o ff-limb loops will\neither be directed toward or away from the observersline-of -\nsight (if loops are not radially directed). This may lead to\nsome enhancements in the line-width. However, as veloci-\nties in line-width measurements add in quadrature, contrib u-\ntion of Doppler velocities due to plasma upflows ( <10 km\ns−1, due to inclination of loops along the line-of-sight) will\nagain be minimal in the non-thermal velocities. Moreover,\nthere might be some enhancement in the non-thermal broad-eningduetothesefactorsbutgiventherangeoferrorbars(2 –4\nkm s−1), their contribution can not be quantified. Measure-\nmentsonAR2whichwereobtainedaftertakingaverageover\nlargerspatial length,showsenhancementsin non-thermalv e-\nlocities by≈2−3 km s−1as compared to measurements on\nAR1. This could possibly be the e ffect of different Doppler\nshifted flows present along several di fferent loop structures\nwhich were summedtogether to obtain the integratedprofile,\nand thus, resulted in larger non-thermal velocities. Hence -\nforth, measured non-thermal velocities along AR2 can only\nbeconsideredasanupperlimit.\nAs mentioned earlier, role of thermal conduction in the\ndampingofslowmagneto-acousticwavesiswellknown. One\nof the possibility of damping of Alfv´ en waves would be that\nAlfv´ en wave energy is being transferred to slow magneto-\nacoustic waves. These slow waves will further get easily\ndissipated via thermal conduction and will finally show up\nas temperature dependent damping of Alfv´ en waves. Za-\nqarashvili et al. (2006) studied wave energy conversion pro-\ncess in the non-linear ideal MHD framework. They demon-\nstratedthatwaveenergycanbeconvertedfromAlfv´ enwaves\nto slow magneto-acoustic waves near the region of corona\nwhereplasma-βapproachesunity. Ascontributionsfromslow\nmagneto-acousticwavesin thecurrentmeasurementsof non-\nthermal velocities are minimal, henceforth, only the damp-\ning of Alfv´ en waves can be inferred from the observed non-\nthermalvelocities.\n5.SUMMARYAND CONCLUSIONS\nWe investigated off-limb active and quiet Sun region using\nspectroscopic data from EIS /Hinode. We studied height de-\npendence of basic plasma parameters such as intensity, elec -\ntron number density, and non-thermal velocity along the ac-\ntive region and quiet Sun. These estimated parameters en-\nabled us to further study height dependence of Alfv´ en wave\nenergyfluxin thebothregions. Mainfindingsofouranalysis\naresummarizedas,\n•We identified several isolated spectral lines with good\nsignaltonoiseratiointheo ff-limbregions. Theselines14 Gupta\nareformedat differenttemperaturesandcoverthetem-\nperaturerangeof1.1–2.8MK.\n•Weobtainedelectrondensitiesandcorrespondingscale\nheights from different spectral line pairs which sug-\ngested that observed active and quiet Sun regions are\nbasically underdense with few measured exceptions\nfromquietSunregion.\n•Non-thermal velocities measured from warm spectral\nlinesfirstshowedincreasewithheightandlatershowed\neither decrease or almost constant value with height in\nthe far off-limbactive regionwhereashot lines showed\ngradual decrease with height. However, those mea-\nsured from various spectral lines in the quiet Sun re-\ngionshowedeithergradualdecreaseoralmostconstant\nvaluewithheight.\n•Calculated Alfv´ en wave energy fluxes were similar to\norslightlylessthantheenergyrequiredtomaintainthe\nactive region corona. Results also showed damping of\nAlfv´ enwaveenergyfluxwithheight.\n•We founddampinglengthsofAlfv´ enwave energyflux\n(Dl)tobeintherangeof25-170Mmasmeasuredfrom\ndifferentspectrallinesformedatdi fferenttemperatures.\n•Variationof dampinglengthsfirst showedincrease and\nlater decrease with increasing temperature. Damping\nlengthpeakedat aroundtemperatureof1.78MKin the\nbothactiveandquietSunregions.\nThis work provides measurements of non-thermal veloci-\nties and Alfv´ en wave energy fluxes at wide range of temper-\nature. Possible interpretation of these results would be ei -\nthertemperaturedependentdampingofAlfv´ enwavesormea-\nsurementsalong di fferent coronal structuresformed at di ffer-\nent temperatures. Possible temperature dependent damping\nmay suggest some important role of thermal conduction in\nthe damping of Alfv´ en waves in the lower corona. This may\neven suggest some non-linear coupling between Alfv´ en and\nslow MHD modes (see Zaqarashvili et al. 2006 ). We believe\nthistobeanimportantresultasthiswillprovidemoreinsig ht\nin to the dissipation mechanism of Alfv´ en waves. Recent 3-\nD MHD models of Van Ballegooijen et al. (2011) explained\nrole of Alfv´ en wave turbulence in the heating of solar chro-\nmosphere and corona. They predicted velocity amplitude of\nAlfv´ enwavesinthecoronatobeintherangeof20-40kms−1\nso as to maintain the typical active region loops. In our anal -\nysis, we found almost similar wave velocity amplitude in the\nactive region. Observeddampingrate of Alfv´ enwave energy\nflux with height is similar or slightly less than to the requir e-\nments of coronal active region. Asgari-Targhi et al. (2014)\nalso measured non-thermal velocities in the range of 25-45\nkm s−1using observationfromEIS /Hinodealong the on-disk\nindividual coronal loop length. Their findings were consis-\ntentwiththepredictionsfromAlfv´ enwaveturbulencemode l.\nHowever, we would also like to point out that model of Van\nBallegooijen et al. (2011) still do not include e ffects of ther-\nmal conduction and radiative losses. Therefore, at present ,\nexactformofanyrelationbetweendampinglengthofAlfv´ en\nwave turbulence and temperature can not be comprehended.\nHenceforth, these results demand for development of more\nsophisticated 3-D MHD models of Alfv´ en wave propagation\nand dissipation including the e ffects of thermal conductionand non-linear coupling between various MHD modes in the\nsolaratmosphere.\nAuthor thanks the referee for the careful reading and con-\nstructivecriticismthathelpedimprovethepaper. GRGissu p-\nportedthroughtheINSPIREFacultyAwardofDepartmentof\nScience and Technology (DST), India. Author thanks T. V.\nZaqarashviliforthehelpfuldiscussionandP.Youngforhel p-\nfulclarifications. HinodeisaJapanesemissiondevelopeda nd\nlaunched by ISAS/JAXA, collaborating with NAOJ as a do-\nmestic partner, NASA and STFC (UK) as international part-\nners. Scientific operationofthe Hinodemission is conducte d\nby the Hinode science team organized at ISAS /JAXA. This\nteammainlyconsistsofscientistsfrominstitutesinthepa rtner\ncountries. Support for the post-launch operation is provid ed\nby JAXA and NAOJ (Japan), STFC (U.K.), NASA (U.S.A.),\nESA,andNSC(Norway).\nREFERENCES\nAlfv´ en, H.1942, Nature, 150, 405\n—.1947, MNRAS, 107,211\nArregui, I. 2015, Philosophical Transactions of theRoyal S ociety of London\nSeries A,373, 40261\nAschwanden, M.J.,Newmark, J.S.,Delaboudini` ere, J.,eta l. 1999, ApJ,\n515, 842\nAsgari-Targhi, M.,van Ballegooijen, A.A.,&Imada, S.2014 , ApJ,786, 28\nBanerjee, D.,Gupta, G.R.,& Teriaca, L.2011, Space Sci. Rev .,158, 267\nBanerjee, D.,P´ erez-Su´ arez, D.,&Doyle, J.G.2009, A&A,5 01, L15\nBanerjee, D.,Teriaca, L.,Doyle, J.G.,& Wilhelm, K.1998, A &A,339,208\nBemporad, A.,& Abbo,L.2012, ApJ,751, 110\nBrooks, D.H.,&Warren, H.P.2011, ApJL,727,L13\n—.2016, ApJ,820, 63\nCranmer, S.R.,Asgari-Targhi, M.,Miralles, M.P.,etal. 20 15,Philosophical\nTransactions of theRoyal Society of London Series A,373, 40 148\nCranmer, S.R.,& van Ballegooijen, A.A. 2005, ApJS,156, 265\nCranmer, S.R.,van Ballegooijen, A.A.,&Edgar, R.J.2007, A pJS,171,\n520\nCulhane, J.L.,Harra, L.K.,James,A.M.,et al. 2007, Sol. Ph ys.,243, 19\nDeMoortel, I.,&Browning, P.2015, Philosophical Transact ions of the\nRoyal Society of London Series A,373, 40269\nDeMoortel, I.,Hood, A.W.,Ireland, J.,&Walsh, R.W.2002, S ol. Phys.,\n209, 89\nDeMoortel, I.,&Nakariakov, V. M.2012, Royal Society of Lon don\nPhilosophical Transactions Series A,370, 3193\nDelZanna, G.2008, A&A,481, L49\nDelZanna, G.,Dere, K.P.,Young, P.R.,Landi, E.,&Mason, H. E.2015,\nA&A,582, A56\nDelZanna, G.,&Mason, H.E.2003, A&A,406, 1089\nDere, K.P.,Landi, E.,Mason, H.E.,Monsignori Fossi, B.C., &Young,\nP.R.1997, A&AS,125, 149\nDoyle, J. G.,Banerjee, D.,& Perez, M. E.1998, Sol. Phys.,18 1, 91\nFreeland, S.L.,&Handy, B. N.1998, Sol. Phys.,182, 497\nGupta, G.R. 2014, A&A,568,A96\nGupta, G.R.,Banerjee, D.,Teriaca, L.,Imada, S.,& Solanki , S.2010, ApJ,\n718, 11\nGupta, G.R.,Tripathi, D.,&Mason, H.E.2015, ApJ, 800,140\nHahn, M.,Landi, E.,&Savin, D.W.2012, ApJ,753, 36\nHahn, M.,&Savin, D.W.2014, ApJ,795, 111\nHarrison, R. A.,Hood, A.W.,&Pike, C.D.2002, A&A,392, 319\nHassler, D.M.,Rottman, G.J.,Shoub, E.C.,&Holzer, T.E.19 90, ApJL,\n348, L77\nJess,D.B.,Mathioudakis, M.,Erd´ elyi, R.,et al. 2009, Sci ence, 323, 1582\nJess,D.B.,Morton, R.J.,Verth, G.,etal. 2015, Space Sci. R ev.,190, 103\nKlimchuk, J. A.2015, Philosophical Transactions of theRoy al Society of\nLondon Series A,373, 40256\nKosugi,T.,Matsuzaki, K.,Sakao, T.,etal. 2007, Sol. Phys. ,243, 3\nKrishnaPrasad, S.,Banerjee, D.,&Van Doorsselaere, T.201 4, ApJ,789,\n118\nLang,J.,Kent, B.J.,Paustian, W.,et al. 2006, Appl. Opt., 4 5,8689\nLee,K.-S.,Imada, S.,Moon, Y.-J.,& Lee,J.-Y. 2014, ApJ,78 0, 177\nLin,H.,Penn, M.J.,&Tomczyk, S.2000, ApJL,541, L8315\nMarkwardt, C. B.2009, in Astronomical Society of the Pacific Conference\nSeries, Vol. 411, Astronomical Data Analysis Software and S ystems\nXVIII, ed. D.A.Bohlender, D.Durand, & P.Dowler, 251\nMason, H.E.,& Monsignori Fossi, B.C. 1994, A&ARev., 6,123\nMcIntosh, S.W.,dePontieu, B.,Carlsson, M.,et al. 2011, Na ture, 475, 477\nMoran, T.G.2001, A&A,374, L9\nMorton, R. J.,Tomczyk, S.,&Pinto, R.2015, Nature Communic ations, 6,\n7813\nMorton, R. J.,Verth, G.,Hillier, A.,&Erd´ elyi, R.2014, Ap J,784, 29\nO’Dwyer, B.,DelZanna, G.,Mason, H. E.,et al. 2011, A&A,525 ,A137\nParker, E.N.1988, ApJ,330, 474\nParnell, C. E.,&DeMoortel, I.2012, Philosophical Transac tions of the\nRoyal Society of London Series A,370, 3217\nSingh, J.,Sakurai, T.,& Ichimoto, K.2006, ApJ,639, 475\nSuzuki, T.K.,&Inutsuka, S. 2005, ApJL,632, L49\nTesta, P.,DePontieu, B.,&Hansteen, V.2016, ApJ,827, 99\nTomczyk, S.,McIntosh, S.W.,Keil, S.L.,et al. 2007, Scienc e, 317,1192Tripathi, D.,Mason, H.E.,Dwivedi, B. N.,del Zanna, G.,&Yo ung, P.R.\n2009, ApJ,694, 1256\nVanBallegooijen, A.A.,Asgari-Targhi, M.,Cranmer, S.R., & DeLuca,\nE.E.2011, ApJ,736, 3\nVanDoorsselaere, T.,Nakariakov, V. M.,Young,P.R.,& Verw ichte, E.\n2008, A&A,487, L17\nVerdini, A.,Velli, M.,Matthaeus, W.H.,Oughton, S.,&Dmit ruk, P.2010,\nApJL,708, L116\nWarren, H.P.,Ugarte-Urra, I.,Doschek, G.A.,Brooks, D.H. ,&Williams,\nD.R. 2008, ApJL,686,L131\nWarren, H.P.,Ugarte-Urra, I.,& Landi, E.2014, ApJS,213, 1 1\nWithbroe, G.L.,&Noyes, R. W.1977, ARA&A,15, 363\nYoung,P.R.,Del Zanna, G.,Mason, H.E.,et al. 2007, PASJ,59 ,857\nZaqarashvili, T.V.,Oliver, R.,& Ballester, J.L.2006, A&A ,456,L13"    },    {        "title": "1701.02475v1.Magnetic_properties_in_ultra_thin_3d_transition_metal_alloys_II__Experimental_verification_of_quantitative_theories_of_damping_and_spin_pumping.pdf",        "content": "1  Magnetic properties in ultra -thin 3d transition metal alloys \nII: Experimental verification of quantitative theories of \ndamping and spin -pumping    \n \nMartin A. W. Schoen,1,2* Juriaan Lucassen,3 Hans T. Nembach,1 Bert Koopmans,3 T. J. Silva,1 Christian \nH. Bac k,2 and Justin M. Shaw1 \n \n1Quantum E lectromagnetics Division, National Institute of Standards and Technology, Boulder, CO \n80305 , USA  \n2Institute of Experimental and Applied Physics, University of Regensburg, 93053 Regensburg, \nGermany  \n3Department of Applied P hysics, Eindhoven University of Technology, 5600 MB Eindhoven, The \nNetherlands  \n \n \nDated: 01/05/2017  \n \n *Corresponding author: martin1.schoen@physik.uni -regensburg.de  \n \n \n \n \nAbstract  \nA systematic experimental study of Gilbert damping is performed via ferromagnet ic \nresonance for the disordered  crystalline  binary 3 d transition metal alloys Ni -Co, Ni -Fe and \nCo-Fe over the full range of alloy compositions. After accounting for inhomogeneous \nlinewidth  broadening , the damping shows clear evidence of both  interfacial da mping \nenhancement (by spin pumping) and radiative damping. We quantify these two extrinsic  \ncontributions  and thereby determine the intrinsic damping. The comparison of the intrinsic \ndamping to multiple theoretical calculation s yields good qualitative  and q uantitative  \nagreement  in most cases . Furthermore, the values of the damping obtained in this study are \nin good agreement with  a wide range of published experimental and theoretical values . \nAdditionally, we find a compositional dependence of the spin mixing  conductance.  \n \n \n \n \n 2  \n \n \n \n1  Introduction  \nThe magnetization dynamics in f erromagnetic films are phen omenologically well described \nby the Landau -Lifshitz -Gilbert formalism (LLG)  where  the damping  is described by a \nphenomenological damping parameter α.4,5 Over the past four decades, there have been \nconsiderable efforts to derive the phenomenological d amping parameter from first principles \ncalculations and to do so in a quantitative manner. One of the early promising  theories was that of \nKamberský, who introduced the so -called breathing Fermi surface model6–8. The name “breathing \nFermi  surface ” stems from the picture that the precessing magnetization, due to spin -orbit coupling, \ndistorts the Fermi surface. Re -populating the Fermi surface is delayed by the scattering time, \nresulting in a phase lag between the precession  and the Fermi sur face distortion. This  lag leads to  a \ndamping that is proportional to the scattering time. Although this approach describes the so-called \nconductivity -like behavior of the damping at low temperatures, it fails to describe  the high \ntemperature behavior of so me materials . The high temperature or resistivity -like behavior is \ndescribed by the so-called  “bubbling Fermi surface ” model. In the case of energetically shifted  \nbands , thermal  broaden ing can lead to a significant  overlap  of the spin-split bands  in 3d \nferromagnets.  A precessing magnetization can induce elec tronic transitions between such \noverlapping bands , leading to spin-flip process es. This process scales with the amount of band \noverlap. Since s uch overlap is further increased with the band broadening th at results from the finite \ntemperature of the sample, this contribution  is expected to increase as the temperature is increased. \nThis model for interband transition mediated damping describes the resistivity -like behavior of the \ndamping at higher temperatu res (shorter scattering times).  These two damping processes are  \ncombined in a torque correlation model by Gilmore , et al.9, as well as Thonig , et al.10, that  describes \nboth the low -temperature  (intra band transitio ns) and high -temperature  (inter band transitions) \nbehavior  of the damping . Another app roach via  scattering theory was successfully implemented by \nBrataas , et al.11 to describe damping in transition metals. Starikov , et al. ,2 applied the scattering \nmatrix approach to calculate the damping  of NixFe1-x alloys and  Liu, et al. ,12 expanded the formalism \nto include the influence of electron -phonon interaction s.  \nA numerical realization of the torque correlation model was performed  by Mankovsky , et \nal., for NixCo1-x, Ni xFe1-x, CoxFe1-x, and FexV1-x1. More recently , Turek , et al. ,3 calculated the \ndamping for NixFe1-x and CoxFe1-x alloys with the torque -correlation model, u tilizing non -local \ntorque correlators. It is important to stress that a ll of these  approaches  consider  only the intrinsic \ndamping. This complicates the quantitative comparison of calculated values for t he damping to \nexperimental data  since there are many extrinsic contributions to the damping that result from \nsample  structure , measurement geometr y, and/or sample properties . While some extrinsic \ncontributions to the damping  and linewidth  were discovered in the 1960 ’s and 1970 ’s, and are well \ndescribed  by the ory, e.g. eddy -current damping13,14, two -magnon scattering15–17, the slow rel axer \nmechanism18,19, or radiative damping20,21, interest in these mechanisms has been re -ignited \nrecently22,23. Further contributions, such as  spin-pumping, both extrinsic24,25 and intrinsic24,26, have 3 been discovered more recently and a re subject to extensive research27–31 for spintronics application.  \nTherefore , in order to allow a quantitative comparison to theoretical calculations  for intrins ic \ndamping, both the measurement and sample geometry must  be designed  to allow both the \ndetermination and possibl e minimization  of all additional contributions to the measured damping.  \nIn this study , we demonstrate methods to determine significant extrins ic contributions to the \ndamping , which  includ es a measurement of the effective spin mixing conductance for both the pure \nelements and select alloys. By precisely accounting for all of these extrinsic contributions, we \ndetermine the intrinsic damping  parame ters of the binary alloys Ni xCo1-x, Ni xFe1-x and Co xFe1-x and \ncompare them  to the calculation s by Mankovsky , et al. ,1, Turek , et al. , and Starikov , et al.2. \nFurthermore, we present the concentration -dependence of the  inhomogeneous linewidth \nbroadening, which for most alloys shows exceptionally small values, indicative  of the high \nhomogeneity of our samples.  \n2 Samples and method  \nWe deposited  NixCo1-x, Ni xFe1-x and Co xFe1-x alloys of varying composition (all compositio ns \ngiven in atomic percent)  with a thickness of 10 nm  on an oxidized (001)  Si substrate with  a Ta(3 \nnm)/Cu(3 nm) seed layer and  a Cu(3 nm) /Ta(3 nm) cap layer. In order to investigate interface \neffects , we also deposited multiple thickness series at 10 nm, 7 nm, 4 nm, 3 nm , and 2 nm of  both \nthe pure elements and select alloy s.  Structural characterization  was performed using X-ray \ndiffraction (XRD).  Field swept vector -network -analyzer  ferromagnetic resonance spectroscopy \n(VNA -FMR) was used in the out -of-plane geometry  to determine the total damping parameter αtot. \nFurther d etails of the deposition conditions, XRD, FMR measurement and fitting of the complex \nsusceptibility to the measured S21 parameter are reported in  Ref [66]. \nAn example of susceptibility fits t o the complex S21 data is shown in Fig.  1 (a) and (b). All \nfits were constrained  to a 3×  linewidth ΔH field window around the resonance field in order to \nminimize the influence of measurement drifts on the error in the susceptibility fits. The total \ndampin g parameter αtot and the inhomogeneous linewidth broadening Δ H0 are then determined  from \na fit to  the linewidth Δ H vs. frequency f plot22, as shown in Fig.  1 (c). \n ∆𝐻= 4𝜋𝛼𝑡𝑜𝑡𝑓\n𝛾𝜇0+ ∆𝐻0, (1) \nwhere γ=gμB/ħ is the gyro -magnetic ratio, μ0 is the vacuum permeability, μB is the Bohr -magneton , \nħ is the reduced Planck constant,  and g is the spectroscopic g-factor reported in Ref [ 66].  \n  4  \nFigure 1: (a) and (b) show respectively the real and imaginary part of the S21 transmission \nparameter (black squares)  measured at 20 GHz  with the complex susceptibility fit (red lines)  for \nthe Ni 90Fe10 sample . (c) The linewidths  from the suscept ibility fits  (symbols) and linear fits (solid \nlines) are plotted against frequency for different Ni -Fe compositions. Concentrations are denoted \non the right -hand  axis. The damping α and the inhomogeneous linewidth broadening Δ H0 for each \nalloy can be extracted from the fits via Eq.  (1). \n3 Results  \nThe first contribution t o the linewidth we discuss  is the  inhomogeneous linewidth broadening \nΔH0, which  is presumably  indicative of  sample inhomogeneity32,33. We plot Δ H0 for all the alloy \nsystem s against the respective concentrations in Fig.  2. For all alloys , ΔH0 is in the range of a few \nmT to 10  mT . There are only a limited  number of reports  for ΔH0 in the literature with which to \ncompare . For Permalloy (Ni 80Fe20) we measure Δ H0 = 0.35 mT, which is close to other reported \nvalues .34 For the  other  NixFe1-x alloys , ΔH0 exhibits a significant peak  near the fcc-to-bcc (face -\ncentered -cubic to body -centered -cubic) phase transition  at 30  % Ni , (see Fig.  2 (b)) which is easily \nseen in the raw data  in Fig. 1 (c). We speculate  that this  increase of  inhomogeneous broadening  in \nthe NixFe1-x is caused by the coexistence of the bcc and fcc  phases at the phase transition.  However,  \nthe CoxFe1-x alloys do not exhibit an increase in ΔH0 at the equivalent phase transition  at 70  % Co . \nThis suggests that the bcc and fcc phases of NixFe1-x tend to segregate near the phase transition, \nwhereas the same phases for CoxFe1-x remain intermixed throughout the transition.  \n5 One possible explanation for inhomogeneous broadening is magnetic anisotropy , as originally \nproposed  in Ref. [35]. However, this explanation does not account for our measured dependence of \nΔH0 on alloy concentration, since the perpendicular magnetic anisotropy , described in Ref [ 66] \neffectively  exhibits opposite behavior with alloy concentration.  For our alloys Δ H0 seems to roughly \ncorrelate to the inverse exchange constant36,37, which co uld be a starting point for future \ninvestigation of a quantitative theory of inhomogeneous broadening.  \n \n \n \n  \nFigure 2: The inhomogeneous linewidth -broadening Δ H0 is plotted vs. alloy composition for (a) \nNi-Co, (b) Ni -Fe and (c) Co -Fe. The alloy phases are denoted by color code described in Ref [ 66] \n \nWe plot the total measured damping αtot vs. composition for NixCo1-x, NixFe1-x and CoxFe1-x in \nFig. 3 (red crosses). The total damping of the NixCo1-x system increases monotonically  with \nincreased Ni content . Such smooth  behavior in the damping is  not surprising  owing to the absence \nof a phase transition for this alloy . In the NixFe1-x system , αtot changes very little from pure Fe to \napproximately 25 % Ni  where the bcc to fcc phase transition occurs . At the phase  transition,  αtot \nexhibits a step, increasing  sharply by approximately 30  %. For higher Ni concentrations , αtot \nincreases monotonically  with increasing Ni concentration . On the other hand, t he CoxFe1-x system \nshows a different behavior in the damping and d isplays a sharp minimum of (2.3 ± 0.1)×10-3 at 25 \n6 % Co as previously reported38. As the system changes to an fcc phase ( ≈ 70 % Co), αtot become \nalmost  constant.  \nWe compare our data to previously published values in Table I. However, direct c omparison \nof our data to previous report s is not trivial , owing to the variation in measurement conditions  and \nsample characteristic s for all the reported measurements . For example , the damping  can depend  on \nthe temperature .9,39 In addition, multiple intrinsic and intrinsic contributions to the total damping  \nare not always accounted for in the literature . This can be seen in the fact that the reported  damping \nin Ni80Fe20 (Permalloy) varies  from α=0.0 055 to α=0.04  at room tem perature  among studies . The \nlarge variation  for these reported data  is possibly the result of  different uncontrolled contributions  \nto the extrinsic damping  that add  to the total damping in the different experiments , e.g. spin-\npumping40–42, or roughness41. Therefore , the value for the intrinsic damping of Ni 20Fe80 is expected \nto be at the low end of this scatter . Our measured value of α=0.007 2 lies within the range of reported \nvalues. Similarly , many of our measured damping values for different alloy compositions lie within \nthe range  of reported values22,43 –48. Our measured  damping of the pure elements and the Ni 80Fe20 \nand Co 90Fe10 alloys is compared to room temperature values found in literature in Table 1, Col umns  \n2 and 3 . Column 5 contains theoretically calculated values .   \n \nTable 1: The total measured damping α tot (Col. 2)  and the intrinsic damping (C ol. 4) f or Ni 80Fe20, \nCo90Fe10, and the pure elements are compared to both experimental (Col. 3) and theoretical (Col. \n5) values from the literature . All values of the damping  are at room temperature if not noted \notherwise . \nMaterial  αtot (this study)  \n Liter ature values  αint (this study)  Calculated literature \nvalues  \nNi 0.029 (fcc)  0.06444 \n0.04549 0.024 (fcc)  0.0179 (fcc) at 0K  \n0.02212 (fcc) at 0K  \n0.0131 (fcc)  \nFe 0.0036 (bcc)  0.001944 \n0.002746 0.0025 (bcc)  0.00139 (bcc) at 0K  \n0.001012 (bcc)  at 0K  \n0.00121 (bcc) at 0K  \nCo 0.0047 (fcc)  0.01144  \n 0.0029 (fcc)  0.00119 (hcp)  at 0K  \n0.0007312 (hcp)  at 0K  \n0.0011 (hcp)  \nNi80Fe20 0.0073 (fcc)  0.00844 \n0.008 -0.0450 \n0.007848 \n0.00751 \n0.00652 \n0.00647 \n0.005553 0.0050 (fcc)  0.00462,54 (fcc) at 0K  \n0.0039 -0.00493 (fcc)  at 0K  \nCo90Fe10 0.0048 (fcc)  0.004344 \n0.004855 0.0030 (fcc)   7  \n \n \n \n  \nFigure 3: (color online) The measured damping αtot of all the alloys is plotted against the alloy \ncompositi on (red crosses) for (a) Ni -Co, (b) Ni -Fe and (c) Co -Fe (the data in (c) are taken from \nRef.[38]). The black squares  are the intrinsic damping  αint after correction for spin pumping and \nradiative contributions to the measured damping. The blue line is the intr insic damping calculated \nfrom the Ebert -Mankovsky theory ,1 where the blue circles are the values for the pure elements  at \n300K . The green  line is the calculated damping for the Ni -Fe alloys  by Starikov , et al.2 The inset \nin (b) depicts the damping in a smaller concentration window in order to better depict the small \nfeatures in the damping around the ph ase transition. The damping  for the Co -Fe alloys, calculated \nby Turek et al.3 is plotted as the orange  line. For the Ni -Co alloys the damp ing calculated by th e \nspin density of the respective alloy weighted bulk damping55 (purple dashed line).  \n8  \n \nThis scatter in the experimental data reported in the literature and its divergence from calculated \nvalues of the damping shows th e necessity to determine  the intrinsic damping  αint by quantification \nof all extrinsic contributions to the measured total damping α tot.  \nThe first extrinsic  contribution to the damping  that we consider  is the radiative damping α rad, \nwhich is caused by ind uctive coupling between sample and  waveguide , which results in energy \nflow from the sample back into the microwave circuit.23 αrad depends directly  on the measurement \nmethod and geometry. The effect is easily understood , since the strength of the inductive coupling \ndepends on the inductance of the FMR mode itself , which is in turn determined by the saturation \nmagnetization, sampl e thickness, sample  length, and waveguide width.  Assuming a homogeneous \nexcitation field , a uniform magnetization profile throughout the sample , and negligible spacing \nbetween the waveguide and sample , αrad is well approximated by23  \n 𝛼rad=𝛾𝑀𝑠𝜇02𝛿𝑙\n16 𝑍0𝑤𝑐𝑐, (2) \nwhere l (= 10 mm  in our case) is the sample length on the waveguide, wcc (= 100µm) is the width \nof the co -planar wave guide center conductor and Z0 (= 50 Ω), the impedance of the waveguide. \nThough inh erently small for most thin films , αrad can become  significant  for alloys with \nexceptionally small intrinsic damping and /or high saturation magnetization.  For example, it plays \na significant role (values of αrad ≈ 5x10-4) for the whole composition range of  the Co -Fe alloy system \nand the Fe -rich side of the Ni -Fe system. On the other hand, for pure Ni and Permalloy (Ni 80Fe20) \nαrad comprises only 3 % and 5  % of αtot, respectively.  \nThe second non -negligible contribution to the damping  that we consider  is the interfacial \ncontribution to the measured  damping , such as  spin-pumping into the adjacent Ta/Cu bilayers . Spin \npumping  is proportional to the reciprocal sample thickness as described in24 \n 𝛼sp= 2𝑔eff↑↓𝜇𝐵𝑔\n4𝜋𝑀s𝑡. (3) \nThe spectroscopic  g-factor and the saturation magnetization Ms of the alloys were reported  in \nRef [66] and the factor of 2 accounts for the presence of two nominally identical interfaces of the \nalloys in the cap and seed layers.  In Fig.  4 (a)-(c) we plot the damping dependence on reciprocal \nthickness  1/t for select alloy concentrations, which allows us to determine  the effective spin mixing \nconductance 𝑔eff↑↓  through fits to Eq. (3) . The effective spin m ixing conductance contains details of \nthe spin transport in the adjacent non -magnetic layers, such as the interfacial spin mixing \nconductance, both the conductivity and spin diffusion for all the non -magnetic layers with a non -\nnegligible spin accumulation,  as well as the details of the spatial profile for the net spin \naccumulation .56,57 The values of 𝑔eff↑↓, are plotted versus the alloy concentration in Fig. 4 (d), and are \nin the range of previously reported values  for samples prepare d under similar growth conditions55–\n59. Intermediate values of 𝑔eff↑↓ are determined by a guide to the eye interpolation [ grey lines, Fig.  4 \n(d)] and αsp is calculated for all alloy concentrations utilizing  those interpolated values.  \nThe data for 𝑔eff↑↓  in the NixFe1-x alloys shows approximately a factor two  increase of 𝑔eff↑↓ between \nNi concentrations of 30  % Ni and 50  % Ni, which we speculate to occur at the fcc to bcc phase \ntransition around 30  % Ni. According to this line of speculation , the previously mentioned step  \nincrease in the measured total damping at the NixFe1-x phase transition can be fully attributed to the \nincrease in spin pumping at the phase transition.  In CoxFe1-x, the presence of a step in  𝑔eff↑↓   at the \nphase transition is not confirmed, given the measurement precision, although  we do observe an \nincrease in the effective spin mixing conductance when transitioning from the bcc to fcc phase.  The 9 concentration dependence of 𝑔eff↑↓  requires further thorough investigation and we therefore restrict \nourselves to reporting the expe rimental findings.  \n \n \n           \n \nFigure 4: The damping for the thickness series at select alloy compositions vs. 1/ t for (a) Ni -Co, \n(b) Ni -Fe and (c) Co -Fe (data points, concentrations denoted in the plots), with linear fits to Eq. \n(3) (solid lines). (d) The extracted effective spin mixing conductance 𝑔eff↑↓  for the measured alloy \nsystems, where the gray lines show the  linear  interpolations for intermediate alloy concentrations. \nThe data for the Co -Fe system are taken from Ref.[38].  \n \n \n Eddy -current damping13,14 is estimated  by use of  the equations given in Ref.  [23]  for films  \n10 nm thick or less . Eddy currents are neglected because they are found to be less than 5  % of the \ntotal damping. Two -magnon scattering  is disregarded  because the mechanism is largely e xcluded \nin the out -of-plane measurement geometry15–17. The total measured damping  is therefore well \napproximated as the sum   \n 𝛼tot≅𝛼int+𝛼rad+𝛼sp, (4) \nWe determine  the intrinsic damping of the material by subtracting α sp and α rad from the measured \ntotal damping , as shown in Fig. 3 .  \n10 The intrinsic damping increases monotonically with Ni concentration  for the NixCo1-x alloys . \nIndicative of the importance of extrinsic sources of damping, αint is approximately 40 % smaller \nthan αtot for the Fe -rich alloy, though the difference decreases to only 15 % for pure Ni. This \nbehavior is expected, given that both αrad and αsp are proportional to  Ms. A comparison of αint to the \ncalculations by Mankovsky , et al. ,1 shows excellent quantitative agreement  to within 30 %.  \nFurthermore, w e compare αint of the NixCo1-x alloys to the spin density weighted average of the \nintrinsic damping of Ni and C o [purple dashed line in Fig. 3 (a)] , which gives good agreement with \nour data, as previously reported .55  \nαint for NixFe1-x (Fig. 3 (b)) also increases with Ni concentration after a small initial decrease \nfrom pure Fe to the first NixFe1-x alloys. The step increase found in αtot at the bcc to fcc phase \ntransition is fully attributed to αsp, as detailed in the previous section, and therefore does not occur \nin αint. Similar to the NixCo1-x system αint is significantly lower than αtot for Fe -rich alloys. With in \nerror bars, a  comparison to the calculations by Mankovsky , et al.1 (blue line) and Starikov , et al.2 \n(green line) exhibit excellent agreement in the fcc phase, with marginally larger deviations in the \nNi rich regime. Starikov , et al.2 calculated the damping over the ful l range of compositions, under \nthe assumption of continuous  fcc phase. This calcu lation deviates further from  our measured αint in \nthe bcc phase exhibiting qualitatively different behavior.  \nAs previously reported, t he dependence of αint on alloy compositio n in the CoxFe1-x alloys \nexhibits strongly non -monotonic behavior, differing from the two previously discussed alloys.38  \nαint displays a minimum at 25 % Co concentration with a, for conducting ferromagnets \nunprecedented, low value of int (5±1.8)  × 10-4. With increasing Co concentration , αint grows up \nto the phase transition, at which point  it increases by 10  % to 20 % unt il it reaches  the value for \npure Co.  It was shown  that αint scales with the density of states  (DOS)  at the Fermi energy n( EF) in \nthe bcc phase38, and the DOS also exhibits a sharp  minimum for Co 25Fe75. This scaling is \nexpected60,61 if the damping is dominated by the breathing Fermi surface process. With the \nbreathing surface model,  the intraband scattering  that leads to damping  directly scales with n( EF). \nThis scaling is particularly pronounc ed in the Co -Fe alloy system due to the small concentration  \ndependence of the spin -orbit coupling on alloy composition. The special properties of the CoxFe1-x \nalloy system are discussed  in greater detail in Ref.[38]. \nComparing αint to the calculations by Mankovsky  et al.1, we find good quantitative \nagreement with the value of the minimum. However, t he concentration of the minimum is \ncalculated to occur  at approximately 10 % to  20% Co, a slightly lower value than 25 % Co t hat we \nfind in this study. Furthermore , the strong concentration dependence around the minimum is not \nreflected in the calculations. More recent calculations by Turek et al.3, for the bcc CoxFe1-x alloys \n[orange line in Fig. 3 (c)] find the a minimum of the damping of 4x10-4 at 25 % Co concentration \nin good agreement with our experiment, but there is some deviation in concentration dependenc e \nof the damping around the minimum. Turek et al.3 also reported on the damping in the NixFe1-x \nalloy system, with similar qualitative and  quantitative results as the other two presented quantitative \ntheories1,2 and the results are therefore not plotted  in Fig.  3 (b) for the sake of comprehensibility of \nthe figure.  For both NixFe1-x and the CoxFe1-x alloys , the calculated spin density weighte d intrinsic \ndamping of the pure elements (not plotted) deviates significantly from the determined intrinsic \ndamping of the alloys, in contrary to the  good agreement  archived  for the  CoxNi1-x alloys. We \nspeculate that this difference between the alloy syste ms is caused by the non -monotonous \ndependence of the density of states at the Fermi Energy in the CoxFe1-x and NixFe1-x systems.  \nOther  calculated damping values for the pure elements and the Ni80Fe20 and Co 90Fe10 alloys \nare compared to the determined intr insic damping in Table 1. Generally , the calculations \nunderestimate the damping significantly, but our data are in good agreement with more recent \ncalculations for Permalloy ( Ni80Fe20).   11 It is important to point out that n one of the theories  considered he re include thermal \nfluctuations . Regardless, we find exceptional agreement with the calculations to αint at intermediate \nalloy concentrations . We speculate that the modeling of atomic disorder in the alloys in the \ncalculations, by the coherent potential approximation (CPA)  could be responsible for this \nexceptional agreement. The effect of disorder on the  electronic band structure possibly dominates \nany effect s due to nonzero temperature. Indeed, both effects cause a broadening of the bands due \nto enhanced  momentum scattering rates. This directly correlates to a change of the damping \nparameter  according to  the theory of Gilmore and Stiles9. Therefore , the inclusion of the inherent  \ndisorder  of solid -solution alloys  in the calculations  by Mankovsky et al1 mimic s the effects of \ntemperature on damping  to some extent . This argument is corroborated by the fact that the \ncalculations  by Mankovsky et al1 diverge for diluted alloys and pure elements  (as shown in Fig. 2 \n(c) for pure Fe) , where no  or to little disorder is introduced to account for temperature effects. \nMankovsky et al.1 performed temperature dependent calculations of the damping for pure bcc Fe, \nfcc Ni and hcp Co  and the values for 300 K are shown in Table 1 and Fig. 3. These calculations  for \nαint at a temperature of 300 K  are approximately a factor of two  less than our measured values , but \nthe agreement is  significantly improved relative to those  obtained by calculations that neglect \nthermal fluctuations . \n \n \n Figure 5: The intrinsic damping α int is plotted against ( g-2)2 for \nall alloys. We do not observe a proportionality between α int and \n(g-2)2. \n12 Finally, i t has been  reported45,64 that there is a  general proportionality between αint and (g-\n2)2 , as contained in the original microscopic BFS model proposed by  Kambersky .62 To examine \nthis relationship, w e plot αint versus (g-2)2 (determined in Ref [66]) for all samples measured here  \nin Figure 5 . While some samples with large values for ( g-2)2 also exhib it large αint, this is not a \ngeneral trend for all the measured samples . Given  that the damping is not purely a function of the \nspin-orbit strength, but also depends on the details of the band structure , the result  in Fig.  5 is \nexpected .  For example , the amount of  band overlap  will determine the amount of interband \ntransition leading to that damping channel.  Furthermore, the density of states at the Fermi energy \nwill affect the intraband contribution to the damping9,10.   Finally , the ratio of inter - to intra -band \nscattering that mediate s damping contributions at a fixed temperature (RT for our measurements) \nchanges  for different elements9,10 and therefore with alloy concentration. None of these f actors are \nnecessarily proportional to the spin -orbit coupling .  Therefore , we conclude that this simple \nrelation, which originally traces to an order of magnitude estimate  for the case of spin relaxation \nin semiconductors65, does not hold for  all magnetic systems  in general.   \n \n4  Summary  \nWe determined  the damping for the full compositi on range of the binary  3d transition  metal all oys \nNi-Co, Ni -Fe, and Co -Fe and showed that the measured damping can be explained by three \ncontributions to the damping: Intrinsic damping, radiative damping and damping due to spin \npumping. By quantifying all  extrinsic contributions to the measured damping, we determine  the \nintrinsic damping over the whole range of alloy compositions .  These values are compared  to \nmultiple theoretical calculations and yield excellent qualitative and good quantitative agreement for \nintermediate alloy concentrations. For pure elements or diluted alloys, the effect of temperature \nseems to play a larger role for the damping and calculations including temperature effects give \nsignificantly better agreement to our data. Furthermore, w e demonstrated a compositional \ndependence of the spin mixing conductance , which can vary by a factor of two . Finally , we showed \nthat the often postulated dependence of the damping on the g-factor does not apply to the \ninvestigated binary alloy systems, as their damping cannot  be described solely by the strength of \nthe spin -orbit interaction . \n \n \n \n \n \n \n \n \n \n \n \n 13 5  References  \n \n1. Mankovsky, S., Ködderitzsch, D., Woltersdorf, G. & Ebert, H. First -principles calculation of the Gilbert d amping \nparameter via the linear response formalism with application to magnetic transition metals and alloys. Phys. Rev. \nB 87, 014430 (2013).  \n2. Starikov, A. A., Kelly, P. J., Brataas, A., Tserkovnyak, Y. & Bauer, G. E. W. Unified First -Principles Study of  \nGilbert Damping, Spin -Flip Diffusion, and Resistivity in Transition Metal Alloys. Phys. Rev. Lett.  105, 236601 \n(2010).  \n3. Turek, I., Kudrnovský, J. & Drchal, V. Nonlocal torque operators in ab initio theory of the Gilbert damping in \nrandom ferromagnetic a lloys. Phys. Rev. B  92, 214407 (2015).  \n4. Gilbert, T. L. A phenomenological theory of damping in ferromagnetic materials. IEEE Trans. Mag. 40 (6): 3443 -\n3449  (2004).  \n5. Landau, L. D. & Lifshitz, E. M. Theory of the dispersion of magnetic permeability in fer romagnetic bodies. Phys. \nZ. Sowietunion, 8, 153  (1935).  \n6. Kamberský, V. FMR linewidth and disorder in metals. Czechoslovak Journal of Physics B  34, 1111 –1124 (1984).  \n7. Kamberský, V. On ferromagnetic resonance damping in metals. Czechoslovak Journal of Ph ysics B  26, 1366 –\n1383 (1976).  \n8. Kambersky, V. & Patton, C. E. Spin -wave relaxation and phenomenological damping in ferromagnetic resonance. \nPhys. Rev. B  11, 2668 –2672 (1975).  \n9. Gilmore, K., Idzerda, Y. U. & Stiles, M. D. Identification of the Dominant Pr ecession -Damping Mechanism in \nFe, Co, and Ni by First -Principles Calculations. Phys. Rev. Lett.  99, 027204 (2007).  \n10. Thonig, D. & Henk, J. Gilbert damping tensor within the breathing Fermi surface model: anisotropy and non -\nlocality. New J. Phys.  16, 0130 32 (2014).  \n11. Brataas, A., Tserkovnyak, Y. & Bauer, G. E. W. Scattering Theory of Gilbert Damping. Phys. Rev. Lett.  101, \n037207 (2008).  \n12. Liu, Y., Starikov, A. A., Yuan, Z. & Kelly, P. J. First -principles calculations of magnetization relaxation in pure  \nFe, Co, and Ni with frozen thermal lattice disorder. Phys. Rev. B  84, 014412 (2011).  \n13. Lock, J. M. Eddy current damping in thin metallic ferromagnetic films. British Journal of Applied Physics  17, \n1645 (1966).  \n14. Pincus, P. Excitation of Spin Waves in Ferromagnets: Eddy Current and Boundary Condition Effects. Phys. Rev.  \n118, 658 –664 (1960).  \n15. Hurben, M. J. & Patton, C. E. Theory of two magnon scattering microwave relaxation and ferromagnetic \nresonance linewidth in magnetic thin films. Journal of Appli ed Physics  83, 4344 –4365 (1998).  \n16. Lindner, J. et al.  Non-Gilbert -type damping of the magnetic relaxation in ultrathin ferromagnets: Importance of \nmagnon -magnon scattering. Phys. Rev. B  68, 060102 (2003).  \n17. Sparks, M., Loudon, R. & Kittel, C. Ferromagn etic Relaxation. I. Theory of the Relaxation of the Uniform \nPrecession and the Degenerate Spectrum in Insulators at Low Temperatures. Phys. Rev.  122, 791 –803 (1961).  \n18. Dillon, J. F. & Nielsen, J. W. Effects of Rare Earth Impurities on Ferrimagnetic Reson ance in Yttrium Iron Garnet. \nPhys. Rev. Lett.  3, 30–31 (1959).  \n19. Van Vleck, J. H. & Orbach, R. Ferrimagnetic Resonance of Dilute Rare -Earth Doped Iron Garnets. Phys. Rev. \nLett. 11, 65–67 (1963).  \n20. Sanders, R. W., Paquette, D., Jaccarino, V. & Rezende, S. M. Radiation damping in magnetic resonance. II. \nContinuous -wave antiferromagnetic -resonance experiments. Phys. Rev. B  10, 132 –138 (1974).  \n21. Wende, G. Radiation damping in FMR measurements in a nonresonant rectangular waveguide. physica status \nsolidi ( a) 36, 557 –567 (1976).  \n22. Nembach, H. T. et al.  Perpendicular ferromagnetic resonance measurements of damping and Lande g -factor in \nsputtered (Co 2Mn) 1-xGex films. Phys. Rev. B  84, 054424 (2011).  \n23. Schoen, M. A. W., Shaw, J. M., Nembach, H. T., Weiler, M . & Silva, T. J. Radiative damping in waveguide -\nbased ferromagnetic resonance measured via analysis of perpendicular standing spin waves in sputtered permalloy \nfilms. Phys. Rev. B  92, 184417 (2015).  \n24. Tserkovnyak, Y., Brataas, A. & Bauer, G. E. W. Enhanc ed Gilbert Damping in Thin Ferromagnetic Films. Phys. \nRev. Lett.  88, 117601 (2002).  \n25. Y. Tserkovnyak, A. Brataas G.E.W. Bauer & Halperin, B. I. Nonlocal magnetization dynamics in ferromagnetic \nheterostructures. Rev. Mod. Phys. 77, 1375  (2005).  \n26. Baryak htar, V., E., K. & D., Y. Sov. Phys. JETP  64, 542 –548 (1986).  14 27. Haertinger, M. et al.  Spin pumping in YIG/Pt bilayers as a function of layer thickness. Phys. Rev. B  92, 054437 \n(2015).  \n28. Lequeux, S. et al.  Increased magnetic damping of a single domain wa ll and adjacent magnetic domains detected \nby spin torque diode in a nanostripe. Applied Physics Letters  107, (2015).  \n29. Li, Y. & Bailey, W. E. Wavenumber -dependent Gilbert damping in metallic ferromagnets. ArXiv e -prints  (2014).  \n30. Nembach, H. T., Shaw, J. M., Boone, C. T. & Silva, T. J. Mode - and Size -Dependent Landau -Lifshitz Damping \nin Magnetic Nanostructures: Evidence for Nonlocal Damping. Phys. Rev. Lett.  110, 117201 (2013).  \n31. Weindler, T. et al.  Magnetic Damping: Domain Wall Dynamics versus Local Ferromagnetic Resonance. Phys. \nRev. Lett.  113, 237204 (2014).  \n32. Celinski, Z. & Heinrich, B. Ferromagnetic resonance linewidth of Fe ultrathin films grown on a bcc Cu substrate. \nJournal of Applied Physics  70, 5935 –5937 (1991).  \n33. Platow, W., Anisimov, A.  N., Dunifer, G. L., Farle, M. & Baberschke, K. Correlations between ferromagnetic -\nresonance linewidths and sample quality in the study of metallic ultrathin films. Phys. Rev. B  58, 5611 –5621 \n(1998).  \n34. Twisselmann, D. J. & McMichael, R. D. Intrinsic damp ing and intentional ferromagnetic resonance broadening \nin thin Permalloy films. Journal of Applied Physics  93, 6903 –6905 (2003).  \n35. McMichael, R. D., Twisselmann, D. J. & Kunz, A. Localized Ferromagnetic Resonance in Inhomogeneous Thin \nFilms. Phys. Rev. L ett. 90, 227601 (2003).  \n36. Hennemann, O. D. & Siegel, E. Spin - Wave Measurements of Exchange Constant A in Ni -Fe Alloy Thin Films. \nphys. stat. sol.  77, 229 (1976).  \n37. Wilts, C. & Lai, S. Spin wave measurements of exchange constant in Ni -Fe alloy films. IEEE Transactions on \nMagnetics  8, 280 –281 (1972).  \n38. Schoen, M. A. W. et al.  Ultra -low magnetic damping of a metallic ferromagnet. Nat. Phys  (2016)  12, 839 –842 \n(2016)  \n39. Heinrich, B., Meredith, D. J. & Cochran, J. F. Wave number and temperature dependent Landau -Lifshitz damping \nin nickel. Journal of Applied Physics  50, 7726 –7728 (1979).  \n40. Lagae, L., Wirix -Speetjens, R., Eyckmans, W., Borghs, S. & Boeck, J. D. Increased Gilbert damping in spin \nvalves and magnetic tunnel junctions. Journal of Magnetism and  Magnetic Materials  286, 291 –296 (2005).  \n41. Rantschler, J. O. et al.  Damping at normal metal/permalloy interfaces. Magnetics, IEEE Transactions on  41, \n3523 –3525 (2005).  \n42. Shaw, J. M., Nembach, H. T. & Silva, T. J. Determination of spin pumping as a sour ce of linewidth in sputtered \nCo90Fe10/Pd multilayers by use of broadband ferromagnetic resonance spectroscopy. Phys. Rev. B  85, 054412 \n(2012).  \n43. Bhagat, S. M. & Lubitz, P. Temperature variation of ferromagnetic relaxation in the 3d transition metals. Phys. \nRev. B  10, 179 (1974).  \n44. Oogane, M. et al.  Magnetic Damping in Ferromagnetic Thin Films. Japanese Journal of Applied Physics  45, \n3889 –3891 (2006).  \n45. Pelzl, J. et al.  Spin-orbit -coupling effects on g -value and damping factor of the ferromagnetic res onance in Co \nand Fe films. Journal of Physics: Condensed Matter  15, S451 (2003).  \n46. Scheck, C., Cheng, L. & Bailey, W. E. Low damping in epitaxial sputtered iron films. Applied Physics Letters  88, \n(2006).  \n47. Shaw, J. M., Silva, T. J., Schneider, M. L. & McMichael, R. D. Spin dynamics and mode structure in nanomagnet \narrays: Effects of size and thickness on linewidth and damping. Phys. Rev. B  79, 184404 (2009).  \n48. Yin, Y. et al.  Tunable permalloy -based films for magnonic devices. Phys. Rev. B  92, 024427 ( 2015).  \n49. Walowski, J. et al.  Intrinsic and non -local Gilbert damping in polycrystalline nickel studied by Ti:sapphire laser \nfs spectroscopy. Journal of Physics D: Applied Physics  41, 164016 (2008).  \n50. Ingvarsson, S. et al.  Role of electron scattering in  the magnetization relaxation of thin Ni 81Fe19 films. Phys. Rev. \nB 66, 214416 (2002).  \n51. Luo, C. et al.  Enhancement of magnetization damping coefficient of permalloy thin films with dilute Nd dopants. \nPhys. Rev. B  89, 184412 (2014).  \n52. Kobayashi, K. et al. Damping Constants for Permalloy Single -Crystal Thin Films. IEEE Transactions on \nMagnetics  45, 2541 –2544 (2009).  \n53. Zhao, Y. et al.  Experimental Investigation of Temperature -Dependent Gilbert Damping in Permalloy Thin Films. \nScientific Reports  6, 22890 – (2016).  \n54. Liu, Y., Yuan, Z., Wesselink, R. J. H., Starikov, A. A. & Kelly, P. J. Interface Enhancement of Gilbert Damping \nfrom First Principles. Phys. Rev. Lett.  113, 207202 (2014).  \n55. Shaw, J. M., Nembach, H. T. & Silva, T. J. Damping phenomena in Co 90Fe10/Ni multilayers and alloys. Applied \nPhysics Letters  99, (2011).  15 56. Boone, C. T., Nembach, H. T., Shaw, J. M. & Silva, T. J. Spin transport parameters in metallic multilayers \ndetermined by ferromagnetic resonance measurements of spin -pumping. Journal of Applied Physics  113, 15, \n101063 (2013).  \n57. Boone, C. T., Shaw, J. M., Nembach, H. T. & Silva, T. J. Spin -scattering rates in metallic thin films measured by \nferromagnetic resonance damping enhanced by spin -pumping. Journal of Applied Physics  117, 22, 1 01063 \n(2015).  \n58. Czeschka, F. D. et al.  Scaling Behavior of the Spin Pumping Effect in Ferromagnet -Platinum Bilayers. Phys. Rev. \nLett. 107, 046601 (2011).  \n59. Weiler, M., Shaw, J. M., Nembach, H. T. & Silva, T. J. Detection of the DC Inverse Spin Hall Eff ect Due to Spin \nPumping in a Novel Meander -Stripline Geometry. Magnetics Letters, IEEE  5, 1–4 (2014).  \n60. Ebert, H., Mankovsky, S., Ködderitzsch, D. & Kelly, P. J. Ab Initio Calculation of the Gilbert Damping Parameter \nvia the Linear Response Formalism. Phys. Rev. Lett.  107, 66603 –66607 (2011).  \n61. Lounis, S., Santos Dias, M. dos & Schweflinghaus, B. Transverse dynamical magnetic susceptibilities from \nregular static density functional theory: Evaluation of damping and g shifts of spin excitations. Phys. Rev . B 91, \n104420 (2015).  \n62. Kamberský, V. On the Landau -Lifshitz relaxation in ferromagnetic metals. Can. J. Phys.  48, 2906 (1970).  \n63. Korenman, V. & Prange, R. E. Anomalous Damping of Spin Waves in Magnetic Metals. Phys. Rev. B  6, 2769 –\n2777 (1972).  \n64. Devolder, T. et al.  Damping of Co xFe80-xB20 ultrathin films with perpendicular magnetic anisotropy. Applied \nPhysics Letters  102, (2013).  \n65. Elliott, R. J. Theory of the Effect of Spin -Orbit Coupling on Magnetic Resonance in Some Semiconductors. Phys. \nRev. 96, 266 –279 (1954).  \n66. Schoen, Martin A. W. , Lucassen, Juriaan , Nembach, Hans T. , Silva, T. J. , Koopmans, Bert , Back, Christian H. , \nShaw, Justin M.  Magnetic properties of ultra -thin 3d transition -metal binary alloys I: spin and orbital mo ments, \nanisotropy, and confirmation of Slater -Pauling behavior . arXiv:1701.02177  (2017 ).  \n \n \n "    },    {        "title": "1701.03083v2.The_Cauchy_problem_for_the_Landau_Lifshitz_Gilbert_equation_in_BMO_and_self_similar_solutions.pdf",        "content": "The Cauchy problem for the Landau–Lifshitz–Gilbert equation\nin BMO and self-similar solutions\nSusana Gutiérrez1and André de Laire2\nAbstract\nWe prove a global well-posedness result for the Landau–Lifshitz equation with Gilbert\ndamping provided that the BMO semi-norm of the initial data is small. As a consequence,\nwe deduce the existence of self-similar solutions in any dimension. In the one-dimensional\ncase, we characterize the self-similar solutions associated with an initial data given by some\n(S2-valued) step function and establish their stability. We also show the existence of multiple\nsolutions if the damping is strong enough.\nOur arguments rely on the study of a dissipative quasilinear Schrödinger equation ob-\ntained via the stereographic projection and techniques introduced by Koch and Tataru.\nKeywords and phrases: Landau–Lifshitz–Gilbert equation, global well-posedness, discontin-\nuous initial data, stability, self-similar solutions, dissipative Schrödinger equation, complex\nGinzburg–Landau equation, ferromagnetic spin chain, heat-flow for harmonic maps.\n2010Mathematics Subject Classification: 35R05, 35Q60, 35A01, 35C06, 35B35, 35Q55,\n35Q56, 35A02, 53C44.\nContents\n1 Introduction and main results 2\n2 The Cauchy problem 6\n2.1 The Cauchy problem for a dissipative quasilinear Schrödinger equation . . . . . . 6\n2.2 The Cauchy problem for the LLG equation . . . . . . . . . . . . . . . . . . . . . 15\n3 Applications 22\n3.1 Existence of self-similar solutions in RN. . . . . . . . . . . . . . . . . . . . . . . 22\n3.2 The Cauchy problem for the one-dimensional LLG equation with a jump initial\ndata . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23\n3.2.1 Existence, uniqueness and stability. Proof of Theorem 1.2 . . . . . . . . . 24\n3.2.2 Multiplicity of solutions. Proof of Theorem 1.3 . . . . . . . . . . . . . . . 28\n3.3 A singular solution for a nonlocal Schrödinger equation . . . . . . . . . . . . . . . 30\n4 Appendix 34\n1School of Mathematics, University of Birmingham, Edgbaston, Birmingham, B15 2TT, United Kingdom.\nE-mail: s.gutierrez@bham.ac.uk\n2Univ. Lille, CNRS, Inria, UMR 8524, Laboratoire Paul Painlevé, F-59000 Lille, France.\nE-mail: andre.de-laire@univ-lille.fr\n1arXiv:1701.03083v2  [math.AP]  20 Mar 20191 Introduction and main results\nWe consider the Landau–Lifshitz–Gilbert (LLG) equation\n@tm=\fm\u0002\u0001m\u0000\u000bm\u0002(m\u0002\u0001m);onRN\u0002R+; (LLG\u000b)\nwhere m= (m1;m2;m3) :RN\u0002R+\u0000!S2is the spin vector, \f\u00150,\u000b\u00150;\u0002denotes the usual\ncross-product in R3, and S2is the unit sphere in R3. This model introduced by Landau and\nLifshitz describes the dynamics for the spin in ferromagnetic materials [26, 16] and constitutes a\nfundamental equation in the magnetic recording industry [36]. The parameters \f\u00150and\u000b\u00150\nare respectively the so-called exchange constant and Gilbert damping, and take into account the\nexchange of energy in the system and the effect of damping on the spin chain. Note that, by\nperforming a time-scaling, we assume w.l.o.g. that\n\u000b2[0;1]and\f=p\n1\u0000\u000b2:\nThe Landau–Lifshitz family of equations includes as special cases the well-known heat-flow for\nharmonic maps and the Schrödinger map equation onto the 2-sphere. In the limit case \f= 0\n(and so\u000b= 1) the LLG equation reduces to the heat-flow equation for harmonic maps\n@tm\u0000\u0001m=jrmj2m;onRN\u0002R+: (HFHM)\nThe case when \u000b= 0(i.e. no dissipation/damping) corresponds to the Schrödinger map equation\n@tm=m\u0002\u0001m;onRN\u0002R+: (SM)\nIn the one-dimensional case N= 1, we established in [17] the existence and asymptotics of the\nfamilyfmc;\u000bgc>0of self-similar solutions of (LLG \u000b) for any fixed \u000b2[0;1], extending the results\nin Gutiérrez, Rivas and Vega [18] in the setting of the Schrödinger map equation and related\nbinormal flow equation. The motivation for the results presented in this paper first originated\nfrom the desire to study further properties of the self-similar solutions found in [17], and in\nparticular their stability. In the case \u000b= 0, the stability of the self-similar solutions of the\nSchrödinger map has been considered in the series of papers by Banica and Vega [5, 6, 7], but\nno stability result is known for these solutions in the presence of damping, i.e. \u000b > 0. One of\nthe key ingredients in the analysis given by Banica and Vega is the reversibility in time of the\nequation in the absence of damping. However, since (LLG \u000b) is a dissipative equation for \u000b>0,\nthis property is no longer available and a new approach is needed.\nIn the one-dimensional case and for fixed \u000b2[0;1], the self-similar solutions of (LLG \u000b)\nconstitute a uniparametric family fmc;\u000bgc>0where mc;\u000bis defined by\nmc;\u000b(x;t) =f\u0012xp\nt\u0013\n;\nfor some profile f:R\u0000!S2, and is associated with an initial condition given by a step function\n(at least when cis small) of the form\nm0\nc;\u000b:=A+\nc;\u000b\u001fR++A\u0000\nc;\u000b\u001fR\u0000; (1.1)\nwhere A\u0006\nc;\u000bare certain unitary vectors and \u001fEdenotes the characteristic function of a set E. In\nparticular, when \u000b>0, the Dirichlet energy associated with the solutions mc;\u000bgiven by\nkrmc;\u000b(\u0001;t)k2\nL2=c2\u00102\u0019\n\u000bt\u00111=2\n; t> 0; (1.2)\n2diverges as t!0+3.\nA first natural question in the study of the stability properties of the family of solutions\nfmc;\u000bgc>0is whether or not it is possible to develop a well-posedness theory for the Cauchy\nproblem for (LLG \u000b) in a functional framework that allows us to handle initial conditions of the\ntype (1.1). In view of (1.1) and (1.2), such a framework should allow some “rough” functions\n(i.e. function spaces beyond the “classical” energy ones) and step functions.\nA few remarks about previously known results in this setting are in order. In the case \u000b>0,\nglobal well-posedness results for (LLG \u000b) have been established in N\u00152by Melcher [31] and by\nLin, Lai and Wang [30] for initial conditions with a smallness condition on the gradient in the\nLN(RN)and the Morrey M2;2(RN)norm4, respectively. Therefore these results do not apply\nto the initial condition m0\nc;\u000b. When\u000b= 1, global well-posedness results for the heat flow for\nharmonic maps (HFHM) have been obtained by Koch and Lamm [22] for an initial condition\nL1-close to a point and improved to an initial data with small BMO semi-norm by Wang [35].\nThe ideas used in [22] and [35] rely on techniques introduced by Koch and Tataru [23] for the\nNavier–Stokes equation. Since m0\nc;\u000bhas a small BMO semi-norm if cis small, the results in [35]\napply to the case \u000b= 1.\nTherearetwomainpurposesinthispaper. Thefirstoneistoadaptandextendthetechniques\ndeveloped in [22, 23, 35] to prove a global well-posedness result for (LLG \u000b) with\u000b2(0;1]for\ndatam0inL1(RN;S2)with small BMO semi-norm. The second one is to apply this result to\nestablish the stability of the family of self-similar solutions fmc;\u000bgc>0found in [17] and derive\nfurther properties for these solutions. In particular, a further understanding of the properties of\nthe functions mc;\u000bwill allow us to prove the existence of multiple smooth solutions of (LLG \u000b)\nassociated with the same initial condition, provided that \u000bis close to one.\nIn order to state the first of our results, we introduce the function space Xas follows:\nX=fv:Rn\u0002R+!R3:v;rv2L1\nloc(RN\u0002R+)andkvkX:=supt>0kv(t)kL1+ [v]X<1g\nwhere\n[v]X:=supt>0p\ntkrvkL1+ sup\nx2RN\nr>0 \n1\nrN\u0002\nBr(x)\u0002[0;r2]jrv(y;t)j2dtdy!1\n2\n;\nandBr(x)denotes the ball with center xand radius r >0inRN. Let us remark that the first\nterm in the definition of [v]Xallows to capture a blow-up rate of 1=p\ntforkrv(t)kL1, ast!0+.\nThis is exactly the blow-up rate for the self-similar solutions (see (3.1) and (3.12)). The integral\nterm in [v]Xis associated with the space BMO as explained in Subsection 2.1, and it is also well\nadapted to the self-similar solutions (see Proposition 3.4 and its proof).\nWe can now state the following (global) well-posedness result for the Cauchy problem for the\nLLG equation:\nTheorem 1.1. Let\u000b2(0;1]. There exist constants M1;M2;M3>0, depending only on \u000band\nNsuch that the following holds. For any m02L1(RN;S2),Q2S2,\u000e2(0;2]and\"0>0such\nthat\"0\u0014M1\u000e6,\ninf\nRNjm0\u0000Qj2\u00152\u000eand [m0]BMO\u0014\"0; (1.3)\nthere exists a unique solution m2X(RN\u0002R+;S2)of(LLG\u000b)with initial condition m0such\nthat\ninf\nx2RN\nt>0jm(x;t)\u0000Qj2\u00154\n1 +M2\n2(M3\u000e4+\u000e\u00001)2and [m]X\u00144M2(M3\u000e4+ 8\u000e\u00002\"0):(1.4)\n3We refer the reader to Theorem A.5 in the Appendix and to [17] for precise statements of these results.\n4See footnote in Section 3.3 for the definition of the Morrey space M2;2(RN).\n3In addition, mis a smooth function belonging to C1(RN\u0002R+;S2). Furthermore, assume that\nnis a solution to (LLG\u000b)fulfilling (1.4), with initial condition n0satisfying (1.3). Then\nkm\u0000nkX\u0014120M2\n\u000e2km0\u0000n0kL1: (1.5)\nAs we will see in Section 2, the proof of Theorem 1.1 relies on the use of the stereographic pro-\njection to reduce Theorem 1.1 to establish a well-posedness result for the associated dissipative\n(quasilinear) Schrödinger equation (see Theorem 2.1). In order to be able to apply Theorem 1.1\nto the study of both the initial value problem related to the LLG equation with a jump initial\ncondition, and the stability of the self-similar solutions found in [17], we will need a more quanti-\ntative version of this result. A more refined version of Theorem 1.1 will be stated in Theorem 2.9\nin Subsection 2.2.\nTheorem 1.1 (or more precisely Theorem 2.9) has two important consequences for the Cauchy\nproblem related to (LLG \u000b) in one dimension:\n8\n<\n:@tm=\fm\u0002@xxm\u0000\u000bm\u0002(m\u0002@xxm);onR\u0002R+;\nm0\nA\u0006:=A+\u001fR++A\u0000\u001fR\u0000;(1.6)\nwhere A\u0006are two given unitary vectors such that the angle between A+andA\u0000is sufficiently\nsmall:\n(a)From the uniqueness statement in Theorem 1.1, we can deduce that the solution to (1.6)\nprovided by Theorem 1.1 is a rotation of a self-similar solution mc;\u000bfor an appropriate\nvalue ofc(see Theorem 3.3 for a precise statement).\n(b)(Stability) From the dependence of the solution with respect to the initial data established\nin (1.5) and the analysis of the 1d-self-similar solutions mc;\u000bcarried out in [17], we obtain\nthe following stability result: For any given m02S2satisfying (1.3) and close enough to\nm0\nA\u0006, the solution mof (LLG\u000b) associated with m0given by Theorem 1.1 must remain\nclose to a rotation of a self-similar solution mc;\u000b, for somec>0. In particular, mremains\nclose to a self-similar solution.\nThe precise statement is provided in the following theorem.\nTheorem 1.2. Let\u000b2(0;1]. There exist constants L1;L2;L3>0,\u000e\u00032(\u00001;0),#\u0003>0such\nthat the following holds. Let A+,A\u00002S2with angle#between them. If\n0<#\u0014#\u0003;\nthen there is c>0such that for every m0satisfying\nkm0\u0000m0\nA\u0006kL1\u0014cp\u0019\n2p\u000b;\nthere existsR2SO(3), depending only on A+,A\u0000,\u000bandc, such that there is a unique global\nsmooth solution mof(LLG\u000b)with initial condition m0that satisfies\ninf\nx2R\nt>0(Rm)3(x;t)\u0015\u000e\u0003and [m]X\u0014L1+L2c: (1.7)\nMoreover,\nkm\u0000Rmc;\u000bkX\u0014L3km0\u0000m0\nA\u0006kL1:\n4In particular,\nk@xm\u0000@xRmc;\u000bkL1\u0014L3p\ntkm0\u0000m0\nA\u0006kL1;\nfor allt>0.\nNotice that Theorem 1.2 provides the existence of a unique solution in the set defined by the\nconditions (1.7), and hence it does not exclude the possibility of the existence of other solutions\nnot satisfying these conditions. In fact, as we will see in Theorem 1.3 below, one can prove the\nexistence of multiple solutions of the initial value problem (1.6), at least in the case when \u000bis\nclose to 1.\nWe point out that our results are valid only for \u000b>0. If we let\u000b!0, then the constants M1\nandM3in Theorem 1.1 go to 0 and M2blows up. Indeed, we use that the kernel associated with\nthe Ginzburg–Landau semigroup e(\u000b+i\f)t\u0001belongs toL1and its exponential decay. Therefore\nour techniques cannot be generalized (in a simple way) to cover the critical case \u000b= 0. In\nparticular, we cannot recover the stability results for the self-similar solutions in the case of\nSchrödinger maps proved by Banica and Vega in [5, 6, 7].\nAs mentioned before, in [30] and [31] some global well-posedness results for (LLG \u000b) with\n\u000b2(0;1]were proved for initial conditions with small gradient in LN(RN)andM2;2(RN),\nrespectively (see footnote in Subsection 3.3 for the definition of the space M2;2(RN)). In view of\nthe embeddings\nLN(RN)\u001aM2;2(RN)\u001aBMO\u00001(RN);\nforN\u00152, Theorem 1.1 can be seen as generalization of these results since it covers the case\nof less regular initial conditions. The arguments in [30, 31] are based on the method of moving\nframes that produces a covariant complex Ginzburg–Landau equation. In Subsection 3.3 we give\nmore details and discuss the corresponding equation in the one-dimensional case and provide\nsome properties related to the self-similar solutions.\nOur existence and uniqueness result given by Theorem 1.1 requires the initial condition to be\nsmall in the BMO semi-norm. Without this condition, the solution could develop a singularity\nin finite time. In fact, in dimensions N= 3;4, Ding and Wang [13] have proved that for some\nsmooth initial conditions with small (Dirichlet) energy, the associated solutions of (LLG \u000b) blow\nup in finite time.\nIn the context of the initial value problem (1.6), the smallness condition in the BMO semi-\nnorm is equivalent to the smallness of the angle between A+andA\u0000. As discussed in [17], in the\none dimensional case N= 1for fixed\u000b2(0;1]there is some numerical evidence that indicates\nthe existence of multiple (self-similar) solutions associated with the same initial condition of the\ntype in (1.6) (see Figures 2 and 3 in [17]). This suggests that the Cauchy problem for (LLG \u000b)\nwith initial condition (1.6) is ill-posed for general A+andA\u0000unitary vectors.\nThe following result states that in the case when \u000bis close to 1, one can actually prove the\nexistenceofmultiplesmoothsolutionsassociatedwiththesameinitialcondition m0\nA\u0006. Moreover,\ngiven any angle #2(0;\u0019)between two vectors A+andA\u00002S2, one can generate any number\nof distinct solutions by considering values of \u000bsufficiently close to 1.\nTheorem 1.3. Letk2N,A+,A\u00002S2and let#be the angle between A+andA\u0000. If\n#2(0;\u0019), then there exists \u000bk2(0;1)such that for every \u000b2[\u000bk;1]there are at least kdistinct\nsmooth self-similar solutions fmjgk\nj=1inX(R\u0002R+;S2)of(LLG\u000b)with initial condition m0\nA\u0006.\nThese solutions are characterized by a strictly increasing sequence of values fcjgk\nj=1, withck!1\nask!1, such that\nmj=Rjmcj;\u000b; (1.8)\n5whereRj2SO(3). In particular\np\ntk@xmj(\u0001;t)kL1=cj;for allt>0: (1.9)\nFurthermore, if \u000b= 1and#2[0;\u0019], then there is an infinite number of distinct smooth self-\nsimilar solutions fmjgj\u00151inX(R\u0002R+;S2)of(LLG\u000b)with initial condition m0\nA\u0006. These\nsolutions are also characterized by a sequence fcjg1\nj=1such that (1.8)and(1.9)are satisfied.\nThis sequence is explicitly given by\nc2`+1=`p\u0019\u0000#\n2p\u0019; c 2`=`p\u0019+#\n2p\u0019;for`\u00150: (1.10)\nIt is important to remark that in particular Theorem 1.3 asserts that when \u000b= 1, given\nA+;A\u00002S2such that A+=A\u0000, there exists an infinite number of distinct solutions fmjgj\u00151\ninX(R\u0002R+;S2)of (LLG\u000b) with initial condition m0\nA\u0006such that [m0\nA\u0006]BMO = 0. This\nparticular case shows that a condition on the size of X-norm of the solution as that given in\n(1.4) in Theorem 1.1 is necessary for the uniqueness of solution. We recall that for finite energy\nsolutions of (HFHM) there are several nonuniqueness results based on Coron’s technique [11] in\ndimensionN= 3. Alouges and Soyeur [2] successfully adapted this idea to prove the existence of\nmultiple solutions of the (LLG \u000b), with\u000b>0, for maps m: \n\u0000!S2, with \na bounded regular\ndomain of R3. In our case, since fcjgk\nj=1is strictly increasing, we have at least kgenuinely\ndifferent smoothsolutions. Notice also that the identity (1.9) implies that the X-norm of the\nsolution is large as j!1.\nStructure of the paper. This paper is organized as follows: in Section 2 we use the ste-\nreographic projection to reduce matters to the study the initial value problem for the resulting\ndissipative Schrödinger equation, prove its global well-posedness in well-adapted normed spaces,\nand use this result to establish Theorem 2.9 (a more quantitative version of Theorem 1.1). In\nSection 3 we focus on the self-similar solutions and we prove Theorems 1.2 and 1.3. In Section 3.3\nwe discuss some implications of the existence of explicit self-similar solutions for the Schrödinger\nequation obtained by means of the Hasimoto transformation. Finally, and for the convenience of\nthe reader, we have included some regularity results for the complex Ginzburg–Landau equation\nand some properties of the self-similar solutions mc;\u000bin the Appendix.\nNotations. We write R+= (0;1). Throughout this paper we will assume that \u000b2(0;1]and\nthe constants can depend on \u000b. In the proofs A.Bstands forA\u0014CBfor some constant\nC > 0depending only on \u000bandN. We denote in bold the vector-valued variables.\nSince we are interested in S2-valued functions, with a slightly abuse of notation, we denote\nbyL1(RN;S2)(resp.X(RN;S2)) the space of function in L1(RN;R3)(resp.X(RN;R3)) such\nthatjmj=1 a.e. on RN.\n2 The Cauchy problem\n2.1 The Cauchy problem for a dissipative quasilinear Schrödinger equation\nOur approach to study the Cauchy problem for (LLG \u000b) consists in analyzing the Cauchy prob-\nlem for the associated dissipative quasilinear Schrödinger equation through the stereographic\nprojection, and then “transferring” the results back to the original equation. To this end, we\nintroduce the stereographic projection from the South Pole P:S2nf(0;0;\u00001)g!Cdefined for\nby\nP(m) =m1+im2\n1 +m3:\n6Letmbe a smooth solution of (LLG \u000b) withm3>\u00001, then its stereographic projection u=\nP(m)satisfies the quasilinear dissipative Schrödinger equation (see e.g. [25] for details)\niut+ (\f\u0000i\u000b)\u0001u= 2(\f\u0000i\u000b)\u0016u(ru)2\n1 +juj2: (DNLS)\nAt least formally, the Duhamel formula gives the integral equation:\nu(x;t) =S\u000b(t)u0+\u0002t\n0S\u000b(t\u0000s)g(u)(s)ds; (IDNLS)\nwhereu0=u(\u0001;0)corresponds to the initial condition,\ng(u) =\u00002i(\f\u0000i\u000b)\u0016u(ru)2\n1 +juj2\nandS\u000b(t)is the dissipative Schrödinger semigroup (also called the complex Ginzburg–Landau\nsemigroup) given by S\u000b(t)\u001e=e(\u000b+i\f)t\u0001\u001e, i.e.\n(S\u000b(t)\u001e)(x) =\u0002\nRNG\u000b(x\u0000y;t)\u001e(y)dy;withG\u000b(x;t) =e\u0000jxj2\n4(\u000b+i\f)t\n(4\u0019(\u000b+i\f)t)N=2:(2.1)\nOne difficulty in studying (IDNLS) is to handle the term g(u). Taking into account that\nj\f\u0000i\u000bj= 1anda\n1 +a2\u00141\n2;for alla\u00150; (2.2)\nwe see that\njg(u)j\u0014jruj2; (2.3)\nso we need to control jruj2. Koch and Taratu dealt with a similar problem when studying the\nwell-posedness for the Navier–Stokes equation in [23]. Their approach was to introduce some new\nspaces related to BMO and BMO\u00001. Later, Koch and Lamm [22] and Wang [35] have adapted\nthese spaces to study some geometric flows. Following these ideas, we define the Banach spaces\nX(RN\u0002R+;F) =fv:RN\u0002R+!F:v;rv2L1\nloc(RN\u0002R+);kvkX<1gand\nY(RN\u0002R+;F) =fv:RN\u0002R+!F:v2L1\nloc(RN\u0002R+);kvkY<1g;\nwhere\nkvkX:= sup\nt>0kvkL1+ [v]X;with\n[v]X:= sup\nt>0p\ntkrvkL1+ sup\nx2RN\nr>0 \n1\nrN\u0002\nQr(x)jrv(y;t)j2dtdy!1\n2\n;and\nkvkY= sup\nt>0tkvkL1+ sup\nx2RN\nr>01\nrN\u0002\nQr(x)jv(y;t)jdtdy:\nHereQr(x)denotes the parabolic ball Qr(x) =Br(x)\u0002[0;r2]andFis either CorR3. The\nabsolute value stands for the complex absolute value if F=Cand for the euclidean norm if\nF=R3. We denote with the same symbol the absolute value in FandF3. Here and in the\nsequel we will omit the domain in the norms and semi-norms when they are taken in the whole\nspace, for example k\u0001kLpstands fork\u0001kLp(RN), forp2[1;1].\n7The spaces XandYare related to the spaces BMO (RN)and BMO\u00001(RN)and are well-\nadapted to study problems involving the heat semigroup S1(t) =et\u0001. In order to establish\nthe properties of the semigroup S\u000b(t)with\u000b2(0;1], we introduce the spaces BMO \u000b(RN)and\nBMO\u00001\n\u000b(RN)as the space of distributions f2S0(RN;F)such that the semi-norm and norm\ngiven respectively by\n[f]BMO\u000b:= sup\nx2RN\nr>0 \n1\nrN\u0002\nQr(x)jrS\u000b(t)fj2dtdy!1\n2\n;and\nkfkBMO\u00001\n\u000b:= sup\nx2RN\nr>0 \n1\nrN\u0002\nQr(x)jS\u000b(t)fj2dtdy!1\n2\n;\nare finite.\nOntheonehand, theCarlesonmeasurecharacterizationofBMOfunctions(see[34, Chapter4]\nand [27, Chapter 10]) yields that for fixed \u000b2(0;1],BMO\u000b(RN)coincides with the classical\nBMO (RN)space5, that is for all \u000b2(0;1]there exists a constant \u0003>0depending only on \u000b\nandNsuch that\n\u0003[f]BMO\u0014[f]BMO\u000b\u0014\u0003\u00001[f]BMO: (2.4)\nOn the other hand, Koch and Tataru proved in [23] that BMO\u00001(or equivalently BMO\u00001\n1,\nusing our notation) can be characterized as the space of derivatives of functions in BMO. A\nstraightforward generalization of their argument shows that the same result holds for BMO\u00001\n\u000b\n(see Theorem A.1). Hence, using the Carleson measure characterization theorem, we conclude\nthat BMO\u00001\n\u000bcoincides with the space BMO\u00001and that there exists a constant ~\u0003>0, depending\nonly on\u000bandN, such that\n~\u0003kfkBMO\u00001\u0014kfkBMO\u00001\n\u000b\u0014~\u0003\u00001kfkBMO\u00001: (2.5)\nThe above remarks allows us to use several of the estimates proved in [22, 23, 35] in the case\n\u000b= 1to study the integral equation (IDNLS) by using a fixed-point approach.\nOur first result concerns the global well-posedness of the Cauchy problem for (IDNLS) with\nsmall initial data in BMO (RN).\nTheorem 2.1. Let\u000b2(0;1]. There exist constants C;K\u00151such that for every L\u00150,\">0,\nand\u001a>0satisfying\n8C(\u001a+\")2\u0014\u001a; (2.6)\nifu02L1(RN;C), with\nku0kL1\u0014Land [u0]BMO\u0014\"; (2.7)\nthen there exists a unique solution u2X(RN\u0002R+;C)to(IDNLS) such that\n[u]X\u0014K(\u001a+\"): (2.8)\nMoreover,\n5\nBMO (RN) =ff:RN\u0002[0;1)!F:f2L1\nloc(RN);[f]BMO <1g;\nwith the semi-norm\n[f]BMO = sup\nx2RN\nr>0 \nBr(x)jf(y)\u0000fx;rjdy;\nwhere fx;ris the average\nfx;r= \nBr(x)f(y)dy=1\njBr(x)j\u0002\nBr(x)f(y)dy:\n8(i)supt>0kukL1\u0014K(\u001a+L).\n(ii)u2C1(RN\u0002R+)and(DNLS) holds pointwise.\n(iii) lim\nt!0+u(\u0001;t) =u0as tempered distributions. Moreover, for every '2S(RN), we have\nk(u(\u0001;t)\u0000u0)'kL1!0;ast!0+: (2.9)\n(iv) (Dependence on the initial data) Assume that uandvare respectively solutions to (IDNLS)\nfulfilling (2.8)with initial conditions u0andv0satisfying (2.7). Then\nku\u0000vkX\u00146Kku0\u0000v0kL1: (2.10)\nAlthough condition (2.6) appears naturally from the fixed-point used in the proof, it may be\nno so clear at first glance. To better understand it, let us define for C > 0\nS(C) =f(\u001a;\")2R+\u0002R+:C(\u001a+\")2\u0014\u001ag: (2.11)\nWe see that if (\u001a;\")2S(C), then\u001a;\"> 0and\n\"\u0014p\u001ap\nC\u0000\u001a: (2.12)\nTherefore the set S(C)is non-empty and bounded. The shape of this set is depicted in Figure 1.\nIn particular, we infer from (2.12) that if (\u001a;\")2S(C), then\n1\n4C\n1\n4C1\nCρε\nFigure 1: The shape of the set S(C).\n\u001a\u00141\nCand\"\u00141\n4C: (2.13)\nIn addition, if ~C\u0015C, then\nS(~C)\u0012S(C): (2.14)\nMoreover, taking for instance \u001a= 1=(32C), Theorem 2.1 asserts that for fixed \u000b2(0;1], we can\ntake for instance \"= 1=(32C)(that depends on \u000bandN, but not on the L1-norm of the initial\ndata) such that for any given initial condition u02L1(RN)with [u0]BMO\u0014\", there exists a\nglobal (smooth) solution u2X(RN\u0002R+;C)of (DNLS). Notice that u0is allowed to have a\nlargeL1-norm as long as [u0]BMOis sufficiently small; this is a weaker requirement that asking\nfor theL1-norm ofu0to be sufficiently small, since\n[f]BMO\u00142kfkL1;for allf2L1(RN): (2.15)\n9Remark 2.2. The smallness condition in (2.8) is necessary for the uniqueness of the solution.\nAs we will see in Subsection 3.2.2, at least in dimension one, it is possible to construct multiple\nsolutions of (IDNLS) in X(RN\u0002R+;C), if\u000bis close enough to 1.\nThe aim of this section is to prove Theorem 2.1 using a fixed-point technique. To this pursuit\nwe write (IDNLS) as\nu(t) =Tu0(u)(t); (2.16)\nwhere\nTu0(u)(t) =S\u000b(t)u0+T(g(u))(t)andT(f)(t) =\u0002t\n0S\u000b(t\u0000s)f(s)ds: (2.17)\nIn the next lemmas we study the semigroup S\u000band the operator Tto establish that the appli-\ncationTu0is a contraction on the ball\nB\u001a(u0) =fu2X(RN\u0002R+;C) :ku\u0000S\u000b(t)u0kX\u0014\u001ag;\nfor some\u001a>0depending on the size of the initial data.\nLemma 2.3. There exists C0>0such that for all f2BMO\u00001\n\u000b(RN),\nsup\nt>0p\ntkS\u000b(t)fkL1(RN)\u0014C0kfkBMO\u00001\n\u000b: (2.18)\nProof.The proof in the case \u000b= 1is done in [27, Lemma 16.1]. For \u000b2(0;1), decomposing\nS\u000b(t) =S\u000b(t\u0000s)S\u000b(s)and using the decay properties of the kernel G\u000bassociated with the\noperatorsS\u000b(t)(see (2.1)), we can check that the same proof still applies.\nLemma 2.4. There exists C1\u00151such that for all f2Y(RN\u0002R+;C),\nkT(f)kX\u0014C1kfkY: (2.19)\nProof.Estimate (2.19) can be proved using the arguments given in [23] or [35]. For the conve-\nnience of the reader, we sketch the proof following the lines in [35, Lemma 3.1]. By scaling and\ntranslation, it suffices to show that\njT(f)(0;1)j+jrT(f)(0;1)j+ \u0002\nQ1(0)jrT(f)j2!1=2\n.kfkY: (2.20)\nLetBr=Br(0). SettingW=T(f), we have\nW(0;1) =\u00021\n0\u0002\nRNG\u000b(\u0000y;1\u0000s)f(y;s)dyds\n= \u00021\n1=2\u0002\nRN+\u00021=2\n0\u0002\nB2+\u00021=2\n0\u0002\nRNnB2!\nG\u000b(\u0000y;1\u0000s)f(y;s)dyds\n:=I1+I2+I3:\nSincejG\u000b(y;1\u0000s)j=e\u0000\u000bjyj2\n4(1\u0000s)\n(4\u0019(1\u0000s))N=2;we obtain\njI1j\u0014\u00021\n1=2\u0002\nRNjG\u000b(\u0000y;1\u0000s)jjf(y;s)jdyds\n\u0014sup\n1\n2\u0014s\u00141kf(s)kL1 \u00021\n1\n2\u0002\nRnjG\u000b(\u0000y;1\u0000s)jdyds!\n.kfkY;\n10jI2j\u0014\u00021=2\n0\u0002\nB2jG\u000b(\u0000y;1\u0000s)jjf(y;s)jdyds\n.sup\n0\u0014s\u00141\n2kG\u000b(\u0001;1\u0000s)kL1(RN)\u0002\nB2\u0002[0;1\n2]jf(y; s)jdyds.kfkY\nand\njI3j\u0014\u00021=2\n0\u0002\nRNnB2jG\u000b(\u0000y;1\u0000s)jjf(y;s)jdyds\n\u0014C\u00021\n2\n0\u0002\nRNnB2e\u0000\u000bjyj2\n4jf(y; s)jdyds\n\u0014C 1X\nk=2kn\u00001e\u0000\u000bk2\n4! \nsup\ny2RN\u0002\nQ1(y)jf(y; s)jdyds!\n.kfkY:\nThe quantityjrT(f)(0;1)jcan be bounded in a similar way. The last term in the l.h.s. of (2.20)\ncan be controlled using an energy estimate. Indeed, Wsatisfies the equation\ni@tW+ (\f\u0000i\u000b)\u0001W=if (2.21)\nwith initial condition W(\u0001;0) = 0. Let\u00112C1\n0(B2)be a real-valued cut-off function such that\n0\u0014\u0011\u00141onRNand\u0011= 1onB1. By multiplying (2.21) by \u0000i\u00112W, integrating and taking\nreal part, we get\n1\n2@t\u0002\nRN\u00112jWj2+\u000b\u0002\nRN\u00112jrWj2+ 2 Re\u0012\n(\u000b+i\f)\u0002\nRN\u0011r\u0011WrW\u0013\n=\u0002\nRN\u00112Re(fW):\nUsing thatj\u000b+i\fj= 1and integrating in time between 0and1, it follows that\n1\n2\u0002\nRN\u00112jW(x;1)j2+\u000b\u0002\nRN\u0002[0;1]\u00112jrWj2\u0014\u0002\nRN\u0002[0;1](2\u0011jr\u0011jjWjjrWj+\u00112jfjjWj):\nFrom the inequality ab\u0014\"a2+b2=(4\");witha=\u0011jrWj,b= 2jr\u0011jjWjand\"=\u000b=2, we deduce\nthat\u000b\n2\u0002\nRN\u0002[0;1]\u00112jrWj2\u0014\u0002\nRN\u0002[0;1]\u00002\n\u000bjr\u0011j2jWj2+\u00112jfjjWj\u0001\n:\nBy the definition of \u0011, this implies that\nkrWk2\nL2(B1\u0002[0;1]).kWk2\nL1(B2\u0002[0;1])+kWkL1(B2\u0002[0;1])kfkL1(B2\u0002[0;1]):(2.22)\nFrom the first part of the proof, we have\nkWkL1(B2\u0002[0;1])\u0014CkfkY:\nUsing also that\nkfkL1(B2\u0002[0;1]).kfkY;\nwe conclude from (2.22) that\nkrWkL2(B1\u0002[0;1]).kfkY;\nwhich finishes the proof.\n11Lemma 2.5. Let\u000b2(0;1]and\u001a;\";L> 0. There exists C2\u00151, depending on \u000bandN, such\nthat for all u02L1(RN)\nkS\u000b(t)u0kX\u0014C2(ku0kL1+ [u0]BMO ): (2.23)\nIf in additionku0kL1\u0014Land[u0]BMO\u0014\", then for all u2B\u001a(u0)we have\nsup\nt>0kukL1\u0014C2(\u001a+L)and [u]X\u0014C2(\u001a+\"): (2.24)\nProof.We first controlkS\u000b(t)u0kX. On the one hand, using the definition of G\u000band the relation\n\u000b2+\f2= 1, we obtain\nkS\u000b(t)u0kL1=kG\u000b\u0003u0kL1\u0014kG\u000bkL1ku0kL1=\u000b\u0000N\n2ku0kL1;8t>0:\nThus\nsup\nt>0kS\u000b(t)u0kL1\u0014\u000b\u0000N\n2ku0kL1: (2.25)\nOn the other hand, using Lemma 2.3, Theorem A.1 and (2.4),\n[S\u000b(t)u0]X= sup\nt>0p\ntkrS\u000b(t)u0kL1+ sup\nx2RN\nr>0 \n1\nrN\u0002\nQr(x)jrS\u000b(t)u0j2dtdy!1\n2\n.kru0kBMO\u00001\n\u000b+ [u0]BMO\u000b\n.[u0]BMO\u000b\n.[u0]BMO:(2.26)\nThe estimate in (2.23) follows from (2.25) and (2.26), and we w.l.o.g. can choose C2\u00151.\nFinally, using (2.25), given u0such thatku0kL1\u0014Land[u0]BMO\u0014\", for allu2B\u001a(u0)we\nhave\nkukL1\u0014ku\u0000S\u000b(t)u0kL1+kS\u000b(t)u0kL1\u0014ku\u0000S\u000b(t)u0kX+kS\u000b(t)u0kL1\u0014C2(\u001a+L);\nand, using (2.26),\n[u]X\u0014[u\u0000S\u000b(t)u0]X+ [S\u000b(t)u0]X\u0014ku\u0000S\u000b(t)u0kX+ [S\u000b(t)u0]X\u0014C2(\u001a+\");\nwhich finishes the proof of (2.24).\nNow we proceed to bound the nonlinear term\ng(u) =\u00002i(\f\u0000i\u000b)\u0016u(ru)2\n1 +juj2:\nLemma 2.6. For allu2X(RN\u0002R+;C), we have\nkg(u)kY\u0014[u]2\nX:\nProof.Letu2X(RN\u0002R+;C). Using (2.3) and the definitions of the norms in YandX, it\nfollows that\nkg(u)kY\u0014\u0012\nsup\nt>0p\ntkrukL1\u00132\n+ sup\nx2RN\nr>01\nrN\u0002\nQr(x)jruj2dtdy\u0014[u]2\nX:\n12Now we have all the estimates to prove that Tu0is a contraction on B\u001a(u0).\nProposition 2.7. Let\u000b2(0;1]and\u001a;\"> 0. Given any u02L1(RN)with [u0]BMO\u0014\", the\noperatorTu0given in (2.17)defines a contraction on B\u001a(u0), whenever \u001aand\"satisfy\n8C1C2\n2(\u001a+\")2\u0014\u001a: (2.27)\nMoreover, for all u;v2X(RN\u0002R+;C),\nkT(g(u))\u0000T(g(v))kX\u0014C1(2[u]2\nX+ [u]X+ [v]X)ku\u0000vkX: (2.28)\nHere,C1\u00151andC2\u00151are the constants in Lemmas 2.4 and 2.5, respectively.\nRemark 2.8. Using the notation introduced in (2.11), the hypothesis (2.27) means that (\u001a;\")2\nS(8C1C2\n2). Therefore, by (2.13),\n\u001a\u00141\n8C1C2\n2;and\"\u00141\n32C1C2\n2; (2.29)\nso\u001aand\"are actually small. Since C1;C2\u00151, we have\nC2(\u001a+\")\u00145\n32: (2.30)\nProof.Letu02L1(RN)withku0kL1\u0014Land[u0]BMO\u0014\", andu2B\u001a(u0). Using Lemma 2.4,\nLemma 2.5 and Lemma 2.6, we have\nkTu0(u)\u0000S\u000b(t)u0kX=kT(g(u))kX\u0014C1kg(u)kY\u0014C1[u]2\nX\u0014C1C2\n2(\u001a+\")2:\nThereforeTu0mapsB\u001a(u0)into itself provided that\nC1C2\n2(\u001a+\")2\u0014\u001a: (2.31)\nNotice that by (2.14), the condition (2.27) implies that (2.31) is satisfied.\nTo prove (2.28), we use the decomposition\ng(u)\u0000g(v) =\u00002i(\f\u0000i\u000b)\u0014\u0012\u0016u\n1 +juj2\u0000\u0016v\n1 +jvj2\u0013\n(ru)2+\u0016v\n1 +jvj2((ru)2\u0000(rv)2))\u0015\n:\nSince \f\f\f\f\u0016u\n1 +juj2\u0000\u0016v\n1 +jvj2\f\f\f\f\u0014ju\u0000vj1 +jujjvj\n(1 +juj2)(1 +jvj2)\u0014ju\u0000vj;\nand using (2.2), we obtain\njg(u)\u0000g(v)j\u00142ju\u0000vjjruj2+jru\u0000rvj(jruj+jrvj):\nTherefore\nkg(u)\u0000g(v)kY\u00142kju\u0000vjjruj2kY+kjru\u0000rvj(jruj+jrvj)kY:=I1+I2:(2.32)\nForI1, it is immediate that\nI1\u00142 sup\nt>0ku\u0000vkL12\n64\u0010\nsup\nt>0p\ntkrukL1\u00112\n+ sup\nx2RN\nr>01\nrN\u0002\nQr(x)jruj2dtdy3\n75\u00142ku\u0000vkX[u]2\nX:\n(2.33)\n13Similarly, using the Cauchy–Schwarz inequality,\nI2\u0014\u0012\nsup\nt>0p\ntkru\u0000rvkL1\u0013\u0012\nsup\nt>0p\nt(krukL1+krvkL1)\u0013\n+ sup\nx2RN\nr>01\nrN\u0000\nkru\u0000rvkL2(Qr(x))\u0001\u0000\nkrukL2(Qr(x))+krvkL2(Qr(x))\u0001\n\u0014ku\u0000vkX([u]X+ [v]X):(2.34)\nUsing Lemma 2.4, (2.32), (2.33) and (2.34), we conclude that\nkT(g(u))\u0000T(g(v))kX\u0014C1(2[u]2\nX+ [u]X+ [v]X)ku\u0000vkX: (2.35)\nLetu;v2B\u001a(u0), by Lemma 2.5 and (2.30)\n[u]X\u0014C2(\u001a+\")\u00145\n32; (2.36)\nso that\n2[u]2\nX+ [u]X+ [v]X\u001437\n16C2(\u001a+\")<3C2(\u001a+\"): (2.37)\nThen (2.35) implies that\nkTu0(u)\u0000Tu0(v)kX\u00143C1C2(\u001a+\")ku\u0000vkX: (2.38)\nFrom (2.29), we conclude that\n3C1C2(\u001a+\")\u001415\n32\u00141\n2; (2.39)\nand then (2.38) yields that the operator Tu0defined in (2.17) is a contraction on B\u001a(u0). This\nconcludes the proof of the proposition.\nProof of Theorem 2.1. Let us setC=C1C2\n2andK=C2, whereC1andC2are the constants\nin Lemma 2.4 and Lemma 2.5 respectively. Since \u001asatisfies (2.6), Proposition 2.7 implies that\nthere exists a solution uof equation (2.16) in the ball B\u001a(u0), and in particular from Lemma 2.5\nsup\nt>0kukL1\u0014K(\u001a+L)and [u]X\u0014K(\u001a+\"):\nTo prove the uniqueness part of the theorem, let us assume that uandvare solutions of (IDNLS)\ninX(RN\u0002R+;C)such that\n[u]X;[v]X\u0014K(\u001a+\"); (2.40)\nwith the same initial condition u0. By the definitions of CandK, (2.6) and (2.40), the estimates\nin (2.29) and (2.30) hold. It follows that (2.36), (2.37) and (2.39) are satisfied. Then, using\n(2.28),\nku\u0000vkX=kT(g(u))\u0000T(g(v))kX\u0014C1(2[u]2\nX+ [u]X+ [v]X)ku\u0000vkX\n\u00141\n2ku\u0000vkX:\nFrom which it follows that u=v.\nTo prove the dependence of the solution with respect to the initial data (part (iv)), consider\nuandvsolutions of (IDNLS) satisfying (2.40) with initial conditions u0andv0. Then, by\ndefinition,u=Tu0(u),v=Tv0(v)and\nku\u0000vkX=kTu0(u)\u0000Tv0(v)kX\u0014kS\u000b(u0\u0000v0)kX+kT(g(u))\u0000T(g(v))kX:\n14Using (2.15), (2.23) and (2.28) and arguing as above, we have\nku\u0000vkX\u0014C2(ku0\u0000v0kL1+ [u0\u0000v0]BMO ) +C1(2[u]2\nX+ [u]X+ [v]X)ku\u0000vkX\n\u00143C2ku0\u0000v0kL1+1\n2ku\u0000vkX:\nThis yields (2.10), since K=C2.\nThe assertions in (ii)and(iii)follow from Theorem A.3.\n2.2 The Cauchy problem for the LLG equation\nBy using the inverse of the stereographic projection P\u00001:C!S2nf0;0;\u00001g, that is explicitly\ngiven by m= (m1;m2;m3) =P\u00001(u), with\nm1=2 Reu\n1 +juj2; m 2=2 Imu\n1 +juj2; m 3=1\u0000juj2\n1 +juj2; (2.41)\nwe will be able to establish the following global well-posedness result for (LLG \u000b).\nTheorem 2.9. Let\u000b2(0;1]. There exist constants C\u00151andK\u00154, such that for any\n\u000e2(0;2],\"0>0and\u001a>0such that\n8K4C\u000e\u00004(\u001a+ 8\u000e\u00002\"0)2\u0014\u001a; (2.42)\nifm0= (m0\n1;m0\n2;m0\n3)2L1(RN;S2)satisfies\ninf\nRNm0\n3\u0015\u00001 +\u000eand [m0]BMO\u0014\"0; (2.43)\nthen there exists a unique solution m= (m1;m2;m3)2X(RN\u0002R+;S2)of(LLG\u000b)such that\ninf\nx2RN\nt>0m3(x;t)\u0015\u00001 +2\n1 +K2(\u001a+\u000e\u00001)2and [m]X\u00144K(\u001a+ 8\u000e\u00002\"0):(2.44)\nMoreover, we have the following properties.\ni)m2C1(RN\u0002R+;S2).\nii)jm(\u0001;t)\u0000m0j\u0000! 0inS0(RN)ast\u0000!0+.\niii) Assume that mandnare respectively smooth solutions to (IDNLS) satisfying (2.44)with\ninitial conditions m0andn0satisfying (2.43). Then\nkm\u0000nkX\u0014120K\u000e\u00002km0\u0000n0kL1: (2.45)\nRemark 2.10. The restriction (2.42) on the parameters is similar to (2.27), but we need to\ninclude\u000e. To better understand the role of \u000e, we can proceed as before. Indeed, setting for\na;\u000e> 0,\nS\u000e(a) =f(\u001a;\"0)2R+\u0002R+:a\u000e\u00004(\u001a+ 8\u000e\u00002\"0)2\u0014\u001ag;\nwe see that its shape is similar to the one in Figure 1. It is simple to verify that for any\n(\u001a;\"0)2S\u000e(a), we have the bounds\n\u001a\u0014\u000e4\naand\"0\u0014\u000e6\n32a; (2.46)\nand the maximum value \"\u0003\n0=\u000e6\n32ais attained at \u001a\u0003=\u000e4\n4a. Also, the sets are well ordered, i.e. if\n~a\u0015a>0, thenS\u000e(~a)\u0012S\u000e(a).\n15We emphasize that the first condition in (2.43) is rather technical. Indeed, we need the\nessential range of m0to be far from the South Pole in order to use the stereographic projection.\nIn the case\u000b= 1, Wang [35] proved the global well-posedness using only the second restriction in\n(2.43). It is an open problem to determinate if this condition is necessary in the case \u000b2(0;1).\nThe choice of the South Pole is of course arbitrary. By using the invariance of (LLG \u000b) under\nrotations, we have the existence of solutions provided that the essential range of the initial\ncondition m0is far from an arbitrary point Q2S2. Precisely,\nCorollary 2.11. Let\u000b2(0;1],Q2S2,\u000e2(0;2], and\"0;\u001a> 0such that (2.42)holds. Given\nm0= (m0\n1;m0\n2;m0\n3)2L1(RN;S2)satisfying\ninf\nRNjm0\u0000Qj2\u00152\u000eand [m0]BMO\u0014\"0;\nthere exists a unique smooth solution m2X(RN\u0002R+;S2)of(LLG\u000b)with initial condition m0\nsuch that\ninf\nx2RN\nt>0jm(x;t)\u0000Qj2\u00154\n1 +K2(\u001a+\u000e\u00001)2and [m]X\u00144K(\u001a+ 8\u000e\u00002\"0):(2.47)\nFor the sake of clarity, before proving Theorem 2.9, we provide a precise meaning of what we\nrefer to as a weak and smooth global solution of the (LLG \u000b) equation. The definition below is\nmotivated by the following vector identities for a smooth function mwithjmj= 1:\nm\u0002\u0001m= div( m\u0002rm);\n\u0000m\u0002(m\u0002\u0001m) = \u0001m+jrmj2m:\nDefinition 2.12. LetT2(0;1]andm02L1(RN;S2). We say that\nm2L1\nloc((0;T);H1\nloc(RN;S2))\nis a weak solution of (LLG\u000b)in(0;T)with initial condition m0if\n\u0000hm;@t'i=\fhm\u0002rm;r'i\u0000\u000bhrm;r'i+\u000bhjrmj2m;'i;\nand\nk(m(t)\u0000m0)'kL1!0;ast!0+;for all'2C1\n0(RN\u0002(0;T)): (2.48)\nIfT=1, and in addition m2C1(RN\u0002R+), we say that mis a smooth global solution of\n(LLG\u000b)inRN\u0002R+with initial condition m0. Hereh\u0001;\u0001istands for\nhf1;f2i=\u00021\n0\u0002\nRNf1\u0001f2dxdt:\nWith this definition, we see the following: Assume that mis a smooth global solution of\n(LLG\u000b) with initial condition m0and consider its stereographic projection P(m). IfP(m)and\nP(m0)are well-defined, then P(m)2C1(RN\u0002R+;C)satisfies (DNLS) pointwise, and\nlim\nt!0+P(m) =P(m0)inS0(RN):\nTherefore,ifinaddition P(m)2X(RN\u0002R;C),thenP(m)isasmoothglobalsolutionof (DNLS)\nwith initial condition P(m0). Reciprocally, suppose that u2X(RN\u0002R+;C)\\C1(RN\u0002R+)\nis a solution of (IDNLS) with initial condition u02L1(RN)such that (2.9) holds. If P\u00001(u)\nandP\u00001(u0)are in appropriate spaces, then P\u00001(u)is a global smooth solution of (LLG \u000b) with\ninitial conditionP\u00001(u0). The above (formal) argument allows us to obtain Theorem 2.9 from\nTheorem 2.1 once we have established good estimates for the mappings PandP\u00001. In this\ncontext, we have the following\n16Lemma 2.13. Letu;v2C1(RN;C),m= (m1;m2;m3);n= (n1;n2;n3)2C1(RN;S2).\na) Assume that inf\nRNm3\u0015\u00001+\u000eandinf\nRNn3\u0015\u00001+\u000efor some constant \u000e2(0;2]. Ifu=P(m)\nandv=P(n), then\nju(x)\u0000v(x)j\u00144\n\u000e2jm(x)\u0000n(x)j; (2.49)\n[u]BMO\u00148\n\u000e2[m]BMO; (2.50)\njru(x)j\u00144\n\u000e2jrm(x)j; (2.51)\nfor allx2RN.\nb) Assume thatkukL1\u0014M,kvkL1\u0014M, for some constant M\u00150. Ifm=P\u00001(u)and\nn=P\u00001(v), then\ninf\nRNm3\u0015\u00001 +2\n1 +M2; (2.52)\njm(x)\u0000n(x)j\u00143ju(x)\u0000v(x)j; (2.53)\njrm(x)j\u00144jru(x)j; (2.54)\njrm(x)\u0000rn(x)j\u00144jru(x)\u0000rv(x)j+ 12ju(x)\u0000v(x)j(jru(x)j+jrv(x)j):(2.55)\nProof.In the proof we will use the notation \u0014m:=m1+im2. To establish (2.49), we write\nu(x)\u0000v(x) =\u0014m(x)\u0000\u0014n(x)\n1 +m3(x)+\u0014n(x)(n3(x)\u0000m3(x))\n(1 +m3(x))(1 +n3(x)):\nHence, sincej\u0014nj\u00141,m3(x) + 1\u0015\u000eandn3(x) + 1\u0015\u000e,8x2RN,\nju(x)\u0000v(x)j\u0014j\u0014m(x)\u0000\u0014n(x)j\n\u000e+jn3(x)\u0000m3(x)j\n\u000e2:\nUsing that\nj\u0014m\u0000\u0014nj\u0014jm\u0000nj (2.56)\nand that\nmax\u001a1\na;1\na2\u001b\n\u00142\na2;for alla2(0;2];\nwe obtain (2.49). The same argument also shows that\nju(y)\u0000u(z)j\u00144\n\u000e2jm(y)\u0000m(z)j;for ally;z2RN: (2.57)\nTo verify (2.50), we recall the following inequalities in BMO (see [10]):\n[f]BMO\u0014sup\nx2RN \nBr(x) \nBr(x)jf(y)\u0000f(z)jdydz\u00142[f]BMO: (2.58)\nEstimate (2.50) is an immediate consequence of this inequality and (2.57). To prove (2.51) it is\nenough to remark that\njruj\u00142\n\u000e2(jrm1j+jrm2j+jrm3j)\u00144\n\u000e2jrmj:\n17We turn into (b). Using the explicit formula for P\u00001in (2.41), we can write\nm3=\u00001 +2\n1 +juj2:\nSincekukL1\u0014M, we obtain (2.52).\nTo show (2.53), we compute\n\u0014m\u0000\u0014n=2u\n1 +juj2\u00002v\n1 +jvj2=2(u\u0000v) + 2uv(\u0016v\u0000\u0016u)\n(1 +juj2)(1 +jvj2); (2.59)\nm3\u0000n3=1\u0000juj2\n1 +juj2\u00001\u0000jvj2\n1 +jvj2=2(jvj2\u0000juj2)\n(1 +juj2)(1 +jvj2): (2.60)\nUsing the inequalities\na\n1 +a2\u00141\n2;1 +ab\n(1 +a2)(1 +b2)\u00141;anda+b\n(1 +a2)(1 +b2)\u00141;for alla;b\u00150;(2.61)\nfrom (2.59) and (2.60) we deduce that\nj\u0014m\u0000\u0014nj\u00142ju\u0000vjandjm3\u0000n3j\u00142ju\u0000vj: (2.62)\nHence\njm\u0000nj=p\nj\u0014m\u0000\u0014nj2+jm3\u0000n3j2\u0014p\n8ju\u0000vj\u00143ju\u0000vj:\nTo estimate the gradient, we compute\nr\u0014m=2ru\n1 +juj2\u00004uRe(\u0016uru)\n(1 +juj2)2; (2.63)\nfrom which it follows that\njr\u0014mj\u0014jruj\u00122\n1 +juj2+4juj2\n(1 +juj2)2\u0013\n\u00143jruj;\nsince4a\n(1+a)2\u00141;for alla\u00150. Form3, we have\nrm3=\u00002 Re(\u0016uru)\n1 +juj2\u00002 Re(\u0016uru)(1\u0000juj2)\n(1 +juj2)2=\u00004 Re(\u0016uru)\n(1 +juj2)2;\nand thereforejrm3j\u00142jruj, since\na\n(1 +a2)2\u00141\n2;for alla\u00150: (2.64)\nHence\njrmj=p\njrm1j2+jrm2j2+jrm3j2\u0014p\n13jruj\u00144jruj;\nwhich gives (2.54).\nIn order to prove (2.55), we start differentiating (2.59)\nr\u0014m\u0000r\u0014n=2r(u\u0000v) +r(uv)(\u0016v\u0000\u0016u) +uvr(\u0016v\u0000\u0016u)\n(1 +juj2)(1 +jvj2)\n\u00004((u\u0000v) +uv(\u0016v\u0000\u0016u))(Re(\u0016uru)(1 +jvj2) + Re(\u0016vrv)(1 +juj2))\n(1 +juj2)2(1 +jvj2)2;\n18Hence, setting R= maxfjru(x)j;jrv(x)jg,\njr\u0014m\u0000r\u0014nj\u00142jru\u0000rvj\u00121 +jujjvj\n(1 +juj2)(1 +jvj2)\u0013\n+ 2Rju\u0000vj\u0012juj+jvj\n(1 +juj2)(1 +jvj2)\u0013\n+ 4Rju\u0000vj\u0012juj(1 +jujjvj)\n(1 +juj2)2(1 +jvj2)+jvj(1 +jujjvj)\n(1 +juj2)(1 +jvj2)2\u0013\n:\nUsing again (2.61), we get\njuj(1 +jujjvj)\n(1 +juj2)2(1 +jvj2)\u0014juj\n(1 +juj2)\u00141\n2:\nBy symmetry, the same estimate holds interchanging ubyv. Therefore, invoking again (2.61),\nwe obtain\njr\u0014m\u0000r\u0014nj\u00142jru\u0000rvj+ 6Rju\u0000vj: (2.65)\nSimilarly, writing juj2\u0000jvj2= (u\u0000v)\u0016u+ (\u0016u\u0000\u0016v)v, from (2.60) we have\njrm3\u0000rn3j\u00142jru\u0000rvj+ 6Rju\u0000vj: (2.66)\nTherefore, sincep\na2+b2\u0014a+b;8a;b\u00150;\ninequalities (2.65) and (2.66) yield (2.55).\nNow we have all the elements to establish Theorem 2.9.\nProof of Theorem 2.9. We continue to use the constants CandKdefined in Theorem 2.1. We\nrecall that they are given by C=C1C2\n2andK=C2, whereC1\u00151andC2\u00151are the constants\nin Lemmas 2.4 and 2.5, respectively. In addition, w.l.o.g. we assume that\nK=C2\u00154; (2.67)\nin order to simplify our computations.\nFirst we notice that by Remark 2.10, any \u001aand\"0fulfilling the condition (2.42), also satisfy\n8C(\u001a+ 8\u000e\u00002\"0)2\u0014\u001a; (2.68)\nsince\u000e4=K4\u00141(notice that K\u00154and\u000e2(0;2]).\nLetm0as in the statement of the theorem and set u0=P(m0). Using (2.50) in Lemma 2.13,\nwe have\nku0kL1\u0014\r\r\r1\n1 +m0\n3\r\r\r\nL1\u00141\n\u000eand [u0]BMO\u00148\"0\n\u000e2:\nTherefore, bearing in mind (2.68), we can apply Theorem 2.1 with\nL:=1\n\u000eand\":= 8\u000e\u00002\"0;\nto obtain a smooth solution u2X(RN\u0002R+;C)to (IDNLS) with initial condition u0. In\nparticularusatisfies\nsup\nt>0kukL1\u0014K(\u001a+\u000e\u00001)and [u]X\u0014K(\u001a+ 8\u000e\u00002\"0): (2.69)\n19Defining m=P\u00001(u), we infer that mis a smooth solution to (LLG \u000b) and, using the fact that\nk(u(\u0001;t)\u0000u0)'kL1!0(see (2.9)) and (2.53),\njm(\u0001;t)\u0000m0j\u0000! 0inS0(RN);ast!0+:\nNotice also that applying Lemma 2.13 we obtain\ninf\nx2RN\nt>0m3(x;t)\u0015\u00001 +2\n1 +K2(\u001a+\u000e\u00001)2and [m]X\u00144[u]X\u00144K(\u001a+ 8\u000e\u00002\"0);\nwhich yields (2.44).\nLet us now prove the uniqueness. Let nbe a another smooth solution of (LLG \u000b) with initial\nconditionu0satisfying\ninf\nx2RN\nt>0n3(x;t)\u0015\u00001 +2\n1 +K2(\u001a+\u000e\u00001)2and [n]X\u00144K(\u001a+ 8\u000e\u00002\"0); (2.70)\nand letv=P(n)be its stereographic projection. Then by (2.51),\n[v]X\u0014\u0010\n1 +K2(\u001a+\u000e\u00001)2\u00112\n[n]X: (2.71)\nWe continue to control the upper bounds for [v]Xand[u]Xin terms of \u000eand the constants\nC1\u00151andC2\u00154. Notice that since \u001aand\"0satisfy (2.42), from (2.46) with a= 8K4C, it\nfollows that\n\u001a\u0014\u000e4\n8K4Cand\"0\u0014\u000e6\n28K4C;\nor equivalently (recall that K=C2andC=C1C2\n2)\n\u001a\u0014\u000e4\n8C1C6\n2and8\"0\n\u000e2\u0014\u000e4\n32C1C6\n2: (2.72)\nHence\nK(\u001a+ 8\u000e\u00002\"0)\u00145\u000e4\n32C1C5\n2: (2.73)\nAlso, using (2.72), we have\n1 +K2(\u001a+\u000e\u00001)2=1 +C2\n2\n\u000e2(\u001a\u000e+ 1)2=C2\n2\n\u000e2\u0012\u000e2\nC2\n2+ (\u001a\u000e+ 1)2\u0013\n\u0014C2\n2\n\u000e2 \n\u000e2\nC2\n2+\u0012\u000e5\n8C1C6\n2+ 1\u00132!\n\u00142C2\n2\n\u000e2;(2.74)\nsinceC1\u00151,C2\u00154and\u000e\u00142.\nFrom the bounds in (2.73) and (2.74), combined with (2.69), (2.70) and (2.71), we obtain\n[u]X\u0014K(\u001a+ 8\u000e\u00002\"0)\u00145\u000e4\n32C1C5\n2\u00145\n211C1\nand\n[v]X\u0014(1+K2(\u001a+\u000e\u00001)2)2[n]X\u0014(1+K2(\u001a+\u000e\u00001)2)24K(\u001a+8\u000e\u00002\"0)\u0014\u0012\n2C2\n2\n\u000e2\u0013220\u000e4\n32C1C5\n2\u00145\n8C1;\n20since\u000e\u00142andC2\u00154. Finally, since uandvare solutions to (IDNLS) with initial condition\nu0, (2.28) and the above inequalities for [u]Xand[v]Xyield\nku\u0000vkX\u0014C1(2[u]2\nX+ [u]X+ [v]X)ku\u0000vkX\n\u0014C1 \n2\u00125\n211C1\u00132\n+5\n211C1+5\n8C1!\nku\u0000vkX;\nwhich implies that u=v, bearing in mind that the constant on the r.h.s. of the above inequality\nis strictly less that one. This completes the proof of the uniqueness.\nIt remains to establish (2.45). Let mandntwo smooth solutions of (LLG \u000b) satisfying (2.44).\nAs a consequence of the uniqueness, we see that mandnare the inverse stereographic projection\nof some functions uandvthat are solutions of (IDNLS) with initial condition u0=P(m0)and\nv0=P(n0), respectively. In particular, uandvsatisfy the estimates in (2.69). Using also (2.53)\nand (2.55), we deduce that\nkm\u0000nkX\u00143 sup\nt>0ku\u0000vkL1+ 4[u\u0000v]X+ 12 sup\nt>0ku\u0000vkL1([u]X+ [v]X])\n\u00144ku\u0000vkX+ 24C2(\u001a+ 8\u000e\u00002\"0)ku\u0000vkX;\n\u00145ku\u0000vkX;\nwhere we have used (2.73) in obtaining the last inequality. Finally, using also (2.43) and (2.49),\nand applying (2.10) in Theorem 2.1,\nkm\u0000nkX\u001430Kku0\u0000v0kL1\n\u0014120K\u000e\u00002km0\u0000n0kL1;\nwhich yields (2.45).\nProof of Corollary 2.11. LetR2SO(3)such thatRQ= (0;0;\u00001), i.e.Ris the rotation that\nmapsQto the South Pole. Let us set m0\nR=Rm0. Then\njm0\u0000Qj2=jR(m0\u0000Q)j2=jm0\nR\u0000(0;0;\u00001)j2= 2(1 +m0\n3;R):\nHence,\ninf\nx2RNm0\n3;R\u0015\u00001 +\u000e\nand\n[m0\nR]BMO = [m0]BMO\u0014\"0:\nTherefore, Theorem2.9providestheexistenceofauniquesmoothsolution mR2X(RN\u0002R+;S2)\nof (LLG\u000b) satisfying (2.44). Using the invariance of (LLG \u000b) and setting m=R\u00001mRwe obtain\nthe existence of the desired solution. To establish the uniqueness, it suffices to observe that if n\nis another smooth solution of (LLG \u000b) satisfying (2.47), then nR:=Rnis a solution of (LLG \u000b)\nwith initial condition m0\nRand it fulfills (2.44). Therefore, from the uniqueness of solution in\nTheorem 2.9, it follows that mR=nRand then m=n.\nProof of Theorem 1.1. In Theorem 2.9 and Corollary 2.11, the constants are given by C=C1C2\n2\nandK=C2. As discussed in Remark 2.10, the value\n\u001a\u0003=\u000e4\n32C1C2\n2\n21maximizes the range for \"0in (2.27) and this inequality is satisfied for any \"0>0such that\n\"0\u0014\u000e6\n256C1C2\n2:\nTaking\nM1=1\n256C1C2\n2; M 2=C2andM3=1\n32C1C2\n2;\nso that\u001a\u0003=M3\u000e4, the conclusion follows from Theorem 2.9 and Corollary 2.11.\nRemark2.14. Wefinallyremarkthatispossibletostatelocal(intime)versionsofTheorems2.1\nand 2.9 as it was done in [23, 22, 35]. In our context, the local well-posedness would concern\nsolutions with initial condition m02VMO (RN), i.e. such that\nlim\nr!0+sup\nx2RN \nBr(x)jm0(y)\u0000m0\nx;rjdy= 0: (2.75)\nMoreover, some uniqueness results have been established for solutions with this kind of initial\ndata by Miura [32] for the Navier–Stokes equation, and adapted by Lin [29] to (HFHM). It is\nalso possible to do this for (LLG \u000b), for\u000b > 0. We do not pursuit here these types of results\nbecause they do not apply to the self-similar solutions mc;\u000b. This is due to the facts that the\nfunction m0\nA\u0006does not belong to VMO (R)and that\nlim\nT!0+sup\n0<t<Tp\ntk@xmc;\u000bkL16= 0:\n3 Applications\n3.1 Existence of self-similar solutions in RN\nThe LLG equation is invariant under the scaling (x;t)!(\u0015x;\u00152t), for\u0015 > 0, that is if m\nsatisfies (LLG \u000b), then so does the function\nm\u0015(x;t) =m(\u0015x;\u00152t); \u0015> 0:\nTherefore is natural to study the existence of self-similar solutions (of expander type), i.e. a\nsolution msatisfying\nm(x;t) =m(\u0015x;\u00152t);8\u0015>0; (3.1)\nor, equivalently,\nm(x;t) =f\u0012xp\nt\u0013\n;\nfor some f:RN\u0000!S2profile of m. In particular we have the relation f(y) =m(y;1), for all\ny2RN. From (3.1) we see that, at least formally, a necessary condition for the existence of a\nself-similar solution is that initial condition m0be homogeneous of degree 0, i.e.\nm0(\u0015x) =m0(x);8\u0015>0:\nSince the norm in X(RN\u0002R+;R3)is invariant under this scaling, i.e.\nkm\u0015kX=kmkX;8\u0015>0;\nwhere m\u0015is defined by (3.1), Theorem 2.9 yields the following result concerning the existence\nof self-similar solutions.\n22Corollary 3.1. With the same notations and hypotheses as in Theorem 2.9, assume also that\nm0is homogeneous of degree zero. Then the solution mof(LLG\u000b)provided by Theorem 2.9 is\nself-similar. In particular there exists a smooth profile f:RN!S2such that\nm(x;t) =f\u0012xp\nt\u0013\n;\nfor allx2RNandt>0, andfsatisfies the equation\n\u00001\n2y\u0001rf(y) =\ff(y)\u0002\u0001f(y)\u0000\u000bf(y)\u0002(f(y)\u0002\u0001f(y));\nfor ally2RN. Herey\u0001rf(y) = (y\u0001rf1(y);:::;y\u0001rfN(y)).\nRemark 3.2. Analogously, Theorem 2.1 leads to the existence of self-similar solutions for\n(DNLS), provided that u0is a homogeneous function of degree zero.\nFor instance, in dimensions N\u00152, Corollary 3.1 applies to the initial condition\nm0(x) =H\u0012x\njxj\u0013\n;\nwithHa Lipschitz map from SN\u00001toS2\\f(x1;x2;x3) :x3\u0015\u00001=2g, provided that the Lipschitz\nconstant is small enough. Indeed, using (2.58), we have\n[m0]BMO\u00144kHkLip;\nso that taking\n\u000e= 1=2; \u001a =\u000e4\n32K4C; \" 0=\u000e6\n256K4CandkHkLip\u0014\"0;\nthe condition (2.42) is satisfied and we can invoke Corollary 3.1.\nOther authors have considered self-similar solutions for the harmonic map flow (i.e. (LLG \u000b)\nwith\u000b= 1) in different settings. Actually, equation (HFHM) can be generalized for maps\nm:M\u0002R+!N, withMandNRiemannian manifolds. Biernat and Bizoń [8] established\nresults whenM=N=Sdand3\u0014d\u00146:Also, Germain and Rupflin [15] have investigated the\ncaseM=RdandN=Sd, ind\u00153. In both works the analysis is done only for equivariant\nsolutions and does not cover the case M=RNandN=S2.\n3.2 The Cauchy problem for the one-dimensional LLG equation with a jump\ninitial data\nThis section is devoted to prove Theorems 1.2 and 1.3 in the introduction. These two results con-\ncern the question of well-posedness/ill-posedness of the Cauchy problem for the one-dimensional\nLLG equation associated with a step function initial condition of the form\nm0\nA\u0006:=A+\u001fR++A\u0000\u001fR\u0000; (3.2)\nwhere A+andA\u0000are two given unitary vectors in S2.\n233.2.1 Existence, uniqueness and stability. Proof of Theorem 1.2\nAs mentioned in the introduction, in [17] we proved the existence of the uniparametric smooth\nfamily of self-similar solutions fmc;\u000bgc>0of (LLG\u000b) for all\u000b2[0;1]with initial condition of the\ntype (3.2) given by\nm0\nc;\u000b:=A+\nc;\u000b\u001fR++A\u0000\nc;\u000b\u001fR\u0000; (3.3)\nwhere A\u0006\nc;\u000b2S2are given by Theorem A.5. For the convenience of the reader, we collect some\nof the results proved in [17] in the Appendix. The results in this section rely on a further\nunderstanding of the properties of the self-similar solutions mc;\u000b.\nIn Proposition 3.4 we show that\nmc;\u000b= (m1;c;\u000b;m2;c;\u000b;m3;c;\u000b)2X(R\u0002R+;S2);\nthatm3;c;\u000bis far from the South Pole and that [mc;\u000b]Xis small, if cis small enough. This\nwill yield that mc;\u000bcorresponds (up to a rotation) to the solution given by Corollary 3.1. More\nprecisely, using the invariance under rotations of (LLG \u000b), we can prove that, if the angle between\nA+andA\u0000is small enough, then the solution given by Corollary 3.1 with initial condition m0\nA\u0006\ncoincides (modulo a rotation) with mc;\u000b, for somec. We have the following:\nTheorem 3.3. Let\u000b2(0;1]. There exist L1;L2>0,\u000e\u00032(\u00001;0)and#\u0003>0such that the\nfollowing holds. Let A+,A\u00002S2and let#be the angle between them. If\n0<#\u0014#\u0003; (3.4)\nthen there exists a solution mof(LLG\u000b)with initial condition m0\nA\u0006. Moreover, there exists\n0< c <p\u000b\n2p\u0019, such that mcoincides up to a rotation with the self-similar solution mc;\u000b, i.e.\nthere existsR2SO(3), depending only on A+,A\u0000,\u000bandc, such that\nm=Rmc;\u000b; (3.5)\nandmis the unique solution satisfying\ninf\nx2R\nt>0m3(x;t)\u0015\u000e\u0003and [m]X\u0014L1+L2c: (3.6)\nIn order to prove Theorem 3.3, we need some preliminary estimates for mc;\u000bin terms of c\nand\u000b. To obtain them, we use some properties of the profile profile fc;\u000b= (f1;c;\u000b;f2;c;\u000b;f3;c;\u000b)\nconstructed in [17] using the Serret–Frenet equations with initial conditions\nf1;c;\u000b(0) = 1; f 2;c;\u000b(0) =f3;c;\u000b(0) = 0:\nAlso,\njf0\nj;c;\u000b(s)j\u0014ce\u0000\u000bs2=4;for alls2R;\nforj2f1;2;3gand\nmc;\u000b(x;t) =fc;\u000b\u0012xp\nt\u0013\n;for all (x;t)2R\u0002R+: (3.7)\nHence, for any x2R,\njf3;c\u000b(x)j=jf3;c\u000b(x)\u0000f3;c\u000b(0)j\u0014\u0002jxj\n0ce\u0000\u000b\u001b2=4d\u001b\u0014cp\u0019p\u000b:\n24Since the same estimate holds for f2;c;\u000b, we conclude that\njm2;c;\u000b(x;t)j\u0014cp\u0019p\u000b;andjm3;c;\u000b(x;t)j\u0014cp\u0019p\u000bfor all (x;t)2R\u0002R+:(3.8)\nMoreover, since\nA\u0006\nc;\u000b= lim\nx!\u00061fc;\u000b(x);\nwe also get\njA\u0006\nj;c;\u000bj\u0014cp\u0019p\u000b;forj2f2;3g: (3.9)\nWe now provide some further properties of the self-similar solutions.\nProposition 3.4. For\u000b2(0;1]andc>0, we have\nkm0\n2;c;\u000bkL1\u0014cp\u0019p\u000b;km0\n3;c;\u000bkL1\u0014cp\u0019p\u000b;sup\nt>0km3;c;\u000bkL1\u0014cp\u0019p\u000b;(3.10)\n[m0\nc;\u000b]BMO\u00142cp\n2\u0019p\u000b; (3.11)\np\ntk@xmc;\u000bk1=c;for allt>0; (3.12)\nsup\nx2R\nr>01\nr\u0002\nQr(x)j@ymc;\u000b(y;t)j2dtdy\u00142p\n2\u0019c2\np\u000b: (3.13)\nIn particular, mc;\u000b2X(R\u0002R+;S2)and\n[mc;\u000b]X\u00144c\n\u000b1\n4: (3.14)\nProof of Proposition 3.4. The estimates in (3.10) follow from (3.8) and (3.9). To prove (3.11),\nwe use (2.58), (3.3), (3.10) and the fact that\nA\u0000\nc;\u000b= (A+\n1;c;\u000b;\u0000A+\n2;c;\u000b;\u0000A+\n3;c;\u000b); (3.15)\n(see Theorem A.5) to get\n[m0\nc;\u000b]BMO\u0014sup\nx2RN \nBr(x) \nBr(x)jm0\nc;\u000b(y)\u0000m0\nc;\u000b(z)jdydz\n\u00142q\n(A+\n2;c;\u000b)2+ (A+\n3;c;\u000b)2sup\nx2RN \nBr(x) \nBr(x)dydz\n\u00142cp\n2\u0019p\u000b:\nFrom (A.12) we obtain the equality in (3.12) and also\nIr;x:=1\nr\u0002\nQr(x)j@ymc;\u000b(y;t)j2dtdy =c2\nr\u0002x+r\nx\u0000r\u0002r2\n0e\u0000\u000by2\n2t\ntdtdy: (3.16)\nPerforming the change of variables z= (\u000by2)=(2t), we see that\n\u0002r2\n0e\u0000\u000by2\n2t\ntdt=E1\u0012\u000by2\n2r2\u0013\n; (3.17)\n25whereE1is the exponential integral function\nE1(y) =\u00021\nye\u0000z\nzdz:\nThis function satisfies that limy!0+E1(y) =1andlimy!1E1(y) = 0(see e.g. [1, Chapter 5]).\nMoreover, taking \u000f>0and integrating by parts,\n\u00021\n\u000fE1(y2)dy=yE1(y2)\f\f1\n\u000f+ 2\u00021\n\u000fe\u0000y2dy; (3.18)\nso L’Hôpital’s rule shows that the first term in the r.h.s. of (3.18) vanishes as \u000f!0+. Therefore,\nthe Lebesgue’s monotone convergence theorem allows to conclude that E1(y2)2L1(R+)and\n\u00021\n0E1(y2) =p\u0019: (3.19)\nBy using (3.16), (3.17), (3.19), and making the change of variables z=p\u000by=(rp\n2), we obtain\nIr;x=c2\nr\u0002x+r\nx\u0000rE1\u0012\u000by2\n2r2\u0013\ndy=p\n2c2\np\u000b\u0002p\u000bp\n2(x\nr+1)\np\u000bp\n2(x\nr\u00001)E1(z2)dz\u0014p\n2c2\np\u000b\u00012p\u0019; (3.20)\nwhich leads to (3.13). Finally, the bound in (3.14) easily follows from those in (3.12) and (3.13)\nand the elementary inequality\n\u0010\n1 +\u00102p\n2\u0019p\u000b\u00111=2\u0011\n\u00141\n\u000b1\n4\u0000\n1 + (2p\n2\u0019)1=2\u0001\n\u00144\n\u000b1\n4; \u000b2(0;1]:\nProof of Theorem 3.3. First, we consider the case when A+=A+\nc;\u000bandA\u0000=A\u0000\nc;\u000b(i.e. when\nm0\nA\u0006=m0\nc;\u000b) for some c >0. We will continue to show that the solution provided by Theo-\nrem 2.9 is exactly mc;\u000b, forcsmall. Indeed, bearing in mind the estimates in Proposition 3.4,\nwe consider\nc\u0014p\u000b\n2p\u0019;\nso that\ninf\nx2Rm0\n3;c;\u000b(x)\u0015\u00001\n2: (3.21)\nIn view of (3.11), (3.21) and Remark 2.10, we set\n\"0:= 4cp\u0019p\u000b; \u000e :=1\n2; \u001a :=\u000e4\n8K4C=1\n27K4C; (3.22)\nwhereC;K\u00151are the constants given by Theorem 2.9. In this manner, from (3.11), (3.21) and\n(3.22), we have\ninf\nRm0\n3\u0015\u00001 +\u000eand [m0]BMO\u0014\"0;\nand the condition (2.42) is fulfilled if\n\"0\u0014\u000e6\n256K4C;\nor equivalently, if c\u0014~c, with\n~c:=p\u000b\n216K4Cp\u0019:\n26Observe that in particular ~c<p\u000b\n2p\u0019.\nFor fixed 0<c< ~c, we can apply Theorem 2.9 to deduce the existence and uniqueness of a\nsolution mof (LLG\u000b) satisfying\ninf\nx2R\nt>0m3(x;t)\u0015\u00001 +2\n1 +K2(\u001a+ 2)2and [m]X\u00144K\u001a+29Kcp\u0019p\u000b:(3.23)\nNow by Proposition 3.4, for fixed 0<c\u0014~c, we have the following estimates for mc;\u000b\n[mc;\u000b]X\u00144c=\u000b1\n4and inf\nx2R\nt>0m3;c;\u000b(x;t)\u0015\u00001\n2;\nso in particular mc;\u000bsatisfies (3.23). Thus the uniqueness of solution implies that m=mc;\u000b,\nprovided that c\u0014~c. Defining the constants L1,L2and\u000e\u0003by\nL1= 4K\u001a; L 2=29Kp\u0019p\u000band\u000e\u0003=\u00001 +2\n1 +K2(\u001a+ 2)2;(3.24)\nthe theorem is proved in the case A\u0006=A\u0006\nc;\u000b.\nFor the general case, we would like to understand which angles can be reached by varying the\nparametercin the range (0;~c]. To this end, for fixed 0< c\u0014~c, let#c;\u000bbe the angle between\nA+\nc;\u000bandA\u0000\nc;\u000b. From Lemma A.6,\n#c;\u000b\u0015arccos\u0012\n1\u0000c2\u0019+ 32c3p\u0019\n\u000b2\u0013\n;for allc2\u0010\n0;\u000b2p\u0019\n32i\n:\nNow, it is easy to see that the function F(c) = arccos\u0010\n1\u0000c2\u0019+ 32c3p\u0019\n\u000b2\u0011\nis strictly increasing\non the interval [0;\u000b2p\u0019\n48]so that\nF(c)>F(0) = 0;for allc2\u0010\n0;\u000b2p\u0019\n48i\n: (3.25)\nLetc\u0003= min(~c;\u000b2p\u0019\n48)and consider the map T\u000b:c\u0000!#c;\u000bon[0;c\u0003]. By Lemma A.6, T\u000bis\ncontinuous on [0;c\u0003],T\u000b(0) = lim c!0+T\u000b(c) = 0and, bearing in mind (3.25), T(c\u0003) =#c\u0003;\u000b>0.\nThus, from the intermediate value theorem we infer that for any #2(0;#c\u0003;\u000b), there exists\nc2(0;c\u0003)such that\n#=T\u000b(c) =#c;\u000b:\nWe can now complete the proof for any A+,A\u00002S2. Let#be the angle between A+and\nA\u0000. From the previous lines, we know that there exists #\u0003:=#c\u0003;\u000bsuch that if #2(0;#\u0003),\nthere exists c2(0;c\u0003)such that#=#c;\u000b. For this value of c, consider the initial value problem\nassociated with m0\nc;\u000band the constants defined in (3.24). We have already seen the existence\nof a unique solution mc;\u000bof the LLG equation associated with this initial condition satisfying\n(3.6). LetR2SO(3)be the rotation on R3such that A+=RA+\nc;\u000bandA\u0000=RA\u0000\nc;\u000b. Then\nm:=Rmc;\u000bsolves (LLG \u000b) with initial condition m0\nA\u0006. Finally, recalling the above definition\nofL1,L2and\u000e\u0003, using the invariance of the norms under rotations and the fact that mc;\u000bis the\nunique solution satisfying (3.23), it follows that mis the unique solution satisfying the conditions\nin the statement of the theorem.\nWe are now in position to give the proof of Theorem 1.2, the second of our main results\nin this paper. In fact, we will see that Theorem 1.2 easily follows from Theorem 3.3 and the\nwell-posedness for the LLG equation stated in Theorem 2.9.\n27Proof of Theorem 1.2. Let#\u0003,\u000e\u0003,L1andL2betheconstantsdefinedintheproofofTheorem3.3.\nGiven A+andA\u0000such that 0<#<#\u0003, Theorem 3.3 asserts the existence of\n0<c<p\u000b\n2p\u0019(3.26)\nandR2SO(3)such thatRmc;\u000bis the unique solution of (LLG \u000b) with initial condition m0\nA\u0006\nsatisfying (3.6), and in particular m0\nA\u0006=Rm0\nc;\u000b. By hypothesis m0satisfies\nkm0\u0000m0\nA\u0006kL1\u0014cp\u0019\n2p\u000b: (3.27)\nHence, defining m0\nR=R\u00001m0, recalling that [f]BMO\u00142kfkL1and using the invariance of the\nnorms under rotations, we deduce from (3.27) that\nkm0\nRkL1\u0014km0\nc;\u000bkL1+cp\u0019\n2p\u000band [m0\nR]BMO\u0014[m0\nc;\u000b]BMO +cp\u0019p\u000b:\nThen, by Proposition 3.4,\nkm0\n3;RkL1\u00142cp\u0019p\u000band [m0\nR]BMO\u00144cp\u0019p\u000b: (3.28)\nFrom (3.26) and (3.28), it follows that\nm0\n3;R(x)\u0015\u00001=2;for allx2R:\nTherefore, as in the proof of Theorem 3.3, we can apply Theorem 2.9 with the values of \"0,\u000e\nand\u001agiven in (3.22) to deduce the existence of a unique (smooth) solution mRof (LLG\u000b) with\ninitial condition m0\nRsatisfying\ninf\nx2R\nt>0m3;R(x;t)\u0015\u00001 +2\n1 +K2(\u001a+ 2)2=\u000e\u0003and [mR]X\u00144K\u001a+29Kcp\u0019p\u000b=L1+L2c:\nSince we have taken the values for \"0,\u000eand\u001aas in the proof Theorem 3.3, Theorem 2.9 also\nimplies that\nkmR\u0000mc;\u000bkX\u0014480Kkm0\nR\u0000m0\nc;\u000bkL1:\nThe conclusion of the theorem follows defining m=RmRandL3= 480K, and using once\nagain the invariance of the norm under rotations.\n3.2.2 Multiplicity of solutions. Proof of Theorem 1.3\nAs proved in [17], when \u000b= 1, the self-similar solutions are explicitly given by\nmc;1(x;t) = (cos(cErf(x=p\nt));sin(cErf(x=p\nt));0);for all (x;t)2R\u0002R+;(3.29)\nfor everyc>0, where Erf(\u0001)is the non-normalized error function\nErf(s) =\u0002s\n0e\u0000\u001b2=4d\u001b:\nIn particular,\n~A\u0006\nc;1= (cos(cp\u0019);\u0006sin(cp\u0019);0)\n28#c;1\n\u0019\nc\nFigure 2: The angle #c;\u000bas a function of cfor\u000b= 1.\nand the angle between A+\nc;1andA\u0000\nc;1is given by\n#c;1= arccos(cos(2 cp\u0019)): (3.30)\nFormula (3.30) and Figure 2 show that there are infinite values of cthat allow to reach any\nangle in [0;\u0019]. Therefore, using the invariance of (LLG \u000b) under rotations, in the case when\n\u000b= 1, one can easily prove the existence of multiple solutions associated with a given initial\ndata of the form m0\nA\u0006for any given vectors A\u00062S2(see argument in the proof included below).\nIn the case that \u000bis close enough to 1, we can use a continuity argument to prove that we still\nhave multiple solutions. More precisely, Theorem 1.3 asserts that for any given initial data of\nthe form m0\nA\u0006with angle between A+andA\u0000in the interval (0;\u0019), if\u000bis sufficiently close\nto one, then there exist at leastk-distinct solutions of (LLG \u000b) associated with the same initial\ncondition, for any given k2N.\nThe rest of this section is devoted to the proof of Theorem 1.3.\nProof of Theorem 1.3. Letk2N,A\u00062S2and#2(0;\u0019)be the angle between A+andA\u0000.\nUsing the invariance of (LLG \u000b) under rotations, it suffices to prove the existence of \u000bk2(0;1)\nsuch that for every \u000b2[\u000bk;1]there exist 0< c1<\u0001\u0001\u0001< cksuch that the angle #cj;\u000bbetween\nA+\ncj;\u000bandA\u0000\ncj;\u000b, satisfies\n#cj;\u000b=#;for allj2f1;:::;kg: (3.31)\nIn what follows, and since we want to show the existence of at least k-distinct solutions, we will\nassume without loss of generality that kis large enough.\nFirst observe that, since A\u0000\nc;\u000b= (A+\n1;c;\u000b;\u0000A+\n2;c;\u000b;\u0000A+\n3;c;\u000b), we have the explicit formula\ncos(#c;\u000b) = 2(A+\n1;c;\u000b)2\u00001;\nand using Lemma A.8 in the Appendix, we get\njcos(#c;\u000b)\u0000cos(#c;1)j=j2((A+\n1;c;\u000b)2\u0000(A+\n1;c;1)2)j\u00144jA+\n1;c;\u000b\u0000A+\n1;c;1j\u00144h(c)p\n1\u0000\u000b;(3.32)\nfor all\u000b2[1=2;1], withh:R+\u0000!R+an increasing function satisfying lims!1h(s) =1.\nForj2N, we setaj= (2j+ 1)p\u0019=2andbj= (2j+ 2)p\u0019=2, so that (3.30) and (3.32) yield\ncos(#aj;\u000b)\u0014\u00001 + 4h(aj)p\n1\u0000\u000band cos(#bj;\u000b)\u00151\u00004h(bj)p\n1\u0000\u000b;8\u000b2[1=2;1]:\n(3.33)\n29Definel= cos(#)and\n\u000bk= max\u0012\n1\u0000\u00101\u0000l\n8h(bk)\u00112\n;1\u0000\u00101 +l\n8h(bk)\u00112\u0013\n:\nNotice that, since #2(0;\u0019), we have\u00001< l < 1and thus\u000bk<1. Also, since hdiverges to\n1, we can assume without loss of generality that \u000bk2[1=2;1), and from the definition of \u000bkwe\nhave\n0<p1\u0000\u000bk<min\u00121\u0000l\n8h(bk);1 +l\n8h(bk)\u0013\n:\nTherefore, from (3.33) and h(aj)< h(bj)\u0014h(bk)(sincehis a strictly increasing function), we\nget\ncos(#aj;\u000b)\u0014\u00001 +l\n2and1 +l\n2\u0014cos(#bj;\u000b);8j2f1;:::;kg;8\u000b2[\u000bk;1];\nand thus\ncos(#aj;\u000b)<l< cos(#bj;\u000b);8j2f1;:::;kg;8\u000b2[\u000bk;1]; (3.34)\nsincel2(\u00001;1).\nLet us fix\u000b2[\u000bk;1]andj2f1;:::;kg. By Lemma A.8, c!cos(#c;\u000b)is a continuous\nfunction on c. Therefore (3.34) and the intermediate value theorem yield the existence of cj2\n[aj;bj]such that\ncos(#cj;\u000b) =l= cos(#);\nor equivalently, such that #cj;\u000b=#.\nFinally, for each j2f1;:::;kg, letRj2SO(3)be such that m0\nA\u0006=Rjm0\ncj;\u000b, and define mj\nbymj=Rjmcj;\u000b. Then mjsolves (LLG \u000b) with initial data m0\nA\u0006and the control of @xmjin\n(1.9) follows from the definition of mjin terms of mcj;\u000band the analogous property established\nin Theorem A.5 for the self-similar solution mcj;\u000b(see (A.12)).\nIn the case when \u000b= 1and#2[0;\u0019], formula (3.30) shows that sequence fcjgj\u00151in (1.10)\nsatisfies#cj;1=#for allj2N\u0003. The result follows by considering the sequence of solutions\nfmjgj\u00151described above.\nRemark 3.5. Notice that the proof given above also shows how close \u000bkneeds to be to 1 in\nterms of the (fixed) angle #2(0;\u0019), and in particular \u000bk!1ask!1, and\u000bk!1as#!0\n(i.e. whenl!1).\nRemark 3.6. For\u000b= 1andc>0, the function\nu(x;t) =P(mc;1) = exp\u0000\nicErf(x=p\nt)\u0001\nis a solution of (DNLS) with initial condition\nu0=eicp\u0019\u001fR++e\u0000icp\u0019\u001fR\u0000:\nTherefore there is also a multiplicity phenomenon for the equation (DNLS).\n3.3 A singular solution for a nonlocal Schrödinger equation\nWe have used the stereographic projection to establish a well-posedness result for (LLG \u000b).\nMelcher [31] showed a global well-posedness result, provided that\nkrm0kLN\u0014\";m0\u0000Q2H1(RN)\\W1;N(RN); \u000b> 0; N\u00153;\n30for some Q2S2and\">0small. Later, Lin, Lan and Wang [30] improved this result and proved\nglobal well-posedness under the conditions\nkrm0kM2;2\u0014\";m0\u0000Q2L2(RN); \u000b> 0; N\u00152;\nfor some Q2S2and\">0small.6In the context of Theorem 1.1 and using the characterization\nofBMO\u00001in Theorem A.1, the second condition in (1.3) says that krm0kBMO\u00001is small. In\nview of the embeddings\nLN(RN)\u001aM2;2(RN)\u001aBMO\u00001(RN);\nforN\u00152, we deduce that Theorem 1.1 includes initial conditions with less regularity, as long\nas their essential range is not S2. The argument in [30, 31] is based on the method of moving\nframes that produces a covariant complex Ginzburg–Landau equation. One of the aims of this\nsubsection is to compare their approach in the context of the self-similar solutions mc;\u000b, and in\nparticular to draw attention to a possible difficulty in using it to study these solutions.\nIn the sequel we consider the one-dimensional case N= 1and\u000b2[0;1]. Then the moving\nframes technique can be recast as a Hasimoto transformation as follows. Assume that mis the\ntangent vector of a curve in R3, i.e.m=@xX, for some curve X(x;t)2R3parametrized by the\narc-length. It can be shown (see [12]) that if mevolves under (LLG \u000b), then the torsion \u001cand\nthe curvature cofXsatisfy\n@t\u001c=\f\u0012\nc@xc +@x\u0010@xxc\u0000c\u001c2\nc\u0011\u0013\n+\u000b\u0012\nc2\u001c+@x\u0010@x(c\u001c) +\u001c@xc\nc\u0011\u0013\n;\n@tc =\f(\u0000@x(c\u001c)\u0000\u001c@xc) +\u000b\u0000\n@xc\u0000c\u001c2\u0001\n:\nHence, defining the Hasimoto transformation [19] (also called filament function)\nv(x;t) = c(x;t)ei\u0001x\n0\u001c(\u001b;t)d\u001b; (3.35)\nwe verify that vsolves the following dissipative Schrödinger (or complex Ginzburg–Landau)\nequation\ni@tv+ (\f\u0000i\u000b)@xxv+v\n2\u0012\n\fjvj2+ 2\u000b\u0002x\n0Im(\u0016v@xv)\u0000A(t)\u0013\n= 0; (3.36)\nwhere\f=p\n1\u0000\u000b2and\nA(t) =\u0012\n\f\u0012\nc2+2(@xxc\u0000c\u001c2)\nc\u0013\n+ 2\u000b\u0012@x(c\u001c) +\u001c@xc\nc\u0013\u0013\n(0;t):\nThe curvature and torsion associated with the self-similar solutions mc;\u000bare (see [17]):\ncc;\u000b(x;t) =cp\nte\u0000\u000bx2\n4tand\u001cc;\u000b(x;t) =\fx\n2p\nt: (3.37)\nTherefore in this case\nA(t) =\fc2\nt(3.38)\n6We recall that v2M2;2(RN)ifv2L2\nloc(RN)and\nkvkM2;2:= sup\nx2RN\nr>01\nr(N\u00002)=2kvkL2(Br(x))<1:\n31and the Hasimoto transformation of mc;\u000bis\nvc;\u000b(x;t) =cp\nte(\u0000\u000b+i\f)x2\n4t:\nIn particular vc;\u000bis a solution of (3.36) with A(t)as in (3.38), for all \u000b2[0;1]andc > 0.\nMoreover, the Fourier transform of this function (w.r.t. the space variable) is\nbvc;\u000b(\u0018;t) = 2cp\n\u0019(\u000b+i\f)e\u0000(\u000b+i\f)\u00182t;\nso thatvc;\u000bis a solution of (3.36) with a Dirac delta as initial condition:\nvc;\u000b(\u0001;0) = 2cp\n\u0019(\u000b+i\f)\u000e:\nHere\u000edenotes the delta distribution at the point x= 0andpzdenotes the square root of a\ncomplex number zsuch that Im(pz)>0.\nIn the limit cases \u000b= 0and\u000b= 1, the first three terms in equation (3.36) lead to a cubic\nSchrödinger equation and to a linear heat equation, respectively. The Cauchy problem with\na Dirac delta for these kind of equations associated with a power type non-linearity has been\nstudied by several authors (see e.g. [4] and the reference therein). We recall two classical results.\nTheorem 3.7 ([9]).Letp\u00152andu2Lp\nloc(R\u0002R+)be a solution in the sense of distributions\nof\n@tu\u0000@xxu+jujpu= 0onR\u0002R+: (3.39)\nAssume that\nlim\nt!0+\u0002\nRu(x;t)'(x)dx= 0;for all'2C0(Rnf0g); (3.40)\nwhereC0(Rnf0g)denotes the space of continuous functions with compact support in Rnf0g.\nThenu2C2;1(R\u0002[0;1))andu(x;0) = 0for allx2R. In particular there is no solution of\n(3.39)such that\nlim\nt!0+\u0002\nRu(x;t)'(x)dx='(0);for all'2C0(RN):\nIn[9]itisalsoprovedthatif 1<p< 2, equation(3.39)hasaglobalsolutionwithaDiracdelta\nas initial condition, as in the case of the linear parabolic equation. Concerning the Schrödinger\nequation, we have the following ill-posedness result due to Kenig, Ponce and Vega [21].\nTheorem 3.8 ([21]).Letp\u00152. Either there is no solution in the sense of distributions of\ni@tu+@xxu+jujpu= 0onR\u0002R+; (3.41)\nwith\nlim\nt!0+u(\u0001;t) =\u000einS0(R);\nin the class u;jujpu2L1(R+;S0(R)), or there is more than one.\nAfter performing an appropriate change of variables, equation (3.36) leads to the following\nequation\ni@tu+ (\f\u0000i\u000b)@xxu+u\n2\u0012\n\fjuj2+ 2\u000b\u0002x\n0Im(\u0016u@yu)dy\u0013\n= 0: (3.42)\nSince\u000b2[0;1], the above equation can be seen as an intermediate model between (3.39) and\n(3.41). Therefore one could expect that when \u000b2(0;1], the solutions of (3.42) share similar\nproperties to those established in Theorem 3.7 for the equation (3.39). We will continue to\nobserve that this is not necessarily the case. To this end, we need the following:\n32Proposition 3.9. For all\u000b2[0;1]and for all c2Cnf0gthe function wc;\u000b:R\u0002R+!Cgiven\nby\nwc;\u000b(x;t) =cp\ntexp\u0012i\fjcj2\n2ln(t) + (i\f\u0000\u000b)x2\n4t\u0013\nis a solution of (3.42). In addition,\ni) If\u000b2[0;1), thenwc;\u000b(\u0001;t)does not converge in S0(R)ast!0+.\nii) If\u000b2(0;1], then\nlim\nt!0+\u0002\nRwc;\u000b(x;t)'(x)dx= 0;for all'2C0(Rnf0g): (3.43)\nProof.A straightforward computation shows that wc;\u000bsatisfies (3.42). The proof that wc;\u000b(\u0001;t)\ndoes not converge in S0(R)ast!0+ifc6= 0is the same as in [21]. Indeed, for '2S(R), by\nParseval’s theorem,\u0002\nR\u0016wc;\u000b(x;t)'(x)dx=1\n2\u0019\u0002\nRbwc;\u000b(\u0018;t)b'(\u0018)d\u0018\n=\u0016ce\u0000i\fjcj2ln(t)=2\np\u0019p\n\u000b+i\f\u0002\nRe\u0000(\u000b+i\f)\u00182tb'(\u0018)d\u0018:\nBy the dominated convergence theorem, the last integral converges:\nlim\nt!0+\u0002\nRe\u0000(\u000b+i\f)\u00182tb'(\u0018)d\u0018=\u0002\nRb'(\u0018)d\u0018= 2\u0019'(0):\nSince\f6= 0,e\u0000i\fjcj2ln(t)=2does not admit a limit at 0inS0(R). We conclude that wc;\u000b(\u0001;t)does\nnot converge in S0(R)ast!0+.\nIt remains to prove (3.43). Since now '2C0(Rnf0g), we cannot proceed as before. However,\nusing the change of variables x=p\nty, we have\nlim\nt!0+\u0002\nRwc;\u000b(x;t)'(x)dx=cei\fjcj2ln(t)=2\u0002\nRe(\u0000\u000b+i\f)y2=4'(p\nty)dy:\nTherefore, since \u000b>0and'(0) = 0, the dominated convergence theorem implies that\nlim\nt!0+\u0002\nRe(\u0000\u000b+i\f)y2=4'(p\nty)dy='(0)\u0002\nRe(\u0000\u000b+i\f)y2=4dy= 0:\nSincejei\fjcj2ln(t)=2j= 1, we obtain (3.43).\nThe results in Proposition 3.9 lead to the following remarks:\n1. Observe that if \u000b2(0;1),wc;\u000bprovides a solution to the dissipative equation (3.42).\nMoreover, form part (ii)in Proposition 3.9, wc;\u000bsatisfies the condition (3.40). However,\nnotice that wc;\u000bcannot be extended to C2;1(R\u0002[0;1))due to the presence of a logarithmic\noscillation. This is in contrast with the properties for solutions of the cubic heat equation\n(3.39) established in Theorem 3.7.\n2. In the case \u000b= 0, equation (3.42) corresponds to (3.41) with p= 2, i.e. to the equation\ncubic NLS equation that is invariant under the Galilean transformation. The proof of the\nill-posedness result given in Theorem 3.8 relies on this invariance and part (i)of Proposi-\ntion 3.9 with \u000b= 0. Although when \u000b > 0, equation (3.42) is no longer invariant under\nthe Galilean transformation, part (i)of Proposition 3.9 could be an indicator that that the\nCauchy problem (3.42) with a delta as initial condition is still ill-posed. This question rests\nopen for the moment and it seems that the use of (3.36) (or (3.42)) can be more difficult\nto formulate a Cauchy theory for (LLG \u000b) including self-similar solutions.\n334 Appendix\nThe characterization of BMO\u00001\n1(RN)as sum of derivatives of functions in BMO was proved by\nKoch and Tataru in [23]. A straightforward generalization of their proof leads to the following\ncharacterization of BMO\u00001\n\u000b(RN).\nTheorem A.1. Let\u000b2(0;1]andf2S0(RN). Thenf2BMO\u00001\n\u000b(RN)if and only if there exist\nf1;:::;fN2BMO\u000b(RN)such thatf=PN\nj=1@jfj. In addition, if such a decomposing holds,\nthen\nkfkBMO\u00001\n\u000b.NX\nj=1[fj]BMO\u000b:\nThe next results provide the equivalence between the weak solutions and the Duhamel for-\nmulation. We first need to introduce for T >0the spaceL1\nuloc(RN\u0002(0;T))defined as the space\nof measurable functions on RN\u0002(0;T)such that the norm\nkfkuloc;T:= sup\nx02RN\u0002\nB(x0;1)\u0002T\n0jf(y;t)jdtdy\nis finite. We refer the reader to Lemarié–Rieusset’s book [27] for more details about these kinds\nof spaces. In particular, we recall the following result corresponding to Lemma 11.3 in [27] in\nthe case\u000b= 1. It is straightforward to check that the same proof still applies if \u000b2(0;1).\nLemma A.2. Let\u000b2(0;1],T2(0;1)andw2L1\nuloc(RN\u0002(0;T)). Then the function\nW(x;t) :=\u0002t\n0S\u000b(t\u0000s)w(x;s)ds\nis well defined and belongs to L1\nuloc(RN\u0002(0;T)). Moreover,\ni@tW+ (\f\u0000i\u000b)\u0001W=winD0(RN\u0002R+);\nand the application\n[0;T]!R\nt7! kW(\u0001;t)kL1(B1(x0))\nis continuous for any x02R, withkW(\u0001;t)kL1(B1(x0))!0, ast!0+, uniformly in x0.\nFollowing the ideas in [27], we can establish now the equivalence between the notions of\nsolutions as well as the regularity.\nTheorem A.3. Let\u000b2(0;1]andu2X(RN\u0002R+;C). Then the following assertions are\nequivalent:\ni) The function usatisfies\niut+ (\f\u0000i\u000b)\u0001u= 2(\f\u0000i\u000b)\u0016u(ru)2\n1 +juj2inD0(RN\u0002R+): (A.1)\nii) There exists u02S0(RN)such thatusatisfies\nu(t) =S\u000b(t)u0\u00002(\f\u0000i\u000b)\u0002t\n0S\u000b(t\u0000s)\u0016u(ru)2\n1 +juj2ds:\n34Moreover, if (ii) holds, then u2C1(RN\u0002R+)and\nk(u(t)\u0000u0)'kL1(RN)!0;ast!0+; (A.2)\nfor any'2S(RN).\nProof.In view of Lemma A.2, we need to prove that the function\ng(u) =\u00002(\f\u0000i\u000b)\u0016u(ru)2\n1 +juj2\nbelongs toL1\nuloc(RN\u0002(0;T)), for allT >0. Indeed, by (2.3) we have\nkg(u)kuloc;T\u0014kjruj2kuloc;T: (A.3)\nIfT\u00141, then\nkjruj2kuloc;T\u0014sup\nx02RN\u0002\nQ1(x0)jru(y;t)j2dtdy\u0014kuk2\nX: (A.4)\nIfT\u00151, using that\njruj\u0014[u]Xp\nt;for anyt>0;\nwe get\nkjruj2kT;uloc\u0014sup\nx02RN\u0002\nQ1(x0)jru(y;t)j2dtdy + sup\nx02RN\u0002T\n1\u0002\nB1(x0)jruj2dydt\n\u0014kuk2\nX+ [u]2\nXjB1(0)j\u0002T\n11\ntdt\n\u0014kuk2\nX(1 +jB1(0)jln(T)):(A.5)\nIn conclusion, we deduce from (A.3), (A.4) and (A.5) that g(u)2L1\nuloc(RN\u0002(0;T))and then\nit follows from Lemma A.2 that (ii) implies (i). The other implication can be established as in\n[27, Theorem 11.2]. Moreover, we deduce that the function\nW(x;t) :=T(g(u))(x;t) =\u0002t\n0S\u000b(t\u0000s)g(u)ds\nsatisfieskW(\u0001;t)kL1(B1(x0))!0, ast!0+, uniformly in x02RN. Let us take '2S(RN)and\na constantC'>0such thatj'(x)j\u0014C'(2 +jxj)\u0000N\u00001. Then\n\u0002\nRNj'(y)W(y;t)jdy\u0014X\nk2ZN\u0002\nB1(k)C'\n(2 +jxj)N+1jW(y;t)jdy\n\u0014sup\nx02RNkW(\u0001;t)kL1(B1(x0))X\nk2ZNC'\n(1 +jkj)N+1;\nso thatk'W(\u0001;t)kL1(RN)!0ast!0+, i.e.\nk(u(t)\u0000S\u000b(t)u0)'kL1(RN)!0;ast!0+: (A.6)\nOn the other hand, since u02L1(RN),\nkS\u000b(t)u0\u0000u0kL1(Br(0))!0;ast!0+; (A.7)\n35for anyr>0(see e.g. [3, Corollary 2.4]). Given \u000f>0, we fixr\u000f>0such that\n2ku0k1k'kL1(Bcr\u000f(0))\u0014\u000f:\nUsing (A.7), we obtain\nlim\nt!0+k(S\u000b(t)u0\u0000u0)'kL1(Br\u000f(0))= 0:\nThen, passing to limit in the inequality\nk(S\u000b(t)u0\u0000u0)'kL1(RN)\u0014k(S\u000b(t)u0\u0000u0)'kL1(Br\u000f(0))+ 2ku0kL1(RN)k'kL1(Bcr\u000f(0));(A.8)\nwe obtain\nlim sup\nt!0+k(S\u000b(t)u0\u0000u0)'kL1(RN)\u0014\u000f: (A.9)\nTherefore\nlim\nt!0+k(S\u000b(t)u0\u0000u0)'kL1(RN)= 0:\nCombining with (A.6), we conclude the proof of (A.2).\nIt remains to prove that uis smooth for t >0. Sinceu2X(RN\u0002R+;C), we get that\nu;ru2L1\nloc(RN\u0002R+). Theng(u)2L2\nloc(RN\u0002R+)so theLp-regularity theory for parabolic\nequations implies that a function usatisfying (A.1) belongs to u2H2;1\nloc(RN\u0002R+)(see [28, 24]\nand [33, Remark 48.3] for notations and more details). Since the space Hk\\L1is stable under\nmultiplication (see e.g. [20, Chapter 6]), we can use a bootstrap argument to conclude that\nu2C1(RN\u0002R+).\nRemark A.4. Several authors have studied further properties of the solutions found by Koch\nand Tataru for the Navier–Stokes equations. For instance, analyticity, decay rates of the higher-\norder derivatives in space and time have been investigated by Miura and Sawada [32], Germain,\nPavlović and Staffilani [14], among others. A similar analysis for the solution uof (DNLS) is\nbeyond the scope of this paper, but it can probably be performed using the same arguments\ngiven in [32, 14].\nWe end this appendix with some properties of the self-similar found in [17].\nTheorem A.5 ([17]).LetN= 1. For every \u000b2[0;1]andc >0, there exists a profile fc;\u000b2\nC1(R;S2)such that\nmc;\u000b(x;t) =fc;\u000b\u0012xp\nt\u0013\n;for all (x;t)2R\u0002R+;\nis a smooth solution of (LLG\u000b)onR\u0002R+. Moreover,\n(i) There exist unitary vectors A\u0006\nc;\u000b= (A\u0006\nj;c;\u000b)3\nj=12S2such that the following pointwise con-\nvergence holds when tgoes to zero:\nlim\nt!0+mc;\u000b(x;t) =8\n<\n:A+\nc;\u000b;ifx>0;\nA\u0000\nc;\u000b;ifx<0;(A.10)\nandA\u0000\nc;\u000b= (A+\n1;c;\u000b;\u0000A+\n2;c;\u000b;\u0000A+\n3;c;\u000b).\n(ii) There exists a constant C(c;\u000b;p ), depending only on c,\u000bandpsuch that for all t>0\nkmc;\u000b(\u0001;t)\u0000A+\nc;\u000b\u001f(0;1)(\u0001)\u0000A\u0000\nc;\u000b\u001f(\u00001;0)(\u0001)kLp(R)\u0014C(c;\u000b;p )t1\n2p; (A.11)\nfor allp2(1;1). In addition, if \u000b>0,(A.11)also holds for p= 1.\n36(iii) Fort>0andx2R, the derivative in space satisfies\nj@xmc;\u000b(x;t)j=cp\nte\u0000\u000bx2\n4t: (A.12)\n(iv) Let\u000b2[0;1]. Then A+\nc;\u000b!(1;0;0)asc!0+.\nLemma A.6. Letc >0,\u000b2(0;1],A+\nc;\u000b;A\u0000\nc;\u000bbe the unit vectors given in Theorem A.5 and\n#c;\u000bthe angle between A+\nc;\u000bandA\u0000\nc;\u000b. Then, for fixed \u000b2(0;1],#c;\u000bis a continuous function\ninc. Also, for 0<c<\u000b2p\u0019=32,\n#c;\u000b\u0015arccos\u0012\n1\u0000c2\u0019+32c3p\u0019\n\u000b2\u0013\n: (A.13)\nRemark A.7. If\u000b2(0;1]andc2(0;\u000b2p\u0019=32), then 1\u0000c2\u0019+32c3p\u0019\n\u000b22(0;1), so that its\narccosis well-defined.\nProof.The continuity was proved in [17]. To show the estimate (A.13), we use Theorem 1.3 in\n[17], to get\n\f\f\f\f\fA+\n2;c;\u000b\u0000cp\n\u0019(1 +\u000b)p\n2\f\f\f\f\f\u0014c2\u0019\n4+c2\u0019\n\u000bp\n2 \n1 +c2\u0019\n8+cp\n\u0019(1 +\u000b)\n2p\n2!\n+\u0012c2\u0019\n2p\n2\u000b\u00132\n:\nSince\u000b2(0;1], we have for any c2(0;1],\n\u0019\n4+\u0019\n\u000bp\n2 \n1 +c2\u0019\n8+cp\n\u0019(1 +\u000b)\n2p\n2!\n+c2\u00192\n8\u000b2\u00141\n\u000b2\u0012\u0019\n4+\u0019p\n2\u0012\n1 +\u0019\n8+p\u0019\n2\u0013\n+\u00192\n8\u0013\n\u00148\n\u000b2:\nWe deduce that for all \u000b;c2(0;1],\nA+\n2;c;\u000b\u0015cp\n\u0019(1 +\u000b)p\n2\u00008c2\n\u000b2\u0015cp\u0019p\n2\u00008c2\n\u000b2:\nIn particular A+\n2;c;\u000b\u00150ifc\u0014\u000b2p\u0019=(8p\n2). Thus\n(A+\n2;c;\u000b)2\u0015c2\u0019\n2\u000016c3p\u0019p\n2\u000b2+64c4\n\u000b4\u0015c2\u0019\n2\u000016c3p\u0019\n\u000b2;\nso that\ncos(#c;\u000b) = 1\u00002((A+\n2;c;\u000b)2+ (A+\n3;c;\u000b)2)\u00141\u0000c2\u0019+32c3p\u0019\n\u000b2;\nwhich implies (A.13).\nThe following lemma is a slightly refinement of Theorem 1.4 in [17].\nLemma A.8 ([17]).Letc >0,\u000b2[0;1]andA+\nc;\u000bbe the unit vector given in Theorem A.5.\nThenA+\nc;\u000bis a continuous function of \u000bin[0;1]and\njA+\nc;\u000b\u0000A+\nc;1j\u0014h(c)p\n1\u0000\u000b;for all\u000b2[1=2;1]; (A.14)\nwhereh:R+!R+is a strictly increasing function satisfying\nlim\ns!1h(s) =1:\n37Proof.Inviewof[17, Theorem1.4], weonlyneedtoprovethattheconstant C(c)inthestatement\nof the Theorem 1.4 (notice that c0in [17] corresponds to cin our notation) is polynomial in c\nwith nonnegative coefficients. Looking at the proof of [17, Theorem 1.4], we see that the constant\nC(c)behaves like the constant in inequality (3.108) in [17]. In view of (3.17), the estimate (3.23)\nin [17] can be written as\njf(s)j\u0014p\n2andjf0(s)j\u0014c\n2e\u0000\u000bs2=4;\nand then (3.18) can be recast as\njgj\u0014 \nc\n4+c2p\n2\n8!\u0012s\n\fe\u0000\u000bs2=4+s2e\u0000\u000bs2=2\u0013\n:\nThen, it can be easily checked that the function his a polynomial with nonnegative coefficients.\nAcknowledgments. A.deLairewaspartiallysupportedbytheLabexCEMPI(ANR-11-LABX-\n0007-01) and the MathAmSud program. S. Gutierrez was partially supported by the EPSRC\ngrant EP/J01155X/1 and the ERCEA Advanced Grant 2014 669689 - HADE.\nReferences\n[1] M.AbramowitzandI.A.Stegun. Handbook of mathematical functions with formulas, graphs,\nand mathematical tables , volume 55 of National Bureau of Standards Applied Mathematics\nSeries. For sale by the Superintendent of Documents, U.S. Government Printing Office,\nWashington, D.C., 1964.\n[2] F.AlougesandA.Soyeur. OnglobalweaksolutionsforLandau-Lifshitzequations: existence\nand nonuniqueness. Nonlinear Anal. , 18(11):1071–1084, 1992.\n[3] J. M. Arrieta, A. Rodriguez-Bernal, J. W. Cholewa, and T. Dlotko. Linear parabolic equa-\ntions in locally uniform spaces. Math. Models Methods Appl. Sci. , 14(2):253–293, 2004.\n[4] V. Banica and L. Vega. On the Dirac delta as initial condition for nonlinear Schrödinger\nequations. Ann. Inst. H. Poincaré Anal. Non Linéaire , 25(4):697–711, 2008.\n[5] V. Banica and L. Vega. On the stability of a singular vortex dynamics. Comm. Math. Phys. ,\n286(2):593–627, 2009.\n[6] V. Banica and L. Vega. Scattering for 1D cubic NLS and singular vortex dynamics. J. Eur.\nMath. Soc. (JEMS) , 14(1):209–253, 2012.\n[7] V. Banica and L. Vega. Stability of the self-similar dynamics of a vortex filament. Arch.\nRation. Mech. Anal. , 210(3):673–712, 2013.\n[8] P. Biernat and P. Bizoń. Shrinkers, expanders, and the unique continuation beyond generic\nblowup in the heat flow for harmonic maps between spheres. Nonlinearity , 24(8):2211–2228,\n2011.\n[9] H. Brezis and A. Friedman. Nonlinear parabolic equations involving measures as initial\nconditions. J. Math. Pures Appl. (9) , 62(1):73–97, 1983.\n38[10] H. Brezis and L. Nirenberg. Degree theory and BMO. I. Compact manifolds without bound-\naries.Selecta Math. (N.S.) , 1(2):197–263, 1995.\n[11] J.-M. Coron. Nonuniqueness for the heat flow of harmonic maps. Ann. Inst. H. Poincaré\nAnal. Non Linéaire , 7(4):335–344, 1990.\n[12] M. Daniel and M. Lakshmanan. Perturbation of solitons in the classical continuum isotropic\nHeisenberg spin system. Physica A: Statistical Mechanics and its Applications , 120(1):125–\n152, 1983.\n[13] S. Ding and C. Wang. Finite time singularity of the Landau-Lifshitz-Gilbert equation. Int.\nMath. Res. Not. IMRN , (4):Art. ID rnm012, 25, 2007.\n[14] P. Germain, N. Pavlović, and G. Staffilani. Regularity of solutions to the Navier-Stokes\nequations evolving from small data in BMO\u00001.Int. Math. Res. Not. IMRN , (21):Art. ID\nrnm087, 35, 2007.\n[15] P. Germain and M. Rupflin. Selfsimilar expanders of the harmonic map flow. Ann. Inst. H.\nPoincaré Anal. Non Linéaire , 28(5):743–773, 2011.\n[16] T. L. Gilbert. A lagrangian formulation of the gyromagnetic equation of the magnetization\nfield.Phys. Rev. , 100:1243, 1955.\n[17] S. Gutiérrez and A. de Laire. Self-similar solutions of the one-dimensional Landau–Lifshitz–\nGilbert equation. Nonlinearity , 28(5):1307, 2015.\n[18] S. Gutiérrez, J. Rivas, and L. Vega. Formation of singularities and self-similar vortex motion\nunder the localized induction approximation. Comm. Partial Differential Equations , 28(5-\n6):927–968, 2003.\n[19] H. Hasimoto. A soliton on a vortex filament. J. Fluid Mech , 51(3):477–485, 1972.\n[20] L. Hörmander. Lectures on nonlinear hyperbolic differential equations , volume 26 of Math-\nématiques & Applications (Berlin) [Mathematics & Applications] . Springer-Verlag, Berlin,\n1997.\n[21] C. E. Kenig, G. Ponce, and L. Vega. On the ill-posedness of some canonical dispersive\nequations. Duke Math. J. , 106(3):617–633, 2001.\n[22] H. Koch and T. Lamm. Geometric flows with rough initial data. Asian J. Math. , 16(2):209–\n235, 2012.\n[23] H. Koch and D. Tataru. Well-posedness for the Navier-Stokes equations. Adv. Math. ,\n157(1):22–35, 2001.\n[24] O. Ladyzhenskaya, V. Solonnikov, and N. Ural’tseva. Linear and quasi-linear equations of\nparabolic type . Amer. Math. Soc., Transl. Math. Monographs. Providence, R.I., 1968.\n[25] M. Lakshmanan and K. Nakamura. Landau-Lifshitz equation of ferromagnetism: Exact\ntreatment of the Gilbert damping. Phys. Rev. Lett. , 53:2497–2499, 1984.\n[26] L. Landau and E. Lifshitz. On the theory of the dispersion of magnetic permeability in\nferromagnetic bodies. Phys. Z. Sowjetunion , 8:153–169, 1935.\n[27] P. G. Lemarié-Rieusset. Recent developments in the Navier-Stokes problem , volume 431\nofChapman & Hall/CRC Research Notes in Mathematics . Chapman & Hall/CRC, Boca\nRaton, FL, 2002.\n39[28] G. M. Lieberman. Second order parabolic differential equations . World Scientific Publishing\nCo., Inc., River Edge, NJ, 1996.\n[29] J. Lin. Uniqueness of harmonic map heat flows and liquid crystal flows. Discrete Contin.\nDyn. Syst. , 33(2):739–755, 2013.\n[30] J. Lin, B. Lai, and C. Wang. Global well-posedness of the Landau-Lifshitz-Gilbert equation\nfor initial data in Morrey spaces. Calc. Var. Partial Differential Equations , 54(1):665–692,\n2015.\n[31] C. Melcher. Global solvability of the Cauchy problem for the Landau-Lifshitz-Gilbert equa-\ntion in higher dimensions. Indiana Univ. Math. J. , 61(3):1175–1200, 2012.\n[32] H. Miura and O. Sawada. On the regularizing rate estimates of Koch-Tataru’s solution to\nthe Navier-Stokes equations. Asymptot. Anal. , 49(1-2):1–15, 2006.\n[33] P. Quittner and P. Souplet. Superlinear parabolic problems . Birkhäuser Advanced Texts:\nBasler Lehrbücher. [Birkhäuser Advanced Texts: Basel Textbooks]. Birkhäuser Verlag,\nBasel, 2007. Blow-up, global existence and steady states.\n[34] E. M. Stein. Harmonic analysis: real-variable methods, orthogonality, and oscillatory inte-\ngrals, volume 43 of Princeton Mathematical Series . Princeton University Press, Princeton,\nNJ, 1993. With the assistance of Timothy S. Murphy, Monographs in Harmonic Analysis,\nIII.\n[35] C. Wang. Well-posedness for the heat flow of harmonic maps and the liquid crystal flow\nwith rough initial data. Arch. Ration. Mech. Anal. , 200(1):1–19, 2011.\n[36] D. Wei. Micromagnetics and Recording Materials . SpringerBriefs in Applied Sciences and\nTechnology. Springer Berlin Heidelberg, 2012.\n40"    },    {        "title": "1701.03201v2.Dynamic_coupling_of_ferromagnets_via_spin_Hall_magnetoresistance.pdf",        "content": "arXiv:1701.03201v2  [cond-mat.mes-hall]  20 Mar 2017Dynamic coupling of ferromagnets via spin Hall magnetoresi stance\nTomohiro Taniguchi\nNational Institute of Advanced Industrial Science and Tech nology (AIST),\nSpintronics Research Center, Tsukuba, Ibaraki 305-8568, J apan\n(Dated: September 17, 2018)\nThe synchronized magnetization dynamics in ferromagnets o n a nonmagnetic heavy metal caused\nby the spin Hall effect is investigated theoretically. The di rect and inverse spin Hall effects near\nthe ferromagnetic/nonmagnetic interface generate longit udinal and transverse electric currents. The\nphenomenon is known as the spin Hall magnetoresistance effec t, whose magnitude depends on the\nmagnetization direction in the ferromagnet due to the spin t ransfer effect. When another ferro-\nmagnet is placed onto the same nonmagnet, these currents are again converted to the spin current\nby the spin Hall effect and excite the spin torque to this addit ional ferromagnet, resulting in the\nexcitation of the coupled motions of the magnetizations. Th e in-phase or antiphase synchronization\nof the magnetization oscillations, depending on the value o f the Gilbert damping constant and the\nfield-like torque strength, is found in the transverse geome try by solving the Landau-Lifshitz-Gilbert\nequation numerically. On the other hand, in addition to thes e synchronizations, the synchroniza-\ntion having a phase difference of a quarter of a period is also f ound in the longitudinal geometry.\nThe analytical theory clarifying the relation among the cur rent, frequency, and phase difference is\nalso developed, where it is shown that the phase differences o bserved in the numerical simulations\ncorrespond to that giving the fixed points of the energy suppl ied by the coupling torque.\nPACS numbers: 85.75.-d, 75.78.-n, 05.45.Xt, 72.25.-b\nI. INTRODUCTION\nDynamic coupling of ferromagnets has been of interest\ninthefieldofmagnetism. Thedipoleinteractionhasbeen\nthe basic interaction to excite the coupled motion of the\nmagnetizations, and is applied to magnetic recording1,2.\nAnother method to realizethe couplingis to usethe spin-\ntransfer effect3,4, where the applicationof an electric cur-\nrent to ferromagnetic/nonmagnetic multilayers results in\nthe magnetization switching and self-oscillation5–14. The\ncoupled dynamics through pure spin current generated\nin spin pumping15,16and nonlocal17geometries have also\nbeen observed. It should be emphasized that these cou-\nplings are strongly restricted by the characteristic length\nscales. Forexample, thedipolecouplingdecaysaccording\nto the inversecube detection law, whereas the spin trans-\nfer effect by a spin-polarized electric or pure spin current\noccurs in a system smaller than the spin diffusion length.\nRecently, physical phenomena, such as the spin\ntorque18–25and magnetoresistance effects26–34, due to\nthe spin Hall effect35–37in bilayers consisting of an insu-\nlating or metallic ferromagnet and a nonmagnetic heavy\nmetal have attracted much attention. The latter, known\nas the spin Hall magnetoresistance, originates from the\ncharge-spin conversion of an external electric current by\nthe direct and inverse spin Hall effects, and has been\nobserved by measuring the longitudinal and transverse\nelectric currents, which are given by\nJcx\nJ0= 1+χ′′+χm2\ny, (1)\nJcy\nJ0=−χmxmy−χ′mz, (2)respectively, where J0is the electric current density gen-\nerated by the external electric field. The definitions of\nthe dimensionless coefficients, χ,χ′, andχ′′, are given\nbelow. It should be emphasized that the currents given\nby Eqs. (1) and (2) depend on the magnetization direc-\ntionm= (mx,my,mz). When another ferromagnet is\nplaced onto the same nonmagnet, these currents will be\nconverted to spin current by the spin Hall effect again,\nand excite spin torque on this additional ferromagnet.\nThen, the magnetization dynamics of two ferromagnets\nwill be coupled through the angular dependencies of the\nelectric current given by Eqs. (1) and (2). This coupling\nis unavoidable whenever several ferromagnets are placed\nonto the same nonmagnet, and is not restricted by the\ndistance between the ferromagnets because it is carried\nby the electric current. Since the structure consisting\nof several ferromagnets on the nonmagnetic heavy metal\nwill be important from the viewpoints of both fundamen-\ntal physics and practical applications based on the spin\nHalleffect, suchasmagneticrandomaccessmemory, spin\ntorque oscillators, and bio-inspired computing38,39, it is\nof interest to clarify the role of this coupling.\nIn this paper, we investigate the coupled dynamics of\nmagnetizations in ferromagnets in the presence of the\nspin Hall effect by solving the Landau-Lifshitz-Gilbert\n(LLG) equation numerically for both longitudinal and\ntransversegeometries. Inadditiontothe externalelectric\ncurrent, the current contributing to the spin Hall mag-\nnetoresistance also excites the spin torque. The strength\nof this additional torque is estimated from the theory of\nthe spin Hall magnetoresistance extended to the system\nconsistingofseveralferromagnets. The conventionalspin\ntorque is proportional to the spin Hall angle ϑ, whereas\nthe new torque is on the order of ϑ3, and therefore, its\nvalue is two orders of magnitude smaller than the con-2\nx\nyzlongitudinal coupling\ntransverse coupling\nExF2\nF1 F3\nEx\n(b)(a)\nF1 F2\nσNEx+j1+j2SHE SHE ISHE ISHEN\nN\nFIG. 1: (a) Schematic view of the system in this study.\nThree ferromagnets F ℓ(ℓ= 1,2,3) are placed onto the same\nnonmagnet N. The external electric field is applied to the x\ndirection. (b) Schematic view of the generations of electri c\ncurrents by the direct and inverse spin Hall effect (SHE and\nISHE)in the longitudinal geometry. The total electric curr ent\nis the sum of the conventional electric current J0=σNExand\nthe current generated near the F 1/N and F 2/N interfaces, j1\nandj2.\nventional spin torque. Nevertheless, it is found that this\nadditional new torque affects the phase difference of the\nmagnetizationin the self-oscillationstate. The numerical\nsimulationrevealsthatthein-phaseorantiphasesynchro-\nnization is observed in the transverse geometry. On the\nother hand, in addition to them, the phase difference be-\ncomes a quarter of a period in the longitudinal geometry.\nIt is found that these phase differences depend on the\nvalues of the Gilbert damping constant and the dimen-\nsionless field-like torque strength. An analytical theory\nclarifying the relation among the current, frequency, and\nphase difference is also developed.\nThe paper is organized as follows. In Sec. II, we de-\nscribe the system under consideration, and discuss the\ntheoretical formula of the spin torque excited by the spin\nHalleffect inthepresenceoftheseveralferromagnetsand\nthe spin Hall magnetoresistance effect. In Sec. III, we\nstudy the phase differences in the synchronized state of\nthe magnetizations for both the longitudinal and trans-\nverse geometries by solving the LLG equation numeri-\ncally. InSec. IV,thetheoryclarifyingtherelationamong\nthe current, frequency, and phase difference is developed\nbased on the LLG equation averaged over constant en-\nergy curves. The summary of the paper is given in Sec.\nV.\nII. SYSTEM DESCRIPTION AND LLG\nEQUATION\nIn this section, we describe the system adopted in this\nstudy, and show the spin torque formulas including the\ncoupling torques between the ferromagnets.A. System description\nThe system we consider is schematically shown in Fig.\n1(a), where three ferromagnets F ℓ(ℓ= 1,2,3) are placed\nonto the same nonmagnet N. We assume that the ma-\nterial parameters in the ferromagnets are identical, for\nsimplicity. The external electric field, Ex, is applied\nto thexdirection, inducing the electric current density\nJ0=σNEx, where σNis the conductivity of the non-\nmagnet. The direct and inverse spin Hall effects produce\nelectriccurrentsinthelongitudinal( x)andtransverse( y)\ndirections. These electric currents are converted to the\nspin current and injected into the F 2and F 3layers due\nto the spin Hall effect, resulting in the excitation of the\nspin torque. Then, the magnetization dynamics in the F ℓ\n(ℓ= 2,3)layerisaffectedbythatintheF 1layer,andvice\nversa. We call the coupling between the F 1and F2layers\nthe longitudinal coupling, whereas that between the F 1\nand F 3layers the transverse coupling. We assume that\nboth the ferromagnet and nonmagnet are metallic be-\ncause metallic bilayers are generally used to measure the\nmagnetization switching and oscillation by the spin Hall\neffect18–21,24. Although the spin Hall magnetoresistance\nwas originally studied for insulating ferromagnets26–29, a\nlarge spin Hall magnetoresistance in metallic system has\nalso been reported recently32–34.\nThe dimensionless coefficients, χ,χ′, andχ′′, in Eqs.\n(1) and (2) for single ferromagnets have been derived for\nan insulating30or metallic34,40ferromagnet, which are\ngiven by\nχ=ϑ2λN\ndN/bracketleftbigg\nReg↑↓\ngN+g↑↓coth(dN/λN)−g∗\ngN/bracketrightbigg\ntanh2/parenleftbiggdN\n2λN/parenrightbigg\n,\n(3)\nχ′=−ϑ2λN\ndNImg↑↓\ngN+g↑↓coth(dN/λN)tanh2/parenleftbiggdN\n2λN/parenrightbigg\n,\n(4)\nχ′′=2ϑ2λN\ndNtanh/parenleftbiggdN\n2λN/parenrightbigg\n−ϑ2λN\ndNReg↑↓\ngN+g↑↓coth(dN/λN)tanh2/parenleftbiggdN\n2λN/parenrightbigg\n,\n(5)\nwheredNandλNare thickness and spin diffusion\nlength of the nonmagnet, respectively, whereas gN/S=\nhσN//parenleftbig\n2e2λN/parenrightbig\nwith the cross section area of the ferro-\nmagnetic/nonmagnetic interface S. The dimensionless\nmixing conductance g↑↓=g↑↓\nr+ig↑↓\niconsists of its real\nand imaginary parts41–43, andg∗is defined as\n1\ng∗=2\n(1−p2g)g+1\ngFtanh(dF/λF)+1\ngNtanh(dN/λN).\n(6)\nHere,g=g↑↑+g↓↓is the sum of the conductances\nof the spin-up and spin-down electrons, whereas pg=3\n(g↑↑−g↓↓)/gis its spin polarization. The ferromag-\nnetic/nonmagnetic interface resistance ris related to\ngviag/S= (h/e2)/r. We also introduce gF/S=\nh/parenleftbig\n1−p2\nσ/parenrightbig\nσF//parenleftbig\n2e2λF/parenrightbig\n, whereσFis the conductivity of\nthe ferromagnet and pσis its spin polarization. The\nthickness and spin diffusion length of the ferromagnet\nare denoted as dFandλF, respectively. The term related\ntog∗is neglected when the ferromagnetis an insulator30,\ni.e.,r→ ∞. The following quantities correspond to\nthe effective spin polarizations of the damping-like (or\nSlonczewski3) torque and the field-like torque, respec-\ntively,\nϑR(I)=ϑtanh/parenleftbiggdN\n2λN/parenrightbigg\nRe(Im)g↑↓\ngN+g↑↓coth(dN/λN).\n(7)\nFor the later discussion, we introduce\nβ=−ϑI\nϑR, (8)\nwhich corresponds to the ratio of the field-like torque to\nthe damping-like torque.\nThe values of the parameters used in the following\ncalculations are derived from recent experiments on the\nW/CoFeB heterostructure34, where ρF= 1/σF= 1.6\nkΩnm,pσ= 0.72,λF= 1.0 nm,ρN= 1/σN= 1.25\nkΩnm,λN= 1.2 nm, and ϑ= 0.27, whereas the thick-\nnesses areassumed to be dF= 2 nm and dN= 3 nm. The\ninterfaceconductanceswerenotevaluatedinRef.34byas-\nsuming a transparent interface. Instead, we use typical\nvalues of the interface conductances obtained from the\nfirst-principles calculations43,r= 0.25 kΩnm2,pg= 0.5,\nandg↑↓\nr/S= 25 nm−2. We note that the imaginary part\nof the mixing conductance, g↑↓\ni, is either positive or neg-\native, depending on the material and thickness43. The\nsign ofg↑↓\nidetermines those of χ′andϑI, or equivalently,\nβ. For example, when g↑↓\ni/S= 1 nm−2,χ≃0.010,\nχ′≃ −0.0002,χ′′≃0.035,θR≃0.167, and θI≃0.002\n(β≃ −0.010); see Appendix A. In the following calcula-\ntions, we study the magnetization dynamics for several\nvalues of β.\nB. Spin torques in longitudinal and transverse\ngeometries\nEquation (1) was derived for a system having single\nferromagnet. In this case, J0is the external electric cur-\nrent density, whereas ( χ′′+χmy)J0is the current density\ngenerated as a result of the charge-spinconversionby the\ndirect and inverse spin Hall effects. In the longitudinal\ngeometryinthepresentstudy, ontheotherhand, twofer-\nromagnetic/nonmagnetic interfaces, i.e., F 1/N and F 2/N\ninterfaces, contribute to the generation of the longitudi-\nnal current through the direct and inverse spin Hall ef-\nfects, as schematically shown in Fig. 1(b). Let us denote\nthe electric current density generated by these effects\nnear the F ℓ/N interface as jℓx(ℓ= 1,2). This currentdensity is determined by the conservation law of the elec-\ntric current, as follows. Considering in a similar manner\nto the case of the single ferromagnet, the electric current\nJ0+j1xis converted to the spin current by the spin Hall\neffect near the F 2/N interface, and this spin current pro-\nduces an additionalelectric current ( χ′′+χm2\n2y)(J0+j1x)\nby the inverse spin Hall effect. Therefore, the total lon-\ngitudinal electric current density near the F 2/N interface\nis (1+χ′′+χm2\n2y)(J0+j1x). Similarly, the electric cur-\nrent density near the F 1/N interface can be expressed\nas (1+χ′′+χm2\n1y)(J0+j2x). These currents should be\nequal to the total electric current density, J0+j1x+j2x,\naccording to the conservation law of the electric current.\nThen, we find that jℓxis given by\njℓx=/parenleftBig\nχ′′+χm2\nℓy/parenrightBig/parenleftBig\n1+χ′′+χm2\nℓ′y/parenrightBig\n1−/parenleftBig\nχ′′+χ′m2\nℓy/parenrightBig/parenleftBig\nχ′′+χm2\nℓ′y/parenrightBigJ0\n≃/parenleftbig\nχ′′+χm2\nℓy/parenrightbig\nJ0,(9)\nwhere we neglect the higher order terms of ϑ. Then, the\nspin torque excited on the F ℓlayer by the spin Hall effect\nin the longitudinal geometry is obtained by replacing the\nexternal electric current density J0=σNExin the previ-\nous work30withJ0+jℓ′x≃/parenleftBig\n1+χ′′+χm2\nℓ′y/parenrightBig\nJ0, and is\ngiven by\nTL\nℓ=−γ/planckover2pi1ϑRJ0\n2eMdF/parenleftbig\n1+χ′′+χm2\nℓ′y/parenrightbig\nmℓ×(ey×mℓ)\n−γ/planckover2pi1ϑIJ0\n2eMdF/parenleftbig\n1+χ′′+χm2\nℓ′y/parenrightbig\ney×mℓ,\n(10)\nwhere (ℓ,ℓ′) = (1,2) or (2,1). The unit vector pointing\nin the magnetization direction of the F ℓlayer ismℓ, and\nγ,M, anddare the gyromagnetic ratio, the saturation\nmagnetization, and the thickness of the ferromagnet, re-\nspectively.\nNote that the terms,\nT(0)\nℓ=−γ/planckover2pi1ϑRJ0\n2eMdFmℓ×(ey×mℓ)−γ/planckover2pi1ϑIJ0\n2eMdFey×mℓ,\n(11)\nin Eq. (10) are the conventional spin torques generated\nby the external electric current J0and the spin Hall ef-\nfect, and are often referred to as the damping-like torque\nandthefield-liketorque, respectively. Ontheotherhand,\nthe terms\n−γ/planckover2pi1ϑRJ0\n2eMdF/parenleftbig\nχ′′+χm2\nℓ′y/parenrightbig\nmℓ×(ey×mℓ)\n−γ/planckover2pi1ϑIJ0\n2eMdF/parenleftbig\nχ′′+χm2\nℓ′y/parenrightbig\ney×mℓ,(12)\nin Eq. (10) originate from the current jℓ′x, and are newly\nintroduced in this study. It should be emphasized that\nEq. (12) depends on the magnetization direction of the4\nother ferromagnet F ℓ′(ℓ′= 1 or 2), mℓ′, resulting in the\ncoupling between the F 1and F2layers.\nLet us next show the spin torque formulasin the trans-\nverse geometry. We denote the electric current den-\nsity flowing in the ydirection generated near the F ℓ/N\n(ℓ= 1,3) interface by the inverse spin Hall effect as jℓy.\nThe conservation law of the total electric current density,\nj1y+j3y, gives (see also Appendix B)\njℓy=−/bracketleftbigg(χmℓxmℓy+χ′mℓz)\n1−(χ′′+χm2\nℓx)(χ′′+χm2\nℓ′x)\n+/parenleftbig\nχ′′+χm2\nℓx/parenrightbig\n(χmℓ′xmℓ′y+χ′mℓ′z)\n1−(χ′′+χm2\nℓx)(χ′′+χm2\nℓ′x)/bracketrightBigg\nJ0\n≃ −(χmℓxmℓy+χ′mℓz)J0.(13)\nInadditiontotheconventionalspintorque,Eq. (11), this\ntransverse electric current also excites the spin torque on\nthe other ferromagnet. In total, the spin torque acting\non the F ℓlayer is given by [( ℓ,ℓ′) = (1,3) or (3,1)]\nTT\nℓ=−γ/planckover2pi1ϑRJ0\n2eMdFmℓ×(ey×mℓ)−γ/planckover2pi1ϑIJ0\n2eMdFey×mℓ\n−γ/planckover2pi1ϑR(χmℓ′xmℓ′y+χ′mℓ′z)J0\n2eMdFmℓ×(ex×mℓ)\n−γ/planckover2pi1ϑI(χmℓ′xmℓ′y+χ′mℓ′z)J0\n2eMdFex×mℓ,\n(14)\nThelasttwotermsrepresentthe couplingtorquebetween\nthe F1and F 3layers. Note that the direction of this\ncoupling torque is different from that of the conventional\ntorque because the currents J0andjℓyflow in different\ndirections.\nIn the following, the torques related to χ,χ′, andχ′′\nin Eqs. (10) and (14) are referred to as coupling torque.\nThe ratio of these new torques to the conventional spin\ntorque is on the order of χ∝ϑ2∼10−2. Since the\nconventional spin torque due is proportional to the spin\nHall angle ϑR∝ϑ, the coupling torque is proportional\ntoϑ3. Although the spin Hall angle is usually a small\nquantity, it will be shown that the coupling torques play\na non-negligible role on the magnetization dynamics, as\nshown below.\nC. LLG equation\nIn the following sections, we study the magnetization\ndynamics excited by the spin torque given by Eq. (10)\nor (14) both numerically and analytically. We neglect\nthe transverse coupling when the role of the longitudinal\ncoupling is studied, and vice versa, for simplicity, which\ncorresponds to considering a system consisting of the F 1\nand F 2layers, or F 1and F 3layers. The magnetization\ndynamics in the F ℓ(ℓ= 1,2,3) is described by the LLG\nequation,\ndmℓ\ndt=−γmℓ×Hℓ+αmℓ×dmℓ\ndt+TL,T\nℓ,(15)where the Gilbert damping constant is denoted as α.\nIn the following calculations, we use the values of the\nmaterial parameters, γ= 1.764×107rad/(Oe s) and\nα= 0.005, derived from the experiments44. For the later\ndiscussion, it is useful to show the explicit forms of the\nLLG equation in the longitudinal and transverse geome-\ntries. Equation (15) for the longitudinal geometry is\n/parenleftbig\n1+α2/parenrightbigdmℓ\ndt=−γmℓ×Hℓ\n−γ(1+χ′′)/planckover2pi1ϑRJ0\n2eMdFmℓ×(ey×mℓ)−αγmℓ×(mℓ×Hℓ)\n−γχm2\nℓ′y/planckover2pi1ϑRJ0\n2eMdFmℓ×(ey×mℓ)\n−γ(1+χ′′)(α+β)/planckover2pi1ϑRJ0\n2eMdFmℓ×ey\n+γ(1+χ′′)αβ/planckover2pi1ϑRJ0\n2eMdFmℓ×(ey×mℓ)\n−γ(α+β)χm2\nℓ′y/planckover2pi1ϑRJ0\n2eMdFmℓ×ey\n+γαβχm2\nℓ′y/planckover2pi1ϑRJ0\n2eMdFmℓ×(ey×mℓ),\n(16)\nwhereas that for the transverse geometry is\n/parenleftbig\n1+α2/parenrightbigdmℓ\ndt=−γmℓ×Hℓ\n−γ/planckover2pi1ϑRJ0\n2eMdFmℓ×(ey×mℓ)−αγmℓ×(mℓ×Hℓ)\n−γ(χmℓ′xmℓ′y+χ′mℓ′z)/planckover2pi1ϑRJ0\n2eMdFmℓ×(ex×mℓ)\n−γ(α+β)/planckover2pi1ϑRJ0\n2eMdFmℓ×ey\n+γαβ/planckover2pi1ϑRJ0\n2eMdFmℓ×(ey×mℓ)\n−γ(α+β)(χmℓ′xmℓ′y+χ′mℓ′z)/planckover2pi1ϑRJ0\n2eMdFmℓ×ex\n+γαβ(χmℓ′xmℓ′y+χ′mℓ′z)/planckover2pi1ϑRJ0\n2eMdFmℓ×(ex×mℓ).\n(17)\nIII. NUMERICAL ANALYSIS OF\nSYNCHRONIZATION\nIn this section, we study the magnetization dynam-\nics in the ferromagnets in the presence of the coupling\ntorques by solving Eq. (15) numerically. The self-\noscillation of the magnetization provides an interesting\nexample to understand the role of the coupling torque.\nNote that the coupling torques are proportional to the\nparameters χ,χ′, andχ′′, and their products to other pa-\nrameters such as αβχ, the values of which are relatively\nsmall, as can be seen in Eqs. (16) and (17). Nevertheless,\nthe coupling torques lead to the phase synchronization,\nas shown below.5\nThe self-oscillation of the magnetization in single fer-\nromagnets by the spin Hall effect has been observed for\nin-plane magnetized ferromagnets19, and is induced by\nthe conventional spin torque given by Eq. (11). There-\nfore, in the following, we assume that the magnetic field,\nHℓ=HKmℓyey−4πMmℓzez, (18)\nconsists of an in-plane anisotropy field HKalong the y\ndirection and a demagnetization field 4 πMin thezdi-\nrection, which we assumeas HK= 200Oe and M= 1500\nemu/c.c.34in the following calculations. It is known for\nthe case of the single ferromagnet45that the in-plane\nself-oscillation can be excited when the electric current\ndensityJ0is in the range of Jc< J0< J∗, where\nJc=2αeMd F\n/planckover2pi1ϑR(HK+2πM), (19)\nJ∗=4αeMd F\nπ/planckover2pi1ϑR/radicalbig\n4πM(HK+4πM),(20)\nwhich in this study are Jc≃26 andJ∗≃33 MA/cm2.\nA. Transverse geometry\nLet us first investigate the magnetization dynamics\nin the transverse geometry because this geometry pro-\nvides a simple example of the coupled motion. We start\nwith solving the LLG equation given by Eq. (17) for\nthe F1and F 3layers. Figure 2(a) shows an exam-\nple of the trajectory of the magnetization dynamics ob-\ntained by solving Eq. (17) numerically, where J0= 30\nMA/cm2. As shown, the in-plane oscillation is observed\nin the steady state. The initial conditions set for m1\nandm3are different as m1(0) = (cos80◦,sin80◦,0) and\nm3(0) = (cos95◦,sin95◦,0). Therefore, the dynamics of\nm1andm3near the initial time are different, as shown\nin Fig. 2(b), where the time evolutions of m1x(t) and\nm3x(t) in 0≤t≤1 ns are shown. Nevertheless, the dy-\nnamics of m1andm3synchronize gradually, and finally,\nsynchronizationof m1x(t) =m3x(t) andm1y(t) =m3y(t)\nis realized, as shown in Figs. 2(c) and 2(d). We empha-\nsize here that the dynamics shown in these figures are\nobtained for β=−0.01.\nThe mutual, as well as self, synchronization of the\nspin torque induced magnetization oscillation by using\nspin waves, electric current, microwave field, or dipole\ncoupling has been an exciting topic from the viewpoints\nof both nonlinear science and practical applications46–62.\nThe key quantity of the synchronization is the phase dif-\nference ∆ ϕbetween each magnetization to enhance the\nemission power from the spin torque oscillators and to\ninvestigate new practical applications such as neuromor-\nphic architectures38,39. The synchronization found here,\ni.e.,mℓx(t) =mℓ′x(t), is called the in-phase synchro-\nnization. We should emphasize here that, although the(a)\nmxmymz\n1.00-1.0\n-1.01.0\n0-1.001.0\ntime (ns) time (ns)999.0 999.2 999.4 999.6 999.8 1000 999.0 999.2 999.4 999.6 999.8 1000:F1\n:F3:F1\n:F3\n01.0\n-1.0 01.0\n0.5mx\nmy(c) (d)\ntime (ns) time (ns)999.0 999.2 999.4 999.6 999.8 1000 999.0 999.2 999.4 999.6 999.8 1000:F1\n:F301.0\n-1.0 01.0\n0.5mx\nmy(e) (f)time (ns)0 0.2 0.4 0.6 0.8 1.001.0\n-1.0mx(b)\nFIG. 2: (a) A typical in-plane self-oscillation trajectory of\nthe magnetization obtained by solving Eq. (15) numerically\nfor the transverse geometry. The dotted lines indicate the\noscillation direction. (b) The time evolutions of m1x(t) and\nm1y(t) near the initial state. The time variations of the x\nandycomponents of the magnetizations in the steady state\nare shown in (c) and (d) for β=−0.01 and (e) and (f) for\nβ= +0.01. The solid red lines correspond to time evolution\nin F1, and dotted blue lines to those in F 3for (b) through (f).\nresults shown in Figs. 2(b) and 2(c) are shown for one\ncertain initial condition, the in-phase synchronizations\nare confirmed for the present set of the parameters even\nwhen the initial conditions are changed.\nOn the other hand, it was shown for the case of the\ncurrent-injectionlockingofthespin torqueoscillatorthat\nthe phase difference between the magnetization oscilla-\ntion and the alternative current depends on the strength\nof the field-like torque53. The field-like torque in the\npresent system can be either positive or negative, as\nmentioned above. These facts motivate us to study the\nmagnetization dynamics for different values of β. When\nβ= +0.01, synchronized dynamics is observed in a sim-\nilar manner, but in this case, the phase difference is an-\ntiphase, i.e., m1x(t) =−m3x(t), as shown in Figs. 2(e)\nand 2(f).\nFigure 3 summarizes the dependences of the phase dif-\nference, ∆ ϕ, between the magnetizations on the field-like\ntorque strength βfor the several values of the damping\nconstant α. The vertical axis in this figure represents\nthe phase difference in the unit of the oscillation period;\ni.e., ∆ϕ= 0correspondsto the in-phase synchronization.\nOn the other hand, ∆ ϕ= 0.50 means that the phase dif-6\nfield-like torque, βphase difference , Δφ\n-0.030 -0.020 -0.010 0.010 0.020 0.030 000.250.50\nα=0.005, 0.010, 0.015,\n0.020, 0.025, 0.030\nFIG. 3: Dependencesofthephase differences, ∆ ϕ, for several\nvalues of αon the field-like torquestrength βin the transverse\ngeometry. ∆ ϕ= 0 and 0 .5 correspond to the in-phase and\nantiphase, respectively. The values of the current density ,J0,\nis increased linearly, where J0= 30 MA/cm2forα= 0.005.\nference is half of a period, and thus, is antiphase. The\nalgorithm evaluating ∆ ϕin the numerical simulation is\nsummarized in Appendix A. The results indicate that\nthe phase difference is antiphase for positive β, whereas\nit becomes in-phase when βbecomes smaller than a cer-\ntainvalue, exceptforthenarrowintermediateregionnear\nβ∼ −α/2, where the phase difference is in between in-\nphase and antiphase.\nB. Longitudinal geometry\nNext, we study the magnetization dynamics in the lon-\ngitudinal geometry between F 1and F 2layers by solv-\ning Eq. (16) numerically. Figure 4(a) and 4(b) show\nmℓx(t) andmℓy(t) in a steady state, where β=−0.01.\nThe initial conditions in these figures are m1(0) =\n(cos80◦,sin80◦,0) and m2(0) = (cos95◦,sin95◦,0).\nAn antiphase synchronization is observed in this case.\nWe notice, however, that the in-phase synchronization\ncan also be realized when the initial conditions are\nchanged. Figure 4(c) and 4(d) show such an exam-\nple, where m1(0) = (cos80◦,sin80◦,0) andm2(0) =\n(cos85◦,sin85◦,0) are assumed. These numerical results\nindicate that both the in-phase and antiphase synchro-\nnizations are stable in this case, and whether the phase\ndifference finally becomes in-phase or antiphase depends\non the initial conditions, material parameters, and cur-\nrent magnitude. We also notice that the phase difference\nis changed when the value of βis changed. Figures 4(e)\nand 4(f) show mℓxandmℓyforβ= +0.01. In this case,\nthe phase difference of the magnetizations is a quarter of\na precession period.\nThe dependences of the phase difference on the field-\nlike torque strength βfor several values of the Gilbert\ndamping constant αare summarized in Fig. 5, where\n∆ϕ= 0.25 in this figure means that the phase differencetime (ns) time (ns)999.0 999.2 999.4 999.6 999.8 1000 999.0 999.2 999.4 999.6 999.8 1000:F1\n:F201.0\n-1.0 01.0\n0.5mx\nmy(a) (b)\ntime (ns) time (ns)999.0 999.2 999.4 999.6 999.8 1000 999.0 999.2 999.4 999.6 999.8 1000:F1\n:F201.0\n-1.0 01.0\n0.5mx\nmy(c) (d)\ntime (ns) time (ns)999.0 999.2 999.4 999.6 999.8 1000 999.0 999.2 999.4 999.6 999.8 1000:F1\n:F201.0\n-1.0 01.0\n0.5mx\nmy(e) (f)\nFIG. 4: The variations of the xandycomponents of the\nmagnetizations in the F 1(solid, red) and F 2(dotted, blue) in\nthe longitudinal geometry are shown in (a), (c) and (b), (d)\nrespectively, where β=−0.01. The initial conditions of m2\nare different between (a), (b) and (c), (d). The magnetizatio n\ndynamics for β= +0.01 are shown in (e) and (f)\nis a quarter of a period. The phase difference is found to\nbecome a quarter of the period for positive β, whereas it\nbecomes in-phase or antiphase for negative β, depending\non the initial states of the magnetizations.\nC. Summary of numerical simulations\nLet us summarize the results of the numerical simu-\nlations here. In the transverse geometry, the coupling\ntorque induces the synchronized oscillation of the mag-\nnetizations and finally stabilizes the configuration in the\nin-phase or antiphase state, depending on the values of\nthe field-liketorque strength βand the damping constant\nα. The phase difference in the stable synchronized state\nin the longitudinal geometry also depends on the values\nofβandα, as well as the initial conditions. In this case,\nhowever, in addition to the in-phase or antiphase state, a\nphase difference with a quarter of a period is generated.\nThe phase difference can be measured from the spin\nHallmagnetoresistanceeffect. AccordingtoEq. (13), the\ntotal electric current density in the transverse direction7\nfield-like torque, β-0.030 -0.020 -0.010 0.010 0.020 0.030 000.250.50\nα=0.005 α=0.010\nα=0.015\nα=0.020α=0.030α=0.025phase difference , Δφ\nFIG. 5: Dependencesofthephase differences, ∆ ϕ, for several\nvalues of αon the field-like torque strength βin the longitu-\ndinal geometry, where ∆ ϕ= 0 and 0 .5 correspond to the\nin-phase and antiphase, respectively. ∆ ϕ= 0.25 means that\nthe phase difference is a quarter of a period. The values of\nthe current density, J0, is increased linearly, where J0= 30\nMA/cm2forα= 0.005.\ntime (ns)999.0 999.2 999.4 999.6 999.8 100000.3\n0.2\n0.1\n-0.3-0.2-0.1 current density (106 A/cm2)(a)\ntime (ns)999.0 999.2 999.4 999.6 999.8 10003233current density (106 A/cm2)(b)\nFIG. 6: (a) The transverse electric current densities given\nby Eq. (21) for in-phase synchronization (solid line) and an -\ntiphase synchronization (dotted line). (b) The longitudin al\nelectric current densities given by Eq. (22) for in-phase or\nantiphase synchronization (solid line) and phase differenc e of\na quarter of a period (dotted line).\nis\nJT\nc=j1y+j3y\n=−(χm1xm1y+χ′m1z)J0\n−(χm3xm3y+χ′m3z)J0.(21)\nThen, an oscillating current appears in the transverse di-\nrection for in-phase synchronization, whereas the trans-\nversecurrentbecomeszeroforantiphasesynchronization,\nas shown in Fig. 6(a). The Fourier transformation of\nEq. (21) for in-phase synchronization has peaks at the\nfrequencies of fn= (2n−1)f0(n= 1,2,3,···), where f0\nis the lowest frequency; see Appendix C. Similarly, the\ntotal electric current density in the longitudinal direction\nis\nJL\nc=J0+j1x+j2x\n=J0+/parenleftbig\nχ′′+χm2\n1y/parenrightbig\nJ0+/parenleftbig\nχ′′+χm2\n2y/parenrightbig\nJ0.(22)\nThe oscillation frequency of this current is different for\nthe synchronizations having the phase difference of in-\nphase or antiphase and that of a quarter of a period,as shown in Fig. 6(b). The Fourier transformation of\nEq.(22) has the peaks at fn= 2nf0for in-phase or an-\ntiphase and fn= 4nf0when the phase difference is a\nquarter of a period.\nAn interesting question regarding these numerical re-\nsults is to clarify the reason why the phase difference\nfinally becomes a certain value for a given set of the\nparameters, i.e., which phase difference is an attrac-\ntor of the limit cycle. It is, however, difficult to an-\nswer this question directly due to the following rea-\nson. We note that the present model includes sev-\neral small-valued parameters, α,β,χ,χ′, andχ′′,\nas shown in Eqs. (16) and (17), and is complicated.\nThe torque related to the lowest order of βin these\nequations is the conventional field-like torque given by\n[γβ/planckover2pi1ϑRJ0/(2eMdF)]mℓ×ey. It should be noted that\nthe phase difference is not determined solely by this term\nbecause this lowest order term of the field-like torque\ndoes not include the coupling between the magnetiza-\ntions. Similarly, the attractor is not determined solely\nby the lowest order term of the coupling torque, which\nis [γ/planckover2pi1ϑRJ0/(2eMdF)]χm2\nℓ′ymℓ×(ey×mℓ) in Eq. (16)\nand [γ/planckover2pi1ϑRJ0/(2eMdF)]χmℓ′xmℓ′ymℓ×(ex×mℓ) in Eq.\n(17), because this torque does not include the field-like\ntorque. Their combinations or the higher order terms\nincluding both βand the coupling torques related to χ,\nχ′, andχ′′should be taken into account to answer the\nquestion, which is difficult due to the nonlinearity and\ncomplexity of the LLG equation.\nNevertheless, it is possible to reveal the relation be-\ntween the current and frequency in the synchronized os-\ncillation state by assuming a certain value of the phase\ndifference between the magnetizations. The current-\nfrequency relation has been often measured in the ex-\nperiments, and therefore, it will be useful to develop a\ntheory clarifying the role of the coupling on the current-\nfrequency relation. In the next section, we will investi-\ngate this subject.\nIV. THEORETICAL ANALYSIS OF\nCURRENT-FREQUENCY RELATION\nThe purpose of this section is to develop an analyti-\ncal theory of the synchronization revealing the relation\namongthe current, frequency, and the phase difference of\nthe magnetizations in the synchronized oscillation state.\nA. Basis of analysis\nHere, let us mention the basis of our theoretical anal-\nysis. It is difficult to solve the LLG equation exactly\nbecause of its nonlinearity. Instead, we employ the aver-\naging technique of the LLG equation on constant energy\ncurves63. This approach has been used to study the spin\ntorque switching in thermally activated regions64–68and\nspin torque oscillators69–77, as well as the microwave as-8\nsisted magnetization reversal78,79, but has not been ap-\nplied to the coupled system. This approach is valid when\nthe magnetic energy is changed slowly compared with\nthe oscillation period. We note that only the lowest or-\nder terms in the LLG equation is necessary to derive the\ncurrent-frequency relation, as far as several parameters\nsuch asβare small. Thus, we use the simplified LLG\nequationmaintainingonlythedominantterms. TheLLG\nequation used in this section for the longitudinal geome-\ntry is\ndmℓ\ndt≃−γmℓ×Hℓ−αγmℓ×(mℓ×Hℓ)\n−γ/planckover2pi1ϑRJ0\n2eMdF/parenleftbig\n1+χ′′+χm2\nℓ′y/parenrightbig\nmℓ×(ey×mℓ),\n(23)\nwhereas that for the transverse geometry is\ndmℓ\ndt≃−γmℓ×Hℓ−αγmℓ×(mℓ×Hℓ)\n−γ/planckover2pi1ϑRJ0\n2eMdFmℓ×(ey×mℓ)\n−γ/planckover2pi1ϑRJ0\n2eMdFχmℓ′xmℓ′ymℓ×(ex×mℓ).(24)\nIn the self-oscillation state, the damping torque dur-\ning the precession is balanced with the spin torque, and\nthetorqueduetothemagneticfield, correspondingtothe\nfirst term on the right hand side of Eq. (15), becomes the\ndominantterm determining the magnetizationdynamics.\nThe dynamic trajectory given by this field torque corre-\nsponds to constant energy curves of the energy density,\nE=−M/integraltext\ndmℓ·Hℓ, where its explicit form is\nE=−MHK\n2m2\nℓy+2πM2m2\nℓz. (25)\nThe minimum and saddle points of Eq. (25) are mmin=\n±eyandmsaddle=±ex, where the corresponding energy\ndensities are Emin=−MHK/2 andEsaddle= 0. The\nsolution of mℓprecessing on a constant energy curve is\ndescribed by the Jacobi elliptic function80as (see also\nAppendix C summarizing the derivations)\nmx=/radicalbigg\n1+2E\nMHKsn/bracketleftbigg4K(k)\nτ(E)t+ϕ0,k/bracketrightbigg\n,(26)\nmy=/radicalBigg\n4πM−2E/M\nHK+4πMdn/bracketleftbigg4K(k)\nτ(E)t+ϕ0,k/bracketrightbigg\n,(27)\nmz=/radicalBigg\nHK+2E/M\nHK+4πMcn/bracketleftbigg4K(k)\nτ(E)t+ϕ0,k/bracketrightbigg\n,(28)\nwhereϕ0is the initial phase. The period of mxandmzis\nτ(E), whereasthat of myisτ(E)/2, wherethe precession\nfrequency f(E) = 1/τ(E) is given by\nf(E) =γ/radicalbig\nHK(4πM−2E/M)\n4K(k),(29)whereK(k) is the first kind of complete elliptic integral\nwith the modulus\nk=/radicalBigg\n4πM(HK+2E/M)\nHK(4πM−2E/M). (30)\nNote that Eq. (29) reproduces the ferromagnetic res-\nonance (FMR) frequency, γ/radicalbig\nHK(HK+4πM)/(2π), in\nthe limit of E→Emin. Identifying Eandϕ0corresponds\nto the determination of the initial condition.\nThe averagedtechnique investigates the energy change\nduring a precession on a constant energy curve, which is\nobtained from the LLG equation as\n/contintegraldisplay\ndtdE\ndt=Ws+WL,T\ns+Wα, (31)\nwhere Wsis the energy change by the conventional spin\ntorque due to the spin Hall effect, whereas Wαis the dis-\nsipation due to the damping torque. The integral range\nis over the precession period. The explicit forms of Ws\nandWαare given by67\nWs=/contintegraldisplay\ndtγ/planckover2pi1ϑRJ0\n2edF[ey·Hℓ−(mℓ·ey)(mℓ·Hℓ)]\n=π/planckover2pi1θRJ0(HK+2E/M)\nedF/radicalbig\nHK(HK+4πM),(32)\nWα=−/contintegraldisplay\ndtαγM/bracketleftBig\nH2\nℓ−(mℓ·Hℓ)2/bracketrightBig\n=−4αM/radicalBigg\n4πM−2E/M\nHK/bracketleftbigg2E\nMK(k)+HKE(k)/bracketrightbigg\n,\n(33)\nwhereE(k)isthesecondkindofcompleteellipticintegral.\nOn the other hand, WL,T\nsrepresents the work done\nby the coupling torque in the longitudinal or transverse\ngeometry, corresponding to the last term in Eq. (23)\nor (24). The explicit forms of WL\nsandWT\nsare shown\nin the following sections. For both cases, the relation\nbetween the current and frequency is clarified as follows.\nInthe self-oscillationstate, sincethespin torquebalances\nthe damping torque, the following condition should be\nsatisfied,\n/contintegraldisplay\ndtdE\ndt= 0. (34)\nThe current density J0satisfying Eq. (34) is the current\nnecessaryto excite the self-oscillationon the constanten-\nergy curve of E, and is denoted as J0(E). The relation\nbetween the current and frequency in the self-oscillation\nstate is given by this J0(E) andf(E) given by Eq. (29).\nIt should be emphasized that the current density J0(E)\ndepends on the phase difference between the magnetiza-\ntions through WL,T\ns. We will therefore study the relation\nbetween the phase difference and the current-frequency\nrelation in line with this deduction.9\nB. Transverse geometry\nIn this section, we investigate the current-frequency\nrelation in the transverse geometry. The work done by\nthe coupling torque is defined as\nWT\ns=/contintegraldisplay\ndtγ/planckover2pi1ϑRχJ0\n2edFmℓ′xmℓ′y[ex·Hℓ−(mℓ·ex)(mℓ·Hℓ)].\n(35)\nBefore advancing the calculation, let us briefly men-\ntion the definition of the phase difference in the present\napproach. If the oscillation trajectory is a circle, the\nphase difference is easily defined, i.e., the antiphase cor-\nresponds to ∆ ϕ=π, whereas ∆ ϕ= 0 is the in-phase.\nIn the present case, on the other hand, the oscillation\ntrajectory is described by the elliptic function, as shown\nin Eqs. (26), (27), and (28). In this case, the phase dif-\nference is defined using the elliptic integral K(k), where\n∆ϕ= 0 for the in-phase synchronization, and the an-\ntiphase synchronization corresponds to ∆ ϕ= 2K(k). It\nis useful to note that ∆ ϕ= 2K(k) becomes πin the limit\nofk→0, corresponding to the case that the oscillation\ntrajectory becomes a circle. Similarly, ∆ ϕ=K(k) means\nthat the phase difference is a quarter of a period.\nEquation (35) for an arbitrary phase difference is eval-\nuated by numerically calculating the integral with the\nsolutions of mℓandmℓ′shown in Appendix D. It is,\nhowever, useful to derive the analytical solutions of Eq.\n(35) for specific values of the phase difference. Equa-\ntion (35) for both the in-phase (∆ ϕ= 0) and antiphase\n[∆ϕ= 2K(k)] becomes\nWT\ns=∓π/planckover2pi1ϑRχJ0\n2edF/radicalbig\nHK(HK+4πM)/parenleftbigg\n−2E\nM/parenrightbigg/parenleftbigg\n1+2E\nMHK/parenrightbigg\n,\n(36)\nwhere the double sign means the upper for the in-phase\n(∆ϕ= 0)synchronizationandthelowerfortheantiphase\n[∆ϕ= 2K(k)] synchronization. Equation (36) is zero at\nE=EminandEsaddle, and is negative (positive) for the\nenergy density Ein the rage of Emin< E < E saddle\nwhen ∆ϕ= 0 [2K(k)]. This means that the coupling\ntorque acts as a damping (an antidamping) torque when\nthe phase differenceis in-phase(antiphase). We alsonote\nthatWT\ns= 0 when ∆ ϕ=K(k); i.e., the phase difference\nis a quarter of a period. The calculations necessary to\nobtain these specific values of WT\nsare also summarized\nin Appendix D. We note that the sign changeof WT\nswith\nrespect to the phase difference is related to the fact that\nthe coupling torque in the transverse geometry, Eq. (14),\nhas the angular dependence of mℓ′xmℓ′ymℓ×(ex×mℓ).\nBecause of this angular dependence, the coupling torque\nacts as an anti-damping (a damping) torque when mℓx\nandmℓ′xhave the opposite (same) signs, resulting in the\nincrease (decrease) of the energy supplied to the ferro-\nmagnets by the coupling torque.\nIn summary, the work done by the coupling torque,\nWT\ns, isnegativeandminimizedatthe in-phase(∆ ϕ= 0),\nzero for ∆ ϕ=K(k), and positive and maximized at the\nantiphase [∆ ϕ= 2K(k)].32\n30\n28\n26\n6.05.04.03.02.0\n01.02.03.04.0current density, J 0(E) (106 A/cm 2)(a)\nfrequnecy (GHz)\nphase difference, Δφ/K(k)\n31\n29\n27\n25current density, J 0(E) (106 A/cm 2)(c) (d)\nphase difference, Δ φ/K(k)0 1.0 2.0 3.0 4.027.0127.0227.03current density, J 0(E) (106 A/cm 2)\n6.05.04.03.02.0\n01.02.03.04.0\nfrequnecy (GHz)\nphase difference, Δφ/K(k)(b)\nphase difference, Δ φ/K(k)0 1.0 2.0 3.0 4.028.0028.1028.2028.30current density, J 0(E) (106 A/cm 2)\nFIG. 7: (a) The current density, J0(E), necessary to excite\nthe self-oscillation in the transverse geometry as a functi on of\nthe oscillation frequency f(E) and the phase difference ∆ ϕ\nof the magnetizations. The phase difference is in the unit\nofK(k). (b) Dependence of J0(E) for the transverse geome-\ntry, on ∆ ϕatf(E) = 4.6 GHz. The dotted line represents\nJ0(E) in the absence of the coupling. (c) The relation among\nJ0(E),f(E), and ∆ ϕin the longitudinal geometry. (d) The\ncurrent density J0(E) forf(E) = 4.6 GHz in the longitudinal\ngeometry.\nThe current J0in the transverse geometry is defined\nas\nJ0(E) =2αeMd F\n/planckover2pi1ϑRN\nDT, (37)\nwhereNandDTare defined as\nN=γ/contintegraldisplay\ndt/bracketleftBig\nH2\nℓ−(mℓ·Hℓ)2/bracketrightBig\n, (38)\nDT=γ/contintegraldisplay\ndt[ey·Hℓ−(mℓ·ey)(mℓ·Hℓ)]\n+γχ/contintegraldisplay\ndtmℓ′xmℓ′y[ex·Hℓ−(mℓ·ex)(mℓ·Hℓ)].\n(39)\nThe explicit form of Nis obtained from Eq. (33) as\nN= 4/radicalBigg\n4πM−2E/M\nHK/bracketleftbigg2E\nMK(k)+HKE(k)/bracketrightbigg\n.(40)\nOn the other hand, DTfor the in-phase or antiphase is\nobtained from Eqs. (32) and (36) as\nDT=2π(HK+2E/M)/radicalbig\nHK(HK+4πM)\n∓πχ(−2E/M)[1+2E/(MHK)]/radicalbig\nHK(HK+4πM),(41)10\nwhere the double sign means the upper for the in-phase\nsynchronization and the lower for the antiphase synchro-\nnization.\nFigure 7(a) shows J0(E) as functions of the oscilla-\ntion frequency f(E) and the phase difference ∆ ϕ. The\ncurrent-frequencyrelationinthetransversegeometrycan\nbe obtained from this figure. To reveal the role of the\nphase difference more clearly, we show J0(E) for a cer-\ntain value of f(E)(= 4.6GHz) in Fig. 7(b). Note that\nJ0(E) is smaller than that in the absence of the coupling,\nwhich is shown by the dotted line, and minimized when\n∆ϕ= 2K(k), i.e., theantiphase. Thisisbecausethework\ndone by the coupling torque is positive and maximized at\nthe antiphase. On the other hand, J0(E) is maximized\nat the in-phase, and is larger than that in the absence\nof the coupling because the work done by the coupling\ntorqueis negativeand minimized at the in-phase. We no-\ntice that the phase differences observed in the numerical\nsimulation in Sec. III, i.e., the in-phase and antiphase,\ncorrespond to ∆ ϕsatisfying\n∂J0\n∂∆ϕ= 0, (42)\nor equivalently,\n∂WT\ns\n∂∆ϕ= 0. (43)\nIn other words, the phase differences observed in the nu-\nmerical simulations correspond to those giving the ex-\ntrema of J0(WT\ns).\nC. Longitudinal geometry\nLet us investigate the theoretical relation between the\ncurrent and frequency in the longitudinal geometry. In\nthis case, the averaged LLG equation is given by\n/contintegraldisplay\ndtdE\ndt= (1+χ′′)Ws+WL\ns+Wα,(44)\nwhere WsandWαare given by Eqs. (32) and (33). On\nthe other hand, WL\nsrepresenting the energy change due\nto the longitudinal coupling is defined as\nWL\ns=/contintegraldisplay\ndtγ/planckover2pi1ϑRχJ0\n2edFm2\nℓ′y[ey·Hℓ−(mℓ·ey)(mℓ·Hℓ)].\n(45)\nFor both the in-phase and antiphase, Eq. (45) becomes\n(see Appendix D)\nWL\ns=π/planckover2pi1ϑRχJ0\n2edF/radicalbig\nHK(HK+4πM)\n×(HK+2E/M)[4πM(HK−2E/M)−2HK(2E/M)]\nHK(HK+4πM).\n(46)Onthe other hand, when the phasedifference isaquarter\nof a period [∆ ϕ=K(k)], Eq. (45) becomes\nWL\ns=π/planckover2pi1ϑRχJ0\nedFHK+2E/M\nHK(HK+4πM)/radicalBigg\n−2E\nM/parenleftbigg\n4πM−2E\nM/parenrightbigg\n.\n(47)\nItshouldbeemphasizedthat WL\nsisalwayspositiveforan\narbitrary phase difference. This is because the coupling\ntorqueinEq. (10)alwaysactsasananti-dampingtorque.\nWe can determine the current density J0(E) satisfying\nEq. (34) in the longitudinal geometry, as in the case of\nthe transverse geometry, by replacing DTin Eq. (37)\nwith\nDL=γ(1+χ′′)/contintegraldisplay\ndt[ey·Hℓ−(mℓ·ey)(mℓ·Hℓ)]\n+γχ/contintegraldisplay\ndtm2\nℓ′y[ey·Hℓ−(mℓ·ey)(mℓ·Hℓ)].\n(48)\nThe explicit form of DLfor both the in-phase and an-\ntiphase is obtained from Eqs. (32) and (46) as\nDL=2π(1+χ′′)(HK+2E/M)/radicalbig\nHK(HK+4πM)\n+πχ(HK+2E/M)[4πM(HK−2E/M)−2HK(2E/M)]\n[HK(HK+4πM)]3/2,\n(49)\nwhereas that when the phase difference is a quarter of a\nperiod is obtained from Eqs. (32) and (47) as\nDL=2π(1+χ′′)(HK+2E/M)/radicalbig\nHK(HK+4πM)\n+2πχ(HK+2E/M)\nHK(HK+4πM)/radicalBigg\n−2E\nM/parenleftbigg\n4πM−2E\nM/parenrightbigg\n.\n(50)\nFigure 7(c) shows J0(E) as functions of f(E) and ∆ϕ.\nThe current-frequency relation in the longitudinal geom-\netry can be obtained from this figure. It is noted that\nJ0(E) is always smaller than that in the absence of the\ncoupling because the coupling torque in the longitudinal\ngeometry always points to the anti-damping direction,\nand therefore, the work done by the coupling torque is\nalways positive. Figure 7(b) shows J0(E) as a function\nof ∆ϕat a certain value of f(E). As shown, J0(E) has\nminima at both the in-phase (∆ ϕ= 0) and the antiphase\n[∆ϕ= 2K(k)], whereas it is maximized when the phase\ndifference is a quarter of a period [∆ ϕ=K(k)]. We again\nnotice that these phase differences found in the numer-\nical simulations in Sec. III correspond to ∆ ϕsatisfying\n∂J0(E)/∂∆ϕ= 0, or equivalently,\n∂WL\ns\n∂∆ϕ= 0. (51)11\nD. Phase differences in stable synchronization and\nfixed points of effective potential\nEquation (31) describes the slow change of the mag-\nnetic energy in the oscillation state. The magnetization\ndynamics is regarded as a motion of a point particle in\nan effective potential given by its right-hand side. Equa-\ntions (43) and (51) correspond to the stability conditions\nofthe point particlein this effective potential. Therefore,\nthe phase difference found in the numerical simulation fi-\nnally converges to one of these values satisfying Eq. (43)\nor Eq. (51). Whether the in-phase, antiphase, or the\nphase difference with a quarter of a period becomes the\nattractor depends on the higher order terms of the small\nparameters, as well as the initial states of the magnetiza-\ntions, asmentioned at the end ofSec. III. This discussion\nis beyond the scope of this paper.\nE. Instability threshold\nAt mentioned at the beginning of Sec. III, the in-plane\nself-oscillation for a single ferromagnet is stabilized when\nthe currentdensity is in the rangeof Jc< J0< J∗, where\nJcandJ∗are given by Eqs. (19) and (20), respectively.\nAt the end of this section, let us briefly discuss the effect\nof the coupling on these scaling currents.\nLet us remindthe readersthat Jcis the currentdensity\nnecessarytodestabilizethemagnetizationinequilibrium,\nwhereas J∗is the current necessary to overcome the en-\nergy barrier, Esaddle−Emin. These current densities are\ntheoretically defined as67\nJc= lim\nE→EminJ0(E), (52)\nJ∗= lim\nE→EsaddleJ0(E). (53)\nIt is confirmed that Eqs. (19) and (20) are reproduced\nby substituting Eqs. (32) and (33) in the definition of\nJ0(E) in the absence of the coupling.\nOn the other hand, in the presence of the transverse\ncoupling, it is confirmed from Eq. (36) that a factor\n[1−(χ/2)] should be multiplied to the denominator of\nEq. (19) when the phase difference between the mag-\nnetizations is in-phase, whereas this factor is replaced\nby [1+( χ/2)] when the phase difference is antiphase.\nThe other scaling current, J∗, is unchanged for these\nphase differences. In the longitudinal geometry, we see\nfrom Eqs. (46) and (47) that the factor (1+ χ+χ′′)\nshould be multiplied to the denominator of Eq. (19)\nwhen the phase difference is in-phase, antiphase, or a\nquarter of a period, whereas, for J∗, the factor becomes\n1+χ′′+(χ/2)[4πM/(HK+4πM)] for in-phase and an-\ntiphase, and 1+ χ′′when the phase difference is a quarter\nof a period.V. CONCLUSION\nIn conclusion, the coupled magnetization dynamics in\nthe ferromagnets through the spin Hall magnetoresis-\ntance effect was investigated. The coupling appears in\nboth the longitudinal and transverse directions of the\nalignment of the ferromagnets. The in-phase or an-\ntiphase synchronization of the magnetization oscillation\nwas found in the transversegeometry by solving the LLG\nequation numerically. On the other hand, in addition to\nthem, the synchronization having the phase difference of\na quarter of a period is also found in the longitudinal ge-\nometry. It wasshownthatthesephasedifferencesdepend\non the values of the damping constant and the field-like\ntorque strength. The analytical theory revealing the re-\nlationamongthe current, frequency, and phasedifference\nwas also developed. It was shown that the phase differ-\nences observed in the numerical simulations correspond\nto that giving the fixed points of the energy supplied by\nthe coupling torque.\nAcknowledgement\nThe author is grateful to Takehiko Yorozu for his con-\ntributions to the analytical calculations, and to Hitoshi\nKubota, Sumito Tsunegi, Yoichi Shiota, Shingo Tamaru,\nTazumi Nagasawa, Kiwamu Kudo, and Yoji Kawamura\nfor valuable discussions. The author is also thankful to\nSatoshiIba, AurelieSpiesser,AtsushiSugihara, Takahide\nKubota, Hiroki Maehara, and Ai Emura for their sup-\nport and encouragement. This work was supported by\nJSPS KAKENHI Grant-in-Aid for Young Scientists (B)\n16K17486.\nAppendix A: Values of parameters in numerical\nsimulations\nThe exact values of the parameters used in the sim-\nulations, evaluated from the parameters found in the\nexperiment34, areϑR= 0.16680863, β=−0.00973617,\nχ= 0.009525272, χ′=−0.000152995, and χ′′=\n0.03516089for g↑↓\ni/S= 1.0 nm−2. In the main text, β=\n−0.01 andβ= +0.01 correspond to β=−0.00973617\nandβ= +0.00973617, respectively. Strictly speaking,\nthe change of the value of g↑↓\niaffects not only βbut\nalso other quantities such as ϑR,χ, andχ′. We, how-\never, change the value of βonly in the numerical simu-\nlation, for simplicity, because the results do not change\nsignificantly unless |g↑↓\ni/g↑↓\nr| ≪1. The LLG equation\nwith these parameters is solved by using the fourth-order\nRunge-Kutta method from t= 0 tot= 1µs with the\ntime step of ∆ t= 20 fs; i.e., the number of the time\nmesh isNt= 5×107.\nThe presentsystem has twostable states at mℓ=±ey.\nFor convention, we assume that the magnetizations ini-\ntially stay near one equilibrium, mℓ= +ey. For in-phase12\nsynchronizations, such as those shown in Figs. 2(c) and\n2(d), the zcomponents are also synchronized with in-\nphase, i.e., mℓz(t) =mℓ′z(t). On the other hand, for\nantiphase synchronization shown in, for example in Figs.\n2(e) and 2(f), the zcomponents are also synchronized\nwith antiphase, mℓz(t) =−mℓ′z(t).\nThe algorithm evaluating the phase differences shown\nin Figs. 3 and 5 from the discrete numerical data\nis as follows. We gathered Ni= 216= 65536 data\nofmℓ(t) (ℓ= 1,2,3) from t= (Nt−Ni+1)∆tto\nt=Nt∆t= 1µs. Then, the averaged periods\nTℓof the oscillation of each magnetization were eval-\nuated from the peaks of mℓ(t) in this time range\nasTℓ= [(tℓ,a−tℓ,a−1)+···+(tℓ,2−tℓ,1)]/(Nℓ−1) =\n(tℓ,a−tℓ,1)/(Nℓ−1), where Nℓis the number of the\npeaks in mℓ(t), whereas tℓ,ais the time corresponding\nto thea-th peak. Then, the phase difference is evalu-\nated as ∆ ϕ=/summationtextN′\na=1|tℓ,a−tℓ′,a|//parenleftbig\nN′¯T/parenrightbig\n, where N′=\nmin[Nℓ,Nℓ′] and¯T= (Tℓ+Tℓ′)/2 with (ℓ,ℓ′) = (1,3)\nfor the transverse geometry, whereas that is (1 ,2) for\nthe longitudinal geometry. For the in-phase synchro-\nnization, this ∆ ϕis zero because tℓ,a=tℓ′,a. When\nthe phase difference is antiphase, ∆ ϕ= 0.50 because\n|tℓ,a−tℓ′,a|=¯T/2 in this case. Similarly, ∆ ϕis 0.25\nwhen the phase difference is a quarter of a period.\nNote that the critical current density to excite the self-\noscillation, given by Eq. (19), is proportional to the\ndamping constant α. Therefore, the value of the current\ndensity should be increased to observe the self-oscillation\nwhenαis varied, as in the case of Figs. 3 and 5. In these\nfigures,J0isassumedas n×30MA/cm2forα=n×0.005\n(n= 1−6).\nThe numerical simulation in Fig. 2(e) indicates that\nthe antiphase synchronization is an attractor for β=\n+0.01. An exception is that if the initial conditions are\nset to be identical, the final state becomes in-phase syn-\nchronization due to the symmetry of the LLG equation\nwith respect to the change of ( ℓ,ℓ′)→(ℓ′,ℓ). Since Eq.\n(43) is satisfied, the phase difference is fixed to in-phase\neven if it is unstable. Similar situations occur in other\ncases for such specific initial conditions.\nAppendix B: Derivation of coupling torque in\ntransverse geometry\nIn a ferromagnetic/nonmagnetic bilayer, the spin cur-\nrent density flowing in the idirection ( i=x,y,z) with\nthe spin polarization in the νdirection is related to the\nelectrochemical potential ¯ µNand the spin accumulation\nδµNvia\nJsiν,N=−/planckover2pi1σN\n2e2∂iδµN,ν−/planckover2pi1ϑσN\n2e2ǫiνj∂j¯µN,(B1)\nwhere∂j¯µN/eis the electric field in the jdirection, and\ntherefore, σN∂j¯µN/eis the electric current density. Weassume that this equation is extended to\nJ(ℓ)\nszν,N=−/planckover2pi1σN\n2e2∂zδµ(ℓ)\nN,ν+/planckover2pi1ϑ\n2e(J0δνy−jℓ′yδνx),(B2)\nin the transverse geometry, where J(ℓ)\nszν,Nis the spin cur-\nrent density flowing near the F ℓ/N interface in the zdi-\nrection with the spin polarization in the νdirection. The\nspin accumulation obeys the diffusion equation, and the\nboundary conditions of the diffusion equation are given\nby the spin current density at the boundaries. Using Eq.\n(B2), the solution of the spin accumulation is given by40\nδµ(ℓ)\nN,ν=2π\n(gN/S)sinh(dN/λN)/braceleftbigg\n−JFℓ/N\nszνcosh/parenleftbiggz+dN\nλN/parenrightbigg\n−/planckover2pi1ϑ\n2e(J0δνy−jℓ′yδνx)/bracketleftbigg\ncosh/parenleftbiggz\nλN/parenrightbigg\n−cosh/parenleftbiggz+dN\nλN/parenrightbigg/bracketrightbigg/bracerightbigg\n,\n(B3)\nwhere we assume that the nonmagnet is in the region of\n−dN≤z≤0. The spin current density, JFℓ/N\nszν, at the\nFℓ/N interface is given by30,40\nJFℓ/N\ns=/planckover2pi1ϑg∗\n2egNtanh/parenleftbiggdN\n2λN/parenrightbigg\n(J0mℓy−jℓ′ymℓx)mℓ\n+/planckover2pi1\n2eJ0[ϑRmℓ×(ey×mℓ)+ϑIey×mℓ]\n−/planckover2pi1\n2ejℓ′y[ϑRmℓ×(ex×mℓ)+ϑIex×mℓ],\n(B4)\nwhere the vector notation in boldface represents the di-\nrection of the spin polarization, whereas the spatial di-\nrection of Eq. (B4) is defined as the positive direction,\ni.e., from the nonmagnet to the ferromagnet.\nOn the other hand, the electric current density in the\nnonmagnet flowing in the idirection is given by\nJci,N=σN\ne∂i¯µN−ϑσN\neǫijν∂jδµN,ν.(B5)\nIn the present case, the electric current density near the\nFℓ/N interface flowing in the ydirection becomes\nJ(ℓ)\ncy,N=jℓ′y−ϑσN\ne∂zδµ(ℓ)\nN,x. (B6)\nSubstituting Eqs. (B3) and (B4) into Eq. (B6),\nand averaging along the zdirection as J(ℓ)\ncy,N=\n(1/dN)/integraltext0\n−dNJ(ℓ)\ncy,Ndz, we find that\nJ(ℓ)\ncy,N=/parenleftbig\n1+χ′′+χm2\nℓx/parenrightbig\njℓ′y\n−(χmℓxmℓy+χ′mℓz)J0.(B7)\nThe conservation law of the electric current along the y\ndirection requires that J(ℓ)\ncy,N=jℓy+jℓ′y. Solving this\nequation for ℓ= 1 and 3, we obtain Eq. (13). The spin\ntorque is defined from Eq. (B4) as\nTℓ=−γ\nMdFmℓ×/parenleftBig\nJFℓ/N\ns×mℓ/parenrightBig\n.(B8)13\nAppendix C: Analytical solution of magnetization on\na constant energy curve\nIn this appendix, we shows the derivation of the ana-\nlyticalsolutionofthe magnetizationonaconstantenergy\ncurve. Forsimplicity, weremovethesubscript ℓ(= 1,2,3)\ndistinguishing the ferromagnets. The magnetization dy-\nnamics on a constant energy curve is described by the\nLandau-Lifshitz (LL) equation\ndm\ndt=−γm×H. (C1)\nThe magnetic field, Eq. (18), is related to the magnetic\nenergy density EviaE=−M/integraltext\ndm·H, as mentioned\nin the main text. Using the relation m2\nx+m2\ny+m2\nz= 1,\nEq. (25) is rewritten as\nm2\nz+HK\nHK+4πMm2\nx=2E/M+HK\nHK+4πM.(C2)\nThisequationindicatesthat mzandmxcanbe expressed\nasmz=v′cosuandmx= (v′/v)sinu, respectively,\nwherevandv′are defined as v2=HK/(HK+4πM)\nandv′= (2E/M+HK)/(HK+4πM). Then, du/dt=\n(du/dsinu)(dsinu/dt) = (1/cosu)[d(v/v′)mx/dt] =\n(v/mz)(dmx/dt), which becomes, from Eq. (C1),\ndu\ndt=γ(HK+4πM)vmy. (C3)\nIntroducing new variable w= sinu, this equation gives\ndw/radicalbig\n(1−w2)(1−k2w2)=γ(HK+4πM)v/radicalbig\n1−v′2dt,\n(C4)\nwhere the modulus kis given by Eq. (30). The mod-\nulus monotonically varies from 0 to 1 by changing the\nenergy density Efrom its minimum Eminto saddle\nEsaddle. We also notice that ( HK+4πM)v√\n1−v′2=/radicalbig\nHK(4πM−2E/M). Equation (C4) indicates that w\nis given by\nw= sn/bracketleftBig\nγ/radicalbig\nHK(4πM−2E/M)t+ϕ0,k/bracketrightBig\n,(C5)\nwheresn( u,k)istheJacobiellipticfunction, and ϕ0isthe\ninitial phase determined by the initial condition. Using\nthe relations sn2(u,k) + cn2(u,k) = 1 and dn2(u,k) =/radicalbig\n1−k2sn2(u,k), we find that the solution of mon the\nconstant energy curve is given by Eqs. (26), (27), and\n(28).\nThe peak frequencies of the Fourier transformation of\nEq. (21) in the transverse geometry are discussed as fol-\nlows. We note that Eq. (21) for in-phase synchronization\nis proportional to mℓx(t)mℓy(t) andmℓz(t). Substituting\nthe following formulas80,\nsn(u,k) =2π\nkK(k)∞/summationdisplay\nm=0qm+1/2\n1−q2m+1sin/bracketleftbigg(2m+1)πu\n2K(k)/bracketrightbigg\n,\n(C6)cn(u,k) =2π\nkK(k)∞/summationdisplay\nm=0qm+1/2\n1+q2m+1cos/bracketleftbigg(2m+1)πu\n2K(k)/bracketrightbigg\n,\n(C7)\ndn(u,k) =π\n2K(k)\n+2π\nK(k)∞/summationdisplay\nm=0qm+1\n1+q2(m+1)cos/bracketleftbigg(m+1)πu\nK(k)/bracketrightbigg\n,\n(C8)\nto Eqs. (26), (27), (28), where q=\nexp/bracketleftbig\n−πK/parenleftbig√\n1−k2/parenrightbig\n/K(k)/bracketrightbig\n, it is found that the\npeak frequencies of Eq. (21) appear at fn= (2n−1)f0\n(n= 1,2,3,···), where the lowest frequency f0is given\nby Eq. (29).\nIn the longitudinal geometry, Eq. (22) is proportional\ntom2\n1y+m2\n2y. When the phase difference of the magne-\ntizations is in-phase or antiphase, it becomes 2 m2\n1y. In\nthis case, using the formula81\ndn2(u,k) =E(k)\nK(k)+2π2\nK2(k)∞/summationdisplay\nm=1mqm\n1−q2mcos/bracketleftbiggmπu\nK(k)/bracketrightbigg\n,\n(C9)\nit is found that the Fourier transformation of Eq.\n(22) has the peaks at fn= 2nf0. On the other\nhand, when the phase difference is a quarter of a pe-\nriod, Eq. (22) is proportional to g(u)≡dn2(u,k) +\ndn[u+K(k),k] = dn2(u,k) +/bracketleftbig/parenleftbig\n1−k2/parenrightbig\n/dn2(u,k)/bracketrightbig\n. We\nnotice that g[u+K(k)] =g(u), indicating that the\nFourier transformation of Eq. (22) in this case has the\npeaks at fn= 4nf0.\nAppendix D: Details of calculations of Eqs. (35) and\n(45)\nEquations (35) and (45) can be calculated by substi-\ntuting the solution of mℓon a constant energy curve to\nthe integrals. As emphasized in the main text, the phase\ndifference ∆ ϕbetween the magnetizations is an impor-\ntant quantity. According to Eqs. (26), (27), and (28), we\nsetmℓandmℓ′as\nmℓx=/radicalbigg\n1+2E\nMHKsn/bracketleftbigg4K(k)\nτ(E)t,k/bracketrightbigg\n,(D1)\nmℓy=/radicalBigg\n4πM−2E/M\nHK+4πMdn/bracketleftbigg4K(k)\nτ(E)t,k/bracketrightbigg\n,(D2)\nmℓz=/radicalBigg\nHK+2E/M\nHK+4πMcn/bracketleftbigg4K(k)\nτ(E)t,k/bracketrightbigg\n,(D3)\nand\nmℓ′x=/radicalbigg\n1+2E\nMHKsn/bracketleftbigg4K(k)\nτ(E)t+∆ϕ,k/bracketrightbigg\n,(D4)14\nmℓ′y=/radicalBigg\n4πM−2E/M\nHK+4πMdn/bracketleftbigg4K(k)\nτ(E)t+∆ϕ,k/bracketrightbigg\n,(D5)\nmℓ′z=/radicalBigg\nHK+2E/M\nHK+4πMcn/bracketleftbigg4K(k)\nτ(E)t+∆ϕ,k/bracketrightbigg\n.(D6)\nThe value of ∆ ϕvaries in the rage of 0 ≤∆ϕ <4K(k).\n∆ϕ= 0 corresponds to the in-phase synchronization,\nwhereas ∆ ϕis 2K(k) for the antiphase synchronization.\nThe analytical formulas of Eqs. (35) and (45) for\nthe in-phase and antiphase synchronizations can be ob-\ntained as follows. First, since the elliptic functions sat-\nisfy sn[u+2K(k),k] =−sn(u,k), cn[u+2K(k),k] =\n−cn(u,k), and dn[ u+2K(k),k] = dn( u,k),WT\nshas\nthe same magnitude but different sign for ∆ ϕ= 0 and\n∆ϕ= 2K(k), whereas WL\nsis the same for the in-phase\nand antiphase. Therefore, it is sufficient to calculate WT\ns\nandWL\nsfor the in-phase case. In this case, it is unneces-\nsary to distinguish mℓandmℓ′. Next, it should be noted\nthat Eq. (35) includes the following two integrals,\n/contintegraldisplay\ndtm2\nℓxm3\nℓy∝/integraldisplay\ndusn2(u,k)dn3(u,k),(D7)\n/contintegraldisplay\ndtm2\nℓxmℓym2\nℓz∝/integraldisplay\ndusn2(u,k)cn2(u,k)dn(u,k).\n(D8)\nBy replacing the integral variable from uwithx=\nsn(u,k), and noting that du=dx//radicalbig\n(1−x2)(1−k2x2),\nthese integrals are calculated as\n/integraldisplay\ndusn2(u,k)dn3(u,k) =/integraldisplay\ndxx2(1−k2x2)√\n1−x2\n=x√\n1−x2/bracketleftbig\n−4+k2/parenleftbig\n3+2x2/parenrightbig/bracketrightbig\n+/parenleftbig\n4−3k2/parenrightbig\nsin−1x\n8,\n(D9)\n/integraldisplay\ndusn2(u,k)cn2(u,k)dn(u,k) =/integraldisplay\ndxx2/radicalbig\n1−x2\n=x√\n1−x2/parenleftbig\n−1+2x2/parenrightbig\n+sin−1x\n8.\n(D10)\nUsing these integrals, Eq. (36) is obtained. On the other\nhand, Eq. (45) includes the following three integrals,\n/contintegraldisplay\ndtm3\nℓy∝/integraldisplay\ndudn3(u,k)\n=/integraldisplay\ndx1−k2x2\n√\n1−x2\n=k2x√\n1−x2+/parenleftbig\n2−k2/parenrightbig\nsin−1x\n2,(D11)\n/contintegraldisplay\ndtm5\nℓy∝/integraldisplay\ndudn5(u,k)\n=/integraldisplay\ndx/parenleftbig\n1−k2x2/parenrightbig2\n√\n1−x2\n=k2x√\n1−x2/bracketleftbig\n8−k2/parenleftbig\n3+2x2/parenrightbig/bracketrightbig\n+/parenleftbig\n8−8k2+3k4/parenrightbig\nsin−1x\n8,\n(D12)/contintegraldisplay\ndtm3\nℓym2\nℓz∝/integraldisplay\ndudn3(u,k)cn2(u,k)\n=/integraldisplay\ndx/radicalbig\n1−x2/parenleftbig\n1−k2x2/parenrightbig\n=x√\n1−x2/bracketleftbig\n4+k2/parenleftbig\n1−2x2/parenrightbig/bracketrightbig\n+/parenleftbig\n4−k2/parenrightbig\nsin−1x\n8.\n(D13)\nUsing these integrals, Eq. (46) is obtained.\nWhen the phase difference is a quarter\nof a period (∆ ϕ=K(k)), the relations,\nsn[u+K(k),k] = cn(u,k)/dn(u,k), cn[u+K(k),k] =\n−√\n1−k2cn(u,k)/dn(u,k), dn[ u+K(k),k] =√\n1−k2/dn(u,k), are used to evaluate Eqs. (35)\nand (45). In this case, we notice that all the integrands\nin Eq. (35) become odd functions of t, and the integrals\nover 0≤t≤τare zero. Therefore, WT\ns= 0 for\n∆ϕ=K(k). On the other hand, the following integrals\nare necessary to obtain Eq. (47),\n/contintegraldisplay\ndtm2\nℓ′ymℓy∝/integraldisplaydu\ndn(u,k)\n=/integraldisplay1\n0dx√\n1−x2(1−k2x2)\n=tan−1/bracketleftBig/radicalbig\n(1−k2)/(1−x2)x/bracketrightBig\n√\n1−k2,(D14)\n/contintegraldisplay\ndtm2\nℓ′ym3\nℓy∝/integraldisplay\ndudn(u,k)\n=/integraldisplaydx√\n1−x2\n= sin−1x,(D15)\n/contintegraldisplay\ndtm2\nℓ′ymℓym2\nℓz∝/integraldisplay\nducn2(u,k)\ndn(u,k)\n=/integraldisplay\ndx√\n1−x2\n1−k2x2\n=sin−1x−√\n1−k2tan−1/bracketleftBig/radicalbig\n(1−k2)/(1−x2)x/bracketrightBig\nk2.\n(D16)15\n1A. Hubert and R. Sch¨ afer, Magnetic Domains (Springer,\nBerlin, 1998).\n2K. Kudo, T. Nagasawa, K. Mizushima, H. Suto, and\nR. Sato, Appl. Phys. Express 3, 043002 (2012).\n3J. C. Slonczewski, J. Magn. Magn. Mater. 159, L1 (1996).\n4L. Berger, Phys. Rev. B 54, 9353 (1996).\n5J. A. Katine, F. J. Albert, R. A. Buhrman, E. B. Myers,\nand D. C. Ralph, Phys. Rev. Lett. 84, 3149 (2000).\n6S. I. Kiselev, J. C. Sankey, I. N. Krivorotov, N. C. Emley,\nR. J. Schoelkopf, R. A. Buhrman, and D. C. Ralph, Nature\n425, 380 (2003).\n7J. Grollier, V. Cros, H. Jaffres, A. Hamzic, J. M. George,\nG. Faini, J. B. Youssef, H. L. Gall, and A. Fert, Phys. Rev.\nB67, 174402 (2003).\n8W. H. Rippard, M. R. Pufall, S. Kaka, S. E. Russek, and\nT. J. Silva, Phys. Rev. Lett. 92, 027201 (2004).\n9H. Kubota, A. Fukushima, Y. Ootani, S. Yuasa,\nK. Ando, H. Maehara, K. Tsunekawa, D. D. Djayaprawira,\nN. Watanabe, and Y. Suzuki, Jpn. J. Appl. Phys. 44,\nL1237 (2005).\n10A. A. Tulapurkar, Y. Suzuki, A. Fukushima, H. Kub-\nota, H. Maehara, K. Tsunekawa, D. D. Djayaprawira,\nN. Watanabe, and S. Yuasa, Nature 438, 339 (2005).\n11D. Houssameddine, U. Ebels, B. Dela¨ et, B. Rodmacq,\nI. Firastrau, F. Ponthenier, M. Brunet, C. Thirion, J.-\nP. Michel, L. Prejbeanu-Buda, et al., Nat. Mater. 6, 447\n(2007).\n12T. Taniguchi and H. Imamura, Phys. Rev. B 78, 224421\n(2008).\n13A. Slavin and V. Tiberkevich, IEEE. Trans. Magn. 45,\n1875 (2009).\n14H. Kubota, K. Yakushiji, A. Fukushima, S. Tamaru,\nM. Konoto, T. Nozaki, S. Ishibashi, T. Saruya, S. Yuasa,\nT. Taniguchi, et al., Appl. Phys. Express 6, 103003 (2013).\n15Y. Tserkovnyak, A. Brataas, and G. E. W. Bauer, Phys.\nRev. B67, 140404 (2003).\n16B. Heinrich, Y. Tserkovnyak, G. Woltersdorf, A. Brataas,\nR. Urban, and G. E. W. Bauer, Phys. Rev. Lett. 90,\n187601 (2003).\n17T. Kimura, Y. Otani, and J. Hamrle, Phys. Rev. Lett. 96,\n037201 (2006).\n18L. Liu, O. J. Lee, T. J. Gudmundsen, D. C. Ralph, and\nR. A. Buhrman, Phys. Rev. Lett. 109, 096602 (2012).\n19L. Liu, C.-F. Pai, D. C. Ralph, and R. A. Buhrman, Phys.\nRev. Lett. 109, 186602 (2012).\n20C.-F. Pai, L. Liu, Y. Li, H. W. Tseng, D. C. Ralph, and\nR. A. Buhrman, Appl. Phys. Lett. 101, 122404 (2012).\n21J. Kim, J. Sinha, M. Hayashi, M. Yamanouchi, S. Fukami,\nT. Suzuki, S. Mitani, and H. Ohno, Nat. Mater. 12, 240\n(2012).\n22P. M. Haney, H.-W.Lee, K.-J. Lee, A.Manchon, andM. D.\nStiles, Phys. Rev. B 87, 174411 (2013).\n23K.-S. Lee, S.-W. Lee, B.-C. Min, and K.-J. Lee, Appl.\nPhys. Lett. 102, 112410 (2013).\n24K. Garello, C. O. Avci, I. M. Miron, M. Baumgartner,\nA. Ghosh, S. Auffret, O. Boulle, G. Gaudin, and P. Gam-\nbardella, Appl. Phys. Lett. 105, 212402 (2014).\n25M. Cubukcu, O. Boulle, M. Drouard, K. Garello, C. O.\nAvci, I. M. Miron, J. Langer, B. Ocker, P. Gambardella,\nand G. Gaudin, Appl. Phys. Lett. 104, 042406 (2014).\n26S. Y. Huang, X. Fan, D. Qu, Y. P. Chen, W. G. Wang,J. Wu, T. Y. Chen, J. Q. Xiao, and C. L. Chien, Phys.\nRev. Lett. 109, 107204 (2012).\n27H. Nakayama, M. Althammer, Y.-T. Chen, K. Uchida,\nY. Kajiwara, D. Kikuchi, T. Ohtani, S. Gepr¨ ags, M. Opel,\nS. Takahashi, et al., Phys. Rev. Lett. 110, 206601 (2013).\n28M. Althammer, S. Meyer, H. Nakayama, M. Schreier,\nS. Altmannshofer, M. Weiler, H. Huebl, S. Gepr¨ ags,\nM. Opel, R. Gross, et al., Phys. Rev. B 87, 224401 (2013).\n29C.Hahn, G.deLoubens, O.Klein, M.Viret, V.V.Naletov,\nand J. BenYoussef, Phys. Rev. B 87, 174417 (2013).\n30Y.-T.Chen, S. Takahashi, H. Nakayama, M. Althammer,\nS. T. B. Goennenwein, E. Saitoh, and G. E. W. Bauer,\nPhys. Rev. B 87, 144411 (2013).\n31J. Liu, T. Ohkubo, S. Mitani, K. Hono, and M. Hayashi,\nAppl. Phys. Lett. 107, 232408 (2015).\n32C. O. Avci, K. Garello, A. Ghosh, M. Gabureac, S. F.\nAlvarado, and P. Gambardella, Nat. Phys. 11, 570 (2015).\n33S. Cho, S.-H. C. Baek, K.-D. Lee, Y. Jo, and B.-G. Park,\nSci. Rep. 5, 14668 (2015).\n34J. Kim, P. Sheng, S. Takahashi, S. Mitani, andM. Hayashi,\nPhys. Rev. Lett. 116, 097201 (2016).\n35M. I. Dyakonov and V. I. Perel, Phys. Lett. A 35, 459\n(1971).\n36J. E. Hirsch, Phys. Rev. Lett. 83, 1834 (1999).\n37Y. K. Kato, R. C. Myers, A. C. Gossard, and D. D.\nAwschalom, Science 306, 1910 (2004).\n38N. Locatelli, V. Cros, and J. Grollier, Nat. Mater. 13, 11\n(2014).\n39J. Grollier, D. Querlioz, and M. D. Stiles, Proc. IEEE 104,\n2024 (2016).\n40T. Taniguchi, Phys. Rev. B 94, 174440 (2016).\n41A. Brataas, Y. V. Nazarov, and G. E. W. Bauer, Phys.\nRev. Lett. 84, 2481 (2000).\n42A. Brataas, Y. V. Nazarov, and G. E. W. Bauer, Eur.\nPhys. J. B 22, 99 (2001).\n43M. Zwierzycki, Y. Tserkovnyak, P. J. Kelly, A. Brataas,\nand G. E. W. Bauer, Phys. Rev. B 71, 064420 (2005).\n44S. Tsunegi, H. Kubota, S. Tamaru, K. Yakushiji,\nM. Konoto, A. Fukushima, T. Taniguchi, H. Arai, H. Ima-\nmura, and S. Yuasa, Appl. Phys. Express 7, 033004 (2014).\n45B. Hillebrands and A. Thiaville, eds., Spin Dynamics in\nConfined Magnetic Structures III (Springer, Berlin, 2006).\n46S. Kaka, M. R. Pufall, W. H. Rippard, T. J. Silva, S. E.\nRussek, and J. A. Katine, Nature 437, 389 (2005).\n47F. B. Mancoff, N. D. Rizzo, B. N. Engel, and S. Tehrani,\nNature437, 393 (2005).\n48J. Grollier, V. Cros, and A. Fert, Phys. Rev. B 73, 060409\n(2006).\n49A. N. Slavin and V. S. Tiberkevich, Phys. Rev. B 74,\n104401 (2006).\n50J. Persson, Y. Zhou, and J. Akerman, J. Appl. Phys. 101,\n09A503 (2007).\n51A. Ruotolo, V. Cros, B. Georges, A. Dussaux, J. Grollier,\nC. Deranlot, R. Guillemet, K. Bouzehouane, S. Fusil, and\nA. Fert, Nat. Nanotechnol. 4, 528 (2009).\n52B. Georges, J. Grollier, V. Cros, and A. Fert, Appl. Phys.\nLett.92, 232504 (2008).\n53Y. Zhou and J. Akerman, Appl. Phys. Lett. 94, 112503\n(2009).\n54S. Urazhdin, P. Tabor, V. Tiberkevich, and A. Slavin,\nPhys. Rev. Lett. 105, 104101 (2010).16\n55S. Sani, J. Persson, S. M. Mohseni, Y. Pogoryelov, P. K.\nMuduli, A. Eklund, G. Malm, M. K¨ all, A. Dmitriev, and\nJ. Akerman, Nat. Commun. 4, 2731 (2013).\n56V. E. Demidov, H. Ulrichs, S. V. Gurevich, S. O. Demokri-\ntov, V. S. Tiberkevich, A. N. Slavin, A. Zholud, and\nS. Urazhdin, Nat. Commun. 5, 3179 (2014).\n57N. Locatelli, A. Hamadeh, F. A. Araujo, A. D. Belanovsky,\nP. N. Skirdkov, R. Lebrun, V. V. Naletov, K. A. Zvezdin,\nM. Munoz, J. Grollier, et al., Sci. Rep. 5, 17039 (2015).\n58G. Khalsa, M. D. Stiles, and J. Grollier, Appl. Phys. Lett.\n106, 242402 (2015).\n59T. Kendziorczyk and T. Kuhn, Phys. Rev. B 93, 134413\n(2016).\n60S.Tsunegi, E.Grimaldi, R.Lebrun, H.Kubota, A.S.Jenk-\nins, K. Yakushiji, A. Fukushima, P. Bortolotti, J. Grollier ,\nS. Yuasa, et al., Sci. Rep. 6, 26849 (2016).\n61A. A. Awad, P. D¨ urrenfeld, A. Houshang, M. Dvornik,\nE. Iacoca, R. K. Dumas, and J. Akerman, Nat. Phys. 13,\n292 (2017).\n62K. Kudo and T. Morie, Appl. Phys. Express 10, 043001\n(2017).\n63G. Bertotti, I. Mayergoyz, and C. Serpico, Nonlinear Mag-\nnetization Dynamics in Nanosystems (Elsevier, Oxford,\n2009).\n64D.M. ApalkovandP.B. Visscher, Phys.Rev.B 72, 180405\n(2005).\n65G. Bertotti, I. D. Mayergoyz, and C. Serpico, J. Appl.\nPhys.99, 08F301 (2006).\n66K. A. Newhall and E. V. Eijnden, J. Appl. Phys. 113,\n184105 (2013).\n67T. Taniguchi, Y. Utsumi, M. Marthaler, D. S. Golubev,and H. Imamura, Phys. Rev. B 87, 054406 (2013).\n68D. Pinna, A. D. Kent, and D. L. Stein, Phys. Rev. B 88,\n104405 (2013).\n69G. Bertotti, C. Serpico, I. D. Mayergoyz, A. Magni,\nM. d’Aquino, and R. Bonin, Phys. Rev. Lett. 94, 127206\n(2005).\n70C. Serpico, M. d’Aquino, G. Bertotti, and I. D. Mayergoyz,\nJ. Magn. Magn. Mater. 290, 502 (2005).\n71U. Ebels, D. Houssameddine, I. Firastrau, D. Gusakova,\nC. Thirion, B. Dieny, and L. D. Buda-Prejbeanu, Phys.\nRev. B78, 024436 (2008).\n72Y. B. Bazaliy and F. Arammash, Phys. Rev. B 84, 132404\n(2011).\n73M. Dykman, ed., Fluctuating Nonlinear Oscillators (Ox-\nford University Press, Oxford, 2012), chap. 6.\n74B. Lacoste, L. D. Buda-Prejbeanu, U. Ebels, and B. Dieny,\nPhys. Rev. B 88, 054425 (2013).\n75T. Taniguchi, Appl. Phys. Express 7, 053004 (2014).\n76D. Pinna, D. L. Stein, and A. D. Kent, Phys. Rev. B 90,\n174405 (2014).\n77T. Taniguchi, Phys. Rev. B 91, 104406 (2015).\n78T. Taniguchi, Phys. Rev. B 90, 024424 (2014).\n79H. Suto, K. Kudo, T. Nagasawa, T. Kanao, K. Mizushima,\nR. Sato, S. Okamoto, N. Kikuchi, and O. Kitakami, Phys.\nRev. B91, 094401 (2015).\n80P. F. Byrd and M. D. Friedman, Handbook of Elliptic In-\ntegrals for Engineers and Scientists (Springer, 1971), 2nd\ned.\n81A. Kiper, Math. Comput. 43, 247 (1984)."    },    {        "title": "1701.03232v3.Blow_up_for_semilinear_wave_equations_with_the_scale_invariant_damping_and_super_Fujita_exponent.pdf",        "content": "arXiv:1701.03232v3  [math.AP]  22 Jun 2017Blow-up for semilinear wave equations\nwith the scale invariant damping\nand super-Fujita exponent\nNing-An Lai∗Hiroyuki Takamura†Kyouhei Wakasa‡\nKeywords: damped wave equation, semilinear, blow-up\nMSC2010: primary 35L71, secondary 35B44\nAbstract\nThe blow-up for semilinear wave equations with the scale inv ariant\ndamping has been well-studied for sub-Fujita exponent. How ever,\nfor super-Fujita exponent, there is only one blow-up result which is\nobtained in 2014 by Wakasugi in the case of non-effective dampi ng. In\nthis paper we extend his result in two aspects by showing that : (I) the\nblow-up will happen for bigger exponent, which is closely re lated to\nthe Strauss exponent, the critical number for non-damped se milinear\nwave equations; (II) such a blow-up result is established fo r a wider\nrange of theconstant than theknownnon-effective onein theda mping\nterm.\n∗Department of Mathematics, Lishui University, Lishui City 323000 , China. e-mail:\nhyayue@gmail.com.\n†Department of Complex and Intelligent Systems, Faculty of System s Information Sci-\nence, Future University Hakodate, 116-2 Kamedanakano-cho, H akodate, Hokkaido 041-\n8655, Japan. e-mail: takamura@fun.ac.jp.\n‡College of Liberal Arts, Mathematical Science Research Unit, Muro ran Institute\nof Technology, 27-1, Mizumoto-cho, Muroran, Hokkaido 050-858 5, Japan. email:\nwakasa@mmm.muroran-it.ac.jp.\n11 Introduction\nIn this paper, we consider the following initial value problem.\n/braceleftBigg\nutt−∆u+µ\n1+tut=|u|pinRn×[0,∞),\nu(x,0) =εf(x), ut(x,0) =εg(x), x∈Rn,(1.1)\nwhereµ>0, f,g∈C∞\n0(Rn) andn∈N. We assume that ε>0 is a “small”\nparameter.\nFirst, we shall outline a background of (1.1) briefly according to the clas-\nsifications by Wirth in [20, 21, 22] for the corresponding linear proble m. Let\nu0be a solution of the initial value problem for the following linear damped\nwave equation.\n/braceleftBigg\nu0\ntt−∆u0+µ\n(1+t)βu0\nt= 0,inRn×[0,∞),\nu0(x,0) =u1(x), u0\nt(x,0) =u2(x), x∈Rn,(1.2)\nwhereµ >0,β∈R,n∈Nandu1,u2∈C∞\n0(Rn). Whenβ∈(−∞,−1),\nwe say that the damping term is “overdamping” in which case the solut ion\ndoes not decay to zero when t→ ∞. Whenβ∈[−1,1), the solution behaves\nlike that of the heat equation, which means that the term u0\nttin (1.2) has\nno influence on the behavior of the solution. In fact, Lp-Lqdecay estimates\nof the solution which are almost the same as those of the heat equat ion are\nestablished. In this case, we say that the damping term is “effective .” In\ncontrast, when β∈(1,∞), it is known that the solution behaves like that\nof the wave equation, which means that the damping term in (1.2) has no\ninfluence on the behavior of the solution. In fact, in this case the so lution\nscatters to that of the free wave equation when t→ ∞, and thus we say that\nwe have “scattering.” When β= 1, the equation in (1.2) is invariant under\nthe following scaling\n/tildewideu0(x,t) :=u0(σx,σ(1+t)−1), σ>0,\nand hence we say that the damping term is “scale invariant.” The rema rkable\nfact in this case is that the behavior of the solution of (1.2) is determ ined\nby the value of µ. Actually, for µ∈(0,1), it is known that the asymptotic\nbehavior of the solution is closely related to that of the free wave eq uation.\nForthisrangeof µ, wesaythatthedampingtermis“non-effective.” However,\nthe threshold of µaccording to the behavior of the solution is still open. We\nconjecture that it may be µ= 1 since we have the following L2estimates:\n/ba∇dblu0(·,t)/ba∇dblL2/lessorsimilar/ba∇dblu1/ba∇dblL2+/ba∇dblu2/ba∇dblH−1×\n\n(1+t)1−µifµ∈(0,1),\nlog(e+t) ifµ= 1,\n1 if µ>1.\n2In this way, we may summarize all the classifications of the damping te rm in\n(1.2) in the following table.\nβ∈(−∞,−1) overdamping\nβ∈[−1,1) effective\nβ= 1scaling invariant\nµ∈(0,1)⇒non-effective\nβ∈(1,∞) scattering\nNext, weconsiderthefollowinginitialvalueproblemforsemilineardamp ed\nwave equation.\n/braceleftBigg\nvtt−∆v+µ\n(1+t)βvt=|v|p,inRn×[0,∞),\nv(x,0) =εf(x), vt(x,0) =εg(x), x∈Rn,(1.3)\nwhereµ>0, β≥ −1, f,g∈C∞\n0(Rn) andn∈N. We assume that ε>0 is\na “small” parameter.\nForthe constant coefficient case, β= 0, Todorova and Yordanov [15] have\nshown that the energy solution of (1.3) exists globally-in-time for “s mall”\ninitial data if p>pF(n), where\npF(n) := 1+2\nn(1.4)\nis the so-called Fujita exponent, the critical exponent for semilinea r heat\nequations. It has been also obtained in [15] that the solution of (1.3) blows-\nup in finite time for some positive data if 1 < p < p F(n). The critical case\np=pF(n) has been studied by Zhang [24] by showing the blow-up result.\nWe note that Li and Zhou [10], or Nishihara [12], have obtained the sha rp\nupper bound of the lifespan which is the maximal existence time of solu tions\nof (1.3) in the case of n= 1,2, orn= 3, respectively. The sharpness of\nthe upper bound has been studied by Li [11] including the result for m ore\ngeneral equations with all n≥1, but for smooth nonlinear terms. The\nsharp lower bound has been obtained by Ikeda and Ogawa [6] for the critical\ncase. Recently, Lai and Zhou [9] have obtained the sharp upper bo und of the\nlifespan in the critical case for n≥4.\nFor the variable coefficient case of the most part of the effective da mping\nwith−1< β <1, Lin, Nishihara and Zhai [13] have obtained the blow-up\nresult if 1< p≤pF(n) and the global existence result if p > pF(n). Later,\nD’Abbicco, Lucente and Reissig [2] have extended the global existen ce result\nto more general equations. For the precise estimates of the lifesp an in this\ncase, see Introduction in Ikeda and Wakasugi [7]. Recently, similar r esults\n3on the remaining part of effective damping with β= 1 have been obtained\nby Wakasugi [19] for the global existence part, and by Fujiwara, I keda and\nWakasugi [5] for the blow-up part. The sharp estimates of the lifes pan are\nalso obtained by [5] except for the upper bound in the critical case.\nNow, let us turn back to our problem (1.1). Wakasugi [18] has obtain ed\nthe blow-up result if 1 <p≤pF(n) andµ>1, or 1<p≤1+2/(n+µ−1)\nand 0<µ≤1. He has also shown in [17] that an upper bound of the lifespan\nis\n\n\nCε−(p−1)/{2−n(p−1)}if 1<p<p F(n) andµ≥1,\nCε−(p−1)/{2−(n+µ−1)(p−1)}if 1<p<1+2\nn+µ−1and 0<µ<1,\nwhereCis a positive constant independent of ε. We note that the both\nproofs in [17] and [18] are based on the so-called “test function met hod”\nintroduced by Zhang [24]. On the other hand, D’Abbicco [1] has obtain ed\nthe global existence result if p>pF(n), butµhas to satisfy\nµ≥\n\n5/3 forn= 1,\n3 forn= 2,\nn+2 forn≥3 (andp≤1+2/(n−2)).\nIt is remarkable that, by the so-called Liouville transform;\nw(x,t) := (1+t)µ/2u(x,t),\n(1.1) can be rewritten as\n\n\nwtt−∆w+µ(2−µ)\n4(1+t)2w=|w|p\n(1+t)µ(p−1)/2inRn×[0,∞),\nw(x,0) =εf(x), wt(x,0) =ε{(µ/2)f(x)+g(x)}, x∈Rn.(1.5)\nWhenµ= 2, D’Abbicco, Lucente and Reissig [3] have obtained the following\nresult. Let\npc(n) := max {pF(n), p0(n+2)}, (1.6)\nwhere\np0(n) :=n+1+√\nn2+10n−7\n2(n−1)(n≥2) (1.7)\nis the so-called Strauss exponent, the positive root of the quadra tic equation,\nγ(p,n) := 2+(n+1)p−(n−1)p2= 0. (1.8)\nWe note that p0(n) is the critical exponent for semilinear wave equations,\nµ= 0 in (1.1). They have shown in [3] that the problem (1.1) admits a\n4global-in-time solution in the classical sense for “small” εifp > p c(n) in\nthe case of n= 2,3 although the radial symmetry is assumed in n= 3,\nand that the classical solution of (1.1) with positive data blows-up in fi nite\ntime if 1< p≤pc(n) andn≥1. In the same year, with radial symmetric\nassumption, D’Abbicco and Lucente [4] extended the global existen ce result\nforp0(n+ 2)< p <1 + 2/(max{2,(n−3)/2}) to odd higher dimensions\n(n≥5). We remark that, in the case of n= 1, Wakasa [16] has studied the\nestimates of the lifespan and has shown that the critical exponent pc(1) =\npF(1) = 3 changes to p0(1 + 2) = 1 +√\n2 when the nonlinearity is a sign-\nchanging type, |u|p−1u, and the initial data is of odd functions. Both results\nin [3] and [16] heavily rely on the special structure of the massless wa ve\nequations,µ= 2 in (1.5). In view of them, µ= 2 may be an exceptional\ncase. Recalling Wirth’ classification in the linear problem, (1.2), one ma y\nregardµ= 1 as a threshold also for the semilinear problem, (1.1). In this\nsense, the blow-up result in Wakasugi [18] says that the solution ma y be\n“heat-like” if µ >1. Here, “heat-like” means that the critical exponent for\n(1.1) is Fujita exponent.\nIn this paper, we claim that the solution of (1.1) is “wave-like” in some\ncase even for µ>1. Here, “wave-like” means that the critical exponent for\n(1.1) is bigger than Fujita exponent and is related to Strauss expon ent. We\nalso conjecture that such a threshold of µdepends on the space dimension\nn. The main tool of our result is Kato’s lemma in Kato [8] on ordinary\ndifferential inequalities which is improved to be applied to semilinear wave\nequations by Takamura [14]. Together with Yordanov and Zhang’s es timate\nin [23], we can prove a new blow-up result for wave-like solutions by mea ns\nof some special transform for the time-derivative of the spatial in tegral of\nunknown functions.\nThis paper is organized as follows. In the next section, we state our main\nresult. In thesection 3, we estimate thespatial integral ofunkno wn functions\nfrom below. Making use of such an estimate, we prove the main result for\nµ≥2 in section 4, and for 0 <µ<2 in section 5.\n2 Main Result\nFirst we define an energy solution of (1.1).\nDefinition 2.1 We say that uis an energy solution of (1.1) on [0,T)if\nu∈C([0,T),H1(Rn))∩C1([0,T),L2(Rn))∩Lp\nloc(Rn×[0,T)) (2.1)\n5satisfies\n/integraldisplay\nRnut(x,t)φ(x,t)dx−/integraldisplay\nRnut(x,0)φ(x,0)dx\n+/integraldisplayt\n0ds/integraldisplay\nRn{−ut(x,s)φt(x,s)+∇u(x,s)·∇φ(x,s)}dx\n+/integraldisplayt\n0ds/integraldisplay\nRnµut(x,s)\n1+sφ(x,s)dx=/integraldisplayt\n0ds/integraldisplay\nRn|u(x,s)|pφ(x,s)dx(2.2)\nwith anyφ∈C∞\n0(Rn×[0,T))and anyt∈[0,T).\nWe note that, employing the integration by parts in (2.2) and letting\nt→T, we have that\n/integraldisplay\nRn×[0,T)u(x,s)/braceleftbigg\nφtt(x,s)−∆φ(x,s)−/parenleftbiggµφ(x,s)\n1+s/parenrightbigg\ns/bracerightbigg\ndxds\n=/integraldisplay\nRnµu(x,0)φ(x,0)dx−/integraldisplay\nRnu(x,0)φt(x,0)dx\n+/integraldisplay\nRnut(x,0)φ(x,0)dx+/integraldisplay\nRn×[0,T)|u(x,s)|pφ(x,s)dxds.\nThis is exactly the definition of a weak solution of (1.1).\nOur main result is the following theorem.\nTheorem 2.1 Letn≥2,\n0<µ<µ 0(n) :=n2+n+2\n2(n+2)andpF(n)≤p<p0(n+2µ).(2.3)\nAssume that both f∈H1(Rn)andg∈L2(Rn)are non-negative and do not\nvanish identically. Suppose that an energy solution uof (1.1) satisfies\nsuppu⊂ {(x,t)∈Rn×[0,T) :|x| ≤t+R} (2.4)\nwith someR≥1. Then, there exists a constant ε0=ε0(f,g,n,p,µ,R )>0\nsuch thatThas to satisfy\nT≤Cε−2p(p−1)/γ(p,n+2µ)(2.5)\nfor0<ε≤ε0, whereCis a positive constant independent of ε.\nRemark 2.1 Theorem 2.1 can be established also for n= 1if one define\nφ1(x) =ex+e−xforx∈Rin Section 3 below. But the result is not new.\nSee the following two remarks.\n6Remark 2.2 In view of (1.4) and (1.8), one can see that\nγ(pF(n),n+2µ) =2\nn2/braceleftbig\nn2+n+2−2(n+2)µ/bracerightbig\n.\nTherefore 0<µ<µ 0(n)is equivalent to\npF(n)<p0(n+2µ).\nWe note that µ0(2) = 1. This means that Theorem 2.1 just covers the non-\neffective range of µforn= 2. Sinceµ0(n)is increasing in n, Theorem 2.1\ngives us the blow-up result on super-Fujita exponent even fo rµin outside of\nthe non-effective range for n≥3. We also note that µ0(n)<2forn= 2,3,4\nandµ0(n)>2forn≥5.\nRemark 2.3 One can see also that\nγ/parenleftbigg\n1+2\nn+µ−1,n+2µ/parenrightbigg\n=2{(n−1)2+(n−3)µ}\n(n+µ−1)2.\nTherefore we have that\n1+2\nn+µ−1<p0(n+2µ)\nforn= 2and0<µ<1, orn≥3andµ>0. This means that Theorem 2.1\nincludes the blow-up result in Wakasugi [18].\nRemark 2.4 Ifβis in the scattering range, (1,∞), for the problem,\nutt−∆u+µ\n(1+t)βut=|u|p,\nthe result will be\nT≤Cε−2p(p−1)/γ(p,n)for1<p<p 0(n)\nfor allµ>0. This estimate coincides with the one for non-damped equati on,\nutt−∆u=|u|p\nexcept for the case of/integraltext\nR2g(x)dx/ne}ationslash= 0inn= 2and1<p≤2. See Introduc-\ntion of Takamura [14] for its summary. The proof of this fact w ill appear in\nour forthcoming paper.\n73 Lower bound of the functional\nLetube an energy solution of (1.1) on [0 ,T). We estimate\nF0(t) :=/integraldisplay\nRnu(x,t)dx\nfrom below in this section. Choosing the test function φ=φ(x,s) in (2.2) to\nsatisfyφ≡1 in{(x,s)∈Rn×[0,t] :|x| ≤s+R}, we get\n/integraldisplay\nRnut(x,t)dx−/integraldisplay\nRnut(x,0)dx\n+/integraldisplayt\n0ds/integraldisplay\nRnµut(x,s)\n1+sdx=/integraldisplayt\n0ds/integraldisplay\nRn|u(x,s)|pdx,\nwhich means that\nF′\n0(t)−F′\n0(0)+/integraldisplayt\n0µF′\n0(s)\n1+sdx=/integraldisplayt\n0ds/integraldisplay\nRn|u(x,s)|pdx.\nAll the quantities in this equation except for F′\n0(t) are differentiable in t, so\nthat so isF′\n0(t). Hence we have\nF′′\n0(t)+µF′\n0(t)\n1+t=/integraldisplay\nRn|u(x,t)|pdx. (3.1)\nIntegrating this equation with a multiplication by (1+ t)µ, we obtain\n(1+t)µF′\n0(t)−F′\n0(0) =/integraldisplayt\n0(1+s)µds/integraldisplay\nRn|u(x,s)|pdx. (3.2)\nIt follows from this equation and the assumption on the initial data th at\nF′\n0(t)≥(1+t)−µF′\n0(0)>0 andF0(t)≥F0(0)>0 fort≥0.(3.3)\nFrom now on, we employ the modified argument of Yordanov and Zhan g\n[23]. Let us define\nF1(t) :=/integraldisplay\nRnu(x,t)ψ1(x,t)dx,\nwhere\nψ1(x,t) :=φ1(x)e−t, φ1(x) :=/integraldisplay\nSn−1ex·ωdSω.\nIn view of (3.2) and the argument of (2.4)-(2.5) in [23], we know that t here\nis a positive constant C1=C1(n,p,R) such that\n(1+t)µF′\n0(t)−F′\n0(0)≥C1/integraldisplayt\n0(1+s)µ+(n−1)(1−p/2)|F1(s)|pds.(3.4)\n8In order to get a lower bound of F1(t), we turn back to (2.2) and obtain\nthatd\ndt/integraldisplay\nRnut(x,t)φ(x,t)dx\n+/integraldisplay\nRn{−ut(x,t)φt(x,t)−u(x,t)∆φ(x,t)}dx\n+/integraldisplay\nRnµut(x,t)\n1+tφ(x,t)dx=/integraldisplay\nRn|u(x,t)|pφ(x,t)dx.\nMultiplying the above equality by(1+ t)µ, we have that\nd\ndt/braceleftbigg\n(1+t)µ/integraldisplay\nRnut(x,t)φ(x,t)dx/bracerightbigg\n+(1+t)µ/integraldisplay\nRn{−ut(x,t)φt(x,t)−u(x,t)∆φ(x,t)}dx\n= (1+t)µ/integraldisplay\nRn|u(x,t)|pφ(x,t)dx.\nIntegrating this equality over [0 ,t], we get\n(1+t)µ/integraldisplay\nRnut(x,t)φ(x,t)dx−ε/integraldisplay\nRng(x)φ(x,0)dx\n−/integraldisplayt\n0ds/integraldisplay\nRn(1+s)µut(x,s)φt(x,s)dx\n=/integraldisplayt\n0ds/integraldisplay\nRn{(1+s)µu(x,s)∆φ(x,s)+(1+s)µ|u(x,s)|pφ(x,s)}dx.\nIt follows from this equation and\n/integraldisplayt\n0ds/integraldisplay\nRn(1+s)µut(x,s)φt(x,s)dx\n= (1+t)µ/integraldisplay\nRnu(x,t)φt(x,t)dx−/integraldisplay\nRnu(x,0)φt(x,0)dx\n−/integraldisplayt\n0ds/integraldisplay\nRnµ(1+s)µ−1u(x,s)φt(x,s)dx\n−/integraldisplayt\n0ds/integraldisplay\nRn(1+s)µu(x,s)φtt(x,s)dx,\n9which follows from integration by parts that\n(1+t)µ/integraldisplay\nRn{ut(x,t)φ(x,t)−u(x,t)φt(x,t)}dx\n−ε/integraldisplay\nRng(x)φ(x,0)dx+ε/integraldisplay\nRnf(x)φt(x,0)dx\n+/integraldisplayt\n0ds/integraldisplay\nRnµ(1+s)µ−1u(x,s)φt(x,s)dx\n=/integraldisplayt\n0ds/integraldisplay\nRn(1+s)µu(x,s){∆φ(x,s)−φtt(x,s)}dx\n+/integraldisplayt\n0ds/integraldisplay\nRn(1+s)µ|u(x,s)|pφ(x,s)dx.\nIf we put\nφ(x,t) =ψ1(x,t) =e−tφ1(x) on supp u,\nwe have\nφt=−φ, φtt= ∆φon suppu.\nHence we obtain that\n(1+t)µF′\n1(t)+2(1+t)µF1(t)−ε/integraldisplay\nRn{f(x)+g(x)}φ(x)dx\n=/integraldisplayt\n0µ(1+s)µ−1F1(s)ds+/integraldisplayt\n0ds/integraldisplay\nRn(1+s)µ|u(x,s)|pφ(t,x)dx,\nwhich yields\nF′\n1(t)+2F1(t)≥Cf,gε\n(1+t)µ+1\n(1+t)µ/integraldisplayt\n0µ(1+s)µ−1F1(s)ds,\nwhere\nCf,g:=/integraldisplay\nRn{f(x)+g(x)}φ1(x)dx>0.\nIntegrating this inequality over [0 ,t] with a multiplication by e2t, we get\ne2tF1(t)≥F1(0)+Cf,gε/integraldisplayt\n0e2s\n(1+s)µds\n+/integraldisplayt\n0e2s\n(1+s)µds/integraldisplays\n0µ(1+r)µ−1F1(r)dr.(3.5)\nWe note that the assumption on fimpliesF1(0)>0. Hence we find that\nthere is no zero point of F1(t) fort>0. Because the continuity of F1implies\n10F1(t)>0 for small t >0. If one assumes that there is a nearest zero point\nt0ofF1to 0, then one has a contradiction in (3.5);\ne2t0F1(t0) = 0≥F1(0)+Cf,gε/integraldisplayt0\n0e2s\n(1+s)µds\n+/integraldisplayt0\n0e2s\n(1+s)µds/integraldisplays\n0µ(1+r)µ−1F1(r)dr.\nThe last term in the right-hand side of this inequality is positive by F1(t)>0\nfor 0<t<t 0. Turning back to (3.5), we obtain\nF1(t)>Cf,gε\n2(1+t)µ(1−e−2t)+e−2tF1(0)≥Cf,0ε\n2(1+t)µfort≥0.\nHere we have used the fact that Cf,g>Cf,0.\nPlugging this estimate into (3.4), we have\n(1+t)µF′\n0(t)−F′\n0(0)>C1Cp\nf,0\n2pεp/integraldisplayt\n0(1+s)µ(1−p)+(n−1)(1−p/2)ds.\nSinceF′\n0(0)>0 and it follows from t≥1 that\n/integraldisplayt\n0(1+s)µ(1−p)+(n−1)(1−p/2)ds≥(2t)µ(1−p)/integraldisplayt\nt/2s(n−1)(1−p/2)ds\n≥2µ(1−p)−(n−1)(1−p/2)+1t1+µ(1−p)+(n−1)(1−p/2),\nwe obtain that\nF′\n0(t)>C2εpt1−µp+(n−1)(1−p/2)fort≥1,\nwhere\nC2:=C1Cp\nf,0\n2(µ+1)p+(n−1)(1−p/2)−1>0.\nHere we have used the fact that\n1+(n−1)/parenleftBig\n1−p\n2/parenrightBig\n>0\nfollows from\np<p0(n+2µ)<\n\np0(n)≤p0(4) = 2 for n≥4,\np0(3) = 1+√\n2<3 for n= 3,\np0(2) = (3+√\n17)/2<4 forn= 2.\n11Integrating this inequality over [1 ,t] and making use of F0(0)>0 and\n/integraldisplayt\n1s1−µp+(n−1)(1−p/2)ds≥t−µp/integraldisplayt\nt/2s1+(n−1)(1−p/2)dsfort≥2,\nwe get\nF0(t)>C3εpt2−µp+(n−1)(1−p/2)fort≥2, (3.6)\nwhere\nC3:=C2\n22+(n−1)(1−p/2)>0.\n4 Proof of Theorem 2.1 for µ≥2\nLet us define\nF(t) :=/integraldisplay\nRnw(x,t)dx= (1+t)µ/2F0(t),\nwherewis the solution of (1.5). We note that (3.1) yields\nF′′(t)+µ(2−µ)\n4(1+t)2F(t) = (1+t)−µ(p−1)/2/integraldisplay\nRn|w(x,t)|pdx.(4.1)\nThen it follows from (2.4) and H¨ older’s inequality that\n/integraldisplay\nRn|w(x,t)|pdx≥ {vol(Bn(0,1))}1−p(t+R)−n(p−1)|F(t)|p.(4.2)\nBy combining (4.1) and (4.2), and noting the assumption R≥1, we come to\nF′′(t)+µ(2−µ)\n4(1+t)2F(t)≥C4(1+t)−(n+µ/2)(p−1)|F(t)|pfort≥0,(4.3)\nwhere\nC4:={vol(Bn(0,1))}1−pR−n(p−1)>0.\nDue to (3.3), we have that\nF(t) = (1+t)µ/2F0(t)>0,\nF′(t) =µ\n2(1+t)µ/2−1F0(t)+(1+t)µ/2F′\n0(t)>0,(4.4)\nwhich implies that\nF(0) =F0(0) =/ba∇dblf/ba∇dblL1(Rn)ε,\nF′(0) =µ\n2F0(0)+F′\n0(0) =/parenleftBigµ\n2/ba∇dblf/ba∇dblL1(Rn)+/ba∇dblg/ba∇dblL1(Rn)/parenrightBig\nε.(4.5)\n12From now on, we focus on the case of µ≥2. Then it follows from (4.3)\nand (4.4) that\nF′′(t)≥C4(1+t)−(n+µ/2)(p−1)|F(t)|p. (4.6)\nWe shall employ the following lemma now.\nLemma 4.1 (Takamura[14]) Letp>1,a>0,q>0satisfy\nM:=p−1\n2a−q\n2+1>0. (4.7)\nAssume that F∈C2([0,T))satisfies\nF(t)≥Atafort≥T0, (4.8)\nF′′(t)≥B(t+R)−q|F(t)|pfort≥0, (4.9)\nF(0)≥0, F′(0)>0, (4.10)\nwhereA,B,R,T 0are positive constants. Then, there exists a positive con-\nstantC0=C0(p,a,q,B)such that\nT <22/MT1 (4.11)\nholds provided\nT1:= max/braceleftbigg\nT0,F(0)\nF′(0),R/bracerightbigg\n≥C0A−(p−1)/(2M). (4.12)\nDue to the lower bound of F0in (3.6) and the definition of F(t) in (4.4),\nwe have\nF(t)>C3εpt2−µp+(n−1)(1−p/2)+µ/2fort≥2, (4.13)\nwhich is (4.8) in Lemma 4.1 with\nA=C3εp, a= 2−µp+(n−1)/parenleftBig\n1−p\n2/parenrightBig\n+µ\n2, T0= 2.\nThe inequality (4.9) with\nB=C4, q=/parenleftBig\nn+µ\n2/parenrightBig\n(p−1)\nfollows from (4.6), and (4.10) is already established by (4.5). The fina l step\nto use Lemma 4.1 is to check the sign of M. By the assumption that p <\np0(n+2µ), we have\nM=γ(p,n+2µ)\n4>0.\n13Set\nT0=C0A−(p−1)/(2M)=C0C−2(p−1)/γ(p,n+2µ)\n3 ε−2p(p−1)/γ(p,n+2µ).\nThen, since F(0)/F′(0) is independent of εby (4.5), one can see that there\nis anε0=ε0(f,g,n,p,µ,R )>0 such that\nT0≥max/braceleftbigg\n2,F(0)\nF′(0)/bracerightbigg\nfor 0<ε≤ε0.\nThis means that T1=T0in (4.12). Therefore the conclusion of Lemma 4.1\nimplies that the maximal existence time TofF(t) has to satisfy\nT≤C5ε−2p(p−1)/γ(p,n+2µ)for 0<ε≤ε0,\nwhere\nC5:= 28/γ(p,n+2µ)C0C−2(p−1)/γ(p,n+2µ)\n4 >0.\nThis completes the proof in the case of µ≥2. ✷\n5 Proof of Theorem 2.1 for 0 < µ <2\nBefore showing the proof of Theorem 2.1 for 0 < µ <2, we first prepare\nthe following lemma:\nLemma 5.1 Suppose that the assumption in Theorem 2.1 is fulfilled. Then\nit holds that\nF′(t)>/radicalBigg\nC4\n2(p+1)(1+t)−(n+µ/2)(p−1)/2F(t)(p+1)/2(5.1)\nfort≥C6ε−2p(p−1)/γ(p,n+2µ), where we set\nC6:=/parenleftbiggµ(2−µ)(p+1)\n2Cp−1\n3C4/parenrightbigg1/X\n>0\nand\nX:= 2−/parenleftBig\nn+µ\n2/parenrightBig\n(p−1)+(p−1)/braceleftBig\n2−µp+(n−1)/parenleftBig\n1−p\n2/parenrightBig\n+µ\n2/bracerightBig\n=γ(p,n+2µ)\n2.\nC3,C4are the one in (3.6), (4.3), respectively,\n14Proof.Multiplying the both sides of (4.3) by (1+ t)2F′(t)>0 and noting\nthat (4.4), we get\n(1+t)2\n2/braceleftBig\n(F′(t))2/bracerightBig′\n+µ(2−µ)\n8{F(t)2}′\n≥C4(1+t)2−(n+µ/2)(p−1)F(t)pF′(t)fort≥0.\nIntegration by parts yields that\n(1+t)2\n2(F′(t))2+µ(2−µ)\n8F(t)2\n>C4/integraldisplayt\n0(1+s)2−(n+µ/2)(p−1)F(s)pF′(s)dsfort≥0.\nNoting the assumption on p\np≥pF(n)>1+2\nn+µ\n2forµ>0,\nit is easy to get that\n2−/parenleftBig\nn+µ\n2/parenrightBig\n(p−1)<0.\nAnd hence we have\n/integraldisplayt\n0(1+s)2−(n+µ/2)(p−1)F(s)pF′(s)ds\n≥(1+t)2−(n+µ/2)(p−1)F(t)p+1−F(0)p+1\np+1.\nSince\npF(n) = 1+2\nn>2\nn+1−µforn≥2 and 0<µ<2,\nand hence\np>2\nn+1−µ.\nThis is equivalent to\np/parenleftBig\n2−µp+(n−1)/parenleftBig\n1−p\n2/parenrightBig\n+µ\n2/parenrightBig\n>γ(p,n+2µ)\n2.\nThus, fort≥C6ε−2p(p−1)/γ(p,n+2µ)≥2 (εsmall enough), we have\nC3εpt2−µp+(n−1)(1−p/2)+µ/2≥2/ba∇dblf/ba∇dblL1(Rn)ε, (5.2)\n15which implies\nF(t)≥2F(0). (5.3)\nHence, it follows from\nF(t)p+1−F(0)p+1≥F(t)p{F(t)−F(0)} ≥1\n2F(t)p+1\nthat\n(1+t)2\n2(F′(t))2+µ(2−µ)\n8F(t)2>C4(1+t)2−(n+µ/2)(p−1)F(t)p+1\n2(p+1)(5.4)\nfort≥C6ε−2p(p−1)/γ(p,n+2µ).\nOn the other hand, for t≥C6ε−2p(p−1)/γ(p,n+2µ), we have\nC4\n4(p+1)(1+t)2−(n+µ/2)(p−1){C3εpt2−µp+(n−1)(1−p/2)+µ/2}p−1≥µ(2−µ)\n8\nwhich gives us\nC4\n4(p+1)(1+t)2−(n+µ/2)(p−1)F(t)p+1≥µ(2−µ)\n8F(t)2(5.5)\nby combining (4.13). Therefore, we get (5.1) from (5.4).\n✷\nBy (5.1), it is easy to see that there is a ε0=ε0(f,g,n,p,µ,R )>0 such\nthat\nF′(t)\nF(t)1+δ>/radicalBigg\nC4\n2(p+1)(1+t)−(n+µ/2)(p−1)/2F(t)(p−1)/2−δ\nwith 0<δ<(p−1)/2 holds for\nt≥T1:=C6ε−2p(p−1)/γ(p,n+2µ)and 0<ε≤ε0.\nHere we use our lower bound of Fin (4.13) again to get\nF′(t)\nF(t)1+δ>C(p−1)/2−δ\n3/radicalBigg\nC4\n2(p+1)εp{(p−1)/2−δ}tYfort≥T1,(5.6)\nwhere\nY:=/parenleftbiggp−1\n2−δ/parenrightbigg/braceleftBig\n2−µp+(n−1)/parenleftBig\n1−p\n2/parenrightBig\n+µ\n2/bracerightBig\n−/parenleftBig\nn+µ\n2/parenrightBigp−1\n2\n=γ(p,n+2µ)\n4−1−/braceleftBig\n2−µp+(n−1)/parenleftBig\n1−p\n2/parenrightBig\n+µ\n2/bracerightBig\nδ.\n16Therefore, taking δsmall enough such that Y+ 1>0, we then have by\nintegrating (5.6) over [ T1,t],\nF(T1)−δ\nδ>C(p−1)/2−δ\n3\nY+1/radicalBigg\nC4\n2(p+1)εp{(p−1)/2−δ}(tY+1−TY+1\n1) fort≥T1.\nMaking use of (4.13) with t=T1in this inequality, we obtain that\n1>C7εp(p−1)/2Tγ(p,n+2µ)/4−(Y+1)\n1 (tY+1−TY+1\n1) fort≥T1,\nwhere\nC7:=δC(p−1)/2\n3\nY+1/radicalBigg\nC4\n2(p+1)>0\nIf one setst=kT1withk>1, then, due to the definition of T1, one has\n1>C7Cγ(p,n+2µ)/4\n6 (kY+1−1).\nTherefore the conclusion of the Theorem 2.1,\nT≤C8ε−2p(p−1)/γ(p,n+2µ)for 0<ε≤ε0,\nis now established, where\nC8:=/parenleftBig\n1+C−1\n7C−γ(p,n+2µ)/4\n6/parenrightBig1/(Y+1)\nC6>0.\nThis completes the proof in the case of 0 <µ<2. ✷\nAcknowledgment\nThe first author is partially supported by NSFC(11501273), high lev el tal-\nent project of Lishui City(2016RC25), the Scientific Research Fo undation\nof the First-Class Discipline of Zhejiang Province(B)(201601), the key lab-\noratory of Zhejiang Province(2016E10007). The second author is partially\nsupported by the Grant-in-Aid for Scientific Research (C) (No.15K 04964),\nJapan Society for the Promotion of Science, and Special Research Expenses\nin FY2016, General Topics (No.B21), Future University Hakodate.\nFinally, all the authors are grateful to Prof.M.Reissig (Technical Un iver-\nsity Bergakademie Freiberg, Germany) for his great advice on the c lassifica-\ntion of the linear problem for which our first manuscript of arXiv:1701 .03232\nwas not appropriate.\n17References\n[1] M.D’Abbicco, The threshold of effective damping for semilinear wave\nequations , Mathematical Methods in Applied Sciences, 38(2015) 1032-\n1045.\n[2] M.D’Abbicco, S.Lucente andM.Reissig, Semi-linear wave equations with\neffective damping , Chin. Ann. Math. Ser. B, 34(2013), 345-380.\n[3] M.D’Abbicco, S.Lucente and M.Reissig, A shift in the Strauss exponent\nfor semilinear wave equations with a not effective damping , J. Differen-\ntial Equations, 259(2015), 5040-5073.\n[4] M.D’Abbicco andS.Lucente, NLWE with a special scale invariant damp-\ning in odd space dimension , Dynamical Systems, Differential Equations\nand Applications AIMS Proceedings, (2015), 312-319.\n[5] K.Fujiwara, M.Ikeda andY.Wakasugi, Estimates of lifespan and blow-up\nrate for the wave equation with a time-dependent damping and a power-\ntype nonlinearity , arXiv:1609.01035.\n[6] M.Ikeda and T.Ogawa, Lifespan of solutions to the damped wave equa-\ntion with a critical nonlinearity , J. Differential Equations, 261(2016),\n1880-1903.\n[7] M.Ikeda and Y.Wakasugi, A note on the lifespan of solutions to the\nsemilinear damped wave equation , Proc. Amer. Math. Soc., 143(2015),\n163-171.\n[8] T.Kato, Blow up of solutions of some nonlinear hyperbolic equations ,\nComm. Pure Appl. Math., 33(1980), 501-505.\n[9] N.-A.Lai and Y.Zhou, The sharp lifespan estimate for semilinear\ndamped wave equation with Fujita critical power in high dime nsions,\narXiv:1702.07073.\n[10] T.T.Li andY.Zhou, Breakdown of solutions to ✷u+ut=|u|1+α, Discrete\nand Continuous Dynamical Systems, 1(1995), 503-520.\n[11] Y.Li, Classical solutions for fully nonlinear wave equations wit h dissipa-\ntion(in Chinese), Chin. Ann. Math. Ser. A, 17(4)(1996), 451-466.\n[12] K.Nishihara, Lp-Lqestimates for the 3-D damped wave equation and\ntheir application to the semilinear problem , Sem. NotesMath. Sci., vol.6,\nIbaraki Univ., 2003, pp.69-83.\n18[13] J.Lin, K.Nishihara and J.Zhai, Critical exponent for the semilinear wave\nequation with time-dependent damping , Discrete and Continuous Dy-\nnamical Systems - Series A, 32(2012), 4307-4320.\n[14] H.Takamura, Improved Kato’s lemma on ordinary differential inequality\nand its application to semilinear wave equations , Nonlinear Analysis,\nTMA,125(2015), 227-240.\n[15] G.Todorova and B.Yordanov Critical exponent for a nonlinear wave\nequation with damping , J. Differential Equations, 174(2001) 464-489.\n[16] K.Wakasa, The lifespan of solutions to semilinear damped wave equa-\ntions in one space dimension , Communications on Pure and Applied\nAnalysis, 15(2016), 1265-1283.\n[17] Y.Wakasugi, On the diffusive structure for the damped wave equation\nwith variable coefficients , Doctoral thesis, Osaka University (2014).\n[18] Y.Wakasugi, Critical exponent for the semilinear wave equation with\nscale invariant damping , Fourier analysis, 375-390, Trends Math.,\nBirkh¨ auser/Springer, Cham, (2014).\n[19] Y.Wakasugi, Scaling variables and asymptotic profiles for the semilinea r\ndamped wave equation with variable coefficients , J. Math. Anal. Appl.,\n447(2017), 452-487.\n[20] J.Wirth, Solution representations for a wave equation with weak diss i-\npation, Math. Methods Appl. Sci., 27(2004), 101-124.\n[21] J.Wirth, Wave equations with time-dependent dissipation. I. Non-\neffective dissipation , J. Differential Equations, 222(2006), 487-514.\n[22] J.Wirth, Wave equations with time-dependent dissipation. II. Effect ive\ndissipation , J. Differential Equations, 232(2007), 74-103.\n[23] B.Yordanov and Q.S.Zhang, Finite time blow up for critical wave equa-\ntions in high dimensions , J. Funct. Anal., 231(2006), 361-374.\n[24] Q.S.Zhang, A blow-up result for a nonlinear wave equation with damp-\ning: the critical case , C. R. Math. Acad. Sci. Paris, S´ er. I, 333(2001)\n109-114.\n19"    },    {        "title": "1701.08076v2.Structural_scale__q__derivative_and_the_LLG_Equation_in_a_scenario_with_fractionality.pdf",        "content": "Structural scale q\u0000derivative and the LLG-Equation in a scenario with\nfractionality\nJ.Weberszpil\u0003\nUniversidade Federal Rural do Rio de Janeiro, UFRRJ-IM/DTL and\nAv. Governador Roberto Silveira s/n- Nova Iguaçú, Rio de Janeiro, Brasil, 695014.\nJ. A. Helayël-Netoy\nCentro Brasileiro de Pesquisas Físicas-CBPF-Rua Dr Xavier Sigaud 150, and\n290-180, Rio de Janeiro RJ Brasil.\n(Dated: November 5, 2018)\nIn the present contribution, we study the Landau-Lifshitz-Gilbert equation\nwith two versions of structural derivatives recently proposed: the scale\u0000\nq\u0000derivative in the non-extensive statistical mechanics and the axiomatic met-\nric derivative, which presents Mittag-Leffler functions as eigenfunctions. The\nuse of structural derivatives aims to take into account long-range forces, possi-\nble non-manifest or hidden interactions and the dimensionality of space. Hav-\ning this purpose in mind, we build up an evolution operator and a deformed\nversion of the LLG equation. Damping in the oscillations naturally show up\nwithout an explicit Gilbert damping term.\nKeywords: Structural Derivatives, Deformed Heisenberg Equation, LLG Equation, Non-extensive\nStatistics, Axiomatic Deformed Derivative\nI. INTRODUCTION\nIn recent works, we have developed connections and a variational formalism to treat deformed or metric\nderivatives, considering the relevant space-time/ phase space as fractal or multifractal [1] and presented a\n\u0003Electronic address: josewebe@gmail.com\nyElectronic address: helayel@cbpf.br arXiv:1701.08076v2  [math-ph]  28 Feb 20172\nvariational approach to dissipative systems, contemplating also cases of a time-dependent mass [2].\nThe use of deformed-operators was justified based on our proposition that there exists an intimate\nrelationship between dissipation, coarse-grained media and a limit energy scale for the interactions. Con-\ncepts and connections like open systems, quasi-particles, energy scale and the change in the geometry\nof space–time at its topological level, nonconservative systems, noninteger dimensions of space–time con-\nnected to a coarse-grained medium, have been discussed. With this perspective, we argued that deformed\nor, we should say, Metric or Structural Derivatives, similarly to the Fractional Calculus (FC), could allows\nus to describe and emulate certain dynamics without explicit many-body, dissipation or geometrical terms\nin the dynamical governing equations. Also, we emphasized that the paradigm we adopt was different\nfrom the standard approach in the generalized statistical mechanics context [3–5], where the modification\nof entropy definition leads to the modification of the algebra and, consequently, the concept of a derivative\n[1, 2]. This was set up by mapping into a continuous fractal space [6–8] which naturally yields the need\nof modifications in the derivatives, that we named deformed or, better, metric derivatives [1, 2]. The\nmodifications of the derivatives, accordingly with the metric, brings to a change in the algebra involved,\nwhich, in turn, may lead to a generalized statistical mechanics with some adequate definition of entropy.\nThe Landau-Lifshitz-Gilbert (LLG) equation sets out as a fundamental approach to describe physics\nin the field of Applied Magnetism. It exhibits a wide spectrum of effects stemming from its non-linear\nstructure, and its mathematical and physical consequences open up a rich field of study. We pursue the\ninvestigation of the LLG equation in a scenario where complexity may play a role. The connection between\nLLG and fractionality, represented by an \u000b\u0000deformation parameter in the deformed differential equations,\nhas not been exploited with due attention. Here, the use of metric derivatives aims to take into account\nlong-range forces, possible non-manifest or hidden interactions and/or the dimensionality of space.\nIn this contribution, considering intrinsically the presence of complexity and possible dissipative effects,\nand aiming to tackle these issues, we apply our approach to study the LLG equation with two metric\nor structural derivatives, the recently proposed scale \u0000q\u0000derivative [2] in the nonextensive statistical\nmechanics and, as an alternative, the axiomatic metric derivative (AMD) that has the Mittag-Leffler\nfunction as eigenfunction and where deformed Leibniz and chain rule hold - similarly to the standard\ncalculus - but in the regime of low-level of fractionality. The deformed operators here are local. We3\nactually focus our attention to understand whether the damping in the LLG equation can be connected\nto some entropic index, the fractionality or even dimensionality of space; in a further step, we go over\ninto anisotropic Heisenberg spin systems in (1+1) dimensions with the purpose of modeling the weak\nanisotropy effects by means of some representative parameter, that depends on the dimension of space or\nthe strength of the interactions with the medium. Some considerations about an apparent paradox in the\nmagnetization or angular damping is given.\nOur paper is outlined as follows: In Section 2, we briefly present the scale \u0000q\u0000derivative in a nonex-\ntensive context, building up the q\u0000deformed Heisenberg equation and applying to tackle the problem of\nthe LLG equation; in Section 3, we apply the axiomatic derivative to build up the \u000b\u0000deformed Heisen-\nberg equation and to tackle again the problem of LLG equation. We finally present our Conclusions and\nOutlook in Section 4.\nII. APPLYING SCALE \u0000q\u0000DERIVATIVE IN A NONEXTENSIVE CONTEXT\nHere, in this Section, we provide some brief information to recall the main forms of scale \u0000q\u0000derivative.\nThe readers may see ref. [1, 2, 6] for more details.\nSome initial claims here coincide with our work of Refs. [1, 2] and the approaches here are in fact based\non local operators [1].\nThe local differential equation,\ndy\ndx=yq; (1)\nwith convenient initial condition, yields the solution given by the q-exponential, y=eq(x)[3–5].\nThe key of our work here is the Scale \u0000q\u0000derivative (Sq-D) that we have recently defined as\nD\u0015\n(q)f(\u0015x)\u0011[1 + (1\u0000q)\u0015x]df(x)\ndx: (2)\nThe eigenvalue equation holds for this derivative operator, as the reader can verify:\nD\u0015\n(q)f(\u0015x) =\u0015f(\u0015x): (3)4\nA.q\u0000deformed Heisenberg Equation in the Nonextensive Statistics Context\nWith the aim to obtain a scale \u0000q\u0000deformed Heisenberg equation, we now consider the scale\u0000q\u0000\nderivative [2]\ndq\ndtq= (1 + (1\u0000q)\u0015xd\ndx(4)\nand the Scale - q\u0000Deformed Schrödinger Equation [2],\ni~D\u0015\nq;t =\u0000~2\n2mr2 \u0000V =H ; (5)\nthat, as we have shown in [2], is related to the nonlinear Schrödinger equation referred to in Refs. [10]\nas NRT-like Schrödinger equation (with q=q0\u00002compared to the q\u0000index of the reference) and can be\nthought as resulting from a time\u0000scale\u0000q\u0000deformed-derivative applied to the wave function  .\nConsidering in eq.(5),  (~r;t) =Uq(t;t0) (~r;t0), theq\u0000evolution operator naturally emerges if we take\ninto account a time\u0000scale\u0000q\u0000deformed-derivative (do not confuse with formalism of discrete scale time\nderivative):\nUq(t;t0) =e(\u0000i\n~MqHqt)\nq: (6)\nHere,Mqis a constant for dimensional regularization reasons. Note that the q-deformed evolution\noperator is neither Hermitian nor unitary, the possibility of a q\u0000unitary asUy\nq(t;t0)\nqUq(t;t0) =1could\nbe thought to come over these facts. In this work, we assume the case where the commutativity of Uqand\nHholds, but the q\u0000unitarity is also a possibility.\nNow, we follow similar reasonings that can be found in Ref.[12] and considering the Sq-D.\nSo, with these considerations, we can now write a nonlinear Scale\u0000q\u0000deformed Heisenberg Equation\nas\nD\u0015\nt;q^A(t) =\u0000i\n~Mq[^A;H]; (7)\nwhere we supposed that UqandHcommute and Mqis some factor only for dimensional equilibrium.5\nB.q\u0000deformed LLG Equation\nTo build up the scale \u0000q\u0000deformed Landau-Lifshitz-Gilbert Equation, we consider eq.(7), with ^A(t) =\n^Sq\nD\u0015\nt;q^Sq(t) =\u0000i\n~Mq[^Sq;H]; (8)\nwhere we supposed that UqandHcommute.\nH=\u0000gq\u0016B\n~Mq^Sq\u000e~Heff: (9)\nHere,~Heffis some effective Hamiltonian whose form that we shall clearly write down in the sequel.\nThe scale\u0000q\u0000deformed momentum operator is here defined as bp\u0015\nq0=\u0000i~Mq0[1 +\u0015(1\u0000q0)x]@q\n@xq:\nConsidering this operator, we obtain a deformed algebra, here in terms of commutation relation between\ncoordinate and momentum\n\u0002\n^xq\ni;^pq\nj\u0003\n={[1 +\u0015(1\u0000q0)x]~Mq0\u000e{jI (10)\nand, for angular momentum components, as\nh\n^Lq\ni;^Lq\nji\n={[1 +\u0015(1\u0000q0)x]~Mq^Lq\nk: (11)\nTheq0factor in ^xq0\n{;^pq0\nj;^Lq0\ni;^Lq0\nj;Mq0is only an index and qis not necessarily equal to q0.\nThe resulting scale \u0000q\u0000deformed LLG equation can now be written as\nD\u0015\nt;q^Sq(t) =\u0000[1 +\u0015(1\u0000q0)x]gq\u0016B\n~Mq^Sq\u0002~Heff: (12)\nTake ^mq\u0011\rq^Sq; \rq0\u0011[1+\u0015(1\u0000q0)x]gq\u0016B\n~Mq.\nIf we consider that the spin algebra is nor affected by any emergent effects, we can take q0= 1.\nConsidering the eq.(7) with ^A(t) = ^Sqand ^mq=j\rqj^Sqandq0= 1; we obtain the q\u0000time deformed\nLLG dynamical equation for magnetization as\nD\u0015\nt;q^mq(t) =\u0000j\rj^mq\u0002~Heff: (13)6\nConsidering ~Heff=H0^k;we have the solution:\nmx;q=\u001acosq(\u00120) cosq(\rH0t) +\u001asinq(\u00120) sinq(\rH0t): (14)\nIn the figure, \u00120= 0:\nFigure 1: Increase/Damping- cosq(x)\n.\nIII. APPLYING AXIOMATIC DERIVATIVE AND THE \u000b\u0000DEFORMED HEISENBERG\nEQUATION\nNow, to compare results with two different local operators, we apply the axiomatic metric derivative.\nFollowing the steps on [12] and considering the axiomatic MD [13], there holds the eigenvalue equation\nD\u000b\nxE\u000b(\u0015x\u000b) =\u0015E\u000b(\u0015x\u000b);whereE\u000b(\u0015x\u000b)is the Mittag-Leffler function that is of crucial importance to\ndescribe the dynamics of complex systems. It involves a generalization of the exponential function and\nseveral trigonometric and hyperbolic functions. The eigenvalue equation above is only valid if we consider\n\u000bvery close to 1:This is what we call low-level fractionality [13]. Our proposal is to allow the use o Leibniz\nrule, even if it would result in an approximation. So, we can build up an evolution operator:\nU\u000b(t;t0) =E\u000b(\u0000i\n~\u000bHt\u000b); (15)\nand for the deformed Heisenberg Equation\nD\u000b\ntAH\n\u000b(t) =\u0000i\n~\u000b[AH\n\u000b;H]; (16)7\nwhere we supposed that U\u000bandHcommute.\nTo build up the deformed Landau-Lifshitz-Gilbert Equation, we use the eq. (16), and considering and\nspin operator ^S\u000b(t), in such a way that we can write the a deformed Heisenberg equation as\nD\u000b\nt^S\u000b(t) =\u0000i\n~\u000b[^S\u000b;H]; (17)\nwhith\nH=\u0000g\u000b\u0016B\n~\u000b^S\u000b\u000e~Heff: (18)\nHere,~Heffis some effective Hamiltonian whose form that we will turn out clear forward.\nNow, consider the deformed momentum operator as [9, 11, 12]\nbp\u000b=\u0000i(~)\u000bMx;\u000b@\u000b\n@x\u000b: (19)\nTakingthisoperator, weobtainadeformedalgebra, hereintermsofcommutationrelationforcoordinate\nand momentum\n\u0002\n^x\u000b\ni;^p\u000b\nj\u0003\n={\u0000(\u000b+ 1)~\u000bM\u000b\u000e{jI (20)\nand for angular momentum components as\nh\n^L\u000b\ni;^L\u000b\nji\n={\u0000(\u000b+ 1)~\u000bM\u000b^L\u000b\nk: (21)\nThe resulting the \u000b\u0000deformed LLG equation can now be written as\nJ\n0D\u000b\nt^S\u000b(t) =\u0000M\u000b\u0000(\u000b+ 1)g\u000b\u0016B\n~\u000b^S\u000b\u0002~Heff: (22)\nIf we take ^m\u000b\u0011\r\u000b^S\u000b,\r\u000b\u0011M\u000b\u0000(\u000b+1)g\u000b\u0016B\n~\u000b, we can re-write the equation as the \u000b\u0000deformed LLG\nJ\n0D\u000b\nt^m\u000b(t) =\u0000j\r\u000bj^m\u000b\u0002~Heff; (23)\nwith~Heff=H0^k. We have the Solution of eq.(23):\nm\u000bx=Acos\u00120E2\u000b(\u0000!2\n0t2\u000b) +Asin\u00120:x:E 2\u000b;1+\u000b(\u0000!2\n0t2\u000b): (24)8\nIn the figure below, the reader may notice the behavior of the magnetization, considering \u00120= 0.\nFigure 2: a) Damping of oscillations. In the figure \f= 1. b) Increase of oscillations\n.\nFor\u000b= 1;the solution reduces to mx=Acos(!0t+\u00120), the standard Simple Harmonic Oscillator\nsolution for the precession of magnetization.\nThe presence of complex interactions and dissipative effects that are not explicitly included into the\nHamiltonian can be seen with the use of deformed metric derivatives. Without explicitly adding up\nthe Gilbert damping term, the damping in the oscillations could reproduce the damping described by\nthe Gilbert term or could it disclose some new extra damping effect. Also, depending on the relevant\nparameter, the q\u0000entropic parameter or for \u000b, the increasing oscillations can signally that it is sensible\nto expect fractionality to interfere on the effects of polarized currents as the Slonczewski term describes.\nWe point out that there are qualitative similarities in both cases, as the damping or the increasing of the\noscillations, depending on the relevant control parameters. Despite that, there are also some interesting\ndifferences, as the change in phase for axiomatic derivative application case.\nHere, we cast some comments about an apparent paradox: If we make, as usually done in the literature\nfor LLG, the scalar product in eq. (13) with, ^mq;we obtain an apparent paradox that the modulus of\n^mqdoes not change. On the other hand, if instead of ^m\u000b;we proceed now with a scalar product with ~Heff\nand we obtain thereby the indications that the angle between ^m\u000band~Heffdoes not change. So, how to\nexplain the damping in osculations for ^mq?This question can be explained by the the following arguments.\nEven the usual LLG equation, with the term of Gilbert, can be rewritten in a form similar to eq. LLG9\nwithout term of Gilbert. See eq. (2.7) in the Ref. [14]. The effective ~Hefffield now stores information\nabout the interactions that cause damping. In our case, when carrying out the simulations, we have taken\n~Heffas a constant effective field. Here, we can argue that the damping term, eq. (2.8) in Ref. [14] being\nsmall, this would cause the effective field ~Heff=\u0000 !H(t) +\u0000 !k(\u0000 !S\u0002\u0000 !H)to be approximately\u0000 !H(t). In this\nway, the scalar product would make dominate over the term of explicit dissipation. This could, therefore,\nexplain the possible inconsistency.\nIV. CONCLUSIONS AND OUTLOOK\nIn short:\nHere, we tackle the problem of LLG equations considering the presence of complexity and dissipation\nor other interactions that give rise to the term proposed by Gilbert or the one by Slonczewski.\nWith this aim, we have applied scale - q\u0000derivative and the axiomatic metric derivative to build up\ndeformed Heisenberg equations. The evolution operator naturally emerges with the use of each case of the\nstructural derivatives. The deformed LLG equations are solved for a simple case, with both structural or\nmetric derivatives.\nAlso, in connection with the LLG equation, we can cast some final considerations for future investiga-\ntions:\nDoes fractionality simply reproduce the damping described by the Gilbert term or could it disclose\nsome new effect extra damping?\nIs it sensible to expect fractionality to interfere on the effects of polarized currents as the Slonczewski\nterm describes?\nThese two points are relevant in connection with fractionality and the recent high precision measure-\nments in magnetic systems may open up a new venue to strengthen the relationship between the fractional\nproperties of space-time and Condensed Matter systems.\n[1] J. Weberszpil, Matheus Jatkoske Lazo and J.A. Helayël-Neto, Physica A 436, (2015) 399–404.10\n[2] Weberszpil, J.; Helayël-Neto, J.A., Physica. A (Print), v. 450, (2016) 217-227; arXiv:1511.02835 [math-ph].\n[3] C. Tsallis, J. Stat. Phys. 52, (1988) 479-487.\n[4] C. Tsallis, Brazilian Journal of Physics, 39, 2A, (2009) 337-356.\n[5] C. Tsallis, Introduction to Nonextensive Statistical Mechanics - Approaching a Complex World (Springer,\nNew York, 2009).\n[6] Alexander S. Balankin and Benjamin Espinoza Elizarraraz, Phys. Rev. E 85, (2012) 056314.\n[7] A. S. Balankin and B. Espinoza, Phys. Rev. E 85, (2012) 025302(R).\n[8] Alexander Balankin, Juan Bory-Reyes and Michael Shapiro, Phys A, in press, (2015)\ndoi:10.1016/j.physa.2015.10.035.\n[9] Weberszpil, J. ; Helayël-Neto, J. A., Advances in High Energy Physics, (2014), p. 1-12.\n[10] F. D. Nobre, M. A. Rego-Monteiro, and C. Tsallis, Phys. Rev. Lett. 106, (2011) 140601.\n[11] J.Weberszpil, C.F.L.Godinho, A.ChermanandJ.A.Helayël-Neto, In: 7thConferenceMathematicalMethods\nin Physics - ICMP 2012, 2012, Rio de Janeiro. Proceedings of Science (PoS). Trieste, Italia: SISSA. Trieste,\nItalia: Published by Proceedings of Science (PoS), 2012. p. 1-19.\n[12] J. Weberszpil and J. A. Helayël-Neto, J. Adv. Phys. 7, 2 (2015) 1440-1447, ISSN 2347-3487.\n[13] J. Weberszpil, J. A. Helayël-Neto, arXiv:1605.08097 [math-ph]\n[14] M. Lakshmanan, Phil. Trans. R. Soc. A (2011) 369, 1280–1300 doi:10.1098/rsta.2010.0319"    },    {        "title": "1701.09110v1.Lack_of_correlation_between_the_spin_mixing_conductance_and_the_ISHE_generated_voltages_in_CoFeB_Pt_Ta_bilayers.pdf",        "content": "arXiv:1701.09110v1  [cond-mat.mes-hall]  31 Jan 2017Lack of correlation between the spin mixing conductance and the ISHE-generated\nvoltages in CoFeB/Pt,Ta bilayers\nA. Conca,1,∗B. Heinz,1M. R. Schweizer,1S. Keller,1E. Th. Papaioannou,1and B. Hillebrands1\n1Fachbereich Physik and Landesforschungszentrum OPTIMAS,\nTechnische Universit¨ at Kaiserslautern, 67663 Kaisersla utern, Germany\n(Dated: June 21, 2021)\nWe investigate spin pumping phenomena in polycrystalline C oFeB/Pt and CoFeB/Ta bilayers\nand the correlation between the effective spin mixing conduc tanceg↑↓\neffand the obtained voltages\ngenerated by the spin-to-charge current conversion via the inverse spin Hall effect in the Pt and Ta\nlayers. For this purpose we measure the in-plane angular dep endence of the generated voltages on\nthe external static magnetic field and we apply a model to sepa rate the spin pumping signal from the\none generated by the spin rectification effect in the magnetic layer. Our results reveal a dominating\nrole of anomalous Hall effect for the spin rectification effect with CoFeB and a lack of correlation\nbetween g↑↓\neffand inverse spin Hall voltages pointing to a strong role of th e magnetic proximity\neffect in Pt in understanding the observed increased damping . This is additionally reflected on the\npresence of a linear dependency of the Gilbert damping param eter on the Pt thickness.\nINTRODUCTION\nIn spin pumping experiments,[1, 2] the magnetization\nof a ferromagnetic layer (FM) in contact with a non-\nmagnetic one (NM) is excited by a microwave field. A\nspin current is generated and injected into the NM layer\nand its magnitude is maximized when the ferromagnetic\nresonance (FMR) condition is fulfilled. The spin cur-\nrent can be detected by using the inverse spin Hall effect\n(ISHE) for conversion into a charge current in appropri-\nate materials. The injected spin current Jsin the NM\nlayer has the form[1]\nJs=/planckover2pi1\n4πg↑↓ˆm×dˆm\ndt(1)\nwhere ˆmis the magnetization unit vector and g↑↓is the\nrealpartofthe spinmixing conductancewhich iscontrol-\nling the intensity of the generated spin current. Its value\nis sensitive to the interface properties. The generation of\nthe spin current opens an additional loss channel for the\nmagnetic system and consequently causes an increase in\nthe measured Gilbert damping parameter α:\n∆αsp=γ/planckover2pi1\n4πMsdFMg↑↓(2)\nThis expression is only valid for thick enough NM lay-\ners where no reflection of the spin current takes place\nat the interfaces. In principle, it allows the estimation\nofg↑↓by measuring the increase in damping compared\nto the intrinsic value. However, other phenomena, like\nthe magnetic proximity effect (MPE) in the case of Pt\nor interface effects depending on the exact material com-\nbination or capping layer material, can have the same\ninfluence, [7, 8] which challenges the measurement of the\ncontribution from the spin pumping. In this sense, it\nis preferable to use an effective value g↑↓\neff. Still, if thespin pumping is the main contribution to the increase\ninα, a correlation between g↑↓\neffand the measured ISHE\nvoltages is expected. A suitable approach in order to un-\nderstand the weight of MPE on the value of g↑↓\neffis the\nuse of FM/NM with varying NM metals, with presence\nand absence of the MPE effect. The measurement of ∆ α\nandg↑↓\nefftogether with the ISHE voltages generated by\nthe spin current in the NM layer can bring clarity to the\nissue.\nHowever, the generation of an additional dc voltage\nby the spin rectification effect,[3–6] which adds to the\nvoltagegeneratedbythe ISHE spin-to-chargeconversion,\ndeters the analysis of the obtained data. The spin recti-\nfication originates from the precession of the magnetiza-\ntion in conducting layers with magnetoresistive proper-\nties, mainly Anisotropic Magnetoresistance (AMR) and\nAnomalous Hall Effect (AHE). Information about the\nphysics behind the measured voltage can only be ob-\ntained after separation of the different contributions. For\nthis purpose, we made use of the different angulardepen-\ndenciesofthecontributionsunderin-planerotationofthe\nexternal magnetic field.\nEXPERIMENTAL DETAILS\nHere, we report on results on polycrystalline\nCo40Fe40B20/Pt,Ta bilayers grown by rf-sputtering on Si\nsubstrates passivated with SiO 2. CoFeB is a material\nchoice for the FM layer due to its low damping proper-\nties and easy deposition.[9, 10] A microstrip-based VNA-\nFMR setup was used to study the damping properties. A\nmore detailed description of the FMR measurement and\nanalysis procedure is shown in previous work.[7, 10] A\nquadrupole-based lock-in setup described elsewhere[11]\nwas used in order to measure the ISHE generated volt-\nage. The dependence of the voltage generated during the\nspin pumping experiment on the in-plane static external2\nfield orientation is recorded for a later separation of the\npure ISHE signal from the spin rectification effect.\nGILBERT DAMPING PARAMETER AND SPIN\nMIXING CONDUCTANCE\nFigure1showsthedependenceoftheeffectivedamping\nparameter αeff(sum ofall contributions)onthe thickness\ndof the NM metal for a CoFeB layer with a fixed thick-\nness of 11 nm. The case d= 0 nm represents the case of\nreference layers with Al capping. From previous studies\nit is known that the use of an Al capping layer induces\na large increase of damping in Fe epitaxial layers.[7] For\npolycrystallineNiFe andCoFeBlayersthis is notthe case\nand it allows the measurement of the intrinsic value α0.\n[8]\nThe observed behavior differs strongly for Pt and\nTa. In the Pt case a large increase in damping is ob-\nserved with a sharp change around d= 1 nm and a\nfast saturation for larger thicknesses. This is quali-\ntatively very similar to our previous report on Fe/Pt\nbilayers.[7] From the measured ∆ αwe extract the value\ng↑↓\neff= 6.1±0.5·1019m−2. This value is larger than the\nonereportedpreviouslyinourgroup[8]forthinnerCoFeB\nlayers with larger intrinsic damping 4 .0±1.0·1019m−2\nandalsolargerthanthevaluereportedbyKim et al.[12],\n5.1·1019m−2. The impact of the Ta layer on damping\nis very reduced and, consequently, a low value for g↑↓\neffof\n0.9±0.3·1019m−2is obtained. This value is now smaller\nthan the one reported by Kim et al.1.5·1019m−2) in-\ndicating that the difference between CoFeB/Pt and Ta\nis larger in our case. A reference has also to be made\nto the work of Liu et al.on CoFeB films thinner than\nin this work. [13] There, no value for the spin mixing\nconductance is provided, but the authors claim a vanish-\nFIG. 1. (Color online) Dependence of the effective Gilbert\ndamping parameter αeffon the thickness of the NM metal.\nA large increase in damping is observed for the Pt case while\na very small but not vanishing increase is observed for Ta.\nFrom the change ∆ αthe effective spin mixing conductance\ng↑↓\neffis estimated using Eq. 2.ing impact on αfor the Ta case. On the contrary the\nincrease due to Pt is almost three times larger than ours,\npointing to a huge difference between both systems. In\nany case, the trend is similar, only the relative difference\nbetween Ta and Pt changes.\nA closer look to the data allows to distinguish a region\nin the Pt damping evolution prior to the sharp increase\nwhere a linear behavior is recognized ( d <1 nm). A lin-\near thickness dependence of αin spin-sink ferromagnetic\nfilms and in polarized Pt has been reported. [14, 15] The\nincreasein damping due to spin currentabsorptionin the\nPt with ferromagnetic order can then be described by:\n∆α= ∆αMPE·dPt/dPt\nc (3)\nwhere ∆αMPEis the total increase in damping due only\nto the magnetic proximity effect in Pt, dPtis the thick-\nness of the Pt layer and dPt\ncis a cutoff thickness which\nis in the order of magnitude of the coherence length in\nferromagnetic layers.[15, 16]\nThe inset in Fig. 1 shows a fit of Eq. 3 from where\ndPt\nc= 0.8nm isobtained assumingavalue ∆ αMPE= 1.2.\nThe value is in qualitative agreement with the reported\nthickness where MPE is present in Pt, ( dPt\nMPE≤1 nm\n[17, 18]) and is lower than the one reported for Py/Pt\nsystems.[14]\n/s45/s52/s48/s48/s52/s48/s56/s48/s49/s50/s48\n/s40/s98/s41\n/s80/s116/s32/s68/s97/s116/s97\n/s32/s70/s105/s116\n/s32/s83/s121/s109/s109/s101/s116/s114/s105/s99\n/s32/s65/s110/s116/s105/s115/s121/s109/s109/s101/s116/s114/s105/s99/s86/s111/s108/s116/s97/s103/s101/s32/s40/s181/s86/s41\n/s84/s97\n/s53/s48 /s49/s48/s48 /s49/s53/s48 /s50/s48/s48/s45/s50/s48/s48/s50/s48\n/s32/s32/s86/s111/s108/s116/s97/s103/s101/s32/s40/s181/s86/s41\n/s181\n/s48/s72/s32/s40/s109/s84/s41/s40/s97/s41\nFIG. 2. (Color online) Voltage spectra measured for (a)\nCoFeB/Ta and (b) CoFeB/Pt at 13 GHz. The solid line is a\nfit to Eq. 4. The symmetric voltage Vsymand antisymmetric\nvoltageVantisymcontributions are separated and plotted inde-\npendently (dashed lines). The voltage signal is dominated b y\nVantisymin the Pt case and by Vsymin the Ta case.3\nThe increase of damping due to spin pumping is de-\nscribed by an exponential dependence and explains the\nsharp increase at dPt= 1 nm. However, the fast increase\ndoes not allow for a deep analysis and it is pointing to a\nspin diffusion length in Pt not larger than 1 nm.\nIn any case, this point has to been treated with care.\nThe contribution of MPE to damping can be easily un-\nderestimated and consequently also the value for dPt\nc. In\nany case, the value can be interpreted as a lower limit\nfor ∆αMPE. If this is substracted, under the assumption\nthat the rest of increase is due to spin pumping, the spin\nmixing conductance due only to the this effect would be\ng↑↓\neff= 4.9±0.5·1019m−2.\nELECTRICAL DETECTION OF SPIN PUMPING\nFigure 2(a),(b) shows two voltage measurements\nrecorded at 13 GHz for a NM thickness of 3 nm and\na nominal microwave power of 33 dBm. The measured\nvoltage is the sum of the contribution of the ISHE effect\nand of spin rectification effect originating from the dif-\nferent magnetoresistive phenomena in the ferromagnetic\nlayer. While the spin rectification effect generates both a\nsymmetric and an antisymmetric contribution, [3–5] the\npure ISHE signal is only symmetric. For this reason a\nseparation of both is carried out by fitting the voltage\nspectra (solid line) to\nVmeas=Vsym(∆H)2\n(H−HFMR)2+(∆H)2+\n+Vantisym−2∆H(H−HFMR)\n(H−HFMR)2+(∆H)2(4)\nwhere ∆ HandHFMRare the linewidth and the reso-\nnance field, respectively. The dotted lines in Fig. 2 show\nthe two contributions. When comparing the data for Pt\nand Ta some differences are observed. First of all, the\nabsolute voltage values are smaller for the Pt cases and,\nmoreimportant, therelativeweightofbothcontributions\nis different. While the first point is related to the differ-\nent conductivity of Ta and Pt, the second one is related\nto the intrinsic effect causing the voltage. We calculate\nthe ratio S/A = Vsym/Vantisymfor all the measurements\nand the results are shown in Fig. 3(a) as a function of the\nNM thickness. While the antisymmetric contribution is\ndominating in the Pt samples with a S/A ratio smaller\nthan 1 for the samples with Pt, the opposite is true for\nthe Ta case. Since the ISHE signal is contributing only\ntoVsymit might be concluded that spin pumping is tak-\ning place stronger in the Ta system. However, since also\nthe spin rectificationeffect has a symmetric contribution,\nthis conclusion cannot be supported. Furthermore, since\nthe spin Hall angle θSHEhas opposite sign in these two\nmaterials, also the ISHE signal should have it. In appar-ent contradiction to this, we observe that both symmet-\nric contributions have the same sign in (a) and (b). This\npoints to the fact that for Pt, Vsymis dominated by the\nspin rectification effect, which does not change sign and\novercompensates a smaller ISHE signal. All these con-\nsiderations have the consequence that it is not possible\nto extract complete information of the origin of the mea-\nsured voltage by analyzing single spectra. For the same\nreason, the large increase in S/A for Ta for d= 5 nm or\nthe change in sign for Pt with the same thickness cannot\nbe correctly explained until the pure ISHE signal is not\nseparated from the spin rectification effect. As already\npointed out in recent papers[3–5, 11, 19], an analysis of\nthe angular dependence (in-plane or out-of-plane) of the\nmeasured voltages can be used to separate the different\ncontributions.\nIn any case, before proceeding it has to be proven that\nallthe measurementswereperformed in the linearregime\nwith small cone angles for the magnetization precession.\nThe measurements performed out of this regime would\nhave a large impact on the linewidth and a Gilbert-like\ndampingwouldnotbeguaranteed. Figure3(b) showsthe\ndependence of the voltage amplitude on the microwave\nnominal power proving indeed that the measurements\nwere carried on in the linear regime.\nFIG. 3. (Color online) (a) Dependence of the ratio S/A\n=Vsym/Vantisymon the thickness of the NM layer. (b)\nDependence of the total voltage on the applied microwave\npower proving the measurements were carried out in the lin-\near regime.4\nSEPARATION OF THE ISHE SIGNAL FROM\nTHE SPIN RECTIFICATION VOLTAGE\nWe performed in-plane angular dependent measure-\nments of the voltage and Eq. 4 was used to extract\nVsym,antisymfor each value of the azimuthal angle φ\nspanned between the direction of the magnetic field and\nthe microstrip antenna used to excite the magnetization.\nWe used a model based on the work of Harder et al.[3] to\nfit the dependence. This model considers two sources for\nthe spin rectification, which are the Anisotropic Mag-\nnetoresistance (AMR) and the Anomalous Hall Effect\n(AHE):\nVsym=Vspcos3(φ)+\n+VAHEcos(Φ)cos( φ)+Vsym\nAMR−⊥cos(2φ)cos(φ)\n+Vsym\nAMR−/bardblsin(2φ)cos(φ)\nVantisym=VAHEsin(Φ)cos( φ) +Vantisym\nAMR−⊥cos(2φ)cos(φ)\n+Vantisym\nAMR−/bardblsin(2φ)cos(φ)\n(5)\nHere,VspandVAHEare the contributions from spin\npumping (pure ISHE) and from AHE, respectively. Φ\nis the phase between the rf electric and magnetic fields\nin the medium. The contribution from the AMR is di-\nvided in one generating a transverse ⊥(with respect to\nthe antenna) or longitudinal /bardblvoltage. In an ideal case\nwith perfect geometry and point-like electrical contacts\nVsym,antisym\nAMR−/bardblshould be close to zero.\nFigure 4 shows the angular dependence of Vsym(top)\nandVantisym(bottom) for the samples with NM thick-\nness of 3 nm. The lines are a fit to the model which\nis able to describe the dependence properly. From the\ndata it can be clearly concluded that while the values\nofVantisymare comparable, with the difference resulting\nfrom the different resistivity of Pt and Ta, the values\nofVsymare much larger for Ta. The values obtained\nfrom the fits for the different contributions are plotted\nin Fig. 5 as a function of the thickness of the NM layer.\nThe value of Φ is ruling the lineshape of the electrically\nmeasured FMR peak[20] which is always a combination\nof a dispersive ( D, antisymmetric) and a Lorentzian ( L,\nsymmetric) contribution in the form D+iL. In order\nto compare the relative magnitudes of the different con-\ntributions independently of Φ we compute the quantities\nVAMR−/bardbl,⊥=/radicalbigg/parenleftBig\nVantisym\nAMR−/bardbl,⊥/parenrightBig2\n+/parenleftBig\nVsym\nAMR−/bardbl,⊥/parenrightBig2\nwhich it\nis equivalent to√\nD2+L2and we show them together\nwithVAHEandVsp. This step is important to allow for\ncomparison of the different contributions independent of\nthe value of Φ.\nSeveral conclusions can be extracted from Fig. 5. First\nof all, the spin rectification effect in CoFeB systems is al-\nmost fully dominated by the AHE. AMR plays a veryFIG. 4. (Color online) Angular dependence of Vsym(top)\nandVantisym(bottom) for CoFeB/Pt,Ta samples with NM\nthickness of 3 nm. The lines are a fit to the model described\nin Eq. 5.\nminor role. This is a difference with respect to NiFe\nor Fe. [4, 11, 20] This is correlated with the very large\nAHE reported in CoFeB films. [21, 22] Second, the volt-\nages generated by the spin pumping via the ISHE are\nlarger in the case of Ta and of opposite sign as expected\nfrom the different sign of θSHEin both materials. This\nsolves the apparent contradiction observed by the posi-\ntive symmetric contributions in both materials as shown\nin Fig. 2(a) and (b) and confirms the interpretation than\ninthecaseofPtthesymmetriccontributionisdominated\nby the spin rectification effect with opposite sign to the\nISHE signal. Again, this shows that the interpretation\nusing single spectra may lead to confusion and that angle\ndependent measurements are required.\nThe evolution of the spin rectification voltages with\nNM thickness shows a saturation behavior in both cases\nfor small thicknesses and a decrease with the NM layer\nthickness compatible with a dominant role of the re-\nsistance of the CoFeB layer. This is expected from\nthe resistivity values for amorphous CoFeB layers, 300-\n600µm·cm,[23] which are much larger than for β-Ta\n(6-10µm·cm) or sputtered Pt (100-200 µm·cm).[24, 25]\nHowever, the dependence does not completely agree with\nthe expected behavior[19] 1 /dNMpointing out to addi-\ntional effects like a variation of the conductivity of Pt for\nthe thinner layers.\nConcerning the correlation of the absolute values of\nthe ISHE-generated voltages and the spin Hall angles in\nboth materials, unfortunately the scatter in θSHEvalues\nin the literature is very large.[26] Howeverthis is reduced\nif we consider works were θSHEwas measured simultane-\nously for Pt and Ta in similar samples. In YIG/Pt,Ta5\nsystems[27, 28] it was determined that |θPt\nSHE|>|θTa\nSHE|\nwith a relative difference of around 30% which it is at\nodds with our results. On the contrary, in CoFeB/Pt,Ta\nbilayers|θTa\nSHE|= 0.15>|θPt\nSHE|= 0.07 is reported.[13]\nHowever the difference is not large enough to cover com-\npletely the difference in our samples. In order to ex-\nplain this point together with the absolute low value in\nCoFeB/Pt we have to take into account the possibility of\na certain loss of spin current at the interface FM/Pt or at\nthe very first nanometer, the latter due to the presence\nof a static magnetic polarization due to the proximity ef-\nfect. With this the spin current effectively being injected\nin Pt would be lower than in the Ta case.\nThe data does not allow for a quantitative estimation\nof the spin diffusion length λsd, but in any case the evo-\nlution is only compatible with a value for Pt not thicker\nthan 1 nm, similarto reportedvalues forsputtered Pt[25]\nand a a value of a few nm for Ta, also compatible with\nliterature.[28]\nAn important point is the lack of correlation of g↑↓\neff\nand the expected generated spin current using Eq. 1 with\nthe absolute measured ISHE voltage that results from\nthe spin-to-charge current conversion, obtained after the\nseparationfromthe overimposedspin rectificationsignal.\nThis is true even if we substract the MPE contribution\nassumed for Eq. 3. The same non-mutually excluding\nexplanations are possible here: ∆ αin Pt in mainly due\nto the MPE, or the spin current pumped into Pt van-\nishes at or close to the interface. The first alternative\nwould render Eq. 2 unuseful since most of the increase\nin damping is not due to spin pumping as long as the\nMPE is present. The second would reduce the validity\nof Eq. 1 to estimate the current injected in Pt and con-\nverted into a charge current by the ISHE. In any case,\nCoFeB/Ta shows very interesting properties, with strong\nspin pumping accompanied by only a minor impact on\nα.\nLet us discuss the limitations of the model defined in\nEq. 5 and the suitability to describe the measurements.\nFirst of all, the model assumes a perfect isotropic mate-\nrial. The anisotropy in CoFeB is known to be small but\nnot zero and a weak uniaxial anisotropy is present. The\neffect onthe angulardependenceisnegligible. Themodel\nassumes also a perfect geometry and point-like electrical\ncontacts to measure the voltages. Our contacts are ex-\ntended (∼200µm) and a small misalignment is possible\n(angle between the antenna and the imaginary line con-\nnecting the electrical contacts may not be exactly 90◦).\nThis is the most probable reason for the non-vanishing\nsmall value for Vsym,antisym\nAMR−/bardbl. Nevertheless, the angular\ndependence of the measured voltage is well described by\nthe model and no large deviations are observed.\nFIG. 5. (Color online) NM thickness dependence of the dif-\nferent contributions to the measured voltages extracted fr om\nthe angular dependence of VsymandVantisymfor Ta (top) and\nPt (bottom).\nCONCLUSIONS\nIn summary, we made use of in-plane angular de-\npendent measurements to separate ISHE-generated from\nspin rectification voltages and we compare the absolute\nvalues and thickness dependence for Pt and Ta. Differ-\nently to other materials, the spin rectification signal in\nCoFeB is almost fully dominated by AHE. No correlation\nbetween the observed spin mixing conductance via FMR\nmeasurement and the spin pumping signal is obtained\npointing to a dominant role of the magnetic proximity\neffect in the increase in damping with Pt.\nACKNOWLEDGEMENTS\nFinancial support by M-era.Net through the\nHEUMEM project and by the Carl Zeiss Stiftung\nis gratefully acknowledged.\n∗conca@physik.uni-kl.de\n[1] Y. Tserkovnyak, A. Brataas, G. E. W. Bauer, and\nB. I. Halperin, Rev. Mod. Phys. 77, No. 4, 1375 (2005).\n[2] Y. Tserkovnyak, A. Brataas, and G. E. W. Bauer,\nPhys. Rev. Lett. 88, 117601 (2002).\n[3] M. Harder, Y. Gui, and C.-M. Hu, arXiv:1605.00710v1\n[cond-mat.mtrl-sci] (2016).6\n[4] W. T. Soh, B. Peng, and C. K. Ong, J. Phys. D:\nAppl. Phys. 47, 285001 (2014).\n[5] Y. Gui, L. Bai, and C. Hu, Sci. China-Phys. Mech. As-\ntron.56, 124 (2013).\n[6] W. Zhang, , B. Peng, F. Han, Q. Wang, W. T. Soh,\nC. K. Ong, and W. Zhang, Appl. Phys. Lett. 108, 102405\n(2016).\n[7] A. Conca, S. Keller, L. Mihalceanu, T. Kehagias,\nG. P. Dimitrakopulos, B. Hillebrands, and E. Th. Pa-\npaioannou, Phys. Rev. B 93, 134405 (2016).\n[8] A. Ruiz-Calaforra, T. Br¨ acher, V. Lauer, P. Pirro,\nB. Heinz, M. Geilen, A. V. Chumak, A. Conca, B. Leven,\nand B. Hillebrands, J. Appl. Phys. 117, 163901 (2015).\n[9] A. Conca, J. Greser, T. Sebastian, S. Klingler, B. Obry,\nB. Leven, and B. Hillebrands, J. Appl.Phys. 113, 213909\n(2013).\n[10] A. Conca, E. Th. Papaioannou, S. Klingler, J. Greser,\nT. Sebastian, B. Leven, J. L¨ osch, and B. Hillebrands,\nAppl. Phys. Lett. 104, 182407 (2014).\n[11] S. Keller et al., in preparation.\n[12] D.-J Kim, S.-I. Kim, S.-Y. Park, K.-D. Lee, and B.-\nG. Park, Current Appl. Phys. 14, 1344 (2014). Please\nnote the different stoichiometry: Co 32Fe48B20.\n[13] L. Liu, C.-F. Pai, Y. Li, H. W. Tseng, D. C. Ralph, and\nR. A. Buhrman, Science 336, 555 (2012).\n[14] M. Caminale, A. Ghosh, S. Auffret, U. Ebels, K. Ollefs,\nF. Wilhelm, A. Rogalev, and W. E. Bailey, Phys. Rev. B.\n94, 014414 (2016).\n[15] A. Ghosh, S. Auffret, U. Ebels, and W. E. Bailey,\nPhys. Rev. Lett. 109, 127202 (2012).\n[16] M. D. Stiles and A. Zangwill, Phys. Rev. B. 66, 014407\n(2002).[17] M. Suzuki, H. Muraoka, Y. Inaba, H. Miyagawa,\nN. Kawamura, T. Shimatsu, H. Maruyanma, N. Ishi-\nmatsu, Y. Isohama, and Y. Sonobe, Phys. Rev. B 72,\n054430 (2005).\n[18] F. Wilhelm, P. Poulopoulos, G. Ceballos, H. Wende,\nK.Baberschke, P.Srivastava, D.Benea, H.Ebert, M.An-\ngelakeris, N. K. Flevaris, D. Niarchos, A. Rogalev, and\nN. B. Brookes, Phys. Rev. Lett. 85, 413 (2000).\n[19] R. Iguchi, and E. Saitoh, arXiv:1607.04716v1 [cond-\nmat.mtrl-sci] (2016).\n[20] M. Harder, Z. X. Cao, Y. S. Gui, X. L. Fan, and C.-M.\nHu, Phys. Rev. B. 84, 054423 (2011).\n[21] T. Zhu, P. Chen, Q. H. Zhang, R. C. Yu, and B. G. Liu,\nAppl. Phys. Lett. 104, 202404 (2014).\n[22] T. Zhu, Chin. Phys. B 23,No 4, 047504 (2014).\n[23] S. U. Jen, Y. D. Yao, Y. T. Chen, J. M. Wu, C. C. Lee,\nT. L. Tsai, Y. C. Chang, J. Appl. Phys. 99, 053701\n(2006).\n[24] K. Stella, D. B¨ urstel, S. Franzka, O. Posth and\nD. Diesing, J. Phys. D: Appl. Phys. 42, 135417 (2009).\n[25] E. Sagasta, Y. Omori, M. Isasa, M. Gradhand,\nL. E. Hueso, Y. Niimi, Y. Otani, and F. Casanova,\nPhys. Rev. B. 44, 060412 (2016).\n[26] J. Sinova, S. O. Valenzuela, J. Wunderlich, C. H. Back,\nandT. Jungwirth, Reb.ofModern Phys. 87, 1213(2015).\n[27] H. L. Wang, C. H. Du, Y. Pu, R. Adur, P. C. Hammel,\nand F. Y. Yang, Phys. Rev. Lett. 112, 197201 (2014).\n[28] C. Hahn, G. de Loubens, O. Klein, M. Viret, V. V. Nale-\ntov, andJ. BenYoussef, Phys.Rev.B. 87, 174417 (2013)."    },    {        "title": "1702.05588v2.Inf_sup_stable_finite_element_methods_for_the_Landau__Lifshitz__Gilbert_and_harmonic_map_heat_flow_equation.pdf",        "content": "arXiv:1702.05588v2  [math.NA]  21 Mar 2017INF-SUP STABLE FINITE-ELEMENT METHODS FOR THE\nLANDAU–LIFSHITZ–GILBERT AND HARMONIC MAP HEAT FLOW\nEQUATION\nJUAN VICENTE GUTI ´ERREZ-SANTACREU†AND MARCO RESTELLI‡\nAbstract. In this paper we propose and analyze a finite element method fo r both the harmonic map\nheat and Landau–Lifshitz–Gilbert equation, the time varia ble remaining continuous. Our starting\npoint is to set out a unified saddle point approach for both pro blems in order to impose the unit\nsphere constraint at the nodes since the only polynomial fun ction satisfying the unit sphere constraint\neverywhere are constants. A proper inf-sup condition is pro ved for the Lagrange multiplier leading to\nthe well-posedness of the unified formulation. A priori energy estimates are shown for the proposed\nmethod.\nWhen time integrations are combined with the saddle point fin ite element approximation some\nextra elaborations are required in order to ensure both a priori energy estimates for the director or\nmagnetization vector depending on the model and an inf-sup c ondition for the Lagrange multiplier.\nThis is due to the fact that the unit length at the nodes is not s atisfied in general when a time\nintegration is performed. We will carry out a linear Euler ti me-stepping method and a non-linear\nCrank–Nicolson method. The latter is solved by using the for mer as a non-linear solver.\n2010 Mathematics Subject Classification. 35K55; 65M12; 65M60.\nKeyword. Finite-element approximation; Inf-sup conditions; Landau–Lifshit z–Gilbert equation;\nHarmonic map heat flow equation.\nContents\n1. Introduction 2\n1.1. The model 2\n2. Statement of the saddle-point problem 4\n2.1. Notation 4\n2.2. Saddle-point formulation 5\n2.3. Inf-sup conditions 5\n3. Spatial discretization 5\n3.1. Finite element spaces 5\n4. Numerical scheme 8\n5. Temporal discretization 11\n6. Implementation details 13\n7. Numerical results 14\n7.1. Convergence test for smooth solutions 14\n7.2. Behaviour for singular solutions 15\n8. Conclusion 20\nReferences 20\nDate: September 15, 2018.\n†Dpto. de Matem´ atica Aplicada I, Universidad de Sevilla, E. T. S. I. Inform´ atica. Avda. Reina Mercedes, s/n.\n41012 Sevilla, Spain. juanvi@us.es . Partially supported by Ministerio de Econom´ ıa y Competit ividad under Spanish\ngrant MTM2015-69875-P with the participation of FEDER.\n‡Numerische Methoden in der Plasmaphysik, Max-Planck-Inst itut f¨ ur Plasmaphysik, Boltzmannstr. 2, 85748 Garch-\ning, Germany ( marco.restelli@ipp.mpg.de ).\n12 JUAN VICENTE GUTI ´ERREZ-SANTACREU†AND MARCO RESTELLI‡\n1.Introduction\n1.1.The model. In this paper we propose and analyze inf-sup stable finite element ap proximations\nfor the harmonic map heat and Landau–Lifshitz–Gilbert equation. T he unified equations are given by\n\n\n∂tu−γ∆u−γ|∇u|2u+αu×∂tu=0in Ω×R+,\n|u|= 1 in Ω ×R+,\n∂nu=0on∂Ω×R+,\nu(0) = u0in Ω,(1)\nwhereu: Ω×R+→SM−1, withM= 2 or 3 being the space dimension, Ω is a bounded domain of\nRMwith boundary ∂Ω,SM−1is the unit ( M−1)-sphere, ∂nis the normal derivative, with nbeing\nthe unit outward normal vector on ∂Ω, andγ,α∈Rwithγ >0 andα≥0; forM= 2 we assume that\nα= 0, so that the last term in the right-hand-side of (1) 1appears only in the three-dimensional case.\nThe normalization condition (1) 2, where| ·|stands for the Euclidean norm for vectors and matrices\nand which will be referred to as the unit sphere constraint , is assumed to be satisfied by the initial\ncondition u0, i.e.|u0|= 1; it can be verified that this assumption, together with (1) 1, implies (1) 2for\nevery time t >0.\nEquations (1) arise in the phenomenological description of widely diffe rent physical systems. Ac-\ncording to the theory developed by Ericksen [20, 21] and Leslie [32, 3 3], system (1) for α= 0 may\ngovern the dynamics of a nematic crystal fluid in the limit of low fluid velo city, where the coupling to\nthe fluid motion is negligible. Here, urepresents the orientation of the liquid crystal molecules, which\nare modeled as elongated rods tending to line up locally along a preferr ed direction, while γstands\nfor a relaxation time constant. According to the theory by Landau and Lifshitz [31] and Gilbert [24]\n(in his apparently unpublished first version in [23]), system (1) may als o govern the dynamics of mag-\nnetization in ferromagnetic materials in the classical continuum appr oximation, where the relativistic\ninteractions are modeled by the damping term αu×∂tuand the thermal fluctuations are negligible.\nTo be more precise, for such a case, the original equation takes th e form\n∂tu+γu×(u×∆u)+αu×∂tu=0, (2)\nwhich can be recast as (1) by using the identity\nu×(u×∆u) =−∆u−|∇u|2u. (3)\nHere,ustands for the magnetization vector without the presence of an a pplied magnetic field, while\nγandαstand for the electron gyromagnetic radius and a damping paramet er, respectively.\nIn constructing a numerical algorithm for approximating (1), one lo oks for an energy law which is\nsatisfied at the continuous level; such an energy law can be obtained as follows. Multiplying (1) 1by\n∂tuand integrating over Ω, we obtain\n/integraldisplay\nΩ|∂tu(x)|2dx−γ/integraldisplay\nΩ∆u(x)·∂tu(x)dx−γ/integraldisplay\nΩ|∇u(x)|2u(x)·∂tu(x)dx= 0.\nSinceu·∂tu= 0 by virtue of (1)2, the third term in this expression vanishes, while the second one\ncan be rewritten as\n−/integraldisplay\nΩγ∆u(x)·∂tu(x)dx=1\n2d\ndt/integraldisplay\nΩγ|∇u(x)|2dx\nby using the Green formula and the homogeneous boundary conditio n (1)3. Thus, the energy law for\n(1) reads/integraldisplay\nΩ|∂tu(x)|2dx+1\n2d\ndt/integraldisplay\nΩγ|∇u(x)|2dx= 0. (4)\nThe fact that, in the derivation of (4), the unit sphere constraint is invoked in a pointwise sense has\nimportant implications at the time of deriving a numerical discretizatio n for (1), where it turns out\nbeing a major source of difficulties. In fact, two contradicting requ irements must be accounted for: on\nthe one hand, using standard piecewise polynomial finite element spa ces, the only possibility to satisfy\nthe unit sphere constraint in a pointwise sense, which would then allow repeating the derivation of (4)INF-SUP CONDITION FOR THE LLG AND HARMONIC MAP HEAT EQUATION 3\nalso at the discrete level, is using piecewise constant functions; on t he other hand, (1) 1calls for more\nregularity in the approximating space for uthan is provided by a piecewise constant function.\nTo this end, we note that various approaches have been considere d in the literature.\nOne first possibility was consideringa projection step which, in its rud imental version [37], consisted\nin enforcing the unit sphere constraint at the sole finite–element no des. This resulted in a numerical\nscheme using first order conforming finite elements which did not enj oying a discrete energy law.\nIn [2], a refined version of the method was proposed, where a finite– element approximation of ∂tuwas\ncomputed in a suitable tangent space and for which convergence to weak solutions could be proved\nunder the assumption that the space and time discretization param eters tend to zero in a specified\nway. Such a restriction on the space and time discretization parame terswas motivated by the use of an\nexplicitfirst-ordertime integrator;[1] then introducedaformulat ionwhichcircumventedthisdrawback\nusing aθ-method. In this latter formulation, for θ∈(1\n2,1], the algorithm was unconditionally energy\nstable and convergent. Yet, the main limitation is that the projectio n step prevented the scheme from\nbeing second order accurate in time; subsequent modifications add ressing this have been considered\nin [3, 4].\nOne second possibility was using closed nodal integration together w ith reformulation (2), in order\nto avoid the projection step, required to enforce the nodal fulfillm ent of the unit sphere constraint.\nThis approach has been successfully used with a Crank–Nicolson time integration to obtain numerical\nschemes which satisfied a discrete energy law and preserved the un it sphere constraint at the nodes\nwhile converging toward weak solutions. The conditional solvability of this approach is the main\ndisadvantage with respect to the projection method. We refer to [8] for the Landau–Lifshitz–Gilbert\nequation and [9] for the harmonic map heat flow equation.\nOne third option [37] was reformulating (1) at the continuous level in troducing a penalization term\nto enforce the unit sphere constraint, which also requires modifyin g the expression of the energy\nlaw (4) including the potential of the penalization term itself. The pen alty method was probably the\nfirst strategy for approximating (1), and the most common penalt y function is the Ginzburg–Landau\nfunction. The key idea is that an energy law can be obtained without e xplicitly using the unit sphere\nconstraint. Yet, a significant drawback of this approach is that ch oosing a “good” value for the penalty\nparameter is far from trivial.\nOne fourth possibility was based on introducing in (1) a Lagrange mult iplier associated with the\nunit sphere constraint, hence obtaining a saddle point formulation. To the best of our knowledge, the\nonly numerical scheme using such an approach can be found in [10], wh ere the multiplier was chosen\nso that the unit sphere restriction was enforced at the nodal poin ts, taking advantage of a closed\nnodal numerical integration rule. Regarding the time integration, a second-order algorithm based on a\nCrank–Nicolson method was used to approximate the primary variab le while the Lagrange multiplier\nwas implicitly computed in terms of the primary variable itself. An uncon ditional energy law was\nobtained and convergence toward weak solutions established. No in f-sup condition was proved in [10],\nbecause at the time the finite element spaces and the estimates for the Lagrange multiplier were not\nwell understood. In fact, the study of the inf-sup condition for t he Lagrange multiplier in the saddle\npoint formulation of (1) is one of the main contributions of the prese nt paper.\nIn addition to the above references, the interested reader is ref erred to [30, 17] for two numerical\nsurveys concerning specific topics for the Landau–Lifshitz–Gilber t equations.\nThe first three methods mentioned above share one main drawback , namely the fact that they\nare can not be easily modified when a coupling term comes into play. For instance, the Ericksen–\nLeslie equations consist of the Navier–Stokes equations with an add itional viscous stress tensor and\na convective harmonic map heat flow equation. In [11], a numerical sc heme is proposed for solving\nthem following the ideas in [8] and [9]. However, the presence of the co nvective term in the harmonic\nmap heat flow equation prevents fulfilling the discrete unit sphere co ndition, despite the possibility\nto obtain a priori energy estimates. Instead, in [6], a saddle point formulation is prese nted for the\nEricksen–Leslie equations enjoying a discrete energy law and allowing a nodal enforcement of the unit\nsphere constraint. Yet, the inf-sup condition was not well unders tood at the time of writing [6]. In this4 JUAN VICENTE GUTI ´ERREZ-SANTACREU†AND MARCO RESTELLI‡\nwork we present some ideas which lead to an inf-sup condition for the associated Lagrange multiplier\nof the scheme described in [6]; thereby, the numerical analysis may b e concluded.\nThe goal of the present paper is to providea saddle point framewor kfor approximating(1) in which,\nusing appropriate numerical tools, an inf-sup stable finite element m ethod can be constructed. In this\nregard, it should be stressed that a proper choice of the finite elem ent spaces for the saddle point\nproblem, namely one which results in favorable estimates for the Lag range multiplier, is extremely\nimportant in order to ensure stability and avoid the so-called lockingof the numerical solution, i.e.\nan unphysical stiffness of the computed ufield [12]. To deal with the contradictory requirements\nmentioned above concerning the regularity of the numerical solutio n on the one hand and, on the\nother hand, the fulfillment of the unit sphere constraint, we propo se to use first order, conforming\nfinite elements and to enforce the unit sphere constraint at the fin ite element nodes. We show that,\nwhen combined with a suitable closed quadrature rule, this ansatz re sults in a discrete version of the\nenergy low (4). Hence, summarizing, we are interested in a numerica l algorithm which uses low-order\nfinite elements, preserves the unit length at the nodal points and s atisfies a discrete energy law and a\ndiscrete inf-sup condition for the discrete Lagrange multiplier.\nMoreover, we discuss two time integrators for our finite element sa ddle point formulation. Indeed,\nit seems that there are very few time integrators available which pre serve a discrete energy law. In\nparticular, we will present one first-order time integrator based o n a semi-implicit Euler method and\none second-order time integrator based on the Crank–Nicolson me thod.\nThe rest of the paper is organized as follows. In section 2, some not ation is introduced, then we\npresent the saddle-point formulation for (1) and prove an energy law for such formulation. Moreover,\nsome inf-sup conditions for the Lagrange multiplier are established. In section 3 we set out our\nassumptions concerning the finite element spaces used to approxim ate the saddle-point formulation,\nand conclude with some results required in proving an inf-sup conditio n at the discrete level. In\nsection 4 we present our numerical scheme discretized in space with the time being continuous. In\nsection5weend up with sometime realizationsofthe semi-discretized schemethat preservethe desired\nproperties. Section 6 deals with some specific implementation aspect s of the fully discretized scheme.\nFinally, section 7 is devoted to various computational experiments.\n2.Statement of the saddle-point problem\n2.1.Notation. We will assume the following notation throughout this paper. Let O ⊂RM, with\nM= 2 or 3, be a Lebesgue-measurable domain and let 1 ≤p≤ ∞. We denote by Lp(O) the space\nof all Lesbegue-measurable real-valued functions, f:O →R, beingpth-summable in Oforp <∞or\nessentially bounded for p=∞, and by /ba∇dblf/ba∇dblLp(O)its norm. When p= 2, the L2(O) space is a Hilbert\nspace whose inner product is denoted by ( ·,·).\nLetα= (α1,α2,...,αd)∈NMbe a multi-index with |α|=α1+α2+...+αM, and let ∂αbe the\ndifferential operator such that\n∂α=/parenleftBig∂\n∂x1/parenrightBigα1\n.../parenleftBig∂\n∂xd/parenrightBigαd.\nForm≥0 and 1≤p≤ ∞, we define Wm,p(O) to be the Sobolev space of all functions whose m\nderivatives are in Lp(O), with the norm\n/ba∇dblf/ba∇dblWm,p(O)=\n/summationdisplay\n|α|≤m/ba∇dbl∂αf/ba∇dblp\nLp(O)\n1/p\nfor 1≤p <∞,\n/ba∇dblf/ba∇dblWm,p(O)= max\n|α|≤m/ba∇dbl∂αf/ba∇dblL∞(Ω), forp=∞,\nwhere∂αis understoodin the distributionalsense. In the particularcaseof p= 2,Wm,p(O) =Hm(O).\nWe also consider C0(¯O) to be the space of continuous functions on ¯O.\nFor any space X, we shall denote the vector space Xdby its bold letter X. For example, ( L2(O))d\nis denoted by L2(O), (Hm(O))dbyHm(O), etc. Consistently, in order to distinguish scalar-valued\nfields from vector-valued ones, we denote them by roman letters a nd bold-face letters, respectively. ToINF-SUP CONDITION FOR THE LLG AND HARMONIC MAP HEAT EQUATION 5\nshorten the notation, the norms /ba∇dbl·/ba∇dblL2(Ω)and/ba∇dbl·/ba∇dblL2(Ω)are abbreviated /ba∇dbl·/ba∇dbl; moreover, the dual space\nofXis denoted by X′, with/an}b∇acketle{t·,·/an}b∇acket∇i}htindicating its dual pairing.\n2.2.Saddle-point formulation. The saddle-point formulation for (1) reads as follows: Find u:\nΩ×R+→SN−1andq: Ω×R+→Rsatisfying\n\n\n∂tu−γ∆u+γqu+αu×∂tu=0in Ω×R+,\n|u|2= 1 in ∂Ω×R+.\n∂nu=0on∂Ω×R+,\nu(0) = u0in Ω.(5)\nThe energy estimate associated with problem (5) was derived in [6]. If we multiply (5) 1by∂tu, and\nintegrate over Ω, we have\n/ba∇dbl∂tu/ba∇dbl2+1\n2d\ndt/ba∇dbl∇u/ba∇dbl2+/integraldisplay\nΩ∂tu(x)·q(x)u(x)dx= 0.\nTo control the third term on the left hand side of the above equatio n, we take the time derivative of\n|u|2= 1. Thus, it follows that ∂tu·u= 0, i.e. ∂tuanduare orthogonal. Therefore,\n/ba∇dbl∂tu/ba∇dbl2+1\n2d\ndt/ba∇dbl∇u/ba∇dbl2= 0. (6)\nThe method under consideration is based on a variational formulatio n for (5) with uandqas\nprimary variables, where the unit sphere constraint is satisfied only at the nodes. This requirement is\nenough to prove a discrete version of an inf-sup condition.\n2.3.Inf-sup conditions. The natural inf-sup condition for problem (5) is\n/ba∇dblq/ba∇dblL∞(Ω)′≤sup\n¯u∈L∞(Ω)\\{0}/an}b∇acketle{tq,u·¯u/an}b∇acket∇i}ht\n/ba∇dbl¯u/ba∇dblL∞(Ω)∀q∈L∞(Ω)′, (7)\nsinceq=−|∇u|2∈L∞(0,T;L1(Ω)) and L1(Ω)⊂L∞(Ω)′. To prove such an inf-sup condition (7) one\nneeds to make the assumption that |u|= 1 holds a.e. in Ω. Under this assumption, let us first see that\nthe mapping u·:L∞(Ω)→L∞(Ω) is surjective. Indeed, let e∈L∞(Ω), then choose ¯u=ue. Clearly,\ne=u·¯u∈L∞(Ω). Next, observe that /ba∇dbl¯u/ba∇dblL∞(Ω)≤ /ba∇dble/ba∇dblL∞(Ω). Thus, we have\n/ba∇dblq/ba∇dblL∞(Ω)′= sup\ne∈L∞(Ω)\\{0}/an}b∇acketle{tq,e/an}b∇acket∇i}ht\n/ba∇dble/ba∇dblL∞(Ω)≤sup\n¯u∈L∞(Ω)\\{0}/an}b∇acketle{tq,u·¯u/an}b∇acket∇i}ht\n/ba∇dbl¯u/ba∇dblL∞(Ω)\nfor allq∈L∞(Ω)′. This inf-sup condition however is not applicable because, due to the presence of\n−∆uin (5)1which can not be bounded in L∞(Ω)′. Therefore, we need to weaken the norm for the\nLagrange multiplier q. Now, the mapping u·:L∞(Ω)∩H1(Ω)→L∞(Ω)∩H1(Ω) is surjective by\nassuming u∈L∞(Ω)∩H1(Ω) such that |u|= 1 a.e. in Ω. Moreover, there exists a positive constant\nC=C(u) such that /ba∇dbl∇¯u/ba∇dbl ≤C/ba∇dbl∇e/ba∇dblfore∈L∞(Ω)∩H1(Ω). Thus, if q∈(H1(Ω)∩L∞(Ω))′, then\none can prove\n/ba∇dblq/ba∇dbl(H1(Ω)∩L∞(Ω))′≤C sup\n¯u∈H1(Ω)∩L∞(Ω)\\{0}/an}b∇acketle{tq,u·¯u/an}b∇acket∇i}ht\n/ba∇dbl∇¯u/ba∇dbl+/ba∇dbl¯u/ba∇dblL∞(Ω). (8)\nFrom a numerical point of view, one must be aware that the difficulty lie s in establishing the\ncounterpart of such an inf-sup condition at the discrete level.\n3.Spatial discretization\n3.1.Finite element spaces. Herein we introduce the hypotheses that will be required along this\nwork.\n(H1) Let Ω be a bounded domainof RMwith apolygonalorpolyhedralLipschitz-continuousbound-\nary.6 JUAN VICENTE GUTI ´ERREZ-SANTACREU†AND MARCO RESTELLI‡\n(H2) Let {Th}h>0be a family of shape-regular, quasi-uniform triangulations of Ω made up of tri-\nangles in two dimensions and tetrahedra in three dimensions, so that Ω =∪K∈ThK, where\nh= max K∈ThhK, withhKbeing the diameter of K. Further, let Nh={ai}i∈Idenote the set\nof all nodes of Th.\n(H3) Conforming finite-element spaces associated with Thare assumed for approximating H1(Ω).\nLetP1(K) be the set oflinear polynomialson K; the space ofcontinuous, piecewise polynomial\nfunctions on This then denoted as\nXh=/braceleftbig\nvh∈C0(Ω) :vh|K∈ P1(K),∀K∈ Th/bracerightbig\n.\nForvh∈Xh, we denote the nodal values by vh(a) =va. Also, we identify the Lagrangianbasis\nfunctions of Xhthrough the node where they do not vanish, using the notation ϕa, so that\nvh=/summationtext\na∈Nvaϕaand, for vector valued functions, vh=/summationtextM\ni=1ei/summationtext\na∈Nvi\naϕa=/summationtext\na∈Nvaϕa,\nwitheibeing the unit vectors of the canonical basis in RM.\n(H4) We assume that u0∈H1(Ω) with |u0|= 1 a.e. in Ω. Then we consider u0h∈Uhsuch that\n|u0h(a)|= 1 for all a∈ Nhand/ba∇dbl∇u0h/ba∇dbl ≤ /ba∇dbl∇u0/ba∇dbl.\nWe choose the following continuous finite-element spaces\nUh=XhandQh=Xh\nto approximate the vector field and the Lagrange multiplier, respec tively.\nIn proving a discrete inf-sup condition we will need to set out some co mmuter properties for the\nnodal projection operator into Uh. Although these properties were already obtained in [28], the proof\nof such properties will be helpful to see that the above assumption s are enough for our purpose.\nTo start with, some inverse inequalities are provided in the following pr oposition (see e.g. [13, Lm\n4.5.3] or [22, Lm 1.138]).\nProposition 3.1. Under hypotheses (H1)–(H3), it follows that, for all xh∈ P1(K),\n/ba∇dbl∇xh/ba∇dblL2(K)≤Cinvh−1\nK/ba∇dblxh/ba∇dblL2(K), (9)\nand\n/ba∇dbl∇xh/ba∇dblL∞(K)≤Cinvh−1\nK/ba∇dblxh/ba∇dblL∞(K), (10)\nwhereCinv>0is a constant independent of handK.\nFor each K∈ Th, letiKbe the local nodal interpolation operator defined from C0(K) intoP1(K),\nand letiXhbe the associated global nodal interpolation operatorfrom C0(¯Ω) intoXh, i.e.iK:=iXh|K,\nfor allK∈ Th. Moreover, let π0\nKbe theL2(K) orthogonal projection operator from L1(Ω) onto P0,\nwhereP0is the set of constant polynomials on K. Next we give some local error estimates for these\ntwo local interpolants. See e.g. [13, Th 4.4.4] or [22, Th 1.103].\nProposition 3.2. Suppose that hypotheses (H1)–(H3)hold. Then the local nodal interpolation operator\niKsatisfies\n/ba∇dblϕ−iKϕ/ba∇dblL2(K)≤Capph2\nK/ba∇dbl∇2ϕ/ba∇dblL2(K)for all ϕ∈H2(K) (11)\nand\n/ba∇dblϕ−iKϕ/ba∇dblL∞(K)≤CapphK/ba∇dbl∇ϕ/ba∇dblL∞(K)for all ϕ∈W1,∞(K), (12)\nwhereCapp>0is a constant independent of KandhK.\nProposition 3.3. Under hypotheses (H1)–(H3), it follows that, for all xh,yh∈Xh,\n/ba∇dblxhyh−iK(xhyh)/ba∇dblL2(K)≤CapphK/ba∇dbl∇(xhyh)/ba∇dblL2(K), (13)\nwhereCapp>0is a constant independent of KandhK.\nProof.Estimate(13)followsreadilyfrom(11)and(9), uponobservingtha tthecomponentsof ∇(xhyh)\nbelong to P1(K). /squareINF-SUP CONDITION FOR THE LLG AND HARMONIC MAP HEAT EQUATION 7\nProposition 3.4. Suppose that hypotheses (H1)–(H3)hold. Then π0\nKsatisfies\n/ba∇dblϕ−π0\nKϕ/ba∇dblL2(K)≤CapphK/ba∇dbl∇ϕ/ba∇dblL2(K)for all ϕ∈H1(K), (14)\nand\n/ba∇dblϕ−π0\nKϕ/ba∇dblL∞(K)≤CapphK/ba∇dbl∇ϕ/ba∇dblL∞(K)for all ϕ∈H1(K), (15)\nwhereCapp>0is constant independent of hK.\nLetπQhdenote the L2(Ω)-orthogonal projection operator from L2(Ω) into Qh. The following\nproposition deals with the stability of πQh. See [14] and [22, Lm 1.131].\nProposition 3.5. Suppose that assumptions (H1)–(H3)are satisfied. Then there exists a positive\nconstant Csta, independent of h, such that\n/ba∇dblπQhϕ/ba∇dblL∞(Ω)≤Csta/ba∇dblϕ/ba∇dblL∞(Ω)for all ϕ∈L∞(Ω), (16)\nand\n/ba∇dbl∇πQhϕ/ba∇dbl ≤Csta/ba∇dbl∇ϕ/ba∇dblfor all ϕ∈H1(Ω). (17)\nWe will prove discrete commuter properties for iXh, following very closely the arguments of [28].\nProposition 3.6. Assume that hypotheses (H1)–(H3)hold and let xh,yh∈Xh. Then there exists a\nconstant Ccom>0, independent of handK, such that\n/ba∇dblxhyh−iK(xhyh)/ba∇dblL∞(K)≤CcomhK/ba∇dblxh/ba∇dblL∞(K)/ba∇dbl∇yh/ba∇dblL∞(K), (18)\nand\n/ba∇dbl∇(xhyh−iK(xhyh))/ba∇dblL2(K)≤Ccom/ba∇dblxh/ba∇dblL∞(K)/ba∇dbl∇yh/ba∇dblL2(K) (19)\nhold for all K∈ Th.\nProof.Using the triangle inequality, we bound\n/ba∇dbliK(xhyh)−xhyh/ba∇dblL∞(K)≤ /ba∇dbliK(xhyh)−iK(xhπ0\nK(yh))/ba∇dblL∞(K)\n+/ba∇dbliK(xhπ0\nK(yh)−xhπ0\nK(yh))/ba∇dblL∞(K)\n+/ba∇dblxhπ0\nK(yh)−xhyh/ba∇dblL∞(K).(20)\nThe first term on the right-hand side of (20) can be estimated as fo llows:\n/ba∇dbliK(xhyh)−iK(xhπ0\nK(yh))/ba∇dblL∞(K)=/ba∇dbliK(xhyh−xhπ0\nK(yh))/ba∇dblL∞(K)\n≤ /ba∇dbliK(xhyh−xhπ0\nK(yh))−(xhyh−xhπ0\nK(yh))/ba∇dblL∞(K)\n+/ba∇dblxhπ0\nK(yh)−xhyh/ba∇dblL∞(K).(21)\nThus, by (12), (15) and (10), we have\n/ba∇dbliK(xhπ0\nK(yh)−xhyh)−(xhπ0\nK(yh)−xhyh)/ba∇dblL∞(K)≤ChK/ba∇dbl∇(xhπ0\nK(yh)−xhyh)/ba∇dblL∞(K)\n≤ChK/ba∇dblxh/ba∇dblL∞(Ω)/ba∇dbl∇yh/ba∇dblL∞(K)(22)\nand\n/ba∇dblxhπ0\nh(yh)−xhyh/ba∇dblL∞(K)≤ChK/ba∇dblxh/ba∇dblL∞(K)/ba∇dbl∇yh/ba∇dblL∞(K).\nTherefore,\n/ba∇dbliK(xhyh)−iK(xhπ0\nK(yh))/ba∇dblL∞(K)≤ChK/ba∇dblxh/ba∇dblL∞(K)/ba∇dbl∇yh/ba∇dblL∞(K).\nNow observe that the second term in the right-hand side of (20) is z ero since iK(xhπ0\nK(yh)) =\nxhπ0\nK(yh). And the third term can be easily estimated as before. In view of th e above computations,\none can conclude that (18) holds.\nSimilarly, we have\n/ba∇dbl∇(iK(xhyh)−xhyhyh)/ba∇dblL2(K)≤ /ba∇dbl∇(iK(xhyh)−iK(xhπ0\nK(yh)))/ba∇dblL2(K)\n+/ba∇dbl∇(iK(xhπ0\nK(yh))−xhπ0\nK(yh))/ba∇dblL2(K)\n+/ba∇dbl∇(xhπ0\nK(yh)−xhyh)/ba∇dblL2(K).(23)\nFrom (9), (13), (10) and (14), we obtain\n/ba∇dbl∇(iK(xhyh)−iK(xhπ0\nK(yh)))/ba∇dblL2(K)≤Ch−1\nK/ba∇dbliK(xhyh)−iK(xhπ0\nK(yh))/ba∇dblL2(K)\n≤C/ba∇dblxh/ba∇dblL∞(K)/ba∇dbl∇yh/ba∇dblL2(K),8 JUAN VICENTE GUTI ´ERREZ-SANTACREU†AND MARCO RESTELLI‡\nwhere we have argued as in estimating (21), but for the L2(K)-norm. The second term on the right-\nhand side of (23) is zero again. To control the last term of (23), we have, by (10) and (14), that\n/ba∇dbl∇(xhπ0\nK(yh)−xhyh)/ba∇dblL2(K)=/ba∇dbl∇(xh(π0\nK(yh)−yh))/ba∇dblL2(K)\n≤ /ba∇dbl∇xh/ba∇dblL∞(K)/ba∇dblπ0\nK(yh)−yh/ba∇dblL2(K)\n+/ba∇dblxh/ba∇dblL∞(K)/ba∇dbl∇(π0\nK(yh)−yh)/ba∇dblL2(K)\n≤C/ba∇dblxh/ba∇dblL∞(K)/ba∇dbl∇yh/ba∇dblL2(K).\n/square\nRemark 3.7. The global version of the above propositions holds due to the assumed quasi-uniformity\nfor the mesh Th.\nLet us define\n(uh,¯uh)h=/integraldisplay\nΩiQh(uh·¯uh) =/summationdisplay\na∈Nhuh(a)·¯uh(a)/integraldisplay\nΩϕa\nfor alluh,¯uh∈Uh, with the induced norm /ba∇dbluh/ba∇dblh=/radicalbig\n(uh,uh)h.\n4.Numerical scheme\nIn this section we will propose our numerical method and will prove a d iscrete energy law and\na discrete inf-sup condition for the Lagrange multiplier similar to (6) a nd (8), respectively, at the\ncontinuous level. The main results are given in Lemma 4.5 and Corollary 4 .8 which is a consequence\nof Lemma 4.7.\nThenumericalapproximationunderconsiderationisbasedonaconf ormingfiniteelementmethodfor\nthevariationalformulationof(5). Thenwewanttofind( uh,qh)∈C∞([0,+∞);Uh)×C∞([0,+∞);Qh)\nsuch that, for all ( ¯uh,¯qh)∈Uh×Qh,\n/braceleftbigg\n(∂tuh,¯uh)h+γ(∇uh,∇¯uh)+γ(qh,iQh(uh·¯uh))+α(uh×∂tuh,¯uh)h= 0,\n(iQh(uh·uh),¯qh) = (1 ,¯qh),(24)\nwith\nuh(0) =u0hin Ω,\nwhereu0h∈Uhis defined as in (H4).\nRemark 4.1. How to obtain an approximate initial condition u0h∈Uhsuch that /ba∇dbl∇u0h/ba∇dbl ≤C/ba∇dbl∇u0/ba∇dbl\nand|u0h(a)|= 1for alla∈ Nhis rarely explicitly mentioned in numerical papers. It seem s that this\ncondition is overlooked. Nevertheless, it is very importan t as it can be checked in the proof of Lemma\n4.2 below. For instance, these conditions can be achieved by appling the nodal interpolation operator\niUhtou0∈C0(¯Ω)and by assuming (H5)in Section 5.\nAvoiding the C0(¯Ω)-regularity to obtain (H4)is an interesting open problem in the numerical frame-\nwork of the Landau–Lifshitz–Gilbert and harmonic map heat fl ow equation.\nNext we consider the local-in-time well-posedness of (24).\nLemma 4.2. There exists Th>0, depending possibly on h, such that there is a unique solution to\nproblem (24) on [0,Th).\nProof.The proof includes two steps: first we show that (24) is equivalent t o a system of ordinary\ndifferential equation, then we show that such a system has a unique solution.\nLet us assume that uh,qhis a solution of (24). Pick ¯a∈ Nhand take ¯uh=uh(¯a,t)ϕ¯a=u¯aϕ¯a\nin (24) 1to get\n(ϕ¯a,ϕ¯a)d\ndt|u¯a|2+γ/summationdisplay\na∈Nhua·u¯a(∇ϕa,∇ϕ¯a)+γ/summationdisplay\na∈Nhqa(ϕa,|u¯a|2ϕ¯a) = 0.\nUsing now (24) 2, we conclude |u¯a|= 1, so that the first term vanishes and we obtain\n/summationdisplay\na∈Nhua·u¯a(∇ϕa,∇ϕ¯a)+/summationdisplay\na∈Nhqa(ϕa,ϕ¯a) = 0. (25)INF-SUP CONDITION FOR THE LLG AND HARMONIC MAP HEAT EQUATION 9\nThe coefficients multiplying qain (25) define a nonsingular matrix (the mass matrix of Qh), so that\nthis equation can be used to compute qauniquely in terms of ua. Next take ¯uh=eiϕ¯ain (24) 1to\nobtain\nµ¯a(ui\n¯a)′+γ/summationdisplay\na∈Nhui\na(∇ϕa,∇ϕ¯a)+γ/summationdisplay\na∈Nhqa(ϕa,ui\n¯aϕ¯a)+αµ¯au¯a×(u¯a)′·ei= 0,\nwhere we have introduced µ¯a=/integraltext\nΩϕ¯a. Observe that\nu¯a×(u¯a)′·ei=U×,¯a(u¯a)′·ei=M/summationdisplay\nj=1(U×,¯a)ij(uj\n¯a)′,\nwhereU×,¯ais the skew-symmetric matrix representing the vector product, i.e .\nU×,¯a=\n0−u3\n¯au2\n¯a\nu3\n¯a0−u1\n¯a\n−u2\n¯au1\n¯a0\n.\nUsing now (25) one arrives at\nµ¯a(I+αU×,¯a)(u¯a)′+γ/summationdisplay\na∈Nh(∇ϕa,∇ϕ¯a)ua−γ/summationdisplay\na∈Nhua·u¯a(∇ϕa,∇ϕ¯a)u¯a=0,(26)\nwithIbeing the identity matrix. One can easily verify that ( I+αU×,¯a) is a nonsingular matrix with\ndeterminant 1+ α2|u¯a|2, so that (26) defines a system of ordinary differential equation wh ich is locally\nLipschitz continuous in the coefficients ui\n¯a, for all¯a∈ Nh.\nConversely, let us assume that uh,qhare defined by the nodal coefficient solving (26) and (25).\nSubstituting (25) in (26) immediately provides (24) 1. To see that in fact also (24) 2is verified, proceed\nas follows. For each ¯a∈ Nh, multiply (26) by u¯aand observe that uT\n¯aU×,¯avanishes, so that\nµ¯a\n2d\ndt/parenleftbig\n1−|u¯a|2/parenrightbig\n+γ(1−|u¯a|2)/summationdisplay\na∈Nhua·u¯a(∇ϕa,∇ϕ¯a) = 0.\nDefine now\ng¯a(t) =2\nγµ¯a/summationdisplay\na∈Nhua(t)·u¯a(t)(∇ϕa,∇ϕ¯a)\nso that\nd\ndt/parenleftbig\n1−|u¯a|2/parenrightbig\n+g¯a(1−|u¯a|2) = 0,\nwith solution\n(1−|u¯a(t)|2) =e/integraldisplayt\n0g¯a(s)ds\n(1−|u¯a(0)|2).\nHence, assuming that |u¯a(0)|= 1, (24) 2holds for any time.\nTo complete the proof we need to show that (26) has a unique solutio n on [0,Th), which follows\nfrom Picard’s theorem. /square\nRemark 4.3. As a result of the a prioriestimates for (uh,qh)in the next section, the approximated\nsolution (uh,qh)will exist globally in time on R+.\nRemark 4.4. Equation (26) gives us a way to compute uhwithout using the Lagrange multiplier qh.\nThis way qhis somehow an approximation of −|∇uh|2.\nIn the following lemma, we prove a pointwise estimate and a priori energy estimates for (24).\nLemma 4.5. Assume that assumptions (H1)–(H4)hold. Then the discrete solution uhof scheme (24)\nsatisfies\n|uh(a)|= 1for alla∈ Nh, (27)10 JUAN VICENTE GUTI ´ERREZ-SANTACREU†AND MARCO RESTELLI‡\nand/integraldisplayt\n0/ba∇dbl∂tuh(s)/ba∇dbl2\nhds+γ\n2/ba∇dbl∇uh(t)/ba∇dbl2=γ\n2/ba∇dbl∇u0h/ba∇dbl2for allt∈R+. (28)\nProof.The nodal equality (27) follows easily from (24) 2sinceiQh(uh·uh)(a) = 1 for all a∈ Nh.\nSelecting ¯uh=∂tuhin (24)1, we obtain\n/ba∇dbl∂tuh/ba∇dbl2\nh+γ\n2d\ndt/ba∇dbl∇uh/ba∇dbl2+γ(qh,iQh(uh·∂tuh)) = 0. (29)\nNow differentiating (24)2with respect to tand then setting ¯ qh=qhyields\n2(iQh(∂tuh·uh),qh) = 0.\nUsing this in (29), we have\n/ba∇dbl∂tuh/ba∇dbl2\nh+γ\n2d\ndt/ba∇dbl∇uh/ba∇dbl2= 0. (30)\nThen we have that (28) holds by integrating (30). /square\nRemark 4.6. The satisfaction of the unit sphere constraint at the nodes a long with the fact that uh\nis a piecewise linear finite element solution implies a unifo rm pointwise estimate for uh, i.e., that\n/ba∇dbluh/ba∇dblL∞(Ω)≤1.\nThe next lemma deals with the discrete inf-sup condition for (24).\nLemma 4.7. Assume that assumptions (H1)–(H4)hold. Let uh∈Uhsuch that |uh(a)|= 1for all\na∈ Nh. Then the following inf-sup condition holds:\nC1\n1+/ba∇dbl∇uh/ba∇dbl≤inf\nqh∈Qhsup\n¯uh∈Uh\\{0}(qh,iQh(uh·¯uh))\n/ba∇dblqh/ba∇dbl(H1(Ω)∩L∞(Ω))′(/ba∇dbl∇¯uh/ba∇dbl+/ba∇dbl¯uh/ba∇dblL∞(Ω)), (31)\nwhereC >0is a constant independent of h.\nProof.Letq∈H1(Ω)∩L∞(Ω). Take ¯uh=iUh(uhπQh(q)), where iUhis the nodal interpolation\noperator into UhandπQhis theL2(Ω) orthogonal projection operator onto Qh, and observe that\n(qh,iQh(uh·¯uh)) = ( qh,/summationdisplay\na∈NhπQh(q)|aua·uaϕa) =/summationdisplay\na∈Nh(qh,πQh(q)|aϕa)\n= (qh,iQh(πQh(q))) = (qh,πQh(q)) = (qh,q).\nThen we obtain\nsup\n¯uh∈Uh\\{0}(qh,iQh(uh·¯uh))\n/ba∇dbl∇¯uh/ba∇dbl+/ba∇dbl¯uh/ba∇dblL∞(Ω)≥ sup\nq∈H1(Ω)∩L∞(Ω)\\{0}(qh,q)\n/ba∇dbl∇iUh(uhπQh(q))/ba∇dbl+/ba∇dbliUh(uhπQh(q))/ba∇dblL∞(Ω).\nMoreover, we have, by (18) and (19), that\n/ba∇dbliUh(uhπQh(q))/ba∇dblL∞(Ω)≤C/ba∇dbluh/ba∇dblL∞(Ω)/ba∇dblq/ba∇dblL∞(Ω) (32)\nand\n/ba∇dbl∇iUh(uhπQh(q))/ba∇dbl ≤C(/ba∇dbl∇uh/ba∇dbl/ba∇dblq/ba∇dblL∞(Ω)+/ba∇dbl∇q/ba∇dbl/ba∇dbluh/ba∇dblL∞(Ω)) (33)\ndue to Remark 3.7. Observe also that we have utilized (16) and (17). Therefore,\n/ba∇dbl∇iUh(uhπQh(q))/ba∇dbl+/ba∇dbliUh(uhπQh(q))/ba∇dblL∞(Ω)≤C(1+/ba∇dbl∇uh/ba∇dbl)(/ba∇dblq/ba∇dblL∞(Ω)+/ba∇dbl∇q/ba∇dbl).(34)\nAs a result, we find\nsup\n¯uh∈Uh\\{0}(qh,iQh(uh·¯uh))\n/ba∇dbl∇¯uh/ba∇dbl+/ba∇dbl¯uh/ba∇dblL∞(Ω)≥C1\n1+/ba∇dbl∇uh/ba∇dblsup\nq∈H1(Ω)∩L∞(Ω)\\{0}(qh,q)\n/ba∇dbl∇q/ba∇dbl+/ba∇dblq/ba∇dblL∞(Ω)\n≥C1\n1+/ba∇dbl∇uh/ba∇dbl/ba∇dblqh/ba∇dbl(H1(Ω)∩L∞(Ω))′.\nThen the proof follows by taking infimum over Qh. /squareINF-SUP CONDITION FOR THE LLG AND HARMONIC MAP HEAT EQUATION 11\nCorollary 4.8. Assume that assumptions (H1)–(H4)hold. The discrete Lagrange multiplier qhof\nscheme (24) satisfies\n/ba∇dblqh/ba∇dblL2(0,T;(H1(Ω)∩L∞(Ω))′)≤C(1+/ba∇dbl∇u0/ba∇dbl)/ba∇dbl∇u0/ba∇dbl. (35)\nProof.From (24) 1, we have\nγ(qh,iQh(uh·¯uh))≤ /ba∇dbl∂tuh/ba∇dblh/ba∇dbl¯uh/ba∇dblh+γ/ba∇dbl∇uh/ba∇dbl/ba∇dbl∇¯uh/ba∇dbl+α/ba∇dbl∂tuh/ba∇dblh/ba∇dbluh/ba∇dblh/ba∇dbl¯uh/ba∇dblL∞(Ω)\n≤/parenleftBig\n(1+α)/radicalbig\nmeas(Ω)/ba∇dbl∂tuh/ba∇dblh+γ/ba∇dbl∇uh/ba∇dbl/parenrightBig/parenleftbig\n/ba∇dbl¯uh/ba∇dblL∞(Ω)+/ba∇dbl∇¯uh/ba∇dbl/parenrightbig\n.\nTherefore,\n(qh,iQh(uh·¯uh))\n/ba∇dbl¯uh/ba∇dblL∞(Ω)+/ba∇dbl∇¯uh/ba∇dbl≤(1+α)\nγ/radicalbig\nmeas(Ω)/ba∇dbl∂tuh/ba∇dblh+/ba∇dbl∇uh/ba∇dbl.\nApplying (31) above, we find\n/ba∇dblqh/ba∇dbl(H1(Ω)∩L∞(Ω))′≤C(1+/ba∇dbl∇uh/ba∇dbl)/parenleftbigg(1+α)\nγ/radicalbig\nmeas(Ω)/ba∇dbl∂tuh/ba∇dblh+/ba∇dbl∇uh/ba∇dbl/parenrightbigg\n.\nThe proof follows by using (28). /square\nRemark 4.9. Observe that if we apply directly to (24)1the argument leading to (31) so as to obtain\nan estimate for qhwe will improve estimate (35) in time. Indeed, let q∈H1(Ω)∩L∞(Ω)and select\n¯uh=iUh(uhπQh(q))in(24)1. Then we find\nγ(qh,q) = (∂tuh,uh·iUh(uhπQh(q)))+γ(∇uh,∇iUh(uhπQh(q)))+α(uh×∂tuh,iUh(uhπQh(q))).\nNoting that both (∂tuh,uh·iUh(uhπQh(q))) = 0and(uh×∂tuh,iUh(uhπQh(q))) = 0, we obtain, by\n(34), that\n(qh,q)≤C/ba∇dbl∇uh/ba∇dbl(1+/ba∇dbl∇uh/ba∇dbl)(/ba∇dblq/ba∇dblL∞(Ω)+/ba∇dbl∇qh/ba∇dbl).\nTherefore,\n/ba∇dblqh/ba∇dblL∞(0,+∞;(H1(Ω)∩L∞(Ω))′)≤C/ba∇dbl∇u0/ba∇dbl(1+/ba∇dbl∇u0/ba∇dbl). (36)\nRemark 4.10. Replacing (qh,iQh(uh·¯uh))with(qh,uh·¯uh)hin (24), we obtain the following scheme.\nFind(uh,qh)∈C∞([0,+∞);Uh)×C∞([0,+∞);Qh)such that, for all (uh,qh)∈Uh×Qh,\n/braceleftbigg\n(∂tuh,¯uh)h+γ(∇uh,∇¯uh)+γ(qh,uh·¯uh)h+α(uh×∂tuh,¯uh)h= 0,\n(uh·uh,¯qh)h= (1,¯qh)h.(37)\nThen, Lemma 4.2 holds, and the nodal enforcement (27) and the energy law (28) are valid for\nscheme (37). Moreover, the inf-sup condition (31) can be pro ved by selecting ¯uh=iUh(uhPh(q))\nwherePhis defined by\n(Ph(uh),¯uh)h= (uh,¯uh)for all ¯uh∈Uh.\n5.Temporal discretization\nIn this section we shall propose two time integrators for (24) which preserve the energy law (28)\nand estimate (36). More precisely, we will construct a linearly implicit E uler and a nonlinearly implicit\nCrank–Nicolson time-stepping algorithm. For the linear one, we will re quire an extra assumption on\nthe mesh Th.\n(H5) Assume Thto satisfy that if uh∈Uhwith|uh(a)| ≥1 for alla∈ Nh, then\n/ba∇dbl∇iUh(uh\n|uh|)/ba∇dbl ≤ /ba∇dbl∇uh/ba∇dbl.\nAssumption (H5) is assured under the condition [7]/integraldisplay\nΩ∇ϕa·∇ϕ˜a≤0 for all a,˜a∈ Nhwitha/ne}ationslash=˜a,\nwhere we remember that {ϕa:a∈ Nh}is the nodal basis of Xh. In particular, such a condition holds\nfor meshes of the Delaunay type in two dimensions and with all dihedra l angles of the tetrahedra being\nat mostπ/2 in three dimensions.12 JUAN VICENTE GUTI ´ERREZ-SANTACREU†AND MARCO RESTELLI‡\nIt is assumed here for simplicity that we have a uniform partition of [0 ,T] intoNpieces. So, the\ntime step size is k=T/Nand the time values ( tn=nk)N\nn=0. To simplify the notation let us denote\nδtun+1=un+1−un\nk.\nFirst we present a first-order linear numerical scheme.\nAlgorithm 1 : Euler time-stepping scheme\nStep(n+1): Given un\nh∈Uh, find (un+1\nh,qn+1\nh)∈Uh×Qhsolving the algebraic\nlinear system\n\n(δtun+1\nh,¯uh)h+γ(∇un+1\nh,∇¯uh)\n+γ(qn+1\nh,iQh(un\nh\n|un\nh|·¯uh))+α(un\nh×δtun+1\nh,¯uh)h= 0,\n(iQh(un\nh·δtun+1\nh),¯qh) = 0,(38)\nfor all (¯uh,¯qh,)∈Uh×Qh.\nTheorem 5.1. Assume that assumptions (H1)–(H5)hold. Let {um\nh}N\nm=1be the numerical solution\nof (38). Then\nm/summationdisplay\nn=0k/parenleftbigg\n/ba∇dblδtun+1\nh/ba∇dbl2\nh+γk\n2/ba∇dbl∇δtun+1\nh/ba∇dbl2/parenrightbigg\n+γ\n2/ba∇dbl∇um+1\nh/ba∇dbl2=γ\n2/ba∇dbl∇u0\nh/ba∇dbl2. (39)\nMoreover, the Lagrange multiplier {qm\nh}N\nm=1satisfies\nmax\nn=1,···,N/ba∇dblqn\nh/ba∇dbl(H1(Ω)∩L∞(Ω))′≤C(1+/ba∇dbl∇u0\nh/ba∇dbl)/ba∇dbl∇u0\nh/ba∇dbl, (40)\nwhereC >0is a constante independent of handk.\nProof.Let¯uh= 2kδtun+1\nhin (38) 1to get\n2k/ba∇dblδtun+1\nh/ba∇dbl2\nh+γ/ba∇dbl∇un+1\nh/ba∇dbl2−γ/ba∇dbl∇un\nh/ba∇dbl2+γk2/ba∇dbl∇δtun+1\nh/ba∇dbl2+2γk(qn+1\nh,iQh(un\n|un\nh|·δtun+1\nh)) = 0,\nwhere the damping term has disappeared. From (38) 2, we infer that un\na·δtun+1\na= 0 for all a∈ Nh.\nTherefore the last term in the above equation vanishes. Thus, it fo llows that (39) holds by summing\novern.\nTo prove the inf-sup condition, we select ¯uh=iUh(un\nh\n|un\nh|πQh(q)) in (38) 1, withq∈H1(Ω)∩L∞(Ω),\nto obtain\n(qn+1\nh,q) = (∇un+1\nh,∇(iUh(un\nh\n|un\nh|πQh(q)))) = (∇un+1\nh,∇(iUh(iUh(un\nh\n|un\nh|)πQh(q)))).\nUsing estimate (33), we have\n(qn+1\nh,q)≤C/ba∇dbl∇un+1\nh/ba∇dbl/parenleftbigg\n/ba∇dbl∇iUh(un\nh\n|un\nh|)/ba∇dbl/ba∇dblq/ba∇dblL∞(Ω)+/ba∇dbl∇q/ba∇dbl/ba∇dbliUh(un\nh\n|un\nh|)/ba∇dblL∞(Ω)/parenrightbigg\n.\nIn view of (38) 2, we deduce that\n0 =un\na·δtun+1\na=|un+1\na|2−|un\na|2−|un+1\na−un\na|;\nhence|un\na| ≥ |un−1\na| ≥1 holds since |u0\na|= 1 for all a∈ Nh. This fact combined with assumption\n(H5) yields\n(qn+1\nh,q)≤C/ba∇dbl∇un+1\nh/ba∇dbl(/ba∇dbl∇un\nh/ba∇dbl/ba∇dblq/ba∇dblL∞(Ω)+/ba∇dbl∇q/ba∇dbl)\n≤C/ba∇dbl∇un+1\nh/ba∇dbl(1+/ba∇dbl∇un\nh/ba∇dbl)(/ba∇dblq/ba∇dblL∞(Ω)+/ba∇dbl∇q/ba∇dbl).\nEstimate (40) then follows by utilizing duality and (39). /square\nEquation (39) is the fully discrete counterpart of (4) and (28).\nNext we deal with a second-order approximation based on a Crank– Nicolson method.INF-SUP CONDITION FOR THE LLG AND HARMONIC MAP HEAT EQUATION 13\nAlgorithm 2 : Crank–Nicolson time-stepping scheme\nStep(n+1):Givenun\nh∈Uh, find (un+1\nh,qn+1\nh)∈Uh×Qhsolving the algebraic\nnonlinear system\n\n(δtun+1\nh,¯uh)h+γ(∇un+1\n2\nh,∇¯uh)\n+γ(qn+1\n2\nh,iQh(un+1\n2\nh\n|un+1\n2\nh|·¯uh))+α(un\nh×δtun+1\nh,¯uh) = 0,\n(iQh(un+1\nh·un+1\nh),¯qh) = (1 ,¯qh),(41)\nfor all (¯uh,¯qh,)∈Uh×Qh.\nTheorem 5.2. Assume that assumptions (H1)–(H4)are satisfied. Let {um\nh}N\nm=1be the numerical\nsolution of (41). Then\nm/summationdisplay\nn=0k/ba∇dblδtun+1\nh/ba∇dbl2\nh+γ\n2/ba∇dbl∇um+1\nh/ba∇dbl2=γ\n2/ba∇dbl∇u0\nh/ba∇dbl2. (42)\nMoreover, the Lagrange multiplier {qm\nh}N\nm=1satisfies\nmax\nn=1,···N/ba∇dblqn\nh/ba∇dbl(H1(Ω)∩L∞(Ω))′≤C(1+/ba∇dbl∇u0\nh/ba∇dbl)/ba∇dbl∇u0\nh/ba∇dbl, (43)\nwhereC >0is a constante independent of handk.\nProof.As in the proof of Theorem (5.1), we substitute ¯uh=δtun+1\nhinto (41) 1and ¯qh=qn+1\nhinto\n(41)2to obtain (42), and then ¯uh=iUh(un+1\n2\nhπQh(¯q)) into (41) 1to get (43). /square\nRemark 5.3. In the next section we will use scheme (38) as a non-linear sol ver for approximating\neach step of scheme (41) when rewritten in the appropriate fa shion.\n6.Implementation details\nThe second order time integrator (41) requires solving at each time step a nonlinear system. In this\nsection, we discuss a possible solution strategy for such a problem, considering for simplicity the case\nα= 0. The first step is rewriting (41) in terms of wh=un+1\n2\nhandsh=qn+1\n2\nhas\n\n\n(wh−un\nh,¯uh)h+γk\n2(∇wh,∇¯uh)+γk\n2(sh,iQh/parenleftbiggwh\n|wh|·¯uh/parenrightbigg\n) = 0\n(iQh/parenleftbigg\nwh·(wh−un\nh)−1−un\nh·un\nh\n4/parenrightbigg\n,¯qh) = 0.(44)\nIn (44) 2, the term involving 1 −un\nh·un\nhshould vanish, thanks to the unit sphere constraint. However,\nsince the nonlinear problem in general can not be solved exactly, we h ave two options: normalize un\nh\nafter each time step, or accept a (small) violation of the unit sphere constraint and include such a\nterm. Notice that (44) is an implicit Euler step for the solution at half t ime levels.\nObserve now that (44) 2amounts to requiring that the argument of iQhvanishes at each node of the\ntriangulation; assuming wh/ne}ationslash=0we can reformulate this constraint as\nγk\n2(iQh/parenleftbiggwh\n|wh|·(wh−un\nh)−1−un\nh·un\nh\n4|wh|/parenrightbigg\n,¯qh) = 0. (45)\nEquations (44) 1and (45) are now taken as the basis for a fixed point iteration: given w(i)\nh, let\np=γk\n2w(i)\nh\n|w(i)\nh|,Γn=γk\n21−|un\nh|2\n4|w(i)\nh|14 JUAN VICENTE GUTI ´ERREZ-SANTACREU†AND MARCO RESTELLI‡\nand compute the next iteration solving\n\n\n(w(i+1)\nh,¯uh)+γk\n2(∇w(i+1)\nh,∇¯uh)+(s(i+1)\nh,iQh(p·¯uh)) = ( un\nh,¯uh),\n(iQh/parenleftBig\np·w(i+1)\nh/parenrightBig\n,¯qh) = (iQh(p·un\nh+Γn),¯qh).(46)\nNotice the analogy between (46) and the linearly implicit method (38). The matrix of the linear\nsystem (46) has a classical/bracketleftbiggA BT\nB/bracketrightbigg\nstructure, where Ais symmetric and positive definite and is block diagonal with each block c orre-\nsponding to one spatial dimension. Hence, it can be solved either usin g a direct method or an iterative\none, such as the Uzawa algorithm [19], which would then naturally lead t o a Newton–Krylov approach\nfor the original nonlinear problem (44).\n7.Numerical results\nWe consider here some numerical experiments aiming at verifying num erically the convergence of\nthe proposed scheme as well as analyzing its behaviour in presence o f singular solutions, including the\ncase of singular solutions in two space dimensions, which is outside the scope of the theory presented\nin this paper.\n7.1.Convergence test for smoothsolutions. Intwospatialdimensions,wecanset u= [cosθ,sinθ]T\nand observe that, for α= 0, (1) implies ∂tθ−γ∆θ= 0 in Ω ×R+with∂nθ= 0 on∂Ω×R+. This\nlets us construct the following exact solution for Ω = ( −1,1)2:\nu=/bracketleftbiggcosθ\nsinθ/bracketrightbigg\n, θ= Θe−γ(k2\nx+k2\ny)tcos(kxx)cos(kyy), q=−(∂xθ)2−(∂yθ)2,\nwith Θ = π,γ= 0.01,kx=π,ky= 2π. To verify the convergence of the proposed discretization,\nwe compare the numerical results with the exact solution at t= 1, using a collection of structured\ntriangular grids with h= 2−i,i= 1,...,8 and the Crank–Nicolson scheme (41) with time-step\nk= 0.1·2−j, forj= 0,...,6. In all the computations, the nonlinear iterations are carried out\nuntil reaching convergence within machine precision, which in practic e amounts to performing O(10)\nnonlinear iterations.\nSince our results indicate that the error resulting from the time disc retization is smaller than the\none resulting from the space discretization for all the considered g rid sizes and time-steps, we can\nanalyze the two effects separately, focusing first on the space dis cretization error. In order to do this,\nwe fixk= 1/640, corresponding to j= 6, and collect the error norms for uandqin Tables 1 and 2,\nrespectively.\nConcerning the /ba∇dbl·/ba∇dbl(H1)′norm appearing in Table 2 as well as in Figure 1, it is computed as follows.\nFirst of all, thanks to the Riestz theorem, given g∈H−1there isrg∈H1\n0such that, for any f∈H1\n0,\n/an}b∇acketle{tg,f/an}b∇acket∇i}ht(H1)′×H1\n0= (rg,f)H1\n0; moreover, /ba∇dblg/ba∇dbl(H1)′=/ba∇dblrg/ba∇dblH1\n0. The difficulty is that, taking g=q−qh,\nrq−qh/∈Qh, so that we can not compute it. This problem can be circumvented co mputing the H1\n0\nprojection of rq−qhonQh, denoted here as Π rq−qh, which is uniquely determined by\n(Πrq−qh,fh)H1\n0= (rq−qh,fh)H1\n0=/an}b∇acketle{tq−qh,fh/an}b∇acket∇i}ht(H1)′×H1\n0\nfor anyfh∈Qh. As shown in [13, Th 5.8.3], /ba∇dblΠrq−qh/ba∇dblH1\n0provides a second order estimate in hof the\ndesired norm.\nForu, second order convergenceis observed in the L1,L2andL∞norms, while first order converge\nis observed in the H1norm. For the Lagrange multiplier q, second order convergence is observed\nin the (H1)′norm, while the L1,L2andL∞norms are bounded and the H1norm diverges. This\nbehaviour of the error for qcan be explained noting that the numerical approximation exhibits gr id\nscale oscillations maintaining a constant amplitude while the grid is refine d. It is important to stress,INF-SUP CONDITION FOR THE LLG AND HARMONIC MAP HEAT EQUATION 15\nTable 1. Computed error norms for u−uhfor a collection of structured triangular\ngrids with h= 2−i,i= 1,...,8. The numerical convergence rates are also reported.\ni/ba∇dblu−uh/ba∇dblL1/ba∇dblu−uh/ba∇dblL2/ba∇dblu−uh/ba∇dblL∞/ba∇dblu−uh/ba∇dblH1\n14.8 –2.0 –1.6 –1.4·101–\n22.2 1.11.0 1.08.6·10−10.91.2·1010.3\n37.0·10−11.73.5·10−11.53.7·10−11.26.5 0.8\n41.6·10−12.27.8·10−22.19.0·10−22.03.0 1.1\n53.8·10−22.11.9·10−22.12.0·10−22.11.4 1.1\n69.3·10−32.04.7·10−32.05.2·10−32.07.0·10−11.0\n72.3·10−32.01.2·10−32.01.3·10−32.03.5·10−11.0\n85.8·10−42.02.9·10−42.03.2·10−42.01.8·10−11.0\nTable 2. Computed error norms for q−qhfor a collection of structured triangular\ngrids with h= 2−i,i= 1,...,8. The negative Sobolev norm error /ba∇dblq−qh/ba∇dbl(H1)′, for\nwhich the numerical convergence rate is also reported, is estimate d with/ba∇dblΠrq−qh/ba∇dblH1\n0as discussed in the text. Notice that, since qhis naturally computed at half time steps,\nthe analytic solution is evaluated ad t= 1−k/2.\ni/ba∇dblq−qh/ba∇dbl(H1)′/ba∇dblq−qh/ba∇dblL1/ba∇dblq−qh/ba∇dblL2/ba∇dblq−qh/ba∇dblL∞/ba∇dblq−qh/ba∇dblH1\n159.0784–131.3211 93.1235 109.1703 852.73\n29.16032.7110.0432 72.2844 76.7221 1022.32\n316.4411-0.855.8291 35.7175 62.8248 1242.74\n44.82471.831.6080 23.3787 71.2639 2123.59\n51.21662.027.2561 20.9493 62.0149 3990.89\n60.30792.026.5922 20.5044 59.6844 7862.59\n70.08031.926.5218 20.4059 59.1055 15666.76\n80.02311.826.5223 20.3821 58.9610 31304.46\nhowever, that such oscillations are consistent with the stability est imates (35) and are not, thus, an\nindication of numerical instability.\nIn order to isolate the error resulting from the time discretization, we proceed by fixing the grid\nsizehand computing a reference solution for a small time-step, which the n allows computing the self\nconvergence rate. Taking as reference time-step kref= 0.1·2−8we observe, for all the considered grid\nsizes, second order convergence for both uhandqh, for all the considered norms (indeed, for fixed hall\nthese normsareequivalent); Figure1showsthe results for /ba∇dblu−uh/ba∇dblH1and/ba∇dblq−qh/ba∇dbl(H1)′. Acomparison\nof this figure with the values reported in Tables 1 and 2 confirms that the time discretization error is\nsmaller than the space discretization one. Second order converge nce is also apparent estimating the\nconvergence rate as log2ρ, where (see [36, Eq (4.7)])\nρ=/ba∇dbluk\nh−uk/2\nh/ba∇dblH1\n/ba∇dbluk/2\nh−uk/4\nh/ba∇dblH1,\nas shown in Table 3.\n7.2.Behaviour for singular solutions. After considering the behaviour of the scheme for problems\nwith smooth solutions, we turn our attention to problems including sin gularities. In fact, singularities\nof the form u∼x−x0\n|x−x0|are very important in the study of liquid crystals, and various relate d test\ncases have been considered in the literature [34, 18, 35, 11, 6, 26, 15].\nAn important distinction here must be done between two- and three -dimensional problems, since\nsuch singularities have a finite energy in the three-dimensional case but not in the two-dimensional one\n(mathematically, they belong to H1(Ω) for Ω ⊂R3but not for Ω ⊂R2). From a practical perspective,16 JUAN VICENTE GUTI ´ERREZ-SANTACREU†AND MARCO RESTELLI‡\n10-310-210-110-610-510-410-310-210-1\np = 2\n10-310-210-110-610-510-410-310-210-1\np = 2\nFigure 1. Computed error norms /ba∇dbluref\nh−uh/ba∇dblH1(left) and /ba∇dblqref\nh−qh/ba∇dbl(H1)′(right),\nfor varying time-step k, withkref= 0.1·2−8, and fixed mesh size. Results for mesh\nsizesh= 2−3(/Circle),h= 2−4(♦),h= 2−5(/square) andh= 2−6(⋆). Notice that, since qh\nis naturally computed at half time steps, the corresponding errors are computed at\nt= 1−k/2 for each time-step k.\nTable 3. Estimated time discretization errors /ba∇dbluk\nh−uk/2\nh/ba∇dblH1fork= 0.1·2−j,j=\n0,...,5, for four triangular grids with h= 2−i,i= 3,4,5,6 (see also Figure 1). The\nresulting convergence rates are also reported.\nj h= 2−3h= 2−4h= 2−5h= 2−6\n03.9·10−3–6.6·10−3–9.9·10−3–9.2·10−2–\n19.7·10−41.9989 1.6·10−32.0018 2.0·10−32.3348 4.1·10−34.5030\n22.4·10−41.9998 4.1·10−42.0004 4.9·10−42.0004 5.1·10−43.0037\n36.1·10−51.9999 1.0·10−42.0001 1.2·10−42.0001 1.3·10−42.0001\n41.5·10−52.0000 2.6·10−52.0000 3.1·10−52.0000 3.2·10−52.0000\n53.8·10−62.0000 6.4·10−62.0000 7.7·10−62.0000 7.9·10−62.0000\nasingularsolution canbe approximatedin the chosenfinite element sp aceboth in twoandthree spatial\ndimensions, for instance by nodal interpolation (provided that non e of the nodes coincides with the\nsingular point x0); the resulting function belongs to H1and can serve as an initial condition for a\ntime dependent computation. Hence, one might be tempted to dismis s the distinction between the\ntwo cases as a merely theoretical argument with no practical implica tions. However, this would be\nincorrect, as the following results demonstrate. Indeed, for thr ee-dimensional problems, the theoretical\nanalysisprovidedaboveholds, and ourmethod isguaranteedto sat isfyourstability estimates when the\ngrid is refined. For the two-dimensional case, on the contrary, th e theoretical analysis does not apply\nand nothing can be said a priori; nevertheless, computational experiments indicate that, althou gh\nitispossible to compute a numerical solution, such a solution critically dep ends on the numerical\ndiscretization, does not converge to a well defined limit when the grid is refined and is thus essentially\nmeaningless.\nBefore discussing the numerical results, it is useful to provide a qu alitative analysis of the problem\nof representing a singular solution with a finite element function. In t wo spatial dimensions, for a\nstructured, triangular grid, the finite element function will resemb le the patterns shown in Figure 2.\nThis implies that there are at least two elements where the numerical solution has O(1) variations\nwithin an O(h) distance, which, in turn, implies that the energy of the discrete so lution/ba∇dbl∇uh/ba∇dbl2\nL2\nundergoes O(1) variations for an O(h) displacement of the singularity. For instance, referring toINF-SUP CONDITION FOR THE LLG AND HARMONIC MAP HEAT EQUATION 17\nFigure 2. Interpolation of a singular solution of the formx−x0\n|x−x0|on a structured,\ntriangular grid for different location of x0with respect to the computational grid.\nFigure 2, and given that we consider linear finite elements, it easy to c heck that the two elements\ncontaining the singularity, K1andK2, contribute to the total energy with /ba∇dbl∇uh/ba∇dbl2\nL2(K1∪K2)= 4 and\n/ba∇dbl∇uh/ba∇dbl2\nL2(K1∪K2)= (22−2√\n5)/5 for the two depicted configurations, independently of h. Since the\nsolution is smooth far from the singularity, such O(1) energy variations for O(h) displacements of\nthe singularity are also present if we consider the total energy /ba∇dbl∇uh/ba∇dbl; this is shown in Figure 3\n(left) where we plot, for various mesh sizes, the total energy of t he nodal interpolant ofx−x0\n|x−x0|on\n−2.0 −1.5 −1.0 −0.5 0.0101520253035\n−2.0 −1.5 −1.0 −0.5 0.0121416182022242628\nFigure 3. Energy/ba∇dbl∇uh/ba∇dbl2, for Ω = ( −2,2)×(−1,1) anduhdefined as the nodal\ninterpolant ofx−x0\n|x−x0|on a structured, triangular mesh, as a function of x0. Left:\nvalues for three isotropic grids with 34 ×17 elements (light gray), 66 ×33 elements\n(gray) and 130 ×65 elements (black) as functions of x0such that x0= (x0,0)T,\nx0∈[−2,0]. Right: values for a single anisotropic grid with 48 ×17 elements and\nx0= (x0,0)T(gray) and x0= (1/24,y0)T(black), with x0∈[−2,0] andy0∈[−1,0].\nThe large-scale variations are due to the fact that, as x0approaches the boundary of\nthe domain, a “large part” of the field lies outside Ω.\nΩ = (−2,2)×(−1,1) as a function of the position of x0, specified as x0= (x0,0)Tforx0∈[−2,0].\nSince the energy can not increase during the time evolution because of (4), the result of this grid\ndependence of the energy itself can be seen as a “potential barrie r” which tends to trap the singularity\nbetween the grid vertexes, in Figure 2 (left). Two important chara cteristics of such a barrier can be18 JUAN VICENTE GUTI ´ERREZ-SANTACREU†AND MARCO RESTELLI‡\nnoted. First of all, it is independent of h, so that refining the grid has no effect on it; this can be\nseen both by noting that, for elements such as K1andK2in Figure 2, /ba∇dbl∇uh/ba∇dbl2∼h−2and the area\nelement is proportional to h2, as well as by considering Figure 3 (left) where the amplitude of the\nsmall-scale oscillations is constant for different resolutions. The sec ond characteristic of the potential\nbarrier is its dependency on the grid anisotropy. This is illustrated in F igure 4, which shows how, for\na uniform, triangular grid with different spacings in the two Cartesian directions, the variation of the\nfinite element solution is more pronounced when the singularity moves fromx0tox′\n0compared to\na displacement from x0tox′′\n0. Again, this qualitative description is confirmed considering a specific\nFigure 4. Interpolation of a singular solution of the formx−x0\n|x−x0|on a structured,\ntriangular grid for different location of x0with respect to the computational grid.\nContrary to the case shown in Figure 2, we consider here an anisotr opic grid, with\ndifferent spacings in the two Cartesian directions.\nexample of a grid composed of 48 ×17 elements for Ω = ( −2,2)×(−1,1) and evaluating the energy of\nthe finite element solution when the singularity is displaced along the tw o Cartesian axes, as shown in\nFigure 3(right) where it is apparent that the amplitude of the grid-in ducedoscillations is much larger\nwhen the singularity is displaced along the direction with the largest gr id spacing.\nFor three spatial dimensions, this qualitative analysis still holds, up t o one important difference:\nin such a case, O(1) variations over an O(h) distance result in O(h) energy contributions, because\nnow it is still /ba∇dbl∇uh/ba∇dbl2∼h−2but the area element is proportional to h3. Hence, for three-dimensional\ncomputations, the discrete potential barriersat the element bou ndaries vanish when the grid is refined.\nSummarizing now the conclusions of the qualitative analysis, we can ex pect that, initializing the\nfinite element computation interpolating a singular field u, the numerical solution will be strongly\ninfluenced by the computational grid. Moreover, while three-dimen sional computation will converge\nto the analytic solution, two-dimensional ones will not show any cons istent limit when the grid is\nrefined. Indeed, this is precisely the outcome of our numerical exp eriments, which we now describe in\nthe remaining of the present section.\nThe initial condition for the numericalexperiments is a modified versio nofthe one consideredin [35]\nand is defined as\nu0(x) =˜u0(x)\n|˜u0(x)|,˜u0(x) =w(x)(x+δ)+(1−w(x))(−(x−δ))\nwith\nw(x) =1\n1+exp(5 x)\nandx= (x,y)T,δ= (δ,0)Tand and x= (x,y,z)T,δ= (δ,0,0)Tin two and three space dimensions,\nrespectively. This corresponds to two singularities located on the xaxis approximately at x=±δ\nhaving opposite sign and thus repelling each other. We take γ= 1 and α= 0, while the compu-\ntational domain is Ω = ( −2,2)×(−1,1) in two dimensions and Ω = ( −2,2)×(−1,1)×(−1,1) in\nthree dimensions. The computational grid is uniform and structure d and is obtained, in two spatial\ndimensions, partitioning Ω into rectangles with dimensions ∆ x,∆yand dividing each rectangle intoINF-SUP CONDITION FOR THE LLG AND HARMONIC MAP HEAT EQUATION 19\ntwo triangles with alternating direction, obtaining a grid analogous to those depicted in Figures 2\nand 4. For the three dimensional case, the construction is similar wit ch each prism being divided into\nsix tetrahedral elements. Grids will be defined also by means of the n umber of subdivisions in each\nCartesian direction, so that a grid with Nx×Nyelements correspond to ∆ x= 4/Nx,∆y= 2/Ny. The\noverall evolution of the numerical solution is determined by the inter play between the large-scale and\nthe grid-scale energy variations associated with a displacement of t he two singularities: the former\ncorresponds to an energy decrease when the two singularities drif t apart, the latter has been analyzed\npreviously in this section and tends to lock the singularities between t he grid vertexes. The initial\nseparation is chosen so that, for all the considered computations , a transient is observed at least in\nthe initial phase, with the large-scale effect overcoming the grid one . In practice, we take δ= 0.0625\nin two dimensions and δ= 0.5 in three dimensions and choose in both cases adequate grid anisotr opy\nlevels.\nThe time evolution of the energy of the finite element solution for the two dimensional case in\nshown in Figure 5 for four levels of grid anisotropy and two levels of gr id refinement. The energy\n0 5 10 15 20020406080100\n0 5 10 15 20020406080100\nFigure 5. Time evolution of /ba∇dbl∇uh/ba∇dbl2for two repelling singularities in two space\ndimensions for different levels of grid anisotropy, defined as ∆ y: ∆x, namely 1 : 1\n(/Circle), 6 : 5 (♦), 7 : 5 (/square) and 8 : 5 ( ⋆). Left: grids with 34 ×17, 41×17, 48×17 and\n54×17 elements. Right: grids with 66 ×33, 79×33, 92×33 and 106 ×33 elements.\ndecreases as the two singularities drift apart and drops to zero if t hey reach the boundary and leave\nthe computational domain. The first observation is that, dependin g on the anisotropy of the grid,\nthe two singularities can reach the boundary (when the anisotropy is such that the potential barrier\nassociated with the grid for displeacemet along xis small) or reach a steady state condition inside the\ngrid after an initial transient. The second observation is that the d rift velocity is strongly affected by\nthe grid anisotropy. A third observation is that the energy time evo lution has a step pattern where\neach step corresponds to the displacement of the singularities ove r one grid element, i.e. to the crossing\nof one potential barrier. A fourth observation is that, when the g rid is refined, the effect of the grid\nis not reduced: in fact, for finer grids, the spread among the comp utations with similar resolution but\ndifferent stretching increases and the numerical steady state is r eached earlier. Finally, we mention\nthat that analogous computations using unstructured grids, not reported here, show that even the\ndirection in which the singularities drift is strongly affected by the com putational grid.\nRepeatingnowtheexperimentinthreespatialdimensions, weobtain theresultsreportedinFigure6.\nThe step pattern observed for two-dimensional computations is s till present, however we notice that:\na) regardless of the grid anisotropy, the singularities leave the com putational domain; b) refining the\ngrid, the amplitute of the steps decreases and a tendency of the s olutions obtained for different levels\nof grid anisotropy to converge to a unique limit can be observed.20 JUAN VICENTE GUTI ´ERREZ-SANTACREU†AND MARCO RESTELLI‡\n0 2 4 6 8 10020406080\n0 2 4 6 8 10020406080\nFigure 6. Time evolution of /ba∇dbl∇uh/ba∇dbl2for two repelling singularities in three space\ndimensions for different levels of grid anisotropy, which can be define d as ∆ y: ∆x\nthanks to the fact that ∆ y= ∆z, namely: 10 : 17 ( /Circle), 10 : 15 ( ♦), 10 : 12 ( /square) and\n1 : 1 (⋆). Left: grids with 20 ×17×17, 23×17×17, 28×17×17 and 34 ×17×17\nelements. Right: grids with 38 ×33×33, 44×33×33, 53×33×33 and 66 ×33×33\nelements.\n8.Conclusion\nIn this paper we have proposed and analyzed a unified saddle-point s table finite element method for\napproximating the harmonic map heat and Landau–Lifshitz–Gilbert e quation. We have mainly proved\nthat the numerical solution satisfies an energy law and a nodal satis faction of the unit sphere, and\nthe associated Lagrange multiplier satisfies an inf-sup condition. Th e key ingredients are using piece-\nwise linear finite element spaces, applying a nodal interpolation to the terms involving the nonlinear\nrestriction and a mass lumping technique to the terms involving time de rivatives.\nThis work has important implications in the context of the Ericksen–L eslie equations which in-\ncorporate a convective term to the harmonic map heat equation. W hile other existing approaches\nin the literature are not readily adapted to these equations due to t he convective term, our proposed\nmethod maybe directly applied to them without anymodification keepin gthe desired properties above\nmentioned.\nConcerning the numerical results we have shown that the finite elem ent solution computed by a\nnonlinear Crank–Nicolson method, which is solved by using semi-implicit E uler iterations, enjoys the\nexpected accurate approximations. Moreover, we have identified that the dynamics of singularity\npoints depends on dimension. That is, we have seen that, depending on the mesh anisotropy, in two\ndimensions, two singularity points can either be trapped among two e lements of the mesh or move\naccording to their sign. Instead, in three dimensions, the trapping effect does not occur. Therefore,\nsome care must be taken in simulating singularities in two dimensions sinc e these do not have finite\nenergy and the limit equation only holds in the sense of measures.\nReferences\n[1]F. Alouges ,A new finite element scheme for Landau–Lifchitz equations , Discrete Contin. Dyn. Syst. Ser. S 1\n(2008), no. 2, 187–196.\n[2]F. Alouges, P. Jaisson ,Convergence of a finite element discretization for the Landa u–Lifshitz equations in mi-\ncromagnetism, Math. Models Methods Appl. Sci. 16 (2006), no. 2, 299–316.\n[3]F. Alouges, E. Kritsikis, J.-C. Toussaint ,A convergent finite element approximation for the Landau–Li fschitz–\nGilbert equation , Physica B 407 (2012). 1345–1349.\n[4]F. Alouges, E. Kritsikis, J. Steiner, J.-C. Toussaint ,A convergent and precise finite element scheme for\nLandau–Lifschitz-Gilbert equation , Numer. Math. 128 (2014), no. 3, 407–430.INF-SUP CONDITION FOR THE LLG AND HARMONIC MAP HEAT EQUATION 21\n[5]S. Badia, F. Guill ´en-Gonz ´alez, J.V. Guti ´errez-Santacreu ,An overview on numerical analyses of nematic\nliquid crystal flows, Arch. Comput. Methods Eng. 18 (2011), no. 3, 285–313.\n[6]S. Badia, F. Guill ´en-Gonz ´alez, J. V. Guti ´errez-Santacreu ,Finite element approximation of nematic liquid\ncrystal flows using a saddle-point structure , J. Comput. Phys. 230 (2011), no. 4, 1686–1706.\n[7]S. Bartels ,Stability and convergence of finite-element approximation schemes for harmonic maps , SIAMJ.Numer.\nAnal. 43 (2005), no. 1, 220-238.\n[8]S. Bartels, A. Prohl ,Convergence of an implicit finite element method for the Land au–Lifshitz–Gilbert equation ,\nSIAM J. Numer. Anal. 44, pp. 1405–1419 (2006).\n[9]S. Bartels, A. Prohl ,Constraint preserving implicit finite element discretizat ion of harmonic map flow into\nspheres, Math. Comp. 76, pp. 1847–1859 (2007).\n[10]S. Bartels, C. Lubich, A. Prohl ,Convergent discretization of heat and wave map flows to spher es using approx-\nimate discrete Lagrange multipliers , Math. Comp. 78 (2009), no. 267, 1269–1292.\n[11]R. Becker, X. Feng, and A. Prohl ,Finite element approximations of the Ericksen–Leslie mode l for nematic\nliquid crystal flow , SIAM J. Numer. Anal., 46 (2008), pp. 1704–1731.\n[12]F. Brezzi, M. Fortin ,Mixed and hybrid finite element methods , Springer Series in Computational Mathematics,\n15. Springer-Verlag, New York, 1991.\n[13]S. C. Brenner, L. R. Scott ,The mathematical theory of finite element methods , Third edition. Texts in Applied\nMathematics, 15. Springer, New York, 2008.\n[14]J. Jr. Douglas, T. Dupont, L. Wahlbin ,The stability in Lqof theL2-projection into finite element function\nspaces, Numer. Math. 23 (1975), 193–197.\n[15]R. C. Cabrales, F. Guill ´en-Gonz ´alez, J. V. Guti ´errez-Santacreu ,A time-splitting finite-element stable ap-\nproximation for the Ericksen-Leslie equations , SIAM J. Sci. Comput. 37 (2015), no. 2, B261–B282.\n[16]I. Cimr´ak,Error estimates for a semi-implicit numerical scheme solvi ng the Landau–Lifshitz equation with an\nexchange field , IMA J. Numer. Anal. 25 (2005), no. 3, 611–634.\n[17]I. Cimr´ak,A survey on the numerics and computations for the Landau–Lif shitz equation of micromagnetism , Arch.\nComput. Methods Eng. 15 (2008), no. 3, 277–309.\n[18]Q. Du, B. Guo, J. Shen ,Fourier spectral approximation to a dissipative system mod eling the flow of liquid crystals ,\nSIAM J. Numer. Anal. 39 (2000), no. 3, 735–762.\n[19]H. C. Elman, G. H. Golub ,Inexact and preconditioned Uzawa algorithms for saddle poi nt problems , SIAM J.\nNumer. Anal. 31 (6) (1994) 1645–1661.\n[20]J. Ericksen ,Conservation laws for liquid crystals , Trans. Soc. Rheology, 5 (1961), 22–34.\n[21]J. Ericksen ,Continuum theory of nematic liquid crystals , Res. Mechanica, 21 (1987), 381–392.\n[22]A. Ern, J.-L. Guermond ,Theory and practice of finite elements , Applied Mathematical Sciences, 159. Springer-\nVerlag, New York, 2004.\n[23]T. L. Gilbert ,A Lagrangian formulation of the gyromagnetic equation of th e magnetization fields , Phys. Rev. 100\n(1955), 1243.\n[24]T.L. Gilbert ,A phenomenological theory of damping in ferromagnetic mate rials, IEEE Trans. Magn. 40 (2004),\n3443–3449.\n[25]F. Guill ´en-Gonz ´alez, J. V. Guti ´errez-Santacreu ,A linear mixed finite element scheme for a nematic Ericksen–\nLeslie liquid crystal model , ESAIM Math. Model. Numer. Anal., 47 (2013), pp. 1433–1464.\n[26]F. Guill ´en-Gonzalez, J. Koko ,A splitting in time scheme and augmented Lagrangian method f or a nematic\nliquid crystal problem , J. Sci. Comput., 65 (2015), pp. 1129–1144.\n[27]Q. Hu, X.-C. Tai, R. Winther ,A saddle point approach to the computation of harmonic maps , SIAM J. Numer.\nAnal. 47 (2009), no. 2, 1500–1523.\n[28]C. Johnson, A. Szepessy, P. Hansbo ,On the convergence of shock-capturing streamline diffusion finite element\nmethods for hyperbolic conservation laws . Mathematics of Computation, 54 (1990), pp. 107–129.\n[29]E. Kritsikis, A. Vaysset, L. D. Buda-Prejbeanu, F. Alouges, J. -C. Toussaint ,Beyond first-order finite element\nschemes in micromagnetics. J. Comput. Phys. 256 (2014), 357–366.\n[30]M. Kruzik, A. Prohl .Recent developments in the modeling, analysis, and numeric s of ferromagnetism. SIAM\nRev. 48 (2006), no. 3, 439–483.\n[31]L. D. Landau, E. M. Lifshitz ,Theory of the dispersion of magnetic permeability in ferrom agnetic bodies , Phys.\nZ. Sowietunion, 8 (1935), 153.\n[32]F. Leslie ,Some constitutive equations for liquid crystals , Arch. Rational Mech. Anal., 28 (1968), 265–283.\n[33]F. Leslie ,Theory of flow phenomena in liquid crystals , in Advances in Liquid Crystals, Vol. 4, G. H. Brown, ed.,\nAcademic Press, New York, 1979,1–81.\n[34]C. Liu, N. J. Walkington ,Approximation of liquid crystal flows , SIAM J. Numer. Anal. 37 (2000), no. 3, 725–741.\n[35]C. Liu, N. J. Walkington ,Mixed methods for the approximation of liquid crystal flows , ESAIM Math. Model.\nNumer. Anal., 36 (2002), pp. 205–222.\n[36]M. Mu, X. Zhu ,Decoupled schemes for a non-stationary mixed Stokes–Darcy model, Math. Comp. 79, pp. 707–731\n(2010).\n[37]A. Prohl ,Computational micromagnetism , Advances in Numerical Mathematics. B. G. Teubner, Stuttga rt, 2001."    },    {        "title": "1702.08408v2.Current_Induced_Damping_of_Nanosized_Quantum_Moments_in_the_Presence_of_Spin_Orbit_Interaction.pdf",        "content": "Current Induced Damping of Nanosized Quantum Moments in the Presence of\nSpin-Orbit Interaction\nFarzad Mahfouzi\u0003and Nicholas Kioussisy\nDepartment of Physics and Astronomy, California State University, Northridge, CA, USA\n(Dated: November 10, 2021)\nMotivated by the need to understand current-induced magnetization dynamics at the nanoscale,\nwe have developed a formalism, within the framework of Keldysh Green function approach, to study\nthe current-induced dynamics of a ferromagnetic (FM) nanoisland overlayer on a spin-orbit-coupling\n(SOC) Rashba plane. In contrast to the commonly employed classical micromagnetic LLG simula-\ntions the magnetic moments of the FM are treated quantum mechanically . We obtain the density\nmatrix of the whole system consisting of conduction electrons entangled with the local magnetic\nmoments and calculate the e\u000bective damping rate of the FM. We investigate two opposite limiting\nregimes of FM dynamics: (1) The precessional regime where the magnetic anisotropy energy (MAE)\nand precessional frequency are smaller than the exchange interactions, and (2) The local spin-\rip\nregime where the MAE and precessional frequency are comparable to the exchange interactions. In\nthe former case, we show that due to the \fnite size of the FM domain, the \\Gilbert damping\"does\nnot diverge in the ballistic electron transport regime, in sharp contrast to Kambersky's breathing\nFermi surface theory for damping in metallic FMs. In the latter case, we show that above a critical\nbias the excited conduction electrons can switch the local spin moments resulting in demagnetization\nand reversal of the magnetization. Furthermore, our calculations show that the bias-induced anti-\ndamping e\u000eciency in the local spin-\rip regime is much higher than that in the rotational excitation\nregime.\nPACS numbers: 72.25.Mk, 75.70.Tj, 85.75.-d, 72.10.Bg\nI. INTRODUCTION\nUnderstanding the current-induced magnetization\nswitching (CIMS) at the nanoscale is mandatory for the\nscalability of non-volatile magnetic random access mem-\nory (MRAM) of the next-generation miniaturized spin-\ntronic devices. However, the local magnetic moments of a\nnanoisland require quantum mechanical treatment rather\nthan the classical treatment of magnetization commonly\nemployed in micromagnetic simulations, which is the cen-\ntral theme of this work.\nThe \frst approach of CIMS employs the spin transfer\ntorque (STT)1,2in magnetic tunnel junctions (MTJ) con-\nsisting of two ferromagnetic (FM) layers (i.e., a switch-\nable free layer and a \fxed layer) separated by an insulat-\ning layer, which involves spin-angular-momentum trans-\nfer from conduction electrons to local magnetization3,4.\nAlthough STT has proven very successful and brings the\nprecious bene\ft of improved scalability, it requires high\ncurrent densities ( \u00151010A/cm2) that are uncomfort-\nably high for the MTJ's involved and hence high power\nconsumption. The second approach involves an in-plane\ncurrent in a ferromagnet-heavy-metal bilayer where the\nmagnetization switching is through the so-called spin-\norbit torque (SOT) for both out-of-plane and in-plane\nmagnetized layers.5{8The most attractive feature of the\nSO-STT method is that the current does not \row through\nthe tunnel barrier, thus o\u000bering potentially faster and\nmore e\u000ecient magnetization switching compared to the\nMTJs counterparts.\nAs in the case of STT, the SO-STT has two compo-\nnents: a \feld-like and an antidamping component. Whilethe \feld-like component reorients the equilibrium direc-\ntion of the FM, the antidamping component provides the\nenergy necessary for the FM dynamics by either enhanc-\ning or decreasing the damping rate of the FM depending\non the direction of the current relative to the magneti-\nzation orientation as well as the structural asymmetry\nof the material. For su\u000eciently large bias the SOT can\novercome the intrinsic damping of the FM leading to ex-\ncitation of the magnetization precession.8The underlying\nmechanism of the SOT for both out-of-plane and in-plane\nmagnetized layers remains elusive and is still under de-\nbate. It results from either the bulk Spin Hall E\u000bect\n(SHE)9{12, or the interfacial Rashba-type spin-orbit cou-\npling,13{16or both17{19.\nMotivated by the necessity of scaling down the size\nof magnetic bits and increasing the switching speed, the\nobjective of this work is to develop a fully quantum me-\nchanical formalism, based on the Keldysh Green function\n(GF) approach, to study the current-induced local mo-\nment dynamics of a bilayer consisting of a FM overlayer\non a SOC Rashba plane, shown in Fig. 1.\nUnlike the commonly used approaches to investigate\nthe magnetization dynamics of quantum FMs, such as the\nmaster equation20, the scattering21or quasi-classical22\nmethods, our formalism allows the study of magnetiza-\ntion dynamics in the presence of nonequilibrium \row of\nelectrons.\nWe consider two di\u000berent regimes of FM dynamics: In\nthe \frst case, which we refer to as the single domain\ndynamics, the MAE and the precession frequency are\nsmaller than the exchange interactions, and the FM can\nbe described by a single quantum magnetic moment, of\na typically large spin, S, whose dynamics are governedarXiv:1702.08408v2  [cond-mat.mes-hall]  27 Apr 20172\nFIG. 1: (Color online) Schematic view of the FM/Rashba\nplane bilayer where the FM overlayer has length Lxand is\nin\fnite (\fnite) along the y-direction for the case of a single\ndomain (nano-island) discussed in Sec. III (IV). The magne-\ntization,~ m, of the FM precesses around the direction denoted\nby the unit vector, ~ nM, with frequency !and cone angle, \u0012.\nThe Rashba layer is attached to two normal (N) leads which\nare semi-in\fnite along the x-direction, across which an exter-\nnal bias voltage, V, is applied.\nmainly by the quantized rotational modes of the magne-\ntization. We show that the magnetic degrees of freedom\nentering the density matrix of the conduction electron-\nlocal moment entagled system simply shift the chemical\npotential of the Fermi-Dirac distribution function by the\nrotational excitations energies of the FM from its ground\nstate. We also demonstrate that the e\u000bective damping\nrate is simply the netcurrent along the the auxiliarym-\ndirection, where m= -S, -S+1, :::, +S, are the eigenval-\nues of the total Szof the FM. Our results for the change\nof the damping rate due to the presence of a bias volt-\nage are consistent with the anti-damping SOT of clas-\nsical magnetic moments,16,23, where due to the Rashba\nspin momentum locking, the anti-damping SOT, to low-\nest order in magnetic exchange coupling, is of the form,\n~ m\u0002(~ m\u0002^y), where ^yis an in-plane unit vector normal\nto the transport direction.\nIn the adiabatic and ballistic transport regimes due to\nthe \fnite S value of the nanosize ferromagnet our formal-\nism yields a \fnite \\Gilbert damping\", in sharp contrast to\nKambersky's breathing Fermi surface theory for damp-\ning in metallic FMs.24On the other hand, Costa and\nMuniz25and Edwards26demonstrated that the prob-\nlem of divergent Gilbert damping is removed by takinginto account the collective excitations. Furthermore, Ed-\nwards points out26the necessity of including the e\u000bect of\nlong-range Coulomb interaction in calculating damping\nfor large SOC.\nIn the second case, which corresponds to an indepen-\ndent local moment dynamics, the FM has a large MAE\nand hence the rotational excitation energy is compara-\nble to the local spin-\rip excitation (exchange energy).\nWe investigate the e\u000bect of bias on the damping rate of\nthe local spin moments. We show that above a criti-\ncal bias voltage the \rowing conduction electrons can ex-\ncite (switch) the local spin moments resulting in demag-\nnetization and reversal of the magnetization. Further-\nmore, we \fnd that, in sharp contrast to the single do-\nmain precessional dynamic, the current-induced damping\nis nonzero for in-plane and out-of-plane directions of the\nequilibrium magnetization. The bias-induced antidamp-\ning e\u000eciency in the local moment switching regime is\nmuch higher than that in the single domain precessional\ndynamics.\nThe paper is organized as follows. In Sec. II we present\nthe Keldysh formalism for the density matrix of the en-\ntagled quantum moment-conduction electron system and\nthe e\u000bective dampin/antdamping torque. In Sec. III we\npresent results for the current-induced damping rate in\nthe single domain regime. In Sec. IV we present results\nfor the current-induced damping rate in the independent\nlocal regime. We conclude in Sec. V.\nII. THEORETICAL FORMALISM\nFig. 1 shows a schematic view of the ferromagnetic\nheterostructure under investigation consisting of a 2D\nferromagnet-Rashba plane bilayer attached to two semi-\nin\fnite normal (N) leads whose chemical potentials are\nshifted by the external bias, Vbias. The magnetization\nof the FM precesses around the axis speci\fed by the\nunit vector, ~ nM, with frequency !and cone angle \u0012.\nThe FM has length, LFM\nx, along the transport direction.\nThe total Hamiltonian describing the coupled conduc-\ntion electron-localized spin moment system in the het-\nerostructure in Fig. 1 can be written as,\nHtot=X\nrr0;\u001b\u001b0Trfsdgh\u0010\n1s^H\u001b\u001b0\nrr0+\u000err0\u000e\u001b\u001b01s\u0016r+\u000err0Jsd~ \u001b\u001b\u001b0\u0001~sd(r) +\u000e\u001b\u001b0\u000err0HM\u0011\n \u0003\nfs0\ndgr0\u001b0 fsdgr\u001bi\n: (1)\nHere,~sd(r) is the local spin moment at atomic position\nr, the trace is over the di\u000berent con\fgurations of the lo-\ncal spin moments, fsdg, fsdgr\u001b=jfsdgi\n e\nr\u001bis the\nquasi-particle wave-function associated with the conduc-tion electron (  e) entangled to the FM states ( jfsdgi),\nJsdis thes\u0000dexchange interaction, 1sis the identity ma-\ntrix in spin con\fguration space, and ^ \u001bx;y;z are the Pauli\nmatrices. We use the convention that, except for r, bold3\nsymbols represent operators in the magnetic con\fgura-\ntion space and symbols with hat represent operators in\nthe single particle Hilbert space of the conduction elec-\ntrons. The magnetic Hamiltonian HMis given by\nHM=\u0000g\u0016BX\nr~Bext(r)\u0001~sd(r) (2)\n\u0000X\nhr;r0iJdd\nrr0\ns2\nd~sd(r0)\u0001~sd(r)\u0000X\nrJsd\nsd~sc(r)\u0001~sd(r);\nwhere, the \frst term is the Zeeman energy due to the\nexternal magnetic \feld, the second term is the magnetic\ncoupling between the local moments and the third term\nis the energy associated with the intrinsic magnetic \feld\nacting on the local moment, ~sd(r), induced by the local\nspin of the conduction electrons, ~sc(r).\nThe Rashba model of a two-dimensional electron gas\nwith spin orbit coupling interacting with a system of\nlocalized magnetic moments has been extensively em-\nployed14,27,28to describe the e\u000bect of enhanced spin-orbit\ncoupling solely at the interface on the current-induced\ntorques in ultrathin ferromagnetic (FM)/heavy metal\n(HM) bilayers. The e\u000bects of (i) the ferromagnet induc-\ning a moment in the HM and (ii) the HM with strong\nspin-orbit coupling inducing a large spin-orbit e\u000bect in\nthe ferromagnet (Rashba spin-orbit coupling) lead to a\nthin layer where the magnetism and the spin-orbit cou-\npling coexist.27\nThe single-electron tight-binding Hamiltonian29for\nthe conduction electrons of the 2D Rashba plane, H\u001b\u001b0\nrr0\nwhich is \fnite along the transport direction xand in\fnite\nalong theydirection is of the form,\n^H\u001b\u001b0\nxx0(kya) = [tcos(kya)\u000e\u001b\u001b0\u0000tsosin(kya)\u001bx\n\u001b\u001b0]\u000exx0(3)\n+t(\u000ex;x0+1+\u000ex+1;x0)\u000e\u001b\u001b0+itso(\u000ex;x0+1\u0000\u000ex+1;x0)\u001by\n\u001b\u001b0:\nHere,x;x0denote atomic coordinates along the trans-\nport direction, ais the in-plane lattice constant, and tso\nis the Rashba SOI strength. The values of the local e\u000bec-\ntive exchange interaction, Jsd= 1eV, and of the nearest-\nneighbor hopping matrix element, t=1 eV, represent a\nrealistic choice for simulating the exchange interaction of\n3dferromagnetic transition metals and their alloys (Fe,\nCo).30{32The Fermi energy, EF=3.1 eV, is about 1 eV\nbelow the upper band edge at 4 eV consistent with the\nab initio calculations of the (111) Pt surface33. Further-\nmore, we have used tso=0.5 eV which yields a Rashba\nparameter, \u000bR=tsoa\u00191.4 eV \u0017A (a=2.77 \u0017A is the in-\nplane lattice constant of the (111) Pt surface) consis-\ntent with the experimental value of about 1-1.5 eV \u0017A34\nand the ab initio value of 1 eV \u0017A28. However, because\nother experimental measurements for Pt/Co/Pt stacks\nreport35a Rashba parameter which is an order of mag-\nnitude smaller, in Fig.3 we show the damping rate for\ndi\u000berent values of the Rashba SOI.. For the results in\nSec. IV, we assume a real space tight binding for propa-\ngation along y-axis.The single particle propagator of the coupled electron-\nspin system is determined from the equation of motion\nof the retarded Green function,\n\u0012\nE\u0000i\u0011\u0000^\u0016\u0000^H\u0000HM\u0000Jsd\n2^~ \u001b\u0001^~sd\u0013\n^Gr(E) =^1;(4)\nwhere,\u0011is the broadening of the conduction electron\nstates due to inelastic scattering from defects and/or\nphonons, and for simplicity we ignore writing the identity\nmatrices ^1 and 1in the expression. The density matrix\nof the entire system consisting of the noninteracting elec-\ntrons (fermionic quasi-particles) and the local magnetic\nspins is determined (see Appendix A for details of the\nderivation for a single FM domain) from the expression,\n^\u001a=ZdE\n\u0019^Gr(E)\u0011f(E\u0000^\u0016\u0000HM)^Ga(E): (5)\nIt is important to emphasize that Eq. (5) is the central\nresult of this formalism which demonstrates that the ef-\nfect of the local magnetic degrees of freedom is to shift the\nchemical potential of the Fermi-Dirac distribution func-\ntion by the eigenvalues, \"m, ofHMjmi=\"mjmi, i.e.,\nthe excitation energies of the FM from its ground state.\nHere,jmiare the eigenstates of the Heisenberg model\ndescribing the FM. The density matrix can then be used\nto calculate the local spin density operator of the con-\nduction electrons, [ ~sc(r)]mm0=P\nss0\u001amm0\nss0;rr~ \u001bss0=2, which\nalong with Eqs. (2), (4), and (5) form a closed set\nof equations that can be solved self consistently. Since,\nthe objective of this work is the damping/anti-damping\n(transitional) behavior of the FM in the presence of bias\nvoltage, we only present results for the \frst iteration.\nEq. (5) shows that the underlying mechanism of the\ndamping phenomenon is the \row of conduction electrons\nfrom states of higher chemical potential to those of lower\none where the FM state relaxes to its ground state by\ntransferring energy to the conduction electrons. There-\nfore, the FM dynamical properties in this formalism is\ncompletely governed by its coupling to the conduction\nelectrons, where conservation of energy and angular mo-\nmentum dictates the excitations as well as the \ructua-\ntions of the FM sate through the Fermi distribution func-\ntion of the electrons coupled to the reservoirs. This is\ndi\u000berent from the conventional Boltzmann distribution\nfunction which is commonly used to investigate the ther-\nmal and quantum \ructuations of the magnetization.\nDue to the fact that the number of magnetic con\fgura-\ntions (i.e. size of the FM Hilbert space) grows exponen-\ntially with the dimension of the system it becomes pro-\nhibitively expensive to consider all possible eigenstates\nof theHMoperator. Thus, in the following sections we\nconsider two opposite limiting cases of magnetic con\fg-\nurations. In the \frst case we assume a single magnetic\nmoment for the whole FM which is valid for small FMs\nwith strong exchange coupling between local moments\nand small MAE. In this case the dynamics is mainly gov-\nerned by the FM rotational modes and local spin \rips can4\nFIG. 2: (Color online) Schematic representation of the quasi-\nparticles of the FM and conduction electron entangled states.\nThe horizontal planes denote the eigenstates, jS;miof the\ntotalSzof the FM with eigenvalues m=\u0000S;\u0000S+ 1;:::; +S\nalong the auxiliarym-direction. Excitation of magnetic state\ninduces a shift of the chemical potential of the Fermi-Dirac\ndistribution function leading to \row of quisiparticles along the\nm-direction which corresponds to the damping rate of the FM.\nThe FM damping involves two processes: (1) An intra-plane\nprocess involving spin reversal of the conduction electron via\nthe SOC; and (2) An inter-plane process involving quasiparti-\ncle \row of majority (minority) spin along the ascending (de-\nscending)m-direction due to conservation of total angular\nmomentum, where the interlayer hopping is accompanied by\na spin \rip of conduction electrons.\nbe ignored. In the second case we ignore the correlation\nbetween di\u000berent local moments and employ a mean \feld\napproximation such that at each step we focus on an indi-\nvidual atom by considering the local moment under con-\nsideration as a quantum mechanical object while the rest\nof the moments are treated classically. We should men-\ntion that a more accurate modeling of the system should\ncontain both single domain rotation of the FM as well\nas the local spin \ripping but also the e\u000bect of nonlocal\ncorrelations between the local moments and conduction\nelectrons, which are ignored in this work.\nIII. SINGLE DOMAIN ROTATIONAL\nSWITCHING\nIn the regime where the energy required for the excita-\ntion of a single local spin moment ( \u0019meV ) is much larger\nthan the MAE (\u0019\u0016eV) the low-energy excited states cor-\nrespond to rotation of the total angular momentum of the\nFM acting as a single domain and the e\u000bects of local spin\n\rips described as the second term in Eq 2, can be ignored.\nIn this regime all of the local moments behave collectively\nand the local moment operators can be replaced by the\naverage spin operator, ~sd(r) =P\nr0~sd(r0)=Nd=sd~S=S,\nwhereNdis the number of local moments and ~Sis the\ntotal angular momentum with amplitude S. The mag-\nnetic energy operator is given by HM=\u0000~B\u0001S, where,\n~B=g\u0016B~Bext+Jsd~ sc. Here, for simplicity we assume\n~ scto be scalar and independent of the FM state. The\neigenstates of HMoperator are then simply the eigen-states,jS;mi, of the total angular momentum Sz, with\neigenvalues m!=\u0000S!;:::; +S!, where!=Bzis the\nLarmor frequency. Thus, the wave function of the cou-\npled electron-spin con\fguration system, shown schemat-\nically in Fig. 2 is of the form,  ms0r(t) =jS;mi\n s0r(t).\nOne can see that the magnetic degrees of freedom corre-\nsponding to the di\u000berent eigenstates of the Szoperator,\nenters as an additional auxiliary dimension for the elec-\ntronic system where the variation of the magnetic energy,\nhS;mjHMjS;mi=m!, shifts the chemical potentials of\nthe electrons along this dimension. The gradient of the\nchemical potential along the auxiliary direction, is the\nLarmor frequency ( \u0016eV\u0019GHz ) which appears as an\ne\u000bective \\electric \feld\"in that direction.\nSubstituting Eq (5) in Eq (A1)(b) and averaging over\none precession period we \fnd that the average rate of\nangular momentum loss/gain, which we refer to as the\ne\u000bective \\ damping rate \"per magnetic moment, can be\nwritten as\nTm=1\n2=(T\u0000\nm\u0000T+\nm); (6)\nwhere,\nT\u0006\nm=Jsd\n2SNdTrel[^\u001b\u0007S\u0006\nm^\u001am;m\u00061]: (7)\nis the current along the auxiliarym-direction in Fig. 2\nfrom them$m+ 1 (\u0006sign) state of the total Szof the\nFM. Here,Trel, is the trace over the conduction electron\ndegrees of freedom, and S\u0006\nm=p\nS(S+ 1)\u0000m(m\u00061)\nare the ladder operators. It is important to note that\nwithin this formalism the damping rate is simply the net\ncurrent across the mth-layer along the auxiliary direction\nassociated with the transition rate of the FM from state\nmto its nearest-neighbor states ( m\u00061).\nFig. 3 shows the damping rate as a function of the pre-\ncession cone angle, \u0012= cos\u00001(m\nS), for di\u000berent values of\nbias and for an in-plane e\u000bective magnetic \feld (a) along\nand (b) normal to the transport direction, and (c) an out-\nof-plane magnetic \feld. For cases (a) and (c) the damp-\ning rate is negative and relatively independent of bias for\nlow bias values. A negative damping rate implies that the\nFM relaxes towards the magnetic \feld by losing its angu-\nlar momentum, similar to the Gilbert damping rate term\nin the classical LLG equation, where its average value\nover the azimuthal precession angle, '=!t, is of the\nform,T=\u0000\u000bsdRd'\n2\u0019~ m\u0002(~ m\u0002~B)\u0001~ nM, which is nonzero\n(zero) when the unit vector ~ nMis along (perpendicular\nto) the e\u000bective magnetic \feld. The dependence of the\ndamping rate on the bias voltage when the e\u000bective mag-\nnetic \feld~Bis inplane and normal to the transport direc-\ntion can be understood by the spin-\rip re\rection mech-\nanism accompanied by Rashba spin-momentum locking\ndescribed in Ref.16. One can see that a large enough bias\ncan result in a sign reversal of the damping rate and hence\na magnetization reversal of the FM. It's worth mention-\ning that due to the zero-point quantum \ructuations of5\nthe magnetization, at \u0012= 0;\u0019(i.e.m=\u0006S) we have\nT 6=0 which is inversely proportional to the size of the\nmagnetic moment, S.\nIn Fig. 4(a) we present the e\u000bective damping rate ver-\nsus bias for di\u000berent values of the Rashba SOC. The re-\nsults show a linear response regime with respect to the\nbias voltage where both the zero-bias damping rate and\nthe slope,dT=dV increases with the Rashba SOC. This\nis consistent with Kambersky's mechanism of Gilbert\ndamping due to the SOC of itinerant electrons,24and the\nSOT mechanism16. Fig. 4(b) shows that in the absence\nof bias voltage the damping rate is proportional to t2\nsoand\nthe e\u000bect of the spin current pumped into the left and\nright reservoirs is negligible. This result of the t2\nsodepen-\ndence of the zero-bias damping rate is in agreement with\nrecent calculations of Costa and Muniz25and Edwards26\nwhich took into account the collective excitations. In the\npresence of an external bias, Tvaries linearly with the\nSOC, suggesting that to the lowest order it can be \ftted\nto\nT= sin2(\u0012)tso(c1tso~!+c2eVbias); (8)\nwherec1andc2are \ftting parameters.\nThe bias-induced e\u000eciency of the anti-damping SOT,\n\u0002\u0011~!(T(Vbias)\u0000T(0))=eVbiasT(0), describes how e\u000e-\ncient is the energy conversion between the magnetization\ndynamics and the conduction electrons. Accordingly, for\na given bias-induced e\u000eciency, \u0002, one needs to apply an\nexternal bias equal to ~!=e\u0002 to overcome the zero-bias\ndamping of the FM. Fig. 5 displays the anti-damping ef-\n\fciency versus the position of the Fermi energy of the FM\nfrom the bottom (-4 t=-4 eV) to the top (4 t=4eV) of the\nconduction electron band for the two-dimensional square\nlattice. The result is independent of the bias voltage and\nthe Larmor frequency in the linear response regime ( i.e.\nVbias;!\u001ct). We \fnd that the e\u000eciency peaks when the\nFermi level is in the vicinity of the bottom or top of the\nenergy band where the transport is driven by electron- or\nhole-like carriers and the Gilbert damping is minimum.\nThe sign reversal of the antidamping SOT is due to the\nelectron- or hole-like driven transport similar to the Hall\ne\u000bect.36\nClassical Regime of the Zero Bias Damping rate |\nIn the following we show that in the case of classical\nmagnetic moments ( S!1 ) and the adiabatic regime\n(!!0), the formalism developed in this paper leads to\nthe conventional expressions for the damping rate. In this\nlimit the system becomes locally periodic and one can\ncarry out a Fourier transformation from m\u0011Szspace\nto azimuthal angle of the magnetization orientation, ',\nspace. Conservation of the angular momentum suggests\nthat the majority- (minority-) spin electrons can propa-\ngate only along the ascending (descending) m-direction,\nwhere the hopping between two nearest-neighbor m-\nlayers is accompanied by a spin-\rip. As shown in Fig.\n2 the existence of spin-\rip hopping requires the presence\nof intralayer SOC-induced noncollinear spin terms which\nrotate the spin direction of the conduction electrons as\n04590135180−10−505B=[|B|,0,0]Damping Rate ( µeV)\n  \n4590135180B=[0,|B|,0]\nCone Angle, θ (deg)4590135180B=[0,0,|B|]\nVbias=−5 mV\nVbias=0\nVbias=5 mV(a) (b) (c)\nStudent Version of MATLABFIG. 3: (Color online) E\u000bective damping rate for a single\nFM domain as a function of the precession cone angle, \u0012, for\nvarious bias values under an e\u000bective magnetic \feld which is\nin-plane (a) along and (b) normal to the transport direction\nand (c) out-of-plane. The length of the FM along the xdi-\nrection isLx= 25awhile it is assumed to be in\fnite in the\ny-direction, ~!= 10\u0016eV, the broadening parameter \u0011= 0,\nkBT= 10meV and the domain magnetic moment S= 200.\nThe results are robust with larger values of Sin either the\nballistic,\u0011\u001c~!, or dirty,\u0011\u001d~!, regimes.\n−2−1012−3−2−101\nBias Voltage (mV)Damping Rate ( µeV)\n  \n−0.5 0 0.5−15−10−505\nSpin Orbit Coupling, tso (eV)  \ntso=0\ntso=0.1 eV\ntso=0.2 eVVbias=−5 mV\nVbias=0\nVbias=5 mV(a) (b)\nStudent Version of MATLAB\nFIG. 4: (Color online) Damping torque versus (a) bias voltage\nand (b) spin-orbit coupling strength, for m= 0 corresponding\nto the precession cone angle of 90o. The precession axis of the\nFM is along the y-direction and the rest of the parameters are\nthe same as in Fig. 3. The zero-bias damping rate versus SOC\nshows at2\nsodependence while the damping rate under non-\nzero bias exhibits nearly linear SOC dependence.\nthey propagate in each m-layer. This is necessary for\nthe persistent \row of electrons along the 'auxiliary di-\nrection and therefore damping of the magnetization dy-\nnamics. Using the Drude expression of the longitudinal\nconductivity along the '-direction for the damping rate,\nwe \fnd that, within the relaxation time approximation,\n\u0011=!!1 , where the relaxation time of the excited con-\nduction electrons is much shorter than the time scale of6\n−4−3−2−101234−3\n−2\n−1\n0\n1\n2\n3\nFermi Energy (eV)Antidamping Efficiency (%)\n−4−2024−40−200Damping Rate ( µeV)\n  \nVbias=5 mV\nVbias=0\nVbias=−5 mV\nStudent Version of MATLAB\nFIG. 5: (Color online) Bias-induced precessional anti-\ndamping e\u000eciency, \u0002 = ~!(T(Vbias)\u0000T(0))=eVbiasT(0), ver-\nsus the Fermi energy of the 2D Rashba plane in Fig.1, where\nthe energy band ranges from -4 eV to +4 eV. The magnetiza-\ntion precesses around the in-plane direction ( y\u0000axis) normal\nto the transport direction and the rest of the parameters of\nthe system are the same as in Fig. 3. Note, for magnetization\nprecession around the xandzaxis,T(Vbias) =T(0) for all\nprecession cone angles and hence \u0002=0. Inset shows the damp-\ning rate versus the Fermi energy for di\u000berent bias values used\nto calculate precessional anti-damping e\u000eciency.\nthe FM dynamics, Tis given by\nT=\u0000!\n\u0011X\nnZdkxdkyd'\n(2\u0019)3(v'\nn~k)2f0(\"n~k(')):(9)\nHere,v'\nn~k=@\"n~k(')=@' is the group velocity along the\n'-direction in Fig. 2, and \"n;~k=\"0(j~kj)\u0006j~h(~k)jfor the\n2D-Rashba plane, where \"0(j~kj) is the spin independent\ndispersion of the conduction electrons and ~h=atso^ez\u0002\n~k+1\n2Jsd~ m, is the spin texture of the electrons due to\nthe SOC and the s\u0000dexchange interaction. For small\nprecession cone angle, \u0012, the Gilbert damping constant\ncan be determined from \u000b=\u0000T=sd!sin2(\u0012), where the\nzero-temperature Tis evaluated by Eq. (9). We \fnd\nthat\n\u000b\u00191\n\u0011t2\nso\u0002\n(k+\nFa)2D+(EF) + (k\u0000\nFa)2D\u0000(EF)\u0003\n(1+cos2(\r));\n(10)\nwhereD+(\u0000)(E) is the density of states of the majority\n(minority) band, \ris the angle between the precession\naxis and the normal to the Rashba plane, and the Fermi\nwave-vectors ( k\u0006\nF) are obtained from, \"0(k\u0006\nF) =EF\u0007\nJsd=2. Eq. (10) shows that the Gilbert damping increases\nas the precession axis changes from in-plane ( \r=\u0019=2) to\nout of plane ( \r= 0),37which can also be seen in Fig. 3.\nIt is important to emphasize that in contrast to Eq. (9)\nthe results shown in Fig. 4 correspond to the ballistic\nregime with \u0011= 0 in the central region and the relaxation\nof the excited electrons occurs solely inside the metallic\nreservoirs. To clarify how the damping rate changes from\n10−810−610−410−2100−80−60−40−20020\nBroadening (eV)Damping Rate ( µeV)\n  \nS=200, Vbias=0\nS=200, Vbias=3 mV\nS=300, Vbias=0\nS=300, Vbias=3 mV\nStudent Version of MATLABFIG. 6: (Color online) Precessional damping rate versus\nbroadening of the states in the presence (solid lines) and ab-\nsence (dashed lines) of bias voltage for two values of the do-\nmain sizeS= 200 and S= 300. In both ballistic, \u0011=!\u00190,\nand di\u000busive, \u0011=!\u001d1, regimes the precessional damping rate\nis independent of the domain size, while in the intermediate\ncase, the amplitude of the minimum of damping rate shows a\nlinear dependence versus S. Note that the value of the broad-\nening at which the damping rate is minimum varies inversely\nproportional to the domain size, S.\nthe ballistic to the di\u000busive regime we present in Fig.\n6 the damping rate versus the broadening, \u0011, of states\nin the presence (solid line) and absence (dashed line) of\nbias voltage. We \fnd that in both ballistic ( \u0011=!\u00190)\nand di\u000busive ( \u0011=!\u001d1) regimes the damping rate is\nindependent of the size of the FM domain, S. On the\nother hand, in the intermediate regime the FM dynamics\nbecome strongly dependent on the e\u000bective domain size\nwhere the minimum of the damping rate varies linearly\nwithS. This can be understood by the fact that the\ne\u000bective chemical potential di\u000berence between the \frst,\nm=\u0000Sand last,m=Slayers in Fig.3 is proportional\ntoSand for a coherent electron transport the conduc-\ntance is independent of the length of the system along\nthe transport direction. Therefore, in this case the FM\nmotion is driven by a coherent dynamics.\nIV. DEMAGNETIZATION MECHANISM OF\nSWITCHING\nIn Sec. III we considered the case of a single FM do-\nmain where its low-energy excitations, involving the pre-\ncession of the total angular momentum, can be described\nby the eigenstates jmiofSzand local spin \rip processes\nwere neglected. However, for ultrathin FM \flms or FM\nnanoclusters, where the MAE per atom ( \u0019meV ) is com-\nparable to the exchange energy between the local mo-\nments (Curie temperature), the low-energy excitations\ninvolve both magnetization rotation and local moments\nspin-\rips due to conduction electron scattering which can\nin turn change also S. In this case the switching is ac-7\nFIG. 7: (Color online) Spatial dependence of the local damp-\ning rate for the spin-1 =2 local moments of a FM island under\ndi\u000berent bias voltages ( \u00060:4V) and magnetization directions.\nFor the parameters we chose the size of the FM island to be\n25\u000225a2, the e\u000bective magnetic \feld, jBj= 20meV , the\nbroadening, \u0011= 0, andkBT= 10 meV.\ncompanied by the excitation of local collective modes that\ne\u000bectively lowers the amplitude of the magnetic ordering\nparameter. For simplicity we employ the mean \feld ap-\nproximation for the 2D FM nanocluster where the spin\nunder consideration at position ris treated quantum me-\nchanically interacting with all remaining spins through\nan e\u000bective magnetic \feld, ~B. The spatial matrix ele-\nments of the local spin operator are\n[^~sd;r]r1r2=~ sd(r1)\u000er1r2(1\u0000\u000er1r)1s+1\n2\u000er1r2\u000er1r~ \u001c;(11)\nwhere,~\u001cs are the Pauli matrices. The magnetic energy\ncan be expressed as, HM(r) =\u0000~B(r)\u0001~\u001c=2, where, the\ne\u000bective local magnetic \feld is given by,\n~B(r) =g\u0016B~Bext+ 4X\nr0Jdd\nrr0~ sd(r0) + 2Jsd~ sc(r):(12)\nThe equation of motion for the single particle propa-\ngator of the electronic wavefunction entangled with the\nlocal spin moment under consideration can then be ob-\ntained from,\n\u0012\nE\u0000^\u0016\u0000HM(r)\u0000^H\u0000Jsd\n2^~ \u001b\u0001^~sd;r\u0013\n^Gr\nr(E) =^1:(13)\nThe density matrix is determined from Eq. (5) which\ncan in turn be used to calculate the spin density of the\nconduction electrons, ~ sc(r) =Tr(^~ \u001b^\u001arr)=2, and the di-\nrection and amplitude of the local magnetic moments,\n~ sd(r) =Tr(~\u001c^\u001arr)=2.\nFig. 7 shows the spatial dependence of the spin-1\n2local\nmoment switching rate for a FM/Rashba bilayer (Fig.\n−101−30−20−100102030\n  Damping Torque (meV)B=[|B|,0,0]\n−101\nBias Voltage (V)B=[0,|B|,0]\n−101B=[0,0,|B|]\n|B|=1 meV\n|B|=20 meV(c) (a) (b)\nStudent Version of MATLABFIG. 8: (Color online) Bias dependence of the average (over\nall sites) damping rate of the FM island for in-plane e\u000bective\nmagnetic \feld (or equilibrium magnetization) (a) along and\n(b) normal to the transport direction and (c) out-of-plane\nmagnetic \feld for two values of jBj.\n1) for two bias values ( Vbias=\u00060:4V) and for an in-\nplane e\u000bective magnetic \feld (a) along and (b) normal\nto the transport direction, and (c) an out-of-plane mag-\nnetic \feld. The size of the FM island is 25 a\u000225a, where\nais the lattice constant. Negative local moment switch-\ning rate (blue) denotes that, once excited, the local mo-\nment relaxes to its ground state pointing along the di-\nrection of the e\u000bective magnetic \feld; however positive\nlocal damping rate (red) denotes that the local moments\nremain in the excited state during the bias pulse dura-\ntion. Therefore, the damping rate of the local moments\nunder bias voltage can be either enhanced or reduced\nand even change sign depending on the sign of the bias\nvoltage and the direction of the magnetization. We \fnd\nthat the bias-induced change of the damping rate is high-\nest when the FM magnetization is in-plane and normal to\nthe transport directions similar to the single domain case.\nFurthermore, the voltage-induced damping rate is peaked\nclose to either the left or right edge of the FM (where the\nreservoirs are attached) depending on the sign of the bias.\nNote that there is also a \fnite voltage-induced damping\nrate when the magnetization is in-plane and and along\nthe transport direction ( x) or out-of-the-plane ( z).\nFig. 8 shows the bias dependence of the average (over\nall sites) damping rate for in- (a and b) and out-of-plane\n(c) directions of the e\u000bective magnetic \feld (direction\nof the equilibrium magnetization) and for two values of\njBj. This quantity describes the damping rate of the\namplitude of the magnetic order parameter. For an in-\nplane magnetization and normal to the transport direc-\ntion (Fig. 8) the bias behavior of the damping rate is lin-\near and \fnite in contrast to the single domain [Fig. 3(a)]\nwhere the damping rate was found to have a negligible\nresponse under bias. On the other hand, the bias behav-\nior of the current induced damping rate shows similar\nbehavior to the single domain case when the equilibrium8\n−4 −2 0 2 4−30−20−100102030\nFermi Energy (eV)Antidamping Efficiency (%)\n  \nB=[20,0,0] meV\nB=[0,20,0] meV\nB=[0,0,20] meV\nStudent Version of MATLAB\nFIG. 9: (Color online) Bias-induced local anti-damping\ne\u000eciency due to local spin-\rip, \u0002 = jBj(T(Vbias)\u0000\nT(0))=eVbiasT(0), versus Fermi energy for di\u000berent equilib-\nrium magnetization orientations. For the calculation we chose\nVbias= 0:2 V and the rest of the Hamiltonian parameters are\nthe same as in Fig. 7.\nmagnetization direction is in-plane and normal to the\ntransport direction (Fig. 8(b)). For an out-of-plane ef-\nfective magnetic \feld [Fig. 8(c)] the damping torque has\nan even dependence on the voltage bias.\nIn order to quantify the e\u000eciency of the voltage in-\nduced excitations of the local moments, we calculate the\nrelative change of the average of the damping rate in the\npresence of a bias voltage and present the result versus\nthe Fermi energy for di\u000berent orientations of the magne-\ntization in Fig 9. We \fnd that the e\u000eciency is maximum\nfor an in-plane equilibrium magnetization normal to the\ntransport direction and it exhibits an electron-hole asym-\nmetry. The bias-induced antidamping e\u000eciency due to\nspin-\rip can reach a peak around 20% which is much\nhigher than the peak e\u000eciency of about 2% in the sin-\ngle domain precession mechanism in Fig. 5 for the same\nsystem parameters.\nFuture work will be aimed in determining the switch-\ning phase diagram16by calculating the local antidamping\nand \feld-like torques self consistently for di\u000berent FM\ncon\fgurations.\nV. CONCLUDING REMARKS\nIn conclusion, we have developed a formalism to in-\nvestigate the current-induced damping rate of nanoscale\nFM/SOC 2D Rashba plane bilayer in the quantum\nregime within the framework of the Kyldysh Green func-\ntion method. We considered two di\u000berent regimes of FM\ndynamics, namely, the single domain FM and indepen-\ndent local moments regimes. In the \frst regime we as-\nsume the rotation of the FM as the only degree of free-\ndom, while the second regime takes into account only\nthe local spin-\rip mechanism and ignores the rotation ofthe FM. When the magnetization (precession axis) is in-\nplane and normal to the transport direction, similar to\nthe conventional SOT for classical FMs, we show that the\nbias voltage can change the damping rate of the FM and\nfor large enough voltage it can lead to a sign reversal. In\nthe case of independent spin-1 =2 local moments we show\nthat the bias-induced damping rate of the local quantum\nmoments can lead to demagnetization of the FM and has\nstrong spatial dependence. Finally, in both regimes we\nhave calculated the bias-induced damping e\u000eciency as a\nfunction of the position of the Fermi energy of the 2D\nRashba plane.\nAppendix A: Derivation of Electronic Density\nMatrix\nUsing the Heisenberg equation of motion for the an-\ngular momentum operator, ~S(t), and the commutation\nrelations for the angular momentum, we obtain the fol-\nlowing Landau-Lifshitz equations of motion,\n\u0007i@\n@tS\u0006(t) =hzS\u0006(t)\u0000h\u0006(t)Sz(t) (A1a)\n\u0000i@\n@tSz(t) =1\n2\u0000\nh+(t)S\u0000(t)\u0000h\u0000(t)S+(t)\u0001\n(A1b)\n~hmm0(t) =1\n~X\nrJsd~smm0\nc(r) +g\u0016B\u000emm0~B(t);(A1c)\nwhere,S\u0006=Sx\u0006Sy(\u001b\u0006=\u001bx\u0006\u001by), is the angu-\nlar momentum (spin) ladder operators, ~smm0\nc(r) =\n1\n2P\n\u001b\u001b0~ \u001b\u001b\u001b0\u001amm0\n\u001b\u001b0;rris the local spin density of the con-\nduction electrons which is an operator in magnetic con-\n\fguration space. Here, \u001ais the density matrix of the\nsystem, and the subscripts, r;m;\u001b refer to the atomic\ncite index, magnetic state and spin of the conduction\nelectrons, respectively. In the following we assume a pre-\ncessing solution for Eq (A1)(a) with a \fxed cone angle\nand Larmor frequency !=hz. Extending the Hilbert\nspace of the electrons to include the angular momentum\ndegree of freedom we de\fne  ms0i(t) =jS;mi\n s0i(t).\nThe equation of motion for the Green function (GF) is\nthen given\n\u0010\nE\u0000i\u0011\u0000^H(k) +n!\u0000n\n2SJsd(k)\u001bz\u0011\n^Gr\nnm(E;k) (A2)\n\u0000p\nS(S+ 1)\u0000n(n+ 1)\n2SJsd(k)\u001b\u0000^Gr\nn+1m(E;k)\n\u0000p\nS(S+ 1)\u0000n(n\u00001)\n2SJsd(k)\u001b+^Gr\nn\u00001m(E;k) =^1\u000enm\nwhere,n= (\u0000S;\u0000S+1;:::;S ) and the gauge transforma-\ntion n\u001bi(t)!ein!t n\u001bi(t) has been employed to remove\nthe time dependence. The density matrix of the system\nis of the form\n^\u001anm=e\u0000i(n\u0000m)!tSX\np=\u0000SZdE\n2\u0019^Gr\nnp2\u0011fp^\u0016^Ga\npm (A3)9\nwhere,fp^\u0016(E) =f(E\u0000p!\u0000^\u0016) is the equilibrium Fermi\ndistribution function of the electrons. Due to the fact\nthatp!are the eigenvalues of HM=\u0000g\u0016B~B\u0001S, one\ncan generalize this expression by transforming into a ba-\nsis where the magnetic energy is not diagonal which in\nturn leads to Eq (5) for the density matrix of the con-\nduction electron-local moment entagled system.\nAppendix B: Recursive Relation for GFs\nSince in this work we are interested in diagonal blocks\nof the GFs and in general for FMs at low temperaturewe haveS\u001d1, we need a recursive algorithm to be able\nto solve the system numerically. The surface Keldysh\nGFs corresponding to ascending ^ gu;r=<, and descending\n^gd;r=<, recursion scheme read,\n^gu;r\nn(E;k) =1\nE\u0000!n\u0000i\u0011n\u0000^H(k)\u0000^\u0006rn(E;k)\u0000n\n2SJsd(k)\u001bz\u0000(S\u0000\nn)2\n4S2Jsd(k)\u001b+^gu;r\nn\u00001(E;k)\u001b\u0000Jsd(k)(B1)\n^\u0006u;<\nn(E;k) =\u0000X\n\u000b\u0010\n2i\u0011n+^\u0006r\nn;\u000b(E;k)\u0000^\u0006a\nn;\u000b(E;k)\u0011\nfn\u000b+(S\u0000\nn)2\n4S2Jsd\u001b+^gu;r\nn\u00001^\u0006u;<\nn\u00001^gu;a\nn\u00001\u001b\u0000Jsd (B2)\n^gd;r\nn(E;k) =1\nE\u0000!n\u0000i\u0011n\u0000^\u0006rn(E;k)\u0000^H(k)\u0000n\n2SJsd(k)\u001bz\u0000(S+\nn)2\n4S2Jsd(k)\u001b\u0000^gu;r\nn+1(E;k)\u001b+Jsd(k)(B3)\n^\u0006d;<\nn(E;k) =\u0000X\n\u000b\u0010\n2i\u0011n+^\u0006r\nn;\u000b(E;k)\u0000^\u0006a\nn;\u000b(E;k)\u0011\nfn\u000b+(S+\nn)2\n4S2Jsd\u001b\u0000^gd;r\nn+1^\u0006d;<\nn+1^gd;a\nn+1\u001b+Jsd (B4)\nwhere, ^\u0006r\nn(E;k) =P\n\u000b^\u0006r\n\u000b(E\u0000!n;k) corresponds to the\nself energy of the leads, \u000b=L;R refers to the left and\nright leads in the two terminal device in Fig. 3 and S\u0006\nm=p\nS(S+ 1)\u0000m(m\u00061). Using the surface GFs we can\ncalculate the GFs as follows,\n^Gr\nn;m(E;k) =1\nE\u0000!n\u0000i\u0011n\u0000^H(k)\u0000^\u0006rn\u0000n\n2SJsd(k)\u001bz\u0000^\u0006r;u\nn\u0000^\u0006r;d\nn; n =m (B5)\n=S+\nn\n2S^gu;r\nn(E;k)Jsd(k)\u001b\u0000^Gr\nn+1;m(E;k); n6=m (B6)\n=S\u0000\nn\n2S^gd;r\nn(E;k)Jsd(k)\u001b+^Gr\nn\u00001;m(E;k); n6=m (B7)\nwhere the ascending and descending self energies are given by,\n^\u0006r;u\nn=(S\u0000\nn)2\n4S2Jsd(k)\u001b+^gu;r\nn\u00001(E;k)\u001b\u0000Jsd(k) (B8)\n^\u0006r;d\nn=(S+\nn)2\n4S2Jsd(k)\u001b\u0000^gd;r\nn+1(E;k)\u001b+Jsd(k) (B9)\nThe average rate of angular momentum loss/gain can be obtained from the real part of the loss of angular momentum\nin one period of precession,\nT0\nn=1\n2(T0\u0000\nn\u0000T0+\nn) =1\n2= X\nkTr[S\u0000\nn\n2S\u001b+Jsd(k)^\u001ann+1(k)\u0000S+\nn\n2S\u001b\u0000Jsd(k)^\u001ann\u00001(k)]!\n(B10)10\nwhich can be interpreted as the current \rowing across the layer n.\nT0\u0000=+\nn =X\nkZdE\n2\u0019iTrnh\n^\u0006d=u;r\nn(E;k)\u0000^\u0006d=u;a\nn(E;k)i\n^G<\nnn(E;k) +^\u0006d=u;<\nn (E)h\n^Gr\nnn(E;k)\u0000^Ga\nnn(E;k)io\n;(B11)\nAcknowledgments\nThe work at CSUN is supported by NSF-Partnership\nin Research and Education in Materials (PREM) GrantDMR-1205734, NSF Grant No. ERC-Translational Ap-\nplications of Nanoscale Multiferroic Systems (TANMS)-\n1160504, and US Army of Defense Grant No. W911NF-\n16-1-0487.\n\u0003Electronic address: Farzad.Mahfouzi@gmail.com\nyElectronic address: nick.kioussis@csun.edu\n1J. C. Slonczewski, Current-driven excitation of magnetic\nmultilayers, J. Magn. Magn. Mater. 159, L1-L7 (1996).\n2L. Berger, Emission of spin waves by a magnetic multilayer\ntraversed by a current, Phys. Rev. B 54, 9353 (1996).\n3D. Ralph and M. Stiles, Spin transfer torques, J. Magn.\nMagn. Mater. 320, 1190 (2008); A. Brataas, A. D. Kent,\nand H. Ohno, Current-induced torques in magnetic mate-\nrials, Nature Mater. 11, 372 (2012).\n4Ioannis Theodonis, Nicholas Kioussis, Alan Kalitsov, Mair-\nbek Chshiev, and W. H. Butler, Anomalous Bias Depen-\ndence of Spin Torque in Magnetic Tunnel Junctions, Phys.\nRev. Lett. 97, 237205 (2006).\n5P. Gambardella and I. M. Miron, Current-induced spinor-\nbit torques, Phil. Trans. R. Soc. A 369, 3175 (2011).\n6I. M. Miron et al. , Nature Mater. 9, 230 (2010) I. M.\nMiron, K. Garello, G. Gaudin, Pierre-Jean Zermatten, M.\nV. Costache, S. Au\u000bret, S. Bandiera, B. Rodmacq, A.\nSchuhl, and P. Gambardella, Perpendicular switching of\na single ferromagnetic layer induced by in-plane current\ninjection, Nature 476, 189 (2011);\n7L. Liu, C. F. Pai, Y. Li, H. W. Tseng, D. C. Ralph, and\nR. A. Buhrman, Spin-torque switching with the giant spin\nHall e\u000bect of tantalum, Science 336, 555 (2012).\n8L. Liu, T. Moriyama, D. C. Ralph, and R. A. Buhrman,\nSpin-Torque Ferromagnetic Resonance Induced by the\nSpin Hall E\u000bect, Phys. Rev. Lett. 106, 036601 (2011);\nLuqiao Liu, O. J. Lee, T. J. Gudmundsen, D. C. Ralph,\nand R. A. Buhrman, Current-Induced Switching of Perpen-\ndicularly Magnetized Magnetic Layers Using Spin Torque\nfrom the Spin Hall E\u000bect, Phys. Rev. Lett. 109, 096602\n(2012).\n9M. I. Dyakonov and V. I. Perel, Current-induced spin ori-\nentation of electrons in semiconductors, Phys. Lett. 35A,\n459 (1971).\n10J. E. Hirsch, Spin Hall E\u000bect, Phys. Rev. Lett. 83, 1834\n(1999).\n11T. Jungwirth, J. Wunderlich and K. Olejn \u0013ik, Spin Hall\ne\u000bect devices, Nature Materials 11, (2012).\n12J. Sinova, S. O. Valenzuela, J. Wunderlich, C. H. Back,\nand T. Jungwirth, Spin Hall E\u000bects, Rev. Mod. Phys. 87,\n1213 (2015).\n13E. van der Bijl and R. A. Duine, Current-induced torques\nin textured Rashba ferromagnets, Phys. Rev. B 86, 094406\n(2012).\n14Kyoung-Whan Kim, Soo-Man Seo, Jisu Ryu, Kyung-JinLee, and Hyun-Woo Lee, Magnetization dynamics in-\nduced by in-plane currents in ultrathin magnetic nanos-\ntructures with Rashba spin-orbit coupling, Phys. Rev. B\n85, 180404(R) (2012).\n15H. Kurebayashi, Jairo Sinova, D. Fang, A. C. Irvine, T.\nD. Skinner, J. Wunderlich, V. Nov\u0013 ak, R. P. Campion, B.\nL. Gallagher, E. K. Vehstedt, L. P. Z^ arbo, K. V\u0013yborn\u0013y,\nA. J. Ferguson and T. Jungwirth, An antidamping spinor-\nbit torque originating from the Berry curvature, Nature\nNanotechnology 9, 211 (2014).\n16F. Mahfouzi, B. K. Nikoli\u0013 c, and N. Kioussis, Antidamping\nspin-orbit torque driven by spin-\rip re\rection mechanism\non the surface of a topological insulator: A time-dependent\nnonequilibrium Green function approach, Phys. Rev. B 93,\n115419 (2016).\n17Frank Freimuth, Stefan Bl ugel, and Yuriy Mokrousov,\nSpin-orbit torques in Co/Pt(111) and Mn/W(001) mag-\nnetic bilayers from \frst principles, Phys. Rev. B 90, 174423\n(2014).\n18J. Kim, J. Sinha, M. Hayashi, M. Yamanouchi, S. Fukami,\nT. Suzuki, S. Mitani and H. Ohno, Layer thickness de-\npendence of the current-induced e\u000bective \feld vector in\nTa|CoFeB|MgO, Nature Materials 12, 240245 (2013).\n19Xin Fan, H. Celik, J. Wu, C. Ni, Kyung-Jin Lee, V. O.\nLorenz, and J. Q. Xiao, Quantifying interface and bulk\ncontributions to spinorbit torque in magnetic bilayers,\n5:3042 | DOI: 10.1038/ncomms4042 (2014).\n20A. Chudnovskiy, Ch. Hubner, B. Baxevanis, and D.\nPfannkuche, Spin switching: From quantum to quasiclas-\nsical approach, Phys. Status Solidi B 251, No. 9, 1764\n(2014).\n21M. Fahnle and C. Illg, Electron theory of fast and ultrafast\ndissipative magnetization dynamics, J. Phys.: Condens.\nMatter 23, 493201 (2011)\n22J. Swiebodzinski, A. Chudnovskiy, T. Dunn, and A.\nKamenev, Spin torque dynamics with noise in magnetic\nnanosystems, Phys. Rev. B 82, 144404 (2010).\n23Hang Li, H. Gao, Liviu P. Zarbo, K. Vyborny, Xuhui\nWang, Ion Garate, Fatih Dogan, A. Cejchan, Jairo Sinova,\nT. Jungwirth, and Aurelien Manchon, Intraband and inter-\nband spin-orbit torques in noncentrosymmetric ferromag-\nnets Phys. Rev. B 91, 134402 (2015).\n24V. Kambersk\u0013 y, On the LandauLifshitz relaxation in fer-\nromagnetic metals, Can. J. Phys. 48, 2906 (1970); V.\nKambersk\u0013y, Spin-orbital Gilbert damping in common\nmagnetic metals Phys. Rev. B 76, 134416 (2007).\n25A. T. Costa and R. B. Muniz, Breakdown of the adiabatic11\napproach for magnetization damping in metallic ferromag-\nnets, Phys. Rev B 92, 014419 (2015).\n26D. M. Edwards, The absence of intraband scattering in\na consistent theory of Gilbert damping in pure metallic\nferromagnets, Journal of Physics: Condensed Matter, 28,\n8 (2016).\n27Paul M. Haney, Hyun-Woo Lee, Kyung-Jin Lee, Aurlien\nManchon, and M. D. Stiles, Current induced torques\nand interfacial spin-orbit coupling: Semiclassical modeling\nPhys. Rev. B 87, 174411 (2013).\n28Jin-Hong Park, Choong H. Kim, Hyun-Woo Lee, and Jung\nHoon Han, Orbital chirality and Rashba interaction in\nmagnetic bands, Phys. Rev. B 87, 041301(R) (2013).\n29Suprio Data, Quantum Transport: Atom to Transistor ,\nCambridge University Press, New York, (2005).\n30S. Zhang and Z. Li, Roles of Nonequilibrium Conduction\nElectrons on the Magnetization Dynamics of Ferromagnets\nPhys. Rev. Lett. 93, 127204 (2004).\n31Maria Stamenova, Stefano Sanvito, and Tchavdar N.\nTodorov, Current-driven magnetic rearrangements in spin-\npolarized point contacts, Phys. Rev. B 72, 134407 (2005).\n32D. E. Eastman, F. J. Himpsel, and J. A. Knapp, Exper-\nimental Exchange-Split Energy-Band Dispersions for Fe,Co, and Ni, Phys. Rev. Lett. 44, 95 (1980).\n33Anton Kokalj and Mauro Caus , Periodic density func-\ntional theory study of Pt(111): surface features of slabs of\ndi\u000berent thicknesses, J. Phys.: Condens. Matter 117463\n(1999).\n34Ioan Mihai Miron, Thomas Moore, Helga Szambolics, Lil-\niana Daniela Buda-Prejbeanu, Stphane Au\u000bret, Bernard\nRodmacq, Stefania Pizzini, Jan Vogel, Marlio Bon\fm,\nAlain Schuhl and Gilles Gaudins, Fast current-induced\ndomain-wall motion controlled by the Rashba e\u000bect, Na-\nture Materials 10, 419?423 (2011).\n35P. P. J. Haazen, E. Mure, J. H. Franken, R. Lavrijsen, H.\nJ. M. Swagten, and B. Koopmans, Domain wall depinning\ngoverned by the spin Hall e\u000bect, Nat. Mat. 12, 4, 299303\n(2013).\n36Charles Kittel, Introduction to Solid State Physics , Wiley,\n(2004).\n37F. Mahfouzi, N. Kioussis, Ferromagnetic Damping/Anti-\ndamping in a Periodic 2D Helical Surface; A Nonequi-\nlibrium Keldysh Green Function Approach, SPIN 06,\n1640009 (2016)."    },    {        "title": "1703.01586v2.On_the_VC_Dimension_of_Binary_Codes.pdf",        "content": "arXiv:1703.01586v2  [cs.IT]  27 Jun 2018On the VC-Dimension of Binary Codes∗\nSihuang Hu†Nir Weinberger‡Ofer Shayevitz‡\nAbstract\nWe investigate the maximal asymptotic rates of length- nbinary codes with\nVC-dimension at most dnand minimum distance at least δn. Two upper bounds\nare obtained, one as a simple corollary of a result by Haussle r and the other via\na shortening approach combining theSauer–Shelah lemma and the linear pro-\ngramming bound. Two lower bounds are given using Gilbert–Va rshamov type\narguments over constant-weight and Markov-type sets.\n1 Introduction\nLetC ⊆ {0,1}nbe a binary code of length nand rate R=1\nnlog2|C|. In this paper,\nwe study the relation between the rate of the code and two fundam ental properties:\nits minimum (Hamming) distance and its Vapnik–Chervonenkis (VC) dime nsion [19].\nRecall that the Hamming distance between two codewords is the num ber of positions in\nwhich they differ; the minimum distance ofC, which is the smallest Hamming distance\nbetween any pair of codewords, plays an important role in coding the ory. Recall that\nthe projection of Conto a coordinate set I⊆[n] :={1,2,...,n}, denoted C|I, is the set\nof all possible values assigned to these coordinates by the codewor ds inC. The code Cis\nsaid toshatterIifC|I={0,1}|I|. TheVC-dimension ofC, which is the maximum size\nof a coordinate set that is shattered byC, plays an important role in statistical learning\ntheory and computational geometry [1, 6, 9].\nOur goal in this paper is to analyze codes of simultaneously large minimu m distance\nand small VC-dimension. Loosely speaking, we note that fixing a rate and striving to\noptimize one of these properties is expected to essentially be the wo rst possible for the\nother property. Indeed, on the one hand, it is well known that ran dom linear codes\n∗This paper was presented in part at 2017 IEEE International Symp osium on Information Theory.\n†Lehrstuhl D f¨ ur Mathematik, RWTH Aachen, Germany (husihuang @gmail.com). This work was\ndone while S. Hu was with Department of Electrical Engineering - Syst ems, Tel Aviv University, Israel.\nResearch supported by ERC grant no. 639573 and the Alexander v on Humboldt Foundation.\n‡Department of Electrical Engineering–Systems, Tel Aviv Universit y, Tel Aviv, Israel\n(nir.wein@gmail.com, ofersha@eng.tau.ac.il). The work of N. Weinberge r was supported by ERC\ngrant no. 639573. The work of O. Shayevitz was supported by ERC grant no. 639573 and ISF grant\nno. 1367/14.\n1achieve the Gilbert-Varshamov bound [7, 20], which is the best known lower bound on\nthe rate of binary codes under a minimum distance constraint, yet c learly their VC-\ndimension is the largest possible (attained by any information set). O n the other hand,\nby the Sauer–Shelah lemma [17, 18], the VC-dimension at any given ra te is essentially\nminimized by any Hamming ball of a suitable radius, yet clearly the minimum distance\nof a Hamming ball is equal to 1, the smallest possible. These extremal observations\ndemonstrate the tension between increasing the minimum distance a nd decreasing the\nVC-dimension.\nBesides being an interesting combinatorial problem, finding codes th at have a large\nminimum distance as well as a small VC-dimension also admits the following coding-\ntheoretic motivation. Suppose that a binary code Cwith minimum distance ∆ and\nVC-dimension Dis used over an errors and erasures channel. Suppose there were e\nerasures, and we are now interested in detecting whether any err ors have fallen in the\nremaining n−ecoordinates. Let te∈ {0,1,...,n−e}be the maximal number of errors\nthat the code can guarantee to detect, and let πe∈ {0,1,...,2n−e}the maximal number\nof distinct error sequences (of length n−e) that the code can guarantee to detect. The\nerror detection threshold pertaining to each of these quantities is the maximal number of\nerasuresesuch that the respective quantity is nonzero. Ife <∆−1, then the minimum\ndistance of the projection of Conto the remaining n−ecoordinates is at least ∆ −e >1.\nHence, the code can correct at least ⌊(∆−e)/2⌋errors and thus in this case te>0.\nSimilarly, if e < n−DthenCcannot shatter the remaining n−e(> D) coordinates.\nThus, there must be error sequences that result in vectors that are not contained in the\nprojection of Conto the remaining n−ecoordinates; such error sequences can clearly be\ndetected, hence πe>0.Adopting this viewpoint, it is interesting to seek codes for which\nboth error detection thresholds are high, namely codes with a large minimum distance\nand a small VC-dimension. We are interested in the maximum size of suc h codes.\nIn what follows, we consider the asymptotic formulation of the prob lem. For any1\nd,δ∈[0,1\n2], we say that a rate Ris (d,δ)-achievable if for any Nthere exists a binary\ncodeCof length n≥N, rate at least R, VC-dimension at most ⌊dn⌋, and minimum\ndistance at least ⌈δn⌉. We are interested in characterizing C(d,δ), which we define to\nbe the supremum of all ( d,δ)-achievable rates. For brevity, we assume throughout that\ndnandδnare integers, as this does not affect the asymptotic behavior.\nIn Section 2 we derive two upper bounds for C(d,δ). The first is obtained as a\nsimple asymptotic corollary of a result by Haussler [8], and the second is derived via a\nshorteningapproachthatcombinestheSauer–Shelahlemma[17,1 8](controllingtheVC-\ndimension) and the linear programming bound [15] (controlling the minim um distance).\nIn Section 3 we present two lower bounds for C(d,δ). Both these bounds are obtained\nvia GV-type arguments (controlling the minimum distance) applied to c onstant-weight\nand Markov-type sets respectively (whose structure controls t he VC-dimension).\n1Ford≥1/2 it is easy to see that the rate Ris always equal to 1, which is not interesting. Therefore\nwe limit din the interval [0 ,1/2].\n22 Upper Bounds\nWe first briefly review upper bounds on C(d,δ) that can be easily deduced from known\nresults. To begin, one can clearly ignore either the minimal distance c onstraint or the\nVC-dimension constraint.\nWhen accounting only for the minimal distance constraint, the best known up-\nper bound is the second MRRW bound given by McEliece, Rodemich, Rumsey, and\nWelch [15] as follows:\nRLP(δ) := min\n0≤u≤1−2δ{1+g(u2)−g(u2+2δu+2δ)}\nwithg(x) :=h((1−√1−x)/2).Here and throughout this paper we define h(x) =\n−xlog2(x)−(1−x)log2(1−x) to be the binary entropy function. The following is\ndirect.\nLemma 1. C(d,δ)≤RLP(δ).\nWhen accounting only for the VC-dimension constraint, the size of a codeCwith\nVC-dimension dncan be upper bounded by the Sauer–Shelah lemma [17, 18]\n|C| ≤dn/summationdisplay\ni=0/parenleftbiggn\ni/parenrightbigg\n(1)\nand so the following is evident.\nLemma 2. C(d,δ)≤h(d).\nIn [8] Haussler directly addressed the problem of bounding the size o f codes with\nrestricted minimal distance and VC-dimension. In his setting, the VC -dimension is a\nbounded constant. However, from the results there the following bound on C(d,δ) can\nstill be deduced. For a number a≥0 we define /an}⌊ra⌋k⌉tl⌉{ta/an}⌊ra⌋k⌉tri}ht:= min(a,1\n2). For a code Cwe\ndefine the unit distance graph UD(C) whose vertex set is all codewords in Cand two\ncodewords x,yare adjacent if their Hamming distance dist( x,y) = 1.\nLemma 3 (Corollary to [8, Theorem 1]) .\nC(d,δ)≤2d\nδ+2d·h/parenleftbigg/angbracketleftbiggδ+2d\n2/angbracketrightbigg/parenrightbigg\n.\nProof.LetCbe a length- nbinary code with VC-dimension at most dnand minimum\ndistance δn. Suppose 0 ≤s≤1.We choose a random subset I⊆[n] :={1,2,...,n}of\nsizesnuniformly. For each codeword u∈ C|I, we define its weight w(u) as the number\nof codewords in Csuch that its projection on Iis equal to u. LetEbe the edge set\nof the unit distance graph UD( C|I), and define the weight of an edge e={u,v}as\nw(e) = min{w(u),w(v)}. PutW=/summationtext\ne∈Ew(e), and note that Wis a random variable\n3depending on the random choice of I. The bound follows by estimating E[W], the\nexpectation of W, in two ways. First, we claim that for any I⊂[n],\nW≤2dn|C|. (2)\nOn the other hand, we can bound E[W] from below:\nE[W]≥sn·δn\nn−sn+1/parenleftBigg\n|C|−dn/summationdisplay\ni=0/parenleftbiggsn\ni/parenrightbigg/parenrightBigg\n. (3)\n(Please refer to [14, Lemma 5.14] for the proof of (2) and (3).) Thu s we have\n/parenleftbigg\n((δ+2d)s−2d)−2d\nn/parenrightbigg\n|C| ≤sδdn/summationdisplay\ni=0/parenleftbiggsn\ni/parenrightbigg\n.\nFor any s >2d\nδ+2dand sufficient large n, we can get |C|=O(/summationtextdn\ni=0/parenleftbigsn\ni/parenrightbig\n), and hence\nC(d,δ)≤s·h(/an}⌊ra⌋k⌉tl⌉{td/s/an}⌊ra⌋k⌉tri}ht).The result follows directly.\nWe shall next combine Lemma 1 and Lemma 2 to obtain an improved uppe r bound.\nThroughout this paper, we define 0 /0 = 0.\nTheorem 1.\nC(d,δ)≤min\n0≤s≤1−2δ/braceleftbigg\ns·h/parenleftbigg/angbracketleftbiggd\ns/angbracketrightbigg/parenrightbigg\n+(1−s)RLP/parenleftbiggδ\n1−s/parenrightbigg/bracerightbigg\n.\nProof.LetCbealength- nbinarycodewithVC-dimension atmost dnandminimum dis-\ntanceδn. Choose s∈[0,1−2δ], andconsider theprojectionof Con[sn] ={1,2,...,sn}.\nOf course the VC-dimension of C|[sn]is also at most dn, and so its rate can be bounded\nby Lemma 2. For any given prefix u∈ C|[sn], we denote the set of its possible suffixes by\nZ(u)⊂ {0,1}(1−s)n, i.e., for any v∈ Z(u) there exists a codeword x∈ Csuch that xis\nthe concatenation of uandv. Clearly, Z(u) is a code of length (1 −s)nand minimal\ndistance δn, and so its rate can be bounded by the second MRRW bound. Then ou r\nresult follows from\n|C|=/summationdisplay\nu∈C|[sn]|Z(u)| ≤/vextendsingle/vextendsingle/vextendsingleC|[sn]/vextendsingle/vextendsingle/vextendsingle·max\nu∈C|[sn]|Z(u)|.\n3 Lower Bounds\nA general procedure to obtain lower bounds on C(d,δ) is the following.\n4(i) Pick some subset Sof the Hamming cube {0,1}nthat has some “nice” structure.\n(ii) Compute a generalized GV bound for subset S, namely a lower bound on the size\nof the largest code of minimum distance at least δnwhere all codewords belong to\nS.\n(iii) FindanupperboundfortheVC-dimension dnofanysubset of Sthathasminimum\ndistance at least δn.\n(iv) Combine the bounds (ii)-(iii).\nIn the following two subsections, we will show two ways to choose “nic e” subsets of\nthe Hamming cube and calculate the corresponding bounds.\n3.1 Constant Weight Codes\nHere we choose subset Sto be the collection of all codewords with some constant weight.\nLemma 4. Suppose δ∈[0,1\n2]andw∈[0,1]. LetCbe a binary code of length n,\nconstant weight wn, and minimum distance δn. Then the VC-dimension of Cis at most\n(w−δ/2)n+1.\nProof.Suppose the VC-dimension of Cisdn. Without loss of generality, we assume that\nthe firstdncoordinates are shattered. Then there exist two codewords x=x1x2···xn\nandy=y1y2···ynsuch that xi= 1 for 1 ≤i≤dnandyi= 1 for 1 ≤i≤dn−1\nandydn= 0. Hence |supp(x)∩supp(y)| ≥dn−1. On the other hand, dist( x,y) =\n2wn−2|supp(x)∩supp(y)|, which is at least δn. Therefore δn≤2wn−2|supp(x)∩\nsupp(y)| ≤2wn−2(dn−1). This proves the result.\nLetA(n,δn,wn ) denote the maximum size of length- nbinary code with constant\nweightwnand minimum distance δn. The following GV-type bound is well-known.\nLemma 5.\nA(n,δn,wn )≥/parenleftbign\nwn/parenrightbig\n/summationtextδn/2−1\ni=0/parenleftbigwn\ni/parenrightbig/parenleftbign−wn\ni/parenrightbig. (4)\nNow we are ready to state our first lower bound for C(d,δ).\nTheorem 2. Letd,δ∈[0,1\n2], and let w=d+δ\n2. Then\nC(d,δ)≥\n\nh(w)−max\n0≤x≤δ/2/bracketleftBig\nwh/parenleftBig\nx\nw/parenrightBig\n+(1−w)h/parenleftBig\nx\n1−w/parenrightBig/bracketrightBig\nifw <1\n2\n1−h(δ) otherwise .\nProof.Ifw <1\n2, plug it into (4) and take the asymptotic form, then the result follow s\ndirectly from Lemma 4. If w≥1\n2then set w=1\n2in (4)which maximizes the lower\nbound.\n53.2 Markov Type\nFor a binary codeword x=x1x2···xn∈ {0,1}n, the number of switches ofxis equal\nto|{i: 1≤i≤n−1,xi⊕xi+1= 1}|,that is the number of length-2 consecutive\nsubsequence 01 or 10. (Here⊕is the XOR operation.) Now we present another lower\nbound for C(d,δ) based on the following observation.\nFact 1. LetSbe the collection of all codewords in the Hamming cube {0,1}nthat has\nat mostdnswitches. Then the VC-dimension of Sor any subset of Sis at most dn+1.\nProof.LetIbe anydn+ 2 coordinates. Let cbe a length-( dn+ 2) vector such that\nci= 0 for odd i∈ {1,2,...,dn+2}andci= 1 for even i∈ {1,2,...,dn+2}. Then the\nnumber of switches of cisdn+1. Hence the projection of Sonto these coordinates S|I\ndoes not contain c, therefore Sdoes not shatter I. This concludes our proof.\nWe refer to an ( S,M,δn)-code as a subset of Swith size Mand minimum distance\nat leastδn. We will prove a GV-type bound for such ( S,M,δn)-codes, and thus get\na lower bound for C(d,δ). Our proof relies on a generalized GV bound provided by\nKolesnik and Krachkovsky [11], and follows the same line of reasoning a s in Sections\nIII-V of [13], where Marcus and Roth developed an improved GV boun d for constrained\nsystems based on stationary Markov chains.\nLemma 6. [11, Lemma 1] Let Sbe a subset of {0,1}n. Then there exists an (S,M,δn)-\ncode such that\nM≥|S|2\n4|BS(δn−1)|\nwhere\nBS(δn−1) :={(w,w′)∈S×S: dist(w,w′)≤δn−1}.\nIn order to compute our lower bound, we shall consider stationary Markov chains on\ngraphs. A labeled graph G= (VG,EG,LG) is a finite directedgraph with vertices VG,\nedgesEG, and a labeling LG:EG→Σ for some finite alphabet Σ. For any vertex u,\nthe set of outgoing edges from uis denoted by E+\nG(u), and the set of incoming edges\ntouisE−\nG(u). A graph Gis called irreducible if there is a path in each direction\nbetween each pair of vertices of the graph. The greatest common divisor of the lengths\nof cycles of a graph Gis called the periodof G. An irreducible graph Gwith period 1 is\ncalledprimitive . Astationary Markov chain on a finite directed graph Gis a function\nP:EG→[0,1] such that\n(i)/summationtext\ne∈EGP(e) = 1;\n(ii)/summationtext\ne∈E+\nG(u)P(e) =/summationtext\ne∈E−\nG(u)P(e) for every u∈VG.\n6Evidently, P(e) represents the probability that the chain will make a transition alon g\nthe edge e.We denote by M(G) the set of all stationary Markov chains on G. For a\nstationary Markov chain P∈ M(G), we introduce two dummy random variables X,Y\nsuch that their joint distribution is defined by\nPr{X=u,Y=v}=/braceleftBigg\nP((u,v)) if (u,v)∈EG\n0 otherwise.\nThen the condition (ii) amounts to saying that the marginal distribut ions ofXandY\nare equal.\nFor a stationary Markov chain P∈ M(G) and a function f:EG→Rk, we denote\nbyEP(f) the expected value of fwith respect to P, that is,\nEP(f) :=/summationdisplay\ne∈EGP(e)f(e).\nFix a vertex u, and let Γ n(G) denote the set of all cycles in Gof length nthat start and\nend atu. For a cycle γ=e1e2...en∈Γn(G), letPγdenote the stationary Markov chain\ndefined by\nPγ(e) :=1\nn|{i∈ {1,2,...,n}:ei=e}|.\nWe refer to Pγas theempirical distribution of the cycle γ, and to\nEPγ(f) =/summationdisplay\ne∈EGPγ(e)f(e)\nas theempirical average offon the cycle γ. (Note that the empirical distribution Pγis\ncloselyrelatedtotheso-called“second-ordertype”ofsequence LG(e1)LG(e2)···LG(en).)\nFor a subset U⊂Rk, letM(G;f,U) denote the set of all stationary Markov chains P\nonGsuch that EP(f)∈U, and let\nΓn(G;f,U) :={γ∈Γn(G) :EPγ(f)∈U}.\nThe following lemma is a consequence of well-known results on second- order types of\nMarkov chains, cf. Boza [2], Davisson, Longo, Sgarro [5], Nataraj an [16], Csisz´ ar, Cover,\nChoi [4], and Csisz´ ar [3]. (Throughout this paper, the base of the lo garithm is |Σ|.)\nLemma 7. [13, Lemma 2] Let Gbe a primitive graph and f:EG→Rkbe a function\non the edges of G. LetUbe an open and nonempty subset of Rk. Then\nlim\nn→∞1\nnlog|Γn(G;f,U)|= sup\nP∈M(G;f,U)HP(Y|X).\nHereafter we will consider the labeled graph Gover alphabet Σ = {0,1}depicted in\nFigure 1. The labeling LGis defined by LG((a,b)) =LG((b,b)) = 0 and LG((b,a)) =\nLG((a,a)) = 1. On the other hand, the function f:EG→Ris defined by f((a,a)) =\nf((b,b)) = 0 and f((a,b)) =f((b,a)) = 1. Then we can verify the following.\n7a b0\n11 0\nFigure 1: labeled graph Gover Σ = {0,1}\nFact 2. For a cycle γ=e1e2···en∈Γn(G), the value nEPγ(f)−f(e1)is equal to the\nnumber of switches of the corresponding binary sequence LG(e1)LG(e2)···LG(en).\nNow we come to our second lower bound for C(d,δ). We will consider the subset\nSn(d) =Sn([0,d]) :={LG(e1)LG(e2)···LG(en) :e1e2···en∈Γn(G;f,[0,d])}.\nBy definition, for any x∈Sn(d) its number of switches is at most dn.\nIn order to use Lemma 6, we introduce the graph G×Gwhose vertex set is VG×G=\nVG×VG={/an}⌊ra⌋k⌉tl⌉{tu,u′/an}⌊ra⌋k⌉tri}ht:u,u′∈VG}and edge set is EG×G=EG×EG={/an}⌊ra⌋k⌉tl⌉{te,e′/an}⌊ra⌋k⌉tri}ht:e,e′∈EG}.\nGiven the function fdefined on the edges of G, we define two functions f(1)andf(2)on\nEG×Gby\nf(1)(/an}⌊ra⌋k⌉tl⌉{te,e′/an}⌊ra⌋k⌉tri}ht) =f(e), f(2)(/an}⌊ra⌋k⌉tl⌉{te,e′/an}⌊ra⌋k⌉tri}ht) =f(e′)\nand a function ∆ : EG×G→Rby\n∆(/an}⌊ra⌋k⌉tl⌉{te,e′/an}⌊ra⌋k⌉tri}ht) =/braceleftBigg\n1 ifLG(e)/n⌉}ationslash=LG(e′)\n0 otherwise.\nNote that the function ∆ is used to count the Hamming distance betw een two binary\nsequences. We collect f(1),f(2)and ∆ to define a function ϕ:EG×G→R3byϕ=\n[f(1),f(2),∆]. For a subset U⊂[0,1] we set\nF(U) := sup\nP∈M(G;f,U)HP(Y|X),\nG(U,δ) := sup\nQ∈M(G×G;ϕ,U×U×[0,δ))HQ(Y|X).\nIn particular, we use F(p) andG(p,δ) as short-hand notations for F({p}) andG({p},δ)\nrespectively, where 0 ≤p≤1. Set\nRMA(d,δ) := sup\np∈[0,d]{2F(p)−G(p,δ)}\n= sup\np∈[0,d]/braceleftBig\n2 sup\nP∈M(G):\nEP(f)=pHP(Y|X)−sup\nQ∈M(G×G):\nEQ(f(i))=p,i=1,2\nEQ(∆)∈[0,δ)HQ(Y|X)/bracerightBig\n.\n8Lemma 8. There exist (Sn(d),M,δn)-codes satisfying\nlogM\nn≥RMA(d,δ)−o(1).\nProof.Forp∈[0,d] andε >0, letUp,ε= (p−ε,p+ε),\nSn(Up,ε) :={LG(e1)LG(e2)···LG(en) :e1e2···en∈Γn(G;f,Up,ε)},\nand\nBSn(Up,ε)(δn−1) :={(w,w′)∈Sn(Up,ε)×Sn(Up,ε) : dist(w,w′)≤δn−1}.\nBy Lemma 7,\nlim\nn→∞1\nnlog|Sn(Up,ε)|= lim\nn→∞1\nnlog|Γn(G;f,Up,ε)|=F(Up,ε),\nand\nlim\nn→∞1\nnlog|BSn(Up,ε)(δn−1)|= lim\nn→∞1\nnlog|Γn(G×G;ϕ;Up,ε×Up,ε×[0,δ))|=G(Up,ε,δ).\nNote that both HP(Y|X) andEP(f) are continuous in P.So if we let ε→0, then by\nLemma 6 there exist ( Sn(d),M,δn)-codes satisfying\nlogM\nn≥2F(p)−G(p,δ)−o(1).\nThen our result follows.\nTheorem 3. C(d,δ)≥RMA(d,δ).\nProof.This follows from Lemma 8 and the fact that any ( Sn(d),M,δn) code has VC-\ndimension at most dn+1.\nUsing convex duality we can compute RMA(d,δ) through an unconstrained optimiza-\ntion problem with convex objective function as follows. For a functio nf:EG→Rk, let\nAG;f(x),x∈Rk, be the matrix function indexed by the states of Gwith entries\n[AG;f(x)]u,v=/braceleftBigg\n2−x·f((u,v))if (u,v)∈EG\n0 otherwise ,\nand letλG;f(x) denote the spectral radius of AG;f(x).(Here the ·operator in the\nexponent is the inner product of two vectors.) Recall the definitions of f,f(1),f(2),∆,ϕ,\nand define ϕ′= [f(1)+f(2),∆] :EG×G→R2.LetGbe the graph of Figure 1. Then\nAG;f(x) =/bracketleftbigga b\na1 2−x\nb2−x1/bracketrightbigg\n9and\nAG×G;ϕ′(x,z) =\n/angbracketlefta,a/angbracketright /angbracketlefta,b/angbracketright /angbracketleft b,a/angbracketright /angbracketleftb,b/angbracketright\n/angbracketlefta,a/angbracketright1 2−x−z2−x−z2−2x\n/angbracketlefta,b/angbracketright2−x2−z2−2x−z2−x\n/angbracketleftb,a/angbracketright2−x2−2x−z2−z2−x\n/angbracketleftb,b/angbracketright2−2x2−x−z2−x−z1\n.\nThrough direct computations, we have λG;f(x) = 2−x+1,and\nλG×G;ϕ′(x,z) =1\n2/parenleftBig\n(4−x+1)(2−z+1)+\n/radicalbig\n(4−x+1)24−z−2(16−x−6·4−x+1)2−z+(4−x+1)2/parenrightBig\n.\nFrom the well-known results in convex duality principle, we can obtain t he following.\nSimilar results are also obtained in [10, 12].\nLemma 9. [13, Lemma 5] Let Gbe a graph and let f:EG→Rk,g:EG→Rlbe\nfunctions on the edges of G. Setφ= [f,g] :EG→Rk+l. Then for any r∈Rkand\ns∈Rl,\nsup\nP∈M(G):\nEP(f)=r\nEP(g)≤sHP(Y|X) = inf\nx∈Rk\nz∈Rl\n≥0{x·r+z·s+logλG;φ(x,z)}.\nTheorem 4.\nRMA(d,δ) = sup\np∈[0,d]/braceleftBig\n2h(p)−inf\nx∈R\nz∈R≥0{2px+δz+logλG×G;ϕ′(x,z)}/bracerightBig\n.\nProof.Applying Lemma 9 to compute F(p), we have\nF(p) = sup\nP∈M(G):\nEP(f)=pHP(Y|X)\n= inf\nx∈R{px+logλG;f(x)}\n= inf\nx∈R{px+log(2−x+1)}\n=h(p).\n10Similarly, we have\nG(p,δ) = sup\nQ∈M(G×G):\nEQ(f(i))=p,i=1,2\nEQ(∆)∈[0,δ)HQ(Y|X)\n= inf\nx,y∈R\nz∈R≥0{px+py+δz+logλG×G;ϕ(x,y,z)}\n≤inf\nx∈R\nz∈R≥0{2px+δz+logλG×G;ϕ(x,x,z)}\n= inf\nx∈R\nz∈R≥0{2px+δz+logλG×G;ϕ′(x,z)}.\nOn the other hand, for ε >0, choose some point ( x′,y′,z′) such that\npx′+py′+δz′+logλG×G;ϕ(x′,y′,z′)≤inf\nx,y∈R\nz∈R≥0{px+py+δz+logλG×G;ϕ(x,y,z)}+ε,\nand let ¯x= (x′+y′)/2. Note that λG×G;ϕ(x,y,z) =λG×G;ϕ(y,x,z) and the function\nlogλG×G;ϕ(x,y,z) is convex (see [13, Remark 2]). Thus\npx′+py′+δz′+logλG×G;ϕ(x′,y′,z′)\n= 2p¯x+δz′+logλG×G;ϕ(x′,y′,z′)\n≥2p¯x+δz′+logλG×G;ϕ(¯x,¯x,z′),\nandG(p,δ) = inf x∈R\nz∈R≥0{2px+δz+logλG×G;ϕ′(x,z)}. This concludes our proof.\n4 Examples\nExample 1. We plot the bounds for d=1\n4and1\n16in Fig. 2. Note that all these bounds\nintersect at R=h(d) whenδ= 0; and our shortening upper bound (Thm. 1) is always\nbetter than the second MRRW bound (hence we do not plot it here). As we can see,\nford=1\n4our shortening upper bound (Thm. 1) is always better than Haussle r’s upper\nbound (Lem. 3), and the constant weight lower bound (Thm. 2) is alw ays better than\nthe Markov type lower bound (Thm. 3). For d=1\n16, the performance of these bounds\nare quite different.\nExample 2. We plot the bounds for δ=1\n4and1\n16in Fig. 3.\nRemark 1. Similarly asin [13], we canslightly improve the lower bounds by considering\nsubsetsTof our chosen set S. For example, when d= 1/16 andδ= 0.1927, both\nTheorem 2 and Theorem 3 give that C(d,δ)≥0.046. On the other hand, let Tbe the\ncollection ofallcodewords intheHamming cube {0.1}nthathasweight 0 .5nandat most\n1/16nswitches, thenthegeneralizedGVboundforsubset Tshowsthat C(d,δ)≥0.0461.\n110 0.1 0.2 0.3 0.4 0.500.10.20.30.40.50.60.70.80.91\nLem. 3 (UB)\nThm. 1 (UB)\nThm. 2 (LB)\nThm. 3 (LB)\n0 0.1 0.2 0.3 0.4 0.500.050.10.150.20.250.30.350.40.450.5\nLem. 3 (UB)\nThm. 1 (UB)\nThm. 2 (LB)\nThm. 3 (LB)\nFigure 2: Bounds for d=1\n4andd=1\n16\n0 0.1 0.2 0.3 0.4 0.5\nd00.10.20.30.40.50.60.70.80.91\nLem. 3 (UB)\nThm. 1 (UB)\nThm. 2 (LB)\nThm. 3 (LB)\n0 0.1 0.2 0.3 0.4 0.5\nd00.10.20.30.40.50.60.70.80.91\nLem. 3 (UB)\nThm. 1 (UB)\nThm. 2 (LB)\nThm. 3 (LB)\nFigure 3: Bounds for δ=1\n4andδ=1\n16\n5 Discussion\nIn this paper, we have studied the maximal size of a binary code with a given minimum\ndistance and a given VC dimension. We gave two lower bounds, based o n the idea of\nrandom GV-type constructions inside structured sets (Hamming b alls, Markov types)\nin a way that simultaneously controls the minimum distance and the VC d imension. It\nmay be interesting to consider other structured sets in order to im prove the bounds, or\nto come up with a different method of construction.\nOur weakest point is arguably the upper bound, which unlike the lower bounds, was\nderived by treating the problem of minimum distance and VC dimension s eparately. It\nstands to reason that a different argument that simultaneously co ntrols both quantities\ncould improve our bound. However, so far we have been unable to co me up with such an\n12argument. One reasonable line of attack could be to take the VC dime nsion constraint\ninto consideration as part of an LP-type argument. However, the VC dimension con-\nstraint is global, and our attempts to embed it in the more local LP-ty pe approach have\nnot been fruitful. Another direction to consider is a blow-up argume nt: Given a code\nwith minimum distance δ, we blow-up the code to include parts of the Hamming balls of\nradiusδ/2 around each codeword. If this can be done in a controlled way such that the\nincrease in the VC dimension can be accounted for, then the Sauer– Shelah lemma can\nbe applied to the blown-up code. This currently appears to be difficult . Lastly, it would\nbe interesting to see if a suitable shifting argument that somehow ke eps the minimum\ndistance in check can be used, to yield a bound in the spirit of the Saue r–Shelah lemma.\nAcknowledgement\nWe would like to thank Ronny Roth for his helpful comments on Remark 1.\nReferences\n[1]A. Blumer, A. Ehrenfeucht, D. Haussler, and M. K. Warmuth ,Learn-\nability and the Vapnik-Chervonenkis dimension , J. Assoc. Comput. Mach., 36\n(1989), pp. 929–965, https://doi.org/10.1145/76359.76371 .\n[2]L. B. Boza ,Asymptotically optimal tests for finite Markov chains , Ann. Math.\nStatist., 42 (1971), pp. 1992–2007.\n[3]I. Csisz´ar,The method of types , IEEE Trans. Inform. Theory, 44 (1998), pp. 2505–\n2523,https://doi.org/10.1109/18.720546 .\n[4]I. Csisz ´ar, T. M. Cover, and B. S. Choi ,Conditional limit theorems un-\nder Markov conditioning , IEEE Trans. Inform. Theory, 33 (1987), pp. 788–801,\nhttps://doi.org/10.1109/TIT.1987.1057385 .\n[5]L. D. Davisson, G. Longo, and A. Sgarro ,The error exponent for the noise-\nless encoding of finite ergodic Markov sources , IEEE Trans. Inform. Theory, 27\n(1981), pp. 431–438, https://doi.org/10.1109/TIT.1981.1056377 .\n[6]R. M. Dudley ,Central limit theorems for empirical measures , Ann. Probab., 6\n(1978), pp. 899–929.\n[7]E. Gilbert ,A comparison of signalling alphabets , Bell\nSystem Technical Journal, The, 31 (1952), pp. 504–522,\nhttps://doi.org/10.1002/j.1538-7305.1952.tb01393.x .\n13[8]D. Haussler ,Sphere packing numbers for subsets of the boolean n-cube wit h\nbounded Vapnik-Chervonenkis dimension , Journal of Combinatorial Theory, Series\nA, 69 (1995), pp. 217–232.\n[9]D. Haussler and E. Welzl ,ǫ-nets and simplex range queries , Discrete Comput.\nGeom., 2 (1987), pp. 127–151, https://doi.org/10.1007/BF02187876 .\n[10]J. Justesen and T. Høholdt ,Maxentropic Markov chains , IEEE Trans. Inform.\nTheory, 30 (1984), pp. 665–667, https://doi.org/10.1109/TIT.1984.1056939 .\n[11]V. D. Kolesnik and V. Y. Krachkovsky ,Generating functions and lower\nbounds on rates for limited error-correcting codes , IEEE Trans. Inform. Theory, 37\n(1991), pp. 778–788, https://doi.org/10.1109/18.79947 .\n[12]B. Marcus and S. Tuncel ,Entropy at a weight-per-symbol and em-\nbeddings of Markov chains , Invent. Math., 102 (1990), pp. 235–266,\nhttps://doi.org/10.1007/BF01233428 .\n[13]B. H. Marcus and R. M. Roth ,Improved Gilbert-Varshamov bound for\nconstrained systems , IEEE Trans. Inform. Theory, 38 (1992), pp. 1213–1221,\nhttps://doi.org/10.1109/18.144702 .\n[14]J. Matouek ,Geometric discrepancy , vol. 18 of Algorithms and Combinatorics,\nSpringer-Verlag, Berlin, 2010, https://doi.org/10.1007/978-3-642-03942-3 .\nAn illustrated guide, Revised paperback reprint of the 1999 original.\n[15]R. J. McEliece, E. R. Rodemich, H. Rumsey, Jr., and L. R. Welch ,New\nupper bounds on the rate of a code via the Delsarte-MacWillia ms inequalities , IEEE\nTrans. Information Theory, 23 (1977), pp. 157–166.\n[16]S. Natarajan ,Large deviations, hypotheses testing, and source coding fo r fi-\nnite Markov chains , IEEE Trans. Inform. Theory, 31 (1985), pp. 360–365,\nhttps://doi.org/10.1109/TIT.1985.1057036 .\n[17]N. Sauer ,On the density of families of sets , J. Combinatorial Theory Ser. A, 13\n(1972), pp. 145–147.\n[18]S. Shelah ,A combinatorial problem; stability and order for models and theories\nin infinitary languages , Pacific J. Math., 41 (1972), pp. 247–261.\n[19]V. N. Vapnik and A. J. ˇCervonenkis ,The uniform convergence of frequencies\nof the appearance of events to their probabilities , Teor. Verojatnost. i Primenen., 16\n(1971), pp. 264–279.\n[20]R. R. Var ˇsamov,The evaluation of signals in codes with correction of errors ,\nDokl. Akad. Nauk SSSR (N.S), 117 (1957), pp. 739–741.\n14"    },    {        "title": "1703.01879v5.Damping_dependence_of_spin_torque_effects_in_thermally_assisted_magnetization_reversal.pdf",        "content": "IEEE TRANSACTIONS ON MAGNETICS, VOL. 53, NO. 10, OCTOBER 2017  1 \nDamping dependence of spin-torque effects in thermally assisted \nmagnetization reversal   \nY.P. Kalmykov,1 D. Byrne,2 W.T. Coffey,3 W. J. Dowling,3 S.V.Titov,4 and J.E. Wegrowe5 \n1Univ. Perpignan Via Domitia, Laboratoire de Mathématiques et Physique, F -66860,  Perpignan, France  \n2School of Physics, University College Dublin, Belfield, Dublin 4, Ireland  \n3Department of Electronic and Electrical Engineering, Trinity College, Dublin 2, Ireland  \n4Kotel’nikov Institute of Radio Engineering and Electronics of the Russia n Academy of Sciences, Vvedenskii Square 1, \nFryazino, Moscow Region, 141120, Russia  \n5Laboratoire des Solides Irradiés, Ecole Polytechnique, 91128 Palaiseau Cedex, France  \nThermal fluctuations of nanomagnets driven by spin -polarized currents are treated via the Landau -Lifshitz -Gilbert equation as \ngeneralized to include both the random thermal noise field and Slonczewski spin -transfer torque (STT) term s. The magnetization \nreversal time of such a nanomagnet is then  evaluated for wide ranges of damping by using a method which  generalizes the solution of \nthe so -called Kramers turnover problem for mechanical Brownian particles  thereby  bridging the very low damping (VLD) and \nintermediate damping (ID) Kramers escape rates , to the analogous magnetic turnover problem. The reversal time is then evaluated for \na nanomagnet with the free energy density given in the standard form of superimposed easy -plane and in -plane easy -axis anisotropies \nwith the dc bias field along the easy axis.  \n \nIndex Terms — Escape rate, Nanomagnets, Reversal time of the magnetization, Spin -transfer t orque .  \n \nI. INTRODUCTION  \nue to the  spin-transfer torque (STT) effect [1 -6], the \nmagnetization of a nanoscale ferromagnet may be altered \nby spin -polarized currents . This phenomenon occurs because \nan electri c current with spin polarization in a ferromagnet has \nan associated flow of angular momentum [3,7] ther eby \nexerting a macroscopic spin torque.  The phenomenon  is the \norigin of the novel  subject of spintronics  [7,8], i.e., current -\ninduced control over magnet ic nanostructures . Common \napplications are very high-speed  current -induced \nmagnetization switching by  (a) reversing the orien tation of \nmagnetic bits [3,9 ] and (b) using spin polarized currents  to \ncontrol steady state microwave oscillations [9 ]. This is \naccomplished via the steady state magnetization precession \ndue to STT representing the conversion of DC input into an \nAC output voltage [3]. Unfortunately , thermal fluctuations \ncannot now be ignored due to the nanometric  size of STT \ndevices, e.g., leading to mainly noise -induced switching at \ncurrents far less than the critical switching current without \nnoise [10] as corroborated by experiments (e.g., [11]) \ndemonstrating that STT near room temperature significantly \nalters thermally activated switching processes . These now \nexhibit a pronounced dependence on both material and \ngeometrical parameters. Consequently, an accurate account of \nSTT switching effects at finite temperatures is necessary in \norder to achieve further improvements in the design and \ninterpretatio n of experiments, in view of the manifold practical applications in spintronics, random access memory \ntechnology, and so on.  \nDuring the last decade, various analytical and numerical \napproaches to the study of STT effects in the thermally \nassisted  magnetiza tion reversal (or switching) time in \nnanoscale ferromagnets  have been developed  [6,7,12 -26]. \nTheir objective being to generalize  methods originally \ndeveloped for zero STT  [12,27 -32] such as  stochastic \ndynamics simulations (e. g., Refs. [21 -25]) and  extensio ns to \nspin Hamiltonians  of the mean first passage time (MFPT) \nmethod (e.g., Refs. [16] and [17] ) in the Kramers escape rate \ntheory  [33,34]. However, unlike zero STT substantial progress \nin escape rate theory including STT effects has so far been \nachieved o nly in  the limit  of very low damping (VLD), \ncorresponding to vanishingly small values of the damping \nparameter \n  in the Landau -Lifshitz -Gilbert -Slonczewski \nequation  (see Eq. (5) below). Here  the pronounced time \nseparation between fast precessional and slow energy changes \nin lightly  damped  closed phase space trajectories (ca lled \nStoner -Wohlfarth orbits) has been exploited in Refs. \n[7,14, 16,17]  to formulate a one -dimensional Fokker -Planck \nequation for the energy distribution function  which may be \nsolved by quadratures. This equation is  essentially similar to \nthat derived by Kramers [ 33] in treating the VLD noise -\nactivated escape rate of a point Brownian particle from a \npotential well although the Hamiltonian of the magnetic \nproblem is no longer separable and additive and the barrier \nheight is now STT depend ent. The Stoner -Wohlfarth orbits \nand steady precession along such an orbit of constan t energy \noccur if the spin -torque is strong enough to cancel out the \ndissipative torque. The origin of the orbits arises from the \nbistable (or, indeed, in general multistable) structure of the \nanisotropy potential. This structure allows one to define a \nnonconservative “effective” potential with damping - and D \nManuscript received April 6, 2017; revised June 27, 2017; accepted July \n24, 2017. Date of publication July 27, 2017; date of current ver -sion \nSeptember 18, 2017. Correspondin g author: Y. P. Kalmykov (e -mail: \nkalmykov@univ -perp.fr).  \nColor versions of one or more of the figures in this paper are available \nonline at http://ieeexplore.ieee.org . \nDigital Object Identifier: 10.1109/TMAG.2017. 2732944  IEEE TRANSACTIONS ON MAGNETICS, VOL. 53, NO. 10, OCTOBER 2017  \n 2 \ncurrent -dependent potential barrier s between stationary self -\noscillatory states of the magnetization, thereby permitting one  \nto estimate the reversal (switching) time between these states . \nThe magnetizat ion reversal time in the VLD  limit  is then \nevaluated  [16,17,35 ] both for zero and nonzero STT. In \nparticular, for nonzero STT , the VLD reversal time has been \nevaluated analytically in Refs. [16,17 ]. Here it has been shown \nthat in the high barrier limit, an asymptotic equation for the \nVLD magnetization reversal time from a single  well in the \npresence of the STT is given by  \n \nVLD\nTST1\nCES . (1) \nIn Eq. (1), \n is the damping parameter  arising from the \nsurroundings , \nTST\nAE\nEfe  is the escape rate render ed by \ntransition state theory (TST) which ignores effects due to the \nloss of spins at the barrier [34], \nAEf  is the well precession \nfrequency, \nE  is the damping and spin -polarized -current \ndependent effective en ergy barrier, and \nCES  is the \ndimensionless action  at the saddle point C (the action is given \nby Eq. (13) below).  \nThe most essential fea ture of the results obtained in Refs. \n[16,17,35 ] and how they pertain to this  paper is that they apply \nat VLD only where  the inequality \n1\nCES  holds meaning \nthat the energy loss per cycle of the almost periodic motion at \nthe critical en ergy is much less than the thermal energy . \nUnfortunately  for typical values of the material parameters \nCES\n may be very high (\n310 ), meaning that this inequality  \ncan be fulfilled only for \n0.001 . In addition, both \nexperimental and theoretical estimates suggest  higher  values \nof  of the order of 0.001 -0.1 ( see, e.g., Refs. [6,36 -38]), \nimplying that the VLD asymptotic results are no longer valid \nas they will now differ substantially from the true  value of the \nreversal time . These considerations suggest that the \nasymptotic calculations for STT should be extended to include \nboth the VLD and intermediate damping (ID) regions. This is \nour primar y objective here . Now like point Brownian particles  \nwhich  are governed by a separable and additive Hamiltonian , \nin the escape rate problem as it pertains to magnetic moments \nof nanoparticles, three regimes of damping appear [ 12,33,34]. \nThese are  (i) very low damping \n( 1)\nCES , (ii) intermediate -\nto-high damping (IHD)  \n( 1)\nCES , and  (iii) a more or less \ncritically damped turnover regime \n( ~ 1)\nCES . Also , Kramers \n[33] obtained his now -famous VLD and IHD escape rate \nformulas for point Brownian particles by assuming in  both \ncases that the energy barrier is much greater than the thermal \nenergy so that the concept of an escape rate applies. He \nmentioned, however, that he could not find a general method \nof attack in order to obtain an escape rate formula valid for \nany damp ing regime. This problem, namely the Kramers \nturnover, was initially solved by Mel’nikov and Meshkov \n[39]. They obtained an escape rate that is valid for all values \nof the damping by a semi heuristic argument, thus constituting a solution of the Kramers tu rnover problem for point particles. \nLater, Grabert [40] and Pollak et al . [41] have presented  by \nusing a coupled oscillator model of the thermal bath , a \ncomplete solution of the Kramers turnover problem and have \nshown that the turnover escape rate formula can be obtained \nwithout  the ad hoc  interp olation between the VLD  and IHD \nregimes  as used by Mel’nikov and Meshkov . Finally, Coffey \net al. [42,43 ] have shown  for classical spins  that at zero STT , \nthe magnetization reversal  time for values of damping up to \nintermediate values, \n1,  can also be evaluated via the \nturnover formula for the escape rate bridging the  VLD  and ID  \nescape rates, namely,  \n \nTST1\n()\nCE AS , (2) \nwhere \n()Az  is the so-called depopulation factor, namely  [39-\n42] \n \n 2\n2\n0ln 1 exp[ ( 1/4)]1\n1/4()z\nd\nA z e\n   \n\n . (3) \nNow the ID reversal time (or the lower bound of the reversal \ntime) may always be evaluated via TST as [32,34]  \n \nID\nTST1 . (4) \nTherefore b ecause \n()\nCCEE A S S  is the energy loss per \ncycle at the critical energy  \n0\nCES  [39] (i.e. , in the VLD \nlimit) , Eq. (2) transparently reduces to the VLD Kramers \nresult, Eq.  (1). Moreover  in the ID range, where \n( ) 1\nCE AS , \nEq. (2) reduces to the TST Eq.  (4). Nevertheless in the high \nbarrier limit \n1,\nCES  \n given by Eq. (2) can substantially  \ndeviate in the damping range \n0.001 1  both from \nID , \nEq. (4), and \nVLD , Eq. (1). Now, the approach of Coffey et al. \n[42,43 ] generalizing the Kramers turnover results to classical \nspins (nanomagnets)  was developed for zero STT, \nnevertheless, it can also be used to account for STT effects. \nHere  we shall  extend th e zero STT results of Refs. \n[14,16,17,39 -42] treating the damping dependence of STT \neffects  in the magnetization reversal of  nano scaled \nferro magnets  via escape rate theory in the most important \nrange of damping comprising  the VLD and ID ranges , \n1.   \nII. MODEL  \nThe object of our study is the role played by STT effects in the  \nthermally assist ed magnetization reversal  using an adaptation \nof the theory of thermal fluctuations in nanomagnets \ndeveloped in the seminal work s of Néel [27] and Brown  \n[28,29]. The Néel -Brown theory i s effect ively  an adaptation of \nthe Kramers theory [ 33,34 ] originally given for point \nBrownian particles to magnetization  relaxa tion governed by a \ngyromagnetic -like equation  which is taken as the Langevin \nequation of the pro cess. Hence, the verification of that theory \nin the pure (i.e., without STT) nanomagnet context nicely \nillustrates the Kramers conception of a thermal relaxation \nprocess as escape over a potential barrier arising from the IEEE TRANSACTIONS ON MAGNETICS, VOL. 53, NO. 10, OCTOBER 2017  \n 3 \nshuttling action of the Brownian m otion. However, it should \nbe recalled throughout that unlike nanomagnets at zero STT \n(where the giant spin escape rate theory may be effectively \nregarded as fully developed),  devices  based on STT , due to the \ninjection of the spin -polarized current, invaria bly represent an \nopen system in an out-of-equilibrium steady state.  This is in \nmarked contrast to the conventional steady state of \nnanostructures characterized by the Boltzmann equilibrium \ndistribution that arises when STT is  omitted .  Hence both the \ngover ning Fokker -Planck and Langevin equations and the \nescape rate theory based on these must be modified .  \nTo facilitate our di scussion, we first describe a schematic \nmodel of the STT effect.  The archetypal model (Fig. 1  (a)) of \na STT device is a nanostructure  compr ising two magnetic \nstrata label ed the free and fixed layers and a nonmagnetic \nconducting spacer. The fixed layer is much more strongly \npinned along its orientation than the free one. If an electric \ncurrent  is passsed  through the fixed layer it become s spin -\npolarized . Thus , the current , as it encounters the free layer, \ninduces a STT . Hence, the magnetization \nM  of the free  layer  \nis altered . Both ferromagnetic layers are assumed to be \nuniformly  magnetized [3,6]. Although th is gia nt coherent spin  \napproximation cannot explain all observations of the \nmagnetization dynamics in spin -torque systems, nevertheless \nmany qualitative features needed to interpret experimental \ndata are satisfactorily reproduced. Indeed,  the current -induced \nmagnetization dynamics in the free layer may be described by \nthe Landau -Lifshitz -Gilbert -Slonczewski equation  including thermal fluctuations , i.e., the usual Landau -Lifshitz -Gilbert \nequation [ 44] incl uding STT, however  augmented by a \nrandom magnetic field  \n()tη  which is regarded as white noise. \nHence it now becomes a magnetic Langevin equation \n[3,6,7,12 ], viz.,  \n \nS             u u H η u u u u I\n . (5) \nHere  \n/SMuM  is the unit vector directed along \nM , \nSM  is \nthe saturation magnetization, and  is the gyromagnetic -type \nconstant . The effective  magnetic field \nH  comprising the \nanisotropy and external applied fields  is defined as  \n \n0SkT E\nvMHu . (6) \nHere  E is the  normalized free energy density of the free layer  \nconstituting a conservative potential, \nv  is the free layer  \nvolume , \n7 2 1\n04 10 JA m    in SI units,  and \nkT  is the \nthermal energy.  For purposes of illustration , we sh all take  \n,)(E\n in the standard form of superimposed easy -plane and \nin-plane easy -axis anisotropies  plus the Zeeman term due to \nthe applied  magnetic  field \n0H  [45] (in our notation):  \n \n22 2, ) sin cos sin cos )( ( 2 cos h E         . (7) \nIn Eq. (7)  and  are the polar and azimuthal angles  in the \nusual spherical polar coordinate system , \n0S/ (2 ) h H M D\n  \nand \n2\n0S / ( ) v M D kT\n  are the external field and anisotropy \nparameters, \n/1DD \n  is the biaxiality parameter \ncharacterized by  \nD\n  and \nD  thereby encompassing both \ndemagnetizing and magnet ocrystalline anisotropy effects \n(since \n  and \n  are determined by both  the volume an d the \nthickness of the free layer, th eir numerical values may vary \nthrough  a very large range, in particular, they can be very \nlarge , > 100  [45]). The form of Eq. (7) implies that  both the \napplied field \n0H  and the unit vector \nPe  identifying the \nmagnetization direction in the fixed layer are directed along \nthe easy X-axis (see Fig. 1(a)) . In general,  \n,()E  as \nrendered by Eq. (7) has two equivalent saddle points  C and \ntwo nonequivalent  wells  at \nA  and \nA  (see Fig.1(b) ). Finally , \nthe STT induced field \nSI  is given by   \n \n0S\nSkT\nvMIu , (8) \nwhere  \n is the normalized non conservative potential due to \nthe spin -polarized current, which in its simplest form i s \n \n ( , )PJ   eu . (9) \nIn Eq.  (9), \n()P J b I e kT\n  is the dimensionless STT \nparameter , I is the spin -polarized current regarded  as positive \nif electrons flow  from the free into the fixed layer, e is the \nelectronic charge, \n  is Planck’s reduced constant , and \nPb is a \nparameter determined  by the spin polarization factor \nP  [1]. \nAccompanying  the magnetic Langevin equation  (5) (i.e., the \nstochastic differential equation of the random magnetization \nprocess) , one has the Fokker -Planck equation  for th e evolution \nof the  associated probability density function \n( , , )Wt  of \norientations of \nM  on the unit sphere, viz., [ 6,12,16 ] \n \nX e   u Z \nY M  \n \neasy axis  H0  \nfixed layer  free layer   I eP \n(a)   \n \n \n       (b)  \nFig. 1. (a)  Geometry of the problem: A STT device consists of two \nferromagnetic strata labelled the free and fixed layers, respectively, and a \nnormal conducting spacer all sandwiched on a pillar between two ohmic \ncontact s [3,6]. Here I is the spin -polarized current, M is the magnetization of \nthe free layer, H0 is the dc bias magnetic field. The magnetization of the \nfixed layer is directed along  the unit vector  eP. (b) Free energy potential of \nthe free layer presented in the standard form of superimposed easy -plane and \nin-plane easy -axis anisotropies, Eq. (7), at  = 20 and h = 0.2 .  IEEE TRANSACTIONS ON MAGNETICS, VOL. 53, NO. 10, OCTOBER 2017  \n 4 \n \nFPLWWt , (10) \nwhere \nFPL  is the Fokker -Planck operator in phase space \n( , )\n defined via [6,12,26]  \n \n1\nN\n1\n11FP1 ( )L sin2 sin\n1 ( ) 1\nsin sin\n( ) ( )sinWEWW\nEW\nEEW   \n    \n\n\n\n\n\n         \n           \n \n\n \n    \n   (11) \nand \n0 N1\nS( ) / (2 ) v M kT     is the free diffus ion time \nof the magnetic moment. If \n0=  (zero STT), Eq. (10) \nbecomes the original Fokker -Planck equation derived by \nBrown  [33] for magnetic nanoparticles . \nIII. ESCAPE RATES AND REVE RSAL TIME IN THE DAM PING \nRANGE \n1  \nThe magnetization  reversal tim e can be calculated  exactly by \nevaluating  the smallest nonvanishing eigenvalue \n1  of the \nFokker -Planck operator L FP in Eq. (10) [32,34 ,42]. Thus \n1  is \nthe inverse of the longest relaxation time of the magnetization \n11/\n, which is usually associated with the reversal time . \nIn the manner of zero STT [42,43], the calculation of \n1  can \nbe approximately accomplished using the Mel’nikov -Meshkov \nformalism  [39]. This relies on the fact that  in the high barrier  \nand underda mped limit s, one may rewrite the Fokker -Planck \nequation, Eq. (10), as an energy -action diffusion equation. \nThis in turn is very similar to that for translating point \nBrownian particles moving along the x-axis in an external \npotential V(x) [7,17,42] . In the under damp ed case, which is the \nrange of interest, for the escape of spins from a single \npotential well with a minimum at a point A of the \nmagnetocristalline anisotropy over a single saddle point C, the \nenergy distribution function  \n()WE  for magnetic moments \nprecessing in the  potential well can  then be found via an \nintegral equation [42], which  can be  solved for \n()WE  by the \nWiener –Hopf method. Then, the flux -over-population method \n[33,34]  yields the decay ( escape ) rate as \n1/CAJN . Here \nconstCJ\n is the probability current density  over the sadd le \npoint and \n()C\nAE\nAEN W E dE  is the well population while the \nescape rate is rendered as the product of the depopulation \nfactor  \n( ),\nCE AS  Eq. (3), and the TST  escape rate \nTST\nAE\nEfe\n. In the preceding equation \nE  is the effective \nspin-polari zed current dependent energy barrier  given by  \n \n1\nAC\nCE\nE EAEVdE E E ES    , (12) \nwhere \nAE  is the energy at the bottom of the potential well, \nCE\n is the energy at the saddle  point, and the dimensionless \naction \nES  and the dimensionless work \nEV  done by the STT are defined as [7,17]  \n \nEEd SE  \nuuu\n , (13) \n \nEE d Vuuu\n , (14) \nrespectively. T he contour integrals in Eqs. (13) and (14) are \ntaken along the energy trajectory  \nconstE  and are to be \nevaluated in the vanishing damping sense.  \nFor the bistable potential, Eq. (7), having  two nonequivalent \nwells  \nA and \nA  with minima  \n( 1 2 ) Eh\n  at \n0A  \nand \nA , respectively,  and two equivalent saddle points C \nwith \n2\nCEh  at \ncosC h  (see Fig. 1(b)) we see that two \nwells and two escape routes over two saddle points  are \ninvolved in the relaxation process . Thus, a finite probability \nfor the magnetic dipole to return to the initi al well having \nalready visited the second one exists. This possibility cannot \nbe ignored in the underdamped regime  because then the \nmagnetic dipole having entered the second well loses its \nenergy so slowly that even after several precessions, thermal \nfluctuations may still reverse it back over the potential barrier. \nIn such a situation, on applying the Mel’nikov -Meshkov \nformalism  [39] to the free energy potential, Eq. (7), and the \nnonconservative potential, Eq. (9), the energy distribution \nfunction s \n()WE  and \n()WE  for magne tic moments \nprecessing in the  two potential well s can then be found by \nsolving two coupled integral equations for \n()WE  and \n()WE\n. These then yield the depopulation factor \n, ()\nCCEE A S S\n via the Mel’nik ov-Meshkov formula for two \nwells, viz., [39]  \n \n( ) ( )\n((), )CC\nCC\nCCEE\nEE\nEEA S A S\nA S SA S S\n\n\n .  \nHere \n()Az  is the depopulation factor for a single well \nintroduced in accordance with Eq. (3) above while \nCES  are the \ndimensionless action s at the energy saddle point s for two \nwells. These are to be calculated via Eq. (13) by integrating \nalong the energy trajectories \nC EE  between two saddle \npoints and are explicitly given by  \n     \n2\n3/2\n12\n21\n12(1 )\n(1(1 2 a4\n(1\nrct)\n)1an )(1 )\n)1 (1CCEEh\nhhhES\nhd\nh\nh \n\n\n\n\n\n      \n   \n \n\nuuu\n  (15) \n(at zero dc bias field, h = 0, these simplify to \nCCEESS  \n4\n). Furthermore, the overall TST escape rate \nTST  for \na bistable potential, Eq. (7), is estimated via the individual  \nescape rates \nTST\n  from each of the two wells as  \n \n TST TSTTST2.EEffee   \n      (16) \nIn Eq. (16), the factor 2 occurs because two magnetization \nescape r outes from each well over the two saddle points exist, \nwhile \nE  are the effective spin -polarized current dependent IEEE TRANSACTIONS ON MAGNETICS, VOL. 53, NO. 10, OCTOBER 2017  \n 5 \nbarrier heights  for two wells (explicit equations for \nE  are \nderived in Appendix A). In addit ion  \n \n01(1 )(1 )2f h h      (17) \nare the corresponding well precession frequencies, where \n1\n0S 2MD\n is a precession time  constant . Thus, the \ndecay rate \n1  becomes  \n2\n2(1 ) ( , )1\n0\n(1 ) ( , )(1 )(1 )\n(( ) ( )\n()\n, 1 )(1 )CC\nCCJh F hEE\nEE\nJh F hA S A S\nAh h e\nhhS\neS\n\n\n\n\n\n\n\n\n  \n\n \n\n \n\n(18) \nwhere both the functions \n( , )Fh  occurring  in each \nexponential are given by the analytical formula:  \n  \n22\n2\n1\n12\n2 1 1 2(1 ) (1( , ) 12 2 2 (1\n(1 21 arctan\n(1)(1 )\n)\n)1\n)(1 ) 1 (1 )hhFhhh h\nhh h\nh h h  \n\n\n\n\n       \n    \n\n\n\n  (19) \nand  0.38 is a numerical par ameter  (see Eq.  (A.6), etc. in \nAppendix A ). For zero STT, J = 0, Eq. (18) reduces to the \nknown results  of the Néel -Brown theory [32,43] for classical \nmagnetic moments with superimposed easy -plane and in -plane \neasy-axis anisotropies  plus the Zeeman term due to the applied  \nmagnetic  field. In contrast to zero STT, for  normalized spin \ncurrents J  0,  depends on \n  not only through the \ndepopulation factors \n()\nCE AS  but also through the spin-\npolarized current dependent  effective barrier heights \nE . \nThis i s so because part s of the arguments of the exponentials \nin Eq.  (18) , namely Eq. (19), are markedly dependent on the \nratio \n/J  and the dc bias field parameter.  The turnover Eq. \n(18) also yields a n asymptotic estimate for the inverse of the \nsmallest nonvanishing eigenvalue of the Fokker -Planck \noperator \nFPL  in Eq. (10). In additio n, one may  estimate two \nindividual  reversal times, namely, \n  from the deeper well \naround the energy minimum at \n0A  and \n  from the \nshallow well around the energy minimum at \nA  (see Fig. \n1(b))  as \n \n2(1 ) ( , )\n02\n( ) (1 )(1 )\nCJh F h\nEe\nA S h h\n\n\n  \n . (20) \nThe individual  times are in general unequal, i.e., \n . In \nderiving Eqs. (18) and (20), all terms of order \n22, , ,JJ  etc. \nare neglected. This hypothesis is true only for the \nunderdamped regime , α < 1, and weak  spin-polarized currents, \nJ<<1. ( Despite these restrictions as we will see below Eqs. \n(18) and (20) still yield accurate estimates for \n  for much \nhigher values of J). Now,  \n can also be calculated  \nnumerically  via the method of statistical moments developed  \nin Ref. [26] whereby t he solution of the Fokker -Planck \nequation (10) in configuration space  is reduced to the task of solving  an infinite hierarchy of differential -recurrence \nequations for the averaged spherical harmonics  \n( , ) ( )lmYt  \ngoverning the magnetization relaxation . (The \n( , )lmY  are the \nspherical harmonics [46 ], and the angular brackets denote the \nstatistical aver aging ). Thus one can evaluate  \n numerically \nvia \n1  of the Fokker -Planck operator L FP in Eq. (10) by using \nmatrix continued fr actions as described in Ref. [47 ]. We \nremark that the r anges of applicability of the escape rate \ntheory and the matrix continued -fraction method are in a sense  \ncomplementary because  escape rate theory cannot be used for  \nlow potential barrie rs, \n3E , while  the matrix continued -\nfraction method encounters substantial computation al \ndifficulties for  very high  potential barriers \n25E  in the \nVLD range, \n410 . Thus , in the foregoing se nse, numerical \nmethods and escape rate theory are very useful for the \ndetermination of τ for low and very high potential barriers, \nrespectively. Nevertheless , in certain  (wide) ranges of model \nparameters both methods yield accurate results for the reversal  \ntime ( here these ranges are \n5 30, 3,     and \n410 ). \nThen the numerically exact  benchmark solution provided by \nthe matrix continued fraction method  allows one  to test the \naccuracy of the analytical es cape rate equations given above.  \nIV. RESULTS AND DISCUSSIO N \nThroughout the calculations, the anisotropy and spin -\npolarization parameters will be taken as \n0.034 D\n , \n20 , \nand \n0.3P  (\n0.3 0.4P  are typical  of ferromagnetic \nmetals)  just as in Ref. 6. Thus for \n5 1 1mA s . 10 , 22  \n300T\nK\n, \n24~10v\n3m , and a current density of the order \nof \n7~ 10\n2A cm  in a 3 nm thick layer of cobalt with \n61\nS 1 1. Am 04 M\n, we have  the following estimates for the \nanisotropy (or inverse temperature ) parameter \n20.2 , \ncharacteristic time \n1\n0S2()MD\n0.48 ps, and spin -\npolarized current parameter \n( ) ~1P J b I e kT\n . In Figs. 2 \nand 3,  we compare  from the asymptotic escape rate Eq . (18) \nwith \n1\n1  of the Fokker –Planck operator as calculated \nnumerically via matrix continued fraction s [26]. Apparently,  \nas rendered by  the turnover equation (18) and \n1\n1  both lie \nvery close to each other in the high barrier limit, where the \nasymptotic Eq. (18) provides an  accurate approximation \nto\n1\n1.  In Fig. 2,  is plotte d as a function of \n  for various J. \nAs far as STT effects are concerned they are governed by the \nratio \n/J  so that by altering \n/J  the ensuing variation of  \nmay exceed  several or ders of magnitude (Fig. 2) . Invariably \nfor J << 1, which is a condition of applicability of the escape \nrate equations (1) and (18), STT effects on the magnetization \nrelaxation are pronounced only at very low damping,  << 1 . \nFor \n1 , i.e. high damping, STT influences  the reversal \nprocess very weak ly because the STT term in Eq. (5) is then \nsmall compared to the damping and random field terms . \nFurthermore,  may greatly exceed or, on the other hand, be  \nvery much less than the value for zero STT , i.e., J = 0 (see Fig. \n2). For example, as J decreases from positive values, \n  IEEE TRANSACTIONS ON MAGNETICS, VOL. 53, NO. 10, OCTOBER 2017  \n 6 \nexponentially increases  attaining a  maximum at a critical \nvalue of the spin -polarized current and then smoothly switches \nover to exponential decrease  as \nJ  is further increased \nthrough negative values of J [26]. Now, t he temperature , \nexternal d.c. bias field,  and dam ping dependence of  can \nreadily be understood in terms of  the effective potential \nbarriers  \nE  in Eq.  (18). For example, for  \n5,  the \ntemperature dependence  of  has the customary Arrhenius \nbehavior \n~,Ee  where \nE , Eq. (19), is markedly \ndependent on \n/J  (see Fig. 3 a). Furthermore, the slope of \n1()T\n significantly decreases as the dc bias field parameter h increases due to lowering  of the barrier height \nE  owing to \nthe action of  the external field  (see Fig. 3b) . Now, although \nthe range of applicability of  Eqs. (18) and (20) is ostensibly \nconfined to weak spin -polarized currents, J << 1, they can still \nyield accurate estimates for the reversal time for much higher \nvalues of J far exceeding this condition (see Fig. 3 a).  \nThus , the turnover formula for , Eqs. (18) and (20), \nbridgi ng the Kramers VLD  and ID escape rates as a function \nof the damping  parameter for point  particles [35,39 -41] as \nextended by Coffey et al.  [42,43] to the magnetization \nrelaxation in nanoscale ferromagnets allows us (via the further \nextension to include STT embodied in Eq. (18)) to accurately  \nevaluate STT effects in the magnetization reversal time of  a \nnanomagnet driven by spin -polarized current in the highly \nrelevant ID to VLD damping range. This (underdamped) range  \nis characterized by  \n1  and the asymptotic escape rates are \nin complete agreement with independent  numerical results  \n[17]. Two particular merits of the escape rate equations  for the \nreversal time are that (i)  they are relatively simple ( i.e., \nexpressed via elementary functions) and (ii) that they can be \nused in those parameter ranges, where numerical methods \n(such as matrix continued fractions [17]) may be no longer \napplicable , e.g., for very high barriers , \n25E . Hence , one \nmay conclude that the damping dependence of the \nmagnetization reversal time is very marked in the \nunderdamped regime \n1 , a fact which may be very \nsignificant  in int erpreting  many  STT experiments.  \nV. APPENDIX A: CALCULATION OF \n( , )Fh  IN EQ. (19) \nFor the bistable potential  given by  Eq. (7), and the \nnonconservative potential, Eq. (9), the spin -polarized current \ndependent effective barrier heights  \nE  for each of the two \nwells are given by (cf. Eq. (12)) \n \n21(1 ) ( , )h J F E h     \n , (A.1) \nwhere  \n \n( , )C\nAVFhSd\n\n \n\n\n , (A.2) \nwith \n/E , \n/ 1 2AAEh \n , \n2/CCEh . The \ndimensionless action  \nS and the dimensionless work  done by \nthe STT \nV for the deeper well can be calculated analytically \nvia elliptic integrals as described in detail in Ref. [17] yielding  \n   \n2 2\n0\n2\n22(1 )\n1\n2 ( )11 ( ) ( )\n)(2\n1\n(1 )( )\n() (142),(1 )( ( ( ) ) 1)p Ehd hpf\nEm hqq q m K m\nq h q mhpq q mS\nm\nKm\n\n\n\n\n   \n \n\n\n\n \n\n\n\n \n         \n   \n    \n\n  \n\n \n uuu\n  (A.3) \n\n54321: J = 0.2\n2: J = 0.1\n3: J = 0\n4: J = 0.1\n5: J = 0.2/ \nh =0.15\n =20\n = 201 \nFig. 2. Reversal time \n0/  vs the damping parameter \n  for various values \nof the  spin-polarized  current  parameter J. Solid lines : numerical calculations \nof the inverse of t he smallest nonvanishing eigenvalue \n1\n01()  of the \nFokker –Planck operator , Eq. (11). Asterisks: the turnover  formula, Eq. (18). \n \n54\n/ 3211: J = 1\n2: J = \n3: J = \n4: J = \n5: J = \nh = 0.1\n = 0.01\n  = 20\n(a)\n \n  (b)\n4\n/ 321 1: h = 0.0\n2: h = 0.1\n3: h = 0.2\n4: h = 0.3\n = 0.01\n  = 20\nJ  = \n\n \nFig. 3 . Reversal time \n0/  vs. the anisotropy  (inverse temperature) \nparameter  for various spin-polarized currents J (a) and dc bias field \nparameters h (b). Solid lines: numerical solution for the inverse of the \nsmallest nonvanishing eigenvalue \n1\n01() of the Fokker –Planck operator , \nEq. (11). Asterisks: Eq. (18). IEEE TRANSACTIONS ON MAGNETICS, VOL. 53, NO. 10, OCTOBER 2017  \n 7 \n    \n\n0\n2 23\n2\n2( 1)\n1 ( 1)\n( | )1\n2\n21 2 1()\n( ) (1\n(|)121 ( ) (,))phV\nm\nm\nqhdhf\nq hpK\nhp q E m m qqq q m K mm\n\n\n\n\n     \n\n\n\n \n\n    \n\n \n    \n\n \n    \n\nueu\n  (A.4) \nwhere  \n \n2\n2\n21( 1)ph\n\n , \n1\n1eqe\n ,  \n \n\n1 (1 )\n(1 ) 1eemee\n , \n2\n( 1)hepph\n\n  , \n()Km\n, \n()Em , and \n( | )am  are the complete  elliptic integrals \nof the fir st, seco nd, and third kinds, respectively [48], and  \nf \nis the precession frequency in the deeper well at a given \nenergy, namely,  \n \n0( 1)(1\n())(1 )\n8p e efKm\n\n     . (A.5) \nThe quantities \nS , \nV , and \nf  for the shallower well are \nobtained simply by replacing  the dc bias field parameter  \nh by \nh\n in all the equations for \nS , \nV , and \nf . We remark that  \nS\n and \nV in Eqs. (A.3) and (A.4) differ by a factor 2 from \nthose given in Ref. [17]. This is so because \nS  and \nV are \nnow calculated between the saddle points and not over the \nprecession period . When \n( , )C     , \nS  in Eqs. (A.3) \nreduces to \nCES , Eq. (15). \nIn the parameter ranges \n01h  and \n1 , the integral in \nEq. (A.2) can be accurately evaluated analytically using an \ninterpolation function for \n/VS  between t he two limiting \nvalues \n/\nAAVS  and \n/\nCCVS  at \n1A h\n  and \n2\nCh , \nnamely  \n \n11\nC AA\nA C AA\nCAV VV V\nS S S S\n \n   \n  \n          , (A.6) \nwhere  0.38 is an interpolation parameter yielding the best \nfit of \n/VS  in the interval \n.AC    These limiting \nvalues can be calculated from Eqs. (A.3) and (A.4) yielding \nafter tedious algebra:  \n \n1\n22A\nAV\nh S\n\n  (A.7) \nand  \n2\n2\n1\n12\n2 1 1 2)(1 ) 1 (112 (1\n(1 21 arct)\n)1an\n(1 (1 )(1 ) ) 1C\nCV h\nh Sh\nhh h\nh h h\n\n\n\n\n\n\n\n\n\n\n \n    \n. (A.8) \nHence with Eqs. (A.2) and (A.6), we have a simple a nalytic \nformula for  the current -dependent parts of the exponentials in \nEq. (18) \n( , )Fh , viz.   \n2( , ) (1 )C A\nACV V\nF h hSS \n    \n\n    , (A.9) \nwhich yields Eq. (19). For zero dc bias field, \n0h , Eq. (A.9) \nbecomes  \n \n1( ,0) ( ,0)2 4 ( 1)FF      . (A.10) \nThe maximum relative deviation bet ween the exact Eqs. (A.2) \nand approximate Eqs. (A.9) and (A.10) is less than 5% in the \nworst cases.  \nREFERENCES  \n[1] J. C. Slonczewski, “Current -driven excita tion of magnetic multilayers ”, \nJ. Magn. Magn. Mater ., vol.  159, p. L1, 1996 . \n[2] L. Berger, “Emission of spin waves by a magnetic multilayer traversed \nby a current ”, Phys. Rev. B , vol.  54, p. 9353 , 1996 . \n[3] M. D. Stiles and J. Miltat, “Spin-Transfer Torque and Dy namics ”, in: \nSpin Dynamics in Confined Magnetic Structures III , p. 225, Eds. B. \nHillebrandsn and A. Thiaville , Springer -Verlag, Berlin, 2006 . \n[4] D. C. Ralph and M. D. Stiles, Spin transfer torques, J. Magn. Magn. \nMater . vol. 320, p. 1190 , 2008 . \n[5]  Y. Suzuki, A.  A. Tulapurkar, and C. Chappert, “Spin-Injection \nPhenomena and Applications ”, Ed. T. Shinjo, Nanomagnetism and \nSpintronics . Elsevier, Amsterdam, 2009 , Chap. 3, p. 94.  \n[6] G. Bertotti, C. Serpico, and I. D. Mayergoyz, Nonlinear Magnetization \nDynamics in Nanosys tems. Elsevier, Amsterdam, 2009 . \n[7] T. Dunn, A. L. Chudnovskiy, and A. Kamenev, “Dynamics of nano -\nmagnetic oscillators ”, in Fluctuating Nonlinear Oscillators, Ed. M. \nDykman . Oxford  University Press, London, 2012 . \n[8] Yu. V. Gulyaev, P. E. Zilberman, A. I. Panas, and E. M. Epshtein, \nSpintronics: exchange switching of ferromagnetic metallic junctions at a \nlow current density , Usp. Fiz. Nauk , vol.  179, p. 359, 2009  [Phys. Usp ., \nvol. 52, p. 335, 2009 ]. \n[9] R. Heindl, W. H. Rippard, S. E. Russek, M. R. Pufall, and A. B. Ko s, \n“Validity of the thermal activation model for spin -transfer torque \nswitching in magnetic tunnel junctions ”, J. Appl. Phys ., vol.  109, p. \n073910 (2011); W. H. Rippard, R. Heindl, M.R. Pufall, S. E. Russek, \nand A. B. Kos, ‘Thermal relaxation rates of magn etic nanoparticles in \nthe presence of magnetic fields and spin -transfer effects ’, Phys. Rev. B , \nvol. 84, p. 064439 , 2011 . \n[10] J. Swiebodzinski, A. Chudnovskiy, T. Dunn, and A. Kamenev, “Spin \ntorque dynamics with noise in magnetic nanosystems ”, Phys. Rev. B , \nvol. 82, p. 144404 , 2010 . \n[11] (a) E. B. Myers, F. J. Albert, J. C. Sankey, E. Bonet, R. A. Buhrman, and \nD. C. Ralph, “Thermally Activated Magnetic Reversal Induced by a \nSpin-Polarized Current ”, Phys. Rev. Lett ., vol.  89, p. 196801 , 2002 ; (b) \nA. Fabian, C. Terrie r, S. Serrano Guisan, X. Hoffer, M. Dubey, L. \nGravier, J. -Ph. Ansermet, and J. -E. Wegrowe, “Current -Induced Two -\nLevel Fluctuations in Pseudo -Spin-Valve ( Co/Cu/Co) Nanostructures ”, \nPhys. Rev. Lett ., vol.  91, p. 257209 , 2003 ; (c) F. B. Mancoff, R. W. \nDave, N . D. Rizzo, T. C. Eschrich, B. N. Engel, and S. Tehrani, \n“Angular dependence of spin -transfer switching in a magnetic \nnanostructure ”, Appl. Phys. Lett ., vol.  83, p. 1596 , 2003 ; (d) I. N. \nKrivorotov, N. C. Emley, A. G. F. Garcia, J. C. Sankey, S. I. Kiselev , D. \nC. Ralph, and R. A. Buhrman, “Temperature Dependence of Spin -\nTransfer -Induced Switching of Nanomagnets ”, Phys. Rev. Lett ., vol.  93, \np. 166603 , 2004 ; (e) R. H. Koch, J. A. Katine, and J. Z. Sun, “Time -\nResolved Reversal of Spin -Transfer Switching in a N anomagnet ”, Phys. \nRev. Lett ., vol.  92, p. 088302 , 2004 ; (f) M. L. Schneider, M. R. Pufall, \nW. H. Rippard, S. E. Russek, and J. A. Katine, “Thermal effects on the \ncritical current of spin torque switching in spin valve nanopillars ”, Appl. \nPhys. Lett ., vol.  90, p. 092504 , 2007 . \n[12] W. T. Coffey and Y. P. Kalmykov, The Langevin Equation , 4th ed. \nWorld Scientific, Singapore, 2017.  \n[13] Z. Li and S. Zhang, “Thermally assisted magnetization reversal in the \npresence of a spin -transfer torque ”, Phys. Rev. B , vol.  69, p. 134416, \n2004 . \n[14] D. M. Apalkov and P. B. Visscher, “Spin-torque switching: Fokker -\nPlanck rate calculation ”, Phys. Rev. B , vol.  72, p. 180405(R) , 2005 . IEEE TRANSACTIONS ON MAGNETICS, VOL. 53, NO. 10, OCTOBER 2017  \n 8 \n[15] G. Bertotti, I. D. Mayergoyz, and C. Serpico, “Analysis of random \nLandau -Lifshitz dynamics by using stochastic processes on graphs ”, J. \nAppl. Phys ., vol.  99, p. 08F301 , 2006 . \n[16] T. Taniguchi and H. Imamura, “Thermally assisted spin transfer torque \nswitching in synthetic free layers ”, Phys. Rev. B , vol.  83, p. 054432 , \n2011 ; “Thermal switching rate of a ferromagnetic ma terial with uniaxial \nanisotropy ”, Phys. Rev. B , vol.  85, p. 184403 , 2012 ; T. Taniguchi, Y. \nUtsumi, and H. Imamura, “Thermally activated switching rate of a \nnanomagnet in the pre sence of spin torque ”, Phys. Rev. B , vol.  88, p. \n214414 , 2013 ; T. Taniguchi, Y. Utsumi, M. Marthaler, D. S. Golubev, \nand H. Imamura, “Spin torque switching of an in -plane magnetized \nsystem in a thermally activated region ”, Phys. Rev. B , vol.  87, p. \n054406 , 2013 ; T. Taniguchi and H. Imamura, “Current dependence of \nspin torque switching rate based on Fokker -Planck app roach ”, J. Appl. \nPhys ., vol. 115, p. 17C708 , 2014 . \n[17] D. Byrne, W.T. Coffey, W. J. Dowling, Y.P. Kalmykov, and S.T. Titov, \n“Spin transfer torque and dc bias magnetic field effects on the \nmagnetization reversal time of nanoscale ferromagnets at very low \ndampin g: Mean first -passage time versus numerical methods ”, Phys. \nRev. B , vol.  93, p. 064413 , 2016 . \n[18] W.H. Butler, T. Mewes, C.K.A. Mewes, P.B. Visscher, W.H. Rippard, \nS.E. Russek, and R. Heindl, “Switching Distributions for Perpendicular \nSpin-Torque Devices Withi n the Macrospin Approximation ”, IEEE \nTrans. Magn. , vol. 48, p. 4684 , 2012 . \n[19] K. A. Newhall and E. Vanden -Eijnden, “Averaged equation for energy \ndiffusion on a graph reveals bifurcation diagram and thermally assisted \nreversal times in spin -torque driven nanom agnets ”. J. Appl. Phys ., vol.  \n113, p. 184105 , 2013 . \n[20] J. M. Lee and S. H. Lim, “Thermally activated magnetization switching \nin a nanostructured synthetic ferrimagnet ”, J. Appl. Phys ., vol.  113, p. \n063914 , 2013 . \n[21] D. V. Berkov and N. L. Gorn, “Magnetization pre cession due to a spin -\npolarized current in a thin nanoelement: Numerical simulation study ”, \nPhys. Rev. B , vol.  72, p. 094401 , 2005 ; “Non-linear magnetization \ndynamics in nanodevices induced by a spin -polarized current: \nmicromagnetic simulation ”, J. Phys. D : Appl. Phys ., vol.  41, p. 164013 , \n2008 . \n[22] K. Ito, “Micromagnetic Simulation on Dynamics of Spin Transfer \nTorque Magnetization Reversal ”, IEEE Trans. Magn. , vol.  41, p. 2630 , \n2005 . \n[23] T. Taniguchi, M. Shibata, M. Marthaler, Y. Utsumi and H. Imamura, \n“Numerical Study on Spin Torque Switching in Thermally Activated \nRegion ”, Appl. Phys. Express , vol.  5, p. 063009 , 2012 . \n[24] D. Pinna, A. D. Kent, and D. L. Stein, “Spin-Transfer Torque \nMagnetization Reversal in Uniaxial Nanomagnets with Thermal Noise ”, \nJ. Appl. Phys ., vol. 114, p. 033901 , 2013 ; “Thermally assisted spin -\ntransfer torque dynamics in energy space ”, Phys. Rev. B , vol.  88, p. \n104405 , 2013 ; “Spin-torque oscillators with thermal noise: A constant \nenergy orbit approach ”, Phys. Rev. B , vol.  90, 174405 , 2014 ; “Large  \nfluctuations and singular behavior of nonequilibrium systems ”, Phys. \nRev. B , vol.  93, p. 012114 , 2016 . \n[25] M. d'Aquino, C. Serpico, R. Bonin, G. Bertotti, and I. D. Mayergoyz, \n“Stochastic resonance in noise -induced transitions between self -\noscillations and eq uilibria in spin -valve nanomagnets ”, Phys. Rev. B , \nvol. 84, p. 214415 , 2011 . \n[26] Y. P. Kalmykov, W. T. Coffey, S. V. Titov, J. E. Wegrowe, and D. \nByrne , “Spin-torque effects in thermally assisted magnetization reversal: \nMethod of statistical moments ”, Phys. Re v. B, vol.  88, p. 144406 , 2013 ; \nD. Byrne , W. T. Coffey, Y. P. Kalmykov, S. V. Titov, and J. E. \nWegrowe, “Spin-transfer torque effects in the dynamic forced response \nof the magnetization of nanoscale ferromagnets in superimposed ac and \ndc bias fields in the  presence of thermal agitation ”, Phys. Rev. B , vol.  91, \np. 174406 , 2015.  \n[27] L. N éel, “Théorie du traînage magnétique des ferromagnétiques en \ngrains fins avec applications aux terres cuites  », Ann. Géophys ., vol.  5, p. \n99, 1949 . \n[28] W. F. Brown, Jr., “Thermal fl uctuations of a single -domain particle ”, \nPhys. Rev ., vol.  130, p. 1677 , 1963 . \n[29] W.F. Brown Jr., “Thermal Fluctuations of Fine Ferromagnetic \nParticles ”, IEEE Trans. Mag ., vol.  15, p. 1196 , 1979 . \n[30] I. Klik and L. Gunther, “First-passage -time approach to overbarr ier \nrelaxation of magnetization ”, J. Stat. Phys ., vol.  60, p. 473, 1990 ; I. Klik and L. Gunther, “Thermal relaxation over a barrier in single domain \nferromagnetic particles ”, J. Appl. Phys ., vol.  67, p. 4505 , 1990 . \n[31] W. T. Coffey, “Finite integral representa tion of characteristic times of \norientation relaxation processes: Application to the uniform bias force \neffect in relaxation in bistable potentials ”, Adv. Chem. Phys ., vol.  103, p. \n259, 1998 . \n[32] W. T. Coffey and Y. P. Kalmykov, “Thermal fluctuations of magnet ic \nnanoparticles ”, J. Appl. Phys ., vol.  112, p. 121301 , 2012 . \n[33] H. A. Kramers, “Brownian motion in a field of force and the diffusion \nmodel of chemical reactions ”, Physica , vol.  7, p. 284, 1940. \n[34] P. Hänggi, P. Talkner, and M. Borkovec, “Reaction -Rate Theory: Fifty \nYears After Kramers ”, Rev. Mod. Phys ., vol.  62, p. 251, 1990 . \n[35] W.T. Coffey, Y.P. Kalmykov, and S.T. Titov, “Magnetization reversal \ntime of magnetic nanoparticles at very low damping ”, Phys. Rev. B , vol.  \n89, p. 054408 , 2014 ; D. Byrne, W.T. Coffey, W. J. Dowling, Y.P. \nKalmykov, and S.T. Titov, “On the Kramers very low damping escape \nrate for point particles and classical spins ”, Adv. Chem. Phys ., vol.  156, \np. 393, 2015 . \n[36] W. Wernsdorfer, E. Bonet Orozco, K. Hasselbach, A. Benoit, D. Mailly, \nO. Kubo, H. Nakano, and B. Barbara, “Macroscopic Quantum Tunneling \nof Magnetization of Single Ferrimagnetic Nanoparticles of Barium \nFerrite ”, Phys. Rev. Lett ., vol.  79, p. 4014 , 1997 ; W.T. Coffey, D.S.F . \nCrothers, J.L. Dormann, Yu.P. Kalmykov, E.C. Kennedy, and W. \nWernsdorfer, “Thermally Activated Relaxation Time of a Single \nDomain Ferromagnetic Particle Subjected to a Uniform Field at an \nOblique Angle to the Easy Axis: Comparison with Experimental \nObser vations ”, Phys. Rev. Lett ., vol.  80, p. 5655 , 1998 . \n[37] M. Oogane, T. Wakitani, S. Yakata, R. Yilgin, Y. Ando, A. Sakuma, and \nT. Miyazaki, “Magnetic Damping in Ferromagnetic Thin Films ”, Jpn. J. \nAppl. Phys ., vol.  45, p. 3889 , 2006 . \n[38] M. C. Hickey and J. S. Moode ra, “Origin of Intrinsic Gilbert Damping ”, \nPhys. Rev. Lett ., vol.  102, p. 137601 (2009).  \n[39] V. I. Mel’nikov and S. V. Meshkov, “Theory of activated rate processes: \nexact solution of the Kramers problem ”, J. Chem. Phys ., vol.  85, p. \n1018 , 1986.  \n[40] H. Grabert, “Escape from a metastable well: The Kramers turnover \nproblem ”, Phys. Rev. Lett. , vol. 61, p. 1683 , 1988 . \n[41] E. Pollak, H. Grabert, and P. Hänggi, “Theory of activated rate processes \nfor arbitrary frequency dependent friction: Solution of the turnover \nproblem ”, J. Chem. Phys ., vol.  91, p. 4073 , 1989 . \n[42] W. T. Coffey, D. A. Garanin, and D. J. McCarthy, “Crossover formulas \nin the Kramers theory of thermally activated escape rates – application \nto spin systems ”, Adv. Chem. Phys . vol. 117, p. 483, 2001 ; P.M. \nDéjardin, D .S.F. Crothers, W.T. Coffey, and D.J. McCarthy, \n“Interpolation formula between very low and intermediate -to-high \ndamping Kramers escape rates for single -domain ferromagnetic \nparticles ”, Phys. Rev. E , vol.  63, p. 021102 , 2001 . \n[43] Yu. P. Kalmykov, W.T. Coffey, B. Ouari, and S. V. Titov, “Damping \ndependence of the magnetization relaxation time of single -domain \nferromagnetic particles ”, J. Magn. Magn. Mater ., vol. 292, p. 372, 2005 ; \nYu. P. Kalmykov and B. Ouari, “Longitudinal complex magnetic \nsusceptibility and re laxation times of superparamagnetic particles with \ntriaxial anisotropy ”, Phys. Rev. B , vol. 71, p. 094410 , 2005 ; B. Ouari  and \nYu. P. Kalmykov , “Dynamics of the magnetization of single domain \nparticles having triaxial anisotropy subjected to a uniform dc ma gnetic \nfield”, J. Appl. Phys ., vol.  100, p. 123912 , 2006 . \n[44] T. L. Gilbert, “A Lagrangian formulation of the gyromagnetic equation \nof the magnetic field ”, Phys. Rev ., vol.  100, p. 1243 , 1955  (Abstract \nonly; full report in: Armour Research Foundation Project N o. A059, \nSupplementary Report, 1956). Reprinted in T.L. Gilbert , A \nphenomenological theory of damping in ferromagnetic materials, IEEE \nTrans. Magn ., vol. 40, p. 3443 , 2004 . \n[45] J. Z. Sun, “Spin-current interaction with a monodomain magnetic body: \nA model study ”, Phys. Rev. B , vol. 62, p. 570, 2000 . \n[46] D. A. Varshalovitch, A. N. Moskalev, and V. K. Khersonskii, Quantum \nTheory of Angular Momentum . World Scientific, Singapore, 1988 . \n[47] Y.P. Kalmykov, “Evaluation of the smallest nonvanishing eigenvalue of \nthe Fokker -Planck equation for the Brownian motion in a potential. II. \nThe matrix continued fraction approach ”, Phys. Rev. E , vol.  62, p. 227, \n2000. \n[48] M. Abramowitz and I. Stegun, Eds., Handbook of Mathematical \nFunctions . Dover, New York, 1964 . IEEE TRANSACTIONS ON MAGNETICS, VOL. 53, NO. 10, OCTOBER 2017  \n 9 \n "    },    {        "title": "1703.03198v3.Material_developments_and_domain_wall_based_nanosecond_scale_switching_process_in_perpendicularly_magnetized_STT_MRAM_cells.pdf",        "content": "Material developments and domain wall based nanosecond-scale switching process in\nperpendicularly magnetized STT-MRAM cells\nThibaut Devolder\u0003and Joo-V on Kim\nCentre de Nanosciences et de Nanotechnologies, CNRS, Univ. Paris-Sud,\nUniversit ´e Paris-Saclay, C2N-Orsay, 91405 Orsay cedex, France\nJ. Swerts, S. Couet, S. Rao, W. Kim, S. Mertens, and G. Kar\nIMEC, Kapeldreef 75, B-3001 Leuven, Belgium\nV . Nikitin\nSAMSUNG Electronics Corporation, 601 McCarthy Blvd Milpitas, CA 95035, USA\nWe investigate the Gilbert damping and the magnetization switching of perpendicularly magnetized FeCoB-\nbased free layers embedded in magnetic tunnel junctions adequate for spin-torque operated magnetic memories.\nWe first study the influence of the boron content in MgO / FeCoB /Ta systems alloys on their Gilbert damping pa-\nrameter after crystallization annealing. Increasing the boron content from 20 to 30% increases the crystallization\ntemperature, thereby postponing the onset of elemental diffusion within the free layer. This reduction of the in-\nterdiffusion of the Ta atoms helps maintaining the Gilbert damping at a low level of 0.009 without any penalty on\nthe anisotropy and the magneto-transport properties up to the 400\u000eC annealing required in CMOS back-end of\nline processing. In addition, we show that dual MgO free layers of composition MgO/FeCoB/Ta/FeCoB/MgO\nhave a substantially lower damping than their MgO/FeCoB/Ta counterparts, reaching damping parameters as\nlow as 0.0039 for a 3 ˚A thick Tantalum spacer. This confirms that the dominant channel of damping is the\npresence of Ta impurities within the FeCoB alloy. On optimized tunnel junctions, we then study the duration of\nthe switching events induced by spin-transfer-torque. We focus on the sub-threshold thermally activated switch-\ning in optimal applied field conditions. From the electrical signatures of the switching, we infer that once the\nnucleation has occurred, the reversal proceeds by a domain wall sweeping though the device at a few 10 m/s.\nThe smaller the device, the faster its switching. We present an analytical model to account for our findings. The\ndomain wall velocity is predicted to scale linearly with the current for devices much larger than the wall width.\nThe wall velocity depends on the Bloch domain wall width, such that the devices with the lowest exchange\nstiffness will be the ones that host the domain walls with the slowest mobilities.\nI. INTRODUCTION\nTunnel magnetoresistance (TMR) and spin transfer torque\n(STT) – the fact that spin-polarized currents manipu-\nlate the magnetization of nanoscale magnets and in par-\nticular magnetic tunnel junction (MTJ) nanopillars – are\nthe basic phenomena underpinning an emerging technol-\nogy called Spin-Transfer-Torque Magnetic Random Access\nMemory (STT-MRAM)1, which combines high endurance,\nlow power requirement2,3, CMOS back-end-of-line (BEOL)\ncompatibility4and potentially large capacity5.\nThe core of an STT-MRAM stack is a magnetic tunnel\njunction composed6of an FeCoB/MgO/FeCoB central block.\nOne of the FeCoB layer is pinned to a high anisotropy syn-\nthetic ferrimagnet to create a fixed reference layer (RL) sys-\ntem while the second FeCoB acts as a free layer (FL). Histor-\nically, the FL is capped with (or deposited on) an amorphous\nmetal such as Ta4,7and more recently capped with a second\nMgO layer to benefit from a second interface anisotropy7–9\nin the so-called ’dual MgO’ configuration. So far, it is un-\nclear whether this benefit of anisotropy can be obtained with-\nout sacrificing the other important properties of the free layer,\nin particular the Gilbert damping.\nIn this paper, we will first tailor the Boron content inside\nthe FeCoB alloy to improve the properties of Ta / FeCoB /\nMgO ’single MgO’ free layers and their resilience to thermal\nannealing. The idea is to postpone the FeCoB crystalliza-tion till the very last stage of the BEOL annealing. Indeed\nmaintaining the amorphous state of FeCoB allows to mini-\nmize the interdiffusion of materials –in our case: tantalum–\nwithin the stack. This interdiffusion is otherwise detrimental\nto the Gilbert damping.\nWe then turn to dual MgO systems comprising a Ta spacer\nlayer in the midst of the FL. This spacer is empirically needed\nto allow proper crystallization and to effectively get perpen-\ndicular magnetic anisotropy (PMA)8,10–14. Unfortunately, the\npresence of heavy elements inside the FeCoB free layer is ex-\npected to alter its damping and to induce some loss of mag-\nnetic moment usually referred as the formation of magneti-\ncally dead layers. We study to what extend the Ta spacer in\nthe dual MgO free layers affects the damping and how this\ndamping compares with the one that can be obtained with sin-\ngle MgO free layers. Once optimized, damping factors as low\nas 0.0039 can be obtained a dual MgO free layer.\nBesides the material issues, the success of STT-MRAM\nalso relies on the capacity to engineer devices in accordance\nwith industry roadmaps concerning speed and miniaturiza-\ntion. To achieve fast switching and design devices accordingly\noptimized, one needs to elucidate the physical mechanism by\nwhich the magnetization switches by STT. Several categories\nof switching modes – macrospin15, domain-wall based16,\nbased on sub-volume nucleation17or based on the spin-wave\namplification18– have been proposed, but single-shot time-\nresolved experimental characterization of the switching patharXiv:1703.03198v3  [cond-mat.mtrl-sci]  4 Sep 20172\nare still scarce19–21. Here we study the nanosecond-scale spin-\ntorque-induced switching in perpendicularly magnetized tun-\nnel junctions with sizes from 50 to 300 nm. Our time-resolved\nexperiments argue for a reversal that happens by the motion\nof a single domain wall, which sweeps through the sample\nat a velocity set by the applied voltage. As a result, the\nswitching duration is proportional to the device length. We\nmodel our finding assuming a single wall moving in a uni-\nform material as a result of spin torque. The wall moves with\na time-averaged velocity that scales with the product of the\nwall width and the ferromagnetic resonance linewidth, such\nthat the devices with the lowest nucleation current densities\nwill be the ones that host the domain walls with the lowest\nmobilities.\nThe paper is split in first a material science part, followed\nby a study of the magnetization reversal dynamics. After a de-\nscription of the samples and the caracterization methods, sec-\ntion II C describes how to choose the optimal Boron content\nin an FeCoB-based free layer for STT-MRAM applications.\nSection II D discusses the benefits of ’dual MgO’ free layers\nwhen compared to ’single MgO’ free layers. Moving to the\nmagnetization switching section, the part III A gathers the de-\nscription of the main properties of the samples and the experi-\nmental methods used to characterize the STT-induced switch-\ning speed. Section III B describes the electrical signatures of\nthe switching mechanism at the nanosecond scale. The latter\nis modeled in section III C in an analytical framework meant\nto clarify the factors that govern the switching speed when the\nreversal involves domain wall motion.\nII. ADVANCED FREE LAYER DESIGNS\nA. Model systems under investigation\nOur objective is to study advanced free layer designs in\nfull STT-MRAM stacks. The stacks were deposited by phys-\nical vapor deposition in a Canon-Anelva EC7800 300 mm\ncluster tool. The MgO tunnel barriers were deposited by\nRF-magnetron sputtering. In dual MgO systems, the top\nMgO layer was fabricated by oxidation of a thin metallic Mg\nfilm. All stacks were post-deposition annealed in a TEL-MSL\nMRT5000 batch furnace in a 1 T perpendicular magnetic field\nfor 30 minutes. Further annealing at 400\u000eC were done in a\nrapid thermal annealing furnace in a N 2atmosphere for a pe-\nriod of 10 minutes.\nWe will focus on several kinds of free layers embod-\nied in state-of-the art bottom-pinned Magnetic Tunnel Junc-\ntions (MTJ) with various reference systems comprising ei-\nther [Co/Ni] and [Co/Pt] based hard layers22,23. Although we\nshall focus here on FLs deposited on [Co/Ni] based synthetic\nantiferromagnet (SAF) reference layers, we have conducted\nthe free layer development also on [Co/Pt] based reference\nlayers. While specific reference layer optimization leads to\nslightly different baseline TMR properties, we have found that\nthe free layer performances were not impacted provided the\nSAF structure is stable with the concerned heat treatment (not\nshown).The first category of samples are the so-called ’single-\nMgO’ free layers. We shall focus on samples with a free\nlayer consists of a 1.4 nm thick Fe 60Co20B20or a 1.6 nm\nthick Fe 52:5Co17:5B30layer sandwiched between the MgO\ntunnel oxide and a Ta (2 nm) metal cap. Note that these\nso-called ”boron 20%” and ”boron 30%” samples have dif-\nferent boron contents but have the same number of Fe+Co\natoms. A sacrificial4Mg layer is deposed before the Ta cap\nto avoid Ta and FeCoB mixing during the deposition, and\navoid the otherwise resulting formation of a dead layer. The\nMg thickness is calibrated so that the Mg is fully sputtered\naway upon cap deposition. This advanced capping method has\nproven to provide improved TMR ratios and lower RA prod-\nucts thanks to an improved surface roughness and a higher\nmagnetic moment4.\nThe second category of free layers are the so-called ’dual\nMgO’ free layers in which the FeCoB layer is sandwiched\nby the MgO tunnel oxide and an MgO cap which concur to\nimprove the magnetic anisotropy. The exact free layer com-\npositions are MgO (1.0 nm) / Fe 60Co20B20(1.1 nm) / spacer\n/ Fe 60Co20B20(0.9 nm) / MgO (0.5 nm). We study shall two\nspacers: a Mg/Ta(3 ˚A) spacer and a Mg/Ta(4 ˚A) spacer, both\ncomprising a sacrificial Mg layer.\nB. Experimental methods used for material quality assessment\nWe studied our samples by current-in-plane tunneling\n(CIPT), vibrating sample magnetometry (VSM) and Vector\nNetwork Ferromagnetic resonance (VNA-FMR)24in out-of-\nplane applied fields. CIPT was performed to extract the tun-\nnel magneto-resistance (TMR) and the resistance-area product\n(RA) of the junction. VSM measurements of the free layer\nminor loops have been used to extract the areal moments. We\nthen use VNA-FMR to identify selectively the properties of\neach subsystem. Our experimental method is explained in\nFig. 1, which gathers some VNAFMR spectra recorded on\noptimized MTJs. The first panel records the permeability of\na single MgO MTJ in the ffield-frequencygparameter space.\nWe systematically investigated a sufficiently large parameter\nspace to detect 4 different modes whose spectral characters\ncan be used to index them22. Three of the modes belong to the\nreference system that comprises 3 magnetic blocks coupled\nby interlayer exchange coupling through Ru and Ta spacers\nas usually done22,23; the properties of these 3 modes are inde-\npendent from the nature of the free layer. While we are not\npresently interested in analyzing the modes of the fixed sys-\ntem – thorough analyses can be found in ref.22,23– we empha-\nsize that it is necessary to detect all modes to unambiguously\nidentify the one belonging to the free layer, in order to study\nit separately. The free layer modes are the ones having V-\nshaped frequency versus field curves [Fig. 1(a)], whose slope\nchanges at the free layer coercivity. in each sample, the free\nlayer modes showed an asymmetric Lorentzian dispersion for\nthe real part of the permeability and a symmetric Lorentzian\ndispersion for the imaginary part [see the examples Fig. 1(b,\nc)]. As we found no signature of the two-layer nature of the\ndual MgO free layers, we modeled each free layer as a sin-3\nSingle MgOfree layerModes of the reference layers\nDual MgOTa spacer\u0000f2f=0.006\u0000f2f=0.016Contrastx 10Permeabilitymap\n↵=12@\u0000f@f=0.0039\nFIG. 1. (Color online). Examples of MTJ dynamical properties to\nillustrate the method of analysis. (a) Microwave permeability versus\nincreasing out-of-plane field and frequency for an MTJ with a sin-\ngle MgO free layer after an annealing of 300\u000eC. Note that the scale\nof the permeability was increased by a factor of 10 above 58 GHz\nfor a better contrast. The apparent vertical bars are the eigenmode\nfrequency jumps at the different switching fields of the MTJ. (b)\nReal and imaginary parts of the experimental (symbols) and modeled\n(lines) permeability for an out-of-plane field of 1.54 T for the same\nMTJ. The model is for an effective linewidth \u0001f=(2f) = 0:016,\nwhich includes both the Gilbert damping and a contribution from the\nsample inhomogeneity. (c) Same but for a dual MgO free layer based\non a 3 ˚A Ta spacer, modeled with \u0001f=(2f) = 0:006. (d) Cross\nsymbols: FMR half frequency linewidth versus FMR frequency for\na dual MgO free layer based on a 3 ˚A Ta spacer. The line is a guide\nto the eye corresponding to a Gilbert damping of 0.0039.\nglemacrospin, disregarding whether it was a single MgO or a\ndual MgO free layer.\nFMR frequency versus field fits [see one example in\nfig. 2(c)] were used to get the effective anisotropy fields\nHk\u0000Msof all free layers25. The curve slopes are \r0, where\n\r0= 230 kHz.m/A is the gyromagnetic factor \rmultiplied\nby the vacuum permeability \u00160. It was consistent with a spec-\ntroscopic splitting Land ´e factor ofg\u00192:08. Damping analy-\nsis was conducted as follows: the free layer composition can\nyield noticeable differences in the FMR linewidths [see for in-\nstance Fig. 1(b) and (c)]. To understand these differences, we\nsystematically separated the Gilbert damping contribution to\nthe linewidth from the contribution of the sample’s inhomo-\ngeneity using standard VNA-FMR modeling25. This is doneby plotting the half FMR linewidth \u0001f=2versus FMR fre-\nquencyfFMR [see one example in Fig. 1(d)]. The Gilbert\ndamping is the curve slope and the line broadening arising\nfrom the inhomogeneity of the effective field within the free\nlayer is the zero frequency intercept1\n2\r0\u0001fjf=0) of the curve.\nC. Boron content and Gilbert damping upon annealing of\nsingle MgO free layers\nDesigning advanced free layer in STT-MRAM stacks re-\nquires to minimize the Gilbert damping of the used raw ma-\nterial. In Ta/FeCoB/MgO ’single MgO’ free layers made of\namorphous FeCoB alloys or made of FeCoB that has been\njust crystallized, a damping of 0.008 to 0.011 can be found\ntypically19,25. (Note that lower values can be obtained but for\nthicknesses and anisotropies that are not adequate for spin-\ntorque application26). The damping of Ta/FeCoB/MgO sys-\ntems generally degrades substantially when further annealing\nthe already crystallized state27. Let us emphasize than even in\nthe best cases26, the damping of FeCoB based free layers are\nstill very substantially above the values of 0.002 or slightly\nless than can be obtained on FeCo of Fe bcc perfect single\ncrystals28,29.\nThere are thus potentially opportunities to improve the\ndamping of free layers by material engineering. We illustrate\nthis in fig. 2 in which we show that a simple increase of the\nBoron content is efficient to maintain the damping unaffected,\neven upon annealing at 400\u000eC in a single MgO free layer. In-\ndeed starting from Ta/FeCoB/MgO ’single MgO’ free layers\nsharing the same damping of 0.009 after annealing at 300\u000eC\n(not shown), an additional 100\u000eC yields\u000b= 0:015for the\nfree layer with 20% of boron, while the boron 30% free lay-\ners keep a damping of \u000b= 0:009[see fig. 2(d)]. Meanwhile\nthe anisotropies of these two free layers remain perpendicu-\nlar [fig. 2(c)] with \u00160(Hk\u0000Ms)being 0.27 and 0.17 T, re-\nspectively, after annealing at 400\u000eC. Let us comment on this\ndifference of damping.\nTwo mechanisms can yield to extra damping: spin-\npumping30and spin-flip impurity scattering of the conduc-\ntion electrons by a spin-orbit process31. Tantalum is known\nto be a poor spin-sink material as this early transition metal\nhas practically no delectrons and therefore its spin-pumping\ncontribution to the damping of an adjacent magnetic layer is\nweak30. We expect a spin pumping contribution to the damp-\ning of Ta (2 nm) / FeCoB (1.4 nm) / MgO ’single MgO’\nfree layers that compares with for instance that measured by\nMizukami et al. on Ta (3 nm) / Fe 20Ni80(3 nm) which was\nundetectable32since below 0.0001; we therefore expect that\nthe spin-pumping contribution to the total free layer damping\nis too negligible to account for the differences observed be-\ntween a free layer and the corresponding perfect single crys-\ntals. The main remaining contribution to the damping is the\nmagnon scattering by the paramagnetic impurities within the\nFeCoB material33. Indeed the Ta atoms within an FeCoB layer\nare paramagnetic impurities that contribute to the damping ac-\ncording to their concentration like any paramagnetic dopant;\nhowever the effect with Ta is particularly large34as Fe and Co4\n/s50/s50 /s50/s52 /s50/s54/s48 /s49\n/s49/s48/s50/s48/s51/s48\n/s49/s48 /s50/s48 /s51/s48/s49/s48 /s50/s48 /s51/s48\n/s48/s46/s51/s48/s46/s54\n/s40/s100/s41/s66/s111/s114/s111/s110/s32/s51/s48/s37\n/s70/s114/s101/s113/s117/s101/s110/s99/s121/s32/s40/s71/s72/s122/s41/s40/s99/s41\n/s40/s98/s41/s40/s97/s41/s66/s111/s114/s111/s110/s32/s50/s48/s37/s84/s114/s97/s110/s115/s118/s101/s114/s115/s101/s32/s112/s101/s114/s109/s101/s97/s98/s105/s108/s105/s116/s121\n/s32/s70/s77/s82/s32/s70/s114/s101/s113/s46/s32/s40/s71/s72/s122/s41/s32/s66/s32/s40/s84/s41\n/s72/s97/s108/s102/s32/s108/s105/s110/s101/s119/s105/s100/s116/s104/s32/s40/s71/s72/s122/s41\n/s32/s84\n/s97/s110/s110/s101/s97/s108/s32/s61/s32/s52/s48/s48/s176/s67\nFIG. 2. (Color online). Properties of single MgO free layers after\nannealing at 400\u000eC. (a) and (b): Real part (narrow lines) and imagi-\nnary part (bold lines) of the free layer permeability in a field of 0.7 T.\nThe lines are macrospin fits. (c) Ferromagnetic resonance frequency\nversus field curves. (d) Half linewidth versus FMR frequencies. The\nlines have slopes of \u000b= 0:009(red, B30%) and \u000b= 0:015(green,\nB20%)\natoms in direct contact with Ta atoms loose part of their mo-\nment and get an extra paramagnetic character, an effect usu-\nally referred as a ”magnetically dead layer”. Qualitatively, the\nTa atoms in the inner structure of the free layer degrade its\ndamping.\nAs the cap of Ta / FeCoB / MgO ’single MgO’ free lay-\ners contain many Ta atoms available for intermixing, a strong\ndegradation of the damping can be obtained in single MgO\nsystems when interdiffusion occurs. To prevent interdiffu-\nsion, we used the following strategy. Amorphous materials\n(including the glassy metals like FeCoB) are known to be ef-\nficient diffusion barriers, as they exhibit atom mobilities that\nare much smaller than their crystalline counterparts. To avoid\nthe diffusion of Ta atoms to the inner part of the FeCoB free\nlayer, a straightforward way is to maintain the FeCoB in an\namorphous state as long as possible during the annealing.\nIn metal-metalloid glasses, the crystallization temperature in-\ncreases with the metalloid content. In our FeCoB free lay-\ners, we find crystallization temperatures of 200, 300, 340 and\n375\u000eC for boron contents of respectively 10%, 20%, 25% and\n30%. Increasing the boron content in FeCo alloys is a way\nto conveniently increase the crystallization temperature and\nthus preserve a low damping. However since to obtain large\nTMR requires the FeCoB to be crystalline35,36, one should en-\ngineer the boron content such that the crystallization tempera-\nture matches with that used in the CMOS final BEOL anneal-\ning of 400\u000eC. In practice, we have found that this situation\nis better approached with a boron content of 30% than 0% to\n25%.D. Gilbert damping in single MgO and dual MgO free layers\nIn our search to further improve the free layers for STT-\nMRAM applications, we have compared the damping of op-\ntimized ’single MgO’ and optimized ’dual MgO’ free layers.\nFor a fair comparison, we first compare samples made from\nFeCoB with the same boron content of 20% and the same\n300\u000eC annealing treatement. From Fig. 1(b) and (c), there\nis a striking improvement of the FMR linewidths when pass-\ning from a single MgO to a dual MgO free layer. To discuss\nthis difference in linewidth, we have separated the Gilbert\ndamping contribution to the linewidth from the contribution\nof the sample’s inhomogeneity. We find that dual MgO sys-\ntems have systematically a substantially lower damping than\nsingle MgO free layers which confirms the trends indepen-\ndently observed by other authors9. Damping values as low\nas low as 0.0039\u00060:005were obtained in Ta 3 ˚A-spacer dual\nMgO stacks [Fig. 1(d)] after 300\u000eC annealing. Samples with\na thicker Ta spacer exhibit an increased damping (not shown).\nThis trend –lower damping in dual MgO systems –is main-\ntained after 400\u000eC annealing; for that annealing temperature,\nthe best damping are obtained for a slightly different internal\nconfiguration of the dual MgO free layer. Indeed a damping of\n0.0048 was obtained (not shown) in MgO / Fe 52:5Co17:5B30\n(1.4 nm) / Ta (0.2 nm) / Fe 52:5Co17:5B30(0.8 nm). This\nshould be compared with that the corresponding single MgO\nfree layer which had a damping of 0.009 for the same an-\nnealing condition [Fig. 2(d)]. This finding is consistent with\nthe results obtained on the single MgO free layer if we as-\nsume that the Ta impurities within an FeCoB layer contribute\nto the damping according to their concentration. Somehow,\nthe number of Tantalum atoms in the initial structure of the\nfree layer sets an upper bound for the maximum degradation\nof the damping upon its interdiffusion that can occur during\nthe annealing. Notably, the single MgO free layers contain\nmuch more Ta atoms (i.e. 2 nm compared to 0.2 to 0.4 nm)\navailable for intermixing: not only the initial number of Ta\nimpurities within the FeCoB layer directly after deposition is\nlarger in the case of single MgO free layer, but in addition a\nmuch stronger degradation of the damping can be obtained in\nsingle MgO systems when interdiffusion occurs, in line with\nour experimental findings. This interpretation – the dominant\nsource of damping is the Ta content – is further strengthened\nby the fact that the thickness of the Ta spacer strongly impacts\nthe damping in dual MgO free layers.\nLet us now study the spin-torque induced switching process\nin nanopillars processed from optimized MTJs.\nIII. SPIN-TORQUE INDUCED SWITCHING PROCESS\nA. Sample and methods for the switching experiments\nIn this section we use two kinds of perpendicularly magne-\ntized MTJ: a ’single MgO’ and a ’dual MgO’ free layer whose\nproperties are detailed respectively in ref.19and20. Note that\nthe devices are made from stacks that do not include all the\nlatest material improvement described in the previous sec-5\ntions and underwent only moderate annealing processes of\n300\u000eC. The ’single MgO’ free layer samples include a 1.4 nm\nFeCoB 20%free layer and a Co/Pt based reference synthetic\nantiferromagnet. Its most significant properties include19an\nareal moment of Mst\u00191:54mA, a damping of 0.01, an ef-\nfective anisotropy field of 0.38 T, a TMR of 150% . The ’dual\nMgO’ devices are made from tunnel junctions with a 2.2 nm\nthick FeCoB-based free layer and a hard reference system also\nbased on a well compensated [Co/Pt]-based synthetic antifer-\nromagnet. The perpendicular anisotropy of the (much thicker)\nfree layer is ensured by a dual MgO encapsulation and an iron-\nrich composition. After annealing, the free layer has an areal\nmoment of Mst\u00191:8mA and an effective perpendicular\nanisotropy field 0.33 T. Before pattering, standard ferromag-\nnetic resonance measurements indicated a Gilbert damping\nparameter of the free layer being \u000b= 0:008. Depending on\nthe size of the patterned device, the tunnel magnetoresistance\n(TMR) is 220 to 250%.\nBoth types of MTJs were etched into pillars of various size\nand shapes, including circles from sub-50 nm diameters to 250\nnm and elongated rectangles with aspect ratio of 2 and foot-\nprint up to 150\u0002300 nm. The MTJs are inserted in series\nbetween coplanar electrodes [Fig. 3(a)] using a device integra-\ntion scheme that minimizes the parasitic parallel capacitance\nso as to ensure an electrical bandwidth in the GHz range. The\njunction properties19,20are such that the quasi-static switching\nthresholds are typically 500 mV . Spin-wave spectroscopy ex-\nperiments similar to ref.37indicated that the main difference\nbetween the two sample series lies in the FL intralayer ex-\nchange stiffness. It is A= 8\u00009pJ/m in the 2.2 nm thick\ndual MgO free layers of the samples of ref.20and more usual\n(\u001920pJ/m) in the 1.4 nm thick ’single MgO’ free layers of\nthe samples of ref.19.\nFor switching experiments, the sample were characterized\nin a set-up whose essential features are described in Fig. 3(a):\na slow triangular voltage ramp is applied to the sample in se-\nries with a 50 \n oscilloscope. As the device impedance is\nmuch larger than the input impedance of the oscilloscope, we\ncan consider that the switching happens at an applied voltage\nthat is constant during the switching. We capture the elec-\ntrical signature of magnetization switching by measuring the\ncurrent delivered to the input of the oscilloscope [Fig. 3(b)].\nWhen averaging several switching events [as conducted in\nFig. 3(b)], the stochasticity of the switching voltage induces\nsome rounding of the electrical signature of the transition.\nHowever, the single shot switching events can also be cap-\ntured (Fig. 4-5). In that case, we define the time origins in\nthe switching as the time at which a perceivable change of the\nresistance suddenly happens (see the convention in Fig. 5).\nThis will be referred hereafter as the ”nucleation” instant.\nThis measurement procedure – slow voltage ramp and time-\nresolved current – entails that the studied reversal regime is\nthe sub-threshold thermally activated reversal switching. This\nsub-threshold thermally-activated switching regime is not di-\nrectly relevant to understand the switching dynamics in mem-\nory devices in which the switching will be forced by short\npulses of substantially higher voltage21. However elucidat-\ning the sub-threshold switching dynamics is of direct inter-\n50 ΩMTJVoltagebias\nOscilloscope50 Ω(a)(b)FIG. 3. (Color online). (a) Sketch of the experimental set-up. Mea-\nsurement procedure: the device is biased with a triangular kHz-rate\nvoltage (green) and the current (red) is monitored by a fast oscillo-\nscope connected in series. (b) The switching transitions are seen as\nabrupt changes of the current (red) followed by a change of the cur-\nrent slope. The resistance (blue) can be computed from the voltage-\nto-current ratio when the current is sufficiently non-zero. In this fig-\nure, the displayed currents and resistances are the averages over 1000\nevents for a 250 nm device with a dual MgO free layer of thickness\n2.2 nm and a weak exchange stiffness.\nest for the quantitative understanding of read disturb errors\nthat may happen at applied voltages much below the writing\npulses. Note finally that sending directly the current to the os-\ncilloscope has a drawback: the current decreases as the MTJ\narea such that the signal-to-noise ratio of our measurement\ndegrades substantially for small device areas (Fig. 5). As a\nresult, the comfortable signal-to-noise ratio allows for a very\nprecise determination of the onset of the reversal in large de-\nvices, but the precision degrades substantially to circa 500 ps\nfor the smallest (40 nm) investigated devices.\nB. Switching results\nIn samples whose (i) reference layers are sufficiently fixed\nto ensure the absence of back-hopping19and (ii) in which the\nstray field from the reference layer is rather uniform20, opti-\nmized compensation of the stray field of the reference layers\nleads to a STT-induced switching with a simple and abrupt\nelectrical signature [Fig. 4(a)]. If examined with a better time\nresolution, the switching event [Fig. 4(b)] appears to induce\na monotonic ramp-like evolution of the device conductance.\nFor a given MTJ stack, the switching voltage is practically in-\ndependent from the device size and shape in our interval of\ninvestigated sizes (not shown). This finding is consistent with\nthe consensual conclusion that the switching energy barrier\nis almost independent from the device area38,39for device ar-\neas above 50 nm. In spite of this quasi-independence of the\nswitching voltage and the device size, the switching duration\nwas found to strongly depend on device size (Fig. 5); we have\nfound that smaller devices switch faster, and the trend is that\nthe switching duration correlates linearly with the longest di-\nmension of the device. This is shown in Fig. 5: 40 nm devices\nswitch in typically 2 to 3 ns whereas devices that are 6 times6\n33PAP\nFIG. 4. (Color online). Single-shot time-resolved absolute value\nof the current during a spin-torque induced switching for parallel to\nantiparallel switching for a circular device of diameter 250 nm made\nwith a weak exchange stiffness, dual MgO 2.2 nm thick free layer.\n(a) Two microsecond long time trace, illustrating that the switching\nis complete, free of back-hopping phenomena, and occurs between\ntwo microwave quiet states. (b) 30 ns long time trace illustrating\nthe regular monotonic change of the device conductance during the\nswitching.\nlarger switch in 10 to 15 ns.\nSuch a reversal path can be interpreted this way: once a do-\nmain is nucleated at one edge of the device, the domain wall\nsweeps irreversibly through the system at a velocity set by\nthe applied voltage [sketch in Fig. 4(b)]. The average domain\nwall speed is then about 20 nm/ns for the low-exchange-free-\nlayers of ref.20. The other devices (not shown but described in\nref.19) based on a ’single MgO’ free layer with a more bulk-\nlike exchange switch with a substantially higher apparent do-\nmain wall velocity, reaching 40 m/s.\nC. Switching Model: domain wall-based dynamics\nTo model the switching, we assume that there is a domain\nwall (DW) which lies at a position qand moves along the\nlongest axis xof the device. The domain wall is assumed\nto be straight along the ydirection, as sketched in Fig. 4(b).\nWe describe the wall in the so-called 1D model40: the wall is\nassumed to be a rigid object of fixed width \u0019\u000epresenting a\ntilt\u001eof its magnetization in the device plane; by convention\n\u001e= 0is for a wall magnetization along x, i.e. a N ´eel wall.\n-10-5051015202501002000510m\nodelduration (ns)switchingd\nevice diam. (nm) \n90 nm \n60 nm 150 nm4\n0 nm80 nmCurrent (norm.)T\nime after nucleation (ns)250 nm  FIG. 5. (Color online). Single-shot time-resolved conductance traces\nfor parallel to antiparallel switching events occurring at at -0.5 V\nfor circular devices of various diameters. The curves are for the de-\nvices whose dual MgO free layer has a thickness of 2.2 nm and has a\nweak exchange stiffness. The curves have been vertically offset and\nvertically normalized to ease the comparison. The time origins and\nswitching durations are chosen at the perceivable onset and end of\nthe conductance change: they are defined by fitting the experimental\nconductance traces by 3 segments (see the sketch labelled ”model”).\nInset: duration of the switching events versus free layer diameter\n(symbols) and linear fit thereof with an inverse slope of 20 m/s.\nThe local current density at the domain wall position is\nwrittenj. The wall is subjected to an out-of-plane field Hz\nassumed to vary slowly in space at the scale of the DW width.\njis assumed to transfer p\u00191spin per electron to the DW by\na pure Slonczewski-like STT. We define\n\u001b=~\n2e\r0\n\u00160MSt(1)\nas the spin-transfer efficiency in unit such that \u001bjis a fre-\nquency. With typical FeCoB parameters, i.e. magnetization\nMs2[1:1;1:4]MA/m and free layer thickness t2[1:4;2:2]\nnm, we have \u001b2[0:018;0:036] Hz / (A/m2) where the low-\nest value corresponds to the largest areal moment Mst. With\nswitching current density of the order of 4\u00021010A/m2, this\nyields\u001bjdcbetween 0.72 and 1.4 GHz.\nFollowing ref.41, the wall position qand and wall tilt \u001eare7\nlinked by the two differential equations:\n_\u001e+\u000b\n\u000e_q=\r0Hz; (2)\n_q\n\u000e\u0000\u000b_\u001e=\u001bjdc+\r0HDW\n2sin(2\u001e) (3)\nin which\u0019\u000eis the width of a Bloch domain wall in an ultra-\nthin film, with \u000e2= 2A=(\u00160MsHeff\nk)whereAis the exchange\nstiffness. A wall parameter \u000e= 12 nm will be assumed for\nthe normal exchange 1.4 nm free layer from various estimates\nincluding ref.37for the exchange stiffness and ref.22for the\nanisotropy of the free layer. The domain wall stiffness field42\nHDWis the in-plane field that one would need to apply to have\nthe wall transformed from a Bloch wall to a N ´eel wall. As it\nexpresses the in-plane demagnetization field within the wall,\nit depends on the wall width \u0019\u000eand on the wall length when\nthe finite size of the device constrains the wall dimensions.\nUsing42, the domain wall stiffness field can be estimated\nto be at the most 20 mT in our devices. In circular devices,\nthe domain wall has to elongate upon its propagation38such\nthat the domain wall stiffness field HDWdepends in princi-\nple on the DW position. It should be maximal when the wall\nis along the diameter of the free layer. However we will see\nthatHDWis not the main determinant of the dynamics. In-\ndeed in the absence of stray field and current, the Walker field\nHWalker is proportional to the domain wall stiffness field times\nthe damping parameter, i.e. HWalker =\u000bH DW=2. As the sam-\nples required for STT switching are typically made of low\ndamped materials with \u000b < 0:01, the Walker field is very\nsmall and likely to be smaller than the stray fields emanating\nfrom either the reference layers or the applied field. This very\nsmall Walker field has implications: in practice as soon as\nthere is some field of some applied current, any domain wall\nin the free layer is bound to move in the Walker regime and\nto make the back-and-forth oscillatory movements that are in-\nherent to this regime. The DW oscillates at a generally fast\n(GHz) frequency43such that only the time-averaged velocity\nmatters to define how much it effectively advances.\nTo see quantitatively the effect of a constant current on the\ndomain wall dynamics, we assume that the sample is invari-\nant along the domain wall propagation direction (x) (like in\nan hypothetical stripe-shaped sample). Solving numerically\nEq. 2 and 3, we find that the Walker regime is maintained for\njdc6= 0(not shown). Two points are worth noticing:\nThe time-averaged domain wall velocity h_qivaries linearly\nwith the applied current density. When in the Walker regime,\nthe current effect can be understood from Eq. 3. Indeed the\nsin(2\u001e)term essentially averages out in a time integration as\n\u001eis periodic, and the term \u000b_\u001eis neligible, such that the time-\naveraged wall velocity reduces to:\nh_qi\u0019\u000e\u001bj dc (4)\nFor\u000e= 12 nm and\u001bjin the range of 1.4 GHz at the switching\nvoltage for the bulk-like exchange stiffness sample with free\nlayer thcikness 1.4 nm, the previous equation would predict atime-averaged domain wall velocity of 17 m/s (or nm/ns) dur-\ning the switching. More compact domain walls are expected\nfor the samples with a weaker exchange stiffness; the twice\nlower\u001bj\u00190:72GHz related to the larger thickness would\nreinforce this trend to a much a lower domain wall velocity (9\nm/s for our material parameters estimates). This expectation\ncompares qualitatively well with our experimental findings of\nslower walls in weakly exchanged materials (Fig. 5).\nWe wish to emphasize that Eq. 4 can be misleading regard-\ning the role of damping. Indeed a too quick look at Eq. 4 could\nlet people wrongly conclude the domain wall velocity is es-\nsentially set by the areal moment Mstand that the wall veloc-\nity under STT from a current perpendicular to the plane (CPP\ncurrent) is independent from the damping factor (see Eq. 1).\nHowever this is not the case as the switching current jdcis\na sweep-rate-dependent and temperature-determined fraction\n\u00112[1\n2;1]of the zero temperature instability current jc0of a\nmacrospin in the parallel state, which reads15,44:\njc0=\u000b4e\n~1 +p2\np\u00160MstHeff\nk\n2(5)\nwherep\u00191is an effective spin polarization.\nUsing Eq. 1, 4 and 5, the time-averaged wall velocity at the\npractical switching voltage is:\nh_qi\u0019\u000b\u000e\r 0Heff\nk\u0011 (6)\nThis expression indicates that the samples performing best\nin term of switching current (minimal damping and easy nu-\ncleation thanks to a small exchange) will host domain walls\nthat are inherently slow when pushed by the CPP current in\nthe Walker regime. The domain wall speed scales with the\ndomain wall width, which may be the reason why the low\nexchange stiffness samples host domain walls that are experi-\nmentally slower.\nTo summarize, once nucleated at the instability of the uni-\nformly magnetized state at jdc=\u0011jc0, the domain wall flows\nin a Walker regime through the device. The switching dura-\ntion varies thus simply with the inverse current:\n\u001cswitch =L\n\u000e\u001bj dc\u0019L\n\u000e\u00021\n\u000b\r0Heff\nk\u00021\n\u0011(7)\nLet us comment on this equation which is the main con-\nclusion of this section. The underlying simplifications are:\n(i) a rigid wall (ii) that does not sense the sample’s edges\n(iii) that moves at a speed equal to its average velocity in the\nWalker regime (iv) at a switching voltage that is independent\nfrom the sample geometry. Under these assumptions, the du-\nration of the switching scales with the length Lof the sam-\nple, as observed experimentally. It also scales with the inverse\nof the zero-field ferromagnetic resonance linewith 2\u000b\r0Heff\nk.\nThe practical switching voltage is below the zero temperature\nmacrospin switching voltage by a factor \u0011, which gathers the\neffect of the thermal activation and of the sweeping rate of the\napplied voltage45.\u0011\u00191=2for quasi-static experiments like\nreported here and \u0011!1for experiments in which the voltage\nrise timeVmax=_Vis short enough compared to the switching\nduration (Eq. 7).8\nIV . SUMMARY AND CONCLUSION\nIn summary, we have investigated the Gilbert damping of\nadvanced free layer designs: they comprise FeCoB alloys with\nvariable B contents from 20 to 30% and are organized in the\nsingle MgO or dual MgO free layer configuration fully em-\nbedded in functional STT-MRAM magnetic tunnel junctions.\nIncreasing the boron content increases the cristallization tem-\nperature, thereby postponing the onset of elemental diffusion\nwithin the free layer. This reduction of the interdiffusion of\nthe Ta atoms helps maintaining the Gilbert damping at a low\nlevel without any penalty on the anisotropy and the transport\nproperties. Thereby, increasing the Boron content to at least\n30% is beneficial for the thermal robustness of the MTJ up\nto the 400\u000erequired in CMOS back-end of line processing.\nIn addition, we have shown that dual MgO free layers have a\nsubstantially lower damping than their single MgO counter-\nparts, and that the damping increases as the thickness of the\nTa spacer within dual MgO free layers. This indicates that\nthe dominant source of extra damping is the presence of Ta\nimpurities within the FeCoB alloy. Using optimized MTJs,\nwe have studied the duration of the switching events as in-\nduced by spin-transfer-torque. Our experimental procedure –\ntime-resolving the switching with a high bandwidth but dur-\ning slow voltage sweep – ensures that we are investigating\nonly sub-threshold thermally activated switching events. In\noptimal conditions, the switching induces a ramp-like mono-\ntonic evolution of the device conductance that we interpret\nas the sweeping of a domain wall through the device. The\nswitching duration is roughly proportional to the device size:\nthe smaller the device, the faster it switches. We studied twoMTJ stacks and found domain wall velocities from 20 to 40\nm/s. A simple analytical model using a rigid wall approxima-\ntion can account for our main experimental findings. The do-\nmain wall velocity is predicted to scale linearly with the cur-\nrent for device sizes much larger than the domain wall widths.\nThe domain wall velocity depends on the material parame-\nters, such that the samples with the thinnest domain walls will\nbe the ones that host the domain walls with the lowest mo-\nbilities. Schematically, material optimization for low current\nSTT-induced switching (i.e. in practice: fast nucleation be-\ncause of low exchange stiffness Aand low damping \u000b) will\ncome together with slow STT-induced domain wall motion at\nleast in the range of device sizes in which the STT-induced re-\nversal proceeds through domain wall motion. If working with\nSTT-MRAM memory cells made in the same range of device\nsizes, read disturb should be minimal (if not absent) provided\nthat the voltage pulse used to read the free layer magnetiza-\ntion state has a duration much shorter than the time needed\nfor a domain wall to sweep through the device at that voltage\n(Eq. 7).\nACKNOWLEDGMENT\nThis work is supported in part by IMEC’s Industrial Affil-\niation Program on STT-MRAM device, in part by the Sam-\nsung Global MRAM Innovation Program and in part by\na public grant overseen by the French National Research\nAgency (ANR) as part of the Investissements dAvenir pro-\ngram (Labex NanoSaclay, reference: ANR-10-LABX-0035).\nT. D. would like to thank Andr ´e Thiaville, Paul Bouquin and\nFelipe Garcia-Sanchez for useful discussions.\n\u0003thibaut.devolder@u-psud.fr\n1A. V . Khvalkovskiy, D. Apalkov, S. Watts, R. Chepulskii,\nR. S. Beach, A. Ong, X. Tang, A. Driskill-Smith, W. H.\nButler, P. B. Visscher, D. Lottis, E. Chen, V . Nikitin, and\nM. Krounbi, “Basic principles of STT-MRAM cell operation\nin memory arrays,” Journal of Physics D: Applied Physics ,\nvol. 46, no. 7, p. 074001, Feb. 2013. [Online]. Available:\nhttp://iopscience.iop.org/0022-3727/46/7/074001\n2E. Chen, D. Apalkov, Z. Diao, A. Driskill-Smith, D. Druist,\nD. Lottis, V . Nikitin, X. Tang, S. Watts, S. Wang, S. A. Wolf,\nA. W. Ghosh, J. W. Lu, S. J. Poon, M. Stan, W. H. Butler, S. Gupta,\nC. K. A. Mewes, T. Mewes, and P. B. Visscher, “Advances and Fu-\nture Prospects of Spin-Transfer Torque Random Access Memory,”\nIEEE Transactions on Magnetics , vol. 46, no. 6, pp. 1873–1878,\nJun. 2010.\n3L. Thomas, G. Jan, J. Zhu, H. Liu, Y .-J. Lee, S. Le, R.-Y . Tong,\nK. Pi, Y .-J. Wang, D. Shen, R. He, J. Haq, J. Teng, V . Lam,\nK. Huang, T. Zhong, T. Torng, and P.-K. Wang, “Perpendicular\nspin transfer torque magnetic random access memories with\nhigh spin torque efficiency and thermal stability for embedded\napplications (invited),” Journal of Applied Physics , vol. 115,\nno. 17, p. 172615, May 2014. [Online]. Available: http://scitation.\naip.org/content/aip/journal/jap/115/17/10.1063/1.48709174J. Swerts, S. Mertens, T. Lin, S. Couet, Y . Tomczak, K. Sankaran,\nG. Pourtois, W. Kim, J. Meersschaut, L. Souriau, D. Radisic,\nS. V . Elshocht, G. Kar, and A. Furnemont, “BEOL compatible\nhigh tunnel magneto resistance perpendicular magnetic tunnel\njunctions using a sacrificial Mg layer as CoFeB free layer cap,”\nApplied Physics Letters , vol. 106, no. 26, p. 262407, Jun. 2015.\n[Online]. Available: http://scitation.aip.org/content/aip/journal/\napl/106/26/10.1063/1.4923420\n5H. Sato, E. C. I. Enobio, M. Yamanouchi, S. Ikeda, S. Fukami,\nS. Kanai, F. Matsukura, and H. Ohno, “Properties of magnetic\ntunnel junctions with a MgO/CoFeB/Ta/CoFeB/MgO recording\nstructure down to junction diameter of 11 nm,” Applied Physics\nLetters , vol. 105, no. 6, p. 062403, Aug. 2014. [Online].\nAvailable: http://scitation.aip.org/content/aip/journal/apl/105/6/\n10.1063/1.4892924\n6D. C. Worledge, G. Hu, D. W. Abraham, J. Z. Sun, P. L.\nTrouilloud, J. Nowak, S. Brown, M. C. Gaidis, E. J. OSullivan,\nand R. P. Robertazzi, “Spin torque switching of perpendicular\nTa?CoFeB?MgO-based magnetic tunnel junctions,” Applied\nPhysics Letters , vol. 98, no. 2, p. 022501, Jan. 2011. [Online].\nAvailable: http://scitation.aip.org/content/aip/journal/apl/98/2/10.\n1063/1.3536482\n7S. Ikeda, K. Miura, H. Yamamoto, K. Mizunuma, H. D. Gan,\nM. Endo, S. Kanai, J. Hayakawa, F. Matsukura, and H. Ohno, “A9\nperpendicular-anisotropy CoFeBMgO magnetic tunnel junction,”\nNature Materials , vol. 9, no. 9, pp. 721–724, Sep. 2010.\n[Online]. Available: http://www.nature.com/nmat/journal/v9/n9/\nabs/nmat2804.html\n8H. Sato, M. Yamanouchi, S. Ikeda, S. Fukami, F. Matsukura,\nand H. Ohno, “Perpendicular-anisotropy CoFeB-MgO magnetic\ntunnel junctions with a MgO/CoFeB/Ta/CoFeB/MgO recording\nstructure,” Applied Physics Letters , vol. 101, no. 2, p. 022414,\nJul. 2012. [Online]. Available: http://scitation.aip.org/content/aip/\njournal/apl/101/2/10.1063/1.4736727\n9M. Konoto, H. Imamura, T. Taniguchi, K. Yakushiji, H. Kubota,\nA. Fukushima, and S. Yuasa, “Effect of MgO Cap Layer\non Gilbert Damping of FeB Electrode Layer in MgO-\nBased Magnetic Tunnel Junctions,” Applied Physics Express ,\nvol. 6, no. 7, p. 073002, Jun. 2013. [Online]. Available:\nhttp://iopscience.iop.org/article/10.7567/APEX.6.073002/meta\n10J. Sinha, M. Gruber, T. Ohkubo, S. Mitani, K. Hono, and\nM. Hayashi, “Influence of boron diffusion on the perpendicular\nmagnetic anisotropy in Ta jCoFeB jMgO ultrathin films,” Journal\nof Applied Physics , vol. 117, no. 4, p. 043913, Jan. 2015. [Online].\nAvailable: http://aip.scitation.org/doi/abs/10.1063/1.4906096\n11M. Gottwald, J. J. Kan, K. Lee, X. Zhu, C. Park, and S. H. Kang,\n“Scalable and thermally robust perpendicular magnetic tunnel\njunctions for STT-MRAM,” Applied Physics Letters , vol. 106,\nno. 3, p. 032413, Jan. 2015. [Online]. Available: http://scitation.\naip.org/content/aip/journal/apl/106/3/10.1063/1.4906600\n12Y .-J. Chang, A. Canizo-Cabrera, V . Garcia-Vazquez, Y .-H.\nChang, and T.-h. Wu, “Effect of Ta thickness on the perpendicular\nmagnetic anisotropy in MgO/CoFeB/Ta/[Co/Pd]n structures,”\nJournal of Applied Physics , vol. 114, no. 18, p. 184303, Nov.\n2013. [Online]. Available: http://scitation.aip.org/content/aip/\njournal/jap/114/18/10.1063/1.4829915\n13M. P. R. Sabino, S. T. Lim, and M. Tran, “Influence of Ta\ninsertions on the magnetic properties of MgO/CoFeB/MgO films\nprobed by ferromagnetic resonance,” Applied Physics Express ,\nvol. 7, no. 9, p. 093002, Aug. 2014. [Online]. Available:\nhttp://iopscience.iop.org/article/10.7567/APEX.7.093002/meta\n14J.-H. Kim, J.-B. Lee, G.-G. An, S.-M. Yang, W.-S. Chung,\nH.-S. Park, and J.-P. Hong, “Ultrathin W space layer-\nenabled thermal stability enhancement in a perpendicular\nMgO/CoFeB/W/CoFeB/MgO recording frame,” Scientific Re-\nports , vol. 5, p. 16903, Nov. 2015.\n15J. Z. Sun, “Spin-current interaction with a monodomain\nmagnetic body: A model study,” Physical Review B , vol. 62,\nno. 1, pp. 570–578, Jul. 2000. [Online]. Available: http:\n//link.aps.org/doi/10.1103/PhysRevB.62.570\n16D. P. Bernstein, B. Brauer, R. Kukreja, J. Stohr, T. Hauet,\nJ. Cucchiara, S. Mangin, J. A. Katine, T. Tyliszczak, K. W. Chou,\nand Y . Acremann, “Nonuniform switching of the perpendicular\nmagnetization in a spin-torque-driven magnetic nanopillar,”\nPhysical Review B , vol. 83, no. 18, p. 180410, May 2011.\n[Online]. Available: http://link.aps.org/doi/10.1103/PhysRevB.\n83.180410\n17J. Sun, R. Robertazzi, J. Nowak, P. Trouilloud, G. Hu,\nD. Abraham, M. Gaidis, S. Brown, E. O Sullivan, W. Gallagher,\nand D. Worledge, “Effect of subvolume excitation and spin-\ntorque efficiency on magnetic switching,” Physical Review B ,\nvol. 84, no. 6, p. 064413, Aug. 2011. [Online]. Available:\nhttp://link.aps.org/doi/10.1103/PhysRevB.84.064413\n18K. Munira and P. B. Visscher, “Calculation of energy-\nbarrier lowering by incoherent switching in spin-transfer torque\nmagnetoresistive random-access memory,” Journal of Applied\nPhysics , vol. 117, no. 17, p. 17B710, May 2015. [Online].\nAvailable: http://scitation.aip.org/content/aip/journal/jap/117/17/10.1063/1.4908153\n19T. Devolder, J.-V . Kim, F. Garcia-Sanchez, J. Swerts, W. Kim,\nS. Couet, G. Kar, and A. Furnemont, “Time-resolved spin-torque\nswitching in MgO-based perpendicularly magnetized tunnel\njunctions,” Physical Review B , vol. 93, no. 2, p. 024420,\nJan. 2016. [Online]. Available: http://link.aps.org/doi/10.1103/\nPhysRevB.93.024420\n20T. Devolder, A. Le Goff, and V . Nikitin, “Size dependence\nof nanosecond-scale spin-torque switching in perpendicularly\nmagnetized tunnel junctions,” Physical Review B , vol. 93,\nno. 22, p. 224432, Jun. 2016. [Online]. Available: http:\n//link.aps.org/doi/10.1103/PhysRevB.93.224432\n21C. Hahn, G. Wolf, B. Kardasz, S. Watts, M. Pinarbasi,\nand A. D. Kent, “Time-resolved studies of the spin-transfer\nreversal mechanism in perpendicularly magnetized magnetic\ntunnel junctions,” Physical Review B , vol. 94, no. 21, p. 214432,\nDec. 2016. [Online]. Available: http://link.aps.org/doi/10.1103/\nPhysRevB.94.214432\n22T. Devolder, S. Couet, J. Swerts, and A. Furnemont, “Evolution\nof perpendicular magnetized tunnel junctions upon annealing,”\nApplied Physics Letters , vol. 108, no. 17, p. 172409, Apr. 2016.\n[Online]. Available: http://scitation.aip.org/content/aip/journal/\napl/108/17/10.1063/1.4948378\n23T. Devolder, S. Couet, J. Swerts, E. Liu, T. Lin, S. Mertens,\nA. Furnemont, and G. Kar, “Annealing stability of magnetic\ntunnel junctions based on dual MgO free layers and [Co/Ni]\nbased thin synthetic antiferromagnet fixed system,” Journal of\nApplied Physics , vol. 121, no. 11, p. 113904, Mar. 2017. [Online].\nAvailable: http://aip.scitation.org/doi/full/10.1063/1.4978633\n24C. Bilzer, T. Devolder, P. Crozat, C. Chappert, S. Cardoso, and\nP. P. Freitas, “Vector network analyzer ferromagnetic resonance\nof thin films on coplanar waveguides: Comparison of different\nevaluation methods,” Journal of Applied Physics , vol. 101, no. 7,\np. 074505, Apr. 2007. [Online]. Available: http://scitation.aip.\norg/content/aip/journal/jap/101/7/10.1063/1.2716995\n25T. Devolder, P.-H. Ducrot, J.-P. Adam, I. Barisic, N. Vernier, J.-V .\nKim, B. Ockert, and D. Ravelosona, “Damping of CoxFe80?xB20\nultrathin films with perpendicular magnetic anisotropy,” Applied\nPhysics Letters , vol. 102, no. 2, p. 022407, Jan. 2013. [Online].\nAvailable: http://scitation.aip.org/content/aip/journal/apl/102/2/\n10.1063/1.4775684\n26X. Liu, W. Zhang, M. J. Carter, and G. Xiao, “Ferromagnetic\nresonance and damping properties of CoFeB thin films as free\nlayers in MgO-based magnetic tunnel junctions,” Journal of\nApplied Physics , vol. 110, no. 3, p. 033910, Aug. 2011. [Online].\nAvailable: http://aip.scitation.org/doi/10.1063/1.3615961\n27C. Bilzer, T. Devolder, J.-V . Kim, G. Counil, C. Chappert,\nS. Cardoso, and P. P. Freitas, “Study of the dynamic\nmagnetic properties of soft CoFeB films,” Journal of Applied\nPhysics , vol. 100, no. 5, p. 053903, Sep. 2006. [Online].\nAvailable: http://scitation.aip.org/content/aip/journal/jap/100/5/\n10.1063/1.2337165\n28M. Oogane, T. Wakitani, S. Yakata, R. Yilgin, Y . Ando,\nA. Sakuma, and T. Miyazaki, “Magnetic Damping in Ferro-\nmagnetic Thin Films,” Japanese Journal of Applied Physics ,\nvol. 45, no. 5A, pp. 3889–3891, May 2006. [Online]. Available:\nhttp://stacks.iop.org/1347-4065/45/3889\n29T. Devolder, T. Tahmasebi, S. Eimer, T. Hauet, and S. Andrieu,\n“Compositional dependence of the magnetic properties of\nepitaxial FeV/MgO thin films,” Applied Physics Letters , vol. 103,\nno. 24, p. 242410, Dec. 2013. [Online]. Available: http://scitation.\naip.org/content/aip/journal/apl/103/24/10.1063/1.4845375\n30Y . Tserkovnyak, A. Brataas, and G. E. W. Bauer, “Spin pumping\nand magnetization dynamics in metallic multilayers,” Physical10\nReview B , vol. 66, no. 22, p. 224403, Dec. 2002. [Online].\nAvailable: http://link.aps.org/doi/10.1103/PhysRevB.66.224403\n31V . Kambersky, “Spin-orbital Gilbert damping in common mag-\nnetic metals,” Physical Review B , vol. 76, no. 13, 2007.\n32S. Mizukami, Y . Ando, and T. Miyazaki, “Ferromagnetic reso-\nnance linewidth for NM/80nife/NM films (NM=Cu, Ta, Pd and\nPt),” Journal of Magnetism and Magnetic Materials , vol. 226230,\nPart 2, pp. 1640–1642, May 2001. [Online]. Available: http://\nwww.sciencedirect.com/science/article/pii/S0304885300010970\n33J. O. Rantscher, R. D. McMichael, A. Castillo, A. J. Shapiro,\nW. F. Egelhoff, B. B. Maranville, D. Pulugurtha, A. P. Chen, and\nL. M. Connors, “Effect of 3d, 4d, and 5d transition metal doping\non damping in permalloy thin films,” Journal of Applied Physics ,\nvol. 101, no. 3, p. 033911, Feb. 2007. [Online]. Available:\nhttp://aip.scitation.org/doi/abs/10.1063/1.2436471\n34T. Devolder, I. Barisic, S. Eimer, K. Garcia, J.-P. Adam,\nB. Ockert, and D. Ravelosona, “Irradiation-induced tailoring\nof the magnetism of CoFeB/MgO ultrathin films,” Journal of\nApplied Physics , vol. 113, no. 20, p. 203912, May 2013. [Online].\nAvailable: http://scitation.aip.org/content/aip/journal/jap/113/20/\n10.1063/1.4808102\n35S. Yuasa, T. Nagahama, A. Fukushima, Y . Suzuki, and\nK. Ando, “Giant room-temperature magnetoresistance in single-\ncrystal Fe/MgO/Fe magnetic tunnel junctions,” Nature Materials ,\nvol. 3, no. 12, pp. 868–871, Dec. 2004. [Online]. Available:\nhttp://www.nature.com/nmat/journal/v3/n12/full/nmat1257.html\n36S. S. P. Parkin, C. Kaiser, A. Panchula, P. M. Rice,\nB. Hughes, M. Samant, and S.-H. Yang, “Giant tunnelling\nmagnetoresistance at room temperature with MgO (100) tunnel\nbarriers,” Nature Materials , vol. 3, no. 12, pp. 862–867, Dec.\n2004. [Online]. Available: http://www.nature.com/nmat/journal/\nv3/n12/full/nmat1256.html\n37T. Devolder, J.-V . Kim, L. Nistor, R. Sousa, B. Rodmacq,\nand B. Dieny, “Exchange stiffness in ultrathin perpendicularly\nmagnetized CoFeB layers determined using the spectroscopy of\nelectrically excited spin waves,” Journal of Applied Physics ,\nvol. 120, no. 18, p. 183902, Nov. 2016. [Online]. Avail-\nable: http://scitation.aip.org/content/aip/journal/jap/120/18/10.\n1063/1.4967826\n38G. D. Chaves-OFlynn, G. Wolf, J. Z. Sun, and A. D.\nKent, “Thermal Stability of Magnetic States in Circular\nThin-Film Nanomagnets with Large Perpendicular MagneticAnisotropy,” Physical Review Applied , vol. 4, no. 2, p. 024010,\nAug. 2015. [Online]. Available: http://link.aps.org/doi/10.1103/\nPhysRevApplied.4.024010\n39J. Z. Sun, S. L. Brown, W. Chen, E. A. Delenia, M. C. Gaidis,\nJ. Harms, G. Hu, X. Jiang, R. Kilaru, W. Kula, G. Lauer,\nL. Q. Liu, S. Murthy, J. Nowak, E. J. OSullivan, S. S. P.\nParkin, R. P. Robertazzi, P. M. Rice, G. Sandhu, T. Topuria,\nand D. C. Worledge, “Spin-torque switching efficiency in\nCoFeB-MgO based tunnel junctions,” Physical Review B ,\nvol. 88, no. 10, p. 104426, Sep. 2013. [Online]. Available:\nhttp://link.aps.org/doi/10.1103/PhysRevB.88.104426\n40A. Thiaville, Y . Nakatani, J. Miltat, and Y . Suzuki,\n“Micromagnetic understanding of current-driven domain\nwall motion in patterned nanowires,” Europhysics Letters ,\nvol. 69, no. 6, p. 990, Mar. 2005. [Online]. Available:\nhttp://iopscience.iop.org/0295-5075/69/6/990\n41J. Cucchiara, S. Le Gall, E. E. Fullerton, J.-V . Kim,\nD. Ravelosona, Y . Henry, J. A. Katine, A. D. Kent, D. Bedau,\nD. Gopman, and S. Mangin, “Domain wall motion in nanopillar\nspin-valves with perpendicular anisotropy driven by spin-transfer\ntorques,” Physical Review B , vol. 86, no. 21, p. 214429,\nDec. 2012. [Online]. Available: http://link.aps.org/doi/10.1103/\nPhysRevB.86.214429\n42A. Mougin, M. Cormier, J. P. Adam, P. J. Metaxas,\nand J. Ferr, “Domain wall mobility, stability and Walker\nbreakdown in magnetic nanowires,” Europhysics Letters , vol. 78,\nno. 5, p. 57007, Jun. 2007. [Online]. Available: http:\n//iopscience.iop.org/0295-5075/78/5/57007\n43A. Thiaville and Y . Nakatani, “Domain-Wall Dynamics in\nNanowiresand Nanostrips,” in Spin Dynamics in Confined Mag-\nnetic Structures III . Hillebrands, B., Thiaville, A., 2006, pp.\n161–205.\n44J. Z. Sun and D. C. Ralph, “Magnetoresistance and spin-transfer\ntorque in magnetic tunnel junctions,” Journal of Magnetism\nand Magnetic Materials , vol. 320, no. 7, pp. 1227–1237, Apr.\n2008. [Online]. Available: http://www.sciencedirect.com/science/\narticle/pii/S0304885307010177\n45R. H. Koch, J. A. Katine, and J. Z. Sun, “Time-Resolved Reversal\nof Spin-Transfer Switching in a Nanomagnet,” Physical Review\nLetters , vol. 92, no. 8, p. 088302, Feb. 2004. [Online]. Available:\nhttp://link.aps.org/doi/10.1103/PhysRevLett.92.088302"    },    {        "title": "1703.07310v2.Using_rf_voltage_induced_ferromagnetic_resonance_to_study_the_spin_wave_density_of_states_and_the_Gilbert_damping_in_perpendicularly_magnetized_disks.pdf",        "content": "Using rf voltage induced ferromagnetic resonance to study the spin-wave density of states and the\nGilbert damping in perpendicularly magnetized disks\nThibaut Devolder\u0003\nCentre de Nanosciences et de Nanotechnologies, CNRS, Univ. Paris-Sud,\nUniversit ´e Paris-Saclay, C2N-Orsay, 91405 Orsay cedex, France\n(Dated: September 18, 2018)\nWe study how the shape of the spinwave resonance lines in rf-voltage induced FMR can be used to extract the\nspin-wave density of states and the Gilbert damping within the precessing layer in nanoscale magnetic tunnel\njunctions that possess perpendicular magnetic anisotropy. We work with a field applied along the easy axis to\npreserve the cylindrical symmetry of the uniaxial perpendicularly magnetized systems. We first describe the\nexperimental set-up to study the susceptibility contributions of the spin waves in the field-frequency space. We\nthen identify experimentally the maximum device size above which the spinwaves confined in the free layer can\nno longer be studied in isolation as the linewidths of their discrete responses make them overlap into a continuous\ndensity of states. The rf-voltage induced signal is the sum of two voltages that have comparable magnitudes: a\nfirst voltage that originates from the linear transverse susceptibility and rectification by magneto-resistance and a\nsecond voltage that arises from the non-linear longitudinal susceptibility and the resultant time-averaged change\nof the exact micromagnetic configuration of the precessing layer. The transverse and longitudinal susceptibility\nsignals have different dc bias dependences such that they can be separated by measuring how the device rectifies\nthe rf voltage at different dc bias voltages. The transverse and longitudinal susceptibility signals have different\nlineshapes; their joint studies in both fixed field-variable frequency, or fixed frequency-variable field configura-\ntions can yield the Gilbert damping of the free layer of the device with a degree of confidence that compares\nwell with standard ferromagnetic resonance. Our method is illustrated on FeCoB-based free layers in which\nthe individual spin-waves can be sufficiently resolved only for disk diameters below 200 nm. The resonance\nline shapes on devices with 90 nm diameters are consistent with a Gilbert damping of 0:011. A single value\nof the damping factor accounts for the line shape of all the spin-waves that can be characterized. This damp-\ning of 0.011 exceeds the value of 0.008 measured on the unpatterned films, which indicates that device-level\nmeasurements are needed for a correct evaluation of dissipation.\nThe frequencies of the magnetization eigenmodes of\nmagnetic body reflect the energetics of the magnetization.\nAs a result the frequency-based methods – the ferromag-\nnetic resonances (FMR)1and more generally the spin-wave\nspectroscopies– are particularly well designed for the metrol-\nogy of the various magnetic interactions. In particular, mea-\nsuring the Gilbert damping parameter \u000bthat describes the\ncoupling of the magnetization dynamics to the thermal bath,\nspecifically requires high frequency measurements. There are\ntwo main variants of these resonance techniques. The so-\ncalled conventional FMR and its modern version the vector\nnetwork analyzer2(VNA)-FMR are established technique to\nharness the coupling of microwave photons to the magneti-\nzation eigenmodes to measure to anisotropy fields1, demag-\nnetizing fields, exchange stiffness3, interlayer exchange4and\nspin-pumping5, most often at film level. More recent meth-\nods, like the increasingly popular spin-transfer-torque-(STT)-\nFMR, are developed6to characterize the magnetization dy-\nnamics of magnetic bodies embodied in electrical devices pos-\nsessing a magneto-resistance of some kind.\nIn conventional FMR or VNA-FMR, the community is well\naware that the line shape of a resonance is more complicated\nthan simple arguments based on the Landau-Lifshitz-Gilbert\nequation would tell. There are for instance substantial contri-\nbutions from microwave shielding effects7(”Eddy currents”)\nfor conductive ferromagnetic films8or ferromagnetic films in\ncontact with (or capacitively coupled to) a conductive layers.\nA hint to these effect is for instance to compare the lineshapes8\nfor the quasi-uniform precession mode and the first perpen-\ndicular standing spin wave modes that occur in different res-onance conditions. Note that the experimental lineshapes are\nalready complex in VNA-FMR despite the fact that the dy-\nnamics is induced by simple magnetic fields supposedly well\ncontrolled.\nIn contrast, STT-FMR methods rely on torques [spin-orbit\ntorques (SOT)9or STT] that have less hindsight that magnetic\nfields or that are the targeted measurements. These torques are\nrelated to the current across the device and the experimental\nanalysis generally assumes that this current is in phase with\nthe applied voltage. This implicitly assumes that the sam-\nple is free of capacitive and inductive responses, even at the\nmicrowave frequencies used for the measurement. A careful\nanalysis is thus needed when the STT-FMR methods analyze\nthe phase of the device response to separate the contribution of\nthe different torques6,10,11. Besides, the quasi-uniform mode\nis often the sole to be analyzed despite that fact that the line\nshapes of the higher frequency modes can be very different10.\nFinally, an external field is generally applied in a direction\nthat is not a principal direction of the magnetization energy\nfunctional12. While this maximizes the signal, this unfortu-\nnately makes numerical simulation unavoidable to model the\nexperimental responses.\nWith the progress in MTJ technologies, much larger\nmagneto-resistance are now available13, such that signals can\nbe measured while maintaining sample symmetries, for in-\nstance with a static field applied collinearly to the magne-\ntization. In addition, high anisotropy materials can now be\nincorporated in these MTJs. This leads to a priori much\nmore uniform magnetic configurations in which analytical de-\nscriptions are more likely to apply. In this paper, we revisitarXiv:1703.07310v2  [cond-mat.mtrl-sci]  4 Sep 20172\nrf-voltage induced FMR in a situation where the symmetry\nis chosen so that all torques should yield a priori the same\ncanonical lineshape for all spinwaves excited in the system.\nWe use PMA MTJ disks of sizes 500 nm, on which a quasi-\ncontinuum of more that 20 different spin-wave modes can be\ndetected, down to sizes of 60 nm where only a few discrete\nspinwave modes can be detected. We discuss the lineshapes\nof the spin-wave signals with the modest objective of deter-\nmining if at least the Gilbert damping of the dynamically ac-\ntive magnetic layer can be reliably extracted. We show that\nthe linear transverse susceptibility and the non-linear longitu-\ndinal susceptibilities must both be considered when a finite dc\nvoltage is applied through the device. We propose a method-\nology and implement it on a nanopillars made with a stan-\ndard MgO/FeCoB/MgO free layer system in which we obtain\na Gilbert damping of 0:011\u00060:0003 . This exceeds the value\nof 0.008 measured on the unpatterned film, which indicates\nthat device-level measurements are needed for a correct eval-\nuation of dissipation.\nThe paper is organized as follows:\nThe first section lists the experimental considerations, includ-\ning the main properties of the sample, the measurement set-\nup and the mathematical post-processing required for an in-\ncreased sensitivity. The second section discusses the origins\nof the measured resonance signals and their main properties.\nThe third section describes how the device diameter affects the\nspin-wave signals in rf-voltage-induced ferromagnetic reso-\nnance. The last section describes how the voltage bias depen-\ndence of the spinwave resonance signals can be manipulated\nto extract the Gilbert damping of the dynamically active mag-\nnetic layer. After the conclusion, an appendix details the main\nfeatures of the spectral shapes expected in ideal perpendicu-\nlarly magnetized systems.\nI. EXPERIMENTAL CONSIDERATIONS\nA. Magnetic tunnel junctions samples\nWe implement our characterization technique on the sam-\nples described in detail in ref. 14. They are tunnel junctions\nwith an FeCoB-based free layer and a hard reference system\nbased on a well compensated [Co/Pt]-based synthetic antifer-\nromagnet. All layers have perpendicular magnetic anisotropy\n(PMA). The perpendicular anisotropy of the thick ( t= 2nm)\nfree layer is ensured by a dual MgO encapsulation and an\niron-rich composition. After annealing, the free layer has an\nareal moment of Mst\u00191:8mA and an effective perpendic-\nular anisotropy field \u00160(Hk\u0000Ms)= 330 mT. Before pat-\ntering, standard ferromagnetic resonance measurements in-\ndicated a Gilbert damping parameter of the free layer being\n\u000b= 0:008. Depending on the size of the patterned device,\nthe tunnel magnetoresistance (TMR) is 220 to 250%, for a\nstack resistance-area product is RA = 12 \n:\u0016m2. The de-\nvices are circular pillars with diameters varied from 60 to 500\nnm. The materials, processing and device rfcircuitry were\noptimized for fast switching14spin-transfer-torque magnetic\nrandom access memories (STT-MRAM15) ; the quasi-static\ndcRF50 ΩPulse modulation ac, 50 kHz\n50 ΩVac ~ 40 mVLI75AamplifierMSG3697synthetiser\nK2400sourcemeter\nSR830lock-inamplifier×100-10 dB attenuation~ 0 dBm~ 14 µAdc10 mVdcMTJ device300 nm~700 ΩFIG. 1. (Color online). Sketch of the experimental set-up with an\n300\u0002300\u0016m2optical micrograph of the device circuitry. The given\nnumbers are the typical experimental parameters for a 300 nm diam-\neter junction. Inset: resistance versus out-of-plane field hysteresis\nloop for a device with 300 nm diameter.\ndcswitching voltage is \u0019600mV . In the present report, the\napplied voltages shall never exceed 100 mV to minimize spin-\ntransfer-torque effects. The fields will always be applied along\n(z) which is the easy magnetization axis. The sample will be\nmaintained in the antiparallel (AP) state.\nB. Measurement set-up\nThe pillars are characterized in a set-up (Fig. 1) inspired\nfrom spin-torque diode experiments6but an electrical band-\nwidth increased to 70 GHz. The objective is to identify the\nregions in theffrequency, fieldgspace in which the magneti-\nzation is responding in a resonant manner. The device is at-\ntacked with an rfvoltageVrf. A 10 dB attenuator is inserted\nat the output port of the synthesizer to improve its impedance\nmatching so as to avoid standing waves in the circuit. This im-\nproves the frequency flatness of the amplitude of the stimulus\narriving at the device. To ease the detection of the sample’s re-\nsponse, the rfvoltage is pulse-modulated at an acfrequency\n!ac=(2\u0019) = 50 kHz (Fig. 1). The current passing through\nthe MTJ has thus frequency components at the two sidebands\n!rf\u0006!ac. The acvoltage which appears across the device\nis amplified and analyzed by a lock-in amplifier. We shall\ndiscuss the origin of this acvoltage in section II. Optionally,\nthe device is biased using a dcsourcemeter supplying Vdcand\nmeasuringIdc.\nFigure 2 shows a representative map of thedVac\ndHzresponse\nobtained on a pillar of diameter 300 nm with Vdc= 10 mV.\nAs positive fields are parallel to the free layer magnetization,\nthe spin waves of the free layer appear with a positive fre-\nquency versus field slope, expected to be the gyromagnetic3\n⦰ 300 nm\nFIG. 2. Field derivative of the rectified voltagedVac\ndHzin the\nffrequency-fieldgparameter space for a 300 nm diameter device in\nthe AP state when the field is parallel to the free layer magnetization.\nThe linear features with positive (resp. negative) slopes correspond\nto free layer (resp. reference layers) confined spin-wave modes.\nBlack and white colors correspond to signals exceeding \u00060:01V/T.\nThe one-pixel high horizontal segments are experimental artefacts\ndue to transient changes of contact resistances.\nratio\r0of the free layer material (see appendix). Conversely,\nthe reference layer eigenmodes appear with a negative slope,\nexpectedly\u0000\r0, where this time \r0is gyromagnetic ratio of\nthe reference layer material combination. Working in the AP\nstate is thus a convenient way to easily distinguish between\nthe spinwaves of the free layer and of the reference layers.\nNote that the gyromagnetic ratios \r0of the free layer mode\nand the reference layer modes differ slightly owing to their\ndifference chemical nature. The free layer has a Land ´e factor\ng= 2:085\u00060:015where the error bar is given by the precision\nof the field calibration; the reference layer modes are consis-\ntent with a 1.2% larger gyromagnetic ratio. The accuracy of\nthis latter number is limited only by the signal-to-noise ratio\nin the measurement of the reference layer properties. Looking\nat Fig. 2, one immediately notices that the linewidths of the\nreference layer modes are much broader than that of the free\nlayer. While the linewidh of the reference layer modes will not\nbe analyzed here, we mention that this increased linewidth is\nto be expected for reference layers that contain heavy metals\n(Pt, Ru) with large spin-orbit couplings, hence larger damping\nfactors16.C. Experimental settings\nIn practice, we choose an applied field interval of\n[\u0000110;110mT]that is narrow enough to stay in a state whose\nresistance is very close to that of the remanent AP state. The\nfrequency!rf=(2\u0019)is varied from 1 to 70 GHz; we gener-\nally could not detect signals above 50 GHz. The practical\nfrequency range 2\u0019\u000250GHz=\r0\u00191:6T is much wider that\nour accessible field range. For wider views of the experimen-\ntal signals (for instance when the spin-wave density of states\nis the studied thing), we shall thus prefer to plot them versus\nfrequency than versus field. The response is recorded pixel by\npixel in in theffrequency, fieldgspace. The typical pixel size\nisf\u000eHz\u0002\u000efg=f1 mT\u000250 MHzg. The field and frequency\nresolutions are thus comparable (indeed 2\u0019\u0002\u000ef=\r 0= 1:7\nmT).\nD. Signal conditioning\n1. Mathematical post-treatments\nFinally, despite all our precautions to suppress the rectify-\ning phenomena that do not originate from magnetization dy-\nnamics, we have to artificially suppress the remaining ones.\nThis was done by mathematical differentiation, and we gener-\nally plotdVac\ndfordVac\ndHzin the experimental figures (Figs. 2-5).\n2. Dynamic range improvement by self-conformal averaging\nA special procedure (Fig. 3) is applied when a better signal\nto noise ratio is desired while the exact signal lineshape and\namplitudes are not to meant to be looked at. This procedure\nharnesses the fact that the normalized shape of the sample’s\nresponse is essentially self-conformal when moving across a\nline withd!\ndHz=\r0in theffrequency, fieldgparameter space\n(see appendix). The procedure consists in calculating the fol-\nlowing primitive:\ns(f0) =1\n2\r0HmaxzZ\ncontourdVac\ndHzdf ; (1)\nin which the integration contour is the segment linking the\npoints (\u0000Hmax\nz;f0\u0000\r0Hmax\nz) and (Hmax\nz;f0+\r0Hmax\nz)\nin theffield, frequencygparameter space. Such contours ap-\npear as pixel columns in Fig. 3(b). This primitive (eq. 1) is\nefficient to reveal the free layer spin-wave modes that yield an\notherwise too small signal. For instance when only 7 modes\ncan be detected in single field spectra [Fig. 3(a)], the aver-\naging procedure can increase this number to typically above\n25. The averaging procedure is also effective in suppressing\nthe signals of the reference layer as these laters average out\nover a contour designed for the free layer mode when in the\nAP state. However as the linewidth of the free layer modes\nis proportional to the frequency, it is not constant across the\ncontour; the higher signal to noise ratio is thus unfortunately4\nf\u0000\u00000Hz2⇡Hz(b)(a)\nFIG. 3. (Color online). Illustration of the dynamic range improve-\nment by self-conformal averaging (section I D 2). The procedure is\nimplemented on a 300 nm diameter device to evidence the free layer\nmodes. Bottom panel: field derivative of the acsignal in the rotated\nframe in which the modes withdf\ndHz=\r0\n2\u0019should appear as vertical\nlines. Top panel: comparison of a single field frequency scan (red)\nwith the average over all scans as performed in the !=\r0Hzdi-\nrection. Note that the signal of the lowest frequency mode (which\ncorresponds to the quasi-uniform precession) disappears near zero\nfield, at 5 mT (see the apparent break in the middle of the most left\nline in the bottom panel).\nobtained at the expense of a distorted (and unphysical) line-\nshape. Note also that this procedure can not be applied to the\nquasi-uniform precession mode as will be explained in section\nII D 2).\nII. ORIGIN AND NATURE OF THE RECTIFIED SIGNAL\nLet us now discuss the origin of the demodulated acvolt-\nage. In this section, we assume that the reference layer mag-\nnetization is static but not necessarily uniformly magnetized.\nWe can thus express any change of the resistance by writing\n\u000eR=\u000eR\n\u000eM\u000eM where\u000ehas to be understood as a functional\nderivative with respect to the free layer magnetization distri-\nbution.\nA. The two origins of the rectified signals\nTheacsignal can contain two components V1;acandV2;ac\nof different physical origins17. The first component is the\n’standard’ STT-FMR signal: the pulse-modulated rfcurrent is\nat the frequency sidebands !rf\u0006!acand it rectifies to acany\noscillation of the resistance \u000eRrfoccurring at the frequency\n!rf. We simply have V1;ac=\u000eRrf\u0002i!rf\u0006!ac.\nThe second acsignal (V2;ac) is related to the change of the\ntime-averaged resistance due to the population of spinwavescreated when the rfcurrent is applied12. Indeed the time-\naveraged magnetization distribution is not the same when the\nrfisonoroff. This change of resistance \u000eRaccan revealed by\nthe (optional) dccurrentIdcpassing through the sample, i.e.\nV2;ac=\u000eRac\u0002Idc.\nNote that a third rectification channel18can be obtained by\na combination of spin pumping and inverse spin Hall effect in\nin-plane magnetized systems19. This third rectification chan-\nnel yields symmetric lorentzian lines when applied to PMA\nsystems in out-of-plane applied fields (see eq. 23 in ref. 18).\nBesides, the spin-pumping is known to be largely suppressed\nby the MgO tunnel barrier20, such that we will consider that\nwe can neglect this third rectification channel from now on. In\nsummary, we have:\nV1;ac=Vrf\nR+ 50\u000eR\n\u000eM\u000eMrf and (2)\nV2;ac=Vdc\nR+ 50\u000eR\n\u000eM\u000eMac (3)\nThis has important consequences.\nB. Compared signal amplitudes in the P and AP states\nThe first important consequence of Eq. 2 and 3 is that the\nsignal amplitude depends on the nature of the micromagnetic\nconfiguration. As intuitive, both V1;acandV2;acscale with\nhow much the instantaneous device resistance depends on its\ninstantaneous micromagnetic configuration. This is expressed\nby the sensitivity factor\u000eR\n\u000eMwhich is essentially a magneto-\nresistance. We expect no signal when the resistance is insen-\nsitive to the magnetization distribution at first order (i.e. when\n\u000eR\n\u000eM\u00110).\nIn our samples, the shape of the hysteresis loop (Fig. 1)\nseems to indicate that the free layer magnetization is very uni-\nform when in the Parallel state. Consistently, the experimental\nrectified signal were found to be weak signals when in the P\nstate. Conversely, there is a pronounced curvature in the AP\nbranch of the R(Hz)hysteresis loop (see one example in the\ninset of Fig. 1). This indicates that the resistance is much de-\npendent on the exact magnetization configuration when in the\nAP state. Consistently, this larger\u000eR\n\u000eMin the AP state is proba-\nbly the reason why the rectified signal is much easier to detect\nin the AP state for our samples.\nC. Bias dependence of the rectified signals\nThe second important consequence of Eqs. 2-3 concerns the\ndependence of the rectified acsignalsV1;acandV2;acon the\ndcandrfstimuli. As\u000eMrfscales with the applied rftorque\naccording to a linear transverse susceptibility ( <e(\u001fxx), see\nappendix),V1;acis expected to scale with the rfpowerVrf2\n(see Eq. 2) independently from the dcbias, i.e. we have\nV1;ac/Vrf2:5\nIn contrast,\u000eMacis related to a longitudinal susceptibility and\nis thus quadratic with the rftorques (see appendix). Using\nEq. 3, we thus expect the following bias dependence\nV2;ac/Vrf2Vdc:\nD. Peculiarities of the quasi-uniform precession (QUP) mode\nThe last important consequence of Eqs. 2-3 concerns\nspecifically the quasi-uniform precession (QUP) mode that\nshows a peculiar acsignal.\n1. Quasi-absence of STT-FMR like signal for the quasi-uniform\nprecession mode\nIn the idealized macrospin case (see appendix) the uniform\nprecession is perfectly circular with no rfvariation of Mzat\nany order. If the fixed layer was uniformly magnetized along\nexactly (z), this would lead V1;ac= 0 such that the signal of\nthe QUP mode would be given by purely V2;ac. This qual-\nitatively ’pure V2;accharacter’ is confirmed experimentally\nby the fact that the signal of the QUP mode systematically\nchanges sign with Vdcin our sample series (not shown).\n2. Strong dependence of the QUP signal amplitude with the\napplied field\nIn addition, the experimental signal of the quasi-uniform\nmode is found to disappear at low fields [see Figs. 2, 3(b) and\n4(a, b)] exactly at the apex of the AP branch of the hysteresis\nloop (Fig. 1), i.e. for the field leading specifically todR\ndHz= 0.\nMoreover, the amplitude of the acsignal of the QUP mode ap-\npears to be essentially linearly correlated with the loop slope\ndR\ndHz(not shown). For instance, the V2;acof the QUP mode\nchanges sign when the applied field crosses the apex of the\nR(Hz)loops (Fig. 1).\nThe reason stems probably from the sensitivity factor\u000eR\n\u000eM\nand its correlation with the loop slopedR\ndHz; in some sense, a\nlarge loop slope should translate in a large sensitivity factor.\nWhile a numerical evaluation of this correlation goes beyond\nthe scope of this paper, we stress that if the magnetization was\nperfectly uniform there would be a one-to-one correlation be-\ntween loop slopedR\ndHzand magnetoresistance sensitivity fac-\ntor\u000eR\n\u000eM. This trend remains qualitatively true for the QUP\nmode. Indeed as the hysteresis loop is monitoring the spatial\naverage of the magnetization, it is more insightful for the uni-\nform mode than for any other (higher order) modes whose dy-\nnamic profiles spatially average to essentially zero21; the cor-\nrelation betweendR\ndHzand\u000eR\n\u000eMis thus expected to be maximal\nfor quasi-uniform changes of the magnetization configuration.\nWhile this property – the disappearance of the QUP mode\nsignal whendR\ndHz= 0 – can be used to distinguish the QUP\nmode from the higher order spin waves, the pronounced field\ndependence of the QUP signal complicates the analysis, as it\nprevents to conveniently analyze the field derivative of the acsignal (I D 1). In the remainder of this paper we shall focus on\nonly higher order modes to avoid such difficulties.\nE. Signals for non-uniform spin-waves\nBefore analyzing the spin-wave density of states (section\nIII), let us comment on the amplitude of the STT-FMR-like\nsignalV1;acfor the non-uniform spin-waves. In the perpen-\ndicular magnetization state, these spin-waves have a circular\nprecession22. By symmetry, the resistance is not expected to\nchange during a period of circular precession when in the per-\nfect collinear cases and for radial spin waves maintaining the\ncylindrical symmetry of the system. In other words, when\nthe dynamical magnetization of the eigenmode maintains the\ncylindrical symmetry and when the free and reference layers\nequilibrium magnetizations follow ~Mfree\u0002~Mref=~0every-\nwhere in the (xy) plane, with \u0002being the conventional vec-\ntor product) the device resistance is not expected to oscillate.\nWhile we can not identify to what extent we depart from this\nideal situation, we speculate that this perfect collinearity does\nnot happen in practice at least because of finite thermal fluctu-\nations. The effect of thermal fluctuations on the device resis-\ntance is not averaged out for non-uniform spin-waves, while\nit could be essentially averaged out for the QUP mode ana-\nlyzed earlier. In practice a finite variation of the resistance\n\u000eRrf6= 0 is always present during a precession period for a\nnon-uniform spin-wave. This provides a finite sensitivity to\nany spin-wave mode. This resistance variation at !rfhas the\nspectral shape of a transverse susceptibility term <e(\u001fxx)(see\nappendix).\nIII. SPIN WA VE DENSITY OF STATES AGAINST\nLATERAL CONFINEMENT\nAny reliable analysis of a spectral lineshape or linewidth re-\nquires to determine priorly how many spin-waves contribute\nto the lineshape under study. Therefore, before discussing the\nlineshapes of the individual spin-wave modes, let us determine\nhow the lateral confinement influences the measured rectified\nsignal. The impact of the device diameter on the spectral sig-\nnals is reported in Fig. 4.\nA. Spin-waves within the references layers\nFig. 4 indicates that the modes of the reference layers have\nfrequencies that are almost not affected by the device diam-\neter. This fact is related to the well compensated character\nof the synthetic antiferromagnet that composes the reference\nlayers. Indeed the internal demagnetizing fields compensate\nto some extent, such that they do not influence the frequency\nof the acoustical mode of a SAF as much as the anisotropies\nand the interlayer exchange couplings do.6\nSignal Spectral Peak-to-peak Full Width Zero crossings\nshape separation at Half Maximum separation\nExpected signals and their stimulus dependence:\nV1;ac/V2\nrf <e(\u001fxx) 2\u000b! - -\nV2;ac/V2\nrfVdc \u0001Mz - 2\u000b! -\nSignal extraction procedure from experiments:\nd\ndHzVexp\n1;acestimated fromdVexp\nac\ndHz\f\f\f\f\nVdc=0d<e(\u001fxx)\nd!- - 2\u000b!\nd\ndHzVexp\n2;acestimated from\"\ndVexp\nac\ndHz\f\f\f\f\nVdc6=0\u0000dVexp\nac\ndHz\f\f\f\f\nVdc=0#\nd=m(\u001fxx)\nd!2p\n3\u000b! - -\nTABLE I. Summary of the expected lineshapes and linewidths for the different signals that can be encountered in rf-voltage-induced FMR\nexperiments. An hyphen is inserted when the concept is not applicable.\nB. Spin-waves within the free layer\nConversely, the frequencies of the modes of the free layer\nare strongly affected by the device diameter (Fig. 4). First,\nthe modes are pushed to higher frequencies as the device\nis shrunk. At remanence, the lowest frequency mode is at\nfQUP= 12:3GHz for a diameter of 500 nm; it reaches 19.5\nGHz for 60 nm devices (not shown). Second, the frequency\nspacing between the free layer modes increases substantially\nwhen downsizing the device.\nThe first effect – increase of fQUPat downscaling – is in-\ndicative of a dependence of some effective fields with the\ndevice diameter. Among the effective fields, the only ones\nthat vary with the diameter are the exchange fields (positive\ncontribution to the frequency fQUPif magnetization is non-\nuniform), the demagnetizing fields (positive contribution to\nthe frequency fQUPat downscaling) and the local effective\nanisotropy fields in case some process damages alter locally\nthe interface anisotropy at the perimeter of the free layer (neg-\native contribution to the frequency fQUPat downscaling) or\nalter the local magnetization of the rim of the free layer (pos-\nitive contribution to the frequency fQUPat downscaling). The\nexchange fields are related to the non uniformities of either the\nstatic configuration –the fact that the AP state is not perfectly\nuniform as inferred previously from the loop in Fig. 1– or non\nuniformities of the dynamic magnetization –i.e. the fact that\nthe quasi-uniform mode is not a strictly uniform mode–. If\nthe frequency increase was due to the sole demagnetizing ef-\nfects, it could be estimated from the demagnetizing factors of\ndisks23which areNz\u00191\u0000(3\u0019=8)t=a, wheretandaare\nthe thickness and radius of the free layer. However a fQUP\nagainst 1=aplot (not shown) has a perceivable curvature near\nall sizes; an unwise linear fit through fQUPagainst the ex-\npected\rNzwould give a slope of \u00192:5T, which is obvi-\nously too large for the magnetization of the free layer. This\nindicates that the sole change of the global shape anisotropy\nwith the device diameter is insufficient to account for the in-crease offQUPat downscaling: exchange contributions or non\nuniformities induced by process damages also contribute to\nthe frequencies. Exchange contributions should not contribute\nfor the largest devices, however even for those devices the ex-\nperimental frequencies are larger than the ones expected from\nglobal shape anisotropy only, which argues for some process\ndamages. Since the TMR is almost independent of the de-\nvice size14, we can reasonably assume that the MgO/FeCoB\ninterface is not substantially affected by the patterning and\nthat consequently the interface anisotropy is essentially pre-\nserved at the rim of the free layer. We conclude that part of\nthe increase of fQUPat downscaling is due to a reduced mag-\nnetization (magnetically ”dead” or weak zone) near the edges\nof the free layer. This interpretation is probably very much\nstack and process technology dependent, hence it should not\nbe considered as general.\nThe second effect – the increased frequency spacing be-\ntween the modes at small diameters – is the expected effect\nof the confinement of the spin waves and the resulting in-\ncrease of the exchange contribution to the mode frequencies17.\nThe eigenmodes of perpendicularly magnetized circular disks\nare well understood and can be described analytically in a\nsemi-quantitative manner21,24–28. The frequency spacing be-\ntween the lowest frequency modes scales with \r0HJwhere\nHJ=2Ak2\n\u00160MSis a generalized exchange field with Athe ex-\nchange stiffness. The effective wavevector kis reminiscent of\nthe lateral confinement and reads k2= (u2\n2\u0000u2\n1)=a2\u00199=a2\nwhereu1andu2are the first zeros of the first and second\nBessel functions21. The lowest frequency spinwave modes\ncan be resolved only if their frequency spacing is comparable\nor greater than their linewidth 2\u000b\r0(Hz+Hk\u0000Ms+HJ)\n(see appendix).\nThis condition can be used to define a critical device diam-\neter:\na2\ncrit=9A\n\u000b\u00160MS(Hz+Hk\u0000Ms)(4)\nFor large devices with a\u001dacritwe expect to observe a quasi-7\ncontinuum of overlapping modes above fQUP, while discrete\nnon-overlapping modes are anticipated in the opposite limit.\nTypical parameters of an FeCoB-based free layer include a\nmagnetization of \u00160Ms= 1:2T and a Gilbert damping of29\n\u000b= 0:01. From the quasi-uniform mode frequency, we can\nget our effective anisotropy which is Hk\u0000Ms= 330 kA/m.\nIf the exchange stiffness of the free layer was bulk-like (i.e.\nA= 22 pJ/m) like in ref. 17, the critical diameter would\nbe2acrit= 444 nm. In practice the small frequency spac-\nings between the modes of our samples indicates that the ex-\nchange stiffness of our free layer is in the range of 6-7 pJ/m\ni.e. well below the bulk value. This estimate of the exchange\nstiffness was deduced assuming perfectly pinned boundary\nconditions for the spin-waves at the device edge, which is a\nquestionable30assumption. However the exchange stiffness is\nanyway weak in the free layer and this can be also qualita-\ntively seen directly from the spin wave spectroscopy: indeed\nthe frequency spacing of the lowest modes of the reference\nlayer system is typically twice larger that the frequency spac-\ning of the lowest modes of the free layer [see for instance\nFig. 3(b)]. While the reason for this small value of the free\nlayer exchange stiffness is not entirely clear, we emphasize\nthat having such a small exchange stiffness is not uncommon\nin magnetic systems that comprise only a small number of\natomic layers, starting for instance from 2 pJ/m for a single\nlayer of iron31. Anyway with these parameters, we expect\na clear separation of the lowest frequency modes at rema-\nnence provided that the device diameter is much smaller than\n2acrit= 250 nm.\nIn practice for 300 and 500 nm devices a fine structure\ncan still be detected in the spin-wave density of states [see\nFig. 4(c)] but it is hard to count the modes and guess their\nfrequencies out of this fine structure. In the remainder of this\npaper, we shall thus only consider devices of diameter less\nthan 200 nm, in which the different spin-wave modes can be\nunambiguously resolved [see Fig. 4(e-f)].\nIV . LINESHAPE EVOLUTIONS WITH BIAS AND\nEXTRACTION OF THE GILBERT DAMPING\nLet us now compare the shapes of the experimental rectified\nsignal with those expected (see appendix). For that purpose\nwe harness the different bias dependences (Table I) of the rec-\ntified signals V1;acandV2;acto isolate each of them in the\nexperimental signal Vac. We identify V1;acto the experimen-\ntal curveVacmeasured at Vdc= 0, and we construct an esti-\nmate ofV2;acby subtracting Vacmeasured at Vdc= 0 from\nthat measured at Vdc6= 0 (see Table I). Note because of this\nsubtraction, any dependence of the spin-wave frequency with\nthedcvoltage will prevent the measurement of the voltage de-\npendence of the damping factor or of the voltage dependence\nof the exciting torques.\nWe illustrate this procedure in Fig. 5 in which we plot the\nfield derivatives of the so-calculated V1;acandV2;acrectified\nsignals in both fixed field or fixed frequency experimental con-\nditions for a device of diameter 90 nm. We center the curves\non the second lowest frequency eigenmode since it provides\n(a)(b)\n(c)(e)(f)(d)FIG. 4. (Color online). Dependence of the field derivative of the ac\nvoltage over the device diameter. Top panels: full spectral depen-\ndence in the window [6;40GHz]\u0002[\u000090;90mT]for 90 nm (left)\nand 500 nm (right) diameter devices. Bottom panels: frequency de-\npendence of the field derivative of the acvoltage after self-conformal\naveraging for various device sizes. The signal amplitudes have been\nnormalized to ease their comparison.\nthe largest signals and it is reasonably separated from both the\nquasi-uniform precession mode and from the other high order\nmodes. The obtained experimental V1;accurves [see Fig. 5\n(a) and (d)] have the expected line shapes (see appendix) with\na negative peak surrounded by two tiny positive halos (areas\nshaded in red). The separation between the two zero crossings\nis9:5\u00060:5mT or 285\u000610MHz. The obtained experimental\nV2;accurves also have the expected line shape of the deriva-\ntive of a Lorentzian distribution (see appendix). As expected,\nthe sign of the response changes with the dcbias voltage. The\nseparation between the positive and negative maxima of the\ndistribution are 6:1\u00060:5mT and 170\u000610MHz.\nThese four different ways of measuring the linewidths\n[Fig. 5 (a, d, b, e)] are consistent with a free layer damping\nof\u000b= 0:011\u00060:0003 . Indeed this value of damping would\npredict linewidths of 2\u000bf= 295 MHz and 2\u00160\u000b!=\r 0=\n9:54mT (materialized as black bars in [Fig. 5 (a, d) and (d)])\nand1:15\u000bf= 171 MHz or 1:15\u000b!=\r 0= 6:21mT (material-\nized as blue bars in [Fig. 5 (b, c, e, f)]).\nThis proposed value of damping is also consistent with the\nlinewidths of higher order spin waves that appear at larger fre-\nquencies but with a lower signal. This is illustrated in fig. 6\nwhere a comparison is drawn between the values of the damp-\ning estimated for each applied field from the second (and most\nintense) mode and from the third mode for a device of diame-\nter 80 nm. The used procedure is a direct fit of the experimen-8\ntal lineshapes to the derivative of Eq. 7 with the damping, the\nresonance frequency and the signal amplitude as free parame-\nters. For this specific device, the estimates of the damping pa-\nrameter are subjected to a random error of standard deviation\n0.0018 around a mean value of \u000b= 0:0119 . It is interesting\nto note that the non-local contributions32,33to the damping ex-\npected for the relatively large wavevectors of the second and\nthird spin-wave modes seem to be too small to be observed\nin our samples. Note that as mentioned earlier, the same pro-\ncedure can in principle not be applied to the quasi-uniform\nmode since it exhibits a strong dependence of the mode am-\nplitude with the field which invalidates the procedure to some\nextent. For the sake of completeness of this paper, we have\nanyway fitted the experimental QUP lineshapes with the field\nderivative of Eq. 8; this is not possible near zero field, as the\ncorresponding signal vanishes. The value of the Gilbert damp-\ning that would be illegitimately deduced would be 0:01, i.e.\n20% lower than the correct value. Besides, the estimates from\nthe QUP mode would exhibit a substantially larger spread in\nthe fit results [compare the histograms in in fig. 6(b)]. For\nthese two reasons, we consider that the reliable estimate of the\ndamping is the one extracted from the non-uniform modes.\nAbove 100 mV of dcbias, the amplitude of the constructed\nexperimental V2;acstart to depart from proportionality with\nVdcand a frequency shift is observed, as expected when dc\nfield-like spin torques are applied. This comes with by a dis-\ntortion of the line shape, probably linked to the modification of\nthe spin waves lifetimes by spin-transfer torque as commonly\nobserved in in-plane magnetized MTJs34.\nV . SUMMARY AND CONCLUSIONS\nIn this paper, we have studied how to use rf-voltage-\ninduced ferromagnetic resonance to study the spin-wave den-\nsity of states and the Gilbert damping in perpendicularly mag-\nnetized disks embodied in magnetic tunnel junctions. We have\napplied the field along the easy axis to preserve the cylindrical\nsymmetry of the magnetization energy functional. The inter-\nest of this configuration is that all the current-induced torques\nthat potentially excite the dynamics yield the same type of\nsusceptibility spectral shape. Additionally, this configuration\nis the sole in which the applied field and the frequency play\nsimilar roles near FMR so that consistency crosschecks be-\ntween variable-field and variable-frequency experiments can\nbe performed to reveal and suppress potential experimental\nartefacts.\nWorking in a situation in which the fixed layer and the free\nlayer are oppositely magnetized in a convenient way to clas-\nsify the spin-waves according to their hosting layer, as the two\nsub-systems have opposite eigenmode frequency-versus-field\nslopes. The dcbias dependence of the signal of the quasi-\nuniform mode is peculiar and can be used to ambiguously\nidentify the free layer quasi-uniform mode in the manifold of\nspin-waves. The non-uniform (higher order) spin-waves are\neasier to analyze, as their amplitudes weakly depend on the\napplied field so that field differentiation can be used safely for\nbackground subtraction. Optionally, the dynamic range of theexperiment can be improved by self-conformal averaging of\nthe resonance spectra.\nThe unambiguous identification of the spin-wave frequen-\ncies requires devices that are sufficiently small to avoid that\nthe spinwave modes overlap into a quasi-continuous density\nof states. The critical device size is set by the exchange\nstiffness, the damping, the magnetization and the effective\nanisotropy field. In practice, device diameters below 200 nm\nare needed in our low-damped FeCoB-based PMA system.\nFor each spin-wave mode, the rf-voltage-induced spin-\nwave spectra contain contributions from two different phys-\nical mechanisms. The first one is the standard STT-FMR-like\nsignal, whose spectral shape is a linear transverse susceptibil-\nity term. It is independent from the dcvoltage applied across\nthe MTJ. The second one is a variation of the time-averaged\nmagnetic configuration when the rfvoltage is applied. It is\nproportional to the dcvoltage applied across the MTJ and it\nhas the spectral shape of a non-linear longitudinal susceptibil-\nity. The bias dependence can be used to separate these two\nsignals. The analysis of their spectral shape yields the Gilbert\ndamping within the precessing layer. A single value of the\ndamping factor is found to account for the lineshapes of all\nstudied spin-waves.\nThe spectra of rf-voltage-induced rectified voltages for a\nvanishing dcvoltage bias are in principle sufficient to get the\nGilbert damping of the dynamically active layer. However\nas microwave methods are prone to artefacts, a consistency\ncheck exploiting the bias dependence of the resonance spectra\nis useful for a consolidation of the numerical estimation of the\ndamping.\nThis work was supported in part by the Samsung Global\nMRAM Innovation Program, who provided also the samples.\nCritical discussions with Vladimir Nikitin, Jean-Paul Adam,\nJoo-V on Kim and Paul Bouquin are acknowledged.\nVI. APPENDIX: SUSCEPTIBILITIES IN AN IDEALIZED\nPMA FILM\nIn this appendix, our aim is to determine the transverse and\nlongitudinal microwave susceptibility versus frequency fand\nstatic fieldHzfor a PMA film as a response to an harmonic\ntransverse field hxcos(!t)~ ex. We shall write the equations\nwith this transverse field hxbut any other effect that yields a\ntorque possessing a component transverse to the static mag-\nnetization will yield similar lineshapes. This includes current-\ninduced Oersted-Ampere fields but also Slonczewski STT and\nfield-like STT as soon as the reference layer magnetization\n~Mrefis not strictly collinear with that of the free layer ~Mfree.\nThe susceptibility tensor will be used to deduce the shape of\nthe line expected in rf-voltage-induced FMR, as summarized\nin Table I. Throughout this appendix, we assume a dcfield\nHzperfectly perpendicular to the plane and a free layer mag-\nnetization~M=Mz~ ez+mx~ ex+my~ ey, where the transverse\nterms are assumed small and written as complex numbers in\nthe frequency space. For conveniency, we will use the nota-\ntionH0=Hz+Hk\u0000Ms. We shall also write the frequencies\nin field units and define !0=!=\r 0. This is meant to empha-9\nddHzVexp2,a cforVdc= 100 mVddHzVexp1,a cforVdc=0\nddHzVexp2,a cforVdc=\u0000100 mV\nFIG. 5. (Color online). Rectified acsignals versus frequency at fixed\napplied field (left panels) or versus field at fixed frequency (right\npanels). The plotted data ared\ndHzV1;ac(black, panels a and d) and\nd\ndHzV2;ac(blue, panels b, c, e and f) as estimated according to the\nformulas of Table I. The arbitrary vertical scale is the same for all\npanels. The dotted black lines are separated by 9.5 mT and 270 MHz.\nThe dotted blue lines are separated by 6.1 mT or 170 MHz. These\ndotted lines correspond to the expected linewidth for a damping of\n0.011. The panel (b) is measured for a device different from that of\nthe other panels.\nsize the fact that the generalized field H0and the generalized\nfrequency!0play very similar roles in the FMR of perpendic-\nularly magnetized macrospin when near resonnance. We shall\nsystematically assume that \u000b\u001c1and only keep the lowest\norder of the damping terms in the equations. In sections VI A\n-VI C we make the macrospin approximation. This approxi-\nmation is discussed in section VI D.\nA. Transverse linear susceptibility\nFollowing the usual procedure, we project the linearized\nLandau-Lifshitz-Gilbert equation along ~ exet~ ey:\n\u001a\nH0(my+\u000bmx) +imx!0=hxMs\u000b\nH0(mx\u0000\u000bmy)\u0000imy!0=hxMs\nWe then invert this system of equations to get the susceptibil-\nitiesmx=\u001fxxhxandmy=\u001fyxhx. They are:\n\u001fxx=MsH0\nH02\u0000!02+ 2i\u000bH0!0(5)\nand\n\u001fyx=\u0000iMs!0\nH02\u0000!02+ 2i\u000bH0!0(6)\n-100- 500 5 01 000.0000.0050.0100.0150.0200\n.0080 .0100 .0120 .014Third modeSecond modeOccurrence (a. u.)D\namping parameterQuasi-uniform mode(\nb)DampingF\nield (mT)(a)FIG. 6. (Color online). Statistical view of the Gilbert damping\nparameters obtained from the fitting of the three lowest spin-wave\nresonances of a device of 90 nm diameter. For each applied field,\nthe spectral line shapes of the quasi-uniform mode is fitted using\nd\ndHzV2;acfunction (red symbols) while the lineshapes of the two\nnext modes are fitted withd\ndHzV1;acfunction (blue and black). The\npanel (a) gathers the values of the damping parameters that best fit\neach spectrum recorded at a given applied field. Panel (b) displays\nthe histogram of the distribution of these estimates of the Gilbert\ndamping for the non-uniform modes (dashed-dotted line histogram\nand its blue Gaussian guide to the eye, of half width 0.0009) and\na Gaussian fit (red curve) of the distribution for the quasi-uniform\nmode, of half width 0.0015.\nSeveral points are worth to remind:\n(i) The dctransverse susceptibilities are \u001fdc\nxx=Ms=H0and\n\u001fdc\nyx= 0.\n(ii) The in-phase transverse susceptibility \u001fxxis peaked at\nthe FMR condition !0=H0. It reaches\u001fFMR\nxx =\u00001\n2i\u000b\u001fdc\nxx.\n(iii) When near the FMR condition, we have \u001fyx\u0019\u0000i\u001fxx.\nHence the two transverse components of the magnetization\nare in quadrature and the forced precession is essentially\ncircular. Note that this holds true despite the fact that\nthe pumping field is linearly polarized (i.e. along (x)only).\nIt would also remain true for other (e.g STT) pumping torques.\nFrom Eq. 5 we deduce the classical expressions for the real\nand imaginary parts of the transverse susceptibility:\n<e(\u001fxx) =MsH0(H02\u0000!02)\n4\u000b2H02!02+ (H02\u0000!02)2(7)\n=m(\u001fxx) =\u00002\u000bMs!0H02\n4\u000b2H02!02+ (H02\u0000!02)2(8)\nThe lineshapes given by the above expressions are shown\nin Fig. 7. Their main properties are summarized in Table I.10\nB. Longitudinal non-linear susceptibility\nLet us now express the non-linear change of the longitu-\ndinal magnetization \u0001Mzthat occurs due to the precession.\nThis can be viewed as an rf-induced reduction of the rema-\nnence. Using the circularity of the precession near the FMR\nresonance and the conservation of the magnetization norm to\nsecond order in mx;y, one gets:\n\u0001Mz\u0019jjhxjj2\n2MSM2\nsH02\n[H02\u0000!02]2+ 4\u000b2H02!02(9)\nwherejjhxjjis the (constant) amplitude of the applied rffield.\nIt is worth noticing that the longitudinal loss of the magneti-\nzation is stationary (constant in time) despite the fact that the\nmagnetization precesses continuously.\nC. Lineshapes and linewidths of the susceptibiliies and their\nderivatives\nThe different susceptibility expressions are plotted in Fig. 7.\nThe lineshape of the functions \u0001Mzand\u001fxxare essentially\ndetermined by their denominator which are the fast varying\nfunctions of eqs. 8 and 9. As \u001fxxand\u0001Mzare two signatures\nof the same resonance process, their denominators are equal\n(see eqs. 8 and 9) and a simple algebra confirms that =m(\u001fxx)\nand\u0001Mzlead to the same frequency or field linewidths (Half\nWidth at Half Maximum) which are:\n\u0001!= 2\u000b!or equivalently, \u0001Hz= 2\u000bH0(10)\nNote that \u0001!(resp. \u0001Hz) is also the frequency (resp. field)\nspacing between the positive maximum and negative maxi-\nmum of<e(\u001fxx)about the FMR condition.\nWe stress that this linewidth is different from that obtained\nby conventional FMR in which people examine the derivative\nof the absorption signald=m(\u001fxx)\ndHzversusHz. The peak-to-\npeak separation of the conventional FMR signal is \u0001Hz=\n2p\n3\u000bH0=2p\n3\u000b!0. The factor 2=p\n3is 1.1547. The spectral\nshapes of<e(\u001fxx)and\u0001Mzas deduced above are to be used\nin the main part of the paper to describe respectively V1;ac\nandV2;ac(Table I).\nD. Linewidth beyond the macrospin approximation\nSome words of caution are needed as the calculations done\nso far the appendix are for the uniform precession mode of a\nmacrospin, while most of our experimental results were ob-\ntained on the higher order (non-uniform) spin-waves. When\nmodeling non-uniform spin-waves, one needs to take into ac-\ncount additional exchange and dipole-dipole terms.\nFor non-uniform spin-waves in perpendicularly magnetized\nsystem, the exchange fields related to the non uniformity of\nthe dynamical magnetizations mxandmycan be added to H0\nto form a new generalized field ~H=H+Hk\u0000Ms+2Ak2\n\u00160MS,\n/s48/s46/s57/s48 /s48/s46/s57/s53 /s49/s46/s48/s48 /s49/s46/s48/s53 /s49/s46/s49/s48/s45/s48/s46/s53/s48/s46/s48/s48/s46/s53/s45/s49/s48/s48/s46/s57/s48 /s48/s46/s57/s53 /s49/s46/s48/s48 /s49/s46/s48/s53 /s49/s46/s49/s48\n/s77\n/s122\n/s82/s101 /s40 /s41\n/s112/s112/s32/s82/s101/s115/s111/s110/s97/s110/s99/s101/s45/s108/s105/s107/s101\n/s71/s101/s110/s101/s114/s97/s108/s105/s122/s101/s100/s32/s102/s105/s101/s108/s100/s32/s111/s114/s32/s102/s114/s101/s113/s117/s101/s110/s99/s121/s32/s40 /s39/s32/s111/s114/s32/s72/s39/s41/s70/s87 /s72/s77/s100/s100/s73/s109 /s40 /s41\n/s100/s100 /s77\n/s122\n/s111/s114/s100/s100/s82/s101 /s40 /s41\n/s73/s109 /s40 /s41/s32\n/s32/s76/s111/s115/s115/s45/s108/s105/s107/s101/s111/s114FIG. 7. (Color online). Transverse susceptibilities, longitudinal sus-\nceptibility and their derivatives in the PMA macrospin model. The\ncurves are plotted for \u000b= 0:02and a resonance condition of unity.\nThe responses amplitudes have been normalized to ease the compar-\nison between the different lineshapes. The black horizontal segment\nin the bottom panel is the FWHM of =m(\u001fxx)and\u0001Mzor equiva-\nlently the peak-to-peak separation of <e(\u001fxx)(also sketched as the\nblack square dots). The blue segment sketches the peak-to-peak sep-\naration ofd=m(\u001fxx)\nd!0 (also sketched as the empty blue square dots).\nThe area shaped in red is the positive halo that surrounds the main\n(negative) peak ind<e(\u001fxx)\nd!0.\nwherekis a generalized wavevector21,26,28. The additional\nexchange contributions act on mxandmyon equal footing,\nhence they maintain the circularity of the precession. The\nGilbert linewidth for a non-uniform spin-wave is simply35,36:\n\u0001!=\u000b!k@!k\n\r0@~H(11)\nWhere this equation holds as ~His a circular term. As a result\nof Eq. 11 the proportionality of an eigemode linewidth to its\nfrequency (Eq. 10) is not broken by the exchange contribu-\ntions in non-uniform spin-waves.\nConversely, the directional nature of the dipole-dipole in-\nteraction is such that the related effective fields act differently\non the two dynamical magnetizations mxandmyfor spin-\nwaves having a non-radial character. As a result the dynamic\ndemagnetizing fields can not be simply added to the circu-\nlar precession H0term and they induce some ellipticity of\nthe precession of the non-uniform spin-waves. Dipole-dipole\ninteractions make the eigenmode frequency non-linear with\nthe field (see Eq. 52 in ref. 22). For the lowest lying non-\nuniform spin-wave the dipole-dipole stiffness field is MSkt=2\nwithk\u0019\u0019=a. It is negligible against the generalized field\n~Hfor the thickness used in practice in PMA systems meant\nfor spin-torque applications. The precession stays therefore\nessentially circular for all the modes observed experimentally11\nhere; we will thus consider that it is legitimate to use Eq. 10\nand deduce the damping from the ratio of the half frequency\nlinewidth to the eigenmode frequency.\nFinally, we would like to mention that our method is notrestricted to the PMA materials only: it should hold when the\nconsidered spin-waves are quasi-circular. In particular, this is\nthe case of exchange-dominated spin waves in in-plane mag-\nnetized systems37.\n\u0003thibaut.devolder@u-psud.fr\n1C. Kittel, Physical Review 73, 155 (1948).\n2C. Bilzer, T. Devolder, P. Crozat, C. Chappert, S. Cardoso, and\nP. P. Freitas, Journal of Applied Physics 101, 074505 (2007).\n3T. Kubota, J. Hamrle, Y . Sakuraba, O. Gaier, M. Oogane,\nA. Sakuma, B. Hillebrands, K. Takanashi, and Y . Ando, Journal\nof Applied Physics 106, 113907 (2009).\n4T. Devolder, Journal of Applied Physics 119, 153905 (2016).\n5S. Mizukami, Y . Ando, and T. Miyazaki, Journal of Magnetism\nand Magnetic Materials Proceedings of the International Confer-\nence on Magnetism (ICM 2000), 226230, Part 2 , 1640 (2001).\n6A. A. Tulapurkar, Y . Suzuki, A. Fukushima, H. Kubota, H. Mae-\nhara, K. Tsunekawa, D. D. Djayaprawira, N. Watanabe, and\nS. Yuasa, Nature 438, 339 (2005).\n7M. Bailleul, Applied Physics Letters 103, 192405 (2013).\n8Z. Lin and M. Kostylev, Journal of Applied Physics 117, 053908\n(2015).\n9L. Liu, Physical Review Letters 106 (2011), 10.1103/Phys-\nRevLett.106.036601.\n10J. C. Sankey, Y .-T. Cui, J. Z. Sun, J. C. Slonczewski, R. A.\nBuhrman, and D. C. Ralph, Nature Physics 4, 67 (2008).\n11H. Kubota, A. Fukushima, K. Yakushiji, T. Nagahama, S. Yuasa,\nK. Ando, H. Maehara, Y . Nagamine, K. Tsunekawa, D. D.\nDjayaprawira, N. Watanabe, and Y . Suzuki, Nature Physics 4,\n37 (2008).\n12X. Cheng, J. A. Katine, G. E. Rowlands, and I. N. Krivorotov,\nApplied Physics Letters 103, 082402 (2013).\n13S. Ikeda, J. Hayakawa, Y . Ashizawa, Y . M. Lee, K. Miura,\nH. Hasegawa, M. Tsunoda, F. Matsukura, and H. Ohno, Applied\nPhysics Letters 93, 082508 (2008).\n14T. Devolder, A. Le Goff, and V . Nikitin, Physical Review B 93,\n224432 (2016).\n15A. V . Khvalkovskiy, D. Apalkov, S. Watts, R. Chepulskii, R. S.\nBeach, A. Ong, X. Tang, A. Driskill-Smith, W. H. Butler, P. B.\nVisscher, D. Lottis, E. Chen, V . Nikitin, and M. Krounbi, Journal\nof Physics D: Applied Physics 46, 074001 (2013).\n16V . Kambersky, Physical Review B 76(2007), 10.1103/Phys-\nRevB.76.134416.\n17T. Devolder, J.-V . Kim, L. Nistor, R. Sousa, B. Rodmacq, and\nB. Dieny, Journal of Applied Physics 120, 183902 (2016).\n18J. N. Kupferschmidt, S. Adam, and P. W. Brouwer, Physical Re-\nview B 74, 134416 (2006).\n19L. Mihalceanu, S. Keller, J. Greser, D. Karfaridis, K. Symeoni-\ndis, G. V ourlias, T. Kehagias, A. Conca, B. Hillebrands, and\nE. T. Papaioannou, arXiv:1702.05298 [cond-mat] (2017), arXiv:\n1702.05298.20A. A. Baker, A. I. Figueroa, D. Pingstone, V . K. Lazarov, G. v. d.\nLaan, and T. Hesjedal, Scientific Reports 6, srep35582 (2016).\n21O. Klein, G. de Loubens, V . V . Naletov, F. Boust, T. Guillet,\nH. Hurdequint, A. Leksikov, A. N. Slavin, V . S. Tiberkevich, and\nN. Vukadinovic, Physical Review B 78, 144410 (2008).\n22B. A. Kalinikos and A. N. Slavin, Journal of Physics C: Solid State\nPhysics 19, 7013 (1986).\n23K. Mizunuma, M. Yamanouchi, H. Sato, S. Ikeda, S. Kanai,\nF. Matsukura, and H. Ohno, Applied Physics Express 6, 063002\n(2013).\n24G. N. Kakazei, P. E. Wigen, K. Y . Guslienko, V . Novosad, A. N.\nSlavin, V . O. Golub, N. A. Lesnik, and Y . Otani, Applied Physics\nLetters 85, 443 (2004).\n25R. E. Arias and D. L. Mills, Physical Review B 79, 144404 (2009).\n26V . V . Naletov, G. de Loubens, G. Albuquerque, S. Borlenghi,\nV . Cros, G. Faini, J. Grollier, H. Hurdequint, N. Locatelli,\nB. Pigeau, A. N. Slavin, V . S. Tiberkevich, C. Ulysse, T. Valet,\nand O. Klein, Physical Review B 84, 224423 (2011).\n27S. V . Nedukh, S. I. Tarapov, D. P. Belozorov, A. A. Kharchenko,\nV . O. Golub, I. V . Kilimchuk, O. Y . Salyuk, E. V . Tartakovskaya,\nS. A. Bunyaev, and G. N. Kakazei, Journal of Applied Physics\n113, 17B521 (2013).\n28K. Munira and P. B. Visscher, Journal of Applied Physics 117,\n17B710 (2015).\n29T. Devolder, J.-V . Kim, F. Garcia-Sanchez, J. Swerts, W. Kim,\nS. Couet, G. Kar, and A. Furnemont, Physical Review B 93,\n024420 (2016).\n30K. Y . Guslienko, S. O. Demokritov, B. Hillebrands, and A. N.\nSlavin, Physical Review B 66, 132402 (2002).\n31N. Romming, A. Kubetzka, C. Hanneken, K. von Bergmann, and\nR. Wiesendanger, Physical Review Letters 114, 177203 (2015).\n32H. T. Nembach, J. M. Shaw, C. T. Boone, and T. J. Silva, Physical\nReview Letters 110, 117201 (2013).\n33W. Wang, M. Dvornik, M.-A. Bisotti, D. Chernyshenko, M. Beg,\nM. Albert, A. Vansteenkiste, B. V . Waeyenberge, A. N. Kuchko,\nV . V . Kruglyak, and H. Fangohr, Physical Review B 92, 054430\n(2015).\n34S. Petit, C. Baraduc, C. Thirion, U. Ebels, Y . Liu, M. Li, P. Wang,\nand B. Dieny, Physical Review Letters 98, 077203 (2007).\n35A. N. Slavin and P. Kabos, IEEE Transactions on Magnetics 41,\n1264 (2005).\n36J.-V . Kim, Physical Review B 73, 174412 (2006).\n37M. Dvornik and V . V . Kruglyak, Physical Review B 84, 140405\n(2011)."    },    {        "title": "1703.09444v2.Temperature_dependent_magnetic_damping_of_yttrium_iron_garnet_spheres.pdf",        "content": "Temperature dependent magnetic damping of yttrium iron garnet spheres\nH. Maier-Flaig,1, 2,\u0003S. Klingler,1, 2C. Dubs,3O. Surzhenko,3R.\nGross,1, 2, 4M. Weiler,1, 2H. Huebl,1, 2, 4and S. T. B. Goennenwein1, 2, 4, 5, 6\n1Walther-Mei\u0019ner-Institut, Bayerische Akademie der Wissenschaften, 85748 Garching, Germany\n2Physik-Department, Technische Universit at M unchen, 85748 Garching, Germany\n3INNOVENT e.V. Technologieentwicklung, 07745 Jena, Germany\n4Nanosystems Initiative Munich, 80799 M unchen, Germany\n5Institut f ur Festk orperphysik, Technische Universit at Dresden, 01062 Dresden, Germany\n6Center for Transport and Devices of Emergent Materials,\nTechnische Universit at Dresden, 01062 Dresden, Germany\n(Dated: June 5, 2017)\nWe investigate the temperature dependent microwave absorption spectrum of an yttrium iron\ngarnet sphere as a function of temperature (5 K to 300 K) and frequency (3 GHz to 43 :5 GHz). At\ntemperatures above 100 K, the magnetic resonance linewidth increases linearly with temperature\nand shows a Gilbert-like linear frequency dependence. At lower temperatures, the temperature\ndependence of the resonance linewidth at constant external magnetic \felds exhibits a characteristic\npeak which coincides with a non-Gilbert-like frequency dependence. The complete temperature and\nfrequency evolution of the linewidth can be modeled by the phenomenology of slowly relaxing rare-\nearth impurities and either the Kasuya-LeCraw mechanism or the scattering with optical magnons.\nFurthermore, we extract the temperature dependence of the saturation magnetization, the magnetic\nanisotropy and the g-factor.\nI. INTRODUCTION\nThe magnetization dynamics of the ferrimagnetic insu-\nlator yttrium iron garnet (YIG) recently gained renewed\ninterest as YIG is considered an ideal candidate for spin-\ntronic applications as well as spin-based quantum infor-\nmation storage and processing1{4due to the exception-\nally low damping of magnetic excitations as well as its\nmagneto-optical properties5{7. In particular, consider-\nable progress has been made in implementing schemes\nsuch as coupling the magnetic moments of multiple YIG\nspheres2,8or interfacing superconducting quantum bits\nwith the magnetic moment of a YIG sphere3,9.\nMagnetization dynamics in YIG have been investi-\ngated in a large number of studies in the 1960s.10{12\nHowever, a detailed broadband study of the magnetiza-\ntion dynamics in particular for low temperatures is still\nmissing for bulk YIG. Nevertheless, these parameters are\nessential for the design and optimization of spintronic\nand quantum devices. Two recent studies13,14consider\nthe temperature dependent damping of YIG thin \flms.\nHaidar et al.13report a large Gilbert-like damping of\nunknown origin, while the low damping thin \flms inves-\ntigated by Jermain et al.14show a similar behavior as\nreported here.\nOur systematic experiments thus provide an impor-\ntant link between the more recent broadband studies on\nYIG thin \flms and the mostly single-frequency studies\nfrom the 1960s: We investigate the magnetostatic spin\nwave modes measured in a YIG sphere using broadband\nmagnetic resonance up to 43.5 GHz in the temperature\nrange from 5 to 300K. We extract the temperature de-\npendent magnetization, the g-factor and the magnetic\nanisotropy of YIG. Additionally, we focus our analysis on\nthe temperature dependent damping properties of YIGand identify the phenomenology of slowly relaxing rare-\nearth impurities and either the Kasuya-LeCraw mecha-\nnism or the scattering with optical magnons as the mi-\ncroscopic damping mechanism.\nThe paper is organized as follows. We \frst give a short\nintroduction into the experimental techniques followed by\na brief review of the magnetization damping mechanisms\nreported for YIG. Finally, we present the measured data\nand compare the evolution of the linewidth with temper-\nature and frequency with the discussed damping models.\nThe complete set of raw data and the evaluation routines\nare publicly available.15\nII. EXPERIMENTAL DETAILS AND\nFERROMAGNETIC RESONANCE THEORY\nThe experimental setup for the investigation of the\ntemperature dependent broadband ferromagnetic reso-\nnance (bbFMR) is shown schematically in Fig. 1. It\nconsists of a coplanar wave guide (CPW) onto which\na 300 µm diameter YIG sphere is mounted above the\n300µm wide center conductor. The [111] direction of\nthe single crystalline sphere is aligned along the CPW\nsurface normal as con\frmed using Laue di\u000braction (not\nshown). We mount a pressed diphenylpicrylhydrazyl\n(DPPH) powder sample in a distance of approximately\n5 mm from the sphere. The identical sample with the\nsame alignment has been used in Ref. 16. This assembly\nis mounted on a dip stick in order to place the YIG sphere\nin the center of a superconducting magnet (Helmholtz\ncon\fguration) in a Helium gas-\row cryostat. End-launch\nconnectors are attached to the CPW and connected to\nthe two ports of a vector network analyzer (VNA) mea-\nsuring the phase sensitive transmission of the setup uparXiv:1703.09444v2  [cond-mat.mtrl-sci]  2 Jun 20172\nliquid Helium\nside view\nCPWP1\nP2H0hMW hMWVNAP1P2\nside viewside view5mm\nDPPH\nYIG\ncom ±1‰ ±1‰\nFIG. 1. The coplanar waveguide (CPW), on which the YIG\nsphere and a DPPH marker are mounted (right), is inserted\ninto a magnet cryostat (left). The microwave transmission\nthrough the setup is measured phase sensitively using a vec-\ntor network analyzer (VNA). As the Oersted \feld hMW(red)\naround the center conductor of the CPW extends into the\nYIG sphere and the DPPH, we can measure the microwave\nresponse spectra of both samples. A superconducting magnet\nprovides the static external magnetic \feld H0(orange) at the\nlocation of the sample. Also shown are the lines correspond-\ning to the speci\fed 1 ‰homogeneity of the \feld for a on-axis\ndeviation from the center of magnet (com).\nto 43:5 GHz.\nThe sphere is placed within the microwave Oersted\n\feldhMWof the CPW's center conductor which is excited\nwith a continuous wave microwave of variable frequency.\nWe apply a static external magnetic \feld H0perpen-\ndicular to the CPW surface and thus hMWis oriented\nprimarily perpendicular to H0. The microwave Oersted\n\feld can therefore excite magnetization precession at fre-\nquencies that allow a resonant drive. The magnetization\nprecession is detected by electromagnetic induction via\nthe same center conductor.17This induction voltage in\ncombination with the purely transmitted microwave sig-\nnal is measured phase sensitively as the complex scatter-\ning parameter Sraw\n21(!) at port 2 of the VNA.\nThe frequency-dependent background is eliminated as\nfollows: A static external magnetic \feld su\u000eciently large\nthat no resonances are expected in the given microwave\nfrequency range is applied and the transmission at this\n\feld is recorded as the background reference SBG\n21. Then,\nthe external \feld is set to the value at which we ex-\npect resonances of YIG in the given frequency range and\nrecord the transmission Sraw\n21. We \fnally divide Sraw\n21by\nSBG\n21. This corrects for the frequency dependent attenu-\nation and the electrical length of the setup. We choose\nthis background removal method over a microwave cal-\nibration because it additionally eliminates the \feld andtemperature dependence of S21that arises from the ther-\nmal contraction and movement of the setup and magnetic\nmaterials in the microwave connectors. In the following\nwe always display S21=Sraw\n21=SBG\n21.\nFor the evaluation of the magnetization dynamics, we\n\ft the transmission data to S21=\u0000ifZ\u001f +A1+A2f\nfor eachH0. Here,A1;2describe a complex-valued back-\nground and\n\u001f(f;H 0) =\u00160Ms\r\n2\u0019\u0000\r\n2\u0019\u00160H0\u0000i\u0001f\u0001\nf2res\u0000f2\u0000if\u0001f(1)\nis the ferromagnetic high-frequency susceptibility.17,18\nThe free parameters of the \ft are the resonance frequency\nfres, the full width at half maximum (FWHM) linewidth\n\u0001fas well as the complex scaling parameter Z, which\nis proportional to the strength of the inductive coupling\nbetween the speci\fc magnetic resonance mode and the\nCPW. For a given \fxed magnetic \feld the \ft parame-\nters\r\n2\u0019(gyromagnetic ratio) and Ms(saturation magne-\ntization) are completely correlated with Zand are thus\n\fxed. They are later determined from \ftting the disper-\nsion curves.19\nIn spheres various so-called magnetostatic modes\n(MSM) arise due to the electromagnetic boundary\nconditions.20These modes can be derived from the\nLandau-Lifshitz equation in the magnetostatic limit ( ~r\u0002\n~H= 0) for insulators.21The lineshape of all modes is\ngiven by Eq. 1. Due to the di\u000berent spatial mode pro\fles\nand the inhomogeneous microwave \feld, the inductive\ncoupling and thus Zis mode dependent.22A detailed re-\nview of possible modes, their distribution and dispersion\nis given in R oschmann and D otsch20. We will only dis-\ncuss the modes (110) and (440) in detail in the following\nas all the relevant characteristics of all other modes can\nbe related to these two modes. Their linear dispersions\nare given by20\nf110\nres=\r\n2\u0019\u00160(H0+Hani) (2)\nf440\nres=\r\n2\u0019\u00160\u0012\nH0+Hani+Ms\n9\u0013\n(3)\nwhereHaniis the magnetic anisotropy \feld and\r\n2\u0019is\nthe gyromagnetic ratio which relates to the g-factor by\n\r\n2\u0019=\u0016B\nhg. It is thus generally assumed that gis the\nsame for all modes. We note that the apparent g-factor\nmay still vary in between modes if the modes experience\na di\u000berent anisotropy.23,24Such an anisotropy contribu-\ntion can be caused by surface pit scattering as it a\u000bects\nmodes that are localized at the surface stronger than bulk\nlike modes25. In our experiment, no such variation in g\ncoinciding with a change in anisotropy was observed and\nwe use a mode number independent gin the following.\nKnowledge of the dispersion relations of the two modes\nallows to determine the saturation magnetization from\n\u00160Ms(T) = 92\u0019\n\r\u0001fM= 92\u0019\n\r\u0000\nf440\nres\u0000f110\nres\u0001\n:(4)3\nThe anisotropy \feld is extracted by extrapolating the\ndispersion relations in Eqs. (2) and (3) to H0= 0.\nThe temperature dependent linewidth \u0001 fof the modes\nis the central result of this work. For a short review of\nthe relevant relaxation processes we refer to the dedicated\nSec. III.\nIn this work, we investigate the T-dependence of Ms,\nHani,gand \u0001f. Accurate determination of the g-factor\nand the anisotropy Hanirequires accurate knowledge of\nH0. In order to control the temperature of the YIG\nsphere and CPW, they are placed in a gas-\row cryo-\nstat as displayed schematically in Fig. 1. The challenge\nin this type of setup is the exact and independent deter-\nmination of the static magnetic \feld and its spatial in-\nhomogeneity. Lacking an independent measure of H0,26\nwe only report the relative change of gandHanifrom\ntheir respective room temperature values which were de-\ntermined separately using the same YIG sphere.16Note\nthat we determine the resonance frequencies directly in\nfrequency space. Our results on linewidth and magneti-\nzation are hence independent of a potential uncertainty\nin the absolute magnitude of H0and its inhomogeneity.\nIII. RELAXATION THEORY\nWhen relaxation properties of ferromagnets are dis-\ncussed today, the most widely applied model is the so-\ncalled Gilbert type damping. This purely phenomeno-\nlogical model is expressed in a damping term of the form\n\u000bM\u0002dM\ndtin the Landau-Lifshitz equation. It describes\na viscous damping, i.e. a resonance linewidth that de-\npends linearly on the frequency. A linear frequency de-\npendence is often found in experiments and the Gilbert\ndamping parameter \u000bserves as a \fgure of merit of the\nferromagnetic damping that allows to compare samples\nand materials. It contains, however, no insight into the\nunderlying physical mechanisms.\nIn order to understand the underlying microscopic re-\nlaxation processes of YIG, extensive work has been car-\nried out. Improvements on both the experimental side\n(low temperatures27, temperature dependence10,28,29,\nseparate measurements of MzandMxy11) and on the\nsample preparation (varying the surface pit size25, pu-\nrifying Yttrium10, doping YIG with silicon30and rare-\nearth elements30{33) led to a better understanding of\nthese mechanisms.\nHowever, despite these e\u000borts the microscopic origin\nof the dominant relaxation mechanism for bulk YIG at\nroom temperature is still under debate. It has been\ndescribed by a two-magnon process by Kasuya and\nLeCraw28(1961). In this process, a uniformly-precessing\nmagnon (k= 0) relaxes under absorption of a phonon\nto ak6= 0 magnon. If the thermal energy kBTis\nmuch larger than the energy of the involved magnons and\nphonons (T > 100 K) and low enough that no higher-\norder processes such as four-magnon scattering play a\nrole (T < 350 K), the Kasuya-LeCraw process yields alinewidth that is linear in frequency and temperature:\n\u0001fKL/T;f.28,30This microscopic process is therefore\nconsidered to be the physical process that explains the\nphenomenological Gilbert damping for low-damping bulk\nYIG. More recently, Cherepanov et al.34pointed out\nthat the calculations by Kasuya and LeCraw28assume\na quadratic magnon dispersion in k-space which is only\ncorrect for very small wave numbers k. Taking into ac-\ncount a more realistic magnon dispersion (quadratic at\nlowk, linear to higher k), the Kasuya-LeCraw mechanism\ngives a value for the relaxation rate that is not in line\nwith the experimental results. Cherepanov therefore de-\nveloped an alternative model that traces back the linear\nfrequency and temperature dependence at high tempera-\ntures (150 K to 300 K) to the interaction of the uniform-\nprecession mode with optical magnons of high frequency.\nRecently, atomistic calculations by Barker and Bauer35\ncon\frmed the assumptions on the magnon spectrum that\nare necessary for the quantitative agreement of the latter\ntheory with experiment.\nBoth theories, the Kasuya-LeCraw theory and the\nCherepanov theory, aim to describe the microscopic ori-\ngin of the intrinsic damping. They deviate in their\nprediction only in the low-temperature ( T < 100 K)\nbehavior.30At these temperatures, however, impurities\ntypically dominate the relaxation and mask the contri-\nbution of the intrinsic damping process. Therefore, the\ndominant microscopic origin of the YIG damping at tem-\nperatures above 150K has not been unambiguously de-\ntermined to date.\nIf rare-earth impurities with large orbital momentum\nexist in the crystal lattice, their exchange coupling with\nthe iron ions introduces an additional relaxation chan-\nnel for the uniform precession mode of YIG. Depend-\ning on the relaxation rate of the rare-earth impurities\nwith respect to the magneto{dynamics of YIG, they are\nclassi\fed into slowly and fast relaxing rare-earth impu-\nrities. This is an important distinction as the e\u000eciency\nof the relaxation of the fundamental mode of YIG via\nthe rare-earth ion to the lattice at a given frequency de-\npends on the relaxation rate of the rare-earth ion and the\nstrength of the exchange coupling. In both the slow and\nthe fast relaxor case, a characteristic peak-like maximum\nis observed in the linewidth vs. temperature dependence\nat a characteristic, frequency-dependent temperature12.\nThe frequency dependence of this peak temperature al-\nlows to distinguish fast and slowly relaxing rare-earth\nions: The model of a fast relaxing impurity predicts that\nthe peak temperature is constant, while in the case of\nslowly relaxing rare-earth ions the peak temperature is\nexperted to increase with increasing magnetic \feld (or\nfrequency). The relaxation rate of rare-earths \u001cREis typ-\nically modeled by a direct magnon to phonon relaxation,\nan Orbach processes36,37that involves two phonons, or\na combination of both. The inverse relaxation rate of\nan Orbach process is described by1\n\u001cOrbach =B\ne\u0001=(kBT)\u00001\nwith the crystal \feld splitting \u0001 and a proportionality\nfactorB. A direct process leads to an inverse relax-4\nation rate of1\n\u001cdirect =1\n\u001c0coth\u000e\n2kBTwith\u001c0, the relaxation\ntime atT= 0 K. It has been found experimentally that\nmost rare-earth impurities are to be classi\fed as slow\nrelaxors.30The sample investigated here is not intention-\nally doped with a certain rare-earth element and the peak\nfrequency and temperature dependence indicates a slow\nrelaxor. We therefore focus on the slow relaxing rare-\nearth impurity model in the following.\nDeriving the theory of the slowly relaxing impurities\nhas been performed comprehensively elsewhere.30The\nlinewidth contribution caused by a slowly relaxing rare-\nearth impurity is given by31:\n\u0001fSR=C\n2\u0019f\u001cRE\n1 + (f\u001cRE)2(5)\nwithC/1\nkBTsech\u0010\n\u000ea\n2kBT\u0011\n. Therein, \u000eais the splitting\nof the rare-earth Kramers doublet which is given by the\ntemperature independent exchange interaction between\nthe iron ions and the rare-earth ions.\nAlso Fe2+impurities in YIG give rise to a process\nthat leads to a linewidth peak at a certain temperature.\nThe physical origin of this so-called valence exchange or\ncharge-transfer linewidth broadening is electron hopping\nbetween the iron ions.30Simpli\fed, it can be viewed as\na two level system just like a rare-earth ion and thus re-\nsults in the same characteristic linewidth maximum as\na slowly relaxing rare-earth ion. For valence exchange,\nthe energy barrier \u0001 hopthat needs to be overcome for\nhopping determines the time scale of the process. The\ntwo processes, valence exchange and rare-earth impurity\nrelaxation, can therefore typically not be told apart from\nFMR measurements only. In the following, we use the\nslow relaxor mechanism exclusively. This model consis-\ntently describes our measurement data and the resulting\nmodel parameters are in good agreement with literature.\nWe would like to emphasize, however, that the valence\nexchange mechanism as the relevant microscopic process\nresulting for magnetization damping can not be ruled out\nfrom our measurements.\nIV. EXPERIMENTAL RESULTS AND\nDISCUSSION\nTwo exemplary S21broadband spectra recorded at two\ndistinct temperatures are shown in Fig. 2. The color-\ncoded magnitude jS21jis a measure for the absorbed mi-\ncrowave power. High absorption (bright color) indicates\nthe resonant excitation of a MSM in the YIG sphere or\nthe excitation of the electron paramagnetic resonance of\nthe DPPH. In the color plot the color scale is truncated\nin order to improve visibility of small amplitude reso-\nnances. In addition, the frequency axis is shifted relative\nto the resonance frequency of a linear dispersion with\ng= 2:0054 (fg=2:0054\nres =g\u0016B\nh\u00160H) for each \feld. In this\nway, modes with g= 2:0054 appear as vertical lines. A\n0.20.40.60.81.01.21.4¹0H0 (T)290 K\nminmax\nAbsorption\n−1.0 −0.5 0.0 0.5 1.0\nf¡f(g=2:0054)\nres  (GHz)0.20.40.60.81.01.21.4¹0H0 (T)20 KΔfAΔfM\nf (GHz)10.05 10.06 10.07 10.08\nIm(S21)\n2\n046810\n0\n-4-224Re(S21)\nȴffres(a)\n(b)FIG. 2. Eigenmode spectra of the YIG sphere at (a) 290 K\nand (b) 20 K. The (110) and (440) MSM are marked with red\ndashed lines. The change in their slope gives the change of the\ng-factor of YIG. Their splitting (\u0001 fM, red arrow) depends lin-\nearly on the YIG magnetization. The increase in Msto lower\ntemperatures is already apparent from the increased splitting\n\u0001fM. Marked in orange is the o\u000bset of the resonance fre-\nquency \u0001fAextrapolated to H0= 0 resulting from anisotropy\n\feldsHanipresent in the sphere. The green marker denotes\nthe position of the DPPH resonance line which increases in\namplitude considerably to lower temperatures. Inset: S21pa-\nrameter (data points) and \ft (lines) at \u00160H= 321 mT and\nT= 20 K.\ndeviatingg-factor is therefore easily visible as a di\u000ber-\nent slope. Comparing the spectra at 290 K [Fig. 2 (a)]\nto the spectra at 20 K [Fig. 2 (b)], an increase of the g-\nfactor is observed for all resonance modes upon reducing\nthe temperature. The rich mode spectrum makes it nec-\nessary to carefully identify the modes and assign mode\nnumbers. Note that the occurrence of a particular mode\nin the spectrum depends on the position of the sphere\nwith respect to the CPW due to its inhomogeneous exci-\ntation \feld. We employ the same method of identifying\nthe modes as used in Ref. 16 and \fnd consistent mode\nspectra. As mentioned before, we do not use the DPPH\nresonance (green arrow in Fig. 2) but the (110) YIG\nmode as \feld reference. For this \feld reference, we take\ng(290 K) = 2 :0054 and\r\n2\u0019\u00160Hani(290 K) = 68 :5 MHz de-\ntermined for the same YIG sphere at room temperature\nin an electromagnet with more accurate knowledge of the\napplied external magnetic \feld.16The discrepancy of the\nDPPHg-value from the literature values of g= 2:0036 is5\nattributed to the non-optimal location of the DPPH spec-\nimen in the homogeneous region of the superconducting\nmagnet coils.\nIn Fig. 2, the \ftted dispersion of the (110) and (440)\nmodes are shown as dashed red lines. As noted previ-\nously, we only analyze these two modes in detail as all\nparameters can be extracted from just two modes. The\n(110) and (440) mode can be easily and unambiguously\nidenti\fed by simply comparing the spectra with the ones\nfound in Ref 16. Furthermore, at high \felds, both modes\nare clearly separated from other modes. This is necessary\nas modes can start hybridizing when their (unperturbed)\nresonance frequencies are very similar (cf. low-\feld re-\ngion of Fig. 2 (b)) which makes a reliable determination of\nthe linewidth and resonance frequency impossible. These\nattributes make the (110) and the (440) mode the ideal\nchoice for the analysis.\nAs described in Sec. II, we simultaneously \ft the (110)\nand the (440) dispersions with the same g-factor in or-\nder to extract Ms,Haniandg. In the \ft, we only take\nthe high-\feld dispersion of the modes into account where\nno other modes intersect the dispersion of the (110) and\n(440) modes. The results are shown in Fig. 3. Note that\nthe statistical uncertainty from the \ft is not visible on the\nscale of any of the parameters Ms,Haniandg. Following\nthe work of Solt38, we model the resulting temperature\ndependence of the magnetization (Fig. 3 (a)) with the\nBloch-law taking only the \frst order correction into ac-\ncount:\nMs=M0\u0010\n1\u0000aT3\n2\u0000bT5\n2\u0011\n: (6)\nThe best \ft is obtained for \u00160M0 =\n249:5(5) mT, a = (23\u00063)\u000210\u00006K\u00003=2and\nb= (1:08\u00060:11)\u000210\u00007K\u00005=2. The obtained \ft\nparameters depend strongly on the temperature window\nin which the data is \ftted. Hence, the underlying\nphysics determining the constants aandbcannot be\nresolved.39Nevertheless, the temperature dependence\nofMsis in good agreement with the results determined\nusing a vibrating sample magnetometer.40\nIn particular, also the room temperature saturation\nmagnetization of \u00160Ms(300 K) = (180 :0\u00060:8) mT is in\nperfect agreement with values reported in literature.41,42\nNote that the splitting of the modes is purely in frequency\nspace and thus errors in the \feld do not add to the uncer-\ntainty. We detect a small non-linearity of the (110) and\n(440) mode dispersions that is most likely due to devia-\ntions from an ideal spherical shape or strain due to the\nYIG mounting. This results in a systematic, temperature\nindependent residual of the linear \fts to these disper-\nsions. This resulting systematic error of the magnetiza-\ntion is incorporated in the uncertainty given above. How-\never, a deviation from the ideal spherical shape, strain in\nthe holder or a misalignment of the static magnetic \feld\ncan also modify the splitting of the modes and hence re-\nsult in a di\u000berent Ms.43This fact may explain the small\ndiscrepancy of the value determined here and the valuedetermined for the same sphere in a di\u000berent setup at\nroom temperature.16\nFrom the same \ft that we use to determine the mag-\nnetization, we can deduce the temperature dependence\nof the anisotropy \feld \u00160Hani[Fig. 3 (b)]. Most notably,\nHanichanges sign at 200 K which has not been observed\nin literature before and can be an indication that the\nsample is slightly strained in the holder. The resonance\nfrequency of DPPH extrapolated to \u00160H0= 0 (\u0001fani, red\nsquares) con\frms that the error in the determined value\nHaniis indeed temperature independent and very close\nto zero. Thus, the extracted value for the anisotropy is\nnot merely given by an o\u000bset in the static magnetic \feld.\nThe evolution of the g-factor with temperature is\nshown in Fig. 3 (c). It changes from 2 :005 at room tem-\nperature to 2 :010 at 10 K where it seems to approach a\nconstant value. As mentioned before, the modes' disper-\nsion is slightly non-linear giving rise to a systematic, tem-\nperature independent uncertainty in the determination of\ngof\u00060:0008. Theg-factor of YIG has been determined\nusing the MSMs of a sphere for a few selected tempera-\ntures before.12Comparing our data to these results, one\n\fnds that the trend of the temperature dependence of g\nagrees. However, the absolute value of gand the mag-\nnitude of the variation di\u000ber. At the same time, we \fnd\na change of the g-factor of DPPH that is on the scale\nof 0:0012. This may be attributed to a movement of\nthe sample slightly away from the center of magnet with\nchanging temperature due to thermal contraction of the\ndip stick. In this case, the YIG g-factor has to be cor-\nrected by this change. The magnitude of this e\u000bect on\nthe YIGg-factor can not be estimated reliably from the\nchange of the DPPH g-factor alone. Furthermore, the\ntemperature dependence of the DPPH g-factor has not\nbeen investigated with the required accuracy in literature\nto date to allow excluding a temperature dependence of\ntheg-factor of DPPH. We therefore do not present the\ncorrected data but conclude that we observe a change in\nthe YIGg-factor from room temperature to 10 K of at\nleast 0.2 %.\nNext, we turn to the analysis of the damping properties\nof YIG. We will almost exclusively discuss the damping of\nthe (110) mode in the following but the results also hold\nquantitatively and qualitatively for the other modes.16\nVarying the applied microwave excitation power P(not\nshown) con\frms that no nonlinear e\u000bects such as a power\nbroadening of the modes are observed with P= 0:1 mW.\nNote that due to the microwave attenuation in the mi-\ncrowave cabling, the microwave \feld at the sample loca-\ntion decreases with increasing frequency for the constant\nexcitation power.\nFirst, we evaluate the frequency dependent linewidth\nfor several selected temperatures [Fig. 4 (a)]. At temper-\natures above 100 K, a linear dependence of the linewidth\nwith the resonance frequency is observed. This depen-\ndence is the usual so-called Gilbert-like damping and\nthe slope is described by the Gilbert damping param-\neter\u000b. A linear frequency dependence of the damping6\n180200220240260¹0M (mT)\n(a)model\n−20246¹0Hani (mT)\n(b)YIG\nDPPH\n0 50 100 150 200 250 300\nTemperature (K)2.0042.0082.0122.0162.020g-factor (unitless)\n0.08%\n0.26%\n(c)\nFIG. 3. (a)YIG magnetization as function of tempera-\nture extracted from the (110) and (440) mode dispersions us-\ning Eq. 4. The purple line shows the \ft to a Bloch model\n(cf. parameters in the main text). (b)YIG anisotropy \feld\n\u00160Hani(T) =2\u0019\n\r\u0001fani. Red squares: Same procedure applied\nto the DPPH dispersion as reference. (c)YIGg-factor (blue\ncircles). For reference, the extracted DPPH g-factor is also\nshown (red squares). The gray numbers indicate the rela-\ntive change of the g-factors from the lowest to the highest\nmeasured temperature (gray horizontal lines). As we use the\nYIG (110) mode as the magnetic \feld reference, the extracted\nvalue ofgandHaniat 300 K are \fxed to the values determined\nin the room temperature setup.16\nin bulk YIG has been described by the theory developed\nby Kasuya and LeCraw28and the theory developed by\nCherepanov et al.34(cf. Sec. III). We extract \u000bfrom\na global \ft of a linear model to the (110) and (440)\nlinewidth with separate parameters for the inhomoge-\nneous linewidths \u0001 f110\n0and \u0001f440\n0and a shared Gilbert\ndamping parameter \u000bfor all modes:16\n\u0001f= 2\u000bf+ \u0001f110;440\n0 (7)\nThe \ft is shown exemplarily for the 290 K (red) data in\nFig. 4 (a).\nThe Gilbert damping parameter \u000bextracted using this\n\ftting routine for each temperature is shown in Fig. 5 (a).\nConsistently with both theories, \u000bincreases with increas-\ning temperature. The error bars in the \fgure correspond\nto the maximal deviation of \u000bextracted from separate\n\fts for each mode. They therefore give a measure of\nhow\u000bscatters in between modes. The statistical error\nof the \ft (typically \u00060:00001) is not visible on this scale.\nThe Gilbert damping parameter \u000blinearly extrapolatedto zero temperature vanishes. Note that this is consis-\ntent with the magnon-phonon process described by Ka-\nsuya and LeCraw28but not with the theory developed\nby Cherepanov et al.34. For room temperature, we ex-\ntract a Gilbert damping of 4 \u000210\u00005which is in excel-\nlent agreement with the literature value.16,44From the\n\ft, we also extract the inhomogeneous linewidth \u0001 f0,\nwhich we primarily associate with surface pit scattering\n(Sec. III, Ref 16). In the data, a slight increase of \u0001 f0\ntowards lower temperatures is present [Fig. 5 (b)]. Such a\nchange in the inhomogeneous linewidth can be caused by\na change in the surface pit scattering contribution when\nthe spin-wave manifold changes with Ms.16,25\nNote that according to Fig. 5 (b) \u0001 f0is higher for the\n(440) mode than for the (110) mode. This is in agreement\nwith the theoretical expectation that surface pit scatter-\ning has a higher impact on \u0001 f0for modes that are more\nlocalized at the surface of the sphere like the (440) mode\ncompared to the more bulk-like modes such as the (110)\nmode25.45\nTurning back to Fig. 4 (a), for low temperatures (20 K,\nblue data points), a Gilbert-like damping model is obvi-\nously not appropriate as the linewidth increases consider-\nably towards lower frequencies instead of increasing lin-\nearly with increasing frequency. Typically, one assumes\nthat the damping at low frequencies is dominated by so-\ncalled low \feld losses that may arise due to domain for-\nmation. The usual approach is then to \ft a linear trend\nto the high-frequency behavior only. Note, however, that\neven though the frequency range we use is already larger\nthan usually reported13,14,46, this approach yields an un-\nphysical, negative damping. We conclude that the model\nof a Gilbert-like damping is only valid for temperatures\nexceeding 100 K (Fig. 5) for the employed \feld and fre-\nquency range.\nThe linewidth data available in literature are typi-\ncally taken at a \fxed frequency and the linewidth is dis-\nplayed as a function of temperature12,29,30. We can ap-\nproximately reproduce these results by plotting the mea-\nsured linewidth at \fxed H0as a function of temperature\n[Fig. 4 (b)].47A peak-like maximum of the linewidth be-\nlow 100 K is clearly visible. For increasing magnetic \feld\n(frequency), the peak position shifts to higher tempera-\ntures. This is the signature of a slowly relaxing rare-earth\nimpurity (Sec. III). A fast relaxing impurity is expected\nto result in a \feld-independent linewidth vs. temperature\npeak and can thus be ruled out. At the peak position,\nthe linewidth shows an increase by 2 :5 MHz which trans-\nlates with the gyromagnetic ratio to a \feld linewidth in-\ncrease of 0:08 mT. For 0.1 at. % Terbium doped YIG, a\nlinewidth increase of 80 mT has been observed48. Con-\nsidering that the linewidth broadening is proportional to\nthe impurity concentration and taking the speci\fed pu-\nrity of the source material of 99.9999% used to grow the\nYIG sphere investigated here, we estimate an increase of\nthe linewidth of 0 :08 mT, in excellent agreement with the\nobserved value.\nModeling the linewidth data is more challenging: The7\n0 5 10 15 20 25 30 35 40 45\nf110\nres (GHz)0¢f023456¢f110 (MHz)(a) 20 K\n290 K\n0 50 100 150 200\nTemperature (K)0.51.01.52.02.53.03.5¢f110 (MHz)\n(b) 341 mT, 9.6 GHz\n1007 mT, 28.3 GHzTmax\n0:3TTmax\n1:0T\nFIG. 4. (a)Full width at half maximum (FWHM) linewidth \u0001 f110\nresof the (110) mode as a function of frequency for di\u000berent\ntemperatures. A linear Gilbert-like interpretation is justi\fed in the high- Tcase (T > 100 K) only. Below 100 K, the slope of\n\u0001f110(f110\nres) is not linear so that a Gilbert type interpretation is no longer applicable. (b)FWHM linewidth as a function of\ntemperature for two di\u000berent \fxed external magnetic \felds. The linewidth peaks at a magnetic \feld dependent temperature\nthat can be modeled using the phenomenology of rare-earth impurities resulting in Tmax(vertical dotted lines).\n01234® (unitless)£10−5\n(a)\n0 50 100 150 200 250 300\nTemperature (K)0123¢f0 (MHz) (b)110\n440\nFIG. 5. (a)Gilbert damping parameter \u000bdetermined from the slope of a linear \ft to the \u0001 f(f;T) data for frequencies\nabove 20 GHz. The red line shows the linear dependence of the linewidth with temperature expected from the Kasuya-LeCraw\nprocess. (b)Inhomogeneous linewidth \u0001 f0(intersect of the aforementioned \ft with f110\nres= 0) as a function of temperature.\nThe inhomogeneous linewidth shows a slight increase with decreasing temperature down to 100 K. In the region where the\nslow relaxor dominates the linewidth (gray shaded area, cf. Fig. 4), the linear \ft is not applicable and unphysical damping\nparameters and inhomogeneous linewidths are extracted.\nmodel of a slowly relaxing rare-earth ion contains the\nexchange coupling of the rare-earth ion and the iron\nsublattice, and its temperature dependent relaxation fre-\nquency as parameters. As noted before, typically a di-\nrect and an Orbach process model the relaxation rate,\nand both of these processes have two free parameters.\nUnless these parameters are known from other experi-\nments for the speci\fc impurity and its concentration in\nthe sample, \ftting the model to the temperature behav-\nior of the linewidth at just one \fxed frequency gives am-\nbiguous parameters. In principle, frequency resolved ex-\nperiments as presented in this work make the determina-\ntion of the parameters more robust as the mechanism re-\nsponsible for the rare-earth relaxation is expected not to\nvary as a function of frequency. The complete frequency\nand \feld dependence of the linewidth is shown in Fig. 6.At temperatures above approx 100 K, the linewidth in-\ncreases monotonically with \feld, in agreement with a\ndominantly Gilbert-like damping mechanisms, which be-\ncomes stronger for higher temperatures. On the same\nlinear color scale, the linewidth peak below 100 K and its\nfrequency evolution is apparent. Fig. 4 (b) corresponds\nto horizontal cuts of the data in Fig. 6 at \u00160H0= 341\nand 1007 mT.\nFor typical YIG spheres, that are not speci\fcally en-\nriched with only one rare-earth element, the composition\nof the impurities is unknown. Di\u000berent rare-earth ions\ncontribute almost additively to the linewidth and have\ntheir own characteristic temperature dependent relax-\nation frequency respectively peak position. This is most\nprobably the case for the YIG sphere of this study. The\nconstant magnitude of the peak above 0 :3 T and the con-8\nstant peak width indicates that fast relaxing rare-earth\nions play a minor role. The evolution of the linewidth\nwithH0andfcan therefore not be \ftted to one set\nof parameters. We thus take a di\u000berent approach and\nmodel just the shift of the peak position in frequency\nand temperature as originating from a single slowly re-\nlaxing rare-earth impurity. For this, we use a value\nfor the exchange coupling energy between the rare-earth\nions and the iron sublattice in a range compatible with\nliterature30of\u000ea= 2:50 meV. To model the rare-earth\nrelaxation rate as a function of temperature, we use the\nvalues determined by Clarke et al.32for Neodymium\ndoped YIG: \u001c0= 2:5\u000210\u000011s for the direct process and\n\u0001 = 10:54 meV and B= 9\u00021011s\u00001for the Orbach pro-\ncess. The model result, i.e. the peak position, is shown as\ndashed line in Fig. 6 and shows good agreement with the\ndata. This indicates that, even though valence exchange\nand other types of impurities cannot be rigourously ex-\ncluded, rare-earth ions are indeed the dominant source\nfor the linewidth peak at low temperatures.\nV. CONCLUSIONS\nWe determined the ferromagnetic dispersion and\nlinewidth of the (110) magnetostatic mode of a polished\nYIG sphere as a function of temperature and frequency.\nFrom this data, we extract the Gilbert damping param-\neter for temperatures above 100 K and \fnd that it varies\nlinearly with temperature as expected according to the\ntwo competing theories of Kasuya and LeCraw28andCherepanov et al.34. At low temperatures, the temper-\nature dependence of the linewidth measured at constant\nmagnetic \feld shows a peak that shifts to higher tempera-\ntures with increasing frequency. This indicates slowly re-\nlaxing impurities as the dominant relaxation mechanism\nfor the magnetostatic modes below 100 K. We model\nthe shift of the peak position with temperature and fre-\nquency with values reported for Neodymium impurities32\nin combination with a typical value for the impurity-ion\nto iron-ion exchange coupling. We \fnd that these param-\neters can be used to describe the position of the linewidth\npeak. We thus directly show the implications of (rare\nearth) impurities as typically present in YIG samples\non the dynamic magnetic properties of the ferrimagnetic\ngarnet material. Furthermore, we extract the temper-\nature dependence of the saturation magnetization, the\nanisotropy \feld and the g-factor.\nACKNOWLEDGEMENTS\nThe authors thank M. S. Brandt for helping out with\nthe microwave equipment. C.D. and S.O. would like to\nacknowledge R. Meyer, M. Reich, and B. Wenzel (IN-\nNOVENT e.V.) for technical assistance in the YIG crys-\ntal growth and sphere preparation. We gratefully ac-\nknowledge funding via the priority program Spin Caloric\nTransport (spinCAT), (Projects GO 944/4 and GR\n1132/18), the priority program SPP 1601 (HU 1896/2-\n1) and the collaborative research center SFB 631 of the\nDeutsche Forschungsgemeinschaft.\n\u0003hannes.maier-\raig@wmi.badw.de\n1X. Zhang, C.-l. Zou, L. Jiang, and H. X. Tang, Physical\nReview Letters 113, 156401 (2014).\n2X. Zhang, C.-l. Zou, N. Zhu, F. Marquardt, L. Jiang, and\nH. X. Tang, Nature Communications 6, 8914 (2015).\n3Y. Tabuchi, S. Ishino, T. Ishikawa, R. Yamazaki, K. Usami,\nand Y. Nakamura, Physical Review Letters 113, 083603\n(2014).\n4L. Bai, M. Harder, Y. P. Chen, X. Fan, J. Q. Xiao, and\nC.-M. Hu, Physical Review Letters 114, 227201 (2015).\n5S. Klingler, H. Maier-Flaig, R. Gross, C.-M. Hu, H. Huebl,\nS. T. B. Goennenwein, and M. Weiler, Applied Physics\nLetters 109, 072402 (2016).\n6S. Viola Kusminskiy, H. X. Tang, and F. Marquardt, Phys-\nical Review A 94, 033821 (2016).\n7R. Hisatomi, A. Osada, Y. Tabuchi, T. Ishikawa,\nA. Noguchi, R. Yamazaki, K. Usami, and Y. Nakamura,\nPhysical Review B 93, 174427 (2016).\n8N. J. Lambert, J. A. Haigh, S. Langenfeld, A. C. Doherty,\nand A. J. Ferguson, Physical Review A 93, 021803 (2016).\n9Y. Tabuchi, S. Ishino, A. Noguchi, T. Ishikawa, R. Ya-\nmazaki, K. Usami, and Y. Nakamura, Science 349, 405\n(2015).\n10E. G. Spencer, R. C. Lecraw, and A. M. Clogston, Physical\nReview Letters 3, 32 (1959).11M. Sparks and C. Kittel, Physical Review Letters 4, 232\n(1960).\n12K. P. Belov, L. A. Malevskaya, and V. I. Sokoldv, Soviet\nPhysics JETP 12, 1074 (1961).\n13M. Haidar, M. Ranjbar, M. Balinsky, R. K. Dumas,\nS. Khartsev, and J. \u0017Akerman, Journal of Applied Physics\n117, 17D119 (2015).\n14C. L. Jermain, S. V. Aradhya, J. T. Brangham, M. R. Page,\nN. D. Reynolds, P. C. Hammel, R. A. Buhrman, F. Y.\nYang, and D. C. Ralph, arXiv preprint arXiv:1612.01954\n(2016).\n15H. Maier-Flaig, \\Temperature dependent damp-\ning of yttrium iron garnet spheres { Measure-\nment data and analysis programs,\" (2017),\nhttps://dx.doi.org/10.17605/OSF.IO/7URPT.\n16S. Klingler, H. Maier-Flaig, C. Dubs, O. Surzhenko,\nR. Gross, H. Huebl, S. T. B. Goennenwein, and M. Weiler,\nApplied Physics Letters 110, 092409 (2017).\n17S. S. Kalarickal, P. Krivosik, M. Wu, C. E. Patton, M. L.\nSchneider, P. Kabos, T. J. Silva, and J. P. Nibarger, Jour-\nnal of Applied Physics 99, 093909 (2006).\n18M. L. Schneider, J. M. Shaw, A. B. Kos, T. Gerrits, T. J.\nSilva, and R. D. McMichael, Journal of Applied Physics\n102, 103909 (2007).9\n50 100 150 200 250\nTemperature (K)0.20.40.60.81.01.21.4¹0H0 (T)\n123456\n¢f110 (MHz)\n5152535\nf110\nres (GHz)\nFIG. 6. Full map of the FWHM linewidth of the (110) mode as function of temperature and \feld resp. resonance frequency\nf110\nres. At low temperatures, only the slow relaxor peak is visible while at high temperatures the Gilbert-like damping becomes\ndominant. The position of the peak in the linewidth modeled by a slow relaxor is shown as dashed orange line. The model\nparameters are taken from Clarke31and taking \u000ea= 2:50 meV. The dotted lines indicate the deviation of the model for 0 :5\u000ea\n(lowerTmax) and 2\u000ea(higherTmax).\n19H. T. Nembach, T. J. Silva, J. M. Shaw, M. L. Schneider,\nM. J. Carey, S. Maat, and J. R. Childress, Physical Review\nB84, 054424 (2011).\n20P. R oschmann and H. D otsch, physica status solidi (b) 82,\n11 (1977).\n21L. R. Walker, Physical Review 105, 390 (1957).\n22Due to the comparable dimensions of center conductor\nwidth and sphere diameter, we expect that the sphere ex-\nperiences an inhomogeneous microwave magnetic \feld with\nits main component parallel to the surface of the CPW and\nperpendicular to its center conductor. As the microwave\nmagnetic \feld is su\u000eciently small to not cause any non-\nlinear e\u000bects, a mode dependent excitation e\u000eciency is the\nonly e\u000bect of the microwave magnetic \feld inhomogeneity.\n23C. Kittel, Physical Review 76, 743 (1949).\n24P. Bruno, Physical Review B 39, 865 (1989).\n25J. Nemarich, Physical Review 136, A1657 (1964).\n26The resonance frequency of the DPPH sample that has\nbeen measured simultaneously was intended as a \feld cal-\nibration but can not be utilized due to the magnetic \feld\ninhomogeneity. In particular, since the homogeneity of our\nsuperconducting magnet system is speci\fed to 1 ‰for an\no\u000b-axis deviation of 2 :5 mm, the spatial separation of 5 mm\nof the DPPH and the YIG sphere already falsi\fes DPPH\nas an independent magnetic \feld standard. Placing DPPH\nand YIG in closer proximity is problematic as the stray\n\feld of the YIG sphere will a\u000bect the resonance frequency\nof the DPPH. Note further that we are not aware of any re-\nports showing the temperature independence of the DPPH\ng-factor with the required accuracy.\n27J. F. Dillon, Physical Review 111, 1476 (1958).\n28T. Kasuya and R. C. LeCraw, Physical Review Letters 67,\n223 (1961).\n29E. G. Spencer, R. C. Lecraw, and J. Linares, Physical\nReview 123, 1937 (1961).\n30M. Sparks, Ferromagnetic-Relaxation Theory , edited by\nW. A. Nierenberg (McGraw-Hill, 1964).\n31B. H. Clarke, Physical Review 139, A1944 (1965).\n32B. H. Clarke, K. Tweedale, and R. W. Teale, Physical\nReview 139, A1933 (1965).33M. Sparks, Journal of Applied Physics 38, 1031 (1967).\n34V. Cherepanov, I. Kolokolov, and V. L'vov, Physics Re-\nports 229, 81 (1993).\n35J. Barker and G. E. W. Bauer, Physical Review Letters\n117, 217201 (2016).\n36R. Orbach, Proceedings of the Physical Society 77, 821\n(1961).\n37B. H. Clarke, Journal of Applied Physics 36, 1211 (1965).\n38I. H. Solt, Journal of Applied Physics 33, 1189 (1962).\n39Note that we failed to reproduce the \ft of Ref. 38 using\nthe data provided in this paper and that the reasonable\nagreement with the there-reported \ft parameters might\nbe coincidence.\n40E. E. Anderson, Physical Review 134, A1581 (1964).\n41P. Hansen, Journal of Applied Physics 45, 3638 (1974).\n42G. Winkler, Magnetic Garnets , Tracts in pure and applied\nphysics; Vol. 5 (Vieweg, 1981).\n43R. L. White, Journal of Applied Physics 31, S86 (1960).\n44P. R oschmann and W. Tolksdorf, Materials Research Bul-\nletin18, 449 (1983).\n45In comparison to Klingler et al.16, here, we do not see\nan increased inhomogeneous linewidth of the (110) mode\nand no secondary mode that is almost degenerate with the\n(110) mode. The di\u000berence can be explained by the ori-\nentation of the sphere which is very di\u000ecult to reproduce\nvery accurately ( <1°) between the experimental setups:\nThe change in orientation either separates the mode that\nis almost degenerate to the (110) mode or makes the degen-\neracy more perfect in our setup. The di\u000berent placement of\nthe sphere on the CPW can also lead to a situation where\nthe degenerate mode is not excited and therefore does not\ninterfere with the \ft.\n46Y. Sun, Y.-Y. Song, H. Chang, M. Kabatek, M. Jantz,\nW. Schneider, M. Wu, H. Schultheiss, and A. Ho\u000bmann,\nApplied Physics Letters 101, 152405 (2012).\n47Naturally, the resonance frequency varies slightly\n(\u00060:9 GHz) between the data points because the magneti-\nzation and the anisotropy changes with temperature.\n48J. F. Dillon and J. W. Nielsen, Physical Review Letters 3,\n30 (1959)."    },    {        "title": "1703.10903v1.Spin_Seebeck_effect_in_Y_type_hexagonal_ferrite_thin_films.pdf",        "content": "arXiv:1703.10903v1  [cond-mat.mtrl-sci]  31 Mar 2017Spin Seebeck effect in Y-type hexagonal ferrite thin films\nJ. Hirschner,1K. Kn´ ıˇ zek,1,∗M. Maryˇ sko,1J. Hejtm´ anek,1R. Uhreck´ y,2\nM. Soroka,2J. Burˇ s´ ık,2A. Anad´ on Barcelona,3and M. H. Aguirre3\n1Institute of Physics ASCR, Cukrovarnick´ a 10, 162 00 Prague 6, Czech Republic.\n2Institute of Inorganic Chemistry ASCR, 250 68 ˇReˇ z near Prague, Czech Republic.\n3Instituto de Nanociencia de Arag´ on, Universidad de Zaragoz a, E-50018 Zaragoza, Spain\nSpin Seebeck effect (SSE) has been investigated in thin films o f two Y-hexagonal ferrites\nBa2Zn2Fe12O22(Zn2Y) and Ba 2Co2Fe12O22(Co2Y) deposited by a spin-coating method on\nSrTiO 3(111) substrate. The selected hexagonal ferrites are both f errimagnetic with similar magnetic\nmoments at room temperature and both exhibit easy magnetiza tion plane normal to c-axis. Despite\nthat, SSE signal was only observed for Zn2Y, whereas no signi ficant SSE signal was detected for\nCo2Y. We tentatively explain this different behavior by a pre sence of two different magnetic ions in\nCo2Y, whose random distribution over octahedral sites inte rferes the long range ordering and en-\nhances the Gilbert damping constant. The temperature depen dence of SSE for Zn2Y was measured\nand analyzed with regard to the heat flux and temperature grad ient relevant to the SSE signal.\nKeywords:\nI. INTRODUCTION\nSpintronics is a multidisciplinary field which involves\nthe study of active manipulation of spin degrees of free-\ndom in solid-state systems [1]. Thermoelectricity con-\ncerns the ability of a given material to produce voltage\nwhen temperature gradient is present, thus converting\nthermal energy to electric energy [2]. The emerging re-\nsearch field of spin caloritronics, which may be regarded\nas interconnection of spintronics and thermoelectricity,\ncombines spin-dependent charge transport with energy\nor heat transport. One of the core elements of spin\ncaloritronics is the spin-Seebeck effect discovered in 2008\nby Uchida et al.[3]. The spin-Seebeck effect (SSE) is\na combination of two phenomena - the generation of a\nspin current by a temperature gradient applied across a\nmagnetic material, and a conversion of the spin current\nto electrical current by means of the inverse spin Hall\neffect (ISHE) [4] in the attached metallic thin layer. A\nnecessarycondition for the observationof SSE is that the\ndirections of the spin current, magnetic moments of the\nmagnetic material, and electrical current in the metal-\nlic layer, are mutually perpendicular. Since the resulting\nelectric field is related to temperature gradient, it is pos-\nsible in the regime of linear response to define a spin\nSeebeck coefficient SSSE=EISHE/∇T.\nAs regards the magnetic material as a source of the\nspincurrent,itismoreconvenienttouseinsulatorsrather\nthan conductors, in order to avoid parasitic signals such\nas a planar or anomalous Nernst effect [5]. There are\nthree main types of magnetic insulators possessing crit-\nical temperature TCabove the room temperature: gar-\nnets, spinels, and hexagonal ferrites. So far, most of the\nSSE experiments employed iron-based garnet because of\ntheir very low Gilbert damping constant, i.e.slow decay\n∗corresponding author: knizek@fzu.czof spin waves, since this decay limits the thickness of the\nmagnetic layer that actively generates the spin flow.\nIn this work we have focused on Y-hexaferrites as\nmagnetic material, namely Ba 2Zn2Fe12O22(Zn2Y) and\nBa2Co2Fe12O22(Co2Y). Their mass magnetizations at\nroomtemperatureare42.0emu/gforZn2Yand34emu/g\nfor Co2Y [6], which are higher than 27.6 emu/g of yt-\ntrium ferrite garnet Y 3Fe5O12[7]. Since a positive cor-\nrelation between SSE and the saturation magnetization\nhas also been proposed [8], Y-hexaferrites appear to be\na suitable material for the spin current generation in the\nspin-Seebeck effect.\nThe crystal structure of Y-hexaferrites belongs to the\ntrigonalspace group R3mand is composed ofalternating\nstacksofS (spinel Me 2Fe4O8, Me =Zn or Coin our case)\nand T (Ba 2Fe8O14) blocks along the hexagonal c-axis.\nThe magnetic configuration of Y-hexaferrites is usually\nferrimagnetic, with spin up orientation in octahedral 3 a,\n3band 18hsites and spin down in tetrahedral 6 cT, 6cS\nand octahedral 6 csites, see Fig. 1.\nMagnetocrystallineanisotropyis observedin all hexag-\nonal ferrites, which means that their induced magnetisa-\ntion has a preferred orientation within the crystal struc-\nture, either with an easy axis of magnetisation in the\nc-direction or with an easy plane of magnetisation per-\npendicular to c-direction, the latter being the case of the\nselected Y-hexaferrites. Due to their direction of easy\ngrow lying in ab-plane, hexaferrites inherently tend to\ngrow with their c-axis perpendicular to the film plane\nwhen deposited as thin films. Since the magnetization\nvector in SSE element should lie in parallel to the film\nsurface, the hexaferrites with an easy plane of magneti-\nsation are more suitable for the SSE experiment.\nThe principal difference between Zn2Y and Co2Y\ncomes from a different site preferences and magnetic\nproperties of Zn2+and Co2+. Zn2+ion is non-magnetic\n(d10) and occupies preferentially the tetrahedral sites.\nSince both Fe3+in tetrahedral sites have spin down ori-\nentation, thesubstitutionofZn2+tothesesitesmaximize2\n    \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n T \nS /g3 /g3 /g127/g374/g1006/g122/g3 /g18/g381/g1006/g122/g3\n/g1007/g258/g3/g3/g894/g381/g272/g410/g895/g3 → /g38/g286/g3 /g38/g286/g853/g18/g381/g3\n/g1010/g272\u0001/g3/g894/g410/g286/g410/g396/g895 /g3← /g38/g286\u0002/g127/g374/g3 /g38/g286/g3\n/g1010/g272/g3/g3/g894/g381/g272/g410/g895/g3 ← /g38/g286/g3 /g38/g286/g853/g18/g381/g3\n/g1005/g1012/g346/g3/g894/g381/g272/g410/g895 /g3→→→ /g38/g286/g3 /g38/g286/g853/g18/g381/g3\n/g1010/g272\u0003/g3/g894/g410/g286/g410/g396/g895 /g3← /g38/g286/g853/g127/g374/g3 /g38/g286/g3\n/g1007/g271/g3/g3/g894/g381/g272/g410/g895/g3 → /g38/g286/g3 /g38/g286/g853/g18/g381/g3\n/g1010/g272\u0000/g3/g894/g410/g286/g410/g396/g895 /g3← /g38/g286/g853/g127/g374/g3 /g38/g286/g3\n/g1005/g1012/g346/g3/g894/g381/g272/g410/g895 /g3→→→ /g38/g286/g3 /g38/g286/g853/g18/g381/g3\n/g1010/g272/g3/g3/g894/g381/g272/g410/g895/g3 ← /g38/g286/g3 /g38/g286/g853/g18/g381/g3\n/g1010/g272\u0004/g3/g894/g410/g286/g410/g396/g895 /g3← /g38/g286/g853/g127/g374/g3 /g38/g286/g3\n/g3 /g3 /g3 /g3\n/g3\nFIG. 1: One formula unit of Ba 2Zn2Fe12O22or\nBa2Co2Fe12O22structure with alternating structural\nblocksSandT. Shown are Fe, Co and Zn polyhedra and Ba\ncations (magenta bullets). The description includes Wycko ff\npositions, types of polyhedra (tetrahedral or octahedral) ,\narrows indicating spin direction of the collinear ferrimag netic\nstructure, and the preferential occupation of sites.\ntheoverallmagneticmomentandthesaturationmagneti-\nsation at low temperature reaches 18.4 µB(theoretical\nlimit considering 5 µBperFe3+would be 20 µB). How-\never, because of the relatively low critical temperature\nTC∼130◦C, the magnetization at room temperature is\nonly about 10.7 µB(42 emu/g). Co2+ion is in the low\nspin state (LS, t6\n2ge1\ng) and occupies preferentially the oc-\ntahedral sites. The resulting magnetic moment depends\non the actual distribution of Co between octahedral sites\noccupied by Fe3+with spin up or spin down orienta-\ntion, nevertheless generally will be much lower than in\nZn2Y and the typical saturation magnetization is around\n10µB. On the other hand, since the critical temperature\nTC∼340◦C of Co2Y is higher, the magnetic moment\natTroomaround 8.6 µBis not so different from that of\nZn2Y, see e.g.the review paper [6].\nSpin-SeebeckeffectinY-hexaferritewasstudiedforthe\ncompound of stoichiometry Ba 2−xSrxZn2Fe12O22(x=\n1.5) [9]. In this study it was observed, that the magni-\ntude of SEE is proportional to bulk magnetization even\nthrough the successive magnetic transitions among vari-\nous helimagnetic and ferrimagnetic phases. M-type hex-\naferrite BaFe 12O19was studied in [10]. Since M-type\nhexaferrite have strong anisotropy with an easy axis of\nmagnetisation in the c-direction, a proper substrate and\ndeposition procedure must be selected in order to grow\nthe thin films with the c-axis oriented parallel to the sur-\nface. The advantage of M-type is its high coercive field,\nwhich makes the resulting SSE element self-biased, thus\nproducing SEE signal even without presence of magnetic\nfield. Spin-Seebeck effect was also studied in Fe 3O4with\nspinel structure, which may be in some context consid-\nered as the simplest structural type of hexagonal ferrites.\nLarge coercive fields and high saturation magnetisationISHEEr\nMr/g115\nsJrΔ/g100/g460/g410/g460/g87/g410\n/g122/g882/g346/g286/g454/g258/g296/g286/g396/g396/g349/g410/g286\n/g94/g396/g100/g349/g75/g1007/g400/g437/g271/g400/g410/g396/g258/g410/g286\n/g4/g367/g69 /g400/g286/g393/g258/g396/g258/g410/g381/g396\n/g346/g286/g258/g410/g286/g396\nFIG. 2: Schema of the longitudinal experimental configura-\ntions. Directions of temperature gradient ( ∇T), magnetiza-\ntion (M), spin current ( Js) and electrical field resulted from\ninverse spin Hall effect ( EISHE) are shown. The meaning of\nparameters Vx,tz,dxand△Tzused in eq. 2 is also indicated.\nmakes Fe 3O4promising magnetic material for the inves-\ntigation of self-biased SSE elements [11–16]\nCurrent researchdescribes the SSE using typical quan-\ntity of spin Seebeck coefficient with unit of µV/K, which\nis in conventional thermoelectric materials used for eval-\nuating the effectiveness of the process. However, in most\nof the experimental setups the temperature sensors mea-\nsuring the temperature difference △Tare attached to\nthe measure cell itself. This implies that △Tdescribes\nnot only the thermal characteristics of the studied ma-\nterial, but the whole measurement cell instead, making\nthe quantity in unit of µV/Kphysically irrelevant to the\nspin Seebeck effect itself. This issue was studied in de-\ntails in Ref. [17]. The authors pointed out, that when\nusing the setup dependent △Tas independent variable\nthe determined SSE can be hardly comparable between\nlaboratories. In order to solve this problem, the authors\ndesigned a measurement system with precise measure-\nment of the heat flux through the sample and proposed\nusing heat flux or thermal gradient at the sample as the\nindependent variable.\nIn this work we have followed this approach and man-\nifested, that the total temperature difference △Tis not\nsuitable independent variable even for measuring within\none setup if the temperature dependent experiment is\nperformed, since the temperature evolution of thermal\nconductivity of the whole setup may be different from\nthat of the sample material itself.\nII. EXPERIMENTAL\nThin films of Ba 2Zn2Fe12O22(Zn2Y) and\nBa2Co2Fe12O22(Co2Y) were prepared by spin-coating\ntechnique on (111)-oriented, epitaxially polished3\nSrTiO 3(STO) single crystals with metalorganic\nprecursor solutions. Commercial 2-ethylhexanoates\nMe(CH 3(CH2)3CH(C2H5)COO) n(n= 2 for Me = Ba,\nCo, Zn; n = 3 for Me = Fe, ABCR, Germany) were used\nas precursors. Calculated amounts of metal precursors\nwere dissolved in iso-butanol, mixed and heated for\nseveral hours at 80◦C to accomplish homogenization.\nSubsequently a suitable amount of 2,2-diethanolamine\n(DEA) used as a modifier was added. The modifier to\nalkali earth metal molar ratio was n(DEA)/n(alkali earth\nmetal) = 2. Prior to the deposition the stock solutions\nwere usually diluted with iso-butanol to obtain films of\ndesired thickness. All reactions and handling were done\nunder dry nitrogen atmosphere to prevent reaction with\nair humidity and preliminary formation of alkaline earth\ncarbonates in solutions. Single crystals of STO were\nwashed in acetone combined with sonication and then\nannealed at 1200◦C in air for 24 hours to heal up the sur-\nface damage caused during polish treatment. Prior the\ndeposition they were treated with plasma (Zepto Plasma\ncleaner, Diener Electronic, Germany). After the drying\nat 110◦C for several minutes and pyrolysis of gel films at\n300◦C for 5 minutes, crystallization annealing was done\nat 1000◦C for 5 minutes in conventional tube furnace\nunder open air atmosphere. The deposition-annealing\ncycle was repeated ten times to obtain the desired film\nwith approximately 300 −350 nm of thickness. Final\nannealing was done in tube furnace under open air\natmosphere at 1050◦C for 5 min (Zn2Y) or 1000◦C for\n60 min (Co2Y).\nSpin Seebeck effect was measured using home-made\napparatus. A longitudinal configuration was used, in\nwhich the directions of the spin current, magnetic mo-\nments and electrical current are mutually perpendicular\n[18], see Fig. 2. AlN plate with high thermal conduc-\ntivity was used to separate the heater and the sample\nin order to uniformly spread the heat flux over the sam-\nple area. The thermal barriers between individual parts\nof the call were treated by appropriate greases (Apiezon\ntype N, Dow Corning Varnish, Ted Pella silver paste).\nThe width of the measured sample was 2 mm, the\nlengthwas7mmandcontactdistancewasapprox. 5mm.\nThickness of the Zn2Y-hexaferrite layers was between\n300−350 nm, the thickness for Co2Y-hexaferrite layers\nranged between 150 −300 nm. Pt layer was deposited\nusing K550X Quorum Technologies sputter coater. The\nthickness of the layer was determined by internal FTM\ndetector (Tool factor 4.7), the final Pt deposition thick-\nness was ∼8 nm. The resistanceof the Pt-layermeasured\nby a 2-point technique was within the range 350 −650 Ω\nat room temperature and linearly decreased by 10 −15%\ndown to 5 K, whereas the resistance of the Y-hexaferrite\nthin layer itself was more than GΩ. Therefore, the con-\ntributionfromthe anomalousNernst effect(ANE) canbe\nconsidered as negligible due to the resistivity difference\nbetween Y-hexaferrite and Pt layers.\nThe magnetic hysteresis loops were measured within\nthe range of magnetic field from −25 to 25 kOe at room25 30 35 45 50 55 60 650100020003000400050006000\n-0.5 0.0 0.5-0.5 0.0 0.5\n00120015\n0027\n0030Co2Y\nZn2Y\n2theta (Cu, K α) counts per sec.\nomega (°) counts per sec.(0012)\n±0.45°counts per sec.\nomega (°) \n(0012)\n±0.58°\nFIG. 3: X-ray diffraction of the Ba 2Zn2Fe12O22(black line,\nZn2Y) and Ba 2Co2Fe12O22(blue line, Co2Y) thin film. The\ninsets show rocking-curve measurements. The diffraction\npeak (111) of the SrTiO 3substrate is skipped.\nFIG. 4: AFM images of surface topography of (a)\nBa2Zn2Fe12O22(calculated roughness r.m.s. = 27 nm) and\n(b) Ba 2Co2Fe12O22(r.m.s = 30 nm).\ntemperature using a SQUID magnetometer (MPMSXL,\nQuantum Design)\nThe phase purity and degree of preferred orientation\nof the thin films was checked by X-ray diffraction over\nthe angular range 10 −100◦2θusing the X-ray pow-\nder diffractometer Bruker D8 Advance (CuK α1,2radia-\ntion, secondary graphite monochromator). Atomic force\nmicroscopy AFM (Explorer, Thermomicroscopes, USA)\nwas used to evaluate surface microstructure of the thin\nfilms.\nIII. RESULTS AND DISCUSSION\nThe X-ray diffraction confirmed single phase purity of\nthe thin film and c-axis preferred orientation, quantified\nby the full-width at the half-maximum (FWHM) of the\nrocking curve as 0.45◦for Zn2Y and 0.58◦for Co2Y, see\nFig. 3. The c-lattice parameters 43.567(7) ˚A for Zn2Y4\n/s45/s49/s46/s48 /s45/s48/s46/s53 /s48/s46/s48 /s48/s46/s53 /s49/s46/s48/s45/s50/s45/s49/s48/s49/s50/s86\n/s83/s83/s69/s32/s40 /s86/s41\n/s77/s97/s103/s110/s101/s116/s105/s99/s32/s102/s105/s101/s108/s100/s32/s40/s84/s41/s32 /s84/s32/s61/s32/s49/s75\n/s32 /s84/s32/s61/s32/s50/s75\n/s32 /s84/s32/s61/s32/s51/s75\n/s32 /s84/s32/s61/s32/s52/s75\n/s32 /s84/s32/s61/s32/s53/s75/s84/s32/s61/s32/s51/s48/s48/s75\n/s116/s32/s61/s32/s51/s48/s48/s32/s110/s109\n/s45/s49/s46/s48 /s45/s48/s46/s53 /s48/s46/s48 /s48/s46/s53 /s49/s46/s48/s45/s53/s48/s45/s52/s48/s45/s51/s48/s45/s50/s48/s45/s49/s48/s48/s49/s48/s50/s48/s51/s48/s52/s48/s53/s48/s77/s97/s103/s110/s101/s116/s105/s122/s97/s116/s105/s111/s110/s32/s77/s32/s40/s101/s109/s117/s47/s103/s41\n/s77/s97/s103/s110/s101/s116/s105/s99/s32/s102/s105/s101/s108/s100/s32/s72/s32/s40/s84/s41/s32/s80/s101/s114/s112/s101/s110/s100/s105/s99/s117/s108/s97/s114\n/s32/s80/s97/s114/s97/s108/s108/s101/s108\nFIG. 5: Spin Seebeck signal (upper panel), and in plane and\nout of plane magnetization (lower panel), in dependence on\nmagnetic field at 300 K for Ba 2Zn2Fe12O22.\nand 43.500(9) ˚A for Co2Y, calculated using cos θ/tanθ\nextrapolation to correct a possible off-centre position of\nthe film during XRD measurement, are in good agree-\nment with literature values [19].\nFig. 4 shows AFM images of surface topography of\nZn2Y and Co2Y. Platelets with hexagonal shape can be\nidentified in both images with similar shape and size.\nCalculated roughness (r.m.s.) values are around 27 −\n30 nm.\nThe magnetic properties ofthe Y-hexaferritesthin lay-\ners werecharacterizedby magnetization curvesmeasured\nat room temperature. The magnetic moment of Zn2Y at\nTroomdetermined from the saturatedvalue of magnetiza-\ntionin parallelorientationis11 µB, seethelowerpanelof\nFig. 5, which is comparable with the expected value [6].\nThe measurement confirms that Zn2Y is a soft magnet\nwith negligible hysteresis. The saturation in the orien-\ntation parallel with the thin layer is attained already at\nlow field, whereas the saturation in the out of plane ori-\nentation, i.e.along the c-direction, is achieved at higher\nfield above 1 ∼T, in agreement with the abeasy plane\norientation.\nThe Spin Seebeck signal of Zn2Y at room tempera-\nture is displayed in the upper panel of Fig. 5 for various/s45/s49/s46/s48 /s45/s48/s46/s53 /s48/s46/s48 /s48/s46/s53 /s49/s46/s48/s45/s48/s46/s49/s48/s46/s48/s48/s46/s49/s86\n/s83/s83/s69/s32/s40 /s86/s41\n/s77/s97/s103/s110/s101/s116/s105/s99/s32/s102/s105/s101/s108/s100/s32/s72/s32/s40/s84/s41/s32/s116/s32/s61/s32/s51/s48/s48/s32/s110/s109\n/s32/s116/s32/s61/s32/s49/s53/s48/s32/s110/s109/s84/s32/s61/s32/s51/s48/s48/s32/s75\n/s84/s32/s61/s32/s53/s32/s75\n/s45/s49/s46/s48 /s45/s48/s46/s53 /s48/s46/s48 /s48/s46/s53 /s49/s46/s48/s45/s52/s48/s45/s51/s48/s45/s50/s48/s45/s49/s48/s48/s49/s48/s50/s48/s51/s48/s52/s48/s77/s97/s103/s110/s101/s116/s105/s122/s97/s116/s105/s111/s110/s32/s77/s32/s40/s101/s109/s117/s47/s103/s41\n/s77/s97/s103/s110/s101/s116/s105/s99/s32/s102/s105/s101/s108/s100/s32/s72/s32/s40/s84/s41/s32/s80/s101/s114/s112/s101/s110/s100/s105/s99/s117/s108/s97/s114\n/s32/s80/s97/s114/s97/s108/s108/s101/s108\nFIG. 6: Spin Seebeck signal (upper panel) for 2 selected\nthin layers, and in plane and out of plane magnetization\n(lower panel), in dependence on magnetic field at 300 K for\nBa2Co2Fe12O22.\ntemperature gradients applied across the thin layer. The\nmeasuredvoltageispositiveinpositiveexternalmagnetic\nfield, in agreement with the positive spin Hall angle of Pt\n[20], and changes sign when switching the polarity of the\nmagneticfield. Thedependence onthe magneticfield has\nthe same shape with negligible hysteresis as the magne-\ntization in parallel orientation. The data clearly show\nlinear dependence on temperature gradient.\nThe magnetic moment of Co2Y at Troomdetermined\nfrom the saturated value of magnetization in parallel ori-\nentation is 10 µB/f.u., see the lower panel of Fig. 6.\nThis value is slightly higher than the expected moment\n[6], presumably due to relatively higher structural pref-\nerence of Co for spin down sites in the case of our thin\nfilms. The difference between the saturation in parallel\nandperpendicularorientationisbiggerinagreementwith\nhigher magnetocrystalline anisotropy of Co2Y compared\nto Zn2Y.\nHowever, despite the similar magnetic properties, the\nSSE signal for Co2Y was not observed, see the upper\npanel of Fig. 6. To explain this different behavior of\nZn2Y and Co2Y, we have considered the difference in\nthe cation distributions of the transition metal cations\nover the structure, see for details the Introduction sec-5\n/s45/s48/s46/s50/s48/s46/s48/s48/s46/s50/s48/s46/s52\n/s45/s48/s46/s53 /s48/s46/s48 /s48/s46/s53 /s49/s46/s48/s45/s48/s46/s50/s48/s46/s48/s48/s46/s50/s48/s46/s52\n/s45/s48/s46/s53 /s48/s46/s48 /s48/s46/s53 /s49/s46/s48/s83/s83/s69/s32/s40 /s86/s47/s75/s41/s32/s84/s32/s61/s32/s53/s75 /s32/s84/s32/s61/s32/s49/s48/s48/s75/s83/s83/s69/s32/s40 /s86/s47/s75/s41\n/s77/s97/s103/s110/s101/s116/s105/s99/s32/s102/s105/s101/s108/s100/s32/s72/s32/s40/s84/s41/s32/s84/s32/s61/s32/s50/s48/s48/s75\n/s77/s97/s103/s110/s101/s116/s105/s99/s32/s102/s105/s101/s108/s100/s32/s72/s32/s40/s84/s41/s32/s84/s32/s61/s32/s51/s48/s48/s75\nFIG. 7: Spin Seebeck signal (SSE) dependence on magnetic\nfield, divided by the total temperature difference △T, for\nBa2Zn2Fe12O22at selected temperatures.\ntion. Since Zn2+ion is a non-magnetic, the structure of\nBa2Zn2Fe12O22only contains one type of magnetic ion.\nZn cation preferentially substitutes Fe3+in two tetrahe-\ndral sites with the same direction of spin polarization,\ntherefore the total spin polarization of the unit cell does\nnot significantly fluctuate across the material. In dis-\ntinction, Ba 2Co2Fe12O22contains two types of magnetic\nions,i.e.Co2+in low spin state in addition to Fe3+\nin high spin state. Co substitute Fe3+in all octahedral\nsites, where Fe may have both directions of spin polar-\nization, without strong preference for a particular sites.\nWe tentatively propose, that this random distribution of\nCo2+over octahedral sites with various spin polariza-\ntions interferes the long range magnetic ordering across\nthe material, enhancesthe Gilbert damping constant and\npossibly results in suppressing of the SSE signal.\nSSE loops of Ba 2Zn2Fe12O22was measured at several\ntemperatures down to 5 K, see the measurements at se-\nlected temperatures 5, 100, 200 and 300 K in Fig. 7. The\noutput power of the heater was the same for all temper-\natures. The character of the loops is not changed with\nloweringtemperature, the magnitude of the signalis only\nvarying.\nIn order to investigate the temperature dependence of\nthe SSE signal of Zn2Y in more details, we have per-\nformed measurementdown to low temperature with 10 K\nstep. The output power of the heater was also kept con-\nstantduringthismeasurement. ThevalueoftheSSEwas\ndetermined by switching the magnetic field to ±0.4 T at\neach temperature and calculating the difference\nSSE=V+0.4T−V−0.4T\n2(1)\nThe resulting temperature dependence is displayed in\nthe Fig. 9 in three ways. In the upper part of the figure,\nFig. 9a, the SSE signal is divided by the total temper-\nature difference △Tdetermined over the whole measur-0 50 100 150 200 250 3000.00.51.01.52.0\n0 100 200 300050100150200250 Thermal conductivity (W/K/m)\nTemperature (K) AlN\n SrTiO3 W/K/m\nT (K)\nFIG. 8: Thermal conductivity of bulk Ba 2Zn2Fe12O22. Inset:\nthermal conductivity of AlN and SrTiO 3.\ning cell. The temperature evolution of △Tshown in the\ninset revealed, that △Tincreased several times during\ncooling. Since the output power of the heater was kept\napproximately constant, this increase should be related\nto a decrease of the thermal conductivity of the materi-\nals between the temperatures probes. To verify this as-\nsumption, we have measured thermal conductivity of the\nrelevant materials, i.e.the bulk sample Ba 2Zn2Fe12O22\nsynthesizedfromtheprecursorsusedforthethinlayerde-\nposition and compacted by isostatic pressing, AlN plate\nused to separate the heater and the sample, and the\nSrTiO 3substrate, see Fig. 8. However, thermal conduc-\ntivities of these materials weighted by their thickness in\nthe measuring cell cannot explain the evolution of △T.\nIt is obvious, that in order to explain the observed tem-\nperature dependence of △T, the thermal resistance of\nthe thermal barriers between the attached parts of the\ncell must be taken into account. We have calculated,\nthat the thermal resistance of the barriers at room tem-\nperature represents more than 50% of the total thermal\nresistance of the cell, and its percentage increases with\ntemperature.\nSince the value of the total temperature difference △T\ncannot be used as independent variable in different mea-\nsurement setups among various laboratories, another less\nsetup dependent parameter should be used instead, in\norder to normalize the measured SSE signal. We used\nthe heat flux through the sample, as it was proposed in\nRef [17]. SSE signal divided by the heat flux through\nthe sample is displayed in Fig. 7b, the corresponding\nheat flux corrected for the heat losses due to radiation,\nis shown in the inset.\nIn order to extract quantity comparable over differ-\nent measurement setups including the geometry of the\nsample, an expression for spin Seebeck effect related to\nsample dimensions and temperature difference over the\nsample itself was defined [17, 21]\nSSSE=Vxtz\ndx△Tz(2)\nwhereVxis the voltage measured, tzis the thickness6\n0.00.20.40.60.8\n0 100 200 30002040\n020406080100\n0 50 100 150 200 250 3000.00.10.20.30.40.50 100 200 300110120130\n0 100 200 3000.00.51.01.5SSE (µV/K)\na)ΔT (K)\nT (K)\nb) SSE (µV/ W)\nc) SSSE (µV/K)\nTemperature (K) Heat flux\n(mW)\nT (K)\n ΔTz(10-3K)\nT (K)\nFIG. 9: Spin Seebeck signal (SSE) dependence on tempera-\nture for Ba 2Zn2Fe12O22, (a) divided by overall temperature\ngradient, inset: temperature difference. (b) divided by hea t\nflux, inset: heat fluux. (c) calculated according to eq. 2, ins et:\ntemperature difference △Tz.\nof the magnetic material, dxis the electric contact dis-\ntance, and △Tzis the temperature difference at the mag-\nnetic material along the thickness tz, see Fig 2. With the\nknowledge of the heat flux and the thermal conductivity\nof the sample material Ba 2Zn2Fe12O22we were able to\ncalculate SSSEaccording to eq. 2, see the temperature\ndependence in Fig. 9c, the evolution of △Tzis displayed\nin the inset.\nThe correct normalization of SSE signal is important\nnot only for comparing among various measurement se-\ntups, but also for the correct determination of the tem-\nperature dependence, as it is evident by comparison of\nvarious temperature evolutions of SSE shown in Fig. 7.\nThe SSE related to the total temperature difference △T\n(Fig. 7a) shows incorrect temperature dependence influ-\nenced by the temperature dependence of the total ther-\nmal conductivity of the measuring setup. We propose\nthat the correct temperature dependence is determined\nby relating SSE to heat flux (Fig. 7b) or to temperature\ndifference at the sample △Tz(Fig. 7c). In this case, SSE\nis almost linearly increasing with lowering temperature.\nThe almost 5 ×increase of SSE at low temperaturecompared to room temperature can be partially ex-\nplained by the increased magnetization (almost 2 ×), but\nthe decrease of Gilbert damping factor αshould be of\ngreater influence in this regard. It was determined in\nthe study of the temperature dependence of SSE signal\nin Y3Fe5O12garnet (YIG) [22], that the effective prop-\nagation length of thermally excited magnons ξis pro-\nportional to T−1, and since at the same time α∼ξ−1\n[23, 24], it means that Gilbert damping factor α, which is\nexpected to suppress the SSE signal, is linearly decreas-\ning with temperature.\nIn distinction to temperature dependence of SSE in\nYIG [22], where a maximum in SSE was observed and\nexplained by the interplay of the increase of magnon ef-\nfective propagationlength and decrease ofthe total num-\nber of thermally excited magnons, we observed no max-\nimum down to low temperature. We ascribe it to the\nlower dispersion of acoustic branches in magnon spectra\nof Y-hexaferrite, which makes the influence of increasing\ntotal number of thermally excited magnons less impor-\ntant.\nFor the confrontation of the normalized room val-\nues between Y-hexaferrite and garnet, we have deter-\nmined values 21 µV/W and SSSE= 0.11µV/K for\nBa2Zn2Fe12O22, which are lower in comparison with\n46.6µV/W and 0.28 µV/K for Y 3Fe5O12[17], despite\nthe higher magnetic moment of Zn2Y. We presume, that\nit is due to the lower Gilbert damping constant αand\nhigher dispersion of acoustic branches in magnon spectra\nof Y3Fe5O12.\nIV. CONCLUSIONS\nSpin Seebeck effect (SSE) has been investigatedin thin\nfilms of two Y-hexagonal ferrites Ba 2Zn2Fe12O22(Zn2Y)\nand Ba 2Co2Fe12O22(Co2Y) deposited by spin-coating\nmethod on SrTiO 3(111) substrate. The SSE signal was\nobservedfor Zn2Y, whereasno significant SSE signal was\ndetected for Co2Y. This can be explained by a pres-\nence of two different magnetic ions in Co2Y, whose ran-\ndom distribution over octahedral sites interferes the long\nrange ordering and enhances the Gilbert damping con-\nstant. The magnitude of spin-Seebeck signal of Zn2Y\nnormalized to the temperature difference at the inves-\ntigated layer and sample dimensions ( SSSE) is compa-\nrable to the results measured on yttrium iron garnet\nY3Fe5O12.SSSEofZn2Yexhibitsmonotonicallyincreas-\ning behaviour with decreasing temperature, as a result\nof the simultaneous increase of the magnetization and\nmagnon effective propagation length.\nAcknowledgement . This work was supported by\nProject No. 14-18392S of the Czech Science Foundation\nand SGS16/245/OHK4/3T/14 of CTU Prague.7\n[1] Y. Xu, D. D. Awschalom, and J. Nitta, Handbook of\nSpintronics (Springer Netherlands, 2015).\n[2] D. M. Rowe, Thermoelectrics Handbook: Macro to Nano\n(Taylor & Francis, 2005).\n[3] K. Uchida, S. Takahashi, K. Harii, J. Ieda, W. Koshibae,\nK. Ando, S. Maekawa, and E. Saitoh, Observation of the\nspin Seebeck effect, Nature 455, 778 (2008).\n[4] E. Saitoh, M. Ueda, H. Miyajima, and G. Tatara, Con-\nversion of spin current into charge current at room tem-\nperature: Inverse spin-Hall effect, Appl. Phys. Lett. 88,\n182509 (2006).\n[5] K. Uchida, J. Xiao, H. Adachi, J. Ohe, S. Takahashi,\nJ. Ieda, T. Ota, Y. Kajiwara, H. Umezawa, H. Kawai,\nG.E.W.Bauer, S.Maekawa, andE.Saitoh, SpinSeebeck\ninsulator, Nat. Mater. 9, 894 (2010).\n[6] R. C. Pullar, Hexagonal ferrites: A review of the synthe-\nsis, properties and applications of hexaferrite ceramics,\nProg. Mater. Sci. 57, 1191 (2012).\n[7] NIST Standard Reference Material 2853, Magnetic Mo-\nment Standard - Yttrium Iron Garnet Sphere (2002).\n[8] K. Uchida, T. Nonaka, T. Kikkawa, Y. Kajiwara, and\nE. Saitoh, Longitudinal spin Seebeck effect in various\ngarnet ferrites, Phys. Rev. B 87, 104412 (2013).\n[9] R. Takagi, Y. Tokunaga, T. Ideue, Y. Taguchi,\nY. Tokura, and S. Seki, Thermal generation of spin cur-\nrent in a multiferroic helimagnet, APL Mater. 4, 32502\n(2016).\n[10] P. Li, D. Ellsworth, H. C. Chang, P. Janantha,\nD. Richardson, F. Shah, P. Phillips, T. Vijayasarathy,\nand M. Z. Wu, Generation of pure spin currents via spin\nSeebeck effect in self-biased hexagonal ferrite thin films,\nAppl. Phys. Lett. 105, 242412 (2014).\n[11] R. Ramos, T. Kikkawa, K. Uchida, H. Adachi, I. Lucas,\nM. H. Aguirre, P. Algarabel, L. Morellon, S. Maekawa,\nE. Saitoh, and M. R. Ibarra, Observation of the spin\nSeebeck effect in epitaxial Fe3O4 thin films, Appl. Phys.\nLett.102, 72413 (2013).\n[12] S. M. Wu, J. Hoffman, J. E. Pearson, and A. Bhat-\ntacharya, Unambiguous separation of the inverse spin\nHall and anomalous Nernst effects within a ferromag-\nnetic metal using the spin Seebeck effect, Appl. Phys.\nLett.105, 92409 (2014).\n[13] R. Ramos, T. Kikkawa, M. H. Aguirre, I. Lucas,\nA. Anadon, T. Oyake, K. Uchida, H. Adachi, J. Shiomi,\nP. A. Algarabel, L. Morellon, S. Maekawa, E. Saitoh, and\nM. R. Ibarra, Unconventional scaling and significant en-\nhancementofthespinSeebeckeffectinmultilayers, Phys.Rev. B92, 220407 (2015).\n[14] A. Anadon, R. Ramos, I. Lucas, P. A. Algarabel,\nL. Morellon, M. R. Ibarra, and M. H. Aguirre, Char-\nacteristic length scale of the magnon accumulation in\nFe3O4/Pt bilayer structures by incoherent thermal ex-\ncitation, Appl. Phys. Lett. 109, 12404 (2016).\n[15] A. J. Caruana, M. D. Cropper, J. Zipfel, Z. X. Zhou,\nG. D. West, and K. Morrison, Demonstration of poly-\ncrystalline thin film coatings on glass for spin Seebeck\nenergy harvesting, Phys. Status Solidi-Rapid Res. Lett.\n10, 613 (2016).\n[16] R. Ramos, A. Anadon, I. Lucas, K. Uchida, P. A. Al-\ngarabel, L. Morellon, M. H. Aguirre, E. Saitoh, and\nM. R. Ibarra, Thermoelectric performance of spin See-\nbeck effect in Fe3O4/Pt-based thin film heterostructures,\nAPL Mater. 4, 104802 (2016).\n[17] A. Sola, M. Kuepferling, V. Basso, M. Pasquale,\nT. Kikkawa, K. Uchida, and E. Saitoh, Evaluation of\nthermalgradientsinlongitudinalspinSeebeckeffectmea-\nsurements, J. Appl. Phys. 117, 17C510 (2015).\n[18] K. Uchida, M. Ishida, T. Kikkawa, A. Kirihara, T. Mu-\nrakami, and E. Saitoh, Longitudinal spin Seebeck effect:\nfrom fundamentals to applications, J. Phys.-Condens.\nMat.26, 343202 (2014).\n[19] H. S. Shin and S.-J. Kwon, X-ray powder diffraction pat-\nterns of two Y-type hexagonal ferrites, Powder Diffr. 8,\n98 (1993).\n[20] J. Sinova, S. O. Valenzuela, J. Wunderlich, C. H. Back,\nand T. Jungwirth, Spin Hall effects, Rev. Mod. Phys. 87,\n1213 (2015).\n[21] C. M. Jaworski, J. Yang, S. Mack, D. D. Awschalom,\nJ. P. Heremans, and R. C. Myers, Observation of the\nspin-Seebeck effect in a ferromagnetic semiconductor,\nNat. Mater. 9, 898 (2010).\n[22] E. J. Guo, J. Cramer, A. Kehlberger, C. A. Ferguson,\nD. A. MacLaren, G. Jakob, and M. Klaui, Influence of\nThickness and Interface on the Low-Temperature En-\nhancement of the Spin Seebeck Effect in YIG Films,\nPhys. Rev. X 6, 031012 (2016).\n[23] U. Ritzmann, D. Hinzke, and U. Nowak, Propagation of\nthermally induced magnonic spin currents, Phys. Rev. B\n89, 024409 (2014).\n[24] U. Ritzmann, D. Hinzke, A. Kehlberger, E. J. Guo,\nM. Klaui, and U. Nowak, Magnetic field control of the\nspin Seebeck effect, Phys. Rev. B 92, 174411 (2015)."    },    {        "title": "1704.01559v1.Relativistic_theory_of_magnetic_inertia_in_ultrafast_spin_dynamics.pdf",        "content": "Relativistic theory of magnetic inertia in ultrafast spin dynamics\nRitwik Mondal,\u0003Marco Berritta, Ashis K. Nandy, and Peter M. Oppeneer\nDepartment of Physics and Astronomy, Uppsala University, P. O. Box 516, SE-75120 Uppsala, Sweden\n(Dated: November 12, 2018)\nThe influence of possible magnetic inertia effects has recently drawn attention in ultrafast mag-\nnetization dynamics and switching. Here we derive rigorously a description of inertia in the\nLandau-Lifshitz-Gilbert equation on the basis of the Dirac-Kohn-Sham framework. Using the Foldy-\nWouthuysen transformation up to the order of 1=c4gives the intrinsic inertia of a pure system\nthrough the 2ndorder time-derivative of magnetization in the dynamical equation of motion. Thus,\nthe inertial damping Iis a higher order spin-orbit coupling effect, \u00181=c4, as compared to the\nGilbert damping \u0000that is of order 1=c2. Inertia is therefore expected to play a role only on ultra-\nshort timescales (sub-picoseconds). We also show that the Gilbert damping and inertial damping\nare related to one another through the imaginary and real parts of the magnetic susceptibility tensor\nrespectively.\nPACS numbers: 71.15.Rf, 75.78.-n, 75.40.Gb\nI. INTRODUCTION\nThe foundation of contemporary magnetization dy-\nnamics is the Landau-Lifshitz-Gilbert (LLG) equation\nwhich describes the precession of spin moment and a\ntransverse damping of it, while keeping the modulus of\nmagnetization vector fixed [1–3]. The LLG equation of\nmotion was originally derived phenomenologically and\nthe damping of spin motion has been attributed to rela-\ntivistic effects such as the spin-orbit interaction [1, 4–6].\nIn recent years there has been a flood of proposals for the\nfundamental microscopic mechanism behind the Gilbert\ndamping: the breathing Fermi surface model of Kamber-\nský, where the damping is due to magnetization preces-\nsion and the effect of spin-orbit interaction at the Fermi\nsurface [4], the extension of the breathing Fermi surface\nmodel to the torque-torque correlation model [5, 7], scat-\ntering theory description [8], effective field theories [9],\nlinear response formalism within relativistic electronic\nstructure theory [10], and the Dirac Hamiltonian theory\nformulation [11].\nFor practical reasons it was needed to extend the orig-\ninal LLG equation to include several other mechanisms\n[12, 13]. To describe e.g. current induced spin-transfer\ntorques, the effects of spin currents have been taken\ninto account [14–16], as well as spin-orbit torques [17],\nand the effect of spin diffusion [18]. A different kind of\nspin relaxation due to the exchange field has been intro-\nduced by Bar’yakhtar et al.[19]. In the Landau-Lifshitz-\nBar’yakhtar equation spin dissipations originate from the\nspatial dispersion of exchange effects through the second\norder space derivative of the effective field [20, 21]. A\nfurther recent work predicts the existence of extension\nterms that contain spatial as well temporal derivatives of\nthe local magnetization [22].\nAnother term, not discussed in the above investiga-\ntions, is the magnetic inertial damping that has recently\n\u0003ritwik.mondal@physics.uu.sedrawnattention[23–25]. Originally, magneticinertiawas\ndiscussed following the discovery of earth’s magnetism\n[26]. Within the LLG framework, inertia is introduced\nas an additional term [24, 27–29] leading to a modified\nLLG equation,\n@M\n@t=\u0000\rM\u0002He\u000b+M\u0002\u0012\n\u0000@M\n@t+I@2M\n@2t\u0013\n;(1)\nwhere \u0000is the Gilbert damping constant [1–3], \rthe gy-\nromagnetic ratio, He\u000bthe effective magnetic field, and I\nis the inertia of the magnetization dynamics, similar to\nthe mass in Newton’s equation. This type of motion has\nthe same classical analogue as the nutation of a spinning\nsymmetric top. The potential importance of inertia is il-\nlustrated in Fig. 1. While Gilbert damping slowly aligns\nthe precessing magnetization to the effective magnetic\nfield, inertial dynamics causes a trembling or nutation of\nthe magnetization vector [24, 30, 31]. Nutation could\nconsequently pull the magnetization toward the equa-\ntor and cause its switching to the antiparallel direction\n[32, 33], whilst depending crucially on the strength of\nthe magnetic inertia. The parameter Ithat character-\nizes the nutation motion is in the most general case a\ntensor and has been associated with the magnetic suscep-\ntibility [29, 31, 33]. Along a different line of reasoning,\nFähnle et al.extended the breathing Fermi surface model\nto include the effect of magnetic inertia [27, 34]. The\ntechnological importance of nutation dynamics is thus\nits potential to steer magnetization switching in memory\ndevices [23–25, 32] and also in skyrmionic spin textures\n[35]. Magnetization dynamics involving inertial dynam-\nics has been investigated recently and it was suggested\nthat its dynamics belongs to smaller time-scales i.e., the\nfemtosecond regime [24]. However, the origin of inertial\ndamping from a fundamental framework is still missing,\nand, moreover, although it is possible to vary the size of\nthe inertia in spin-dynamics simulations, it is unknown\nwhat the typical size of the inertial damping is.\nNaturally the question arises whether it is possible\nto derive the extended LLG equation including iner-arXiv:1704.01559v1  [cond-mat.other]  20 Mar 20172\nMHeff\nPrecession\nNutation\nFigure 1. (Color online) Schematic illustration of magnetiza-\ntion dynamics. The precessional motion of Maround He\u000bis\ndepicted by the blue solid-dashed curve and the nutation is\nshown by the red curve.\ntia while starting from the fully relativistic Dirac equa-\ntion. Hickey and Moodera [36] started from a Dirac\nHamiltonian and obtained an intrinsic Gilbert damping\nterm which originated from spin-orbit coupling. How-\never they started from only a part of the spin-orbit cou-\npling Hamiltonian which was anti-hermitian [37, 38]. A\nrecent derivation based on Dirac Hamiltonian theory for-\nmulation [11] showed that the Gilbert damping depends\nstrongly on both interband and intraband transitions\n(consistent with Ref. [39]) as well as the magnetic sus-\nceptibility response function, \u001fm. This derivation used\nthe relativistic expansion to the lowest order 1=c2of the\nhermitian Dirac-Kohn-Sham (DKS) Hamiltonian includ-\ning the effect of exchange field [40].\nIn this article we follow an approach similar to that of\nRef. [11] but we consider higher order expansion terms\nof the DKS Hamiltonian up to the order of 1=c4. This is\nshown to lead to the intrinsic inertia term in the modi-\nfied LLG equation and demonstrates that it stems from\na higher-order spin-orbit coupling term. A relativistic\norigin of the spin nutation angle, caused by Rashba-like\nspin-orbit coupling, was previously concluded, too, in the\ncontext of semiconductor nanostructures [41, 42].\nIn the following, we derive in Sec. II the relativistic\ncorrection terms to the extended Pauli Hamiltonian up\nto the order of 1=c4, which includes the spin-orbit inter-\naction and an additional term. Then the corresponding\nmagnetization dynamics is computed from the obtained\nspin Hamiltonian in Sec. III, which is shown to contain\nthe Gilbert damping and the magnetic inertial damping.\nFinally, we discuss the size of the magnetic inertia in re-\nlation to other earlier studies.II. RELATIVISTIC HAMILTONIAN\nFORMULATION\nWe start our derivation with a fully relativistic par-\nticle, a Dirac particle [43] inside a material and in the\npresence of an external field, for which we write the DKS\nHamiltonian:\nH=c\u000b\u0001(p\u0000eA) + (\f\u0000 1)mc2+V 1\n=O+ (\f\u0000 1)mc2+E; (2)\nwhereVis the effective crystal potential created by the\nion-ion, ion-electron and electron-electron interactions,\nA(r;t)is the magnetic vector potential from the external\nfield,cis the speed of light, mis particle’s mass and 1\nis the 4\u00024unit matrix. \u000band\fare the Dirac matrices\nthat have the form\n\u000b=\u0012\n0\u001b\n\u001b0\u0013\n; \f =\u0012\n10\n0\u00001\u0013\n;\nwhere\u001bis the Pauli spin matrix vector and 1is2\u00022unit\nmatrix. TheDiracequationisthenwrittenas i~@ (r;t)\n@t=\nH (r;t)for a Dirac bi-spinor  . The quantityO=c\u000b\u0001\n(p\u0000eA)defines the off-diagonal, or odd terms in the\nmatrix formalism and E=V 1are the diagonal, i.e., even\nterms. The latter have to be multiplied by a 2\u00022block\ndiagonal unit matrix in order to bring them in a matrix\nform. To obtain the nonrelativistic Hamiltonian and the\nrelativistic corrections one can write down the Dirac bi-\nspinor in double two component form as\n (r;t) =\u0012\n\u001e(r;t)\n\u0011(r;t)\u0013\n;\nand substitute those into the Dirac equation. The up-\nper two components represent the particle and the lower\ntwo components represent the anti-particle. However the\nquestion of separating the particle’s and anti-particle’s\nwave functions is not clear for any given momentum. As\nthe part\u000b\u0001pis off-diagonal in the matrix formalism, it\nretains the odd components and thus links the particle-\nantiparticle wave function. One way to eliminate the an-\ntiparticle’s wave function is by an exact transformation\n[44] which gives terms that require a further expansion in\npowers of 1=c2. Another way is to search for a represen-\ntation where the odd terms become smaller and smaller\nand one can ignore those with respect to the even terms\nand retain only the latter [45]. The Foldy-Wouthuysen\n(FW) transformation [46, 47] was the very successful at-\ntempt to find such a representation.\nIt is an unitary transformation obtained by suitably\nchoosing the FW operator,\nUFW=\u0000i\n2mc2\fO: (3)\nThe minus sign in front of the operator is because \fand\nOanti-commute with each other. The transformation of\nthewavefunctionadoptstheform  0(r;t) =eiUFW (r;t)3\nsuch that the probability density remains the same,\nj j2=j 0j2. The time-dependent FW transformation\ncan be expressed as [45, 48]\nHFW=eiUFW\u0012\nH\u0000i~@\n@t\u0013\ne\u0000iUFW+i~@\n@t:(4)\nThe first term can be expanded in a series as\neiUFWHe\u0000iUFW=H+i[UFW;H] +i2\n2![UFW;[UFW;H]]\n+::::+in\nn![UFW;[UFW;:::[UFW;H]:::]] +::: :(5)\nThe time dependency enters through the second term of\nEq. (4) and for a time-independent transformation one\nworks with@UFW\n@t= 0. It is instructive to note that the\naim of the whole procedure is to make the odd termssmaller and one can notice that as it goes higher and\nhigher in the expansion, the corresponding coefficients\ndecrease of the order 1=c2due to the choice of the unitary\noperator. After a first transformation, the new Hamilto-\nnian will contain new even terms, E0, as well as new odd\nterms,O0of1=c2or higher. The latter terms can be used\nto perform a next transformation having the new unitary\noperator as U0\nFW=\u0000i\n2mc2\fO0. After a second transfor-\nmation the new Hamiltonian, H0\nFWis achieved that has\nthe odd terms of the order 1=c4or higher. The trans-\nformation is a repetitive process and it continues until\nthe separation of positive and negative energy states are\nguaranteed.\nAfter a fourth transformation we derive the new trans-\nformed Hamiltonian with all the even terms that are cor-\nrect up to the order of1\nm3c6as [48–50]\nH000\nFW= (\f\u0000 1)mc2+\f\u0012O2\n2mc2\u0000O4\n8m3c6\u0013\n+E\u00001\n8m2c4h\nO;[O;E] +i~_Oi\n+\f\n16m3c6fO;[[O;E];E]g+\f\n8m3c6n\nO;h\ni~_O;Eio\n+\f\n16m3c6n\nO;(i~)2Oo\n: (6)\nNote that [A;B]defines the commutator, while fA;Bgrepresents the anti-commutator for any two operators Aand\nB. A similar Foldy-Wouthuysen transformation Hamiltonian up to an order of 1=m3c6was derived by Hinschberger\nand Hervieux in their recent work [51], however there are some differences, for example, the first and second terms in\nthe second line of Eq. (6) were not given. Once we have the transformed Hamiltonian as a function of odd and even\nterms, the final form is achieved by substituting the correct form of odd terms Oand calculating term by term.\nEvaluating all the terms separately, we derive the Hamiltonian for only the positive energy solutions i.e. the upper\ncomponents of the Dirac bi-spinor as a 2\u00022matrix formalism [40, 51, 52]:\nH000\nFW=(p\u0000eA)2\n2m+V\u0000e~\n2m\u001b\u0001B\u0000(p\u0000eA)4\n8m3c2\u0000e~2\n8m2c2r\u0001Etot+e~\n8m3c2n\n(p\u0000eA)2;\u001b\u0001Bo\n\u0000e~\n8m2c2\u001b\u0001[Etot\u0002(p\u0000eA)\u0000(p\u0000eA)\u0002Etot]\n\u0000e~2\n16m3c4f(p\u0000eA);@tEtotg\u0000ie~2\n16m3c4\u001b\u0001[@tEtot\u0002(p\u0000eA) + (p\u0000eA)\u0002@tEtot]; (7)\nwhere@t\u0011@=@tdefines the first-order time derivative.\nThehigherorderterms( 1=c6ormore)willinvolvesimilar\nformulations and more and more time derivatives of the\nmagnetic and electric fields will appear that stem from\nthe time derivative of the odd operator O[48, 51].\nThe fields in the last Hamiltonian (7) are defined as\nB=r\u0002A, the external magnetic field, Etot=Eint+\nEextare the electric fields where Eint=\u00001\nerVis the\ninternal field that exists even without any perturbation\nandEext=\u0000@A\n@tis the external field (only the temporal\npart is retained here because of the Coulomb gauge).\nThe spin Hamiltonian\nThe aim of this work is to formulate the magnetiza-\ntion dynamics on the basis of this Hamiltonian. Thus,we split the Hamiltonian into spin-independent and spin-\ndependentpartsandconsiderfromnowonelectrons. The\nspin Hamiltonian is straightforwardly given as\nHS(t) =\u0000e\nmS\u0001B+e\n4m3c2n\n(p\u0000eA)2;S\u0001Bo\n\u0000e\n4m2c2S\u0001[Etot\u0002(p\u0000eA)\u0000(p\u0000eA)\u0002Etot]\n\u0000ie~\n8m3c4S\u0001[@tEtot\u0002(p\u0000eA) + (p\u0000eA)\u0002@tEtot];\n(8)\nwhere the spin operator S= (~=2)\u001bhas been used. Let\nus briefly explain the physical meaning behind each term\nthat appears inHS(t). The first term defines the Zee-\nman coupling of the electron’s spin with the externally\napplied magnetic field. The second term defines an indi-\nrect coupling of light to the Zeeman interaction of spin\nand the optical B-field, which can be be shown to have4\nthe form of a relativistic Zeeman-like term. The third\nterm implies a general form of the spin-orbit coupling\nthat is gauge invariant [53], and it includes the effect of\nthe electric field from an internal as well as an external\nfield. Thelasttermisthenewtermofrelevanceherethat\nhas only been considered once in the literature by Hin-\nschberger et al.[51]. Note that, although the last term\nin Eq. (8) contains the total electric field, only the time-\nderivative of the external field plays a role here, because\nthe time derivative of internal field is zero as the ionic\npotential is time independent. In general if one assumes\na plane-wave solution of the electric field in Maxwell’s\nequation asE=E0ei!t, the last term can be written as\ne~!\n8m3c4S\u0001(E\u0002p)and thus adopts the form of a higher-\norder spin-orbit coupling for a general E-field.\nThe spin-dependent part can be easily rewritten in a\nshorter format using the identities:\nA\u0002(p\u0000eA)\u0000(p\u0000eA)\u0002A= 2A\u0002(p\u0000eA)\n+i~r\u0002A (9)\nA\u0002(p\u0000eA) + (p\u0000eA)\u0002A=\u0000i~r\u0002A(10)\nfor any operator A. This allows us to write the spin\nHamiltonian as\nHS=\u0000e\nmS\u0001B+e\n2m3c2S\u0001B\u0014\np2\u00002eA\u0001p+3e2\n2A2\u0015\n\u0000e\n2m2c2S\u0001\u0002\nEtot\u0002(p\u0000eA)\u0003\n+ie~\n4m2c2S\u0001@tB\n+e~2\n8m3c4S\u0001@ttB: (11)\nHere, the Maxwell’s equations have been used to derive\nthe final form that the spatial derivative of the electric\nfield will generate a time derivative of a magnetic field\nsuch that r\u0002Eext=\u0000@B\n@t, whilst the curl of a internal\nfield results in zero as the curl of a gradient function is\nalways zero. The final spin Hamiltonian (11) bears much\nimportance for the strong laser field-matter interaction\nas it takes into account all the field-spin coupling terms.\nIt is thus the appropriate fundamental Hamiltonian to\nunderstand the effects of those interactions on the mag-\nnetization dynamics described in the next section.\nIII. MAGNETIZATION DYNAMICS\nIn general, magnetization is given by the magnetic mo-\nment per unit volume in a magnetic solid. The magnetic\nmomentisgivenby g\u0016BhSi,wheregistheLandég-factor\nand\u0016Bis the unit of Bohr magneton. The magnetization\nis then written\nM(r;t) =X\njg\u0016B\n\nhSji; (12)\nwhere \nis the suitably chosen volume element, the sum\njgoes over all electrons in the volume element, and h::i\nis the expectation value. To derive the dynamics, wetake the time derivative in both the sides of Eq. (12)\nand, withintheadiabaticapproximation, wearriveatthe\nequation of motion for the magnetization as [36, 54, 55]\n@M\n@t=X\njg\u0016B\n\n1\ni~h\u0002\nSj;HS(t)\u0003\ni:(13)\nNow the task looks simple, one needs to substitute the\nspin Hamiltonian (11) and calculate the commutators in\norder to find the equation of motion. Note that the dy-\nnamics only considers the local dynamics as we have not\ntaken into account the time derivative of particle density\noperator (for details, see [11]). Incorporating the latter\nwould give rise the local as well as non-local processes\n(i.e., spin currents) within the same footing.\nThe first term in the spin Hamiltonian produces the\ndynamics as\n@M(1)\n@t=\u0000\rM\u0002B; (14)\nwith\r=gjej=2mdefines the gyromagnetic ratio and\nthe Landé g-factor g\u00192for spins, the electronic charge\ne < 0. Using the linear relationship of magnetization\nwith the magnetic field B=\u00160(H+M), the latter is\nreplaced in Eq. (14) to get the usual form in the Landau-\nLifshitz equations, \u0000\r0M\u0002H, where\r0=\u00160\ris the\neffective gyromagnetic ratio. This gives the Larmor pre-\ncessionofmagnetizationaroundaneffectivefield H. The\neffective field will always have a contribution from a ex-\nchange field and the relativistic corrections to it, which\nhas not been explicitly taken into account in this article,\nas they are not in the focus here. For detailed calcula-\ntions yet including the exchange field see Ref. [11].\nThe second term in the spin Hamiltonian Eq. (11) will\nresult in a relativistic correction to the magnetization\nprecession. Within an uniform field approximation (A=\nB\u0002r=2), the corresponding dynamics will take the form\n@M(2)\n@t=\r\n2m2c2M\u0002BD\np2\u0000eB\u0001L+3e2\n8(B\u0002r)2E\n;\n(15)\nwithL=r\u0002pthe orbital angular momentum. The\npresence of \r=2m2c2implies that the contribution of this\ndynamics to the precession is relatively small, while the\nleading precession dynamics is given by Eq. (14). For\nsake of completeness we note that a relativistic correc-\ntion to the precession term of similar order 1=m2c2was\nobtained previously for the exchange field [11].\nThe next term in the Hamiltonian is a bit tricky to\nhandle as the third term in Eq. (11) is not hermitian, not\neven the fourth term which is anti-hermitian. However\ntogether they form a hermitian Hamiltonian [11, 37, 38].\nTherefore one has to work together with those terms and\ncannot only perform the dynamics with an individual\nterm. In an earlier work [11] we have shown that taking\nanuniformmagneticfieldalongwiththegauge A=B\u0002r\n2\nwill preserve the hermiticity. The essence of the uniform\nfield lies in the assumption that the skin depth of the5\nelectromagnetic field is longer than the thickness of the\nthin-film samples used in experiments. The dynamical\nequation of spin motion with the second and third terms\nthus thus be written in a compact form for harmonic ap-\nplied fields as [11]\n@M(3;4)\n@t=M\u0002\u0012\nA\u0001@M\n@t\u0013\n; (16)\nwith the intrinsic Gilbert damping parameter Athat is\na tensor defined by\nAij=\r\u00160\n4mc2X\nn;kh\nhripk+pkrii\u0000hrnpn+pnrni\u000eiki\n\u0002\u0010\n1+\u001f\u00001\nm\u0011\nkj:(17)\nHere\u001fmis the magnetic susceptibility tensor of rank 2\n(a3\u00023matrix) and 1is the 3\u00023unit matrix. Note that\nfor diagonal terms i.e., i=kthe contributions from the\nexpectation values of rkpicancel each other. The damp-\ning tensor can be decomposed to have contributions from\nan isotropic Heisenberg-like, anisotropic Ising-like and\nDzyaloshinskii-Moriya-like tensors. The anti-symmetric\nDzyaloshinskii-Moriya contribution has been shown to\nlead to a chiral damping of the form M\u0002(D\u0002@M=@t)\n[11]. Experimental observations of chiral damping have\nbeen reported recently [56]. The other cross term having\nthe formE\u0002Ain Eq. (11) is related to the angular mo-\nmentum of the electromagnetic field and thus provides\na torque on the spin that has been at the heart of an-\ngular magneto-electric coupling [53]. A possible effect in\nspin dynamics including the light’s angular momentum\nhas been investigated in the strong field regime and it\nhas been shown that one has to include this cross term\nin the dynamics in order to explain the qualitative and\nquantitative strong field dynamics [57].\nFor the last term in the spin Hamiltonian (11) it is\nrather easy to formulate the spin dynamics because it is\nevidently hermitian. Working out the commutator with\nthe spins gives a contribution to the dynamics as\n@M(5)\n@t=\u000eM\u0002@2B\n@t2; (18)\nwith the constant \u000e=\r~2\n8m2c4.\nLet us work explicitly with the second-order time\nderivative of the magnetic induction by the relation B=\n\u00160(H+M), using a chain rule for the derivative:\n@2B\n@t2=@\n@t\u0010@B\n@t\u0011\n=\u00160@\n@t\u0010@H\n@t+@M\n@t\u0011\n=\u00160\u0010@2H\n@t2+@2M\n@t2\u0011\n: (19)\nThis is a generalized equation for the time-derivative of\nthe magnetic induction which can be used even for non-\nharmonic fields. The magnetization dynamics is then\ngiven by\n@M(5)\n@t=\u00160\u000eM\u0002\u0010@2H\n@t2+@2M\n@t2\u0011\n:(20)Thus the extended LLG equation of motion will have\nthese two additional terms: (1) a field-derivative torque\nand (2) magnetization-derivative torque, and they ap-\npear with their 2ndorder time derivative. It deserves to\nbe noted that, in a previous theory we also obtained a\nsimilar term–a field-derivative torque in 1storder-time\nderivative appearing in the generalized Gilbert damping.\nSpecifically, the extended LLG equation for a general\ntime-dependent field H(t)becomes\n@M\n@t=\u0000\r0M\u0002H+M\u0002h\n\u0016A\u0001\u0010@H\n@t+@M\n@t\u0011i\n+\u00160\u000eM\u0002\u0010@2H\n@t2+@2M\n@t2\u0011\n; (21)\nwhere \u0016AisamodifiedGilbertdampingtensor(fordetails,\nsee [11]).\nHowever for harmonic fields, the response of the ferro-\nmagnetic materials is measured through the differential\nsusceptibility, \u001fm=@M=@H, because there exists a net\nmagnetization even in the absence of any applied field.\nWith this, the time derivative of the harmonic magnetic\nfield can be further written as:\n@2H\n@t2=@\n@t\u0010@H\n@M@M\n@t\u0011\n=@\n@t\u0010\n\u001f\u00001\nm\u0001@M\n@t\u0011\n=@\u001f\u00001\nm\n@t\u0001@M\n@t+\u001f\u00001\nm\u0001@2M\n@t2: (22)\nIn general the magnetic susceptibility is a spin-spin re-\nsponse function that is wave-vector and frequency depen-\ndent. Thus, Eq. (18) assumes the form with the first and\nsecond order time derivatives as\n@M(5)\n@t=M\u0002\u0012\nK\u0001@M\n@t+I\u0001@2M\n@t2\u0013\n;(23)\nwhere the parameters Iij=\u00160\u000e\u0000\n1+\u001f\u00001\nm\u0001\nijandKij=\n\u00160\u000e@t(\u001f\u00001\nm)ijare tensors. The dynamics of the second\nterm is that of the magnetic inertia that operates on\nshorter time scales [25].\nHaving all the required dynamical terms, finally the\nfull magnetization dynamics can be written by joining\ntogether all the individual parts. Thus the full magneti-\nzation dynamics becomes, for harmonic fields,\n@M\n@t=M\u0002\u0012\n\u0000\r0H+ \u0000\u0001@M\n@t+I\u0001@2M\n@t2\u0013\n:(24)\nNote that the Gilbert damping parameter \u0000has two con-\ntributions, one from the susceptibility itself, Aij, which\nis of order 1=c2and an other from the time derivative of\nit,Kijof order 1=c4. Thus, \u0000ij=Aij+Kij. However we\nwill focus on the first one only as it will obviously be the\ndominant contribution, i.e., \u0000ij\u0019Aij. Even though we\nconsider only the Gilbert damping term of order 1=c2in\nthe discussions, we shall explicitly analyze the other term\nof the order 1=c4. For an ac susceptibility i.e., \u001f\u00001\nm/ei!t\nwe find thatKij/\u00160\u000e@t(\u001f\u00001\nm)ij/i\u00160!\u000e\u001f\u00001\nm, which\nsuggests again that the Gilbert damping parameter of6\nthe order 1=c4will be given by the imaginary part of the\nsusceptibility,Kij/\u0000\u00160!\u000eIm\u0000\n\u001f\u00001\nm\u0001\n.\nThe last equation (24) is the central result of this\nwork, as it establishes a rigorous expression for the in-\ntrinsic magnetic inertia. Magnetization dynamics in-\ncluding inertia has been discussed in few earlier articles\n[24, 30, 31, 58]. The very last term in Eq. (24) has been\nassociated previously with the inertia magnetization dy-\nnamics [32, 59, 60]. As mentioned, it implies a magne-\ntization nutation i.e., a changing of the precession angle\nas time progresses. Without the inertia term we obtain\nthe well-known LLG equation of motion that has already\nbeen used extensively in magnetization dynamics simu-\nlations (see, e.g., [61–65]).\nIV. DISCUSSIONS\nMagnetic inertia was discussed first in relation to the\nearth’s magnetism [26]. From a dimensional analysis,\nthe magnetic inertia of a uniformly magnetized sphere\nundergoing uniform acceleration was estimated to be of\nthe order of 1=c2[26], which is consistent with the here-\nobtained relativistic nature of magnetic inertia.\nOur derivation based on the fundamental Dirac-Kohn-\nSham Hamiltonian provides explicit expressions for both\nthe Gilbert and inertial dampings. Thus, a comparison\ncan be made between the Gilbert damping parameter\nand the magnetic inertia parameter of a pure system.\nAs noticed above, both the parameters are given by the\nmagnetization susceptibility tensor, however it should be\nnoted that the quantiy hr\u000bp\fiis imaginary itself, because\n[11],\nhr\u000bp\fi=\u0000i~\n2mX\nn;n0;kf(Enk)\u0000f(En0k)\nEnk\u0000En0kp\u000b\nnn0p\f\nn0n:(25)\nThus the Gilbert damping parameter should be given by\nthe imaginary part of the susceptibility tensor [36, 66].\nOn the other hand the magnetic inertia tensor must be\ngiven by the real part of the susceptibility [31]. This is\nin agreement with a recent article where the authors also\nfound the same dependence of real and imaginary parts\nof susceptibility to the nutation and Gilbert damping re-\nspectively [33]. In our calculation, the Gilbert damping\nand inertia parameters adopt the following forms respec-tively,\n\u0000ij=i\r\u00160\n4mc2X\nn;k[hripk+pkrii\u0000hrnpn+pnrni\u000eik]\n\u0002Im\u0000\n\u001f\u00001\nm\u0001\nkj\n=\u0000\u00160\r~\n4mc2X\nn;k\u0014hripk+pkrii\u0000hrnpn+pnrni\u000eik\ni~\u0015\n\u0002Im\u0000\n\u001f\u00001\nm\u0001\nkj\n=\u0000\u0010X\nn;k\u0014hripk+pkrii\u0000hrnpn+pnrni\u000eik\ni~\u0015\n\u0002Im\u0000\n\u001f\u00001\nm\u0001\nkj;(26)\nIij=\u00160\r~2\n8m2c4h\n1+Re\u0000\n\u001f\u00001\nm\u0001\nkji\n=\u0010~\n2mc2h\n1+Re\u0000\n\u001f\u00001\nm\u0001\nkji\n; (27)\nwith\u0010\u0011\u00160\r~\n4mc2. Note that the change of sign from damp-\ningtensortotheinertiatensorthatisalsoconsistentwith\nRef. [33], and also a factor of 2 present in inertia. How-\never, most importantly, the inertia tensor is ~=mc2times\nsmallerthan the damping tensor as is revealed in our\ncalculations. Considering atomic units we can evaluate\n\u0010\u0018\u00160\n4c2\u00180:00066\n4\u00021372\u00188:8\u000210\u00009;\n\u0010~\n2mc2\u0018\u0010\n2c2\u00188:8\u000210\u00009\n2\u00021372\u00182:34\u000210\u000013:\nThis implies that the intrinsic inertial damping is typi-\ncally 4\u0002104times smaller than the Gilbert damping and\nit is not an independently variable parameter. Also, be-\ncause of its smallness magnetic inertial dynamics will be\nmore significant on shorter timescales [24].\nA further analysis of the two parameters can be made.\nOnecanusetheKramers-Kronigtransformationtorelate\nthe real and imaginary parts of a susceptibility tensor\nwith one another. This suggests a relation between the\ntwo parameters that has been found by Fähnle et al.[34],\nnamelyI=\u0000\u0000\u001c,where\u001cisarelaxationtime. Weobtain\nhere a similar relation, I/\u0000 \u0000\u0016\u001c, where \u0016\u001c=~=mc2has\ntime dimension.\nEven though the Gilbert damping is c2times larger\nthan the inertial damping, the relative strength of the\ntwo parameters also depends on the real and imaginary\nparts of the susceptibility tensor. In special cases, when\nthe real part of the susceptibility is much higher than\nthe imaginary part, their strength could be comparable\nto each other. We note in this context that there exist\nmaterials where the real part of the susceptibility is 102\u0000\n103times larger than the imaginary part.\nFinally, we emphasize that our derivation provides the\nintrinsic inertial damping of a pure, isolated system. For\nthe Gilbert damping it is already well known that en-\nvironmental effects, such as interfaces or grain bound-\naries, impurities, film thickness, and even interactions of7\nthe spins with quasi-particles, for example, phonons, can\nmodify the extrinsic damping (see, e.g., [67–69]). Simi-\nlarly, it can be expected that the inertial damping will\nbecome modified through environmental influences. An\nexample of environmental effects that can lead to mag-\nnetic inertia have been considered previously, for the case\nof a local spin moment surrounded by conduction elec-\ntrons, whose spins couple to the local spin moment and\naffect its dynamics [31, 32].\nV. CONCLUSIONS\nIn conclusion, we have rigorously derived the magne-\ntization dynamics from the fundamental Dirac Hamilto-\nnian and have provided a solid theoretical framework for,\nand established the origin of, magnetic inertia in pure\nsystems. We have derived expressions for the Gilbert\ndamping and the magnetic inertial damping on the same\nfooting and have shown that both of them have a rela-\ntivistic origin. The Gilbert damping stems from a gen-\neralized spin-orbit interaction involving external fields,\nwhiletheinertialdampingisduetohigher-order(in 1=c2)\nspin-orbit contributions in the external fields. Both have\nbeen shown to be tensorial quantities. For general time\ndependent external fields, a field-derivative torque with\na 1storder time derivative appears in the Gilbert-type\ndamping, and a 2ndorder time-derivative field torque ap-\npears in the inertial damping.\nIn the case of harmonic external fields, the expressions\nof the magnetic inertia and the Gilbert damping scalewith the real part and the imaginary part, respectively,\nof the magnetic susceptibility tensor, and they are op-\nposite in sign. Alike the Gilbert damping, the magnetic\ninertia tensor is also temperature dependent through the\nmagnetic response function and also magnetic moment\ndependent. Importantly, we find that the intrinsic iner-\ntial damping is much smaller than the Gilbert damping,\nwhich corroborates the fact that magnetic inertia was\nneglected in the early work on magnetization dynamics\n[1–3, 19]. This suggests, too, that the influence of mag-\nnetic inertia will be quite restricted, unless the real part\nof the susceptibility is much larger than the imaginary\npart. Another possibility to enhance the magnetic iner-\ntia would be to use environmental influences to increase\nits extrinsic contribution. Our theory based on the Dirac\nHamiltonian leads to exact expressions for both the in-\ntrinsic Gilbert and inertial damping terms, thus provid-\ning a solid base for their evaluation within ab initio elec-\ntronic structure calculations and giving suitable values\nthat can be used in future LLG magnetization dynamics\nsimulations.\nACKNOWLEDGMENTS\nWe thank Danny Thonig and Alex Aperis for use-\nful discussions. This work has been supported by the\nSwedishResearchCouncil(VR),theKnutandAliceWal-\nlenberg Foundation (Contract No. 2015.0060), and the\nSwedish National Infrastructure for Computing (SNIC).\n[1] L. D. Landau and E. M. Lifshitz, Phys. Z. Sowjetunion\n8, 101 (1935).\n[2] T. L. Gilbert, Ph.D. thesis, Illinois Institute of Technol-\nogy, Chicago, 1956.\n[3] T. L. Gilbert, IEEE Trans. Magn. 40, 3443 (2004).\n[4] V. Kamberský, Can. J. Phys. 48, 2906 (1970).\n[5] V. Kamberský, Czech. J. Phys. B 26, 1366 (1976).\n[6] J. Kuneš and V. Kamberský, Phys. Rev. B 65, 212411\n(2002).\n[7] V. Kamberský, Phys. Rev. B 76, 134416 (2007).\n[8] A. Brataas, Y. Tserkovnyak, and G. E. W. Bauer, Phys.\nRev. Lett. 101, 037207 (2008).\n[9] M. Fähnle and C. Illg, J. Phys.: Condens. Matter 23,\n493201 (2011).\n[10] H. Ebert, S. Mankovsky, D. Ködderitzsch, and P. J.\nKelly, Phys. Rev. Lett. 107, 066603 (2011).\n[11] R.Mondal, M.Berritta, andP.M.Oppeneer,Phys.Rev.\nB94, 144419 (2016).\n[12] L. Berger, Phys. Rev. B 54, 9353 (1996).\n[13] G. P. Zhang, W. Hübner, G. Lefkidis, Y. Bai, and T. F.\nGeorge, Nature Phys. 5, 499 (2009).\n[14] S. Zhang and Z. Li, Phys. Rev. Lett. 93, 127204 (2004).\n[15] G. Tatara and H. Kohno, Phys. Rev. Lett. 92, 086601\n(2004).\n[16] R. A. Duine, A. S. Núñez, J. Sinova, and A. H. Mac-Donald, Phys. Rev. B 75, 214420 (2007).\n[17] F. Freimuth, S. Blügel, and Y. Mokrousov, Phys. Rev.\nB92, 064415 (2015).\n[18] Y. Tserkovnyak, E. M. Hankiewicz, and G. Vignale,\nPhys. Rev. B 79, 094415 (2009).\n[19] V. G. Bar’yakhtar, Sov. Phys. JETP 60, 863 (1984).\n[20] V. G. Bar’yakhtar and A. G. Danilevich, Low Temp.\nPhys. 39, 993 (2013).\n[21] V.G.Bar’yakhtar,V.I.Butrim, andB.A.Ivanov,JETP\nLett.98, 289 (2013).\n[22] Y. Li and W. E. Bailey, Phys. Rev. Lett. 116, 117602\n(2016).\n[23] A. V. Kimel, B. A. Ivanov, R. V. Pisarev, P. A. Usachev,\nA. Kirilyuk, and T. Rasing, Nature Phys. 5, 727 (2009).\n[24] M.-C. Ciornei, J. M. Rubí, and J.-E. Wegrowe, Phys.\nRev. B 83, 020410 (2011).\n[25] M.-C. Ciornei, Ph.D. thesis, Ecole Polytechnique, Uni-\nversidad de Barcelona, 2010.\n[26] G. W. Walker, Proc. Roy. Soc. London A 93, 442 (1917).\n[27] M. Faehnle and C. Illg, J. Phys.: Condens. Matter 23,\n493201 (2011).\n[28] J.-E. Wegrowe and M.-C. Ciornei, Am. J. Phys. 80, 607\n(2012).\n[29] E. Olive, Y. Lansac, M. Meyer, M. Hayoun, and J.-E.\nWegrowe, J. Appl. Phys. 117, 213904 (2015).8\n[30] D.BöttcherandJ.Henk,Phys.Rev.B 86,020404(2012).\n[31] S. Bhattacharjee, L. Nordström, and J. Fransson, Phys.\nRev. Lett. 108, 057204 (2012).\n[32] T. Kikuchi and G. Tatara, Phys. Rev. B 92, 184410\n(2015).\n[33] D. Thonig, M. Pereiro, and O. Eriksson,\narXiv:1607.01307v1 (2016).\n[34] M. Fähnle, D. Steiauf, and C. Illg, Phys. Rev. B 88,\n219905(E) (2013).\n[35] F. Büttner, C. Moutafis, M. Schneider, B. Krüger, C. M.\nGünther, J. Geilhufe, C. von Korff Schmising, J. Mo-\nhanty, B. Pfau, S. Schaffert, A. Bisig, M. Foerster,\nT. Schulz, C. A. F. Vaz, J. H. Franken, H. J. M. Swagten,\nM. Kläui, and S. Eisebitt, Nature Phys. 11, 225 (2015).\n[36] M. C. Hickey and J. S. Moodera, Phys. Rev. Lett. 102,\n137601 (2009).\n[37] A. Widom, C. Vittoria, and S. D. Yoon, Phys. Rev. Lett.\n103, 239701 (2009).\n[38] M. C. Hickey, Phys. Rev. Lett. 103, 239702 (2009).\n[39] K. Gilmore, Y. U. Idzerda, and M. D. Stiles, Phys. Rev.\nLett.99, 027204 (2007).\n[40] R. Mondal, M. Berritta, K. Carva, and P. M. Oppeneer,\nPhys. Rev. B 91, 174415 (2015).\n[41] S. Y. Cho, Phys. Rev. B 77, 245326 (2008).\n[42] R. Winkler, Phys. Rev. B 69, 045317 (2004).\n[43] P. A. M. Dirac, Proc. Roy. Soc. London A 117, 610\n(1928).\n[44] T. Kraft, P. M. Oppeneer, V. N. Antonov, and H. Es-\nchrig, Phys. Rev. B 52, 3561 (1995).\n[45] P. Strange, Relativistic Quantum Mechanics: With Ap-\nplications in Condensed Matter and Atomic Physics\n(Cambridge University Press, 1998).\n[46] L. L. Foldy and S. A. Wouthuysen, Phys. Rev. 78, 29\n(1950).\n[47] W. Greiner, Relativistic Quantum Mechanics. Wave\nEquations (Springer, 2000).\n[48] A. J. Silenko, Phys. Rev. A 93, 022108 (2016).\n[49] F.Schwabl, R.Hilton, andA.Lahee, Advanced Quantum\nMechanics , Advanced Texts in Physics Series (Springer\nBerlin Heidelberg, 2008).\n[50] A. Y. Silenko, Theor. Math. Phys. 176, 987 (2013).\n[51] Y. Hinschberger and P.-A. Hervieux, Phys. Lett. A 376,\n813 (2012).\n[52] Y. Hinschberger and P.-A. Hervieux, Phys. Rev. B 88,\n134413 (2013).\n[53] R. Mondal, M. Berritta, C. Paillard, S. Singh, B. Dkhil,\nP. M. Oppeneer, and L. Bellaiche, Phys. Rev. B 92,\n100402 (2015).\n[54] J. Ho, F. C. Khanna, and B. C. Choi, Phys. Rev. Lett.\n92, 097601 (2004).\n[55] R. M. White, Quantum Theory of Magnetism: Magnetic\nProperties of Materials (Springer, Heidelberg, 2007).\n[56] E. Jué, C. K. Safeer, M. Drouard, A. Lopez, P. Balint,\nL. Buda-Prejbeanu, O. Boulle, S. Auffret, A. Schuhl,\nA. Manchon, I. M. Miron, and G. Gaudin, Nat. Mater.\n15, 272 (2016).\n[57] H. Bauke, S. Ahrens, C. H. Keitel, and R. Grobe, New\nJ. Phys. 16, 103028 (2014).\n[58] M. Fähnle, D. Steiauf, and C. Illg, Phys. Rev. B 84,\n172403 (2011).\n[59] Y. Li, A.-L. Barra, S. Auffret, U. Ebels, and W. E.\nBailey, Phys. Rev. B 92, 140413 (2015).\n[60] E. Olive, Y. Lansac, and J.-E. Wegrowe, Appl. Phys.\nLett.100, 192407 (2012).[61] M. Djordjevic and M. Münzenberg, Phys. Rev. B 75,\n012404 (2007).\n[62] U. Nowak, in Handbook of Magnetism and Advanced\nMagnetic Materials , Vol. 2, edited by H. Kronmüller and\nS. Parkin (John Wiley & Sons, Chichester, 2007).\n[63] B. Skubic, J. Hellsvik, L. Nordström, and O. Eriksson,\nJ. Phys.: Condens. Matter 20, 315203 (2008).\n[64] R. F. L. Evans, W. J. Fan, P. Chureemart, T. A. Ostler,\nM. O. A. Ellis, and R. W. Chantrell, J. Phys.: Condens.\nMatter 26, 103202 (2014).\n[65] D. Hinzke, U. Atxitia, K. Carva, P. Nieves,\nO. Chubykalo-Fesenko, P. M. Oppeneer, and U. Nowak,\nPhys. Rev. B 92, 054412 (2015).\n[66] K. Gilmore, Ph.D. thesis, Montana State University,\n2007.\n[67] B. K. Kuanr, R. Camley, and Z. Celinski, J. Magn.\nMagn. Mater. 286, 276 (2005).\n[68] J. Walowski, M. D. Kaufmann, B. Lenk, C. Hamann,\nJ.McCord, andM.Münzenberg,J.Phys.D:Appl.Phys.\n41, 164016 (2008).\n[69] H.-S. Song, K.-D. Lee, J.-W. Sohn, S.-H. Yang, S. S. P.\nParkin, C.-Y. You, and S.-C. Shin, Appl. Phys. Lett.\n102, 102401 (2013)."    },    {        "title": "1704.03326v1.CoFeAlB_alloy_with_low_damping_and_low_magnetization_for_spin_transfer_torque_switching.pdf",        "content": "arXiv:1704.03326v1  [cond-mat.mtrl-sci]  11 Apr 2017CoFeAlB alloy with low damping and low magnetization for spi n transfer torque\nswitching\nA. Conca,1,∗T. Nakano,2T. Meyer,1Y. Ando,2and B. Hillebrands1\n1Fachbereich Physik and Landesforschungszentrum OPTIMAS,\nTechnische Universit¨ at Kaiserslautern, 67663 Kaisersla utern, Germany\n2Department of Applied Physics, Tohoku University, Japan\n(Dated: June 29, 2021)\nWe investigate the effect of Al doping on the magnetic propert ies of the alloy CoFeB. Comparative\nmeasurements of the saturation magnetization, the Gilbert damping parameter αand the exchange\nconstantasafunctionoftheannealingtemperature forCoFe B andCoFeAlBthinfilmsare presented.\nOur results reveal a strong reduction of the magnetization f or CoFeAlB in comparison to CoFeB.\nIf the prepared CoFeAlB films are amorphous, the damping para meterαis unaffected by the Al\ndoping in comparison to the CoFeB alloy. In contrast, in the c ase of a crystalline CoFeAlB film, α\nis found to be reduced. Furthermore, the x-ray characteriza tion and the evolution of the exchange\nconstant with the annealing temperature indicate a similar crystallization process in both alloys.\nThe data proves the suitability of CoFeAlB for spin torque sw itching properties where a reduction\nof the switching current in comparison with CoFeB is expecte d.\nThe alloy CoFeB is widely used in magnetic tunnel-\ning junctions in combination with MgO barriers due to\nthe large magnetoresistance effect originating in the spin\nfiltering effect [1–4]. For the application in magnetic ran-\ndom accessmemories, the switching ofthe magnetization\nof the free layer via spin transfer torque (STT) with spin\npolarised currents is a key technology. However, the re-\nquired currents for the switching process are still large\nand hinder the applicability of this technique. The criti-\ncal switching current density for an in-plane magnetized\nsystem is given by [5]\nJc0=2eαMStf(HK+Hext+2πMS)\n/planckover2pi1η(1)\nwhereeis the electron charge, αis the Gilbert damping\nparameter, MSis the saturation magnetization, tfis the\nthickness of the free layer, Hextis the external field, HK\nis the effective anisotropy field and ηis the spin transfer\nefficiency. Fromtheexpressionitisclearthat, concerning\nmaterial parameters, Jc0is ruled by the product αM2\nS.\nFor out-of-plane oriented layers, the term 2 πMSvanishes\nand then JC0is proportional to αMS[6]. Even in the\ncase of using pure spin currents created by the Spin Hall\neffect, the required currents are proportional to factors\nof the form αnMSwithn= 1,1/2 [7]. A proper strat-\negy to reduce the critical switching currents is then de-\nfined by reducing the saturation magnetization. This can\nbe achieved by the development of new materials or the\nmodification of known materials with promising prop-\nerties. Since the compatibility with a MgO tunneling\nbarrier and the spin filtering effect must be guaranteed\ntogether with industrial applicability, the second option\nis clearly an advantage by reducing MSin the CoFeB al-\nloy. In this case, a critical point is that this reduction\nmust not be associated with an increase of the damping\nparameter α.In the last years, several reports on doped CoFeB al-\nloys have proven the potential of this approach. The\nintroduction of Cr results in a strong reduction of MS\n[8–10], however, it is sometimes also causing an increase\nof the damping parameter [8]. The reduction of MSby\ndoping CoFeB with Ni is smaller compared to a doping\nwith Cr but it additionally leads to a reduction of α[8].\nFIG. 1. (Color online) θ/2θ-scans for 40 nm thick films of\nCo40Fe40B20(top) and Co 36Fe36Al18B10(bottom) showing\nthe evolution of crystallization with the annealing temper a-\nture.2\nFIG. 2. (Color online) Evolution of the saturation magnetiz a-\ntion for CoFeB and CoFeAlB with the annealing temperature\nTann.\nIn constrast, the reduction of magnetization with V is\ncomparable to Cr [9] but to our knowledge no values for\nαhave been published. In the case of doping of CoFeB\nby Cr or by V, a reduction of the switching current has\nbeen shown [8, 9].\nIn this Letter, we report on results on Al doped CoFeB\nalloy thin films characterized by ferromagnetic resonance\nspectroscopy. The dependence of MS, the Gilbert damp-\ning parameter αand the exchange constant on the an-\nnealing temperature is discussed together with the crys-\ntalline structure of the films and the suitability for STT\nswitching devices.\nThe samples are grown on Si/SiO 2substrates us-\ning DC (for metals) and RF (for MgO) sput-\ntering techniques. The layer stack of the sam-\nples is Si/SiO 2/Ta(5)/MgO(2)/FM(40)/MgO(2)/Ta(5)\nwhere FM = Co 40Fe40B20(CoFeB) or Co 36Fe36Al18B10\n(CoFeAlB). Here, the values in brackets denote the layer\nthicknesses in nm. In particular, the FM/MgO interface\nis chosen since it is widely used for STT devices based\non MTJs. This interface is also required to promote the\ncorrect crystallizationof the CoFeB layerupon annealing\nsince the MgO layer acts as a template for a CoFe bcc\n(100)-oriented structure [1–3] with consequent B migra-\ntion.\nThe dynamic properties and material parameters were\nstudied by measuring the ferromagnetic resonance using\na strip-line vector network analyzer (VNA-FMR). For\nthis, the samples were placed face down and the S 12\ntransmission parameter was recorded. A more detailed\ndescription of the FMR measurement and analysis pro-\ncedure is shown in previous work [11, 12]. Brillouin light\nspectroscopy (BLS) was additionally used for the mea-\nsurement of the exchange constant. The crystalline bulk\nproperties of the films were studied by X-ray diffractom-\netry (XRD) using the Cu-K αline.\nFigure 1 shows the θ/2θ-scans for CoFeB (top) andFIG. 3. (Color online) Linewidth at a fixed frequency of 18\nGHz (a) and Gilbert dampingparameter αdependenceon the\nannealing temperature T ann(b). The αvalue for Tann= 500◦\nis only a rough estimation since the large linewidth value do es\nnot allow for a proper estimation. The inset shows the linear\ndependence of the linewidth on the frequency exemplarily fo r\nCoFeAlB annealed at 350◦C and 400◦C. The red lines are a\nlinear fit.\nCoFeAlB (bottom) samples annealed at different tem-\nperatures T ann. The appearance of the CoFe diffractions\npeaks, as shown by the arrowsin Fig. 1 indicate the start\nof crystallization at high annealing temperatures of more\nthan 400◦C. In the case of lower annealing temperatures\nor the as-deposited samples, the FM layer is in an amor-\nphous state. The first appearance of the (200) diffraction\npeak occurs at the same point for both alloys showing a\nverysimilarthermalevolution. Thissimplifiesasubstitu-\ntion ofCoFeB by the Al alloyin tunneling junctions since\nthe same annealing recipes can be applied. This is criti-\ncal since the used values must be also optimized for the\nquality of the tunneling barrier itself or the perpendicu-\nlar anisotropy induced by the FM/MgO interface. The\n(110) CoFe peak is also present for both material compo-\nsitions owing to a partial texturing of the film. However,\nthe larger intensity of the (200) peak is not compatible\nwith a random crystallite orientation but with a domi-\nnant (100) oriented film [13, 14]. This is needed since the\nspin filtering effect responsible for the large magnetore-\nsistance effect in MgO-based junctions requires a (100)3\nFIG.4. (Color online)Dependenceoftheproduct αM2\nSonthe\nannealing temperature T annfor CoFeB and CoFeAlB. This\nquantityisrulingtheswitchingcurrentinin-planemagnet ized\nSTT devices as shown in Eq. 1.\norientation.\nThe dependence of the FMR frequency on the external\nmagnetic field is described by Kittel’s formula [15]. The\nvalue ofMeffextracted from the Kittel fit is related with\nthe saturation magnetization of the sample and the in-\nterfacial properties by Meff=MS−2K⊥\nS/µ0MSdwhere\nK⊥\nSis the interface perpendicular anisotropy constant.\nFor the thickness used in this work (40 nm) and physi-\ncally reasonable K⊥\nSvalues, the influence of the interface\nis negligible and therefore Meff≈MS. For details about\nthe estimation of Meffthe reader is referred to [12].\nFigure 2 shows the obtained values for MSfor all sam-\nples. AstrongreductionforCoFeAlBin comparisonwith\nstandard CoFeB is observed and the relative difference is\nmaintained for all T ann. The evolution with annealing is\nvery similar for both alloys. Significantly, the increase in\nMSstartsforvaluesofT annlowerthan expectedfromthe\nappearance of the characteristic CoFe diffraction peaks\nin the XRD data (see Fig. 1). This shows that the mea-\nsurement of MSis the more sensitive method to probe\nthe change of the crystalline structure.\nFor CoFeB a saturationvalue around MS≈1500kA/m\nis reached at T ann= 450◦C. This is compatible with val-\nues reported for CoFe (1350-1700 kA/m) [16, 17] and\nCoFeB (1350-1500 kA/m) [17, 18]. On the contrary, for\nCoFeAlB the introduction of Al reduces the magnetiza-\ntion of the samples and the annealing does not recover\nto CoFe-like values.\nFigure 3(a) shows the dependence of the magnetic\nfield linewidth on T annmeasured at a fixed frequency\nof 18 GHz. From the linear dependence of this linewidth\non the FMR frequency, the Gilbert damping parameter\nis extracted (as exemplarily shown for the CoFeAlB al-\nloy in the inset in Fig. 3(b)) and the results are shown\nin Fig. 3(b). For T annvalues up to 350◦C, where theFIG. 5. (Color online) Dependence of the exchange con-\nstantAexon the annealing temperature T annfor CoFeB and\nCoFeAlB. The top panels show typical BLS spectra for ma-\nterials (see text).\namorphousphaseisstill dominating, almostnodifference\nbetween both alloys is observed. With increasing tem-\nperature the damping increases for both alloys but the\nevolution is different. For CoFeAlB the increase starts\nalmost abruptly at T ann= 400◦C, reaches a maximum\naroundα= 0.02 and then decreases again to α= 0.012\nfor Tann= 500◦C. In contrast, the increase for CoFeB\nis more smoothly with T annand increases stadily with\nhigher T ann. In fact, due to the large linewidths reached\nfor Tann= 500◦C, the value of αcannot be properly\nestimated and only a lower limit of 0.03-0.04 can be\ngiven. This situation is represented by the dashed line\nin Fig. 3(b). It is important to note here that when the\ncrystallization process is fulfilled (i.e. for T ann= 500◦C)\nαis much lower for the Al doped alloy. This is rele-\nvant for the application in tunneling junctions where a\nfull crystallization is required for the presence of the spin\nfiltering effect originating large magnetoresistance values\nin combination with MgO barriers [4].\nFor further comparison of both alloys, the quantity\nαM2\nShasbeen calculatedand plotted in Fig. 4. As shown\nin Eq. 1, this value is ruling the critical switching current\nin in-plane magnetized systems. We observe for the al-\nloys showing a mostly amorphous phase (T ann<400◦C)\na slight improvement for CoFeAlB in comparison with\nCoFeB due to the lower MS. However, for fully crys-\ntalline films (T ann= 500◦C), the CoFeAlB shows a much\nsmaller value for αM2\nS. Since a full crystalline phase is\nneeded for any application of this alloy in MTJ-based de-\nvices, this denotes a major advantage of this compound\ncompared to standard.\nThe exchange constant Aexis a critical parameter that\nis strongly influenced by the introduction of Al. Its esti-\nmationinrequiredformodelingthespintorqueswitching\nbehaviorofthe alloys. The accessto the constantisgiven4\nby the dependence of the frequency of the perpendicular\nstanding spin-wave (PSSW) modes on the external static\nmagnetic field [19]. As shown in previous works [12, 20],\nitispossibletoobservethePSSWmodesinmetallicfilms\nwith a standard VNA-FMR setup. However, the signal\nis strongly reduced compared to the FMR peak. For the\nsamples presented in this paper, the PSSW peak could\nnot be observed for T ann>400◦C since the increased\ndamping leads to a broadening and lowering of the peak\nwhich prevents the estimation of Aex. For this reason,\nBLS spectroscopy is used for the measurement of the fre-\nquency position of the PSSW modes. This technique has\nalargersensitivityforthePSSWmodesthanVNA-FMR.\nFigure 5(c) shows the evolution of Aexupon annealing\nfor both alloys. For the films dominated by the amor-\nphous phase the value is much lower for CoFeAlB which\nis also compatible with the lower magnetization. How-\never, asthe crystallizationevolves,theexchangeconstant\nincreases stronger than for CoFeB and the same value is\nobtained for the fully crystallized films. This fact points\nto a similar role of Al and B during the crystallization\nprocess: when the CoFe crystallitesform, the light atoms\nare expelled forming a Al-B-rich matrix embedding the\nmagnetic crystallites. This explains also the similar evo-\nlution observed in the XRD data shown in Fig. 1. The\nlower maximal magnetization obtained for the CoFeAlB\ncan be explained by the reduced CoFe content but also\na certain number of residual Al and B atoms in the crys-\ntallites, which may differ for both alloys.\nTheAexvalues for as-deposited CoFeB films are very\nsimilar to previous reports [12, 20, 21]. Concerning the\nvalues for the crystallized samples, since the properties\nare strongly dependent on the B content and of the ra-\ntio between Co and Fe as well as on the exact annealing\nconditions, a comparison with literature has to be made\ncarefully. Nevertheless, the maximal value and the evo-\nlution with T annfor CoFeB is similar to the one reported\nby some of the authors [12]. Also results for alloys with\nthe same B content arecompatiblewith ourdata [22, 23].\nCoFeB films with reduced B content show larger values\n[17], the same is true for CoFe alloys with values between\n3.84-2.61 ×1011J/m depending on the exact stoichiome-\ntry [16, 17]. This may again be a hint that a rest of Al\nor B is present in the CoFe crystallites.\nIn summary, the presented experimental results show\nthat CoFeAlB is a good candidate as alternative to\nCoFeB for spin torque switching devices due to the re-\nduction of the factor αM2\nSwhich dominates the critical\nswitching current. This reduction was found to originate\nfrom a strong reduction of the saturation magnetization\nandadecreaseddampingparameter αforfullycrystalline\nCoFeAlB films. Furthermore, the results reveal a larger\nthermal stability of the damping properties in CoFeAlB\ncompared to CoFeB. The absolute values of MSand the\nexchange constant Aexfor crystalline films point to a for-\nmation of CoFe crystallines with a non-vanishing contentof the lights atoms embedded in a B or Al matrix.\nFinancial support by M-era.Net through the\nHEUMEM project, the DFG in the framework of\nthe research unit TRR 173 Spin+X and by the JSPS\nCore-to-Core Program is gratefully acknowledged.\n∗conca@physik.uni-kl.de\n[1] S.YuasaandD.D.Djayaprawira, J.Phys.D:Appl.Phys.\n40, R337-R354 (2007).\n[2] S. Yuasa,Y. Suzuki, T. Katayama, and K. Ando,\nAppl. Phys. Lett. 87, 242503 (2005).\n[3] Y. S. Choi, K. Tsunekawa, Y. Nagamine, and\nD. Djayaprawira, J. Appl. Phys. 101, 013907 (2007).\n[4] X.-G. Zhang and W. H. Butler, J. Phys.: Condens. Mat-\nter15R1603, (2003).\n[5] Z. Diao, Z. Li, S. Wang, Y. Ding, A. Panchula, E. Chen,\nL.-C. Wang, and Y. Huai, J. Phys. D: Appl. Phys. 19,\n165209 (2007).\n[6] K. L. Wang, J. G. Alzate, and P.K. Amiri, J. Phys. D:\nAppl. Phys. 46, 074003 (2013).\n[7] T. Taniguchi, S. Mitani, and M. Hayashi, Phys. Rev. B\n92, 024428 (2015).\n[8] K. Oguz, M. Ozdemir, O. Dur, and J. M. D. Coey,\nJ. Appl. Phys. 111, 113904 (2012).\n[9] H. Kubota, A. Fukushima, K. Yakushiji, S. Yakata,\nS. Yuasa, K. Ando, M. Ogane, Y.Ando, andT. Miyazaki,\nJ. Appl. Phys. 105, 07D117 (2009).\n[10] Y. Cui, M. Ding, S. J. Poon, T. P. Adl, S. Keshavarz,\nT. Mewes, S. A. Wolf, and J. Lu, J. Appl. Phys. 114,\n153902 (2013).\n[11] A. Conca, S. Keller, L. Mihalceanu, T. Kehagias,\nG. P. Dimitrakopulos, B. Hillebrands, and E. Th. Pa-\npaioannou, Phys. Rev. B 93, 134405 (2016).\n[12] A. Conca, E. Th. Papaioannou, S. Klingler, J. Greser,\nT. Sebastian, B. Leven, J. L¨ osch, and B. Hillebrands,\nAppl. Phys. Lett. 104, 182407 (2014).\n[13] G. Concas, F. Congiu, G. Ennas, G. Piccaluga, and\nG. Spano. J. of Non-Crystalline Solids 330, 234 (2003).\n[14] C. Y. You, T. Ohkubo, Y. K. Takahashi, and K. Hono,\nJ. Appl. Phys. 104, 033517 (2008).\n[15] C. Kittel, Phys. Rev. 73, 155 (1948).\n[16] X. Liu, R. Sooryakumar, C. J. Gutierrez, and\nG. A. Prinz, J. Appl. Phys. 75, 7021 (1994).\n[17] C. Bilzer, T. Devolder, J.-V. Kim, G. Counil, C. Chap-\npert, S. Cardoso, and P. P. Freitas, J. Appl. Phys. 100,\n053903 (2006).\n[18] X. Liu, W. Zhang, M. J. Carter, and G. Xiao, J. Appl.\nPhys.110, 033910 (2011).\n[19] S. O. Demokritov, B. Hillebrands, Spin Dynamics in\nConfined Magnetic Structures I , Springer, Berlin, (2002).\n[20] A. Conca, J. Greser, T. Sebastian, S. Klingler, B. Obry,\nB. Leven, and B. Hillebrands, J. Appl. Phys. 113, 213909\n(2013).\n[21] J. Cho, J.Jung, K.-E.Kim, S.-I.Kim, S.-Y.Park, andM.-\nH. Jung, C.-Y. You, J. of Magn and Magn. Mat. 339, 36\n(2013).\n[22] A. Helmer, S. Cornelissen, T. Devolder, J.-V. Kim,\nW. van Roy, L. Lagae, and C. Chappert, Phys. Rev. B\n81, 094416 (2010).\n[23] H.Sato, M.Yamanouchi, K.Miura, S.Ikeda, R.Koizumi,5\nF. Matsukura, and H. Ohno, IEEE Magn. Lett., 3,\n3000204 (2012)."    },    {        "title": "1704.05718v1.Refractive_index_of_dense_materials.pdf",        "content": "Refractive index of dense materials  \nGilbert Zalczer  \nSPEC, CEA, CNRS, Université Paris Sac lay, CEA Saclay,  91191 Gif sur Yvette, France  \nWe show that applying the Lorentz -Lorenz transformation to the refractive index of metals , \nsemiconductors and insulators allows for  a less empirical modeling of this refractive index.  \nINTRODUCTION  \nOptical devices are ubiquitous in our environment. They use many materials which interact \nwith light thro ugh a comp lex, frequency depende nt property  : their refractive  indices. These have \nbeen widely measured and tabulated . Several ways of empirical mod eling have  been proposed, \nusually  limit ed to a category of materials  and implying intricate formulas[ 1-5]. We have noticed that \nthe “polarizability ” computed by applyin g blindly the Lorentz -Lorenz formula can be fitted much \nmore accurately. Indeed the imaginary part can be very well described by a few Gaussian functions \nand the real part computed using Kramers -Kronig relationship.  \nMODEL AND FITS  \nThe dielectric constant ε of an assembly of polarizable point particle is well described by the \nClausius -Mossoti  formula  \n \n𝜖−1\n𝜖+2=𝐶 𝜌𝛼 \n \nwhere ρ is the density of particles  α their polarizability  and C a constant . Both ε and  α are complex \nnumbers and frequency dependent. In the optical domain this equation can be rewritten as  \n𝑛2−1\n𝑛2+2=𝐶 𝜌𝛼 \n \nknown as Lorentz -Lorenz formula. This formula has no basis for being applied to metals, but an \nempir ical use reveals an interesting phenomenon.  The data we use originate from the book of \nPalik [6]. \nThe most striking case is that of silver. The real and imaginary part of the so computed \n“polarizability” are shown in fig 1 a s a function of the wavevector (the data are plotted in red, the fit \nin blue) :   \nFigure 1 : Real (top) and imaginary part of the polarizabilty calculated by application of the Lorentz -Lorenz \nformula to the refractive index of silver  as a function of the wavevector (in cm-1) \n \n The imaginary  part is readily seen as the sum of two features : one shar p and intense near \n2.8 104 cm-1 and a broad one centered near 4.5 104 cm-1 . These can be quite accurately fitted by \nGauss curves. The fit parameters for the center wavevector (s0), the width (larg) and the amplitude \n(amp) of these Gauss functions (G1 and G2) are shown beside the plots.  The real part cannot be \ndeduced from the imaginary one using the Kramers -Kronig relation  (the K -K transform of a Gaussian \nis a Dawson function)  because we have to take into account at least one other line far enough in the \nUV not to appear in the figure. By convention we ascribe it a width of 1 and the other parameters are \nreported labeled as D1. The refractive index of transparent media is due to such a line. We shall see \nbelow that the polarizability of many dense material s can be described accurately by just a small \nnumber of Gaussian shaped absorption lines. Those lying out of the observed range are seen only to \nthe long tail of their real part. Their width cannot be determined and has been fixed to 1.  \nThe case of gold ha s some similarities :  \n \nFigure 2 Polarizability of gold  \nA sharp peak at low wavevector and a broad one at the high side.  However the sharp peak is \nmuch less intense and a wide continuum appears between them which can be accurately fitted only \nby introducing two additional gaussians. The same holds for copper  \n \nFigure 3 Polarizability of copper  \nThe f it is a little worse. It could perhaps be improved by adding more lines but their relevance \nis questionable. A common feature of these three spectra is a range of zero α” at small wavevector.  \nPlatinum and nickel can be described by one and two lines coveri ng the visible range  : \n \nFigure 4 Polarizability of platinum  \n \nFigure 5 Polarizability of nickel  \nSemiconductors also can be fitted with 3 lines fo r silicon and 4 for germanium  : \n \nFigure 6 : Polarizability of crystalline silicon.  \non \n \nFigure 7 Polarizability of germanium  \n  \nAs noted above a transparent medium such as silica is due to a UV band  \n \nFigure 8 Polarizability  of silica glass. The imaginary part is zero  \nWhile for zirconia (  ZrO2 ) we have also to take into account an infrared one  : \n \nFigure 9 Polarizability of zirconia glass. The imaginary part is zero  \nCONCLUSION  \nThe refractive index of dielectrics, semiconductors and metals can be interpreted and \ndescribed by transforming their refractive index into a polarizability using the Lorentz -Lorenz \nformula. The imaginary part of this polarizability appears as a sum of a f ew Gaussian functions, the \nparameters of wh ich can be readily fitted. S mall distort ions may be attributed to weaker bands but \ntheir relevance is questionable. The real part can be deduced from the Kramers -Kronig transform of \nthe imaginary part but one has t o take into account lines in the UV or in the IR, invisible in the \nimaginary part but who have a significant but smooth contribution in the real part. In some cases the \nfit for the imaginary part is good but not that of the real part. An apparent violation  of the Kramers \nKronig relation might signal imperfect measurements.  \nREFERENCES  \n1 W. Sellmeier  Annalen der Physik un d Chemie , 219, 272  (1871) \n2 R. Brendel, and D. Bormann, J. Appl. Phys. 71, 1 (1992).  \n3 D. Rioux, S. Vallieres, S. Besner, P. Munoz, E. Mazur, and M. Meunier, Adv Opt Mater 2, 176 (2014).  \n4 A.B. Djuri sic, E.H. Li , D. Raki c, M.L. Majewski , Appl. Phys. A 70, 29  (2000 ) \n5 M. Ruzi , C. Ennis , and E. G. Robertson, AIP Advances 7, 015042 (2017 ) \n6 Edward D. Palik , Handbook of Optical Constants of Solids , Elsevier (1997 ) \n "    },    {        "title": "1704.07006v1.Spin_injection_into_silicon_detected_by_broadband_ferromagnetic_resonance_spectroscopy.pdf",        "content": "1 \n Spin injection into silicon  \ndetected by broadband ferromagnetic resonance spectroscopy  \n \nRyo Ohshima,1 Stefan Klingler,2,3, Sergey Dushenko,1 Yuichiro Ando,1 Mathias Weiler,2,3 \nHans Huebl,2,3,4 Teruya Shinjo,1 Sebastian T. B. Goennenwein,2,3,4 and Masashi Shiraishi1* \n \n1Department of Electronic Science and Engineering, Kyoto Univ., 615 -8510 Kyoto, Japan.  \n2Walther -Meißner -Institut, Bayerische Akademie der Wissenschaften, 85748 Garching,  \nGermany.  \n3Physik -Department, Technische Universität München, 85748 Garching, Germany  \n4Nanosystems Initiative Munich , 80799 München, Germany  \n \n \nWe studied the spin injection in a NiFe(Py)/Si system using broadband \nferromagnetic resonance spectroscopy. The Gilbert damping parameter  of the Py layer \non top of the Si channel was determined as a function of the Si doping concentration  and \nPy layer thickness . For fixed Py thickness w e observe d an increase of the Gilbert damping \nparameter with decreasing resistivity of the Si channel . For a fixed Si doping \nconcentration we measured an increasing Gilbert damping parameter for decreasing Py \nlayer thickness.  No increase of the Gilbert damping param eter was found Py/Si samples \nwith an insulating interlayer . We attribute our observations to an enhanced spin injection \ninto the low -resistivity Si  by spin pumping . \n  2 \n Spin injection into semiconductors was relentlessly studied in recent years in hope to \nharness their long spin relaxation time, and gate tunability to realize spin metal -oxide -\nsemiconductor field -effect -transistors (MOSFET s). A central obstacle for a spin injection into \nsemiconductors was the conductance mismatch1 between ferromagnetic metals (used for the \nspin injection) and semiconductor channels. In an electrical spin injection method —widely \nused from the early years of the non -local spin transport studies —tunnel barriers between the \nsemiconductor and ferromagne t were formed to avoid the conductance mismatch problem2–5. \nUnfortunately, it complicated the production process of the devices , as high quality tunnel \nbarriers are not easy to grow , and presence of impurities, defects and pinholes take s a heavy \ntoll on th e spin injection efficiency and/or induces spurious effects. Meanwhile, the fabrication \nof electrical Si spin devices, like spin MOSFETs, with different resistivities is a time -\nconsuming process, which so far prevented systematic studies of the spin inject ion properties \n(such as spin lifetime, spin injection efficiency etc.). However,  such a systematic study is \nnecessary for further progress towards practical applications of spin MOSFETs.  \nIn 2002, a dynamical spin injection method, known as spin pumping, was introduced \nto the scene of spintronics research6,7. While the method was initially used in the metallic \nmultilayer sy stems, it was later  implemented to inject spin current s into semiconductors. In \ncontrast to electrical spin injection, spin pumping doe s not require the application of an electric \ncurrent across the ferromagnet/semiconductor interface. Devices that operate using spin \ncurrent s instead of charge current s can potentially reduce heat generation and power \nconsumption problems of modern electro nics. From a technological point of view, spin \npumping is also appealing because it does not require a tunnel barrier.  Spin injection —using \nspin pumping —into semiconductors from an adjacent ferromagnetic metal was achieved \ndespite the existence of conducti vity mismatch8-12. However, so far there was no systematic \nstudy of the spin pumping based spin injection in dependence on the resistivity of the Si channel.  3 \n In this letter, we focus on the study of spin injection  by spin pumping  in the NiFe(Py)/Si system  \nwith different resistivities  of the Si channel  using broadband ferromagnetic resonance (FMR) . \nThe broadband FMR method allows for a  precise determination of  the Gilbert damping \nparameter 𝛼, which increases in the presence of spin pumping, and  thus, spin injection . By \ntracking the change of the Gilbert damping parameter in various  Py/Si system s, we determine d \nthe spin pumping  efficiency  in the broad range of resistivities of the Si channel.   \nFor a first set of sample s 7nm-thick Py film s were  deposited  by electron beam \nevaporation  on top of various Si substrates  (1×1 cm2 in size) with resistivit ies in the range from \n10-3 to 103 ･cm (see Table 1 for the list of the prepared samples ). The oxidized surface of the \nSi substrates  was removed using 10% hydrofluoric acid (HF) prior to the Py evaporation. For \na second set of samples  Py films with  thickness es 𝑑Py between 5nm and 80nm  were deposited \non P -doped SOI  (silicon on insulator)  with the same technique . As a control  experiment , \nPy/AlO x and Py/ TiO x films were grown  on Si, P -doped SOI and SiO 2 substrates , as spin \npumping should be suppressed  in systems with an insulati ng barrier13 (see Table 2) . Both Al (3 \nnm, thermal deposition) and Ti  (2 nm, electron beam evaporation) were evaporated on the non -\ntreated substrates and left in the air for one day for oxidation of the surface (for the Al layer , \nthe process was repeated 3 times, with 1  nm of Al evaporated and oxidized at each step). After \noxidation, we evaporated 7nm thick  Py film s on the top of the tunnel barrier s. The properties \nof the prepared samples  are summarized in Table s 1 and 2 . \nA sketch of the broadband ferromagnetic res onance setup is shown in Fig. 1 (a). T he \nsamples were placed  face down on the center conductor  of a coplanar waveguide  (CPW), which \nwas located between the pole shoes of an electromagnet . A static magnetic field  |𝜇0𝐻|≤2.5 \nT was applied perpendicular to the surface of the samples to avoid extra damping due to two -\nmagnon scattering14. One end of the CPW was connected to a microwave source , where \nmicrowaves with frequenc y f < 40 GHz were generated . The other end of the CPW was 4 \n connected to a microwave diode and a lock -in amplifier to measure the rectified microwave \nvoltage  as a function of the applied magnetic field . All measurements were carried out at room \ntemperature.  \nThe microwave current in the  CPW generates an oscillating magnetic field around the \ncenter conductor  which results  in an oscillating torque on the s ample ’s magnetization. For \n𝜇0𝐻=𝜇0𝐻FMR  this torque results in a n absorption of microwave power . The resonance \ncondition i s given by the out -of-plane Kittel equation15,16: \n ℎ𝑓\n𝑔𝜇B=𝜇0𝐻FMR −𝜇0𝑀eff. (1)  \nHere,  ℎ is the Plan ck constant,  𝑔 is the Landé g-factor, 𝜇B is the Bohr magneton, 𝜇0 is \nthe vacuum permeability, and 𝑀eff is the effective saturation magnetization of Py.   \nWe use  the Gilbert damping model, which phenomenologically models the viscous \ndamping of the magnetic resonance. The linear relat ion between the full width at half maximum \n𝛥𝐻 of the resonance and the applied microwave frequency f is given by the Gilbert damping \nequation 17: \n 𝜇0𝛥𝐻=𝜇0𝛥𝐻0+2𝛼ℎ𝑓\n𝑔𝜇B. (2)  \nHere,  𝛥𝐻0 corresponds to  frequency independent scattering processes  and 𝛼 is the Gilbert  \ndamping parameter18,19: \n 𝛼=𝛼0+𝛼SP+𝛼EC. (3)  \nHere, 𝛼0 is the intrinsic Gilbert damping , 𝛼SP=𝑔𝜇𝐵𝑔r↑↓4𝜋𝑀S𝑑Py ⁄  is the damping due to \nspin pumping20, 𝑔r↑↓ is the real part of the spin mixing conductance,  and 𝛼EC=𝐶EC𝑑Py2 is the \neddy -current damping . The parameter  𝐶EC describes efficiency of  the eddy -current damping . \nTo realize a net spin injection via spin pumping, the following conditions should be \nfulfilled in the system: (i ) carriers should be present in the underlying channel, (ii) the spin \nrelaxation time in the channel should be small enough. The available carriers in the channel \ntransfer spin angular momentum away from the spin injection interface, allowing propagation 5 \n of the spin current. On the other hand, the long spin relaxation time in the channel leads to a \nlarge spin accumulation at the interface and generates a diffusive spin backflow in the direction \nopposite to the spin pumping current20 (see Figs. 1(b) and 1(c )). Thus, the spin backflow \neffectively cancels out the spin pumping current for long spin relaxation time, and the spin \npumping contribution to the Gilbert damping parameter should no longer be present in the \nsystem.  \nIn addition to spin pumping,  charge  currents can be induced in the Si channel  and Py  \nlayer due to  the Faraday ’s law  and Py magnetization precession, which results  in a Gilbert -like \ndamping contribution . These processes are  refer red to as radiative damping  and eddy -current \ndamping18. An e nhancement of the Gilbert damping parameter due to these processes  is \nexpected to be especially large for the Si channels with low resistivit ies and thick Py films , \nsince energy dissipation through eddy currents scales linearly  with the conductivity  of the  Si \nlayer  and quadratically with the Py layer thickness . Hence, both spin pumping and eddy current \ndamping are expected to be most efficien t for low -resistivity Si. I n contrast  to spin pumping , \nthe radiative damping  contribution doe s not  require a direct electrical contact between Py and \nSi, and is hence unaffected by the tunnel barrie r. \nFigure 1(d ) show s a typical FMR spectr um of a 7nm thick Py film on a phosphorous \nP-doped Si on insulator (SOI) substrate , where t he microwave frequency was fixed at 30 GHz  \nduring the sweep of the magnetic field. A single FMR  signal  was observed  (Fig. 1(d) red filled \ncircles) , from which 𝜇0𝐻FMR  and 𝜇0𝛥𝐻  were extracted  by a fit of the magnetic ac \nsusceptibility  (Fig. 1(d) black line) 16,21 (see Supplemental Material for additional fitting \nexamples).  An excellent agreement of the fit with the measurement is achieved.  \nFigures 2(a) and (b) show 𝐻FMR and 𝛥𝐻 versus the applied microwave frequency f \nfor the Py/P-doped SOI, Py/SOI and Py/SiO 2 samples. From the f itting of the frequency \ndependence of HFMR with Eq.(1) , 𝑔  and 𝜇0𝑀eff  of the Py /P-doped SOI ( Py/SOI) were 6 \n estimated to be 2.049 (2.051) and 0.732 T (0.724 T), and those of the Py /SiO 2 were estimated \nto be 2.038 and 0.935 T, respectively  (Supplemental Material  for fitting  and data from other \nsamples ). The difference in the 𝑔  and 𝑀eff  between the Py /Si and the Py/ SiO 2 sample  is \nattributed to the inter -diffusion of the Fe/Ni and Si at the  interface , which is  always present to \nsome extent during the growth at room temperature22,23. The Gilbert damping  𝛼 of the Py /P-\ndoped SOI , Py/SOI and Py/SiO 2 were estimated to be 1.25 ×10-2, 9.02 ×10-3 and 8.49 ×10-3, \nrespectively , from the linewidth vs. frequency evolution . The intrinsic Gilbert damping \nparameter 𝛼0  is determined from the linewidth evolution of the Py/SiO 2 and Py/quartz \nsamples to be 8.5×10−3 and 8.6×10−3, respectively, since no spin pumping contribution is \nexpected in these insulating materials ( 𝛼=𝛼0). From this we can see an increasing Gilbert \ndamping with decreasing resistivity. We additionally measured the samples with a n insulating \ntunnel barrier between the Si channel and the Py film. We found Gilbert damping parameters \nof 𝛼 = 8.8×10-3 for the Py/ AlO x/Si samples and 𝛼 = 7.5×10-3 for the Py/TiO x/Si samples, \nindependent of the Si resistivity.  The damping  values are in agreement with the intrinsic \ndamping extracted from the Py/SiO 2 sample , indicating that radiative damping is negligible in \nour samples.  \nFigure 3 (a) summarizes  the dependence of the Gilbert damping parameter 𝛼 on the \nresistivity of the Si channel (see Supplemental Material D for the g -factor, the effective \nsaturation magnetization and the frequency independent term ), including the measured control \nsamples.  The dashed lines  show  the intrinsic contributions 𝛼0  to the Gilbert damping \nparameter 𝛼 measured from the Py/SiO 2 and Py/quartz samples (red dashed), Py/AlO x/SiO 2 \n(blue dashed) and Py/TiO x/SiO 2 (green dashed). All  samples with Py on top of the conductive \nsubstrates without an additional tunnel barrier exhibited the Gilbert damping parameter 𝛼 \nlarger than the intrinsic contribution 𝛼0.  \nThe experimentally measured Gilbert damping parameter decreases logarithmically 7 \n with the resistivity. This result is in agreement with condition (i) for the spin pumping. In the \nSi channels with a small resistivity more carriers were available to transfer the injected angular \nmomentum, leading to an effective spin pumping. Additionally, both electron spin resonance25–\n28 and non -local 4 -terminal Hanle precession29,30 experiments showed, that the spin lifetime in \nSi is increas ing with increasing resistivity . Our samples with low resistivities have a large \ndoping concentration (see Table 1), leading to shorter spin relaxation time. In accordance with \nthe spin pumping condition (ii), the decrease of the spin relaxation time should lead to the \nincrease of the spi n pumping contribution, as now observe d experimentally. While the spin \npumping shows a logarithmic dependence on the resistivity of the channel, we note that an \nincreased Gilbert damping parameter  is observed even for the Si channel with high resistivit ies. \nWe comment on the Sb -doped sample, where the experimentally measured 𝛼SP was lower \nthan one expected from the logarithmic trend of the other samples. We speculate that this might \noriginate from the different doping profile, compared to the other samples . We note, that further \nstudies are necessary to separate the influence of the number of carriers in the channel and the \nspin relaxation time on the spin pumping process.  \nFinally, we show that the Py damping is increased in a broad range of Si resistivitie s \nand attribute this effect to the enhanced spin injection via spin pumping ( a discussion  of the \nspin mixing conductance  for various Si  resistivities is given in the Supplemental M aterial A). \nFigure 3(b) shows the Py thickness dependence of the 𝛼 for Py/P -doped SOI samples. The \nsolid line shows a fit of Eq.(3) to the measured data and a very good agreement of the spin \npumping theory with our measurements is achieved. From the fit we estimate 𝛼0=6.1×10−3, \n𝑔r↑↓=1.2×1019 m-2 and 𝐶EC=2.9×1011 m-2. Both the intrinsic damping and the real part \nof the spin mixing conductance are in good agreement with previous measurements30. The blue \ndashed line indicates 𝛼0+𝛼SP  and the green dashed line shows 𝛼0+𝛼EC . Dominant \ninfluence of spin pumping to the total damping is observed in samples with  small Py thickness , 8 \n while  eddy current contribution is dominant in samples with  thick Py  layer . For the 7 nm -thick \nPy sample, we find 𝛼SP = 3.6 ×10-3 and 𝛼EC = 1.4 ×10-5. Thus, the eddy -current damping in \nour 7 nm Py samples is negligibly small and cannot explain the increase of the damping with \ndecreasing resistivity. The Py thickness dependence of the Gilbert damping indicates  spin \npumping into the Si substrates.  \nIn conclusion , we studied spin pumping based spin injection from a Py layer into Si \nchannels with various resistivit ies using broadband ferromagnetic resonance . We determine d \nthe spin pumping contribution from the  change of the Gilbert damping parameter. The observed \nlogarithmic decrease of the Gilbert damping parameter with  increasing resistivity of the Si \nchannel is attribute  to the decrease in the number of carriers in the channel, and the increase in \nthe spin lifetime. De spite  the reduction of the spin pumping contribution to the Gilbert damping \nparameter with the increasing  resistivity  of the Si channel , we observe  spin pumping even for \nthe channels with high resistivity . We furthermore observe an increase of the Gilbert damping \nparameter for decreasing Py thickness which is in agreement with the spin pumping theory. \nOur results show that spin pumping can be potentially used in  a spin transistors, where low \ndoping concentration in the channel is necessary for the gate control of the device.  \n \nSupplement al Material  \n See Supplementary M aterial for a discussion  of the spin mixing conductance  for \nvarious Si resistivities and additional  fitting examples.  \n \nACKNOWLEGEMENTS  \nThis research was supported in part by a Gran t-in-Aid for Scientific Research from \nthe Ministry of Education, Culture, Sports, Science and Technology (MEXT) of Japan, \nInnovative Area “Nano Spin Conversion Science ” (No. 2 6103003), Scientific Research (S) 9 \n “Semico nductor Spincurrentronics ” (No. 16H0633) and JSPS KAKENHI Grant  (No. \n16J00485).  R.O. acknowledges JSPS Research Fellowship. S.D. acknowledges support by \nJSPS Postdoctoral Fellowship and JSPS KAKENHI Grant No. 16F16064.  \n  10 \n References  \n1 A. Fert and H. Jaffrès, Phys. Rev. B 64, 184420 (2001).  \n2 E.I. Rashba, Phys. Rev. B 62, R16267 (2000).  \n3 I. Appelbaum, B. Huang, and D.J. Monsma, Nature 447, 295 (2007).  \n4 O.M.J. van ’t Erve, A.T. Hanbicki, M. Holub, C.H. Li, C. Aw o-Affouda, P.E. Thompson, \nand B.T. Jonker, Appl. Phys. Lett. 91, 212109 (2007).  \n5 T. Sasaki, T. Oikawa, T. Suzuki, M. Shiraishi, Y. Suzuki, and K. Tagami, Appl. Phys. \nExpress 2, 53003 (2009).  \n6 Y. Tserkovnyak, A. Brataas, and G.E.W. Bauer, Phys. Rev. B 66, 224403 (2002).  \n7 S. Mizukami, Y. Ando, and T. Miyazaki, Phys. Rev. B 66, 104413 (2002).  \n8 K. Ando, S. Takahashi, J. Ieda, H. Kurebayashi, T. Trypiniotis, C.H.W. Barnes, S. \nMaekawa, and E. Saitoh, Nat. Mater. 10, 655 (2011).  \n9 K. Ando and E. Saitoh, Nat. Commun. 3, 629 (2012).  \n10 E. Shikoh, K. Ando, K. Kubo, E. Saitoh, T. Shinjo, and M. Shiraishi, Phys. Rev. Lett. 110, \n127201 (2013).  \n11 A. Yamamoto, Y. Ando, T. Shinjo, T. Uemura, and M. Shiraishi, Phys. Rev. B 91, 24417 \n(2015).  \n12 S. Dushenko, M. Koike, Y. Ando, T. Shinjo, M. Myronov, and M. Shiraishi, Phys. Rev. \nLett. 114, 196602 (2015).  \n13 C.H. Du, H.L. Wang, Y. Pu, T.L. Meyer, P.M. Woodward, F.Y. Yang, and P.C. Hammel, \nPhys. Rev. Lett. 111, 247202 (2013).  \n14 K. Zakeri, J. Lindner, I. Barsukov, R. Meckenstock, M. Farle, U. von Hörsten, H. Wende, \nW. Keune, J. Rocker, S.S. Kalarickal, K. Lenz, W. Kuch, K. Baberschke, and Z. Frait, Phys. \nRev. B 76, 104416 (2007).  \n15 K. Ando and E. Saitoh, J. Appl. Phys. 108, 113925 (2010).  11 \n 16 K. Ando, S. Takahashi, J. Ieda, Y. Kajiwara, H. Nakayama, T. Yoshino, K. Harii, Y. \nFujikawa, M. Matsuo, S. Maekawa, and E. Saitoh, J. Appl. Phys. 109, 103913 (2011).  \n17 H.T. Nembach, T. J. Silva, J.M. Shaw, M.L. Schneider, M. J. Carey, S. M aat, and J.R. \nChildress, Phys. Rev. B 84, 054424 (2011).  \n18 M.A.W. Schoen, J. M. Shaw, H.T. Nembach, M. Weiler and T.J. Silva, Phys. Rev. B 92, \n184417 (2015).  \n19 H. Nakayama, K. Ando, K. Harii, T. Yoshino, R. Takahashi, Y. Kajiwara, K. Uchida, Y. \nFujikwa and E. Saitoh, Phys. Rev. B 85, 144408 (2012).  \n20 Y. Tserkovnyak, A. Brataas, G.E.W. Bauer, and B.I. Halperin, Rev. Mod. Phys. 77, 1375 \n(2005).  \n21 L. Dreher, M. Weiler, M. Pernpeintner, H. Huebl, R. Gross, M.S. Brandt, and S.T.B. \nGoennenwein, Phys. Rev. B 86, 134415 (2012).  \n22 J.M. Gallego, J.M. García, J. Alvarez, and R. Miranda, Phys. Rev. B 46, 13339 (1992).  \n23 N. Kuratani, Y. Murakami, O. Imai, A. Ebe, S. Nishiyama, and K. Ogata, Thin Solid Films \n281–282, 352 (1996).  \n24 Y. Ochiai and E. Matsuura, Phys. Status Solidi A 38, 243 (1976).  \n25 J.H. Pifer, Phys. Rev. B 12, 4391 (1975).  \n26 V. Zarifis and T. Castner, Phys. Rev. B 57, 14600 (1998).  \n27 R. Jansen, Nat. Mater. 11, 400 (2012).  \n28 T. Suzuki, T. Sasaki, T. Oikawa, M. Shiraishi, Y. Suzuki, and K. Noguchi, Appl. Phys. \nExpress 4, 23003 (2011).  \n29 T. Tahara, Y. Ando, M. Kameno, H. Koike, K. Tanaka, S. Miwa, Y. Suzuki, T. Sasaki, T. \nOikawa, and M. Shiraishi, Phys. Rev. B 93, 214406 (2016).  \n30 Note, that the sample set with various Py thicknesses was grown in a different batch than \nthe samples with v arious Si doping. Hence, small deviations in the damping and the spin 12 \n mixing conductance are due to slightly different growth conditions . \n \nFigure 1: (a) Experimental setup for the broadband FMR measurement. The samples were \nplaced with the Py layer facing down on a coplanar waveguide. External magnetic field and \nmicrowave field from the waveguide induce the FMR of the Py and spins are injec ted into Si \nvia spin pumping. Schematic images of spin injection and dephasing in Si that have (b) long  \nand (c) short spin lifetimes. 𝜏1 and 𝜏2 are the spin lifetim e of Si in the case of (b) and (c), \nrespectively. Spin injection efficiency becomes large in the case of (c) because of a reduction \nof the backflow of spins. (d ) The derivative of the FMR signal of Py at 30 GHz microwave \nfrequency.  I is the microwave absorption intensity.  \n \nFigure 2: Frequency dependence of the (a) resonance field 𝐻FMR and (b) full width at half \nmaximum 𝛥𝐻 of the FMR spectra obtained from Py on top of P -doped SOI, SOI and  SiO 2. \nThe solid lines show fitting using  Eqs. (1) and (2) of 𝐻FMR and 𝛥𝐻, respectively.  \n \nFigure  3: (a) Si resistivity dependence of the Gilbert damping parameter  𝛼. The damping of \nthe samples with an insulating layer represents the intrinsic damping of the Py layer and is \nshown by the dashed line s. Red, blue and green coloration represents Py, Py/AlO x and Py/TiO x \nsamples, respectively. The damping of the Py/ P-doped SOI is an averaged value extracted from \nthe two Py/P-doped SOI samples fabricated at different times.  (b) Py thickness dependence of  \n𝛼. The solid line shows  a fit of Eq. (3) to the data . The b lue line shows 𝛼0+𝛼SP, whereas the \ngreen line shows 𝛼0+𝛼EC. \n \n \n 13 \n Fig. 1 R. Ohshima et al . \n \n \nFig. 2  R. Ohshima et al.  \n \n14 \n Fig. 3  R. Ohshima et al.  \n \n \nTable 1: Sample summary  \nName  Dopant  Doping \ndensity (cm-3) Structure  Resistivity \n(･cm) Gilbert damping \nparameter  Mixing \nconductance (m-2) \nPy/P -\ndoped SOI  P 6.5×1019 Py(7 nm)/Si(100 nm)/  \nSiO 2(200 nm)/Si  1.3×10−3 1.1×10−2 5.7×1018 \nPy/Sb -\ndoped Si  Sb 1×1019 Py(7 nm)/Si  5.0×10−3 9.3×10−3 2.3×1018 \nPy/N -\ndoped Si  N 1×1019 Py(7 nm)/Si  1.0×10−1 9.5×10−3 2.6×1018 \nPy/SOI  N/A 1×1015 Py(7 nm)/Si(100 nm)/  \nSiO 2(200 nm)/Si  4.5 9.0×10−3 1.3×1018 \nPy/P -\ndoped Si  P 1×1013 Py(7 nm)/Si  1.0×103 8.7×10−3 5.1×1017 \nPy/SiO 2 - - Py(7 nm)  \n/SiO 2(500 nm)/Si  - 8.5×10−3 - \nPy/Quartz  - - Py(7 nm) /Quartz  - 8.6×10−3 - \n \n \n \n \n \n15 \n Table 2: List of the samples for the control experiment  \nName  Dopant  Doping \ndensity (cm-3) Structure  Resistivity \n(･cm) Gilbert damping \nparameter  \nPy/AlO x/P-\ndoped SOI  P 6.5×1019 Py(7 nm)/AlO x(3 \nnm)/Si(100 nm)/  \nSiO 2(200 nm)/Si  1.3×10−3 8.6×10−3 \nPy/AlO x/P-\ndoped Si  P 1×1013 Py(7 nm)/AlO x(3 nm)/Si  1.0×103 8.5×10−3 \nPy/AlO x/ \nSiO 2 - - Py(7 nm)/ AlO x(3 nm)/  \nSiO 2(500 nm)/Si  - 8.8×10−3 \nPy/TiO x/P-\ndoped SOI  P 6.5×1019 Py(7 nm)/TiO x(2 \nnm)/Si(100 nm)/  \nSiO 2(200 nm)/Si  1.3×10−3 7.5×10−3 \nPy/TiO x/P-\ndoped Si  P 1×1013 Py(7 nm)/TiO x(2 nm)/Si  1.0×103 7.9×10−3 \nPy/TiO x/ \nSiO 2 - - Py(7 nm)/TiO x(2 nm)/  \nSiO 2(500 nm)/Si  - 7.8×10−3 \n \n "    },    {        "title": "1705.03416v1.Low_spin_wave_damping_in_the_insulating_chiral_magnet_Cu___2__OSeO___3__.pdf",        "content": "Low spin wave damping in the insulating chiral magnet Cu 2OSeO 3\nI. Stasinopoulos,1S. Weichselbaumer,1A. Bauer,2J. Waizner,3\nH. Berger,4S. Maendl,1M. Garst,3, 5C. P\reiderer,2and D. Grundler6,\u0003\n1Physik Department E10, Technische Universit at M unchen, D-85748 Garching, Germany\n2Physik Department E51, Technische Universit at M unchen, D-85748 Garching, Germany\n3Institute for Theoretical Physics, Universit at zu K oln, D-50937 K oln, Germany\n4Institut de Physique de la Mati\u0012 ere Complexe, \u0013Ecole Polytechnique F\u0013 ed\u0013 erale de Lausanne, 1015 Lausanne, Switzerland\n5Institut f ur Theoretische Physik, Technische Universit at Dresden, D-01062 Dresden, Germany\n6Institute of Materials and Laboratory of Nanoscale Magnetic Materials and Magnonics (LMGN),\n\u0013Ecole Polytechnique F\u0013 ed\u0013 erale de Lausanne (EPFL), Station 12, 1015 Lausanne, Switzerland\n(Dated: October 2, 2018)\nChiral magnets with topologically nontrivial spin order such as Skyrmions have generated enor-\nmous interest in both fundamental and applied sciences. We report broadband microwave spec-\ntroscopy performed on the insulating chiral ferrimagnet Cu 2OSeO 3. For the damping of magnetiza-\ntion dynamics we \fnd a remarkably small Gilbert damping parameter of about 1 \u000210\u00004at 5 K. This\nvalue is only a factor of 4 larger than the one reported for the best insulating ferrimagnet yttrium\niron garnet. We detect a series of sharp resonances and attribute them to con\fned spin waves in\nthe mm-sized samples. Considering the small damping, insulating chiral magnets turn out to be\npromising candidates when exploring non-collinear spin structures for high frequency applications.\nPACS numbers: 76.50.+g, 74.25.Ha, 4.40.Az, 41.20.Jb\nThe development of future devices for microwave ap-\nplications, spintronics and magnonics [1{3] requires ma-\nterials with a low spin wave (magnon) damping. In-\nsulating compounds are advantageous over metals for\nhigh-frequency applications as they avoid damping via\nspin wave scattering at free charge carriers and eddy\ncurrents [4, 5]. Indeed, the ferrimagnetic insulator yt-\ntrium iron garnet (YIG) holds the benchmark with a\nGilbert damping parameter \u000bintr= 3\u000210\u00005at room\ntemperature [6, 7]. During the last years chiral mag-\nnets have attracted a lot of attention in fundamental\nresearch and stimulated new concepts for information\ntechnology [8, 9]. This material class hosts non-collinear\nspin structures such as spin helices and Skyrmions be-\nlow the critical temperature Tcand critical \feld Hc2\n[10{12]. Additionally, Dzyaloshinskii-Moriya interaction\n(DMI) is present that induces both the Skyrmion lattice\nphase and nonreciprocal microwave characteristics [13].\nLow damping magnets o\u000bering DMI would generate new\nprospects by particularly combining complex spin order\nwith long-distance magnon transport in high-frequency\napplications and magnonics [14, 15]. At low tempera-\ntures, they would further enrich the physics in magnon-\nphoton cavities that call for materials with small \u000bintrto\nachieve high-cooperative magnon-to-photon coupling in\nthe quantum limit [16{19].\nIn this work, we investigate the Gilbert damping in\nCu2OSeO 3, a prototypical insulator hosting Skyrmions\n[20{23]. This material is a local-moment ferrimagnet\nwithTc= 58 K and magnetoelectric coupling [24] that\ngives rise to dichroism for microwaves [25{27]. The\nmagnetization dynamics in Cu 2OSeO 3has already been\nexplored [13, 28, 29]. A detailed investigation on thedamping which is a key quality for magnonics and spin-\ntronics has not yet been presented however. To eval-\nuate\u000bintrwe explore the \feld polarized state (FP)\nwhere the two spin sublattices attain the ferrimagnetic\narrangement[21]. Using spectra obtained by two di\u000ber-\nent coplanar waveguides (CPWs), we extract a minimum\n\u000bintr=(9.9\u00064.1)\u000210\u00005at 5 K, i.e. only about four times\nhigher than in YIG. We resolve numerous sharp reso-\nnances in our spectra and attribute them to modes that\nare con\fned modes across the macroscopic sample and\nallowed for by the low damping. Our \fndings substanti-\nate the relevance of insulating chiral magnets for future\napplications in magnonics and spintronics.\nFrom single crystals of Cu 2OSeO 3we prepared two\nbar-shaped samples exhibiting di\u000berent crystallographic\norientations. The samples had lateral dimensions of\n2:3\u00020:4\u00020:3 mm3. They were positioned on CPWs that\nprovided us with a dynamic magnetic \feld hinduced by\na sinusoidal current applied to the signal surrounded by\ntwo ground lines. We used two di\u000berent CPWs with ei-\nther a broad [30] or narrow signal line width of ws= 1 mm\nor 20\u0016m, respectively [31]. The central long axis of the\nrectangular Cu 2OSeO 3rods was positioned on the central\naxis of the CPWs. The static magnetic \feld Hwas ap-\nplied perpendicular to the substrate with Hkh100iand\nHkh111ifor sample S1 and S2, respectively. The direc-\ntion ofHde\fned the z-direction. The dynamic \feld com-\nponent h?Hprovided the relevant torque for excita-\ntion. Components hkHdid not induce precessional mo-\ntion in the FP state of Cu 2OSeO 3. We recorded spectra\nby a vector network analyzer using the magnitude of the\nscattering parameter S12. We subtracted a background\nspectrum recorded at 1 T to enhance the signal-to-noisearXiv:1705.03416v1  [cond-mat.str-el]  9 May 20172\nratio (SNR) yielding the displayed \u0001 jS12j. In Ref. [7],\nKlingler et al. have investigated the damping of the in-\nsulating ferrimagnet YIG and found that Gilbert param-\neters\u000bintrevaluated from both the uniform precessional\nmode and standing spin waves con\fned in the macro-\nscopic sample provided the same values. For Cu 2OSeO 3\nwe evaluated \u000bin two ways[32]. When extracting the\nlinewidth \u0001 Hfor di\u000berent resonance frequencies fr, the\nGilbert damping parameter \u000bintrwas assumed to vary\naccording to [33, 34]\n\u00160\r\u0001\u0001H= 4\u0019\u000bintr\u0001fr+\u00160\r\u0001\u0001H0; (1)\nwhere\ris the gyromagnetic factor and \u0001 H0the contri-\nbution due to inhomogeneous broadening. Equation (1)\nis valid when viscous Gilbert damping dominates over\nscattering within the magnetic subsystem [35]. When\nperforming frequency-swept measurements at di\u000berent\n\feldsH, the obtained linewidth \u0001 fwas considered to\nscale linearly with the resonance frequency as [36]\n\u0001f= 2\u000bintr\u0001fr+ \u0001f0; (2)\nwith the inhomogeneous broadening \u0001 f0. The conver-\nsion from Eq. (1) to Eq. (2) is valid when frscales linearly\nwithHandHis applied along a magnetic easy or hard\naxis of the material [37, 38]. In Fig. 1 (a) to (d) we show\nspectra recorded in the FP state of the material using the\ntwo di\u000berent CPWs. For the same applied \feld Hwe ob-\nserve peaks residing at higher frequency fforHkh100i\ncompared to Hkh111i. From the resonance frequencies,\nwe extract the cubic magnetocrystalline anisotropy con-\nstantK= (\u00000:6\u00060:1)\u0001103J/m3for Cu 2OSeO 3[31].\nThe magnetic anisotropy energy is found to be extremal\nforh100iandh111ire\recting easy and hard axes, respec-\ntively [31]. The saturation magnetization of Cu 2OSeO 3\namounted to \u00160Ms= 0:13 T at 5 K[22].\nFigure 1 summarizes spectra taken with two di\u000ber-\nent CPWs on two di\u000berent Cu 2OSeO 3crystals exhibit-\ning di\u000berent crystallographic orientation in the \feld H.\nFor the narrow CPW [Fig. 1 (a) and (c)], we observed a\nbroad peak superimposed by a series of resonances that\nall shifted to higher frequencies with increasing H. The\n\feld dependence excluded them from being noise or arti-\nfacts of the setup. Their number and relative intensities\nvaried from sample to sample and also upon remounting\nthe same sample in the cryostat (not shown). They disap-\npeared with increasing temperature Tbut the broad peak\nremained. For the broad CPW [Fig. 1 (b) and (d)], we\nmeasured pronounced peaks whose linewidths were sig-\nni\fcantly smaller compared to the broad peak detected\nwith the narrow CPW. We resolved resonances below\nthe large peaks [arrows in Fig. 1 (b)] that shifted with\nHand exhibited an almost \feld-independent frequency\no\u000bset from the main peaks that we will discuss later. It\nis instructive to \frst follow the orthodox approach and\nanalyze damping parameters from modes re\recting the\n69121518-0.4-0.20.0(d)Δ  |S12|f\n (GHz)H\n || 〈111〉 \n69121518-6-30(c)Δ |S12| (10-2)f\n (GHz)\n-0.6-0.4-0.20.0H\n || 〈100〉 (b)broad CPWΔ |S12|\n-0.3-0.2-0.10.00\n.35 T(a)narrow CPWΔ |S12|0\n.25 T0\n.45 T0.55 TFIG. 1. (Color online) Spectra \u0001 jS12jobtained at T = 5 K\nfor di\u000berent Husing (a) a narrow and (b) broad CPW when\nHjjh100ion sample S1. Corresponding spectra taken on sam-\nple S2 for Hjjh111iare shown in (c) and (d), respectively.\nNote the strong and sharp resonances in (b) and (d) when us-\ning the broad CPW that provides a much more homogeneous\nexcitation \feld h. Arrows mark resonances that have a \feld-\nindependent o\u000bset with the corresponding main peaks and are\nattributed to standing spin waves. An exemplary Lorentz \ft\ncurve is shown in blue color in (b).\nexcitation characteristics of the CPW [29]. Second, we\nfollow Ref. [7] and analyze con\fned modes.\nLorentz curves (blue) were \ftted to the spectra\nrecorded with the broad CPW to determine resonance\nfrequencies and linewidths. Note that the corresponding\nlinewidths were larger by a factor ofp\n3 compared to the\nlinewidth \u0001 fthat is conventionally extracted from the\nimaginary part of the scattering parameters [39]. The\nextracted linewidths \u0001 fwere found to follow linear \fts\nbased on Eq. (2) at di\u000berent temperatures (details are\nshown in Ref. [31]). In Fig. 2 (a) we show a resonance\ncurve that was obtained as a function of Htaken with\nthe narrow CPW at 15 GHz. The curve does not show\nsharp features as Hwas varied in \fnite steps (symbols).\nThe linewidth \u0001 H(symbols) is plotted in Fig. 2 (b) for\ndi\u000berent resonance frequencies and temperatures. The\ndata are well described by linear \fts (lines) based on\nEq. (1). Note that the resonance peaks measured with\nthe broad CPW were extremely sharp. The sharpness\ndid not allow us to analyze the resonances as a function\nofH. We refrained from \ftting the broad peaks of Fig. 1\n(a) and (c) (narrow CPW) as they showed a clear asym-\nmetry attributed to the overlap of subresonances at \fnite\nwavevector k, as will be discussed below.\nIn Fig. 3 (a) and (b) we compare the parameter \u000bintr\nobtained from both di\u000berent CPWs (circles vs. stars) and\nthe two evaluation routes [40]. For Hkh100i[Fig. 3 (a)],\nbetween 5 and 20 K the lowest value for \u000bintramounts to\n(3.7\u00060.4)\u000210\u00003. This value is three times lower com-\npared to preliminary data presented in Ref. [29]. Beyond3\nFIG. 2. (Color online) (a) Lorentz curve (magenta line) \ftted\nto a resonance (symbols) measured at f= 15 GHz as a func-\ntion ofHat 5 K. (b) Frequency dependencies of linewidths\n\u0001H(symbols) for four di\u000berent T. We performed thep\n3-\ncorrection. The slopes of linear \fts (straight lines) following\nEq. 1 are considered to re\rect the intrinsic damping parame-\nters\u000bintr.\n04812H || 〈100〉 αintr (10-3)Δ H  narrow CPWΔ\n f    broad   CPW\nH || 〈111〉 \n1020304050T\n (K)\n10203040500.00.20.40.60.8Δf0 (GHz)T\n (K)(b)( a)(\nd)( c)\nFIG. 3. (Color online) (a) and (b) Intrinsic damping param-\neters\u000bintrand inhomogeneous broadening \u0001 f0for two di\u000ber-\nent \feld directions (see labels) obtained from the slopes and\nintercepts at fr= 0 of linear \fts to the linewidth data (see\nFig. 2 (b) and Ref. [31]). Dashed lines are guides to the eyes.\n20 K the damping is found to increase. For Hkh111i\n[Fig. 3 (b)] we extract (0.6 \u00060.6)\u000210\u00003as the smallest\nvalue. Note that these values for \u000bintrstill contain an ex-\ntrinsic contribution and thus represent upper bounds for\nCu2OSeO 3, as we will show later. For the inhomogeneous\nbroadening \u0001 f0in Fig. 3 (c) and (d) the datasets are\nconsistent (we have used the relation \u0001 f0=\r\u0001H0=2\u0019\nto convert \u0001 H0into \u0001f0). We see that \u0001 f0increases\nwithTand is small for the broad CPW, independent\nof the crystallographic direction of H. For the narrow\nCPW the inhomogeneous broadening is largest at small\nTand then decreases by about 40 % up to about 50\nK. Note that a CPW broader than the sample is as-\nsumed to excite homogeneously at fFMR [41] transfer-\nring a wave vector k= 0 to the sample. Accordinglywe ascribe the intense resonances of Fig. 1 (b) and (d) to\nfFMR. UsingfFMR= 6 GHz and \u000bintr= 3:7\u000210\u00003at 5\nK [Fig. 3 (a)], we estimate a minimum relaxation time of\n\u001c= [2\u0019\u000bintrfr]\u00001= 6:6 ns.\nIn the following, we examine in detail the additional\nsharp resonances that we observed in spectra of Fig. 1.\nIn Fig. 1 (b) taken with the broad CPW for Hkh100i,\nwe identify sharp resonances that exhibit a characteris-\ntic frequency o\u000bset \u000efwith the main resonance at all\n\felds (black arrows). We illustrate this in Fig. 4(a) in\nthat we shift spectra of Fig. 1 (b) so that the positions of\ntheir main resonances overlap. The additional small res-\nonances (arrows) in Fig. 1 (b) are well below the uniform\nmode. This is characteristic for backward volume magne-\ntostatic spin waves (BVMSWs). Standing waves of such\nkind can develop if they are re\rected at least once at the\nbottom and top surfaces of the sample. The resulting\nstanding waves exhibit a wave vector k=n\u0019=d , with\norder number nand sample thickness d= 0:3 mm. The\nBVMSW dispersion relation f(k) of Ref. [13] provides a\ngroup velocity vg=\u0000300 km/s at k=\u0019=d[triangles in\nFig. 4 (b)]. Hence, the decay length ld=vg\u001camounts\nto 2 mm considering \u001c= 6:6 ns. This is larger than\ntwice the relevant lateral sizes, thereby allowing stand-\ning spin wave modes to form in the sample. Based on\nthe dispersion relation of Ref. [13], we calculated the fre-\nquency splitting \u000ef=fFMR\u0000f(n\u0019=d ) [open diamonds\nin Fig. 4 (b)] assuming n= 1 andt= 0:4 mm for the\nsample width tde\fned in Ref. [13]. Experimental val-\nues (\flled symbols) agree with the calculated ones (open\nsymbols) within about 60 MHz. In case of the narrow\nCPW, we observe even more sharp resonances [Fig. 1 (a)\nand (c)]. A set of resonances was reported previously\nin the \feld-polarized phase of Cu 2OSeO 3[26, 28, 42, 43].\nMaisuradze et al. assigned secondary peaks in thin plates\nof Cu 2OSeO 3to di\u000berent standing spin-wave modes [43]\nin agreement with our analysis outlined above.\n0.30.40.51.101.151.201.25-\n500-300-100100δf (GHz)(b)/s61549\n0H (T)v\ng (km/s)\n-10 -0.8-0.6-0.4-0.20.0f\n - f (0)  (GHz)H || 〈100〉 (a)b\nroad CPWΔ |S12|δ\nf\nFIG. 4. (Color online) (a) Spectra of Fig. 1 (b) replotted as\nf\u0000fFMR(H) for di\u000berent Hsuch that all main peaks are at\nzero frequency and the \feld-independent frequency splitting\n\u000efbecomes visible. The numerous oscillations seen particu-\nlarly on the bottom most curve are artefacts from the cali-\nbration routine. (b) Experimentally evaluated (\flled circles)\nand theoretically predicted (diamonds) splitting \u000efusing dis-\npersion relations for a platelet. Calculated group velocity vg\natk=\u0019=(0:3 mm). Dashed lines are guides to the eyes.4\nThe inhomogeneous dynamic \feld hof the narrow\nCPW provides a much broader distribution of kcom-\npared to the broad CPW. This is consistent with the\nfact that the inhomogeneous broadening \u0001 f0is found to\nbe larger for the narrow CPW compared to the broad\none [Fig. 3 (c) and (c)]. Under these circumstances, the\nexcitation of more standing waves is expected. We at-\ntribute the series of sharp resonances in Fig. 1 (a) and\n(c) to such spin waves. In Fig. 5 (a) and (b) we highlight\nprominent and particularly narrow resonances with #1,\n#2 and #3 recorded with the narrow CPW. We trace\ntheir frequencies fras a function of HforHkh100iand\nHkh111i, respectively. They depend linearly on Hsug-\ngesting a Land\u0013 e factor g= 2:14 at 5 K.\nWe now concentrate on mode #1 for Hk h100iat\n5 K that is best resolved. We \ft a Lorentzian line-\nshape as shown in Fig. 5(c) for 0.85 T, and summarize\nthe corresponding linewidths \u0001 fin Fig. 5(d). The inset\nof Fig. 5(d) shows the e\u000bective damping \u000be\u000b= \u0001f=(2fr)\nevaluated directly from the linewidth as suggested in Ref.\n[29]. We \fnd that \u000be\u000bapproaches a value of about 3.5\n\u000210\u00004with increasing frequency. This value includes\nboth the intrinsic damping and inhomogeneous broad-\nening but is already a factor of 10 smaller compared to\n\u000bintrextracted from Fig. 3 (a). Note that Cu 2OSeO 3\nexhibiting 3.5\u000210\u00004outperforms the best metallic thin-\n\flm magnet [44]. To correct for inhomogeneous broad-\nening and determine the intrinsic Gilbert-type damping,\nwe apply a linear \ft to the linewidths \u0001 fin Fig. 5(d) at\nfr>10:6 GHz and obtain (9.9 \u00064.1)\u000210\u00005. Forfr\u0014\n10.6 GHz the resonance amplitudes of mode #1 were\nsmall reducing the con\fdence of the \ftting procedure.\nFurthermore, at low frequencies, we expect anisotropy to\nmodify the extracted damping, similar to the results in\nRef. [45]. For these reasons, the two points at low frwere\nleft out for the linear \ft providing (9.9 \u00064.1)\u000210\u00005.\nWe \fnd \u0001fand the damping parameters of Fig. 3 to\nincrease with T. It does not scale linearly for Hkh100i\n[31]. A deviation from linear scaling was reported for\nYIG single crystals as well and accounted for by the con-\n\ruence of a low- kmagnon with a phonon or thermally\nexcited magnon [5]. In the case of Hkh111i(cf. Fig. 3\n(b)) we obtain a clear discrepancy between results from\nthe two evaluation routes and CPWs used. We relate\nthis observation to a misalignment of Hwith the hard\naxish111i. The misalignment motivates a \feld-dragging\ncontribution [38] that can explain the discrepancy. For\nthis reason, we concentrated our standing wave analysis\non the case Hkh100i. We now comment on our spectra\ntaken with the broad CPW that do not show the very\nsmall linewidth attributed to the con\fned spin waves.\nThe sharp mode #1 yields \u0001 f= 15:3 MHz near 16 GHz\n[Fig. 5 (d)]. At 5 K the dominant peak measured at 0.55 T\nwith the broad CPW provides however \u0001 f= 129 MHz.\n\u0001fobtained by the broad CPW is thus increased by a\nfactor of eight and explains the relatively large Gilbert\nFIG. 5. (Color online) (a)-(b) Resonance frequency as a func-\ntion of \feld Hof selected sharp modes labelled #1 to #3 (see\ninsets) for Hkh100iandHkh111iat T = 5 K. (c) Exemplary\nLorentz \ft of sharp mode #1 for Hkh100iat 0.85 T. (d) Ex-\ntracted linewidth \u0001f as a function of resonance frequency fr\nalong with the linear \ft performed to determine the intrinsic\ndamping\u000bintrin Cu 2OSeO 3. Inset: Comparison among the\nextrinsic and intrinsic damping contribution. The red dotted\nlines mark the error margins of \u000bintr= (9:9\u00064:1)\u000210\u00005.\ndamping parameter in Fig. 3 (a) and (b). We con\frmed\nthis larger value on a third sample with Hkh100iand ob-\ntained (3.1\u00060.3)\u000210\u00003[31] using the broad CPW. The\ndiscrepancy with the damping parameter extracted from\nthe sharp modes of Fig. 5 might be due to the remaining\ninhomogeneity of hover the thickness of the sample lead-\ning to an uncertainty in the wave vector in z-direction.\nFor a standing spin wave such an inhomogeneity does\nnot play a role as the boundary conditions discretize k.\nAccordingly, Klingler et al. extract the smallest damp-\ning parameter of 2 :7(5)\u000210\u00005reported so far for the\nferrimagnet YIG when analyzing con\fned magnetostatic\nmodes [7].\nTo summarize, we investigated the spin dynamics in\nthe \feld-polarized phase of the insulating chiral mag-\nnet Cu 2OSeO 3. We detected numerous sharp reso-\nnances that we attribute to standing spin waves. Their\ne\u000bective damping parameter is small and amounts to\n3:5\u000210\u00004. A quantitative estimate of the intrinsic\nGilbert damping parameter extracted from the con\fned\nmodes provides even \u000bintr=(9.9\u00064.1)\u000210\u00005at 5 K. The\nsmall damping makes an insulating ferrimagnet exhibit-\ning Dzyaloshinskii-Moriya interaction a promising can-\ndidate for exploitation of complex spin structures and\nrelated nonreciprocity in magnonics and spintronics.\nWe thank S. Mayr for assistance with sample prepa-\nration. Financial support through DFG TRR80, DFG\n1143, DFG FOR960, and ERC Advanced Grant 291079\n(TOPFIT) is gratefully acknowledged.5\n\u0003Electronic mail: dirk.grundler@ep\r.ch\n[1] I. Zutic and H. Dery, Nat. Mater. 10, 647 (2011).\n[2] M. Krawczyk and D. Grundler, J. Phys.: Condens. Mat-\nter26, 123202 (2014).\n[3] A. V. Chumak, V. I. Vasyuchka, A. A. Serga, and\nB. Hillebrands, Nat. Phys. 11, 453 (2015).\n[4] A. G. Gurevich and G. A. Melkov, Magnetization Oscil-\nlations and Waves (CRC Press, 1996).\n[5] M. Sparks, Ferromagnetic-Relaxation Theory (McGraw-\nHill, 1964).\n[6] A. A. Serga, A. V. Chumak, and B. Hillebrands, Journal\nof Physics D: Applied Physics 43, 264002 (2010).\n[7] S. Klingler, H. Maier-Flaig, C. Dubs, O. Surzhenko,\nR. Gross, H. Huebl, S. T. B. Goennenwein, and\nM. Weiler, Applied Physics Letters 110, 092409 (2017),\nhttp://dx.doi.org/10.1063/1.4977423.\n[8] A. Fert, V. Cros, and J. Sampaio, Nat. Nanotechn. 8,\n152 (2013).\n[9] N. Nagaosa and Y. Tokura, Nat. Nanotechn. 8, 899\n(2013).\n[10] S. M uhlbauer, B. Binz, F. Jonietz, C. P\reiderer,\nA. Rosch, A. Neubauer, R. Georgii, and P. B oni, Sci-\nence323, 915 (2009).\n[11] X. Z. Yu, Y. Onose, N. Kanazawa, J. H. Park, J. H. Han,\nY. Matsui, N. Nagaosa, and Y. Tokura, Nature (London)\n465, 901 (2010).\n[12] S. Seki, X. Z. Yu, S. Ishiwata, and Y. Tokura, Science\n336, 198 (2012).\n[13] S. Seki, Y. Okamura, K. Kondou, K. Shibata, M. Kubota,\nR. Takagi, F. Kagawa, M. Kawasaki, G. Tatara, Y. Otani,\nand Y. Tokura, Phys. Rev. B 93, 235131 (2016).\n[14] M. Mochizuki and S. Seki, J. Physics: Condens. Matter\n27, 503001 (2015).\n[15] M. Garst, J. Waizner, and D. Grundler,\nhttps://arxiv.org/abs/1702.03668 (2017).\n[16] H. Huebl, C. W. Zollitsch, J. Lotze, F. Hocke, M. Greifen-\nstein, A. Marx, R. Gross, and S. T. B. Goennenwein,\nPhys. Rev. Lett. 111, 127003 (2013).\n[17] Y. Tabuchi, S. Ishino, T. Ishikawa, R. Yamazaki, K. Us-\nami, and Y. Nakamura, Phys. Rev. Lett. 113, 083603\n(2014).\n[18] X. Zhang, C.-L. Zou, L. Jiang, and H. X. Tang, Phys.\nRev. Lett. 113, 156401 (2014).\n[19] M. Goryachev, W. G. Farr, D. L. Creedon, Y. Fan,\nM. Kostylev, and M. E. Tobar, Phys. Rev. Applied 2,\n054002 (2014).\n[20] K. Kohn, J. Phys. Soc. Jpn 42, 2065 (1977).\n[21] M. Belesi, I. Rousochatzakis, H. C. Wu, H. Berger, I. V.\nShvets, F. Mila, and J. P. Ansermet, Phys. Rev. B 82,\n094422 (2010).\n[22] T. Adams, A. Chacon, M. Wagner, A. Bauer, G. Brandl,B. Pedersen, H. Berger, P. Lemmens, and C. P\reiderer,\nPhys. Rev. Lett. 108, 237204 (2012).\n[23] S. Seki, J.-H. Kim, D. S. Inosov, R. Georgii, B. Keimer,\nS. Ishiwata, and Y. Tokura, Phys. Rev. B 85, 220406\n(R) (2012).\n[24] S. Seki, S. Ishiwata, and Y. Tokura, Phys. Rev. B 86,\n060403 (2012).\n[25] Y. Okamura, F. Kagawa, M. Mochizuki, M. Kubota,\nS. Seki, S. Ishiwata, M. Kawasaki, Y. Onose, and\nY. Tokura, Nat. Commun. 4, 2391 (2013).\n[26] Y. Okamura, F. Kagawa, S. Seki, M. Kubota,\nM. Kawasaki, and Y. Tokura, Phys. Rev. Lett. 114,\n197202 (2015).\n[27] M. Mochizuki, Phys. Rev. Lett. 114, 197203 (2015).\n[28] Y. Onose, Y. Okamura, S. Seki, S. Ishiwata, and\nY. Tokura, Phys. Rev. Lett. 109, 037603 (2012).\n[29] T. Schwarze, J. Waizner, M. Garst, A. Bauer,\nI. Stasinopoulos, H. Berger, C. P\reiderer, and\nD. Grundler, Nature Mater. 14, 478 (2015).\n[30] Model B4350-30C from Southwest Microwave, Inc.,\nwww.southwestmicrowave.com.\n[31] See Supplemental Material at [URL] for experimental de-\ntails and additional data.\n[32] Y. Wei, S. L. Chin, and P. Svedlindh, J. Phys. D: Appl.\nPhys. 48, 335005 (2015).\n[33] B. Heinrich, J. F. Cochran, and R. Hasegawa, J. Appl.\nPhys. 57(1985).\n[34] S. S. Kalarickal, P. Krivosik, M. Wu, C. E. Patton, M. L.\nSchneider, P. Kabos, T. J. Silva, and J. P. Nibarger, J.\nAppl. Phys. 99, 093909 (2006).\n[35] K. Lenz, H. Wende, W. Kuch, K. Baberschke, K. Nagy,\nand A. J\u0013 anossy, Phys. Rev. B 73, 144424 (2006).\n[36] C. E. Patton, J. Appl. Phys. 39(1968).\n[37] B. Kuanr, R. E. Camley, and Z. Celinski, Appl. Phys.\nLett.87, 012502 (2005).\n[38] M. Farle and H. Zabel, Magnetic Nanostructures Spin\nDynamics and Spin Transport , Vol. 246 (Springer Tracts\nin Modern Physics, 2013).\n[39] D. D. Stancil and A. Prabhakar, Spin Waves Theory and\nApplications (Springer, 2009).\n[40] We call it \u000bintrat this point as the parameter is extracted\nfrom linear slopes. Later we will show that standing spin\nwaves provide the lowest \u000bintr.\n[41] Y. Iguchi, S. Uemura, K. Ueno, and Y. Onose, Phys.\nRev. B 92, 184419 (2015).\n[42] M. I. Kobets, K. G. Dergachev, E. N. Khatsko, A. I.\nRykova, P. Lemmens, D. Wulferding, and H. Berger,\nLow Temp. Phys. 36(2010).\n[43] A. Maisuradze, A. Shengelaya, H. Berger, D. M. Djoki\u0013 c,\nand H. Keller, Phys. Rev. Lett. 108, 247211 (2012).\n[44] M. A. W. Schoen, D. Thonig, M. L. Schneider, T. J.\nSilva, H. T. Nembach, O. Eriksson, O. Karis, and J. M.\nShaw, Nat. Phys. 12, 839 (2016).\n[45] T. J. Silva, C. S. Lee, T. M. Crawford, and C. T. Rogers,\nJ. Appl. Phys. 85(1999)."    },    {        "title": "1705.03661v1.Negative_mobility_of_a_Brownian_particle__strong_damping_regime.pdf",        "content": "Negative mobility of a Brownian particle:\nstrong damping regime\nA. S lapik, J.  Luczka, J. Spiechowicz\nInstitute of Physics, University of Silesia, 40-007 Katowice, Poland\nSilesian Center for Education and Interdisciplinary Research, University of Silesia,\n41-500 Chorz\u0013 ow, Poland\nAbstract\nWe study impact of inertia on directed transport of a Brownian particle under\nnon-equilibrium conditions: the particle moves in a one-dimensional periodic\nandsymmetric potential, is driven by both an unbiased time-periodic force and\na constant force, and is coupled to a thermostat of temperature T. Within\nselected parameter regimes this system exhibits negative mobility, which means\nthat the particle moves in the direction opposite to the direction of the constant\nforce. It is known that in such a setup the inertial term is essential for the\nemergence of negative mobility and it cannot be detected in the limiting case\nof overdamped dynamics. We analyse inertial e\u000bects and show that negative\nmobility can be observed even in the strong damping regime. We determine\ntheoptimal dimensionless mass for the presence of negative mobility and reveal\nthree mechanisms standing behind this anomaly: deterministic chaotic, thermal\nnoise induced and deterministic non-chaotic. The last origin has never been\nreported. It may provide guidance to the possibility of observation of negative\nmobility for strongly damped dynamics which is of fundamental importance\nfrom the point of view of biological systems, all of which in situ operate in\n\ructuating environments.\nKeywords: Brownian motion, periodic symmetric systems, negative mobility,\nstrong damping regime\n1. Introduction\nWhen a system at thermal equilibrium is exposed to a weak external static\nforce, its response is in the same direction as this of applied bias towards a new\nequilibrium. This restriction is no longer valid under nonequilibrium conditions\nwhen already an unperturbed system may exhibit a current due to the ratchet\nEmail address: jerzy.luczka@us.edu.pl (J.  Luczka)\nPreprint submitted to Elsevier November 7, 2018arXiv:1705.03661v1  [cond-mat.stat-mech]  10 May 2017e\u000bect [1]. Another example is the seemingly paradoxical situation of the nega-\ntive mobility phenomenon when the system response is opposite to the applied\nconstant force [2]. Such anomalous transport behaviour was predicted theoreti-\ncally in 2007 in a system consisting of an inertial Brownian particle moving in a\none-dimensional periodic symmetric potential [3]. Within a year of this discov-\nery, negative mobility was con\frmed experimentally in the experiment involv-\ning determination of current-voltage characteristics of the microwaved-driven\nJosephson junction [4]. Yet further examples of this phenomenon has been de-\nscribed theoretically in companionship of coloured noise [5], white Poissonian\nnoise [6, 7], dichotomous process [8] and for Brownian motion with presence of\ntime-delayed feedback [9, 10], non-uniform space-dependent damping [11] and\npotential phase modulation [12]. Other illustrations include a vibrational motor\n[13], two coupled resistively shunted Josephson junctions [14, 15], active Janus\nparticles in a corrugated channel [16], entropic electrokinetics [17] as well as\nnonlinear response of inertial tracers in steady laminar \rows [18].\nModelling systems and understanding their generic properties discloses which\ncomponents of the setup are crucial and which elements may be sub-relevant.\nFor instance, transport in the micro-world is strongly in\ruenced by \ructuations\nand random perturbations. In some systems, like biological cells [19], they can\neven play a dominant role and a typical situation is that motion of particles\nis strongly damped. This fact justi\fes the use of an overdamped dynamics for\nwhich the particle inertial term Mxcan be formally neglected in comparison\nto the dissipation term \u0000 _ x(Mis the particle mass, \u0000 is the friction coe\u000ecient\nand dot denotes a di\u000berentiation with respect to time t). Omission of the in-\nertial term enormously simpli\fes the modelling and in many cases allows for\nan analytical solutions of the corresponding Fokker-Planck equation. However,\nproperties and features which are allowed to occur in systems with inertia can\ncompletely disappear when the inertial term is put to zero. Certainly a more\ncorrect approach in such a situation is to include the inertial term and use a\ntechnique of mathematical sequences of smaller and smaller dimensionless mass.\nOur main objective is to investigate impact of inertia on negative mobility of a\nBrownian particle moving in one-dimensional periodic systems. It is known that\nin such setups the inertial term is one of the key ingredients for the occurrence\nof this form of anomalous transport [20, 21] and negative mobility is absent for\nthe overdamped dynamics when Mx= 0 . We address the question whether\nit is still possible to observe the negative mobility phenomenon in strongly dis-\nsipative systems. In doing so, we \frst formulate the model and introduce the\nquantities of interest. Then we investigate the general transport behaviour as\na function of model parameters and detect the optimal dimensionless mass for\nthe presence of negative mobility. In the next part we demonstrate three mech-\nanisms responsible for the emergence of this anomalous transport phenomenon:\ndeterministic chaotic, thermal noise induced and deterministic non-chaotic. Fi-\nnally, we discuss impact of inertia on the directed long time particle velocity\nand provide some conclusions.\n22. Model\nThe model of a Brownian particle moving in a one-dimensional periodic\nlandscape has been already well established in the literature [22]. It has been\nused to explore a wide range of phenomena including ratchet e\u000bects [25, 26],\nnoise induced transport [23, 24], the negative mobility [3], the enhancement\nof transport [7] and di\u000busion [27], the anomalous di\u000busion [28, 29] and the\nnon-monotonic temperature dependence of di\u000busion [30, 31]. Here, we consider\nexactly the same model as in [3]: a classical inertial Brownian particle of mass\nM, which moves in a spatially periodic potential U(x) =U(x+L) of period L\nand is subjected to both an unbiased time-periodic force Acos (!t) of amplitude\nAand angular frequency \n and an external static force F. Dynamics of such a\nparticle is described by the following Langevin equation [3]\nMx+ \u0000 _x=\u0000U0(x) +Acos (\nt) +F+p\n2\u0000kBT\u0018(t); (1)\nwhere prime denotes a di\u000berentiation with respect to the particle coordinate x.\nThermal \ructuations due to the coupling of the particle with the thermal bath\nof temperature Tare modelled by Gaussian white noise of zero mean and unity\nintensity, namely\nh\u0018(t)i= 0;h\u0018(t)\u0018(s)i=\u000e(t\u0000s): (2)\nThe noise intensity factor 2\u0000 kBT(wherekBis the Boltzmann constant) follows\nfrom the \ructuation-dissipation theorem [32] and ensures the canonical equilib-\nrium Gibbs state when A= 0 andF= 0. The potential U(x) is assumed to be\nin asymmetric form with the period Land the barrier height 2\u0001 U, namely,\nU(x) = \u0001Usin\u00122\u0019\nLx\u0013\n: (3)\nThere exists a wealth of physical systems that can be described by the Langevin\nequation (1). An important cases that come to mind are the semiclassical dy-\nnamics of a phase di\u000berence across a resistively and capacitively shunted Joseph-\nson junction [33] and a cold atom moving in an optical lattice [1, 34]. Other\nexamples include superionic conductors [35], dipoles rotating in external \feld\n[36], charge density waves [37] and adatoms on a periodic surface [38].\n2.1. Scaling and dimensionless Langevin equation\nSince only relations between scales of length, time and energy are relevant for\nthe observed phenomena, not their absolute values, we next formulate the above\npresented equation of motion in its dimensionless form. This can be achieved\nin several ways [39]. Because investigation of impact of the particle inertia on\nthe system dynamics is our main goal, in the present consideration we propose\nthe use of the following scales as the characteristic units of length and time [39]\n^x=x\nL;^t=t\n\u001c0; \u001c 0=\u0000L2\n\u0001U: (4)\n3Under such a procedure the Langevin equation (1) takes the dimensionless form\n[30]\nm^x+_^x=\u0000^U0(^x) +acos (!^t) +f+p\n2D^\u0018(^t): (5)\nIn this scaling, the dimensionless mass is\nm=\u001c1\n\u001c0=M\u0001U\n\u00002L2; (6)\nwhere the second characteristic time is \u001c1=M=\u0000 and the dimensionless fric-\ntion coe\u000ecient is \r= 1. Other parameters are: a= (L=\u0001U)A,!=\u001c0\n,\nf= (L=\u0001U)F. The rescaled thermal noise reads ^\u0018(^t) = (L=\u0001U)\u0018(t) = (L=\u0001U)\u0018(\u001c0^t)\nand assumes the same statistical properties as \u0018(t), namelyh^\u0018(^t)i= 0 and\nh^\u0018(^t)^\u0018(^s)i=\u000e(^t\u0000^s). The dimensionless noise intensity D=kBT=\u0001Uis the\nratio of thermal energy and half of the activation energy the particle needs\nto overcome the nonrescaled potential barrier. The dimensionless potential\n^U(^x) = sin(2\u0019^x) possesses the period ^L= 1 and the barrier height \u0001 ^U= 2.\nFrom now on, we will use only the dimensionless variables and therefore, in order\nto simplify the notation, we will omit the hatnotation in the above equation.\n2.2. Quantities of interest\nIn the present study we are particularly interested in the impact of inertia\non properties of directed transport of particles in the stationary state. In the\ndimensionless formulation (5) it can be realized by changing the dimensionless\nmass (6). The case m= 0 corresponds to overdamped dynamics and the setting\nm\u001c1 represents the strong damping regime, which means that \u001c1\u001c\u001c0.\nThe characteristic time \u001c1is obtained from a particular form of Eq. (1), i.e.\nM_v+ \u0000v= 0 and has the interpretation of the relaxation time of the velocity\nof the free Brownian particle. The parameter \u001c0is extracted from the equation\n\u0000 _x=\u0000U0(x) which can be viewed as the characteristic time to travel a distance\nfrom a maximum of the potential U(x) to its minimum in the overdamped case\n(it is not exactly this time which is in\fnite in the considered case but \u001c0scales\nit). It is remarkable that parameters of the potential U(x) such as its barrier\nheight \u0001Uand period Lare crucial for controlling the regimes of weak or strong\ndamping. For instance, if Mand \u0000 are \fxed and the system is in a weak\ndamping regime m\u001d1, the transition to the strong damping case m\u001c1 can\nbe achieved by lowering the barrier height and lengthening the period of U(x).\nWe have checked that for values m\u00180:1 and smaller the system (5) can be\nconsidered to be in the strong damping regime.\nDue to the presence of the external time-periodic driving acos (!t), as well\nas the friction term _ x, the particle velocity _ x(t) approaches a unique non-\nequilibrium asymptotic long time state, in which it is characterized by a tem-\nporally periodic probability density. This latter function has the same period\nas the driving T= 2\u0019=! [40]. Therefore, the \frst statistical moment of the in-\nstantaneous particle velocity h_x(t)iassumes for an asymptotic long time regime\nthe form of a Fourier series over all possible harmonics [40]\nlim\nt!1h_x(t)i=hvi+v!(t) +v2!(t) +::: (7)\n4wherehviis the directed (time independent) velocity, while vn!(t) denote time\nperiodic higher harmonics of vanishing time-average over the fundamental pe-\nriodT= 2\u0019=!. The observable of foremost interest in this study is the directed\ntransport component hvi, which due to the mentioned particular decomposition\ncan be obtained in the following way\nhvi= lim\nt!1!\n2\u0019Zt+2\u0019=!\ntdsh_x(s)i; (8)\nwhereh\u0001iindicates averaging over all realizations of thermal noise as well as\nover initial conditions for the position x(0) and the velocity _ x(0). The latter\nis obligatory for the deterministic limit D/T!0 when dynamics may be\nnon-ergodic and results can be a\u000bected by speci\fc choice of initial conditions\n[29].\nDue to the multidimensionality of the parameter space of the considered\nmodel, as well as its nonlinearity, the force-velocity curve hvi=hvi(f) is typ-\nically a nonlinear function of the applied bias f. From the symmetries of the\nunderlying Langevin equation (5) it follows that this observable is odd as a func-\ntion of the external static force f, i.e.hvi(\u0000f) =\u0000hvi(f) and in consequence\nhvi(f= 0)\u00110 [1]. This is in clear contrast to the case of a ratchet mechanism,\nwhich exhibits the \fnite directed transport hvi6= 0 even at the vanishing static\nbias whenf= 0 [2]. Since the observable of our interest is symmetric around\nf= 0, we limit our consideration to the positive bias f > 0. Then, for su\u000e-\nciently small values of the external force fthe directed transport velocity hviis\nusually its increasing function. Such regimes correspond to the normal, expected\ntransport behaviour. However, in the parameter space there are also regimes for\nwhich the particle moves on average in the direction opposite to the applied bias,\ni.e.hvi<0 forf > 0, exhibiting anomalous transport behaviour in the form\nof the negative mobility phenomenon [3, 20]. It has been already shown that\nthere are two fundamentally various mechanisms responsible for negative mobil-\nity in this setup, (i) generated by chaotic dynamics and (ii) induced by thermal\nequilibrium \ructuations [3]. The latter situation is nevertheless rooted in the\nsophisticated evolution of the corresponding deterministic system described by\nEq. (5) with D= 0. Its three-dimensional phase space fx;_x;!tgis minimal for\nchaotic evolution, which is important for negative mobility to occur.\nFor the considered deterministic system with D= 0 there are three Lya-\npunov exponents \u00151,\u00152and\u00153. It can be easily checked that the system is\ndissipative, i.e. the phase space volume is contracting during the time evolu-\ntion. Therefore the sum of all Lyapunov exponents must be negative [41]\n\u00151+\u00152+\u00153<0: (9)\nOne of the exponents, say \u00153= 0 and the other, say \u00152<0. If the system is\nchaotic\u00151must be positive indicating divergence of the trajectories. Therefore\nto detect chaotic behaviour of the system it is su\u000ecient to calculate the maximal\nLyapunov exponent \u0015=\u00151and check whether it is larger than zero [42].\n5Figure 1: The Brownian particle asymptotic long time directed velocity hvias a function of the\namplitudeaand the angular frequency !of the external unbiased harmonic driving acos (!t)\nis shown for di\u000berent values of the bias fwithD= 0 andm= 0:1. Panel (a) f= 0:2, (b)\nf= 0:4, (c)f= 0:6, (d)f= 0:8.\n3. Numerical simulation\nUnluckily, the Fokker-Planck equation corresponding to the Langevin equa-\ntion (5) cannot be handled by any known analytical methods. For this reason,\nin order to analyse transport properties of the system, we carried out compre-\nhensive numerical simulations. We integrated the Langevin equation (5) by\nemploying a weak version of the stochastic second order predictor corrector al-\ngorithm with a time step typically set to about 10\u00002\u00022\u0019=!. We chose the\ninitial coordinates x(0) and velocities _ x(0) equally distributed over the intervals\n[0;1] and [\u00002;2], respectively. The quantities of interest were ensemble averaged\nover 103\u0000104di\u000berent trajectories, which evolved over 103\u0000104periods of the\nexternal harmonic driving. All numerical calculations were performed by the\nuse of CUDA environment implemented on a modern desktop GPU. This gave\nus possibility to speed up the computations up to a factor of the order 103times\nas compared to a common present day CPU method. Details on this promising\nscheme can be found in Ref. [43].\nDynamics described by Eq. (5) is characterized by a 5-dimensional parame-\nter spacefm;a;!;f;Dg, the detailed exploration of which is a very challenging\ntask even for our innovative computational method. However, we focus on the\nimpact of the particle inertia on the anomalous transport processes occurring in\n6Figure 2: The directed velocity hviversus the amplitude aand the angular frequency !is\ndepicted for di\u000berent values of thermal noise intensity Dwithm= 0:1 andf= 0:5. Panel\n(a)D= 0, (b)D= 10\u00005, (c)D= 10\u00003, (d)D= 10\u00002.\nthis setup. This task is very tractable numerically with the currently available\nhardware. We start our analysis by looking at the deterministic system D= 0.\nWe set the bias to a low value f= 0:5 and check how the directed velocity hvi\ndepends on the remaining parameters. In doing so we performed scans of the\nfollowing area m\u0002a\u0002!2[0:01;10]\u0002[0;20]\u0002[0;20] at a resolution of 200\npoints per dimension to determine the general behaviour of the system. Our\nresults reveal that negative mobility is not present for !>18 and!<2. This\nis in agreement with the approximate solutions of Eq. (5). In the limit of low\nfrequencies an adiabatic approximation is valid [44], while for high frequencies\na solution can be formulated in terms of Bessel functions [ ?]. Moreover, there\nis no net transport for a<4.\n4. Results\n4.1. General behaviour of the system\nThe study of various aspects of transport in the system (5) has been pre-\nsented elsewhere [3, 20]. Here, we focus our analysis primarily on the relation-\nship between inertia and negative mobility. In Fig. 1 we present the asymptotic\nlong time directed velocity hvidepicted as a function of the amplitude aand\nangular frequency !of the external harmonic driving acos (!t), for the strongly\n7Figure 3: The directed velocity hviversus the amplitude aand the angular frequency !is\npresented for di\u000berent values of the particle mass mwithD= 0 andf= 0:5. Panel (a)\nm= 0:05, (b)m= 0:1, (c)m= 0:15, and (d) m= 0:2.\ndamped Brownian particle with the dimensionless mass m= 0:1 and di\u000berent\nvalues of the external bias f. Surprisingly, despite the fact that the inertia is one\norder smaller than the dimensionless friction coe\u000ecient \r= 1, there are regions\nin the parameter space of the system where negative mobility occurs. They\nform a band-like structure. The stripes of negative velocity are interspersed\nwith the stripes of positive velocity and the di\u000berence in its magnitude in the\nneighbouring regions can be signi\fcant. This suggest that the system may be\nvery sensitive to a small change of values of parameters. It can be observed\nthat larger values of flead to reduction of the negative mobility areas towards\nthe lower!anda. At the same time the bands of negative velocity become\nwider and more intense. In contrast, the regions of positive mobility increase\npopulation and supersede their negative counterparts. Overall, as the bias f\nis increased the band structure seems to zoom out until the directed velocity\npoints to the direction given by fand becomes almost constant in the whole\nmap. In Fig. 2 we depict the same characteristic but for di\u000berent values of\nthermal noise intensity Dwith \fxedm= 0:1 andf= 0:5. One can expect that\nthermal noise perturbs deterministic dynamics. We observe that larger areas of\nnegative mobility are relatively stable with respect to increasing temperature,\nwhile the smaller ones disappear more quickly. Thermal noise \frst blurs the\nband-like structure of negative mobility areas, erasing the \fner details of the\n80.000.020.040.060.08\n0.01 0 .1 1 10\nm0.000.010.030.040.06\n0.01 0 .1 1 10\nmf= 0.25\nf= 0.5\nf= 0.8(a)D= 10−3\nD= 10−4\nD= 10−5\nD= 0(b)Figure 4: Fraction of the negative mobility area in the analysed parameter space, a2[0;20],\n!2[0;20] is presented in panel (a) for D= 0 and di\u000berent values of the bias fand (b) for\nf= 0:5 and various temperatures D.\nregions visible in the deterministic case D= 0. It seems to be obvious since\nthermal noise enables random transitions between deterministically coexisting\nbasins of attraction. For high enough temperatures, negative mobility disap-\npears completely. A careful inspection of Fig. 2 reveals that there are regions\nin the parameter space where the directed velocity hviis zero or positive in the\ndeterministic case, but becomes negative upon the introduction of noise. This\nfact suggests that thermal \ructuations may induce negative mobility or reverse\nits sign even for the strongly damped Brownian particle. Finally, in Fig. 3 we\npresent the directed velocity hviversus the amplitude aand the angular fre-\nquency!for di\u000berent values of the mass mwithD= 0 andf= 0:5. The\nstripes of negative mobility move towards lower values of !and higher values of\naasmis increased. The band-like structure changes its inclination and the re-\ngions becomes more horizontally oriented. Moreover, for larger masses msome\nnew negative mobility bands appear, while at the same time the negative ve-\nlocity tends to disappear in other regions. This e\u000bect suggest that there should\nexist an optimal mass mfor which the occurrence of negative mobility is mostly\npronounced.\n4.2. Optimal mass for the presence of negative mobility\nSince the impact of the mass mseems to be non-trivial, an interesting task\nis to \fnd the value of mfor which the presence of negative mobility is the most\ncommon. Our numerical scans of the parameter space allowed us to determine\nthis value. The result is shown in Fig. 4. In panel (a) we depict fraction of\nthe negative mobility area in the analysed parameter space for the deterministic\nsystemD= 0 and di\u000berent values of the bias f. A perhaps surprising \fnding\nis that for the small values of the static load f= 0:25 the optimal mass for\nthe presence of negative mobility is m\u00190:13. It is signi\fcantly less than the\nmagnitude of dimensionless friction coe\u000ecient, which for the employed scaling\nis equal to unity \r= 1. This fact indicates that the friction plays prevalent role\nfor the emergence of negative mobility. Moreover, even for the very strongly\n9−0.8−0.5−0.20.10.4\n0.1 0 .3 0 .5 0 .7 0 .9\nf−0.3−0.10.10.30.50.7\n10−710−510−310−1\nD\n−0.8−0.400.40.8\n0.1 0 .3 0 .5 0 .7 0 .9\nf00.61.21.82.43\n0.1 0 .3 0 .5 0 .7 0 .9\nf/angbracketleftv/angbracketright(a)\n/angbracketleftv/angbracketright(b)\n/angbracketleftv/angbracketright (c) λ (d)Figure 5: The negative mobility of the strongly damped Brownian particle m\u001c1 induced\nby the deterministic chaotic dynamics. Panel (a) the directed velocity hvi, (c) bifurcation\ndiagram of the directed velocity hvi, (d) the maximal Lyapunov exponent \u0015as the function\nof the external static bias fwithD= 0. Panel (b) the directed velocity hviversus thermal\nnoise intensity Dforf= 0:66. Other parameters are m= 0:0555,a= 8:55,!= 12:38.\ndamped Brownian particle m\u001c\r= 1 the area of negative mobility is non-zero\nand relatively large. In both limiting regimes of the overdamped m!0 and the\nunderdamped m!1 motion there are no regions of negative mobility. Ipso\nfacto we con\frmed numerically the no-go theorem formulated in Ref. [20]. This\nconclusion is valid also for larger values of the bias f, however, then the optimal\nmassmfor the presence of negative mobility becomes shifted towards higher,\ne.g.m\u00190:47 forf= 0:8. Moreover, as the static load fis increased the overall\noccurrence of negative mobility in the analysed parameter space is decreased\nand the presented curves come to be more \rattened. In panel (b) we show the\nsame characteristic but for the \fxed bias f= 0:5 and di\u000berent values of thermal\nnoise intensity D. As temperature grows the regions of negative mobility in the\nparameter space tend to contract which is illustrated in the \fgure. Apart from\nthis fact for stronger thermal noise, c.f. the case D= 10\u00003, the curve becomes\nnoticeably more bimodal. This observation is most likely due to parameter\nregimes for which negative mobility is induced by thermal \ructuations.\n4.3. The mechanisms of negative mobility\nTo gain further insight into the nature of negative mobility in this system,\nas the next step we identify, exemplify and analyse the mechanisms standing\n10behind emergence of this phenomenon. In Fig. 5 we present the regime of\nparameters for which negative mobility occurs on grounds which are rooted\nsolely in the complex, strongly damped and deterministic chaotic dynamics.\nPanel (a) depicts the directed velocity hviversus the external static bias f. It is\na nonlinear function without any obvious relation to the magnitude of the force\nf. Clearly, there are two windows of the latter parameter for which negative\nmobilityhvi<0 is observed. The \frst starts at f\u00190:1 and ends at f\u00190:7.\nThe second is present for the bias larger than approximately f\u00190:9. In panel\n(b) we present the directed velocity hviof the Brownian particle as a function\nof thermal noise intensity Dfor the \fxed bias f= 0:66 which corresponds to\nthe sharp minimum of hvidepicted in the panel (a). In the limiting case of\nthe very low temperatures D!0 the measured directed velocity is less than\nzero indicating that negative mobility has its origin in the complex deterministic\ndynamics. For increasing temperature the directed velocity grows as well up to\nthe critical thermal noise intensity D\u001910\u00003, for which the Brownian particle\nresponse reverses its sign hvi>0. In the high temperature limit D! 1\n(not depicted) all forces in the right hand side of Eq. (5) become negligible in\ncomparison to thermal noise and thus the directed velocity vanishes completely\nhvi= 0. Panel (c) of the same \fgure presents the bifurcation diagram of the\nlatter quantityhviillustrated as the function of the external bias ffor the\ndeterministic system with D= 0. Each blue dot represents an attractor for\nthe asymptotic long time directed velocity hvi. For almost all values in the\nconsidered range of the bias fthere is the continuum of the directed velocity\nsolutions. This fact suggests that the system is predominantly chaotic in this\ninterval. We con\frm this hypothesis in panel (d) where we depict the maximal\nLyapunov exponent \u0015for the deterministic system described by Eq. (5) with\nD= 0 versus the biasing force f. Accordingly, this quantity is positive in almost\nentire considered interval of the parameter f. In particular, it is so for the values\noffcorresponding to negative mobility. Therefore, we conclude that in the\npresented parameter regime this phenomenon is induced solely by the chaotic\ndeterministic dynamics of the system given by Eq. (5). Such a mechanism has\nbeen already reported in literature [3, 20], however, here we prove that it may\noperate also for the strongly damped Brownian particle m\u001c\r= 1.\nThe second mechanism of the emergence of negative mobility is exempli\fed\nin Fig. 6. In panel (a) we present the directed velocity hviof the strongly\ndamped Brownian particle m\u001c\r= 1 versus the external static bias ffor\nthermal \ructuations intensity D= 0:0009. In this case a very small amount of\nnoise yields negative mobility in the linear response regime, i.e. for small forces\nf. For larger values of the bias fthe directed velocity is positive hvi>0 and the\nparticle moves in the direction pointed by the static force f. Interestingly, there\nis an optimal value of f\u00190:5 for which negative mobility is most pronounced.\nTo gain further insight into the origin of the discussed anomaly in the presented\nregime, in the neighbouring panel (b) we study the directed velocity hvias a\nfunction of thermal noise intensity D/T. Contrary to the previously presented\ncase, here at low temperature D!0 the Brownian particle velocity is positive.\nThe above described negative mobility manifests itself only in \fnite interval of\n11−2−1.3−0.60.10.81.5\n0.1 0 .3 0 .5 0 .7 0 .9\nf−2−1.3−0.60.10.81.5\n10−610−410−2100\nD/angbracketleftv/angbracketright(a)\n/angbracketleftv/angbracketright(b)Figure 6: The negative mobility of the strongly damped Brownian particle m\u001c1 induced by\nthermal equilibrium \ructuations. The directed velocity hviis presented versus external static\nbiasfin panel (a) and versus thermal noise intensity Din panel (b). Parameters are the\nsame as in Fig. 5, except now m= 0:1047,f= 0:5 andD= 0:0009.\ntemperature D2(2:8\u000210\u00004;3:2\u000210\u00003). Further increase of thermal noise\nintensity leads to disappearance of this phenomenon. Although a solely noise\ninduced negative mobility can occur only under impact of thermal \ructuations,\nthe underlying relevant mechanism is strongly in\ruenced by the deterministic\ndynamics as it was already shown in Ref. [3].\nFinally, in Fig. 7 we present a case of negative mobility induced by the deter-\nministic non-chaotic dynamics of the system. Panel (a) illustrates the directed\nvelocityhvias a function of the external bias ffor the deterministic case D= 0\nand in the strong damping regime m\u001c1. For very small values of the force\nfthe directed velocity oscillates around zero. When the bias is of moderate\nmagnitude there is a window for which the particle response is opposite to the\napplied constant perturbation, so we detect there negative mobility. Further\nincrease of the external force rapidly reverses the particle current and causes its\nmonotonic growth. In panel (b) we study the impact of thermal \ructuations\nonhviin the parameter regime with f= 0:5 corresponding to the minimal\nplateau depicted in the plot (a). Indeed, for the deterministic limit of the dy-\nnamicsD!0 the directed velocity hviis negative suggesting that the observed\nphenomenon of negative mobility has the deterministic origin. An increase of\nthermal noise intensity ceases this e\u000bect. For this parameter regime there is a\nsurprisingly simple structure of attractors for the directed velocity which is vi-\nsualized in the panel (c) in the deterministic case with D= 0. In the considered\ninterval of the external force fthere are two asymptotically stable solutions cor-\nresponding tohvi=\u00062. Notably, in the bias window where negative mobility\nis observed only the attractor hvi=\u00002 survives. This unexpected simplicity\nof solutions suggests that in the considered parameter regime the deterministic\ndynamics is non-chaotic, nonetheless still exhibits negative mobility. Our \fnd-\ning is con\frmed in panel (d) where we depict the maximal Lyapunov exponent\n\u0015versus the bias ffor the system with D= 0. An interesting observation is\nthat the dynamics is generally chaotic ( \u0015 > 0) when two attractors hvi=\u00062\n12−2−1.2−0.40.41.2\n0.1 0 .3 0 .5 0 .7 0 .9\nf−2−1.2−0.40.4\n10−610−410−2100\nD\n−2−1.2−0.40.41.22\n0.1 0 .3 0 .5 0 .7 0 .9\nf−0.10.20.50.8\n0.1 0 .3 0 .5 0 .7 0 .9\nf/angbracketleftv/angbracketright(a)\n/angbracketleftv/angbracketright(b)\n/angbracketleftv/angbracketright (c) λ (d)Figure 7: The negative mobility of the strongly damped Brownian particle m\u001c1 induced by\nthe deterministic non-chaotic dynamics. Panel (a) the directed velocity hvi, (c) bifurcation\ndiagram of the directed velocity hvi, (d) the maximal Lyapunov exponent \u0015as the function\nof the external static bias fwithD= 0. Panel (b) the directed velocity hviversus thermal\nnoise intensity Dforf= 0:5. Other parameters are the same as in Fig. 5 and m= 0:1.\ncoexist and it is non chaotic with \u0015= 0 in the window where negative mobility\nemerges. To the best of the authors knowledge such a mechanism has never\nbeen reported. It may potentially open a possibility of observation of the nega-\ntive mobility phenomenon for the discontinuous or non-ergodic one-dimensional\nnonequilibrium overdamped dynamics, corresponding to the formal substitution\nm= 0 in Eq. (5) [20].\n4.4. Impact of inertia\nTo conclude this section we present in Fig. 8 the representative dependence\nof the directed velocity hvion the inertia mfor the deterministic and noisy\nsystem. In panel (a) the amplitude aand frequency !of the periodic driving\nare the same as in the previous \fgures 5-7. Here, the most pronounced negative\nmobility is observed for the mass m\u00190:1. For speci\fcally tailored parameter\nsets this phenomenon could be detected for even smaller mass. We exemplify\nthis situation in panel (b) where it is observed for m\u00190:03 which is indeed the\nregime of very strong damping.\nWe observe that the system response is very sensitive to even smallest\nchanges of the particle mass m. Moreover, there are multiple reversals of the\nsign of the directed velocity which are characteristic for a massive setup driven\n13−2−1.2−0.40.41.22\n0.01 0 .1 1\nm−3−1.8−0.60.61.83\n0.01 0 .1 1\nm/angbracketleftv/angbracketright\n/angbracketleftv/angbracketright(a)\n/angbracketleftv/angbracketright(b)Figure 8: The directed velocity hviof the driven Brownian particle versus its inertia m. Panel\n(a):a= 8:55,!= 12:38 andf= 0:5. The blue curve is for the deterministic case D= 0\nand the red curve is for the noisy system with D= 0:0009. Panel (b): a= 9:845,!= 16:64,\nf= 0:25 andD= 10\u00005.\nby the external harmonic force [45, 46]. This \fnding can be utilized to particle\nsorting [47]. For instance, one can see from the above \fgure that particles with\ndi\u000berent masses can easily be guided into opposite direction by a suitable choice\nof the system parameters. In addition, we want to point out that the occur-\nrence of the three presented mechanisms of negative mobility are controlled by\nthe magnitude of the particle inertia. Depending on its value the determinis-\ntic chaotic, thermal noise induced and the deterministic non-chaotic anomalous\ntransport can be observed. It is worth to explicitly note that a tiny change\nfromm= 0:1 tom= 0:1047 transforms the nature of negative mobility e\u000bect\nfrom deterministic non-chaotic to thermal noise induced, c.f. Fig. 6 and Fig. 7.\nTherefore, the outlined mechanisms of the occurrence of negative mobility are\nalso very sensitive to alteration of the particle inertia.\n5. Conclusion\nIn this work we investigated the impact of inertia on transport properties of\na Brownian particle moving in a periodic symmetric structure, which in addition\nis exposed to a harmonic ac driving as well as a constant bias. The parame-\nter space is 5-dimensional and its complete numerical exploration is far beyond\nthe scope of this work. We \frst analysed the general behaviour of the directed\nvelocityhvias a function of the amplitude aand the angular frequency !of\nthe driving, for selected values of the remaining system parameters: the particle\ninertiam, the external bias fand thermal noise intensity D. These results\nreveal especially that the negative mobility phenomenon emerges also for the\nstrongly damped motion of a Brownian particle when the dissipation dominates\nover inertia. Our scans of the parameter space allowed us to determine the\noptimal mass m\u00190:35 for the presence of the negative mobility phenomenon.\nBy observing that the fraction of the negative mobility area in the analysed pa-\nrameter space disappears for both the limiting cases of the overdamped m!0\n14and underdamped m!1 motion, we con\frmed with precise numerics the no-\ngo theorem formulated in Ref. [20]. We gained further insights into the origin\nof negative mobility in this system by revealing three classes of mechanisms\nresponsible for this anomalous transport process. It can (i) be caused by the\ncomplex deterministic chaotic dynamics of the system or (ii) induced by the\nthermal noise, or (iii) associated with the deterministic, yet non-chaotic system\nevolution. In particular, according to the best of authors knowledge, the latter\norigin has never been reported before. It may provide guidance to the possi-\nbility of observation of the negative mobility phenomenon for the discontinuous\nor non-ergodic one-dimensional nonequilibrium overdamped dynamics when the\nparticle inertia is \fxed to zero m= 0. This case is of fundamental importance\nfrom the point of view of biological systems, all of which in situ operate in\nstrongly \ructuating environments. Finally, we depict the illustrative impact of\nthe particle inertia on its transport properties to unravel its spectacular sensi-\ntiveness to variation of this parameter. We detect the phenomenon of multiple\nvelocity reversals which may constitute a cornerstone of particle sorting. More-\nover, a small change in the particle inertia can radically alter the mechanism of\nnegative mobility.\nOur results can readily be experimentally tested with a single Josephson\njunction device or cold atoms moving in an dissipative optical lattice. Suit-\nable parameter values, for which the above e\u000bects are predicted to occur, are\naccessible experimentally.\n6. Acknowledgement\nThe work was supported by the NCN grant 2015/19/B/ST2/02856 (J. S. &\nJ.  L.).\nReferences\n[1] Denisov S, Flach S, H anggi P. Tunable transport with broken space-time\nsymmetries. Phys. Rep. 2014;538:77-120\n[2] H anggi P and Marchesoni F. Arti\fcial Brownian motors: Controlling trans-\nport on the nanoscale. Rev. Mod. Phys. 2009;81:387\n[3] Machura  L et al. Absolute negative mobility induced by thermal equilib-\nrium \ructuations. Phys. Rev. Lett. 2007;98:40601\n[4] Nagel J et al. Observation of negative Absolute Resistance in a Josephson\nJunction. Phys. Rev. Lett. 2008;100:217001\n[5] Kostur M,  Luczka J and H anggi P. Negative mobility induced by colored\nthermal \ructuations. Phys. Rev. E 2009;80:051121\n[6] Spiechowicz J,  Luczka J and H anggi P. Absolute negative mobility induced\nby white Poissonian noise. J. Stat. Mech. 2013;P02044\n15[7] Spiechowicz J, H anggi P and  Luczka J. Brownian motors in the microscale\ndomain: Enhancement of e\u000eciency by noise. Phys. Rev. E 2014;90:032104\n[8] Spiechowicz J,  Luczka J and Machura  L. E\u000eciency of transport in periodic\npotentials: dichotomous noise contra deterministic force. J. Stat. Mech.\n2016;054038\n[9] Hennig D. Current control in a tilted washboard potential via time-delayed\nfeedback. Phys. Rev. E 2009;79:041114\n[10] Du L and Mei D. Time delay control of absolute negative mobility and\nmultiple current reversals in an inertial Brownian motor. J. Stat. Mech.\n2011;P11016\n[11] Mulhern C. Persistence of uphill anomalous transport in inhomogeneous\nmedia. Phys. Rev. E 2013;88:022906\n[12] Dandogbessi B, Kenfack A. Absolute negative mobility induced by potential\nphase modulation. Phys. Rev. E 2015;92:062903\n[13] Du L and Mei D. Absolute negative mobility in a vibrational motor. Phys.\nRev. E 2012;85:011148\n[14] Januszewski M and  Luczka J. Indirect control of transport and interaction-\ninduced negative mobility in an overdamped system of two coupled parti-\ncles. Phys. Rev. E 2011;83:051117\n[15] Machura  L et al. Two coupled Josephson junctions: dc voltage controlled\nby biharmonic current. J. Phys. Condens. Matt. 2012;24:085702\n[16] Ghosh PK, H anggi P, Marchesoni F and Nori F. Giant negative mobility\nof Janus particles in a corrugated channel. Phys. Rev. E 2014;89:062115\n[17] Malgaretti P, Pagonabarraga I and Rubi JM. Entropic Electrokinetics: Re-\ncirculation, Particle Separation, and Negative Mobility. Phys. Rev. Lett.\n2014;113:128301\n[18] Sarracino A, Cecconi F, Puglisi A and Vulpiani A. Nonlinear Response\nof Inertial Tracers in Steady Laminar Flows: Di\u000berential and Absolute\nNegative Mobility. Phys. Rev. Lett. 2016;117:174501\n[19] Bresslo\u000b PC and Newby JM. Stochastic models of intracellular transport.\nRev. Mod. Phys. 2013;85:135\n[20] Speer D, Eichhorn R and Reimann P. Transient chaos induces anomalous\ntransport properties of an underdamped Brownian particle. Phys. Rev. E\n2007;76:051110\n[21] Spiechowicz J, Kostur M and  Luczka J. Comment on \"Absolute nega-\ntive mobility in a one-dimensional overdamped system\". Phys. Lett. A\n2016;380:1499\n16[22] Risken H. The Fokker-Planck Equation. Berlin: Springer-Verlag; 1984\n[23]  Luczka J, Bartussek R and H anggi P. White noise induced transport in\nperiodic structures. Europhys. Lett. 1995;31:431-436\n[24] Spiechowicz J and  Luczka J. Poissonian noise assisted transport in periodic\nsystems. 2015;165:014015\n[25] Spiechowicz J, H anggi P and  Luczka J. Josephson junction ratchet: The\nimpact of \fnite capacitances. Phys. Rev. B 2014;90:054520\n[26] Spiechowicz J and  Luczka J. E\u000eciency of a SQUID ratchet driven by ex-\nternal current. New J. Phys. 2015;17:023504\n[27] Spiechowicz J and  Luczka J. Josephson phase di\u000busion in the SQUID\nratchet. Chaos 2015;25:053110\n[28] Spiechowicz J and  Luczka J. Di\u000busion anomalies in ac-driven Brownian\nratchets. Phys. Rev. E 2015;91:062104\n[29] Spiechowicz J,  Luczka J and H anggi P. Transient anomalous di\u000busion in\nperiodic systems: ergodicity, symmetry breaking and velocity relaxation.\nSci. Rep. 2016;6:30948\n[30] Spiechowicz J, Talkner P, H anggi P and  Luczka J. Non-monotonic tem-\nperature dependence of chaos-assisted di\u000busion in driven periodic systems.\nNew J. Phys. 2016;18:123029\n[31] Spiechowicz J, Kostur M and  Luczka J. Brownian ratchets: How stronger\nthermal noise can reduce di\u000busion. Chaos 2017;27:023111\n[32] Marconi UMB, Puglisi A, Rondoni L and Vulpiani A. Fluctuationdissipa-\ntion: response theory in statistical physics. Phys. Rep. 2008;461:111-195\n[33] Blackburn JA, Cirillo M and Gronbech-Jensen N. A survey of classical and\nquantum interpretations of experiments on Josephson junctions at very low\ntemperatures. Phys. Rep. 2016;611:1\n[34] Lutz E and Renzoni F. Beyond Boltzmann-Gibbs statistical mechanics in\noptical lattices. Nat. Phys. 2013;9:615\n[35] Fulde P et al. Problem of Brownian motion in a periodic potential. Phys.\nRev. Lett. 1975;35:1776\n[36] Co\u000bey W, Kalmykov Y and Waldron J. The Langevin equation. Singapore:\nWorld Scienti\fc; 2004\n[37] Gruner G, Zawadowski A and Chaikin P. Nonlinear conductivity and\nnoise due to charge-density-wave depinning in NbSe 3. Phys. Rev. Lett.\n1981;46:511\n17[38] Guantes R, Vega J and Miret-Artes S. Chaos and anomalous di\u000busion of\nadatoms on solid surfaces. Phys. Rev. B 2001;64:245415\n[39] Machura  L, Kostur M and  Luczka J. Transport Characteristics of Molecular\nMotors. Biosystems 2008;05:033\n[40] Jung P. Periodically driven stochastic systems. Phys. Rep. 1993;234:175\n[41] Baker GL and Gollub JP. Chaotic Dynamics: An Introduction. Cambridge:\nCambridge University Press; 1996\n[42] Bo\u000betta G, Cencini M, Falcioni M and Vulpiani A. Predictability: a way\nto characterize complexity. Phys. Rep. 2002;356:367\n[43] Spiechowicz J, Kostur M and Machura  L. GPU accelerated Monte Carlo\nsimulation of Brownian motors dynamics with CUDA. Comp. Phys. Com-\nmun. 2015;191:140\n[44] Chi CC and Vanneste C. Onset of chaos and dc current-voltage character-\nistics of rf-driven Josephson junctions in the low-frequency regime. Phys.\nRev. B 1990;42:9875\n[45] Bartussek R, H anggi P and Kissner JG. Periodically rocked thermal ratch-\nets. Europhys. Lett. 1994;28:459\n[46] Jung P, Kissner JG and H anggi P. Regular and chaotic transport in asym-\nmetric periodic potentials: inertia ratchets. Phys. Rev. Lett. 1996;76:3436\n[47] Eichhorn R, Regtmeier J, Anselmetti D and Reimann P. Negative mobility\nand sorting of colloidal particles. Soft Matter 2010;6:1858\n18"    },    {        "title": "1705.07489v2.Dynamical_depinning_of_chiral_domain_walls.pdf",        "content": "Dynamical depinning of chiral domain walls\nSimone Moretti,\u0003Michele Voto, and Eduardo Martinez\nDepartment of Applied Physics, University of Salamanca, Plaza de los Caidos, Salamanca 37008, Spain.\nThe domain wall depinning \feld represents the minimum magnetic \feld needed to move a domain\nwall, typically pinned by samples' disorder or patterned constrictions. Conventionally, such \feld\nis considered independent on the Gilbert damping since it is assumed to be the \feld at which the\nZeeman energy equals the pinning energy barrier (both damping independent). Here, we analyse\nnumerically the domain wall depinning \feld as function of the Gilbert damping in a system with per-\npendicular magnetic anisotropy and Dzyaloshinskii-Moriya interaction. Contrary to expectations,\nwe \fnd that the depinning \feld depends on the Gilbert damping and that it strongly decreases for\nsmall damping parameters. We explain this dependence with a simple one-dimensional model and\nwe show that the reduction of the depinning \feld is related to the \fnite size of the pinning barriers\nand to the domain wall internal dynamics, connected to the Dzyaloshinskii-Moriya interaction and\nthe shape anisotropy.\nI. INTRODUCTION\nMagnetic domain wall (DW) motion along ferromag-\nnetic (FM) nanostructures has been the subject of in-\ntense research over the last decade owing to its po-\ntential for new promising technological applications1,2\nand for the very rich physics involved. A consider-\nable e\u000bort is now focused on DW dynamics in systems\nwith perpendicular magnetic anisotropy (PMA) which\npresent narrower DWs and a better scalability. Typ-\nical PMA systems consist of ultrathin multi-layers of\nheavy metal/FM/metal oxide (or heavy metal), such as\nPt=Co=Pt3,4or Pt=Co=AlOx5{7, where the FM layer has\na thickness of typically 0 :6\u00001 nm. In these systems,\nPMA arises mainly from interfacial interactions between\nthe FM layer and the neighbouring layers (see Ref.8and\nreferences therein). Another important interfacial ef-\nfect is the Dzyaloshinskii-Moriya interaction (DMI)9,10,\npresent in systems with broken inversion symmetry such\nas Pt/Co/AlOx. This e\u000bect gives rise to an internal in-\nplane \feld that \fxes the DW chirality (the magnetization\nrotates always in the same direction when passing from\nup to down and from down to up domains) and it can\nlead to a considerably faster domain wall motion10and to\nnew magnetic patterns such as skyrmions11or helices12.\nNormally, DWs are pinned by samples' intrinsic disorder\nand a minimum propagation \feld is needed in order to\novercome such pinning energy barrier and move the DW.\nSuch \feld is the DW depinning \feld ( Hdep) and it repre-\nsents an important parameter from a technological point\nof view since a low depinning \feld implies less energy\nrequired to move the DW and, therefore, a energetically\ncheaper device.\nFrom a theoretical point of view, DW motion can be\ndescribed by the Landau-Lifshitz-Gilbert (LLG) equa-\ntion13which predicts, for a perfect sample without dis-\norder, the velocity vs\feld curve depicted in Fig. 1 and\nlabelled as Perfect . In a disordered system, experi-\nments have shown that a DW moves as a general one-\ndimensional (1D) elastic interface in a two-dimensional\ndisordered medium3,4and that it follows a theoreticalvelocityvsdriving force curve, predicted for such inter-\nfaces14,15(also shown in Fig. 1 for T= 0 andT= 300K).\nMoreover, this behaviour can be reproduced by including\ndisorder in the LLG equation16{18. At zero temperature\n(T= 0) the DW does not move as long as the applied\n\feld is lower than Hdep, while, at T6= 0, thermal ac-\ntivation leads to DW motion even if H < H dep(the so\ncalled creep regime). For high \felds ( H >> H dep) the\nDW moves as predicted by the LLG equation in a per-\nfect system. Within the creep theory, the DW is con-\nsidered as a simple elastic interface and all its internal\ndynamics are neglected. Conventionally, Hdepis consid-\nered independent of the Gilbert damping because it is as-\nsumed to be the \feld at which the Zeeman energy equals\nthe pinning energy barrier19,20(both damping indepen-\ndent). Such assumption, consistently with the creep the-\nory, neglects any e\u000bects related to the internal DW dy-\nnamics such as DW spins precession or vertical Bloch\nlines (VBL) formation21. The damping parameter, for\nits part, represents another important parameter, which\ncontrols the energy dissipation and a\u000bects the DW veloc-\nity and Walker Breakdown22. It can be modi\fed by dop-\ning the sample23or by a proper interface choice as a con-\nsequence of spin-pumping mechanism24. Modi\fcations of\nthe DW depinning \feld related to changes in the damping\nparameter were already observed in in-plane systems23,25\nand attributed to a non-rigid DW motion23,25. Oscilla-\ntions of the DW depinning \feld due to the internal DW\ndynamics were also experimentally observed in in-plane\nsimilar systems26. Additional dynamical e\u000bects in soft\nsamples, such as DW boosts in current induced motion,\nwere numerically predicted and explained in terms of DW\ninternal dynamics and DW transformations27,28.\nHere, we numerically analyse the DW depinning \feld\nin a system with PMA and DMI as function of the Gilbert\ndamping. We observe a reduction of Hdepfor low damp-\ning and we explain this behaviour by adopting a simple\n1D model. We show that the e\u000bect is due to the \fnite\nsize of pinning barriers and to the DW internal dynam-\nics, related to the DMI and shape anisotropy \felds. This\narticle is structured as follows: in Section II we present\nthe simulations method, the disorder implementation andarXiv:1705.07489v2  [cond-mat.mes-hall]  25 Aug 20172\ntheHdepcalculations. The main results are outlined and\ndiscussed in Section III, where we also present the 1D\nmodel. Finally, the main conclusions of our work are\nsummarized in Section IV.\n●●●●\n●\n●\n●\n●●●●●●●●●●�=��\n�=����\n●�������\n������� ������� ��������\n����★\nFIG. 1. DW velocity vsapplied \feld as predicted by the LLG\nequation in a perfect system and by the creep law atT= 0\nandT= 300K.\nII. MICROMAGNETIC SIMULATIONS\nWe consider a sample of dimensions\n(1024\u00021024\u00020:6) nm3with periodic bound-\nary conditions along the ydirection, in order to simulate\nan extended thin \flm. Magnetization dynamics is\nanalysed by means of the LLG equation13:\ndm\ndt=\u0000\r0\n1 +\u000b2(m\u0002He\u000b)\u0000\r0\u000b\n1 +\u000b2[m\u0002(m\u0002He\u000b)];\n(1)\nwhere m(r;t) =M(r;t)=Msis the normalized magneti-\nzation vector, with Msbeing the saturation magnetiza-\ntion.\r0is the gyromagnetic ratio and \u000bis the Gilbert\ndamping. He\u000b=Hexch+HDMI+Han+Hdmg+Hz^uz\nis the e\u000bective \feld, including the exchange, DMI, uni-\naxial anisotropy, demagnetizing and external \feld con-\ntributions13respectively. Typical PMA samples param-\neters are considered: A= 17\u000210\u000012J=m,Ms= 1:03\u0002\n106A=m,Ku= 1:3\u0002106J=m3andD= 0:9 mJ=m2,\nwhereAis the exchange constant, Dis the DMI constant\nandKuis the uniaxial anisotropy constant. Disorder is\ntaken into account by dividing the sample into grains\nby Voronoi tessellation29,30, as shown in Fig. 2(a). In\neach grain the micromagnetic parameters fMs;Dc;Kug\nchange in a correlated way in order to mimic a normally\ndistributed thickness31:\ntG=N(t0;\u000e)!8\n<\n:MG= (MstG)=t0\nKG= (Kut0)=tG\nDG= (Dct0)=tG; (2)\nwhere the subscript Gstands for grain, t0is the aver-\nage thickness ( t0= 0:6nm) and\u000eis the standard devi-\nation of the thickness normal distribution. The sample\nis discretized in cells of dimensions (2 \u00022\u00020:6)nm3,smaller than the exchange length lex\u00185nm. Grain size\nis GS=15 nm, reasonable for these materials, while the\nthickness \ructuation is \u000e= 7%. Eq. (1) is solved by the\n\fnite di\u000berence solver MuMax 3.9.329.\nA DW is placed and relaxed at the center of the sample\nas depicted in Fig. 2(b). Hdepis calculated by applying\na sequence of \felds and running the simulation, for each\n\feld, until the DW is expelled from the sample, or until\nthe system has reached an equilibrium state (i.e. the DW\nremains pinned): \u001cmax<\u000f(\u000b).\u001cmaxindicates the maxi-\nmum torque, which rapidly decreases when the system is\nat equilibrium. It only depends on the system parame-\nters and damping. For each value of \u000b, we choose a spe-\nci\fc threshold, \u000f(\u000b), in order to be sure that we reached\nan equilibrium state (see Supplementary Material32for\nmore details). The simulations are repeated for 20 dif-\nferent disorder realizations. Within this approach, Hdep\ncorresponds to the minimum \feld needed to let the DW\npropagate freely through the whole sample. In order to\navoid boundaries e\u000bects, the threshold for complete de-\npinning is set tohmzi>0:8, wherehmziis averaged over\nall the realizations, i.e. hmzi=PN\ni=1hmzii=N, where\nN= 20 is the number of realizations. We checked that,\nin our case, this de\fnition of Hdepcoincides with tak-\ningHdep= MaxfHi\ndepg, withHi\ndepbeing the depinning\n\feld of the single realization. In other words, Hdepcor-\nresponds to the minimum \feld needed to depin the DW\nfrom any possible pinning site considered in the 20 real-\nizations33.\nFollowing this strategy, the DW depinning \feld is nu-\nmerically computed with two di\u000berent approaches:\n(1) by Static simulations, which neglect any precessional\ndynamics by solving\ndm\ndt=\u0000\r0\u000b\n1 +\u000b2[m\u0002(m\u0002He\u000b)]: (3)\nThis is commonly done when one looks for a minimum\nof the system energy and it corresponds to the picture\nin whichHdepsimply depends on the balance between\nZeeman and pinning energies.34\n(2) by Dynamic simulations, which include precessional\ndynamics by solving the full Eq. (1). This latter method\ncorresponds to the most realistic case. Another way to\nestimate the depinning \feld is to calculate the DW veloc-\nityvs\feld curve at T= 0 and look for minimum \feld at\nwhich the DW velocity is di\u000berent from zero. For these\nsimulations we use a moving computational region and\nwe run the simulations for t= 80ns (checking that longer\nsimulations do not change the DW velocity, meaning that\nwe reached a stationary state). This second setup re-\nquires more time and the calculations are repeated for\nonly 3 disorder realizations.\nUsing these methods, the depinning \feld Hdepis cal-\nculated for di\u000berent damping parameters \u000b.3\n(a) (b)\nxy\n(c)\nFIG. 2. (a) Grains structure obtained by Voronoi tassellation.\n(b) Initial DW state. (c) Sketch of the internal DW angle \u001e.\nIII. RESULTS AND DISCUSSION\nA. Granular system\nOur \frst result is shown in Fig. 3(a)-(b), which depicts\nthe \fnal average magnetization hmzias function of the\napplied \feld for di\u000berent damping parameters. In the\nStatic simulations (Fig. 3(a)) Hdepdoes not depend on\ndamping, so that a static depinning \feld can be de\fned.\nConversely, in the Dynamic simulations (Fig. 3(b)), Hdep\ndecreases for low damping parameters. The depinning\n\feld is indicated by a star in each plot and the static\ndepinning \feld is labelled as Hs. The same result is ob-\ntained by calculating Hdepfrom the DW velocity vsap-\nplied \feld plot, shown in Fig. 3(c). The stars in Fig. 3(c)\ncorrespond to the depinning \felds calculated in the pre-\nvious simulations and they are in good agreement with\nthe values predicted by the velocity vs\feld curve. The\ndynamical depinning \feld \u00160Hd, normalized to the static\ndepinning \feld \u00160Hs= (87\u00061)mT, with \u00160being the\nvacuum permeability, is shown in Fig. 3(d) as function of\nthe damping parameter \u000b.Hdsaturates for high damp-\ning (in this case \u000b\u00150:5) while it decreases for low damp-\ning untilHd=Hs\u00180:4 at\u000b= 0:02. This reduction must\nbe related to the precessional term, neglected in the static\nsimulations. The same behaviour is observed with di\u000ber-\nent grain sizes (GS=5 and 30 nm) and with a di\u000berent\ndisorder model, consisting of a simple variation of the Ku\nmodule in di\u000berent grains. This means that the e\u000bect is\nnot related to the grains size or to the particular disorder\nmodel we used.\nAdditionally, Fig. 4 represents the DW energy35as\nfunction of DW position and damping parameter for\n\u00160Hz= 70 mT. At high damping, the average DW en-\nergy density converges to \u001b1\u001810 mJ=m2, in good agree-\nment with the analytical value \u001b0= 4pAK0\u0000\u0019D=\n10:4 mJ=m2, whereK0is the e\u000bective anisotropy K0=\nKu\u0000\u00160M2\ns=2. On the contrary, for low damping, the\nDW energy increases up to \u001b(0:02)\u001814 mJ=m2. This\nincrease, related to DW precessional dynamics, reduces\nthe e\u000bective energy barrier and helps the DW to over-\n●●●●●●●●●●●●●●●●●\n○○○○○○○○○○○○○○○○○\n■■■■■■■■■■■■■■■■■\n●α=����\n○α=���\n■α=���\n���������������������<��> ★(�) ������\n��\n●●●●●●●●●●●●●●●●●\n○○○○○○○○○○○○○○○○○\n■■■■■■■■■■■■■■■■■\n�� �� �� ��������������������������\n������� �����(��)<��>★★★(�) �������\n●●●●●●●●●○○○○○○○○○■■■■■■■■■●α=����\n○α=���\n■α=���\n�� �� �� ��������������\n������� �����(��)�� ��������(�/�)★★★(�)\n●●●●● ● ●\n������������������������������������\n�������α��/��(�)FIG. 3. Average hmzias function of applied \feld for dif-\nferent damping parameters for the (a) Static simulations and\n(b)Dynamic simulations. (c) DW velocity vs applied \feld for\ndi\u000berent damping. (d) Dynamical depinning \feld, normalized\ntoHs, as function of damping.\ncome the pinning barriers. Fig. 4(c) shows the total en-\nergy of the system (including Zeeman). As expected36,\nthe energy decreases as the DW moves.\nFinally, Fig. 5 shows the DW motion as function of\ntime for\u000b= 0:02 and\u000b= 0:5, along the same grain\npattern (and therefore along the same pinning barriers).\nThe applied \feld is \u00160Hz= 70mT, which satis\fes\nHd(0:02)<Hz<Hd(0:5). The initial DW con\fguration\nis the same but, for \u000b= 0:02, VBL start to nucleate and\nthe DW motion is much more turbulent (see Supplemen-\ntary Material32for a movie of this process). At t= 4 ns\nthe DW has reached an equilibrium position for \u000b= 0:5,\nwhile it has passed through the (same) pinning barriers\nfor\u000b= 0:02. Thus, one might think that the reduction\nof the depinning \feld could be related to the presence4\n●●●●●●●● ●●●●●●●●●\n●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●\n●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●\n●\n●\n●\n●○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○\n○○○○○○○○○○○■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■■ ■ ■ ■ ■ ■ ■ ■\n●α=����\n○α=���\n■α=���\n��������������������������\n�� ��������(μ�)σ��(��/��)(�)\n●\n●●\n�����������������������������\n�������α<σ��> (��/��)(�)\n●●●●●●●●●●●●\n●●●●●●●●●●●●○○○○○○○○○○○○○○○○○○○○○○○■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■■ ■ ■ ■ ■ ■ ■ ■ ■ ■\n���������������-��-�������\n�� ��������(��)����� ������ �������(��/��)\n(�)\nFIG. 4. (a) DW energy density as function of DW posi-\ntion for di\u000berent damping. The \fnal drop corresponds to\nthe expulsion of the DW. (b) Average DW density as funci-\nton of damping. Dashed line represents the analytical value\n\u001b1\u001810 mJ=m2. (c) Total energy density of the system as\nfunction of DW position for di\u000berent damping parameters.\nof VBL and their complex dynamics21. Further insights\nabout this mechanism are given by analysing the DW\ndepinning at a single energy barrier as described in the\nnext subsection.\nB. Single barrier\nIn order to understand how the DW precessional dy-\nnamics reduces Hdep, we micromagnetically analysed the\nDW depinning from a single barrier as sketched in Fig. 6.\nWe considered a strip of dimensions (1024 \u0002256\u00020:6)nm3\nand we divided the strip into two regions, R1andR2,\nwhich are assumed to have a thickness of t1= 0:58 and\nt2= 0:62 nm respectively. Their parameters vary ac-\ncordingly (see Sec. II), generating the DW energy bar-\nrier (\u000e\u001b) shown in Fig. 6(b). A DW is placed and re-\nlaxed just before the barrier. The \fnite size of the DW\n(\u0019\u0001DW\u001815 nm, with \u0001 DWbeing the DW width pa-\nrameter) smooths the abrupt energy step and, in fact,\nthe energy pro\fle can be successfully \ftted by using theBloch pro\fle22\n\u001bDW=\u001b0+\n+\u0012\u000e\u001b\n2\u0013\u001a\n1 + cos\u0012\n2 arctan\u0014\nexp\u0012x0\u0000x\n\u0001DW\u0013\u0015\u0013\u001b\n;\n(4)\nwherex0= 20 nm is the step position, while \u001b0and\n\u001b1are the DW energies at the left and right side of the\nbarrier as represented in Fig. 6(b). This means that\nthe pinning energy barrier has a spatial extension which\nis comparable to the DW width. By performing the\nsame static and dynamic simulations, we obtain a static\ndepinning \feld of \u00160Hs= 120 mT and, when decreasing\nthe damping parameter, we observe the same reduction\nof the depinning \feld as in the granular system (see\nFig. 6(c)). In this case the DW behaves like a rigid\nobject whose spins precess coherently and no VBL\nnucleation is observed. Hence, Hdepreduction does not\ndepend directly on the presence of VBL but on the more\ngeneral mechanism of spins' precession already present\nin this simpli\fed case.\nNevertheless, an important characteristic of these single\nbarrier simulations is that the barrier is localized and it\nhas a \fnite size which is of the order of the DW width.\nNote that the same holds for the granular system:\ndespite a more complex barrier structure, the dimension\nof the single barrier between two grains has the size of\nthe DW width.\nThus, in order to understand the interplay between the\nDW precessional dynamics and the \fnite size of the bar-\nrier, we considered a 1D collective-coordinate model with\na localized barrier. The 1D model equations, describing\nthe dynamics of the DW position qand the internal angle\n\u001e(sketched in Fig. 2(c)), are given by16\n(1 +\u000b2)_\u001e=\r0[(Hz+Hp(q))\n\u0000\u000b\u0012\nHKsin 2\u001e\n2\u0000\u0019\n2HDMIsin\u001e\u0013\n|{z }\nHint(\u001e)];(5)\n(1 +\u000b2)_q\n\u0001DW=\r0[\u000b(Hz+Hp(q))\n+\u0012\nHKsin 2\u001e\n2\u0000\u0019\n2HDMIsin\u001e\u0013\u0015\n;(6)\nwhereHK=MsNxis the shape anisotropy \feld, favour-\ning Bloch walls, with Nx=t0log 2=(\u0019\u0001DW)37being the\nDW demagnetizing factor along the xaxis.HDMI =\nD=(\u00160Ms\u0001DW) is the DMI \feld. Hint(\u001e) represents\nthe internal DW \feld, which includes DMI and shape\nanisotropy. Hintfavours Bloch ( \u001e=\u0006\u0019=2) or N\u0013 eel wall\n(\u001e= 0 or\u001e=\u0019) depending on the relative strength\nofHKandHDMI. In our system, the DMI dominates\nover shape anisotropy since \u00160HDMI\u0018170 mT while\n\u00160HK\u001830 mT. Hence, the DW equilibrium angle is5\nOut[64]=\nOut[60]=\n… (a) 𝜶=𝟎.𝟎𝟐time 0 0.1 ns 0.2 ns 0.3 ns 4 ns\ntime 0 0.1 ns 0.2 ns 0.3 ns 4 ns(b) 𝜶=𝟎.𝟓\n… \nOut[395]=mx\nOut[395]=mx\nOut[62]=\nOut[65]=\nFIG. 5. (a) Snapshots of the magnetization dynamics at subsequent instants under \u00160Hz= 70mT, for two di\u000berent damping:\n(a)\u000b= 0:02 and (b) \u000b= 0:5. The grains pattern, and therefore the energy barrier, is the same for both cases. In order to let\nthe DW move across more pinning sites, these simulations were performed on a larger sample with Lx= 2048 nm.\n\u001e=\u0019(\u001e= 0 or\u001e=\u0019additionally depends on the sign\nof the DMI). Hp(q) is the DW pinning \feld, obtained\nfrom the DW energy pro\fle (Eq. (4)) as follows: the max-\nimum pinning \feld is taken from the static simulations\nwhile the shape of the barrier is taken as the normalized\nDW energy gradient (see Supplementary Material32for\nmore details),\nHp(q) =Hs\u0012@\u001bDW(x)\n@x\u0013\nN=\n= 2Hsexp\u0010\nx0\u0000q\n\u0001DW\u0011\nsinh\n2 arctan\u0010\nexp\u0010\nx0\u0000q\n\u0001DW\u0011\u0011i\n1 + exp\u0010\n2(x0\u0000q)\n\u0001DW\u0011 :(7)\nThe corresponding pinning \feld is plotted in Fig. 7(a).38\nThe results for the dynamical Hdep, obtained with this\nmodi\fed 1D model, are plotted in Fig. 6(c) and they\nshow a remarkable agreement with the single barrier mi-\ncromagnetic simulations. This indicates that the main\nfactors responsible for the reduction of Hdepare already\nincluded in this simple 1D model. Therefore, additional\ninsights might come from analysing the DW dynamics\nwithin this 1D model. Fig. 7(b) and (c) represents the\nDW internal angle \u001eand the DW position qas function\nof time for di\u000berent damping. The plots are calculated\nwith\u00160Hz= 55 mT which satis\fes Hdep(0:02)< Hz<\nHdep(0:1)< H dep(0:5). As shown in Fig. 7(b) and (c),\nbelow the depinning \feld ( \u000b= 0:1,\u000b= 0:5), both the\ninternal angle and the DW position oscillate before reach-\ning the same \fnal equilibrium state. However, the am-plitude of these oscillations (the maximum displacement)\ndepends on the damping parameter. Fig. 7(d) shows the\n\fnal equilibrium position as function of the applied \feld\nfor di\u000berent damping. The equilibrium position is the\nsame for all damping and it coincides with the position\nat whichHz=Hp(q). Conversely, the maximum dis-\nplacement, shown in Fig. 7(e), strongly increases for low\ndamping parameters. For applied \feld slightly smaller\nthan the depinning \feld, the DW reaches the boundary\nof the pinning barrier, meaning that a further increase\nof the \feld is enough to have a maximum displacement\nhigher than the barrier size and depin the DW. In other\nwords, the decrease of the depinning \feld, observed in\nthe single barrier simulations, is due to DW oscillations\nthat depend on \u000band that can be larger than the bar-\nrier size, leading to DW depinning for lower \feld. The\nDW dynamics and the depinning mechanism are further\nclari\fed in Fig. 7(f) and Fig. 7(g). Fig. 7(f) represents\nthe DW coordinates fq;\u001egfor\u00160Hz= 55 mT and dif-\nferent damping. Before reaching the common equilib-\nrium state, the DW moves in orbits (in the fq;\u001egspace)\nwhose radius depends on the damping parameter. For\n\u000b= 0:5 (black line) the DW rapidly collapse into the \f-\nnal equilibrium state. Conversely, for \u000b= 0:1 (red open\ncircles), the DW orbits around the equilibrium state be-\nfore reaching it. If the radius of the orbit is larger than\nthe barrier size the DW gets depinned, as in the case\nof\u000b= 0:02 (blue full circles). This mechanism is also\nrepresented in Fig. 7(g), where the DW orbits are placed\nin the energy landscape. The energy is calculated as6\n●●●●●●●●●●●●●●●●●μ� ����������� ��●�� ��������������������������������������������\n�������α��/��○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○�� �������������(�)-���������������������������������������(��)σ��(��/��)��δσσ�σ�(b)\n(c)\nR1R2yx(a)\nFIG. 6. (a) Sketch of the two regions implemented for the\nsingle barrier (SB) micromagnetic simulations. (b) DW en-\nergy as function of DW position along the strip. Blue solid\nline represents the analytical value, red points the DW con-\nvoluted energy (due to the \fnite size of the DW) while black\ndashed line a \ft using Eq. 4. (c) Dynamical depinning \feld,\nnormalized to the static depinning \feld, for the single bar-\nrier simulations as function of damping, obtained from full\nmicromagnetic simulations and the 1D model.\n\u001b(q;\u001e) =\u001bDW(q;\u001e)\u00002\u00160MsHzq, where\u001bDWis given by\nEq. (4). Fig. 7(g) shows that the equilibrium state cor-\nresponds to the new minimum of the energy landscape.\nFurthermore, it con\frms that the applied \feld is below\nthe static depinning \feld, at which the pinning barrier\nwould have been completely lifted. Nevertheless, while\nreaching the equilibrium state, the DW moves inside the\nenergy potential and, if the radius of the orbit is larger\nthan the barrier size, the DW can overcome the pinning\nbarrier, as shown for \u000b= 0:02 in Fig. 7(g).\nAt this point we need to understand why the amplitude\nof the DW oscillations depends on damping. By solving\nEq. (5) and Eq.(6) for the equilibrium state ( _ q= 0, _\u001e=\n0) we obtain\n_q= 0)jHp(q)j=Hz+Hint(\u001e)\n\u000b\n\u0019Hz\u0000\u0019\n2HDMI\n\u000bsin\u001e; (8)\n_\u001e= 0)jHp(q)j=Hz\u0000\u000bHint(\u001e)\n\u0019Hz+\u000b\u0019\n2HDMIsin\u001e; (9)\nsince\u00160HDMI\u001d\u00160HKand, therefore, Hint\u0019\n\u0000(\u0019=2)HDMIsin\u001e. These equations have a single com-\nmon solution which corresponds to jHp(q)j=Hzand\n\u001e=\u001e0=\u0019(at whichHint(\u0019) = 0). However, at t= 0,the DW starts precessing under the e\u000bect of the applied\n\feld and, if \u001e6=\u0019whenjHp(q)j=Hz, the DW does not\nstop at the \fnal equilibrium position but it continues its\nmotion, as imposed by Eq. (8) and (9). In other words,\nthe DW oscillations in Fig. 7(b) are given by oscillations\nof the DW internal angle \u001e, around its equilibrium value\n\u001e0=\u0019. These oscillations lead to a modi\fcation of the\nDW equilibrium position due to the DW internal \feld\n(Hint(\u001e)), which exerts an additional torque on the DW\nin order to restore the equilibrium angle. As previously\ncommented, if the amplitude of these oscillations is large\nenough, the DW gets depinned. From Eq. (8) we see\nthat the new equilibrium position (and therefore the am-\nplitude of the oscillations) depends on the DMI \feld, the\nvalue of the DW angle \u001eand the damping parameter.\nIn particular, damping has a twofold in\ruence on this\ndynamics: one the one hand, it appears directly in\nEq. (8), dividing the internal \feld, meaning that for the\nsame deviation of \u001efrom equilibrium, we have a stronger\ninternal \feld for smaller damping. On the other hand,\nthe second in\ruence of damping is on the DW internal\nangle: once the DW angle has deviated from equilibrium,\nthe restoring torque due to DMI is proportional to the\ndamping parameter (see Eq. (9)). Hence, a lower damp-\ning leads to lower restoring torque and a larger deviation\nof\u001efrom equilibrium. The maximum deviation of \u001efrom\nequilibrium ( \u000e\u001e=\u001emax\u0000\u001e0) is plotted in Fig. 8(b) as\nfunction of damping for \u00160Hz= 40 mT. As expected, a\nlower damping leads to a larger deviation \u000e\u001e.\nIn this latter section, the DW was set at rest close to\nthe barrier and, therefore, the initial DW velocity is zero.\nNevertheless, one might wonder what happens when the\nDW reaches the barrier with a \fnite velocity. We simu-\nlated this case by placing the DW at an initial distance\nd1= 200 nm from the barrier. The depinning is further\nreduced in this case (see Supplementary Material32for\nmore details). However, in the static simulations, the de-\npinning \feld remains constant, independently from the\nvelocity at which the DW reaches the barrier, meaning\nthat the reduction of Hdepis again related to the DW\nprecession. When the DW starts from d1it reaches the\nbarrier precessing, thus with a higher deviation from its\nequilibrium angle, leading to a higher e\u000bect of the inter-\nnal \feld.\nC. Di\u000berent DMI and pinning barriers\nFinally, by using the 1D model it is possible to ex-\nplore the dependence of Hdepon the pinning potential\namplitudeHs(related to the disorder strength) and on\nthe DMI constant D. The depinning \feld as function of\ndamping for di\u000berent values of Hsis plotted in Fig. 9(a).\nThe reduction of Hdepis enhanced for larger values of\nHs(strong disorder). This is consistent with our expla-\nnation, since for strong disorder we need to apply larger\n\felds that lead to larger oscillations of \u001e.\nFig. 9(b) represents the dynamical Hdepas function of7\n●●●●●●●●●●●●●●●●●●●●●●●●\n○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○\n▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼�����������������������������������(��)�� ��������(��)●●●●●●●●■■■■■■■■■■■◆◆◆◆◆◆◆◆◆◆◆◆◆▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼○○○○○○○○○○○○○○○○○○○○○○●α=����■α=���◆α=���▲α=����▼α=���○α=�������������������������������������������������\n������� �����μ���(��)���������������(��)������������(��)●●●●●●●■■■■■■■■■◆◆◆◆◆◆◆◆◆◆◆◆▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼○○○○○○○○○○○○○○○○○○○○○●α=����■α=���◆α=���▲α=����▼α=���○α=�������������������������������������������������\n������� �����μ���(��)�����������(��)������������(��)-�������������-���-��-�������������(��)μ���(��)\nMax Displacement\nEq. Position(a)(b)(c)(d)\n(e)(f)\n(g)●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●\n○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○ 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▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼●α=����○α=���α=������������������������������������\n�(��)ϕ(°)\nFIG. 7. (a) Pinning \feld obtained from Eq. (7) as function of DW position. DW position internal angle \u001eas function of\ntime for di\u000berent damping parameter and \u00160Hz= 55 mT. (c) DW position qas function of time for di\u000berent damping and\n\u00160Hz= 55 mT. (d) Equilibrium position as function of applied \feld for di\u000berent damping. (e) Maximum DW displacement as\nfunction of the applied \feld for di\u000berent damping. (f) DW coordinates fq;\u001egfor\u00160Hz= 55 mT and di\u000berent damping. (g)\nDW coordinatesfq;\u001eginside the energy landscape: \u001b=\u001bDW(q;\u001e)\u00002\u00160MsHzq.\n������������������������������\n�������αδϕ(°)\nFIG. 8. Maximum deviation of \u001efrom its equilibrium posi-\ntion as function of damping.\ndamping for \u00160Hs= 120 mT and di\u000berent DMI con-\nstants (expressed in term of the critical DMI constant\nDc= 4pAK0=\u0019= 3:9 mJ=m2)39. In this case, the reduc-\ntion ofHdepis enhanced for low DMI, until D= 0:05Dc,\nbut a negligible reduction is observed for D= 0. This\nnon-monotonic behaviour can be explained by looking at\nthe dependence of \u000e\u001eandHinton the DMI constant.\nFig. 10(a) shows the maximum \ructuation \u000e\u001eas func-\ntion of DMI for \u00160Hz= 30 mT. \u000e\u001eincreases for low\nDMI and it has a maximum at \u0019HDMI =HK, which\nin our case corresponds to D= 0:014Dc. The increase\nof\u000e\u001efor small values of Dis due to the smaller restor-\ning torque in Eq. (9). This holds until \u0019HDMI =HK,\nwhere shape anisotropy and DMI are comparable and\nthey both a\u000bect the DW equilibrium con\fguration. As a\nconsequence, the reduction of Hdepis enhanced by de-creasingDuntilD\u00180:014Dc, while it is reduced if\n0< D < 0:014Dc. Another contribution is given by\nthe amplitude of the internal \feld, Hint. Fig. 10(b) de-\npicts\u00160Hintas function of \u000e\u001eandD. The maximum\n\u000e\u001e, obtained at \u00160Hz= 30 mT, is additionally marked\nin the plot. The internal \feld decreases with the DMI\nbut this reduction is compensated by an increase in \u000e\u001e,\nwhich leads to an overall increase of \u00160Hint, as discussed\nin the previous part. However, at very low DMI, the in-\nternal \feld is dominated by shape anisotropy and, inde-\npendently on the DW angle displacement, it is too small\nto have an e\u000bect on the depinning mechanism. Note,\nhowever, that the amplitude of Hintshould be compared\nwith the amplitude of the pinning barrier Hs. Fig. 9(b)\nis calculated with \u00160Hs= 120 mT and the internal \feld,\ngiven by shape anisotropy ( HK=2\u001815 mT), has indeed\na negligible e\u000bect. However, larger e\u000bects are observed,\nin the caseD= 0, for smaller Hs, with reduction of Hdep\nup toHd=Hs\u00180:6, as shown in Fig. 9(c), which is calcu-\nlated with\u00160Hs= 30 mT. In other words, the reduction\nof the depinning \feld depends on the ratio between the\npinning barrier and the internal DW \feld.\nFinally, it is interesting to see what happens for\nweaker disorder and di\u000berent DMI in the system with\ngrains. Fig. 11 shows the dynamical Hdep, for di\u000berent\npinning potential and di\u000berent DMI, obtained in the\ngranular system. The results are in good agreement with\nwhat predicted by the 1D model for di\u000berent disorder\nstrengths. However, we observe a smaller dependence\non the DMI parameter. This is due to two reasons:8\n●●●●●●● ● ● ● ● ● ● ● ● ● ● ● ● ●\n■■■■■■■■■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■\n◆◆◆◆◆◆◆◆◆◆◆◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆\n▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲ ▲ ▲\n●μ���=�� ��\n■μ���=�� ��\n◆μ���=�� ��\n▲μ���=��� ��\n������������������������������������\n�������α��/��(�)\n●●●●●●●●●●●●●●●●●●●●\n■■■■■■■■■■■■■■■■■■■■\n◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆\n▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲\n▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼\n○○○○○○○○○○○○○○○○○○○○\n□□□□□□□□□□□□□□□□□□□□\n●�=���� � �■�=��� � �\n◆�=��� � �▲�=��� � �\n▼�=��� � �○�=��� � � □�=���\n���������������������������������������\n�������α��/��(�) μ���=��� ��\n●●●●●●●●●●●●●●●●●●●●\n■■■■■■■■■■■■■■■■■■■■\n◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆\n▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲\n▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼\n○○○○○○○○○○○○○○○○○○○○\n□□□□□□□□□□□□□□\n□□□□□□\n●�=���� � �■�=���� � �\n◆�=���� � �▲�=���� � �\n▼�=���� � �○�=���� � � □�=���\n���������������������������������������\n�������α��/��(�) μ���=�� ��\nFIG. 9. (a) Dynamical Hdepas function of damping for di\u000ber-\nentHs(disorder strength). (b) Dynamical Hdepas function\nof damping for di\u000berent DMI constant and \u00160Hs= 120 mT.\n(c) Dynamical Hdepas function of damping for di\u000berent DMI\nconstant and \u00160Hs= 30 mT.\n(1) in the system with grains the static pinning barrier\nis\u00160Hs= 87 mT and the dependence of the depinning\n\feld with DMI is smaller for smaller barriers, as shown\nin Fig. 9(c). (2) The DW motion in the granular\nsystem presents the formation of VBL which might also\ncontribute to the reduction of the depinning \feld. The\nmechanism is the same: a VBL is a non-equilibrium\ncon\fguration for the DW (as a deviation of \u001efrom\nequilibrium) that generates additional torques on the\nDW, which contribute to the DW depinning.\n����� ���� ��� �������������\n�/��δϕ(°)(�)\nπ����=��\nμ�����(��)\n� ��� ��� ��� ���\n������������������������������\n�/��δϕ(°)(�)FIG. 10. (a) Max DW angle \ructuation \u000e\u001e=\u001emax\u0000\u001eeq\nas function of DMI for \u00160Hz= 30 mT. (b) Internal DW\n\feld\u00160Hintas function of DMI and \u000e\u001e. The green points\ncorrespond the max \ructuation plotted in (a). Note that the\nscale is logarithmic in (a).\n●●●●● ● ●\n□□□□□ □\n●μ���=�� ��\n□μ���=�� ����������������/��(�)\n●●●●● ● ●\n◇◇◇◇◇ ◇\n○○○○○ ○\n●�=��� ��/��~���� �\n◇�=��� ��/��~���� �\n○�=�\n������������������������������\n�������α��/��(�)\nFIG. 11. (a) Dynamical Hdepas function of damping for dif-\nferentHs(disorder strength). (b) Dynamical Hdepas function\nof damping for di\u000berent DMI constants.9\nIV. CONCLUSIONS\nTo summarize, we have analysed the DW depinning\n\feld in a PMA sample with DMI and we found that Hdep\ndecreases with the damping parameter with reductions\nup to 50%. This decrease is related to the DW inter-\nnal dynamics and the \fnite size of the barrier: due to\nDW precession, the DW internal angle ( \u001e) deviates from\nequilibrium and triggers the internal DW \feld (DMI and\nshape anisotropy) which tries to restore its original value.\nAt the same time, the internal \feld pushes the DW above\nits equilibrium position within the energy barrier. This\nmechanism leads to DW oscillations and, if the ampli-\ntude of the oscillations is higher than the barrier size,\nthe DW gets depinned for a lower \feld. Deviations of \u001e\nfrom equilibrium and DW oscillations are both damping\ndependent and they are enhanced at low damping.\nIn the system with grains the mechanism is the same\nbut deviations from the internal DW equilibrium include\nthe formation of VBL with more complex dynamics.\nThe e\u000bect is enhanced for low DMI (providing that\u0019HDMI> H K) and for stronger disorder since we need\nto apply larger external \felds, which lead to larger DW\noscillations. These results are relevant both from a tech-\nnological and theoretical point of view, since they \frstly\nsuggest that a low damping parameter can lead to a\nlowerHdep. Furthermore, they show that micromagnetic\ncalculations of the depinning \feld, neglecting the DW\nprecessional dynamics can provide only an upper limit\nforHdep, which could actually be lower due to the DW\nprecessional dynamics.\nV. ACKNOWLEDGEMENT\nS.M. would like to thank K. Shahbazi, C.H. Mar-\nrows and J. Leliaert for helpful discussions. This work\nwas supported by Project WALL, FP7- PEOPLE-2013-\nITN 608031 from the European Commission, Project No.\nMAT2014-52477-C5-4-P from the Spanish government,\nand Project No. SA282U14 and SA090U16 from the\nJunta de Castilla y Leon.\n\u0003Corresponding author: simone.moretti@usal.es\n1D. Allwood, Science 309, 1688 (2005).\n2S. S. P. Parkin, M. Hayashi, and L. Thomas, Science 320,\n190 (2008).\n3P. J. Metaxas, J. P. Jamet, A. Mougin, M. Cormier,\nJ. Ferr\u0013 e, V. Baltz, B. Rodmacq, B. Dieny, and R. L.\nStamps, Physical Review Letters 99, 217208 (2007).\n4J. Gorchon, S. Bustingorry, J. Ferr\u0013 e, V. Jeudy, A. B.\nKolton, and T. Giamarchi, Physical Review Letters 113,\n027205 (2014), arXiv:1407.7781.\n5T. A. Moore, I. M. Miron, G. Gaudin, G. Serret, S. Auf-\nfret, B. Rodmacq, A. Schuhl, S. Pizzini, J. Vogel, and\nM. Bon\fm, Applied Physics Letters 93, 262504 (2008),\narXiv:0812.1515.\n6I. M. Miron, T. Moore, H. Szambolics, L. D. Buda-\nPrejbeanu, S. Au\u000bret, B. Rodmacq, S. Pizzini, J. Vogel,\nM. Bon\fm, A. Schuhl, and G. Gaudin, Nature materials\n10, 419 (2011).\n7E. Ju\u0013 e, A. Thiaville, S. Pizzini, J. Miltat, J. Sampaio,\nL. D. Buda-Prejbeanu, S. Rohart, J. Vogel, M. Bon\fm,\nO. Boulle, S. Au\u000bret, I. M. Miron, and G. Gaudin, Phys-\nical Review B 93, 014403 (2016).\n8F. Hellman, A. Ho\u000bmann, Y. Tserkovnyak, G. S. Beach,\nE. E. Fullerton, C. Leighton, A. H. MacDonald, D. C.\nRalph, D. A. Arena, H. A. D urr, P. Fischer, J. Grollier,\nJ. P. Heremans, T. Jungwirth, A. V. Kimel, B. Koop-\nmans, I. N. Krivorotov, S. J. May, A. K. Petford-Long,\nJ. M. Rondinelli, N. Samarth, I. K. Schuller, A. N.\nSlavin, M. D. Stiles, O. Tchernyshyov, A. Thiaville, and\nB. L. Zink, Reviews of Modern Physics 89, 025006 (2017),\narXiv:1607.00439.\n9A. Cr\u0013 epieux and C. Lacroix, Journal of Magnetism and\nMagnetic Materials 182, 341 (1998).\n10A. Thiaville, S. Rohart, \u0013E. Ju\u0013 e, V. Cros, and A. Fert, EPL\n(Europhysics Letters) 100, 57002 (2012).11S. Rohart and A. Thiaville, Physical Review B 88, 184422\n(2013).\n12N. Perez, E. Martinez, L. Torres, S.-H. Woo, S. Emori,\nand G. S. D. Beach, Applied Physics Letters 104, 092403\n(2014).\n13G. Bertotti, Hysteresis in Magnetism (Academic Press,\n1998).\n14A. B. Kolton, A. Rosso, and T. Giamarchi, Physical Re-\nview Letters 94, 047002 (2005).\n15E. Agoritsas, V. Lecomte, and T. Giamarchi, Physica B:\nCondensed Matter 407, 1725 (2012), arXiv:1111.4899.\n16E. Martinez, L. Lopez-Diaz, L. Torres, C. Tristan, and\nO. Alejos, Physical Review B 75, 174409 (2007).\n17M. Voto, L. Lopez-Diaz, and L. Torres, Journal of Physics\nD: Applied Physics 49, 185001 (2016).\n18M. Voto, L. Lopez-Diaz, L. Torres, and S. Moretti, Phys-\nical Review B 94, 174438 (2016).\n19P. J. Metaxas, Solid State Physics - Advances in Research\nand Applications , Vol. 62 (2010) pp. 75{162.\n20J. H. Franken, M. Hoeijmakers, R. Lavrijsen, and H. J. M.\nSwagten, Journal of Physics: Condensed Matter 24,\n024216 (2012), arXiv:1112.1259.\n21Y. Yoshimura, K.-j. Kim, T. Taniguchi, T. Tono, K. Ueda,\nR. Hiramatsu, T. Moriyama, K. Yamada, Y. Nakatani,\nand T. Ono, Nature Physics 12, 157 (2015).\n22B. Hillebrands and A. Thiaville, Spin Dynamics in Con-\n\fned Magnetic Structures III (Springer, 2005).\n23T. A. Moore, P. M ohrke, L. Heyne, A. Kaldun,\nM. Kl aui, D. Backes, J. Rhensius, L. J. Heyderman, J.-\nU. Thiele, G. Woltersdorf, A. Fraile Rodr\u0013 \u0010guez, F. Nolt-\ning, T. O. Mente\u0018 s, M. \u0013A. Ni~ no, A. Locatelli, A. Potenza,\nH. Marchetto, S. Cavill, and S. S. Dhesi, Physical Review\nB82, 094445 (2010).\n24Y. Tserkovnyak, A. Brataas, and G. E. W. Bauer, Physical\nReview Letters 88, 117601 (2002).10\n25U.-H. Pi, Y.-J. Cho, J.-Y. Bae, S.-C. Lee, S. Seo, W. Kim,\nJ.-H. Moon, K.-J. Lee, and H.-W. Lee, Physical Review\nB84, 024426 (2011).\n26L. Thomas, M. Hayashi, X. Jiang, R. Moriya, C. Rettner,\nand S. S. P. Parkin, Nature 443, 197 (2006).\n27H. Y. Yuan and X. R. Wang, Physical Review B 92, 054419\n(2015).\n28H. Yuan and X. Wang, The European Physical Journal B\n88, 214 (2015).\n29A. Vansteenkiste, J. Leliaert, M. Dvornik, M. Helsen,\nF. Garcia-Sanchez, and B. Van Waeyenberge, AIP Ad-\nvances 4, 107133 (2014), arXiv:1406.7635.\n30J. Leliaert, B. Van de Wiele, A. Vansteenkiste, L. Laurson,\nG. Durin, L. Dupr\u0013 e, and B. Van Waeyenberge, Journal of\nApplied Physics 115, 233903 (2014).\n31A. Hrabec, J. Sampaio, M. Belmeguenai, I. Gross, R. Weil,\nS. M. Ch\u0013 erif, A. Stashkevich, V. Jacques, A. Thiaville,\nand S. Rohart, Nature Communications 8, 15765 (2017),\narXiv:1611.00647.\n32See Appendices.\n33This de\fnition is preferred over the average of Hi\ndepsince\nit is more independent on the sample size. In fact, by in-creasing the sample dimension along the xdirection, we\nincrease the probability of \fnding the highest possible hj\nin the single realization and the average of Hi\ndepwill tend\nto the maximum.\n34This is solved by the Relax solver of MuMax with the as-\nsumption\u000b=(1 +\u000b2) = 1.\n35The DW energy is calculated as the energy of the system\nwith the DW minus the energy of the system without the\nDW (uniform state). The pro\fle is obtained by moving the\nDW with an external applied \feld and then subtracting the\nZeeman energy.\n36X. Wang, P. Yan, J. Lu, and C. He, Annals of Physics\n324, 1815 (2009), arXiv:0809.4311.\n37S. Tarasenko, a. Stankiewicz, V. Tarasenko, and J. Ferr\u0013 e,\nJournal of Magnetism and Magnetic Materials 189, 19\n(1998).\n38The same results are obtained with a Gaussian barrier,\nmeaning that the key point is the \fnite size of the barrier\nrather than its shape.\n39ForD >D c, DW have negative energies and the systems\nspontaneously breaks into non-uniform spin textures.\nAppendix A: Maximum torque and equilibrium state\nIn this section we show in more detail how the maximum torque represents an indicator of the equilibrium state.\nMaximu torque is de\fned as\n\u001cmax\n\r0= Maxf\u00001\n1 +\u000b2mi\u0002He\u000b;i\u0000\u000b\n1 +\u000b2mi\u0002(mi\u0002He\u000b;i)g=1\n\r0Max\u0012dmi\ndt\u0013\n; (A1)\nover all cells with label i=f1;:::;N =Nx\u0001Nyg. MuMax3.9.329can provide this output automatically if selected.\nWe perform the same simulations as indicated in the main text, without any stopping condition, but simply running\nfort= 20 ns. Fig. 12(a) shows the average mzcomponent for \u000b= 0:2 andBz= 10 mT, while Fig. 12(b) depicts the\ncorresponding maximum torque. We can see that, once the system has reached equilibrium, the maximum torque has\ndropped to a minimum value. The same results is obtained for di\u000berent damping but the \fnal maximum torque is\ndi\u000berent. Numerically this value is never zero since it is limited by the code numerical precision and by the system\nparameters, in particular by damping.\nFig. 12(c) represents the maximum torque as function of applied \feld for di\u000berent damping. The maximum torque\nis clearly independent on the applied \feld but depends on the damping value. Finally, Fig. 12(d) shows the max\ntorque as function of damping. The maximum torque decreases with damping and it saturates for \u000b\u00150:5 since we\nhave reached the minimum numerical precision of the code29. For higher damping the maximum torque oscillates\naround this minimum sensibility value, as shown in the inset of Fig. 12(d). The value obtained with these preliminary\nsimulations is used to set a threshold \u000f(\u000b) for the depinning \feld simulations in order to identify when the system has\nreached an equilibrium. Furthermore, additional tests were performed, without putting any max torque condition,\nbut simply running the simulations for a longer time ( t= 80;160 ns) and calculating the depinning \feld in order\nto ensure that the results obtained with these two method were consistent, i.e., that we have actually reached an\nequilibrium state with the maximum torque condition.11\n������������������-���-���-���\n����(��)<��>⨯��-�α=���(�)\n���������������������������\n����(��)��� ������/γ �(��)α=���(�)\n○ ○ ○ ○ ○ ○ ○ ○\n□ □ □ □ □ □ □ □\n◇◇◇◇◇◇◇◇\n○α=����□α=���◇α=���\n� �� �� �� ��������������������\nμ���(��)��� ������/γ �(��)(�)\n●\n●●\n��������������������-�������������������\n�������α��� ������/γ �(��)(�)\n���������������������\n����(��)τ���/γ�(��)α=���\nFIG. 12. (a) average mzas function of time. (b) Max torque/ \r0(\u001cmax) as function of time. \u001cmaxrapidly decreases when the\nsystem is at equilibrium. (c) Max torque as function of applied \feld for di\u000berent damping. (d) Max torque at equilibrium as\nfunction of damping. The inset shows the max torque as function of time for \u000b= 0:5.\nAppendix B: 1D energy barrier\nAs commented in the main text, the pinning \feld implemented in the 1D model simulations is obtained by using the\nshape of the DW energy pro\fle derivative @\u001b(x)=@x(beingxthe DW position) and the amplitude of the depinning\n\feld obtained in the full micromagnetic simulations Hsfor the single barrier case. Namely\nHdep=Hs\u0012@\u001b(x)\n@x\u0013\nN; (B1)\nwhere we recall that Nstands for the normalized value. This choice might sound unusual and needs to be justi\fed.\nIn fact, having the DW energy pro\fle, the depinning \feld could be simply calculated as20\nHdep=1\n2\u00160Ms@\u001b(x)\n@x: (B2)\nThis expression is derived by imposing that the derivative of the total DW energy E(x) = 2\u00160MsHzx+\u001b(x) (Zeeman\n+ internal energy) must be always negative. However, in our case also Ms(x) depends on the DW position and the\nresults obtained with Eq. B2 is di\u000berent from the depinning \feld measured in the static single barrier simulations.\nFor this reason we use Eq. B1 which keep the correct barrier shape and has the measured static value.\nFinally, we recall that equivalent results are obtained by using a simple Gaussian shape for the pinning \feld, meaning\nthat the key point is the localized shape of the barrier, rather than its exact form.\nAppendix C: Dynamical depinning for a moving Domain Wall\nIn this section we show the results for the dynamical depinning \feld when the DW is placed at an initial distance of\nd1= 200 nm from the barrier. In this way the DW hits the pinning with an initial velocity. The d0case corresponds to\nthe DW at rest relaxed just before the barrier and extensively analysed in the main text. Also for this con\fguration\nwe performed static and dynamic simulations, neglecting or including the DW precessional dynamics respectively.\nThe depinning \feld for the d1case is further reduces at small damping, reaching Hd=Hs\u00180:08 (Hd= 9 mT and\nHs= 120 mT) at \u000b= 0:02. Nevertheless, the depinning \feld remains constant in the static simulations independently\non the velocity at which the DW hits the barrier. This suggests that, rather than related to the DW velocity, the\nreduction is again related to the DW precession. When the DW starts from d1it reaches the barrier precessing, thus\nwith a higher displacement from its equilibrium angle, leading to a higher e\u000bect of the internal \feld.12\n●●●●●●\n○○○○○○ ◆◆◆◆◆ ◆ □□□□□ □\n●��(�������)\n○��(�������)◆��(������)\n□��(������)\n��� ��� ��� ��� ������������������������\n�������α��/��\nFIG. 13. Dynamical depinning \feld as function of damping for static and dynamic simulations for the d0andd1cases."    },    {        "title": "1706.00777v2.Power_Loss_for_a_Periodically_Driven_Ferromagnetic_Nanoparticle_in_a_Viscous_Fluid__the_Finite_Anisotropy_Aspects.pdf",        "content": "Power Loss for a Periodically Driven Ferromagnetic Nanoparticle in a Viscous Fluid:\nthe Finite Anisotropy Aspects\nT. V. Lyutyy\u0003, O. M. Hryshko, A. A. Kovner\nSumy State University, 2 Rimsky-Korsakov Street, UA-40007 Sumy, Ukraine\nAbstract\nThe coupled magnetic and mechanical motion of a ferromagnetic nanoparticle in a viscous \ruid is considered within the\ndynamical approach. The equation based on the total momentum conservation law is used for the description of the\nmechanical rotation, while the modi\fed Landau-Lifshitz-Gilbert equation is utilized for the description of the internal\nmagnetic dynamics. The exact expressions for the particles trajectories and the power loss are obtained in the linear\napproximation. The comparison with the results of other widespread approaches, such as the model of \fxed particle\nand the model of rigid dipole, is performed. It is established that in the small oscillations mode the damping precession\nof the nanopartile magnetic moment is the main channel of energy dissipation, but the motion of the nanoparticle easy\naxis can signi\fcantly in\ruence the value of the resulting power loss.\nKeywords: Ferro\ruid, \fnite anisotropy, spherical motion, damping precession, Landau-Lifshitz-Gilbert equation\n1. Introduction\nThe correct description of a ferromagnetic nanoparti-\ncle dynamics in a viscous carrier \ruid is a key to under-\nstanding the ferro\ruid dynamics for all possible applica-\ntions. Up to now, for ferro\ruids composed of small enough\nnanoparticles, the response to a time-periodic magnetic\n\feld was considered \frstly within the concept of com-\nplex magnetic susceptibility, which is well described in [1].\nHowever, when the nanoparticle magnetic energy is com-\nparable with the thermal one, the response of a nanopar-\nticle will be based mainly on the individual trajectories of\neach nanoparticle. For example, the regular viscous rota-\ntion is considered as the main energy dissipation channel\nfor large enough nanoparticles driven by an external al-\nternating \feld [2]. This gives reason to believe that the\nanalytical description of the single nanoparticle motion is\ndemanded.\nTwo components of the nanoparticle dynamics should\nbe considered simultaneously for the trajectory descrip-\ntion: 1) the mechanical rotation (or the so-called spher-\nical motion) of a nanoparticle with respect to a viscous\n\ruid, 2) the internal motion of the nanoparticle magne-\ntization in the framework of the crystal lattice. Since\nthe simultaneous description is faced with some di\u000ecul-\nties, two approximations are utilized instead: 1) the rigid\ndipole approach [3], when the nanoparticle magnetic mo-\nment is supposed to be locked in the nanoparticle crys-\ntal lattice, 2) the \fxed nanoparticle approach [4], when\n\u0003Corresponding author\nEmail address: lyutyy@oeph.sumdu.edu.ua (T. V. Lyutyy)a nanoparticle is assumed to be immobilized because of\nthe rigid bound with a media carrier. Despite the restric-\ntions, both approaches are widely used for the description\nof the response to an alternating \feld of a ferromagnetic\nparticle in a viscous \ruid, including the power loss calcula-\ntion problem, which is closely related to the magnetic \ruid\nhyperthermia for cancer therapy [5, 6]. Thus, the model\nof rigid dipole was applied successfully for the dynamical\nand stochastic approximations: the power loss was found\nfor a circularly-polarized [7, 8, 9] and a linearly-polarized\n[10, 9] \felds. The e\u000bective Langevin equation and the\nkey characteristics of the rotational dynamics were estab-\nlished in [11]. The power loss calculation within the \fxed\nnanoparticle model, where only a damping precession of\nthe magnetic moment is taken into account, was performed\nin [12, 13, 14]. Finally, this problem was investigated in\n[15, 16] for the nanoparticles ensemble.\nThe coupled dynamics of a nanoparticle cannot be de-\nscribed by a simple superposition of these two types of\nmotion because of the essential changes in the equations\nof motion. The coupled motion of the particle magnetic\nmoment and the whole particle was \frstly described in\n[17]. Despite this, the discussion about the basic equations\nof motion in the case of the coupled dynamics is contin-\nued till now [18, 19, 20, 21]. It is especially important in\nthe context of a ferro\ruid heating by an alternating \feld,\nwhen both these types of motion can produce heat. One of\nthe \frst successful attempts concerning the energy absorp-\ntion description was reported in [22]. There the power loss\nwas obtained in the dynamical approximation by lineariza-\ntion of the Lagrangian equation in some speci\fc cases.\nBut within this approach the equations of motion were\nPreprint submitted to Journal of Magnetism and Magnetic Materials October 1, 2018 arXiv:1706.00777v2  [cond-mat.other]  20 Jul 2017not used. The study of the forced coupled dynamics in\na circularly-polarized magnetic \feld using the simpli\fed\nequations of motion was presented in [23], but the energy\nabsorption problem was not considered. The power loss\nwas calculated in recent studies [18, 19]. Unfortunately,\nthe correct explicit form of the equations of motion was\nnot applied there that facilitates the discussion about the\nbasic model equations [20, 21]. And only recently the es-\nsential progress in the description of energy absorption by\na viscously coupled nanoparticle with a \fnite anisotropy\nwas achieved [24]. Here the microwave absorption spec-\ntra was investigated using the linear response approach.\nBut the viscous term as well as in [17] was not taken into\naccount that motivates further research.\nTherefore, we use the correct equations set, presented\nin [20] to investigate the nanoparticle response to an ex-\nternal alternating \feld. Absorption of the \feld energy,\nwhich further is transformed into heating, is in our main\nfocus. In particular, the in\ruence of the easy axis mo-\nbility on the resonance dependencies of the power loss on\nthe \feld frequency is examined. Then, we consider the\nresults obtained for the same conditions within the \fxed\nnanoparticle and the rigid dipole approximations. In this\nway we reveal the role of both the viscous rotation of a\nwhole particle and the damping precession of its magnetic\nmoment inside in the energy dissipation process. We make\na conclusion about the complex character of the coupled\ndynamics and indistinguishability of the contribution of\neach type of motion into the mutual heating in the dy-\nnamical approximation.\n2. Description of the Model\nLet us consider a uniform spherical single-domain fer-\nromagnetic nanoparticle of radius R, magnetization ( M,\njMj=M= const) and density \u001a. This particle performs\nthe spherical motion (or motion with the \fxed center of\nmass) with respect to a \ruid of viscosity \u0011. Then, we as-\nsume that the nanoparticle is driven by the external time-\nperiodic \feld of the following type:\nH(t) =exHcos(\nt) +ey\u001bHsin(\nt); (1)\nwhere ex,eyare the unit vectors of the Cartesian coor-\ndinates,His the \feld amplitude, \n is the \feld frequency,\ntis the time, and \u001bis the factor which determines the\npolarization type ( \u001b=\u00061 corresponds to the circularly\npolarized \feld, 0 <j\u001bj<1 corresponds to the elliptically\npolarized \feld, and \u001b= 0 corresponds to the linearly po-\nlarized \feld). Both the interaction with the viscous media\nand the damping precession of Minside the particle lead to\ndissipation of the nanoparticle energy and further heating\nof the surrounding environment. These losses are compen-\nsated by the energy absorption of the external \feld of type\n(1) and can be characterized by the dimensionless powerloss per period calculated as [14]\nq=\n2\u0019MH aZ2\u0019\n\n0dtH@M\n@t: (2)\n2.1. Equations of Motion\nThree approaches will be considered: 1) the model\nof viscously coupled nanoparticle with a \fnite anisotropy\n(FA-model), 2) the model of \fxed particle (FP-model), 3)\nthe model of rigid dipole (RD-model). Let us start with\nthe \frst one as more important and novel. As follows\nfrom [20], the coupled magnetic dynamics and mechanical\nmotion in the deterministic case obey a pair of coupled\nequations\n_n=!\u0002n;\nJ_!=\r\u00001V_M+VM\u0002H\u00006\u0011V!; (3)\nwhere nis the unit vector which determines the anisotropy\naxis direction, !is the angular velocity of the particle,\nJ(= 8\u0019\u001aR5=15) is the moment of inertia of the particle, \r\nis the gyromagnetic ratio, Vis the nanoparticle volume,\nand dots over symbols represent derivatives with respect\nto time.\nIn fact, the \frst equation in the set (3) is the condi-\ntion of spherical motion for a rigid body, and the second\none is the classical torque equation, where the \frst term\nconstitutes the main di\u000berence from the same equation for\nthe FP-model. This term is originated from the motion\nof magnetization inside the nanoparticle with respect to\nits crystal lattice. In turn, the magnetization dynamics is\ndescribed by the modi\fed Landau-Lifshitz-Gilbert (LLG)\nequation\n_M=\u0000\rM\u0002Heff+\u000bM\u00001\u0010\nM\u0002_M\u0000!\u0002M\u0011\n;(4)\nwhere\u000bis the damping parameter, Heffis the e\u000bective\nmagnetic \feld, which accounts the uniaxial anisotropy \feld\nof magnitude Ha\nHeff=H+HaM\u00001(Mn)n: (5)\nHere, the di\u000berence from the original LLG equation\nconsists in the term which is proportional to M\u0002!\u0002M\nthat excludes the component of Mrotating together with\nthe crystal lattice. As a rule, the inertia term in (3) can be\nneglected even for large enough nanoparticles ( R> 20 nm)\nin a wide frequency range. Therefore, for further analysis\nwe transform the equations of motion (3) and (4) into the\nconvenient form\n_ n=MH a[_ m\u0002n=\nr+ (m\u0002h)\u0002n]=6\u0011;\n_ m(1 +\f) =\u0000\nrm\u0002h1\neff+\u000bm\u0002_ m;(6)\n2where \n r=\rHais the ferromagnetic resonance fre-\nquency,\f=\u000bM= 6\r\u0011,\nh1\neff= (exhcos \nt+eyh\u001bsin \nt) (1 +\f) + (mn)n;(7)\nand, \fnally, m=M=M,h=H=H aare the dimension-\nless magnetic moment and \fled amplitude, respectively.\nThe FP-model is described by the well-known LLG\nequation\n_M=\u0000\rM\u0002Heff+\u000bM\u00001M\u0002_M (8)\nor in the dimensionless form\n_ m=\u0000\nrm\u0002heff+\u000bm\u0002_ m; (9)\nwhere heff=Heff=Ha.\nFinally, the RD-model is described by the set of equa-\ntions similar to the Eqs. (3), but without the term propor-\ntional to _M\n_n=!\u0002n;\nJ_!=VMn\u0002H\u00006\u0011V!: (10)\nWhen the inertia momentum is neglected, Eqs. (10) are\ntransformed into a simple form\n_n=\u0000\ncrn\u0002(n\u0002h); (11)\nwhere \n cr=MH a=6\u0011is the characteristic frequency\nof the uniform mechanical rotation.\n2.2. Validation of the Dynamical Approximation\nThe used systems of equations are valid if thermal \ruc-\ntuations do not signi\fcantly in\ruence the obtained trajec-\ntories. There are two principal issues in this regard needed\nto be considered. Firstly, the magnetic energy should be\nmuch larger than the thermal energy, or \u0000 \u001d1, where\n\u0000 =MHV= (kBT),Tis the thermodynamic temperature,\nkBis the Boltzmann constant. In this case, primarily small\ndeviations from the dynamical trajectories take place. Sec-\nondly, the requirement to the relaxation time \u001cNexists.\nHere, relaxation time is the time, during which the rare,\nbut large \ructuations can occur. When the period of an\nexternal \feld is much smaller than the relaxation time,\nor \n\u00001\u001c\u001cN, the probability of such \ructuation is neg-\nligible, and the dynamical approach remains valid. Fol-\nlowing Brown [4], the relaxation time \u001cNcan be found\nas\u001cN= (\u0000=\u0019)\u00001=2exp(\u0000)(2\u000b\rH )\u00001. Both these factors\ntogether impose the requirements to the nanoparticle size\nand values of the \feld frequency and amplitude. For exam-\nple, \u0000\u001911:9 for the real nanoparticles of maghemite [25]\nwith the following parameters: average radius R= 20 nm,\nHa= 910 Oe,M= 338 G, temperature T= 315 K and ex-\nternal \feld amplitude H= 0:05Ha. Then, the frequency\nshould to be larger than \u001c\u00001\nN, which for the parameters\nstated above and \u000b= 0:05 is equal to \n N\u00191:11\u0001103Hz.These conditions are su\u000ecient for the FP-model. But\nwhen we consider the mechanical rotation in addition to\nthe magnetic dynamics within the FA-model, one needs\nto take into account the conditions of stable spherical mo-\ntion. The signi\fcant changes in the angular coordinates\ncan occur due to thermal excitation, when the observa-\ntion time is much more than the Brownian relaxation time\n\u001cB= 3\u0011V=(kBT) [26]. It imposes the existence of an-\nother characteristic frequency \n B=\u001c\u00001\nB=kBT=(3\u0011V).\nFor the above-mentioned maghemite nanoparticles of ra-\ndiusR= 20 nm and water at temperature of T= 315 K\nand viscosity of \u0011= 0:006 P this frequency is equal to\n\nB\u00192:26\u0001105Hz. One more requirement to the fre-\nquency arises from the condition which represents the va-\nlidity of the Stokes approximation for the friction momen-\ntum [27]: Re = \u001al\nSR2=\u0011\u001810. Here Re is the so-called\nReynolds number, \u001alis the liquid density, \n Sis the corre-\nsponding characteristic frequency, which de\fnes the upper\nlimit of the \feld frequency. Straightforward calculations\ngive \n S\u00181012Hz in our case. Summarizing, one can\nobtain that max[\n B;\nN]\u001c\n\u001c\nS. Therefore, the fre-\nquency interval, where the dynamical approach is valid for\nthe calculation, is \n = (105\u00001012) Hz which includes the\nfrequencies acceptable for the magnetic \ruid hyperthermia\nmethod.\nFinally, the conditions of using the RD-model include\nall the stated above for the FA-model and contain ad-\nditionally the requirement to the \feld amplitudes, which\nshould be much smaller than the e\u000bective anisotropy \feld\n(H\u001cHa). The last inequality satis\fes the above cal-\nculations and corresponds to the limitations of the linear\napproximation utilized for the processing of the equations\nof motion.\nThe importance of the dynamical approximation is not\nrestricted by its validity in a certain interval of the sys-\ntem parameters. The dynamical approximation reveals\nthe main microscopic mechanisms of the ferro\ruid sensi-\ntivity to external \felds. In this way we can estimate the\nupper limits of such important performance criteria as the\nmagnetic susceptibility or the power loss. It is very impor-\ntant in a light of \fctionalization of ferro\ruids and creation\nof the properties demanded in the applications.\n3. Results\nThe solution of the set of equations (6), (11) and (9)\ncan be found in the linear approximation for the small\noscillations mode. In this mode, vectors mandnare\nrotated in a small vicinity around the initial position of\nthe easy axis which, in turn, is de\fned by the angles \u00120\nand'0(see Fig. 1). This takes place for small enough\n\feld amplitudes ( h\u001c1). The linearization procedure\nused here is similar to the reported in [14] and consists in\nthe following. Let us introduce a new coordinate system\nx0y0z0in the way depicted in Fig. 1. Actually, it is rotated\nwith respect to the laboratory system xyzby the angles\n3Figure 1: (Color online) Schematic representation of the model and\nthe coordinate systems.\n\u00120and'0. In this new coordinate system, vectors mand\nncan be represented in the linear approximation as\nm=ex0mx0+ey0my0+ez0; (12)\nn=ex0nx0+ey0ny0+ez0; (13)\nwhere ex0;ey0;ez0are the unit vectors of the coordinate\nsystemx0y0z0. In this system, the external \feld (1) can be\nwritten using the rotation matrix as\nh0=C\u00010\n@hcos \nt\n\u001bhsin \nt\n01\nA; (14)\nC=0\n@cos\u00120cos'0cos\u00120sin'0\u0000sin\u00120\n\u0000sin'0 cos'0 0\nsin\u00120cos'0sin\u00120sin'0 cos\u001201\nA;(15)\nh0=0\n@hcos\u00120cos'0cos \nt+\u001bhcos\u00120sin'0sin \nt\n\u0000hsin'0cos \nt+\u001bhcos'0sin \nt\nhsin\u00120cos'0cos \nt+\u001bhsin\u00120sin'0sin \nt1\nA:\n(16)\nAll the above allows to analyze the features of the\nnanoparticle response to the external \feld (1) for three\napproximations in an uniform manner. The analytical so-\nlutions obtained below describe the principal di\u000berence be-\ntween the coupled and separated motion of the magnetic\nmoment and the whole particle that constitutes our main\nresults.\n3.1. Coupled Oscillations of the Easy Axis and the Mag-\nnetic Moment\nWe start from the most complicated, however, the most\ninteresting case: the case when both the mechanical rota-\ntion and internal magnetic dynamics occur simultaneously,\nor the FA-model. Using (16), assuming nx0;ny0;mx0;my0\u0018\nh, and neglecting all the nonlinear terms with respect to h,we, \fnally, derive from (6) the linearized system of equa-\ntions for mandnin the following form:\n_nx0=MH a( _my0=\nr+hx0)=6\u0011;\n_ny0=\u0000MH a( _mx0=\nr\u0000hy0)=6\u0011;\n(1 +\f) _mx0=\u0000\nr(my0\u0000hy0\u0000ny0)\u0000\u000b_my0;\n(1 +\f) _my0= \nr(mx0\u0000hx0\u0000nx0)\u0000\u000b_mx0:(17)\nSolution of this set of linear equations can be written\nin the standard form\nnx0=ancos \nt+bnsin \nt;\nny0=cncos \nt+dnsin \nt;\nmx0=amcos \nt+bmsin \nt;\nmy0=cmcos \nt+dmsin \nt;(18)\nwherean,bn,cn,dn,am,bm,cm, anddmare the con-\nstant coe\u000ecients which should be de\fned. Substituting\n(18) into (17) and using the linear independence of the\ntrigonometric functions, we obtain the system of linear al-\ngebraic equations for the coe\u000ecients corresponding to m\n(1 +\f)~\nam=dm+\u000ebm\u0000\u000b~\ncm\u0000Am;\n(1 +\f)~\nbm=\u0000cm\u0000\u000eam\u0000\u000b~\ndm+Bm;\n(1 +\f)~\ncm=\u0000bm+\u000edm+\u000b~\nam\u0000Cm;\n(1 +\f)~\ndm=am\u0000\u000ecm+\u000b~\nbm+Dm(19)\nand the explicit expressions for the coe\u000ecients corre-\nsponding to n\nan=\u000ecm\u0000\u001b~\n\u00001hcos\u00120sin'0;\nbn=\u000edm+~\n\u00001hcos\u00120cos'0;\ncn=\u0000\u000eam\u0000\u001b~\n\u00001hcos'0;\ndn=\u0000\u000ebm\u0000~\n\u00001hsin'0:(20)\nHere ~\n = \n=\nr,\u000e=\f=\u000b and\nAm=\u001bh(1 +\f) cos'0\u0000~\n\u00001hsin'0;\nBm=\u0000h(1 +\f) sin'0\u0000\u001b~\n\u00001hcos'0;\nCm=\u0000\u001bh(1 +\f) cos\u00120sin'0\u0000~\n\u00001hcos\u00120cos'0;\nDm=\u0000h(1 +\f) cos\u00120cos'0+\u001b~\n\u00001hcos\u00120sin'0:\nFrom (19) one straightforwardly obtains the unknown\nconstantsam,bm,cm, anddmas follows\nam=Z\u00001h\n~\n1Dm+~\n2Bm+~\n3Cm+~\n4Ami\n;\nbm=Z\u00001h\n~\n1Cm+~\n2Am\u0000~\n3Dm\u0000~\n4Bmi\n;\ncm=Z\u00001h\n\u0000~\n1Bm+~\n2Dm\u0000~\n3Am+~\n4Cmi\n;\ndm=Z\u00001h\n\u0000~\n1Am+~\n2Cm+~\n3Bm\u0000~\n4Dmi\n;\n(21)\n4where\nZ=~\n4\u000b4+ 2~\n4\u000b2\f2+~\n4\f4+ 4~\n4\u000b2\f+\n+ 4~\n4\f3+ 2~\n4\u000b2+ 6~\n4\f2\u00002~\n2\u000b2\u000e2+\n+ 2~\n2\f2\u000e2+ 4~\n4\f+ 8~\n2\u000b\f\u000e+\n+ 4~\n2\f\u000e2+~\n4+ 2~\n2\u000b2+ 8~\n2\u000b\u000e\u0000\n\u00002~\n2\f2+ 2~\n2\u000e2+\u000e4\u00004~\n2\f\u00002~\n2+\n+ 2\u000e2+ 1;(22)\n~\n1=\u0000~\n2\u000b2\u00002~\n2\u000b\f\u000e\u00002~\n2\u000b\u000e+\n+~\n2\f2+ 2~\n2\f+~\n2\u0000\u000e2\u00001;\n~\n2=\u0000~\n2\u000b2\u000e+ 2~\n2\u000b\f+ 2~\n2\u000b+\n+~\n2\f2\u000e+ 2~\n2\f\u000e+~\n2\u000e+\u000e3+\u000e;\n~\n3=~\n3\u000b3+~\n3\u000b\f2+ 2~\n3\u000b\f+\n+~\n3\u000b\u0000~\n\u000b\u000e2+~\n\u000b+ 2~\n\f\u000e+\n+ 2~\n\u000e;\n~\n4=\u0000~\n3\u000b2\f\u0000~\n3\u000b2\u0000~\n3\f3\u0000\n\u00003~\n3\f2\u00003~\n3\f\u0000~\n3\u00002~\n\u000b\u000e\u0000\n\u0000~\n\f\u000e2+~\n\f\u0000~\n\u000e2+~\n:\nUsing (21), one can easily derive the set of constants\n(20), which de\fne the dynamics of the whole particle.\nThe obtained expressions for the nanoparticle trajec-\ntories let us to write the analytical relation for the power\nlossq. Direct integration of (2) with substitution of (18),\n(21) and (20) yields the following formula:\nq= 0:5~\n\nr(bmhcos\u00120cos'0\u0000am\u001ahcos\u00120sin'0\u0000\n\u0000dmhsin'0\u0000cm\u001ahcos'0+bman\u0000ambn+dmcn\u0000\n\u0000cmdn):\n(23)\nThe dependence of qon the system parameters, espe-\ncially on the external \feld frequency, is of great interest\nand will be analyzed below. But, the comparison of this\nresult with similar one obtained within other approxima-\ntions, such as the FP-model and the RD-model, is no less\ninteresting.\n3.2. Oscillations of the Magnetic Moment in a Fixed Nanopar-\nticle\nAs the next stage, let us consider the magnetic dynam-\nics only within the FP-model. As in the previous case,\nthe linearized equations are written under the assumption\nmx0;my0\u0018h, and all the nonlinear terms with respect to\nhare dropped. Using (16), we, \fnally, obtain from (9) the\nlinearized system of equations for mas follows\n_mx0=\u0000\nr( _my0\u0000hy0)\u0000\u000b_my0;\n_my0= \nr( _mx0\u0000hx0)\u0000\u000b_mx0:(24)\nThen, the general form of the solution of (24) can be\neasily written in the standard form\nmx0=afpcos \nt+bfpsin \nt;\nmy0=cfpcos \nt+dfpsin \nt;(25)whereafp,bfp,cfp, anddfpare the oscillation am-\nplitudes for magnetic the moment inside the immobilized\nnanoparticle. Substitution of (25) into (24) yields the sys-\ntem of linear algebraic equations for the desired amplitudes\n~\nafp= (dfp\u0000\u001bhcos'0)\u0000\u000b~\ncfp;\n~\nbfp=\u0000(cfp+hsin'0)\u0000\u000b~\ndfp;\n~\ncfp=\u0000(bfp\u0000\u001bhcos\u00120sin'0) +\u000b~\nafp;\n~\ndfp= (afp\u0000hcos\u00120cos'0) +\u000b~\nbfp:(26)\nAfter the calculations, we \fnd the solution of (26)\nafp=\u0000Z\u00001\nfph\n~\nfp\n1Bfp+~\nfp\n2Afpi\n;\nbfp=Z\u00001\nfph\n~\nfp\n1Cfp+~\nfp\n2Dfpi\n;\nafp=Z\u00001\nfph\n~\nfp\n1Afp\u0000~\nfp\n2Bfpi\n;\ndfp=Z\u00001\nfph\n\u0000~\nfp\n1Dfp+~\nfp\n2Cfpi\n;(27)\nwhere\nZfp= 4\u000b2~\n4+\u0010\n(\u000b2\u00001)~\n\u00002+ 1\u00112\n; (28)\n\nfp\n1= 2\u000b~\n2;\n\nfp\n2= (\u000b2\u00001)~\n2+ 1;\nAfp=\u001bh~\n (\u000bcos\u00120sin'0\u0000cos'0)\u0000hcos\u00120cos'0;\nBfp=\u001bh~\n (cos'0sin'0+\u000bcos'0) +hsin'0;\nCfp=h~\n (cos\u00120cos'0\u0000\u000bsin'0) +\u001bhcos'0;\nDfp=h~\n (\u000bcos\u00120cos'0+ sin'0) +\u001bhcos\u00120sin'0:\nThe power loss in this case can be also found by direct\nintegration of (2) with substitution of (25) and (27)\nq= 0:5h~\n\nrZ\u00001\nfpf~\nfp\n1[2\u001bhcos\u00120+\n+h~\nD] +~\nfp\n2\u000bh~\nDg;(29)\nwhere\nD= cos2\u00120(cos2'0+\u001b2sin2'0) +\u001b2cos2'0+ sin2'0:\n(30)\nThe obtained expression (29) is similar to the reported\nin [14], but accounts an arbitrary orientation of the nanopar-\nticle easy axis. Despite the quantitative di\u000berence caused\nby the turn of the easy axis, the qualitative character of\nthe frequency behavior of qremains.\n3.3. Oscillations of a Nanoparticle with the Locked Mag-\nnetic Moment\nAnd \fnally, we consider within the same framework\nthe widely used approach when the nanoparticle magnetic\nmoment is rigidly bound with the nanoparticle crystal lat-\ntice. In this so-called RD-model, the linearized equations\nhave the simplest form. Expanding the vector equation\n5(11) and taking into account (16), we write the linearized\nsystem of equations for nas follows\n_nx0= \ncrhx0;\n_ny0= \ncrhy0:(31)\nAs in the previous case, we use the trigonometric rep-\nresentation of the solution of (31)\nnx0=ardcos \nt+brdsin \nt;\nny0=crdcos \nt+drdsin \nt:(32)\nAfter direct substitution of (32) into (31), one can eas-\nily obtain the unknown constants, which are the ampli-\ntudes of the vector n\nard=h\ncrsin'0=\n;\nbrd=h\ncrcos\u00120cos'0=\n;\ncrd=\u0000h\ncrcos'0=\n;\ndrd=h\ncrcos\u00120sin'0=\n:(33)\nAnd at last, we can directly \fnd the power loss from\n(2) substituting (32) and (33)\nq= 0:5\ncrh2D: (34)\nIt is remarkable that qdoes not depend on the fre-\nquency, because while \n increases, the coe\u000ecients (33) de-\ncrease proportionally that compensates the possible growth\nof the power loss.\n4. Discussion and Conclusions\nWe have considered the response of a uniaxial ferro-\nmagnetic nanoparticle placed into a viscous \ruid to an al-\nternating \feld in the linear approximation for three mod-\nels, namely, the FA-model (viscously coupled nanoparticle\nwith a \fnite anisotropy), the FP-model (\fxed particle),\nand the RD-model (rigid dipole). As a result, we have\nobtained the expressions for the nanoparticle trajectories\nand for the power loss produced by both the rotation of\na nanoparticle in a viscous media and the internal damp-\ning precession of the nanoparticles magnetic moment. Our\nmain aims were the understanding of 1) the power loss be-\nhavior depending on di\u000berent parameters; 2) the role of\ndissipation mechanisms when they both are present; and\n3) the correlations between the mechanical rotation of a\nnanoparticle and the internal motion of its magnetic mo-\nment. The analysis of three approximations simultane-\nously helps us comprehend the restrictions and applicabil-\nity limits that, in turn, allows to systematize the results\nobtained by other authors. The relevance of our \fndings is\nclosely bounded with the application issues, such as heat-\ning rate during magnetic \ruid hyperthermia or absorption\nfrequency range of the microwave absorbing materials.\nThe comparison of the expressions for the power loss\nderived in the previous section allows a number of con-\nclusions, and some of them are rather unexpected at \frstsight. Firstly, the role of the internal magnetic motion is\nprimary. As follows from (23) and (29), the dependencies\nof the dimensionless power loss on the reduced frequency\nq(~\n) for the model of \fxed particle and for the model of\nviscously coupled nanoparticle with a \fnite anisotropy are\nsimilar: they both demonstrate a resonant behavior. At\nthe same time, for the model of rigid dipole it remains\nconstant (see (34)). Therefore, the dynamics of the mag-\nnetic moment represented by unit vector mdetermines\nthe resulting power loss in a wide range of realistic pa-\nrameters. But the quantitative comparison of these de-\npendencies lets us assume that the easy axis oscillations\ncan considerably modify the power loss induced by damp-\ning precession. The reasons for that are in the character\nof the collective motion of easy axis, which is represented\nby vector n, and magnetic moment m. Although the har-\nmonic motion takes place, the ratio of their phases and\namplitudes can lead to quite di\u000berent values of the en-\nergy dissipation in the system. Further we consider the\nbehavior of q(~\n) in context of the features of the mandn\nmotion.\nThe behavior of q(~\n) is caused by the features of co-\ne\u000ecientsam(~\n),bm(~\n),cm(~\n), anddm(~\n), which de-\ntermine the mdynamics, and an(~\n),bn(~\n),cn(~\n), and\ndn(~\n) de\fning the dynamics of n(see Fig. 2). As seen,\nfor frequencies far from the resonance one, vectors nand\nmalmost coincide and are rotated synchronously. Here,\nthe model of viscously coupled nanoparticle with a \fnite\nanisotropy and the model of \fxed particle yield very close\nvalues of the power loss. But near the resonance, in the\nvicinity of ~\n = 1, coe\u000ecients am(~\n),bm(~\n),cm(~\n), and\ndm(~\n) have the pronounced maxima and change the signs,\nwhile coe\u000ecients an(~\n),bn(~\n),cn(~\n), anddn(~\n) remain\nthe same. Therefore, vectors nandmare rotated in an\nasynchronous way now that leads to a larger angle between\nthe magnetic moment and the resulting or e\u000bective \feld\nheff. Together with increasing precession angle of m, this\ncauses the growth of the power loss compared with the\ncase of \fxed particle (see Fig. 3).\nFor small viscosity, vector nbecomes more suscepti-\nble to the external \feld, and the rotating magnetic mo-\nment can easily involve a whole nanoparticle into rotation.\nBut this does not induce a more intense motion in result.\nFirstly, a considerable decrease in the coe\u000ecients am(~\n),\nbm(~\n),cm(~\n), anddm(~\n) near the resonance takes place\nin comparison with the case of larger viscosity. Then, only\nbm(~\n) anddm(~\n) change the signs now (see Fig. 4). Fi-\nnally, the dependencies an(~\n),bn(~\n),cn(~\n), anddn(~\n)\nget the local maxima (Fig. 4) and slightly decrease in ab-\nsolute values in the vicinity of ~\n = 1. Therefore, the ef-\nfect of the pronounced asynchronous rotation of nandm,\nwhich is actual for the foregoing case, eliminates now, and\nthey become almost parallel for a whole range of frequen-\ncies. Since the angle between the magnetic moment and\nthe resulting \feld is reduced, the model of viscously cou-\npled nanoparticle with a \fnite anisotropy predicts lower\nvalues of the power loss than the model of \fxed particle\n6Figure 2: (Color online) The dependencies of the amplitudes of\ncoupled oscillations of the magnetic moment (21) and the easy axis\n(20) on the \feld frequency. The parameters used are M= 338 G,\nHa= 910 Oe,\u0011= 0:006 P,\u000b= 0:05 that corresponds to maghemite\nnanoparticles ( \r\u0000Fe2O3) in water at the temperature of 42\u000eC,\n\u001b=\u00001,h= 0:01,\u00120= 0:4\u0019,'0= 0:125\u0019.\nFigure 3: (Color online) The frequency dependencies of the power\nloss for the cases of rigid dipole (RD-model), \fxed particle (FP-\nmodel), and viscously coupled nanoparticle with a \fnite anisotropy\n(FA-model). The parameters used are the same as in the caption to\nFig. 2.\nFigure 4: (Color online) The dependencies of the amplitudes of\ncoupled oscillations of the magnetic moment (21) and the easy axis\n(20) on the \feld frequency. The parameters used are the same as in\nthe caption to Fig. 2, but \u0011= 4:0\u00005P.\nnear the resonance (Fig. 5).\nThe situation described above is an origin for extreme\nsensitivity of the power loss to the system parameters,\nwhich may be useful in the applications and can be uti-\nlized in a number of cases. In contrary, in other cases such\nsensitivity can be very undesirable, and we have to take\nmeasures to prevent it. Independently of the further pur-\nposes, one needs to investigate the in\ruence of the main\nparameters in detail. It is especially important for the de-\nsign of the nanoparticle ensembles with the speci\fed prop-\nerties for key applications, such as microwave absorbing\nor magnetic \ruid hyperthermia, where the heating or/and\nabsorbing rates are the primary characteristics.\nIn this regard, the similar parameters \u000band\u0011are the\nmost interesting. In Fig. 6a, the comparison of the power\nloss for two values of \u000bare plotted using the \fxed particle\napproximation and the approximation of viscously cou-\npled nanoparticle with a \fnite anisotropy. As expected,\nthe decrease in \u000bleads to the proportional increase in the\npower loss for both approximations. At the same time, the\nchange in\u0011results in di\u000berent behavior of the power loss\nobtained using the rigid dipole approximation and the ap-\nproximation of viscously coupled nanoparticle with a \fnite\nanisotropy (see Fig. 6b). For the \frst case, the propor-\ntional growth of q(~\n) with decreasing \u0011takes place. But\nfor the second case, account of the \fnite anisotropy leads\nto the opposite results. Here we report a nonlinear growth\ninq(~\n) with increasing viscosity \u0011. As it was explained\n7Figure 5: (Color online) The frequency dependencies of the power\nloss for the cases of rigid dipole (RD-model), \fxed particle (FP-\nmodel), and viscously coupled nanoparticle with a \fnite anisotropy\n(FA-model). The parameters used are the same as in the caption to\nFig. 2, but \u0011= 4:0\u00005P.\nabove, the origin of this e\u000bect lies in the relative motion\nof vectors nandm. Then, to estimate the applicability of\nthe model of rigid dipole, one needs to compare the power\nloss values for these two cases. As seen from Fig. 6b, vari-\nous situations are possible because there are two di\u000berent\nbehavior types when mis unlocked. The \frst type is the\nasynchronous oscillations of mandn, wherein the values\nofq(~\n) for the model of viscously coupled nanoparticle\nwith a \fnite anisotropy can be considerably larger than\nthe values predicted by the model of rigid dipole. The\nsecond type is the synchronous motion of the magnetic\nmoment and the easy axis. Here, both dissipation mecha-\nnisms are suppressed because the amplitudes of nandm\noscillations become smaller. As a result, the power loss for\nthe \fnite anisotropy case can be substantially lower than\nthe value obtained for the model of rigid dipole. This al-\nlows us to conclude about a low applicability of the model\nof rigid dipole in a high frequency limit.\nAnother important issue which needs to be accounted\nis the in\ruence of the external \feld orientation with re-\nspect to the nanoparticle position. As follows from (23),\n(29), (34), this orientation is de\fned by the polarization\ntype and the initial position of the easy axis. The model\nof rigid dipole predicts the di\u000berence of the power loss not\nmore than two times when \u001bvaries in the range of [ \u00001:::1].\nIn accordance with two other models, the dependence of\nthe power loss on \u001bis more strong. As seen from Fig. 7a,\nq(~\n) can be at least 10 times di\u000berent depending on \u001bfor\nthe model of viscously coupled nanoparticle with a \fnite\nanisotropy. Here we need to note that this dependence is\nnot linear and the lowest curve q(~\n) does not correspond\nto\u001b= 0 or\u001b=\u00061. The initial position of the easy\naxis given by angle \u00120essentially in\ruences the power loss\nas well. As seen from Fig. 7b, this di\u000berence may be at\nleast 20 times. Since nanoparticles in real ferro\ruids are\nnon-uniformly distributed, one can highlight the following.\nFirstly, the dipole interaction, which tries to arrange the\nensemble, can considerably in\ruence the power loss. Sec-\nondly, an external magnetic \feld gradient, which is used\nFigure 6: (Color online) The sensitivity of the power loss to the\nattenuation parameters. Plot (a): \fxed particle (FP-model) and\nviscously coupled nanoparticle with a \fnite anisotropy (FA-model)\nand di\u000berent values of the magnetic damping parameter \u000b. Plot (b):\nrigid dipole (RD-model) and viscously coupled nanoparticle with a\n\fnite anisotropy (FA-model) and di\u000berent values of the \u0011and results\nobtained for the cases rigid dipole (RD-model) and viscously coupled\nnanoparticle with a \fnite anisotropy (FA-model). The parameters\nused here and not stated in the \fgure legend are the same as in the\ncaption to Fig. 2, but \u00120= 0:25\u0019.\n8Figure 7: (Color online) The sensitivity of the power loss to the\norientation of the nanoparticle with respect to the external \feld for\nthe case of viscously coupled nanoparticle with a \fnite anisotropy\n(FA-model). The parameters used here and not stated in the \fgure\nlegend are the same as in the caption to Fig. 2, but \u00120= 0:25\u0019for\nthe plot (a) and \u001b= 1 for the plot (b).\nfor the ferro\ruid control during hyperthermia, also de\fnes\nthe power loss. And, thirdly, we can easily control the\npower loss in a wide range of values by a permanent exter-\nnal \feld, which speci\fes the direction of the nanoparticle\neasy axis.\nWe summarize our \fndings as follows. 1) The small os-\ncillations mode is considered for the coupled magnetic and\nmechanical motion for the viscously coupled nanoparticle\nwith a \fnite anisotropy. This mode takes place when the\namplitude of the external alternating \feld is much smaller\nthan the value of the nanoparticle uniaxial anisotropy \feld\n(H\u001cHa). 2) The damping precession of the magnetic\nmoment inside the nanoparticle primarily determines the\nvalue of the power loss and the resonance character of its\nfrequency dependence. 3) The power loss can be signi\f-\ncantly changed by the nanoparticle easy axis motion. For\nthe realistic system parameters, the power loss obtained\nfor the model of viscously coupled nanoparticle with a \f-\nnite anisotropy is larger than the value obtained for the\n\fxed particle model. 4) The decrease in the \ruid car-\nrier viscosity leads to the nonproportional decrease in the\npower loss, which near the resonance can be much smaller\nthan the value obtained for the \fxed particle model. Such\ncomplicated correlation between the magnetic dynamics\nand the mechanical motion does not allow to separate the\ncontributions of these two mechanisms into dissipation. 5)\nThe power loss is extremely sensitive to the system pa-rameters and the nanoparticle initial position. It should\nbe taken into account and can be used, for example, for\nthe control of the heating and absorbing rates. Although\nthe results are obtained in the dynamical approach, they\nestablish the limitation for more precise models which ac-\ncount thermal \ructuations and inter-particle interaction.\nAcknowledgements\nThe authors are grateful to the Ministry of Education\nand Science of Ukraine for partial \fnancial support under\nGrant No. 0116U002622.\nReferences\nReferences\n[1] R. Rosensweig, Heating magnetic \ruid with alternating mag-\nnetic \feld, Journal of Magnetism and Magnetic Materials 252\n(2002) 370 { 374, proceedings of the 9th International Con-\nference on Magnetic Fluids. doi:http://dx.doi.org/10.1016/\nS0304-8853(02)00706-0 .\nURL http://www.sciencedirect.com/science/article/pii/\nS0304885302007060\n[2] W. Andr a, H. Nowak, Magnetism in Medicine: A Handbook.\ndoi:10.1002/9783527610174 .\nURL http://onlinelibrary.wiley.com/book/10.1002/\n9783527610174\n[3] M. I. Shliomis, Magnetic \ruids, Soviet Physics Uspekhi 17 (2)\n(1974) 153.\nURL http://stacks.iop.org/0038-5670/17/i=2/a=R02\n[4] W. F. Brown, Thermal \ructuations of a single-domain particle,\nPhys. Rev. 130 (1963) 1677{1686. doi:10.1103/PhysRev.130.\n1677.\nURL https://link.aps.org/doi/10.1103/PhysRev.130.1677\n[5] A. Jordan, R. Scholz, P. Wust, H. Fhling, R. Felix, Magnetic\n\ruid hyperthermia (mfh): Cancer treatment with fACgmag-\nnetic \feld induced excitation of biocompatible superparamag-\nnetic nanoparticles, Journal of Magnetism and Magnetic Mate-\nrials 201 (1-3) (1999) 413 { 419. doi:http://dx.doi.org/10.\n1016/S0304-8853(99)00088-8 .\nURL http://www.sciencedirect.com/science/article/pii/\nS0304885399000888\n[6] Q. A. Pankhurst, J. Connolly, S. K. Jones, J. Dobson, Appli-\ncations of magnetic nanoparticles in biomedicine, Journal of\nPhysics D: Applied Physics 36 (13) (2003) R167.\nURL http://stacks.iop.org/0022-3727/36/i=13/a=201\n[7] Y. L. Raikher, V. I. Stepanov, Energy absorption by a mag-\nnetic nanoparticle suspension in a rotating \feld, Journal of\nExperimental and Theoretical Physics 112 (1) (2011) 173{177.\ndoi:10.1134/S1063776110061160 .\nURL http://dx.doi.org/10.1134/S1063776110061160\n[8] Y. L. Raikher, V. I. Stepanov, Power losses in a suspension of\nmagnetic dipoles under a rotating \feld, Phys. Rev. E 83 (2011)\n021401. doi:10.1103/PhysRevE.83.021401 .\nURL https://link.aps.org/doi/10.1103/PhysRevE.83.\n021401\n[9] V. V. Reva, T. V. Lyutyy, Microwave absorption by a rigid\ndipole in a viscous \ruid, in: 2016 II International Young Scien-\ntists Forum on Applied Physics and Engineering (YSF), 2016,\npp. 104{107. doi:10.1109/YSF.2016.7753812 .\n[10] B. U. Felderhof, R. B. Jones, Mean \feld theory of the nonlin-\near response of an interacting dipolar system with rotational\ndi\u000busion to an oscillating \feld, Journal of Physics: Condensed\nMatter 15 (23) (2003) 4011.\nURL http://stacks.iop.org/0953-8984/15/i=23/a=313\n9[11] T. V. Lyutyy, S. I. Denisov, V. V. Reva, Y. S. Bystrik, Ro-\ntational properties of ferromagnetic nanoparticles driven by a\nprecessing magnetic \feld in a viscous \ruid, Phys. Rev. E 92\n(2015) 042312. doi:10.1103/PhysRevE.92.042312 .\nURL http://link.aps.org/doi/10.1103/PhysRevE.92.042312\n[12] I. N\u0013 andori, J. R\u0013 acz, Magnetic particle hyperthermia: Power\nlosses under circularly polarized \feld in anisotropic nanoparti-\ncles, Phys. Rev. E 86 (2012) 061404. doi:10.1103/PhysRevE.\n86.061404 .\nURL http://link.aps.org/doi/10.1103/PhysRevE.86.061404\n[13] J. R\u0013 acz, P. F. de Ch^ atel, I. A. Szab\u0013 o, L. Szunyogh, I. N\u0013 andori,\nImproved e\u000eciency of heat generation in nonlinear dynamics of\nmagnetic nanoparticles, Phys. Rev. E 93 (2016) 012607. doi:\n10.1103/PhysRevE.93.012607 .\nURL http://link.aps.org/doi/10.1103/PhysRevE.93.012607\n[14] T. V. Lyutyy, S. I. Denisov, A. Y. Peletskyi, C. Binns, En-\nergy dissipation in single-domain ferromagnetic nanoparticles:\nDynamical approach, Phys. Rev. B 91 (2015) 054425. doi:\n10.1103/PhysRevB.91.054425 .\nURL http://link.aps.org/doi/10.1103/PhysRevB.91.054425\n[15] C. Haase, U. Nowak, Role of dipole-dipole interactions for hy-\nperthermia heating of magnetic nanoparticle ensembles, Phys.\nRev. B 85 (2012) 045435. doi:10.1103/PhysRevB.85.045435 .\nURL https://link.aps.org/doi/10.1103/PhysRevB.85.\n045435\n[16] G. T. Landi, Role of dipolar interaction in mag-\nnetic hyperthermia, Phys. Rev. B 89 (2014) 014403.\ndoi:10.1103/PhysRevB.89.014403 .\nURL https://link.aps.org/doi/10.1103/PhysRevB.89.\n014403\n[17] A. O. Tsebers, Simultaneous rotational di\u000busion of the magnetic\nmoment and the solid matrix of a single-domain ferromagnetic\nparticle, Magnetohydrodynamics 11 (3) (1975) 273{278.\nURL http://mhd.sal.lv/contents/1975/3/MG.11.3.2.R.html\n[18] H. Mamiya, B. Jeyadevan, Hyperthermic e\u000bects of dissipa-\ntive structures of magnetic nanoparticles in large alternating\nmagnetic \felds, Scienti\fc Reports 1 (2011) 157 EP, article.\ndoi:10.1038/srep00157 .\nURL http://dx.doi.org/10.1038/srep00157\n[19] N. A. Usov, B. Y. Liubimov, Dynamics of magnetic nanopar-\nticle in a viscous liquid: Application to magnetic nanoparti-\ncle hyperthermia, Journal of Applied Physics 112 (2) (2012)\n023901. arXiv:http://dx.doi.org/10.1063/1.4737126 ,doi:\n10.1063/1.4737126 .\nURL http://dx.doi.org/10.1063/1.4737126\n[20] K. D. Usadel, C. Usadel, Dynamics of magnetic single do-\nmain particles embedded in a viscous liquid, Journal of Applied\nPhysics 118 (23) (2015) 234303. arXiv:http://dx.doi.org/10.\n1063/1.4937919 ,doi:10.1063/1.4937919 .\nURL http://dx.doi.org/10.1063/1.4937919\n[21] N. Usov, B. Y. Liubimov, Magnetic nanoparticle motion in ex-\nternal magnetic \feld, Journal of Magnetism and Magnetic Ma-\nterials 385 (2015) 339 { 346. doi:http://dx.doi.org/10.1016/\nj.jmmm.2015.03.035 .\nURL http://www.sciencedirect.com/science/article/pii/\nS0304885315002565\n[22] H. Xi, K.-Z. Gao, Y. Shi, S. Xue, Precessional dynamics of\nsingle-domain magnetic nanoparticles driven by small ac mag-\nnetic \felds, Journal of Physics D: Applied Physics 39 (22) (2006)\n4746.\nURL http://stacks.iop.org/0022-3727/39/i=22/a=002\n[23] J. C\u0016 \u0010murs, A. C\u0016 ebers, Dynamics of anisotropic superparamag-\nnetic particles in a precessing magnetic \feld, Phys. Rev. E 87\n(2013) 062318. doi:10.1103/PhysRevE.87.062318 .\nURL https://link.aps.org/doi/10.1103/PhysRevE.87.\n062318\n[24] H. Keshtgar, S. Streib, A. Kamra, Y. M. Blanter, G. E. W.\nBauer, Magnetomechanical coupling and ferromagnetic res-\nonance in magnetic nanoparticles, Phys. Rev. B 95 (2017)\n134447. doi:10.1103/PhysRevB.95.134447 .\nURL https://link.aps.org/doi/10.1103/PhysRevB.95.134447\n[25] M. I. Dar, S. A. Shivashankar, Single crystalline magnetite,\nmaghemite, and hematite nanoparticles with rich coercivity,\nRSC Adv. 4 (2014) 4105{4113. doi:10.1039/C3RA45457F .\nURL http://dx.doi.org/10.1039/C3RA45457F\n[26] Y. L. Raikher, M. I. Shliomis, The e\u000bective \feld method in\nthe orientational kinetics of magnetic \ruids and liquid crystals,\nAdvances in Chemical Physics 87 (1994) 595{751. doi:http:\n//dx.doi.org/10.1002/9780470141465.ch8 .\nURL http://dx.doi.org/10.1002/9780470141465.ch8\n[27] J. Frenkel, Kinetic Theory of Liquids, Dover Publications,\nDover, 1955.\nURL https://books.google.com.ua/books?id=ORdSQwAACAAJ\n10"    },    {        "title": "1706.01185v1.Consistent_microscopic_analysis_of_spin_pumping_effects.pdf",        "content": "Consistent microscopic analysis of spin pumping e\u000bects\nGen Tatara\nRIKEN Center for Emergent Matter Science (CEMS),\n2-1 Hirosawa, Wako, Saitama, 351-0198 Japan\nShigemi Mizukami\nWPI - Advanced Insitute for Materials Research,\nTohoku University Katahira 2-1-1, Sendai, Japan\n(Dated: October 20, 2018)\nAbstract\nWe present a consistent microscopic study of spin pumping e\u000bects for both metallic and insulating\nferromagnets. As for metallic case, we present a simple quantum mechanical picture of the e\u000bect as\ndue to the electron spin \rip as a result of a nonadiabatic (o\u000b-diagonal) spin gauge \feld. The e\u000bect of\ninterface spin-orbit interaction is brie\ry discussed. We also carry out \feld-theoretic calculation to\ndiscuss on the equal footing the spin current generation and torque e\u000bects such as enhanced Gilbert\ndamping constant and shift of precession frequency both in metallic and insulating cases. For thick\nferromagnetic metal, our study reproduces results of previous theories such as the correspondence\nbetween the dc component of the spin current and enhancement of the damping. For thin metal\nand insulator, the relation turns out to be modi\fed. For the insulating case, driven locally by\ninterfacesdexchange interaction due to magnetic proximity e\u000bect, physical mechanism is distinct\nfrom the metallic case. Further study of proximity e\u000bect and interface spin-orbit interaction would\nbe crucial to interpret experimental results in particular for insulators.\n1arXiv:1706.01185v1  [cond-mat.mes-hall]  5 Jun 2017I. INTRODUCTION\nSpin current generation is of a fundamental importance in spintronics. A dynamic method\nusing magnetization precession induced by an applied magnetic \feld, called the spin pumping\ne\u000bect, turns out to be particularly useful1and is widely used in a junction of a ferromagnet\n(F) and a normal metal (N)(Fig. 1). The generated spin current density (in unit of A/m2)\nhas two independent components, proportional to _nandn\u0002_n, wherenis a unit vector\ndescribing the direction of localized spin, and thus is represented phenomenologically as\njs=e\n4\u0019(Arn\u0002_n+Ai_n); (1)\nwhereeis the elementally electric charge and ArandAiare phenomenological constants\nhaving unit of 1 =m2. Spin pumping e\u000bect was theoretically formulated by Tserkovnyak et al.2\nby use of scattering matrix approach3. This approach, widely applied in mesoscopic physics,\ndescribes transport phenomena in terms of transmission and re\rection amplitudes (scattering\nmatrix), and provides quantum mechanical pictures of the phenomena without calculating\nexplicitly the amplitudes. Tserkovnyak et al. applied the scattering matrix formulation\nof general adiabatic pumping4,5to the spin-polarized case. The spin pumping e\u000bect was\ndescribed in Ref.2in terms of spin-dependent transmission and re\rection coe\u000ecients at the\nFN interface, and it was demonstrated that the two parameters, ArandAi, are the real\nand the imaginary part of a complex parameter called the spin mixing conductance. The\nspin mixing conductance, which is represented by transmission and re\rection coe\u000ecients,\nturned out to be a convenient parameter for discussing spin current generation and other\ne\u000bects like the inverse spin-Hall e\u000bect. Nevertheless, scattering approach hides microscopic\nphysical pictures of what is going on, as the scattering coe\u000ecients are not fundamental\nmaterial parameters but are composite quantities of Fermi wave vector, electron e\u000bective\nmass and the interface properties.\nE\u000bects of slowly-varying potential is described in a physically straightforward and clear\nmanner by use of a unitary transformation that represents the time-dependence. (See Sec.\nII A for details.) The laboratory frame wave function under time-dependent potential, j (t)i,\nis written in terms of a static ground state ('rotated frame' wave function) j\u001eiand a unitary\nmatrixU(t) asj (t)i=U(t)j\u001ei. The time-derivative @tis then replaced by a covariant\nderivative,@t+ (U\u00001@tU), and the e\u000bects of time-dependence are represented by (the time-\ncomponent of) an e\u000bective gauge \feld, A \u0011 \u0000i(U\u00001@tU) (See Eq. (12)). In the same\n2FIG. 1. Spin pumping e\u000bect in a junction of ferromagnet (F) and normal metal (N). Dynamic\nmagnetization n(t) generates a spin current jsthrough the interface.\nmanner as the electromagnetic gauge \feld, the e\u000bective gauge \feld generates a current if\nspatial homogeneity is present (like in junctions) and this is a physical origin of adiabatic\npumping e\u000bect in metals.\nIn the perturbative regime or in insulators, a simple picture instead of e\u000bective gauge \feld\ncan be presented. Let us focus on the case driven by an sdexchange interaction, Jsdn(t)\u0001\u001b,\nwhereJsdis a coupling constant and \u001bis the electron spin. Considering the second-order\ne\u000bect of the sdexchange interaction, the electron wave function has a contribution of a\ntime-dependent amplitude\nU(t1;t2) = (Jsd)2(n(t1)\u0001\u001b)(n(t2)\u0001\u001b) = (Jsd)2[(n(t1)\u0001n(t2)) +i[n(t1)\u0002n(t2)]\u0001\u001b];(2)\nwheret1andt2are the time of the interactions. The \frst term on the right-hand side,\nrepresenting the amplitude for charge degrees of freedom, is neglected. The spin contribution\nvanishes for static spin con\fguration, as is natural, while for slowly varying case, it reads\nU(t1;t2)'\u0000i(t1\u0000t2)(Jsd)2(n\u0002_n)(t1)\u0001\u001b: (3)\nAs a result of this amplitude, spin accumulation and spin current is induced proportional to\nn\u0002_n. The fact indicates that n\u0002_nplays a role of an e\u000bective scalar potential or voltage in\nelectromagnetism, as we shall demonstrate in Sec. VII B for insulators. (The factor of time\ndi\u000berence is written in terms of derivative with respect to energy or angular frequency in a\nrigorous derivation. See for example, Eqs. (129)(132).) The essence of spin pumping e\u000bect\nis therefore the non-commutativity of spin operators. The above picture in the perturbative\nregime naturally leads to an e\u000bective gauge \feld in the strong coupling limit6.\n3The same scenario applies for cases of spatial variation of spin, and an equilibrium spin\ncurrent proportional to n\u0002rinemerges, where idenotes the direction of spatial variation7.\nThe spin pumping e\u000bect is therefore the time analog of the equilibrium spin current induced\nby vector spin chirality. Moreover, charge current emerges from the third-order process from\nthe identity6\ntr[(n1\u0001\u001b)(n2\u0001\u001b)(n3\u0001\u001b)] = 2in1\u0001(n2\u0002n3); (4)\nand this factor, a scalar spin chirality, is the analog of the spin Berry phase in the pertur-\nbative regime. The spin pumping e\u000bect and spin Berry's phase and spin motive force have\nthe same physical root, namely the non-commutative spin algebra.\nFrom the scattering matrix theory view point the cases of metallic and insulating fer-\nromagnet make no di\u000berence as what conduction electrons in the normal metal see is the\ninterface. From physical viewpoints, such treatment appears too crude. Unlike the metallic\ncase discussed above, in the case of insulator ferromagnet, the coupling between the mag-\nnetization and the conduction electron in normal metal occurs due to a magnetic proximity\ne\u000bect at the interface. Thus the spin pumping by an insulator ferromagnet seems to be a\nlocally-induced perturbative e\u000bect rather than a transport induced by a driving force due to\na generalized gauge \feld. We therefore need to apply di\u000berent approaches for the two cases\nas brie\ry argued above. In the insulating case, one may think that magnon spin current is\ngenerated inside the ferromagnet because magnon itself couples to an e\u000bective gauge \feld8\nsimilarly to the electrons in metallic case. This is not, however, true, because the gauge \feld\nfor magnon is abelian (U(1)). Although scattering matrix approach apparently seems to\napply to both metallic and insulating cases, it would be instructive to present in this paper\na consistent microscopic description of the e\u000bects to see di\u000berent physics governing the two\ncases.\nA. Brief overview of theories and scope of the paper\nBefore carrying out calculation, let us overview history of theoretical studies of spin\npumping e\u000bect. Spin current generation in a metallic junction was originally discussed by\nSilsbee9before Tserkovnyak et al. It was shown there that dynamic magnetization induces\nspin accumulation at the interface, resulting in a di\u000busive \row of spin in the normal metal.\n4Although of experimental curiosity at that time was the interface spin accumulation, which\nenhances the signal of conduction electron spin resonance, it would be fair to say that Silsbee\npointed out the `spin pumping e\u000bect'.\nIn Ref.2, spin pumping e\u000bect was originally argued in the context of enhancement of\nGilbert damping in FN junction, which had been a hot issue after the study by Berger10,\nwho studied the case of FNF junction based on a quantum mechanical argument. Berger\ndiscussed that when a normal metal is attached to a ferromagnet, the damping of ferromagnet\nis enhanced as a result of spin polarization formed in the normal metal, and the e\u000bect was\nexperimentally con\frmed by Mizukami11. Tserkovnyak et al. pointed out that the e\u000bect has\na di\u000berent interpretation of the counter action of spin current generation, because the spin\ncurrent injected into the normal metal indicates a change of spin angular momentum or a\ntorque on ferromagnet. In fact, the equation of motion for the magnetization of ferromagnet\nreads\n_n=\u0000\rB\u0002n\u0000\u000bn\u0002_n\u0000a3\neSdjs; (5)\nwhere\ris the gyromagnetic ratio, \u000bis the Gilbert damping coe\u000ecient, dis the thickness\nof the ferromagnet, Sis the magnetude of localized spin, and ais the lattice constant. Spin\ncurrent of Eq. (1) thus indicates that the gyromagnetic ratio and the the Gilbert damping\ncoe\u000ecient are modi\fed by the spin pumping e\u000bect to be2\n~\u000b=\u000b+a3\n4\u0019SdAr\n~\r=\r\u0014\n1 +a3\n4\u0019SdAi\u0015\u00001\n: (6)\nThe spin pumping e\u000bect is therefore detected by measuring the e\u000bective damping constant\nand gyromagnetic ratio. The formula (6) is, however, based on a naive picture neglecting\nthe position-dependence of the damping torque and the relation between the pumped spin\ncurrent amplitude and damping or \rwould not be so simple in reality. (See Sec. V.)\nThe issue of damping in FN junction was formulated based on linear-response theory by\nSimanek and Heinirch12,13. They showed that the damping coe\u000ecient is given by the \frst-\norder derivative with respect to the angular frequency !of the imaginary part of the spin\ncorrelation function and argued that the damping e\u000bect is consistent with the Tserkovnyak's\nspin pumping e\u000bect. Recently, a microscopic formulation of spin pumping e\u000bect in metallic\njunction was provided by Chen and Zhang14and one of the author15by use of the Green's\n5functions, and a transparent microscopic picture of pumping e\u000bect was provided. Scattering\nrepresentation and Green's function one are related14because the asymptotic behaviors of\nthe Green's functions at long distance are governed by transmission coe\u000ecient16. In the\nstudy of Ref.15, the uniform ferromagnet was treated as a dot having only two degrees\nof freedom of spin. Such simpli\fcation neglects the dependence on electron wave vectors\nin ferromagnets and thus cannot discuss the the case of inhomogeneous magnetization or\nposition-dependence of spin damping.\nThe aim of this paper is to provide a microscopic and consistent theoretical formula-\ntion of spin pumping e\u000bect for metallic and insulating ferromagnets. We do not rely on\nthe scattering approach. Instead we provide elementary quantum mechanical argument to\ndemonstrated that spin current generation is a natural consequence of magnetization dy-\nnamics (Sec. II). Based on the formulation, the e\u000bect of interface spin-orbit interaction is\ndiscussed in Sec. III. We also provide a rigorous formulation based on \feld-theoretic ap-\nproach emploied in Ref.15in Sec. IV. We also reproduce within the same framework Berger's\nresult10that the spin pumping e\u000bect is equivalent to the enhancement of the spin damping\n(Sec. V). E\u000bect of inhomogeneous magnetization is brie\ry discussed in Sec. VI.\nCase of insulating ferromagnet is studied in Sec. VII assuming that the pumping is\ninduced by an interface exchange interaction between the magnetization and conduction\nelectron in normal metal, namely, by magnetic proximity e\u000bect. The interaction is treated\nperturbatively similarly to Refs.17,18. The dominant contribution to the spin current, the\none linear in the interface exchange interaction, turns out to be proportional to _n, while the\none proportional to n\u0002_nis weaker if the proximity e\u000bect is weak.\nThe contribution from the magnon, magnetization \ructuation, is also studied. As has\nbeen argued8, a gauge \feld for magnon emerges from magnetization dynamics. It is, however,\nan adiabatic one diagonal in spin, which acts as chemical potential for magnon giving rise\nonly to adiabatic spin polarization proportional to n. This is in sharp contrast to the\nmetallic case, where electrons are directly driven by spin-\rip component of spin gauge \feld,\nresulting in perpendicular spin accumulation, i.e., along _nandn\u0002_n. The excitation\nin ferromagnet when magnetization is time-dependent is therefore di\u000berent for metallic and\ninsulating cases. We show that magnon excitation nevertheless generates perpendicular spin\ncurrent,n\u0002_n, in the normal metal as a result of annihilation and creation at the interface,\nwhich in turn \rips electron spin. The result of magnon-driven contribution agrees with the\n6one in previous study19carried out in the context of thermally-driven spin pumping ('spin\nSeebeck' e\u000bect). It is demonstrated that the magnon-induced spin current depends linearly\non the temperature at high temperature compared to magnon energy. The amplitude of\nmagnon-driven spin current provides the magnitude of magnetic proximity e\u000bect.\nIn our analysis, we calculate consistently the pumped spin current and change of the\nGilbert damping and resonant frequency and obtain the relations among them. It is shown\nthat the spin mixing conductance scenario saying that the magnitude of spin current pro-\nportional ton\u0002_nis given by the enhancement factor of the Gilbert damping constant2,\napplies only the case of thick ferromagnetic metal. For thin metallic case and insulator case,\ndi\u000berent relations hold (See Sec. VIII.).\nII. QUANTUM MECHANICAL DESCRIPTION OF METALLIC CASE\nIn this section, we derive the spin current generated by the magnetization dynamics\nof metallic ferromagnet by a quantum mechanical argument. It is sometimes useful for\nintuitive understanding, although the description may lack clearness as it cannot handle\nmany-particle nature like particle distributions. In Sec. IV we formulate the problem in the\n\feld-theoretic language.\nA. Electrons in ferromagnet with dynamic magnetization\nThe model we consider is a junction of metallic ferromagnet (F) and a normal metal (N).\nThe magnetization (or localized spins) in the ferromagnet is treated as spatially uniform but\nchanging with time slowly. As a result of strong sdexchange interaction, the conduction\nelectron's spin follows instantaneous directions of localized spins, i.e., the system is in the\nadiabatic limit. The quantum mechanical Hamiltonian for the ferromagnet is\nHF=\u0000r2\n2m\u0000\u000fF\u0000Mn(t)\u0001\u001b; (7)\nwheremis the electron's mass, \u001bis a vector of Pauli matrices, Mrepresents the energy split-\nting due to the sdexchange interaction and n(t) is a time-dependent unit vector denoting\nthe localized spin direction. The energy is measured from the Fermi energy \u000fF.\n7As a result of the sdexchange interaction, the electron's spin wave function is given by20\njni\u0011cos\u0012\n2j\"i+ sin\u0012\n2ei\u001ej#i (8)\nwherej\"iandj#irepresent the spin up and down states, respectively, and ( \u0012;\u001e) are polar\ncoordinates for n. To treat slowly varying localized spin, we switch to a rotating frame where\nthe spin direction is de\fned with respect to instantaneous direction n7. This corresponds\nto diagonalizing the Hamiltonian at each time by introducing a unitary matrix U(t) as\njn(t)i\u0011U(t)j\"i; (9)\nwhere\nU(r) =0\n@cos\u0012\n2sin\u0012\n2e\u0000i\u001e\nsin\u0012\n2ei\u001e\u0000cos\u0012\n21\nA; (10)\nwhere states are in vector representation, i.e., j\"i=0\n@1\n01\nAandj#i=0\n@0\n11\nA. The rotated\nHamiltonian is diagonalized as (in the momentum representation)\neHF\u0011U\u00001HFU=\u000fk\u0000M\u001bz; (11)\nwhere\u000fk\u0011k2\n2m\u0000\u000fFis the kinetic energy in the momentum representation (Fig. 2). In general,\nFIG. 2. Unitary transformation Ufor conduction electron in ferromagnet converts the original\nHamiltonian HFinto a diagonalized uniformly spin-polarized Hamiltonian eHFand an interaction\nwith spin gauge \feld, As;t\u0001\u001b.\nwhen a statej ifor a time-dependent Hamiltonian H(t), satisfying the Schr odinger equation\ni@\n@tj i=H(t)j i, is written in terms of a state j iconnected by a unitary transformation\nj\u001ei\u0011U\u00001j i, the new state satis\fes a modi\fed Schr odinger equation\n\u0012\ni@\n@t+iU\u00001@\n@tU\u0013\nj\u001ei=~Hj\u001ei; (12)\n8where ~H\u0011U\u00001HU. Namely, there arises a gauge \feld \u0000iU\u00001@\n@tUin the new frame j\u001ei. In\nthe present case of dynamic localized spin, the gauge \feld has three components (su\u000ex t\ndenotes the time-component);\nAs;t\u0011\u0000iU\u00001@\n@tU\u0011As;t\u0001\u001b; (13)\nexplicitly given as7\nAs;t=1\n20\nBBB@\u0000@t\u0012sin\u001e\u0000sin\u0012cos\u001e@t\u001e\n@t\u0012cos\u001e\u0000sin\u0012sin\u001e@t\u001e\n(1\u0000cos\u0012)@t\u001e1\nCCCA: (14)\nIncluding the gauge \feld in the Hamiltonian, the e\u000bective Hamiltonian in the rotated frame\nreads\neHe\u000b\nF\u0011eHF+As;t\u0001\u001b=0\n@\u000fk\u0000M\u0000Az\ns;tA\u0000\ns;t\nA+\ns;t\u000fk+M+Az\ns;t1\nA (15)\nwhereA\u0006\ns;t\u0011Ax\ns;t\u0006iAy\ns;t. We see that the adiabatic ( z) component of the gauge \feld, Az\ns;t,\nacts as a spin-dependent chemical potential (spin chemical potential) generated by dynamic\nmagnetization, while non-adiabatic ( xandy) components causes spin mixing. In the case\nof uniform magnetization we consider, the mixing is between the electrons with di\u000berent\nspin\"and#but having the same wave vector k, because the gauge \feld A\u0006\ns;tcarries no\nmomentum. This leads to a mixing of states having an excitation energy of Mas shown in\nFig. 3. In low energy transport e\u000bects, what concern are the electrons at the Fermi energy;\nThe wave vector kshould be chosen as kF+andkF\u0000, the Fermi wave vectors for \"and#\nelectrons, respectively. (E\u000bects of \fnite momentum transfer is discussed in Sec. VI. )\nThe Hamiltonian Eq. (15) is diagonalized to obtain energy eigenvalues of ~ \u000fk\u001b=\u000fk\u0000\n\u001bq\n(M+Az\ns;t)2+jA?\ns;tj2, wherejA?\ns;tj2\u0011A+\ns;tA\u0000\ns;tand\u001b=\u0006represents spin (\"and#cor-\nrespond to + and \u0000, respectively). We are interested in the adiabatic limit, and so the\ncontribution lowest-order, namely, the \frst order, in the perpendicular component, A?\ns;t, is\nsu\u000ecient. In the present rotating-frame approach, the gauge \feld is treated as a static po-\ntential, since it already include time-derivative to the linear order (Eq. (14)). Moreover, the\nadiabatic component of the gauge \feld, Az\ns;t, is neglected, as it modi\fes the spin pumping\nonly at the second-order of time-derivative. The energy eigenvalues, \u000fk\u001b'\u000fk\u0000\u001bM, are\n9thus una\u000bected by the gauge \feld, while the eigenstates to the linear order read\njk\"iF\u0011jk\"i\u0000A+\ns;t\nMjk#i\njk#iF\u0011jk#i+A\u0000\ns;t\nMjk\"i; (16)\ncorresponding to energy of \u000fk+and\u000fk\u0000, respectively. For low energy transport, states we\nneed to consider are the following two having spin-dependent Fermi wave vectors, kF\u001bfor\n\u001b=\";#, namely\njkF\"\"iF=jkF\"\"i\u0000A+\ns;t\nMjkF\"#i\njkF##iF=jkF##i+A\u0000\ns;t\nMjkF#\"i: (17)\nFIG. 3. For uniform magnetization, the non-adiabatic components of the gauge \feld, A\u0006\ns;t, induces\na spin \rip conserving the momentum.\nB. Spin current induced in the normal metal\nSpin pumping e\u000bect is now studied by taking account of the interface hopping e\u000bects on\nstates in Eq. (17). The interface hopping amplitude of electron in F to N with spin \u001bis\ndenoted by ~t\u001band the amplitude from N to F is ~t\u0003\n\u001b. We assume that the spin-dependence\nof electron state in F is governed by the relative angle to the magnetization vector, and\nhence the spin \u001bis the one in the rotated frame. Assuming moreover that there is no spin\n\rip scattering at the interface, the amplitude ~t\u001bis diagonal in spin. (Interface spin-orbit\ninteraction is considered in Sec. III.) The spin wave function formed in the N region at the\n10interface as a result of the state in F (Eq. (17)) is then\njkF\"iN\u0011~tjkF\"i=~t\"jkF\"i\u0000 ~t#A+\ns;t\nMjkF#i\njkF#iN\u0011~tjkF#i=~t#jkF#i+~t\"A\u0000\ns;t\nMjkF\"i; (18)\nwherekFis the Fermi wave vector of N electron. The spin density induced in N region at\nthe interface is therefore\nes(N)=1\n2(NhkF\"j\u001bjkF\"iN\u0017\"+NhkF#j\u001bjkF#iN\u0017#) (19)\nwhere\u0017\u001bis the spin-dependent density of states of F electron at the Fermi energy. It reads\nes(N)=1\n2X\n\u001b\u0017\u001bT\u001b\u001b^z\u0000\u0017\"\u0000\u0017#\nM\u0000\nRe[T\"#]A?\ns;t+ Im[T\"#](^z\u0002A?\ns;t)\u0001\n(20)\nwhereA?\ns;t= (Ax\ns;t;Ay\ns;t;0) =As;t\u0000^zAz\ns;tis the transverse (non-adiabatic) components of\nspin gauge \feld and\nT\u001b\u001b0\u0011~t\u0003\n\u001b~t\u001b0: (21)\nSpin density of Eq. (20) is in the rotated frame. The spin polarization in the laboratory\nframe is obtained by a rotation matrix Rij, de\fned by\nU\u00001\u001biU\u0011Rij\u001bj; (22)\nas\ns(N)\ni=Rijes(N)\nj: (23)\nExplicitly,Rij= 2mimj\u0000\u000eij, wherem\u0011\u0000\nsin\u0012\n2cos\u001e;sin\u0012\n2sin\u001e;cos\u0012\n2\u00017. Using\nRij(A?\ns;t)j=\u00001\n2(n\u0002_n)i\nRij(^z\u0002A?\ns;t)j=\u00001\n2_ni; (24)\nandRiz=ni, the induced interface spin density is \fnally obtained as\ns(N)=\u0010s\n0n+ Re[\u0010s](n\u0002_n) + Im[\u0010s]_n (25)\n11where\n\u0010s\n0\u00111\n2X\n\u001b\u0017\u001bT\u001b\u001b\n\u0010s\u0011\u0017\"\u0000\u0017#\n2MT\"#: (26)\nSince the N electrons contributing to induced spin density is those at the Fermi energy,\nthe spin current is simply proportional to the induced spin density as jsN=kF\nms(N), resulting\nin\nj(N)\ns=kF\nm\u0010s\n0n+kF\nmRe[\u0010s](n\u0002_n) +kF\nmIm[\u0010s]_n: (27)\nThis is the result of spin current at the interface. The pumping e\u000eciency is determined\nby the product of hopping amplitudes t\"andt\u0003\n#. The spin mixing conductance de\fned in\nRef.2corresponds to iT\"#. If spin mixing e\u000bects due to spin-orbit interaction is neglected at\nthe interface, the hopping amplitudes t\u001bare chosen as real, and Im[ \u0010s] = 0. If spin current\nproportional to _nis measured, it would be useful tool to estimate the strength of interface\nspin-orbit interaction, as discussed in Sec. III.\nIt should be noted that the spin pumping e\u000bect at the linear order in time-derivative is\nmapped to a static problem of spin polarization formed by a static spin-mixing potential in\nthe rotated frame as was mentioned in Ref.15. The rotate frame approach employed here\nprovides clear physical picture, as it grasps the low energy dynamics in a mathematically\nproper manner. In this approach, as we have seen, it is clearly seen that pumping of spin\ncurrent arises as a result of o\u000b-diagonal components of the spin gauge \feld that cause\nelectron spin \rip. Important role of nonadiabaticity is also indicated in a recent analysis\nbased on the full counting statistics21. In the strict sense, spin pumping e\u000bect is a result of a\nnon-adiabatic process including state change. The same goes for general adiabatic pumping;\nSome sort of state change is necessary for current generation, although the nonadiabaticity\nis obscured in the conventional \\adiabatic\\ argument focusing on the wave function in\nthe laboratory frame. In the case of slowly-varying external potential with frequency \nacting on electrons, the state change is represented by the Fermi distribution di\u000berence,\nf(!+ \n)\u0000f(!)'\nf0(!), where!is the electron frequency3,4. The existence of a factor of\nf0clearly indicates that a state change or nonadiabaticity is necessary for current pumping.\n12III. EFFECTS OF INTERFACE SPIN-ORBIT INTERACTION\nIn this section, we discuss the e\u000bect of spin-orbit interaction at the interface, which\nmodi\fes hopping amplitude ~t\u001b. We particularly focus on that linear in the wave vector,\nnamely the interaction represented in the continuum representation by a Hamiltonian\nHso=a2\u000e(x)X\nij\rijki\u001bj; (28)\nwhere\rijis a coe\u000ecient having the unit of energy representing the spin-orbit interaction,\nais the lattice constant, and the interface is chosen as at x= 0. Assuming that spin-\norbit interaction is weaker than the sdexchange interaction in F, we carry out a unitary\ntransformation to which diagonalize the sdinteraction to obtain\nHso=a2\u000e(x)X\nije\rijki\u001bj; (29)\nwheree\rij\u0011P\nl\rilRlj, withRijbeing a rotation matrix de\fned by Eq. (22). This spin-\norbit interaction modi\fes diagonal hopping amplitude ~tiin the direction iat the interface\nto become a complex as\neti=~t0\ni\u0000iX\nje\rij\u001bj: (30)\n(In this section, we denote the total hopping amplitude including the interface spin-orbit\ninteraction by etand the one without by et0.) We consider the hopping amplitude perpendic-\nular to the interface, i.e., along the xdirection, and suppress the su\u000ex irepresenting the\ndirection. In the matrix representation for spin the hopping amplitude is\net(\u0011etx) =0\n@et\"et\"#\net#\"et#1\nA; (31)\nwhere\net\"=~t0\n\"\u0000ie\rxzet#=~t0\n#+ie\rxz\net\"#=i(e\rxx+ie\rxy) et#\"=i(e\rxx\u0000ie\rxy): (32)\nLet us discuss how the spin pumping e\u000bect discussed in Sec. II B is modi\fed when the\nhopping amplitude is a matrix of Eq. (31). The spin pumping e\u000eciency is written as in Eqs.\n13(21)(26). In the absence of spin-orbit interaction hopping amplitude ~tis chosen as real, and\nthus the contribution proportional to n\u0002_nin Eq. (27) is dominant. Spin-orbit interaction\nenhances the other contribution proportional to _nbecause it gives rise to an imaginary part.\nMoreover, it leads to spin mixing at the interface, modifying the spin accumulation formed\nin the N region at the interface.\nThe electron states in the N region at the interface are now given instead of Eq. (18) by\nthe following two states (choosing basis as0\n@jkF\"i\njkF#i1\nA)\njkF\"iN\u0011etjkF\"\"iF=0\n@et\"\u0000et\"#A+\ns;t\nM\net#\"\u0000et#A+\ns;t\nM1\nA\njkF#iN\u0011etjkF##iF=0\n@et\"#+et\"A\u0000\ns;t\nM\net#+et#\"A\u0000\ns;t\nM1\nA: (33)\nThe pumped (i.e., linear in the gauge \feld) spin density for these two states are\nNhkF\"j\u001bjkF\"iN=\u00002\nM(A?\ns;tRe[Ttot\n\"#] + (^z\u0002A?\ns;t)Im[Ttot\n\"#]\n+Re[(et\"#)\u0003et#\"]0\nBBB@Ax\ns;t\n\u0000Ay\ns;t\n01\nCCCA+ Im[(et\"#)\u0003et#\"]0\nBBB@Ay\ns;t\nAx\ns;t\n01\nCCCA1\nCCCA\n\u0000^z(Ax\ns;tRe[(et\")\u0003et\"#\u0000et#(et#\")\u0003]\u0000Ay\ns;tIm[(et\")\u0003et\"#\u0000et#(et#\")\u0003]) (34)\nNhkF#j\u001bjkF#iN=2\nM(A?\ns;tRe[Ttot\n\"#] + (^z\u0002A?\ns;t)Im[Ttot\n\"#]\n+Re[(et\"#)\u0003et#\"]0\nBBB@Ax\ns;t\n\u0000Ay\ns;t\n01\nCCCA+ Im[(et\"#)\u0003et#\"]0\nBBB@Ay\ns;t\nAx\ns;t\n01\nCCCA1\nCCCA\n+^z(Ax\ns;tRe[(et\")\u0003et\"#\u0000et#(et#\")\u0003]\u0000Ay\ns;tIm[(et\")\u0003et\"#\u0000et#(et#\")\u0003]) (35)\nWe here focus on the linear e\u000bect of interface spin-orbit interaction and neglect the\nspin polarization along the magnetization direction, n. The expression for the pumped\nspin current then agrees with Eq. (27) with the amplitude \u0010swritten in terms of hopping\nincluding the interface spin-orbit,\nT\"#= ((~t0\n\")\u0003+i(e\rxz)\u0003)(~t0\n#+ie\rxz): (36)\n14If bulk spin-orbit interaction is neglected, bare hopping amplitude ~t0\n\u001bis real and we may\nreasonably assume that e\rijis real. The interface spin-orbit then leads to an imaginary part\nas (usinge\rxz=ni\rxi)\nIm[\u0010s] =\u0017\"\u0000\u0017#\n2M(~t0\n\"+~t0\n#)\rxini: (37)\nThe amplitude of spin current proportional to _nthus works as a probe for interface spin-orbit\ninteraction strength, \rxi.\nLet us discuss some examples. Of recent particular interest is the interface Rashba\ninteraction, represented by antisymmetric coe\u000ecient\n\r(R)\nij=\u000fijk\u000bR\nk; (38)\nwhere\u000bRis a vector representing the Rashba \feld. In the case of interface, \u000bRis perpen-\ndicular to the interface, i.e., \u000bRk^x. Therefore the interface Rashba interaction leads to\n\r(R)\nxj= 0 and does not modify spin pumping e\u000bect at the linear order. (It contributes at\nthe second order as discussed in Ref.14.) In other words, vector coupling between the wave\nvector and spin in the form of k\u0002\u001bexists only along the x-direction, and does not a\u000bect\nthe interface hopping (i.e., does not include kx).\nIn contrast, a scalar coupling \u0011(D)(k\u0001\u001b) (\u0011(D)is a coe\u000ecient), called the Dirac type\nspin-orbit interaction, leads to \r(D)\nij=\u0011(D)\u000eij. The spin current along _nthen reads\nj_n\ns=\u0011(D)kF(\u0017\"\u0000\u0017#)\n2mM(~t0\n\"+~t0\n#)nx_n: (39)\nFor the case of in-plane easy axis along the zdirection and magnetization precession given\nbyn(t) = (sin\u0012cos!t;sin\u0012sin!t;cos\u0012), where\u0012is the precession angle and !is the angular\nfrequency, we expect to have a dc spin current along the ydirection, as nx_n=\u0000!\n2sin2\u0012^y\n(nx_ndenotes time average).\nIV. FIELD THEORETIC DESCRIPTION OF METALLIC CASE\nHere we present a \feld-theoretic description of spin pumping e\u000bect of metallic ferromag-\nnet. The many-body approach has an advantage of taking account of particle distributions\nautomatically. Moreover, it describes propagation of particle density in terms of the Green's\n15functions, and thus is suitable for studying spatial propagation as well as for intuitive under-\nstanding of transport phenomena. All the transport coe\u000ecients are determined by material\nconstants.\nThe formalism presented here is essentially the same as in Ref.15, but treating the fer-\nromagnet of a \fnite size and taking account of electron states with di\u000berent wave vectors.\nInterface spin-orbit interaction is not considered here.\nConduction electron in ferromagnetic and normal metals are denoted by \feld operators\nd,dyandc,cy, respectively. These operators are vectors with two spin components, i.e.,\nd\u0011(d\";d#). The Hamiltonian describing the F and N electrons is HF+HN, where\nHF\u0011Z\nFd3rdy\u0012\n\u0000r2\n2m\u0000\u000fF\u0000Mn(t)\u0001\u001b\u0013\nd\nHN\u0011Z\nNd3rcy\u0012\n\u0000r2\n2m\u0000\u000fF\u0013\nc: (40)\nWe set the Fermi energies for ferromagnet and normal metal equal. The hopping through\nthe interface is described by the Hamiltonian\nHI\u0011Z\nIFd3rZ\nINd3r0\u0000\ncy(r0)t(r0;r;t)d(r) +dy(r)t\u0003(r0;r;t)c(r0)\u0001\n; (41)\nwheret(r0;r;t) represents the hopping amplitude of electron from rin ferromagnetic regime\nto a siter0in the normal region and the integrals are over the interface (denoted by I F\nand I Nfor F and N regions, respectively). The hopping amplitude is generally a matrix\ndepending on magnetization direction n(t), and thus depends on time t. Hopping is treated\nas energy-conserving. Assuming sharp interface at x= 0, the momentum perpendicular to\nthe interface is not conserved on hopping.\nWe are interested in the spin current in the normal region, given by\nj\u000b\ns;i(r;t) =\u00001\n4m(r(r)\u0000r(r0))itr[\u001b\u000bG<\nN(r;t;r0;t)jr0=r; (42)\nwhereG<\nN(r;t;r0;t0)\u0011i\nc(r;t)cy(r0;t0)\u000b\ndenotes the lesser Green's function for the normal\nregion. It is calculated from the Dyson's equation for the path-ordered Green's function\nde\fned for a complex time along a complex contour C\nGN(r;t;r0;t0) =gN(r\u0000r0;t\u0000t0)\n+Z\ncdt1Z\ncdt2Z\nd3r1Z\nd3r2gN(r\u0000r1;t\u0000t1)\u0006N(r1;t1;r2;t2)GN(r2;t2;r0;t0);\n(43)\n16whereg<\nNdenotes the Green's function without interface hopping and \u0006 N(r1;t1;r2;t2) is the\nself-energy for N electron, given by the contour-ordered Green's function in the ferromagnet\nas\n\u0006N(r1;t1;r2;t2)\u0011Z\nIFd3r3Z\nIFd3r4t(r1;r3;t1)G(r3;t1;r4;t2)t\u0003(r2;r4;t2): (44)\nHerer1andr2are coordinates at the interface I Nin N region and r3andr4are those in\nIFfor F.Gis the contour-ordered Green's function for F electron in the laboratory frame\nincluding the e\u000bect of spin gauge \feld. We denote Green's functions of F electron by G\nandgwithout su\u000ex and those of N electron with su\u000ex N. The lesser component of the\nnormal metal Green's function is obtained from Eq. (43) as (suppressing the time and space\ncoordinates)\nG<\nN= (1 +Gr\nN\u0006r\nN)g<\nN(1 + \u0006a\nNGa\nN) +Gr\nN\u0006<\nNGa\nN: (45)\nFor pumping e\u000bects, the last term on the right-hand side is essential, as it contains the\ninformation of excitation in F region. We thus consider the second term only;\nG<\nN'Gr\nN\u0006<\nNGa\nN; (46)\nand neglect spin-dependence of the normal region Green's functions, Gr\nNandGa\nN. The\ncontribution is diagramatically shown in Fig. 4.\nA. Rotated frame\nTo solve for the Green's function in the ferromagnet, rotated frame we used in Sec. II A\nis convenient. In the \feld representation, the unitary transformation is represented as (Fig.\n5(c))\nd=U~d; c =U~c; (47)\nwhereUis the same 2\u00022 matrix de\fned in Eq. (10). We rotate N electrons as well as F\nelectrons, to simplify the following expressions. The hopping interaction Hamiltonian reads\nHI=Z\nIFd3rZ\nINd3r0\u0010\n~cy(r0)~t(r0;r)~d(r) +~dy(r)~t\u0003(r0;r)~c(r0)\u0011\n; (48)\n17FIG. 4. (a) Schematic diagramatic representations of the lessor Green's function for N electron\nconnecting the same position r,G<\nN(r;r)'Gr\nN\u0006<\nNGa\nNrepresenting propagation of electron density.\nIt is decomposed into a propagation of N electron from rto the interface at r2, then hopping to r4\nin the F side, a propagation inside F, followed by a hopping to N side (to r1) and propagation back\ntor. (Position labels are as in Eqs. (43)(44).) (b): The self energy \u0006<\nNrepresents all the e\u000bects\nof the ferromagnet. (c) Standard Feynman diagram representation of lessor Green's function for\nN atr, Eqs. (46) and (44).\nFIG. 5. Unitary transformation Uof F electron converts the original system with \feld operator\nd(shown as (a)) to the rotated one with \feld operator ~d\u0011U\u00001d(b). The hopping amplitude\nfor representation in (b) is modi\fed by U. If N electrons are also rotated as ~ c\u0011U\u00001c, hopping\nbecomes ~t\u0011U\u00001tU, while the N electron spin rotates with time, as shown as (c).\nwhere\n~t(r0;r)\u0011Uy(t)t(r0;r;t)U(t); (49)\nis the hopping amplitude in the rotated frame. The rotated amplitude (neglecting interface\nspin-orbit interaction) is diagonal in spin;\n~t=0\n@~t\"0\n0~t#1\nA: (50)\nIncluding the interaction with spin gauge \feld, the Hamiltonian for F and N electrons in\n18the momentum representation is\nHF+HN=X\nk~dy\nk0\n@\u000fk\u0000M\u0000Az\ns;tA\u0000\ns;t\nA+\ns;t\u000fk+M+Az\ns;t1\nA~dk+X\nk\u000f(N)\nk~cy\nk~ck (51)\nAs for the hopping, we consider the case the interface is atomically sharp. The hopping\nHamiltonian is then written in the momentum space as\nHI=X\nkk0\u0010\n~cy(k)~t(k;k0)~d(k0) +~dy(k0)~t\u0003(k;k0)~c(k)\u0011\n; (52)\nwherek= (kx;ky;kz),k0= (k0\nx;ky;kz), choosing the interface as the plane of x= 0. Namely,\nthe wave vectors parallel to the interface are conserved while kxandk0\nxare uncorrelated.\nB. Spin density induced by magnetization dynamics in F\nPumped spin current in N is calculated by evaluating \u0006<\nNand using Eqs. (42)(45)(46).\nBefore discussing the spin current, let us calculate spin density in ferromagnet induced by\nmagnetization dynamics neglecting the e\u000bect of interface, HI. (E\u000bects of HIare discussed\nin Sec. V.) The spin accumulation in N is discussed by extending the calculation here as\nshown in Sec. IV C.\nThe lessor Green's function in F in the rotated frame including the spin gauge \feld to\nthe linear order is calculated from the Dyson's equation\nG<=g<+gr(As;t\u0001\u001b)g<+g<(As;t\u0001\u001b)ga; (53)\nwhereg\u000b(\u000b=<;r,a) represents Green's functions without spin gauge \feld. The lessor\nGreen's function satis\fes for static case g<=F(ga\u0000gr), whereF\u00110\n@f\"0\n0f#1\nAis spin-\ndependent Fermi distribution function. We thus obtain the Green's function at the linear\norder as15\n\u000eG<=gr[As;t\u0001\u001b;F]ga+gaF(As;t\u0001\u001b)ga\u0000gr(As;t\u0001\u001b)Fgr: (54)\nThe last two terms of the right-hand side are rapidly oscillating as function of position and\nare neglected. The commutator is calculated as (sign \u0006denotes spin\"and#)\n[As;t\u0001\u001b;F] = (f+\u0000f\u0000)X\n\u0006(\u0006)A\u0006\ns;t\u001b\u0007: (55)\n19FIG. 6. Feynman diagram for electron spin density of ferromagnet induced by magnetization\ndynamics (represented by spin gauge \feld As) neglecting the e\u000bect of normal metal. The amplitude\nis essentially given by the spin \rip correlation function \u001f\u0006(Eq. (58)).\nIn the rotated frame, the spin density in F pumped by the spin gauge \feld is therefore\n(diagrams shown in Fig. 6)\n~s(F)\n\u000b(k;k0)\u0011\u0000iZd!\n2\u0019tr[\u001b\u000b\u000eG<(k;k0;!)]\n=\u0000iZd!\n2\u0019X\nk00(fk00+\u0000fk00\u0000)X\n\u0006(\u0006)A\u0006\ns;ttr[\u001b\u000bgr(k;k00;!)\u001b\u0007ga(k00;k0;!)]\n=8\n<\n:\u0007iRd!\n2\u0019P\nk00(fk00+\u0000fk00\u0000)A\u0006\ns;tgr\n\u0007(k;k00;!)ga\n\u0006(k00;k0;!) (\u000b=\u0006)\n0 ( \u000b=z): (56)\nLet us here neglect the e\u000bects of interface in dicussing spin polarization of F electrons; Then\nthe Green's functions are translationally invariant, i.e., ga(k;k0) =\u000ek;k0ga(k) (a= r;a).\nUsing the explicit form of the free Green's function, ga\n\u001b(k;!) =1\n!\u0000\u000fk;\u001b\u0000i0, and\nZd!\n2\u0019gr\n\u0007(k;k00;!)ga\n\u0006(k00;k0;!) =i\n\u000fk;\u0006\u0000\u000fk;\u0007+i0; (57)\nthe spin density in the rotated frame then reduces to\n~s(F)\n\u0006(k) =\u0000A\u0006\ns;t\u001f\u0006; (58)\nwhere\n\u001f\u0006\u0011\u0000X\nkfk;\u0006\u0000fk;\u0007\n\u000fk;\u0006\u0000\u000fk;\u0007+i0; (59)\nis the spin correlation function with spin \rip, + i0 meaning an in\fnitesimal positive imaginary\npart. Since we focus on adiabatic limit and spatially uniform magnetization, the correlation\nfunction is at zero momentum- and frequency-transfer. We thus easily see that\n\u001f\u0006=n+\u0000n\u0000\n2M; (60)\n20wheren\u0006=P\nkfk\u0006is spin-resolved electron density.\nThe spin polarization of Eq. (58) in the rotated frame is proportional to A?\ns;t, and\nrepresents a renormalization of total spin in F. In fact, it corresponds in the laboratory\nframe tos(F)/n\u0002_n, and exerts a torque proportional to _nonn.\nIt may appear from Eq. (60) that a damping of spin, i.e., a torque proportional to\nn\u0002_n, arises when the imaginary part for the Green's function becomes \fnite, because\n1\nMis replaced by1\nM\u0007i\u0011i, where\u0011iis the imaginary part. This is not always the case. For\nexample, nonmagnetic impurities introduce a \fnite imaginary part inversely proportional to\nthe elastic lifetime ( \u001c),i\n2\u001c. They should not, however, cause damping of spin. The solution\nto this apparent controversy is that Eq. (56) is not enough to discuss damping even including\nlifetime. In fact, there is an additional process called vertex correction contributing to the\nlesser Green's function, and it gives rise to the same order of small correction as the lifetime\ndoes, and the sum of the two contributions vanishes. Similarly, we expect damping does not\narise from spin-conserving component of spin gauge \feld, Az\ns;t. This is indeed true as we\nexplicitly demonstrate in Appendix A. We shall show in Sec. V that damping arises from\nthe spin-\rip components of the self energy.\nC. Spin polarization and current in N\nFIG. 7. Feynman diagram for electron spin density of normal metal driven by the spin gauge \feld\nof ferromagnetic metal, As. The spin current is represented by the same diagram but with spin\ncurrent vertex.\nThe spin polarization of N electron lesser Green's function including the self-energy to\n21the linear order is calculated from Eqs. (46) (54)(55) as (diagram shown in Fig. 7)\n\u0000itr[\u001b\u0006G<\nN(r;t;r0;t)] =\u0000iX\nkk0k00eik\u0001re\u0000ik0\u0001r0gr\nN(k;!)ga\nN(k0;!)\n\u0002X\n\u0006(fk00\u0006\u0000fk00\u0007)A\u0006\ns;t~t\u0007(k;k00)~t\u0003\n\u0006(k00;k0)gr\n\u0007(k00;!)ga\n\u0006(k00;!): (61)\nWe assume that dependence of N Green's functions on !is weak and useP\nkeik\u0001rgr\nN(k;!) =\n\u0000i\u0019\u0017NeikFxe\u0000jxj=`\u0011gr\nN(r), where`is elastic mean free path, \u0017NandkFare the density of\nstates at the Fermi energy and Fermi wave vector, respectively, whose !-dependences are ne-\nglected. (For in\fnitely wide interface, the Green's function becomes one-dimensional.) As a\nresult of summation over wave vectors, the product of hopping amplitudes ~t\u0007(k;k00)~t\u0003\n\u0006(k00;k0)\nis replaced by the average over the Fermi surface, ~t\u0007~t\u0003\n\u0006\u0011T\u0006\u0007, i.e.,\n~t\u0007(k;k00)~t\u0003\n\u0006(k00;k0)!T\u0006\u0007: (62)\nThe spin polarization of N electron induced by magnetization dynamics (the spin gauge\n\feld) is therefore obtained in the rotated frame as\n~s(N)\n\u0006(r;t) =\u0000jgr\nN(r)j2X\n\u0006A\u0006\ns;t\u001f\u0006T\u0006\u0007; (63)\nor using\u001f\u0003\n+=\u001f\u0000\n~s(N)(r;t) =\u00002jgr\nN(r)j2\u0002\nA?\ns;tRe[\u001f+T+\u0000] + (^z\u0002A?\ns;t)Im[\u001f+T+\u0000]\u0003\n: (64)\nIn the laboratory frame, we have (using s(N)\ni=Rij~s(N)\nj)\ns(N)(r;t) =jgr\nN(r)j2\u0002\nRe[\u001f+T+\u0000](n\u0002_n) + Im[\u001f+T+\u0000]_n\u0003\n: (65)\nThe spin current induced in N region is similarly given by (neglecting the contribution\nproportional to n)\njs(r;t) =kF\nmjgr\nN(r)j2\u0002\nRe[\u001f+T+\u0000](n\u0002_n) + Im[\u001f+T+\u0000]_n\u0003\n=e\u0000jxj=`(Re[\u0010s](n\u0002_n) + Im[\u0010s]_n); (66)\nwhere\n\u0010s\u0011\u00192kF\u00172\nN\n2mM(n+\u0000n\u0000)T+\u0000: (67)\n22The coe\u000ecient \u0010sis essentially the same as the one in Eq. (27) derived by quantum mechan-\nical argument, as quantum mechanical dimensionless hopping amplitude corresponds to \u0017N~t\nof \feld representation.\nFor 3d ferromagnet, we may estimate the spin current by approximating roughly M\u0018\n1=\u0017N\u0018\u000fF\u00181eV andn\u001b\u0018kF3. The hopping amplitude jT+\u0000jin metallic case would be\norder of\u000fF. The spin current density then is of the order of (including electric charge eand\nrecovering ~),js\u0018e~kF\nmh~!\n\u000fF\u00185\u00021011A/m2if precession frequency is 10 GHz.\nV. SPIN ACCUMULATION IN FERROMAGNET\nThe spin current pumping is equivalent to the increase of spin damping due to magne-\ntization precession, as was discussed in Refs.2,10. In this section, we con\frm this fact by\ncalculating the torque by evaluating the spin polarization of the conduction electron spin in\nF region.\nThere are several ways to evaluate damping of magnetization. One way is to calculate the\nspin-\rip probability of the electron as in Ref.10, which leads to damping of localized spin in\nthe presence of strong sdexchange interaction. The second is to estimate the torque on the\nelectron by use of equation motion22. The relation between the damping and spin current\ngeneration is clearly seen in this approach. In fact, the total torque acting on conduction\nelectron is ( ~times) the time-derivative of the electron spin density,\nds\ndt=i\u0000\n[H;dy]\u001bd\u000b\n+\ndy\u001b[H;d]\u000b\u0001\n: (68)\nAt the interface, the right-hand side arises from the interface hopping. Using the hopping\nHamiltonian of Eq. (41), we have\nds\ndt\f\f\f\f\ninterface=i\u0000\ncyt\u001bd\u000b\n\u0000\ndy\u001btyc\u000b\u0001\n; (69)\nas the interface contribution. As is natural, the the right -hand side agrees with the de\fnition\nof the spin current passing through the interface. Evaluating the right-hand side, we obtain in\ngeneral a term proportional to n\u0002_n, which gives the Gilbert damping, and term proportional\nto_n, which gives a renormalization of magnetization. In contrast, away from the interface,\nthe commutator [ H;d] arises from the kinetic term H0\u0011R\nd3rjrdj2\n2mdescribing electron\n23propagation, resulting in\nds\u000b\ndt=i\u0000\n[H0;dy]\u001bd\u000b\n+\ndy\u001b[H;d]\u000b\u0001\n=r\u0001j\u000b\ns (70)\nwherej\u000b\ns(r)\u0011\u0000i\n2m(rr\u0000r r0)\ndy(r0)\u001b\u000bd(r)\u000b\njr0=ris the spin current. Away from the inter-\nface, the damping therefore occurs if the spin current has a source or a sink at the site of\ninterest.\nHere we use the third approach and estimate the torque on the localized spin by calculat-\ning the spin polarization of electrons as was done in Refs.7,23. The electron spin polarization\nat positionrin the ferromagnet at time tiss(F)(r;t)\u0011\ndy\u001bd\u000b\n, which reads in the rotated\nframes(F)\n\u000b=R\u000b\f~s(F)\n\f, where\n~s(F)\n\f(r;t) =\u0000itr[\u001b\fG<(r;r;t;t)]; (71)\nwhereG<\n\u001b\u001b0(r;r0;t;t0)\u0011iD\n~dy\n\u001b0~d\u001bE\nis the lesser Green's function in F region, which is a\nmatrix in spin space ( \u001b;\u001b0=\u0006). We are interested in the e\u000bect of the N region arising from\nthe hopping. We must note that the hopping interaction of Eq. (48) is not convenient for\nintegrating out N electrons, since the ~ celectrons' spins are time-dependent as a result of a\nunitary transformation, U(t). We thus use the following form (Fig. 5(b)),\nHI=Z\nIFd3rZ\nINd3r0\u0010\ncy(r0)U~t(r0;r)~d(r) +~dy(r)~t\u0003(r0;r)Uyc(r0)\u0011\n; (72)\nnamely, the hopping amplitude between ~dandcelectrons includes unitary matrix U.\nLet us argue in the rotated frame why the e\u000bect of damping arising from the interface.\nIn the totally rotated frame of Fig. 5(c), the spin of F electron is static, while that of N\nelectron varies with time. When F electron hops to N region and comes back, therefore,\nelectron spin gets rotated with the amount depending on the time it stayed in N region. This\ne\u000bect is in fact represented by a retardation e\u000bect of the matrices UandU\u00001in Eq. (72). If\no\u000b-diagonal nature of UandU\u00001are neglected, the interface e\u000bects are all spin-conserving\nand do not induce damping for F electron (See Sec. A).\nWe now proceed calculation of induced spin density in the ferromagnetic metal. Diagra-\nmatic representation of the contribution is in Fig. 8. Writing spatial and temporal positions\nexplicitly, the self-energy of F electron arising from the hopping to N region reads ( r1and\n24FIG. 8. Diagramatic representation of the spin accumulation in ferromagnetic metal induced as\na result of coupling to the normal metal (Eqs. (71)(73)). Conduction electron Green's functions\nin ferromagnet and normal metal are denoted by gandgN, respectively. Time-dependent matrix\nU(t), de\fned by Eq. (10), represents the e\u000bect of dynamic magnetization. Expanding UandU\u00001\nwith respect to slow time-dependence of magnetization, we obtain gauge \feld representation, Eq.\n(75).\nr2are in F)\n\u0006a(r1;r2;t1;t2) =Z\nINd3r0\n1Z\nINd3r0\n2~t(r1;r0\n1)U\u00001(t1)ga\nN(r0\n1;r0\n2;t1\u0000t2)U(t2)~ty(r2;r0\n2) (73)\nwherea= r;a;<. We assume the Green's function in N region is spin-independent; i.e., we\nneglect higher order contribution of hopping. Moreover, we treat the hopping to occur only\nat the interface, i.e., at x= 0. The self-energy is then represented as\n\u0006a(r1;r2;t1;t2) =a2\u000e(x1)\u000e(x2)~tU\u00001(t1)U(t2)~tyX\nkga\nN(k;t1\u0000t2); (74)\nwhereais the interface thickness, which we assume to be the order of the lattice constant.\nDiagramatic representation of Eqs. (71)(73) are in Fig. 8. Expanding the matrix using\nspin gauge \feld as U\u00001(t1)U(t2) = 1\u0000i(t1\u0000t2)As;t+O((As;t)2), we obtain the gauge \feld\ncontribution of the self-energy as\n\u0006a(r1;r2;t1;t2) =a2\u000e(x1)\u000e(x2)Zd!\n2\u0019de\u0000i!(t1\u0000t2)\nd!~tAs;t~tyX\nkga\nN(k;!)\n=\u0000a2\u000e(x1)\u000e(x2)Zd!\n2\u0019e\u0000i!(t1\u0000t2)~tAs;t~tyX\nkd\nd!ga\nN(k;!) (75)\n25The linear contribution of the lessor component of the o\u000b-diagonal self-energy is\nG<(r;t;r0;t) =gr\u0006rga+gr\u0006<ga+g<\u0006aga\n=a2Zd!\n2\u0019X\nk\u0014\ngr(r;!)dgr\nN(k;!)\nd!~tAs;t~tyg<(\u0000r;!)\n+gr(r;!)dg<\nN(k;!)\nd!~tAs;t~tyga(\u0000r;!) +g<(r;!)dga\nN(k;!)\nd!~tAs;t~tyga(\u0000r;!)\u0015\n(76)\nFor \fnite distance from the interface, r, dominant contribution arises from the terms contain-\ning bothgr(r;!) andga(\u0000r;!), as they do not contain a rapid oscillation like ei(kF++kF\u0000)r\nande2ikF\u001br. Using an approximationP\nkgr\nN(k;!)\u0018\u0000i\u0019\u0017Nand partial integration with\nrespect to!, Eq. (76) reduces to\nG<(r;t;r0;t) = 2\u0019i\u0017Na2Zd!\n2\u0019f0\nN(!)gr(r;!)~tAs;t~tyga(\u0000r;!) (77)\nFor damping, o\u000b-diagonal contributions, A\u0006\ns;t, are obviously essential. The result of the spin\ndensity in F in the rotated frame, Eq. (71), is\n~s(F)\n\u000b(r;t) = 2\u0019i\u0017Na2Zd!\n2\u0019f0\nN(!)A\f\ns;ttr[\u001b\u000bgr(r;!)~t\u001b\f~tyga(\u0000r;!)]\n= 2\u0019i\u0017Na2Zd!\n2\u0019f0\nN(!)A\f\ns;tX\nkk0ei(k\u0000k0)\u0001rtr[\u001b\u000bgr(k;!)~t\u001b\f~tyga(k0;!)] (78)\nEvaluating the trace in spin space, we obtain\n~s(F)(r;t) =\u0000\u0017N\u0014\nA?\ns;t\r1(r) + ( ^z\u0002A?\ns;t)\r2(r)\u0015\n(79)\nwhere\n\r1(r)\u0011X\n\u001b~t\u0000\u001b~ty\n\u001bgr\n\u0000\u001b(r)ga\n\u001b(\u0000r)\n\r2(r)\u0011X\n\u001b(\u0000i\u001b)~t\u0000\u001b~ty\n\u001bgr\n\u0000\u001b(r)ga\n\u001b(\u0000r): (80)\nWe consider an interface with in\fnite area and consider spin accumulation averaged over\nthe plane parallel to the interface. The wave vector contributing is then those with \fnite kx\nbut withky=kz= 0 and Green's function become one-dimensional like\nX\nkeik\u0001rgr\n\u001b(k) =im\nkF\u001beikF\u001bjxje\u0000jxj=(2`\u001b); (81)\n26where`\u001b\u0011vF\u001b\u001c\u001b(vF\u001b\u0011kF\u001b=m) is electron mean free path for spin \u001b. The induced spin\ndensity in the ferromagnet is \fnally obtained from Eq. (79) as\ns(F)(r;t) =m2\u0017Na2\n2kF+kF\u0000X\n\u001b\u0014\n(n\u0002_n)T\u001b;\u0000\u001be\u0000i\u001b(kF+\u0000kF\u0000)x+_n(\u0000i\u001b)T\u001b;\u0000\u001be\u0000i\u001b(kF+\u0000kF\u0000)x\u0015\n=m2\u0017Na2\n2kF+kF\u0000X\n\u001b\u0014\n(n\u0002_n)\u0002\nRe[T\";#] cos((kF+\u0000kF\u0000)x) + Im[T\";#] sin((kF+\u0000kF\u0000)x)\u0003\n+_n\u0002\nIm[T\";#] cos((kF+\u0000kF\u0000)x)\u0000Re[T\";#] sin((kF+\u0000kF\u0000)x)\u0003\n(82)\nand the torque on localized spin, \u0000Mn\u0002s(F), is\n\u001c(r;t) =\u0000m2\u0017Na2M\n2kF+kF\u0000X\n\u001b\u0014\n\u0000_n\u0002\nRe[T\";#] cos((kF+\u0000kF\u0000)x) + Im[T\";#] sin((kF+\u0000kF\u0000)x)\u0003\n+ (n\u0002_n)\u0002\nIm[T\";#] cos((kF+\u0000kF\u0000)x)\u0000Re[T\";#] sin((kF+\u0000kF\u0000)x)\u0003\n: (83)\nA. Enhanced damping and spin renormalization of ferromagnetic metal\nThe total induced spin accumulation density in ferromagnet is\ns(F)\u00111\ndZ0\n\u0000ddxs(F)(x)\n=1\nM\u0010\n(n\u0002_n)h\n\u0000Im[\u000e](1\u0000cos~d) + Re[\u000e] sin~di\n+_nh\nRe[\u000e](1\u0000cos~d) + Im[\u000e] sin~di\u0011\n;\n(84)\nwhere ~d\u0011(kF+\u0000kF\u0000)d,dis the thickness of ferromagnet and\n\u000e\u0011m2\u0017Na2M\nkF+kF\u0000(kF+\u0000kF\u0000)dT\";#: (85)\nAs a result of this induced electron spin density, s(F), the equation of motion for the averaged\nmagnetization is modi\fed to be10\n_n=\u0000\u000bn\u0002_n\u0000\rB\u0002n\u0000Mn\u0002s(F); (86)\nwhereBis the external magnetic \feld.\nLet us \frst discuss thin ferromagnet case, d\u001djkF+\u0000kF\u0000j\u00001, where oscillating part with\nrespect to ~dis neglected. The spin density then reads s(F)'1\nM(\u0000Im[\u000e](n\u0002_n) + Re[\u000e]_n)\nand the equation of motion becomes\n(1 + Im\u000e)_n=\u0000~\u000bn\u0002_n\u0000\rB\u0002n; (87)\n27where\n~\u000b\u0011\u000b+ Re\u000e; (88)\nis the Gilbert damping including the enhancement due to the spin pumping e\u000bect. The\nprecession angular frequency !Bis modi\fed by the imaginary part of T\";#, i.e., by the spin\ncurrent proportional to _n, as\n!B=\rB\n1 + Im\u000e: (89)\nThis is equivalent to the modi\fcation of the gyromagnetic ratio ( \r) or theg-factor.\nFor most 3d ferromagnets, we may approximatem2\u0017NaM\u000fF2\n2kF+kF\u0000(kF+\u0000kF\u0000)'O(1) (askF+\u0000\nkF\u0000/M), resulting in \u000e\u0018a\ndT\";#. As discussed in Sec. III, when interface spin-orbit\ninteraction is taken into account, we have T\";#=~t0\n\"~t0\n#+ie\rxz(~t0\n\"+~t0\n#) +O((e\r)2), where ~t0\n\u001b\nande\rxzare assumed to be real. Moreover, ~t0\n\u001bcan be chosen as positive and thus T\";#>0.\n(~t0\n\u001bhere is \feld-representaion, and has unit of energy.) Equations (88) and (89) indicates\nthat the strength of the hopping amplitude ~t0\n\u001band interface spin-orbit interaction e\rxzare\nexperimentally accessible by measuring Gilbert damping and shift of resonance frequency\nas has been known2. A signi\fcant consequence of Eq. (88) is that the enhancement of the\nGilbert damping,\n\u000e\u000b\u0018a\nd1\n\u000fF2~t0\n\"~t0\n#; (90)\ncan exceed in thin ferromagnets the intrinsic damping parameter \u000b, as the two contributions\nare governed by di\u000berent material parameters. In contrast to the positive enhancement of\ndamping, the shift of the resonant frequency or g-factor can be positive or negative, as it is\nlinear in the interface spin-orbit parameter e\rxz.\nExperimentally, enhancement of the Gilbert damping and frequency shift has been mea-\nsured in many systems11. In the case of Py/Pt junction, enhancement of damping is observed\nto be proportional to 1 =din the range of 2nm <d< 10nm, and the enhancement was large,\n\u000e\u000b=\u000b'4 atd= 2 nm11. These results appears to be consistent with our analysis. Same 1 =d\ndependence was observed in the shift of g-factor. The shift was positive and magnitude is\nabout 2% for Py/Pt and Py/Pd with d= 2nm, while it was negative for Ta/Pt11. The exis-\ntence of both signs suggests that the shift is due to the linear e\u000bect of spin-orbit interaction,\nand the interface spin-orbit interaction we discuss is one of possible mechanisms.\n28For thin ferromagnet, ~d.1, the spin accumulation of Eq. (84) reads\ns(F)=1\nM((n\u0002_n)Re[\u000ethin] +_nIm[\u000ethin]); (91)\nwhere\n\u000ethin\u0011\u000e~d=m2\u0017Na2M\n2kF+kF\u0000T\";#: (92)\nEquation (91) indicates that the roles of imaginary and real part of T\";#are interchanged\nfor thick and thin ferromagnet, resulting in\n~\u000b=\u000b+ Im\u000ethin\n!B=\rB\n1\u0000Re\u000ethin; (93)\nfor thin ferromagnet. Thus, for weak interface spin-orbit interaction, positive shift of reso-\nnance frequency is expected (as Re \u000ethin>0). Signi\fcant feature is that the damping can be\nsmallened or even be negative if strong interface spin-orbit interaction exists with negative\nsign of Im\u000ethin. Our result indicates that 'spin mixing conductance' description of Ref.2\nbreaks down in thin metallic ferromagnet (and insulator case as we shall see in Sec. VII D).\nIn this section, we have discussed spin accumulation and enhanced Gilbert damping in\nferromagnet attached to a normal metal. In the \feld-theoretic description, the damping\nenhancement arises from the imaginary part of the self-energy due to the interface. Thus\na randomness like the interface scattering changing the electron momentum is essential for\nthe damping e\u000bect, which sounds physically reasonable. The same is true for the reaction,\nnamely, spin current pumping e\u000bect into N region, and thus spin current pumping requires\nrandomness, too. (In the quantum mechanical treatment of Sec. II, change of electron\nwave vector at the interface is essential.) The spin current pumping e\u000bect therefore ap-\npears di\u000berent from general pumping e\u000bects, where randomness does not play essential roles\napparently3.\nSpin accumulation and enhanced Gilbert damping was discussed by Berger10based on a\nquantum mechanical argument. There 1 =ddependence was pointed out and the damping\ne\u000bect was calculated by evaluating the decay rate of magnons. Comparison of enhanced\nGilbert damping with experiments was carried out in Ref.2but in a phenomenological man-\nner.\n29VI. CASE WITH MAGNETIZATION STRUCTURE\nField theoretic approach has an advantage that generalization of the results is straightfor-\nward. Here we discuss brie\ry the case of ferromagnet with spatially-varying magnetization.\nThe excitations in metallic ferromagnet consist of spin waves (magnons) and Stoner excita-\ntion. While spin waves usually have gap as a result of magnetic anisotropy, Stoner excitation\nis gapless for \fnite wave vector, ( kF+\u0000kF\u0000)<jqj<(kF++kF\u0000), and it may be expected\nto have signi\fcant contribution for magnetization structures having wavelength larger than\nkF+\u0000kF\u0000. Let us look into this possibility.\nOur result of spin accumulation in ferromagnet, represented in the rotated frame, Eq.\n(63), indicates that when the magnetization has a spatial pro\fle, the accumulation is deter-\nmined by the spin gauge \feld and spin correlation function depending on the wave vector q\nas\nX\nqA\u0006\ns;t(q)\u001f\u0006(q;0); (94)\nwhere\n\u001f\u0006(q;\n)\u0011\u0000X\nkfk+q;\u0006\u0000fk;\u0007\n\u000fk+q;\u0006\u0000\u000fk;\u0007+ \n +i0; (95)\nis the correlation function with \fnite momentum transfer qand \fnite angular frequency \n.\nFor the case of free electron with quadratic dispersion, the correlation function is24\n\u001f\u0006(q;\n) =Aq+i\nBq\u0012st(q) +O(\n2); (96)\nwhere\nAq=ma3\n8\u00192\u0014\n(kF++kF\u0000)\u0012\n1 +(kF+\u0000kF\u0000)2\nq2\u0013\n+1\n2q3((kF++kF\u0000)2\u0000q2)(q2\u0000(kF+\u0000kF\u0000)2) ln\f\f\f\fq+ (kF++kF\u0000)\nq\u0000(kF++kF\u0000)\f\f\f\f\u0015\nBq=m2a3\n4\u0019jqj; (97)\nand\n\u0012st(q)\u00118\n<\n:1 (kF+\u0000kF\u0000)<jqj<(kF++kF\u0000)\n0 otherwise: (98)\n30describes the wave vectors where Stoner excitation exists. As we see from Eq. (96), the\nStoner excitation contribution vanishes to the lowest order in \n, and thus the spin pumping\ne\u000bect in the adiabatic limit (\n !0) is not a\u000bected. Moreover, the real part of the correlation\nfunction,Aq, is a decreasing function of qand thus the spin pumping e\u000eciency would\ndecrease when ferromagnet has a structure. However, for rigorous argument, we need to\ninclude the spatial component of the spin gauge \feld arising form the spatial derivative of\nthe magnetization pro\fle.\nAs for the e\u000bect of the Stoner excitation on spin damping (Gilbert damping), it was\ndemonstrated for the case of a domain wall that the e\u000bect is negligibly small for a wide wall\nwith thickness \u0015\u001d(kF+\u0000kF\u0000)\u00001(Refs.24,25). Simanek and Heinrich presented a result of\nthe Gilbert damping as the linear term in the frequency of the imaginary part of the spin\ncorrelation function integrated over the wave vector12. The result is, however, obtained for a\nmodel where ferromagnet is atomically thin layer (a sheet), and would not be applicable for\nmost experimental situations. Discussion of Gilbert damping including \fnite wave vector\nand the impurity scattering was given in Ref.26. Inhomogenuity e\u000bects of damping of a\ndomain wall was studied recently in detail27.\nVII. INSULATOR FERROMAGNET\nIn this section, we discuss the case of ferromagnetic insulator. It turns out that the\ngeneration mechanisms for spin current in the insulating and metallic cases are distinct.\nA. Magnon and adiabatic gauge \feld\nThe Lagrangian for the insulating ferromagnet is\nLIF=Z\nd3r\u0014\nS_\u001e(cos\u0012\u00001)\u0000J\n2(rS)2\u0015\n\u0000HK; (99)\nwhereJis the exchange interaction between the localized spin, S, andHKdenotes the\nmagnetic anisotropy energy.\nWe \frst study low energy magnon dynamics induced by slow magnetization dynamics.\nFor separating the classical variable and \ructuation (magnon), rotating coordinate descrip-\ntion used in the metallic case is convenient. For magnons described by the Holstein-Primakov\n31boson, the unitary transformation is a 3 \u00023 matrix de\fned as follows28.\nS=UeS; (100)\nwhere\nU=0\nBBB@cos\u0012cos\u001e\u0000sin\u001esin\u0012cos\u001e\ncos\u0012sin\u001ecos\u001esin\u0012sin\u001e\n\u0000sin\u0012 0 cos\u00121\nCCCA=\u0010\ne\u0012e\u001en\u0011\n: (101)\nThe diagonalized spin eSis represented in terms of annihilation and creation operators for\nthe Holstein-Primakov boson, bandby, as29\neS=0\nBBB@q\nS\n2(by+b)\niq\nS\n2(by\u0000b)\nS\u0000byb1\nCCCA: (102)\nWe neglect the terms that are third- and higher-order in boson operators. Derivatives of the\nlocalized spin then read\n@\u0016S=U(@\u0016+iAU;\u0016)eS; (103)\nwhere\nAU;\u0016\u0011\u0000iU\u00001r\u0016U; (104)\nis the spin gauge \feld represented as a 3 \u00023 matrix. The spin Berry's phase of the Lagrangian\n(99) is written in terms of magnon as (derivation is in Sec. B)\nLm= 2S\r2Z\nd3ri[by(@t+iAz\ns;t)b\u0000by( \n@t\u0000iAz\ns;t))b]; (105)\nnamely, magnons interacts with the adiabatic component of the same spin gauge \feld for\nelectrons,Az\ns;t, de\fned in Eq. (14). As magnon is a single-component \feld, the gauge \feld\nis also single-component, i.e., a U(1) gauge \feld. This is a signi\fcant di\u000berence between\ninsulating and metallic ferromagnet; In the metallic case, conduction electron couples to\nan SU(2) gauge \feld with spin-\rip components, which turned out to be essential for spin\ncurrent generation. In contrast, in the insulating case, the magnon has diagonal gauge \feld,\ni.e., a spin chemical potential, which simply induces diagonal spin polarization. Pumping of\nmagnon was discussed in a di\u000berent approach by evaluating magnon source term in Ref.30.\n32The exchange interaction at the interface is represented by a Hamiltonian\nHI=JIZ\nd3rIS(r)\u0001cy\u001bc; (106)\nwhereJIis the strength of the interface sdexchange interaction and the integral is over\nthe interface. We consider a sharp interface at x= 0. Using Eq. (100), the interaction is\nrepresented in terms of magnon operators up to the second order as\nHI=JIZ\nd3rI\"\n(S\u0000byb)cy(n\u0001\u001b)c+r\nS\n2\u0002\nbycy\b\u0001\u001bc+bcy\b\u0003\u0001\u001bc\u0003#\n; (107)\nwhere\n\b\u0011e\u0012+ie\u001e=0\nBBB@cos\u0012cos\u001e\u0000isin\u001e\ncos\u0012sin\u001e+icos\u001e\n\u0000sin\u00121\nCCCA: (108)\nEquation (107) indicates that there are two mechanisms for spin current generation; namely,\nthe one due to the magnetization at the interface (the term proportional to n) and the one\ndue to the magnon spin scattering at the interface (described by the term linear in magnon\noperators).\nLet us brie\ry demonstrate based on the expression of Eq. (107) that spin-\rip processes\ndue to magnon creation or annihilation lead to generation of spin current in the normal\nmetal. At the second order, the interaction induces a factor on the electron wave function\n(\b\u0003(t)\u0001\u001b)(\b(t0)\u0001\u001b) for magnon creation and ( \b(t)\u0001\u001b)(\b\u0003(t0)\u0001\u001b) for annihilation (we\nallow an in\fnitesimal di\u000berence in time tandt0). The factor for the creation has charge\nand spin contributions, ( \b\u0003(t)\u0001\u001b)(\b(t0)\u0001\u001b) =\b\u0003(t)\u0001\b(t0) +i\u001b\u0001(\b\u0003(t)\u0002\b(t0)). For\nmagnon annihilation, we have ( \b\u0003(t)\u0002\b(t0))\u0003, and thus the sum of the magnon creation\nand annihilation processes give arise to a factor\nX\nq[(nq+ 1)(\b\u0003(t)\u0002\b(t0)) +nq(\b\u0003(t)\u0002\b(t0))\u0003] =X\nq[(2nq+ 1)Re[ \b\u0003(t)\u0002\b(t0)] +iIm[\b\u0003(t)\u0002\b(t0)]:\n(109)\nFor adiabatic change the amplitude is expanded as\n(\b\u0003(t)\u0002\b(t0)) = 2i(1 +i(t\u0000t0) cos\u0012_\u001e)n\u0000(t\u0000t0)(n\u0002_n\u0000i_n) +O((@t)2); (110)\n33where we see that an retardation e\u000bect from the adiabatic change of magnetization (rep-\nresented by the second term on the right-hand side) gives rise to a magnon state change\nproportional to n\u0002_nand _n. The retardation contribution for the spin part (Eq. (109)) is\n(t\u0000t0)X\nq[\u0000(2nq+ 1)(n\u0002_n) +i_n]: (111)\nWe therefore expect that a spin current proportional to n\u0002_nemerges proportional to the\nmagnon creation and annihilation number,P\nq(2nq+ 1). (As we shall see below, the factor\nt\u0000t0reduces to a derivative with respect to angular frequency of the Green's function.) A\nrigorous estimation using Green's function method is presented in Sec. VII C.\nIn Eq. (111), the last term proportional to _nis an imaginary part arising from the\ndi\u000berence of magnon creation and annihilation probabilities of vacuum, nq+ 1 andnq.\nThe term is, however, unphysical one corresponding to a real energy shift due to magnon\ninteraction, and is removed by rede\fnition of the Fermi energy.\nB. Spin current pumped by the interface exchange interaction\nHere we study the spin current pumped by the classical magnetization at the interface,\nnamely, the one driven by the term proportional to Snin Eq. (107). We treat the ex-\nchange interaction perturbatively to the second order as the exchange interaction between\nconduction electron and insulator ferromagnet is localized at the interface and is expected\nto be weak. The weak coupling scheme employed here is in the opposite limit as the strong\ncoupling (adiabatic) approach used in the metallic ferromagnet (Sec. IV).\nIn the perturbative regime, the issue of adiabaticity needs to be argued carefully. In\nthe strong sdcoupling limit, the adiabaticity is trivially satis\fed, as the time needed for\nthe electron spin to follow the localized spin is the fastest timescale. In the weak coupling\nlimit, this timescale is long. Nevertheless, the adiabatic condition is satis\fed if the electron\nspin relaxation is strong so that the electron spin relaxes quickly to the local equilibrium\nstate determined by the localized spin. Thus the adiabatic condition is expected to be\nMI\u001csf=~\u001c1, whereMIand\u001csfare the interface spin splitting energy, and conduction\nelectron spin relaxation time, respectively. In the following calculation, we consider the case\nof\u000fF\u001csf=~\u001d1, i.e., ~(\u001csf)\u00001\u001c\u000fF, as the spin \rip lifetime is by de\fnition longer than the\nelastic electron lifetime \u001c, which satis\fes \u000fF\u001c=~\u001d1 in metal. Our results therefore cover\n34both adiabatic and nonadiabatic limits.\nThe calculation is carried out by evaluating Feynmann diagrams of Fig. 9, similar to the\nstudy of Refs.17,18. A di\u000berence is that while Refs.17,18assumed a smooth magnetization\nstructure and used a gradient expansion, the exchange interaction we consider is localized.\nFIG. 9. T\nhe Feynmann diagrams for spin current pumped by interface sdexchange interaction.\nThe lesser Green's function for normal metal including the interface exchange interaction\nto the linear order is\nG(1)<\nN(r;t;r;t) =MIZd!\n2\u0019Zd\n2\u0019X\nkk0e\u0000i\ntei(k0\u0000k)\u0001r\u0002\n(f(!+ \n)\u0000f(!))gr\nk0;!+\nga\nk!\u0000f(!)gr\nk0;!+\ngr\nk!+f(!+ \n)ga\nk0;!+\nga\nk!\u0003\n(n\n\u0001\u001b);\n(112)\nwhereMI\u0011JISis the local spin polarization at the interface. Expanding the expression\nwith respect to \n and keeping the dominant contribution at long distance, i.e., the terms\ncontaining both gaandgr. UsingP\nkga\nk!eik\u0001r'im\nkFeikre\u0000jxj\n`(\u0011ga(r)), the result of spin\ncurrent is\nj(1)\ns(r;t) =\u0000MIm\nkF_ne\u0000jxj=`: (113)\nThe second-order contribution is similarly calculated to obtain\nG(2)<\nN(r;t;r;t)'(MI)2Zd!\n2\u0019Zd\n1\n2\u0019Zd\n2\n2\u0019\n\u0002X\nkk0k00e\u0000i(\n1+\n2)tei(k0\u0000k)\u0001rf0(!)gr\nk0;!ga\nk!(\n1ga\nk00!+ \n 2gr\nk00!)(n\n1\u0001\u001b)(n\n2\u0001\u001b)\n=\u00002\u0019i\u0017(MI)2jgr(r)j2(n\u0002_n)\u0001\u001b: (114)\nThe corresponding spin current at the interface ( x= 0) is thus\nj(2)\ns(x= 0;t) =\u0017(MI)2m\nkF(n\u0002_n); (115)\n35and the total spin current reads\njs(x= 0;t) =\u0000MIm\nkF_n\u00002\u0017(MI)2m\nkF(n\u0002_n): (116)\nIn the perturbation regime, the spin current proportional to _nis dominant (larger by a\nfactor of (\u0017M I)\u00001) compared to the one proportional to n\u0002_n.\nExpression of spin current induced by the interface exchange interaction was presented\nin Ref.31in the limit of strong spin relaxation, MI\u001csf\u001c1, where\u001csfis the spin relaxation\ntime of electron. By solving the Landau-Lifshitz-Gilbert equation for the electron spin, they\nobtained an result of Eq. (116) with \u0017M Ireplaced by MI\u001csf.\nC. Calculation of magnon-induced spin current\nHere magnon-induced spin current due to the magnon interaction in Eq. (107) is cal-\nculated. As magnon is a small \ructuation of magnetization, the contribution here is a\nsmall correction to the contribution of Eq. (116). Nevertheless, the magnon contribution\nhas a typical linear dependence on the temperature, and is expected to be experimentally\nidenti\fed easily.\nSpin current induced in normal metal is evaluated by calculating the self-energy arising\nfrom the interface magnon scattering of Eq. (107). The contribution to the path-ordered\nGreen's function of N electron from the magnon scattering to the second order is\nGN(r;t;r0:t0) =Z\nCdt1Z\nCdt2X\nr1r2gN(r;t;r1;t1)\u0006I(r1;t1;r2;t2)gN(r2;t2;r0;t0); (117)\nwhere\n\u0006I(r1;t1;r2;t2)\u0011iSJ2\nI\n2D\u000b\f(r1;t1;r2;t2)\u001b\u000bgN(r1;t1;r2;t2)\u001b\f; (118)\nrepresents the self energy. Here\nD\u000b\f(r1;t1;r2;t2)\u0011\u0000ihTCB\u000b(r1;t1)B\f(r2;t2)i; (119)\nis the Green's function for magnon dressed by the magnetization structure ( \bis de\fned in\nEq. (108)),\nB\u000b(r;t)\u0011\b\u000b(t)by(r;t) + \by\n\u000b(t)b(r;t): (120)\n36FIG. 10. T\nhe Feynmann diagrams for spin current pumped by magnons at the interface. Green's functions\nfor magnons and electrons in the normal metal are denoted by DandgN, respectively. \b\nrepresents the e\u000bects of magnetization dynamics (Eq. (108)).\nDiagramatic representation is in Fig. 10. In the present approximation including the inter-\nface scattering to the second order, the electron Green's function in Eq. (118) is treated as\nspin-independent, resulting in a self energy (de\fned on complex time contour)\n\u0006I(r1;t1;r2;t2) =iSJ2\nI\n2(\u000e\u000b\f+i\u000f\u000b\f\r\u001b\r)D\u000b\f(r1;t1;r2;t2)gN(r1;t1;r2;t2): (121)\nWe focus on the spin-polarized contribution,\n\u0006I;\r(r1;t1;r2;t2)\u0011\u0000SJ2\nI\n2eD\r(r1;t1;r2;t2)gN(r1;t1;r2;t2); (122)\nwhereeD\r\u0011\u000f\u000b\f\rD\u000b\f. The spin-dependent contribution of lessor Green's function, Eq. (117),\nreads (time and spatial coordinates partially suppressed)\nG<\nN(r;t;r0;t0) =\u001b\rZ1\n\u00001dt1Z1\n\u00001dt2\u0014\ngr\nN(t\u0000t1)\u0006r\nI;\r(t1;t2)g<\nN(t2\u0000t0) +gr\nN\u0006<\nI;\rga\nN+g<\nN\u0006a\nI;\rga\nN\u0015\n\u0011\u001b\rG<\nN;\r(r;t;r0:t0):\n(123)\nFor the self energy type of the Green's functions, depending on two time as g(t1\u0000t2)D(t1\u0000t2)\n(Eq. (122)), real-time components are written as (suppressing time and su\u000ex of N) (See\nSec. C)\n[g(t1\u0000t2)D(t1\u0000t2)]r=grD<+g>Dr=g<Dr+grD>\n[g(t1\u0000t2)D(t1\u0000t2)]a=gaD>+g<Da=gaD<+g>Da\n[g(t1\u0000t2)D(t1\u0000t2)]<=g<D<: (124)\nThe Green's function eDis that of a composite \feld B\u000bde\fned in Eq. (120), and is decom-\n37posed to elementary magnon Green's function, D, as\neD\r(r1;t1;r2;t2) = [\by(t1)\u0002\b(t2)]\rD(r1;t1;r2;t2)\u0000[\by(t2)\u0002\b(t1)]\rD(r2;t2;r1;t1);\n(125)\nwhere\nD(r1;t1;r2;t2)\u0011\u0000i\nTCb(r1;t1)by(r2;t2)\u000b\n: (126)\nThe spin-dependent factor in Eq. (125) is calculated for adiabatic dynamics as\n\by(t1)\u0002\b(t2) = 2in(t1) + (t2\u0000t1)[\t+i_n] +O((@t)2); (127)\nwhere\n\t\u00112 cos\u0012_\u001en+n\u0002_n: (128)\nThe real-time Green's functions are therefore ( D(1;2)\u0011D(r1;t1;r2;t2))\neD<\n\r(r1;t1;r2;t2) = 2in(t1)[D<(r1;t1;r2;t2)\u0000D>(r2;t2;r1;t1)]\n+ (t2\u0000t1)\u0014\n\t[D<(r1;t1;r2;t2) +D>(r2;t2;r1;t1)] +i_n[D<(r1;t1;r2;t2)\u0000D>(r2;t2;r1;t1)]\u0015\neDr\n\r(1;2) =\u0012(t1\u0000t2)(eD<\n\r(1;2)\u0000eD>\n\r(1;2))\neDa\n\r(1;2) =\u0000\u0012(t2\u0000t1)\u000f\u000b\f\r(D<\n\u000b\f(1;2)\u0000D>\n\u000b\f(1;2)); (129)\nandeD<\n\ris obtained by exchanging <and>ineD<\n\r. Elementary Green's functions are\ncalculated as\nD<(r1;t1;r2;t2) =\u0000iX\nqeiq\u0001(r1\u0000r2)nqe\u0000i!q(t1\u0000t2)\nD>(r1;t1;r2;t2) =\u0000iX\nqeiq\u0001(r1\u0000r2)(nq+ 1)e\u0000i!q(t1\u0000t2); (130)\nwhere!qis magnon energy and nq\u00111\ne\f!q\u00001. In our model, the interface is atomically \rat\nand has an in\fnite area, and thus ri(i= 1;2) are atx= 0. Fourier components de\fned as\n(a= r;a;<;> )\neDa\n\r(x1= 0;t1;x2= 0;t2)\u0011X\nqZd\n2\u0019e\u0000i\n(t1\u0000t2)eDa\n\r(q;\n); (131)\n38are calculated from Eq. (129) as\neD<\n\r(q;\n) =\u0000i\u0014\n2n(D<\n\u0000\u0000D>\n+) +d\nd\n\u0002\n\t(D<\n\u0000+D>\n+) +i_n(D<\n\u0000\u0000D>\n+)\u0003\u0015\neDr\n\r(q;\n) =\u0000i\u0014\n2n(Dr\n\u0000+Dr\n+) +d\nd\n\u0002\n\t(Dr\n\u0000\u0000Dr\n+) +i_n(Dr\n\u0000+Dr\n+)\u0003\u0015\neDa\n\r(q;\n) =\u0000i\u0014\n2n(Da\n\u0000+Da\n+) +d\nd\n\u0002\n\t(Da\n\u0000\u0000Da\n+) +i_n(Da\n\u0000+Da\n+)\u0003\u0015\n; (132)\nwhere\nDa\n\u0006\u00111\n\n\u0006!q\u0000i0; Dr\n\u0006\u00111\n\n\u0006!q+i0\nD<\n\u0000\u0011nq(Da\n\u0000\u0000Dr\n\u0000); D>\n+\u0011(1 +nq)(Da\n+\u0000Dr\n+): (133)\nThe spin part of the Green's function, Eq. (123), is\nG<\nN;\r(r;t;r0;t) =\u0000SJ2\nI\n2Zd!\n2\u0019Zd\n2\u0019X\nkk0X\nk00q\u0014\ngr\nN;k!\u0010\neDr\n\r(q;\n)g>\nN;k00;!\u0000\n+eD<\n\r(q;\n)gr\nN;k00;!\u0000\n\u0011\ng<\nN;k0!\n+gr\nN;k!eDr\n\r(q;\n)g>\nN;k00;!\u0000\nga\nN;k0!+g<\nN;k!\u0010\neDa\n\r(q;\n)g>\nN;k00;!\u0000\n+eD<\n\r(q;\n)ga\nN;k00;!\u0000\n\u0011\nga\nN;k0!\u0015\n:\n(134)\nThe contribution survives at long distance is the one containing gr\nN;!(r) andga\nN;!(\u0000r), i.e.,\nG<\nN;\r(r;t;r0;t)'Zd!\n2\u0019X\nkk0gr\nN;k!ga\nN;k0!eik\u0001re\u0000ik0\u0001r0e\u0006I;\r; (135)\nwhere\ne\u0006I;\r\u0011\u0000SJ2\nI\n2Zd\n2\u0019X\nk00q\u0014\u0010\nfk0eDr\n\r(q;\n)\u0000fkeDa\n\r(q;\n)\u0011\n(fk00\u00001)(ga\nN;k00;!\u0000\n\u0000gr\nN;k00;!\u0000\n)\n+eD<\n\r(q;\n)(fk0gr\nN;k00;!\u0000\n\u0000fkga\nN;k00;!\u0000\n+fk00(ga\nN;k00;!\u0000\n\u0000gr\nN;k00;!\u0000\n))\u0015\n: (136)\nWe focus on the pumped contribution, containing derivative with respect to \n in Eq. (132).\nThe result is, using partial integration with respect to \n ( e\u0006Iis a vector representation of\ne\u0006I;\r),\ne\u0006I'\u0000iSJ2\nI\n2Zd\n2\u0019X\nk00q\u0014\n\u0000\nfk0[\t(Dr\n\u0000\u0000Dr\n+) +i_n(Dr\n\u0000+Dr\n+)]\u0000fk[\t(Da\n\u0000\u0000Da\n+) +i_n(Da\n\u0000+Da\n+)]\u0001\n(fk00\u00001)d\nd\n(ga\nN;k00;!\u0000\n\u0000gr\nN;k00;!\u0000\n)\n+ [\t(D<\n\u0000+D>\n+) +i_n(D<\n\u0000\u0000D>\n+)]d\nd\n[(fk00\u0000fk)ga\nN;k00;!\u0000\n\u0000(fk00\u0000fk0)gr\nN;k00;!\u0000\n]\u0015\n:\n(137)\n39Usingd\nd\nga\nk00;!\u0000\n= (ga\nk00;!)2+O(\n) and an approximation, we obtainP\nk00(ga\nk00;!)2'\u0000\u0019i\u0017\n2\u000fF,\ne\u0006I'\u0019\u0017\n\u000fFSJ2\nI\n2Zd\n2\u0019X\nqk00\u0014\n\t\u0012\n(fk00\u00001)[fk0(Dr\n\u0000\u0000Dr\n+)\u0000fk(Da\n\u0000\u0000Da\n+)] +1\n2(2fk00\u0000fk\u0000fk0)(D<\n\u0000+D>\n+)\u0013\n+i_n\u0012\n(fk00\u00001)[fk0(Dr\n\u0000+Dr\n+)\u0000fk(Da\n\u0000+Da\n+)] +1\n2(2fk00\u0000fk\u0000fk0)(D<\n\u0000\u0000D>\n+)\u0013\u0015\n:\n(138)\nAs argued for Eq. (111), only the imaginary part of self energy contributes to the induced\nspin current, as the real part, the shift of the chemical potential, is compensated by redis-\ntribution of electrons. The result is thus\ne\u0006I'i\t\u0019\u0017\n\u000fFSJ2\nI\n2X\nqk00(1 + 2nq)(2fk00\u0000fk\u0000fk0): (139)\nWe further note that the component of \tproportional to n(Eq. (128)) does not contribute\nto the current generation, as a result of gauge invariance. (In other words, the contribution\ncancels with the one arising from the e\u000bective gauge \feld for magnon.)\nThe \fnal result of the spin current pumped by the magnon scattering is therefore\njm\ns(r;t) =\u0019\u0017\n\u000fFSJ2\nI\n2jgr(r)j2X\nq(1 + 2nq)(n\u0002_n): (140)\nAt high temperature compared to magnon energy, \f!q\u001c1, 1 + 2nq'2kBT\n!q, and the\nmagnon-induced spin current depends linearly on temperature. The result (140) agrees\nwith previous study carried out in the context of thermally-induced spin current19.\nD. Correction to Gilbert damping in the insulating case\nIn this subsection, we calculate the correction to the Gilbert damping and g-factor of\ninsulating ferromagnet as a result of spin pumping e\u000bect. We study the torque on the\nferromagnetic magnetization arising from the e\u000bect of conduction electron of normal metal,\ngiven by\n\u001cI=BI\u0002n=MI(n\u0002sI); (141)\n40where\nBI\u0011\u0000\u000eHI\n\u000en=\u0000MIsI; (142)\nis the e\u000bective magnetic \feld arising from the interface electron spin polarization, sI(t)\u0011\n\u0000itr[\u001bG<\nN(0;t)]. The contribution to the electron spin density linear in the interface ex-\nchange interaction, Eq. (106), is\ns(1);\u000b\nI(t) =\u0000iZ\ndt1MIn\f(t1)tr[\u001b\u000bgN(t;t1)\u001b\fgN(t1;t)]<; (143)\nwhere the Green's functions connect positions at the interface, i.e., from x= 0 tox= 0,\nand are spin unpolarized. (The Feynman diagrams for the spin density are the same as the\none for the spin current, Fig. 9 with the vertex jsreplaced by the Pauli matrix.) Pumped\ncontribution proportional to the time variation of magnetization is obtained as\ns(1)\nI(t) =\u0000MI_nZd!\n2\u0019X\nkk0f0(!)(ga\nN;k0\u0000gr\nN;k0)(ga\nN;k\u0000gr\nN;k)\n=\u0000MI(\u0019\u0017)2_n: (144)\nThe second order contribution similarly reads\ns(2);\u000b\nI(t) =\u0000i\n2Z\ndt1Z\ndt2(MI)2n\f(t1)n\r(t2)tr[\u001b\u000bgN(t;t1)\u001b\fgN(t1;t2)\u001b\rgN(t2;t)]<\n'\u00002(MI)2(\u0019\u0017)3(n\u0002_n): (145)\nThe interface torque is therefore\n\u001cI=\u0000(MI\u0019\u0017)2(n\u0002_n) + 2(MI\u0019\u0017)3_n: (146)\nIncluding this torque in the LLG equation, _n=\u0000\u000bn\u0002_n\u0000\rB\u0002n+\u001c, we have\n(1\u0000\u000eI)_n=\u0000\u000bI(n\u0002_n)\u0000\rB\u0002n; (147)\nwhere\n\u000eI= 2\u0016d(\u0019M I\u0017)3\n\u000bI=\u000b+\u0016d(\u0019M I\u0017)2; (148)\nwhere\u0016d\u0018dmp=dis the ratio of the length of magnetic proximity ( dmp) and thickness of\nthe ferromagnet, d. The Gilbert damping constant therefore increases as far as the interface\n41spin-orbit interaction is neglected. The resonance frequency is !B=\rB\n1\u0000\u000eI, and the shift can\nhave both signs depending on the sign of interface exchange interaction, MI.\nThere may be a possibility that magnon excitation induce torque that corresponds to\ne\u000bective damping. In fact, such torque arises of hbior\nby\u000b\nare \fnite, i.e., if magnon Bose\ncondensation glows, as seen from Eq. (102). Such condensation can in principle develop from\nthe interface interaction of magnon creation or annihilation induced by electron spin \rip,\nEq. (107). However, conventional spin relaxation processes arising from the second order\nof random spin scattering do not contribute to such magnon condensation and additional\ndamping.\nComparing the result of pumped spin current, Eq. (116), and that of damping coe\u000ecient,\nEq. (148), we notice that the 'spin mixing conductance' argument2, where the coe\u000ecients\nfor the spin current component proportional to n\u0002_nand the enhancement of the Gilbert\ndamping constant are governed by the same quantity (the imaginary part of 'spin mixing\nconductance') does not hold for the insulator case. In fact, our result indicates that the\nspin current component proportional to n\u0002_narises from the second order correction to\nthe interaction (the second diagram of Fig. 9), while the damping correction arises from the\n\frst order process (the \frst diagram of Fig. 9). Although the magnitudes of the two e\u000bects\nhappen to be both second order of interface spin splitting, MI, physical origins appear to\nbe distinct. From our analysis, we see that the 'spin mixing conductance' description is not\ngeneral and applies only to the case of thick metallic ferromagnet (see Sec. V A for metallic\ncase).\nVIII. DISCUSSION\nOur results are summarized in table I. Let us discuss experimental results in the light of\nour results. In the early ferromagnetic resonance (FMR) experiments, consistent studies of\ng-factor and the Gilbert damping were carried out on metallic ferromagnets11. The results\nappear to be consistent with theories (Refs.2,10and the present paper). Both the damping\nconstant and the gfactor have 1 =d-dependence on the thickness of ferromagnet in the range\nof 2nm<d< 10nm11. The maximum additional damping reaches \u000e\u000b\u00180:1 atd=2nm, which\nexceeds the original value of \u000b\u00180:01. Theg-factor modulation is about 1% at d= 2nm, and\nits sign depends on the material; the g-factor increases for Pd/Py/Pd and Pt/Py/Pt while\n42Ferromagnet(F) Ai Ar \u000e\u000b \u000e! B Assumption Equations\nMetal ImT+\u0000 ReT+\u0000ReT+\u0000\nImT+\u0000ImT+\u0000\nReT+\u0000Thick F\nThin F(27)(66) (88)(89)\n(93)\nInsulator MI\u0017 (MI\u0017)2(MI\u0017)2(MI\u0017)3Weak spin relaxation\u0003(116) (148)\n- (MI\u0017)2P\nq(1 + 2nq) - - Magnon (140)\nTABLE I. Summary of essential parameters determining spin current js, corrections to the Gilbert\ndamping\u000e\u000band resonance frequency shift \u000e!Bfor metallic and insulating ferromagnets. Coe\u000e-\ncientsAiandArare for the spin current, de\fned by Eq. (1). Label \u0000indicates that it is not\ndiscussed in the present paper.\u0003: For strong spin relaxation case, the density of states \u0017is\nreplaced by inverse of electron spin-\rip time, \u001csf.31\ndecreases for Ta/Py/Ta. These results appear consistent with ours, because \u000e!Bis governed\nby ImT+\u0000, whose sign depends on the sign of interface spin-orbit interaction. In contrast,\ndamping enhancement proportional to Re T+\u0000is positive for thick metals. However, other\npossibilities like the e\u000bect of a large interface orbital moment playing a role in the gfactor,\ncannot be ruled out at present.\nRecently, inverse spin Hall measurement has become common for detecting the spin cur-\nrent. In this method, however, only the dc component proportional to n\u0002_nis accessible so\nfar and there remains an ambiguity for qualitative estimates because another phenomeno-\nlogical parameter, the conversion e\u000eciency from spin to charge, enters. Qualitatively, the\nvalues ofArobtained by the inverse spin Hall measurements32and FMR measurements are\nconsistent with each other.\nThe cases of insulating ferromagnets have been studied recently. In the early experiments,\norders of magnitude smaller value of Arcompared to metallic cases were reported31, while\nthose small values are now understood as due to poor interface quality. In fact, FMR\nmeasurements on epitaxially grown samples like YIG/Au/Fe turned out to show Arof 1\u0018\n5\u00021018m\u00002(Refs.33,34), which is the same order as in the metallic cases. Inverse spin Hall\nmeasurements on YIG/Pt reports similar values35, and the value is consistent with the \frst\nprinciples calculation36. Systematic studies of YIG/NM with NM=Pt, Ta, W, Au, Ag, Cu,\nTi, V, Cr, Mn etc. were carried out with the result of Ar\u00181017\u00001018m\u00002(Refs.37{40). If we\nuse naive phenomenological relation, Eq. (6), Ar= 1018m\u00002corresponds to \u000e\u000b= 3\u000210\u00004if\n43a= 2\u0017A,S= 1 andd= 20\u0017A. Assuming interface sdexchange interaction, the value indicates\nMI\u0017\u00180:01, which appears reasonable at least by the order of magnitude from the result of\nx-ray magnetic circular dichroism (XMCD) suggesting spin polarization of interface Pt of\n0:05\u0016B41.\nOn the other hand, FMR frequency shift of insulators cannot be explained by our theory.\nIn fact, the shift for YIG/Pt is \u000e!B=!B\u00181:6\u000210\u00002, which is larger than \u000e\u000b\u00182\u000210\u00003,\nwhile our perturbation theory assuming weak interface sdinteraction predicts \u000e!B=!B<\u000e\u000b .\nWe expect that the discrepancy arises from the interface spin-orbit interaction that would\nbe present at insulator-metal interface, which modi\fes the magnetic proximity e\u000bect and\ndamping torque signi\fcantly. It would be necessary to introduce anomalous sdcoupling at\nthe interface like the one discussed in Ref.42. Experimentally, in\ruence of interface spin-\norbit interaction43and proximity e\u000bect needs to be carefully characterized by using the\nmicroscopic technique, such as MCD, to compare with theories.\nIX. SUMMARY\nWe have presented a microscopic study of spin pumping e\u000bects, generation of spin cur-\nrent in ferromagnet-normal metal junction by magnetization dynamics, for both metallic and\ninsulating ferromagnets. As for the case of metallic ferromagnet, a simple quantum mechan-\nical picture was developed using a unitary transformation to diagonal the time-dependent\nsdexchange interaction. The problem of dynamic magnetization is thereby mapped to the\none with static magnetization and o\u000b-diagonal spin gauge \feld, which mixes the electron\nspin. In the slowly-varying limit, spin gauge \feld becomes static, and the conventional spin\npumping formula is derived simply by evaluating the spin accumulation formed in the nor-\nmal metal as a result of interface hopping. The e\u000bect of interface spin-orbit interaction was\ndiscussed. Rigorous \feld-theoretical derivation was also presented, and the enhancement of\nspin damping (Gilbert damping) in the ferromagnet as a result of spin pumping e\u000bect was\ndiscussed. The case of insulating ferromagnet was studied based on a model where spin\ncurrent is driven locally by the interface exchange interaction as a result of magnetic prox-\nimity e\u000bect. The dominant contribution turns out to be the one proportional to _n, while\nmagnon contribution leads to n\u0002_n, whose amplitude depends linearly on the temperature.\nOur analysis clearly demonstrate the di\u000berence in the spin current generation mechanism\n44for metallic and insulating ferromagnet.\nThe in\ruence of atomic-scale interface structure on the spin pumping e\u000bect are open and\nurgent issues, in particular for the case of ferrimagnetic insulators which have two sub-lattice\nmagnetic moments.\nACKNOWLEDGMENTS\nGT thanks H. Kohno, C. Uchiyama, K. Hashimoto and A. Shitade for valuable discus-\nsions. This work was supported by a Grant-in-Aid for Exploratory Research (No.16K13853)\nand a Grant-in-Aid for Scienti\fc Research (B) (No. 17H02929) from the Japan Society for\nthe Promotion of Science and a Grant-in-Aid for Scienti\fc Research on Innovative Areas\n(No.26103006) from The Ministry of Education, Culture, Sports, Science and Technology\n(MEXT), Japan.\nAppendix A: E\u000bect of spin-conserving spin gauge \feld on spin density\nHere we calculate contribution of spin-conserving spin gauge \feld, Az\ns;t, on the interface\ne\u000bects of spin density in F. It turns out that spin-conserving spin gauge \feld combined with\ninterface e\u000bects does not induce damping. This result is consistent with a naive expectation\nthat only the nonadiabatic components of spin current should contribute to damping.\nFIG. 11. Diagramatic representation of the contribution to the lessor Green's function for F\nelectron arising from the interface hopping (represented by tandt\u0003) and spin gauge \feld ( As;t).\nThe diagram (c) includes the spin gauge \feld implicitly in unitary matrices UandU\u00001.\nThe contribution to the lesser Green's function in F from the interface hopping (lowest,\nthe second-order in the hopping) at the linear order in the spin gauge \feld reads (diagra-\n45matically shown in Fig. 11)\n\u000eG<=\u000eG<\n(a)+\u000eG<\n(b)+\u000eG<\n(c)\n\u000eG<\n(a)=gr(As;t\u0001\u001b)gr\u0006r\n0g<+gr(As;t\u0001\u001b)gr\u0006<\n0ga+gr(As;t\u0001\u001b)g<\u0006a\n0ga+g<(As;t\u0001\u001b)ga\u0006a\n0ga\n\u000eG<\n(b)=gr\u0006r\n0gr(As;t\u0001\u001b)g<+gr\u0006<\n0g<(As;t\u0001\u001b)ga+gr\u0006a\n0ga(As;t\u0001\u001b)ga+g<\u0006a\n0ga(As;t\u0001\u001b)ga\n\u000eG<\n(c)=gr\u0006rg<+gr\u0006<ga+g<\u0006aga: (A1)\nHere\n\u0006a\u0011~tU\u00001ga\nNU~ty;(a= a;r;<)\n\u0006a\n0\u0011~tga\nN; (A2)\nare self energy due to the interface hopping, where \u0006ais the full self energy including the\ntime-dependent unitary matrix U, which includes spin gauge \feld. \u0006a\n0is the contribution of\n\u0006awith the spin gauge \feld neglected. We here focus on the contribution of the adiabatic ( z)\ncomponent,Az\ns;t. Usingg<=F(ga\u0000gr) for F (Fis a 2\u00022 matrix of the spin-polarized Fermi\ndistribution function) and g<\nN=fN(ga\nN\u0000gr\nN) and noting that all the angular frequencies of\nthe Green's function are equal, we obtain\n\u000eG<\n(a)+\u000eG<\n(b)'Az\ns;t\u001bz\u0002\n\u00002F[(gr)3\u0006r\n0\u0000(ga)3\u0006a\n0]\u0000(F\u0000fN)[(gr)2ga+gr(ga)2](\u0006a\n0\u0000\u0006r\n0)\u0015\n:\n(A3)\nThe contribution \u000eG<\n(c)is calculated noting that\n~tU\u00001ga\nNU~ty=ga\nN~t~ty\u0000dga\nN\nd!~t(As;t\u0001\u001b)~ty+O((As;t)2): (A4)\nThe linear contribution with respect to the zcomponent of the gauge \feld turns out to be\n\u000eG<\n(c)'Az\ns;t\u001bz\u0014\nF\u0014\n(gr)2@\n@!\u0006r\n0\u0000(ga)2@\n@!\u0006a\n0\u0015\n+ (F\u0000fN)grga@\n@!(\u0006a\n0\u0000\u0006r\n0)\u0015\n: (A5)\nWe therefore obtain the e\u000bect of spin-conserving gauge \feld as\n\u000eG<=Az\ns;t\u001bz@\n@!\u0014\nF\u0002\n(gr)2\u0006r\n0\u0000(ga)2\u0006a\n0\u0003\n+ (F\u0000fN)grga(\u0006a\n0\u0000\u0006r\n0)\u0015\n; (A6)\nwhich vanishes after integration over !. Therefore, contribution from spin-conserving gauge\n\feld and interface hopping vanishes in the spin density, leaving the damping una\u000bected.\n46Appendix B: Magnon representation of spin Berry's phase term\nHere we derive the expression for the spin Berry's phase term of the Lagrangian (99)\nin terms of magnon operator. The time-integral of the term is written by introducing an\narti\fcial variable uas44\nZ\ndtL B=SZ\ndt_\u001e(cos\u0012\u00001) =S\u00002Z\ndtZ1\n0duS\u0001(@tS\u0002@uS); (B1)\nwhereS(t;u) is extended to a function of tandu, but onlyS(t;u= 1) is physical. Noting\nthat the unitary transformation matrix element of Eq. (101) is written as\nUij= (ej)i; (B2)\nwherer1\u0011e\u0012,e2\u0011e\u001eande3\u0011n, we obtain\nS\u0001(@tS\u0002@uS) =eS\u0001[(@t+iAU;t)eS\u0002(@u+iAU;u)eS)]: (B3)\nEvaluating to the second order in the magnon operators, we have\n@teS\u0002@ueS= 2i\r^z[(@uby)(@tb)\u0000(@tby)(@ub)]: (B4)\nUsing the explicit form of AU;\u0016, the gauge \feld contribution is\n@ueS\u0001[eS\u0002iAU;teS)] =S2\r[(@uby)(\u0000sin\u0012_\u001e+i_\u0012) + (@ub)(\u0000sin\u0012_\u001e\u0000i_\u0012)]\u00002S\r2cos\u0012(@t\u001e)@u(byb):\n(B5)\nThe terms linear in the boson operators vanish by the equation of motion, and the second-\norder contribution is\nS\u0001(@tS\u0002@uS) = 2S\r2\u0014\ni@u[by(@tb)\u0000(@tby)b]\n\u0000@u[cos\u0012(@t\u001e)byb] +@t[cos\u0012(@u\u001e)byb] + sin\u0012((@t\u0012)(@u\u001e)\u0000(@u\u0012)(@t\u001e))byb\u0015\n:\n(B6)\nIntegrating over tandu, the term total derivative with respect to tof Eq. (B6) vanishes,\nresulting in\nZ\ndtZ1\n0duS\u0001(@tS\u0002@uS) = 2S\r2Z\ndt\u0014\ni[by(@tb)\u0000(@tby)b]\u0000cos\u0012(@t\u001e)byb\n+ sin\u0012((@t\u0012)(@u\u001e)\u0000(@u\u0012)(@t\u001e))byb\u0015\n: (B7)\n47The last term of Eq. (B7) represents the renormalization of spin Berry's phase term, i.e.,\nthe e\u000bectS!S\u0000byb, which we neglect below. The Lagrangian for magnon thus reads\nLm= 2S\r2Z\nd3ri[by(@t+iAz\ns;t)b\u0000by( \n@t\u0000iAz\ns;t))b]; (B8)\nnamely, magnons interacts with the adiabatic component of spin gauge \feld, Az\ns;t.\nAppendix C: Decomposition of contour-ordered self energy\nHere we summarize decomposition formula of self energy. Obviously, we have\n[gD]<=g<D<: (C1)\nRetarded component is de\fned as\n[gD]r\u0011[gD]t\u0000[gD]<; (C2)\nwhere the time-ordered one is\n[g(t1\u0000t2)D(t1\u0000t2)]t\u0011\u0012(t1\u0000t2)g>D>+\u0012(t2\u0000t1)g<D<\n=grDr+grD<+g<Dr+g<D<: (C3)\nWe thus obtain\n[gD]r=grDr+grD<+g<Dr: (C4)\nNoting that grDa= 0, we can write it as\n[gD]r=grD<+g>Dr=g<Dr+grD>: (C5)\nThe advanced component is similarly written as\n[gD]a=\u0000gaDa+gaD<+g<Da\n=gaD>+g<Da=gaD<+g>Da: (C6)\n1E. Saitoh, M. Ueda, H. Miyajima, and G. Tatara, Appl. Phys. Lett. 88, 182509 (2006).\n482Y. Tserkovnyak, A. Brataas, and G. E. W. Bauer, Phys. Rev. Lett. 88, 117601 (2002).\n3M. V. Moskalets, Scattering matrix approach to non-stationary quantum transport (Imperial\nCollege Press, 2012).\n4M. B uttiker, H. Thomas, and A. Pretre, Zeitschrift f ur Physik B Condensed Matter 94, 133\n(1994).\n5P. W. Brouwer, Phys. Rev. B 58, R10135 (1998).\n6G. Tatara and H. Kohno, Phys. Rev. B 67, 113316 (2003).\n7G. Tatara, H. Kohno, and J. Shibata, Physics Reports 468, 213 (2008).\n8V. K. Dugaev, P. Bruno, B. Canals, and C. Lacroix, Phys. Rev. B 72, 024456 (2005).\n9R. H. Silsbee, A. Janossy, and P. Monod, Phys. Rev. B 19, 4382 (1979).\n10L. Berger, Phys. Rev. B 54, 9353 (1996).\n11S. Mizukami, Y. Ando, and T. Miyazaki, Japanese J. Appl. Phys. 40, 580 (2001).\n12E.\u0014Sim\u0013 anek and B. Heinrich, Phys. Rev. B 67, 144418 (2003).\n13E.\u0014Sim\u0013 anek, Phys. Rev. B 68, 224403 (2003).\n14K. Chen and S. Zhang, Phys. Rev. Lett. 114, 126602 (2015).\n15G. Tatara, Phys. Rev. B 94, 224412 (2016).\n16D. S. Fisher and P. A. Lee, Phys. Rev. B 23, 6851 (1981).\n17A. Takeuchi and G. Tatara, J. Phys. Soc. Japan 77, 074701 (2008).\n18K. Hosono, A. Takeuchi, and G. Tatara, Journal of Physics: Conference Series 150, 022029\n(2009).\n19H. Adachi, J.-i. Ohe, S. Takahashi, and S. Maekawa, Phys. Rev. B 83, 094410 (2011).\n20J. J. Sakurai, Modern Quantum Mechanics (Addison Wesley, 1994).\n21K. Hashimoto, G. Tatara, and C. Uchiyama, arXiv:1706.00583 (2017).\n22G. Tatara and P. Entel, Phys. Rev. B 78, 064429 (2008).\n23H. Kohno, G. Tatara, and J. Shibata, J. Phys. Soc. Japan 75, 113706 (2006).\n24G. Tatara and H. Fukuyama, J. Phys.l Soc. Japan 63, 2538 (1994).\n25G. Tatara and H. Fukuyama, Phys. Rev. Lett. 72, 772 (1994).\n26N. Umetsu, D. Miura, and A. Sakuma, J. Phys. Soc. Japan 81, 114716 (2012).\n27H. Y. Yuan, Z. Yuan, K. Xia, and X. R. Wang, Phys. Rev. B 94, 064415 (2016).\n28G. Tatara, Phys. Rev. B 92, 064405 (2015).\n29C. Kittel, Quantum theory of solids (Wiley, New York, 1963).\n4930K. Nakata and G. Tatara, J. Phys. Soc. Japan 80, 054602 (2011).\n31Y. Kajiwara, K. Harii, S. Takahashi, J. Ohe, K. Uchida, M. Mizuguchi, H. Umezawa, H. Kawai,\nK. Ando, K. Takanashi, S. Maekawa, and E. Saitoh, Nature 464, 262 (2010).\n32F. D. Czeschka, L. Dreher, M. S. Brandt, M. Weiler, M. Althammer, I.-M. Imort, G. Reiss,\nA. Thomas, W. Schoch, W. Limmer, H. Huebl, R. Gross, and S. T. B. Goennenwein, Phys.\nRev. Lett. 107, 046601 (2011).\n33B. Heinrich, C. Burrowes, E. Montoya, B. Kardasz, E. Girt, Y.-Y. Song, Y. Sun, and M. Wu,\nPhys. Rev. Lett. 107, 066604 (2011).\n34C. Burrowes, B. Heinrich, B. Kardasz, E. A. Montoya, E. Girt, Y. Sun, Y.-Y. Song, and M. Wu,\nApplied Physics Letters 100, 092403 (2012).\n35Z. Qiu, K. Ando, K. Uchida, Y. Kajiwara, R. Takahashi, H. Nakayama, T. An, Y. Fujikawa,\nand E. Saitoh, Applied Physics Letters 103, 092404 (2013).\n36X. Jia, K. Liu, K. Xia, and G. E. W. Bauer, EPL (Europhysics Letters) 96, 17005 (2011).\n37H. L. Wang, C. H. Du, Y. Pu, R. Adur, P. C. Hammel, and F. Y. Yang, Phys. Rev. Lett. 112,\n197201 (2014).\n38C. Du, H. Wang, F. Yang, and P. C. Hammel, Phys. Rev. Applied 1, 044004 (2014).\n39C. Du, H. Wang, F. Yang, and P. C. Hammel, Phys. Rev. B 90, 140407 (2014).\n40H. Wang, C. Du, P. C. Hammel, and F. Yang, Applied Physics Letters 110, 062402 (2017).\n41Y. M. Lu, Y. Choi, C. M. Ortega, X. M. Cheng, J. W. Cai, S. Y. Huang, L. Sun, and C. L.\nChien, Phys. Rev. Lett. 110, 147207 (2013).\n42K. Xia, W. Zhang, M. Lu, and H. Zhai, Phys. Rev. B 55, 12561 (1997).\n43M. Caminale, A. Ghosh, S. Au\u000bret, U. Ebels, K. Ollefs, F. Wilhelm, A. Rogalev, and W. E.\nBailey, Phys. Rev. B 94, 014414 (2016).\n44A. Auerbach, Intracting Electrons and Quantum Magnetism (Springer Verlag, 1994).\n50"    },    {        "title": "1706.02296v2.Adiabatic_and_nonadiabatic_spin_torques_induced_by_spin_triplet_supercurrent.pdf",        "content": "Adiabatic and nonadiabatic spin torques induced by spin-triplet supercurrent\nRina Takashima,1,\u0003Satoshi Fujimoto,2and Takehito Yokoyama3\n1Department of Physics, Kyoto University, Kyoto 606-8502, Japan\n2Department of Materials Engineering Science, Osaka University, Toyonaka 560-8531, Japan\n3Department of Physics, Tokyo Institute of Technology, Tokyo 152-8551, Japan\n(Dated: September 18, 2018)\nWe study spin transfer torques induced by a spin-triplet supercurrent in a magnet with the superconduct-\ning proximity e \u000bect. By a perturbative approach, we show that spin-triplet correlations realize new types of\ntorques, which are analogous to the adiabatic and nonadiabatic ( \f) torques, without extrinsic spin-flip scatter-\ning. Remarkable advantages compared to conventional spin-transfer torques are highlighted in domain wall\nmanipulation. Oscillatory motions of a domain wall do not occur for a small Gilbert damping, and the thresh-\nold current density to drive its motion becomes zero in the absence of extrinsic pinning potentials due to the\nnonadiabatic torque controlled by the triplet correlations.\nE\u000ecient manipulation of magnetization is of great techno-\nlogical importance. Spin-transfer torques (STT), which can\ncontrol magnetization with an electric current, have attracted\nattention[1–7], and STT can be applied to the so-called race-\ntrack memory using magnetic domain walls[8]. In a smooth\nmagnetic texture n, spin-polarized currents exert STT on n,\nwhich is given by [9–11]\n\u001cSTT=\u0000(js\u0001r)n+\fn\u0002(js\u0001r)n: (1)\nHerejs=\u0000(Pa3=2eS)j, and\fis a dimensionless parameter,\nwherejis a charge current density, Pis the spin polariza-\ntion of current, ais the lattice constant, Sis the spin size,\nand\u0000eis the electron charge. The first term in Eq. (1) arises\nwhen the electron spins follow the texture adiabatically. The\nsecond term, often referred to as the non-adiabatic torque, is\nknown to have two origins [10, 12, 13]. It appears from spin-\nflip impurity scatterings or the spin-orbit coupling. It also oc-\ncurs when electrons fail to follow magnetic textures because\nthe texture is not smooth enough. As demonstrated in several\nworks [10, 14], the nonadiabatic torque plays a crucial role\nin magnetization dynamics. For \f,0, the threshold current\ndensity for a steady motion of a domain wall becomes zero in\nthe absence of pinning potentials.\nRecently, superconductivity has opened up new possibil-\nities for spintronics with suppressed Joule heating[15, 16].\nIt has been pointed out that the Josephson current exerts\na spin torque on magnetization in ferromagnetic Joseph-\nson junctions[17–23]. Furthermore, with spin valves using\nsuperconductors[24–28], one can change the resistance dras-\ntically by a magnetic field, and the lifetime of spin density\nis enhanced in a superconducting state relative to a normal\nstate[29–31]. Such an interplay of superconductivity and\nmagnetic moments is important especially with spin-triplet\nCooper pairs due to the coupling between triplet order pa-\nrameters and localized moments[32–37]. Triplet pairs can\narise in the interface between a ferromagnet and singlet su-\nperconductor when there is magnetic inhomogeneity[38] or\nspin-orbit couplings [39, 40]. Experimentally, the proxim-\nity e\u000bect of triplet pairs has been observed in fully-spin\n\u0003Electronic address: takashima@scphys.kyoto-u.ac.jppolarized metals[41, 42] and multilayers with noncollinear\nmagnets[43, 44]. The spin-triplet proximity e \u000bect to a ferro-\nmagnet from Sr 2RuO 4, a candidate of a triplet superconductor,\nhas been also observed[45].\nGiven the experimental advances in the proximity-induced\ntriplet Cooper pairs in magnets, triplet supercurrent-induced\nSTT is a promising way to realize an e \u000ecient control of mag-\nnetization. Utilization of a supercurrent suppresses Joule heat-\ning and the tunablity of STT may be enhanced by pairing de-\ngrees of freedom. However, while several works showed that a\nsupercurrent exerts a spin-torque in ferromagnetic Josepshon\njunctions[17–23], it still remains unclear how a triplet super-\ncurrent acts on a localized moment, and how STT is changed\nby triplet-paring correlation.\nIn this work, we microscopically study STT induced by\ntriplet supercurrents considering the spin-triplet proximity ef-\nfect. We show that the derived STT have two parts, analogous\nto the adiabatic and non-adiabatic torques, which can be tuned\nby the triplet correlations. Remarkable advantages compared\nto conventional STT are highlighted in domain wall manipu-\nlation. In contrast to the non-adiabatic STT in normal metals,\nthe supercurrent-induced STT do not require extrinsic scatter-\ning processes, and hence, is more easy to control. Further-\nmore, a domain wall does not show oscillatory motions for\na small Gilbert damping, and hence an e \u000ecient manipulation\ncan be realized.\nLet us consider a thin-film magnet with the proximity e \u000bect\nofp-wave triplet superconductivity modeled by the Hamilto-\nnian :\nHel=\u0000tX\nhi;jicy\ni\u000bcj\u000b\u0000\u0016X\nicy\ni\u000bci\u000b\n\u0000JsdSX\nin(ri)\u0001\u001b\u000b\fcy\ni\u000bci\f\n+\u00010\n2X\ni;jeiQ\u0001(ri+rj)h\n(di j\u0001\u001b)i\u001byi\n\u000b\fcy\ni\u000bcy\nj\f+H:c:(2)\nHere cy\ni\u000b(ci\u000b) is the electron creation (annihilation) operator at\nsiteiwith spin\u000bon a square lattice, hi;jiis taken over the\nnearest neighbor pairs, tis the hopping amplitude, \u0016is the\nchemical potential, and \u001b=(\u001bx;\u001by;\u001bz) are the Pauli matri-\nces. Electrons couple to localized spins given by Sn(ri)=arXiv:1706.02296v2  [cond-mat.mes-hall]  16 Sep 20172\nTriplet SuperconductorDomain wall motion\nSupercurrentFM\nFIG. 1: (Color online) Proposed setup. A ferromagnet (FM) with a\ndomain wall is attached on a superconductor, which produces a spin-\ntriplet proximity e \u000bect. We apply a supercurrent in the xyplane and\ndrive the domain wall motion.\nS(sin\u0012cos\u001e;sin\u0012cos\u001e;cos\u0012) with the coupling constant Jsd.\nThe spin texture n(r) varies smoothly with the length scale\n`(\u001d\u0018SC) where\u0018SCdenotes the superconducting coherence\nlength. The last term in Eq. (2) is the proximity-induced triplet\np-wave pairing, given by di j=(dx\ni j;dy\ni j;dz\ni j), wheredi j=\u0000dji,\nand\u00010is the pairing amplitude.\nThe phase gradient of the paring function describes a super-\ncurrent. We consider a phase given by Q\u0001(ri+rj), which re-\nsults in the supercurrent density j'\u00002tenea2Q(forjQja\u001c1\nand at low temperature), where neis an electron density[56]\nthat participates in the supercurrent. Here, the supercurrent\ncan be supplied from an external dc current source. We also\nnote that we can restore the gauge invariance by redefining\nQin the current so as to include the vector potential. As\nshown in Fig. 1, a relevant experimental setup of the above\nmodel is a heterostructure composed of a metallic magnet and\na triplet superconductor (e.g., Sr 2RuO 4). A triplet supercon-\nductor can be replaced by a singlet superconductor (e.g., Nb)\nwith a conical magnetic layer such as Ho [44] or the spin-orbit\ncoupling[39, 40], which produces the triplet proximity e \u000bect.\nTo derive STT, we calculate the spin density induced by\na supercurrent to the linear order of j/Q. In the follow-\ning calculation, we perturbatively treat the spatial derivative\nofn(ri). This treatment can be simplified by rewriting the\nHamiltonian with electron operators ai\u000b, the spin quantization\naxis of which is parallel to n(ri)[10]. It is defined by ci\u000b=\n(U(ri))\u000b\fai\f, where U(ri)=m(ri)\u0001\u001bis a unitary matrix, and\nm(ri)=(sin(\u0012=2)cos\u001e;sin(\u0012=2)sin\u001e;cos(\u0012=2)) with\u0012and\n\u001ebeing the angles of n(ri). This satisfies Uy(n\u0001\u001b)U=\u001bz.\nThe Hamiltonian Eq. (2) is rewritten as\nHel'X\nk(\u0018kI2\u0000JsdS\u001bz)\u000b\u000bay\nk\u000bak\u000b\n+X\nk;q3\u0017\nk+q=2Aa\n\u0017(q)\u001ba\n\u000b\fay\nk+q\u000bak\f\n+1\n2X\nk\u0001\u000b\nkay\nk+Q\u000bay\n\u0000k+Q\u000b+H.c.; (3)\nwhere\u0018k=\u00002t(cos( kxa)+cos(kya))\u0000\u0016is the kinetic energy,\n3\u0017\nk=@\u0018k=@k\u0017is the velocity, I2=diag(1;1) , and ak\u000bis the\nFourier transform of ai\u000b. The second term arises from the\nhopping in the presence of a noncollinear texture. Here we\ndefine the spin gauge field Aa\n\u0017i\u001ba=\u0000iU(ri)@\u0017U(ri), where\nwe denote Ab\n\u0017(q)=N\u00001P\niAb\n\u0017ie\u0000iq\u0001riwith\u0017=fx;y;zgandb=fx;y;zg.Nis the total number of sites. Assuming a large\nexchange splitting \u00182JsdS\u001dj\u00010j, we focus on the equal spin\npairing given by\n\u0001\u000b\nk= \u0001 0(Uy(d(k)\u0001\u001b)i\u001byU\u0003)\u000b\u000b; (4)\n=\u0000\u00010\u0010\n(Rabdb(k)\u001ba)i\u001by\u0011\n\u000b\u000b; (5)\nwhered(k)=N\u00001P\nhi jidi je\u0000ik\u0001(ri\u0000rj). We neglect pairings be-\ntween spin-split bands, which correspond to the components\nofd(k) parallel ton.Rab=2mamb\u0000\u000eab(a;b=fx;y;zg) is a\nSO(3) rotation matrix corresponding to the unitary matrix U.\nIn the last term in Eq. (3), we neglect the spatial dependence\nof\u0012;\u001e, terms of the order of \u0018SC=`, assuming thatj\u00010(k)j=tis\nalso a small parameter.\nThe spin expectation value of electrons sa\ni=\n1\n2(\u001ba)\u000b\fhcy\ni\u000bci\fican be described by the operator ai\u000bas\nsa\ni=Rab(ri)˜sb\ni; (6)\nwhere ˜ sa\ni=1\n2(\u001ba)\u000b\fhay\ni\u000bai\fi. Noting this relation, we obtain\nthe spin density induced by a supercurrent as\n\u000e˜sa\nqB \nlim\nQ!0˜sa\nq\u0000˜sa\nqjQ=0\nQ\u0011!j\u0011\n(\u00002tenea2); (7)\n=\u0019ab\n\u0017\u0011Ab\n\u0017(q)j\u0011\n2ene; (8)\nwhere\u000e˜sq=N\u00001P\ni\u000e˜sie\u0000iq\u0001riand\n\u0019ab\n\u0017\u0011=lim\nq!0\u0000T\n4Nta2X\nn;k@2\u0018k\n@k\u0017@k\u0011Tr[SaGk+q(i\u000fn)SbGk(i\u000fn)]:(9)\nSee Supplemental Materials (SM) for detail [55]. Here,\nGk(i\u000fn) is the Green function in Nambu representation, the\nbasis of which is ( ak\";ak#;ay\n\u0000k\";ay\n\u0000k#)T, and Sais a spin ma-\ntrix in the Nambu representation, which is given by\nSa= \n\u001ba\n\u0000(\u001ba)T!\n: (10)\nIts inverse is defined by ( Gk(i\"n))\u00001=i\u000fnI4\u0000HBdG(k), where\nHBdG(k)= \n\u0018kI2\u0000M\u001bz\u0001(k)\n\u0001\u0003(k)\u0000\u0018kI2+M\u001bz!\n; (11)\nI4=diag(1;1;1;1),\u000fn=\u0019T(2n+1) is Matsubara frequency\nwith temperature T, and \u0001(k)=diag(\u0001\"\nk;\u0001#\nk).\nIn Eq. (9), we have neglected terms which are vanishingly\nsmall at low temperatures compared to the critical temper-\nature, in a system with a full gap or point nodes, see SM\n[55]. Also, we have taken the limit q!0assuming that the\nmomentum transfer from a smooth magnetic texture is small\ncompared to the Fermi momentum [10]. We note that siis\ninvariant by a unitary transformation U!Uei'\u001bzwith an\narbitrary spin rotational angle '(r) aroundn, while ˜siand\nRabchange their forms. In the following, we explicitly use\n@2\u0018k\n@k\u0017@k\u0011/\u000e\u0017\u0011and define\u0019ab\n\u0017\u0011=\u0019ab\n\u0017\u000e\u0017\u0011.3\nFrom Eqs. (6) and (8), we obtain the local STT, \u001cSTT=\n2Jsdn\u0002\u000esi, as\n\u001cSTT=X\n\u0017=x;y\u0000˜P\u0017a3\n2eSj\u0017\u0010\n\u0000@\u0017n+˜\f\u0017n\u0002@\u0017n\u0011\n: (12)Here ˜P\u0017and ˜\f\u0017are the analogs of the spin polarization of a\ncurrent Pand\fin Eq. (1), and they are given by\n˜P\u0017=JsdS\nnea3\"1\n2\u0010\n\u0019xx\n\u0017+\u0019yy\n\u0017\u0011\n+1\nj@\u0017nj2\u0010\n\u0000\u0019(1)\n\u0017\u0010\n(@\u0017\u0012)2\u0000sin2\u0012(@\u0017\u001e)2\u0011\n+2\u0019(2)\n\u0017sin\u0012@\u0017\u0012@\u0017\u001e\u0011#\n; (13)\n˜\f\u0017=\u0000JsdS\nnea31\n˜P\u00171\nj@\u0017nj2\u0010\n\u0019(2)\n\u0017\u0010\n(@\u0017\u0012)2\u0000sin2\u0012(@\u0017\u001e)2\u0011\n+2\u0019(1)\n\u0017sin\u0012@\u0017\u0012@\u0017\u001e\u0011\n; (14)\nwhere\n\u0019(1)\n\u0017=cos(2\u001e)1\n2\u0010\n\u0019xx\n\u0017\u0000\u0019yy\n\u0017\u0011\n+sin(2\u001e)\u0019xy\n\u0017; (15)\n\u0019(2)\n\u0017=sin(2\u001e)1\n2\u0010\n\u0019xx\n\u0017\u0000\u0019yy\n\u0017\u0011\n\u0000cos(2\u001e)\u0019xy\n\u0017: (16)\nThese are the central results of this paper, which are ap-\nplicable to any smooth magnetic textures. Notably, in the\nlow density limit and at low temperature, ˜P\u0017and ˜\f\u0017are\ngiven by the spin susceptibility perpendicular to nsince\n\u0019ab\n\u0017is equivalent to the bare spin susceptibility of the ak\u000b\nfield. To make ˜\f\u0017finite, anisotropy such as \u0019xx\n\u0017,\u0019yy\n\u0017or\n\u0019xy\n\u0017,0 is necessary. As is known in the spin susceptibil-\nity [46], such anisotropy naturally arises with a triplet pair-\ning. They depend on the relative phase between \u0001\"\nkand\n\u0001#\nkas1\n2\u0010\n\u0019xx\n\u0017\u0000\u0019yy\n\u0017\u0011\n=limq!0N\u00001P\nkRe(\u0001\u0003\"\nk\u0001#\nk)f\u0017(k;q) and\n\u0019xy\n\u0017=limq!0N\u00001P\nkIm(\u0001\u0003\"\nk\u0001#\nk)f\u0017(k;q);where f\u0017(k;q) is\npresented in the SM [55]. Therefore, a triplet pairing can\nmake ˜\f\u0017finite and cause the non-adiabatic torque without ex-\ntrinsic scattering processes. Furthermore, ˜P\u0017and˜\f\u0017depend on\nthe spatial position through the coupling between the dvector\nandn. This is important for a domain wall dynamics as we\nsee below.\nNow, we demonstrate domain wall dynamics[47] induced\nby the obtained STT. We consider the Hamiltonian Htot=\nHel+Hspin, where Helis given in Eq. (3) and\nHspin=S2\n2\n\u0002X\ni\u0010\n\u0000J(@\u0017n(ri;t))2\u0000Knz(ri;t)2+K?ny(ri;t)2\u0011\n:(17)\nHere, Jis the ferromagnetic exchange coupling, and K;K?\nare the onsite anisotropies that satisfy K?\u001c K\u001c\nJa\u00002. We consider a domain wall configuration given by\nn(r;t)=(cos\u001e0(t) sin\u0012(x;t);sin\u001e0(t) sin\u0012(x;t);cos\u0012(x;t)),\nwhere cos\u0012(x;t)=tanh\u0010x\u0000X(t)\n\u0015\u0011\n,\u0015=pJ=K, and X(t) is the\ndomain wall center. A schematic figure of a domain wall for\n\u001e0=\u0019is shown in Fig. 1. With this configuration, the STT\nin Eq. (12) is characterized by ˜Px=JsdS\nnea3h1\n2\u0010\n\u0019xx\nx+\u0019yy\nx\u0011\n\u0000\u0019(1)\nxiand˜\fx=\u0000JsdS\nnea3\u0019(2)\nx\n˜Px. Including the e \u000bects of damping, we ob-\ntain the equations of motion of \u001e(t) and X(t) as\n@tX=3c\n(1+\u000b2)(\u001c(\u001e0)jx+\u000bF(\u001e0)jx+sin 2\u001e0); (18)\n@t\u001e0=\u00001\n(1+\u000b2)t0(\u000b\u001c(\u001e0)jx\u0000F(\u001e0)jx+\u000bsin 2\u001e0);(19)\nwhere\u000bis the Gilbert damping constant, 3c=K?\u0015S=2, and\nt0=\u0015=3c.\u001c(\u001e0) and F(\u001e0) denote the coupling to the current\nvia STT. They read\n\u001c(\u001e0)=\u0000a\n23cX\ni˜Pxa3\n2eS@xnz; (20)\nF(\u001e0)=a\n23cX\ni˜Pxa3\n2eS˜\fx@xnz: (21)\nIn the above equations of motions, we have assumed that the\nelectron spin density induced by ˙ ndoes not change the dy-\nnamics qualitatively when the spin size Sis large. Also, we\ndid not consider pinning potentials for simplicity.\nIn the following, we consider triplet pairing given by\nd(k)=(\u0000sinkya;sinkxa;\u000esinkxa). Such a pairing can be sta-\nbilized by the spin-orbit coupling g(k)\u0001\u001b(g(k)=\u0000g(\u0000k))\nin a system without inversion symmetry; d(k)kg(k) is en-\nergetically favored[48]. The Rashba type spin-orbit coupling\ncan stabilized(k)=(\u0000sin(kya);sin(kxa);0), and in our case,\nit can originate from the boundary between the superconduc-\ntor and ferromagnet. Furthermore, we add dz(k)=\u000esin(kxa),\nwhich can be attributed to the additional spin-orbit coupling\ndue to the broken mirror symmetry about the xzplane.\nIn numerical calculations, we set \u0016=t=\u00001:8;\u00010=t=5\u0002\n10\u00002;JsdS=t=1, and T=t=5\u000210\u00003. Using the above d(k),\nwe first show ˜Pxand˜\fxin the domain wall configuration for\n\u001e0=\u0019=4 (Fig. 2 (a)-(c)). E \u000bective spin polarization ˜Pxis al-\nmost constant in space. On the other hand, ˜\fxhighly depends\non the spatial position. Importantly, ˜Px(˜\fx) is symmetric (an-\ntisymmetric) under x!\u0000 xfor\u000e=0, while ˜Pxand ˜\fxare\nslightly shifted for \u000e,0. Because of such symmetries, F(\u001e0)\n(Eq. (21)) vanishes for \u000e=0, where we note @xnzis an even\nfunction of x. On the other hand, for \u000e,0 such symmetries4\n×10-1\n×10-40.50.3(a)\n(b)\n(c)(d)\n(e)×10-4\n0.50.3\nFIG. 2: (Color online) (a) ˜Pxand (b) ˜\fxin a domain wall configu-\nration with \u001e0=\u0019=4 as functions of the position xfor di \u000berent\u000e,\nwhere dvector is d(k)=(\u0000sinkya;sinkxa;\u000esinkxa). (c) The profile\nof the domain wall. (d) \u001c(\u001e0) and (e) F(\u001e0) as functions of \u001e0for\ndi\u000berent\u000e, where we define j0=S e3ca\u00003.\n(a) (b)\n(c) (d)2\n5\n10\n152\n5\n10\n15\nFIG. 3: (Color online) Velocity of a domain wall center for \u000e=0\nin (a) and (c), and for \u000e=0:5 in (b) and (d). (a) shows the velocity\naveraged over the oscillation period after su \u000eciently long time, and\n(b) shows the velocity after su \u000eciently long time. In (c) and (d),\nthe time evolution of the velocity for di \u000berent jx=j0(indicated by\ndi\u000berent colors) is presented. We have set \u000b=10\u00004, and the initial\nconditions are \u001e0(t=0)=\u0019and ˙X(t=0).\nof˜Pxand ˜\fxare broken, and hence we obtain finite F(\u001e0).\nSimilar arguments apply to other \u001e0values. In the following,\nwe will show the resulting domain wall dynamics.\nWe next solve the equation of motion. For \u000e=0,\u001c(\u001e0)\nis well fitted by \u001c(\u001e0)'\u001c0+\u001c1cos 2\u001e0, and F(\u001e0)=0 as\nshown in Fig. 2 (d), (e). Note that such \u001e0dependence of \u001c(\u001e0)\narises from the triplet pairing. With this \u001c(\u001e0), we have solved\nEqs. (18) and (19). Fig. 3 (a) shows the averaged velocity\nafter su \u000eciently long time, and there is a threshold current\ndensity. It is given by jc=1p\n\u001c2\n0\u0000\u001c2\n1, which is obtained from\nEqs. (18) and (19) with \u001c(\u001e0)=\u001c0+\u001c1cos 2\u001e0andF(\u001e0)=\n0. As shown in Fig. 3 (c), ˙Xis zero after su \u000eciently longtime for jx<jcbecause of the intrinsic pinning due to the\nanisotropy K?. With a larger current density ( jx>jc), the\ndomain wall center oscillates with a finite drift velocity. The\nabove behavior is similar to a domain wall motion in normal\nferromagnetic metals without the non-adiabatic torque.\nWe now consider the case with \u000e,0. As shown in Fig. 2\n(d) and (e), \u001c(\u001e0) and F(\u001e0) are well fitted by\n\u001c(\u001e0)'\u001c0+\u001c1cos 2\u001e0; (22)\nF(\u001e0)'F0cos\u001e0: (23)\nImportantly, we have finite F(\u001e0) when\u000e,0. As shown\nin SM [55], finite dx(k)dz(k) and dy(k)dz(k) are necessary to\nhave finite F(\u001e0).\nThis F(\u001e0) changes the domain wall motion drastically. As\nshown in Fig. 3(b), there is no threshold current density for\ndriving the steady motion, which is similar to a situation in\nnormal metals with the non-adiabatic torque. An important\ndi\u000berence from conventional STT is that a domain wall shows\nno oscillatory motions ( ¨X=˙\u001e=0) even for a large current\ndensity depending on \u000b. According to Eqs. (19), oscillation\noccurs for j>jmax=max\u001e0f\u000bsin 2\u001e0=(\u000b\u001c(\u001e0)\u0000F(\u001e0))g.\nWith the numerically obtained parameters in Eqs. (22) and\n(23), jmaxis infinite when \u000bis small enough ( \u000b.10\u00003\u0018\nj\u00010j2=t2) because\u000b\u001c(\u001e0)\u0000F(\u001e0) can be zero. In contrast, for\nconventional STT, jmaxis always finite since \u001c(\u001e0) and F(\u001e0)\ndo not depend on \u001e0, and oscillatory motion always appears\nfor a current density larger than jmax. The absence of oscil-\nlatory motion is important for an e \u000ecient manipulation of a\ndomain wall. Let us estimate the required supercurrent den-\nsity. For example, in a ferromagnetic nanowire Ni 81Fe19, an\nexperimental value is S2K?\u0015a\u00003\u00180:05 J/m2[49], and hence\n3c'3\u0002102m/s and j0\u00184\u00021013A/m2. The required current\ndensity to achieve ˙X'0:4\u0016m=s isjx'105A/m2, which is\nlower than the critical current density in typical ferromagnetic\nJosephson junctions[50].\nTo summarize, we have microscopically derived STT in-\nduced by triplet supercurrents. We showed that spin-triplet\npairings give novel types of STT, which can be used for an ef-\nficient control of a domain wall. The results can be applied to\ndi\u000berent dvectors and magnetic textures such as a skyrmion,\nand we expect the possibilities for more interesting aspects of\ntriplet supercurrent-induced STT.\nThere are several comments and discussions. In nor-\nmal metals, a voltage drop occurs due to the domain wall\nmotion[51–54] in addition to the resistance of a sample. For\na superconducting system, while a supercurrent (dc current)\nis not accompanied by the voltage drop, the motion of a do-\nmain wall would also cause time evolution of the phase, and it\nmight result in a finite voltage drop. In this work, we have as-\nsumed such fluctuation is small compared to the overall phase\ngradient.\nIn this paper, we did not consider the Abrikosov vortices in\na superconductor, which might be induced by the stray field of\nthe ferromagnet. These vortices can cause voltage drop due to\ntheir dynamics and a non-uniform current pattern. To suppress\nthe vortices, we can use junctions with a ferromagnet with a\nsmall stray field, e.g., Sr 2RuO 4/permalloy junction with mag-\nnetization oriented in-plane. We can also use a singlet super-5\nconductor with a high critical field [38] such as niobium, and\nhence Nb /Ho/permalloy junction is another possible setup.\nWhen the superconducting pairing is proximity-induced, in\ngeneral, singlet pairing is expected to be mixed. However, the\ndecay length of the singlet proximity e \u000bect is much shorter\nthan that of the equal spin triplet pairing for a large exchange\ncoupling. Furthermore, the contribution from singlet pairing\nto STT is much smaller than that from triplet pairing in the\nadiabatic regime, which is justified in a smooth magnetic tex-ture.\nAcknowledgements. We would like to thank M. S. Anwar,\nH. Kohno, T. Nomoto, Y . Shiomi, and Y . Yanase for fruitful\ndiscussions. This work was supported by Grants-in-Aid for\nScientific Research [Grants No. 25220711, No. 17K05517,\nNo. JP15H05852, No. JP16H00988, (KAKENHI on Innova-\ntive Areas from JSPS of Japan “Topological Materials Sci-\nence), and No. JP17H05179 (KAKENHI on Innovative Areas\n“Nano Spin Conversion Science).\n[1] J. Slonczewski, J. Magn. Magn. Mater. 159, L1 (1996).\n[2] L. Berger, Phys. Rev. B 54, 9353 (1996).\n[3] J. A. Katine, F. J. Albert, R. A. Buhrman, E. B. Myers, and\nD. C. Ralph, Phys. Rev. Lett. 84, 3149 (2000).\n[4] M. Kl ¨aui, C. A. F. Vaz, J. A. C. Bland, W. Wernsdorfer, G. Faini,\nE. Cambril, and L. J. Heyderman, Appl. Phys. Lett. 83, 105\n(2003).\n[5] J. Grollier, P. Boulenc, V . Cros, A. Hamzi ´c, A. Vaur `es, A. Fert,\nand G. Faini, J. Appl. Phys. 95, 6777 (2004).\n[6] C. K. Lim, T. Devolder, C. Chappert, J. Grollier, V . Cros,\nA. Vaur `es, A. Fert, and G. Faini, Appl. Phys. Lett. 84, 2820\n(2004).\n[7] A. Yamaguchi, T. Ono, S. Nasu, K. Miyake, K. Mibu, and\nT. Shinjo, Phys. Rev. Lett. 92, 077205 (2004).\n[8] S. S. P. Parkin, M. Hayashi, and L. Thomas, Science 320, 190\n(2008).\n[9] P. M. Haney, R. A. Duine, A. S. N ´u˜nez, and A. H. MacDonald,\nJ. Magn. Magn. Mater. 320, 1300 (2008).\n[10] G. Tatara, H. Kohno, and J. Shibata, Phys. Rep. 468, 213\n(2008).\n[11] Y . Tserkovnyak, A. Brataas, and G. E. W. Bauer, J. Magn.\nMagn. Mater. 320, 1282 (2008).\n[12] S. Zhang and Z. Li, Phys. Rev. Lett. 93, 127204 (2004).\n[13] J. Xiao, A. Zangwill, and M. D. Stiles, Phys. Rev. B 73, 054428\n(2006).\n[14] A. Thiaville, Y . Nakatani, J. Miltat, and Y . Suzuki, Europhys.\nLett. 69, 990 (2005).\n[15] J. Linder and J. W. A. Robinson, Nat. Phys. 11, 307 (2015).\n[16] M. Eschrig, Reports Prog. Phys. 78, 104501 (2015).\n[17] X. Waintal and P. W. Brouwer, Phys. Rev. B 65, 054407 (2002).\n[18] Y . Tserkovnyak and A. Brataas, Phys. Rev. B 65, 094517\n(2002).\n[19] E. Zhao and J. A. Sauls, Phys. Rev. B 78, 174511 (2008).\n[20] V . Braude and Y . M. Blanter, Phys. Rev. Lett. 100, 207001\n(2008).\n[21] F. Konschelle and A. Buzdin, Phys. Rev. Lett. 102, 017001\n(2009).\n[22] J. Linder and T. Yokoyama, Phys. Rev. B 83, 012501 (2011).\n[23] J. Linder, A. Brataas, Z. Shomali, and M. Zareyan, Phys. Rev.\nLett. 109, 237206 (2012).\n[24] A. I. Buzdin, A. V . Vedyayev, and N. V . Ryzhanova, Europhys.\nLett. 48, 686 (1999).\n[25] L. R. Tagirov, Phys. Rev. Lett. 83, 2058 (1999).\n[26] P. V . Leksin, N. N. Garif’Yanov, I. A. Garifullin, Y . V . Fominov,\nJ. Schumann, Y . Krupskaya, V . Kataev, O. G. Schmidt, and\nB. B ¨uchner, Phys. Rev. Lett. 109, 057005 (2012).\n[27] N. Banerjee, C. B. Smiet, R. G. J. Smits, A. Ozaeta, F. S. Berg-\neret, M. G. Blamire, and J. W. A. Robinson, Nat. Commun. 5,\n3048 (2014).\n[28] A. Singh, S. V oltan, K. Lahabi, and J. Aarts, Phys. Rev. X 5,\n021019 (2015).[29] H. Yang, S.-H. Yang, S. Takahashi, S. Maekawa, and S. S. P.\nParkin, Nat. Mater. 9, 586 (2010).\n[30] C. H. L. Quay, D. Chevallier, C. Bena, and M. Aprili, Nat.\nPhys. 9, 84 (2013).\n[31] F. Hubler, M. J. Wolf, D. Beckmann, and H. Lohneysen, Phys\nRev. Lett. 109, 207001 (2012).\n[32] B. Kastening, D. K. Morr, D. Manske, and K. Bennemann,\nPhys. Rev. Lett. 96, 047009 (2006).\n[33] T. Yokoyama, Phys. Rev. B 84, 132504 (2011).\n[34] P. M. R. Brydon, D. Manske, and M. Sigrist, J. Phys. Soc. Jpn\n77, 103714 (2008).\n[35] P. M. R. Brydon, Phys. Rev. B 80, 224520 (2009).\n[36] P. M. R. Brydon, Y . Asano, and C. Timm, Phys. Rev. B 83,\n180504 (2011).\n[37] T. Yokoyama, Phys. Rev. B 92, 174513 (2015).\n[38] F. S. Bergeret, A. F. V olkov, and K. B. Efetov, Rev. Mod. Phys.\n77, 1321 (2005).\n[39] F. S. Bergeret and I. V . Tokatly, Phys. Rev. Lett. 110, 117003\n(2013).\n[40] F. S. Bergeret and I. V . Tokatly, Phys. Rev. B 89, 134517 (2014).\n[41] R. S. Keizer, S. T. B. Goennenwein, T. M. Klapwijk, G. Miao,\nG. Xiao, and A. Gupta, Nature 439, 825 (2006).\n[42] M. S. Anwar, F. Czeschka, M. Hesselberth, M. Porcu, and\nJ. Aarts, Phys. Rev. B 82, 100501 (2010).\n[43] T. S. Khaire, M. A. Khasawneh, W. P. Pratt, and N. O. Birge,\nPhys. Rev. Lett. 104, 137002 (2010).\n[44] J. W. A. Robinson, J. Witt, and M. Blamire, Science 329, 59\n(2010).\n[45] M. S. Anwar, S. R. Lee, R. Ishiguro, Y . Sugimoto, Y . Tano,\nS. J. Kang, Y . J. Shin, S. Yonezawa, D. Manske, H. Takayanagi,\nT. W. Noh, and Y . Maeno, Nat. Commun. 7, 13220 (2016).\n[46] A. Leggett, Rev. Mod. Phys. 47, 331 (1975).\n[47] G. Tatara and H. Kohno, Phys. Rev. Lett. 92, 086601 (2004).\n[48] P. A. Frigeri, D. F. Agterberg, A. Koga, and M. Sigrist, Phys.\nRev. Lett. 92, 097001 (2004).\n[49] A. Yamaguchi, K. Yano, H. Tanigawa, S. Kasai, and T. Ono,\nJpn. J. Appl. Phys. 45, 3850 (2006).\n[50] V . A. Oboznov, V . V . Bol’ginov, A. K. Feofanov, V . V .\nRyazanov, and A. I. Buzdin, Phys. Rev. Lett. 96, 197003\n(2006).\n[51] L. Berger, Phys. Rev. B 33, 1572 (1986).\n[52] G. E. V olovik, J. Phys. C 20, L83 (1987).\n[53] S. E. Barnes and S. Maekawa, Phys. Rev. Lett. 98, 246601\n(2007).\n[54] R. A. Duine, Phys. Rev. B 79, 014407 (2009).\n[55] See Supplemental Material for the derivation of Eq. (9) and the\nexplicit form of F(\u001e0) (Eq. (21)).\n[56] For a high density, neshould be replaced with an e \u000bective elec-\ntron density modified by lattice e \u000bects.1\nSupplemental Material for\n“Adiabatic and non-adiabatic spin-torque induced by spin-triplet supercurrent”\nI. DERIVATION OF EQ. (9)\nWe start from the action:\nS=\u00001\n2X\nn;k;q\ty\nk+q(i\u000fn)(G\u00001\ntot)k+q;k(i\u000fn)\tk(i\u000fn); (S1)\nwhere\n\tk(i\u000fn)=0BBBBBBBBBBBBBB@ak+Q\"(i\u000fn)\nak+Q#(i\u000fn)\nay\n\u0000k+Q\"(\u0000i\u000fn)\nay\n\u0000k+Q#(\u0000i\u000fn)1CCCCCCCCCCCCCCA; (S2)\n(G\u00001\ntot)k+q;k(i\u000fn)\n=0BBBBB@\u0010\u0010\ni\u000fn\u0000\u0018k+Q\u0011\n1+JsdS\u001bz\u0011\n\u000eq;0\u00003\u0017\nk+q=2+QAa\n\u0017(q)\u001ba\u0000\u0001(k)\u000eq;0\n\u0000\u0001y(k)\u000eq;0\u0010\u0010\ni\u000fn+\u0018\u0000k+Q\u0011\n1\u0000JsdS\u001bz\u0011\n\u000eq;0+3\u0017\n\u0000k\u0000q=2+QAa\n\u0017(q)\u001baT1CCCCCA;(S3)\n'G\u00001\nk\u000eq;0+U(1)\nk+q;k+U(2)\nk+q;k: (S4)\nHere we define\nU(1)\nk+q;k=\u00003\u0017\nkQ\u0017\u000eq;0I4\u00003\u0017\nk+q=2Aa\n\u0017(q) \n\u001ba0\n0\u001bT\na!\n; (S5)\nU(2)\nk+q;k=\u0000@\u0018k+q=2\n@k\u0017@k\u0011Q\u0011Aa\n\u0017(q)Sa; (S6)\nandG\u00001\nkis defined in the main text. To restore the gauge invariance, we need to include the vector potential Aand redefine\n~Q=Q+e\ncA. Assuming the supercurrent, j/˜Q, is homogeneous, we can apply the perturbation with respect to ˜Qin the same\nway.\nThe spin density under a superconducting current is given by ˜ sa\nq=T\n2N1\n2P\nk;ntr[SaGtot;k+q;k(i\"n)]. We calculate it to the linear\norder of Q\u0011andAa\n\u0017(q), and obtain \u000e˜sa\nq=\u0019ab\n\u0017\u0011Ab\n\u0017(q)j\u0011\n2ene. Here\n\u0019ab\n\u0017\u0011=lim\nq!0\u0000T\n4Nta2X\nn;k@2\u0018k\n@k\u0017@k\u0011Tr[SaGk+q(i\u000fn)SbGk(i\u000fn)]+\u000eabLa\n\u0017\u0011; (S7)\nwhere\nLx\n\u0017\u0011=Ly\n\u0017\u0011=1\n2JsdS ta21\nNX\nk3\u0017\nk3\u0011\nk @nF(\")\n@\"\f\f\f\f\f\"=E\"\nk\u0000@nF(\")\n@\"\f\f\f\f\f\"=E#\nk!\n; (S8)\nLz\n\u0017\u0011=\u00001\n2ta21\nNX\nk3\u0017\nk3\u0011\nkX\n\u001b=\";#\u0018\u001b\nk\nE\u001b\nk@nF(\")\n@\"\f\f\f\f\f\"=E\u001b\nk\u0010\n2nF(E\u001b\nk)\u00001\u0011\n; (S9)\nwith E\"\nk=q\n(\u0018k\u0000JsdS)2+j\u0001\"\nkj2,E#\nk=q\n(\u0018k+JsdS)2+j\u0001#\nkj2, and nF(\")=(e\"=T+1)\u00001.La\n\u0017\u0011are the contributions from the\nFermi surface; they are proportional to the derivative of nF(\") and vanishingly small at low temperatures in systems with a full\ngap or point nodes. In the main text, we neglect La\n\u0017\u0017considering a low temperature compared to the superconducting critical\ntemperature. We note that for \u0001(k)=0, two terms in Eq. (S7) cancel each other and \u0019ab\n\u0017\u0011=0.\nAccording to Eq. (14) in the main text, finite1\n2(\u0019xx\n\u0017\u0000\u0019yy\n\u0017) or\u0019xy\n\u0017are necessary for ˜\f\u0017,0, and they depend on the relative2\nphase between \u0001\"\nkand\u0001#\nkas\n1\n2(\u0019xx\n\u0017\u0000\u0019yy\n\u0017)=lim\nq!0T\nNta2X\nk;n@2\u0018k+q=2\n@2k\u00170BBBBBB@Re(\u0001#\nk+q\u0001\"\u0003\nk)\n(\u000f2n+E#2\nk+q)(\u000f2n+E\"2\nk)1CCCCCCA; (S10)\n=lim\nq!01\nNX\nkRe(\u0001#\nk\u0001\"\u0003\nk)f\u0017(k;q); (S11)\n\u0019xy\n\u0017=lim\nq!0T\nNta2X\nk;n@2\u0018k+q=2\n@2k\u00170BBBBBB@Im(\u0001#\nk+q\u0001\"\u0003\nk)\n(\u000f2n+E#2\nk+q)(\u000f2n+E\"2\nk)1CCCCCCA: (S12)\n=lim\nq!01\nNX\nkIm(\u0001#\nk\u0001\"\u0003\nk)f\u0017(k;q); (S13)\nwhere\nf\u0017(k;q)=T\nta2X\nn@2\u0018k+q=2\n@2k\u00171\n(\u000f2n+E#2\nk+q)(\u000f2n+E\"2\nk): (S14)\nII. F(\u001e0)FOR A DOMAIN WALL\nLet us consider F(\u001e0) in a domain wall configuration when d\u0017(k)2R. We have\nF(\u001e0)=j\u00010j2a4\n2e3cSX\ni@xnzlim\nq!01\nNX\nkfx(k;q)\u0002dy(k)dz(k) sin\u001e0\u0000dx(k)dz(k) cos\u001e0\u0003sin\u0012\n+j\u00010j2a4\n2e3cSX\ni@xnzlim\nq!01\nNX\nkfx(k;q)\"\n\u00001\n2\u0010\ndx(k)2\u0000dy(k)2\u0011\nsin(2\u001e0)+dx(k)dy(k) cos(2\u001e0)#\ncos\u0012; (S15)\nwhere we have used Eq. (5) in the main text.\nIn the following, we show that F(\u001e0)=0 ford(k)=(\u0000sin(kya);sin(kxa);0). The first line in Eq. (S15) is zero since\ndz(k)=0. We note that in a domain wall configuration, \u0012(x)=\u0019\u0000\u0012(\u0000x) is satisfied. Since fx(k;q), which depends on n\nthroughj\u0001\"\nkj=j\u0001#\nkj=j\u00010j2jd(k)\u0002njinE\u001b\nk, and@xnzare invariant under the spatial reflection ( \u0012!\u0019\u0000\u0012), the second line in\nEq. (S15) vanishes after the spatial summationP\ni.\nForF(\u001e0),0, finite dx(k)dz(k) ordy(k)dz(k) is necessary. In this case, the first line in Eq. (S15) is nonzero, and jd(k)\u0002nj\nchanges its value under \u0012!\u0019\u0000\u0012so that the second term is also nonzero in general. As we show in the main text, d(k)=\n(\u0000sin(kya):sin(kxa);\u000esin(kxa)) is one way to satisfy this condition."    },    {        "title": "1706.04670v2.Temperature_dependent_Gilbert_damping_of_Co2FeAl_thin_films_with_different_degree_of_atomic_order.pdf",        "content": "1 \n Temperature -dependent Gilbert damping of Co 2FeAl thin films with different degree of \natomic order  \nAnkit Kumar1*, Fan  Pan2,3, Sajid  Husain4, Serkan  Akansel1, Rimantas  Brucas1, Lars \nBergqvist2,3, Sujeet Chaudhary4, and Peter  Svedlindh1# \n \n1Department of Engineering Sciences, Uppsala University, Box 534, SE -751 21 Uppsala, \nSweden \n2Department of Applied Physics, School of Engineering Sciences, KTH Royal Institute of \nTechnology, Electrum 229, SE -16440 Kista, Sweden \n3Swedish e -Science Research Center, KTH Roy al Institute of Technology, SE -10044 \nStockholm, Sweden \n4Department of Physics, Indian Institute of Technology Delhi, New Delhi -110016, India  \n \nABSTRACT  \nHalf-metallicity and low magnetic damping are perpetually sought for in spintronics materials \nand full He usler alloys in this respect provide outstanding properties . However, it is \nchallenging to obtain the well -ordered half-metallic phase in as -deposited full Heusler alloys \nthin films and theory has struggled to establish  a fundamentals understanding of the \ntemperature dependent Gilbert damping in these systems. Here we present a study of the \ntemperature dependent Gilbert damping of differently ordered as -deposited Co 2FeAl full \nHeusler alloy thin films. The sum of inter - and intraband electron scattering in conjunction \nwith the finite electron lifetime in Bloch states govern the Gilbert damping for the well -\nordered phase in contrast to the damping of partially -ordered and disordered phases which is \ngoverned by interband electronic scat tering alone. These results, especially the ultralow room \ntemperature intrinsic damping observed for the well -ordered phase provide new fundamental \ninsight s to the physical origin of the Gilbert damping in full Heusler alloy thin films.   \n 2 \n INTRODUCTION \nThe Co-based full Heusler alloys have gained massive attention over the last decade due to \ntheir high Curie temperature and half-metallicity; 100% spin polarization of the density of \nstates at the Fermi level [1 -2]. The room temperature half- metallicity and lo w Gilbert \ndamping make them ideal candidates for magnetoresistive and thermoelectric spintronic \ndevices [3]. Co2FeAl (CFA), which is one of the most studied Co-based Heusler alloys , \nbelongs to the 𝐹𝐹𝐹𝐹 3𝐹𝐹 space group, exhibits half-metallicity  and a high C urie temperature \n(1000 K) [2, 4]. In CFA, half-metallicity is the result of hybridization between the d orbitals \nof Co and Fe. The d orbitals of Co hybridize resulting in bonding (2e g and 3t 2g) and non-\nbonding hybrids (2e u and 3t 1u). The bonding hybrids of Co further hybridise with the d \norbitals of Fe yielding bonding and anti -bonding hybrids. However, the non-bonding hybrids \nof Co cannot hybridise with the d orbitals of Fe. The half-metallic gap arises from the \nseparation of non-bonding states, i.e. the conduction band of e u hybrids and the valence band \nof t 1u hybrids [5, 6]. However, chemical or atomic disorder modifies the band hybridization \nand results in a reduc ed half-metallicity in CFA. The ordered phase of CFA is the L2 1 phase, \nwhich is half -metallic [7]. The partially ordered B2 phase forms when the Fe and Al atoms \nrandomly share their sites, while the disordered phase forms when Co, Fe, and Al atoms \nrandomly share all the sites [5-8]. These chemical disorders strongly influence the physical \nproperties and result in additional states at the Fermi level therefore reducing the half-\nmetallicity or spin polarization [7, 8]. It is challenging to obtain the ordered L2 1 phase of \nHeusler alloys in as-deposited films, which is expecte d to possess the lowest Gilbert damping \nas compared to the other phases [4, 9-11]. Therefore, in the last decade several attempts have \nbeen made to grow the ordered phase of CFA thin films employing different methods [4, 9-\n13]. The most successful attempts  used post -deposition annealing to reduce the anti -site \ndisorder by a thermal activation process [4]. The observed value of the Gilbert damping for \nordered thin films was found to lie in the range of 0.001-0.004 [7-13]. However, the \nrequirement of post -deposition annealing might not be compatible with the process constraints \nof spintronics and CMOS devices. The annealing treatment requirement for the formation of \nthe ordered phase can be circumvented by employing energy enhanced growth mechanisms \nsuch as io n beam sputtering where the sputtered species carry substantially larger  energy, ~20 \neV , compared to other deposition techniques [14, 15]. This higher energy of the sputtered \nspecies enhances the ad -atom mobility during coalescence of nuclei in the initial  stage of the \nthin film growth, therefore enabling the formation of the ordered phase. Recently we have 3 \n reported growth of the ordered CFA phase on potentially advantageous Si substrate  using ion \nbeam sputtering. The samples deposited in the range of 300°C to 500°C substrate temperature \nexhibited nearly equivalent I(002)/I(004) Bragg diffraction intensity peak ratio, which \nconfirms at least B2 order ed phase as it is difficult to identify  the formation of the L2 1 phase \nonly by X -ray diffraction analysis [16] .  \nDifferent theoretical approaches have been employed to calculate the Gilbert damping in Co -\nbased full Heusler alloys, including first principle calculations on the ba sis of (i) the torque \ncorrelation  model [17], (ii) the fully relativistic Korringa -Kohn-Rostoker model in \nconjunction with the coherent potential approximation and the linear response formalism [8] , \nand (iii) an approach considering different exchange correlation effects using both the local \nspin density approximation including the Hubbard  U and the local spin density approximation \nplus the dynamical mean field theory approximation [7]. However, very little is known about \nthe temperature dependence of the Gilbert damping in differently ordered Co-based Heusler \nalloys and a unifying conse nsus between theoretical and experimental results is still lacking. \nIn this study we report the growth of differently ordered  phases, varying from disordered to \nwell-ordered phases, of as -deposited CFA thin films grown on Si employing ion beam \nsputtering a nd subsequently the detailed temperature dependent measurements of the Gilbert \ndamping. The observed increase in intrinsic Gilbert damping with decreasing temperature in \nthe well -ordered sample is in contrast to the continuous decrease in intrinsic Gilbert  damping \nwith decreasing temperature observed for partially ordered and disordered phases. These \nresults are satisfactorily explained by employing spin polarized relativistic Korringa -Kohn-\nRostoker band structure calculations in combination with the local spin density \napproximation.  \nSAMPLES & METHODS  \nThin films of CFA were deposited on Si substrates at various growth temperatures using ion \nbeam sputtering system operating at 75W RF ion-source power ( 𝑃𝑃𝑖𝑖𝑖𝑖𝑖𝑖). Details of the \ndeposition process as well as st ructural and magnetic properties of the films have been \nreported elsewhere [16]. In the present work to study the temperature dependent Gilbert \ndamping of differently ordered phases (L2 1 and B2) we have chosen CFA thin films deposited \nat 573K, 673K and 773K substrate temperature ( 𝑇𝑇𝑆𝑆) and the corresponding samples are named \nas LP573K, LP673K and LP773K, respectively. The sample thickness was kept constant at 50 \nnm and the samples were capped with a 4 nm thick Al layer. The capping layer protects the \nfilms by forming a 1.5 nm thin protective layer of Al 2O3. To obtain the A2 disordered CFA 4 \n phase, the thin film was deposited at 300K on Si employing 100W ion-source power, this \nsample is referred to  as HP300K. Structural and magnetic properties of this film are presented \nin Ref. [18]. The absence of the (200) diffraction peak in the HP300K sample [18] reveals that \nthis sample exhibits the A2 disordered structure. The appearance of the (200) pea k in the LP \nseries samples clearly indicates at least formation of B2 order [16]. Employing the Webster model along with the analysis approach developed by Takakura et al. [19] we have calculated \nthe degree of B2 ordering in the samples, S\nB2= �I200 I220⁄\nI200full orderI220full order�� , where \nI200 I220⁄  is the experimentally obtained intensity ratio of the (200) and (220) diffractions and \nI200full orderI220full order⁄  is the theoretically calculated intensity ratio for fully ordered B2 structure \nin polycrystalline films  [20]. The estimated values of SB2 for the LP573, LP673, and LP773 \nsamples are found to be ∼  90 %, 90% and 100%, respectively , as presented in Ref.  [20]. The \nI200 I400⁄  ratio of the (200) and (400) diffraction peaks for all LP series samples is ∼  30 %, \nwhich compares well with  the theoretical value for perfect B2 order [21, 22]. Here it is \nimportant to note that the L21 ordering parameter, SL21, will take different values depending \non the degree of B2 ordering. S L21 can be calculated from the I111 I220⁄  peak ratio in \nconjunction with the SB2 ordering parameter [19]. However, in the recorded grazing incident \nXRD spectra on the polycrystalline LP samples (see Fig. 1 of Ref. [16]) we did not observe \nthe (111) peak. This could be attributed to the fact t hat theoretical intensity of this peak is only \naround two percent of the (220) principal peak. The appearance of this peak is typically \nobserved in textured/columnar thicker films [19, 23 ]. Therefore, here using the experimental \nresults of the Gilbert damping, Curie temperature and saturation magnetization, in particular employing the temperature dependence of the Gilbert damping that is very sensitive to the \namount of site disorder in CFA films, and comparing with corresponding results obtained \nfrom first principle calculations, we provide a novel method for determining the type of \ncrystallographic ordering in full Heusler alloy thin films.  \nThe observed values of the saturation magnetization ( µ0MS) and coercivity ( µ0Hci), taken \nfrom Refs. [16, 18] are presented in Table I. The temperature dependence of the magnetization \nwas recorded in the high temperature region (300–1000K) using a vibrating sample \nmagnetometer i n an external magnetic field of 𝜇𝜇 0𝐻𝐻=20 mT. An ELEXSYS EPR \nspectrometer from Bruker equipped with an X -band resonant cavity was used for angle \ndependent in-plane ferromagnetic resonance (FMR) measurements . For studying the 5 \n temperature dependent spin dynamics in the magnetic thin films, an in-house built out -of-\nplane FMR setup was used. The set up, using a Quantum Design Physical Properties \nMeasurement System covers the temperature range 4 – 350 K and the magnetic field range \n±9T. The system employs an Agilent N5227A PNA network analyser covering the frequency \nrange 1 – 67 GHz and an in-house made coplanar waveguide. The layout of the system is \nshown in Fig. 1. The complex transmission coefficient ( 𝑆𝑆21) was recorded as a function of \nmagnetic field for different frequencies in the range 9-20 GHz and different temperatures in \nthe range 50-300 K.  All FMR measurements were  recorded keeping constant 5 dB power.  \nTo calculate the Gilbert damping, we have the used the torque –torque correlation model [7, \n24], which includes both intra - and interband transitions. The electronic structure was \nobtained from the spin polarized relativistic Korringa -Kohn-Rostoker (SPR- KKR) band \nstructure method [24, 25] and the local spin density approximation (LSDA) [26] was used for \nthe exchange correlation potential. Relativistic effects were taken into account by solving the Dirac equation for the electronic states, and the atomic sphere approximation (ASA) was employed for the shape of potentials. The experimental bulk value of the lattice constant [27] \nwas used. The angular momentum cut -off of 𝑙𝑙\n𝑚𝑚𝑚𝑚𝑚𝑚 =4 was used in the mu ltiple -scattering \nexpansion.  A k-point grid consisting of ~1600 points in the irreducible Brillouin zone was \nemploye d in the self -consistent calculation while a substantially more dense grid of ~60000  \npoints was employe d for the Gilbert damping calculation.  The exchange parameters 𝐽𝐽 𝑖𝑖𝑖𝑖 \nbetween the atomic magnetic moments were calculated using the magnetic force theorem \nimplemented in the Liechtenstein -Katsnelson -Antropov-Gubanov (LKAG) formalism [28, 29] \nin order to construct a parametrized mod el Hamiltonian. For the B2 and L2 1 structures, the \ndominating exchange interactions were found to be between the Co and Fe atoms, while in A2 the Co-Fe and Fe -Fe interactions are of similar size. Finite temperature properties such as the \ntemperature dependent magnetization was obtained by performing Metropolis Monte Carlo \n(MC) simulations [30] as implemented in the UppASD software [31,  32] using the \nparametrized Hamiltoni an. The coherent potential approximation (CPA) [33, 34] was ap plied \nnot only for the treatment of the chemical disorder of the system, but also used to include the \neffects of quasi -static lattice displacement and spin fluctuations in the calculation of the \ntemperature dependent Gilbert damping [35–37] on the basis of linear response theory [38].  \nRESULTS & DISCUSSION  \nA. Magnetization vs. temperature measurements  6 \n Magnetization measurements were performed with the ambition to extract values for the \nCurie temperature ( 𝑇𝑇𝐶𝐶) of CFA films with different degree of atomic order; the results a re \nshown in Fig. 2(a). Defining 𝑇𝑇𝐶𝐶 as the inflection point in the magnetization vs. temperature \ncurve, the observed values are found to be 810  K, 890 K and 900 K for the LP573K, LP773K \nand LP673K samples, respectively. The 𝑇𝑇𝐶𝐶 value for the HP300K sample is similar to the \nvalue obtained for LP573. Using the theoretically calculated exchange interactions, 𝑇𝑇𝐶𝐶 for \ndifferent degree of atomic order in CFA varying from B2 to L2 1 can be calculated using MC \nsimul ations. The volume was kept fixed as the degree of order varied between B2 and L2 1 \nand the data presented here represent the effects of differently ordered CFA phases. To obtain \n𝑇𝑇𝐶𝐶 for the different phases, the occupancy of Fe atoms on the Heusler alloy 4a sites was varied \nfrom 50% to 100%, corresponding to changing the structure from B2 to L2 1. The estimated \n𝑇𝑇𝐶𝐶 values , cf. Fig. 2 (b), monotonously increases from 𝑇𝑇 𝐶𝐶=810 K (B2) to 𝑇𝑇𝐶𝐶=950 K \n(L2 1). A direct comparison between experimental and calculated  𝑇𝑇𝐶𝐶 values is hampered by the \nhigh temperature (beyond 800K) induced structural transition from well -ordered to partially -\nordered CFA phase which interferes with the magnetic transition [39, 40]. The irreversible \nnature of the recorded magnetization vs . temperature curve  indicates a distortion of structure \nfor the ordered phase during measurement , even though interface alloying at elevated \ntemperature cannot be ruled out . The experimentally observe d 𝑇𝑇𝐶𝐶 values are presented in \nTable I.  \nB. In-plane angle dependent FMR measurements  \nIn-plane angle dependent FMR measurements were performed at 9.8 GHz frequency for all \nsamples; the resonance field 𝐻𝐻𝑟𝑟 vs. in -plane angle  𝜙𝜙𝐻𝐻 of the applied magnetic field is plotted \nin Fig. 3. The  experimental results have been fitted using the expression [41],  \n𝑓𝑓=\n𝑔𝑔∥𝜇𝜇𝐵𝐵𝜇𝜇0\nℎ��𝐻𝐻𝑟𝑟cos(𝜙𝜙𝐻𝐻−𝜙𝜙𝑀𝑀)+2𝐾𝐾𝑐𝑐\n𝜇𝜇0𝑀𝑀𝑠𝑠cos4(𝜙𝜙𝑀𝑀−𝜙𝜙𝑐𝑐)+2𝐾𝐾𝑢𝑢\n𝜇𝜇0𝑀𝑀𝑠𝑠cos2(𝜙𝜙𝑀𝑀−𝜙𝜙𝑢𝑢)��𝐻𝐻𝑟𝑟cos(𝜙𝜙𝐻𝐻−\n𝜙𝜙𝑀𝑀)+ 𝑀𝑀𝑒𝑒𝑒𝑒𝑒𝑒+𝐾𝐾𝑐𝑐\n2𝜇𝜇0𝑀𝑀𝑠𝑠(3+cos4(𝜙𝜙𝑀𝑀−𝜙𝜙𝑐𝑐)+2𝐾𝐾𝑢𝑢\n𝜇𝜇0𝑀𝑀𝑠𝑠cos2 (𝜙𝜙𝑀𝑀−𝜙𝜙𝑢𝑢)��12�\n, (1) \nwhere 𝑓𝑓 is resonance frequency , 𝜇𝜇𝐵𝐵 is the Bohr magneton  and ℎ is Planck constant . 𝜙𝜙𝑀𝑀, 𝜙𝜙𝑢𝑢 \nand 𝜙𝜙𝑐𝑐 are the in -plane directions of the magnetization, uniaxial anisotropy and cubic \nanisotropy, respectively , with respect to the [100] direction of the Si substrate . 𝐻𝐻𝑢𝑢=2𝐾𝐾𝑢𝑢\n𝜇𝜇0𝑀𝑀𝑠𝑠 and \n𝐻𝐻𝑐𝑐=2𝐾𝐾𝑐𝑐\n𝜇𝜇0𝑀𝑀𝑠𝑠 are the in-plane uniaxial and cubic anisotropy fields , respectively, and 𝐾𝐾𝑢𝑢 and 𝐾𝐾𝑐𝑐 7 \n are the uniaxial and cubic magnetic anisotrop y constant s, respectively, 𝑀𝑀𝑠𝑠 is the saturation \nmagnetization and 𝑀𝑀𝑒𝑒𝑒𝑒𝑒𝑒 is the effective magnetization . By considering 𝜙𝜙 𝐻𝐻 ∼ 𝜙𝜙𝑀𝑀, 𝐻𝐻𝑢𝑢 and 𝐻𝐻𝑐𝑐 \n<<𝐻𝐻𝑟𝑟<< 𝑀𝑀𝑒𝑒𝑒𝑒𝑒𝑒, equation (1) can be simplified as:  \n𝐻𝐻𝑟𝑟=�ℎ𝑒𝑒\n𝜇𝜇0𝑔𝑔∥𝜇𝜇𝐵𝐵�21\n𝑀𝑀𝑒𝑒𝑒𝑒𝑒𝑒−2𝐾𝐾𝑐𝑐\n𝜇𝜇0𝑀𝑀𝑠𝑠cos4(𝜙𝜙𝐻𝐻−𝜙𝜙𝑐𝑐)−2𝐾𝐾𝑢𝑢\n𝜇𝜇0𝑀𝑀𝑠𝑠cos2(𝜙𝜙𝐻𝐻−𝜙𝜙𝑢𝑢) . (2) \nThe extracted cubic anisotropy fields µ 0Hc ≤ 0.22mT are negligible for all the samples. The \nextracted in -plane Landé splitting factors g∥ and the uniaxial anisotropy fields  µ0Hu are \npresented in T able I. The purpose of the angle dependent FMR measurements was only to \ninvestigate the symmetry of the in -plane magnetic anisotropy. Therefore, care was not taken \nto have the same in -plane orientation of the samples during angle dependent FMR \nmeasurements, which explains why the maxima appear at diffe rent angles for the different \nsamples.  \nC. Out-of-plane FMR measurements  \nField -sweep out -of-plane FMR measurements were performed at different constant \ntemperatures in the range 50K – 300K and at different constant frequencies in the range of 9-\n20 GHz. Figure 1(b) shows the amplitude  of the complex transmission coefficient 𝑆𝑆21(10 \nGHz) vs. field measured for the LP673K thin film at different temperatures. The recorded \nFMR spectra were fitted using the equation [42],  \n𝑆𝑆21=𝑆𝑆�∆𝐻𝐻2��2\n(𝐻𝐻−𝐻𝐻𝑟𝑟)2+�∆𝐻𝐻2��2+𝐴𝐴�∆𝐻𝐻2��(𝐻𝐻−𝐻𝐻𝑟𝑟)\n(𝐻𝐻−𝐻𝐻𝑟𝑟)2+�∆𝐻𝐻2��2+𝐷𝐷∙𝑡𝑡,  (3) \nwhere 𝑆𝑆 represents the coefficient describing the transmitted microwave power, 𝐴𝐴 is used to \ndescribe a waveguide induced phase shift  contribution  which is, however, minute , 𝐻𝐻 is \napplied magnetic field, ∆𝐻𝐻 is the full-width of half maxim um, and 𝐷𝐷∙𝑡𝑡 describes the linear \ndrift in time  (𝑡𝑡) of the recorded signal. The extracted ∆ 𝐻𝐻 vs. frequency at different constant  \ntemperatures  are shown in Fig. 4 for all the samples. For brevity only data at a few \ntemperatures are plotted. The Gilbert damping was estimated using the equation [42 ],  \n∆𝐻𝐻=∆𝐻𝐻0+2ℎ𝛼𝛼𝑒𝑒\n𝑔𝑔⊥𝜇𝜇𝐵𝐵𝜇𝜇0   (4) \nwhere ∆𝐻𝐻0 is the inhomogeneous line -width broadening, 𝛼𝛼 is the experimental Gilbert \ndamping constant , and 𝑔𝑔⊥ is the Landé  splitting factor measured employing out -of-plane \nFMR. The insets in the figures show the temperature dependence of  𝛼𝛼. The effective 8 \n magnetization ( 𝜇𝜇0𝑀𝑀𝑒𝑒𝑒𝑒𝑒𝑒) was estimated  from the 𝑓𝑓 vs. 𝐻𝐻𝑟𝑟 curves using out -of-plane Kittel’s \nequation [43],  \n𝑓𝑓=𝑔𝑔⊥𝜇𝜇0𝜇𝜇𝐵𝐵\nℎ�𝐻𝐻𝑟𝑟−𝑀𝑀𝑒𝑒𝑒𝑒𝑒𝑒�,     (5) \nas shown in Fig. 5 . The temperature dependence of  𝜇𝜇0𝑀𝑀𝑒𝑒𝑒𝑒𝑒𝑒 and 𝜇𝜇0∆𝐻𝐻0 are shown as  insets  in \neach figure . The observed room temperature values of  𝜇𝜇0𝑀𝑀𝑒𝑒𝑒𝑒𝑒𝑒 are closely equal  to the  𝜇𝜇0𝑀𝑀𝑠𝑠 \nvalues obtained  from static magnetizat ion measurements, presented in T able I. The extracted \nvalues of g⊥ at different temperatures are within error limits  constant for all samples. \nHowever , the difference between estimated values of g ∥ and g⊥ is ≤ 3%. This difference c ould \nstem from the limited frequency range  used since  these values are quite sensitive to the value \nof 𝑀𝑀𝑒𝑒𝑒𝑒𝑒𝑒, and even a minute uncertainty in this quantity can result in the observed small \ndifference between  the g∥ and g⊥ values.  \nTo obtain  the intrinsic Gilbert damping (𝛼𝛼 𝑖𝑖𝑖𝑖𝑖𝑖) all extrinsic contributions to the experimental 𝛼𝛼 \nvalue  need to be  subtracted. In metallic ferromagnets , the intrinsic Gilbert damping is mostly \ncaused by electron magnon scattering, but  several other extrinsic co ntributions can also \ncontribute to the experimental value of the damping constant. One  contribution is two -\nmagnon scattering which is however minimized for the  perpendicular geometry used in this \nstudy and therefore this contribution is disregarded [44].  Another contribution is spin-\npumping into the  capping layer as the LP573K, LP673K and LP773K samples are capped \nwith 4 nm of Al that naturally forms a thin top layer consisting of  Al2O3. Since spin pumping \nin low spin-orbit coupling materials with  thickness less than the spin-diffusion length is quite \nsmall this contribution is also disregarded in all samples. However, the HP300K sample is \ncapped with Ta and therefore a spin-pumping contribution have been subtracted from the \nexperimental 𝛼𝛼 value ; 𝛼𝛼𝑠𝑠𝑠𝑠= 𝛼𝛼𝐻𝐻𝐻𝐻300𝐾𝐾(with  Ta capping )−𝛼𝛼𝐻𝐻𝐻𝐻300𝐾𝐾(without capping )≈\n1×10−3. The third contribution arises  from the inductive coupling between  the precessi ng \nmagnetization and the CPW , a reciprocal phenomenon of FMR, known as radiative damping \n𝛼𝛼𝑟𝑟𝑚𝑚𝑟𝑟 [45]. This damping is directly proportional to the magnetization and thickness of the thin \nfilms samples and therefore usually dominates in thicker and/or high magnetization samples. \nThe l ast contribution is eddy current damping ( αeddy) caused  by eddy current s in metallic \nferromagnetic  thin films [ 45, 46]. As per Faraday’s law the time varying magnetic flux density  \ngenerate s an AC voltage in the metallic ferromagnetic  layer and therefore result s in the eddy 9 \n current damping . Thi s damping is directly proportional to the square of the film thickness  and \ninversely proportional to the resistivity of the sample [ 45].  \nIn contrast to eddy -current damping, 𝛼𝛼𝑟𝑟𝑚𝑚𝑟𝑟 is independent of the conductivity of the \nferromagnetic layer, hence this damping mechanism is also operati ve in ferromagnetic \ninsulators. Assuming a uniform magnetization of the sample the radiative damping can be \nexpressed as [45],  \n𝛼𝛼𝑟𝑟𝑚𝑚𝑟𝑟= 𝜂𝜂𝜂𝜂𝜇𝜇02𝑀𝑀𝑆𝑆𝛿𝛿𝛿𝛿\n2 𝑍𝑍0𝑤𝑤 ,   (6) \nwhere 𝛾𝛾=𝑔𝑔𝜇𝜇𝐵𝐵ℏ� is the gyromagnetic ratio, 𝑍𝑍0 = 50 Ω is the waveguide impedance, 𝑤𝑤 = 240 \nµm is the width of the waveguide, 𝜂𝜂 is a dimensionless parameter which accounts for the \nFMR mode profile and depends on boundary conditions, and 𝛿𝛿 and 𝑙𝑙 are the thickness and \nlength of the sample on the waveguide, respectively. The strength of this inductive coupling \ndepends on the inductance of the FMR mode which is determined by the waveguide width, \nsample length over waveguide, sample saturation magnetization and sample thickness.  The \ndimensions of the LP573K, LP673K  and LP773K samples were 6.3×6.3 mm2, while the \ndimensions of the HP300K sample were  4×4 mm2. Th e 𝛼𝛼𝑟𝑟𝑚𝑚𝑟𝑟 damping was estimated  \nexperimentally as explained by Schoen et al.  [45] by  placing a 200 µm thick glass spacer \nbetween  the waveguide and the sample , which decreases  the radiative damping by more than \none order magnitude as shown in Fig. 6(a). The measured radiative damping by placing the \nspacer  between the waveguide and the LP773 sample, \n𝛼𝛼𝑟𝑟𝑚𝑚𝑟𝑟=𝛼𝛼𝑤𝑤𝑖𝑖𝑖𝑖ℎ𝑖𝑖𝑢𝑢𝑖𝑖  𝑠𝑠𝑠𝑠𝑚𝑚𝑐𝑐𝑒𝑒𝑟𝑟  −𝛼𝛼𝑤𝑤𝑖𝑖𝑖𝑖ℎ 𝑠𝑠𝑠𝑠𝑚𝑚𝑐𝑐𝑒𝑒𝑟𝑟≈ (2.36 ±0.10×10−3) − (1.57 ±0.20×10−3)=\n0.79±0.22×10−3. The estimated value matches  well with the calculated value using Eq. \n(6); 𝛼𝛼𝑟𝑟𝑚𝑚𝑟𝑟= 0.78 ×10−3. Our results are also analogous to previously reported results  on \nradiative damping [45].  The estimated temperature dependent radiative damping values for all \nsamples are shown in Fig. 6(b).   \nSpin wave precession in ferromagnetic layers induces an AC current in the conducting \nferromagnetic layer which results in  eddy current damping. It can be  expressed as [45, 46], \n𝛼𝛼𝑒𝑒𝑟𝑟𝑟𝑟𝑒𝑒 = 𝐶𝐶𝜂𝜂𝜇𝜇02𝑀𝑀𝑆𝑆𝛿𝛿2\n16 𝜌𝜌 ,   (7) \nwhere 𝜌𝜌 is the resistivity of the sample and 𝐶𝐶 accounts for the eddy current distribution in the \nsample ; the smaller the value of 𝐶𝐶 the larger is the localization of eddy currents in the sample. \nThe measured  resistivity values between 300  K to 50 K temperature range fall in the ranges \n1.175 – 1.145  µΩ-m, 1 .055 – 1.034 µΩ -m, 1 .035 – 1.00 µΩ -m, and 1. 45 – 1.41 µΩ -m for the \nLP573K, LP673 , LP773 and HP300K samples, respect ively. The parameter 𝐶𝐶 was obtained 10 \n from thickness dependent experimental Gilbert damping constants measured for B2 ordered  \nfilms,  by line ar fitting of 𝛼𝛼−𝛼𝛼𝑟𝑟𝑚𝑚𝑟𝑟≈𝛼𝛼𝑒𝑒𝑟𝑟𝑟𝑟𝑒𝑒 vs. 𝛿𝛿2 keeping other parameters  constant  (cf. Fig. \n6(c)). The fit to the data yield ed 𝐶𝐶 ≈ 0.5±0.1. These results are concurrent to those  \nobtained for permalloy  thin films [45]. Since the variations of the resistivity and \nmagnetization for the samples  are small , we have used the same 𝐶𝐶 value  for the estimation of \nthe eddy current damping in all the samples. The estimated temperature dependent values of \nthe eddy current damping are presented in Fig. 6(d).  \nAll these contributions have  been subtracted from the experimentally observed values of 𝛼𝛼. \nThe estimated intrinsic Gilbert damping 𝛼𝛼𝑖𝑖𝑖𝑖𝑖𝑖 values so obtained are plotted in Fig. 7(a) for all \nsamples.   \nD. Theoretical results: first principle calculations  \nThe calculated temperature dependent intrinsic Gilbert damping for Co 2FeAl phases with \ndifferent degree of atomic order are shown in Fig. 7(b). The temperature dependent Gilbert \ndamping indicates that the lattice displacements and spin fluctuations contribute differently in \nthe A2, B2 and L2 1 phases. The torque correlation model [47, 48] describes qualitatively two \ncontributions to the Gilbert damping. The first one is the intraband scattering where the band \nindex is always conserved. Since it has a linear dependence on the electron lifetime, in the \nlow temperature regime this term increases rapid ly, it is also known as the conductivity like \nscattering. The second mechanism is due to interband transitions where the scattering occurs \nbetween bands with different indices. Opposite to the intraband scattering, the resistivity like \ninterband scattering with an inverse depe ndence on the electron lifetime  increases with \nincreas ing temperature. The sum of the intra - and interband electron scattering contributions \ngives rise to a non-monotonic dependence of the Gilbert damping on temperature for the L2 1 \nstructure. In contrast to the case for L2 1, only interband scattering is present in the A2 and B2 \nphases, which results in a monotonic increase of the intrinsic Gi lbert damping with increas ing \ntemperature. This fact is also supported by a previous study [37 ] which showed that even a \nminute chemical disorder can inhibit the intraband scattering of the system. Our theoretical results manifest that the L2\n1 phase has the lowest Gilbert damping around 4.6 × 10−4 at 300 \nK, and that the value for the B2 phase is only slightly larger at room temperature. According \nto the torque correlation model, the two main contributions to damping are the spin orbit \ncoupling and the density of states (DOS) at the Fermi level [47 , 48]. Since the spin orbit \nstrength is the same for the different phases it is enough to focus the discussion on the DOS 11 \n that provide s a qualitative explanation  why damping is found lower in B2 and L2 1 structure s \ncompare d to A2 structure. The DOS at the Fermi level of the B2 phase (24.1 states/Ry/f.u;  f.u \n= formula unit ) is only slightly  larger to that of the L2 1 phase (20.2 states/Ry/f.u.) , but both \nare significantly smaller than for the A2 phase (59.6 states/Ry/f.u.) as shown in Fig. 8. The \ngap in the mi nority spin channel of the DOS for the B2 and L2 1 phases indicate half-\nmeta llicity, while the A2 phase is metallic. The atomically resolved spin polarized DOS \nindica tes that the Fermi -level states mostly have  contr ibutions from Co and Fe atoms. For \ntransition elements such as Fe and Ni, it has been reported that  the intrinsic Gilbert damping \nincreases significantly below 100K with decreas ing temperature [37]. The present electronic \nstructure calculations were performed using Green’s functions, which do rely on a \nphenomenological relaxation time parameter,  on the expense that the different contributions to \ndamping cannot  be separated eas ily. The reported results in Ref. [37] are by some means \nsimilar to our findings of the temperature dependent Gilbert damping in full Heusler alloy films with different degr ee of atomic order. The intermediate states of B2 and L2\n1 are more \nclose to the trend of B2 than L2 1, which indicates that even a tiny atomic orde r induced by the \nFe and Al site disorder will inhibit the conductivity -like channel in  the low temperature \nregion. The theoretically calculated Gilbert damping constants are matching qualitatively with \nthe experimentally observed  𝛼𝛼𝑖𝑖𝑖𝑖𝑖𝑖 values as shown in Fig. 7. However, the theoretically \ncalculated 𝛼𝛼𝑖𝑖𝑖𝑖𝑖𝑖 for the L2 1 phase increases rapidly below 100K, in co ntrast to the \nexperimental results for the well -ordered CFA thin film (LP673K ) indicating that  \n𝛼𝛼𝑖𝑖𝑖𝑖𝑖𝑖 saturates at low temperature. This discrepancy between the theoretical and experimental \nresults can be understood taking into account the low temperature behaviour of the life time τ \nof Bloch states. The present theoretical model assum ed that the Gilbert damping has a linear \ndependence on the electron lifetime in intraband transitions which is however correct only in \nthe limit of small lifetime, i.e., 𝑞𝑞𝑣𝑣𝐹𝐹𝜏𝜏≪1, where q is the magnon wave vector and 𝑣𝑣𝐹𝐹 is the \nelectron Fermi velocity. However, in the low temperature limit the lifetime  𝜏𝜏 increases and as \na result of the anomalous skin effect the intrinsic Gilbert damping saturates  \n𝛼𝛼𝑖𝑖𝑖𝑖𝑖𝑖∝ tan−1𝑞𝑞𝑣𝑣𝐹𝐹𝜏𝜏𝑞𝑞𝑣𝑣𝐹𝐹�  at low temperature [37], which is evident from our experimental \nresults.  \nRemaining discrepancies between theoretical and experimental values of the intrinsic Gilbert \ndamping might stem from the fact that the samples used in the present study are 12 \n polycrystalline and because of sample imperfections these fil ms exhibit significant \ninhomogenous line -width broadening due to superposition of local resonance fields.  \nCONCLUSION  \nIn summary , we report temperature dependent FMR measurements on as -deposited Co 2FeAl \nthin films with different degree of atomic order. The degree of atomic ordering is established \nby comparing experimental and theoretical results for the temperature dependent intrinsic \nGilbert damping constant. It is evidenced that the experimentally observed intrinsic Gilbert \ndamping in samples with atomic  disorder (A2 and B2 phase samples) decreases with \ndecreasing temperature. In contrast, the atomically well -ordered sample, which we identify at \nleast partial L21 phase, exhibits an intrinsic Gilbert damping constant that increases with \ndecreasing temperat ure. These temperature dependent results are explained employing the \ntorque correction model including interband transitions and both interband as well as \nintraband transitions for samples with atomic disorder and atomically ordered phases, \nrespectively.  \nACKNOWLEDGEMENT \nThis work is supported by the Knut and Alice Wallenberg (KAW) Foundation,  Grant No.  \nKAW 2012.0031  and from Göran Gustafssons Foundation (GGS), Grant No.  GGS1403A. The \ncomputations were performed on resources provided by SNIC (Swedish National \nInfrastructure for Computing) at NSC (National Supercomputer Centre) in Linköping, \nSweden.  S. H. acknowledges the Department of Science and Technology India for providing \nthe INSPIRE fellow (IF140093) grant. Daniel Hedlund is acknowledged for performing \nmagnetization versus temperature measurements.  \nAuthor Information  \nCorresponding Authors E -mails: ankit.kumar@angstrom.uu.se , peter.svedlindh@angstrom.uu.se  \nREFERENECS  \n1. S. Picozzi, A.  Continenza, and A. J.  Freeman, Phys. Rev. B.  69, 094423 (2004).  \n2. I. Galanakis, P. H. Dederichs, and N. Papanikolaou, P hys. Rev. B  66, 174429 \n(2002).  \n3. Z. Bai, L. Shen, G. Han, Y . P. Feng, Spin  02, 1230006 (2012).  \n4. M. Belmeguenai,  H. Tuzcuoglu,  M. S. Gabor,  T. Petrisor jr,  C. Tuisan,  F. Zighem,  S. \nM. Chérif,  P. Moch, J. Appl. Phys.  115, 043918 (2014).  13 \n 5. I. Galanakis, P. H. Dederichs, and N. Papanikolaou, Phys. Rev. B 66, 174429 \n(2002).  \n6. S. Skaftouros,  K. Ozdogan,  E. Sasioglu,  I. Galanakis, Physical Review B 87, \n024420 (2013).  \n7. J. Chico, S. Keshavarz, Y . Kvashnin, M. Pereiro, I. D. Marco, C. Etz, O. Eriksson, \nA. Bergman, and L . Bergqvist, Phys. Rev. B  93, 214439 (2016).  \n8. B. Pradines, R. Arras, I. Abdallah, N. Biziere, and L. Calmels, Phys. Rev. B 95 , \n094425 (2017).  \n9. X. G. Xu, D. L. Zhang, X. Q. Li, J. Bao, Y . Jiang, , and M. B. A. Jalil, J. Appl. Phys. \n106, 123902 (2009).  \n10. M. Bel meguenai, H. Tuzcuoglu, M. S. Gabor, T. Petrisor, Jr., C. Tiusan, D. Berling, \nF. Zighem, T. Chauveau, S. M. Chérif, and P. Moch, Phys. Rev. B 87 , 184431 \n(2013).  \n11. S. Qiao, S. Nie, J. Zhao, Y . Huo, Y . Wu, and X. Zhang, Appl. Phys. Lett. 103, \n152402 (2013).  \n12. B. S. Chun, K. H. Kim, N. Leibing, S. G. Santiago, H. W. Schumacher, M. Abid, I. \nC. Chu, O. N. Mryasov, D. K. Kim, H. C. Wu, C. Hwang, and Y . K. Kim, Acta \nMater.  60, 6714 (2012).  \n13. H. Sukegawa, Z. C. Wen, K. Kondou, S. Kasai, S. Mitani, and K. Inomata, Appl. \nPhys. Lett.  100, 182403 (2012).  \n14. G. Aston, H. R. Kaufman,  and P. J.  Wilbur, AIAAA journal 16, 516– 524 (1978).  \n15. A. Kumar, D. K. Pandya, and S. Chaudhary, J. Appl. Phys.  111, 073901 (2012).  \n16. S. Husain, S. Akansel, A. Kumar, P. Svedlindh , and S. Chaudhary, Scientific \nReports 6, 28692 (2016).  \n17. A Sakuma, J. Phys. D: Appl. Phys. 48 , 164011 (2015).  \n18. S. Husain, A. Kumar, S. Chaudhary, and P. Svedlindh, AIP Conference Proceedings \n1728, 020072 (2016).  \n19.  Y . Takamura, R. Nakane, and S. Sugahare, J. Appl. Phys. 105 , 07B109 (2009).  \n20.  S. Husain, A. Kumar, S. Akansel, P. Svedlindh, and S. Chaudhary, J. Mag. Mag. \nMater. 442, 288 –294 (2017).  \n21. K. Inomata, S. Okamura, A. Miyazaki1, M. Kikuchi, N. Tezuka, M. Wojcik and E. \nJedryka, J. Phys. D: Appl. Phys. 39 , 816 –823 (2006).  \n22. M. Oogane , Y . Sakuraba, J. Nakata1, H. Kubota, Y . Ando, A. Sakuma, and T. \nMiyazaki, J. Phys. D: Appl. Phys. 39 , 834- 841 (2006).  \n23.  S. Okamura, A. Miyazaki, N. Tezuka, S. Sugimoto, and K. Inomata, Materials \nTransactions 47, 15 - 19 (2006).  \n24.  H. Ebert, D. Ködderitzsch , and J. Min´ar, Rep. Prog. Phys. 74, 096501 (2011).  \n25. H. Ebert, http://ebert.cup.uni -muenchen.de/SPRKKR.  \n26. S. H. V osko, L. Wilk, and M. Nusair, Canadian J. Phys. 58 , 1200 (1980).  \n27. M. Belmeguenai, H. Tuzcuoglu, M. Gabor, T. Petrisor, C. Tiusan, D. Berling , F. \nZighem, and S. M. Chrif, J. Mag. Mag. Mat. 373 , 140 (2015).  \n28. A. Liechtenstein, M. Katsnelson, V . Antropov, and V . Gubanov, J. Mag. Mag. Mat. \n67, 65 (1987).   \n29. H. Ebert, and S. Mankovsky, Phys. Rev. B 79, 045209 (2009).  14 \n 30. N. Metropolis, A. W. Rosenbluth,  M. N. Rosenbluth, A. H. Teller, and E. Teller, J. \nChem. Phys. 21, 1087 (1953).   \n31. B. Skubic, J. Hellsvik, L. Nordström, and O. Eriksson, J. Phys.: Cond. Matt. 20, \n315203 (2008).  \n32. http://physics.uu.se/uppasd.  \n33. P. Soven, Phys. Rev. 156 , 809 (1967).  \n34. G. M. St ocks, W. M. Temmerman, and B. L. Gyorffy, Phys. Rev. 41 , 339 (1978).  \n35. A. Brataas, Y . Tserkovnyak, and G. E. W. Bauer, Phys. Rev. Lett. 101 , 037207 \n(2008).  \n36. H. Ebert, S. Mankovsky, D. Ködderitzsch, and P. J. Kelly, Phys. Rev. Lett. 107, \n066603 (2011).  \n37. S. Mankovsky, D. Ködderitzsch, G. Woltersdorf, and H. Ebert, Phys. Rev. B 87, \n014430 (2013).  \n38. H. Ebert, S. Mankovsky, K. Chadova, S. Polesya, J. Min´ar, and D. Ködderitzsch, \nPhys. Rev. B 91, 165132 (2015).  \n39. Rie Y . Umetsu, A. Okubo, M. Nagasako, M. Ohtsuka,  R. Kainuma, and K. Ishida, Spin \n04, 1440018 (2014).  \n40. D. Comtesse, B. Geisler, P. Entel, P. Kratzer, and L. Szunyogh, Phy s. Rev. B 89, \n094410 (2014).  \n41. H. Kurebayashi, T. D. Skinner, K. Khazen, K. Olejník, D. Fang, C. Ciccarelli , R. P. \nCampion, B. L. Gallagher, L. Fleet, A. Hirohata, and A. J. Ferguson, Appl. Phys. Lett. \n102, 062415 (2013).  \n42. Y . Zhao, Qi Song , S.-H. Yang , T. Su, W. Yuan, S. S. P. Parkin, J Shi & W. Han, \nScientific Reports 6 , 22890 (2016).  \n43. J. M. Shaw , Hans T. Nembach , T. J. Silva , C. T. Boone, J. Appl. Phys.  114, 243906 \n(2013).  \n44. P. Landeros, R. E. Arias, and D. L. Mills, Phys. Rev. B 77, 214405 (2008).  \n45. M. A. W. Schoen, J. M. Shaw, H. T. Nembach, M. Weiler, and T. J. Silva,  Phys. Rev. \nB 92, 184417 (2015).  \n46. Y . Li and W. E. Bailey, Phys. Rev. Lett. 116 , 117602 (2016).  \n47. V.  Kambersk´y, Czech. J. Phys.  B 26, 1366 (1976).  \n48. K. Gilmore, Y . U. Idzerda, and M. D. Stiles, Phys. Rev. Lett. 99 , 027204 (2007) .  \n  15 \n Table I P arameters describing magnetic properties of the different CFA samples.  \nSample  𝜇𝜇0𝑀𝑀𝑆𝑆(𝜇𝜇0𝑀𝑀𝑒𝑒𝑒𝑒𝑒𝑒) \n(T) 𝜇𝜇0𝐻𝐻𝑐𝑐𝑖𝑖 \n(mT)  𝑔𝑔∥(𝑔𝑔⊥) \n 𝜇𝜇0𝐻𝐻𝑢𝑢 \n(mT)  𝑇𝑇𝐶𝐶 \n(K) 𝛼𝛼𝑖𝑖𝑖𝑖𝑖𝑖 \n(× 10-3) \nLP573K  1.2±0.1 (1.09 1±0.003)  0.75 2.06 (2.0)  1.56 810 2.56 \nLP673K  1.2±0.1 (1.11 0±0.002)  0.57 2.05 (2.0)  1.97 >900  0.76 \nLP773K  1.2±0.1 (1.081 ±0.002)  0.46 2.05 (2.0)  1.78 890 1.46 \nHP300K  0.9±0.1 (1.066 ±0.002)  1.32 2.01 (2.0)  3.12 -- 3.22  \n 16 \n Figure 1  \nFig. 1.  (a) Layout of the in -house made VNA-based out -of-plane ferromagnetic resonance  \nsetup . (b) Out -plane ferromagnetic resonance spectra recorded for the well -ordered LP673K \nsample at different temperatures  𝑓𝑓=10 GHz . \n  \n \n17 \n Figure 2  \nFig. 2. (a) Magnetization vs. temperature plots measured on the CFA films with different \ndegree of atomic order. (b) Theoretically calculated magnetization vs. temperature curves for \nCFA phases with different degree of atomic order, where 50 % (100 %) Fe atoms on Heusler \nalloy 4a sites indicate B2 (L2 1) ordered phase, and the rest are intermediate B2 & L2 1 mixed \nordered  phases.  \n  \n \n \n18 \n Figure 3  \nFig. 3.  Resonance field vs. in -plane orientation of the applied magnetic field of (a) 𝑇𝑇𝑆𝑆=\n300℃ , 𝑃𝑃𝐼𝐼𝑖𝑖𝑖𝑖=75 𝑊𝑊 deposited, (b) 𝑇𝑇𝑆𝑆=400℃ , 𝑃𝑃𝐼𝐼𝑖𝑖𝑖𝑖=75 𝑊𝑊 deposited, (c) 𝑇𝑇 𝑆𝑆=500℃ , \n𝑃𝑃𝐼𝐼𝑖𝑖𝑖𝑖=75 𝑊𝑊 deposited, and (d) 𝑇𝑇 𝑆𝑆=27℃ , 𝑃𝑃𝐼𝐼𝑖𝑖𝑖𝑖=100  𝑊𝑊 deposited films. Red lines \ncorrespond to fits to the data using Eq. (1).  \n    \n \n \n19 \n  \nFigure 4  \nFig. 4.   Line-width vs. frequency of (a) 𝑇𝑇𝑆𝑆=300℃ , 𝑃𝑃𝐼𝐼𝑖𝑖𝑖𝑖=75 𝑊𝑊 deposited, (b) 𝑇𝑇𝑆𝑆=400℃ , \n𝑃𝑃𝐼𝐼𝑖𝑖𝑖𝑖=75 𝑊𝑊 deposited, (c) 𝑇𝑇 𝑆𝑆=500℃ , 𝑃𝑃𝐼𝐼𝑖𝑖𝑖𝑖=75 𝑊𝑊 deposited, and (d) 𝑇𝑇𝑆𝑆=27℃ , \n𝑃𝑃𝐼𝐼𝑖𝑖𝑖𝑖=100  𝑊𝑊 deposited samples. Red lines correspond to fits to the data to extract the \nexperimental Gilbert damping constant and inhomogeneous line -width. Respective insets \nshow the experimentally determined temperature dependent Gilbert damping constants.  \n  \n  \n \n \n20 \n Figure 5  \nFig. 5.  Frequency vs. applied field of (a) 𝑇𝑇𝑆𝑆=300℃ , 𝑃𝑃𝐼𝐼𝑖𝑖𝑖𝑖=75 𝑊𝑊 deposited, (b) 𝑇𝑇𝑆𝑆=\n400℃ , 𝑃𝑃𝐼𝐼𝑖𝑖𝑖𝑖=75 𝑊𝑊 deposited, (c) 𝑇𝑇𝑆𝑆=500℃ , 𝑃𝑃𝐼𝐼𝑖𝑖𝑖𝑖=75 𝑊𝑊 deposited, and (d) 𝑇𝑇𝑆𝑆=27℃ , \n𝑃𝑃𝐼𝐼𝑖𝑖𝑖𝑖=100  𝑊𝑊 deposited samples. Red lines correspond to Kittel’s fits to the data. Respective \ninsets show the temperature dependent effective magnetization a nd inhomogeneous line -width \nbroadening values.  \n  \n  \n \n21 \n Figure 6  \nFig. 6.  (a) Linewidth vs. frequency with and without a glass spacer between the waveguide \nand the sample.  Red lines correspond to fits using Eq. (4).  (b) Temperature dependent values \nof the radiative damping using Eq.  (6). The lines are guide to the eye. (c) 𝛼𝛼−𝛼𝛼𝑟𝑟𝑚𝑚𝑟𝑟≈𝛼𝛼𝑒𝑒𝑟𝑟𝑟𝑟𝑒𝑒 \nvs 𝛿𝛿2. The red line corresponds to a fit  using Eq. (7) to extract the value of the correction \nfactor  𝐶𝐶. (d) Temperature dependent values of eddy current dampi ng using Eq. (7). The lines \nare guide to the eye.  \n  \n  \n \n \n22 \n Figure 7  \nFig. 7.  Experimental (a) and theoretical (b) results for the temperature dependent intrinsic \nGilbert damping constant for CFA samples with different degree of atomic order . The B2 & \nL21 mixed phase corresponds to the 75 % occupancy of Fe atoms on the Heusler alloy 4a \nsites. The lines are guide to the eye.  \n  \n \n \n23 \n Figure 8  \nFig. 8.  Total and atom -resolved spin polarized density of states plots for various \ncompositional CFA phases; (a) A2, (b) B2 and (c) L2 1. \n  \n \n \n"    },    {        "title": "1706.04959v1.Generalized_Voltage_based_State_Space_Modelling_of_Modular_Multilevel_Converters_with_Constant_Equilibrium_in_Steady_State.pdf",        "content": "1\nGeneralized V oltage-based State-Space\nModelling of Modular Multilevel Converters\nwith Constant Equilibrium in Steady-State\nGilbert Bergna-Diaz, Julian Freytes, Xavier Guillaud, Member, IEEE,\nSalvatore D’Arco, and Jon Are Suul, Member, IEEE\nAbstract\nThis paper demonstrates that the sum and difference of the upper and lower arm voltages are suitable variables for\nderiving a generalized state-space model of an MMC which settles at a constant equilibrium in steady-state operation,\nwhile including the internal voltage and current dynamics. The presented modelling approach allows for separating the\nmultiple frequency components appearing within the MMC as a first step of the model derivation, to avoid variables\ncontaining multiple frequency components in steady-state. On this basis, it is shown that Park transformations at three\ndifferent frequencies ( +!,\u00002!and+3!) can be applied for deriving a model formulation where all state-variables\nwill settle at constant values in steady-state, corresponding to an equilibrium point of the model. The resulting model\nis accurately capturing the internal current and voltage dynamics of a three-phase MMC, independently from how the\ncontrol system is implemented. The main advantage of this model formulation is that it can be linearised, allowing for\neigenvalue-based analysis of the MMC dynamics. Furthermore, the model can be utilized for control system design\nby multi-variable methods requiring any stable equilibrium to be defined by a fixed operating point. Time-domain\nsimulations in comparison to an established average model of the MMC, as well as results from a detailed simulation\nmodel of an MMC with 400 sub-modules per arm, are presented as verification of the validity and accuracy of the\ndeveloped model.\nIndex Terms\nHVDC transmission, modular multilevel converter, Park Transformations, State-Space Modelling.\nThis manuscript is partly based on the following conference publication: Gilbert Bergna, Jon Are Suul, Salvatore\nD’Arco, \"State-Space Modeling of Modular Multilevel Converters for Constant Variables in Steady-State,\" presented\nat the 17th IEEE Workshop on Control and Modeling for Power Electronics, COMPEL 2016, Trondheim, Norway,\n27-30 June 2016. Compared to the initial conference paper, the text has been thoroughly revised, the presented\nmodel has been generalized and new results have been included.\nG. Bergna-Diaz is with the Norwegian University of Science and Technology (NTNU), Trondheim, Norway (gilbert.bergna@ntnu.no)\nJ. Freytes and X. Guillaud are with the Université Lille, Centrale Lille, Arts et Métiers, HEI - EA 2697, L2EP - Lille, France\n(julian.freytes@centralelille.fr, xavier.guillaud@centralelille.fr)\nS. D’Arco and J. A. Suul are with SINTEF Energy Research, Trondheim, Norway, (salvatore.darco@sintef.no, jon.a.suul@sintef.no)arXiv:1706.04959v1  [cs.SY]  15 Jun 20172\nI. I NTRODUCTION\nThe Modular Multilevel Converter (MMC) is emerging as the preferred topology for V oltage Source Converter\n(VSC) -based HVDC transmission schemes [1], [2]. Especially in terms of its low losses, modularity, scalability\nand low harmonic content in the output ac voltage, the MMC topology provides significant advantages for HVDC\napplications compared to two- or three-level VSCs. However, the MMC is characterized by additional internal\ndynamics related to the circulating currents and the internal capacitor voltages of the upper and lower arms of each\nphase [3], [4]. Thus, the modelling, control and analysis of the MMC is more complicated than for other VSC\ntopologies.\nDifferent types of studies are necessary for design and analysis of MMC-based HVDC transmission systems,\nrequiring various detailing levels in the modelling. A general overview of MMC modelling approaches suitable\nfor different types of studies is shown in Fig. 1. The most detailed models allow for simulating the switching\noperations of the individual sub-modules of the MMC, as shown to the right of the figure. Such models can be used\nfor studying all modes of operation and all the control loops of the MMC, including the algorithms for balancing\nthe sub-module voltages. If equal voltage distribution among the sub-modules in each arm of an MMC can be\nassumed, average arm models (AAM) can be introduced. The AAM modelling approach allows for representing\neach arm of the MMC by a controllable voltage source associated with a corresponding equivalent capacitance,\nand is introducing a significant reduction of complexity while still maintaining an accurate representation of the\ninternal dynamics. [3]–[6].\nAverage modelling by the AAM representation, or by equivalent energy-based models, are suitable for simplified\nsimulations and analysis, and have been widely used as basis for control system design [4], [6], [7]. However, the\nvariables of such models are Steady State Time Periodic (SSTP), with the currents and capacitor voltages in each\narm of the MMC containing multiple frequency components [8]. This prevents a straightforward application of the\nPark transformation for obtaining state-space models of three-phase MMCs represented in a single Synchronously\nRotating Reference Frame (SRRF), according to the modelling approaches commonly applied for control system\ndesign and small-signal stability analysis of two-level VSCs [9]–[11]. However, for obtaining a linearized small-\nsignal model of an MMC that can be analyzed by traditional techniques for eigenvalue-based stability analysis, it is\nnecessary to derive a SRRF state-space model with a Steady-State Time Invariant (SSTI) solution. As indicated in\nFig. 1, such a SRRF dqzmodel must be derived from an equivalent average model in the stationary abccoordinates.\nIf a non-linear model with a SSTI solution, corresponding to defined equilibrium point, can be obtained, a Linear\nTime Invariant (LTI) model suitable for eigenvalue analysis can be directly obtained by linearization.\nSeveral approaches for obtaining LTI state-space models of MMCs have been recently proposed in the literature,\nmotivated by the need for representing MMC HVDC transmission systems in eigenvalue-based small-signal stability\nstudies. The simplest approach has been to neglect parts of the internal dynamics of the MMC, and model mainly\nthe ac-side dynamics in a SRRF together with a simplified dc-side representation, as in the models proposed in\n[12]–[14]. However, if the dynamics associated with the internal equivalent capacitor voltages of the MMC and the\ninteraction with the circulating currents are ignored, such models will imply significant inaccuracies. Especially if3\nfsw(t)\nSteady-State Time-Periodic (SSTP)\nMMC Switching/\nDiscontinous\nModels\nIGBT+\nDiode\nIdeal\nSwitches\nSwitching\nFunction\nON/OFF\nResistors\nAveraging\n(Simplification)\nMMC Averaged Continous\nModel in `abc' Coordinates\nmU;L\nj(t)\nContinous\nInsertion Indexes\nContinous non-linear\nState-Space representation\nd\ndtxabc=f\u0000\nxabc;uabc; t\u0001\nTransformation\nSimplifications\nMinor\nNon-Linear State-Space\nModels with SSTI\nd\ndtxdqz=f\u0000\nxdqz;udqz\u0001\n(focus of this paper )\nLinearized Small-Signal\nState-Space Models\nd\ndt\u0001xdqz=A(x0)\u0001xdqz\nSteady-State Time-Invariant (SSTI)\nLinearization\n\u000fExplicit representation of sub-\nmodules and switching events\n{Computationally intensive\n{Th\u0013 evenin equivalents for faster\nsimulation\n\u000fSuitable for Electro-Magnetic\nTransient (EMT) simulations\n{Assessment of component\nstress\n{Design validation\n{Evaluation of sub-module con-\ntrol strategies\n{Fault studies\n\u000fAssumes ideal capacitor voltage\nbalancing\n\u000fEquivalent arm capacitance and\ncontinuoustime averaged arm volt-ages\n{Faster and less computationally\nintensive time-domain simula-tions\n{Explicit analytical expressions\nfor arm voltage and energy bal-ance\n\u000fSuitable for analysis and design of\ncontrollers\n{Current control\n{Energy control\n{Outer loop power \row con-\ntrollers\n\u000fAnalysis of state-space models in\ntimeperiodic framework\n\u000fDerived from average model in\nstationary `abc' coordinates\n{Transformed into syn-\nchronously rotating `dqz'\nreference frames\n{Similar to stationary frame\naverage models in accuracy\nand computational require-\nments\n\u000fNon-linear system analysis and\ncontrol design requiring con-\nstant variables in steady-state\n{Calculation of steady-state\noperation\n{Estimation of region of at-\ntraction\n{Passivity based control\n\u000fPrerequisite for linearization at\nequilibrium\n\u000fValid for a small region\naround the linearization point\n\u000fAnalysis of system dynamics\nand stability by linear meth-\nods`abc' coordinates\n{Small-signal stability\nassessment by eigenvalue\nanalysis\n{Participation factor analy-\nsis\n{Parametric sensitivity\n\u000fCompatible with models used\nfor analysis of small-signal sta-\nbility in large-scale power sys-\ntems\n: : :+B(x0)\u0001udqz\nSolution\nFigure 1: Overview of MMC modelling approaches and their areas of application\na power balance between the ac- and dc-sides of the converter is assumed in the same way as for a two-level VSC\nmodel, like in [12], [14], the model will only be suitable for representing very slow transients. Therefore, more\ndetailed dynamic state-space models have been proposed in [15]–[21]. These available models have been developed\nfor representing two different cases:\n1) The approaches presented in [15], [16] are based on the assumption that the modulation indices for the\nMMC arms are calculated to compensate for the voltage oscillations in the internal equivalent arm capacitor\nvoltages, referred to as Compensated Modulation (CM). This strategy for control system implementation\nlimits the coupling between the internal variables of the MMC and the ac- and dc-side variables. Thus,\nCM-based control allows for simplified modelling of the MMC, where only the aggregated dynamics of the\nzero-sequence circulating current and the total energy stored in the capacitors of the MMC are represented.\nAs a result, these models can provide accurate representation of the ac- and dc-side terminal behavior of\nMMCs, but imply that the dynamics of the internal variables cannot be analyzed.\n2) The approaches proposed in [17]–[19], [21] consider all the internal variables of the MMC, under the\nassumption of a control system with a Circulating Current Suppression Controller (CCSC) implemented\nin a negative sequence double frequency SRRF [22]. Indeed, the methods proposed in [17], [18], [21] model\nthe MMC by representing the internal second harmonic circulating currents and the corresponding second\nharmonic arm voltage components in a SRRF rotating at twice the fundamental frequency. However, the\nharmonic superposition principles assumed in the modelling, corresponding to phasor-based representation,4\ncould affect the information about the non-linear characteristics of the MMC, and correspondingly limit\nthe applicability of the models in non-linear techniques for analysis and control system design. A similar\napproximation was also made when separately modelling the fundamental frequency and the second harmonic\nfrequency dynamics of the upper and lower arm capacitor voltages in [19].\nThe main contribution of this paper is to present a linearizable SSTI state-space representation of an MMC with\nas few simplifications as possible in the derivation of the model. Indeed, the presented approach is intended for\npreserving the fundamental non-linearity of the stationary frame average model of the MMC that is used as starting\npoint for the presented derivations. This is achieved by utilizing the information about how the different variables of\nthe MMC contain mainly combinations of dc-components, fundamental frequency components and double frequency\noscillations in steady state operation. By using the sum ( \u0006) and difference ( \u0001) between the variables of the upper\nand lower arms of the MMC as variables, a natural frequency separation can be obtained where the \u0001variables\ncontain only a fundamental frequency component while \u0006variables contain dc and 2 !components. This frequency\nseparation allows for applying appropriate Park transformations to each set of variables, resulting in an SSRF model\nwhere all state variables settle to a constant equilibrium point in steady-state operation. Thus, the obtained model\nis suitable for non-linear control system design, for instance by applying passivity theory [23], [24], but can also\nbe directly linearized to obtain a detailed small-signal model representing the dynamic characteristics of an MMC.\nThe first contribution to this modelling approach was presented in [20], but this paper will extends the derivations\nfrom [20] to obtain a model that is applicable independently from the applied approach for calculating the modulation\nindices of the MMC. Furthermore, the model derivation has been expanded to include the effect of the zero-sequence\nof the difference between upper and lower insertion indexes mzin the MMC dynamics, which was neglected in\n[20]. This extension of the model can be useful when third harmonic injection is used for increasing the voltage\nutilization [25], [26], and in case a zero sequence component in the output voltage is utilized to control the energy\ndistribution within the MMC.\nThe applied modelling approach and the derivations required for obtaining the presented generalized voltage-\nbased state-space model of an MMC with SSTI characteristics are presented in detail, since similar techniques can\nalso be useful for modelling and analysis of MMC control strategies implemented in the stationary frame. The\nvalidity of the derived model is demonstrated by time-domain simulations in comparison to the average model\nused as starting point for the derivations, and the validity of the obtained results are confirmed in comparison to a\ndetailed simulation model of an MMC with 400 sub-modules per arm.\nII. MMC M ODELLING IN THE STATIONARY REFERENCE FRAME : TOPOLOGY ,\u0006-\u0001VECTOR REPRESENTATION\nAND FREQUENCY ANALYSIS\nA. Average model representation of the MMC topology\nThe basic topology of a three-phase MMC is synthesized by the series connection of Nsub-modules (SMs)\nwith independent capacitors Cto constitute one arm of the converter as indicated by Fig. 2. The sub-modules\nin one arm are connected to a filter inductor with equivalent inductance Larm and resistance Rarm to form the\nconnection between the dc terminal and the ac-side output. Two identical arms are connected to the upper and lower5\nSMU\n1a\nSMU\n2a\nSMU\nNa\nRf\nLf\nRf\nLf\nRf\nLf\nS1\nS1\nS2\nC\nvCi\nvxi\nvdc\nidc\nidc\nRarm\nLarm\nRarm\nLarm\nRarm\nLarm\nLarm\nRarm\nLarm\nRarm\nLarm\nRarm\nvU\nma\nvL\nma\niU\na\niU\nb\niU\nc\niL\na\niL\nb\niL\nc\nvG\nc\nvG\nb\nvG\na\ni\u0001\nc\ni\u0001\nb\ni\u0001\na\niL\nmc\nCarm\nvL\nCc\nmL\nc\nAAM\nSMU\n1b\nSMU\n2b\nSMU\nNb\nSMU\n1c\nSMU\n2c\nSMU\nNc\nSML\n1a\nSML\n2a\nSML\nNa\nSML\n1b\nSML\n2b\nSML\nNb\nSML\n1c\nSML\n2c\nSML\nNc\nFigure 2: MMC Topology and AAM for the lower arm (phase C)\ndc-terminals respectively to form one leg for each phase j(j=a;b;c ). The AC side is modeled with an equivalent\nresistance and inductance RfandLfrespectively [27].\nAssuming that all the SMs capacitors voltages are maintained in a close range, the series connection of submodules\nin each arm can be replaced by a circuit-based average model, corresponding to the well-known Arm Averaged\nModel (AAM) as indicated in Fig. 2 for the lower arm of phase c[4], [7]. If the MMC is modelled by the AAM\nrepresentation, each arm is appearing as a controlled voltage source in the three-phase topology, while a power\nbalance is established between the arm and its equivalent capacitance. Thus, each arm can be represented by a\nconventional power-balance-based average model of a VSC, with a modulated voltage source interfacing the filter\ninductor, and a current source interfacing the capacitor-side.\nThe output of the controlled voltage and current sources of the AAM, are here referred as the modulated voltages\nvU\nmjandvL\nmjand modulated currents iU\nmjandiL\nmj, for the upper ( U) and lower ( L) arms of a generic phase j, and\nare described by the following equations:\nvU\nmj=mU\njvU\nCj; vL\nmj=mL\njvL\nCj; iU\nmj=mU\njiU\nj; iL\nmj=mL\njiL\nj(1)\nwherevU\nCjandvL\nCjare respectively the voltages across the upper and lower arm equivalent capacitors; mU\njand\nmL\njare the corresponding insertion indexes for the upper and lower arms, and iU\njandiL\njare the currents in the\nupper and lower arms, respectively.\nB. Modelling of the MMC with \u0006-\u0001variables in the stationary abcframe\nAs mentioned in the introduction, the proposed approach adopts the \u0006-\u0001representation as opposed to the more\ncommon one based on the Upper-Lower (U-L) arm notation, to ease the derivation of an MMC model with SSTI\nsolution. More precisely, under this \u0006-\u0001representation, it is possible to initially classify the 11 states and 6 control\nvariables for an average model of a three-phase MMC into two frequency groups; i.e., the \u0001variables which are6\nassociated to the fundamental frequency !, and the \u0006variables which are in turn associated to \u00002!, and will be\nfurther discussed in section II-C. It is therefore useful to redefine the voltages and currents that are defined in Fig.\n2 using this nomenclature, resulting in (2). Indeed, i\u0001\njis the current flowing through the AC-side grid, whereas i\u0006\nj\nis the well-known circulating current of the MMC. Moreover, v\u0001\nCjandv\u0006\nCjare respectively the difference and the\nsum of voltages across the upper and lower equivalent capacitors.\ni\u0001\njdef=iU\nj\u0000iL\nj; i\u0006\njdef=\u0000\niU\nj+iL\nj\u0001\n=2; v\u0001\nCjdef= (vU\nCj\u0000vL\nCj)=2; v\u0006\nCjdef= (vU\nCj+vL\nCj)=2; (2)\nIn addition, it is also useful to define the modulated voltages given in (1) in the \u0006-\u0001representation as in (3), as\nwell as modulation indexes as in (4).\nv\u0001\nmjdef=\u0000vU\nmj+vL\nmj\n2; v\u0006\nmjdef=vU\nmj+vL\nmj\n2(3)\nm\u0001\njdef=mU\nj\u0000mL\nj; m\u0006\njdef=mU\nj+mL\nj (4)\n1) AC-grid current dynamics: The three-phase ac-grid currents dynamics i\u0001\nabcare expressed using vector nomen-\nclature in the stationary frame as in (5),\nLac\neqdi\u0001\nabc\ndt=v\u0001\nmabc\u0000vG\nabc\u0000Rac\neqi\u0001\nabc; (5)\nwhere vG\nabcis the grid voltage vector defined as [vG\navG\nbvG\nc]>, whereas v\u0001\nmabc is the modulated voltage driving\nthe ac-grid current defined as [v\u0001\nmav\u0001\nmbv\u0001\nmc]>, or more precisely as:\nv\u0001\nmabc =\u00001\n2\u0000\nm\u0001\nabc\nv\u0006\nCabc +m\u0006\nabc\nv\u0001\nCabc\u0001\n(6)\nwhere the upper andlower modulation indexes and voltage variables were replaced by their \u0006-\u0001equivalents for\nconvenience. It is worth noticing that the operator “ \n” will be used here to represent the element-wise multiplication\nof vectors (e.g. [a\nb]\n[c\nd] = [ac\nbd]). Furthermore, Rac\neqandLac\neqare the equivalent ac resistance and inductance,\nrespectively defined as Rf+Rarm=2andLf+Larm=2.\n2) Circulating current dynamics: The three-phase circulating currents dynamics in the stationary frame can be\nwritten by using vector notation as:\nLarmdi\u0006\nabc\ndt=vdc\n2\u0000v\u0006\nmabc\u0000Rarmi\u0006\nabc; (7)\nwhere vdcis defined as [vdcvdcvdc]>andv\u0006\nmabc is the modulated voltage driving the circulating current defined\nas[v\u0006\nmav\u0006\nmbv\u0006\nmc]>, or more precisely as:\nv\u0006\nmabc =1\n2\u0000\nm\u0006\nabc\nv\u0006\nCabc +m\u0001\nabc\nv\u0001\nCabc\u0001\n; (8)\nwhere the upper andlower modulation indexes and voltage variables were replaced by their \u0006-\u0001equivalents for\nconvenience here as well.\n3) Arm capacitor voltage dynamics: Similarly, the dynamics of the voltage sum and difference between the\nequivalent capacitors of the AAM can be expressed respectively as in (9) and (10).\n2Carmdv\u0006\nCabc\ndt=m\u0001\nabc\ni\u0001\nabc\n2+m\u0006\nabc\ni\u0006\nabc (9)\n2Carmdv\u0001\nCabc\ndt=m\u0006\nabc\ni\u0001\nabc\n2+m\u0001\nabc\ni\u0006\nabc (10)7\nC. Simplified frequency analysis of the \u0006-\u0001variables\nIt is well known that under normal operating conditions the grid current of the MMC i\u0001\nabcoscillates at the grid\nfrequency!, whereas the circulating current consists of a dc value or a dc value in addition to oscillating signals\nat\u00002!, depending on whether the second harmonic component is eliminated by control or not. Therefore, the\nfollowing analysis will only focus on the remaining voltage states v\u0006\nCabc andv\u0001\nCabc .\nA simplified analysis can be performed by assuming that mU\njis phase-shifted approximately 180° with respect\ntomL\nj, resulting in m\u0006\nj\u00191andm\u0001\nj\u0019^mcos (!t). By inspecting the right-side of (9), it can be seen that in\nsteady-state, the first product m\u0001\nji\u0001\nj=2gives a dc value in addition to an oscilatory signal at \u00002!, while the second\nproductm\u0006\nji\u0006\njgives a dc value in case a constant value of i\u0006\nabcis imposed by control (e.g. by CCSC [22]), or a\n2!signal otherwise, resulting for both cases in 2!oscillations in v\u0006\nCj.\nSimilarly for v\u0001\nCj, the first product on the right-side of (10), m\u0006\nji\u0001\nj=2, oscillates at !, while the second product\nm\u0001\nji\u0006\njoscillates at !in the case the CCSC is used or will result in a signal oscillating at !superimposed to one\nat3!otherwise. Note that if the assumption m\u0006\nj\u00191is no longer considered, but instead m\u0006\nabcis allowed to have\na second harmonic component superimposed to its dc value, the first term of (10) will also produce an additional\ncomponent at 3!.\nAs will be shown in the remainder of the paper, this additional 3rd harmonic in the \u0001variable does not\nsignificantly affect the initial frequency classification of the variables as it will be captured and isolated by the\nzero-sequence component after the application of Park’s transformation at !without affecting its corresponding\ndqcomponents. This is similar to the case for the \u0006variables, as in addition to the \u00002!signals, they present\na dc value. Here too, this additional (dc) value is isolated by the zero-sequence after the application of Park’s\ntransformation at \u00002!, without affecting its dqcomponents.\nThis initial classification of the state and control variables according to their main oscilatory frequency is\nsummarized in Table I and is considered the base for the methodology which is presented in the next section.\nTable I: MMC variables in \u0006-\u0001representation\nVariables oscillating at ! Variables oscillating at \u00002!\ni\u0001\nj=iU\nj\u0000iL\nji\u0006\nj= (iU\nj+iL\nj)=2\nv\u0001\nmj= (\u0000vU\nmj+vL\nmj)=2v\u0006\nmj= (vU\nmj+vL\nmj)=2\nm\u0001\nj=mU\nj\u0000mL\njm\u0006\nj=mU\nj+mL\nj\nIII. N ON-LINEAR TIME -INVARIANT MMC MODEL WITH \u0006-\u0001REPRESENTATION IN dqz FRAME AND\nVOLTAGE -BASED FORMULATION\nIn this section, the derivations needed for obtaining the state-space time-invariant representation of the MMC\nwith voltage-based formulation is presented in detail on basis of the approach from [20].8\n+!\n3!\nv\u0001\nCabc\nP!\nv\u0006\nCabc\nP\u00002!\nv\u0001\nCz\nT3!\n?90\u000e\nv\u0001?90\u000e\nCz\nv\u0006\nCdqz\nv\u0001\nCdq\nv\u0001\nCZ\ni\u0001\nabc\nP!\ni\u0006\nabc\nP\u00002!\ni\u0006\ndqz\ni\u0001\ndq\nm\u0006\nabc\nP\u00002!\nm\u0006\ndqz\nm\u0001\nabc\nP!\nm\u0001\nz\nT3!\n?90\u000e\nm\u0001?90\u000e\nz\nm\u0001\ndq\nm\u0001\nZ\n\u00002!\nDC\nMMC Internal State Variables\nControl Variables\nFigure 3: The proposed modelling approach based on three Park transformations to achieve SSTI control and state\nvariables\nThe formulation of the MMC variables such that the initial separation of frequency components can be achieved\nconstitutes the basis for the proposed modelling approach, as illustrated in Fig. 3. This figure indicates that Park\ntransformations at different frequencies will be used to derive dynamic equations for equivalent dqz variables\nthat are SSTI in their respective reference frames. More precisely, the \u0001-variables ( v\u0001\nCabc ,i\u0001\nabcandm\u0001\nabc) are\ntransformed into their dqzequivalents by means of a Park transformation P!at the grid fundamental frequency !.\nBy contrast, the \u0006-variables ( v\u0006\nCabc ,i\u0006\nabcandm\u0006\nabc) are transformed into their dqzequivalents by means of a Park\ntransformation P\u00002!at twice the grid frequency in negative sequence, \u00002!. In addition, a Park transformations at\n3!will be used to ensure a SSTI representation of the zero sequence of the voltage difference v\u0001\nCz, as well as for\nthe zero sequence of the modulation index difference m\u0001\nz.\nIn the remainder of this section, the mathematical derivation of dynamic equations with SSTI solution representing\nthe dynamics of a three phase MMC will be expressed by using the approach illustrated by Fig. 3. The mathematical\nreformulation consists in expressing the vector variables in the stationary abcframe as a function of their dqz\nequivalents at their respective rotating frequencies.\nA. Voltage difference SSTI dynamics derivation\n1) Initial formulation: The SSTI dynamics for the voltage difference is derived in the following. The starting\npoint is indeed the SSTP dynamics of the variable given in (10), and recalled in (11) for convenience. The first step\nconsists in expressing the abcvectors in the stationary frame as functions of their respective dqzequivalents. This9\ncan be seen in the second line of (11), where v\u0001\nCabc ,m\u0006\nabc,i\u0001\nabc,m\u0001\nabcandi\u0006\nabchave been respectively replaced\nbyP\u00001\n!v\u0001\nCdqz ,P\u00001\n\u00002!m\u0006\ndqz,P\u00001\n!i\u0001\ndqz,P\u00001\n!m\u0001\ndqzandP\u00001\n\u00002!i\u0006\ndqz. Notice that the choice of using the inverse Park\ntransformation matrix at !(P\u00001\n!) or at 2!(P\u00001\n\u00002!) is according to the frequency separation of the variables given\nin Table I and Fig. 3.\n2Carmdv\u0001\nCabc\ndt=m\u0006\nabc\ni\u0001\nabc\n2+m\u0001\nabc\ni\u0006\nabc=:::\n2CarmdP\u00001\n!\ndtv\u0001\nCdqz + 2CarmP\u00001\n!dv\u0001\nCdqz\ndt| {z }\n\b\u0001\nA=P\u00001\n\u00002!m\u0006\ndqz\nP\u00001\n!i\u0001\ndqz\n2|{z}\n\b\u0001\nB+P\u00001\n!m\u0001\ndqz\nP\u00001\n\u00002!i\u0006\ndqz|{z}\n\b\u0001\nC(11)\nThe equation expressed in (11), must be multiplied by the Park transformation matrix at the angular frequency\n!, so that it can be possible to solve for dv\u0001\nCdqz=dt.\nMultiplying \b\u0001\nAbyP!, gives:\nP!\b\u0001\nA= 2CarmJ!v\u0001\nCdqz + 2Carmdv\u0001\nCdqz\ndt(12)\nwhere J!is defined as in (13):\nJ!def=2\n66640!0\n\u0000!0 0\n0 0 03\n7775(13)\nFurthermore, multiplying \b\u0001\nBbyP!gives:\nP!\b\u0001\nB=P! \nP\u00001\n\u00002!m\u0006\ndqz\nP\u00001\n!i\u0001\ndqz\n2!\n=M\u0001\n\bBh\ni\u0001\ndi\u0001\nqi\u0001\nzi>\n(14)\nwhere M\u0001\n\bBis expressed in (15). For simplicity, it will be considered that the system under study does not allow for\nthe existence of the zero-sequence grid current; i.e., i\u0001\nz= 0, highlighted in gray in (14). Under this assumption, only\nthedqcomponent of (14) is time-invariant, as the 3!oscillatory signals that appear in M\u0001\n\bBare either multiplying\ni\u0001\nz(third column of the matrix) or appear in the last row. However, it is possible to rewrite also the dynamics of\nv\u0001\nCzin SSTI form by means of additional mathematical manipulations, as will be shown further.\nM\u0001\n\bB=1\n42\n6664m\u0006\nd+ 2m\u0006\nz \u0000m\u0006\nq m\u0006\ndcos(3!t)\u0000m\u0006\nqsin(3!t)\n\u0000m\u0006\nq \u0000m\u0006\nd+ 2m\u0006\nzm\u0006\nqcos(3!t) +m\u0006\ndsin(3!t)\nm\u0006\ndcos(3!t)\u0000m\u0006\nqsin(3!t)m\u0006\nqcos(3!t) +m\u0006\ndsin(3!t) 2 m\u0006\nz3\n7775(15)\nFinally, multiplying \b\u0001\nCbyP!gives:\nP!\b\u0001\nC=P!\u0010\nP\u00001\n!m\u0001\ndqz\nP\u00001\n\u00002!i\u0006\ndqz\u0011\n=M\u0001\n\bCh\ni\u0006\ndi\u0006\nqi\u0006\nzi>\n(16)\nM\u0001\n\bC=1\n22\n6664m\u0001\nd+ 2m\u0001\nzcos(3!t) \u0000m\u0001\nq\u00002m\u0001\nzsin(3!t)m\u0001\nd\n\u0000m\u0001\nq+ 2m\u0001\nzsin(3!t) \u0000m\u0001\nd+ 2m\u0001\nzcos(3!t)m\u0001\nq\nm\u0001\ndcos(3!t) +m\u0001\nqsin(3!t)m\u0001\nqcos(3!t)\u0000m\u0001\ndsin(3!t)m\u0001\nz3\n7775(17)\nwhere M\u0001\n\bCis expressed in (17). Here, M\u0001\n\bCrequires further mathematical manipulation to achieve the desired\nSSTI performance, as the 3!signals also appear. Moreover, they affect not only the zero-sequence as in the10\nprevious case, yet the dqcomponents as well. Replacing the definitions given in (12), (14) and (16) in P\u00001\n!\b\u0001\nA=\nP\u00001\n!\b\u0001\nB+P\u00001\n!\b\u0001\nCand solving for the voltage difference dynamics in their dqzcoordinates results in (18):\ndv\u0001\nCdqz\ndt=1\n2Carm\u0012\nM\u0001\n\bBh\ni\u0001\ndi\u0001\nqi\u0001\nzi>\n+M\u0001\n\bCh\ni\u0006\ndi\u0006\nqi\u0006\nzi>\u0013\n\u0000J!v\u0001\nCdqz (18)\nSince neither M\u0001\n\bBorM\u0001\n\bCare SSTI, equation (18) is not directly providing a SSTI solution. This issue is\ntreated in the remainder of this section.\n2) Deriving the SSTI dq dynamics of (18) :First, thedqdynamics of (18) are addressed. As discussed earlier,\nsince it is assumed that i\u0001\nz= 0, only M\u0001\n\bCis hindering a SSTI representation for the dqdynamics due to the\nappearance of the cos(3!t)andsin(3!t)in the 2\u00022sub-matrix at the upper left corner of M\u0001\n\bCin (17), referred\nto asM\u00012\u00022\n\bC. One possible solution is to assume that the MMC control will always set m\u0001\nzto zero, as was done\nin [20], asm\u0001\nzis multiplying all of the 3!oscillating signals. However, this lead to a restrictive model from a\ncontrol perspective, and therefore such assumption is avoided here. Taking inspiration from common engineering\npractices to increase controllability in VSCs [25], the proposed solution is to redefine m\u0001\nzas a third harmonic\ninjection, as given in (19), where m\u0001\nZdandm\u0001\nZqare two SSTI variables that will define the amplitude and phase\nangle of third harmonic oscilllations in m\u0001\nz.\nm\u0001\nzdef=m\u0001\nZdcos(3!t) +m\u0001\nZqsin(3!t) (19)\nReplacing the new definition (19) in (17), results in the sub-matrix (20).\nM\u00012\u00022\n\bC=1\n22\n4+\u0000\nm\u0001\nd+m\u0001\nZd\u0001\n\u0000\u0010\nm\u0001\nq+m\u0001\nZq\u0011\n\u0000\u0010\nm\u0001\nq\u0000m\u0001\nZq\u0011\n\u0000\u0000\nm\u0001\nd\u0000m\u0001\nZd\u00013\n5+2\n4+ cos (6!t) + sin (6!t)\n+ sin (6!t)\u0000cos (6!t)3\n52\n4+m\u0001\nZd+m\u0001\nZq\n+m\u0001\nZq\u0000m\u0001\nZd3\n5\n| {z }\n\u00190\n(20)\nFurthermore, the oscillatory signals at 3!are replaced by signals at 6!, which can be neglected as will be\nconfirmed via time-domain simulations.\n3) Deriving SSTI expressions for the zero-sequence dynamics of (18):The zero sequence dynamics equation of\n(18), is given again in (21) for convenience.\ndv\u0001\nCz\ndt=1\nCarm\u00141\n8\u0000\nm\u0006\ndi\u0001\nd+m\u0006\nqi\u0001\nq+ 2m\u0001\ndi\u0006\nd+ 2m\u0001\nqi\u0006\nq\u0001\ncos(3!t) +::: (21)\n\u0001\u0001\u0001+1\n8\u0000\n\u0000m\u0006\nqi\u0001\nd+m\u0006\ndi\u0001\nq+ 2m\u0001\nqi\u0006\nd\u00002m\u0001\ndi\u0006\nq\u0001\nsin(3!t) +m\u0001\nzi\u0006\nz\u0015\nBy replacing the new definition of (19) into (21), the zero-sequence dynamics of v\u0001\nCzcan be written as:\ndv\u0001\nCz\ndt=1\nCarm[\tdcos(3!t) + \t qsin(3!t)] (22)\nwhere \tdand\tqare defined as below.\n\td=1\n8\u0000\n+m\u0006\ndi\u0001\nd+m\u0006\nqi\u0001\nq+ 2m\u0001\ndi\u0006\nd+ 2m\u0001\nqi\u0006\nq+ 4m\u0001\nZdi\u0006\nz\u0001\n\tq=1\n8\u0010\n\u0000m\u0006\nqi\u0001\nd+m\u0006\ndi\u0001\nq+ 2m\u0001\nqi\u0006\nd\u00002m\u0001\ndi\u0006\nq+ 4m\u0001\nZqi\u0006\nz\u001111\nSince the zero sequence dynamics in (22) are still time-varying in steady state, further reformulation is necessary\nto obtain the desired model with SSTI solution. This can be obtained by defining an auxiliary virtual state v\u0001\nCZ\f,\nshifted 90° with respect to the original \"single-phase\" time-periodic voltage difference signal v\u0001\nCzaccording to\nthe approach from [20]. This approach is conceptually similar to the commonly applied strategy of generating a\nvirtual two-phase system for representing single-phase systems in a SRRF [28]. However, since the amplitudes of\nthe different sine and cosine components \tdand\tqare defined by SSTI variables, the virtual signal v\u0001\nCZ\fcan be\nidentified without any additional delay.\nThe real and virtual voltage difference zero-sequence variables can be labelled as v\u0001\nCZ\u000bandv\u0001\nCZ\f, and together\nthey define an orthogonal \u000b\f-system. This \u000b\f-system can be expressed by (23a)-(23b), where the first equation\nis exactly the same as (22), while the second equation replaces the cos(3!t)andsin(3!t)that appear in (22) by\nsin(3!t)and\u0000cos(3!t), respectively.\ndv\u0001\nCZ\u000b\ndt=1\nCarm[\tdcos(3!t) + \t qsin(3!t)] (23a)\ndv\u0001\nCZ\f\ndt=1\nCarm[\tdsin(3!t)\u0000\tqcos(3!t)] (23b)\nDefining v\u0001\nCZ\u000b\fdef= [v\u0001\nCZ\u000bv\u0001\nCZ\f]>, the equations (23a) and (23b) are written in a compact form as shown in (24).\ndv\u0001\nCZ\u000b\f\ndt=1\nCarm\u001a\nT3!h\n\td\tqi>\u001b\n(24)\nwhere T3!can be viewed as a Park transformation at 3!as defined in (25).\nT3!def=2\n4cos(3!t) sin(3!t)\nsin(3!t)\u0000cos(3!t)3\n5 (25)\nFurthermore, by defining v\u0001\nCZdef= [v\u0001\nCZdv\u0001\nCZq]>which verifies:\nv\u0001\nCZ\u000b\f=T\u00001\n3!v\u0001\nCZ; (26)\nreplacing (26) into (24), multiplying by T3!and solving for the dynamics of v\u0001\nCZgives:\ndv\u0001\nCZ\ndt=1\nCarm\u001ah\n\td\tqi>\n\u0000CarmJ3!v\u0001\nCZ\u001b\n(27)\nwhere J3!is defined as in (28).\nJ3!def=2\n40\u00003!\n3! 03\n5: (28)\nEquation (27) will produce now a SSTI solution. The original oscillating zero-sequence component v\u0001\nCzcan\nalways be re-created as a function of v\u0001\nCZdandv\u0001\nCZqby means of (26), as:\nv\u0001\nCz=v\u0001\nCZdcos(3!t) +v\u0001\nCZqsin(3!t) (29)\n4) Final formulation: It is useful to redefine a new augmented vector for the SSTI voltage difference states\nv\u0001\nCdqZ (with capital Z), as:\nv\u0001\nCdqZdef=h\nv\u0001\nCdv\u0001\nCqv\u0001\nCZdv\u0001\nCZqi>\n;(30)12\nas well as for the “ \u0001” modulation indexes, as:\nm\u0001\ndqZdef=h\nm\u0001\ndm\u0001\nqm\u0001\nZdm\u0001\nZqi>\n:(31)\nWith the new definitions v\u0001\nCZd,v\u0001\nCZqand their associated dynamics given (27) as well as taking into account the\nmodified (sub-)matrix M\u00012\u00022\n\bCgiven in (20); the SSTP dynamics of v\u0001\nCdqz from (18) may be now expressed in\ntheir SSTI equivalents, by means of the 4\u00021state vector v\u0001\nCdqZ as shown in (32), with JGdefined in (33).\ndv\u0001\nCdqZ\ndt=\u0000JGv\u0001\nCdqZ +1\nCarm8\n>>>>>><\n>>>>>>:1\n82\n6666664\u0000\nm\u0006\nd+ 2m\u0006\nz\u0001\n\u0000m\u0006\nq\n\u0000m\u0006\nq\u0000\n\u0000m\u0006\nd+ 2m\u0006\nz\u0001\n+m\u0006\nd m\u0006\nq\n\u0000m\u0006\nq m\u0006\nd3\n7777775i\u0001\ndq+::: (32)\n\u0001\u0001\u0001+1\n42\n6666664+\u0000\nm\u0001\nd+m\u0001\nZd\u0001\n\u0000\u0010\nm\u0001\nq+m\u0001\nZq\u0011\nm\u0001\nd\n\u0000\u0010\nm\u0001\nq\u0000m\u0001\nZq\u0011\n\u0000\u0000\nm\u0001\nd\u0000m\u0001\nZd\u0001\nm\u0001\nq\nm\u0001\nd m\u0001\nq 2m\u0001\nZd\nm\u0001\nq \u0000m\u0001\nd 2m\u0001\nZq3\n7777775i\u0006\ndqz9\n>>>>>>=\n>>>>>>;:\nJGdef=2\n4J!02\u00022\n02\u00022J3!3\n5 (33)\nB. Voltage sum SSTI dynamics derivation\n1) Initial formulation: The SSTI dynamics for the voltage sum can be derived in a similar way as for the voltage\ndifference. The starting point is indeed the SSTP dynamics of the variable given in (9) and recalled in (34a) for\nconvenience. The first step consist in expressing the stationary frame abcvectors present in (34a) as functions\nof their respective dqz equivalents. This is done in (34b), where v\u0006\nCabc ,m\u0001\nabc,i\u0001\nabc,m\u0006\nabcandi\u0006\nabchave been\nrespectively replaced by P\u00001\n\u00002!v\u0006\nCdqz ,P\u00001\n!m\u0001\ndqz,P\u00001\n!i\u0001\ndqz,P\u00001\n\u00002!m\u0006\ndqz andP\u00001\n\u00002!i\u0006\ndqz. Notice that here too, the\nchoice of using the inverse Park transformation at !(P\u00001\n!) or at 2!(P\u00001\n\u00002!) is according to the frequency separation\nof the variables given in Table I and Fig. 3.\nEquation (34b) can be divided in three parts: \b\u0006\nA,\b\u0006\nBand\b\u0006\nC, as indicated in (34b). These three parts are\ntreated consecutively in the following.\n2Carmdv\u0006\nCabc\ndt=m\u0001\nabc\ni\u0001\nabc\n2+m\u0006\nabc\ni\u0006\nabc (34a)\n2CarmdP\u00001\n\u00002!\ndtv\u0006\nCdqz + 2CarmP\u00001\n\u00002!dv\u0006\nCdqz\ndt| {z }\n\b\u0006\nA=P\u00001\n!m\u0001\ndqz\nP\u00001\n!i\u0001\ndqz\n2|{z}\n\b\u0006\nB+P\u00001\n\u00002!m\u0006\ndqz\nP\u00001\n\u00002!i\u0006\ndqz|{z}\n\b\u0006\nC(34b)\nThe equation expressed in (34b), needs to be multiplied by Park’s transformation at \u00002!, so that it can be solved\nfordv\u0006\nCdqz=dt. Multiplying \b\u0006\nAbyP\u00002!gives (35), where J\u00002!is defined as 2J!.\nP\u00002!\b\u0006\nA= 2CarmJ\u00002!v\u0006\nCdqz + 2Carmdv\u0006\nCdqz\ndt(35)13\nFurthermore, multiplying \b\u0006\nBbyP\u00002!gives (36), where M\u0006\n\bBis expressed in (37).\nP\u00002!\b\u0006\nB=P\u00002! \nP\u00001\n!m\u0001\ndqz\nP\u00001\n!i\u0001\ndqz\n2!\n=M\u0006\n\bBh\ni\u0001\ndi\u0001\nqi\u0001\nzi>\n(36)\nAs mentioned earlier, it is assumed for simplicity in this work that there cannot be any zero-sequence grid current;\ni.e.,i\u0001\nz= 0 (highlighted in gray).\nM\u0006\n\bB=1\n42\n6664m\u0001\nd+ 2m\u0001\nzcos (3!t)\u0000m\u0001\nq+ 2m\u0001\nzsin (3!t) +2\u0000\nm\u0001\ndcos(3!t) +m\u0001\nqsin(3!t)\u0001\nm\u0001\nq+ 2m\u0001\nzsin (3!t) +m\u0001\nd\u00002m\u0001\nzcos (3!t)\u00002\u0000\nm\u0001\nqcos(3!t)\u0000m\u0001\ndsin(3!t)\u0001\nm\u0001\nd m\u0001\nq 2m\u0001\nz3\n7775(37)\nEquation (14) does not yet produce a SSTI solution, as the elements in the upper left 2\u00022sub-matrix of M\u0006\n\bB\nin (37), contain sine and cosine signals oscillating at 3!. Note that this is also the case for the terms highlighted\nin gray, but since these are being multiplied by i\u0001\nz= 0, they are not considered in this work. To overcome this\nobstacle, the same solution used in the previous section is applied: as all the oscillating terms are being multiplied\nbym\u0001\nz, it is convenient to redefine m\u0001\nzby a third harmonic injection as in (19), as a function of the SSTI virtual\nvariablesm\u0001\nZdandm\u0001\nZq. Replacing (19) into (37) allows for re-writing (36) as in (38).\nP\u00002!\b\u0006\nB=1\n42\n6664+\u0000\nm\u0001\nd+m\u0001\nZd\u0001\ni\u0001\nd\u0000\u0010\nm\u0001\nq\u0000m\u0001\nZq\u0011\ni\u0001\nq\n\u0000\u0010\nm\u0001\nq+m\u0001\nZq\u0011\ni\u0001\nd\u0000\u0000\nm\u0001\nd\u0000m\u0001\nZd\u0001\ni\u0001\nq\nm\u0001\ndi\u0001\nd+m\u0001\nqi\u0001\nq3\n7775+1\n42\n6664cos(6!t)\u0000sin(6!t) 0\n\u0000sin(6!t)\u0000cos(6!t) 0\n0 0 03\n77752\n6664+m\u0001\nZdi\u0001\nd\u0000m\u0001\nZqi\u0001\nq\n\u0000m\u0001\nZqi\u0001\nd\u0000m\u0001\nZdi\u0001\nq\n03\n7775\n| {z }\n\u00190\n(38)\nEquation (38) will become time-invariant only if it is assumed that the oscillatory signals at 6!can be neglected,\nwhich has been confirmed via time-domain simulations.\nIn a similar fashion, \b\u0006\nC; i.e., the second component on the right side of (34b), is multiplied by P\u00002!, resulting\nin (39), which can be considered SSTI if the sixth harmonic components are neglected. Here again, the validity of\nthe approximation was confirmed via time-domain simulations.\nP\u00002!\b\u0006\nC=1\n22\n66642m\u0006\nzi\u0006\nd+ 2m\u0006\ndi\u0006\nz\n2m\u0006\nzi\u0006\nq+ 2m\u0006\nqi\u0006\nz\nm\u0006\ndi\u0006\nd+m\u0006\nqi\u0006\nq+ 2m\u0006\nzi\u0006\nz3\n7775+1\n22\n6664\u0000\nm\u0006\ndi\u0006\nd\u0000m\u0006\nqi\u0006\nq\u0001\ncos(6!t) +\u0000\nm\u0006\nqi\u0006\nd+m\u0006\ndi\u0006\nq\u0001\nsin(6!t)\n\u0000\nm\u0006\ndi\u0006\nd\u0000m\u0006\nqi\u0006\nq\u0001\nsin(6!t)\u0000\u0000\nm\u0006\nqi\u0006\nd+m\u0006\ndi\u0006\nq\u0001\ncos(6!t)\n03\n7775\n| {z }\n\u00190(39)\n2) Final formulation : The SSTI dynamics of the voltage sum vector v\u0006\nCdqz are found by replacing the SSTI\nequations (35), (38) and (39) in (34b) and solving for dv\u0006\nCdqz=dt, resulting in (40).\ndv\u0006\nCdqz\ndt=\u0000J\u00002!v\u0006\nCdqz +::: (40)\n:::+1\nCarm8\n>>><\n>>>:1\n42\n66642m\u0006\nz 0 2m\u0006\nd\n0 2m\u0006\nz2m\u0006\nq\nm\u0006\ndm\u0006\nq2m\u0006\nz3\n7775i\u0006\ndqz+1\n82\n6664+\u0000\nm\u0001\nd+m\u0001\nZd\u0001\n\u0000\u0010\nm\u0001\nq\u0000m\u0001\nZq\u0011\n\u0000\u0010\nm\u0001\nq+m\u0001\nZq\u0011\n\u0000\u0000\nm\u0001\nd\u0000m\u0001\nZd\u0001\nm\u0001\nd m\u0001\nq3\n7775i\u0001\ndq9\n>>>=\n>>>;14\nC. Circulating current SSTI dynamics derivation\nThe SSTI dynamics for the circulating current are derived in the following. First, the equation of the dynamics\nin stationary frame given in (7) and recalled in (41a), is rewritten by expressing the abcvectors in the equation as\na function of their dqzequivalents, as indicated in (41b).\nLarmdi\u0006\nabc\ndt=vdc\n2\u0000v\u0006\nmabc\u0000Rarmi\u0006\nabc (41a)\nLarmdP\u00001\n\u00002!\ndti\u0006\ndqz+LarmP\u00001\n\u00002!di\u0006\ndqz\ndt=vdc\n2\u0000P\u00001\n\u00002!v\u0006\nmdqz\u0000RarmP\u00001\n\u00002!i\u0006\ndqz (41b)\nBy further multiplying (41b) by P\u00002!and solving for di\u0006\ndqz=dtgives:\nLarmdi\u0006\ndqz\ndt=h\n0 0vdc\n2i>\n\u0000v\u0006\nmdqz\u0000Rarmi\u0006\ndqz\u0000LarmJ\u00002!i\u0006\ndqz; (42)\nwhere v\u0006\nmdqz =P\u00001\n\u00002!v\u0006\nmabc , andv\u0006\nmabc is defined in (8). Nonetheless, in order to assess if (42) is SSTI, v\u0006\nmdqz\nneeds to be rewritten by expressing the abcvectors in the equation as a function of their dqz equivalents, as\nindicated in (43).\nv\u0006\nmdqz =1\n2P\u00002!\u0010\nP\u00001\n\u00002!m\u0006\ndqz\nP\u00001\n\u00002!v\u0006\nCdqz +P\u00001\n!m\u0001\ndqz\nP\u00001\n!v\u0001\nCdqz\u0011\n=M\u0006\n\tBh\nv\u0006\nCdv\u0006\nCqv\u0006\nCzi>\n+M\u0006\n\tCh\nv\u0001\nCdv\u0001\nCqv\u0001\nCzi>\n;(43)\nwhere M\u0006\n\tBandM\u0006\n\tCare expressed in (44) and (45), respectively.\nM\u0006\n\tB=1\n42\n66642m\u0006\nz 0 2m\u0006\nd\n0 2m\u0006\nz2m\u0006\nq\nm\u0006\ndm\u0006\nq2m\u0006\nz3\n7775+2\n66666664m\u0006\ndcos(6!t) +m\u0006\nqsin(6!t)\u0000m\u0006\nqcos(6!t) +m\u0006\ndsin(6!t) 0\n\u0000m\u0006\nqcos(6!t) +m\u0006\ndsin(6!t)\u0000m\u0006\ndcos(6!t)\u0000m\u0006\nqsin(6!t) 0\n0 0 0\n| {z }\n\u001903\n77777775\n(44)\nM\u0006\n\tC=1\n42\n6664m\u0001\nd+ 2m\u0001\nzcos(3!t)\u0000m\u0001\nq+ 2m\u0001\nzsin(3!t) 2m\u0001\ndcos(3!t) + 2m\u0001\nqsin(3!t)\nm\u0001\nq+ 2m\u0001\nzsin(3!t) +m\u0001\nd\u00002m\u0001\nzcos(3!t) 2m\u0001\ndsin(3!t)\u00002m\u0001\nqcos(3!t)\nm\u0001\nd m\u0001\nq 2m\u0001\nz3\n7775(45)\nIf the sixth harmonic components are neglected, M\u0006\n\tBgiven in (44) can be considered as SSTI. This is confirmed\nvia time-domain simulations. However, this is not the case for M\u0006\n\tCgiven in (45), as it presents non-negligible\nthird harmonic oscillations. To overcome this obstacle, it is necessary to replace into (44) and in (43) the new\ndefinitions of both m\u0001\nzandv\u0001\nCzgiven in (19) and (29), respectively. Doing so, results in the SSTI definition of\nv\u0006\nmdqz in (46), where M\u0006?\n\tCis given in (47) and is SSTI if the sixth harmonic components are neglected.\nv\u0006\nmdqz =M\u0006\n\tBh\nv\u0006\nCdv\u0006\nCqv\u0006\nCzi>\n+M\u0006?\n\tCh\nv\u0001\nCdv\u0001\nCqv\u0001\nCZdv\u0001\nCZqi>\n;(46)15\nM\u0006?\n\tC=1\n42\n6664m\u0001\nd+m\u0001\nZdm\u0001\nZq\u0000m\u0001\nqm\u0001\ndm\u0001\nq\nm\u0001\nq+m\u0001\nZqm\u0001\nd\u0000m\u0001\nZd\u0000m\u0001\nqm\u0001\nd\nm\u0001\ndm\u0001\nqm\u0001\nZdm\u0001\nZq3\n7775+::: (47)\n2\n66666664m\u0001\nZdcos(6!t) +m\u0001\nZqsin(6!t)m\u0001\nZdsin(6!t)\u0000m\u0001\nZqcos(6!t)m\u0001\ndcos(6!t) +m\u0001\nqsin(6!t)\u0000m\u0001\nqcos(6!t) +m\u0001\ndsin(6!t)\nm\u0001\nZdsin(6!t)\u0000m\u0001\nZqcos(6!t)\u0000m\u0001\nZdcos(6!t)\u0000m\u0001\nZqsin(6!t)m\u0001\ndsin(6!t)\u0000m\u0001\nqcos(6!t)\u0000m\u0001\ndcos(6!t)\u0000m\u0001\nqsin(6!t)\n0 0 m\u0001\nZdcos(6!t) +m\u0001\nZqsin(6!t)m\u0001\nZdsin(6!t)\u0000m\u0001\nZqcos(6!t)\n| {z }\n\u001903\n77777775\nFinally, replacing (44) and (47) in (46), and further in (42) gives the SSTI dynamics of the circulating current\n(48), provided the sixth harmonic components are neglected.\ndi\u0006\ndqz\ndt=1\nLarm8\n>>><\n>>>:2\n66640\n0\nvdc\n23\n7775\u0000Rarmi\u0006\ndqz\u00001\n42\n66642m\u0006\nz 0 2m\u0006\nd\n0 2m\u0006\nz2m\u0006\nq\nm\u0006\ndm\u0006\nq2m\u0006\nz3\n7775v\u0006\nCdqz +::: (48)\n:::\u00001\n42\n6664m\u0001\nd+m\u0001\nZdm\u0001\nZq\u0000m\u0001\nqm\u0001\ndm\u0001\nq\nm\u0001\nq+m\u0001\nZqm\u0001\nd\u0000m\u0001\nZd\u0000m\u0001\nqm\u0001\nd\nm\u0001\ndm\u0001\nqm\u0001\nZdm\u0001\nZq3\n7775v\u0001\nCdqZ9\n>>>=\n>>>;\u0000J\u00002!i\u0006\ndqz\nD. Grid currents SSTI dynamics derivation\nFinally, the derivation of SSTI expressions for the grid current dynamics are presented in the following. The\nbeginning of the proof is the SSTP dynamics equation of the grid current in the stationary reference frame given in\n(5)-(6), and recalled in (49a) for convenience. As for the previous cases, the dynamics are rewritten by expressing\ntheabcvectors present in (49a) as a function of their dqzequivalents, as indicated in (49b).\nLac\neqdi\u0001\nabc\ndt=v\u0001\nmabc\u0000vG\nabc\u0000Rac\neqi\u0001\nabc; (49a)\nLac\neqdP\u00001\n!\ndti\u0001\ndqz+Lac\neqP\u00001\n!di\u0001\ndqz\ndt=P\u00001\n!v\u0001\nmdqz\u0000P\u00001\n!vG\ndqz\u0000Rac\neqP\u00001\n!i\u0001\ndqz (49b)\nBy further multiplying (49b) by P!and solving for di\u0001\ndqz=dtgives (50).\nLac\neqdi\u0001\ndqz\ndt=v\u0001\nmdqz\u0000vG\ndqz\u0000Rac\neqi\u0001\ndqz\u0000Lac\neqJ!i\u0001\ndqz (50)\nwhere vG\ndqz= [vG\ndvG\nq0]>,v\u0001\nmdqzdef=P!v\u0001\nmabc andv\u0001\nmabc is defined in (6). Nonetheless, v\u0001\nmdqz needs to be\nassessed in order to verify if (50) produces a SSTI solution. For this purpuse, v\u0001\nmdqz is rewritten by expressing\ntheabcvectors in its definition as a function of their dqzequivalents, as indicated in (51a).\nv\u0001\nmdqz =\u0000P!1\n2\u0010\nP\u00001\n!m\u0001\ndqz\nP\u00001\n\u00002!v\u0006\nCdqz +P\u00001\n\u00002!m\u0006\ndqz\nP\u00001\n!v\u0001\nCdqz\u0011\n(51a)\nv\u0001\nmdqz =M\u0001\n\tBh\nv\u0006\nCdv\u0006\nCqv\u0006\nCzi>\n+M\u0001\n\tCh\nv\u0001\nCdv\u0001\nCqv\u0001\nCzi>\n(51b)\nwhere M\u0001\n\tBandM\u0001\n\tCare expressed in (52) and (53), respectively. Both these matrices present non-negligible\nthird order harmonic components preventing the possibility of considering SSTI solutions from (50) . As was done16\nin the previous section, it is necessary to replace into (52) and (53) and (51a) the new definitions of m\u0001\nzandv\u0001\nCz\ngiven in given in (19) and (29), respectively.\nM\u0001\n\tB=1\n42\n6664\u0000m\u0001\nd\u00002m\u0001\nzcos(3!t) \u0000m\u0001\nq\u00002m\u0001\nzsin(3!t) \u00002m\u0001\nd\n+m\u0001\nq\u00002m\u0001\nzsin(3!t) \u0000m\u0001\nd+ 2m\u0001\nzcos(3!t) \u00002m\u0001\nq\n\u0000m\u0001\ndcos(3!t)\u0000m\u0001\nqsin(3!t)\u0000m\u0001\ndsin(3!t) +m\u0001\nqcos(3!t)\u00002m\u0001\nz3\n7775(52)\nM\u0001\n\tC=1\n42\n6664\u0000m\u0006\nd\u00002m\u0006\nz \u0000m\u0006\nq \u00002m\u0006\ndcos(3!t)\u00002m\u0006\nqsin(3!t)\n\u0000m\u0006\nq \u00002m\u0006\nz+m\u0006\nd +2m\u0006\nqcos(3!t)\u00002m\u0006\ndsin(3!t)\n\u0000m\u0006\nqsin(3!t)\u0000m\u0006\ndcos(3!t)m\u0006\nqcos(3!t)\u0000m\u0006\ndsin(3!t) \u00002m\u0006\nz3\n7775\n(53)\nBy doing so, the (reduced) definition of the modulation voltage v\u0001\nmdq can be expressed as in (54), where M\u0001?\n\tB\nandM\u0001?\n\tCare given in (55) and (56), and will result in SSTI solutions if the sixth harmonic are neglected.\nv\u0001\nmdq =M\u0001?\n\tBh\nv\u0006\nCdv\u0006\nCqv\u0006\nCzi>\n+M\u0001?\n\tCh\nv\u0001\nCdv\u0001\nCqv\u0001\nCZdv\u0001\nCZqi>\n(54)\nM\u0001?\n\tB=1\n42\n4\u0000m\u0001\nd\u0000m\u0001\nZd\u0000m\u0001\nq\u0000m\u0001\nZq\u00002m\u0001\nd\nm\u0001\nq\u0000m\u0001\nZq\u0000m\u0001\nd+m\u0001\nZd\u00002m\u0001\nq3\n5+::: (55)\n:::+2\n4\u0000m\u0001\nZdcos(6!t)\u0000m\u0001\nZqsin(6!t)\u0000m\u0001\nZdsin(6!t) +m\u0001\nZqcos(6!t) 0\nm\u0001\nZqcos(6!t)\u0000m\u0001\nZdsin(6!t)m\u0001\nZqsin(6!t) +m\u0001\nZdcos(6!t) 03\n5\n| {z }\n\u00190\nM\u0001?\n\tC=1\n42\n4\u0000m\u0006\nd\u00002m\u0006\nz \u0000m\u0006\nq \u0000m\u0006\nd\u0000m\u0006\nq\n\u0000m\u0006\nq \u00002m\u0006\nz+m\u0006\ndm\u0006\nq\u0000m\u0006\nd3\n5+::: (56)\n:::+2\n40 0 \u0000m\u0006\nqsin(6!t)\u0000m\u0006\ndcos(6!t)m\u0006\nqcos(6!t)\u0000m\u0006\ndsin(6!t)\n0 0m\u0006\nqcos(6!t)\u0000m\u0006\ndsin(6!t)m\u0006\nqsin(6!t) +m\u0006\ndcos(6!t)3\n5\n| {z }\n\u00190\nFinally, replacing (55) and (56) in (54) and further in (50) gives the SSTI dynamics of the grid current (57),\nprovided the sixth harmonic components are neglected.\ndi\u0001\ndq\ndt=1\nLaceq8\n<\n:\u0000vG\ndq\u0000Rac\neqi\u0001\ndq+2\n4\u0000m\u0001\nd\u0000m\u0001\nZd\u0000m\u0001\nq\u0000m\u0001\nZq\u00002m\u0001\nd\nm\u0001\nq\u0000m\u0001\nZq\u0000m\u0001\nd+m\u0001\nZd\u00002m\u0001\nq3\n5v\u0006\nCdqz +::: (57)\n:::+2\n4\u0000m\u0006\nd\u00002m\u0006\nz \u0000m\u0006\nq \u0000m\u0006\nd\u0000m\u0006\nq\n\u0000m\u0006\nq \u00002m\u0006\nz+m\u0006\ndm\u0006\nq\u0000m\u0006\nd3\n5v\u0001\nCdqZ9\n=\n;\u0000J!i\u0001\ndq\nE. MMC Model with SSTI Solution Summary\nTo summarize, the MMC SSTI dynamics can be represented by means of equations (30), (40), (48) and (57),\ncorresponding to the 12 SSTI state variables of the arm voltages difference v\u0001\nCdqZ , arm voltages sum v\u0006\nCdqz ,\ncirculating currents i\u0006\ndqzand grid currents i\u0001\ndq. Moreover, this model accepts 7 SSTI control inputs represented\nby the sum and difference of the modulation indices m\u0006\ndqzandm\u0001\ndqZ. In addition, the model receives 3 physical17\nGrid currents\nvG\ndq\nvdc\ni\u0001\ndq\ni\u0006\ndqz\nCalculations\nCalculations\nCalculations\nv\u0001\nmdq\nv\u0006\nmdqz\nv\u0006\nmdqz\nv\u0001\nmdq\nv\u0006\nCdqz\nv\u0006\nCdqz\nCalculations\nv\u0001\nCdqZ\nm\u0006\ndqz\nm\u0001\ndqZ\nv\u0006\nCdqz\nv\u0006\nCdqz\nInputs\nStates\nCirculating\ncurrents\ni\u0001\ndq\ni\u0006\ndqz\nv\u0001\nCdqZ\nv\u0001\nCdqZ\nv\u0001\nCdqZ\nAC Source\ni\u0001\ndq\nvG\ndq\nDC Source\nvdc\n3i\u0006\nz\nAlgebraic equations\nDi\u000berential equations\nSource equations\nInternal variables\nFigure 4: Summary of the MMC equations in dqzframe\nSSTI inputs represented by the voltage at the dc terminals vdcand thedqcomponents of the grid voltage, vG\ndq.\nFinally, the proposed MMC model with SSTI solution is graphically represented in Fig. 4.\nIV. M ODEL VALIDATION BY TIME -DOMAIN SIMULATION\nTo validate the developed modelling approach, results from time-domain simulation of the following three different\nmodels will be shown and discussed in this section.\n1) The proposed time-invariant MMC model derived in section III and represented by equations (30), (40), (48)\nand (57), corresponding to the SSTI dynamics of the arm voltages difference, arm voltages sum, circulating\ncurrents and grid currents. Simulations result obtained with this model are identified in the legend by a ?\nsymbol as a superscript for each variable.\n2) The AAM of a three-phase MMC, where each arm is represented by a controlled voltage source and where the\ninternal arm voltage dynamics is represented by an equivalent arm capacitance as indicated in the lower right\npart of Fig. (2) [4], [5], [29]. This model includes non-linear effects except for the switching operations and\nthe dynamics of the sub-module capacitor voltage balancing algorithm, as indicated in Fig. 1. Since this model\nis well-established for analysis and simulation of MMCs and has been previously verified in comparison to\nexperimental results [4], [5], it will be used as a benchmark reference for verifying the validity of the derived\nmodel with SSTI solution. The model is simulated in Matlab/Simulink with the SimPowerSystem toolbox.\nSimulation results obtained with this model are identified in the legend by “ AAM ”.\n3) The system from Fig. 2 implemented in EMTP-RV for an MMC with 400 sub-modules per arm, with a\ncapacitance of 0:01302Feach. The MMC is modeled with the so-called “Model # 2: Equivalent Circuit-Based\nModel ” from [27]. This model includes non-linear effects and the switching operations and the dynamics of18\n!Lac\neq\n!Lac\neq\nvG\nd\nvG\nq\ni\u0001\u0003\nd\ni\u0001\u0003\nq\ni\u0001\nd\ni\u0001\nq\n+\n\u0000\n\u0000\n+\nv\u0001\u0003\nmd\nv\u0001\u0003\nmq\n+\n+\n\u0000\n+\n+\n+\nPIi\u0001\nPIi\u0001\nm\u0001\ndqz\nm\u0006\ndqz\nGrid currents control\ni\u0006\nd\n+\n\u0000\n\u0000\n+\nv\u0006\u0003\nmd\nv\u0006\u0003\nmq\nCCSC\ni\u0006\nq\ni\u0006\u0003\nd= 0\ni\u0006\u0003\nq= 0\n2!Larm\n2!Larm\n\u0000\n+\n\u0000\n\u0000\nv\u0006\u0003\nmz= 1\nPIi\u0006\nPIi\u0006\n+\n\u0000\nKp\n1\nTis\nRef.\nMeas.\n+\nOut\nGeneric PI strutcure\n\u0018: State of the integral part\n\u0018\n1\nvdc\nv\u0001\u0003\nmz= 0\nFigure 5: Circulating Current Suppression Control (CCSC) and standard SRRF grid current vector control\nthe sub-module capacitor voltage balancing algorithm from [22], as indicated in Fig. 1. Simulation results\nobtained with this model are identified in the legend by “ EMT ”.\nIt is worth mentioning that the verification of the scientific contribution represented by the proposed modelling\napproach should be done first and foremost with respect to the model it has been derived from; i.e., the AAM.\nThis initial comparison, where the AAM is considered as the reference model, is enough to evaluate the accuracy\nof the modelling proposal and the simplifications it entails. Thus, the analysis of simulation results that will follow\nis mainly focused on these two modelling approaches. Nonetheless, for a more practical-oriented comparison, the\ndetailed switching model has been included as well, to provide an indication to the reader on the accuracy of both\nthe well-established AAM and the proposed modelling approach with respect to a detailed switching model of the\nMMC.\nAll simulations are based on the MMC HVDC single-terminal configuration shown in Fig. 2, with the parameters\ngiven in Table II under the well known Circulating Current Suppression Control (CCSC) technique described in\n[22], and with standard SRRF vector control for the grid current, similarly to what was presented in [20], and\nshown in Fig. 5. For comparing the models, it should be considered that the reference model is a conventional\ntime-domain simulation model of a three-phase MMC, while the derived model with SSTI solution represents the\nMMC dynamics by variables transformed into a set of SRRFs. Nonetheless, comparison of transient and steady-state\nresponse is simpler when the variables have SSTI representation. Thus, in most cases, the results obtained from the\nreference model are transformed into the appropriate SRRFs to ease the comparison. However, the results from the\nmodels with SSTI solution can also be transformed to the stationary phase coordinates, although this would imply19\n  \nEMT AAM vΣ⋆\nCq vΣ⋆\nCdvΣ\nCdq[pu]\nTime [s]0 0 .05 0.1 0.15 0 .2 0.25 0.3 0 .35 0 .4 0 .45 0.5−0.0100.010.02\n  \nEMT AAM vΣ⋆\nCzvΣ\nCz[pu]\nTime [s]0 0.05 0.1 0 .15 0.2 0.25 0 .3 0 .35 0 .4 0.45 0.50.9811.02\nFigure 6: V oltage Sum\ncomparison of signals with sinusoidal or multi-frequency oscillations in steady-state. All results are plotted in per\nunit quantities.\nTo excite the MMC dynamics in the different models, first the reactive power reference is set from zero to \u00000:1\np.u.att= 0:05s. Second, the ac-side active power reference is reduced from 1p.u.to0:5p.u.att= 0:15s.\nThe dynamics of the voltage sum v\u0006\nCdqz for the above described case scenario are illustrated in Fig. 6. More\nprecisely, the dqcomponents of this variable is given in the upper sub-figure while its zero-sequence is shown in the\nlower one, due to the differences in scale between them. From Fig. 6, it can be seen how the variables calculated\nwith the AAM-MMC used as reference are overlapping those calculated with the model with SSTI solution derived\nin this paper. This is true for both transient and steady-state conditions. Notice that the steady-state value of v\u0006\nCz\nchanges with respect to each of the reference steps, as only the CCSC is implemented assuming no regulation of\nthe capacitive energy stored in the MMC. Furthermore, the non-zero steady-state values of v\u0006\nCdq reflect the 2!\noscillations that this variable has in the stationary abcreference frame.\nSimilarly, the dynamics of the energy difference v\u0001\nCdqz are depicted in Fig. 7. More precisely, the upper figure is\nillustrating the dqcomponents behaviour of this variable under the above described case scenario while the lower\nfigure does the same for the zero-sequence. In terms of accuracy, both of the sub-figures show how the proposed\nmodel with SSTI solution accurately captures the behaviour of the AAM-MMC model used as reference. This is\nparticularly true for the case of v\u0001\nCdq as almost no distinction can be made between the voltage waveforms resulting20\n  \nEMT AAM v∆⋆\nCq v∆⋆\nCdv∆\nCdq[pu]\nTime [s]0 0 .05 0.1 0.15 0 .2 0.25 0.3 0 .35 0 .4 0 .45 0.5−0.0500.05\n  \nEMT AAM v∆⋆\nCzv∆\nCz·10−3[pu]\nTime [s]0 0.05 0.1 0 .15 0.2 0.25 0 .3 0 .35 0 .4 0.45 0.5−202\nFigure 7: V oltage difference\n  \nv∆⋆\nCZq v∆⋆\nCZdv∆\nCZ·10−3[pu]\nTime [s]0 0.05 0.1 0 .15 0.2 0.25 0 .3 0 .35 0 .4 0.45 0.5−202\nFigure 8: SSTI representation of the voltage difference zero sequence\nfrom the two models. For v\u0001\nCzhowever, it is possible to notice a slight mismatch between the derived model and\nthe AAM, particularly during the transient behaviour between t= 0:15sandt= 0:2s. This is indeed associated to\nthe neglected sixth harmonic components in the mathematical derivation of the proposed model with SSTI solution.\nNonetheless, the error is very small and is not having noticeable influence on the general dynamics of the model.\nNotice that the comparison between the reference and the proposed MMC model with SSTI solution has been\ndone using the SSTP signal v\u0001\nCzinstead of its equivalent SSTI version v\u0001\nCZdefined in section III. This is done\nfor simplicity, as the dynamics of the virtual system used to create v\u0001\nCZdo not directly exist in the reference21\n  \nEMT AAM iΣ⋆\nq iΣ⋆\ndiΣdq[pu]\nTime [s]0 0.05 0.1 0 .15 0.2 0.25 0 .3 0 .35 0 .4 0.45 0.5−0.0500.05\n  \nEMT AAM 3·iΣ\nz3·iΣz[pu]\nTime [s]0 0 .05 0 .1 0.15 0.2 0 .25 0.3 0.35 0 .4 0 .45 0 .5−0.500.511.5\nFigure 9: Circulating current\nAAM-MMC model. However, for the sake of completeness, the dynamics of the SSTI v\u0001\nCZobtained with the\nproposed model are depicted in Fig. 8, where it can be confirmed that both the v\u0001\nCZdandv\u0001\nCZqsub-variables reach\na constant value in steady-state operation.\nThe dynamics of the circulating currents i\u0006\ndqz are shown in Fig. 9, where the upper sub-figure depicts the\ndynamics of the dqcomponents while the lower figure shows the zero-sequence components multiplied by three,\nsince this signal corresponds to the dc current idcflowing into the dc terminals of the MMC. From the figure it can\nbe also concluded that the proposed model with SSTI solution replicates quite accurately the dynamic behaviour of\nthe reference model. It can be noticed that the accuracy of the model is very good for the zero-sequence component\ni\u0006\nz. However, for the dqcomponent, the 6th order harmonics have been neglected in the modelling. Although these\ncomponents are very small, they are still present in the reference model, and can be noticed in the figure. Still,\nthe proposed model captures most of the current dynamics, and is accurately representing the average value of the\ncurrent components as shown in the zoom of the steady-state operation of i\u0006\ndqdepicted in Fig. 10. In this figure, it\nis possible to see that the reference AAM-MMC model still presents its sixth harmonic components in both the d\nandqcomponents of the circulating current, whereas the same variables calculated with the proposed MMC model\nwith SSTI solution only capture the average behaviour. However, given the small value of these oscillations (notice\nthe scale) it can still be considered that the presented modelling approach is sufficiently accurate for most purposes.\nFinally, the dynamics of the dqcomponents of the grid current are shown in Fig. 11. It is possible to see that\nfor this variable the reference model and the proposed model with SSTI dynamics are practically overlapping.22\n  \niΣ\nqEMTiΣ\ndEMTiΣ\nqAAMiΣ\ndAAMiΣ⋆\nq iΣ⋆\ndiΣdq·10−3[pu]\nTime [s]0 0.002 0.004 0 .006 0.008 0.01 0 .012 0 .014 0 .016 0.018 0.02−505\nFigure 10: Circulating current zoom\n  \nEMT AAM i∆⋆\nq i∆⋆\ndi∆dq[pu]\nTime [s]0 0.05 0.1 0 .15 0.2 0.25 0 .3 0 .35 0 .4 0.45 0.500.51\nFigure 11: Grid Current\nV. C ONCLUSION\nThis paper presents a modelling approach for obtaining a state-space representation of an MMC with Steady-State\nTime-Invariant (SSTI) solution. The presented approach can be considered independent from the modulation and\ncontrol strategy adopted, as only the physical equations of the MMC have been mathematically manipulated, gaining\na more generalized model compared to previous efforts. Results from time-domain simulation of a detailed MMC\nmodel with 400 sub-modules per arm are presented as point of reference to illustrate the validity of the derived\nmodel. These results demonstrate how the state-space model with SSTI solution accurately captures the MMC\ninternal dynamics while imposing that all state variables settle to a constant equilibrium in steady-state operation.\nThis was achieved by a voltage-current \u0006-\u0001formulation which enabled separation of the MMC variables according\nto their oscillation frequencies as part of the initial model formulation. A procedure for deriving equivalent SSTI\ndqzrepresentation of all state variables by applying three different Park transformations was presented, referring the\nvariables to three different rotating reference frames, rotating at once, twice and three times the grid fundamental\nfrequency. The resulting model can be suited for detail-oriented studies of MMC control strategies, as it captures\nthe dynamics of the second harmonic circulating currents and the internal energy dynamics of the MMC.23\nUtilization of the presented model can enable a wide range of studies related to analysis and control system design\nfor MMCs. Since the derived model can be linearised, it can also be utilized for studies of multi-variable control\ntechniques and optimization methods. Furthermore, the model can be utilized for small-signal stability studies by\neigenvalue analysis, considering an individual MMC HVDC terminal, or an HVDC terminal integrated in a larger\npower system configuration. Since the developed MMC model with SSTI dynamics preserves the mathematical\ninformation about system non-linearities, it is also suited for application of techniques for multi-variable non-linear\nanalysis and control. Moreover, the presented derivations can be useful for developing small-signal models of MMC\ncontrol systems implemented for operation with individual phase- or arm-quantities.\nAPPENDIX\nThe main parameters are listed in Table II.\nTable II: Nominal Values & Parameters\nU1n 320[kV]Rf 0:512[\n]\u001ci\u00010:0019 [ms]\nfn 50[Hz]Lf58:7[mH]\u001ci\u00060:0149 [ms]\nN 400[-]Rarm 1:024[\n]kp\u0006 0:1253 [pu]\nCarm 32:55[\u0016F]Larm 48:9[mH]kp\u0001 0:8523 [pu]\nREFERENCES\n[1] A. Lesnicar and R. Marquardt, “An innovative modular multilevel converter topology suitable for a wide power range,” in Power Tech\nConference Proceedings, 2003 IEEE Bologna , vol. 3, June 2003, pp. 6 pp. V ol.3–.\n[2] R. Adapa, “High-wire act: Hvdc technology: The state of the art,” IEEE Power and Energy Magazine , vol. 10, no. 6, pp. 18–29, Nov\n2012.\n[3] A. Antonopoulos, L. Angquist, and H. P. Nee, “On dynamics and voltage control of the modular multilevel converter,” in 2009 13th\nEuropean Conference on Power Electronics and Applications , Sept 2009, pp. 1–10.\n[4] L. Harnefors, A. Antonopoulos, S. Norrga, L. Angquist, and H. P. Nee, “Dynamic analysis of modular multilevel converters,” IEEE\nTransactions on Industrial Electronics , vol. 60, no. 7, pp. 2526–2537, July 2013.\n[5] S. Rohner, J. Weber, and S. Bernet, “Continuous model of modular multilevel converter with experimental verification,” in 2011 IEEE\nEnergy Conversion Congress and Exposition , Sept 2011, pp. 4021–4028.\n[6] P. Delarue, F. Gruson, and X. Guillaud, “Energetic macroscopic representation and inversion based control of a modular multilevel\nconverter,” in Power Electronics and Applications (EPE), 2013 15th European Conference on , Sept 2013, pp. 1–10.\n[7] H. Saad, X. Guillaud, J. Mahseredjian, S. Dennetière, and S. Nguefeu, “Mmc capacitor voltage decoupling and balancing controls,” IEEE\nTransactions on Power Delivery , vol. 30, no. 2, pp. 704–712, April 2015.\n[8] K. Ilves, A. Antonopoulos, S. Norrga, and H.-P. Nee, “Steady-state analysis of interaction between harmonic components of arm and line\nquantities of modular multilevel converters,” IEEE Transactions on Power Electronics , vol. 27, no. 1, pp. 57–68, January 2012.\n[9] N. R. Chaudhuri, R. Majumder, B. Chaudhuri, and J. Pan, “Stability analysis of vsc mtdc grids connected to multimachine ac systems,”\nIEEE Transactions on Power Delivery , vol. 26, no. 4, pp. 2274–2784, October 2011.\n[10] G. O. Kalcon, G. P. Adam, O. Anaya-Lara, S. Lo, and K. Uhlen, “Small-signal stability analysis of multi-terminal vsc-based dc transmission\nsystems,” IEEE Transactions on Power Systems , vol. 27, no. 4, pp. 1818–1830, Nov 2012.\n[11] J. Beerten, S. D’Arco, and J. A. Suul, “Identification and small-signal analysis of interaction modes in vsc mtdc systems,” IEEE Transactions\non Power Delivery , vol. 31, no. 2, pp. 888–897, April 2016.24\n[12] S. Liu, Z. Xu, W. Hua, G. Tang, and Y . Xue, “Electromechanical transient modeling of modular multilevel converter based multi-terminal\nhvdc systems,” IEEE Transactions on Power Systems , vol. 29, no. 1, pp. 72–83, Jan 2014.\n[13] D. C. Ludois and G. Venkataramanan, “Simplified terminal behavioral model for a modular multilevel converter,” IEEE Transactions on\nPower Electronics , vol. 29, no. 4, pp. 1622–1631, April 2014.\n[14] N. T. Trinh, M. Zeller, K. Wuerflinger, and I. Erlich, “Generic model of mmc-vsc-hvdc for interaction study with ac power system,” IEEE\nTransactions on Power Systems , vol. 31, no. 1, pp. 27–34, Jan 2016.\n[15] G. B. Diaz, J. A. Suul, and S. D’Arco, “Small-signal state-space modeling of modular multilevel converters for system stability analysis,”\nin2015 IEEE Energy Conversion Congress and Exposition (ECCE) , Sept 2015, pp. 5822–5829.\n[16] J. Freytes, L. Papangelis, H. Saad, P. Rault, T. V . Cutsem, and X. Guillaud, “On the modeling of mmc for use in large scale dynamic\nsimulations,” in 2016 Power Systems Computation Conference (PSCC) , June 2016, pp. 1–7.\n[17] A. a. J. Far and D. Jovcic, “Circulating current suppression control dynamics and impact on mmc converter dynamics,” in 2015 IEEE\nEindhoven PowerTech , June 2015, pp. 1–6.\n[18] A. Jamshidifar and D. Jovcic, “Small-signal dynamic dq model of modular multilevel converter for system studies,” IEEE Transactions\non Power Delivery , vol. 31, no. 1, pp. 191–199, Feb 2016.\n[19] V . Najmi, M. N. Nazir, and R. Burgos, “A new modeling approach for modular multilevel converter (mmc) in d-q frame,” in 2015 IEEE\nApplied Power Electronics Conference and Exposition (APEC) , March 2015, pp. 2710–2717.\n[20] G. Bergna, J. A. Suul, and S. D’Arco, “State-space modelling of modular multilevel converters for constant variables in steady-state,” in\n2016 IEEE 17th Workshop on Control and Modeling for Power Electronics (COMPEL) , June 2016, pp. 1–9.\n[21] T. Li, A. M. Gole, and C. Zhao, “Harmonic instability in mmc-hvdc converters resulting from internal dynamics,” IEEE Transactions on\nPower Delivery , vol. 31, no. 4, pp. 1738–1747, Aug 2016.\n[22] Q. Tu, Z. Xu, and L. Xu, “Reduced switching-frequency modulation and circulating current suppression for modular multilevel converters,”\ninTransmission and Distribution Conference and Exposition (T D), 2012 IEEE PES , May 2012, pp. 1–1.\n[23] M. Perez, R. Ortega, and J. Espinoza, “Passivity-based pi control of switched power converters,” in 2003 European Control Conference\n(ECC) , Sept 2003, pp. 542–547.\n[24] M. Hernandez-Gomez, R. Ortega, F. Lamnabhi-Lagarrigue, and G. Escobar, “Adaptive pi stabilization of switched power converters,” IEEE\nTransactions on Control Systems Technology , vol. 18, no. 3, pp. 688–698, May 2010.\n[25] J. A. Houldsworth and D. A. Grant, “The use of harmonic distortion to increase the output voltage of a three-phase pwm inverter,” IEEE\nTransactions on Industry Applications , vol. IA-20, no. 5, pp. 1224–1228, Sept 1984.\n[26] S. Norrga, L. Ängquist, and K. Ilves, “Operating region extension for multilevel converters in hvdc applications by optimisation methods,”\nin10th IET International Conference on AC and DC Power Transmission (ACDC 2012) , Dec 2012, pp. 1–6.\n[27] H. Saad, S. Dennetiére, J. Mahseredjian, P. Delarue, X. Guillaud, J. Peralta, and S. Nguefeu, “Modular multilevel converter models for\nelectromagnetic transients,” IEEE Transactions on Power Delivery , vol. 29, no. 3, pp. 1481–1489, June 2014.\n[28] R. Teodorescu, M. Liserre, and P. Rodríguez, Grid converters for photovoltaic and wind power systems . IEEE/John Wiley & Sons, 2011,\nvol. 29.\n[29] A. Christe and D. Duji ´c, “State-space modeling of modular multilevel converters including line frequency transformer,” in Power Electronics\nand Applications (EPE’15 ECCE-Europe), 2015 17th European Conference on , Sept 2015, pp. 1–10."    },    {        "title": "1706.08488v1.Perpendicular_magnetic_anisotropy_in_insulating_ferrimagnetic_gadolinium_iron_garnet_thin_films.pdf",        "content": "Perpendicular magnetic anisotropy in insulating ferrimagnetic gadolinium iron garnet\nthin \flms\nH. Maier-Flaig,1, 2S. Gepr ags,1Z. Qiu,3, 4E. Saitoh,3, 4, 5, 6, 7R. Gross,1, 2, 8\nM. Weiler,1, 2H. Huebl,1, 2, 8and S. T. B. Goennenwein1, 2, 8, 9, 10\n1Walther-Mei\u0019ner-Institut, Bayerische Akademie der Wissenschaften, Garching, Germany\n2Physik-Department, Technische Universit at M unchen, Garching, Germany\n3WPI Advanced Institute for Materials Research, Tohoku University, Sendai, Japan\n4Spin Quantum Recti\fcation Project, ERATO, Japan Science and Technology Agency, Sendai, Japan\n5Institute for Materials Research, Tohoku University, Sendai, Japan\n6PRESTO, Japan Science and Technology Agency, Saitama, Japan\n7Advanced Science Research Center, Japan Atomic Energy Agency, Tokai, Japan\n8Nanosystems Initiative Munich, M unchen, Germany\n9Institut f ur Festk oper- und Materialphysik, Technische Universit at Dresden, Dresden, Germany\n10Center for Transport and Devices of Emergent Materials, Technische Universit at Dresden, 01062 Dresden\n(Dated: June 27, 2017)\nWe present experimental control of the magnetic anisotropy in a gadolinium iron garnet (GdIG)\nthin \flm from in-plane to perpendicular anisotropy by simply changing the sample temperature.\nThe magnetic hysteresis loops obtained by SQUID magnetometry measurements unambiguously\nreveal a change of the magnetically easy axis from out-of-plane to in-plane depending on the sam-\nple temperature. Additionally, we con\frm these \fndings by the use of temperature dependent\nbroadband ferromagnetic resonance spectroscopy (FMR). In order to determine the e\u000bective mag-\nnetization, we utilize the intrinsic advantage of FMR spectroscopy which allows to determine the\nmagnetic anisotropy independent of the paramagnetic substrate, while magnetometry determines\nthe combined magnetic moment from \flm and substrate. This enables us to quantitatively evalu-\nate the anisotropy and the smooth transition from in-plane to perpendicular magnetic anisotropy.\nFurthermore, we derive the temperature dependent g-factor and the Gilbert damping of the GdIG\nthin \flm.\nControlling the magnetization direction of magnetic\nsystems without the need to switch an external static\nmagnetic \feld is a challenge that has seen tremendous\nprogress in the past years. It is of considerable interest\nfor applications as it is a key prerequisite to store infor-\nmation in magnetic media in a fast, reliable and energy\ne\u000ecient way. Two notable approaches to achieve this in\nthin magnetic \flms are switching the magnetization by\nshort laser pulses[1, 2] and switching the magnetization\nvia spin orbit torques[3{5]. For both methods, materials\nwith an easy magnetic anisotropy axis oriented perpen-\ndicular to the \flm plane are of particular interest. While\nall-optical switching requires a magnetization component\nperpendicular to the \flm plane in order to transfer angu-\nlar momentum[2], spin orbit torque switching with per-\npendicularly polarized materials allows fast and reliable\noperation at low current densities[3]. Therefore great ef-\nforts have been undertaken to achieve magnetic thin \flms\nwith perpendicular magnetic anisotropy.[6] However, re-\nsearch has mainly been focused on conducting ferromag-\nnets that are subject to eddy current losses and thus of-\nten feature large magnetization damping. Magnetic gar-\nnets are a class of highly tailorable magnetic insulators\nthat have been under investigation and in use in appli-\ncations for the past six decades.[7{9] The deposition of\ngarnet thin \flms using sputtering, pulsed laser deposition\nor liquid phase epitaxy, and their properties are very well\nunderstood. In particular, doping the parent compound\n(yttrium iron garnet, YIG) with rare earth elements is apowerful means to tune the static and dynamic magnetic\nproperties of these materials.[7, 10, 11]\nHere, we study the magnetic properties of a gadolin-\nium iron garnet thin \flm sample using broadband fer-\nromagnetic resonance (FMR) and SQUID magnetome-\ntry. By changing the temperature, we achieve a transi-\ntion from the typical in-plane magnetic anisotropy (IPA),\ndominated by the magnetic shape anisotropy, to a per-\npendicular magnetic anisotropy (PMA) at about 190 K.\nWe furthermore report the magnetodynamic properties\nof GdIG con\frming and extending previous results.[7]\nI. MATERIAL AND SAMPLE DETAILS\nWe investigate a 2 :6µm thick gadolinium iron gar-\nnet (Gd 3Fe5O3, GdIG) \flm grown by liquid phase epi-\ntaxy (LPE) on a (111)-oriented gadolinium gallium gar-\nnet substrate (GGG). The sample is identical to the\none used in Ref. 12 and is described there in detail.\nGdIG is a compensating ferrimagnet composed of two\ne\u000bective magnetic sublattices: The magnetic sublattice\nof the Gd ions and an e\u000bective sublattice of the two\nstrongly antiferromagnetically coupled Fe sublattices.\nThe magnetization of the coupled Fe sublattices shows a\nweak temperature dependence below room temperature\nand decreases from approximately 190 kA m\u00001at 5 K to\n140 kA m\u00001at 300 K.[8] The Gd sublattice magnetization\nfollows a Brillouin-like function and decreases drasticallyarXiv:1706.08488v1  [cond-mat.mtrl-sci]  26 Jun 20172\nfrom approximately 800 kA m\u00001at 5 K to 120 kA m\u00001at\n300 K.[8] As the Gd and the net Fe sublattice magneti-\nzations are aligned anti-parallel, the remanent magneti-\nzations cancel each other at the so-called compensation\ntemperature Tcomp = 285 K of the material.[13] Hence,\nthe remanent net magnetization Mof GdIG vanishes at\nTcomp.\nThe typical magnetic anisotropies in thin garnet \flms\nare the shape anisotropy and the cubic magnetocrys-\ntalline anisotropy, but also growth induced anisotropies\nand magnetoelastic e\u000bects due to epitaxial strain have\nbeen reported in literature.[14, 15] We \fnd that our ex-\nperimental data can be understood by taking into ac-\ncount only shape anisotropy and an additional anisotropy\n\feld perpendicular to the \flm plane. A full determina-\ntion of the anisotropy contributions is in principle pos-\nsible with FMR. Angle dependent FMR measurements\n(not shown) indicate an anisotropy of cubic symmetry\nwith the easy axis along the crystal [111] direction in\nagreement with literature.[16] The measurements sug-\ngest that the origin of the additional anisotropy \feld\nperpendicular to the \flm plane is the cubic magnetocrys-\ntalline anisotropy. However, the low signal amplitude and\nthe large FMR linewidth towards Tcomp in combination\nwith a small misalignment of the sample, render a com-\nplete, temperature dependent anisotropy analysis impos-\nsible. In the following, we therefore focus only on shape\nanisotropy and the additional out-of-plane anisotropy\n\feld.\nII. SQUID MAGNETOMETRY\nSQUID magnetometry measures the projection of the\nmagnetic moment of a sample on the applied magnetic\n\feld direction. For thin magnetic \flms, however, the\nbackground signal from the comparatively thick sub-\nstrate can be on the order of or even exceed the magnetic\nmomentmof the thin \flm and hereby impede the quan-\ntitative determination of m. Our 2:6µm thick GdIG \flm\nis grown on a 500 µm thick GGG substrate warranting a\ncareful subtraction of the paramagnetic background sig-\nnal of the substrate. In our experiments, H0is applied\nperpendicular to the \flm plane and thus, the projection\nof the net magnetization M=m=Vto the out-of-plane\naxis is recorded as M?. Fig. 1 shows M?of the GdIG \flm\nas function of the externally applied magnetic \feld H0.\nIn the investigated small region of H0, the magnetization\nof the paramagnetic substrate can be approximated by\na linear background that has been subtracted from the\ndata. The two magnetic hysteresis loops shown in Fig. 1\nare typical for low temperatures ( T.170 K) and for\ntemperatures close to Tcomp. The hysteresis loops unam-\nbiguously evidence hard and easy axis behavior, respec-\ntively. Towards low temperatures ( T= 170 K, Fig. 1 (a))\nthe net magnetization M=jMjincreases and hence, the\nanisotropy energy associated with the demagnetization\n\feldHshape =\u0000M?[17] dominates and forces the mag-\n−150 −100 −50 0 50 100 150\nµ0H0(mT)-40-2002040M⊥(kA/m)a\n170 K (1)(2)(3)\nMH0z\n−100 mT 10 mTz\n−150 −100 −50 0 50 100 150\nµ0H0(mT)-4-2024M⊥(kA/m)b\n250 K(2)(3)\n10 mTz − HC+ HC\n−100 mTH0M z\n(1)FIG. 1. Out-of-plane magnetization component M?mea-\nsured by SQUID magnetometry. For di\u000berent temperatures,\nmagnetically hard (170K, (a)) and easy (250K, (b)) axis loops\nare observed. The arrows on the data indicate the sweep di-\nrection ofH0. The insets schematically show the magnetiza-\ntion direction MandH0=H0zwith the \flm normal zat\nthe indicated values of H0.\nnetization to stay in-plane. At these low temperatures,\nthe anisotropy \feld perpendicular to the \flm plane, Hk,\ncaused by the additional anisotropy contribution has a\nconstant, comparatively small magnitude. We therefore\nobserve a hard axis loop in the out-of-plane direction:\nUpon increasing H0from\u0000150 mT to +150 mT, Mcon-\ntinuously rotates from the out-of-plane (oop) direction to\nthe in-plane (ip) direction and back to the oop direction\nagain. The same continuous rotation happens for the op-\nposite sweep direction of H0with very little hysteresis.\nFor temperatures close to Tcomp (T= 250 K, Fig. 1 (b)),\nHshape becomes negligible due to the decreasing Mwhile\nHkincreases as shown below. Hence, the out-of-plane di-\nrection becomes the magnetically easy axis and, in turn,\nan easy-axis hysteresis loop is observed: After applying\na large negative H0[(1) in Fig. 1 (a)] MandH0are\n\frst parallel. Sweeping to a positive H0,M\frst stays\nparallel to the \flm normal and thus M?remains con-\nstant [(2) in Fig. 1 (a)] until it suddenly \rips to being\naligned anti-parallel to the \flm normal at H0>+Hc\n[(3) in Fig. 1 (a)]. These loops clearly demonstrate that\nthe nature of the anisotropy changes from IPA to PMA\non varying temperature.3\n0.0 0.5 1.0\nµ0H0(T)0102030ωres/2π (GHz)110K 190K 240Ka\n0.70 0.75µ0H0(T)\n−40040∆S21×103\n110 K0.30.40.5\nµ0H0(T)−0.50.00.5\n∆S21×103\n240 K\n0 50 100 150 200 250 300\nT(K)0.00.5−µ 0Hi(T)b\nHani=−Meff\nHshape =−M⊥\nHk=M⊥−MeffIPA PMA\nRe ImReIm\nFIG. 2. Broadband FMR spectroscopy data reveiling a smooth transition from in-plane to perpendicular anisotropy. (a)\nFMR resonance frequency plotted against H0taken for three di\u000berent temperatures (symbols) and \ft to Eq. (3) (solid lines).\nFor an IPA, a positive e\u000bective magnetization Me\u000b(positivex-axis intercept) is extracted, whereas Me\u000bis negative for a PMA.\n(inset) Exemplary resonance spectra (symbols) at 14 :5 GHz recorded at 110 K and 240 K as well as the \fts to Eq. (1) used\nto determine !res(solid lines). A complex o\u000bset S0\n21has been subtracted for visual clarity, plotted is \u0001 S21=S21\u0000S0\n21.(b)\nAnisotropy \feld Hani=\u0000Me\u000bas a function of temperature (open squares). Prediction for shape anisotropy Hshape based on\nSQUID magnetometry data (solid line) from Ref. 18. The additional perpendicular anisotropy \feld Hk=M?\u0000Me\u000b(red dots)\nincreases to approximately 0 :18 T at 250 K where its value is essentially identical to Hanidue to the vanishing M?.\nIII. BROADBAND FERROMAGNETIC\nRESONANCE\nIn order to quantify the transition from in-plane to per-\npendicular anisotropy found in the SQUID magnetome-\ntry data, broadband FMR is performed as a function of\ntemperature with the external magnetic \feld H0applied\nalong the \flm normal.[19] For this, H0is swept while\nthe complex microwave transmission S21of a coplanar\nwaveguide loaded with the sample is recorded at vari-\nous \fxed frequencies between 10 GHz and 25 GHz. We\nperform \fts of S21to[20]\nS21(H0)j!=\u0000iZ\u001f(H0) +A+B\u0001H0 (1)\nwith the complex parameters AandBaccounting for a\nlinear \feld-dependent background signal of S21, the com-\nplex FMR amplitude Z, and the Polder susceptibility[21,\n22]\n\u001f(H0) =Me\u000b(H\u0000Me\u000b)\n(H\u0000Me\u000b)2\u0000H2\ne\u000b+i\u0001H\n2(H\u0000Me\u000b):(2)\nHere,\ris the gyromagnetic ratio, He\u000b=!=(\r\u00160), and\n!is the microwave frequency and the e\u000bective magneti-\nzationMe\u000b=Hres\u0000!res=(\r\u00160). From the \ft, the res-\nonance \feld Hresand the full width at half-maximum\n(FWHM) linewidth \u0001 His extracted. Exemplary data\nforS21(data points) and the \fts to Eq. (1) (solid lines)\nat two distinct temperatures are shown in the two in-\nsets of Fig. 2 (a). We obtain excellent agreement of the\n\fts and the data. The insets furthermore show that the\nsignal amplitude is signi\fcantly smaller for T= 240 K\nthan for 110 K. This is expected as the signal amplitude\nis proportional to the net magnetization Mof the sam-\nple which decreases considerably with increasing temper-\nature (cf. Fig. 2 (b)). At the same time, the linewidthdrastically increases as discussed in the following section.\nThese two aspects prevent a reliable analysis of the FMR\nsignal in the temperature region 250 K <T < 300 K (i.e.\naround the compensation temperature). Therefore we do\nnot report data in this temperature region. Nevertheless,\nFMR is ideally suited to investigate the magnetic prop-\nerties of the GdIG \flm selectively, i.e. independent of\nthe substrate, for temperatures below Tcomp.\nAs all measurements are performed in the high \feld\nlimit of FMR, the dispersions shown in Fig. 2 (a) are\nlinear and we can use the Kittel equation\n!res=\r\u00160(Hres\u0000Me\u000b) (3)\nto extract\randMe\u000b. It is customary to describe the\nmagnetic anisotropy using Me\u000bwhich can be related to\nan anisotropy \feld Hanialong + zasMe\u000b=\u0000Hani=\nM?\u0000Hkfor positive H0. Here,Haniis given by the\ndemagnetization \feld Hshape =\u0000M?(along\u0000z) and\nthe anisotropy \feld Hkof the additional perpendicular\nanisotropy (along + z). Evidently, Me\u000bcan be deter-\nmined by linearly extrapolating the data to !res= 0.\nThe FMR dispersion and the \ft to Eq. (3) (solid lines) are\nshown for three selected temperatures in Fig. 2 (a). At\n110 K (blue curve) Me\u000bis positive. Therefore, M >H k\nindicating that shape anisotropy dominates, and the \flm\nplane is a magnetically easy plane while the oop direction\nis a magnetically hard axis. At 240 K (red curve) Me\u000bis\nnegative and hence, the oop direction is a magnetically\neasy axis. Figure 2 (b) shows the extracted Me\u000b(T).\nAt 190 K, Me\u000bchanges sign. Above this tempera-\nture (marked in red), the oop axis is magnetically easy\n(PMA) and below this temperature (marked in blue),\nthe oop axis is magnetically hard (IPA). The knowledge\nofM?(T) obtained from SQUID measurements allows\nto separate the additional anisotropy \feld HkfromMe\u000b4\n1.61.82.02.2g\na\n10−310−210−1αb\n0 50 100 150 200 250\nT(K)100030005000∆ω0/2π(MHz)c\nFIG. 3. Key parameters characterizing the magnetiza-\ntion dynamics of GdIG as a function of T:(a)g-factor\ng=\r~=\u0016B,(b)Gilbert damping constant \u000band(c)inhomo-\ngeneous linewidth \u0001 !0=(2\u0019)\n(red dots in Fig. 2 (b)). Hk=M?\u0000Me\u000bincreases\nconsiderably for temperatures close to Tcomp while at\nthe same time the contribution of the shape anisotropy,\nHshape =\u0000M?trends to zero. For T'180 K,Hkex-\nceedsHshape which is indicated by the sign change of\nMe\u000b. Above this temperature, we thus observe PMA. We\nuse the magnetization Mdetermined using SQUID mag-\nnetometry from Ref. 18 normalized to the here recorded\nMe\u000bat 10 K in order to quantify Hk. The maximal value\n\u00160Hk= 0:18 T is obtained at 250 K which is the highest\nmeasured temperature due to the decreasing signal-to-\nnoise ratio towards Tcomp.\nWe can furthermore extract the g-factor and damp-\ning parameters from FMR. The evolution of the g-factor\ng=\r~\n\u0016Bwith temperature is shown in Fig. 3 (a). We ob-\nserve a substantial decrease of gtowardsTcomp. This is\nconsistent with reports in literature for bulk GIG and can\nbe explained considering that the g-factors of Gd and Fe\nions are slightly di\u000berent such that the angular momen-\ntum compensation temperature is larger than the mag-\nnetization compensation temperature.[23] The linewidth\n\u0001!=\r\u0001Hcan be separated into a inhomogeneous con-\ntribution \u0001 !0= \u0001!(H0= 0) and a damping contribu-\ntion varying linear with frequency with the slope \u000b:\n\u0001!= 2\u000b\u0001!res+ \u0001!0: (4)\nClose toTcomp = 285 K, the dominant contribution to\nthe linewidth is \u0001 !0which increases by more than an\norder of magnitude from 390 MHz at 10 K to 6350 MHz\nat 250 K [Fig. 3 (c)]. This temperature dependence of the\nlinewidth has been described theoretically by Clogston\net al.[24, 25] in terms of a dipole narrowing of the in-homogeneous broadening and was reported experimen-\ntally before[7, 16]. As opposed to these single frequency\nexperiments, our broadband experiments allow to sepa-\nrate inhomogeneous and intrinsic damping contributions\nto the linewidth. We \fnd that in addition to the in-\nhomogeneous broadening of the line, also the Gilbert-\nlike (linearly frequency dependent) contribution to the\nlinewidth changes signi\fcantly: Upon approaching Tcomp\n[Fig. 3 (b)], the Gilbert damping parameter \u000bincreases\nby an order of magnitude. Note, however, that due to\nthe large linewidth and the small magnetic moment of\nthe \flm, the determination of \u000bhas a relatively large\nuncertainty.[26] A more reliable determination of the\ntemperature evolution of \u000busing a single crystal GdIG\nsample that gives access to the intrinsic bulk damping\nparameters remains an important task.\nIV. CONCLUSIONS\nWe investigate the temperature evolution of the mag-\nnetic anisotropy of a GdIG thin \flm using SQUID mag-\nnetometry as well as broadband ferromagnetic resonance\nspectroscopy. At temperatures far away from the com-\npensation temperature Tcomp, the SQUID magnetome-\ntry reveals hard axis hysteresis loops in the out-of-plane\ndirection due to shape anisotropy dominating the mag-\nnetic con\fguration. In contrast, at temperatures close to\nthe compensation point, we observe easy axis hysteresis\nloops. Broadband ferromagnetic resonance spectroscopy\nreveals a sign change of the e\u000bective magnetization (the\nmagnetic anisotropy \feld) which is in line with the mag-\nnetometry measurements and allows a quantitative anal-\nysis of the anisotropy \felds. We explain the qualitative\nanisotropy modi\fcations as a function of temperature by\nthe fact that the magnetic shape anisotropy contribu-\ntion is reduced considerably close to Tcomp due to the re-\nduced net magnetization, while the additional perpendic-\nular anisotropy \feld increases considerably. We conclude\nthat by changing the temperature the nature of the mag-\nnetic anisotropy can be changed from an in-plane mag-\nnetic anisotropy to a perpendicular magnetic anisotropy.\nThis perpendicular anisotropy close to Tcomp in combi-\nnation with the small magnetization of the material may\nenable optical switching experiments in insulating fer-\nromagnetic garnet materials. Furthermore, we analyze\nthe temperature dependence of the FMR linewidth and\ntheg-factor of the GdIG thin \flm where we \fnd values\ncompatible with bulk GdIG[7, 25]. The linewidth can\nbe separated into a Gilbert-like and an inhomogeneous\ncontribution. We show that in addition to the previously\nreported increase of the inhomogeneous broadening, also\nthe Gilbert-like damping increases signi\fcantly when ap-\nproachingTcomp5\nV. ACKNOWLEDGMENTS\nWe gratefully acknowledge funding via the priority pro-\ngram Spin Caloric Transport (spinCAT), (Projects GO\n944/4 and GR 1132/18), the priority program SPP 1601(HU 1896/2-1) and the collaborative research center SFB\n631 of the Deutsche Forschungsgemeinschaft.\nVI. BIBLIOGRAPHY\n[1] C. D. Stanciu, F. Hansteen, A. V. Kimel, A. Kirilyuk,\nA. Tsukamoto, A. Itoh, and T. Rasing, Physical Review\nLetters 99, 1 (2007).\n[2] C.-H. Lambert, S. Mangin, B. S. D. C. S. Varaprasad,\nY. K. Takahashi, M. Hehn, M. Cinchetti, G. Malinowski,\nK. Hono, Y. Fainman, M. Aeschlimann, and E. E. Fuller-\nton, Science 345, 1337 (2014).\n[3] K. Garello, C. O. Avci, I. M. Miron, M. Baumgartner,\nA. Ghosh, S. Au\u000bret, O. Boulle, G. Gaudin, and P. Gam-\nbardella, Applied Physics Letters 105, 212402 (2014).\n[4] I. M. Miron, K. Garello, G. Gaudin, P.-J. Zermatten,\nM. V. Costache, S. Au\u000bret, S. Bandiera, B. Rodmacq,\nA. Schuhl, and P. Gambardella, Nature 476, 189 (2011).\n[5] A. Brataas, A. D. Kent, and H. Ohno, Nature Materials\n11, 372 (2012).\n[6] S. Ikeda, K. Miura, H. Yamamoto, K. Mizunuma, H. D.\nGan, M. Endo, S. Kanai, J. Hayakawa, F. Matsukura,\nand H. Ohno, Nature Materials 9, 721 (2010).\n[7] B. Calhoun, J. Overmeyer, and W. Smith, Physical Re-\nview107(1957).\n[8] G. F. Dionne, Journal of Applied Physics 42, 2142\n(1971).\n[9] J. D. Adam, L. E. Davis, G. F. Dionne, E. F. Schloemann,\nand S. N. Stitzer, IEEE Transactions on Microwave The-\nory and Techniques 50, 721 (2002).\n[10] K. P. Belov, L. A. Malevskaya, and V. I. Sokoldv, Soviet\nPhysics JETP 12, 1074 (1961).\n[11] P. R oschmann and W. Tolksdorf, Materials Research\nBulletin 18, 449 (1983).\n[12] H. Maier-Flaig, M. Harder, S. Klingler, Z. Qiu, E. Saitoh,\nM. Weiler, S. Gepr ags, R. Gross, S. T. B. Goennen-\nwein, and H. Huebl, Applied Physics Letters 110, 132401\n(2017).\n[13] G. F. Dionne, Magnetic Oxides (Springer US, Boston,\nMA, 2009).\n[14] S. A. Manuilov, S. I. Khartsev, and A. M. Grishin, Jour-\nnal of Applied Physics 106, 123917 (2009).\n[15] S. A. Manuilov and A. M. Grishin, Journal of Applied\nPhysics 108, 013902 (2010).\n[16] G. P. Rodrigue, H. Meyer, and R. V. Jones, Journal of\nApplied Physics 31, S376 (1960).\n[17] We use the demagnetization factors of a in\fnite thin \flm:\nNx;y;z= (0;0;1).\n[18] S. Gepr ags, A. Kehlberger, F. D. Coletta, Z. Qiu, E.-J.\nGuo, T. Schulz, C. Mix, S. Meyer, A. Kamra, M. Al-\nthammer, H. Huebl, G. Jakob, Y. Ohnuma, H. Adachi,\nJ. Barker, S. Maekawa, G. E. W. Bauer, E. Saitoh,\nR. Gross, S. T. B. Goennenwein, and M. Kl aui, Nature\nCommunications 7, 10452 (2016).\n[19] The alignment of the sample is con\frmed at low temper-\natures by performing rotations of the magnetic \feld di-\nrection at \fxed magnetic \feld magnitude while recording\nthe frequency of resonance !res. As the shape anisotropydominates at low temperatures, !resgoes through an\neasy-to-identify minimum when the sample is aligned\noop.\n[20] H. Maier-Flaig, S. T. B. Goennenwein, R. Ohshima,\nM. Shiraishi, R. Gross, H. Huebl, and M. Weiler, arXiv\npreprint arXiv:1705.05694 .\n[21] J. M. Shaw, H. T. Nembach, and T. J. Silva, Physical\nReview B 87, 054416 (2013).\n[22] H. T. Nembach, T. J. Silva, J. M. Shaw, M. L. Schneider,\nM. J. Carey, S. Maat, and J. R. Childress, Physical\nReview B 84, 054424 (2011).\n[23] R. K. Wangsness, American Journal of Physics 24, 60\n(1956).\n[24] A. M. Clogston, Journal of Applied Physics 29, 334\n(1958).\n[25] S. Geschwind and A. M. Clogston, Physical Review 108,\n49 (1957).\n[26] For the given signal-to-noise ratio and the large\nlinewidth, \u000band \u0001!0are correlated to a non-\nnegligible degree with a correlation coe\u000ecient of\nC(intercept;slope) = \u00000:967."    },    {        "title": "1707.02425v1.Nonlinear_dynamics_of_damped_DNA_systems_with_long_range_interactions.pdf",        "content": "arXiv:1707.02425v1  [physics.bio-ph]  8 Jul 2017Nonlinear dynamics of damped DNA systems with long-range in teractions\nJ. Brizar Okalya,c,∗,Alain Mvogoa,c,R. Laure Woulach´ eb,c,T. Cr´ epin Kofan´ eb,c\naLaboratory of Biophysics, Department of Physics, Faculty o f Science, University of Yaounde I, P.O. Box 812, Yaounde, Ca meroon\nbLaboratory of Mechanics, Department of Physics, Faculty of Science, University of Yaounde I, P.O. Box 812, Yaounde, Cam eroon\ncAfrican Center of Excellence in Information and Communicat ion Technologies, University of Yaounde I, P.O. Box 812, Yao unde,\nCameroon.\nAbstract\nWeinvestigatethenonlineardynamicsofadampedPeyrard-BishopD NAmodeltakingintoaccountlong-rangeinteractions\nwith distancedependence |l|−son the elasticcoupling constantbetween different DNA basepairs. C onsideringboth Stokes\nand long-range hydrodynamical damping forces, we use the discre te difference operator technique and show in the short\nwavelength modes that the lattice equation can be governed by the complex Ginzburg-Landau equation. We found\nanalytically that the technique leads to the correct expression for the breather soliton parameters. We found that the\nviscosity makes the amplitude of the breather to damp out. We comp are the approximate analytic results with numerical\nsimulations for the value s= 3 (dipole-dipole interactions).\nKeywords: DNA, long-range interactions, damping forces, breather soliton.\n1. Introduction\nThe DNA molecule is known to be very important and essential in the pr otection, transport and transmission of the\ngenetic code. Many theoretical models have been proposed to des cribe the nonlinear dynamics of DNA. The first nonlinear\nDNA model was suggested by Englander et al.[1]. Thereafter, simplified models were proposed to describe the angu lar\ndistortion of DNA [ 2,3,4] and the micromanipulation experiments were investigated, showing the great importance of\nradial displacementsof basesduring the processesofreplication a nd transcription[ 5,6,7]. The Peyrard-Bishop(PB) DNA\nmodel [5], which has been successfully used to analyze experiments on short DNA sequences [ 8] has gained a popularity\nin that direction.\nDNA double helix spontaneously denatures locally and the breathing m ode occurs when it is locally excited with large\namplitude [ 9]. The amplitude of this excitation depends on the surrounding enviro nment-DNA interactions and the inner\nmechanism of the DNA molecule depending on the relative motion of par ticles. It is therefore of physical importance to\ntake into account these two effects in the nonlinear dynamics of DNA molecule. Some authors emphasized the influence\nof the viscosity on the dynamical properties of the DNA molecule [ 10,11]. In particular, it is shown that the viscosity\nof the medium damps out the amplitude of the nonlinear wave propaga ting through the molecule [ 10,11]. In the above\nstudies, the analysis deal with the PB DNA model with nearest neighb or interactions between base pairs. The long-range\ninteractions (LRI) play a crucial role in molecular systems [ 12,13,14]. The importance of LRI in DNA molecule is due\nto the presence of phosphate groups along the strands [ 12]. The LRI therefore allow to take into account the screening\n∗Corresponding author\nEmail addresses: okalyjoseph@yahoo.fr (J. Brizar Okaly ),mvogal_2009@yahoo.fr (Alain Mvogo ),rwoulach@yahoo.com (R. Laure\nWoulach´ e ),tckofane@yahoo.com (T. Cr´ epin Kofan´ e )\nPreprint submitted to : September 11, 2018of the interactions or an indirect coupling between base pairs (e.g. v ia water filaments) [ 13]. Rau and Parsegian, in their\nexperimental studies in the direct measurements of the intermolec ular forces between counterion-condensed DNA double\nhelices, have shown the importance of long-range attractive hydr ogen forces and emphasize that many forces could be\nresponsible for LRI in the DNA molecule [ 14]. In fact, the charged groups in the molecular chains and DNA molecu les\ninteract through long-range dipole-dipole interactions. Along the s ame line, it has been shown that the study of the\npower-law LRI in nonlinear lattice models is very relevant mainly in molecu lar chains and DNA molecules where Coulomb\nand dipole-dipole interactions are of a great physical importance [ 15,16,17]. Recent works by Mvogo et al.[18,19] also\nindicated qualitative effects of the power-law LRI in molecular system s.\nTo the best of our knowledge, no work has been reported on the st udy of DNA dynamics taking into account both\ndamping and LRI effects. Our aim in this paper is to study the DNA dyna mics taking into account LRI between different\nDNA base pairs and both Stokes and long-range hydrodynamical da mping effects. To address this issue, our analytical\nstudy has been inspired by the one recently developed by Miloshevich et al.[20] to investigate traveling solitons in\nlong-range oscillator chains. Due to the non analytical properties o f the dispersion relation, Miloshevich et al.[20] have\nshown that the discrete difference operator (DDO) technique is mo re appropriate to study physical systems with LRI. In\nthis paper, we use the DDO and show that the DNA models with LRI can be reduced to a specific form of the complex\nGinzburg-Landau (CGL) equation, where the dispersion coefficient is complex and the nonlinearity coefficient is real. A\nsimilarequationhasbeen obtainedby Zdravkov´ ıc et al.[10,11] while studyingthe effect ofviscosityon the dynamicsofthe\nPeyrard-Bishop-Dauxois(PBD) DNA model in the absence of hydro dynamical damping and LRI forces. The investigators\nstate that the CGL equation cannot be solved analytically like a nonline ar Schrodinger (NLS) equation [ 10,11]. In this\npaper, following the work by Pereira and Stenflo [ 21], we analytically solve the CGL equation.\nThe paper is organized as follows. In Section 2, we propose the Hamiltonian model and derive the discrete equation s\nof motion for the in-of-phase and out-of-phase motions. In Sect ion3, we use the DDO and show that the out-of-phase\ndynamical equation can be reduced to the CGL equation. The envelo pe soliton solution of this equation is reported and\nthe breather solution of the discrete equation of motion for the ou t-of-phase motion is derived. In Section 4, we perform\nthe numerical simulations with emphasis on the effects of the LRI and damping forces. Section 5concludes the work.\n2. Model and equations of motion\nWe consider the PB model [ 5] for DNA denaturation where the degrees of freedom xnandynassociated to each base\npair correspond to the displacements of the bases from their equilib rium positions along the direction of the hydrogen\nbonds that connect the two bases in a pair. A LRI coupling between t he base pairs due to the presence of phosphate\ngroup along the DNA strands is assumed so that the Hamiltonian for t he model is given by\nH=N/summationdisplay\nn/braceleftBig1\n2m(˙x2\nn+ ˙y2\nn)+1\n2/summationdisplay\nl=1Jl[(xn−xn−l)2+(yn−yn−l)2]+V(xn,yn)/bracerightBig\n, (1)\nwheremis the average mass of the nucleotides and Nrepresents the number of the base pairs of the DNA molecule. The\ninteractions between hydrogen bonds in a pair is modeled by the Mors e potential V(xn,yn) given by\nV(xn,yn) =D/bracketleftBig\ne−a(xn−yn)−1/bracketrightBig2\n, (2)\n2whereDis the depth of the Morse potential well, which may depend on the typ e of base pair and ais the width of the\nwell. The quantity\nJl=J|l|−s, (3)\nis the power-law dependence of the elastic coupling constant, wher esand|l|are the LRI parameter and the normalized\ndistance betweenbasepairs, respectively. In practice, tokeep t he spatialhomogeneityin a finite DNA systemwith periodic\nboundary conditions, usually the LRI is limited in each direction to1\n2(N−1), ifNis odd, or1\n2(N−2), ifNis even,\nand 1≤ |l| ≤1\n2(N−1) [22]. The parameter scan be used to model Coulomb interactions between charged partic les of a\nchain (s= 1), dipole-dipole interactions ( s= 3). Below s= 1 the energy diverges and above s= 3 the system becomes\nshort-range. In this paper, we use the case s= 3, where multiple solutions exist [ 23].\nThe values of parameters used to perform our analysis are those f rom the dynamical and denaturation properties of\nDNA. They are [ 24]:m= 300 amu, J= 0.06 eV/˚A2,D= 0.03eVanda= 4.5˚A−1. Our system of units (amu, ˚A, eV)\ndefines a time unit ( t.u.) equal to 1 .018×10−14s.\nTo describe the motions of the two strands, we introduce the new v ariablesunandvnsuch that\nun=xn+yn√\n2andvn=xn−yn√\n2, (4)\nwhereunandvnrepresent the in-phase and the out-of-phase motions. Taking int o account Eq. ( 2) and Eq. ( 4), the\nHamiltonian of the system can be rewritten as\nH=N/summationdisplay\nn/braceleftBig1\n2m˙u2\nn+1\n2/summationdisplay\nl=1Jl(un−un−l)2/bracerightBig\n+N/summationdisplay\nn/braceleftBig1\n2m˙v2\nn+1\n2/summationdisplay\nl=1Jl(vn−vn−l)2+D/parenleftBig\ne−a√\n2vn−1/parenrightBig2/bracerightBig\n.(5)\nThe equations of motions of the system then read\nm¨un=/summationdisplay\nl=1Jl(un+l−2un+un−l), (6)\nm¨vn=/summationdisplay\nl=1Jl(vn+l−2vn+vn−l)+2√\n2aDe−a√\n2vn/parenleftBig\ne−a√\n2vn−1/parenrightBig\n. (7)\nFor a more realistic study of dynamical properties of DNA, one must take into account its environment. In the present\nwork, we take into account the Stokes ( Fst) and the long-range hydrodynamical ( Fhy) damping forces in the equations\nof motion of the system. These forces account respectively for D NA molecules in a viscous environment and their inner\nmechanism. The Stokes damping forces is given by\nFst\nn=−mγst˙qn, (8)\nwhereγstis the Stokes damping constant. The coordinate qncan be replaced by unorvn. In previous works in discrete\nlattices, investigators assume the hydrodynamical damping force in the nearest neighbor interactions [ 25,26,27,28]. In\nthis work, the DNA molecule is considered as a collection of nucleotides linked to the neighbors of the same strand by\nspring. Each of them is assumed to be point masses of mass m. Thus, the displacement of one base pair causes a more or\nless significant displacement of the other base pairs of the chain acc ording to whether they are closed or distanced from\nthe initial base pair. This displacement gives rise to the hydrodynamic al viscous forces which influence the motion of the\n3initial nucleotide. Since our work focuses on LRI between base pairs , we have introduced the LRI in the hydrodynamic\ndissipation Fhyin order to takes into account the contribution of all base pairs of t he chain so that,\nFhy\nn=m/summationdisplay\nl=1γl(˙qn−l−2˙qn+ ˙qn+l), (9)\nwhereγl=γhy|l|−s′andγhyis the hydrodynamical damping coupling constant. Taking into accou ntFst\nnandFhy\nnas\ndefined above, we obtain the following equations of motion:\n¨un=/summationdisplay\nl=1Jl\nm(un+l−2un+un−l)−γSt˙un+/summationdisplay\nl=1γl(˙un+l−2˙un+ ˙un−l) (10)\n¨vn=/summationdisplay\nl=1Jl\nm(vn+l−2vn+vn−l)+2√\n2aD\nme−a√\n2vn(e−a√\n2vn−1)−γSt˙vn+/summationdisplay\nl=1γl(˙vn+l−2˙vn+ ˙vn−l).(11)\nThe solution un(t) of Eq. ( 10) is an ordinary solution of a damped linear schr ¨odinger equation and represents a plane\nwave in a viscous medium in the presence of LRI. In what follows, the s ystem will be considered heavily damped. Our\ninvestigationswillbelimitedtotheanalysisofthedynamicalbehavioro fthestretchingmotionofeachbasepairrepresented\nby the solution of Eq. ( 11), in the presence of LRI and the“big viscosity ” [ 10,11].\n3. Discrete difference operator technique\nIn this section, the DDO technique which is appropriate for long-ran ge interacting systems [ 20] is used to study the\ndynamics of DNA breathing. Assuming as usual small amplitude oscillat ion of the nucleotide around the bottom of the\nMorse potential, we obtain up to the third order of the Morse poten tial the following equation of motion:\n¨vn=/summationdisplay\nl=1Jl\nm(vn+l−2vn+vn−l)−ω2\ng(vn+αv2\nn+βv3\nn)−γSt˙vn+/summationdisplay\nl=1γl(˙vn+l−2˙vn+ ˙vn−l). (12)\nwhereω2\ng=4a2D\nm,α=−3a√\n2andβ=7a2\n3. Eq. (12) describes the dynamics of the out-of-phase motion of the DNA in\nviscous medium in the presence of LRI forces. Introducing the dist ance of neighboring bases rand assuming plane wave\nsolutions of the form\nvn=F1ei(qnr−ωt)+c.c. (13)\nand substituting them into the equations of motion, we obtain the no nlinear dispersion relation in rotating wave approxi-\nmation for the normal mode frequencies ωnand wave numbers qn\nω2\nn=ω2\ng(1+3β|F1|2)+4/summationdisplay\nl=1Jl\nmsin2(q0\nnlr/2)−iωnγn, (14)\nwhereγnis the damping coefficient given by:\nγn=γst+4/summationdisplay\nl=1γlsin2(q0\nnrl/2). (15)\nAfter some algebras, this dispersion relation can be rewritten as th e sum of its real part ωrand imaginary part ωi. That\nis\nωn=ωr+iωi, ω r=ω0,n/radicalbig\n1−δ2n, ω i=−γn\n2, (16)\n4withδn=γn\n2ω0,nandω0,nthe optical frequency of vibrations of base pairs in the absence of damping forces given by\nω2\n0,n=ω2\ng(1+3β|F1|2)+4/summationdisplay\nl=1Jl\nmsin2(q0\nnrl/2). (17)\nWe plot in Figure 1the real and imaginary parts of the angular frequency of the wave (Eq. (16)), the real and imaginary\nparts of the dispersion coefficient in the linear limit |F1| →0 for discretized values of the wave vector q0\nnrfors= 3.00 and\nγ0= 0.15. In the panel (a) we observe that the real part of the angular frequency is equal to zero for qnr∈]π\n12,23π\n12[ and\ndifferent from zero otherwise, namely qnr∈[0,π\n12] andqnr∈[23π\n12,2π]. Then the vibration can appears and propagates in\nthe DNA molecule only if the carrier wave vector qnris selected in a finite interval/braceleftBig\n[0,π\n12]∪[23π\n12,2π]/bracerightBig\n.\nThe contribution of the long range decays of the stacking and visco us interactions are not the same, since the origins of\nthe two forces are physically different, but nevertheless for seek of simplicity the same exponent is assumed that is s=s′.\nAlso, as in [ 29], we set γ0=γst=γhy. At small wavelength, the soliton solution of Eq. ( 12) is found as an expansion in\nnormal modes and may be found in the form [ 30].\nvn=ε[F1(z1,τ)ei(q0rn−ω0t)+c.c.], (18)\nwith\nF1(z1,τ) =N/summationdisplay\nn=1Bnei(δqnz1−δωnτ), q n=q0+εδqn, ω n=ω0+ε2δωn. (19)\nThe function F1is a slowly varying function in space z1=εrnand time τ=ε2t. The parameter q0is the wavenumber of\nthe wave packet and the associate frequency ω0≡ω(q0,F1= 0) in the limit F1→0\nThe time derivative of the wave amplitude Eq. ( 19) reads:\n∂F1(z1,τ)\n∂τ= [−iδωn]N/summationdisplay\nn=1Bnei(δqnz1−δωnτ)= [−iδωn]F1. (20)\nFrom Eqs. ( 14), (19) and (18) it is seen that the term ε2δωnis an evolution function of two variables: the wavenumber q0\nand the slowly varying wave amplitude |εF1|2. The Taylor expansion of this term around the value q0and|εF1= 0|and\nneglecting higher order terms ( >2), give us\nε2δωn(∂qn,|εF1|2) =δω0(q0)+(qn−q0)∂ω0(q0)\n∂q0+1\n2(qn−q0)2∂2ω0(q0)\n(∂q0)2+|εF1|2∂ωn(q0)\n∂(|F1|2)/vextendsingle/vextendsingle/vextendsingle\nF1=0. (21)\nThe first term of the right hand site of Eq. ( 21) is assuming to be very close to zero. From Eq. ( 19) we have εδqn=qn−q0.\nThe above considerations in Eq. ( 21) lead to:\nδωn(∂qn,|F1|2) =(εδqn)\nε2∂ω0(q0)\n∂q0+1\n2(εδqn)2\nε2∂2ω0(q0)\n∂q02+|F1|2∂ωn(q0)\n∂(|F1|2)/vextendsingle/vextendsingle/vextendsingle\nF1=0. (22)\nThe discrete difference operator is used instead of the continuous derivatives which can cause divergences. Therefore we\nhave:\n∂ω0(q0)\n∂q0=ω0(q0+h)−ω0(q0)\nh,∂2ω0(q0)\n∂q02=ω0(q0+h)−2ω0(q0)+ω0(q0−h)\nh2, (23)\nand finally we get,\nδωn(∂qn,|F1|2) =2/summationdisplay\nν=1(εδqn)ν\nε2ν!∆(ν)\nh[ω0(q0)]\nhν+|F1|2∂ωn(q0)\n∂(|F1|2)/vextendsingle/vextendsingle/vextendsingle\nF1=0, (24)\n5where ∆(ν)\nhis the difference operator of order νwith step size h= 2π/Nin the limit F1= 0 and given below,\n∆(1)\nh[ω0] =ω0(q0+h)−ω0(q0),∆(2)\nh[ω0] =ω0(q0+h)−2ω0(q0)+ω0(q0−h). (25)\nFrom Eq. ( 19) the term ( δqn)νcan be expressed as follows:\n(iδqn)νF1=∂νF1\n∂zν\n1. (26)\nUsing Eqs. [ 26-23] into Eq. ( 20) give the nonlinear equation of evolution of the envelope function wr itten as\ni/bracketleftBig∂F1\n∂τ+vg\nε∂F1\n∂z1/bracketrightBig\n+P∂2F1\n∂z2+Q|F1|2F1= 0 (27)\nwhere the parameters vg,PandQare the group velocity, the dispersion and the nonlinearity coefficien ts given by\nvg=∆(1)\nh[ω0]\nh, P=∆(2)\nh[ω0]\n2h2, Q=−∂ωn(q0)\n∂(|F1|2)/vextendsingle/vextendsingle/vextendsingle\nF1=0. (28)\nThe above parameters can be rewritten as:\nvg=vgr+ivgi, v gr=∆(1)\nh[ω0\nr]\nh, v gi=∆(1)\nh[ω0\ni]\nh\nP=Pr+iPi, P r=∆(2)\nh[ω0\nr]\n2h2, P i=∆(2)\nh[ω0\ni]\n2h2,\nQ=Qr+iQi, Q r=−3βω2\ng\n2ω0r, Q i= 0.(29)\nSettingξ1=z1−εvgtin the co-moving reference frame with a rescaled time tsuch ast→ε2t, Eq. (24) becomes\ni∂F1\n∂t+(Pr+iPi)∂2F1\n∂ξ2\n1+Q|F1|2F1= 0. (30)\nIt should be noted that Eq. ( 30) is the well-known CGL equation for the evolution of the envelope whe re the dispersion\ncoefficient is complex and the nonlinearity coefficient is real. Similar equa tion was found in Ref. [ 10,11] where the authors\nstudied the dynamics of a damped DNA in the absence of hydrodynam ical damping and LRI forces using the semi-discrete\napproximation. In their study they found the dispersion coefficient real and the nonlinearity coefficient complex contrary\nof the one obtained in this work. The nonlinearity coefficient Qand the dispersion coefficient Pnot only depend on\nthe wave vector q0\nnr, and the Stokes viscous forces as previously mentioned by these a uthors, but also depend on the\nhydrodynamic damping and the LRI forces.\nSeveral methods related to soliton solutions for the specific forms of CGL equation have been developed [ 21,31,32].\nA key problem in this paper is to give an analytical soliton solution of the CGL equation (Eq. ( 30)), and use it to study\nthe effect of viscosity and LRI on the DNA opening state configurat ion. The character of this solution is determined by\nthe sign of QandPrwhile the stability of the plane wave solution through the Benjamin-Fe ir instability depends on the\nsign of the product PrQ. ForPrQ <0, the plane wave solution is stable and for PrQ >0 it is unstable. Particularly,\nsince the nonlinear coefficient Qis always negative the sign of the constant PrQdepends on real part of the dispersion\ncoefficient Prwhich can take positive or negative values depending on the range of variations of the wave vector. Here,\nonly localized solutions in space for any wave carrier whose wavenumb er is in the positive range of PrQare considered.\n6In Figure 2, the product PrQis represented as a function of the wave vector for s= 3.00 andγ= 0.15. We observe\nin the plots that PrQ >0 is always positive. From Eqs. ( 15), (16) and (29), we observe that the imaginary parts of the\nsolitonic parameters strongly depend on the damping forces of the system, therefore their absolute values decrease with\nthe decreasing of the damping constant and vanish when the dampin g is switched off.\nAs in [21,31,32], an analytical solution of Eq. ( 31), is found in the form\nF1=A/bracketleftBig\nsech(ηξ1)/bracketrightBig1+iσ\ne−iφt, (31)\nwhereA,φ,η−1, andσare parameters to be determined and represent respectively the complex amplitude, the complex\n“angular frequency ”, the width and the chirp of the soliton. By intr oducing Eq. ( 31) into Eq ( 30) and after canceling the\nterms in [sech( ηξ1)/bracketrightBig1+iσ\n, the real and imaginary part of the phase of the soliton is written\nφr=−η2/bracketleftBig\n(1−σ2)Pr−2σPi/bracketrightBig\n, φ i=−η2/bracketleftBig\n(1−σ2)Pi+2σPr/bracketrightBig\n. (32)\nFrom the annihilation of the terms in [sech( ηξ1)/bracketrightBig3+iσ\n, the width and the chirp of the soliton is given by\n|A|=η/radicalBigg/vextendsingle/vextendsingle/vextendsingle(2−σ2)Pr−3σPi\nQr/vextendsingle/vextendsingle/vextendsingle, σ=3Pr+√\n∆\n2Pi(33)\nwhere ∆ = 9 P2\nr+8P2\niandσa solution of the following quadratic equation\nPiσ2−3Prσ−2Pi= 0. (34)\nThis choice of σimplies that the soliton is strongly chirped.\nNow to determine the complex amplitude Aγof the soliton, let us consider the system in the non-viscouslimit ( γ0= 0).\nIn that case Pivanishes and Eq. ( 30) becomes the standard NLS equation\ni∂G1\n∂t+P′∂2G1\n∂ξ2\n1+Q′|G1|2G1= 0, (35)\nwhere the associated group velocity, dispersion coefficient and non linearity coefficient are\nv′\ng≡vgr/vextendsingle/vextendsingle/vextendsingle\nγ=0=∆(1)\nh[ω0\n0]\nh, P′≡Pr/vextendsingle/vextendsingle/vextendsingle\nγ=0=∆(2)\nh[ω0\n0]\n2h2, Q′≡Qr/vextendsingle/vextendsingle/vextendsingle\nγ=0=−3βω2\ng\n2ω0. (36)\nThe solution of Eq. ( 35) is the well known modulated solitonic wave called breather [ 24,33] and given by\nG1=A′sech/bracketleftBig\nL(ξ1−uet)/bracketrightBig\neiue\n2P′(ξ1−uct), (37)\nwhereueanducare real parameters representing respectively the velocities of t he envelope and the carrier wave of the\nsoliton. The amplitude of the envelope A′and its inverse width L′are given by the relations\nL′=/radicalbig\nu2e−2ueuc\n2P′, A′=/radicalBigg\nu2e−2ueuc\n2P′Q′. (38)\nLet us assume that at the initial time ( t= 0), for γ0= 0, the soliton solution Eqs. ( 31) and (38) should be equivalent,\nA/vextendsingle/vextendsingle/vextendsingle\nγ=0=A′eiue\n2P′ξ1,0, (39)\n7whereξ1,0is the initial position of the soliton. Taking into account this new expre ssion of the amplitude A, the one soliton\nsolution of Eq. ( 30) is\nF1=η/radicalBigg/vextendsingle/vextendsingle/vextendsingle(2−σ2)Pr−3σPi\nQr/vextendsingle/vextendsingle/vextendsingleeφit/bracketleftBig\nsech(ηξ1)/bracketrightBig1+iσ\nei(ue\n2Pξ1,0−φrt). (40)\nNow considering the fact that the angular frequency of the soliton is complex (see Eq. ( 16)), one can note that the term\neωitenters inside the amplitude of the soliton. Thus, the complete expre ssionof the envelope soliton written in the original\ntemporal ( t) and spacial ( z) variables is finally given by\nF′\n1=F1eωit=η/radicalBigg/vextendsingle/vextendsingle/vextendsingle(2−σ2)Pr−3σPi\nQr/vextendsingle/vextendsingle/vextendsinglee−Γt/bracketleftBig\nsech/bracketleftbig\nη(z1−εvgrt)/bracketrightbig/bracketrightBig1+iσ\neiΘ (41)\nwith\nΘ =ue\n2Pξ1,0+η2/bracketleftBig\n(1−σ2)Pr−2σPi/bracketrightBig\nt,Γ =γn\n2+η2/bracketleftBig\n(1−σ2)Pi+2σPr/bracketrightBig\n, η=/radicalbig\nu2e−2ueuc/vextendsingle/vextendsingle/vextendsingle(2−σ2)Pr−3σPi/vextendsingle/vextendsingle/vextendsingle(42)\nwhere Γ is the effective damping constant of the medium.\nTo obtain the solution of the EOM of the out-of-phase motion vngiven by Eq. ( 11), some results of the previous\nsection are used. Using Eqs. ( 16), (28) and (40) the soliton solution describing the out-of-phase motion of the DNA in\nthe viscous medium takes the form\nvn(t) =2ε|A|e−Γtsech/bracketleftbig\nεη(nr−vgrt)/bracketrightbig\ncos(qγnr−̟t) (43)\nwhere\n̟=ω0\nr−η2/bracketleftBig\n(1−σ2)Pr−2σPi/bracketrightBig\n, q γ=q0+ue\n2P/parenleftBigξ1,0\nnr/parenrightBig\n+(σ/nr)log/vextendsingle/vextendsingle/vextendsinglesech/bracketleftbig\nεη(nr−vgrt)/bracketrightbig/vextendsingle/vextendsingle/vextendsingle. (44)\nThe solution Eq. ( 43) represents the breather solution in the DNA molecule suggested b y the Infrared and Raman\nexperiments [ 34]. This solution is represented in Figure 3. It can be seen that due to the damping effect, the amplitude\nof the soliton is a decreasing function of time and hence it will propaga tes only for a limited distance and vanishes.\nProhofsky et al.have shown in their studies [ 34] that, the breather can be strongly located, or distributed on a w ide\nzone of the DNA molecule and could be at the origin of the localization of the energy in the molecule which lead to the\nlocal denaturation.\nIt is noteworthy to mention that if one uses the semi-discrete appr oach [10,11], the results fails in the derivation\nof the breather soliton profile Eq. ( 43). In fact, we can obtain a similar CGL equation Eq. ( 30), but with a different\ndispersion coefficient containing continuous derivatives instead of d ifference operators as given in equation Eq. ( 28). Using\nthe semi-discrete approach, we obtain:\nPr=1\n2ωr/bracketleftBig/summationdisplay\nl=1/parenleftBigJl\nm+ωiγl/parenrightBig\n(rl)2cos(qlr)−|vgγ|2/bracketrightBig\n, Pi=−1\n2/summationdisplay\nl=1γl(rl)2cos(qrl),\nωr=ω/radicalbigg\n1−/parenleftBigγ\n2ω/parenrightBig2\n, ω2=ω2\ng+4/summationdisplay\nl=1Jl\nmsin2(qrl/2), ωi=−γ\n2, γ=γst+4/summationdisplay\nl=1γlsin2(qrl/2),\nvgr=r\nωr/summationdisplay\nl=1/parenleftBigJl\nm+ωiγl/parenrightBig\nlsin(qrl), vgi=−r/summationdisplay\nl=1γllsin(qrl), ς=ω2\ng+4\nm/summationdisplay\nl=1Jlsin2(qrl),\nQr=−ω2\ngα\nωr(−2α+3\n2β/α+BC), Qi=−ω2\ngα\nωrCD1B= 4ω2\nrς, D1= 2γωr, C=ω2\ngα\nB2+D2\n1.(45)\n8InEq.(45), theappearanceofcontinuousderivativescancauseadivergen ceofboththegroupvelocity vgandthedispersion\ncoefficient P. The coefficients oscillate for different values of chain length and doe s not converge to a definite value.\n4. Numerical investigations.\nThe results discussed in the previous section are obtained from the CGL equation (Eq. ( 30)) derived after some\napproximations and hypothesis and not from the discrete EOM. In o rder to verify the analytical predictions and check if\nthe above analytical breather soliton can survive in the discrete lat tice, the numerical simulations of the discrete EOM\n(Eq. (11)) is carried out by means of a standard fourth-order Runge-Kut ta computational scheme with periodic boundary\nconditions. In our simulations we use the initial condition given by Eq. ( 43) and time step h= 2π/N t.u. withN= 600.\nFigure4presents the 3 Dand 2Dtime-evolution of the solution for a value of LRI parameter s= 3.00, and we observe\nthe decreasing of the amplitude of the soliton due to the damping for ces (Figure 4a). Figure 4b presents the aspect of the\nnumerical solution at t= 115. As predicted by the analytical results, we observe that the breathing mode appears in DNA\nmolecule when the wave vector is small, namely qnr≤π\n12which corresponds to the domain where the real parts of the\nangular frequency is different from zero. In Figure 5and Figure 6, we have depicted the 2 Dschematic representation of\nthe analytical and numerical solutions at few time positions for two d ifferent wave vectors ( qnr=π/16 andqnr=π/18):\nt= 0,t= 50,t= 200 and t= 300. We observe that when the wave vector increases, the width of the soliton grows by\nincreasingthe number of basepairs in the bubble. At the same time, t he waveamplitude must increasein orderto keep the\nsoliton within the lattice length limits. Therefore, small wavenumber le ads to a more localized solution. The decreasing\nof the amplitude of the soliton in time is observed. Also we notice that t he shape, the decay of the amplitude and the\nnumber of base pairs in the bubble of both solutions after a limited time propagation are the same. But after a long time\npropagation the number of base pairs which form the bubble remains constant, the shape of the numerical solution is\nslightly modified also the decay in the amplitude of the numerical solutio n is less than the theoretical expectations, due to\nthe discreteness effects which usually tend to slow down the motion [ 9]. These results confirm that our analytical solution\nis stable and suitable to predict formation of breather solitons in the DNA molecular chain in the presence of damping\nand LRI forces.\n5. Conclusion\nIn this work, we have studied the dynamics of breather solitons in a lo ng-range version of the Peyrard-Bishop DNA\nmodel taking into Stokes and hydrodynamical viscous forces. Usin g the discrete difference operator technique, we have\nshown that the out-of-phase motion can be described by the CGL e quation. As compared to the semi-discrete approach,\nthis technique can lead to the correct expression for the soliton pa rameters. The breather soliton which represents the\nopening of base pairs experimentally observed in the DNA molecular ch ain in the form of bubble, has been found stable\nwhen it propagates, however its amplitude decreases due to dampin g effect. Our numerical simulations have confirmed\nthe validity of the analytical approximate results.\n9References\nReferences\n[1] S.W. Englander et al., Proc. Natl. Acad. Sci. U.S.A. 77, 7222 (1980).\n[2] S. Yomosa, Phys. Rev. A 27, 2120 (1983).\n[3] S. Homma and S. Takeno, Prog. Theor. Phys. 70, 308 (1983);.\n[4] S. Takeno and S. Homma, Prog. Theor. Phys. 72, 679 (1984).\n[5] M. Peyrard and A.R. Bishop, Phys. Rev. Lett. 62, 2755 (1989).\n[6] T. Dauxois, M. Peyrard, and A.R. Bishop, Phys. Rev. E 47, R44 (1993).\n[7] T. Dauxois and M. Peyrard, Phys. Rev. E 51, 4027 (1995).\n[8] A. Campa and A. Giansanti, Phys. Rev. E 58, 3585 (1998).\n[9] T. Dauxois, Phys. Lett. A. l59, 390 (l991).\n[10] S. Zdravkovi´ c and M.V. Satari´ c, Chin. Phys. Lett. 24, 1210 (2003).\n[11] S. Zdravkovi´ c M.V. Satari´ c and L. Hadˇ zievski, Chaos 20, 043141 (2010).\n[12] C. Calladine et al., Understanding DNA, 3rd edition (Academic Press, London, 2004) .\n[13] N.N. Shafranovskaya et al., Pis’ma. Zh. Eksp. Teor. Fiz. 15, 404 (1972).\n[14] D.C. Rau and V.A. Parsegian, Biophysics J. 61, 246 (1992).\n[15] Y. Ishimori, Prog. Theor. Phys. 68, 402 (1982).\n[16] Yu. B. Gaididei, S.F. Mingaleev, P.L. Christiansen and K. ∅. Rasmussen, Phys. Lett. A. 222, 152 (1996).\n[17] S. F. Mingaleev, Yu.B. Gaididei, and F.G. Mertens, Phys. Rev. E. 58, 3833 (1998).\n[18] A. Mvogo, G.H. Ben-Bolie and T.C. Kofan´ e, Phys. Lett. A. 378, 2509 (2014).\n[19] A. Mvogo, G.H. Ben-Bolie and T.C. Kofan´ e, Chaos 25, 063115 (2015).\n[20] G. Miloshevich, J.P. Nguenang, T. Dauxois, R. Khomeriki and S. R uffo, J. Phys. A: Math. Theor. 50, 12LT02\n(2017).\n[21] N.R. Pereira and L. Stenflo, Phys. Fluids 20, 1733 (1977).\n[22] S. Flach, Phys. Rev. E 58, R4116 (1998).\n[23] Y.B. Gaididei, S.F. Mingaleev, P.L. Christiansen and K. ∅Rasmussen, Phys. Rev. E 55, 6141 (1997).\n[24] M. Peyrard, Nonlinearity 17, R1 (2004).\n10[25] E. Ar´ evalo and F.G. Mertens, Phys. Rev. E. 67, 016610 (2003).\n[26] C. Brunhuber and F.G. Mertens, Phys. Rev. E. 73, 016614 (2006).\n[27] I. Daumont and M. Peyrard, Chaos 13, 624 (2003).\n[28] M. Peyrard and I. Daumont, Europhys. Lett., 59834 (2002).\n[29] E. Ar´ evalo Yu. Gaididei and F.G. Mertens, Eur. Phys. J. B 27, 63 (2002)\n[30] J.W. Boyle, S. A. Nikitov, A.D. Boardman, J.G. Booth and K. Booth , Phys. Rev. B 53, 12173 (1996)\n[31] L.M. Hocking and K. Stewartson, Proc. R. Soc. London, Ser. A 326, 289 (1972).\n[32] N. Akhmediev and A. Ankiewicz: Solitons of the Complex Ginzburg-Landau Equation in Spatial Solitons,\nEdited by S. Trillo, p. 311 (Springer, New York, 2002).\n[33] A.C. Scott, F.Y.F. Chu and D.W. McLaughlin, Proc. IEEE. E. 61, 1443 (1973).\n[34] B.F. Putnam, L.L. Van Zandt, E.W. Prohofsky, K.C. Lu, and W. N. Mei, Biophys. J. 35, 271 (1981); E.W.\nProhofsky, K.C. Lu, L.L. Van Zandt and B.F. Putnam, Phys. Lett. A 70, 492 (1979).\n0π/4π/2 3 π/4π5π/4 3π/2 7π/4 2 π\nqnr00.0050.010.0150.020.0250.030.0350.040.0450.05ωr (t.u.-1)(a)\n0π/4π/2 3 π/4π5π/4 3π/2 7π/4 2 π\nqnr-0.4-0.35-0.3-0.25-0.2-0.15-0.1-0.05ωi (t.u.-1)(b)\n0π/4π/2 3π/4π5π/4 3π/2 7π/4 2 π\nqnr-16-14-12-10-8-6-4-20Pr (Å2/t.u.)(c)\n0π/4π/2 3 π/4π5π/4 3π/2 7π/4 2 π\nqnr-0.5-0.4-0.3-0.2-0.100.1Pi (Å2/t.u.)(d)\nFigure 1: (Color online) (a) and (b) the real and imaginary pa rt of the dispersion relation (Eq. ( 16)) in the linear limit F1→0. (c) and (d) the\nreal and imaginary part of the second discrete derivative of the dispersion relation (Eq. ( 29)) (Dispersion coefficient) in terms of the discretized\nvalues of the wave vector qnr= 2πn/N,N= 600 for s= 3.00,γ0= 0.15t.u−1.\n11π/4 π/2 3 π/4 π5π/4 3 π/2 7 π/4\nqnr00.10.20.30.40.50.60.70.80.91PrQ\nFigure 2: (Color online) The product PrQin terms of the the discretized values of the wave vector qnr= 2πn/N,N= 600, for s= 3.00,\nγ0= 0.15t.u−1\n0 100 200 300 400 500 600\nBase pairs (n)-0.06-0.04-0.0200.020.040.060.08vn (Å)(b): t=115\nFigure 3: (Color online) (a) Analytical stretching of the nu cleotide pair as a function of the time and the number of base p airs. (b) stretching\nof the nucleotide pair as a function of the number of base pair s att= 115 for s= 3.00,ε= 0.9,ue= 1,γ0= 0.15t.u−1,uc= 0.45ueand\nq0\nnr=π\n16.\n120 100 200 300 400 500 600\nBase pairs (n)-0.08-0.06-0.04-0.0200.020.040.060.08vn (Å)(b): t=115\nFigure 4: (Color online) (a) Numerical stretching of the nuc leotide pair as a function of the time and the number of base pa irs. (b) stretching\nof the nucleotide pair as a function of the number of base pair s att= 115 for s= 3.00,ε= 0.9,ue= 1,γ0= 0.15t.u−1,uc= 0.45ueand\nq0\nnr=π\n16.\n0 100 200 300 400 500 600\nBase pairs (n)-0.08-0.06-0.04-0.0200.020.040.060.080.1vn (Å)(a): t=0\n0 100 200 300 400 500 600\nBase pairs (n)-0.08-0.06-0.04-0.0200.020.040.060.080.1vn (Å)(b): t=50\n0 100 200 300 400 500 600\nBase pairs (n)-0.08-0.06-0.04-0.0200.020.040.060.08vn (Å)(c): t=200\n0 100 200 300 400 500 600\nBase pairs (n)-0.08-0.06-0.04-0.0200.020.040.060.08vn (Å)(d): t=300\nFigure 5: (Color online) Comparison between analytical and numerical solution of the EOM Eq. ( 11) at different time positions for s= 3.00,\nε= 0.9,ue= 1,γ0= 0.15t.u−1,uc= 0.45ueandq0\nnr=π\n16. (solid blue line) the numerical solution. (dash red line) t he analytical solution.\n130 100 200 300 400 500 600\nBase pairs (n)-0.04-0.0200.020.040.060.080.10.12vn (Å)(a): t=0\n0 100 200 300 400 500 600\nBase pairs (n)-0.04-0.0200.020.040.060.080.10.12vn (Å)(b): t=50\n0 100 200 300 400 500 600\nBase pairs (n)-0.0500.050.1vn (Å)(c): t=200\n0 100 200 300 400 500 600\nBase pairs (n)-0.0500.050.1vn (Å)(d): t=300\nFigure 6: (Color online) Comparison between analytical and numerical solution of the EOM Eq. ( 11) at different time positions for s= 3.00,\nε= 0.9,ue= 1,γ0= 0.15t.u−1,uc= 0.45ueandq0\nnr=π\n18. (solid blue line) the numerical solution. (dash red line) t he analytical solution.\n14"    },    {        "title": "1707.09583v3.Blow_up_for_semilinear_damped_wave_equations_with_sub_Strauss_exponent_in_the_scattering_case.pdf",        "content": "arXiv:1707.09583v3  [math.AP]  25 Aug 2017Blow-up for semilinear damped wave equations\nwith sub-Strauss exponent\nin the scattering case\nNing-An Lai∗Hiroyuki Takamura†\nKeywords: damped wave equation, semilinear, blow-up, lifespan\nMSC2010: primary 35L71, secondary 35B44\nAbstract\nIt is well-known that the critical exponent for semilinear d amped\nwave equations is Fujita exponent when the damping is effectiv e. Lai,\nTakamura and Wakasa in 2017 have obtained a blow-up result no t\nonly for super-Fujita exponent but also for the one closely r elated\nto Strauss exponent when the damping is scaling invariant an d its\nconstant is relatively small, whichhas beenrecently exten dedby Ikeda\nand Sobajima.\nIntroducing a multiplier for the time-derivative of the spa tial in-\ntegral of unknown functions, we succeed in employing the tec hnics on\nthe analysis for semilinear wave equations and proving a blo w-up re-\nsult for semilinear damped wave equations with sub-Strauss exponent\nwhen the damping is in the scattering range.\n1 Introduction\nIn this paper, we consider the following initial value problem\n/braceleftBigg\nutt−∆u+µ\n(1+t)βut=|u|pinRn×[0,∞),\nu(x,0) =εf(x), ut(x,0) =εg(x), x∈Rn,(1.1)\n∗Department of Mathematics, Lishui University, Lishui City 323000 , China. e-mail:\nhyayue@gmail.com.\n†Department of Complex and Intelligent Systems, Faculty of System s Information Sci-\nence, Future University Hakodate, 116-2 Kamedanakano-cho, H akodate, Hokkaido 041-\n8655, Japan. e-mail: takamura@fun.ac.jp.\n1whereµ>0, n∈N, β∈Randβ >1. We assume that ε>0 is a “small”\nparameter.\nFirst, we shall outline a background of (1.1) briefly according to the clas-\nsifications of Wirth [31, 32, 33] for the corresponding linear problem . Let\nu0be a solution of the initial value problem for the following linear damped\nwave equation\n/braceleftBigg\nu0\ntt−∆u0+µ\n(1+t)βu0\nt= 0,inRn×[0,∞),\nu0(x,0) =u1(x), u0\nt(x,0) =u2(x), x∈Rn,(1.2)\nwhereµ >0,β∈R,n∈Nandu1,u2∈C∞\n0(Rn). Whenβ∈(−∞,−1),\nwe say that the damping term is “overdamping”, in which case the solu tion\ndoes not decay to zero when t→ ∞. Whenβ∈[−1,1), the solution behaves\nlike that of the heat equation, which means that the term u0\nttin (1.2) has\nno influence on the behavior of the solution. In fact, Lp-Lqdecay estimates\nof the solution which are almost the same as those of the heat equat ion are\nestablished. In this case, we say that the damping term is “effective .” In\ncontrast, when β∈(1,∞), it is known that the solution behaves like that\nof the wave equation, which means that the damping term in (1.2) has no\ninfluence on the behavior of the solution. In fact, in this case the so lution\nscatters to that of the free wave equation when t→ ∞, and thus we say that\nwe have “scattering.” When β= 1, the equation in (1.2) is invariant under\nthe following scaling\n/tildewideu0(x,t) :=u0(σx,σ(1+t)−1), σ>0,\nand hence we say that the damping term is “scale invariant.” The rema rkable\nfact in this case is that the behavior of the solution of (1.2) is determ ined\nby the value of µ. Actually, for µ∈(0,1), it is known that the asymptotic\nbehavior of the solution is closely related to that of the free wave eq uation.\nForthisrangeof µ, wesaythatthedampingtermis“non-effective.” However,\nthe threshold of µaccording to the behavior of the solution is still open. In\nthis way, we may summarize all the classifications of the damping term in\n(1.2) in the following table.\nβ∈(−∞,−1) overdamping\nβ∈[−1,1) effective\nβ= 1scaling invariant\nµ∈(0,1)⇒non-effective\nβ∈(1,∞) scattering\n2Ifβ= 0, then we say that the damping term in (1.1) is classical, or of\nconstant coefficient case. In this case the equation is a good model to de-\nscribe the wave propagationwith the friction, such as the telegrap h equation,\nthe elastic vibration with the damping proportional to the velocity an d the\nheat conduction with the finite speed of the propagation. There is e xtensive\nliteratures on the question of the blow-up in a finite time or global-in-t ime\nexistence of the Cauchy problem of semilinear damped wave equation with\nconstant coefficients. Based on these works we now know that it ad mits a\ncritical power, the so-called Fujita exponent defined by\npF(n) := 1+2\nn, (1.3)\nwhich means that the solution will blow up in a finite time for p∈(1,pF(n)],\nand there is a global solution for p > p F(n) with small data. We list the\nrelated results (but maybe not all of them) in the following table.\n1<p<p F(n) p=pF(n) p>pF(n)\nLi & Zhou [21]\n(n= 1,2),\nTodorova &\nYordanov [26]Li & Zhou [21] ( n= 1,2),\nZhang [35], or indep.,\nKirane & Qafsaoui [18]Todorova &\nYordanov [26]\n(p≤n/(n−2)\nforn≥3)\nIf the solution blows up in a finite time, people then are interested in th e\nlifespan estimate, the maximal existence time of the energy solution of (1.1)\nfor arbitrarily fixed ( f,g). We denote it by T(ε). Now we know the estimate\nof lifespan will be\nT(ε)∼/braceleftbigg\nexp/parenleftbig\nCε−(p−1)/parenrightbig\nforp=pF(n),\nCε−2(p−1)/(2−n(p−1))for 1<p<p F(n),(1.4)\nwhereT(ε)∼A(C,ε) stands for the fact that the following estimate holds\nwith positive constants C1,C2independent of ε;\nA(C1,ε)≤T(ε)≤A(C2,ε).\nTo our best knowledge, we have the following table,\nβ= 0 1< p < p F(n) p=pF(n)\nn≤2U: Li & Zhou [21], or indep.,\nIkeda & Wakasugi [12]\nL: Fujiwara & Ikeda & Wakasugi [6]U: Li & Zhou [21]\nL: Ikeda & Ogawa [10]\nn= 3U: Ikeda & Wakasugi [12]\nL: Fujiwara & Ikeda & Wakasugi [6]U: Nishihara [23]\nL: Ikeda & Ogawa [10]\nn≥4U: Ikeda and Wakasugi [12]\nL: Fujiwara & Ikeda & Wakasugi [6]U: Lai & Zhou [20]\nL: Ikeda & Ogawa [10]\n3where “U”, or “L”, denotes the upper bound, or the lower bound, of the\nlifespan estimate respectively.\nRecently, the Cauchy problem for the damped wave equation with va ri-\nable coefficients and the power type source term, (1.1), attracts more and\nmore attention. We want to see whether this equation still has a diffu sive\nstructure. Generally speaking, based on the classification for the linear prob-\nlem, (1.2), mentioned above, one may expect that if the coefficient µ/(1+t)β\ndecays not so fast, then thedamping is effective, i.e. the solution be haves like\nthat of the corresponding nonlinear heat equation. And if µ/(1+t)βdecays\nsufficiently fast, then the damping becomes non-effective, i.e. the s olution\nbehaves like that of the nonlinear wave equation. Essentially, it is tot ally\ndetermined by the constants µandβ. Till now, most of the known results\nfocus on the range β∈(−∞,1].\nWhenβ∈(−1,1), the global existence, i.e. T(ε) =∞, has been obtained\nby D’Abbicco, Lucente and Reissig [4] for pF(n)<p<n/ [n−2]+, where\nm\n[n−2]+:=/braceleftbigg∞forn= 1,2,\nm/(n−2) forn≥3.\nNishihara [24]hasextended thisupper boundto( n+2)/[n−2]+forβ∈[0,1).\nLin, Nishihara and Zhai [22] have removed this restriction on β. When\nβ=−1, Wakasugi [30] has obtained the global existence for pF(n)< p <\nn/[n−2]+. Inthe counter parts on β∈[−1,1), we have the following lifespan\nestimates by Ikeda and Ogawa [10], Fujiwara, Ikeda and Wakasugi [6 ], Ikeda\nand Inui [9].\nT(ε) 1<p<p F(n) p=pF(n)\nβ=−1∼exp/parenleftbig\nCε−2(p−1)/(2−n(p−1))/parenrightbig\n[6]∼exp/parenleftbig\nexp(Cε−(p−1))/parenrightbig\nL: [6], U: [9]\nβ∈(−1,1)\n(β/ne}ationslash= 0)∼Cε−2(p−1)/{(2−n(p−1))(1+β)}[6]∼exp/parenleftbig\nCε−(p−1)/parenrightbig\nL: [6], U: [9]\nSo we can conclude that the critical exponent is still pF(n) in (1.3) which is\nthe one for semilinear heat equation, ut−∆u=|u|p. Whenβ <−1, Ikeda\nand Wakasugi [13] have recently proved that the global existence holds for\nanyp>1.\nWhenβ= 1, the situation is a bit complicated. It seems to be the\nthreshold between effective and non-effective damping. Actually, t he other\nconstantµalso plays a crucial role in this case. D’Abbicco and Lucente [2],\nand D’Abbicco [1] have showed that the critical power is pF(n) when\nµ≥\n\n5/3 forn= 1,\n3 forn= 2,\nn+2 forn≥3.\n4Noting that µ= 2 is an exceptional case, since after making the so-called\nLiouville transform\nw(x,t) := (1+t)µ/2u(x,t),\nthen problem (1.1) can be rewritten as\n\n\nwtt−∆w+µ(2−µ)\n4(1+t)2w=|w|p\n(1+t)µ(p−1)/2inRn×[0,∞),\nw(x,0) =εf(x), wt(x,0) =ε{(µ/2)f(x)+g(x)}, x∈Rn.(1.5)\nWhenβ= 1 andµ= 2, it is natural to think that the critical power is\nrelated to the so-called Strauss exponent pS(n) which is defined for n>1 as\na positive root of the quadratic equation,\nγ(p,n) := 2+(n+1)p−(n−1)p2= 0. (1.6)\nWe note that\npF(n)<pS(n) =n+1+√\nn2+10n−7\n2(n−1)forn≥2\nand thatpS(n) is the critical power for semilinear wave equations, utt−∆u=\n|u|p. D’Abbicco, Lucente and Reissig [5] have determined a critical power\npc(n) := max {pF(n),pS(n+2)}forn≤3.\nD’Abbicco and Lucente [3] have also showed the global existence for pS(n+\n2)< p <1 + 2/max{2,(n−3)/2}to odd and higher dimensions( n≥5)\nunder the spherically symmetric assumption. In the blow-up case, w e have\nresults about the lifespan estimate as in the following table.\nT(ε) 1<p<p c(n) p=pc(n)\nn= 1∼\n\nCε−(p−1)/(3−p)\nforIf,g/ne}ationslash= 0,\nCε−p(p−1)/(3−p)\nforIf,g= 0 and 2<p,\nCb(ε)\nforIf,g= 0 andp= 2,\nCε−2p(p−1)/γ(p,3)\nforIf,g= 0 andp<2∼\n\nexp/parenleftbig\nCε−(p−1)/parenrightbig\nforIf,g/ne}ationslash= 0,\nexp/parenleftbig\nCε−p(p−1)/parenrightbig\nforIf,g= 0\nn= 2 ∼\n\nCε−(p−1)/(4−2p)\nforIf,g/ne}ationslash= 0,\nCε−2p(p−1)/γ(p,4)\nforIf,g= 0∼\n\nexp/parenleftbig\nCε−(p−1)/parenrightbig\nforIf,g/ne}ationslash= 0,\nexp/parenleftbig\nCε−p(p−1)/parenrightbig\nforIf,g= 0\nn= 3 ∼Cε−2p(p−1)/γ(p,n+2)∼exp/parenleftbig\nCε−p(p−1)/parenrightbig\n5whereb=b(ε) is a positive number satisfying bε2log2(b+1) = 1 and\nIf,g:=/integraldisplay\nRn{f(x)+g(x)}dx.\nSee Wakasa [27], and Kato, Takamura, Wakasa [17] for n= 1, Imai, Kato,\nTakamura and Wakasa [15] for n= 2, Kato and Sakuraba [16] for n= 3.\nWhenβ= 1 andµ/ne}ationslash= 2, due to the work of Wakasugi [29], we may\nbelieve that the solution is “heat-like”(the critical power is Fujita ex ponent)\nforµ>1. Actually he established the following lifespan estimates,\n\n\nT(ε)≤Cε−(p−1)/{2−n(p−1)}\nfor 1<p<p F(n) andµ≥1,\nT(ε)≤Cε−(p−1)/{2−(n+µ−1)(p−1)}\nfor 1<p<p F(n+µ−1) and 0<µ<1(1.7)\nas well asT(ε)<∞forp=pF(n) andµ≥1, and forp=pF(n+µ−1)\nand 0< µ <1. Surprisingly, the authors and Wakasa [19] found that the\nsolution is “wave-like”(the critical exponent is bigger than Fujita ex ponent\nand is related to the Strauss exponent) in some case even for µ>1, by using\nthe improved Kato’s lemma introduced by the second author [25]. Act ually\nwe have the estimate,\nT(ε)≤Cε−2p(p−1)/γ(p,n+2µ)\nforp<pS(n+2µ) and 0<µ<n2+n+2\n2(n+2).(1.8)\nWe note that (1.8) is stronger than (1.7) for n≥2. Very recently, we have\nbeen informed that Ikeda and Sobajima [11] extend this result to\nT(ε)≤\n\nexp/parenleftbig\nCε−p(p−1)/parenrightbig\nforp=pS(n+µ),\nCδε−2p(p−1)/γ(p,n+µ)−δforpS(n+2+µ)≤p<pS(n+µ),\nCδε−1−δforpF(n)<p<p S(n+2+µ)\nwhen\nn≥2 and 0 ≤µ<n2+n+2\nn+2,\nand\nT(ε)≤\n\nexp/parenleftbig\nCε−p(p−1)/parenrightbig\nforp=pS(1+µ),\nCδε−2p(p−1)/γ(p,1+µ)−δfor max{3,2/µ} ≤p<pS(1+µ),\nCδε−2(p−1)/µ−δfor 0<µ<2/3,3≤p<2/µ\nwhenn= 1 and 0<µ<4/3, with arbitrary small δ >0.\n6Remark 1.1 For the scale invariant case, β= 1, it is still open to determine\nthe critical exponent. we also remark that for β= 1, the case of µ≈1is\nrelated to the semilinear generalized Tricomi equation, wh ich comes from the\ngas dynamics, see [7, 8].\nIn this paper, we are devoted to studying the Cauchy problem (1.1) with\nβ >1. Due to the authors’ best knowledge, this problem is completely\nopen. As mentioned above, the corresponding linear problem belong s to\nthe scattering case. We then expect that the solution behaves like that\nof the semilinear wave equation without the damping term. We are main ly\nconcerned abouttheblow-upresultandupper boundofthelifespa nestimate.\nThe novelty is that we introduce a multiplier of exponential type, whic h is\nbounded from above and below. Then we get the lower bound of the t ime-\nderivative of the spatial integral of the unknown function by the s pace-time\nintegralofthenonlinearterm. Finally, thedesiredblow-upresultan dlifespan\nestimate for sub-Strauss exponent are established by an iteratio n argument.\nRemark 1.2 Compared to the scale invariant case, β= 1, the main diffi-\nculty is that we cannot use Liouville transform to rewrite th e equation in a\nform of nonlinear wave or Klein-Gordon equation. See (1.5). We overcome\nthis obstacle by introducing a “good” multiplier.\nWe organize this paper infive sections. In Section 2, we give main resu lts.\nIn Section 3 the key multiplier is introduced, and the lower bound of th e\nnonlinear term is obtained. In Section 4, we obtain the blow-up result and\nthe upper bound of the lifespan for sub-Strauss exponent by an it eration\nargument. We also give improvements of estimates of the lifespan fo r one\ndimensional case and two dimensional case with low powers in Section 5\nunder an additional assumption on the initial speed.\n2 Main Result\nAs in the work [19], we first define the energy and weak solution of the\nCauchy problem (1.1).\nDefinition 2.1 We say that uis an energy solution of (1.1)over[0,T)if\nu∈C([0,T),H1(Rn))∩C1([0,T),L2(Rn))∩Lp\nloc(Rn×(0,T)) (2.1)\n7satisfies\n/integraldisplay\nRnut(x,t)φ(x,t)dx−/integraldisplay\nRnut(x,0)φ(x,0)dx\n+/integraldisplayt\n0ds/integraldisplay\nRn{−ut(x,s)φt(x,s)+∇u(x,s)·∇φ(x,s)}dx\n+/integraldisplayt\n0ds/integraldisplay\nRnµut(x,s)\n(1+s)βφ(x,s)dx=/integraldisplayt\n0ds/integraldisplay\nRn|u(x,s)|pφ(x,s)dx(2.2)\nwith anyφ∈C∞\n0(Rn×[0,T))and anyt∈[0,T).\nEmploying the integration by parts in (2.2) and letting t→T, we have\nthat\n/integraldisplay\nRn×[0,T)u(x,s)/braceleftbigg\nφtt(x,s)−∆φ(x,s)−/parenleftbiggµφ(x,s)\n(1+s)β/parenrightbigg\ns/bracerightbigg\ndxds\n=/integraldisplay\nRnµu(x,0)φ(x,0)dx−/integraldisplay\nRnu(x,0)φt(x,0)dx\n+/integraldisplay\nRnut(x,0)φ(x,0)dx+/integraldisplay\nRn×[0,T)|u(x,s)|pφ(x,s)dxds.\nThis is exactly the definition of the weak solution of (1.1).\nOur main results are stated in the following three theorems.\nTheorem 2.1 Letβ >1and\n1<p</braceleftbiggpS(n)forn≥2,\n∞forn= 1.\nAssume that both f∈H1(Rn)andg∈L2(Rn)are non-negative, and fdoes\nnot vanish identically. Suppose that an energy solution uof (1.1) satisfies\nsuppu⊂ {(x,t)∈Rn×[0,T) :|x| ≤t+R} (2.3)\nwith someR≥1. Then, there exists a constant ε0=ε0(f,g,n,p,µ,β,R )>0\nsuch thatThas to satisfy\nT≤Cε−2p(p−1)/γ(p,n)(2.4)\nfor0<ε≤ε0, whereCis a positive constant independent of ε.\nRemark 2.1 In (2.4)for n= 1, wenote that γ(p,1) = 2+2pbyits definition\n(1.6).\n8In low dimensions, we have improvements on the lifespan estimates as\nfollows.\nTheorem 2.2 Letn= 2and1<p<2. Assume that the initial data satisfy\nthe same condition as that in Theorem 2.1. If\n/integraldisplay\nR2g(x)dx/ne}ationslash= 0, (2.5)\nthen (2.4) is replaced by\nT≤Cε−(p−1)/(3−p). (2.6)\nRemark 2.2 (2.6) is stronger than (2.4) by the fact that 1< p <2is\nequivalent to\np−1\n3−p<2p(p−1)\nγ(p,2).\nTheorem 2.3 Letn= 1andp>1. Assume that the initial data satisfy the\nsame condition as that in Theorem 2.1. If\n/integraldisplay\nRg(x)dx/ne}ationslash= 0, (2.7)\nthen (2.4) is replaced by\nT≤Cε−(p−1)/2. (2.8)\nRemark 2.3 (2.8) is stronger than (2.4) by the fact that p>1is equivalent\nto\np−1\n2<2p(p−1)\nγ(p,1).\nRemark 2.4 These results in the theorems above are the same as those of\nCauchy problem of semilinear wave equations, utt−∆u=|u|p, except for the\ncase of\nn=p= 2and/integraldisplay\nR2g(x)dx/ne}ationslash= 0.\nFor example, see the introductions in Takamura [25] and Imai, Kato, Taka-\nmura and Wakasa [14].\n93 Lower Bound of the Functional\nIn this section we introduce the multiplier for our problem. Let\nm(t) := exp/parenleftbigg\nµ(1+t)1−β\n1−β/parenrightbigg\n. (3.1)\nThen it is easy to see that for β >1 we have\n1≥m(t)≥m(0) fort≥0, (3.2)\nwhich means that m(t) is bounded from both above and below.\nRemark 3.1 If one puts\nm(t) = exp(µlog(1+t)) = (1+t)µ\ninstead of (3.1) in the proof below, it will give us a simple pr oof of the result\nof Lai, Takamura and Wakasa [19] which is cited in (1.8).\nSet\nF0(t) :=/integraldisplay\nRnu(x,t)dx.\nChoosing the test function φ=φ(x,s) in (2.2) to satisfy φ≡1 in{(x,s)∈\nRn×[0,t] :|x| ≤s+R}, we get\n/integraldisplay\nRnut(x,t)dx−/integraldisplay\nRnut(x,0)dx\n+/integraldisplayt\n0ds/integraldisplay\nRnµut(x,s)\n(1+s)βdx=/integraldisplayt\n0ds/integraldisplay\nRn|u(x,s)|pdx,\nwhich means that\nF′′\n0+µ\n(1+t)βF′\n0=/integraldisplay\nRn|u(x,t)|pdx. (3.3)\nMultiplying the both sides of (3.3) with m(t) yields\n{m(t)F′\n0}′=m(t)/integraldisplay\nRn|u(x,t)|pdx. (3.4)\nWe then get the lower bound of F′\n0(t) by integrating (3.4) over [0 ,t]\nF′\n0(t)≥m(0)/integraldisplayt\n0ds/integraldisplay\nRn|u(x,s)|pdxfort≥0, (3.5)\n10where we used the fact that F0(0)>0,F′\n0(0)≥0 and (3.2).\nLet\nF1(t) :=/integraldisplay\nRnu(x,t)ψ1(x,t)dx,\nwhere\nψ1(x,t) :=e−tφ1(x), φ1(x) :=\n\n/integraldisplay\nSn−1ex·ωdSωforn≥2,\nex+e−xforn= 1.\nThis is a test function introduced by Yordanov and Zhang [34].\nLemma 3.1 (Inequality (2.5) of Yordanov and Zhang [34])\n/integraldisplay\n|x|≤t+R[ψ1(x,t)]p/(p−1)dx≤C(1+t)(n−1){1−p/(2(p−1))},(3.6)\nwhereC=C(n,p,R)>0.\nBy H¨ older’s inequality and (3.6), one has\n/integraldisplay\nRn|u(x,t)|pdx≥C1(1+t)(n−1)(1−p/2)|F1(t)|pfort≥0,(3.7)\nwhereC1=C1(n,p,R)>0. Now we are left with the lower bound of F1(t).\nWe start with the definition of the energy solution (2.2), which yields t hat\nd\ndt/integraldisplay\nRnut(x,t)φ(x,t)dx\n+/integraldisplay\nRn{−ut(x,t)φt(x,t)−u(x,t)∆φ(x,t)}dx\n+/integraldisplay\nRnµut(x,t)\n(1+t)βφ(x,t)dx=/integraldisplay\nRn|u(x,t)|pφ(x,t)dx.\nMultiplying the both sides of the above equality with m(t), we have that\nd\ndt/braceleftbigg\nm(t)/integraldisplay\nRnut(x,t)φ(x,t)dx/bracerightbigg\n+m(t)/integraldisplay\nRn{−ut(x,t)φt(x,t)−u(x,t)∆φ(x,t)}dx\n=m(t)/integraldisplay\nRn|u(x,t)|pφ(x,t)dx.\n11Integrating this equality over [0 ,t], we get\nm(t)/integraldisplay\nRnut(x,t)φ(x,t)dx−m(0)ε/integraldisplay\nRng(x)φ(x,0)dx\n−/integraldisplayt\n0ds/integraldisplay\nRnm(s)ut(x,s)φt(x,s)dx\n=/integraldisplayt\n0ds/integraldisplay\nRn{m(s)u(x,s)∆φ(x,s)+m(s)|u(x,s)|pφ(x,s)}dx.\nIf we put\nφ(x,t) =ψ1(x,t) =e−tφ1(x) on supp u,\nwe have\nφt=−φ, φtt= ∆φon suppu.\nHence we obtain that\nm(t){F′\n1(t)+2F1(t)}−m(0)ε/integraldisplay\nRn{f(x)+g(x)}φ(x)dx\n=/integraldisplayt\n0m(s)µ\n(1+s)βF1(s)ds+/integraldisplayt\n0ds/integraldisplay\nRnm(s)|u(x,s)|pdx,\nwhich yields\nF′\n1(t)+2F1(t)≥m(0)\nm(t)Cf,gε+1\nm(t)/integraldisplayt\n0m(s)µ\n(1+s)βF1(s)ds\n≥m(0)Cf,gε+1\nm(t)/integraldisplayt\n0m(s)µ\n(1+s)βF1(s)ds,\nwhere\nCf,g:=/integraldisplay\nRn{f(x)+g(x)}φ1(x)dx>0.\nIntegrating the above inequality over [0 ,t] with a multiplication by e2t, we\nget\ne2tF1(t)≥F1(0)+m(0)Cf,gε/integraldisplayt\n0e2sds\n+/integraldisplayt\n0e2s\nm(s)ds/integraldisplays\n0m(r)µ\n(1+r)βF1(r)dr.(3.8)\nDue to a comparison argument, we have that F1(t)>0 fort≥0. Actually,\nF1(0)>0 and the continuity of F1(t) intyield thatF1(t)>0 for small\nt >0. If there is the nearest zero point t0tot= 0 ofF1, then (3.8) gives a\ncontradiction at t0.\n12Therefore we obtain that\ne2tF1(t)≥F1(0)+m(0)Cf,gε/integraldisplayt\n0e2sds\n≥m(0)F1(0)+1\n2m(0)Cf,0ε(e2t−1)\n>1\n2m(0)Cf,0εe2t\nbecause of\nF1(0) =Cf,0ε, C f,g≥Cf,0,\nwhich in turn gives us the lower bound of F1(t),\nF1(t)>1\n2m(0)Cf,0εfort≥0. (3.9)\n4 Proof for Theorem 2.1\nBy H¨ older inequality with (2.3), it is easy to get\n/integraldisplay\nRn|u(x,t)|pdx≥C2(1+t)−n(p−1)|F0(t)|pfort≥0,(4.1)\nwhereC2=C2(n,p,R)>0. Then it follows from (3.5) and (4.1) that\nF0(t)>C3/integraldisplayt\n0ds/integraldisplays\n0(1+r)−n(p−1)F0(r)pdrfort≥0,(4.2)\nwhere\nC3:=C2m(0)>0.\nPlugging (3.9) into (3.5) with (3.7), we have\nF′\n0(t)≥C4εp/integraldisplayt\n0(1+s)(n−1)(1−p/2)dsfort≥0, (4.3)\nwhere\nC4:=C1m(0)/parenleftbigg1\n2m(0)Cf,0/parenrightbiggp\n.\nIntegrating (4.3) over [0 ,t], we have\nF0(t)>C4εp/integraldisplayt\n0dr/integraldisplayr\n0(1+s)(n−1)(1−p/2)ds\n≥C4εp(1+t)−(n−1)p/2/integraldisplayt\n0dr/integraldisplayr\n0sn−1ds\n=C4\nn(n+1)εp(1+t)−(n−1)p/2tn+1fort≥0.(4.4)\n13Next we will begin our iteration argument. First we may assume that\nF0(t) satisfies\nF0(t)>Dj(1+t)−ajtbjfort≥0 (j= 1,2,3···) (4.5)\nwith positive constants, Dj,aj,bj, which will be determined later. Due to\n(4.4), noting that (4.5) is true with\nD1=C4\nn(n+1)εp, a1= (n−1)p\n2, b1=n+1. (4.6)\nPlugging (4.5) into (4.2), we have\nF0(t)>C3Dp\nj/integraldisplayt\n0ds/integraldisplays\n0(1+r)−n(p−1)−pajrpbjdr\n>C3Dp\nj(1+t)−n(p−1)−paj/integraldisplayt\n0ds/integraldisplays\n0rpbjdr\n>C3Dp\nj\n(pbj+2)2(1+t)−n(p−1)−pajtpbj+2fort≥0.\nSo we can define the sequences {Dj},{aj},{bj}by\nDj+1≥C3Dp\nj\n(pbj+2)2, aj+1=paj+n(p−1), bj+1=pbj+2 (4.7)\nto establish\nF0(t)>Dj+1(1+t)−aj+1tbj+1fort≥0.\nIt follows from (4.6) and (4.7) that\naj=pj−1/parenleftBig\n(n−1)p\n2+n/parenrightBig\n−nforj= 1,2,3,···\nand\nbj=pj−1/parenleftbigg\nn+1+2\np−1/parenrightbigg\n−2\np−1forj= 1,2,3,···.\nIf we employ the inequality\nbj+1=pbj+2≤pj/parenleftbigg\nn+1+2\np−1/parenrightbigg\nin (4.7), we have\nDj+1≥C5Dp\nj\np2j, (4.8)\n14where\nC5:=C3/parenleftbigg\nn+1+2\np−1/parenrightbigg2.\nFrom (4.8) it holds that\nlogDj≥plogDj−1−2(j−1)logp+logC5\n≥p2logDj−2−2/parenleftbig\np(j−2)+(j−1)/parenrightbig\nlogp+(p+1)logC5.\nRepeating this procedure, we have\nlogDj≥pj−1logD1−j−1/summationdisplay\nk=12klogp−logC5\npk,\nwhich yields that\nDj≥exp/braceleftbig\npj−1(logD1−Sp(j))/bracerightbig\n,\nwhere\nSp(j) :=j−1/summationdisplay\nk=12klogp−logC5\npk.\nBy d’Alembert’s criterion we know that Sp(j) converges for p>1 asj→ ∞.\nHence we obtain that\nDj≥exp/braceleftbig\npj−1(logD1−Sp(∞))/bracerightbig\n.\nTurning back to (4.5), we have\nF0(t)≥(1+t)nt−2/(p−1)exp/parenleftbig\npj−1J(t)/parenrightbig\nfort>0,(4.9)\nwhere\nJ(t) =−/parenleftBig\n(n−1)p\n2+n/parenrightBig\nlog(1+t)+/parenleftbigg\nn+1+2\np−1/parenrightbigg\nlogt\n+logD1−Sp(∞).\nFort≥1, by the definition of J(t) we have\nJ(t)≥ −/parenleftBig\n(n−1)p\n2+n/parenrightBig\nlog(2t)+/parenleftbig\nn+1+2\np−1/parenrightbig\nlogt\n+logD1−Sp(∞)\n=γ(p,n)\n2(p−1)logt+logD1−/parenleftBig\n(n−1)p\n2+n/parenrightBig\nlog2−Sp(∞)\n= log/parenleftbig\ntγ(p,n)/{2(p−1)}D1/parenrightbig\n−C6,\n15where\nC6:=/parenleftBig\n(n−1)p\n2+n/parenrightBig\nlog2+Sp(∞)>0.\nThus, if\nt>C7ε−2p(p−1)/γ(p,n)\nwith\nC7:=/parenleftBign(n+1)eC6+1\nC4/parenrightBig2(p−1)/γ(p,n)\n,\nwe then get J(t)>1, and this in turn gives that F0(t)→ ∞by letting\nj→ ∞in (4.9). Therefore we get the desired upper bound,\nT≤C7ε−2p(p−1)/γ(p,n),\nand hence we finish the proof of Theorem 2.1.\n5 Proof for Theorem 2.2 and Theorem 2.3\nDue to (2.5), (3.2) and (3.5), integrating (3.4) over [0 ,t] yields\nF′\n0(t)≥m(t)F′\n0(t)≥C8ε,\nwhere\nC8:=m(0)/integraldisplay\nRng(x)dx.\nThe above inequality implies that\nF0(t)≥C9ε(1+t) fort≥0, (5.1)\nwhere\nC9:= min/braceleftbigg\nC8,/integraldisplay\nRnf(x)dx/bracerightbigg\n.\nFirst we prove Theorem 2.2 for n= 2. Due to the assumption on g(x),\nwe note that /integraldisplay\nR2g(x)dx>0\nwhich yields C8,C9>0. By (4.1) and (5.1), we have\n/integraldisplay\nR2|u(x,t)|pdx≥C10εp(1+t)2−p, (5.2)\n16withC10:=C2Cp\n9>0. Plugging (5.2) into (3.5) and integrating it over [0 ,t]\nwe come to\nF0(t)≥C11εp/integraldisplayt\n0dr/integraldisplayr\n0(1+s)2−pds\n≥C11εp(1+t)1−p/integraldisplayt\n0dr/integraldisplayr\n0sds\n=C11\n6εp(1+t)1−pt3fort≥0(5.3)\nwithC11:=C10m(0)>0. Noting that the above inequality improves the\nlower bound of (4.4) for n= 2 and 1< p <2, and this is the key point to\nprove Theorem 2.2.\nAs in section 4, we define our iteration sequence, {/tildewideDj},{/tildewideaj},{/tildewidebj}, as\nF0(t)≥/tildewideDj(1+t)−/tildewideajt/tildewidebjfort≥0, j= 1,2,3··· (5.4)\nwith positive constants, /tildewideDj,/tildewideaj,/tildewidebj, and\n/tildewideD1=C11\n6εp,/tildewidea1=p−1,/tildewideb1= 3.\nCombining (4.2) and (5.4), we have\nF0(t)≥C3/tildewideDp\nj/integraldisplayt\n0dr/integraldisplayr\n0(1+s)2−2p−p/tildewideajsp/tildewidebjds\n≥C3/tildewideDp\nj\n(p/tildewidebj+2)2(1+t)2−2p−p/tildewideajtp/tildewidebj+2fort≥0.\nSo the sequences satisfy\n/tildewideaj+1=−p/tildewideaj−2(p−1),\n/tildewidebj+1=p/tildewidebj+2,\n/tildewideDj+1≥C3/tildewideDp\nj\n(p/tildewidebj+2)2≥C12/tildewideDp\nj\np2j,\nwhereC12:=C11/{3+2/(p−1)}2>0, which means that\n/tildewideaj=pj−1(p+1)−2,\n/tildewidebj=/parenleftbig\n3+2\np−1/parenrightbig\npj−1−2\np−1,\nlog/tildewideDj≥pj−1log/tildewideD1−j−1/summationdisplay\nk=12klogp−logC12\npk.\n17Then, as in Section 4, we have\nF0(t)≥/tildewideDj(1+t)−pj−1(p+1)+2tpj−1{3+2/(p−1)}−2/(p−1)\n≥(1+t)2t−2/(p−1)exp/parenleftbig\npj−1/tildewideJ(t)/parenrightbig\n,\nwhere\n/tildewideJ(t) :=−(p+1)log(1+ t)+/parenleftbigg\n3+2\np−1/parenrightbigg\nlogt+log/tildewideD1−/tildewideSp(∞)\nand\n/tildewideSp(∞) :=∞/summationdisplay\nk=12klogp−logC12\npk.\nEstimating /tildewideJ(t) as\n/tildewideJ(t)≥ −(p+1)log(2t)+/parenleftbigg\n3+2\np−1/parenrightbigg\nlogt+log/tildewideD1−/tildewideSp(∞)\n=/parenleftbigg\n−p(p−3)\np−1/parenrightbigg\nlogt+log/tildewideD1−/tildewideSp(∞)−(p+1)log2,\nwe obtain that\n/tildewideJ(t)≥log/parenleftBig\nt−p(p−3)/(p−1)/tildewideD1/parenrightBig\n−C13fort≥1,\nwhereC13:=/tildewideSp(∞)+(p+1)log2>0. By the definition of /tildewideD1, we then get\nthe lifespan estimate in Theorem 2.2 by the same way as that in section 4.\nNext we prove Thorem 2.3 for n= 1. The proof can be shown along the\nsame way as that of Theorem 2.2 for n= 2. First we note that (5.1) is also\navailable in this case. Then the first iteration in (5.3) for n= 2 becomes\nF0(t)≥C11εp/integraldisplayt\n0dr/integraldisplayr\n0(1+s)ds≥C11\n6εpt3fort≥0.\nWe then may assume the iteration sequences, {Dj},{aj},{bj}, as\nF0(t)>Dj(1+t)−ajtbjfort≥0 (j= 1,2,3···)\nwith non-negative constants, Dj,aj,bj, and\nD1=C11\n6εp,a1= 0,b1= 3.\nThe left steps to get the lifespan in Theorem 2.3 are exactly the same as that\nof Theorem 2.2.\n18Acknowledgment\nThefirstauthorispartiallysupportedbyNSFC(11501273, 117713 59),high\nlevel talent project of Lishui City(2016RC25), the Scientific Rese arch Foun-\ndation of the First-Class Discipline of Zhejiang Province(B)(201601 ), the key\nlaboratoryof ZhejiangProvince(2016E10007). Thesecond auth or ispartially\nsupported by the Grant-in-Aid for Scientific Research(C) (No.15K 04964),\nJapan Society for the Promotion of Science, and Special Research Expenses\nin FY2017, General Topics(No.B21), Future University Hakodate.\nFinally, the authors are grateful to Prof.M.Ikeda for his pointing ou t the\nmistake intheproofforthecritical caseinourfirst draftatarXiv:1 707.09583.\nReferences\n[1] M.D’Abbicco, The threshold of effective damping for semilinear wave\nequations , Mathematical Methods in Applied Sciences, 38(2015) 1032-\n1045.\n[2] M.D’AbbiccoandS.Lucente, A modifiedtest function methodfor damped\nwave equations , Adv. Nonlinear Stud., 13(2013), 867-892.\n[3] M.D’Abbicco andS.Lucente, NLWE with a special scale invariant damp-\ning in odd space dimension , Dynamical Systems, Differential Equations\nand Applications AIMS Proceedings, 2015, 312-319.\n[4] M.D’Abbicco, S.Lucente andM.Reissig, Semi-linear wave equations with\neffective damping , Chin. Ann. Math. Ser. B, 34(2013), 345-380.\n[5] M.D’Abbicco, S.Lucente and M.Reissig, A shift in the Strauss exponent\nfor semilinear wave equations with a not effective damping , J. Differen-\ntial Equations, 259(2015), 5040-5073.\n[6] K.Fujiwara, M.Ikeda andY.Wakasugi, Estimates of lifespan and blow-up\nrate for the wave equation with a time-dependent damping and a power-\ntype nonlinearity , arXiv:1609.01035.\n[7] D.Y.He, I.Witt and H.C.Yin, On the global solution problem for semi-\nlinear generalized Tricomi equations, I , Calc. Var. Partial Differential\nEquations, 56(21)(2017).\n[8] D.Y.He, I.Witt and H.C.Yin, On the global solution problem for semi-\nlinear generalized Tricomi equations, II , arXiv:1611.07606.\n19[9] M.Ikeda andT.Inui, The sharp estimate of the lifespan for the semilinear\nwave equation with time-dependent damping , arXiv:1707.03950.\n[10] M.Ikeda and T.Ogawa, Lifespan of solutions to the damped wave equa-\ntion with a critical nonlinearity , J. Differential Equations, 261(2016),\n1880-1903.\n[11] M.Ikeda and M.Sobajima, Life-span of solutions to semilinear wave\nequation with time-dependent critical damping for special ly localized ini-\ntial data, preprint.\n[12] M.Ikeda and Y.Wakasugi, A note on the lifespan of solutions to the\nsemilinear damped wave equation , Proc. Amer. Math. Soc., 143(2015),\n163-171.\n[13] M.Ikeda and Y.Wakasugi, A remark on the global existence for the semi-\nlinear damped wave equation in the overdamping case , in preparation.\n[14] T.Imai, M.Kato, H.Takamura and K.Wakasa, The sharp lower bound\nof the lifespan of solutions to semilinear wave equations wi th low pow-\ners in two space dimensions , Proceeding of the international confer-\nence ”Asymptotic Analysis for Nonlinear Dispersive and Wave Equa-\ntions” of a volume in Advanced Study of Pure Mathematics, to appea r\n(arXiv.1610.05913).\n[15] T.Imai, M.Kato, H.Takamura and K.Wakasa, The lifespan of solutions\nof semilinear wave equations with the scale invariant dampi ng in two\nspace dimensions , in preparation.\n[16] M.Kato and M.Sakuraba, Global existence and blow-up for semilinear\ndamped wave equations in three space dimensions , in preparation.\n[17] M.Kato, H.Takamura and K.Wakasa, The lifespan of solutions of semi-\nlinear wave equations with the scale invariant damping in on e space\ndimension , in preparation.\n[18] M.KiraneandM.Qafsaoui, Fujita’s exponent for a semilinear wave equa-\ntion with linear damping , Adv. Nonlinear Stud., 2(1)(2002), 41-49.\n[19] N.-A.Lai, H.Takamura and K. Wakasa, Blow-up for semilinear wave\nequations with the scale invariant damping and super-Fujit a exponent ,\nJ. Differential Equations, 263(9)(2017), 5377-5394.\n20[20] N.-A.LaiandY.Zhou, The sharp lifespan estimate for semilinear damped\nwave equation with Fujita critical power in high dimensions ,\narXiv:1702.07073.\n[21] T.T.Li andY.Zhou, Breakdown of solutions to ✷u+ut=|u|1+α, Discrete\nand Continuous Dynamical Systems, 1(1995), 503-520.\n[22] J.Lin, K.Nishihara and J.Zhai, Critical exponent for the semilinear wave\nequation with time-dependent damping , Discrete and Continuous Dy-\nnamical Systems - Series A, 32(2012), 4307-4320.\n[23] K.Nishihara, Lp-Lqestimates for the 3-D damped wave equation and\ntheir application to the semilinear problem , Sem. NotesMath. Sci., vol.6,\nIbaraki Univ., 2003, 69-83.\n[24] K.Nishihara, Asymptotic behavior of solutions to the semilinear wave\nequation with time-dependent damping , Tokyo J. Math., 34(2011), 327-\n343.\n[25] H.Takamura, Improved Kato’s lemma on ordinary differential inequality\nand its application to semilinear wave equations , Nonlinear Analysis,\nTMA,125(2015), 227-240.\n[26] G.Todorova and B.Yordanov Critical exponent for a nonlinear wave\nequation with damping , J. Differential Equations, 174(2001) 464-489.\n[27] K.Wakasa, The lifespan of solutions to semilinear damped wave equa-\ntions in one space dimension , Communications on Pure and Applied\nAnalysis, 15(2016), 1265-1283.\n[28] Y.Wakasugi, On the diffusive structure for the damped wave equation\nwith variable coefficients , Doctoral thesis, Osaka University (2014).\n[29] Y.Wakasugi, Critical exponent for the semilinear wave equation with\nscale invariant damping , Fourier analysis, 375-390, Trends Math.,\nBirkh¨ auser/Springer, Cham, (2014).\n[30] Y.Wakasugi, Scaling variables and asymptotic profiles for the semilinea r\ndamped wave equation with variable coefficients , J. Math. Anal. Appl.,\n447(2017), 452-487.\n[31] J.Wirth, Solution representations for a wave equation with weak diss i-\npation, Math. Methods Appl. Sci., 27(2004), 101-124.\n21[32] J.Wirth, Wave equations with time-dependent dissipation. I. Non-\neffective dissipation , J. Differential Equations, 222(2006), 487-514.\n[33] J.Wirth, Wave equations with time-dependent dissipation. II. Effect ive\ndissipation , J. Differential Equations, 232(2007), 74-103.\n[34] B.Yordanov and Q.S.Zhang, Finite time blow up for critical wave equa-\ntions in high dimensions , J. Funct. Anal., 231(2006), 361-374.\n[35] Q.S.Zhang, A blow-up result for a nonlinear wave equation with damp-\ning: the critical case , C. R. Math. Acad. Sci. Paris, S´ er. I, 333(2001)\n109-114.\n22"    },    {        "title": "1708.01126v2.Evolution_of_the_interfacial_perpendicular_magnetic_anisotropy_constant_of_the_Co__2_FeAl_MgO_interface_upon_annealing.pdf",        "content": "arXiv:1708.01126v2  [cond-mat.mtrl-sci]  23 Jan 2018Evolution of the interfacial perpendicular magnetic aniso tropy constant of the\nCo2FeAl/MgO interface upon annealing\nA. Conca,1,∗A. Niesen,2G. Reiss,2and B. Hillebrands1\n1Fachbereich Physik and Landesforschungszentrum OPTIMAS,\nTechnische Universit¨ at Kaiserslautern, 67663 Kaisersla utern, Germany\n2Center for Spinelectronic Materials and Devices,\nPhysics Department, Bielefeld University, 100131 Bielefe ld, Germany\n(Dated: May 7, 2021)\nWe investigate a series of films with different thickness of th e Heusler alloy Co 2FeAl in order to\nstudy the effect of annealing on the interface with a MgO layer and on the bulk magnetic properties.\nOur results reveal that while the perpendicular interface a nisotropy constant K⊥\nSis zero for the as-\ndeposited samples, its value increases with annealing up to a value of 1 .14±0.07 mJ/m2for the\nseries annealed at 320oC and of 2 .01±0.7 mJ/m2for the 450oC annealed series owing to a strong\nmodification of the interface during the thermal treatment. This large value ensures a stabilization\nof a perpendicular magnetization orientation for a thickne ss below 1.7 nm. The data additionally\nshows that the in-plane biaxial anisotropy constant has a di fferent evolution with thickness in as-\ndeposited and annealed systems. The Gilbert damping parame terαshows minima for all series for\na thickness of 40 nm and an absolute minimum value of 2 .8±0.1×10−3. The thickness dependence\nis explained in terms of an inhomogeneous magnetization sta te generated by the interplay between\nthe different anisotropies of the system and by the crystalli ne disorder.\nINTRODUCTION\nIn order to achieve efficient spin torque switching, ma-\nterialswithacertainsetofpropertiesarerequired. These\nproperties are a combination of low damping and low\nmagnetization, together with the presence of a robust\nperpendicular magnetic anisotropy(PMA). Additionally,\nthese materials should have a high spin polarisation and\nbe compatible with standard tunneling barrier materials\nsuch as MgO or MgAl 2O4. A high Curie temperature is\nalso desirable to guarantee temperature stability.\nIn the wide family of the Heusler compounds, some\ncandidates can be found which fulfill the aforementioned\nrequirements. For instance, large tunneling magnetore-\nsistance (TMR) ratios have been reported for several\ncompounds [1–6]. Heusler films have been successfully\nemployed in systems with PMA [7–11] and show also\nlow damping properties [12]. For the PMA properties\nof thin Heusler films, the interface-induced perpendicular\nanisotropy plays a critical role and its strength is given\nby the value of the perpendicular interfacial anisotropy\nconstant K⊥\nS. The interfacial properties, and therefore\nthe value of the constant, are strongly modified by the\nexact conditions of the annealingtreatment for the stack,\nwhich is required to improve the crystalline order of the\nHeusler films [6, 13, 14] and to achieve large TMR val-\nues [35]. The alloy Co 2FeAl belongs to the materials\nfor which large TMR [15] have been reported, even for\ntextured films on a SiO 2amorphous substrate [16]. Low\ndamping[17–19]andPMA[18,20]havealsobeenproven.\nIn this work, we study the evolution with annealing of\nK⊥\nSin systems with a MgO interface, by measuring dif-\nferent thickness series. Since the in-plane anisotropies\nand the Gilbert damping parameter change with varyingthicknessandannealingtemperature, alsotheirevolution\nis reported. The relevance of the study is not limited to\nCo2FeAl but it is a model for all TMR systems with\nCo-based Heusler alloys and an interface with a MgO\ntunneling barrier.\nSAMPLE PREPARATION\nThickness series (7-80 nm) of Co 2FeAl (CFA) epitax-\nial films were prepared and a microstrip-based VNA-\nFMR setup was used to study their magnetic proper-\nties. The dependence of the in-plane anisotropies and\nthe Gilbert damping parameter on the thickness and the\ndetermination of the interface perpendicular anisotropy\nconstant K⊥\nSfor the CFA/MgO interface is presented for\nFIG. 1. (Color online) X-ray diffraction patterns of 20nm\nthinCFAlayersas-deposited, annealedat320oCandannealed\nat 450oC. The (002) superlattice and the fundamental (004)\npeak of the CFA are clearly visible, confirming the partial B2\ncrystalline order.2\nFIG. 2. (Color online) X-Ray reflectometry data correspond-\ning to samples with a CFA thickness of 20 nm and different\nannealing temperatures.\nas-deposited samples and for two different values of the\nannealing temperature.\nThe stack layer structure is MgO(100)(subs)/\nMgO(5)/CFA( d)/MgO(7)/Ru(2) with d= 7, 9, 11, 15,\n20, 40 and 80 nm. Rf-sputtering was used for the MgO\ndeposition and dc-sputtering for the rest. The values of\nthe annealing temperature for the two series with ther-\nmal treatment are 320oC and 450oC. The layer stacking\nis symmetrical around CFA so that a similar interface is\nexpected for both sides. The samples were all deposited\nat room temperature and annealed afterwardsunder vac-\nuum conditions.\nX-RAY CHARACTERIZATION\nCrystallographic properties of the CFA thin films were\ndetermined using x-ray diffraction (XRD) measurements\nin a Philips X’Pert Pro diffractometer equipped with a\nCu anode. The (002) superlattice and the fundamental\n(004) peak of the CFA can be observed (see Fig. 1) al-\nready for the as-deposited state. In-plane performed φ\nscan measurements reveal the absence of the (111) su-\nperlattice reflection in these films. Therefore, partial\nB2 crystalline order is verified. Epitaxial, 45orotated\ngrowth, relative to the MgO buffer layer, was verified us-\ning aφscan of the reflection from the (202) planes (not\nshown here). The epitaxial relationship CFA (001)[100]\n//MgO(001)[110]wasthereforeconfirmedforthesefilms,\ni.e. CFA grows with the same crystalline orientation as\nthe substrate but the unit cell is rotated 45oin plane\nrespect to the MgO unit cell.\nX-ray reflectometry (XRR) has been performed on the\n20 nm thick films and it is shown in Fig. 2. The esti-\nmation of the RMS value is only possible with a certain\nuncertainty due to the number of layers which increases\nthe number of fitting parameters but it is possible to say\nthat it lays around 0.1-0.3 nm for the three samples. In\nany case, it is evident that the interface is very smoothin all cases and that the annealing is not modifying the\nroughness properties.\nRESULTS AND DISCUSSION\nFrom the dependence of HFMRon the resonance fre-\nquencyfFMR, the effective magnetization Meffis ex-\ntracted using a fit to Kittel’s formula [23]. For a more\ndetailed description of the FMR measurement and anal-\nysis procedure please see Ref. [24]. Meffis related to the\nsaturation magnetization of CFA by [25–27]\nMeff=Ms−H⊥\nK=Ms−2K⊥\nS\nµ0Msd(1)\nwhereK⊥\nSis the perpendicular surface (or interfacial)\nanisotropy constant.\nFig. 3 shows the dependence of Meffon 1/dfor the\nthree CFA series. The lines are a fit to Eq. 1. Let us\nfirst discuss the case of the as-deposited series shown in\nFig. 3(a). An almost constant value for Meffis observed\nfor the low thickness range (15-7 nm) where the inter-\nface properties should become dominating. The fit gives\na value for K⊥\nSof 0.03±0.1 mJ/m2compatible with\nzero (hollow values in Fig. 3 not considered for the fit).\nThis implies that it is not possible to obtain a stable per-\npendicular magnetization orientation for any thickness\nvalue based only on the interface effect. However, it has\nto be commented that a non-vanishing volume perpen-\ndicular anisotropy has also been reported for CFA [20]\nwhich may indeed stabilize an out-of-plane orientation.\nConcerning the relative decrease of Mefffor large thick-\nnesses, we attribute this to a inhomogeneous magnetiza-\ntion state which is sometimes observedin thick films [34].\nThis point will be later commented when analyzing the\ndamping properties.\nFigs. 3(b) and (c) show the evolution of the situation\nwhen the annealingstep is applied. The interfaceproper-\nties changewith the thermaltreatment and K⊥\nSincreases\nto avalue of1 .14±0.07mJ/m2forthe 320oCcaseand of\n2.07±0.7 mJ/m2for 450oC. The larger error bar in the\nlater value is due to a larger scattering of values for Meff.\nArecentstudyoftheperpendicularanisotropyproperties\non CFA thin films has been published where a novel TiN\nbuffer layer is employed [7]. In- and out-of-plane hystere-\nsis loopsareused to determine the value of K⊥\nSinstead of\nthe FMR measurements used here. However, the largest\nobtained values for K⊥\nSare in both cases in accordance\nwith ours (0 .86±0.16 mJ/m2). For comparison it has\nto be taken into account that due to the presence of two\nCFA/MgO interfaces, the values presented here are ex-\npected to be a factor of two larger. Both values are then\nin good agreement. The different annealing temperature\nrange does not allow for a comparison of the evolution\nofK⊥\nSwith that parameter but a remarkable difference3\nFIG. 3. (Color online) Dependence of Meffextracted from\nthe Kittel fit on the inverse thickness 1 /dfor three sample\nseries: (a) as-deposited, (b) annealed at 320oC, (c) annealed\nat 450oC. The lines are a fit to Eq. 1, the hollow data points\nwere not considered.\ncan be found in the as-deposited samples. A compara-\ntively smaller but, contrary to our case, non-zero value\nis reported. This reveals the role of the TiN buffer layer\nin improving the interface quality.\nAlthough it cannot be quantified by XRD, the exis-\ntence of a certain level of stress in the films cannot be ex-\ncluded. This stress is changing upon annealing together\nwith the crystalline orderat the interface and therefore it\nis reasonableto admit that it playsa role in the evolution\nofK⊥\nS. However, it is not possible to separate the con-\ntribution to the evolution of the PMA due to these two\neffects. First principle calculations of K⊥\nSfor stress-free\nCFA/MgO interfaces [36] has provided a value for K⊥\nS\nof 1.31 mJ/m2for Co-terminated interfaces while FeAl-\ntermination does induce in-plane orientation. This value\nis compatible with our results for the 450oC case takinginto account that our samples have two CFA/MgO inter-\nfaces. In any case, our results are more compatible with\na Co-termination at the MgO interfaces following this\ncalculation. Other experimental results using XMCD at-\ntribute, contrarily to the previous calculation, a PMA\ncontribution to the Fe atoms at the interface [37]. The\nexact atomic origin of the PMA is then still under dis-\ncussion and therefore also the actual impact of stress.\nAs already shown in Fig. 2, the roughness remains un-\nchanged after the annealing process. The increase of K⊥\nS\nis then due to a more subtle change of the atomic order-\ning at the immediate interface and is not connected to a\nroughness modification, or at least not in a large degree.\nBy setting d=∞in Eq. 1 it is possible to extract a\nvalue for Msof 1140±30 kA/m from the linear fit for\nthe as-deposited samples. This value is larger than the\nones reported in [21, 28] (1000-1030 kA/m) but similar\nto a FMR study [32] on very thick (140 nm) CFA poly-\ncrystalline films providing a value of Meff= 1200 kA/m.\nThe saturation magnetization Msfor TiN buffered\nCFA, deposited and investigated by the same group,\nwas measured to be 1140 ±60 kA/m, which is in excel-\nlent agreement with the value obtained from the FMR\ndata. The saturation magnetization for TiN buffered\nCFA was obtained using alternating gradient magne-\ntometer (AGM) measurements and verified using vibrat-\ning sample magnetometry (VSM) on a 10 nm thin CFA\nlayer [7].\nThe value of Msalso increases upon annealing up to\n1213±8 kA/m for the 320oC series and 1340 ±70 kA/m\nfor the 450oC one. This increase can be attributed to an\nimprovement of the crystalline order with annealing.\nFrom the extrapolation of the linear fits to Meff= 0\nit is possible to extract the thickness at which the in-\nterfacial perpendicular anisotropy is able to stabilize an\nout-of-plane configuration by overcoming the demagne-\ntization field and allowing the magnetic easy axis to be\nout-of-plane. This thickness is 1.2 nm and 1.7 nm for\n320oC and 450oC annealing temperature, respectively.\nThe relative difference between both values for the crit-\nical thickness is smaller than the relative difference for\nK⊥\nSfor the respective temperature values. This is ex-\nplained by the larger Msvalue for the 450oC case for\nwhich a larger demagnetizing field must be overcome to\nachieve PMA.\nBelmeguenai et al.presented data very similar to\nthe one shown in Fig. 3(a) for (110)-ordered textured\nfilms [21] and for (100)-oriented epitaxial films grown on\nMgO(100) substrates [22]. The annealing temperature is\n600oC. The data is given for thickness values not smaller\nthan 10 nm. However, the interpretation of the data is\ncompletely opposite to ours, resulting in a negative value\nK⊥\nS=−1.8 mJ/m2. The negative value indicates that\nthe interface anisotropy is favoring an in-plane orienta-\ntion of the magnetization. PMA with Ta/CFA/MgO (or\nCrorRu)systemshavebeenindeedachieved[29–31]with4\nFIG. 4. (Color online) Dependence of the Gilbert damping\nparameter αon the thickness dfor three sample series: as-\ndeposited, annealed at 320oC, and annealed at 450oC. The\ninset shows the dependence of the linewidth ∆ Hon the fre-\nquency for the 80 nm samples. The lines are a linear fit used\nto extract the damping parameter α.\nvaluesof K⊥\nS= +0.6mJ/m2fortheTacase,+1.0mJ/m2\nfor Cr and +2.0mJ/m2for Ru. This shows how sensitive\nK⊥\nSis to the exact growth properties which are modified\nby the different seed layer. The values reported in this\nwork for both annealed series are very similar to the Cr\nand Ru buffered systems. The fact that K⊥\nSvanishes\nin the as-deposited series shows also how important the\nannealing step is for adjusting the interface properties.\nFigure 4 shows the dependence of the Gilbert damping\nparameter αon the thickness dfor the as-deposited sam-\nplesandthe annealedseries. Theinsetshowsexemplarily\nfor the 80 nm samples the dependence of the linewidth\n∆H on the frequency and the linear fits to obtain α. For\nthe three series we observe a minimum in the αvalue for\nd= 40 nm. The smallest value obtained for this series\nisα= 2.8±0.1×10−3. When comparing to the liter-\nature it has to be taken into account that the value of\nαis very sensitive to the growth conditions and to the\nannealing temperature. Therefore the scatter of values is\nlarge. The smallest reported value [33] is around 1 ×10−3\nbut for films annealed at 600oC. The damping increases\nwhen the annealing temperature is lower, up to values\nsimilar to the ones reported here at ∼450oC.\nThe reasons for the increased damping are different\nfor the thicker and the thinner films. Concerning the\nlarge damping value for the 80 nm samples, it is a com-\nmon behavior in soft magnetic thin films that the damp-\ning increases strongly with thickness starting at a cer-\ntain value. An example of this can be seen for NiFe\nin the literature [34]. In this case the damping of the\nfilms strongly increases starting at d= 90 nm. The rea-\nson for that is a non-homogeneous magnetization state\nfor thicker films which open new loss channels in addi-tion to two-magnon scattering responsible for Gilbert-\nlike behavior in in-plane magnetized films. Nevertheless,\nthe value of αdecreases with the annealing temperature\npointing to a overall improvement of the uniformity of\nthe film and of the crystalline order.\nFor the thinner samples down to 11 nm we also ob-\nserve a reduction of αupon annealing, however this sit-\nuation is inverted for d <11 nm and provides a hint\nto one of the posible reasons for the increase of damping\nwith decreasingthickness. When the thicknessis reduced\nand the effect of the interface anisotropy is becoming\nlarger the magnetization state is becoming more inho-\nmogeneous due to the counterplay between the demag-\nnetization field and the anisotropy field. However, this is\nnot the only reason explaining the αincrease since this\nis also observable in the as-deposited sample series where\nK⊥\nS≈0, although to a lower degree, and additional ef-\nfects, e.g. due to roughness, play also a role.\nA comment has to be done concerning the exact mean-\ning of the concept of inhomogeneous magnetization used\nto describe our films. In an ideal thin film with smooth\ninterfaces and in the case of K⊥\nS= 0, the demagnetizing\nfield due to the shape anisotropy would induce a perfect\nin-plane orientation of the magnetization and a homoge-\nneous state with an external applied field. For the case\nof a large enough K⊥\nS>0 for a thickness below a criti-\ncal value ( d < dmin) the magnetization would again be\nhomogeneous but with out-of-plane orientation and for\nd > d maxan homogeneous in-plane state is expected.\nHowever, for a transition region dmin< d < d maxdiffer-\nent inhomogeneous states can be formed. Some of them\ncan be modelled by a simple analytical model or by mi-\ncromagnetic simulations as for instance in [38]. On the\nother limit case, for very large thickness, the situation is\nsimilar although the origin is different. For large thick-\nness values, the demagnetizing field responsible for the\nin-plane orientation is weakened allowing for the forma-\ntionofinhomogeneousstatessimilartothepreviousones.\nThe in-plane anisotropies were studied by measuring\nthe dependence of the resonant field HFMRon the az-\nimuthal angle φ. Fig. 5(a) shows exemplarily this depen-\ndence for a thickness of 11 nm in the range 0-180oat\n18 GHz for the as-deposited sample and the 450oC an-\nnealed one. An overall four-fold anisotropy, as expected\nfrom the cubic lattice of CFA and the (100) growth di-\nrection is observed. The easy axes correspond to 0oand\n90o. Overimposed to this, an additional weaker two-fold\nuniaxial anisotropy is also observed ( HFMRat 0oand\n90oare slighty different). The uniaxial anisotropy may\nbeinduced bystressinthe filmorbythe vicinalstructure\nin the substrate surface induced by miscut.\nIn order to extract the anisotropy fields the following\nformula was used:\nHFMR=¯HFMR+Hbcos(4φ)+Hucos(2φ+ϕ) (2)5\nFIG. 5. (Color online) (a) Dependence of HFMRon the az-\nimuthal angle ϕfor 11 nm thick films for the as-deposited and\nthe 450oC annealed samples. The lines are a fit to Eq. 2. (b)\nDependence of the in-plane biaxial anisotropy constant Kb\n(filled points) and the in-plane uniaxial anisotropy consta nt\nKu(hollow points) on the thickness dfor the as-deposited and\nthe annealed series.\nHereHbandHuarethebiaxialanduniaxialanisotropy\nfields,φis the in-plane azimuthal angle and ¯HFMRis the\naveragedvalue. The angle ϕallows for a misalignment of\nthe uniaxial and biaxial contributions, i.e. the easy axis\nof both contributions may be at different angles. The\nlines in Fig. 5(a) are fits to this formula. These field\nvalues are related to the anisotropy constants Hb,u=\n2Kb,u\nMs.\nThe results for KbandKufrom the fits are plotted\nin Fig. 5(b). For the calculation of the anisotropy con-\nstant the magnetization values obtained from the fits in\nFig. 3 are used. For Kbwe observe a different thick-\nness dependence for the as-deposited series and the se-\nries annealed at 320oC compared to the series annealed\nat 450oC. The value of Kbshows minor variation for the\nas-deposited samples with a small reduction for the thin-\nner films. The evolution is similar for the 320oC case.\nOn the contrary, the anisotropy constant increases con-\ntinously and strongly with decreasing thickness in the\nannealed series. However, the values converge for thick\nfilms and for 80 nm the difference vanishes. This points\nto an important role of the stress in the films, which\nnormally relaxes with thickness, in the evolution of Kb.\nThe absolutevalues arein agreementwith literaturedata\n[22]. The values of Kuare a order of magnitude smaller\nand the absolutevalues andthe thicknessdependence arevery similar for the three cases.\nCONCLUSIONS\nIn summary, we measured the evolution of the in-\nterface induced perpendicular anisotropy for epitaxial\nCFA/MgO interfaces and we observed a strong increase\nwith the annealing temperature up to a value of K⊥\nS=\n2.01±0.7mJ/m2foranannealingtemperatureof450oC.\nA stabilization of a perpendicular magnetization orienta-\ntion is then expected for films thinner than 1.7 nm. We\nstudied the thickness dependent magnetic properties of\nCFA for as-deposited and annealed series. We obtained\nminimum values for αfor a thickness of 40 nm for all\nseries and a different evolution with annealing for thin-\nner or thicker films. We correlate this with interface and\nbulk changes upon annealing, respectively. The study\nof the in-plane anisotropy constant shows a much larger\nthickness dependence on the annealed samples compared\nto the as-deposited ones.\nACKNOWLEDGEMENTS\nFinancial support by M-era.Net through the\nHEUMEM project is gratefully acknowledged.\n∗conca@physik.uni-kl.de\n[1] S. Tsunegi, Y. Sakuraba, M. Oogane, K. Takanashi, and\nY. Ando, Appl. Phys. Lett. 93, 112506 (2008).\n[2] N. Tezuka, N. Ikeda, S. Sugimoto, and K. Inomata, Jpn.\nJ. Appl. Phys. 46, L454 (2007).\n[3] W. Wang, H. Sukegawa, R. Shan, S. Mitani, and K. In-\nomata, Appl. Phys. Lett. 95, 182502 (2009).\n[4] T. Ishikawa, S. Hakamata, K. Matsuda, T. Uemura, and\nM. Yamamoto, J. Appl. Phys. 103, 07A919 (2008).\n[5] D. Ebke, V. Drewello, M. Sch¨ afers, G. Reiss, and\nA. Thomas, Appl. Phys. Lett. 95, 232510 (2009).\n[6] V. Drewello, D. Ebke, M. Sch¨ afers, Z. Kugler, G. Reiss,\nand A. Thomas, J. Appl. Phys. 111, 07C701 (2012).\n[7] A. Niesen, J. Ludwig, M. Glas, R. Silber, J.-M. Schmal-\nhorst, E. Arenholz, and G. Reiss, J. Appl. Phys. 121,\n223902 (2017).\n[8] Y. Takamura, T. Suzuki, Y. Fujino, and S. Nakagawa,\nJ. Appl. Phys. 115, 17C732 (2014).\n[9] T. Kamada, T. Kubota, S. Takahashi, Y. Sonobe, and\nK. Takanashi, IEEE Trans. onMagn. 50, 2600304 (2014).\n[10] B. M. Ludbrook, B. J. Ruck, and S. Granville,\nJ. Appl. Phys. 120, 013905 (2016).\n[11] B. M. Ludbrook, B. J. Ruck, and S. Granville,\nAppl. Phys. Lett. 110, 062408 (2017).\n[12] M. Oogane, R. Yilgin, M. Shinano, S. Yakata,\nY. Sakuraba, Y. Ando, and T. Miyazaki, J. Appl. Phys.\n101, 09J501 (2007).6\n[13] M. Cinchetti, J-P. W¨ ustenberg, M. S´ anchez Albaneda,\nF. Steeb, A. Conca, M. Jourdan, and M. Aeschlimann,\nJ. Phys. D: Appl. Phys. 40, 1544 (2007).\n[14] A. Conca, M. Jourdan, and H. Adrian, J. Phys. D:\nAppl. Phys. 401534 (2007).\n[15] W. Wang, H. Sukegawa, R. Shan, S. Mitani, and K. In-\nomata, Appl. Phys. Lett. 95, 182502 (2009).\n[16] Z. Wen, H. Sukegawa, S. Mitani, and K. Inomata,\nAppl. Phys. Lett. 98, 192505 (2011).\n[17] S. Mizukami, D. Watanabe, M. Oogane, Y. Ando,\nY. Miura, M. Shirai, and T. Miyazaki, J. Appl. Phys.\n105, 07D306 (2009).\n[18] Y. Cui, B. Khodadadi, S. Sch¨ afer, T. Mewes, J. Lu, and\nS. A. Wolf, Appl. Phys. Lett. 102, 162403 (2013).\n[19] Y. Cui, J. Lu, S. Sch¨ afer, B. Khodadadi, T. Mewes,\nM. Osofsky, and S. A. Wolf, J. Appl. Phys. 116, 073902\n(2014).\n[20] Z. Wen, H. Sukegawa, S. Mitani, and K. Inomata,\nAppl. Phys. Lett. 98, 242507 (2011).\n[21] M. Belmeguenai, H. Tuzcuoglu, M. Gabor, T. Petrisor,\nC. Tiusan, D. Berling, F. Zighem, and S. M. Ch´ erif, J.\nof Magn. and Magn. Mat. 373, 140 (2015).\n[22] M. Belmeguenai, H. Tuzcuoglu, M. S. Gabor, T. Petrisor,\nJr., C. Tiusan, D. Berling, F. Zighem, T. Chauveau,\nS. M. Ch´ erif, and P. Moch, Phys. Rev. B 87, 184431\n(2013).\n[23] C. Kittel, Phys. Rev. 73, 155 (1948).\n[24] A. Conca, S. Keller, L. Mihalceanu, T. Kehagias,\nG. P. Dimitrakopulos, B. Hillebrands, and E. Th. Pa-\npaioannou, Phys. Rev. B 93, 134405 (2016).\n[25] X. Liu, W. Zhang, M. J. Carter, and G. Xiao,\nJ. Appl. Phys. 110, 033910 (2011).[26] V. P. Nascimento, E. B. Saitovitch, F. Pelegrini,\nL. C. Figueiredo, A. Biondo, and E. C. Passamani,\nJ. Appl. Phys. 99, 08C108 (2006).\n[27] J.-M. L. Beaujour, W. Chen, A. D. Kent, and J. Z. Sun,\nJ. Appl. Phys. 99, 08N503 (2006).\n[28] G. Ortiz, M.S. Gabor, T. Petrisor, Jr., F. Boust, F. Issa c,\nC. Tiusan, M. Hehn, and J. F. Bobo, J. Appl. Phys. 109,\n07D324 (2011).\n[29] M. S. Gabor, T. Petrisor Jr., C. Tiusan, and T. Petrisor,\nJ. Appl. Phys. 114, 063905 (2013).\n[30] Z. Wen, H. Sukegawa, S. Mitani and K. Inomata,\nAppl. Phys. Lett. 98, 242507 (2011).\n[31] Z. C. Wen, H. Sukegawa, T. Furubayashi, J. Koo, K. Ino-\nmata, S. Mitani, J. P. Hadorn, T. Ohkubo, and K. Hono,\nAdv. Mater. 26, 6483 (2014).\n[32] A. Yadav, and S. Chaudhary, J. Appl. Phys. 115, 133916\n(2014).\n[33] S. Mizukami, D. Watanabe, M. Oogane, Y. Ando,\nY. Miura, M. Shirai, and T. Miyazaki, J. Appl. Phys.\n105, 07D306 (2009).\n[34] Y. Chen, D. Hung, Y. Yao, S. Lee, H. Ji, and C. Yu,\nJ. Appl. Phys. 101, 09C104 (2007).\n[35] O. Schebaum, D. Ebke, A. Niemeyer, G. Reiss,\nJ. S. Moodera, and A. Thomas, J. Appl. Phys. 107,\n09C717 (2010).\n[36] R. Vadapoo, A. Hallal, H. Yang, and M. Chshiev,\nPhys. Rev. B 94, 104418 (2016).\n[37] Z. Wen, J. P. Hadorn, J. Okabayashi, H. Sukegawa,\nT. Ohkubo, K. Inomata, S. Mitani, and K. Hono, Appl.\nPhys. Express 10, 013003 (2017).\n[38] N. A. Usov, and O. N. Serebryakova, J. Appl. Phys. 121,\n133905 (2017)."    },    {        "title": "1708.02008v2.Chiral_damping__chiral_gyromagnetism_and_current_induced_torques_in_textured_one_dimensional_Rashba_ferromagnets.pdf",        "content": "arXiv:1708.02008v2  [cond-mat.mes-hall]  31 Aug 2017Chiral damping, chiral gyromagnetism and current-induced torques in textured\none-dimensional Rashba ferromagnets\nFrank Freimuth,∗Stefan Bl¨ ugel, and Yuriy Mokrousov\nPeter Gr¨ unberg Institut and Institute for Advanced Simula tion,\nForschungszentrum J¨ ulich and JARA, 52425 J¨ ulich, German y\n(Dated: May 17, 2018)\nWe investigate Gilbert damping, spectroscopic gyromagnet ic ratio and current-induced torques\nin the one-dimensional Rashba model with an additional nonc ollinear magnetic exchange field. We\nfind that the Gilbert damping differs between left-handed and right-handed N´ eel-type magnetic\ndomain walls due to the combination of spatial inversion asy mmetry and spin-orbit interaction\n(SOI), consistent with recent experimental observations o f chiral damping. Additionally, we find\nthat also the spectroscopic gfactor differs between left-handed and right-handed N´ eel- type domain\nwalls, which we call chiral gyromagnetism. We also investig ate the gyromagnetic ratio in the Rashba\nmodel with collinear magnetization, where we find that scatt ering corrections to the gfactor vanish\nfor zero SOI, become important for finite spin-orbit couplin g, and tend to stabilize the gyromagnetic\nratio close to its nonrelativistic value.\nI. INTRODUCTION\nIn magnetic bilayer systems with structural inversion\nasymmetry the energies of left-handed and right-handed\nN´ eel-type domain walls differ due to the Dzyaloshinskii-\nMoriya interaction (DMI) [1–4]. DMI is a chiral interac-\ntion, i.e., it distinguishes between left-handed and right-\nhanded spin-spirals. Not only the energy is sensitive to\nthe chirality of spin-spirals. Recently, it has been re-\nported that the orbital magnetic moments differ as well\nbetween left-handed and right-handed cycloidal spin spi-\nrals in magnetic bilayers [5, 6]. Moreover, the experi-\nmental observation of asymmetry in the velocity of do-\nmain walls driven by magnetic fields suggests that also\nthe Gilbert damping is sensitive to chirality [7, 8].\nIn this work we show that additionally the spectro-\nscopic gyromagnetic ratio γis sensitive to the chirality\nof spin-spirals. The spectroscopic gyromagnetic ratio γ\ncan be defined by the equation\ndm\ndt=γT, (1)\nwhereTis the torque that acts on the magnetic moment\nmand dm/dtis the resulting rate of change. γenters\nthe Landau-Lifshitz-Gilbert equation (LLG):\ndˆM\ndt=γˆM×Heff+αGˆM×dˆM\ndt,(2)\nwhereˆMis a normalized vector that points in the direc-\ntionofthemagnetizationandthetensor αGdescribesthe\nGilbert damping. The chiralityofthe gyromagneticratio\nprovides another mechanism for asymmetries in domain-\nwall motion between left-handed and right-handed do-\nmain walls.\nNot only the damping and the gyromagnetic ratio\nexhibit chiral corrections in inversion asymmetric sys-\ntems but also the current-induced torques. Amongthese torques that act on domain-walls are the adia-\nbatic and nonadiabatic spin-transfer torques [9–12] and\nthe spin-orbit torque [13–16]. Based on phenomenologi-\ncal grounds additional types of torques have been sug-\ngested [17]. Since this large number of contributions\nare difficult to disentangle experimentally, current-driven\ndomain-wall motion in inversion asymmetric systems is\nnot yet fully understood.\nThe two-dimensionalRashbamodel with an additional\nexchange splitting has been used to study spintronics\neffects associated with the interfaces in magnetic bi-\nlayer systems [18–22]. Recently, interest in the role of\nDMI in one-dimensional magnetic chains has been trig-\ngered [23, 24]. For example, the magnetic moments in\nbi-atomic Fe chains on the Ir surface order in a 120◦\nspin-spiral state due to DMI [25]. Apart from DMI, also\nother chiral effects, such as chiral damping and chiral\ngyromagnetism, are expected to be important in one-\ndimensional magnetic chains on heavy metal substrates.\nThe one-dimensional Rashba model [26, 27] with an ad-\nditional exchange splitting can be used to simulate spin-\norbit driven effects in one-dimensional magnetic wires on\nsubstrates [28–30]. While the generalized Bloch theo-\nrem[31]usuallycannotbeusedtotreatspin-spiralswhen\nSOI is included in the calculation, the one-dimensional\nRashba model has the advantage that it can be solved\nwith the help of the generalized Bloch theorem, or with a\ngauge-field approach [32], when the spin-spiral is of N´ eel-\ntype. WhenthegeneralizedBlochtheoremcannotbeem-\nployed one needs to resort to a supercell approach [33],\nuse open boundary conditions [34, 35], or apply pertur-\nbation theory [6, 9, 36–39] in order to study spintronics\neffects in noncollinear magnets with SOI. In the case of\nthe one-dimensional Rashba model the DMI and the ex-\nchangeparameterswerecalculatedbothdirectlybasedon\nagauge-fieldapproachandfromperturbationtheory[38].\nThe results from the two approaches were found to be in\nperfect agreement. Thus, the one-dimensional Rashba2\nmodel provides also an excellent opportunity to verify\nexpressions obtained from perturbation theory by com-\nparisonto the resultsfromthe generalizedBlochtheorem\nor from the gauge-field approach.\nIn this work we study chiral gyromagnetism and chi-\nral damping in the one-dimensional Rashba model with\nan additional noncollinear magnetic exchange field. The\none-dimensional Rashba model is very well suited to\nstudy these SOI-driven chiral spintronics effects, because\nit can be solved in a very transparent way without the\nneed for a supercell approach, open boundary conditions\nor perturbation theory. We describe scattering effects by\nthe Gaussian scalar disorder model. To investigate the\nrole of disorder for the gyromagnetic ratio in general, we\nstudyγalso in the two-dimensional Rashba model with\ncollinear magnetization. Additionally, we compute the\ncurrent-induced torques in the one-dimensional Rashba\nmodel.\nThis paper is structured as follows: In section IIA we\nintroduce the one-dimensional Rashba model. In sec-\ntion IIB we discuss the formalism for the calculation\nof the Gilbert damping and of the gyromagnetic ratio.\nIn section IIC we present the formalism used to calcu-\nlate the current-induced torques. In sections IIIA, IIIB,\nand IIIC we discuss the gyromagnetic ratio, the Gilbert\ndamping, and the current-induced torques in the one-\ndimensionalRashbamodel, respectively. Thispaperends\nwith a summary in section IV.\nII. FORMALISM\nA. One-dimensional Rashba model\nThe two-dimensional Rashba model is given by the\nHamiltonian [19]\nH=−/planckover2pi12\n2me∂2\n∂x2−/planckover2pi12\n2me∂2\n∂y2+\n+iαRσy∂\n∂x−iαRσx∂\n∂y+∆V\n2σ·ˆM(r),(3)\nwhere the first line describes the kinetic energy, the first\ntwotermsin thesecondline describethe RashbaSOI and\nthe last term in the second line describes the exchange\nsplitting. ˆM(r) is the magnetization direction, which\nmay depend on the position r= (x,y), andσis the\nvector of Pauli spin matrices. By removing the terms\nwith the y-derivatives from Eq. (3), i.e., −/planckover2pi12\n2me∂2\n∂y2and\n−iαRσx∂\n∂y, one obtains a one-dimensional variant of the\nRashba model with the Hamiltonian [38]\nH=−/planckover2pi12\n2me∂2\n∂x2+iαRσy∂\n∂x+∆V\n2σ·ˆM(x).(4)\nEq. (4) is invariant under the simultaneous rotation\nofσand of the magnetization ˆMaround the yaxis.Therefore, if ˆM(x) describes a flat cycloidal spin-spiral\npropagating into the xdirection, as given by\nˆM(x) =\nsin(qx)\n0\ncos(qx)\n, (5)\nwe can use the unitary transformation\nU(x) =/parenleftBigg\ncos(qx\n2)−sin(qx\n2)\nsin(qx\n2) cos(qx\n2)/parenrightBigg\n(6)\nin order to transform Eq. (4) into a position-independent\neffective Hamiltonian [38]:\nH=1\n2m/parenleftbig\npx+eAeff\nx/parenrightbig2−m(αR)2\n2/planckover2pi12+∆V\n2σz,(7)\nwherepx=−i/planckover2pi1∂/∂xis thexcomponent of the momen-\ntum operator and\nAeff\nx=−m\ne/planckover2pi1/parenleftbigg\nαR+/planckover2pi12\n2mq/parenrightbigg\nσy (8)\nis thex-component of the effective magnetic vector po-\ntential. Eq. (8) shows that the noncollinearity described\nbyqacts like an effective SOI in the special case of the\none-dimensional Rashba model. This suggests to intro-\nduce the concept of effective SOI strength\nαR\neff=αR+/planckover2pi12\n2mq. (9)\nBased on this concept of the effective SOI strength\none can obtain the q-dependence of the one-dimensional\nRashba model from its αR-dependence at q= 0. That a\nnoncollinear magnetic texture provides a nonrelativistic\neffective SOI has been found also in the context of the\nintrinsic contribution to the nonadiabatic torque in the\nabsence of relativistic SOI, which can be interpreted as\na spin-orbit torque arising from this effective SOI [40].\nWhile the Hamiltonian in Eq. (4) depends on position\nxthrough the position-dependence of the magnetization\nˆM(x) in Eq. (5), the effective Hamiltonian in Eq. (7) is\nnot dependent on xand therefore easy to diagonalize.\nB. Gilbert damping and gyromagnetic ratio\nIn collinear magnets damping and gyromagnetic ratio\ncan be extracted from the tensor [16]\nΛij=−1\nVlim\nω→0ImGR\nTi,Tj(/planckover2pi1ω)\n/planckover2pi1ω, (10)\nwhereVis the volume of the unit cell and\nGR\nTi,Tj(/planckover2pi1ω) =−i∞/integraldisplay\n0dteiωt/angbracketleft[Ti(t),Tj(0)]−/angbracketright(11)3\nis the retarded torque-torque correlation function. Tiis\nthei-th component of the torque operator [16]. The dc-\nlimitω→0 in Eq. (10) is only justified when the fre-\nquency of the magnetization dynamics, e.g., the ferro-\nmagnetic resonance frequency, is smaller than the relax-\nationrateoftheelectronicstates. In thin magneticlayers\nand monoatomicchains on substratesthis is typically the\ncase due to the strong interfacial disorder. However, in\nvery pure crystalline samples at low temperatures the\nrelaxation rate may be smaller than the ferromagnetic\nresonance frequency and one needs to assume ω >0 in\nEq. (10) [41, 42]. The tensor Λdepends on the mag-\nnetization direction ˆMand we decompose it into the\ntensorS, which is even under magnetization reversal\n(S(ˆM) =S(−ˆM)), and the tensor A, which is odd un-\nder magnetization reversal ( A(ˆM) =−A(−ˆM)), such\nthatΛ=S+A, where\nSij(ˆM) =1\n2/bracketleftBig\nΛij(ˆM)+Λij(−ˆM)/bracketrightBig\n(12)\nand\nAij(ˆM) =1\n2/bracketleftBig\nΛij(ˆM)−Λij(−ˆM)/bracketrightBig\n.(13)\nOne can show that Sis symmetric, i.e., Sij(ˆM) =\nSji(ˆM), while Ais antisymmetric, i.e., Aij(ˆM) =\n−Aji(ˆM).\nThe Gilbert damping may be extracted from the sym-\nmetric component Sas follows [16]:\nαG\nij=|γ|Sij\nMµ0, (14)\nwhereMis the magnetization. The gyromagnetic ratio\nγis obtained from Λ according to the equation [16]\n1\nγ=1\n2µ0M/summationdisplay\nijkǫijkΛijˆMk=1\n2µ0M/summationdisplay\nijkǫijkAijˆMk.\n(15)\nIt is convenient to discuss the gyromagnetic ratio in\nterms of the dimensionless g-factor, which is related to\nγthrough γ=gµ0µB//planckover2pi1. Consequently, the g-factor is\ngiven by\n1\ng=µB\n2/planckover2pi1M/summationdisplay\nijkǫijkΛijˆMk=µB\n2/planckover2pi1M/summationdisplay\nijkǫijkAijˆMk.(16)\nDue to the presence of the Levi-Civita tensor ǫijkin\nEq. (15) and in Eq. (16) the gyromagnetic ratio and the\ng-factoraredetermined solelyby the antisymmetriccom-\nponentAofΛ.\nVarious different conventions are used in the literature\nconcerning the sign of the g-factor [43]. Here, we define\nthe sign of the g-factor such that γ >0 forg >0 and\nγ <0 forg <0. According to Eq. (1) the rate of change\nofthemagneticmomentisthereforeparalleltothetorqueforpositive gandantiparalleltothetorquefornegative g.\nWhile we are interested in this work in the spectroscopic\ng-factor, and hence in the relation between the rate of\nchange of the magnetic moment and the torque, Ref. [43]\ndiscusses the relation between the magnetic moment m\nandtheangularmomentum Lthatgeneratesit, i.e., m=\nγstaticL. Since differentiation with respect to time and\nuse ofT= dL/dtleads to Eq. (1) our definition of the\nsigns ofgandγagrees essentially with the one suggested\nin Ref. [43], which proposes to use a positive gwhen the\nmagnetic moment is parallel to the angular momentum\ngeneratingitandanegative gwhenthemagneticmoment\nis antiparallel to the angular momentum generating it.\nCombining Eq. (14) and Eq. (15) we can express the\nGilbert damping in terms of AandSas follows:\nαG\nxx=Sxx\n|Axy|. (17)\nIntheindependentparticleapproximationEq.(10)can\nbe written as Λij= ΛI(a)\nij+ΛI(b)\nij+ΛII\nij, where\nΛI(a)\nij=1\nh/integraldisplayddk\n(2π)dTr/angbracketleftbig\nTiGR\nk(EF)TjGA\nk(EF)/angbracketrightbig\nΛI(b)\nij=−1\nh/integraldisplayddk\n(2π)dReTr/angbracketleftbig\nTiGR\nk(EF)TjGR\nk(EF)/angbracketrightbig\nΛII\nij=1\nh/integraldisplayddk\n(2π)d/integraldisplayEF\n−∞dEReTr/angbracketleftbigg\nTiGR\nk(E)TjdGR\nk(E)\ndE\n− TidGR\nk(E)\ndETjGR\nk(E)/angbracketrightbigg\n.(18)\nHere,dis the dimension ( d= 1 ord= 2 ord= 3),GR\nk(E)\nis the retarded Green’s function and GA\nk(E) = [GR\nk(E)]†.\nEFis the Fermi energy. ΛI(b)\nijis symmetric under the\ninterchange of the indices iandjwhile ΛII\nijis antisym-\nmetric. The term ΛI(a)\nijcontains both symmetric and\nantisymmetric components. Since the Gilbert damping\ntensor is symmetric, both ΛI(b)\nijand ΛI(a)\nijcontribute to\nit. Since the gyromagnetic tensor is antisymmetric, both\nΛII\nijand ΛI(a)\nijcontribute to it.\nIn order to account for disorder we use the Gaus-\nsian scalardisordermodel, wherethe scatteringpotential\nV(r) satisfies /angbracketleftV(r)/angbracketright= 0 and /angbracketleftV(r)V(r′)/angbracketright=Uδ(r−r′).\nThis model is frequently used to calculate transport\nproperties in disordered multiband model systems [44],\nbut it has also been combined with ab-initio electronic\nstructure calculations to study the anomalous Hall ef-\nfect [45, 46] and the anomalous Nernst effect [47] in tran-\nsition metals and their alloys.\nIn the clean limit, i.e., in the limit U→0, the an-\ntisymmetric contribution to Eq. (18) can be written as4\nAij=Aint\nij+Ascatt\nij, where the intrinsic part is given by\nAint\nij=/planckover2pi1/integraldisplayddk\n(2π)d/summationdisplay\nn,m[fkn−fkm]ImTi\nknmTj\nkmn\n(Ekn−Ekm)2\n= 2/planckover2pi1/integraldisplayddk\n(2π)d/summationdisplay\nn/summationdisplay\nll′fknIm/bracketleftbigg∂/angbracketleftukn|\n∂ˆMl∂|ukn/angbracketright\n∂ˆMl′/bracketrightbigg\n×\n×/summationdisplay\nmm′ǫilmǫjl′m′ˆMmˆMm′.\n(19)\nThe second line in Eq. (19) expresses Aint\nijin terms of\nthe Berry curvature in magnetization space [48]. The\nscattering contribution is given by\nAscatt\nij=/planckover2pi1/summationdisplay\nnm/integraldisplayddk\n(2π)dδ(EF−Ekn)Im/braceleftBigg\n−/bracketleftbigg\nMi\nknmγkmn\nγknnTj\nknn−Mj\nknmγkmn\nγknnTi\nknn/bracketrightbigg\n+/bracketleftBig\nMi\nkmn˜Tj\nknm−Mj\nkmn˜Ti\nknm/bracketrightBig\n−/bracketleftbigg\nMi\nknmγkmn\nγknn˜Tj\nknn−Mj\nknmγkmn\nγknn˜Ti\nknn/bracketrightbigg\n+/bracketleftBigg\n˜Ti\nknnγknm\nγknn˜Tj\nkmn\nEkn−Ekm−˜Tj\nknnγknm\nγknn˜Ti\nkmn\nEkn−Ekm/bracketrightBigg\n+1\n2/bracketleftbigg\n˜Ti\nknm1\nEkn−Ekm˜Tj\nkmn−˜Tj\nknm1\nEkn−Ekm˜Ti\nkmn/bracketrightbigg\n+/bracketleftBig\nTj\nknnγknm\nγknn1\nEkn−Ekm˜Ti\nkmn\n−Ti\nknnγknm\nγknn1\nEkn−Ekm˜Tj\nkmn/bracketrightBig/bracerightBigg\n.\n(20)\nHere,Ti\nknm=/angbracketleftukn|Ti|ukm/angbracketrightare the matrix elements of\nthe torque operator. ˜Ti\nknmdenotes the vertex corrections\nof the torque, which solve the equation\n˜Ti\nknm=/summationdisplay\np/integraldisplaydnk′\n(2π)n−1δ(EF−Ek′p)\n2γk′pp×\n×/angbracketleftukn|uk′p/angbracketright/bracketleftBig\n˜Ti\nk′pp+Ti\nk′pp/bracketrightBig\n/angbracketleftuk′p|ukm/angbracketright.(21)\nThe matrix γknmis given by\nγknm=−π/summationdisplay\np/integraldisplayddk′\n(2π)dδ(EF−Ek′p)/angbracketleftukn|uk′p/angbracketright/angbracketleftuk′p|ukm/angbracketright\n(22)\nand the Berry connection in magnetization space is de-\nfined as\niMj\nknm=iTj\nknm\nEkm−Ekn. (23)\nThe scattering contribution Eq. (20) formally resembles\nthe side-jump contribution to the AHE [44] as obtainedfrom the scalar disorder model: It can be obtained by\nreplacing the velocity operators in Ref. [44] by torque\noperators. We find thatin collinearmagnetswithoutSOI\nthis scattering contribution vanishes. The gyromagnetic\nratio is then given purely by the intrinsic contribution\nEq. (19). This is an interesting difference to the AHE:\nWithout SOI all contributions to the AHE are zero in\ncollinear magnets, while both the intrinsic and the side-\njump contributions are generally nonzero in the presence\nof SOI.\nIn the absence of SOI Eq. (19) can be expressed in\nterms of the magnetization [48]:\nAint\nij=−/planckover2pi1\n2µB/summationdisplay\nkǫijkMk. (24)\nInserting Eq. (24) into Eq. (16) yields g=−2, i.e., the\nexpected nonrelativistic value of the g-factor.\nTheg-factor in the presence of SOI is usually assumed\nto be given by [49]\ng=−2Mspin+Morb\nMspin=−2M\nMspin,(25)\nwhereMorbis the orbital magnetization, Mspinis the\nspin magnetization and M=Morb+Mspinis the total\nmagnetization. The g-factor obtained from Eq. (25) is\nusually in good agreementwith experimental results [50].\nWhen SOI is absent, the orbital magnetization is zero,\nMorb= 0, and consequently Eq. (25) yields g=−2 in\nthat case. Eq. (16) can be rewritten as\n1\ng=Mspin\nMµB\n2/planckover2pi1Mspin/summationdisplay\nijkǫijkAijˆMk=Mspin\nM1\ng1,(26)\nwith\n1\ng1=µB\n2/planckover2pi1Mspin/summationdisplay\nijkǫijkAijˆMk. (27)\nFrom the comparison of Eq. (26) with Eq. (25) it follows\nthat Eq. (25) holds exactly if g1=−2 is satisfied. How-\never, Eq. (27) usually yields g1=−2 only in collinear\nmagnets when SOI is absent, otherwise g1/negationslash=−2. In the\none-dimensionalRashbamodel the orbitalmagnetization\nis zero,Morb= 0, and consequently\n1\ng=µB\n2/planckover2pi1Mspin/summationdisplay\nijkǫijkAijˆMk. (28)\nThe symmetric contribution can be written as Sij=\nSint\nij+SRR−vert\nij+SRA−vert\nij, where\nSint\nij=1\nh/integraldisplayddk\n(2π)dTr/braceleftbig\nTiGR\nk(EF)Tj/bracketleftbig\nGA\nk(EF)−GR\nk(EF)/bracketrightbig/bracerightbig\n(29)5\nand\nSRR−vert\nij=−1\nh/integraldisplayddk\n(2π)dTr/braceleftBig\n˜TRR\niGR\nk(EF)TjGR\nk(EF)/bracerightBig\n(30)\nand\nSAR−vert\nij=1\nh/integraldisplayddk\n(2π)dTr/braceleftBig\n˜TAR\niGR\nk(EF)TjGA\nk(EF)/bracerightBig\n,\n(31)\nwhereGR\nk(EF) =/planckover2pi1[EF−Hk−ΣR\nk(EF)]−1is the retarded\nGreen’s function, GA\nk(EF) =/bracketleftbig\nGR\nk(EF)/bracketrightbig†is the advanced\nGreen’s function and\nΣR(EF) =U\n/planckover2pi1/integraldisplayddk\n(2π)dGR\nk(EF) (32)\nis the retarded self-energy. The vertex corrections are\ndetermined by the equations\n˜TAR=T+U\n/planckover2pi12/integraldisplayddk\n(2π)dGA\nk(EF)˜TAR\nkGR\nk(EF) (33)\nand\n˜TRR=T+U\n/planckover2pi12/integraldisplayddk\n(2π)dGR\nk(EF)˜TRR\nkGR\nk(EF).(34)\nIn contrast to the antisymmetric tensor A, which be-\ncomes independent of the scattering strength Ufor suf-\nficiently small U, i.e., in the clean limit, the symmetric\ntensorSdepends strongly on Uin metallic systems in\nthe clean limit. Sint\nijandSscatt\nijdepend therefore on U\nthrough the self-energy and through the vertex correc-\ntions.\nIn the case of the one-dimensional Rashba model, the\nequations Eq. (19) and Eq. (20) for the antisymmet-\nric tensor Aand the equations Eq. (29), Eq. (30) and\nEq. (31) for the symmetric tensor Scan be used both\nfor the collinear magnetic state as well as for the spin-\nspiral of Eq. (5). To obtain the g-factor for the collinear\nmagnetic state, we plug the eigenstates and eigenvalues\nof Eq. (4) (with ˆM=ˆez) into Eq. (19) and into Eq. (20).\nIn the case of the spin-spiral of Eq. (5) we use instead the\neigenstates and eigenvalues of Eq. (7). Similarly, to ob-\ntain the Gilbert damping in the collinear magnetic state,\nwe evaluate Eq. (29), Eq. (30) and Eq. (31) based on\nthe Hamiltonian in Eq. (4) and for the spin-spiral we use\ninstead the effective Hamiltonian in Eq. (7).\nC. Current-induced torques\nThe current-induced torque on the magnetization can\nbe expressed in terms of the torkance tensor tijas [15]\nTi=/summationdisplay\njtijEj, (35)whereEjis thej-th component of the applied elec-\ntric field and Tiis thei-th component of the torque\nper volume [51]. tijis the sum of three terms, tij=\ntI(a)\nij+tI(b)\nij+tII\nij, where [15]\ntI(a)\nij=e\nh/integraldisplayddk\n(2π)dTr/angbracketleftbig\nTiGR\nk(EF)vjGA\nk(EF)/angbracketrightbig\ntI(b)\nij=−e\nh/integraldisplayddk\n(2π)dReTr/angbracketleftbig\nTiGR\nk(EF)vjGR\nk(EF)/angbracketrightbig\ntII\nij=e\nh/integraldisplayddk\n(2π)d/integraldisplayEF\n−∞dEReTr/angbracketleftbigg\nTiGR\nk(E)vjdGR\nk(E)\ndE\n− TidGR\nk(E)\ndEvjGR\nk(E)/angbracketrightbigg\n.(36)\nWe decompose the torkance into two parts that are,\nrespectively, even and odd with respect to magnetiza-\ntion reversal, i.e., te\nij(ˆM) = [tij(ˆM) +tij(−ˆM)]/2 and\nto\nij(ˆM) = [tij(ˆM)−tij(−ˆM)]/2.\nIn the clean limit, i.e., for U→0, the even torkance\ncan be written as te\nij=te,int\nij+te,scatt\nij, where [15]\nte,int\nij= 2e/planckover2pi1/integraldisplayddk\n(2π)d/summationdisplay\nn/negationslash=mfknImTi\nknmvj\nkmn\n(Ekn−Ekm)2(37)\nis the intrinsic contribution and\nte,scatt\nij=e/planckover2pi1/summationdisplay\nnm/integraldisplayddk\n(2π)dδ(EF−Ekn)Im/braceleftBigg\n/bracketleftBig\n−Mi\nknmγkmn\nγknnvj\nknn+Aj\nknmγkmn\nγknnTi\nknn/bracketrightBig\n+/bracketleftBig\nMi\nkmn˜vj\nknm−Aj\nkmn˜Ti\nknm/bracketrightBig\n−/bracketleftBig\nMi\nknmγkmn\nγknn˜vj\nknn−Aj\nknmγkmn\nγknn˜Ti\nknn/bracketrightBig\n+/bracketleftBig\n˜vj\nkmnγknm\nγknn˜Ti\nnn\nEkn−Ekm−˜Ti\nkmnγknm\nγknn˜vj\nknn\nEkn−Ekm/bracketrightBig\n+1\n2/bracketleftBig\n˜vj\nknm1\nEkn−Ekm˜Ti\nkmn−˜Ti\nknm1\nEkn−Ekm˜vj\nkmn/bracketrightBig\n+/bracketleftBig\nvj\nknnγknm\nγknn1\nEkn−Ekm˜Ti\nkmn\n−Ti\nknnγknm\nγknn1\nEkn−Ekm˜vj\nkmn/bracketrightBig/bracerightBigg\n.\n(38)\nis the scattering contribution. Here,\niAj\nknm=ivj\nknm\nEkm−Ekn=i\n/planckover2pi1/angbracketleftukn|∂\n∂kj|ukm/angbracketright(39)\nis the Berry connection in kspace and the vertex correc-\ntions of the velocity operator solve the equation\n˜vi\nknm=/summationdisplay\np/integraldisplaydnk′\n(2π)n−1δ(EF−Ek′p)\n2γk′pp×\n×/angbracketleftukn|uk′p/angbracketright/bracketleftbig\n˜vi\nk′pp+vi\nk′pp/bracketrightbig\n/angbracketleftuk′p|ukm/angbracketright.(40)6\nThe odd contribution can be written as to\nij=to,int\nij+\ntRR−vert\nij+tAR−vert\nij, where\nto,int\nij=e\nh/integraldisplayddk\n(2π)dTr/braceleftbig\nTiGR\nk(EF)vj/bracketleftbig\nGA\nk(EF)−GR\nk(EF)/bracketrightbig/bracerightbig\n(41)\nand\ntRR−vert\nij=−e\nh/integraldisplayddk\n(2π)dTr/braceleftBig\n˜TRR\niGR\nk(EF)vjGR\nk(EF)/bracerightBig\n(42)\nand\ntAR−vert\nij=e\nh/integraldisplayddk\n(2π)dTr/braceleftBig\n˜TAR\niGR\nk(EF)vjGA\nk(EF)/bracerightBig\n.(43)\nThe vertex corrections ˜TAR\niand˜TRR\niof the torque op-\nerator are given in Eq. (33) and in Eq. (34), respectively.\nWhile the even torkance, Eq. (37) and Eq. (38), be-\ncomes independent of the scattering strength Uin the\nclean limit, i.e., for U→0, the odd torkance to\nijdepends\nstrongly on Uin metallic systems in the clean limit [15].\nIn the case of the one-dimensional Rashba model, the\nequations Eq. (37) and Eq. (38) for the even torkance\nte\nijand the equations Eq. (41), Eq. (42) and Eq. (43) for\nthe odd torkance to\nijcan be used both for the collinear\nmagnetic state as well as for the spin-spiral of Eq. (5).\nTo obtain the even torkance for the collinear magnetic\nstate, we plug the eigenstates and eigenvalues of Eq. (4)\n(withˆM=ˆez) into Eq. (37) and into Eq. (38). In the\ncase of the spin-spiral of Eq. (5) we use instead the eigen-\nstates and eigenvalues of Eq. (7). Similarly, to obtain the\nodd torkance in the collinear magnetic state, we evaluate\nEq. (41), Eq. (42) and Eq. (43) based on the Hamilto-\nnian in Eq. (4) and for the spin-spiral we use instead the\neffective Hamiltonian in Eq. (7).\nIII. RESULTS\nA. Gyromagnetic ratio\nWe first discuss the g-factor in the collinear case, i.e.,\nwhenˆM(r) =ˆez. Inthis casetheenergybandsaregiven\nby\nE=/planckover2pi12k2\nx\n2m±/radicalbigg\n1\n4(∆V)2+(αRkx)2.(44)\nWhen ∆ V/negationslash= 0 orαR/negationslash= 0 the energy Ecan become\nnegative. The band structure of the one-dimensional\nRashba model is shown in Fig. 1 for the model param-\netersαR=2eV˚A and ∆ V= 0.5eV. For this choice of\nparameters the energy minima are not located at kx= 0\nbut instead at\nkmin\nx=±/radicalBig\n(αR)4m2−1\n4/planckover2pi14(∆V)2\n/planckover2pi12αR,(45)-0.4 -0.2 0 0.2 0.4\nk-Point kx [Å-1]00.511.5Band energy [eV]\nFIG. 1: Band structure of theone-dimensional Rashbamodel.\nand the corresponding minimum of the energy is given\nby\nEmin=−m(αR)4+1\n4/planckover2pi14\nm(∆V)2\n2/planckover2pi12(αR)2. (46)\nThe inverse g-factor is shown as a function of the SOI\nstrength αRin Fig. 2 for the exchange splitting ∆ V=\n1eV and Fermi energy EF= 1.36eV. At αR= 0 the\nscattering contribution is zero, i.e., the g-factor is de-\ntermined completely by the intrinsic Berry curvature ex-\npression, Eq. (24). Thus, at αR= 0 it assumes the value\n1/g=−0.5, which is the expected nonrelativistic value\n(see the discussion below Eq. (24)). With increasing SOI\nstrength αRthe intrinsic contribution to 1 /gis more and\nmore suppressed. However, the scattering contribution\ncompensates this decrease such that the total 1 /gis close\nto its nonrelativistic value of −0.5. The neglect of the\nscattering corrections at large values of αRwould lead in\nthis case to a strong underestimation of the magnitude\nof 1/g, i.e., a strong overestimation of the magnitude of\ng.\nHowever, at smaller values of the Fermi energy, the\ngfactor can deviate substantially from its nonrelativis-\ntic value of −2. To show this we plot in Fig. 3 the in-\nverseg-factor as a function of the Fermi energy when\nthe exchange splitting and the SOI strength are set to\n∆V= 1eV and αR=2eV˚A, respectively. As discussed in\nEq. (44) the minimal Fermi energyis negativ in this case.\nThe intrinsic contribution to 1 /gdeclines with increas-\ning Fermi energy. At large values of the Fermi energy\nthis decline is compensated by the increase of the vertex\ncorrections and the total value of 1 /gis close to −0.5.\nPrevious theoretical works on the g-factor have not\nconsidered the scattering contribution [52]. It is there-\nfore important to find out whether the compensation\nof the decrease of the intrinsic contribution by the in-7\n00.511.52\nSOI strength αR [eVÅ]-0.5-0.4-0.3-0.2-0.101/gscattering\nintrinsic\ntotal\nFIG. 2: Inverse g-factor vs. SOI strength αRin the one-\ndimensional Rashba model.\n0 1 2 3 4 5 6\nFermi energy [eV]-0.6-0.4-0.201/gscattering\nintrinsic\ntotal\nFIG. 3: Inverse g-factor vs. Fermi energy in the one-\ndimensional Rashba model.\ncrease of the extrinsic contribution as discussed in Fig. 2\nand Fig. 3 is peculiar to the one-dimensional Rashba\nmodel or whether it can be found in more general cases.\nFor this reason we evaluate g1for the two-dimensional\nRashba model. In Fig. 4 we show the inverse g1-factor\nin the two-dimensional Rashba model as a function of\nSOI strength αRfor the exchange splitting ∆ V= 1eV\nand the Fermi energy EF= 1.36eV. Indeed for αR<\n0.5eV˚A the scattering corrections tend to stabilize g1at\nits non-relativistic value. However, in contrast to the\none-dimensional case (Fig. 2), where gdoes not deviate\nmuch from its nonrelativistic value up to αR= 2eV˚A,\ng1starts to be affected by SOI at smaller values of αR\nin the two-dimensional case. According to Eq. (26) the\nfullgfactor is given by g=g1(1+Morb/Mspin). There-\nfore, when the scattering corrections stabilize g1at its00.511.52\nSOI strength αR [eVÅ]-0.5-0.4-0.3-0.2-0.101/g1\nscattering\nintrinsic\ntotal\nFIG. 4: Inverse g1-factor vs. SOI strength αRin the two-\ndimensional Rashba model.\nnonrelativistic value the Eq. (25) is satisfied. In the two-\ndimensional Rashba model Morb= 0 when both bands\nare occupied. For the Fermi energy EF= 1.36eV both\nbands are occupied and therefore g=g1for the range of\nparameters used in Fig. 4.\nThe inverse g1of the two-dimensional Rashba model\nis shown in Fig. 5 as a function of Fermi energy for the\nparameters ∆ V= 1eV and αR= 2eV˚A. The scattering\ncorrection is as large as the intrinsic contribution when\nEF>1eV. While the scattering correction is therefore\nimportant, it is not sufficiently large to bring g1close to\nits nonrelativistic value in the energy range shown in the\nfigure, which is a major difference to the one-dimensional\ncase illustrated in Fig. 3. According to Eq. (26) the g\nfactor is related to g1byg=g1M/Mspin. Therefore, we\nshow in Fig. 6 the ratio M/Mspinas a function of Fermi\nenergy. AthighFermienergy(whenbothbandsareoccu-\npied) the orbital magnetization is zeroand M/Mspin= 1.\nAt low Fermi energy the sign of the orbital magnetiza-\ntionis oppositeto the signofthe spin magnetizationsuch\nthat the magnitude of Mis smaller than the magnitude\nofMspinresulting in the ratio M/Mspin<1.\nNext, we discuss the g-factor of the one-dimensional\nRashba model in the noncollinear case. In Fig. 7 we\nplot the inverse g-factor and its decomposition into the\nintrinsic and scattering contributions as a function of\nthe spin-spiral wave vector q, where exchange splitting,\nSOI strength and Fermi energy are set to ∆ V= 1eV,\nαR= 2eV˚A andEF= 1.36eV, respectively. Since\nthe curves are not symmetric around q= 0, the g-\nfactor at wave number qdiffers from the one at −q, i.e.,\nthegyromagnetism in the Rashba model is chiral . At\nq=−2meαR//planckover2pi12theg-factorassumesthevalueof g=−2\nand the scattering corrections are zero. Moreover, the\ncurves are symmetric around q=−2meαR//planckover2pi12. These8\n0 2 4 6\nFermi energy [eV]-0.5-0.4-0.3-0.2-0.101/g1\nscattering\nintrinsic\ntotal\nFIG. 5: Inverse g1-factor 1 /g1vs. Fermi energy in the two-\ndimensional Rashba model.\n-2 0 2 4 6\nFermi energy [eV]00.511.52M/Mspin\nFIG. 6: Ratio of total magnetization and spin magnetization ,\nM/Mspin, vs. Fermi energy in the two-dimensional Rashba\nmodel.\nobservationscan be explained by the concept of the effec-\ntive SOI introduced in Eq. (9): At q=−2meαR//planckover2pi12the\neffective SOI is zero and consequently the noncollinear\nmagnet behaves like a collinear magnet without SOI at\nthis value of q. As we have discussed above in Fig. 2, the\ng-factor of collinear magnets is g=−2 when SOI is ab-\nsent, which explains why it is also g=−2 in noncollinear\nmagnets with q=−2meαR//planckover2pi12. If only the intrinsic con-\ntribution is considered and the scattering corrections are\nneglected, 1 /gvaries much stronger around the point of\nzero effective SOI q=−2meαR//planckover2pi12, i.e., the scattering\ncorrections stabilize gat its nonrelativistic value close to\nthe point of zero effective SOI.-2 -1 0 1\nWave vector q [Å-1]-0.8-0.6-0.4-0.201/g\nscattering\nintrinsic\ntotal\nFIG. 7: Inverse g-factor 1 /gvs. wave number qin the one-\ndimensional Rashba model.\n0 1 2 3 4\nScattering strength U [(eV)2Å]-0.4-0.200.20.4Gilbert Damping αG\nxx\nRR-Vertex\nAR-Vertex\nintrinsic\ntotal\nFIG. 8: Gilbert damping αG\nxxvs. scattering strength Uin the\none-dimensional Rashba model without SOI. In this case the\nvertex corrections and the intrinsic contribution sum up to\nzero.\nB. Damping\nWe first discuss the Gilbert damping in the collinear\ncase, i.e., we set ˆM(r) =ˆezin Eq. (4). The xxcom-\nponent of the Gilbert damping is shown in Fig. 8 as\na function of scattering strength Ufor the following\nmodel parameters: exchange splitting ∆ V=1eV, Fermi\nenergyEF= 2.72eV and SOI strength αR= 0. All\nthree contributions are individually non-zero, but the\ncontribution from the RR-vertex correction (Eq. (30)) is\nmuchsmallerthanthe onefromthe AR-vertexcorrection\n(Eq. (31)) and much smaller than the intrinsic contribu-\ntion (Eq. (29)). However, in this case the total damping\nis zero, because a non-zero damping in periodic crystals\nwith collinear magnetization is only possible when SOI\nis present [53].9\n1 2 3 4\nScattering strength U [(eV)2Å]050100150200250300Gilbert Damping αG\nxx\nRR-Vertex\nAR-Vertex\nintrinsic\ntotal\nFIG. 9: Gilbert damping αG\nxxvs. scattering strength Uin the\none-dimensional Rashba model with SOI.\nIn Fig. 9 we show the xxcomponent of the Gilbert\ndamping αG\nxxas a function of scattering strength Ufor\nthe model parameters ∆ V= 1eV, EF= 2.72eV and\nαR= 2eV˚A. ThedominantcontributionistheAR-vertex\ncorrection. The damping as obtained based on Eq. (10)\ndiverges like 1 /Uin the limit U→0, i.e., proportional\nto the relaxation time τ[53]. However, once the relax-\nation time τis larger than the inverse frequency of the\nmagnetization dynamics the dc-limit ω→0 in Eq. (10)\nis not appropriate and ω >0 needs to be used. It has\nbeenshownthattheGilbertdampingisnotinfinite inthe\nballistic limit τ→ ∞whenω >0 [41, 42]. In the one-\ndimensional Rashba model the effective magnetic field\nexerted by SOI on the electron spins points in ydirec-\ntion. Since a magnetic field along ydirection cannot lead\ntoatorquein ydirectionthe yycomponentoftheGilbert\ndamping αG\nyyis zero (not shown in the Figure).\nNext, we discuss the Gilbert damping in the non-\ncollinear case. In Fig. 10 we plot the xxcomponent\nof the Gilbert damping as a function of spin spiral\nwave number qfor the model parameters ∆ V= 1eV,\nEF= 1.36eV,αR= 2eV˚A, and the scattering strength\nU= 0.98(eV)2˚A. The curves are symmetric around\nq=−2meαR//planckover2pi12, because the damping is determined by\nthe effective SOI defined in Eq. (9). At q=−2meαR//planckover2pi12\nthe effective SOI is zero and therefore the total damp-\ning is zero as well. The damping at wave number qdif-\nfers from the one at wave number −q, i.e.,the damp-\ning is chiral in the Rashba model . Around the point\nq=−2meαR//planckover2pi12the damping is described by aquadratic\nparabola at first. In the regions -2 ˚A−1< q <-1.2˚A−1\nand 0.2˚A−1< q <1˚A−1this trend is interrupted by a W-\nshape behaviour. In the quadratic parabola region the\nlowest energy band crosses the Fermi energy twice. As\nshown in Fig. 1 the lowest band has a local maximum at-2-1.5-1-0.500.51\nWave vector q [Å-1]05101520Gilbert damping αxxG RR-Vertex\nAR-Vertex\nintrinsic\ntotal\nFIG. 10: Gilbert damping αG\nxxvs. spin spiral wave number q\nin the one-dimensional Rashba model.\nq= 0. In the W-shape region this local maximum shifts\nupwards, approaches the Fermi level and finally passes it\nsuch that the lowest energy band crosses the Fermi level\nfour times. This transition in the band structure leads to\noscillations in the density of states, which results in the\nW-shape behaviour of the Gilbert damping.\nSince the damping is determined by the effective SOI,\nwe can use Fig. 10 to draw conclusions about the damp-\ning in the noncollinear case with αR= 0: We only need\nto shift all curves in Fig. 10 to the right such that they\nare symmetric around q= 0 and shift the Fermi energy.\nThus, for αR= 0 the Gilbert damping does not vanish\nifq/negationslash= 0. Since for αR= 0 angular momentum transfer\nfrom the electronic system to the lattice is not possible,\nthe damping is purely nonlocal in this case, i.e., angular\nmomentum is interchanged between electrons at differ-\nent positions. This means that for a volume in which\nthe magnetization of the spin-spiral in Eq. (5) performs\nexactly one revolution between one end of the volume\nand the other end the total angular momentum change\nassociated with the damping is zero, because the angu-\nlar momentum is simply redistributed within this volume\nand there is no net change of the angular momentum.\nA substantial contribution of nonlocal damping has also\nbeen predicted for domain walls in permalloy [35].\nIn Fig. 11 we plot the yycomponent of the Gilbert\ndamping as a function of spin spiral wave number qfor\nthe model parameters ∆ V= 1eV,EF= 1.36eV,αR=\n2eV˚A, and the scattering strength U= 0.98(eV)2˚A. The\ntotaldampingiszerointhiscase. Thiscanbeunderstood\nfrom the symmetry properties of the one-dimensional\nRashba Hamiltonian, Eq. (4): Since this Hamiltonian is\ninvariant when both σandˆMare rotated around the\nyaxis, the damping coefficient αG\nyydoes not depend on\nthe position within the cycloidal spin spiral of Eq. (5).10\n-3 -2 -1 0 1 2\nWave vector q [Å-1]-0.4-0.200.20.4Gilbert Damping αG\nyyRR-Vertex\nAR-Vertex\nintrinsic\ntotal\nFIG. 11: Gilbert damping αG\nyyvs. spin spiral wave number q\nin the one-dimensional Rashba model.\nTherefore, nonlocal damping is not possible in this case\nandαG\nyyhas to be zero when αR= 0. It remains to be\nshown that αG\nyy= 0 also for αR/negationslash= 0. However, this fol-\nlows directly from the observation that the damping is\ndetermined by the effective SOI, Eq. (9), meaning that\nany case with q/negationslash= 0 and αR/negationslash= 0 can always be mapped\nonto a case with q/negationslash= 0 and αR= 0. As an alternative\nargumentation we can also invoke the finding discussed\nabovethat αG\nyy= 0 in the collinearcase. Since the damp-\ning is determined by the effective SOI, it follows that\nαG\nyy= 0 also in the noncollinear case.\nC. Current-induced torques\nWe first discuss the yxcomponent of the torkance. In\nFig. 12 we show the torkance tyxas a function of the\nFermi energy EFfor the model parameters ∆ V= 1eV\nandαR= 2eV˚A when the magnetization is collinear and\npoints in zdirection. We specify the torkance in units of\nthe positive elementary charge e, which is a convenient\nchoice for the one-dimensional Rashba model. When\nthe torkance is multiplied with the electric field, we ob-\ntain the torque per length (see Eq. (35) and Ref. [51]).\nSince the effective magnetic field from SOI points in\nydirection, it cannot give rise to a torque in ydirec-\ntion and consequently the total tyxis zero. Interest-\ningly, the intrinsic and scattering contributions are indi-\nvidually nonzero. The intrinsic contribution is nonzero,\nbecause the electric field accelerates the electrons such\nthat/planckover2pi1˙kx=−eEx. Therefore, the effective magnetic\nfieldBSOI\ny=αRkx/µBchanges as well, i.e., ˙BSOI\ny=\nαR˙kx/µB=−αRExe/(/planckover2pi1µB). Consequently, the electron\nspin is no longer aligned with the total effective magnetic\nfield (the effective magnetic field resulting from both SOI-1 0 1 2 3 4 5 6\nFermi energy [eV]-0.2-0.100.10.2Torkance tyx [e]scattering\nintrinsic\ntotal\nFIG. 12: Torkance tyxvs. Fermi energy EFin the one-\ndimensional Rashba model.\nand from the exchange splitting ∆ V), when an electric\nfield is applied. While the total effective magnetic field\nlies in the yzplane, the electron spin acquires an xcom-\nponent, because it precesses around the total effective\nmagnetic field, with which it is not aligned due to the\napplied electric field [54]. The xcomponent of the spin\ndensity results in a torque in ydirection, which is the\nreason why the intrinsic contribution to tyxis nonzero.\nThe scattering contribution to tyxcancels the intrinsic\ncontribution such that the total tyxis zero and angular\nmomentum conservation is satisfied.\nUsing the concept of effective SOI, Eq. (9), we con-\nclude that tyxis also zero for the noncollinear spin-spiral\ndescribed by Eq. (5). Thus, both the ycomponent of the\nspin-orbit torque and the nonadiabatic torque are zero\nfor the one-dimensional Rashba model.\nTo show that tyx= 0 is a peculiarity of the one-\ndimensional Rashba model, we plot in Fig. 13 the\ntorkance tyxin the two-dimensional Rashba model. The\nintrinsic and scattering contributions depend linearly on\nαRfor small values of αR, but the slopes are opposite\nsuch that the total tyxis zero for sufficiently small αR.\nHowever, for largervalues of αRthe intrinsic and scatter-\ning contributions do not cancel each other and therefore\nthe total tyxbecomes nonzero, in contrast to the one-\ndimensional Rashba model, where tyx= 0 even for large\nαR. Several previous works determined the part of tyx\nthat is proportionalto αRin the two-dimensionalRashba\nmodel and found it to be zero [21, 22] for scalar disor-\nder, which is consistent with our finding that the linear\nslopes of the intrinsic and scattering contributions to tyx\nare opposite for small αR.\nNext, we discuss the xxcomponent of the torkance\nin the collinear case ( ˆM=ˆez). In Fig. 14 we plot\nthe torkance txxvs. scattering strength Uin the one-11\n00.511.52\nSOI strength αR [eVÅ]-0.00500.0050.01Torkance tyx [e/Å]\nscattering\nintrinsic\ntotal\nFIG. 13: Nonadiabatic torkance tyxvs. SOI parameter αRin\nthe two-dimensional Rashba model.\ndimensional Rashba model for the parameters ∆ V=\n1eV,EF= 2.72eV and αR= 2eV˚A. The dominant con-\ntribution is the AR-type vertex correction (see Eq. (43)).\ntxxdiverges like 1 /Uin the limit U→0 as expected for\nthe odd torque in metallic systems [15].\nIn Fig. 15 and Fig. 16 we plot txxas a function of\nspin-spiral wave number qfor the model parameters\n∆V= 1eV,EF= 2.72eV and U= 0.18(eV)2˚A. In Fig. 15\nthe case with αR= 2eV˚A is shown, while Fig. 16 illus-\ntrates the case with αR= 0. In the case αR= 0 the\ntorkance txxdescribes the spin-transfer torque (STT). In\nthe case αR/negationslash= 0 the torkance txxis the sum of contribu-\ntions from STT and spin-orbit torque (SOT). The curves\nwithαR= 0 andαR/negationslash= 0 are essentially related by a shift\nof ∆q=−2meαR//planckover2pi12, which can be understood based on\nthe concept of the effective SOI, Eq. (9). Thus, in the\nspecial case of the one-dimensional Rashba model STT\nand SOT are strongly related.\nIV. SUMMARY\nWe study chiral damping, chiral gyromagnetism and\ncurrent-induced torques in the one-dimensional Rashba\nmodel with an additional N´ eel-type noncollinear mag-\nnetic exchange field. In order to describe scattering ef-\nfects we use a Gaussian scalar disorder model. Scat-\ntering contributions are generally important in the one-\ndimensional Rashba model with the exception of the gy-\nromagnetic ratio in the collinear case with zero SOI,\nwhere the scattering correctionsvanish in the clean limit.\nIn the one-dimensional Rashba model SOI and non-\ncollinearity can be combined into an effective SOI. Us-\ning the concept of effective SOI, results for the mag-\nnetically collinear one-dimensional Rashba model can be\nused to predict the behaviour in the noncollinear case.1 2 3 4\nScattering strength U [(eV)2Å]-6-4-20Torkance txx [e]\nRR-Vertex\nAR-Vertex\nintrinsic\ntotal\nFIG. 14: Torkance txxvs. scattering strength Uin the one-\ndimensional Rashba model.\n-2 -1 0 1\nWave vector q [Å-1]-4-2024Torkance txx [e]RR-Vertex\nAR-Vertex\nintrinsic\ntotal\nFIG. 15: Torkance txxvs. wave vector qin the one-\ndimensional Rashba model with SOI.\nIn the noncollinear Rashba model the Gilbert damp-\ning is nonlocal and does not vanish for zero SOI. The\nscattering corrections tend to stabilize the gyromagnetic\nratio in the one-dimensional Rashba model at its non-\nrelativistic value. Both the Gilbert damping and the\ngyromagnetic ratio are chiral for nonzero SOI strength.\nThe antidamping-like spin-orbit torque and the nonadi-\nabatic torque for N´ eel-type spin-spirals are zero in the\none-dimensional Rashba model, while the antidamping-\nlike spin-orbit torque is nonzero in the two-dimensional\nRashba model for sufficiently large SOI-strength.\n∗Corresp. author: f.freimuth@fz-juelich.de12\n-1-0.500.51\nWave vector q [Å-1]-4-2024Torkance txx [e]RR-Vertex\nAR-Vertex\nintrinsic\ntotal\nFIG. 16: Torkance txxvs. wave vector qin the one-\ndimensional Rashba model without SOI.\n[1] T. Moriya, Phys. Rev. 120, 91 (1960).\n[2] I. Dzyaloshinsky, Journal of Physics and Chemistry of\nSolids4, 241 (1958).\n[3] M. Heide, G. Bihlmayer, and S. Bl¨ ugel, Phys. Rev. B 78,\n140403 (2008).\n[4] P. Ferriani, K. von Bergmann, E. Y. Vedmedenko,\nS. Heinze, M. Bode, M. Heide, G. Bihlmayer, S. Bl¨ ugel,\nand R. Wiesendanger, Phys. Rev. Lett. 101, 027201\n(2008).\n[5] K. Yamamoto, A.-M. Pradipto, K. Nawa, T. Akiyama,\nT. Ito, T. Ono, and K. Nakamura, AIP Advances 7,\n056302 (2017).\n[6] F. R. Lux, F. Freimuth, S. Bl¨ ugel, and Y. Mokrousov,\nArXiv e-prints (2017), 1706.06068.\n[7] E. Ju´ e, C. K. Safeer, M. Drouard, A. Lopez, P. Balint,\nL. Buda-Prejbeanu, O. Boulle, S. Auffret, A. Schuhl,\nA. Manchon, et al., Nature materials 15, 272 (2015).\n[8] C. A. Akosa, I. M. Miron, G. Gaudin, and A. Manchon,\nPhys. Rev. B 93, 214429 (2016).\n[9] Y.Tserkovnyak,H.J.Skadsem, A.Brataas, andG.E.W.\nBauer, Phys. Rev. B 74, 144405 (2006).\n[10] K. Gilmore, I. Garate, A. H. MacDonald, and M. D.\nStiles, Phys. Rev. B 84, 224412 (2011).\n[11] I. Garate, K. Gilmore, M. D. Stiles, and A. H. MacDon-\nald, Phys. Rev. B 79, 104416 (2009).\n[12] K.-W.Kim, K.-J.Lee, H.-W.Lee, andM. D.Stiles, Phys.\nRev. B92, 224426 (2015).\n[13] A. V. Khvalkovskiy, V. Cros, D. Apalkov, V. Nikitin,\nM. Krounbi, K. A. Zvezdin, A. Anane, J. Grollier, and\nA. Fert, Phys. Rev. B 87, 020402 (2013).\n[14] E. van der Bijl and R. A. Duine, Phys. Rev. B 86, 094406\n(2012).\n[15] F. Freimuth, S. Bl¨ ugel, and Y. Mokrousov, Phys. Rev. B\n90, 174423 (2014).\n[16] F. Freimuth, S. Bl¨ ugel, and Y. Mokrousov, Phys. Rev. B\n92, 064415 (2015).\n[17] K. M. D. Hals and A. Brataas, Phys. Rev. B 88, 085423\n(2013).\n[18] A. Manchon and S. Zhang, Phys. Rev. B 78, 212405\n(2008).[19] A. Manchon, H. C. Koo, J. Nitta, S. M. Frolov, and R. A.\nDuine, Nature materials 14, 871 (2015).\n[20] D. A. Pesin and A. H. MacDonald, Phys. Rev. B 86,\n014416 (2012).\n[21] A. Qaiumzadeh, R. A. Duine, and M. Titov, Phys. Rev.\nB92, 014402 (2015).\n[22] I. A. Ado, O. A. Tretiakov, and M. Titov, Phys. Rev. B\n95, 094401 (2017).\n[23] V. Kashid, T. Schena, B. Zimmermann, Y. Mokrousov,\nS. Bl¨ ugel, V. Shah, and H. G. Salunke, Phys. Rev. B 90,\n054412 (2014).\n[24] B. Schweflinghaus, B. Zimmermann, M. Heide,\nG. Bihlmayer, and S. Bl¨ ugel, Phys. Rev. B 94,\n024403 (2016).\n[25] M. Menzel, Y. Mokrousov, R. Wieser, J. E. Bickel,\nE. Vedmedenko, S. Bl¨ ugel, S. Heinze, K. von Bergmann,\nA. Kubetzka, and R. Wiesendanger, Phys. Rev. Lett.\n108, 197204 (2012).\n[26] I. Barke, F. Zheng, T. K. R¨ ugheimer, and F. J. Himpsel,\nPhys. Rev. Lett. 97, 226405 (2006).\n[27] T. Okuda, K. Miyamaoto, Y. Takeichi, H. Miyahara,\nM. Ogawa, A.Harasawa, A.Kimura, I.Matsuda, A.Kak-\nizaki, T. Shishidou, et al., Phys. Rev. B 82, 161410\n(2010).\n[28] A. B. Shick, F. Maca, and P. M. Oppeneer, Journal of\nMagnetism and Magnetic Materials 290, 257 (2005).\n[29] M. Komelj, D. Steiauf, and M. F¨ ahnle, Phys. Rev. B 73,\n134428 (2006).\n[30] P. Gambardella, A. Dallmeyer, K. Maiti, M. C. Malagoli,\nS. Rusponi, P. Ohresser, W. Eberhardt, C. Carbone, and\nK. Kern, Phys. Rev. Lett. 93, 077203 (2004).\n[31] L. M. Sandratskii, physica status solidi (b) 136, 167\n(1986).\n[32] T. Fujita, M. B. A. Jalil, S. G. Tan, and S. Murakami,\nJournal of applied physics 110, 121301 (2011).\n[33] H. Yang, A. Thiaville, S. Rohart, A. Fert, and\nM. Chshiev, Phys. Rev. Lett. 115, 267210 (2015).\n[34] Z. Yuan and P. J. Kelly, Phys. Rev. B 93, 224415 (2016).\n[35] Z. Yuan, K. M. D. Hals, Y. Liu, A. A. Starikov,\nA. Brataas, and P. J. Kelly, Phys. Rev. Lett. 113, 266603\n(2014).\n[36] F. Freimuth, R. Bamler, Y. Mokrousov, and A. Rosch,\nPhys. Rev. B 88, 214409 (2013).\n[37] F. Freimuth, S. Bl¨ ugel, and Y. Mokrousov, Journal of\nphysics: Condensed matter 26, 104202 (2014).\n[38] F. Freimuth, S. Bl¨ ugel, and Y. Mokrousov, Phys. Rev. B\n95, 184428 (2017).\n[39] H. Kohno, Y. Hiraoka, M. Hatami, and G. E. W. Bauer,\nPhys. Rev. B 94, 104417 (2016).\n[40] K.-W. Kim, K.-J.Lee, H.-W.Lee, andM.D. Stiles, Phys.\nRev. B92, 224426 (2015).\n[41] D. M. Edwards, Journal of Physics: Condensed Matter\n28, 086004 (2016).\n[42] A. T. Costa and R. B. Muniz, Phys. Rev. B 92, 014419\n(2015).\n[43] J. M. Brown, R. J. Buenker, A. Carrington, C. D. Lauro,\nR. N. Dixon, R. W. Field, J. T. Hougen, W. H¨ uttner,\nK. Kuchitsu, M. Mehring, et al., Molecular Physics 98,\n1597 (2000).\n[44] A. A. Kovalev, J. Sinova, and Y. Tserkovnyak, Phys.\nRev. Lett. 105, 036601 (2010).\n[45] J. Weischenberg, F. Freimuth, J. Sinova, S. Bl¨ ugel, an d\nY. Mokrousov, Phys. Rev. Lett. 107, 106601 (2011).\n[46] P. Czaja, F. Freimuth, J. Weischenberg, S. Bl¨ ugel, and13\nY. Mokrousov, Phys. Rev. B 89, 014411 (2014).\n[47] J. Weischenberg, F. Freimuth, S. Bl¨ ugel, and\nY. Mokrousov, Phys. Rev. B 87, 060406 (2013).\n[48] Z. Qian and G. Vignale, Phys. Rev. Lett. 88, 056404\n(2002).\n[49] C. Kittel, Phys. Rev. 76, 743 (1949).\n[50] M. A. W. Schoen, J. Lucassen, H. T. Nembach, T. J.\nSilva, B. Koopmans, C. H. Back, and J. M. Shaw, Phys.\nRev. B95, 134410 (2017).\n[51] In Ref. [15] and Ref. [16] we consider the torque per\nunit cell and use therefore charge times length as unit\nof torkance. In this work we consider instead the torque\nper volume. Consequently, the unit of torkance used inthis work is charge times length per volume. In the one-\ndimensional Rashba model the volume is given by the\nlength and consequently the unit of torkance is charge.\n[52] S. Lounis, M. dos Santos Dias, and B. Schweflinghaus,\nPhys. Rev. B 91, 104420 (2015).\n[53] I. Garate and A. MacDonald, Phys. Rev. B 79, 064404\n(2009).\n[54] H. Kurebayashi, J. Sinova, D. Fang, A. C. Irvine, T. D.\nSkinner, J. Wunderlich, V. Nov´ ak, R. P. Campion, B. L.\nGallagher, E. K. Vehstedt, et al., Nature nanotechnology\n9, 211 (2014)."    },    {        "title": "1708.03424v1.Gradient_expansion_formalism_for_generic_spin_torques.pdf",        "content": "Gradient expansion formalism for generic spin torques\nAtsuo Shitade\nRIKEN Center for Emergent Matter Science, 2-1 Hirosawa, Wako, Saitama 351-0198, Japan\n(Dated: August 23, 2021)\nWe propose a new quantum-mechanical formalism to calculate spin torques based on the gradient\nexpansion, which naturally involves spacetime gradients of the magnetization and electromagnetic\n\felds. We have no assumption in the small-amplitude formalism or no di\u000eculty in the SU(2) gauge\ntransformation formalism. As a representative, we calculate the spin renormalization, Gilbert damp-\ning, spin-transfer torque, and \f-term in a three-dimensional ferromagnetic metal with nonmagnetic\nand magnetic impurities being taken into account within the self-consistent Born approximation.\nOur results serve as a \frst-principles formalism for spin torques.\nI. INTRODUCTION\nSpin torques have been investigated both theoreti-\ncally and experimentally in the \feld of magnetic spin-\ntronics since the celebrated discovery of the current-\ninduced magnetization reversal by the spin transfer\ntorque (STT)1{5. When an electric \feld is applied to\na ferromagnetic metal with magnetic structures such as\ndomain walls and skyrmions, the spin-polarized current\n\rows, and electron spin is transferred to the magnetiza-\ntion via the exchange interaction. Furthermore, the so-\ncalled\f-term arises from spin relaxation6{11. Electronic\ncontributions to spin torques in a ferromagnetic metal\nwithout spin-orbit interactions (SOIs) are expressed by\n~ \u001c=\u0000~s_~ n\u0000~\u000b~ n\u0002_~ n\u0000(~js\u0001~@)~ n\u0000\f~ n\u0002(~js\u0001~@)~ n;(1)\nin which~ nis the magnetization which is dynamical and\nnonuniform. sand\u000bare the spin renormalization and\nelectronic contribution to the Gilbert damping, respec-\ntively. The third and fourth terms are the STT and \f-\nterm driven by the spin-polarized current ~js. In the pres-\nence of SOIs, another spin torque called the spin-orbit\ntorque is allowed even without magnetic structures12{14.\nIn real materials, both magnetic structures and SOIs do\nexist, and hence a systematic formalism to calculate these\nspin torques is desired15.\nTo calculate spin torques quantitatively, a quantum-\nmechanical formalism is desirable. It is di\u000ecult to take\ninto account spin relaxation systematically in the semi-\nclassical Boltzmann theory8,10,13{15, and phenomenolog-\nical treatment may even lead to incorrect results on the\n\f-term8. The small-amplitude formalism, in which small\ntransverse \ructuations around a uniform state are as-\nsumed, is quantum-mechanical but cannot be applied\nto the \fnite-amplitude dynamics except for simple cases\nwithout SOIs9. The SU(2) gauge transformation formal-\nism, where a magnetic structure is transformed to a uni-\nform state, is also quantum-mechanical and correct5,11,12.\nHowever, we should be careful when we deal with mag-\nnetic impurities11. Magnetic impurities become dynami-\ncal and nonuniform by the SU(2) gauge transformation,\nwhich yields the additional SU(2) gauge \feld. If this con-\ntribution is not taken into account, the Gilbert damping\nvanishes.Here we propose a new quantum-mechanical formalism\nto calculate generic spin torques based on the gradient ex-\npansion. As a representative, we calculate four terms in\nEq. (1) in a three-dimensional (3d) ferromagnetic metal\nwith nonmagnetic and magnetic impurities. The gradient\nexpansion is a perturbation theory with respect to space-\ntime gradients16,17as well as electromagnetic \felds18{20\nin terms of the Wigner representations of the Keldysh\nGreen's functions. The former two terms in Eq. (1) are\nlinear responses of electron spin to a temporal gradient\nof the magnetization, and the latter two are the second-\norder responses to a spatial gradient and an electric \feld.\nAs mentioned in Ref. 10, it is a natural extension of the\nsemiclassical Boltzmann theory8,10,13{15. We do not have\nto pay any attention to the SU(2) gauge \feld even in the\npresence of magnetic impurities, SOIs, and sublattice de-\ngrees of freedom as in antiferromagnets.\nII. GRADIENT EXPANSION\nIn this Section, we review the gradient expansion of\nthe Keldysh Green's function with external gauge \felds\nbeing taken into account. We do not rely on any spe-\nci\fc form of the Hamiltonian, which may be disordered\nor interacting. Furthermore, gauge \felds may be abelian\nor nonabelian. The gradient expansion was already car-\nried out up to the in\fnite order in the absence of gauge\n\felds16,17and in the abelian case18,19and up to the \frst\norder in the nonabelian case20. Although we are inter-\nested in the abelian case, we give rigorous derivation up\nto the fourth order in the nonabelian case with the help\nof the nonabelian Stokes theorem21,22.\nA. Locally covariant Keldysh Green's function\nWhen we carry out the gradient expansion, it is es-\nsential to keep the local gauge covariance. First, let us\nexplain its meaning here. Under a gauge transformation\n 0(x) =V(x) (x) for a \feld  (x), gauge \feldsA\u0016(x), a\nlocally gauge-covariant quantity ~A(x), and the Keldysh\nGreen's function ^G(x1;x2) transform as\nA0\n\u0016(x) =V(x)A\u0016(x)Vy(x)\u0000i~[@\u0016V(x)]Vy(x);(2a)arXiv:1708.03424v1  [cond-mat.mes-hall]  11 Aug 20172\n~A0(x) =V(x)~A(x)Vy(x); (2b)\n^G0(x1;x2) =V(x1)^G(x1;x2)Vy(x2): (2c)\nThe Green's function ^G(x1;x2) with the hat symbol is\ngauge-covariant in the sense of Eq. (2c). However, in\nthe Wigner representation de\fned later in Eq. (6), the\ncenter-of-mass coordinate X12\u0011(x1+x2)=2 is the only\ncoordinate, and hence the Green's function should be de-\n\fned as locally gauge-covariant with respect to X12. It\ncan be achieved by introducing the Wilson line,\nW(x1;x2)\u0011Pexp\u0014\n\u00001\ni~Zx1\nx2dy\u0016A\u0016(y)\u0015\n; (3)\nwhich transforms in the same way as the Green's func-\ntion, i.e.,W0(x1;x2) =V(x1)W(x1;x2)Vy(x2).Pis\nthe path-ordered product. The locally gauge-covariant\nGreen's function ~G(x1;x2) with the tilde symbol is then\nde\fned by18{20\n~G(x1;x2)\u0011W(X12;x1)^G(x1;x2)W(x2;X12);(4)\nwhich transforms as\n~G0(x1;x2) =V(X12)~G(x1;x2)Vy(X12); (5)\ninstead of Eq. (2c). Similarly to Eq. (4), all the two-\npoint quantities with the hat symbol should be replaced\nby those with the tilde symbol.\nB. Gauge-covariant Wigner representation\nNext, we de\fne the Wigner representation of the lo-\ncally gauge-covariant Green's function18{20,\n~G(X12;p12)\u0011Z\ndDx12ep12\u0016x\u0016\n12=i~~G(x1;x2);(6)whereX12\u0011(x1+x2)=2 andx12\u0011x1\u0000x2are the center-\nof-mass and relative coordinates, respectively, and p12is\nthe relative momentum. Dis the spacetime dimension.\nDynamics of the Green's function is determined by the\nDyson equation involving convolution, which is a two-\npoint quantity de\fned by\n\\A\u0003B(x1;x2)\u0011Z\ndDx3^A(x1;x3)^B(x3;x2);(7)\nfor any two-point quantities ^Aand ^B. Therefore, we have\nto \fnd the Wigner representation of the locally gauge-\ncovariant convolution,\n~A(X12;p12)?~B(X12;p12)\u0011^A\u0003B(X12;p12):(8)\nSince the Wigner representation is just the Fourier trans-\nformation with respect to x12, convolution turns into the\nsimple product ~A(p12)~B(p12) for a translationally invari-\nant system in the absence of gauge \felds; otherwise, it\nbecomes noncommutative and is called the Moyal prod-\nuct. It is evaluated by expanding Eq. (8) with respect\nto the relative coordinates x13andx32as in Appendix A\nand is expressed by\n~A?~B=~A~B+ (i~=2)PD(~A;~B) + (i~=2)PF(~A;~B)\n+ (1=2!)(i~=2)2PD2(~A;~B) + (i~=2)2PD\u0003F(~A;~B) + (1=2!)(i~=2)2PF2(~A;~B); (9a)\nPD(~A;~B)\u0011DX\u0015~A@p\u0015~B\u0000@p\u0015~ADX\u0015~B; (9b)\nPF(~A;~B)\u0011(F\u0016\u0017@p\u0016~A@p\u0017~B+ 2@p\u0016~AF\u0016\u0017@p\u0017~B+@p\u0016~A@p\u0017~BF\u0016\u0017)=4; (9c)\nPD2(~A;~B)\u0011DX\u00151DX\u00152~A@p\u00151@p\u00152~B\u00002DX\u00151@p\u00152~A@p\u00151DX\u00152~B+@p\u00151@p\u00152~ADX\u00151DX\u00152~B; (9d)\nPD\u0003F(~A;~B)\u0011[F\u0016\u0017(DX\u0015@p\u0016~A@p\u0015@p\u0017~B\u0000@p\u0015@p\u0016~ADX\u0015@p\u0017~B)\n+ 2(DX\u0015@p\u0016~AF\u0016\u0017@p\u0015@p\u0017~B\u0000@p\u0015@p\u0016~AF\u0016\u0017DX\u0015@p\u0017~B)\n+ (DX\u0015@p\u0016~A@p\u0015@p\u0017~B\u0000@p\u0015@p\u0016~ADX\u0015@p\u0017~B)F\u0016\u0017]=4; (9e)\nPF2(~A;~B)\u0011(F\u00161\u00171F\u00162\u00172@p\u00161@p\u00162~A@p\u00171@p\u00172~B+ 4@p\u00161@p\u00162~AF\u00161\u00171F\u00162\u00172@p\u00171@p\u00172~B\n+@p\u00161@p\u00162~A@p\u00171@p\u00172~BF\u00161\u00171F\u00162\u00172+ 4F\u00161\u00171@p\u00161@p\u00162~AF\u00162\u00172@p\u00171@p\u00172~B\n+ 4@p\u00161@p\u00162~AF\u00161\u00171@p\u00171@p\u00172~BF\u00162\u00172+ 2F\u00161\u00171@p\u00161@p\u00162~A@p\u00171@p\u00172~BF\u00162\u00172)=42: (9f)3\nHere a covariant derivative and a \feld strength are de-\n\fned by\nDX\u0016~A(X;p)\u0011@X\u0016~A(X;p)\n+ [A\u0016(X);~A(X;p)]=i~; (10a)\nF\u0016\u0017(X)\u0011@X\u0016A\u0017(X)\u0000@X\u0017A\u0016(X)\n+ [A\u0016(X);A\u0017(X)]=i~: (10b)\nFor simplicity, the arguments X;p are omitted in Eq. (9)\nand below.\nAfter all, the Moyal product is regarded as a pertur-\nbation theory with respect to spacetime gradients as well\nas \feld strengths, but not to gauge \felds. Thus, the\ngauge covariance of the results is guaranteed. PDand\nPFdenote the \frst-order contributions with respect to\nspacetime gradients Dand \feld strengths F, respectively.\nPD\u0003Fis the mixed second-order contribution involving D\nandF. We also write down the second order with respect\ntoFin Eq. (9f), which may be useful for studying other\nnonlinear responses in the future. In order to derive PF2,\nwe need the fourth order with respect to x13andx32and\nobtain many other terms. All the terms up to the fourth\norder are written in Eq. (A8).\nC. Gradient expansion up to the second order\nHere we derive the gradient expansion of the Keldysh\nGreen's function. We focus on the abelian case and as-\nsume a static and uniform \feld strength. Similarly to\nEq. (9a), we expand the Green's function and self-energy\nas23,24\n~G=~G0+ (~=2)~GD+ (~=2)~GF+ (1=2!)(~=2)2~GD2\n+ (~=2)2~GD\u0003F+ (1=2!)(~=2)2~GF2; (11a)\n~\u0006 =~\u00060+ (~=2)~\u0006D+ (~=2)~\u0006F+ (1=2!)(~=2)2~\u0006D2\n+ (~=2)2~\u0006D\u0003F+ (1=2!)(~=2)2~\u0006F2: (11b)\nNote that ~G0is the unperturbed Green's function with\ndisorder or interactions being taken into account. ~GP\nand ~GP\u0003Q(P;Q =D;F) are the \frst and second orders\nwith respect to spacetime gradients or \feld strengths,\nrespectively. By substituting these into the left Dyson\nequation,\n(~L\u0000~\u0006)?~G= 1; (12)\nin which ~Lis the Lagrangian, we get ~G0= (~L\u0000 ~\u00060)\u00001\nand\n~G\u00001\n0~GP=~\u0006P~G0\u0000iPP(~G\u00001\n0;~G0); (13a)~G\u00001\n0~GP\u0003Q=~\u0006P\u0003Q~G0\u0000i2PP\u0003Q(~G\u00001\n0;~G0)\n+ [~\u0006Q~GP+iPP(~\u0006Q;~G0)\u0000iPP(~G\u00001\n0;~GQ)\n+ (P$Q)]: (13b)\nThe self-energies are determined self-consistently.\nTo calculate the expectation values, the lesser Green's\nfunction is necessary. In the real-time representation, the\nGreen's function and self-energy are of matrix forms23,24,\n~G=\u0014\nGR2G<\n0GA\u0015\n; (14a)\n~\u0006 =\u0014\n\u0006R2\u0006<\n0 \u0006A\u0015\n; (14b)\nin which R, A, and <indicate the retarded, advanced,\nand lesser components, respectively. For the \frst order,\nthe lesser one can be written as\nG<\nP=\u0006[(GR\nP\u0000GA\nP)f(\u0000p0) +G<(1)\nPf0(\u0000p0)];(15a)\n\u0006<\nP=\u0006[(\u0006R\nP\u0000\u0006A\nP)f(\u0000p0) + \u0006<(1)\nPf0(\u0000p0)];(15b)\nin which the upper and lower signs indicate boson and\nfermion, respectively, and f(\u0018) = (e\u0018=T\u00071)\u00001is the\ndistribution function at temperature T. By introducing\nPP(~A;~B)\u0011\u0011IJ\nP@I~A@J~B(I;J=X\u0016;p\u0017) with\u0011X\u0016p\u0017\nD =\n\u0000\u0011p\u0017X\u0016\nD =\u000e\u0016\n\u0017and\u0011p\u0016p\u0017\nF =F\u0016\u0017, we obtain an equivalent\nform of Eq. (13a)23,24,\n(GR\n0)\u00001GR\nP=\u0006R\nPGR\n0\u0000i\u0011IJ\nP@I(GR\n0)\u00001@JGR\n0; (16a)\n(GA\n0)\u00001GA\nP=\u0006A\nPGA\n0\u0000i\u0011IJ\nP@I(GA\n0)\u00001@JGA\n0; (16b)\n(GR\n0)\u00001G<(1)\nP=\u0006<(1)\nPGA\n0+i\u0011Ip0\nPf@I(GR\n0)\u00001(GR\n0\u0000GA\n0)\n\u0000[(GR\n0)\u00001\u0000(GA\n0)\u00001]@IGA\n0g: (16c)\nSimilarly, for the second order, we obtain the lesser\nGreen's function,\nG<\nP\u0003Q=\u0006[(GR\nP\u0003Q\u0000GA\nP\u0003Q)f(\u0000p0) +G<(1)\nP\u0003Qf0(\u0000p0)\n+G<(2)\nP\u0003Qf00(\u0000p0)]; (17a)\n\u0006<\nP\u0003Q=\u0006[(\u0006R\nP\u0003Q\u0000\u0006A\nP\u0003Q)f(\u0000p0) + \u0006<(1)\nP\u0003Qf0(\u0000p0)\n+ \u0006<(2)\nP\u0003Qf00(\u0000p0)]; (17b)\nand an equivalent form of Eq. (13b),\n(GR\n0)\u00001GR\nP\u0003Q=\u0006R\nP\u0003QGR\n0+ [\u0006R\nQGR\nP+i\u0011IJ\nP@I\u0006R\nQ@JGR\n0\u0000i\u0011IJ\nP@I(GR\n0)\u00001@JGR\nQ+ (P$Q)]\n+\u0011IJ\nP\u0011KL\nQ@I@K(GR\n0)\u00001@J@LGR\n0; (18a)4\n(GA\n0)\u00001GA\nP\u0003Q=\u0006A\nP\u0003QGA\n0+ [\u0006A\nQGA\nP+i\u0011IJ\nP@I\u0006A\nQ@JGA\n0\u0000i\u0011IJ\nP@I(GA\n0)\u00001@JGA\nQ+ (P$Q)]\n+\u0011IJ\nP\u0011KL\nQ@I@K(GA\n0)\u00001@J@LGA\n0; (18b)\n(GR\n0)\u00001G<(1)\nP\u0003Q=\u0006<(1)\nP\u0003QGA\n0+\u0010\n\u0006R\nQG<(1)\nP+ \u0006<(1)\nQGA\nP+i\u0011IJ\nP@I\u0006<(1)\nQ@JGA\n0\u0000i\u0011Ip0\nP[@I\u0006R\nQ(GR\n0\u0000GA\n0)\u0000(\u0006R\nQ\u0000\u0006A\nQ)@IGA\n0]\n\u0000i\u0011IJ\nP@I(GR\n0)\u00001@JG<(1)\nQ+i\u0011Ip0\nPf@I(GR\n0)\u00001(GR\nQ\u0000GA\nQ)\u0000[(GR\n0)\u00001\u0000(GA\n0)\u00001]@IGA\nQg\n\u0000\u0011Ip0\nP\u0011KL\nQf@I@K(GR\n0)\u00001@L(GR\n0\u0000GA\n0) +@L[(GR\n0)\u00001\u0000(GA\n0)\u00001]@I@KGA\n0g+ (P$Q)\u0011\n; (18c)\n(GR\n0)\u00001G<(2)\nP\u0003Q=\u0006<(2)\nP\u0003QGA\n0+ [i\u0011Ip0\nP\u0006<(1)\nQ@IGA\n0+i\u0011Ip0\nP@I(GR\n0)\u00001G<(1)\nQ+ (P$Q)]\n+\u0011Ip0\nP\u0011Kp0\nQf@I@K(GR\n0)\u00001(GR\n0\u0000GA\n0) + [(GR\n0)\u00001\u0000(GA\n0)\u00001]@I@KGA\n0g: (18d)\nNote that the left and right Dyson equations are equiv-\nalent as explicitly proved in Appendix B. Generally, the\nnth-order lesser Green's function with respect to a tem-\nporal gradient or an electric \feld involves the nth deriva-\ntive of the distribution function.\nD. Spin torques\nSpin torques are proportional to the spin expectation\nvalue. The spin expectation value is given by\nh~ \u001bi=\u0006i~ZdDp\n(2\u0019~)Dtr~ \u001bG<\n=\u0006i~ZdDp\n(2\u0019~)Dtr~ \u001b[G<\n0+ (~=2)G<\nD+ (~=2)G<\nF\n+ (1=2!)(~=2)2G<\nD2+ (~=2)2G<\nD\u0003F\n+ (1=2!)(~=2)2G<\nF2]: (19)\nAmong many terms in Eq. (19), G<\nDyields the spin renor-\nmalization and Gilbert damping. G<\nFyields the spin-\norbit torque in the presence of SOIs. In order to calculate\nthe STT and \f-term driven by an electric \feld, G<\nD\u0003Fis\nnecessary. Equations (15)-(19) are our central results for\ncalculating spin torques in generic systems.\nIII. APPLICATION TO A 3D\nFERROMAGNETIC METAL\nAs an example, we explicitly calculate spin torques in\na 3d ferromagnetic metal,\nH(X;~ p) =~ p2=2m\u0000\u0016\u0000J~ n(X)\u0001~ \u001b; (20)\nwith the chemical potential \u0016and [~ n(X)]2= 1. We take\ninto account nonmagnetic and magnetic impurities,\nVimp(~X) =NiX\njvi\u000e(~X\u0000~Xij)+NsX\njvs~ mj\u0001~ \u001b\u000e(~X\u0000~Xsj): (21)\nMagnetic impurities are assumed to be isotropic, namely,\nma\ni= 0;ma\nimb\nj=\u000eab=3 after average over the magnetiza-\ntion directions. The same system was studied within the\nBorn approximation in the literature9,11. Here we em-\nploy the self-consistent Born approximation, but it does\nnot change any results quantitatively.\nWe only have to calculate the momentum integrals of\nG<\n0,G<\nD,G<\nF, andG<\nD\u0003Fwith Eqs. (15)-(18) in order.\nFirst, the unperturbed Green's function is given by\n(GR\n0)\u00001(X;\u0018;~ p ) =\u0018\u0000H(X;~ p)\u0000\u0006R\n0(X;\u0018)\n=\u0018R(\u0018)\u0000x+JR(\u0018)~ n(X)\u0001~ \u001b; (22a)\nGR\n0(X;\u0018;~ p ) =\u0000R\n0(\u0018;~ p)[\u0018R(\u0018)\u0000x\n\u0000JR(\u0018)~ n(X)\u0001~ \u001b]; (22b)\n\u0000R\n0(\u0018;~ p) =f[\u0018R(\u0018)\u0000x]2\u0000[JR(\u0018)]2g\u00001;(22c)\nwith\u0018R(\u0018)\u0011\u0018+\u0016\u0000\u0006R\n00(\u0018),JR(\u0018)\u0011J\u0000\n\u0006R\n03(\u0018), andx\u0011~ p2=2m. Note that the self-energy\nis expressed as \u0006R\n0(X;\u0018) = \u0006R\n00(\u0018) + \u0006R\n03(\u0018)~ n(X)\u0001\n~ \u001bdue to the spin rotation symmetry. For conve-\nnience, we de\fne gas the momentum integral of\nGmutiplied by 4 \u0019(~2=2m)3=2and introduce \ri\u0011\nniv2\ni(2m=~2)3=2=4\u0019;\rs\u0011nsv2\ns(2m=~2)3=2=4\u0019. The mo-\nmentum integral and self-energy are obtained by self-\nconsistently solving\ngR\n0(X;\u0018)\u00114\u0019\u0012~2\n2m\u00133=2Zd3p\n(2\u0019~)3GR\n0(X;\u0018;~ p )\n=IR\n11(\u0018)\u0000IR\n01(\u0018)~ n(X)\u0001~ \u001b; (23a)\n\u0006R\n0(X;\u0018) =\rigR\n0(X;\u0018) +\rs~ mi\u0001~ \u001bgR\n0(X;\u0018)~ mj\u0001~ \u001b\n=(\ri+\rs)gR\n00(\u0018)\n+(\ri\u0000\rs=3)gR\n03(\u0018)~ n(X)\u0001~ \u001b; (23b)\nin which we de\fne\nIR\nmn(\u0018)\u0011Z\u0003\n0pxdx\n\u0019[\u0018R(\u0018)\u0000x]m[JR(\u0018)]2n\u0000m\u00001\nf[\u0018R(\u0018)\u0000x]2\u0000[JR(\u0018)]2gn:(24)5\nSecond, the \frst-order Green's functions are given by\nGR\nD(X;\u0018;~ p ) =GR\n0(X;\u0018;~ p )\u0006R\nD(X;\u0018)GR\n0(X;\u0018;~ p )\n+ 2JR(\u0018)[\u0000R\n0(\u0018;~ p)]2fJR(\u0018)\u0018R0(\u0018)\u0000[\u0018R(\u0018)\u0000x]JR0(\u0018)g~ n(X)\u0002_~ n(X)\u0001~ \u001b\n+ 2[JR(\u0018)]2[\u0000R\n0(\u0018;~ p)]2(pi=m)~ n(X)\u0002@Xi~ n(X)\u0001\u001b; (25a)\nG<(1)\nD(X;\u0018;~ p ) =GR\n0(X;\u0018;~ p )\u0006<(1)\nD(X;\u0018)GA\n0(X;\u0018;~ p )\n\u0000i[JR(\u0018) +JA(\u0018)]j\u0000R\n0(\u0018;~ p)j2[j\u0018R(\u0018)\u0000xj2\u0000jJR(\u0018)j2]_~ n(X)\u0001~ \u001b\n+ [JR(\u0018) +JA(\u0018)]j\u0000R\n0(\u0018;~ p)j2f[\u0018R(\u0018)\u0000x]JA(\u0018)\u0000[\u0018A(\u0018)\u0000x]JR(\u0018)g~ n(X)\u0002_~ n(X)\u0001~ \u001b\n\u0000i@X0[GR\n0(X;\u0018;~ p ) +GA\n0(X;\u0018;~ p )]; (25b)\nGR\nF(X;\u0018;~ p ) =GR\n0(X;\u0018;~ p )\u0006R\nF(X;\u0018)GR\n0(X;\u0018;~ p ); (25c)\nG<(1)\nF(X;\u0018;~ p ) =GR\n0(X;\u0018;~ p )\u0006<(1)\nF(X;\u0018)GA\n0(X;\u0018;~ p )\n+iFj0(pj=m)f2GR\n0(X;\u0018;~ p )GA\n0(X;\u0018;~ p )\u0000[GR\n0(X;\u0018;~ p )]2\u0000[GA\n0(X;\u0018;~ p )]2g: (25d)\nThe self-energies are expressed as \u0006R\nD(X;\u0018)\u0011\u0006R\nD2(\u0018)~ n(X)\u0002_~ n(X)\u0001~ \u001b;\u0006<(1)\nD(X;\u0018)\u0011\u0006<(1)\nD1(\u0018)_~ n(X)\u0001~ \u001b+\u0006<(1)\nD2(\u0018)~ n(X)\u0002\n_~ n(X)\u0001~ \u001band obtained by solving the following sets of linear equations,\ngR\nD2(\u0018) =IR\n01(\u0018)\u0006R\nD2(\u0018)=JR(\u0018) + 2[\u0018R0(\u0018)IR\n02(\u0018)\u0000JR0(\u0018)IR\n12(\u0018)]=JR(\u0018); (26a)\n\u0006R\nD2(\u0018) =(\ri\u0000\rs=3)gR\nD2(\u0018); (26b)\"\ng<(1)\nD1(\u0018)\ng<(1)\nD2(\u0018)#\n=\"\nJ<(1)\n1(\u0018)\u0000J<(1)\n2(\u0018)\nJ<(1)\n2(\u0018)J<(1)\n1(\u0018)#\"\n\u0006<(1)\nD1(\u0018)\n\u0006<(1)\nD2(\u0018)#\n\u00002i\"\n[JR(\u0018) +JA(\u0018)]J<(1)\n1(\u0018)=2\u0000[IR\n01(\u0018) + c:c:]=2\n[JR(\u0018) +JA(\u0018)]J<(1)\n2(\u0018)=2#\n; (26c)\n\"\n\u0006<(1)\nD1(\u0018)\n\u0006<(1)\nD2(\u0018)#\n=(\ri\u0000\rs=3)\"\ng<(1)\nD1(\u0018)\ng<(1)\nD2(\u0018)#\n: (26d)\nHere we de\fne\nJ<(1)\n1(\u0018)\u0011Z\u0003\n0pxdx\n\u0019j\u0018R(\u0018)\u0000xj2\u0000jJR(\u0018)j2\nj[\u0018R(\u0018)\u0000x]2\u0000[JR(\u0018)]2j2; (27a)\nJ<(1)\n2(\u0018)\u0011Z\u0003\n0pxdx\n\u0019i[\u0018R(\u0018)\u0000x]JA(\u0018)\u0000i[\u0018A(\u0018)\u0000x]JR(\u0018)\nj[\u0018R(\u0018)\u0000x]2\u0000[JR(\u0018)]2j2; (27b)\nJ<(1)\n3(\u0018)\u0011J<(1)\n2(\u0018)\u00002\n3[iJR(\u0018)\u0000iJA(\u0018)]Z\u0003\n0pxdx\n\u0019x\nj[\u0018R(\u0018)\u0000x]2\u0000[JR(\u0018)]2j2; (27c)\nJ<(1)\n4(\u0018)\u0011Z\u0003\n0pxdx\n\u00192\nj[\u0018R(\u0018)\u0000x]2\u0000[JR(\u0018)]2j2\u0000\njJR(\u0018)j2\n+2\n3x\u001aJR(\u0018)\n[\u0018R(\u0018)\u0000x]2\u0000[JR(\u0018)]2+JA(\u0018)\n[\u0018A(\u0018)\u0000x]2\u0000JA2(\u0018)\u001b\nf[\u0018R(\u0018)\u0000x]JA(\u0018) + [\u0018A(\u0018)\u0000x]JR(\u0018)g\u0013\n:\n(27d)\nThese momentum integrals in Eqs. (24) and (27) are explicitly calculated in Appendix C. The other components\nvanish, i.e., \u0006R\nF(X;\u0018) =gR\nF(X;\u0018) = 0 and \u0006<(1)\nF(X;\u0018) =g<(1)\nF(X;\u0018) = 0.\nThird, the second-order Green's functions are given by\nGR\nD\u0003F(~X;\u0018;~ p ) =GR\n0(~X;\u0018;~ p )\u0006R\nD\u0003F(~X;\u0018)GR\n0(~X;\u0018;~ p )\n\u00002[\u0000R\n0(\u0018;~ p)]2\u0000\nfJR(\u0018)\u0018R0(\u0018)\u0000[\u0018R(\u0018)\u0000x]JR0(\u0018)g\u000eij\n\u0000JR0(\u0018)pipj=m\u0001\nFj0@Xi~ n(~X)\u0001~ \u001b=m; (28a)\nG<(1)\nD\u0003F(~X;\u0018;~ p ) =GR\n0(~X;\u0018;~ p )\u0006<(1)\nD\u0003F(~X;\u0018)GA\n0(~X;\u0018;~ p )\n\u00002j\u0000R\n0(\u0018;~ p)j2\u0000\nf[\u0018R(\u0018)\u0000x]JA(\u0018)\u0000[\u0018A(\u0018)\u0000x]JR(\u0018)g\u000eij\n\u0000[JR(\u0018)\u0000JA(\u0018)]pipj=m\u0001\nFj0@Xi~ n(~X)\u0001~ \u001b=m\n+ 4ij\u0000R\n0(\u0018;~ p)j2\u0000\njJR(\u0018)j2\u000eij+ [\u0000R\n0(\u0018;~ p)JR(\u0018) + \u0000A\n0(\u0018;~ p)JA(\u0018)]6\n\u0002f[\u0018R(\u0018)\u0000x]JA(\u0018) + [\u0018A(\u0018)\u0000x]JR(\u0018)gpipj=m\u0001\nFj0~ n(~X)\u0002@Xi~ n(~X)\u0001~ \u001b=m; (28b)\nwhere we drop the X0dependence in ~ n(X) because we are interested in the STT and \f-term only. The self-\nenergies are expressed as \u0006R\nD\u0003F(~X;\u0018)\u0011\u0006R\nD\u0003F1(\u0018)Fi0@Xi~ n(~X)\u0001~ \u001b=m; \u0006<(1)\nD\u0003F(~X;\u0018)\u0011\u0006<(1)\nD\u0003F1(\u0018)Fi0@Xi~ n(~X)\u0001~ \u001b=m +\n\u0006<(1)\nD\u0003F2(\u0018)Fi0~ n(~X)\u0002@Xi~ n(~X)\u0001~ \u001b=m and obtained by solving the following sets of linear equations,\ngR\nD\u0003F1(\u0018) =IR\n01(\u0018)\u0006R\nD\u0003F1(\u0018)=JR(\u0018)\n\u00002f[3JR(\u0018)\u0018R0(\u0018)\u00002\u0018R(\u0018)JR0(\u0018)]IR\n02(\u0018)\u0000JR(\u0018)JR0(\u0018)IR\n12(\u0018)g=3JR3(\u0018); (29a)\n\u0006R\nD\u0003F1(\u0018) =(\ri\u0000\rs=3)gR\nD\u0003F1(\u0018); (29b)\"\ng<(1)\nD\u0003F1(\u0018)\ng<(1)\nD\u0003F2(\u0018)#\n=\"\nJ<(1)\n1(\u0018)\u0000J<(1)\n2(\u0018)\nJ<(1)\n2(\u0018)J<(1)\n1(\u0018)#\"\n\u0006<(1)\nD\u0003F1(\u0018)\n\u0006<(1)\nD\u0003F2(\u0018)#\n+ 2i\"\nJ<(1)\n3(\u0018)\nJ<(1)\n4(\u0018)#\n; (29c)\n\"\n\u0006<(1)\nD\u0003F1(\u0018)\n\u0006<(1)\nD\u0003F2(\u0018)#\n=(\ri\u0000\rs=3)\"\ng<(1)\nD\u0003F1(\u0018)\ng<(1)\nD\u0003F2(\u0018)#\n: (29d)\nThe spin expectation value is given by\nh~ \u001bi(X) =\u0006i~Zd\u0018\n2\u0019~Zd3p\n(2\u0019~)3tr~ \u001bG<(X;\u0018;~ p )\n=h~ \u001bi0(X)\u0000~\n4\u00192J\u00122mJ\n~2\u00133=2\n[\u001c\u000b\u0000\u001cren~ n(X)\u0002]_~ n(X)\n\u00001\n4\u00192J\u00122mJ\n~2\u00131=2\nFi0[\u001c\f\u0000\u001cSTT~ n(~X)\u0002]@Xi~ n(~X); (30a)\nh~ \u001bi0(X)\u00111\n2\u00192\u00122m\n~2\u00133=2\n~ n(X)Z\nd\u0018f(\u0018)[\u0000=gR\n03(\u0018)]; (30b)\n\u001c\u000b\u0011J\u00001=2Z\nd\u0018[\u0000f0(\u0018)][ig<(1)\nD1(\u0018)=2]; (30c)\n\u001cren\u0011J\u00001=2Z\nd\u0018ff(\u0018)[\u0000=gR\nD2(\u0018)]\u0000[\u0000f0(\u0018)][ig<(1)\nD2(\u0018)=2]g; (30d)\n\u001c\f\u0011\u0000J1=2Z\nd\u0018ff(\u0018)[\u0000=gR\nD\u0003F1(\u0018)] + [\u0000f0(\u0018)][g<(1)\nD\u0003F1(\u0018)=2i]g; (30e)\n\u001cSTT\u0011J1=2Z\nd\u0018[\u0000f0(\u0018)][g<(1)\nD\u0003F2(\u0018)=2i]; (30f)\nand spin torques by\n~ \u001c(X) =~ n(X)\u0002Jh~ \u001bi(X)\n=\u0000~\n4\u00192\u00122mJ\n~2\u00133=2\n[\u001cren+\u001c\u000b~ n(X)\u0002]_~ n(X)\n\u00001\n4\u00192\u00122mJ\n~2\u00131=2\n\u0002Fi0[\u001cSTT+\u001c\f~ n(~X)\u0002]@Xi~ n(~X): (31)\n\u001crenand\u001c\u000bare the dimensionless spin renormalization\nand Gilbert damping, while \u001cSTT and\u001c\fare the STT\nand\f-term. To evaluate Eqs. (30c)-(30f), we carry out\nnumerical integrals by putting the energy unit J= 1, the\nmomentum cuto\u000b \u0003 = 103, and temperature T= 10\u00003\nand dividing the energy interval j\u0018j<5 into 217subinter-\nvals. In Fig. 1, we show their chemical-potential depen-dences for di\u000berent \riand\rs. We also show the previous\nresults obtained by the small-amplitude9and the SU(2)\ngauge transformation formalisms11at zero temperature,\n\u001cren=(\u00163=2\n+\u0000\u00163=2\n\u0000)=3J3=2; (32a)\n\u001c\u000b=\rs(\u00161=2\n++\u00161=2\n\u0000)2=3J3=2; (32b)\n\u001cSTT=1\n3J1=2X\n\u001b\u001b\u00163=2\n\u001b\n\ri\u00161=2\n\u001b+\rs(2\u00161=2\n\u0000\u001b+\u00161=2\n\u001b)=3\n!2J1=2(3\ri+ 5\rs)=9(\ri+\rs)2\u0011\u001cSTT1;(32c)\n\u001c\f=2\rs(\u00161=2\n++\u00161=2\n\u0000)\u001cSTT=3J; (32d)\nwith\u0016\u001b\u0011\u0016+\u001bJ. Note that \u001cSTT1in Eq. (32c) is \u001cSTT\nfor the\u0016!1 limit. Our results completely coincide\nwith these previous ones.7\n 0 1 2 3 4 5 6\n-1-0.5  0 0.5  1 1.5  2 2.5  3(a)\n3τren\nµγi=4e-3, γs=4e-3\nγi=4e-3, γs=8e-3\nγi=8e-3, γs=4e-3\nγi=8e-3, γs=8e-3\n 0 2 4 6 8 10 12\n-1-0.5  0 0.5  1 1.5  2 2.5  3(b)\n3τα/γs\nµ 0 0.2 0.4 0.6 0.8 1 1.2 1.4\n-1-0.5  0 0.5  1 1.5  2 2.5  3(c)\nτSTT/τSTT∞\nµ 0 0.5 1 1.5 2 2.5 3 3.5\n-1-0.5  0 0.5  1 1.5  2 2.5  3(d)\n3τβ/2γsτSTT\nµ\nFIG. 1. (Color online) Chemical-potential dependences of (a) 3 \u001cren, (b) 3\u001c\u000b=\rs, (c)\u001cSTT=\u001cSTT1, and (d) 3 \u001c\f=2\rs\u001cSTTfor\ndi\u000berent\riand\rs. Points are obtained by the gradient expansion formalism and numerical calculation of Eqs. (30c)-(30f),\nwhile lines are obtained by the small-amplitude9and the SU(2) gauge transformation formalisms11.\nIV. DISCUSSION AND SUMMARY\nLet us clarify how the SU(2) gauge transformation\nformalism corresponds to the gradient expansion for-\nmalism. In the former, the Green's function is diag-\nonalized by a unitary matrix U\u0011~ u\u0001~ \u001bwith~ u\u0011\n[sin\u0012=2 cos\u001e;sin\u0012=2 sin\u001e;cos\u0012=2], which transforms the\nmagnetization ~ n\u0011[sin\u0012cos\u001e;sin\u0012sin\u001e;cos\u0012] to~ zand\nyields the SU(2) gauge \feld A\u0016\u0011 \u0000iUy@X\u0016U=~ u\u0002\n@X\u0016~ u\u0001~ \u001b5,11,12. Thus, spacetime gradients of the magne-\ntization are described by this SU(2) gauge \feld. In the\nlatter,@X\u0016(GR\n0)\u00001/@X\u0016~ n\u0001~ \u001bin Eqs. (16) and (18) is\ntransformed to Uy@X\u0016(GR\n0)\u00001U/2~ z\u0002~A\u0016\u0001~ \u001band thus\nplays the same role as the SU(2) gauge \feld.\nWe emphasize that there is another source of the SU(2)\ngauge \feld in the SU(2) gauge transformation formal-\nism11. Magnetic impurities, which are quenched in the\noriginal frame, become dynamical and nonuniform in the\nadiabatic frame and yield the SU(2) gauge \feld. If this\ncontribution is not taken into account, the Gilbert damp-\ning vanishes, and the \f-term is not fully reproduced when\nmagnetic impurities are anisotropic. In the gradient ex-\npansion, we do not rely on the SU(2) gauge transforma-tion and hence do not encounter such di\u000eculty.\nIn summary, we demonstrated how the gradient ex-\npansion works for calculating spin torques quantum-\nmechanically. We derived the gradient expansion up to\nthe fourth order with respect to the relative coordinates\nwith abelian or nonabelian gauge \felds being taken into\naccount, which enables us to investigate nonlinear re-\nsponses with respect to spacetime gradients as well as\n\feld strengths. We applied this formalism to a 3d ferro-\nmagnetic metal with nonmagnetic and magnetic impuri-\nties and successfully reproduced the previous results on\nthe spin renormalization, Gilbert damping, STT, and \f-\nterm. The greatest advantage of our formalism is that we\ndo not assume any assumptions such as small transverse\n\ructuations or su\u000ber from the SU(2) gauge \feld arising\nfrom magnetic impurities or SOIs. Our central results\nEqs. (15)-(19) serve as \frst-principles formulas of spin\ntorques.\nACKNOWLEDGMENTS\nWe thank J. Fujimoto and G. Tatara for discussion\nand reading our manuscript. This work was supported\nby RIKEN Special Postdoctoral Researcher Program.\nAppendix A: Derivation of Eq. (9)and more\nIn this Appendix, we evaluate the Wigner representation of convolution Eq. (8),\n^A\u0003B(X12;p12) =Z\ndDx12Z\ndDx3ep12\u0016x\u0016\n12=i~W(X12;x1)^A(x1;x3)^B(x3;x2)W(x2;X12)\n=Z\ndDx12Z\ndDx3ZdDp13\n(2\u0019~)DZdDp32\n(2\u0019~)Dep12\u0016x\u0016\n12=i~e\u0000p13\u0016x\u0016\n13=i~e\u0000p32\u0016x\u0016\n32=i~\n\u0002W(X12;x1)W(x1;X13)~A(X13;p13)W(X13;x3)\n\u0002W(x3;X32)~B(X32;p32)W(X32;x2)W(x2;X12) (A1)\n=Z\ndDx12Z\ndDx3ZdDp13\n(2\u0019~)DZdDp32\n(2\u0019~)Dep12\u0016x\u0016\n12=i~e\u0000p13\u0016x\u0016\n13=i~e\u0000p32\u0016x\u0016\n32=i~\n\u0002[W(X12;x1)W(x1;X13)W(X13;X12)][W(X12;X13)~A(X13;p13)W(X13;X12)]8\n(a)\nX12\nX32X13x1\nx3 x2(b)\nX12\nX32X13x1\nx3 x2(c)v\n-1\n-11\n1\nu\nS2S3S1\nFIG. 2. Transformation from (a) the Wilson line corresponding to Eq. (A1) to (b) that to Eq. (A2). The plane spanned by\nx1;x2;x3is divided into three planes, S1,S2, andS3. (c) Three planes are parametrized by y=X12+ux32=2 +vx13=2.\n\u0002[W(X12;X13)W(X13;x3)W(x3;X32)W(X32;X12)][W(X12;X32)~B(X32;p32)W(X32;X12)]\n\u0002[W(X12;X32)W(X32;x2)W(x2;X12)]: (A2)\nHere we insert the identities W(X13;X12)W(X12;X13) =W(X32;X12)W(X12;X32) = 1, which corresponds to trans-\nformation of the Wilson line as shown in Fig. 2. We de\fne three planes S1,S2, andS3as in Fig. 2(b), whose origin\nisX12. To evaluate the \frst factor of the integrand in Eq. (A2), we use the nonabelian Stokes theorem21,22,\nW(X12;x1)W(x1;X13)W(X13;X12) =Sexp\u0014\n\u00001\n2i~Z\nS1dy\u0016dy\u0017W(X12;y)F\u0016\u0017(y)W(y;X 12)\u0015\n=Sexp\u00141\n8i~Z1\n0duZu\n0dv(x\u0016\n13x\u0017\n32\u0000x\u0017\n13x\u0016\n32)W(X12;y)F\u0016\u0017(y)W(y;X 12)\u0015\n;(A3)\nin whichSis the surface-ordered product21,22, andy=X12+ux32=2+vx13=2. By using the Taylor expansion around\nX12, we obtain\nF\u0016\u0017(y) =e(ux\u0015\n32+vx\u0015\n13)@X\u0015\n12=2F\u0016\u0017\n=F\u0016\u0017+ (ux\u0015\n32+vx\u0015\n13)@X\u0015\n12F\u0016\u0017=2\n+ (ux\u00151\n32+vx\u00151\n13)(ux\u00152\n32+vx\u00152\n13)@X\u00151\n12@X\u00152\n12F\u0016\u0017=8 +O(x3); (A4a)\nW(y;X 12) =1\u00001\n2i~Z1\n0dw(ux\u0016\n32+vx\u0016\n13)[A\u0016+w(ux\u0017\n32+vx\u0017\n13)@X\u0017\n12A\u0016=2]\n+1\n4(i~)2Z1\n0dw1Zw1\n0dw2(ux\u00161\n32+vx\u00161\n13)(ux\u00162\n32+vx\u00162\n13)A\u00161A\u00162+O(x3)\n=1\u0000(ux\u0016\n32+vx\u0016\n13)A\u0016=2i~\u0000(ux\u0016\n32+vx\u0016\n13)(ux\u0017\n32+vx\u0017\n13)@X\u0017\n12A\u0016=8i~\n+ (ux\u00161\n32+vx\u00161\n13)(ux\u00162\n32+vx\u00162\n13)A\u00161A\u00162=8(i~)2+O(x3); (A4b)\nW(X12;y)F\u0016\u0017(y)W(y;X 12) =F\u0016\u0017+ (ux\u0015\n32+vx\u0015\n13)(@X\u0015\n12F\u0016\u0017+ [A\u0015;F\u0016\u0017]=i~)=2\n+ (ux\u00151\n32+vx\u00151\n13)(ux\u00152\n32+vx\u00152\n13)[@X\u00151\n12@X\u00152\n12F\u0016\u0017+ [@X\u00152\n12A\u00151;F\u0016\u0017]=i~\n+ 2[A\u00151;@X\u00152\n12F\u0016\u0017]=i~+ [A\u00151;[A\u00152;F\u0016\u0017]]=(i~)2]=8 +O(x3)\n=F\u0016\u0017+ (ux\u0015\n32+vx\u0015\n13)DX\u0015\n12F\u0016\u0017=2\n+ (ux\u00151\n32+vx\u00151\n13)(ux\u00152\n32+vx\u00152\n13)DX\u00151\n12DX\u00152\n12F\u0016\u0017=8 +O(x3)\n=e(ux\u0015\n32+vx\u0015\n13)DX\u0015\n12=2F\u0016\u0017: (A4c)\nHere and below we omit the argument X12for simplicity. Then, we express Eq. (A3) as\nW(X12;x1)W(x1;X13)W(X13;X12) =1 +1\n8i~Z1\n0duZu\n0dv(x\u0016\n13x\u0017\n32\u0000x\u0017\n13x\u0016\n32)[F\u0016\u0017+ (ux\u0015\n32+vx\u0015\n13)DX\u0015\n12F\u0016\u0017=29\n+ (ux\u00151\n32+vx\u00151\n13)(ux\u00152\n32+vx\u00152\n13)DX\u00151\n12DX\u00152\n12F\u0016\u0017=8]\n+1\n64(i~)2Z1\n0du1Zu1\n0dv1Zu1\n0du2Zu2\n0dv2\n\u0002(x\u00161\n13x\u00171\n32\u0000x\u00171\n13x\u00161\n32)(x\u00162\n13x\u00172\n32\u0000x\u00172\n13x\u00162\n32)F\u00161\u00171F\u00162\u00172+O(x5)\n=1 + (x\u0016\n13x\u0017\n32\u0000x\u0017\n13x\u0016\n32)F\u0016\u0017=16i~+ (x\u0016\n13x\u0017\n32\u0000x\u0017\n13x\u0016\n32)(2x\u0015\n32+x\u0015\n13)DX\u0015\n12F\u0016\u0017=96i~\n+ (x\u00161\n13x\u00171\n32\u0000x\u00171\n13x\u00161\n32)(x\u00162\n13x\u00172\n32\u0000x\u00172\n13x\u00162\n32)F\u00161\u00171F\u00162\u00172=512(i~)2\n+ (x\u0016\n13x\u0017\n32\u0000x\u0017\n13x\u0016\n32)[6x\u00151\n32x\u00152\n32+ 3(x\u00151\n32x\u00152\n13+x\u00152\n32x\u00151\n13) + 2x\u00151\n13x\u00152\n13]\n\u0002DX\u00151\n12DX\u00152\n12F\u0016\u0017=1536i~+O(x5): (A5)\nThe other factors in Eq. (A2) are similarly obtained as\nW(X12;X13)~A(X13;p13)W(X13;X12) =ex\u0015\n32DX\u0015\n12=2~A(p13); (A6a)\nW(X12;X32)~B(X32;p32)W(X32;X12) =e\u0000x\u0015\n13DX\u0015\n12=2~B(p32); (A6b)\nW(X12;X13)W(X13;x3)W(x3;X32)W(X32;X12) =1 + (x\u0016\n13x\u0017\n32\u0000x\u0017\n13x\u0016\n32)F\u0016\u0017=8i~\n+ (x\u0016\n13x\u0017\n32\u0000x\u0017\n13x\u0016\n32)(x\u0015\n32\u0000x\u0015\n13)DX\u0015\n12F\u0016\u0017=32i~\n+ (x\u00161\n13x\u00171\n32\u0000x\u00171\n13x\u00161\n32)(x\u00162\n13x\u00172\n32\u0000x\u00172\n13x\u00162\n32)F\u00161\u00171F\u00162\u00172=128(i~)2\n+ (x\u0016\n13x\u0017\n32\u0000x\u0017\n13x\u0016\n32)[4x\u00151\n32x\u00152\n32\u00003(x\u00151\n32x\u00152\n13+x\u00152\n32x\u00151\n13) + 4x\u00151\n13x\u00152\n13]\n\u0002DX\u00151\n12DX\u00152\n12F\u0016\u0017=768i~+O(x5); (A6c)\nW(X12;X32)W(X32;x2)W(x2;X12) =1 + (x\u0016\n13x\u0017\n32\u0000x\u0017\n13x\u0016\n32)F\u0016\u0017=16i~\n\u0000(x\u0016\n13x\u0017\n32\u0000x\u0017\n13x\u0016\n32)(x\u0015\n32+ 2x\u0015\n13)DX\u0015\n12F\u0016\u0017=96i~\n+ (x\u00161\n13x\u00171\n32\u0000x\u00171\n13x\u00161\n32)(x\u00162\n13x\u00172\n32\u0000x\u00172\n13x\u00162\n32)F\u00161\u00171F\u00162\u00172=512(i~)2\n+ (x\u0016\n13x\u0017\n32\u0000x\u0017\n13x\u0016\n32)[2x\u00151\n32x\u00152\n32+ 3(x\u00151\n32x\u00152\n13+x\u00152\n32x\u00151\n13) + 6x\u00151\n13x\u00152\n13]\n\u0002DX\u00151\n12DX\u00152\n12F\u0016\u0017=1536i~+O(x5): (A6d)\nThus, the integrand in Eq. (A2) is given by\nO(1) = ~A(p13)~B(p32); (A7a)\nO(x) =[x\u0015\n32DX\u0015\n12~A(p13)~B(p32)\u0000x\u0015\n13~A(p13)DX\u0015\n12~B(p32)]=2; (A7b)\nO(x2) =(x\u0016\n13x\u0017\n32\u0000x\u0017\n13x\u0016\n32)[F\u0016\u0017~A(p13)~B(p32) + 2 ~A(p13)F\u0016\u0017~B(p32) +~A(p13)~B(p32)F\u0016\u0017]=16i~ (A7c)\n+ [x\u00151\n32x\u00152\n32DX\u00151\n12DX\u00152\n12~A(p13)~B(p32)\u00002x\u00151\n32x\u00152\n13DX\u00151\n12~A(p13)DX\u00152\n12~B(p32)\n+x\u00151\n13x\u00152\n13~A(p13)DX\u00151\n12DX\u00152\n12~B(p32)]=8; (A7d)\nO(x3) =(x\u0016\n13x\u0017\n32\u0000x\u0017\n13x\u0016\n32)fF\u0016\u0017[x\u0015\n32DX\u0015\n12~A(p13)~B(p32)\u0000x\u0015\n13~A(p13)DX\u0015\n12~B(p32)]\n+ 2[x\u0015\n32DX\u0015\n12~A(p13)F\u0016\u0017~B(p32)\u0000x\u0015\n13~A(p13)F\u0016\u0017DX\u0015\n12~B(p32)]\n+ [x\u0015\n32DX\u0015\n12~A(p13)~B(p32)\u0000x\u0015\n13~A(p13)DX\u0015\n12~B(p32)]F\u0016\u0017g=32i~ (A7e)\n+ (x\u0016\n13x\u0017\n32\u0000x\u0017\n13x\u0016\n32)[(2x\u0015\n32+x\u0015\n13)DX\u0015\n12F\u0016\u0017~A(p13)~B(p32) + 3(x\u0015\n32\u0000x\u0015\n13)~A(p13)DX\u0015\n12F\u0016\u0017~B(p32)\n\u0000(x\u0015\n32+ 2x\u0015\n13)~A(p13)~B(p32)DX\u0015\n12F\u0016\u0017]=96i~ (A7f)\n+ [x\u00151\n32x\u00152\n32x\u00153\n32DX\u00151\n12DX\u00152\n12DX\u00153\n12~A(p13)~B(p32)\u00003x\u00151\n32x\u00152\n32x\u00153\n13DX\u00151\n12DX\u00152\n12~A(p13)DX\u00153\n12~B(p32)\n+ 3x\u00151\n32x\u00152\n13x\u00153\n13DX\u00151\n12~A(p13)DX\u00152\n12DX\u00153\n12~B(p32)\u0000x\u00151\n13x\u00152\n13x\u00153\n13~A(p13)DX\u00151\n12DX\u00152\n12DX\u00153\n12~B(p32)]=48; (A7g)\nO(x4) =(x\u00161\n13x\u00171\n32\u0000x\u00171\n13x\u00161\n32)(x\u00162\n13x\u00172\n32\u0000x\u00172\n13x\u00162\n32)\n\u0002[F\u00161\u00171F\u00162\u00172~A(p13)~B(p32) + 4 ~A(p13)F\u00161\u00171F\u00162\u00172~B(p32) +~A(p13)~B(p32)F\u00161\u00171F\u00162\u00172\n+ 4F\u00161\u00171~A(p13)F\u00162\u00172~B(p32) + 4 ~A(p13)F\u00161\u00171~B(p32)F\u00162\u00172+ 2F\u00161\u00171~A(p13)~B(p32)F\u00162\u00172]=512(i~)2(A7h)\n+ (x\u0016\n13x\u0017\n32\u0000x\u0017\n13x\u0016\n32)fF\u0016\u0017[x\u00151\n32x\u00152\n32DX\u00151\n12DX\u00152\n12~A(p13)~B(p32)\u00002x\u00151\n32x\u00152\n13DX\u00151\n12~A(p13)DX\u00152\n12~B(p32)10\n+x\u00151\n13x\u00152\n13~A(p13)DX\u00151\n12DX\u00152\n12~B(p32)] + 2[x\u00151\n32x\u00152\n32DX\u00151\n12DX\u00152\n12~A(p13)F\u0016\u0017~B(p32)\n\u00002x\u00151\n32x\u00152\n13DX\u00151\n12~A(p13)F\u0016\u0017DX\u00152\n12~B(p32) +x\u00151\n13x\u00152\n13~A(p13)F\u0016\u0017DX\u00151\n12DX\u00152\n12~B(p32)]\n+ [x\u00151\n32x\u00152\n32DX\u00151\n12DX\u00152\n12~A(p13)~B(p32)\u00002x\u00151\n32x\u00152\n13DX\u00151\n12~A(p13)DX\u00152\n12~B(p32)\n+x\u00151\n13x\u00152\n13~A(p13)DX\u00151\n12DX\u00152\n12~B(p32)]F\u0016\u0017g=128i~ (A7i)\n+ (x\u0016\n13x\u0017\n32\u0000x\u0017\n13x\u0016\n32)f(2x\u00151\n32+x\u00151\n13)DX\u00151\n12F\u0016\u0017[x\u00152\n32DX\u00152\n12~A(p13)~B(p32)\u0000x\u00152\n13~A(p13)DX\u00152\n12~B(p32)]\n+ 3(x\u00151\n32\u0000x\u00151\n13)[x\u00152\n32DX\u00152\n12~A(p13)DX\u00151\n12F\u0016\u0017~B(p32)\u0000x\u00152\n13~A(p13)DX\u00151\n12F\u0016\u0017DX\u00152\n12~B(p32)]\n\u0000(x\u00151\n32+ 2x\u00151\n13)[x\u00152\n32DX\u00152\n12~A(p13)~B(p32)\u0000x\u00152\n13~A(p13)DX\u00152\n12~B(p32)]DX\u00151\n12F\u0016\u0017g=96i~ (A7j)\n+ (x\u0016\n13x\u0017\n32\u0000x\u0017\n13x\u0016\n32)f[6x\u00151\n32x\u00152\n32+ 3(x\u00151\n32x\u00152\n13+x\u00152\n32x\u00151\n13) + 2x\u00151\n13x\u00152\n13]DX\u00151\n12DX\u00152\n12F\u0016\u0017~A(p13)~B(p32)\n+ [8x\u00151\n32x\u00152\n32\u00006(x\u00151\n32x\u00152\n13+x\u00152\n32x\u00151\n13) + 8x\u00151\n13x\u00152\n13]~A(p13)DX\u00151\n12DX\u00152\n12F\u0016\u0017~B(p32)\n+ [2x\u00151\n32x\u00152\n32+ 3(x\u00151\n32x\u00152\n13+x\u00152\n32x\u00151\n13) + 6x\u00151\n13x\u00152\n13]~A(p13)~B(p32)DX\u00151\n12DX\u00152\n12F\u0016\u0017g=1536i~ (A7k)\n+ [x\u00151\n32x\u00152\n32x\u00153\n32x\u00154\n32DX\u00151\n12DX\u00152\n12DX\u00153\n12DX\u00154\n12~A(p13)~B(p32)\u00004x\u00151\n32x\u00152\n32x\u00153\n32x\u00154\n13DX\u00151\n12DX\u00152\n12DX\u00153\n12~A(p13)DX\u00154\n12~B(p32)\n+ 6x\u00151\n32x\u00152\n32x\u00153\n13x\u00154\n13DX\u00151\n12DX\u00152\n12~A(p13)DX\u00153\n12DX\u00154\n12~B(p32)\u00004x\u00151\n32x\u00152\n13x\u00153\n13x\u00154\n13DX\u00151\n12~A(p13)DX\u00152\n12DX\u00153\n12DX\u00154\n12~B(p32)\n+x\u00151\n13x\u00152\n13x\u00153\n13x\u00154\n13~A(p13)DX\u00151\n12DX\u00152\n12DX\u00153\n12DX\u00154\n12~B(p32)]=384: (A7l)\nNow we can carry out the integrals by putting p13=p12+q13;p32=p12+q32. Sinceq13andq32appear in the forms\nof~A(p12+q13) and ~B(p12+q32),x\u0016\n13andx\u0016\n32can be replaced with i~@p12\u0016acting on ~Aand ~B, respectively. Finally,\nwe obtain\n~A?~B=~A~B+ (i~=2)PD(~A;~B) + (i~=2)PF(~A;~B) + (1=2!)(i~=2)2PD2(~A;~B)\n+ (i~=2)2PD\u0003F(~A;~B) + (1=3)(i~=2)2PDF(~A;~B) + (1=3!)(i~=2)3PD3(~A;~B)\n+ (1=2!)(i~=2)2PF2(~A;~B) + (1=2!)(i~=2)3PD2\u0003F(~A;~B) + (1=3)(i~=2)3PD\u0003DF(~A;~B)\n+ (1=2!)(1=3)(i~=2)3PD2F(~A;~B) + (1=4!)(i~=2)4PD4(~A;~B); (A8a)\nPD(~A;~B)\u0011DX\u0015~A@p\u0015~B\u0000@p\u0015~ADX\u0015~B; (A8b)\nPF(~A;~B)\u0011(F\u0016\u0017@p\u0016~A@p\u0017~B+ 2@p\u0016~AF\u0016\u0017@p\u0017~B+@p\u0016~A@p\u0017~BF\u0016\u0017)=4; (A8c)\nPD2(~A;~B)\u0011DX\u00151DX\u00152~A@p\u00151@p\u00152~B\u00002DX\u00151@p\u00152~A@p\u00151DX\u00152~B+@p\u00151@p\u00152~ADX\u00151DX\u00152~B; (A8d)\nPD\u0003F(~A;~B)\u0011[F\u0016\u0017(DX\u0015@p\u0016~A@p\u0015@p\u0017~B\u0000@p\u0015@p\u0016~ADX\u0015@p\u0017~B)\n+ 2(DX\u0015@p\u0016~AF\u0016\u0017@p\u0015@p\u0017~B\u0000@p\u0015@p\u0016~AF\u0016\u0017DX\u0015@p\u0017~B)\n+ (DX\u0015@p\u0016~A@p\u0015@p\u0017~B\u0000@p\u0015@p\u0016~ADX\u0015@p\u0017~B)F\u0016\u0017]=4; (A8e)\nPDF(~A;~B)\u0011[DX\u0015F\u0016\u0017(2@p\u0016~A@p\u0015@p\u0017~B+@p\u0015@p\u0016~A@p\u0017~B)\n+ 3(@p\u0016~ADX\u0015F\u0016\u0017@p\u0015@p\u0017~B\u0000@p\u0015@p\u0016~ADX\u0015F\u0016\u0017@p\u0017~B)\n\u0000(@p\u0016~A@p\u0015@p\u0017~B+ 2@p\u0015@p\u0016~A@p\u0017~B)DX\u0015F\u0016\u0017]=4; (A8f)\nPD3(~A;~B)\u0011DX\u00151DX\u00152DX\u00153~A@p\u00151@p\u00152@p\u00153~B\u00003DX\u00151DX\u00152@p\u00153~A@p\u00151@p\u00152DX\u00153~B\n+ 3DX\u00151@p\u00152@p\u00153~A@p\u00151DX\u00152DX\u00153~B\u0000@p\u00151@p\u00152@p\u00153~ADX\u00151DX\u00152DX\u00153~B; (A8g)\nPF2(~A;~B)\u0011(F\u00161\u00171F\u00162\u00172@p\u00161@p\u00162~A@p\u00171@p\u00172~B+ 4@p\u00161@p\u00162~AF\u00161\u00171F\u00162\u00172@p\u00171@p\u00172~B\n+@p\u00161@p\u00162~A@p\u00171@p\u00172~BF\u00161\u00171F\u00162\u00172+ 4F\u00161\u00171@p\u00161@p\u00162~AF\u00162\u00172@p\u00171@p\u00172~B\n+ 4@p\u00161@p\u00162~AF\u00161\u00171@p\u00171@p\u00172~BF\u00162\u00172+ 2F\u00161\u00171@p\u00161@p\u00162~A@p\u00171@p\u00172~BF\u00162\u00172)=42; (A8h)\nPD2\u0003F(~A;~B)\u0011[F\u0016\u0017(DX\u00151DX\u00152@p\u0016~A@p\u00151@p\u00152@p\u0017~B\u00002DX\u00151@p\u00152@p\u0016~A@p\u00151DX\u00152@p\u0017~B\n+@p\u00151@p\u00152@p\u0016~ADX\u00151DX\u00152@p\u0017~B) + 2(DX\u00151DX\u00152@p\u0016~A@p\u00151F\u0016\u0017@p\u00152@p\u0017~B\n\u00002DX\u00151@p\u00152@p\u0016~AF\u0016\u0017@p\u00151DX\u00152@p\u0017~B+@p\u00151@p\u00152@p\u0016~AF\u0016\u0017DX\u00151DX\u00152@p\u0017~B)\n+ (DX\u00151DX\u00152@p\u0016~A@p\u00151@p\u00152@p\u0017~B\u00002DX\u00151@p\u00152@p\u0016~A@p\u00151DX\u00152@p\u0017~B11\n+@p\u00151@p\u00152@p\u0016~ADX\u00151DX\u00152@p\u0017~B)F\u0016\u0017]=4; (A8i)\nPD\u0003DF(~A;~B)\u0011fDX\u00151F\u0016\u0017[2(DX\u00152@p\u0016~A@p\u00151@p\u00152@p\u0017~B\u0000@p\u00152@p\u0016~ADX\u00152@p\u00151@p\u0017~B)\n+ (DX\u00152@p\u00151@p\u0016~A@p\u00152@p\u0017~B\u0000@p\u00151@p\u00152@p\u0016~ADX\u00152@p\u0017~B)]\n+ 3[(DX\u00152@p\u0016~ADX\u00151F\u0016\u0017@p\u00151@p\u00152@p\u0017~B\u0000@p\u00152@p\u0016~ADX\u00151F\u0016\u0017DX\u00152@p\u00151@p\u0017~B)\n\u0000(DX\u00152@p\u00151@p\u0016~ADX\u00151F\u0016\u0017@p\u00152@p\u0017~B\u0000@p\u00151@p\u00152@p\u0016~ADX\u00151F\u0016\u0017DX\u00152@p\u0017~B)]\n\u0000[(DX\u00152@p\u0016~A@p\u00151@p\u00152@p\u0017~B\u0000@p\u00152@p\u0016~ADX\u00152@p\u00151@p\u0017~B)\n+ 2(DX\u00152@p\u00151@p\u0016~A@p\u00152@p\u0017~B\u0000@p\u00151@p\u00152@p\u0016~ADX\u00152@p\u0017~B)]DX\u00151F\u0016\u0017g=4; (A8j)\nPD2F(~A;~B)\u0011fDX\u00151DX\u00152F\u0016\u0017[6@p\u0016~A@p\u00151@p\u00152@p\u0017~B+ 3(@p\u00152@p\u0016~A@p\u00151@p\u0017~B+@p\u00151@p\u0016~A@p\u00152@p\u0017~B)\n+ 2@p\u00151@p\u00152@p\u0016~A@p\u0017~B] + [8@p\u0016~ADX\u00151DX\u00152F\u0016\u0017@p\u00151@p\u00152@p\u0017~B\u00006(@p\u00152@p\u0016~ADX\u00151DX\u00152F\u0016\u0017@p\u00151@p\u0017~B\n+@p\u00151@p\u0016~ADX\u00151DX\u00152F\u0016\u0017@p\u00152@p\u0017~B) + 8@p\u00151@p\u00152@p\u0016~ADX\u00151DX\u00152F\u0016\u0017@p\u0017~B] + [2@p\u0016~A@p\u00151@p\u00152@p\u0017~B\n+ 3(@p\u00152@p\u0016~A@p\u00151@p\u0017~B+@p\u00151@p\u0016~A@p\u00152@p\u0017~B) + 6@p\u00151@p\u00152@p\u0016~A@p\u0017~B]DX\u00151DX\u00152F\u0016\u0017g=16; (A8k)\nPD4(~A;~B)\u0011DX\u00151DX\u00152DX\u00153DX\u00154~A@p\u00151@p\u00152@p\u00153@p\u00154~B\u00004DX\u00151DX\u00152DX\u00153@p\u00154~A@p\u00151@p\u00152@p\u00154DX\u00154~B\n+ 6DX\u00151DX\u00152@p\u00153@p\u00154~A@p\u00151@p\u00152DX\u00153DX\u00154~B\u00004DX\u00151@p\u00152@p\u00153@p\u00154~A@p\u00151DX\u00152DX\u00153DX\u00154~B\n+@p\u00151@p\u00152@p\u00153@p\u00154~ADX\u00151DX\u00152DX\u00153DX\u00154~B: (A8l)\nThe arguments X;p are omitted. Equations (A8f), (A8g), and (A8i)-(A8l) are not written in Eq. (9).\nAppendix B: Equivalence of the left and right Dyson equations\nEquations (13), (16), and (18) are derived from the left Dyson equation. From the right Dyson equation,\n~G?(~L\u0000~\u0006) = 1; (B1)\nwe also obtain\n~GP~G\u00001\n0=~G0~\u0006P\u0000iPP(~G0;~G\u00001\n0); (B2a)\n~GP\u0003Q~G\u00001\n0=~G0~\u0006P\u0003Q\u0000i2PP\u0003Q(~G0;~G\u00001\n0) + [ ~GP~\u0006Q+iPP(~G0;~\u0006Q)\u0000iPP(~GQ;~G\u00001\n0) + (P$Q)]; (B2b)\nGR\nP(GR\n0)\u00001=GR\n0\u0006R\nP\u0000i\u0011IJ\nP@IGR\n0@J(GR\n0)\u00001; (B3a)\nGA\nP(GA\n0)\u00001=GA\n0\u0006A\nP\u0000i\u0011IJ\nP@IGA\n0@J(GA\n0)\u00001; (B3b)\nG<(1)\nP(GA\n0)\u00001=GR\n0\u0006<(1)\nP+i\u0011Ip0\nPf@IGR\n0[(GR\n0)\u00001\u0000(GA\n0)\u00001]\u0000(GR\n0\u0000GA\n0)@I(GA\n0)\u00001g; (B3c)\nand\nGR\nP\u0003Q(GR\n0)\u00001=GR\n0\u0006R\nP\u0003Q+ [GR\nP\u0006R\nQ+i\u0011IJ\nP@IGR\n0@J\u0006R\nQ\u0000i\u0011IJ\nP@IGR\nQ@J(GR\n0)\u00001+ (P$Q)]\n+\u0011IJ\nP\u0011KL\nQ@I@KGR\n0@J@L(GR\n0)\u00001; (B4a)\nGA\nP\u0003Q(GA\n0)\u00001=GA\n0\u0006A\nP\u0003Q+ [GA\nP\u0006A\nQ+i\u0011IJ\nP@IGA\n0@J\u0006A\nQ\u0000i\u0011IJ\nP@IGA\nQ@J(GA\n0)\u00001+ (P$Q)]\n+\u0011IJ\nP\u0011KL\nQ@I@KGA\n0@J@L(GA\n0)\u00001; (B4b)\nG<(1)\nP\u0003Q(GA\n0)\u00001=GR\n0\u0006<(1)\nP\u0003Q+\u0010\nG<(1)\nP\u0006A\nQ+GR\nP\u0006<(1)\nQ+i\u0011IJ\nP@IGR\n0@J\u0006<(1)\nQ\u0000i\u0011Ip0\nP[@IGR\n0(\u0006R\nQ\u0000\u0006A\nQ)\u0000(GR\n0\u0000GA\n0)@I\u0006A\nQ]\n\u0000i\u0011IJ\nP@IG<(1)\nQ@J(GA\n0)\u00001+i\u0011Ip0\nPf@IGR\nQ[(GR\n0)\u00001\u0000(GA\n0)\u00001]\u0000(GR\nQ\u0000GA\nQ)@I(GA\n0)\u00001g\n\u0000\u0011Ip0\nP\u0011KL\nQf@I@KGR\n0@L[(GR\n0)\u00001\u0000(GA\n0)\u00001] +@L(GR\n0\u0000GA\n0)@I@K(GA\n0)\u00001g+ (P$Q)\u0011\n; (B4c)\nG<(2)\nP\u0003Q(GA\n0)\u00001=GR\n0\u0006<(2)\nP\u0003Q+ [\u0000i\u0011Ip0\nP@IGR\n0\u0006<(1)\nQ\u0000i\u0011Ip0\nPG<(1)\nQ@I(GA\n0)\u00001+ (P$Q)]\n+\u0011Ip0\nP\u0011Kp0\nQf@I@KGR\n0[(GR\n0)\u00001\u0000(GA\n0)\u00001] + (GR\n0\u0000GA\n0)@I@K(GA\n0)\u00001g: (B4d)12\nThus, the left and right Dyson equations seem di\u000berent from each other but in fact are equivalent. Both equations\nlead to\nGR\nP=GR\n0\u0006R\nPGR\n0+i\u0011IJ\nPGR\n0@I(GR\n0)\u00001GR\n0@J(GR\n0)\u00001GR\n0; (B5a)\nGA\nP=GA\n0\u0006A\nPGA\n0+i\u0011IJ\nPGA\n0@I(GA\n0)\u00001GA\n0@J(GA\n0)\u00001GA\n0; (B5b)\nG<(1)\nP=GR\n0\u0006<(1)\nPGA\n0\u0000i\u0011Ip0\nPfGR\n0@I[(GR\n0)\u00001+ (GA\n0)\u00001]GA\n0+@I(GR\n0+GA\n0)g: (B5c)\nand\nGR\nP\u0003Q=GR\n0\u0006R\nP\u0003QGR\n0+fGR\n0\u0006R\nQGR\n0\u0006R\nPGR\n0+i\u0011IJ\nPGR\n0[\u0000@I\u0006R\nQGR\n0@J(GR\n0)\u00001\u0000@I(GR\n0)\u00001GR\n0@J\u0006R\nQ\n+ \u0006R\nQGR\n0@I(GR\n0)\u00001GR\n0@J(GR\n0)\u00001+@I(GR\n0)\u00001GR\n0\u0006R\nQGR\n0@JGR\n0+GR\n0@I(GR\n0)\u00001GR\n0@J(GR\n0)\u00001GR\n0\u0006R\nQ]GR\n0\n+ (P$Q)g+\u0011IJ\nP\u0011KL\nQGR\n0f\u0000@I@K(GR\n0)\u00001GR\n0@J@L(GR\n0)\u00001\n\u0000@J(GR\n0)\u00001GR\n0@I@K(GR\n0)\u00001GR\n0@L(GR\n0)\u00001\u0000@L(GR\n0)\u00001GR\n0@I@K(GR\n0)\u00001GR\n0@J(GR\n0)\u00001\n+@I@K(GR\n0)\u00001GR\n0[@J(GR\n0)\u00001GR\n0@L(GR\n0)\u00001+@L(GR\n0)\u00001GR\n0@J(GR\n0)\u00001]\n+ [@J(GR\n0)\u00001GR\n0@L(GR\n0)\u00001+@L(GR\n0)\u00001GR\n0@J(GR\n0)\u00001]GR\n0@I@K(GR\n0)\u00001\n\u0000@I(GR\n0)\u00001GR\n0@J(GR\n0)\u00001GR\n0@K(GR\n0)\u00001GR\n0@L(GR\n0)\u00001\u0000@I(GR\n0)\u00001GR\n0@K(GR\n0)\u00001GR\n0@J(GR\n0)\u00001GR\n0@L(GR\n0)\u00001\n\u0000@I(GR\n0)\u00001GR\n0@K(GR\n0)\u00001GR\n0@L(GR\n0)\u00001GR\n0@J(GR\n0)\u00001\u0000@K(GR\n0)\u00001GR\n0@L(GR\n0)\u00001GR\n0@I(GR\n0)\u00001GR\n0@J(GR\n0)\u00001\n\u0000@K(GR\n0)\u00001GR\n0@I(GR\n0)\u00001GR\n0@L(GR\n0)\u00001GR\n0@J(GR\n0)\u00001\u0000@K(GR\n0)\u00001GR\n0@I(GR\n0)\u00001GR\n0@J(GR\n0)\u00001GR\n0@L(GR\n0)\u00001g\n\u0002GR\n0; (B6a)\nGA\nP\u0003Q=GA\n0\u0006A\nP\u0003QGA\n0+fGA\n0\u0006A\nQGA\n0\u0006A\nPGA\n0+i\u0011IJ\nPGA\n0[\u0000@I\u0006A\nQGA\n0@J(GA\n0)\u00001\u0000@I(GA\n0)\u00001GA\n0@J\u0006A\nQ\n+ \u0006A\nQGA\n0@I(GA\n0)\u00001GA\n0@J(GA\n0)\u00001+@I(GA\n0)\u00001GA\n0\u0006A\nQGA\n0@JGA\n0+GA\n0@I(GA\n0)\u00001GA\n0@J(GA\n0)\u00001GA\n0\u0006A\nQ]GA\n0\n+ (P$Q)g+\u0011IJ\nP\u0011KL\nQGA\n0f\u0000@I@K(GA\n0)\u00001GA\n0@J@L(GA\n0)\u00001\n\u0000@J(GA\n0)\u00001GA\n0@I@K(GA\n0)\u00001GA\n0@L(GA\n0)\u00001\u0000@L(GA\n0)\u00001GA\n0@I@K(GA\n0)\u00001GA\n0@J(GA\n0)\u00001\n+@I@K(GA\n0)\u00001GA\n0[@J(GA\n0)\u00001GA\n0@L(GA\n0)\u00001+@L(GA\n0)\u00001GA\n0@J(GA\n0)\u00001]\n+ [@J(GA\n0)\u00001GA\n0@L(GA\n0)\u00001+@L(GA\n0)\u00001GA\n0@J(GA\n0)\u00001]GA\n0@I@K(GA\n0)\u00001\n\u0000@I(GA\n0)\u00001GA\n0@J(GA\n0)\u00001GA\n0@K(GA\n0)\u00001GA\n0@L(GA\n0)\u00001\u0000@I(GA\n0)\u00001GA\n0@K(GA\n0)\u00001GA\n0@J(GA\n0)\u00001GA\n0@L(GA\n0)\u00001\n\u0000@I(GA\n0)\u00001GA\n0@K(GA\n0)\u00001GA\n0@L(GA\n0)\u00001GA\n0@J(GA\n0)\u00001\u0000@K(GA\n0)\u00001GA\n0@L(GA\n0)\u00001GA\n0@I(GA\n0)\u00001GA\n0@J(GA\n0)\u00001\n\u0000@K(GA\n0)\u00001GA\n0@I(GA\n0)\u00001GA\n0@L(GA\n0)\u00001GA\n0@J(GA\n0)\u00001\u0000@K(GA\n0)\u00001GA\n0@I(GA\n0)\u00001GA\n0@J(GA\n0)\u00001GA\n0@L(GA\n0)\u00001g\n\u0002GA\n0; (B6b)\nG<(1)\nP\u0003Q=GR\n0\u0006<(1)\nP\u0003QGA\n0+\u0010\nGR\n0(\u0006R\nQGR\n0\u0006<(1)\nP+ \u0006<(1)\nQGA\n0\u0006A\nP)GA\n0+i\u0011IJ\nPGR\n0[\u0000@I\u0006<(1)\nQGA\n0@J(GA\n0)\u00001\u0000@I(GR\n0)\u00001GR\n0@J\u0006<(1)\nQ\n+ \u0006<(1)\nQGA\n0@I(GA\n0)\u00001GA\n0@J(GA\n0)\u00001+@I(GR\n0)\u00001GR\n0@J(GR\n0)\u00001GR\n0\u0006<(1)\nQ+@I(GR\n0)\u00001GR\n0\u0006<(1)\nQGA\n0@J(GA\n0)\u00001]GA\n0\n+i\u0011Ip0\nPGR\n0f@I(\u0006R\nQ+ \u0006A\nQ)\u0000\u0006R\nQGR\n0@I[(GR\n0)\u00001+ (GA\n0)\u00001]\u0000@I[(GR\n0)\u00001+ (GA\n0)\u00001]GA\n0\u0006A\nQgGA\n0\n+\u0011Ip0\nP\u0011KL\nQGR\n0f\u0000@I@K[(GR\n0)\u00001+ (GA\n0)\u00001]GA\n0@L(GA\n0)\u00001\u0000@K(GR\n0)\u00001GR\n0@I@L[(GR\n0)\u00001+ (GA\n0)\u00001]\n+@K(GR\n0)\u00001GR\n0@I[(GR\n0)\u00001+ (GA\n0)\u00001]GA\n0@L(GA\n0)\u00001+@I[(GR\n0)\u00001+ (GA\n0)\u00001]GA\n0@K(GA\n0)\u00001GA\n0@L(GA\n0)\u00001\n+@K(GR\n0)\u00001GR\n0@L(GR\n0)\u00001GR\n0@I[(GR\n0)\u00001+ (GA\n0)\u00001]gGA\n0\u0000i\u0011Ip0\nP@I(GR\nQ+GA\nQ) + (P$Q)\u0011\n; (B6c)\nG<(2)\nP\u0003Q=GR\n0\u0006<(2)\nP\u0003QGA\n0+fi\u0011Ip0\nPGR\n0[@I(GR\n0)\u00001GR\n0\u0006<(1)\nQ\u0000\u0006<(1)\nQGA\n0@I(GA\n0)\u00001]GA\n0+ (P$Q)g\n+\u0011Ip0\nP\u0011Kp0\nQ\u0000\nGR\n0f\u0000@I@K[(GR\n0)\u00001\u0000(GA\n0)\u00001]\u0000@I[(GR\n0)\u00001+ (GA\n0)\u00001]GA\n0@K(GA\n0)\u00001\n\u0000@K[(GR\n0)\u00001+ (GA\n0)\u00001]GA\n0@I(GA\n0)\u00001+@I(GR\n0)\u00001GR\n0@K[(GR\n0)\u00001+ (GA\n0)\u00001]\n+@K(GR\n0)\u00001GR\n0@I[(GR\n0)\u00001+ (GA\n0)\u00001]gGA\n0+@I@K(GR\n0\u0000GA\n0)\u0001\n: (B6d)13\nAppendix C: Momentum integrals in Eqs. (24)and (27)\nIn this Appendix, we give the explicit forms of the momentum integrals de\fned above. Here we introduce\nD0(E)\u0011\u0000Z\u0003\n0pxdx\n\u00191\nE\u0000x= 2[\u00031=2\u0000E1=2tanh\u00001(\u0003=E)1=2]=\u0019; (C1a)\nD1(E) =Z\u0003\n0pxdx\n\u00191\n(E\u0000x)2= [\u00031=2=(\u0003\u0000E)\u0000E\u00001=2tanh\u00001(\u0003=E)1=2]=\u0019; (C1b)\nandER\n\u0006(\u0018)\u0011\u0018R(\u0018)\u0006JR(\u0018). Equation (24) is given by\nIR\n01(\u0018) =[D0(ER\n+(\u0018))\u0000D0(ER\n\u0000(\u0018))]=2; (C2a)\nIR\n11(\u0018) =\u0000[D0(ER\n+(\u0018)) +D0(ER\n\u0000(\u0018))]=2; (C2b)\nIR\n02(\u0018) =\u0000[D0(ER\n+(\u0018))\u0000D0(ER\n\u0000(\u0018))\u0000JR(\u0018)D1(ER\n+(\u0018))\u0000JR(\u0018)D1(ER\n\u0000(\u0018))]=4; (C2c)\nIR\n12(\u0018) =\u0000[JR(\u0018)D1(ER\n+(\u0018))\u0000JR(\u0018)D1(ER\n\u0000(\u0018))]=4; (C2d)\nIR\n22(\u0018) =[D0(ER\n+(\u0018))\u0000D0(ER\n\u0000(\u0018)) +JR(\u0018)D1(ER\n+(\u0018)) +JR(\u0018)D1(ER\n\u0000(\u0018))]=4; (C2e)\nand Eq. (27) by\nJ<(1)\n1(\u0018) =1\n2\u0014D0(ER\n+(\u0018))\nER\n+(\u0018)\u0000EA\n\u0000(\u0018)+D0(ER\n\u0000(\u0018))\nER\n\u0000(\u0018)\u0000EA\n+(\u0018)\u0015\n+ c:c:; (C3a)\nJ<(1)\n2(\u0018) =i\n2\u0014D0(ER\n+(\u0018))\nER\n+(\u0018)\u0000EA\n\u0000(\u0018)\u0000D0(ER\n\u0000(\u0018))\nER\n\u0000(\u0018)\u0000EA\n+(\u0018)\u0015\n+ c:c:; (C3b)\nJ<(1)\n3(\u0018) =J<(1)\n2(\u0018)\u00001\n3[iJR(\u0018)\u0000iJA(\u0018)]\n\u0002\u001aER\n+(\u0018)D0(ER\n+(\u0018))\nJR(\u0018)[ER\n+(\u0018)\u0000EA\n+(\u0018)][ER\n+(\u0018)\u0000EA\n\u0000(\u0018)]\u0000ER\n\u0000(\u0018)D0(ER\n\u0000(\u0018))\nJR(\u0018)[ER\n\u0000(\u0018)\u0000EA\n\u0000(\u0018)][ER\n\u0000(\u0018)\u0000EA\n+(\u0018)]+ c:c:\u001b\n;(C3c)\nJ<(1)\n4(\u0018) =1\n3\u001a[ER\n+(\u0018)]2\u0000j\u0018R(\u0018)j2+ 2jJR(\u0018)j2\nJR(\u0018)[ER\n+(\u0018)\u0000EA\n+(\u0018)][ER\n+(\u0018)\u0000EA\n\u0000(\u0018)]D0(ER\n+(\u0018))\n\u0000[ER\n\u0000(\u0018)]2\u0000j\u0018R(\u0018)j2+ 2jJR(\u0018)j2\nJR(\u0018)[ER\n\u0000(\u0018)\u0000EA\n\u0000(\u0018)][ER\n\u0000(\u0018)\u0000EA\n+(\u0018)]D0(ER\n\u0000(\u0018))\n\u0000ER\n+(\u0018)\nER\n+(\u0018)\u0000EA\n+(\u0018)D1(ER\n+(\u0018))\u0000ER\n\u0000(\u0018)\nER\n\u0000(\u0018)\u0000EA\n\u0000(\u0018)D1(ER\n\u0000(\u0018))\u001b\n+ c:c: (C3d)\n1L. Berger, J. Appl. Phys. 55, 1954 (1984).\n2L. Berger, J. Appl. Phys. 71, 2721 (1992).\n3J. Slonczewski, J. Magn. Magn. Mater. 159, L1 (1996).\n4L. Berger, Phys. Rev. B 54, 9353 (1996).\n5G. Tatara and H. Kohno, Phys. Rev. Lett. 92, 086601\n(2004).\n6S. Zhang and Z. Li, Phys. Rev. Lett. 93, 127204 (2004).\n7A. Thiaville, Y. Nakatani, J. Miltat, and Y. Suzuki, Eu-\nrophys. Lett. 69, 990 (2005).\n8Y. Tserkovnyak, H. J. Skadsem, A. Brataas, and G. E. W.\nBauer, Phys. Rev. B 74, 144405 (2006).\n9H. Kohno, G. Tatara, and J. Shibata, J. Phys. Soc. Jpn.\n75, 113706 (2006).\n10F. Pi\u0013 echon and A. Thiaville, Phys. Rev. B 75, 174414(2007).\n11H. Kohno and J. Shibata, J. Phys. Soc. Jpn. 76, 063710\n(2007).\n12K. Obata and G. Tatara, Phys. Rev. B 77, 214429 (2008).\n13A. Manchon and S. Zhang, Phys. Rev. B 78, 212405 (2008).\n14A. Manchon and S. Zhang, Phys. Rev. B 79, 094422 (2009).\n15E. van der Bijl and R. A. Duine, Phys. Rev. B 86, 094406\n(2012).\n16J. Rammer, Quantum Field Theory of Non-equilibrium States\n(Cambridge University Press, New York, 2007).\n17A. Kamenev, Field Theory of Non-Equilibrium Systems\n(Cambridge University Press, New York, 2011).\n18M. Levanda and V. Fleurov, J. Phys.: Condens. Matter 6,\n7889 (1994).14\n19M. Levanda and V. Fleurov, Ann. Phys. 292, 199 (2001).\n20C. Gorini, P. Schwab, R. Raimondi, and A. L. Shelankov,\nPhys. Rev. B 82, 195316 (2010).\n21I. Y. Aref'eva, Theor. Math. Phys. 43, 353 (1980).22N. E. Brali\u0013 c, Phys. Rev. D 22, 3090 (1980).\n23S. Onoda, N. Sugimoto, and N. Nagaosa, Prog. Theor.\nPhys. 116, 61 (2006).\n24A. Shitade, Prog. Theor. Exp. Phys. 2014 , 123I01 (2014)."    },    {        "title": "1709.02295v1.Tunable_spin_pumping_in_exchange_coupled_magnetic_trilayers.pdf",        "content": "arXiv:1709.02295v1  [cond-mat.mes-hall]  7 Sep 2017Tunable spin pumping in exchange coupled magnetic trilayer s\nM. Fazlali,1M. Ahlberg,1M. Dvornik,1and J.˚Akerman1,2\n1Department of Physics, University of Gothenburg, 412 96, Go thenburg, Sweden\n2Department of Materials and Nano Physics, School of Informa tion and Communication Technology,\nKTH Royal Institute of Technology, Electrum 229, 164 40 Kist a, Sweden\n(Dated: September 8, 2017)\nMagnetic thin films at ferromagnetic resonance (FMR) leak an gular momentum, which may be\nabsorbedbyadjacent layers. This phenomenon, knownas spin pumping, is manifested byanincrease\nin the resonance linewidth (∆ H), and the closely related Gilbert damping. Another effect of this\ntransfer of spin currents is a dynamical and long-range coup ling that can drive two magnetic layers\ninto a collective precession when their FMR frequencies coi ncide. A collective behavior is also\nfound in magnetic trilayers with interlayer exchange coupl ing (IEC). In this study we investigate\nthe interplay between IEC and spin pumping, using Co/Cu/Py p seudo-spin values. We employ\nbroadband FMR spectroscopy to explore both the frequency an d coupling-strength dependence\nof ∆H. Our observations show that there exists a cut-off frequency , set by the IEC strength,\nbelow which the precession is truly collective and the spin p umping is suppressed. These results\ndemonstrate that it is possible to control the spin pumping e fficiency by varying the frequency or\nthe interlayer exchange coupling.\nPseudo-spin valves are the building blocks of many\nspintronicdevices, suchasnanocontactspin-torquenano-\noscillators [1–6]. These devices not only show great\npromise for microwave [7, 8] and magnonic [9] applica-\ntions, but have also allowed for the exploration of spin\ntransfer torque driven propagating spin waves [10, 11]\nand magnetodynamical solitons [11–15]. Two corner-\nstones for further development towards applications, and\nto open new routes to answer fundamental questions are\nto tailor the magnetic damping and to control the flow of\nspin currents. The concept of spin pumping describes\nhow the leakage of angular momentum (spin current)\nfrom a precessing magnetic film may be absorbed at the\ninterface to another magnetic/non-magnetic layer, which\nprovides an additional damping term [16–18]. The di-\nmensionless damping coefficient is then given by α=\nα(0)+αsp, whereα(0)is the intrinsic damping of the pre-\ncessing layer and αspis the spin-pumping-induced term.\nWhile this effect has been studied extensively [19], the\nmajority of those investigation has focused on the regime\nwherethe staticcouplingbetweenthe layersisveryweak.\nThe staticinterlayerexchangecoupling(IEC, J′), is an\noscillatorycouplingpresentwhentwoferromagneticfilms\n(FMs) are separated by a sufficiently thin nonmagnetic\nlayer (NM) [20, 21]. The coupling will either promote\na parallel or antiparallel configuration of the magnetiza-\ntion in the films, depending on the spacerlayerthickness.\nIt can be described within the Ruderman-Kittel-Kasuya-\nYosida framework [22] and is therefore often referred to\nas RKKY-interaction. For sufficiently strong J′, the IEC\ncan separate the ferromagnetic resonance (FMR) of a\nFM/NM/FM trilayer into two modes, called the acoustic\n(in-phase) and optical (out-of-phase) mode [23]. While\nthe static IEC is oscillating and short-ranged in nature,\nthere also exists a dynamic and long-ranged coupling be-\ntween magnetic layers. Two magnetic films that precessat the same frequency can synchronize and display a col-\nlective behavior in the presence of spin pumping [18, 21].\nThe exploration of the interplay between the static and\ndynamic interlayer exchange coupling hence has great\nprospects of finding new exciting physics.\nWhile the literature includes studies on spin pumping\nin coupled layers [24–28], most focus on one or a few\nfrequencies. Here we investigate spin pumping in a wide\nfrequency range of 3-37 GHz using a broadband FMR\nsetup and examine four regimes: strong, intermediate,\nweak,andzeroIEC.ThesamplesarebasedonCo/Cu/Py\ntrilayers, where the thickness of the Cu spacer sets the\nstrength of the interlayer interaction.\nWe show that the mode hybridization between the lay-\ners, leading to acoustical and optical modes, is not only\ndependent ontheIECbut alsoonthe field andfrequency.\nThe collective nature of the precession is clear at low\nfields, as reflected by the relative amplitude of the sig-\nnals and the field dependence of the resonance frequency\n(fr). At higher applied fields the layers instead behave\nas single films subjected to an effective field, which scales\nwith the interlayercoupling. This transition, from collec-\ntive to single layer precession, is accompanied by changes\nin the slope of ∆ Hvs.fr, i.e. the damping, and we at-\ntribute those changes to the spin pumping between the\nlayers. The results demonstrate that it is possible to\nengineer a cut-off frequency , below which the spin pump-\ning is effectively turned off. At higher frequencies the\nspin pumping and the concomitant damping gradually\nincreases until it reach a constant value. This effect can\nbe used to tailorthe behaviorof spintronic devices by the\nstrength of the IEC.\nThe samples were prepared on oxidized Si-substrates\nusing magnetronsputtering and havethe following struc-\nture: substrate/seed/Co(80 ˚A)/Cu(dCu)/Py(45 ˚A)/cap,\nwheredCu= 0−40˚A. Single Py (Ni 80Fe20) and Co films2\nwere also prepared, using the same seed and cap layers.\nThe seed layer consisted of Pd(80 ˚A)/Cu(150 ˚A) and the\ncap layer was Cu(30 ˚A)/Pd(30 ˚A). These layers were in-\ncluded in the sample structure for three reasons: i) to\nfacilitate the growth of the Co film, ii) to avoid oxida-\ntion, and iii) to stay close to a material stack commonly\nused in the fabrication of spin-torque nano-oscillators.\nUsing Py and Co with well separated resonance fre-\nquencies also allows us to observe both the acoustic and\noptical modes in the coupled regimes – identical layers\nonly display one resonance [23]. Moreover, by drawing\nthe analogy with classical harmonic oscillators, we ex-\npect the collective modes to gradually split from the free\nrunningresonancesofthe individual layers(see, e.g., Sec-\ntion 2.2 in [29]). Thus, for weak coupling, the acoustical\nand optical modes will have their intensities mostly con-\ncentrated in the Py and Co layer, respectively. As the\ncoupling increases, the collective modes span more over\nboth layers, which leads to an enhancement of the acous-\ntical mode net response and a suppression of the opti-\ncal mode intensity. The measurements were done using\na NanOsc Instruments PhaseFMR-40 broadband FMR\nspectrometer. The external in-plane field was swept at\nfixed frequencies, varied step-wise from 3 to 37 GHz, and\nthe acquired instrument signal represents the derivative\nofthe absorptionpeaks. The derivativeofanasymmetric\nLorentzian[30,31]wasfit tothe signal,providingthe res-\nonance field ( µ0Hr), the full width half maximum (∆ H)\nand the amplitude ( A) of the absorption peak. The Kit-\ntel equation [32] was subsequently fit to the resonance\nfrequencies of the single layer samples:\nfr=γµ0\n2π/radicalbig\n(Hr+Hadd)(Hr+Hadd+Meff) (1)\nwhereµ0is the permabillity of free space, γis the gyro-\nmagnetic ratio, Meffis the effective magnetization, Hr\nis the (applied) resonance field, and Haddis the sum\nof additional in-plane fields, mainly represented by the\nanisotropy.\nTheresultsofthefitsgavethefollowingmagneticprop-\nerties of Py: γ/2π=29.0GHz/T and µ0Meff=0.89T, and\nof Co:γ/2π=30.8 GHz/T and µ0Meff=1.56 T. The ad-\nditional fields are in the order of 2 mT for both samples.\nThere are examples in the literature where the Kittel\nequation has also been used to fit the FMR of exchange\ncoupled layers, and it has been claimed that the magni-\ntude of the IEC can be extracted from the fitted value\nofHadd[27, 33]. However, we observed that this method\nnot only gave quite poor fits, but the J′values deter-\nmined independently from HPy\naddandHCo\naddalso differed\nsignificantly.\nInstead, we used an approach where the relation be-\ntweenfrandHris derived from the free energy of the\nsystem, giving the following expression [34–36]:\naω4+cω2+eω= 0 (2)\nFigure 1. FMR frequency vs. field for a trilayer with a\n7.5˚A Cu spacer. The data (filled symbols) is compared to\nthree different models. The behavior at all fields is well de-\nscribed by a fit of Eq. (2) to the data, using J′= 0.4 mJ/m2\n(solid lines). The transition from collective to single-la yer-like\nprecession is illustrated by two fits using the Kittel equati on,\nEq. (1), where MSandγare fixed either to the values of sin-\ngle Py and Co (dashed lines), or to the volume mean of these\nlayers (dotted lines). The additional field ( Hadd) was used as\na free parameter in those fits.\nwhereω= 2πfr, and the coefficients a,c, andecontain\nthe interlayer coupling, the magnetic properties, as well\nas the thickness of the magnetic layers. We have followed\nthe equations presented in Ref. [35] and [36], and the\nresults show a clear correspondence with the data.\nNevertheless, the Kittel equation still sheds some light\non the nature of the oscillations. In Fig. 1 the fits of\nEq. (2) are therefore supplemented with calculations us-\ning the Kittel equation. At high fields, frfollows the\npredictions of Eq. (1) with magnetizations and gyromag-\nneticratiosequaltothoseofthesinglefilms; atlowfields,\nfrinstead matches the behavior of an effective medium\nwith an Meffandγgiven by the volume average of the\ntwo materials. These limiting cases hence imply a transi-\ntion from a high frregion, where the inherent properties\nof each layer dominates, to a low frregion governed by\ncollective motion.\nTheextractedvaluesoftheIECarepresentedinFig.2.\nThe layer thicknesses and magnetizations were fixed dur-\ningthefitstoEq.(2), while J′,γPy, andγCowereallowed\nto vary. The resulting values of the IEC were essentially\nunchangedifthegyromagneticratiosweretreatedascon-\nstants, but the goodness-of-fit was worse. The coupling\nis ferromagnetic for all tCuin contrast to the familiar\nbehavior where J′is expected to oscillate between posi-\ntive and negative values. We can therefore conclude that\nthe interactions between the layers are not only given by\nRKKYcontributions, but alsoincludes e.g., N´ eel (orange3\nFigure 2. IEC strength ( J′) vs. spacer layer thickness, as\ndetermined from fits of the field dependence of frto Eq. (2).\nInset) FMR spectra of the Cu-7.5 ˚A sample in Fig.1 (thick\nblack lines) together with fits of two asymmetric Lorentzian s\n(cyan lines), and the spectra of the single layer Co (dashed\nlines) and Py (dotted lines). The left and right plots show\nthe absorption at 20 and 35 GHz, respectively.\npeel) coupling [21]. This interpretation is strengthened\nby the fact that we observe a minimum at the thickness\n(10˚A) where a negative maximum should occur [37, 38].\nThe linewidth of the Py and Co resonances for rep-\nresentative samples are shown in Fig. 3(a) and (b), re-\nspectively. The Cu-0 ˚A and the Cu-0.5 ˚A samples have\nstrongIEC and display only the acoustic mode, while the\nnet response of their optical mode is suppressed. The\nfrequency dependence of the linewidth is linear at all\nfrequencies. For samples in the intermediate and weak\ncoupling regimes (Cu-7.5–8.8 ˚A; Cu-12.5–19 ˚A) two reso-\nnances are observedand ∆ Hvs.frhave a rather concave\nshape, which is more pronounced for Co. The linewidth\nof the single Co layer is also non-Gilbert-like, as it sat-\nurates at a constant value at low fr. This behavior is\ninherited in the trilayer samples. However, ∆ HCoof the\ncoupled layers not only flattens out, but also increases\nat low frequencies. This implies that the mode becomes\nmore optical-like, since the optical mode is expected to\nhave a much larger linewidth compared to a single layer\ndue to mutual, out-of-phase, spin pumping [25, 39, 40].\nThere is also some additional variation of αCovs. Cu\nthickness, probably due to strain induced effects beyond\nour control. We have consequently chosen to not dwell\non the linewidth and damping associated with Co, but\ndo note that a large ∆ His consistent with an optical\ncharacter of the mode.\nIf we follow ∆ HPyof the samples with intermedi-\nate/weak IEC from high to low frequencies, we see that\ntheinitialconstantslopeisreducedatacertainfrequency\n(finfl) marked by a vertical line in Fig. 3(a). It is note-\nworthy that finflalso corresponds to the inflection point\nof ∆HCo(Fig. 3(b)). The relative resonance intensities\nFigure 3. ∆ Hvs.frfor the (a) Py and b) Co resonances.\nThe vertical lines (in both (a) and (b)) mark the inflection\npoint of the Co linewidth, which is marks the transition from\ncollective to single-layer-like behavior. The arrows show the\nCo frequency at zero applied field. Below this frequency ther e\nis only one resonance in the system. The insets are schematic\nillustrations of the magnetodynamics in the Py (gray) and\nCo (red) layers at different frequencies, for intermediate I EC\n(Cu-7.5˚A).\n(see Fig. 4 and inset of Fig. 2) also change drastically\naroundfinfl. Both these effects are clear signs of a tran-\nsition from a high-frequency region where the modes are\nassociated with the individual Py and Co layers, to a\nlow-frequency region where the precession is truly col-\nlective. ∆ HPydecreases since it predominantly repre-\nsents acoustic, in-phase, oscillations and the spin cur-\nrents hence cancel, while they instead add up in the opti-\ncal out-of-phase mode, resulting in an increasing ∆ HCo.\nWhen the Co/optical mode disappears, marked by ar-\nrows in Fig. 3(a), the slope of ∆ HPyagain becomes con-\nstantand followsthe single film behavior. The absenceof\nspin-pumping enhanced damping implies that both lay-\nersprecessin-phase andthat the sum ofthe spin currents\nis virtually zero [18].\nThe Cu-40 ˚A represents the samples without IEC and\nin those systems ∆ His linearin f, as expected for a pure\nspin pumping effect. The IEC not only influences the\nlinewidth, but also the signal amplitude. The measured4\nspectra (3-37GHz) of the Cu-40 ˚A and Cu-8.8 ˚A samples\nare presented in Fig. 4(a) and 4(b), respectively. The\nintensity of the signal is dependent on both the probed\nmagnetic moment and the frequency. We have there-\nfore normalized both resonances at each frequency to the\nhighest amplitude. The Co mode of the Cu-40 ˚A sample\nis stronger than the Py mode at all frequencies, as ex-\npected considering the higher magnetization and greater\nthickness of the Co layer. In contrast, the presence of\nIEC in the Cu-8.8 ˚A sample gives rise to a different pic-\nture, asitsComodequicklydecreasesinamplitudebelow\n≈15 GHz. This reveals the transition to a region where\nthe Py and Co show distinct acoustic and optical mode\ncharacteristics, in accordance with the interpretation of\nthe frequency dependence of the linewidth and the shape\nof thefrvs.Hrcurves.\nThe damping parameter αis determined by the rela-\ntion ∆H(fr) = ∆H0+ 4παfr/γ, where ∆ H0is a zero-\nfrequency offset [41]. The top panel of Fig. 5 shows αPy\nextracted from ∆ Hin the linear region at high frequen-\ncies. The damping increaseswith IEC strength, asshown\nby the peaks around tCu= 16 and 8 ˚A, since the precess-\ning Py layer is dragged by the exchange field from the\nstatic Co layer. For even thinner tCuonly the acoustic\nmode is present and αis low. Nonetheless, all samples\nshowing two resonances has a higher damping compared\ntothesinglelayer,and αisconstantforthesampleswith-\nout coupling. Hence, the main source of the increased\ndamping must be spin pumping.\nThe results are summarized in the bottom panel of\nFig. 5, which illustrates how the evolution of the reso-\nnances from Co and Py-like to truly acoustic and optical\nmodes is associated with a reduction, and eventually a\ncancellation, of the spin pumping effects. Different fre-\nquency regimes are therefore characterized by dissimilar\ndamping parameters and different levels of spin currents\nexchanged by the magnetic layers. This effect can be\nused to tailor the behavior of spintronic devices by the\nstrength of the IEC. It is worth noting that the IEC is\nnot only set within the growth process, but can also be\ntunedex situ, for example by loading the sample with\nhydrogen [42]. It could therefore be possible to switch on\nand off the spin pumping in a device under operation.\nIn summary, we have investigated spin pumping in ex-\nchange coupled magnetic layers using broadband FMR\nspectroscopy. We observe a frequency dependence of the\nnature of the resonance modes. They display a single-\nlayer like behavior at high frequencies and transform to\ncharacteristiccollectivemodesas frdecreases. Thistran-\nsition is accompanied by a reduction of the effective spin\npumping and damping. The results demonstrate that\nit is possible to engineer a cut-off frequency, using the\nstrength of the IEC, below which the spin pumping is\nminimized.\nWe acknowledge financial support from ERC Start-\ning Grant 307144 “Mustang”, the Swedish Foundation\nFigure 4. Color map of the normalized FMR response for (a)\na sample with zero, and (b) weak, interlayer coupling. The\ndata broadening along the x-axis is an artifact arising from\nthe conversion of line scans to a field/frequency/amplitude\nmatrix. (a) For decoupled layers, the Py amplitude is lower\nthan that of Co at all frequencies. (b) With IEC the relative\nintensities change significantly below 15 GHz. The modifica-\ntion of the intensities mirrors the transition from single- layer\nlike behavior at high frequencies/fields, to a region where t he\nprecession is truly collective with acoustic (high amplitu de)\nand optical (low amplitude) modes.\nFigure 5. Top) The damping parameter ( α) of the Py lay-\ners. Only data at frequencies well above finflwas used to ex-\ntractα. Bottom) The inflection point of the Co/optical-mode\nlinewidth ( finfl, cyan squares), which marks the transition\nfrom ordinary to reduced spin pumping, and the Co/optical-\nmode frequency at zero field ( f0, blue circles), below which\nthe spin pumping between the magnetic layers is absent.\nfor Strategic Research (SSF) Successful Research Lead-\ners program, the Swedish Research Council (VR), the\nG¨ oran Gustafsson foundation, and the Knut and Alice\nWallenberg Foundation.5\n[1] M. Tsoi, A. G. M. Jansen, J. Bass, W.-C. Chiang,\nM. Seck, V. Tsoi, and P. Wyder, Excitation of a Mag-\nnetic Multilayer by an Electric Current, Phys. Rev. Lett.\n80, 4281 (1998).\n[2] J. Slonczewski, Excitation of spin waves by an electric\ncurrent, J. Magn. Magn. Mater. 195, L261 (1999).\n[3] T. J. Silva and W. H. Rippard, Developments in nano-\noscillators based upon spin-transfer point-contact de-\nvices, J. Magn. Magn. Mater. 320, 1260 (2008).\n[4] A. Slavin and V. Tiberkevich, Nonlinear Auto-Oscillato r\nTheory of Microwave Generation by Spin-Polarized Cur-\nrent, IEEE Trans. Magn. 45, 1875 (2009).\n[5] R. Dumas, S. Sani, S. Mohseni, E. Iacocca, Y. Pogo-\nryelov, P. Muduli, S. Chung, P. D¨ urrenfeld, and\nJ.˚Akerman, Recent Advances in Nanocontact Spin-\nTorque Oscillators, IEEE Trans. Magn. 50, 4100107\n(2014).\n[6] T. Chen, R. Dumas, A. Eklund, P. Muduli, A. Houshang,\nA. Awad, P. D¨ urrenfeld, B. Malm, A. Rusu, and\nJ.˚Akerman, Spin-Torque and Spin-Hall Nano-\nOscillators, Proc. IEEE 104, 1919 (2016).\n[7] S. Bonetti, P. Muduli, F. Mancoff, and J. ˚Akerman, Spin\ntorque oscillator frequency versus magnetic field angle:\nThe prospect of operation beyond 65 GHz, Appl. Phys.\nLett.94, 102507 (2009).\n[8] S. Banuazizi, S. Sani, A. Eklund, M. Naiini, S. Mohseni,\nS. Chung, P. D¨ urrenfeld, B. Malm, and J. ˚Akerman,\nOrder of magnitude improvement of nano-contact spin\ntorque nano-oscillator performance, Nanoscale 9, 1896\n(2017).\n[9] S. Bonetti and J. ˚Akerman, Nano-Contact Spin-Torque\nOscillators as Magnonic Building Blocks, Top. Appl.\nPhys.125, 177 (2013).\n[10] M. Madami, S. Bonetti, G. Consolo, S. Tacchi, G. Car-\nlotti, G. Gubbiotti, F. B. Mancoff, M. A. Yar, and\nJ.˚Akerman, Direct observation of a propagating spin\nwave induced by spin-transfer torque, Nat. Nanotechnol.\n6, 635 (2011).\n[11] S. Bonetti, V. Tiberkevich, G. Consolo, G. Finocchio,\nP. Muduli, F. Mancoff, A. Slavin, and J. ˚Akerman, Ex-\nperimental Evidence of Self-Localized and Propagating\nSpin Wave Modes in Obliquely Magnetized Current-\nDriven Nanocontacts, Phys. Rev. Lett. 105, 217204\n(2010).\n[12] A. Slavin and V. Tiberkevich, Spin Wave Mode Excited\nby Spin-Polarized Current in a Magnetic Nanocontact is\na Standing Self-Localized Wave Bullet, Phys. Rev. Lett.\n95, 237201 (2005).\n[13] V. E. Demidov, S. Urazhdin, and S. O. Demokritov, Di-\nrect observation and mapping of spin waves emitted by\nspin-torque nano-oscillators, Nat. Mater. 9, 984 (2010).\n[14] R. Dumas, E. Iacocca, S. Bonetti, S. Sani, S. Mohseni,\nA. Eklund, J. Persson, O. Heinonen, and J. ˚Akerman,\nSpin-wave-mode coexistence on the nanoscale: A con-\nsequence of the oersted-field-induced asymmetric energy\nlandscape, Phys. Rev. Lett. 110, 257202 (2013).\n[15] S. Mohseni, S. Sani, J. Persson, T. Anh Nguyen,\nS. Chung, Y. Pogoryelov, P. Muduli, E. Iacocca, A. Ek-\nlund, R. Dumas, S. Bonetti, A. Deac, M. Hoefer, and\nJ.˚Akerman, Spintorque-generatedmagneticdropletsoli-\ntons, Science 339, 1295 (2013).[16] R. Urban, G. Woltersdorf, and B. Heinrich, Gilbert\ndamping in single and multilayer ultrathin films: Role\nof interfaces in nonlocal spin dynamics, Phys. Rev. Lett.\n87, 217204 (2001).\n[17] Y. Tserkovnyak, A. Brataas, and G. E. W. Bauer, En-\nhanced Gilbert Damping in Thin Ferromagnetic Films,\nPhys. Rev. Lett. 88, 117601 (2002).\n[18] B. Heinrich, Y.Tserkovnyak, G. Woltersdorf, A.Brataa s,\nR. Urban, and G. E. W. Bauer, Dynamic Exchange Cou-\npling in Magnetic Bilayers, Phys. Rev. Lett. 90, 187601\n(2003).\n[19] Y. Tserkovnyak, A. Brataas, G. E. W. Bauer, and B. I.\nHalperin, Nonlocal magnetization dynamics in ferromag-\nnetic heterostructures, Rev. Mod. Phys. 77, 1375 (2005).\n[20] S. S. P. Parkin, N. More, and K. P. Roche, Oscillations in\nexchange coupling and magnetoresistance in metallic su-\nperlattice structures: Co/Ru, Co/Cr, and Fe/Cr, Phys.\nRev. Lett. 64, 2304 (1990).\n[21] B. Heinrich, Exchange Coupling in Magnetic Multilayers\n(Springer Berlin Heidelberg, Berlin, Heidelberg, 2008),\npp. 185–250.\n[22] P. Bruno and C. Chappert, Oscillatory coupling between\nferromagnetic layers separated by a nonmagnetic metal\nspacer, Phys. Rev. Lett. 67, 1602 (1991).\n[23] B. Heinrich, Radio Frequency Techniques (Springer\nBerlin Heidelberg, Berlin, Heidelberg, 1994), pp. 195–\n296.\n[24] K. Lenz, T. Toli´ nski, E. Kosubek, J. Lindner, and\nK. Baberschke, Oscillations of the interlayer exchange\ncoupling in trilayers with non-collinear easy axes, Phys.\nStatus Solidi C 1, 3260 (2004).\n[25] K. Lenz, T. Toli´ nski, J. Lindner, E. Kosubek, and\nK. Baberschke, Evidence of spin-pumping effect in the\nferromagnetic resonance of coupled trilayers, Phys. Rev.\nB69, 144422 (2004).\n[26] A. A. Timopheev, Y. G. Pogorelov, S. Cardoso, P. P.\nFreitas, G. N. Kakazei, and N. A. Sobolev, Dynamic\nexchange via spin currents in acoustic and optical modes\nofferromagnetic resonanceinspin-valvestructures, Phys .\nRev. B89, 144410 (2014).\n[27] A. A. Baker, A. I. Figueroa, C. J. Love, S. A. Cavill,\nT. Hesjedal, and G. Van Der Laan, Anisotropic Absorp-\ntion of Pure Spin Currents, Phys. Rev. Lett. 116, 047201\n(2016).\n[28] A. A. Baker, A. I. Figueroa, D. Pingstone, V. K. Lazarov,\nG. van der Laan, and T. Hesjedal, Spin pumping in\nmagnetic trilayer structures with an MgO barrier, Sci.\nRep.6, 35582 (2016).\n[29] Y. S. Joe, A. M. Satanin, and C. S. Kim, Classical anal-\nogy of fano resonances, Phys. Scr. 74, 259 (2006).\n[30] G.Woltersdorf, Spin-Pumping and Two-Magnon Scatter-\ning in Magnetic Multilayers , PhD thesis, Simon Fraser\nUniversity, 2004.\n[31] Y. Yin, F. Pan, M. Ahlberg, M. Ranjbar, P. D¨ urrenfeld,\nA. Houshang, M. Haidar, L. Bergqvist, Y. Zhai, R. K.\nDumas, A. Delin, and J. ˚Akerman, Tunable permalloy-\nbased films for magnonic devices, Phys. Rev. B 92,\n024427 (2015).\n[32] C. Kittel, On the theory of ferromagnetic resonance ab-\nsorption, Phys. Rev. 73, 155 (1948).\n[33] G. B. G. Stenning, L. R. Shelford, S. A. Cavill, F. Hoff-\nmann, M. Haertinger, T. Hesjedal, G. Woltersdorf, G. J.\nBowden, S. A. Gregory, C. H. Back, P. A. J. De Groot,\nand G. V. Van Der Laan, Magnetization dynamics in an6\nexchange-coupledNiFe/CoFe bilayer studied byx-rayde-\ntected ferromagnetic resonance, New J. Phys. 17, 013019\n(2015).\n[34] J. Lindner and K. Baberschke, Ferromagnetic resonance\nin coupled ultrathin films, J. Phys.: Condens. Matter\n15, S465 (2003).\n[35] A. Layadi, Effect of biquadratic coupling and in-plane\nanisotropy on the resonance modes of a trilayer system,\nPhys. Rev. B 65, 104422 (2002).\n[36] Y. Wei, S. Jana, R. Brucas, Y. Pogoryelov, M. Ran-\njbar, R. K. Dumas, P. Warnicke, J. ˚Akerman, D. A.\nArena, O. Karis, and P. Svedlindh, Magnetic coupling\nin asymmetric FeCoV/Ru/FeNi trilayers, J. Appl. Phys.\n115, 17D129 (2014), Note:There is a typo in the\nexpression describing the third coefficient in Eq. (4),\ne= [t2MAMB...] should read e= [t2MAMBhA\n2hB\n2...].\n[37] P. J. H. Bloemen, M. T. Johnson, M. T. H. van de Vorst,\nR. Coehoorn, J. J. de Vries, R. Jungblut, J. aan de\nStegge, A. Reinders, and W. J. M. de Jonge, Magnetic\nlayer thickness dependence of the interlayer exchange\ncoupling in (001) Co/Cu/Co, Phys. Rev. Lett. 72, 764\n(1994).[38] T. Luci´ nski, F. Stobiecki, D. Elefant, D. Eckert, G. Re iss,\nB. Szyma´ nski, J. Dubowik, M. Schmidt, H. Rohrmann,\nand K. Roell, The influence of sublayer thickness on\nGMR and magnetisation reversal in permalloy/Cu mul-\ntilayers, J. Magn. Magn. Mater. 174, 192 (1997).\n[39] S. Takahashi, Giant enhancement of spin pumping in the\nout-of-phase precession mode, Appl. Phys. Lett. 104,\n052407 (2014).\n[40] B. Heinrich, G. Woltersdorf, R. Urban, and E. Simanek,\nRole of dynamic exchange coupling in magnetic relax-\nations ofmetallic multilayerfilms(invited), J.Appl.Phys\n93, 7545 (2003).\n[41] B. Heinrich, Spin Relaxation in Magnetic Metallic Lay-\ners and Multilayers (Springer Berlin Heidelberg, Berlin,\nHeidelberg, 2005), pp. 143–210.\n[42] B. Hj¨ orvarsson, J. A. Dura, P. Isberg, T. Watanabe, T. J .\nUdovic, G. Andersson, and C. F. Majkrzak, Reversible\ntuning of the magnetic exchange coupling in Fe/V (001)\nsuperlattices using hydrogen, Phys. Rev. Lett. 79, 901\n(1997)."    },    {        "title": "1709.03735v2.Temperature_effects_on_MIPs_in_the_BGO_calorimeters_of_DAMPE.pdf",        "content": " \n Temperature e ffects on MIP sin the BGO \ncalorimeter sof DAMPE① \n \nYuan -Peng Wang (王远鹏 )1,2, Si-Cheng Wen (文思成 )1,2,Wei Jiang  (蒋维 )1,2, \nChuan Yue  (岳川 )1,2,Zhi-Yong Zhang  (张志永 )3, Yi-Feng Wei (魏逸\n丰)3,Yun-LongZhang  (张云龙 )3, Jing-Jing Zang (藏京京 )1②,JianWu (伍健 )1\n \n1Key Laboratory of Dark Matter and Space Astronomy, Purple Mountain Observatory , Nanjing  210008 , China  \n2University of Chinese Academy of Sciences , Yuquan Road 19, Beijing 100049, China  \n3State Key Laboratory of Particle Detection and Electronics, University of Science and Technology of China , Hefei 230026, China  \nAbstract In this paper, we present a study of temperature effects on BGO calorimeter s using proton MIP s \ncollected in the  first yea r of operation of DAMPE. By directly comparing MIP calibration constants used by the \nDAMPE data production pipe line, we findan experimental relation  between the temperature and signal \namplitudes of each BGO bar:  a general  deviation of −1.162 %/℃,and−0.47%/℃ to −1.60%/℃statistically for \neach detector  element. During 2016, DAMPE’s temperature changed by ~8℃ due to solar elevation angle , and \nthe corresponding energy scal e bias is about 9%. By frequent MIP calibration operation, this kind of bias is \neliminated to  an acceptable value.  \nKeyword s  BGO, MIP, temperature effect  \nPACS  29.40.Vj  \n1 Introduction  \nThe DArk Matter Particle Explorer (DAMPE) is  an orbital telescope with highresolution  and wide energy \nband aimingat detectin g cosmic rays and gamma -rays of 0.5GeV – 100Te V[1][2][3]. Apartfrom the supporting \nstructur es of the satellite, the telescope itself  consists of four sub -detectors .From top to bottom (along the direction \ntowards t he center of the Earth) are plastic scintillator detectors (PSD , 82 units ), a silicon tungsten trac ker (STK , \n768 units ), bismuth germinate oxide detectors (BGO , 308 units ), and neutron detectors (NUD , 4 units ),where the \nnumber of units means the number of corresponding crystals, for example , there are 308 BGO crystal bars on \nDAMPE. Each of the four sub -detectors has its own targets : PSD for charge, BGO for energy /PID , STKfor \ndirection /charge , and NUD for particle identifi cation . Together, they make DAMPE an orbital telescope with the \nability to measure high -energy particles up to 100TeV  with good angular resolution [1]. \nTo measure the above parameters, DAMPE has to deal with thousands of electronic channels that connect the \nPMTs of physical detectors with the data processor  to store the information wanted .Calibration is required  to \nconvert the charge excited by interaction between particles an d DAMPE  to the energy of the particles .A detailed \ndescription of the procedure can be found elsewhere [4], and only part of it is described here: the effects of \ntemperature on the response toMIPs of the BGO calorimeter s of DAMPE . \nMIPs, short for minim um-ioniz ation  particles , wh ose energy loss in material  can be quantified by the \nBethe -Block formula, are the kind of particles that can penetrate materials while depositing energy only by \nionization rather than  suffering  nuclear reactio ns. This means that the energy they losemakes up nearly a fixed \npercentage  of their total energy .Assisted by  simulation with the same structure and (nearly )the same energy \nspectrum  (Section 2  below  introduces it briefly) , the exact energy of a MIP is acquired, andis then utilized to \ncalibrate the BGO and PSD by matching th at energy with the magnitude of the ADC ( Analog toDigital \nConversion③)signal collected bythe PMTs when a MIP hit sthem [5].A thorough understanding of the particle s \n                                                                 \n①The DAMPE mission was founded by National Key Program for Research and Development (No. 2016YFA0400200). This work is also \nsupported in part by the strategic priority science and technology projects in space science of the Chinese Academy of Sciences (No. \nXDA04040000) and by NSFC under grants No. 11303105 and No.11673021.  \n2009 Chinese Physical Society and the Institute of High Energy Physics of the Ch inese Academy of Sciences and the Institute  \nof Modern Physics of the Chinese Academy of Sciences and IOP Publishing Ltd  \n② E-mail: zangjj@pmo.ac.cn  \n③ In many other cases  one may expect \"Convertor\" instead of \"Conversion\". Here we use the latter because \"ADC \" hereafter is used as a \nunit measuring the charge rather than the convert or itself ..  \n2 \n hitting DAMPE requires much more effort than MIP calibration (\"MIP calibration\" is used  hereafter for \"the \nprocedure of calibrating the ADC values of MIPs to acquire their ADCs for future use\") alone . For example , the \nrelation between different  dynodes  of a PMT should also be calibrated, which can also be found in Ref.[5]. \nHowever, the magnitude of charge read from a PMT when a particle hits BGO is actually determined not only \nby the nature of that particle but also by  the PMT itself [6]. So the environment of the PMTs will affect their \nbehavior .Temperature is one of the most important factors which is affected by the angle of the satellite , as it  gets \nheated by sunshine . Aprevious experimental  estimation [7]tells us that its influence on the behavior of PMT s is \napproximately  −1%/℃,but the  precision of that estimation wa sfar from acceptable. So this paper aims at a more \naccurate evaluation  of this effect by comparing the data from  DAMPE from about  one year , covering a range of \ntemperature of  2  − 10 ℃. After the comparison, a linear fit is applied , allowing more precise evaluation of this \neffect . \nSimulation is usedin many partsof  our analysis, and this paper introduces it briefly in the next section , \nfollowed by our procedure for selecting MIP  samples anda brief fitting result . After that is  the major topic , the \ntemperature effects of MIPs, which is arranged in three parts : the variation of temperature and MIP result s on \ndifferent dates, the global relation between temperature and MIP calibration constants, andfinally the results of \neach BGO calorimeter . Finally we  come to the conclusio n. \n2 The DAMPE  simulation  \nA detailed description of simulation in DAMPE is in preparation , and here only a brief introduction is \ngiven ,with special attention paid to MIPs.  \nDAMPE uses GEANT4 as thebasis for simulat ion[8][9]. First we take the spectrum from AMS02 as the input \nof GEANT4. Now the energy of MIPs is affected by the angular distribution, so DAMPE's orbit must be taken. A \ntechniquecalled  \"back -tracing\" is therefore  applied to target the particles according to the orbital parameters of \nDAMPE (time and position). This technique  enabl es a particle in GEANT4 to swirl in a magn etic field according \nto the geomagn etic field at the same coordinates as DAMPE when it is being considered, so we can then re -model \nthe spectrum for DAMPE .Because the tracks of particles coming from earth will even tually collide with the earth \nif we reverse -extend them, abandoning these particles offers us a fine spectrum with orbital information of \nDAMPE. The difference between the real spectrum and that from our simulation  is quite acceptable, as can be \nseen from Ref. [10]. \nWith simulation data, we can find perfect MIP samples whose exact energy is known from input. Using a \ndigitization  technique , the ADCs of the MIPs are also provided . These values are used as default s, for example in \nthe case of a new environment where no calibration for MIP sis available , because they are independent of real data.  \nLater in the following analysis, they are also used as a reference to give a normalized relation where the real \nvalues are not concerned.  \n3 Proc edure forselecting MIP s \nMIPsare defined, in this work, as particles which lose energy only by electromagnetic reactions with \nmaterials.Consequently , the ratio of MIPs when penetrating materials is determ ined by thegeometric structure  and \nthematerials  themselves .Apart from ionization energy loss, MIPs  hardly interact with materials, making the ir track \nquite straight and  clear when penetrating DAMPE . These facts pave the way for determining whether a particle can \ncontribute to MIP  calibration .Five differe nt filters for MIPshave been developed accordingly and are listed \nbelow .Of these, onespecific filter is designed only for DAMPE.  All filters below are listed in the order they are \narranged in the code . If one filter rejects a particle  when it is being checked , it will be thrown away immediately \nwithout check ingthe rest  of the filters .Reference [11] details the software framework we use.  \n Trigger mode  \nExposed directly to cosmic rays, DAMPE gets hit by millions of parti cles every  minute , the majority of \nwhich are low -energy and thus low priority . This makes the  data selecti onprocedure extremely important.The  \ntrigger mode of DAMPE decides what to record. Several trigger modes are designed for DAMPE because it has \nmany scientific tasks .Some mode sare designed to increase the number  of MIPs due to their significance in \ncalibration. In this way, selecting certain trigger modes can help preclude a large number of particles when only \nMIPsareneed ed.MIPmodes are  enabled only at  low latitudes ( 20°S − 20°N). Details o fthe DAMPE trigger \nsystem are available elsewhere. This is a DAMPE -specific filter.  \n Penetration  \nThis filter , making sure that a particle hasdeposited energy inboth the top and bottom of the BGO calorimeter , \naims at selecting MIPsthat penetrate DAMPE thoroughly  by throwing away those enter ing or leav ingDAMPE \nthrough one side of this satellite .It is currently a compromise  to sacrifice the efficiency of MIP selecti on for the \nsake of energy estimation , because otherwise the ratio of the energy from the BGO calorimeter to that of the particle  \n3 \n cannot be easily decided, making it impossible to acquire a good  evaluation of the energy of the particle . Now that \nthe energy spectrum of MIP is necessary when calibrating, th ese \"crippled\" MIPs are abandoned①. \n Total energy  \nThe proportion  of its total energy that a MIP deposits in material during interaction  is approximately fixed, \nalthough that ratio varies with the energy of the particle. This gives us a third filter  toabandon particles that have \ndeposited more than 40 times the energy of a typical MIP (or 2 1.8MeV ). \n Number of hits per layer  \nMIPsinteract with materials in a simple  way without  break ing apart, sothey hardly interact with BGO units \nthey don’t penetrate. This is the basis for the  fourth  filter :if too many detectors  on a plane respond to  a particle , it is \nprobably not a MIP. To avoid noise , only the units the particle pen etrates and those adjacent are considered , and a \nunit is considered  to have responded  only when more than 0.2 times the typical MIP energy (4.36MeV) has been \ndeposited . \n 𝜒2 of the track  \nAMIP penetrate s materials in a simple  way, so it must follow  a clean track, which means that the𝜒2 of th e \ntrack is expected not to be very large . This gives  us our fifth filter. In this filter the track is calculated in a rough \nway by linearly fitt ingthe energy -weighted coordinates of the responding BGO unit and the two next to it oneach  \nlayer, and then omit ting particles with too high a 𝜒2 (>2 for now , ~ 1 typical ).Here we use energy for \nweight ing because it helps improve the resolution. Given its total energy, the energy a MIP deposit sin one unit \nmay not exceed the noise so much  that considering all units unbiasedly would bend our track far off the real one , \nwhile we can expect a higher precision if energy -weighted coordinates are used instead because they can increase \nthe significance of units with high energy and decrease that of some noise units. Details o f track reconstruction go \nbeyond the scope  of this paper. \nThese are the five filters developed to pick up MIP samples from DAMPE  data.Approximately 30000 MIP s \nare selected each day from the 5 million particles recorded by DAMPE . This, however, doesn't mean the efficiency \nof our MIP selection is only ~0.6% ( 30k / 5M ), because the satellite doesn otonly measur eMIPs .The denominator \nshould  be the real number of MIPs  each day, which seems completely in accessible at present . We therefore \nestimat ethe efficiency based on simulation  and conclude  anefficiency of 88%, whic h is the only result we can trust . \nIn order to get adequate samples to fit, data from  5 continuous days is accumulated for calibration each time .This, \nhowever, assumes t hat a span of time does no harm to our resolution. Fortunately for us, it is observed that the \ntemperature changes by no more than  0.25℃/day, so this stability serves as the foundation for our method \nconsidering the temperature effect of  ~−1%/℃. \nSelecting MIPs is only the first step .The second step is to fit them to get an estimat ion of MIP energy deposit . \nFirst we use the track of a MIP to correct path length by a factor of cos𝜃, then a convolution of Landau and a \nGaussian function  is used to fit the ir spectrum . After the fitting, the peak of the fitting convolution is used when \nreconstructing the energy of particles instead of the Most Probable Value (MPV) of the L andau  distribution②, but \nhereafter we use \"MPV\" in our discussion . Whether peaks or MPVs should be used is still in debate, and currently \npeaks are used (MPV was c hosen before in early versions ). Plotted in Fig.1is a sample fitting a MIP histogram  \nduring an instance of calibration .This shows  that our procedure  works well. After al l, rather than individual MIPs, \nwe care more about their spectrum , forthe value we require, peak or MPV  of the function ,comes from statistical \nanalysis .However good the sample looks, it is rather complicated to fit with such a convolution. Details of our \nfitting are provided later in Section 4 ,where more results are shown . \n \nFig.1Asample MIP fitting histogram and its fitting function . The horizontal coordinates , marked ADC, show  the charge read from the \nPMT, and the vertical the counts. We use a convolution of a Landau function and Gaussian function to fit the data  with an empirical \nrange , and  then draw the result onto the data  as the red line . The total number of counts in each histogram is ~ 4000, adequate for fitting, \n                                                                 \n① A deeper understanding of this procedure requires  that a MIP penetrate the top and bottom surface of the BGO unit that is being \ncalibrated , because we are calibrating t he BGO unit instead of the whole DAMPE.  \n② Each bar has its own peaks, so we need to select the peak of the bar a particle hits.   \n4 \n as can be seen from th is histogram , where the red  function looks fine.  \n4 Results  \nMIPsarecalibrated every day to make up for the  possible daily change of temperature , however slight , and \nthis offers us about four hundred results up to now. With thermistors and other similar electronic device suniformly \ndistributed inside DAMPE, the temperature field of DAMPE can be calculated. The relation between the MIP \nparameters and temperature  can then be studied . \nThe variation of average global temperature of DAMPE is shown in Fig.2. This is  the average of all BGO \nunits ,giving  an average temperature of the field of DAMPE , averaged over  of 5 days ,because each calibration is \ndone using 5 days ’ worth of MIPs . In Fig.2, parame tersare marked as the first day of the 5 days.  \n \nFig.2Variation of averaged global temperature of DAMPE .It covers a range of roughly 2℃−10℃since launch. This tendency is largely \ndecided by the orbit, which affects the exposure of the satellite to the sun.  \nFig.2shows  that the temperature  change d by at most ~8℃overthis f irst year. The temperature is always \nchanging but with varying trends.I t kept decreasing  until around the end of February, then increasedcontinually \nuntil about Apr. 25, before decreasing  again until about May 2.After this it started to wobble around  7.2℃, until the \nmiddle of August  when  it climb ed to its peak, after which the temperature decreased until  October, before  finally \nclimb ing again  until the end of the year . This can be explained by the direction  of the satellite (as well as solar \nelevation angle ),as this  affects the efficiency with which it absorbs sunshine and heats up.The direction  is decided \nby its orbit :DAMPE follows a solar -synchronized orbit whose inclination is about  97°, which  gives DAMPE a \nperiodic variation of the time it  isexpose dto the s un, giving rise to this pattern of temperature variation . \nPlotted in Fig.3 is the variation of MPV of MIPs as a whole . When calibration is done, 308 MPVs are \ncalculated  for each of the 308 BGO units. To acquire a point on the diagram, each of those 308 MPVs  are divided \nseparately by th oseof the same unit but calculated instead using simulation samples , and 308 ratios  result . Then a \nGaussian function is used to fit these 3 08 ratios to get its mean and sigma, which are finally plotted on to the  \ndiagram  as a point and its error bar . In this way, the 308 MPVs acquired eachtime are condensed into one point \nonto the diagram representing the global effect of the result of calibration  at one time as a whole instead of one \nparticular  BGO unit . This also explains whythe error bar of any point in the diagram is quite small: it can be taken \nas the sigma of 308 \"repeated\" measurement s.The MIPsare calibrated with 5 days ’ worth of data  each time, and to \nplot them, the first date was chosen , as is the case for the  temperature above . \n \n5 \n  \nFig.3Variation of MIPs as a whole .The vertical coordinates represent a statistical evaluat ion of MIP calibration each time , and the \nhorizontal ones are dates of calibration where a timestamp from the first day is plotted . There are some gaps in the diagram, for \nexample the two isolated dots to the left of 09 -01, but it is quite safe to ignore them  (see the text) .  \nBy roughly comparing the two diagram s one can see that theymatch well, with minor deviations . Some gaps \non Fig. 3, such as the twolocated to the left of 0 9-01, however wide they look, don ot come from the unavailability \nof data on that date  but from the way of selecting their horizontal coordinates : a timestamp from the first day is \nselected but in an arbitrary way . A better  algorithm would help improve the performance by, for example, \nsmoothing some unnecessary sub -structures in the diagram , and it is currently being developed . \nThe effect of global temperature MIPs is shown in Fig. 4 . It is plottedusing  the value s for each day  from  the \ntwo diagrams .Temperature is shown on t he horizontal axis,and  the MPV of the MIPs on the vertical.The standard \ndeviation  of each MPV is also plotted but is very small . Thered line shows the linear fitting function whose \nparameters are giveni n the top right corner.  \n \n6 \n  \nFig.4Variation of M PVs versus temperature , showing  perfect linearity .Due to thermal imbalance on either peak, the fitting range is not as \nwide as the data. The cluster of points near  7℃ is due to the change  of temperature of the satellite, as can be seen above in \ntemperature plot in Fig.2. \nFrom Fig.4 we can see a perfect linearity between the te mperature of DAMPE and the MPV of the MIPs. It \ngives  adeviationof  −1.162 %/℃globally , that is to say,  a temperature change of  1℃ brings about a g lobal \ndeviation of MPV of −1.162 %.The fitting range of this diagram is only 2 − 9℃ instead of the whole data range , \nbecause when it is  too hotor too cold, the thermal equilibrium hasn't been  reached, making it difficult or even \nimpossible to estimate the temperature field.  \nThe final result  is the temperature effect on each BGO detector. For the sake of convenience, each of the 308 \nBGO units  is afterwards referred to as one \"BGO bar\" or only \"one bar\"(\"bar\" is useddue to its dimensions  of \n600mm×25mm×25mm ).Following the global case, temperature sh ould come first. It is however omitted due to \nthe similarity between different bars :different bars show  only minor changesin  the shape of the \ntemperat urevariation, while the overall trend is the same as the global one concluded from Fig.2. This is easily  \nunderst oodin terms of thermal equilibrium . The temperature diagrams are therefore  omitted here and here we start \nwith the MIP ADCs. Results for 4 bars are shown in Fig. 5 .The different bars have been normalized (so the peak of \neach diagram is 1, and errors are scaled accordingly) so that one can focus on the trend s rather than the absolute \nvalue s. This is because t he absolute values of each diagram vary from 200 to 700, which makes it inconvenient to \nevaluate the trends . The density of points  makes it difficult to plot them all on the same figure ,so there are four \nindividual diagrams. We will come back to real ADCs when linearity is concerned . Also, the MPV from the fitting \nfunction of the ADCs is plotted each time, and vertical axes are thus labeled accordingly.  \n \n7 \n  \nFig.5Variation of 4selected bars. Vertical axes are all normalized to the same scale  with 1 the largest , to allow clearer comparison of the \nshapes . Deviation here is quite large compared to temperature becausefitting errors, statistics, and initial valuesof fitting data of one bar \naffect the result more  than in an averaged analysis. Shapes roughly  match th ose of temperature  plots . Bar number sare given in the title \nof each histogram.  \nHere the MPVs  ofeach bar fluctuate more than in the  global case .This can be understood by the fitting \nprocedure. Fitting relies on the initial values of each parameter .This dependency dominates in the case of the \nconvolution of a Landau and Gaussian function , where a minor deviation in the initial value may lead to visible \nmismatch , and MIP calibration depends on this convolution . Initially, r ecursive efforts were made to improve the \ninitial conditions as well as the fitting range for better results, then a two -phase fitting process was developed, \nwhere  the first trial aims only at pr oviding the initial condition s for the real fitting in the second phase . In doing \nthis, only the MPV of the first trial is used to construct the initial conditions for the second fitting. We use only \nMPV, which can be roughly estimated as the peak of the data , because not too many step s are needed to acquire \nthis parameter, because  of efficiency , and because it is observed that the MPV is still reliable even if the fitting \nfails. This two -phase fitting method recovers many failures from the previous fitting method  because the new \ninitia l conditions it uses are more reasonable.If the reduced 𝜒2 =𝜒2/𝑁𝐷𝐹 is larger than 3, we consider this \ntwo-phase fitting to have failed as well, and then a third fit is performed . For this third trial, the MPV of the \nsecond fit is still consulted to give the initial conditions but in a slightly different way than that in the second trial \nto avoid further failure, and these differences also come from our recursive trial. In other word s, each histogram is \nfitted at least twice but at most three times, and t hese efforts give us the smooth ness in  fitting function slike Fig.1 \nwith few exceptions in the end . The global relation s tated above is concluded by a statistical study of 308 \nbarswhere individual deviations are largely tolerated .  \nThe final part is linearity of temperature and MIPs, shown for four bars  inFig.6.A statistical analysis  for all \nthe bars is shown in Fig.7 and 8. \n \n8 \n  \nFig.6 Relation between temperature and MIP constants of the same 4 BGO bars as chosen above. The top two diagrams show good \nlinearity  while the bottom  two are less good . The r elation here deviates from linearity due to the relatively large deviation of MPVs of \nMIPs of each bar as displayed and explained in the text .It remains true, however, that higher  temperature brings about lower MIPs.As in \nthe global case, not all the data points are used, and the ranges here correspond with 2 – 9℃ in the global case (see the text).  \nFollowing therestricted  range, the first step is find the corresponding range for each unit. 2 - 9℃is not suitable \nhere because thermal equilibr ium doesn ot mean that all bars share the same temperature . It is possible to calculate \nusing  linear interpolation, provided that  the same tendency holds for all bars with minor deviation, as mentioned \nbefore. It is an easy method with good resolution, much  easier than the most reliable way which requires a \nthorough calculation for either condition. From these figures  one can also obtaina  standard for selecting bars: \nnormal bars with good linearity  (such as the Fig. 6(a) and (b),  ~ 83% in all bars ), and abno rmal bars with bad \nlinearity  (such as theFig. 6(c) and (d)  ~ 17% ) where good linearity means a bar with a fitting function whose 𝜒2/\n𝑁𝐷𝐹 <1.19 (this \"1.19\" is actually an empirical standard which we set by using a typical ambiguous bar from \nearly analysi s). A closer look at those abnormal bars gives another hint : they are usually located on or near the \nedge  of the308BGO units  of DAMPE①.It may explain their behaviorto understand the lack of statistics compared \nwith the rest.However abnormal a bar appears, their slopes are always negative.  \nFig.7and 8show  the distribution of fitting parameters, where a \"ref\" in the title  (short for  \"reference ) means \nthat the values inse rted have been divided by those from simulation.  \n                                                                 \n① All 308 BGO bars are arranged in 14 layers  of 22 BGO bars  each.The  index of bars, either of its layer or its bar, starts with 0, for \nexample, layer 0 bar 0 is the first bar, layer 8 bar 10 gives the 11th bar on layer 9,  and so on . \n \n9 \n  \nFig.7Distribution of slopes of linearity between temperature and MIP constants of 308 BGO bars. All slopes are negative. (a) Theab solute \nvalues, calculated directly by linearly fitting the MIP constants and temperature instead of a ratio. (b) The relative value s, or ratio of the \nvalue of each bar to that acquired by simulation results. The a verage  ratio is -1.16%, which  matches the global result  well.. \nThe absolute value of the average of slope s in Fig.7(a), −4.057, is very large compared to the other figure \nwhere values have been divided . This direct insertion of slope looks quite loose and is less statistically important \nthan Fig. 7(b). The v alues in Fig.7(b) have been divided by simulation  results . It gives a mean relative slope of \n-0.01166 , indicat ing an average temperature effect of about  −1.166 %/℃, inaccordance with the global result \n-1.162% presented above. The behavior of different BGO bars varies from −0.47%/℃ to −1.60%/℃. If this \neffect were  ignored, the energy resolution would definitely suffer : considering the temperature change of \nabout  8℃, ignor ing temperature would induce a deviation inenergy of about 9%. \n(a) \n(b)  \n10 \n For completeness, t he distribution of intercepts is displayed inFig.8. The mean of the relative intercept, 1.05 1, \nis in accordance with the global mean in Fig.4. \n \nFig.8Distribution of intercepts.  \n5 Conclusion  \nData from DAMPE has been used to calibrate MIPsforits BGO detectors .  Thermistors unif ormly installed \nin DAMPE enable  us to estimate its temperature field.  Combination of thecalibration results with the temperature \nfieldmakes it possible  to analy ze the temperature effects on MIPs of BGO bars (as well as PMTs ). \nThis analy sishas been done not only on DAMPE as a whole  to analyze  the global behavior , but also on each \nof its 308 BGO bars. The global analy sis gives a temperature effect of  −1.162 %/℃, which means that every \ndegree Celsius brings about a global deviation  of -1.162%to thebehavior of DAMPE ’s BGO calorimeter s. In terms \nof one bar alone, the average of all 308 bars gives  -1.166%, and individual behavior  differs from one bar to \nanother from  −0.47%/℃to  −1.60%/℃.FromFig.2, therehave been changes  in temperature of ~ 8℃ since the \nlaunch  of DAMPE , which  would have introduced a global energy biasof ~9% on DAMPE  if the temperature effect \nwas ignored , and would be even worse for individual BGO bars . \nAcknowledge : \nThe author ssincerely thank  the whole DAMPE collaboration, without whom it would be completely \nimpossible to perform this analy sis. We especially thank Pro fJin Chang, the ch ief scientist of  DAMPE, who \nestablished this project and had gathered this team . We also thank Prof Ming-Sheng Cai and his DAMPE payload \ngroup for offering us an excellent detector system. We also thank the Scientific Application System for providing \nus with reli ably reconstructed flight data.  All diagrams have been  plotted with ROOT  (https://root.cern.ch/) . \nReference:  \n[\n1]  J. Chang, Chin. J. Spac. Sci. , 34 (5): 550 -557 (2014)  \n[\n2]  J. Chang, L. Feng, J. Guo et al SCIENTIA SINICA Physica, Mechanica & Astronomica , 45 (11):  \n119510 (2015) (in Chinese)  \n[\n3]  J. Chang and C. DAMPE, arXiv: 1706.08453, 2017.  \n[\n4]  Y . Zhang, B. Li and C. Feng, Chinese Physcis C, vol. 36, p. 71, 2012.  \n[\n5]  Z. Zhang, C. Wang, J. Dong and Y . Wei, Nucl. Instrum. Methods Phys. Res., vol. 836, pp. 98 -104, 11 11 \n2016.  \n[\n6]  Y . Hu, J. Chang, D. Chen, J. Guo, Y . Zhang and C. Feng, Chinese Physics C, vol. 40, no. 11, p. 116003, \n2016.  \n[\n7]  Y . Wei, Z. Zhang, Y . Zhang and S. Wen, Journal of Instrumentation, vol. 11, no. 7, p. T07003, 2016.  \n[\n8]  S. Agostinelli, Nuclear Instrumen ts and Methods in Physics Research A, vol. 506, pp. 250 -303, 27 10 \n2003.  \n[\n9]  J. Allison, Nuclear Instruments and Methods in Physics Research A, vol. 835, pp. 186 -225, 2016.  \n[ C. Yue, J. Zang, T. Dong, X. Li, Z. Zhang, Z. Stephan, J. Wei, Y . Zhang and D. Wei, Nuclear \n(b) (a)  \n11 \n 10]  Instruments and Methods in Physics Research Section A, vol. 856, pp. 11 -16, 2017.  \n[\n11]  C. Wang, D. Liu, Y . Wei, Z. Zhang, Y . Zhang, X. Wang and Xu Zizong, arXiv, p. 1604.03219, 2016.  \n "    },    {        "title": "1709.03775v1.Green_s_function_formalism_for_spin_transport_in_metal_insulator_metal_heterostructures.pdf",        "content": "Green’s function formalism for spin transport in metal-insulator-metal\nheterostructures\nJiansen Zheng,1Scott Bender,1Jogundas Armaitis,2Roberto E. Troncoso,3,4and Rembert A. Duine1,5\n1Institute for Theoretical Physics and Center for Extreme Matter and Emergent Phenomena,\nUtrecht University, Leuvenlaan 4, 3584 CE Utrecht, The Netherlands\n2Institute of Theoretical Physics and Astronomy,\nVilnius University, Saul˙ etekio Ave. 3, LT-10222 Vilnius, Lithuania\n3Department of Physics, Norwegian University of Science and Technology, NO-7491 Trondheim, Norway\n4Departamento de Física, Universidad Técnica Federico Santa María, Avenida España 1680, Valparaíso, Chile\n5Department of Applied Physics, Eindhoven University of Technology,\nPO Box 513, 5600 MB Eindhoven, The Netherlands\n(Dated: September 13, 2017)\nWe develop a Green’s function formalism for spin transport through heterostructures that contain\nmetallic leads and insulating ferromagnets. While this formalism in principle allows for the inclusion\nof various magnonic interactions, we focus on Gilbert damping. As an application, we consider\nballistic spin transport by exchange magnons in a metal-insulator-metal heterostructure with and\nwithout disorder. For the former case, we show that the interplay between disorder and Gilbert\ndamping leads to spin current fluctuations. For the case without disorder, we obtain the dependence\nof the transmitted spin current on the thickness of the ferromagnet. Moreover, we show that the\nresults of the Green’s function formalism agree in the clean and continuum limit with those obtained\nfrom the linearized stochastic Landau-Lifshitz-Gilbert equation. The developed Green’s function\nformalism is a natural starting point for numerical studies of magnon transport in heterostructures\nthat contain normal metals and magnetic insulators.\nPACS numbers: 05.30.Jp, 03.75.-b, 67.10.Jn, 64.60.Ht\nI. INTRODUCTION\nMagnons are the bosonic quanta of spin waves, oscil-\nlations in the magnetization orientation in magnets1,2.\nInterest in magnons has recently revived as enhanced ex-\nperimental control has made them attractive as potential\ndata carriers of spin information over long distances and\nwithoutOhmicdissipation3. Ingeneral, magnonsexistin\ntwo regimes. One is the dipolar magnon with long wave-\nlengths that is dominated by long-range dipolar interac-\ntions and which can be generated e.g. by ferromagnetic\nresonance4,5. The other type is the exchange magnon6,\ndominated by exchange interactions and which generally\nhas higher frequency and therefore perhaps more poten-\ntial for applications in magnon based devices3. In this\npaper, we focus on transport of exchange magnons.\nThermally driven magnon transport has been widely\ninvestigated, and is closely related to spin pumping of\nspin currents across the interface between insulating fer-\nromagnets (FMs) and normal metals (NM)7–9and de-\ntection of spin current by the inverse spin Hall Effect10.\nThe most-often studied thermal effect in this context is\nthe spin Seebeck effect, which is the generation of a spin\ncurrent by a temperature gradient applied to a magnetic\ninsulator that is detected in an adjacent normal metal\nvia the inverse spin Hall effect11,12. Here, thermal fluc-\ntuations in the NM contacts drive spin transport into the\nFM, while the dissipation of spin back into the NM by\nmagnetic dynamics is facilitated by the above mentioned\nspin-pumping mechanism.\nThe injection of spin into a FM can also be accom-plished electrically, via the interaction of spin polarized\nelectrons in the NM and the localized magnetic mo-\nments of the FM. Reciprocal to spin-pumping is the spin-\ntransfer torque, which, in the presence of a spin accu-\nmulation (typically generated by the spin Hall effect)\nin the NM, drives magnetic dynamics in the FM13,14.\nSpin pumping likewise underlies the flow of spin back\ninto the NM contacts, which serve as magnon reservoirs.\nIn two-terminal set-ups based on YIG and Pt, the char-\nacteristic length scales and device-specific parameter de-\npendence of magnon transport has attracted enormous\nattention, both in experiments and theory. Cornelis-\nsenet al.15studied the excitation and detection of high-\nfrequency magnons in YIG and measured the propagat-\ning length of magnons, which reaches up to 10\u0016m in\na thin YIG film at room temperature. Other experi-\nments have shown that the polarity reversal of detected\nspins of thermal magnons in non-local devices of YIG\nare strongly dependent on temperature, YIG film thick-\nness, and injector-detector separation distance16. That\nthe interfaces are crucial can e.g. be seen by changing the\ninterface electron-magnon coupling, which was found to\nsignificantly alter the longitudinal spin Seebeck effect17.\nA linear-response transport theory was developed for\ndiffusive spin and heat transport by magnons in mag-\nnetic insulators with metallic contacts. Among other\nquantities, this theory is parameterized by relaxation\nlengths for the magnon chemical potential and magnon-\nphonon energy relaxation18,19. In a different but closely-\nrelated development, Onsager relations for the magnon\nspin and heat currents driven by magnetic field andarXiv:1709.03775v1  [cond-mat.mes-hall]  12 Sep 20172\ntemperature differences were established for insulating\nferromagnet junctions, and a magnon analogue of the\nWiedemann-Franz law was is also predicted20,21. Wang\net al.22consider ballistic transport of magnons through\nmagnetic insulators with magnonic reservoirs — rather\nthanthemoreexperimentallyrelevantsituationofmetal-\nlic reservoirs considered here — and use a nonequilib-\nrium Green’s function formalism (NEGF) to arrive at\nLanduaer-Bütikker-type expressions for the magnon cur-\nrent. Theabove-mentionedworksareeitherinthelinear-\nresponse regime or do not consider Gilbert damping\nand/or metallic reservoirs. So far, a complete quantum\nmechanical framework to study exchange magnon trans-\nport through heterostructures containing metallic reser-\nvoirs that can access different regimes, ranging from bal-\nlistic to diffusive, large or small Gilbert damping, and/or\nsmall or large interfacial magnon-electron coupling, and\nthat can incorporate Gilbert damping, is lacking.\nFigure 1: Illustration of the system where magnon transport\nin a ferromagnet (orange region) is driven by a spin accu-\nmulation difference \u0001\u0016L\u0000\u0001\u0016Rand temperature difference\nTL\u0000TRbetween two normal-metal leads (blue regions). Spin-\nflip scattering at the interface converts electronic to magnonic\nspin current. Here, Sis the local spin density in equilibrium.\nIn this paper we develop the non-equilibrium Green’s\nfunctionformalism23forspintransportthroughNM-FM-\nNM heterostructures (see Fig. 1). In principle, this for-\nmalism straightforwardly allows for adding arbitrary in-\nteractions, such as scattering of magnons with impuri-\nties and phonons, Gilbert damping, and magnon-magnon\ninteractions, and provides a suitable platform to study\nmagnon spin transport numerically, in particular beyond\nlinear response. Here, we apply the formalism to ballistic\nmagnon transport through a low-dimensional channel in\nthe presence of Gilbert damping. For that case, we com-\npute the magnon spin current as a function of channel\nlength both numerically and analytically. For the clean\ncase in the continuum limit we show how to recover our\nresults from the linearized stochastic Landau-Lifshitz-\nGilbert (LLG) equation24used previously to study ther-\nmal magnon transport in the ballistic regime25that ap-\nplies to to clean systems at low temperatures such that\nGilbert damping is the only relaxation mechanism. Us-\ning this formalism we also consider the interplay between\nGilbert damping and disorder and show that it leads to\nspin-current fluctuations.\nThis paper is organized as follows. In Sec. II, we\ndiscuss the non-equilibrium Green’s function approachto magnon transport and derive an expression for the\nmagnon spin current. Additionally a Landauer-Büttiker\nformula for the magnon spin current is derived. In\nSec. III, we illustrate the formalism by numerically con-\nsidering ballistic magnon transport and determine the\ndependence of the spin current on thickness of the ferro-\nmagnet. To further understand these numerical results,\nwe consider the formalism analytically in the continuum\nlimit in Sec. IV, and also show that in that limit we ob-\ntain the same results using the stochastic LLG equation.\nWe give a further discussion and outlook in section V.\nII. NON-EQUILIBRIUM GREEN’S FUNCTION\nFORMALISM\nIn this section we describe our model and, using\nKeldysh theory, arrive at an expression for the density\nmatrix of the magnons from which all observables can be\ncalculated. The reader interested in applying the final re-\nsult of our formalism may skip ahead to Sec. IIE where\nwe give a summary on how to implement it.\nA. Model\nj j/primej j/primej j/prime∆µL\nTL∆µR\nTR\nTFMNNM FM NM\nJLJRρj,j/prime G(±)\nj,j/prime(t,t/prime)\nGk,k/prime(t,t/prime) Gk,k/prime(t,t/prime)\nΣFM,(±)ΣL,(±)ΣR,(±)\nSelf-energy\nFigure 2: Schematic for the NM-FM-NM heterostructure and\nnotationfortheGreen’sfunctionsandself-energies. Thearray\nof circles denotes the localized magnetic moments, while the\ntwo regions outside the parabolic lines denote the leads, i.e.,\nreservoirs of polarized electrons. Moreover, JL=R\nj;kk0denotes the\ninterface coupling, and TL=Rand\u0001\u0016L=Rdenote the temper-\nature and spin accumulation for the leads. The properties of\nthe magnons are encoded in G(+)\nj;j0(t;t0), the retarded magnon\nGreen’s function, and the magnon density matrix \u001aj;j0. The\nnumber of sites in the spin-current direction is N. The self-\nenergies \u0006FM; (\u0006),\u0006L;(\u0006),\u0006R;(\u0006)are due to Gilbert damping,\nand the left and right lead, respectively.\nWe consider a magnetic insulator connected to two\nnonmagnetic metallic leads, as shown in Fig. 2. For3\nour formalism it is most convenient to consider both the\nmagnons and the electrons as hopping on the lattice for\nthe ferromagnet. Here, we consider the simplest versions\nof such cubic lattice models; extensions, e.g. to multi-\nple magnon and/or electron bands, and multiple leads\nare straightforward. The leads have a temperature TL=R\nand a spin accumulation \u0001\u0016L=Rthat injects spin cur-\nrent from the non-magnetic metal into the magnetic in-\nsulator. This nonzero spin accumulation could, e.g., be\nestablished by the spin Hall effect.\nThe total Hamiltonian is a sum of the uncoupled\nmagnon and lead Hamiltonians together with a coupling\nterm:\n^Htot=^HFM+^HNM+^HC: (1)\nHere, ^HFMdenotesthefreeHamiltonianforthemagnons,\n^HFM=\u0000X\n<j;j0>Jj;j0by\nj0bj+X\nj\u0001jby\njbj\u0011X\n<j;j0>hj0;jby\nj0bj:\n(2)\nwherebj(by\nj)is a magnon annihilation (creation) opera-\ntor. This hamiltonian can be derived from a spin hamil-\ntonian using the Holstein-Primakoff transformation26,27\nand expanding up to second order in the bosonic fields.\nEq. (2) describes hopping of the magnons with amplitude\nJj;j0between sites labeled by jandj0on the lattice, with\nan on-site potential energy \u0001jthat, if taken to be homo-\ngeneous, would correspond to the magnon gap induced\nby a magnetic field and anisotropy. We have taken the\nexternal field in the \u0000zdirection, so that one magnon,\ncreated at site jby the operator ^by\nj, corresponds to spin\n+~.\nThe Hamiltonian for the electrons in the leads is\n^HNM=\u0000X\nr2fL;RgX\n<k;k0>X\n\u001b2\";#tr^ y\nk\u001br^ k0\u001br+h:c:(3)\nwhere the electron creation (  y\nk\u001br) and annihilation\n( k\u001br) operators are labelled by the lattice position k,\nspin\u001b, and an index rdistinguishing (L)eft and (R)ight\nleads. The hopping amplitude for the electrons is de-\nnotedbytrandcouldinprinciplebedifferentfordifferent\nleads. Moreover, terms to describe hopping beyond near-\nestneighborcanbestraightforwardlyincluded. Belowwe\nwill show that microscopic details will be incorporated in\na single parameter per lead that describes the coupling\nbetween electrons and magnons.\nFinally, the Hamiltonian that describes the coupling\nbetween metal and insulator, ^HC, is given by28\n^HC=X\nr;j;kk0\u0010\nJr\nj;kk0^by\nj^ y\nk#r^ k0\"r+ h:c:\u0011\n;(4)\nwith the matrix elements Jr\nj;kk0that depend on the mi-\ncroscopic details of the interface. An electron spin that\nflips from up to down at the interface creates one magnon\nwithspin +~inthemagneticinsulator. Thisformofcou-\npling between electrons and magnons derives from inter-\nface exchange coupling between spins in the insulators\nwith electronic spins in the metal28.\nGk/prime,k/prime/prime;↑\nGk/prime/prime/prime,k;↓t/prime, j/primet, jFigure 3: Feynman diagram for the spin-flip processes emit-\nting and absorbing magnons that are represented by the wavy\nlines. The two vertices indicate the exchange coupling at one\nof the interfaces of the magnetic insulator (sites j;j0) and nor-\nmal metal (sites k;k0;k00;k000).Gk0k00;\"andGk000k;#denotes\nthe electron Keldysh Green’s function of one of the leads.\nB. Magnon density matrix and current\nOur objective is to calculate the steady-state magnon\nGreen’s function iG<\nj;j0(t;t0) =h^by\nj0(t0)^bj(t)i, from which\nall observables are calculated (note that time-dependent\noperators refer to the Heisenberg picture). This “lesser”\nGreen’s function follows from the Keldysh Green’s func-\ntion\niGj;j0(t;t0)\u0011Trh\n^\u001a(t0)TC1\u0010\n^bj(t)^by\nj0(t0)\u0011i\n;(5)\nwith ^\u001a(t0)the initial (at time t0) density matrix, and\nC1the Keldysh contour, and Tr[:::]stands for perform-\ning a trace average. The time-ordering operator on this\ncontour is defined by\nTC1\u0010\n^O(t)^O0(t0)\u0011\n\u0011\u0012(t;t0)^O(t)^O0(t0)\u0006\u0012(t0;t)^O0(t0)^O(t);\n(6)\nwith\u0012(t;t0)the corresponding Heaviside step function\nand the +(\u0000)sign applies when the operators have\nbosonic (fermionic) commutation relations. In Fig. 2\nwe schematically indicate the relevant quantities enter-\ning our theory.\nAtt= 0, the spin accumulation in the two leads is\nWe compute the magnon self energy due the coupling\nbetween magnons and electrons to second order in the\ncoupling matrix elements Jj;kk0. This implies that the\nmagnons acquire a Keldysh self-energy due to lead r\ngiven by\n~\u0006r\nj;j0(t;t0) =\u0000i\n~X\nkk0k00k000Jr\nj;kk0(Jr)\u0003\nj0;k00k000\n\u0002Gk0k00;r\"(t;t0)Gk000k;r#(t0;t);(7)\nwhereGk0k00;r\u001b(t;t0)denotes the electron Keldysh\nGreen’s function of lead r, that reads\nGkk0;r\u001b(t;t0) =\u0000ihTC1^ kr\u001b(t)^ y\nk0r\u001b(t0)i:(8)4\nThe Feynman diagram for this self-energy is shown in\nFig. 3. While this self-energy is computed to second\norder inJr\nj;kk0, the magnon Green’s function and the\nmagnon spin current, both of which we evaluate below,\ncontain all orders in Jr\nj;kk0, which therefore does not need\nto be small. In this respect, our approach is different\nfrom the work of Ohnuma et al.[29], who evaluate the\ninterfacial spin current to second order in the electron-\nmagnon coupling. Irreducible diagrams other than that\nin Fig. 3 involve one or more magnon propagators as in-\nternal lines and therefore correspond to magnon-magnon\ninteractions at the interface induced by electrons in the\nnormal metal. For the small magnon densities of interest\nto use here these can be safely neglected and the self-\nenergy in Eq. (7) thus takes into account the dominant\nprocess of spin transfer between metal and insulator.\nThe lesser and greater component of the electronic\nGreen’s functions can be expressed in terms of the spec-\ntral functions Akk0;r(\u000f)via\n\u0000iG<\nkk0;r\u001b=Akk0;r(\u000f)NF\u0012\u000f\u0000\u0016r\u001b\nkBTr\u0013\n;\niG>\nkk0;r\u001b=Akk0;r(\u000f)\u0014\n1\u0000NF\u0012\u000f\u0000\u0016r\u001b\nkBTr\u0013\u0015\n;(9)\nwithNF(x) = [ex+ 1]\u00001the Fermi distribution function,\nTrthe temperature of lead r(kBbeing Boltzmann’s con-\nstant) and\u0016\u001b;rthe chemical potential of spin projection\n\u001bin leadr. As we will see later on, the lead chemical\npotential are taken spin-dependent to be able to imple-\nment nonzero spin accumulation. The spectral function\nis related to the retarded Green’s function via\nAkk0;r(\u000f) =\u00002Imh\nG(+)\nkk0;r(\u000f)i\n; (10)\nwhich does not depend on spin as the leads are taken to\nbe normal metals. While the retarded Green’s function\nof the leads can be determined explicitly for the model\nthat we consider here, we will show below that such a\nlevel of detail is not needed but that, instead, we can pa-\nrameterize the electron-magnon coupling by an effective\ninterface parameter.\nAs mentioned before, all steady-state properties of the\nmagnon system are determined by the magnon lesser\nGreen’s function leading to the magnon density matrix.\nIt is ultimately given by the kinetic equation23,30\n\u001aj;j0\u0011h^by\nj0(t)^bj(t)i=Zd\u000f\n(2\u0019)h\nG(+)(\u000f)i~\u0006<(\u000f)G(\u0000)(\u000f)i\nj;j0;\n(11)\nwhere ~\u0006j;j0(t;t0)is the total magnon self-energy dis-\ncussed in detail below, of which the \"lesser\" component\nenters in the above equation. In the above and what\nfollows, quantities with suppressed site indexes are in-\nterpreted as matrices, and matrix multiplication applies\nfor products of these quantities. The retarded (+)and\nadvanced (\u0000)magnon Green’s functions satisfy\nh\n\u000f\u0006\u0000h\u0000~\u0006(\u0006)(\u000f)i\nG(\u0006)(\u000f) = 1;(12)where\u000f\u0006=\u000f\u0006i0. The magnon self-energies have con-\ntributions from the leads, as well as a contribution from\nthe bulk denoted by ~\u0006FM:\n~\u0006(\u000f) =~\u0006FM(\u000f) +X\nr2fL;Rg~\u0006r(\u000f):(13)\nFrom Eq. (7) we find that for the retarded and advanced\ncomponent, the contribution due to the leads is given by\n~\u0006r;(\u0006)\nj;j0(\u000f) =X\nkk0k00k000Jr\nj;kk0(Jr)\u0003\nj0;k00k000Zd\u000f0\n(2\u0019)Zd\u000f00\n(2\u0019)\n\u0002Ak0k00;r(\u000f0)Ak000k;r(\u000f00)NF\u0010\n\u000f0\u0000\u0016r\"\nkBTr\u0011\n\u0000NF\u0010\n\u000f00\u0000\u0016r#\nkBTr\u0011\n\u0000\u000f\u0006+\u000f0\u0000\u000f00\n;\n(14)\nwhereas the \"lesser\" self-energy can be shown to be of\nthe form:\n~\u0006r;<\nj;j0(\u000f) = 2iNB\u0012\u000f\u0000\u0001\u0016r\nkBTr\u0013\nImh\n~\u0006r;(+)\nj;j0(\u000f)i\n;(15)\nwithNB(x) = [ex\u00001]\u00001the Bose-Einstein distribution\nfunction and \u0001\u0016r=\u0016r\"\u0000\u0016r#the spin accumulation in\nleadr.\nHaving established the contributions due to the leads,\nwe consider the bulk self-energy ~\u0006FM, which in princi-\nple could include various contributions, such as magnon\nconserving and nonconsering magnon-phonon interac-\ntions, or magnon-magnon interactions. Here, we consider\nmagnon non-conserving magnon-phonon coupling as the\nsource of the bulk self-energy and use the Gilbert damp-\ning phenomenology to parameterize it by the constant\n\u000bwhich for the magnetic insulator YIG is of the order\nof10\u00004. Gilbert damping corresponds to a decay of the\nmagnons into phonons with a rate proportional to their\nenergy. This thus leads to the contributions\n~\u0006FM;<\nj;j0(\u000f) = 2NB\u0012\u000f\nkBTFM\u0013\n~\u0006FM;(+)\nj;j0(\u000f) ;\n~\u0006FM;(+)\nj;j0(\u000f) =\u0000i\u000b\u000f\u000ej;j0; (16)\nwhereTFMis the temperature of phonon bath. We note\nthat in principle the temperature could be taken position\ndependent to implement a temperature gradient, but we\ndo not consider this situation here.\nWith the results above, the density-matrix elements\n\u001aj;j0can be explicitly computed from the magnon re-\ntarded and advanced Green’s function and the “lesser”\ncomponent of the total magnon self-energy using\nEq. (11). The magnon self-energy is evaluated using the\nexplicitexpressionfortheretardedandadvancedmagnon\nself-energies due to leads and Gilbert damping ~\u0006FM, see\nEq. (16).\nWe are interested in the computation of the magnon\nspin currenthjm;jj0iin the bulk of the FM from site j5\nto sitej0, which in terms of the magnon density matrix\nreads,\nhjm;jj0i=\u0000i(hj;j0\u001aj0;j\u0000c:c:); (17)\nand follows from evaluating the change in time of the lo-\ncal spin density, ~dh^by\nj^bji=dt, using the Heisenberg equa-\ntions of motion. The magnon spin current in the bulk\nthus follows straightforwardly from the magnon density\nmatrix.\nWhile the formalism presented so far provides a com-\nplete description of the magnon spin transport driven by\nmetallic reservoirs, we discuss two simplifying develop-\nments below. First, we derive a Landauer-Bütikker-like\nformula for the spin current from metallic reservoirs to\nthe magnon system. Second, we discuss how to replace\nthe matrix elements Jr\nj;k;k0by a single phenomenologi-\ncal parameter that characterizes the interface between\nmetallic reservoirs and the magnetic insulator.\nC. Landauer-Büttiker formula\nIn this section we derive a Landauer-Büttiker formula\nfor the magnon transport. Using the Heisenberg equa-\ntions of motion for the local spin density, we find that\nthe spin current from the left reservoir into the magnon\nsystem is given by\njL\ns\u0011\u0000~\n2*\nd\ndtX\nk\u0010\n^ y\nk\"L k\"L\u0000 y\nk#L k#L\u0011+\n=\u00002\n~X\nj;kk0Re[\u0000\nJL\u0001\u0003\nj;kk0g<\nj;kk0(t;t0)];(18)\nin terms of the Green’s function\ng<\nj;kk0(t;t0)\u0011ih^ y\nk0\"L(t0)^ k#L(t0)^bj(t)i:(19)\nThis “lesser” coupling Green’s function g<\nj;kk0(t;t0)is cal-\nculated using Wick’s theorem and standard Keldysh\nmethods as described below.\nWe introduce the spin-flip operator for lead r\n^dy\nkk0;r(t) =^ y\nk0\"r(t)^ k#r(t); (20)\nso that the coupling Green’s function becomes\ng<\nj;kk0(t;t0)\u0011ih^dy\nkk0;L(t0)^bj(t)i: (21)\nThe Keldysh Green’s function for the spin-flip operator\nis given by\n\u0005r\nkk0k00k000(t;t0) =\u0000ihTC1^dkk0;r(t)^dy\nk00k000;r(t0)i(22)\nand using Wick’s theorem we find that\n\u0005r;>\nkk0k00k000(t;t0) =\u0000iG>\nkk000;r#(t;t0)G<\nk0k00;r\"(t0;t) ;\n\u0005r;<\nkk0k00k000(t;t0) =\u0000iG>\nk0k00;r\"(t0;t)G<\nkk000;r#(t;t0) ;\n\u0005r;(+)\nkk0k00k000(t;t0)\n=\u0000i\u0012(t\u0000t0)h\nG>\nkk000;r#(t;t0)G<\nk0k00;r\"(t0;t)\n\u0000G>\nk0k00;r\"(t0;t)G<\nkk000;r#(t;t0)i\n;(23)where we used the definition for the electron Green’s\nfunction in Eq. (8).\nApplying the Langreth theorem30and Fourier trans-\nforming, we write down the lesser coupling Green’s\nfunction in terms of the spin-flip Green’s function and\nmagnon Green’s function\ng<\nj;kk0(\u000f) =X\nj0;k00k000JL\nj0;k00k000\u0010\nG(+)\nj;j0(\u000f)\u0005L;<\nkk0k00k000(\u000f)\n+G<\nj0;j(\u000f)\u0005L;(\u0000)\nkk0k00k000(\u000f)\u0011\n; (24)\nwhere the retarded and “lesser\" magnon Green’s function\nare given by Eq. (11) and Eq. (12). Using these results,\nwe ultimately find that\njL\ns=Zd\u000f\n2\u0019\u0014\nNB\u0012\u000f\u0000\u0001\u0016L\nkBTL\u0013\n\u0000NB\u0012\u000f\u0000\u0001\u0016R\nkBTR\u0013\u0015\nT(\u000f)\n+Zd\u000f\n2\u0019\u0014\nNB\u0012\u000f\u0000\u0001\u0016L\nkBTL\u0013\n\u0000NB\u0012\u000f\nkBTFM\u0013\u0015\n\u0002Trh\n~\u0000L(\u000f)G(+)(\u000f)~\u0000FM(\u000f)G(\u0000)(\u000f)i\n; (25)\nwith the transmission function\nT(\u000f)\u0011Trh\n~\u0000L(\u000f)G(+)(\u000f)~\u0000R(\u000f)G(\u0000)(\u000f)i\n:(26)\nIn the above, the rates ~\u0000L=R(\u000f)are defined by\n~\u0000r(\u000f)\u0011\u00002Imh\n~\u0006r;(+)(\u000f)i\n; (27)\nand\n~\u0000FM(\u000f)\u0011\u00002Imh\n~\u0006FM;(+)(\u000f)i\n;(28)\nand correspond to the decay rates of magnons with en-\nergy\u000fdue to interactions with electrons in the normal\nmetal at the interfaces, and phonons in the bulk, respec-\ntively. This result is similar to the Laudauer-Büttiker\nformalism23for electronic transport using single-particle\nscattering theory. In the present context, a Landauer-\nBüttiker-like for spin transport was first derived by Ben-\nderet al.[28] for a single NM-FM interface. In the\nabsence of Gilbert damping, the spin current would cor-\nrespond to the expected result from Landauer-Bütikker\ntheory, i.e., the spin current from left to the right lead is\nthen given by the first line of Eq. (25). The presence of\ndamping gives leakage of spin current due to the coupling\nwith the phononic reservoir, as the second term shows.\nFinally,wenotethatthespincurrentfromtherightreser-\nvoir into the system is obtained by interchanging labels L\nand R in the first term, and the label L replaced by R in\nthe second one. Due to the presence of Gilbert damping,\nhowever, we have in general that jL\ns6=\u0000jR\ns.\nD. Determining the interface coupling\nWe now proceed to express the magnon spin current\n(Eq.(25))intermsofamacroscopic,measurablequantity6\nratherthantheinterfacialexchangeconstants Jr\nj;k;k0. For\n\u0001\u0016r\u001c\u000fF(with\u000fFthe Fermi energy of the metallic\nleads), which is in practice always obeyed, we have for\nlow energies and temperatures that\n~\u0006r;(\u0006)\nj;j0(\u000f)'\u0007i1\n4\u0019X\nkk0k00k000Jr\nj;kk0(Jr)\u0003\nj0;k00k000\nAk0;k00;r(\u000fF)Ak000;k;r(\u000fF)(\u000f\u0000\u0001\u0016r):(29)\nHere, we also neglected the real part of this self-energy\nwhich provides a small renormalization of the magnon\nenergies but is otherwise unimportant. The expansion\nfor small energies in Eq. (29) is valid as long as \u000f\u001c\u000fF,\nwhich applies since \u000fis a magnon energy, and therefore\nat most on the order of the thermal energy. Typically,\nthe above self-energy is strongly peaked for j;j0at the\ninterface because the magnon-electron interactions occur\nat the interface. For j;j0at the interface we have that\nthe self-energy depends weakly on varying j;j0along the\ninterface provided that the properties of the interface do\nnot vary substantially from position to position. We can\nthus make the identification:\n~\u0006r;(\u0006)\nj;j0(\u000f)'\u0007i\u0011r(\u000f\u0000\u0001\u0016r)\u000ej;j0\u000ej;jr;(30)\nwithjrthe positions of the sites at the r-th interface,\nand\u0011rparametrizing the coupling between electrons and\nmagnons at the interface. Note that \u0011rcan be read off\nfrom Eq. (29). Rather than evaluating this parameter\nin terms of the matrix elements Jr\nj;kk0and the electronic\nspectral functions of the leads Ak;k0;r(\u000f), we determine it\nin terms of the real part of the spin-mixing conductance\ng\"#;r, a phenomenological parameter that characterizes\nthespin-transferefficiencyattheinterface31. Thiscanbe\ndonebynotingthatintheclassicallimittheself-energyin\nEq. (30) leads to an interfacial contribution, determined\nby the damping constant \u0011r=N, to the Gilbert damping\nofthehomogeneousmode, where Nisthenumberofsites\nof the system perpendicular to the leads, as indicated in\nFig. 2. In terms of the spin-mixing conductance, we have\nthat this contribution is given by32g\"#;r=4\u0019srN, withsr\nthe saturation spin density per area of the ferromagnet\nat the interface with the r-th lead. Hence, we find that\n\u0011r=g\"#;r\n4\u0019sr; (31)\nwhich is used to express the reservoir contributions to the\nmagnon self-energies in terms of measurable quantities.\nThe spin-mixing conductance can be up to 5~nm\u00002for\nYIG-Pt interfaces33, leading to the conclusion that \u0011can\nbe of the order 1\u000010for that case.\nE. Summary on implementation\nWe end this section with some summarizing remarks\non implementation that may facilitate the reader who is\ninterested in applying the formalism presented here.Table I: Parameters chosen for numerical calculations based\non the NEGF formalism (unless otherwise noted).\nQuantity Value\nJ 0:05eV\n\u0001\u0016L=J 2:0\u000210\u00005\n\u0001\u0016R=J 0:0\n\u0011 8\n\u0001=J 2:0\u000210\u00003\nkBTFM=J0:60\nFirst, one determines the retarded and advanced\nmagnon Green’s functions. This can be done given\na magnon hamiltonian characterized by matrix ele-\nmentshj;j0in Eq. (2), mixing conductances for the\nmetal-insulator interfaces g\"#;r, and a value for the\nGilbert damping constant \u000b, from which one computes\nthe retarded self-energies at the interfaces in Eq. (30)\nwith Eq. (31), and Eq. (16). The retarded and ad-\nvanced magnon Green’s functions are then computed via\nEq. (12), which amounts to a matrix inversion. The next\nstepistocalculatethedensitymatrixforthemagnonsus-\ning Eq. (11), with as input the expressions for the “lesser”\nself-energies in Eqs. (15) and (16). Finally, the spin cur-\nrent is evaluated using Eq. (17) in the bulk of the FM or\nEq. (25) at the NM-FM interface. In the next sections,\nwe discuss some applications of our formalism.\nIII. NUMERICAL RESULTS\nIn this section, we present results of numerical calcula-\ntions using the formalism presented in the previous sec-\ntion.\nA. Clean system\nFor simplicity, we consider now the situation where\nthe leads and magnetic insulators are one dimensional.\nThe values of various parameters are displayed in Ta-\nble I, where we take the hopping amplitudes Jj;j0=\nJ(\u000ej;j0+1+\u000ej;j0\u00001), i.e.,Jj;j0is equal toJbetween near-\nest neighbours, and zero otherwise. We focus on trans-\nport driven by spin accumulation in the leads and set\nall temperatures equal, i.e., TL=TR=TFM\u0011T.\nWe also assume both interfaces to have equal proper-\nties, i.e., for the magnon-electron coupling parameters to\nobey\u0011L=\u0011R\u0011\u0011. First we consider the case without\ndisorder and take \u0001j= \u0001.\nWe are interested in how the Gilbert damping affects\nthe magnon spin current. In particular, we calculate the\nspincurrentinjectedintherightreservoirasafunctionof\nsystem size. The results of this calculation are shown in\nFig. 4 for various temperatures, which indicates that for7\na certain fixed spin accumulation, the injected spin cur-\nrent decays with the thickness of the system for N > 25,\nfor the parameters we have chosen. We come back to the\nvarious regimes of thickness dependence when we present\nanalyticalresultsforcleansystemsinthecontinuumlimit\nin Sec. IV. From these results we define a magnon relax-\n0 20 40 60 80 100 120 140 160\nsystem/uni00A0size/uni00A0(d/a)10/uni00AD410/uni00AD310/uni00AD210/uni00AD1100magnon/uni00A0spin/uni00A0current/uni00A0(J)kBT/J=0.012\nkBT/J=0.024\nkBT/J=0.048\nkBT/J=0.108\nkBT/J=0.192\nFigure 4: System-size dependence of spin current ejected in\nthe right reservoir for \u000b= 6:9\u000210\u00002;\u0011= 8:0and various\ntemperatures.\nation length drelaxusing the definition\njm(d)/exp(\u0000d=drelax); (32)\napplied to the region N > 25and where d= Nawith\nathe lattice constant. The magnon relaxation length\ndepends on system temperature and is shown in Fig. 5.\nWe attempt to fit the temperature dependence with\ndrelax(T\u0003) =a(\r0+\r1p\nT\u0003+\r2\nT\u0003);(33)\nwith\r0;\r1;\r2constantsand T\u0003definedasthedimension-\nless temperature T\u0003\u0011kBT=J. The term proportional to\n\r1is expected for quadratically dispersing magnons with\nGilbert damping as the only relaxation mechanism15,25.\nThe terms proportional to \r0and\r2are added to charac-\nterize the deviation from this expected form. Our results\nshow that the relaxation length has not only \u00181=p\nT\nbehaviour. This is due to the finite system size, the con-\ntact resistance that the spin current experiences at the\ninterface between metal and magnetic insulator, and the\ndeviation of the magnon dispersion from a quadratic one\ndue to the presence of the lattice.\nB. Disordered system\nWe now consider the effects of disorder on the spin\ncurrent as a function of the thickness of the FM. We con-\nsider a one-dimensional system with a disorder potential\ndrelax/a=γ0+γ1\nT*+γ2\nT*\nNumerical result\nFitted curve\n0.0 0.2 0.4 0.6 0.8115120125130135140145150\nT*=kBT/Jdrelax/aFigure 5: Magnon relaxation length as a function of dimen-\nsionlesstemperature T\u0003for\u000b= 6:9\u000210\u00002;\u0011= 8:0. Thefitted\nparameters are obtained as \r0= 114:33;\r1= 0:96;\r2= 0:32.\nimplemented by taking \u0001j= \u0001(1+\u000ej), where\u000ejis a ran-\ndom number evenly distributed between \u0000\u000eand\u000e(with\n\u000e\u001c1andpositive)thatisuncorrelatedbetweendifferent\nsites. In one dimension, all magnon states are Anderson\nlocalized34. Since this is an interference phenomenon, it\nis expected that Gilbert damping diminishes such local-\nization effects. The effect of disorder on spin waves was\ninvestigated using a classical model in Ref. [35], whereas\nRef. [36] presents a general discussion of the effect of\ndissipation on Anderson localization. Very recently, the\neffect of Dzyaloshinskii-Moriya interactions on magnon\nlocalization was studied37. Here we consider how the in-\nterplay between Gilbert damping and the disorder affects\nthe magnon transport.\nFor a system without Gilbert damping the spin current\ncarried by magnons is conserved and therefore indepen-\ndent of position regardless of the presence or absence of\ndisorder. DuetothepresenceofGilbertdampingthespin\ncurrent decays as a function of position. Adding disorder\non top of the dissipation due to Gilbert damping causes\nthe spin current to fluctuate from position to position.\nFor large Gilbert damping, however, the effects of dis-\norder are suppressed as the Gilbert damping suppresses\nthe localization of magnon states. In Fig. 6 we show nu-\nmerical results of the position dependence of the magnon\ncurrent for different combinations of disorder and Gilbert\ndamping constants. The plots clearly show that the spin\ncurrent fluctuates in position due to the combined ef-\nfect of disorder and Gilbert damping, whereas it is con-\nstant without Gilbert damping, and decays in the case\nwith damping but without disorder. Note that for the\ntwo cases without Gilbert damping the magnitude of the\nspin current is different because the disorder alters the\nconductance of the system and each curve in Fig. 6 cor-\nresponds to a different realization of disorder.\nTo characterize the fluctuations in the spin current, we8\n0 20 40 60 80\nsite j1.01.52.02.53.03.54.04.5magnon spin current (J)1e7\n=0.0,=0.0\n=0.0,=0.0015\n=0.0069,=0.0\n=0.0069,=0.0015\nFigure 6: Spatial dependence of local magnon current for the\ncase without Gilbert damping and disorder ( \u000b= 0;\u000e= 0),\nwithout disorder ( \u000b= 6:9\u000210\u00003;\u000e= 0), without Gilbert\ndamping (\u000b= 0;\u000e= 1:5\u000210\u00003), and both disorder and\nGilbert damping ( \u000b= 6:9\u000210\u00003;\u000e= 1:5\u000210\u00003). The inter-\nface coupling parameter is taken equal to \u0011= 0:8.\ndefine the correlation function\nCj=vuut\u0000\njm;j;j+1\u0000jm;j;j+1\u00012\n\u0000\njm;j;j+1\u00012; (34)\nwhere the bar stands for performing averaging over the\nrealizations of disorder. Fig. (7) shows this correlation\nfunction for j=N\u00001as a function of Gilbert damp-\ning for various strengths of the disorder. As we expect,\nbasedonthepreviousdiscussion, thefluctuationsbecome\nsmall as the Gilbert damping becomes very large or zero,\nleaving an intermediate range where there are sizeable\nfluctuations in the spin current.\nIV. ANALYTICAL RESULTS\nIn this section we analytically compute the magnon\ntransmission function in the continuum limit a!0\nfor a clean system. We consider again the situation\nof a magnon hopping amplitude Jj;j0that is equal to\nJand nonzero only for nearest neighbors, and a con-\nstant magnon gap \u0001j= \u0001. We compute the magnon\ndensity matrix, denoted by \u001a(x;x0), and retarded and\nadvanced Green’s functions, denoted by G(\u0006)(x;x00;\u000f).\nHere, the spatial coordinates in the continuum are de-\nnoted byx;x0;x00;\u0001\u0001\u0001. We take the system to be trans-\nlationally invariant in the y\u0000z-plane and the current\ndirection as shown in Fig. 1 to be x.\nIn the continuum limit, the imaginary part of the vari-\n0.000 0.005 0.010 0.015 0.020\nα012345CN−1\nδ=0.0005\nδ=0.001\nδ=0.0015Figure 7: Correlation function Cjthat characterizes the fluc-\ntuations in the spin curent for j=N\u00001as a function of the\nGilbert damping constant, for three strengths of the disorder\npotential. The curves are obtained by performing averaging\nover 100 realizations of the disorder. The interface coupling\nparameter is taken equal to \u0011= 0:8.\nous self-energies acquired by the magnons have the form:\nImh\n~\u0006r;(+)(x;x0;\u000f)i\n=\n\u0000~\u0011r(\u000f\u0000\u0016r)\u000e(x\u0000xr)\u000e(x\u0000x0) ;\nImh\n~\u0006FM;(+)(x;x0;\u000f)i\n=\u0000\u000b\u000f\u000e(x\u0000x0);(35)\nwherexris the position of the r-th lead, and where ~\u0011ris\nthe parameter that characterizes the interfacial coupling\nbetween magnons and electrons. We use a different nota-\ntion for this parameter as in the continuum situation its\ndimension is different with respect to the discrete case.\nTo express ~\u0011rin terms of the spin-mixing conductance\nwe have that ~\u0011r=g\"#=4\u0019~srwhere ~sris now the three-\ndimensional saturated spin density of the ferromagnet.\nWe proceed by evaluating the magnon transmission\nfunction from Eq. (26). We compute the rates in Eq. (27)\nfrom the self-energies Eqs. (35) and find for the transmis-\nsion function in the first instance that\nT(\u000f) = 4~\u0011L~\u0011R(\u000f\u0000\u0001\u0016L)(\u000f\u0000\u0001\u0016R)\n\u0002Zdq\n(2\u0019)2g(+)(xL;xR;q;\u000f)g(\u0000)(xR;xL;q;\u000f);(36)\nwhere qis the two-dimensional momentum that results\nfromFouriertransforminginthe y\u0000z-plane. TheGreen’s\nfunctionsg(\u0006)(x;x0;q;\u000f)obey [compare Eq. (12)]\n\u0014\n(1\u0006i\u000b)\u000f+Ad2\ndx2\u0000Aq2\u0000\u0001\n\u0006iX\nr2fL;Rg~\u0011r(\u000f\u0000\u0001\u0016r)\u000e(x\u0000xr)3\n5g(\u0006)(x;x0;q;\u000f)\n=\u000e(x\u0000x0); (37)9\nwhereA=Ja2. This Green’s function is evaluated using\nstandard techniques for inhomogeneous boundary value\nproblems (see Appendix A) to ultimately yield\nT(\u000f) = 4~\u00112(\u000f\u0000\u0001\u0016L)(\u000f\u0000\u0001\u0016R)Zdq\n(2\u0019)2jt(q;\u000f)j2;(38)\nwith\nt(q;\u000f) =A\u0014\u0002\u0000\nA2\u00142\u0000~\u00112(\u000f\u0000\u0001\u0016L)(\u000f\u0000\u0001\u0016R)\u0001\nsinh(\u0014d)\n\u0000iA~\u0011\u0014(2\u000f\u0000\u0001\u0016L\u0000\u0001\u0016R) cosh(\u0014d)]\u00001; (39)\nwith\u0014=p\n(Aq2+ \u0001\u0000\u000f\u0000i\u000b\u000f)=Aand whered=xR\u0000\nxL. Note that we have at this point taken both interfaces\nequal for simplicity, so that ~\u0011L= ~\u0011R\u0011~\u0011. In terms of an\ninterfacial Gilbert damping parameter \u000b0we have that\n~\u0011=d\u000b0.\nLet us identify the magnon decay length\nl\u0011\u0015\n\u000b;\nwhere\u0015=p\nA=kBTis proportional to the thermal de\nBroglie wavelength. Equipped with a closed, analytic\nexpression, we may now, in an analogous way as Hoffman\net al.[25], investigate the behavior of Eq. (38) in the thin\nFM (d\u001cl) and thick FM ( d\u001dl) regimes. In order\nto do so, we take \u0016L= 0so that the second term in\nEq. (25) vanishes and the spin current is fully determined\nby the transmission coefficient T(\u000f). Before analyzing\nthe result for the spin current more closely, we remark\nthat the result for the transmission function may also be\nobtained from the linearized stochastic Landau-Lifshitz-\nGilbert equation, as shown in Appendix B.\nA. Thin film regime ( d\u001cl)\nIn the thin film regime, the transmission coefficient\nT(\u000f)exhibits scattering resonances near \u000f=\u000fnqfor given\nq, where\n\u000fnq\nA=q2+1\n\u00182+n2\u00192\nd2\nandnisanintegerandwhere \u0018=p\nA=\u0001isthecoherence\nlength of the ferromagnet. When the ferromagnet is suf-\nficiently thin ( d\u001c\u0015=\u000b1=2=p\u000bl), one finds that these\npeaks are well separated, and the transmission coefficient\nis approximated as a sum of Lorentzians: T =P1\nn=0Tn,\nwhere:\nTn(\u000f)\u0019Anq\u0000L\nn\u0000R\nn\n\u0000Ln+ \u0000Rn+ \u0000FMn(40)\nwith\nAnq(\u000f) =\u0000n\n(\u000f\u0000\u000fnq)2+ (\u0000n=2)2(41)as the spin wave spectral density. The broadening rates\nare given by \u0000FM\nn= 2\u000b\u000f,\u0000L\n0= 2\u000b0\u000f,\u0000R\n0= 2\u000b0(\u000f\u0000\u0016R),\n\u0000L\nn6=0= 4\u000b0\u000f,\u0000R\nn6=0= 4\u000b0(\u000f\u0000\u0016R)and\u0000n= \u0000FM\nn+ \u0000L\nn+\n\u0000R\nn. Intheextremesmalldissipationlimit(i.e. neglecting\nspectral broadening by the Gilbert damping), one has:\nAnq(\u000f)!2\u0019\u000e(\u000f\u0000\u000fnq); (42)\nand the current has the simple form, jL\ns=P1\nn=0jn,\nwhere\njn=a2Zd2q\n(2\u0019)2\u0000L\nn\u0000R\nn\n\u0000n\u0014\nNB\u0012\u000fnq\nkBT\u0013\n\u0000NB\u0012\u000fnq\u0000\u0001\u0016R\nkBT\u0013\u0015\n(43)\nwhere \u0000L\nn,\u0000R\nnand \u0000FM\nnare all evaluated at \u000f=\u000fnq.\nEq. (43) allows one to estimate the thickness dependence\nof the signal. Supposing \u0016R.\u000fnq, whend\u001cg\"#=s\u000b,\nthen\u000b0\u001d\u000b, and \u0000L\nn\u0000R\nn=\u0000FM\nn\u0018jL\ns;cl\u00181=d; whend\u001d\ng\"#=s\u000b, then\u000b0\u001c\u000b, andjL\ns;cl\u00181=d2. The enhancement\nof the spin current for small dis in rough agreement with\nour numerical results in the previous section as shown in\nFig. 4.\nB. Thick film regime ( d\u001dl)\nIn the thick film regime, the transmission function be-\ncomes\nT(\u000f)\u0019(4Ad)2\u0000L\nx\u0000R\nxp\n(\u000f\u0000\u000f0q)2+ (\u0000FMx=2)2e\u00002\u0014rd\nj(4A\u0014)2\u0000(d)2\u0000Lx\u0000Rx\u0000i4dA\u0000Rx\u0014S(\u0014r)j2\nwhere \u0000L=R= FM\nx = \u0000L=R= FM\nn6=0,\u0014r= Re[\u0014], andS(\u0014r)isthe\nsign of\u0014r. For\u000b\u001c1, we have\u0014=ikx\u0000\n1 +i\u000b\u000f=2Ak2\nx\u0001\n,\nwherekx=p\nq2+\u0018\u00002\u0000\u000f=A. For energies \u000f > A (q2+\n\u0018\u00002),kxis imaginary, and the contribution to the spin\ncurrent decays rapidly with d. When, however, \u000f <\nA(q2+\u0018\u00002),kxis real, and \u0014r=\u0000\u000b\u0000\nq2+\u0018\u00002\u0001\n=2kx\u0018\n\u000b=\u0015(for thermal magnons), so that the signal decays\nover a length scale l/1=p\nT, in agreement with our\nnumerical results as shown in Fig. 5.\nC. Comparison with numerical results\nIn order to compare the numerical with the analyti-\ncal results we plot in Fig. 8 the transmission function\nas a function of energy. Here, the numerical result is\nevaluated for a clean system using Eq. (26) while the an-\nalytical result is that of Eq. (38). While they agree in\nthe appropriate limit ( N!1;a!0), for finite Nthere\nare substantial deviations that are due to the increased\nimportance of interfacing coupling relative to the Gilbert\ndamping for small systems and the deviations of the dis-\npersion from a quadratic one.10\nΔ/J=0.2N=20\nAnalytic\nNumerical\n0 1 2 3 4 50.000.020.040.060.08\nϵ/JT(ϵ)\nFigure 8: Magnon transmission function as a function of en-\nergy. The parameters are chosen to be \u0001=J= 0:2;\u000b=\n0:069;\u0011= 8:0.\nV. DISCUSSION AND OUTLOOK\nWe have developed a NEGF formalism for exchange\nmagnon transport in a NM-FM-NM heterostructure. We\nhave illustrated the formalism with numerical and ana-\nlytical calculations and determined the thickness depen-\ndence of the magnon spin current. We have also con-\nsidered magnon disorder scattering and shown that the\ninterplay between disorder and Gilbert damping leads to\nspin-current fluctuations.\nWehavealsodemonstratedthatforacleansystem,i.e.,\nwithout disorder, in the continuum limit the results ob-\ntained from the NEGF formalism agree with those fromthe stochastic LLG formalism. The latter is suitable for\na clean system in the continuum limit where the vari-\nous boundary conditions on the solutions of the stochas-\ntic equations are easily imposed. The NEGF formal-\nism is geared towards real-space implementation, such\nthat, e.g., disorder scattering due to impurities are more\nstraightforwardly included as illustrated by our example\napplication. The NEGF formalism is also more flexi-\nble for systematically including self-energies due to ad-\nditional physical processes, such as magnon-conserving\nmagnon-phonon scattering and magnon-magnon scatter-\ning, or, for example, for treating strong-coupling regimes\nintowhichthestochasticLandau-Lifshitz-Gilbertformal-\nism has no natural extension.\nUsing our formalism, a variety of mesoscopic transport\nfeatures of magnon transport can be investigated includ-\ning, e.g., magnon shot noise38. The generalization of our\nformalism to elliptical magnons and magnons in antifer-\nromagnets is an attractive direction for future research.\nAcknowledgments\nThis work was supported by the Stichting voor Funda-\nmenteel Onderzoek der Materie (FOM), the Netherlands\nOrganization for Scientific Research (NWO), and by the\nEuropean Research Council (ERC) under the Seventh\nFramework Program (FP7). J. Z. would like to thank\nthe China Scholarship Council. J. A. has received fund-\ning from the European Union’s Horizon 2020 research\nand innovation programme under the Marie Skłodowska-\nCurie grant agreement No 706839 (SPINSOCS).\nAppendix A: Evaluation of magnon Green’s function in the continuum limit\nIn this appendix we evaluate the magnon Green’s function in the continuum limit that is determined by Eq. (37).\nFor simplicity we take the momentum qequal to zero and suppress it in the notation, as it can be trivially restored\nafterwards. The Green’s function is then determined by\n2\n4\u000f\u0006i\u000b\u000f\u0006iX\nr2fL;Rg~\u0011r(\u000f\u0000\u0001\u0016r)\u000e(x\u0000xr) +Ad2\ndx2\u0000H3\n5g(\u0006)(x;x0;\u000f) =\u000e(x\u0000x0): (A1)\nTo determine this Green’s function we first solve for the states \u001f\u0006(x)that obey:\n2\n4\u000f\u0006i\u000b\u000f\u0006iX\nr2fL;Rg~\u0011r(\u000f\u0000\u0001\u0016r)\u000e(x\u0000xr) +Ad2\ndx2\u0000H3\n5\u001f\u0006(x) = 0: (A2)\nIntegrating this equation across x=xLandx=xRleads to the boundary conditions:\nx=xL:\u0006i~\u0011L(\u000f\u0000\u0001\u0016L)\u001f\u0006(xL) +Ad\u001f\u0006(x)\ndxjx=xL= 0; (A3)\nx=xR:\u0006i~\u0011R(\u000f\u0000\u0001\u0016R)\u001f\u0006(xR)\u0000Ad\u001f\u0006(x)\ndxjx=xR= 0: (A4)11\nForxL<x<xR, the general solution is:\n\u001f\u0006(x) =Beik\u0006x+Ce\u0000ik\u0006x; (A5)\nwithk\u0006=p\n(\u000f\u0006i\u000b\u000f\u0000H)=A. We write the solution obeying the boundary condition at x=xLas\n\u001f\u0006\nL(x) =eik\u0006x+C\u0006e\u0000ik\u0006x; (A6)\nWith the boundary condition at x=xL( Eq.A4), we find that\nC\u0006=\u0014Ak\u0006\u0006~\u0011L(\u000f\u0000\u0001\u0016L)\nAk\u0006\u0007~\u0011L(\u000f\u0000\u0001\u0016L)\u0015\ne2ik\u0006xL:\nFor the solution obeying the boundary condition at x=xR, we write\n\u001f\u0006\nR(x) =B\u0006eik\u0006x+e\u0000ik\u0006x: (A7)\nWith the boundary condition at x=xR( Eq.A4), we have:\nA(iB\u0006k\u0006eik\u0006xR\u0000ik\u0006e\u0000ik\u0006xR) =\u0006i~\u0011R(\u000f\u0000\u0016R)(B\u0006eik\u0006xR+e\u0000ik\u0006xR);\nso that\nB\u0006=\u0014Ak\u0006\u0006~\u0011R(\u000f\u0000\u0001\u0016R)\nAk\u0006\u0007~\u0011R(\u000f\u0000\u0001\u0016R)\u0015\ne\u00002ik\u0006xR:\nThe Green’s function is now given by39\ng(\u0006)(x;x0;\u000f) =8\n<\n:\u001f(\u0006)\nL(x0)\u001f(\u0006)\nR(x)\nAW(\u0006)(x0)forx>x0;\n\u001f(\u0006)\nL(x)\u001f(\u0006)\nR(x0)\nAW(\u0006)(x0)forx<x0:(A8)\nwith the Wronskian\nW\u0006(x0) =\u001f\u0006\nL(x0)d\u001f\u0006\nR(x0)\ndx0\u0000\u001f\u0006\nR(x0)d\u001f\u0006\nL(x0)\ndx0:\nInserting the result for the Green’s function in Eq. (36) and using that k+=i\u0014andk\u0000= (k+)\u0003, we obtain Eqs. (38)\nand (39) after restoring the q-dependence and taking ~\u0011R= ~\u0011L= ~\u0011.\nAppendix B: Stochastic Formalism for Spin Transport in a Ferromagnet\nHere, we show how to recover our analytical results from the stochastic Landau-Lifshitz-Gilbert equation, general-\nizing the results of Ref. [25] to the case of nonzero spin accumulation in the metallic reservoirs. The dynamics of the\nspin density unit vector nis governed by:\n(1 +\u000bn\u0002)~_n+n\u0002(H+h)\u0000An\u0002r2n= 0; (B1)\nwhere H= \u0001^zis the effective applied magnetic field (in units of energy) and his the bulk stochastic field24. We\nassume a spin accumulation \u00160= \u0001\u0016Rzin the right normal metal, while the spin accumulation in the left lead is\ntaken zero. The boundary condition at x= 0reads\njs(x= 0) =\u0000A~sn\u0002@xnjx=0\n=\u0014g\"#\n4\u0019(n\u0002(n\u0002\u00160) +n\u0002~_n) +n\u0002h0\nL\u0015\nx=0(B2)\nand atx=d:\njs(x=d) =\u0000A~sn\u0002@xnjx=d\n=\u0000\u0014g\"#\n4\u0019(n\u0002~_n) +n\u0002h0\nR\u0015\nx=d: (B3)12\nDefining (x;t) =n(x;t)p\n~s=2, wheren\u0011nx\u0000iny, we linearize the dynamics around the equilibrium orientation\nn=\u0000z. Fourier transforming:\n (x;q;\u000f) =Zdt\n2\u0019~Zd2r?\n2\u0019ei\u000ft=~e\u0000ir?\u0001q (x;r?;t);\nthe bulk equation of motion reads:\nA\u0000\n@2\nx\u0000\u00142\u0001\n =hp\n~s : (B4)\nThe bulk transformed stochastic field h=hx\u0000ihyobeys the fluctuation dissipation theorem:\nhh\u0003(x;q;\u000f)h(x0;q0;\u000f0)i= 2 (2\u0019)3\u000b(~2=~s)\u000f\n\u0002\u000e(x\u0000x0)\u000e(q\u0000q0)\u000e(\u000f\u0000\u000f0)\ntanh [\u000f=2kBT]: (B5)\nThe boundary conditions, Eqs. (B2) and (B3), become respectively:\nA@x \u0000ig\"#\n4\u0019~s(\u000f\u0000\u0001\u0016R) =hRp\n2~s(B6)\natx= 0and\nA@x +ig\"#\n4\u0019~s\u000f =hLp\n2~s(B7)\natx=d, where we have taken the coupling at both interfaces equal. Similarly, the interfacial stochastic fields obey:\nD\nh0\u0003\nR(q;\u000f)h0\nR(q0;\u000f0)E\n=2 (2\u0019)3\u000b0d~2~s(\u000f\u0000\u0001\u0016R)\u000e(q\u0000q0)\u000e(\u000f\u0000\u000f0)\ntanh [(\u000f\u0000\u0001\u0016R)=2kBT](B8)\nand\nD\nh0\u0003\nL(q;\u000f0)h0\nL(q0;\u000f)E\n=2 (2\u0019)3\u000b0d~2~s\u000f\u000e(q\u0000q0)\u000e(\u000f\u0000\u000f0)\ntanh [\u000f=2kBT]: (B9)\nUsing Eqs. (B4)-(B9), one finds the current on the left side of the structure: jL\ns\u0011z\u0001js(x= 0)to be of the form:\njL\ns=Zd\u000f\n2\u0019\u0014\nNB\u0012\u000f\nkBT\u0013\n\u0000NB\u0012\u000f\u0000\u0001\u0016R\nkBT\u0013\u0015\nT(\u000f) (B10)\nwhere T(\u000f)is the transmission coefficient in Eq. (38). Hence, for a clean system and in the continuum limit the results\nof the stochastic Landau-Lifshitz-Gilbert equation coincide with those of the NEGF formalism given by Eq. (25).\n1Lev Davidovich Landau and Evgenii Mikhailovich Lifshitz.\nStatistical physics. 1958.\n2Charles Kittel. Introduction tosolidstate, volume 162.\nJohn Wiley & Sons, 1966.\n3AV Chumak, VI Vasyuchka, AA Serga, and B Hillebrands.\nMagnon spintronics. NaturePhysics, 11(6):453–461, 2015.\n4T Kasuya and RC LeCraw. Relaxation mechanisms in fer-romagnetic resonance. Physical ReviewLetters, 6(5):223,\n1961.\n5M Sparks. Ferromagnetic resonance in thin films. i. theory\nof normal-mode frequencies. Physical ReviewB, 1(9):3831,\n1970.\n6Christian W Sandweg, Yosuke Kajiwara, Andrii V Chu-\nmak, Alexander A Serga, Vitaliy I Vasyuchka, Mat-13\ntias Benjamin Jungfleisch, Eiji Saitoh, and Burkard Hille-\nbrands. Spin pumping by parametrically excited exchange\nmagnons. Physical reviewletters, 106(21):216601, 2011.\n7Nynke Vlietstra, BJ van Wees, and FK Dejene. Detection\nof spin pumping from yig by spin-charge conversion in a\nau/ni 80 fe 20 spin-valve structure. Physical ReviewB,\n94(3):035407, 2016.\n8ATalalaevskij,MDecker,JStigloher,AMitra,HSKörner,\nOCespedes, CHBack, andBJHickey. Magneticproperties\nof spin waves in thin yttrium iron garnet films. Physical\nReviewB, 95(6):064409, 2017.\n9J Holanda, O Alves Santos, RL Rodríguez-Suárez,\nA Azevedo, and SM Rezende. Simultaneous spin pump-\ning and spin seebeck experiments with thermal control of\nthe magnetic damping in bilayers of yttrium iron garnet\nand heavy metals: Yig/pt and yig/irmn. Physical Review\nB, 95(13):134432, 2017.\n10E Saitoh, M Ueda, H Miyajima, and G Tatara. Conver-\nsion of spin current into charge current at room temper-\nature: Inverse spin-hall effect. Applied PhysicsLetters,\n88(18):182509, 2006.\n11K Uchida, S Takahashi, K Harii, J Ieda, W Koshibae,\nKazuya Ando, S Maekawa, and E Saitoh. Observation of\nthe spin seebeck effect. Nature, 455(7214):778–781, 2008.\n12Jiang Xiao, Gerrit EW Bauer, Ken-chi Uchida, Eiji Saitoh,\nSadamichi Maekawa, et al. Theory of magnon-driven spin\nseebeck effect. Physical ReviewB, 81(21):214418, 2010.\n13L Berger. Emission of spin waves by a magnetic multilayer\ntraversed by a current. Physical ReviewB, 54(13):9353,\n1996.\n14John C Slonczewski. Current-driven excitation of mag-\nnetic multilayers. Journal ofMagnetism andMagnetic\nMaterials, 159(1-2):L1–L7, 1996.\n15LJ Cornelissen, J Liu, RA Duine, J Ben Youssef, and\nBJ Van Wees. Long-distance transport of magnon spin\ninformation in a magnetic insulator at room temperature.\nNaturePhysics, 11(12):1022–1026, 2015.\n16XJZhou, GYShi, JHHan, QHYang, YHRao, HWZhang,\nLL Lang, SM Zhou, F Pan, and C Song. Lateral transport\nproperties of thermally excited magnons in yttrium iron\ngarnetfilms. AppliedPhysicsLetters, 110(6):062407, 2017.\n17Er-Jia Guo, Joel Cramer, Andreas Kehlberger, Ciaran A\nFerguson, Donald A MacLaren, Gerhard Jakob, and Math-\nias Kläui. Influence of thickness and interface on the low-\ntemperature enhancement of the spin seebeck effect in yig\nfilms.Physical ReviewX, 6(3):031012, 2016.\n18Ludo J Cornelissen, Kevin JH Peters, GEW Bauer,\nRA Duine, and Bart J van Wees. Magnon spin transport\ndriven by the magnon chemical potential in a magnetic\ninsulator. Physical ReviewB, 94(1):014412, 2016.\n19Benedetta Flebus, SA Bender, Yaroslav Tserkovnyak, and\nRA Duine. Two-fluid theory for spin superfluidity in mag-\nnetic insulators. Physical reviewletters, 116(11):117201,\n2016.\n20Kouki Nakata, Pascal Simon, and Daniel Loss.Wiedemann-franz law for magnon transport. Physical\nReviewB, 92(13):134425, 2015.\n21Kouki Nakata, Pascal Simon, and Daniel Loss. Spin cur-\nrentsandmagnondynamicsininsulatingmagnets. Journal\nofPhysicsD:AppliedPhysics, 50(11):114004, 2017.\n22Baigeng Wang, Jian Wang, Jin Wang, and DY Xing.\nSpin current carried by magnons. Physical ReviewB,\n69(17):174403, 2004.\n23Massimiliano Di Ventra. Electrical transport innanoscale\nsystems, volume 14. Cambridge University Press Cam-\nbridge, 2008.\n24William Fuller Brown Jr. Thermal fluctuations of a single-\ndomain particle. Physical Review, 130(5):1677, 1963.\n25Silas Hoffman, Koji Sato, and Yaroslav Tserkovnyak.\nLandau-lifshitz theory of the longitudinal spin seebeck ef-\nfect.Physical ReviewB, 88(6):064408, 2013.\n26T Holstein and Hl Primakoff. Field dependence of the\nintrinsic domain magnetization of a ferromagnet. Physical\nReview, 58(12):1098, 1940.\n27Assa Auerbach. Interacting electrons andquantum\nmagnetism. Springer Science & Business Media, 2012.\n28Scott A Bender, Rembert A Duine, and Yaroslav\nTserkovnyak. Electronic pumping of quasiequilibrium\nbose-einstein-condensed magnons. Physical reviewletters,\n108(24):246601, 2012.\n29Y. Ohnuma, M. Matsuo, and S. Maekawa. Theory of spin\nPeltier effect. ArXive-prints, June 2017.\n30Joseph Maciejko. An introduction to nonequilibrium\nmany-body theory. LectureNotes, 2007.\n31Arne Brataas, Yu V Nazarov, and Gerrit EW Bauer.\nFinite-element theory of transport in ferromagnet–normal\nmetalsystems. Physical ReviewLetters, 84(11):2481, 2000.\n32Yaroslav Tserkovnyak, Arne Brataas, and Gerrit EW\nBauer. Enhanced gilbert damping in thin ferromagnetic\nfilms.Physical reviewletters, 88(11):117601, 2002.\n33Jia, Xingtao, Liu, Kai, Xia, Ke, and Bauer, Gerrit E.\nW. Spin transfer torque on magnetic insulators. EPL,\n96(1):17005, 2011.\n34P. W. Anderson. Absence of diffusion in certain random\nlattices. Phys.Rev., 109:1492–1505, Mar 1958.\n35Martin Evers, Cord A. Müller, and Ulrich Nowak. Spin-\nwave localization in disordered magnets. Phys.Rev.B,\n92:014411, Jul 2015.\n36I. Yusipov, T. Laptyeva, S. Denisov, and M. Ivanchenko.\nLocalization in open quantum systems. Phys.Rev.Lett.,\n118:070402, Feb 2017.\n37M. Evers, C. A. Müller, and U. Nowak. Weak localiza-\ntion of magnons in chiral magnets. ArXive-prints, August\n2017.\n38AkashdeepKamraandWolfgangBelzig. Magnon-mediated\nspin current noise in ferromagnet| nonmagnetic conductor\nhybrids. Physical ReviewB, 94(1):014419, 2016.\n39Russell. L. Herman. Introduction to partial differential\nequations. 2015."    },    {        "title": "1709.04911v2.Intrinsic_Damping_Phenomena_from_Quantum_to_Classical_Magnets_An_ab_initio_Study_of_Gilbert_Damping_in_Pt_Co_Bilayer.pdf",        "content": "Intrinsic Damping Phenomena from Quantum to Classical Magnets:\nAn ab-initio Study of Gilbert Damping in Pt/Co Bilayer\nFarzad Mahfouzi,1,\u0003Jinwoong Kim,1, 2and Nicholas Kioussis1,y\n1Department of Physics and Astronomy, California State University, Northridge, CA, USA\n2Department of Physics and Astronomy, Rutgers University, NJ, USA\nA fully quantum mechanical description of the precessional damping of Pt/Co bilayer is presented\nin the framework of the Keldysh Green function approach using ab initio electronic structure cal-\nculations. In contrast to previous calculations of classical Gilbert damping ( \u000bGD), we demonstrate\nthat\u000bGDin the quantum case does not diverge in the ballistic regime due to the \fnite size of\nthe total spin, S. In the limit of S!1 we show that the formalism recovers the torque correla-\ntion expression for \u000bGDwhich we decompose into spin-pumping and spin-orbital torque correlation\ncontributions. The formalism is generalized to take into account a self consistently determined de-\nphasing mechanism which preserves the conservation laws and allows the investigation of the e\u000bect\nof disorder. The dependence of \u000bGDon Pt thickness and disorder strength is calculated and the\nspin di\u000busion length of Pt and spin mixing conductance of the bilayer are determined and compared\nwith experiments.\nPACS numbers: 72.25.Mk, 75.70.Tj, 85.75.-d, 72.10.Bg\nI. INTRODUCTION\nMagnetic materials provide an intellectually rich arena\nfor fundamental scienti\fc discovery and for the invention\nof faster, smaller and more energy-e\u000ecient technologies.\nThe intimate relationship of charge transport and mag-\nnetic structure in metallic systems on one hand, and the\nrich physics occurring at the interface between di\u000berent\nmaterials in layered structures on the other hand, are the\nhallmark of the \rourishing research \feld of spintronics.1{5\nRecently, intense focus has been placed on the signi\f-\ncant role played by spin-orbit coupling (SOC) and the ef-\nfect of interfacial inversion symmetry breaking on the dy-\nnamics of the magnetization in ferromagnet (FM)-normal\nmetal (NM) bilayer systems. Of prime importance to this\n\feld is the (precessional) magnetization damping phe-\nnomena, usually treated phenomenologically by means\nof a parameter referred to as Gilbert damping constant,\n\u000bGD, in the LandauLifshitzGilbert (LLG) equation of\nmotiond~ m=dt =\r~ m\u0002~B+\u000bGD~ m\u0002d~ m=dt , which de-\nscribes the rate of the angular momentum loss of the\nFM.6Here,~ mis the unit vector along the magnetization\ndirection and ~Bis an e\u000bective magnetic \feld.\nIn FM/NM bilayer devices the e\u000bect of the NM on the\nGilbert damping of the FM is typically considered as an\nadditive e\u000bect, where the total Gilbert damping can be\nseparated into an intrinsic bulk contribution and an inter-\nfacial component due to the presence of the NM.7,8While\nthe interfacial Gilbert damping is usually attributed to\nthe loss of angular momentum due to pumped spin cur-\nrent into the NM,9,10in metallic bulk FMs the intrin-\nsic Gilbert damping constant is described by the cou-\npling between the conduction electrons and the (time-\ndependent) magnetization degree of freedom.11\nThe conventional approach to determine the Gilbert\ndamping constant involves calculating the imaginary part\nof the time-dependent susceptibility of the FM in thepresence of conduction electrons in the linear response\nregime.12{14In this case, the time-dependent magneti-\nzation term in the electronic Hamiltonian leads to the\nexcitation of electrons close to the Fermi surface trans-\nferring angular momentum to the conduction electrons.\nThe excited electrons in turn relax to the ground state\nby interacting with their environment, namely through\nphonons, photons and/or collective spin/charge excita-\ntions. These interactions are typically parameterized\nphenomenologically by the broadening of the energy lev-\nels,\u0011=~=2\u001c, where\u001cis the relaxation time of the elec-\ntrons close to the Fermi surface. The phenomenological\ntreatment of the electronic relaxation is valid when the\nenergy broadening is small which corresponds to clean\nsystems, i.e., \u0011D(EF)\u00141, whereD(EF) is the den-\nsity of states per atom at the Fermi energy. In the case\nof large\u0011[\u0011D(EF)&1)] however, this approach vio-\nlates the conservation laws and a more accurate descrip-\ntion of the relaxation mechanism that preserves the en-\nergy, charge and angular momentum conservation laws\nare required.15The importance of including the vertex\ncorrections has already been pointed out in the literature\nwhen the Gilbert damping is dominated by the interband\ncontribution,16{18i.e.,when there is a signi\fcant number\nof states available within the energy window of \u0011around\nthe Fermi energy.\nIn this paper we investigate the magnetic damping phe-\nnomena through a di\u000berent Lens in which the FM is as-\nsumed to be small and quantum mechanical. We show\nthat in the limit of large magnetic moments we recover\ndi\u000berent conventional expressions for the Gilbert damp-\ning of a classical FM. We calculate the Gilbert damping\nfor a Pt/Co bilayer system versus the energy broaden-\ning,\u0011and show that in the limit of clean systems and\nsmall magnetic moments the FM damping is governed\nby a coherent dynamics. We show that in the limit of\nlarge broadening \u0011 > 1meV which is typically the case\nat room temperature, the relaxation time approximationarXiv:1709.04911v2  [cond-mat.mes-hall]  14 Nov 20172\nfails. Hence, we employ a self consistent approach pre-\nserving the conservation laws. We calculate the Gilbert\ndamping versus the Pt and Co thicknesses and by \ftting\nthe results to spin di\u000busion model we calculate the spin\ndi\u000busion length and spin mixing conductance of Pt.\nII. THEORETICAL FORMALISM OF\nMAGNETIZATION DAMPING\nFor a metallic FM the magnetization degree of freedom\nis inherently coupled to the electronic degrees of freedom\nof the conduction electrons. It is usually convenient to\ntreat each degree of freedom separately with the corre-\nsponding time-dependent Hamiltonians that do not con-\nserve the energy. However, since the total energy of the\nsystem is conserved, it is possible to consider the total\nHamiltonian of the combined system and solve the corre-\nsponding stationary equations of motion. For an isolated\nmetallic FM the wave function of the coupled electron-\nmagnetic moment con\fguration system is of the form,\njm\u000b~ki=jS;mi\nj\u000b~ki, where the parameter Sdenotes\nthe total spin of the nano-FM ( S! 1 in the classi-\ncal limit),m=\u0000S:::; +S, are the eigenvalues of the\ntotal Szof the nano-FM,\nrefers to the Kronecker prod-\nuct, and\u000bdenotes the atomic orbitals and spin of the\nelectron Bloch states. The single-quasi-particle retarded\nGreen function and the corresponding density matrix can\nbe obtained from,19\n\u0012\nE\u0000i\u0011\u0000^H~k\u0000HM\u00001\n2S^\u0001~k^~ \u001b\u0001~S\u0013\n^Gr\n~k(E) =^1;(1)\nand\n^\u001a~k=ZdE\n\u0019^Gr\n~k(E)\u0011f(E\u0000HM)^Ga\n~k(E): (2)\nHere,HM=\r~B\u0001~S, is the Hamiltonian of the nano-\nFM in the presence of an external magnetic \feld ~Bwith\neigenstates,jS;mi,\ris the gyromagnetic ratio, f(E) is\nthe Fermi-Dirac distribution function, ^~ \u001bis the vector of\nthe Pauli matrices, ^H~kis the non-spin-polarized Hamilto-\nnian matrix in the presence of spin orbit coupling (SOC),\nand^\u0001~kis the~k-dependent exchange splitting matrix, dis-\ncussed in detail in Sec. III. We employ the notation that\nbold symbols operate on jS;mibasis set and symbols\nwith hat operate on the j\u000b~kis. Here, for simplicity we\nignore explicitly writing the identity matrices ^1 and 1as\nwell as the Kronecker product symbol in the expressions.\nA schematic description of the FM-Bloch electron en-\ntangled system and the damping process of the nano-FM\nis shown in Fig. 1. The presence of the magnetic Hamil-\ntonian in the Fermi distribution function in Eq. (2) act-\ning as a chemical potential leads to transition between\nmagnetic statesjS;mialong the direction in which the\nmagnetic energy is minimized19. The transition rate of\nthe FM from the excited states, jS;mi, to states with\nFIG. 1: (Color online) Schematic representation of the com-\nbined FM-Bloch electron system. The horizontal planes de-\nnote the eigenstates, jS;miof the total Szof the nano-FM\nwith eigenvalues m=\u0000S;\u0000S+ 1;:::; +S. For more details\nsee Fig. 2 in Ref.19\n.\nlower energy ( i.e.the damping rate) can be calculated\nfrom19,\nTm=1\n2=(T\u0000\nm\u0000T+\nm); (3)\nwhere,\nT\u0006\nm=1\n2SNX\n~kTrel[^\u0001~k^\u001b\u0007S\u0006\nm^\u001a~k;m;m\u00061]: (4)\nHere,Nis the number of ~k-points in the \frst Brillouin\nzone,Trel, is the trace over the Bloch electron degrees\nof freedom,S\u0006\nm=p\nS(S+ 1)\u0000m(m\u00061), and ^\u001b\u0007\u0011\n^\u001bx\u0007i^\u001by.\nThe precessional Gilbert damping constant can be de-\ntermined from conservation of the total angular mo-\nmentum by equating the change of angular momen-\ntum per unit cell for the Bloch electrons, Tm, and\nthe magnetic moment obtained from LLG equation,\n\u000bGDMtotsin2(\u0012)=2, which leads to,\n\u000bGD(m) =\u00002\nMtot!sin2(\u0012m)Tm\n\u0011\u0000S2\nMtot!(S(S+ 1)\u0000m2)Tm: (5)\nHere, cos(\u0012m) =mp\nS(S+1), is the cone angle of precession\nandMtotis the total magnetic moment per unit cell in\nunits of1\n2g\u0016Bwithgand\u0016Bbeing the Land\u0013 e factor and\nmagneton Bohr respectively. The Larmor frequency, !,\ncan be obtained from the e\u000bective magnetic \feld along\nthe precession axis, ~!=\rBz.\nThe exact treatment of the magnetic degree of freedom\nwithin the single domain dynamical regime o\u000bers a more\naccurate description of the damping phenomena that can\nbe used even when the classical equation of motion LLG\nis not applicable. However, since in most cases of in-\nterest the FM behaves as a classical magnetic moment,\nwhere the adiabatic approximation can be employed to\ndescribe the magnetization dynamics, in the following\ntwo sections we consider the S!1 limit and close to\nadiabatic regime for the FM dynamics.3\nA. Classical Regime: Relaxation Time\nApproximation\nThe dissipative component of the nonequilibrium elec-\ntronic density matrix, to lowest order in @=@t, can be\ndetermined by expanding the Fermi-Dirac distribution\nin Eq. (2) to lowest order in [ HM]mm0=\u000emm0m~!.\nPerforming a Fourier transformation with respect to the\ndiscrete Larmor frequency modes, m!\u0011i@=@t , we \fnd\nthat, ^\u001adis\nneq(t) =1\n\u0019~\u0011^Gri@^Ga=@t, where ^Gr=\u0002\nEF\u0000i\u0011\u0000\n^H(t)\u0003\u00001and ^Ga= (^Gr)yare the retarded and advanced\nGreen functions calculated at the Fermi energy, EF, and\na \fxed time t.\nThe energy absorption rate of the electrons can\nbe determined from the expectation value of the\ntime derivative of the electronic Hamiltonian, E0\ne=\n<(Tr(^\u001adis\nneq(t)@^H=@t )), where<() refers to the real part.\nCalculating the time-derivative of the Green function and\nusing the identity, \u0011^Gr^Ga=\u0011^Ga^Gr==(^Gr), where,=()\nrefers to the anti-Hermitian part of the matrix, the torque\ncorrelation (TC) expression for the energy excitation rate\nof the electrons is of the form,\nE0\ne=~\n\u0019NX\nkTrh\n=(^Gr)@^H\n@t=(^Gr)@^H\n@ti\n: (6)\nIn the case of semi-in\fnite NM leads attached to the FM,\nusing,=(^Gr) =^Gr^\u0000^Ga=^Ga^\u0000^Gr, Eq.(6) can be written\nas\nE0\ne=~\n\u0019NX\nkTrh\n^\u0000@^Gr\n@t^\u0000@^Ga\n@ti\n(7)\nwhere, ^\u0000 =\u0011^1 + ( ^\u0006r\u0000^\u0006a)=2i, with ^\u0006r=abeing the\nretarded=advanced self energy due to the NM lead at-\ntached to the FM which describes the escape rate of\nelectrons from/to the reservoir. It is useful to separate\nthe dissipation phenomena into local andnonlocal compo-\nnents as follows. Applying the unitary operator, ^U(t) =\nei!^\u001bzt=2ei\u0012^\u001bx=2e\u0000i!^\u001bzt=2= cos(\u0012\n2)^1 +isin(\u0012\n2)(^\u001b+ei!t+\n^\u001b\u0000e\u0000i!t), to \fx the magnetization orientation along z\nwe \fnd,\n@(^U^Gr\n0^Uy)\n@t\u0019!\n2sin(\u0012)\u0010\n^G0ei!t+^G0ye\u0000i!t\u0011\n;(8)\nwhere we have ignored higher order terms in \u0012and,\n^G0= [^Gr\n0;^\u001b+]\u0000^Gr\n0[^H0;^\u001b+]^Gr\n0: (9)\nHere, [;] refers to the commutation relation, ^H0is the\ntime independent terms of the Hamiltonian, and ^Gr=a\n0\nrefers to the Green function corresponding to magnetiza-\ntion alongz-axis. Using Eq. (7) for the average energyabsorption rate we obtain,\nE0\ne=~!2\n2\u0019Nsin2(\u0012)X\nkTr\u0010\n^\u0000^G0^\u0000^G0y\u0011\n=\u0000~!2\n2\u0019Nsin2(\u0012)X\nk<\u0010\nTr\u0010\n^\u0000[^Gr\n0;^\u001b+]^\u0000[^Ga\n0;^\u001b\u0000]\n+=(^Gr)[^H0;^\u001b+]=(^Gr)[^H0;^\u001b\u0000]\n\u00002 [=(^Gr\n0);^\u001b+]^\u0000^Ga\n0[^H0;^\u001b\u0000]\u0011\u0011\n: (10)\nIn the absence of the SOC, the \frst term in Eq.\n(10) is the only non-vanishing term which corresponds\nto the pumped spin current into the reservoir [i.e.\nISz=~Tr(^\u001bz^\u0000^\u001adis\nneq)=2] dissipated in the NM (no back\n\row). This spin pumping component is conventionally\nformulated in terms of the spin mixing conductance20,\nISz=~g\"#sin2(\u0012)=4\u0019, which acts as a nonlocal dissi-\npation mechanism. The second term, referred to as the\nspin-orbital torque correlation11,21(SOTC) expression for\ndamping, is commonly used to calculate the intrinsic con-\ntribution to the Gilbert damping constant for bulk metal-\nlic FMs. The third term arises when both SOC and the\nreservoir are present. It is important to note that the\nformalism presented above is valid only in the limit of\nsmall\u0011(ballistic regime). On the other hand, in the case\nof large\u0011, typical in experiments at room temperature,\nthe results may not be reliable due to the fact that in\nthe absence of metallic leads a \fnite \u0011acts as a \fctitious\nreservoir that yields a nonzero dissipation of spin cur-\nrent even in the absence of SOC. A simple approach to\nrectify the problem is to ignore the e\u000bect of \fnite \u0011in\nthe spin pumping term in calculating the Gilbert damp-\ning constant. A more accurate approach is to employ\na dephasing mechanism that preserves the conservation\nlaws, which we refer it to as conserving torque correlation\napproach discussed in the following subsection.\nB. Classical Regime: Conserving Dephasing\nMechanism\nRather than using the broadening parameter, \u0011, as a\nphenomenological parameter, we determine the self en-\nergy of the Bloch electrons interacting with a dephas-\ning bath associated with phonons, disorder, etc. using a\nself-consistent Green function approach22. Assuming a\nmomentum-relaxing self energy given by,\n^\u0006r=a\nint(E;t) =1\nNX\nk^\u0015k^Gr=a\nk(E;t)^\u0015y\nk; (11)\nwhere ^\u0015kis the interaction coupling matrix, the dressed\nGreen function, ^Gr=a\nk(E;t) , and corresponding self en-\nergy, ^\u0006r=a\nint(E;t), are calculated self-consistently. This\nwill in turn yield a renormalized broadening matrix,\n^\u0000int==(^\u0006r\nint), which is the vertex correction modi\f-\ncation of the in\fnitesimal initial broadening \u00110.4\nThe nonequilibrium density matrix is calculated from\n^\u001adis\nneq(k;t) =~\n\u0019^Gr\nk^\u0000int^Ga\nk\u0010@^Hk(t)\n@t+^Saa\nt\u0011\n^Ga\nk;(12)\nwhere the time derivative vertex correction term is\n^Saa\nt=1\nNX\nk^\u0015k^Ga\nk\u0010@^Hk(t)\n@t+^Saa\nt\u0011\n^Ga\nk^\u0015y\nk: (13)\nThe energy excitation rate for the Bloch electrons then\nreads,\nE0\ne=~\n\u0019NX\nk<h\nTr\u0010\u0010@^Hk(t)\n@t+^Sar\nt\u0011\n^\u001adis\nneq(k;t)\u0011i\n;(14)\nwhere\n^Sar\nt=1\nNX\nk^\u0015k^Ga\nk\u0010@^Hk(t)\n@t+^Sar\nt\u0011\n^Gr\nk^\u0015y\nk: (15)\nThe vertex correction Eqs. (13) and (15) can be solved\neither exactly by transforming them into a system of lin-\near equations or by solving them self consistently. Due to\nthe large number of orbitals and atoms per unit cell for\nthe Co/Pt bilayer the latter approach is computationally\nmore e\u000ecient. In the following numerical calculations we\nassume ^\u0015k=\u0015int^1 to be a constant independent of kand\nof orbitals, which can be viewed as the root mean square\nvalue of a random on-site potential, \u0015int=p\nhV2\nrandi,\nwhereh:::idenotes an ensemble averaging, where the self\nenergy in Eq. (11) corresponds to the self consistent Born\napproximation.\nC. Gilbert Damping Calculation\nHaving determined the energy absorption rate of the\nBloch electrons due to the precessing FM, from conserva-\ntion of energy one can deduce the energy dissipation rate\nof the FM from, E0\nM=\u0000E0\ne. Using the LLG equation\nof motion the energy dissipation rate per unit cell of the\nprecessing FM can be obtained from\nE0\nM=1\n2Mtot~!@mz\n@t=\u00001\n2\u000bGDMtot~!2sin2(\u0012);(16)\nwhere~ mis the unit vector along the magnetization of\nthe FM. The Gilbert damping parameter can then be\nobtained from\n\u000bGD=2E0\ne\nMtot~!2sin2(\u0012): (17)\nIII. COMPUTATIONAL SCHEME\nThe spin-polarized density functional theory calcula-\ntions for the hcp Co(0001)/fcc Pt(111) bilayer were car-\nried out using the Vienna ab initio simulation package(VASP)23,24. The pseudopotential and wave functions\nare treated within the projector-augmented wave (PAW)\nmethod25,26. Structural relaxations were carried using\nthe generalized gradient approximation as parameter-\nized by Perdew et al.27when the largest atomic force\nis smaller than 0.01 eV/ \u0017A. The plane wave cuto\u000b energy\nis 500 eV and a 14 \u000214\u00021kpoint mesh is used in the\n2D BZ sampling. The Pt( m)/Co(n) bilayer is modeled\nemploying the slab supercell approach along the [111]\nconsisting of mfcc Pt monolayers (MLs) (ABC stacking)\n(m=1, 2,:::, 6),nhcp Co MLs (AB stacking) Co ( n= 6),\nand a 25 \u0017A thick vacuum region separating the periodic\nslabs. The in-plane lattice constant of the hexagonal unit\ncell was set to the experimental value of 2.505 \u0017A for bulk\nCo.\nThe Gilbert damping constant was calculated us-\ning the tight-binding parameters obtained from VASP-\nWannier90 calculations28with a 250\u0002250k-mesh for\nthe bilayer and 250 \u0002250\u0002250k-mesh for bulk Co. The\nelectron Hamiltonian, ^H~k, and exchange splitting, ^\u0001~k,\nmatrices in Eq. (1) in the Wannier basis have the form\n^H~k=^HSOC+1\n2X\n~ n1\nD~ n(^H\"\"\n~ n+^H##\n~ n)e2i\u0019~ n\u0001~k(18)\n^\u0001~k=1\n2X\n~ n1\nD~ n(^H\"\"\n~ n\u0000^H##\n~ n)e2i\u0019~ n\u0001~k; (19)\nwhere ^HSOC is the SOC Hamiltonian matrix, ^H\"\"\n~ nand\n^H##\n~ nare the spin-majority and spin-minority matrices,\n~ n= (n1;n2;n3) are integers denoting the lattice vectors,\nD~ nis the degeneracy of the Wigner-Seitz grid point, and\nki2[0;1].\nThe ^H\"\"\n~ nand ^H##\n~ nare determined from spin-polarized\nVASP-Wannier90 calculations without SOC. On the\nother hand, ^HSOC, is determined from VASP-Wannier90\nnon-spin-polarized calculations with SOC as the follow-\ning.\nUsing the identity, Tr[^Li^Lj] =\u000eijl(l+1)(2l+1)\n3, where\n^Liis the angular momentum operator of orbital land\ni;j=x;y;z , the SOC strength of the Ith atom can be\ncalculated from\n\u0018I\nl=3Tr[^HP\nll;II^Li^\u001bi]\nl(l+ 1)(2l+ 1): (20)\nHere, the superscript Pdenotes the paramagnetic Hamil-\ntonian,IIare the on-site Hamiltonian matrix elements\nfor atomIat~ n= (0;0;0), andllis the block Hamiltonian\nmatrix corresponding to orbital l. The result is indepen-\ndent of the direction of the angular momentum operator.\nWe \fnd that \u0018Pt\nd=0.5 eV and \u0018Co\nd=70 meV for the d-\norbitals of Pt and Co, respectively, that are somewhat\nsmaller than the values considered in the literature7(i.e.\n\u0018Pt\nd=0.65 eV,\u0018Co\nd=85 meV). The SOC Hamiltonian can\nin turn be written as,\nhI;lmsj^HSOCjI0;l0m0s0i=1\n2\u000ell0\u000eII0\u0018I\nlX\nihlmj^Lijlm0i^\u001bi\nss0:\n(21)5\n0 45 90 135 1800.0280.02850.0290.02950.030.03050.0310.03150.032\nCone Angle (deg)Gilbert Damping\nStudent Version of MATLAB\nFIG. 2: (Color online). Gilbert damping versus precessional\ncone angle calculated from Eq. (5) for Pt(1 ML)/Co(6 ML)\nbilayer system for S=60 and\u0011=1 meV, respectively.\nIV. RESULTS AND DISCUSSION\nThe Gilbert damping constant (calculated from\nEq. (5))versus the precessional cone angle, \u0012m=\ncos\u00001\u0010\nmp\nS(S+1)\u0011\n, for the Pt(1 ML)/Co(6 ML) bilayer\nis shown in Fig. 2 with \u0011=1 meV and S=60. We \fnd\nthat the Gilbert damping is relatively independent of the\ncone angle. The small angular dependence of the Gilbert\ndamping is material dependent and it could increase or\ndecrease upon increasing the cone angle, depending on\nthe material.\nIn order to see the transition from quantum mechanical\nto classical dynamical regimes, in Fig. 3 we present the\nGilbert damping constant of the Pt(1 ML)/Co(6 ML)\nbilayer versus the broadening parameter, \u0011, for di\u000ber-\nent values of the total spin Sof the FM. For the case\nof \fniteSwe used Eq. (5) while for S=1we used\nthe TC expression Eq. (6). We \fnd that for \fnite S\nthe Gilbert damping value exhibits a peak in the small \u0011\nregime (clean system) where the peak value increases lin-\nearly withSand shifts to smaller broadening value with\nincreasingS. The underlying origin of the \u000bGD(\u0011) behav-\nior withSin coherent regime can be understood in terms\nof the coherent transport of quasi-particles along the aux-\niliary direction min Fig. 1, where the auxiliary current\n\row (damping rate) depends linearly on the chemical po-\ntential di\u000berence between the \frst ( m= +S) and last\nlayers (m=\u0000S) which is simply 2 S. This suggests that\nin the limit of in\fnite Sand ballistic regime \u0011!0 the\nintrinsic Gilbert damping diverges. It was shown that\nthe problem of in\fnite Gilbert damping in the ballistic\nregime can be removed by taking into account the collec-\ntive excitations.29,30\nAs we discussed in Sec. II A, the relaxation time ap-\nproximation is valid only in the small \u0011limit and it vi-\nolates the conservation law when \u0011is large. In order\n10−410−210010−310−210−1100\nBroadening, η, (eV)Gilbert Damping\n  \nS=10S=30S=60S=∞\nStudent Version of MATLABFIG. 3: (Color online) Gilbert damping versus broadening\nparameter for the Pt(1 ML)/Co(6 ML) bilayer for di\u000berent\nvalues of the total spin Sof the FM nanocluster. Eqs. (5)\nand (6) were used to calculate the Gilbert damping for \fnite\nand in\fnite S, respectively.\nto quantify the validity of the relaxation time approxi-\nmation, in Fig. 4 we display the Gilbert damping ver-\nsus broadening, \u0011;or interaction parameter, \u0015int, for\nthe Pt(1 ML)/Co(6 ML) bilayer with S=1, using\n(i)torque correlation(TC) expression (Eq. (6)); (ii)the\nspin-orbital torque correlation (SOTC) expression (sec-\nond term in Eq. (10);) and (iii)the conserving TC expres-\nsion (Eq. (14)). The upper horizontal-axis refers to the\ninteraction strength \u0015intof the conserving TC method\nand the lower one refers to the broadening parameter,\n\u0011. The calculations show that for \u0011 >20 meV the TC\nresults deviate substantially from those of the conserv-\ning TC method. Ignoring the spin pumping contribu-\ntion to the Gilbert damping in Eq. (10) and considering\nonly the SOTC component increases the range of the va-\nlidity of the relaxation time approximation. Therefore,\nthe overestimation of the Gilbert damping using the TC\nmethod can be attributed to the disappearance of elec-\ntrons (pumped spin current) in the presence of the \fnite\nnon-Hermitian term, i\u0011^1, in the Hamiltonian.\nWe have used the conserving TC approach to calculate\nthe e\u000bect of \u0015inton the Gilbert damping as a function of\nthe Pt layer thickness for the Pt( m)/Co(6 ML) bilayer.\nAs an example, we display in Fig. 5 the results of Gilbert\ndamping versus Pt thickness for \u0015int= 1eVwhich yields\na Gilbert damping value of 0.005 for bulk Co ( m= 0 ML)\nand is in the range of 0.00531,32to 0.01133{35reported\nexperimentally. Note that this large \u0015intvalue describes\nthe Gilbert damping in the resistivity-like regime which\nmight not be appropriate to experiment, where the bulk\nGilbert damping decreases with temperature, suggesting\nthat it is in the conductivity regime.36\nFor a given \u0015intwe \ftted the ab initio calculated\nGilbert damping versus Pt thickness to the spin di\u000bu-6\n10−410−210010−310−210−1100\nBroadening, η, (eV)Gilbert Damping10−1100Interaction Strength, λint, (eV)\nConserving TC MethodSOTC MethodTC Method\nStudent Version of MATLAB\nFIG. 4: (Color online). Gilbert damping of Pt(1 ML)/Co(6\nML) bilayer versus the broadening parameter \u0011(lower ab-\nscissa) and interaction strength, \u0015int, (upper abscissa), using\nthe torque correlation (TC), spin-orbital torque correlation\n(SOTC), and conserving TC expressions given by Eqs. (6),\n(10) and (14), respectively.\nsion model,37{39\n\u000bPt=Co =\u000bCo+ge\u000b\n\"#VCo\n2\u0019MCodCo(1\u0000e\u00002dPt=Lsf\nPt):(22)\nHere,ge\u000b\n\"#is the e\u000bective spin mixing conductance, dCo\n(dPt) is the thickness of Co (Pt), VCo= 10:5\u0017A3\n(MCo= 1:6\u0016B) is the volume (magnetic moment) per\natom in bulk Co, and Lsf\nPtis the spin di\u000busion length\nof Pt. The inset of Fig. 5 shows the variation of the ef-\nfective spin mixing conductance and spin di\u000busion length\nwith the interaction strength \u0015int. In the di\u000busive regime\n\u0015int>0:2eV,Lsf\nPtranges between 1 to 6 nm in agree-\nment with experiment \fndings which are between 0.5 and\n10 nm33,40. Moreover, the e\u000bective spin mixing conduc-\ntance is relatively independent of \u0015intoscillating around\n20 nm\u00002, which is approximately half of the experimen-\ntal value of\u001935 - 40 nm\u00002.33,41On the other hand,\nin the ballistic regime ( \u0015int<0.2 eV), although the er-\nrorbar in \ftting to the di\u000busion model is relatively large,\nthe value of Lsf\nPt\u00190.5 nm is in agreement with Ref.7and\nexperimental observation40.\nV. CONCLUDING REMARKS\nWe have developed an ab initio -based electronic struc-\nture framework to study the magnetization dynamics ofa nano-FM where its magnetization is treated quantum\nmechanically. The formalism was applied to investigate\nthe intrinsic Gilbert damping of a Co/Pt bilayer as a\n0 1 2 300.0050.010.0150.02\nPt Thickness, dPt (nm)Gilbert Damping\n  \n10−210−110002468\nInteraction Strength, λint (eV)Spin Diffusion Length (nm)100101102103\ng↑↓eff (nm−2)\nStudent Version of MATLAB\nFIG. 5: (Color online). Ab initio values (circles) of Gilbert\ndamping versus Pt thickness for Pt( mML)/Co(6 ML) bilayer\nwheremranges between 0 and 6 and \u0015int= 1eV. The dashed\ncurve is the \ft of the Gilbert damping values to Eq. (22).\nInset: spin di\u000busion length (left ordinate) and e\u000bective spin\nmixing conductance, ge\u000b\n\"#, (right ordinate) versus interaction\nstrength. The errorbar for ge\u000b\n\"#is equal to the root mean\nsquare deviation of the damping data from the \ftted curve.\nfunction of energy broadening. We showed that in the\nlimit of small Sand ballistic regime the FM damping is\ngoverned by coherent dynamics, where the Gilbert damp-\ning is proportional to S. In order to study the e\u000bect of\ndisorder on the Gilbert damping we used a relaxation\nscheme within the self-consistent Born approximation.\nTheab initio calculated Gilbert damping as a function of\nPt thickness were \ftted to the spin di\u000busion model for a\nwide range of disorder strength. In the limit of large dis-\norder strength the calculated spin di\u000busion length and\ne\u000bective spin mixing conductance are in relative agree-\nment with experimental observations.\nAcknowledgments\nThe work is supported by NSF ERC-Translational Ap-\nplications of Nanoscale Multiferroic Systems (TANMS)-\nGrant No. 1160504 and by NSF-Partnership in Research\nand Education in Materials (PREM) Grant No. DMR-\n1205734.7\n\u0003Electronic address: Farzad.Mahfouzi@gmail.com\nyElectronic address: nick.Kioussis@csun.edu\n1Ioan Mihai Miron, Gilles Gaudin, Stphane Au\u000bret,\nBernard Rodmacq, Alain Schuhl, Stefania Pizzini, Jan Vo-\ngel and Pietro Gambardella, Current-driven spin torque in-\nduced by the Rashba e\u000bect in a ferromagnetic metal layer,\nNat. Mater. 9, 230234 (2010).\n2Ioan Mihai Miron, Kevin Garello, Gilles Gaudin, Pierre-\nJean Zermatten, Marius V. Costache, Stphane Au\u000bret,\nSbastien Bandiera, Bernard Rodmacq, Alain Schuhl and\nPietro Gambardella, Perpendicular switching of a single\nferromagnetic layer induced by in-plane current injection,\nNature 476, 189193 (2011).\n3Luqiao Liu, O. J. Lee, T. J. Gudmundsen, D. C. Ralph,\nand R. A. Buhrman, Current-Induced Switching of Perpen-\ndicularly Magnetized Magnetic Layers Using Spin Torque\nfrom the Spin Hall E\u000bect, Phys. Rev. Lett. 109, 096602\n(2012).\n4L. Liu, C. F. Pai, Y. Li, H. W. Tseng, D. C. Ralph, and R.\nA. Buhrman, Spin-Torque Switching with the Giant Spin\nHall E\u000bect of Tantalum, Science 336, 555 (2012).\n5Igor\u0014Zuti\u0013 c, Jaroslav Fabian, and S. Das Sarma, Spintronics:\nFundamentals and applications, Rev. Mod. Phys. 76, 323\n(2004).\n6T. L. Gilbert, A phenomenological theory of damping\nin ferromagnetic materials, IEEE Trans. Mag. 40, 3443\n(2004).\n7E. Barati, M. Cinal, D. M. Edwards, and A. Umerski,\nGilbert damping in magnetic layered systems, Phys. Rev.\nB90, 014420 (2014).\n8D. L. Mills, Emission of spin waves by a magnetic multi-\nlayer traversed by a current, Phys. Rev. B 54, 9353 (1996).\n9Yaroslav Tserkovnyak, Arne Brataas, and Gerrit E. W.\nBauer, Enhanced Gilbert Damping in Thin Ferromagnetic\nFilms, Phys. Rev. Lett. 88, 117601 (2002)\n10Maciej Zwierzycki, Yaroslav Tserkovnyak, Paul J. Kelly,\nArne Brataas, and Gerrit E. W. Bauer, First-principles\nstudy of magnetization relaxation enhancement and spin-\ntransfer in thin magnetic \flms, Phys. Rev. B 71, 064420\n(2005)\n11V. Kambersky, Spin-orbital Gilbert damping in common\nmagnetic metals, Phys. Rev. B 76, 134416 (2007).\n12L. Berger, Ferromagnetic resonance relaxation in ultrathin\nmetal \flms: The role of the conduction electrons, Phys.\nRev. B 68, 014419 (2003).\n13E. Simanek, B. Heinrich, Gilbert damping in magnetic\nmultilayers, Phys. Rev. B 68, 014419 (2003).\n14Ion Garate, K. Gilmore, M. D. Stiles, and A. H. MacDon-\nald, Nonadiabatic spin-transfer torque in real materials,\nPhys. Rev. B 79, 104416 (2009).\n15Gordon Baym and Leo P. Kadano\u000b, Conservation Laws\nand Correlation Functions, Phys. Rev. 124, 287 (1961).\n16Ion Garate and Allan MacDonald, Gilbert damping in\nconducting ferromagnets. II. Model tests of the torque-\ncorrelation formula, Phys. Rev. B 79, 064404 (2009).\n17S. Mankovsky, D. Kdderitzsch, G. Woltersdorf, and H.\nEbert, First-principles calculation of the Gilbert damping\nparameter via the linear response formalism with applica-\ntion to magnetic transition metals and alloys, Phys. Rev.\nB87, 014430 (2013).\n18I. Turek, J. Kudrnovsky and V. Drchal, Nonlocal torqueoperators in ab initio theory of the Gilbert damping in ran-\ndom ferromagnetic alloys, Phys. Rev. B 92, 214407 (2015).\n19Farzad Mahfouzi and Nicholas Kioussis, Current-induced\ndamping of nanosized quantum moments in the presence\nof spin-orbit interaction, Phys. Rev. B 95, 184417 (2017).\n20Y. Tserkovnyak, A. Brataas, and B. I. Halperin, Nonlocal\nmagnetization dynamics in ferromagnetic heterostructures,\nRev. Mod. Phys. 77, 1375 (2005).\n21V. Kambersky, Can. J. Phys. On Ferromagnetic Resonance\nDamping in Metals, 48, 2906 (1970).\n22Roksana Golizadeh-Mojarad and Supriyo Datta, Nonequi-\nlibrium Greens function based models for dephasing in\nquantum transport, Phys. Rev. B 75, 081301(R) (2007).\n23G. Kresse and J. Furthm uller, E\u000ecient iterative schemes\nfor ab initio total-energy calculations using a plane-wave\nbasis set, Phys. Rev. B 54, 11169 (1996).\n24G. Kresse and J. Furthm uller, E\u000eciency of ab-initio total\nenergy calculations for metals and semiconductors using a\nplane-wave basis set, Comput. Mater. Sci. 6, 15 (1996).\n25P. E. Bl ochl, Projector augmented-wave method, Phys.\nRev. B 50, 17953 (1994).\n26G. Kresse and D. Joubert, From ultrasoft pseudopotentials\nto the projector augmented-wave method, Phys. Rev. B\n59, 1758 (1999).\n27J. P. Perdew, K. Burke, and M. Ernzerhof, Generalized\nGradient Approximation Made Simple, Phys. Rev. Lett.\n77, 3865 (1996).\n28A. A. Mosto\f, J. R. Yates, G. Pizzi, Y.-S. Lee, I. Souza, D.\nVanderbilt, and N. Marzari, An updated version of wan-\nnier90: A tool for obtaining maximally-localised Wannier\nfunctions, Comput. Phys. Commun. 185, 2309 (2014).\n29A. T. Costa and R. B. Muniz, Breakdown of the adiabatic\napproach for magnetization damping in metallic ferromag-\nnets, Phys. Rev B 92, 014419 (2015).\n30D. M. Edwards, The absence of intraband scattering in\na consistent theory of Gilbert damping in pure metallic\nferromagnets, Journal of Physics: Condensed Matter, 28,\n8 (2016).\n31S. M. Bhagat and P. Lubitz, Temperature variation of fer-\nromagnetic relaxation in 3D transition metals, Phys. Rev.\nB,10, 179185 (1974).\n32T. Kato, Y. Matsumoto, S. Kashima, S. Okamoto, N.\nKikuchi, S. Iwata, O. Kitakami and S. Tsunashima, Per-\npendicular Anisotropy and Gilbert Damping in Sputtered\nCo/Pd Multilayers, IEEE Transactions on Magnetics, 48,\nNO. 11, 3288 (2012).\n33J.-C. Rojas-S\u0013 anchez, N. Reyren, P. Laczkowski, W. Savero,\nJ.-P. Attan\u0013 e, C. Deranlot, M. Jamet, J.-M. George, L. Vila,\nand H. Ja\u000br\u0013 es, Spin Pumping and Inverse Spin Hall E\u000bect\nin Platinum: The Essential Role of Spin-Memory Loss at\nMetallic Interfaces, Phys. Rev. Let. 112, 106602 (2014).\n34Nam-Hui Kim, Jinyong Jung, Jaehun Cho, Dong-Soo Han,\nYuxiang Yin, June-Seo Kim, Henk J. M. Swagten, and\nChun-Yeol You, Interfacial Dzyaloshinskii-Moriya inter-\naction, surface anisotropy energy, and spin pumping at\nspin orbit coupled Ir/Co interface, Appl. Phys. Lett. 108,\n142406 (2016).\n35S. Pal, B. Rana, O. Hellwig, T. Thomson, and A. Barman,\nTunable magnonic frequency and damping in [Co/Pd] 8\nmultilayers with variable Co layer thickness, Applied\nPhysics Letters 98, 082501 (2011).8\n36Miren Isasa, Estitxu Villamor, Luis E. Hueso, Martin\nGradhand, and Felix Casanova, Temperature dependence\nof spin di\u000busion length and spin Hall angle in Au and Pt,\nPhys. Rev. B 91, 024402 (2015).\n37J. M. Shaw, H. T. Nembach, and T. J. Silva, Determina-\ntion of spin pumping as a source of linewidth in sputtered\nmultilayers by use of broadband ferromagnetic resonance\nspectroscopy, Phys. Rev. B., 85, 054412 (2012).\n38J. Foros, G. Woltersdorf, B. Heinrich, and A. Brataas,\nScattering of spin current injected in Pd(001), J. Appl.\nPhys., 97, 10A714 (2005).\n39A. Ghosh, S. Au\u000bret, U. Ebels, and W. E. Bailey, Pen-\netration Depth of Transverse Spin Current in UltrathinFerromagnets, Phys. Rev. Let. 109, 127202 (2012).\n40C. T. Boone, Hans T. Nembach, Justin M. Shaw, and T.\nJ. Silva, Spin transport parameters in metallic multilay-\ners determined by ferromagnetic resonance measurements\nof spin-pumping, Journal of Applied Physics 113, 153906\n(2013).\n41S. Azzawi, A. Ganguly, M. Tokac, R. M. Rowan-Robinson,\nJ. Sinha, A. T. Hindmarch, A. Barman, and D. Atkinson,\nEvolution of damping in ferromagnetic/nonmagnetic thin\n\flm bilayers as a function of nonmagnetic layer thickness,\nPhys. Rev. B 93, 054402 (2016)."    },    {        "title": "1709.07483v1.Low_Gilbert_Damping_Constant_in_Perpendicularly_Magnetized_W_CoFeB_MgO_Films_with_High_Thermal_Stability.pdf",        "content": "1 \n Low Gilbert Damping Constant in Perpendicularly Magnetized \nW/CoFeB/MgO Films with High Thermal Stability  \nDustin M. Lattery1†, Delin Zhang2†, Jie Zhu1, Jian-Ping Wang2*, and Xiao jia Wang1* \n1Department of Mechanical  Engineering, University of Minnesota, Minneapolis, MN 55455, USA  \n2Department of Electrical and Computer Engineering , University of Minnesota, Minneapolis, MN \n55455, USA   \n†These authors contributed equally to this work.  \n*Corres ponding a uthor s: wang4940@umn.edu  & jpwang@umn.edu   \n \nAbstract:  Perpendicular magnetic materials with low damping constant and high thermal stability \nhave great potential for realizing high -density, non -volat ile, and low -power consumption \nspintronic devices, which can  sustain operation reliability for high processing temperatures. In this \nwork, we study the Gilbert damping constant ( α) of perpendicularly magnetized W/CoFeB/MgO \nfilms with a high perpendicular magnetic anisotropy (PMA) and superb thermal stability. The α of \nthese PMA films annealed at different temperatures ( Tann) is determined via an all -optical T ime-\nResolved Magneto -Optical Kerr Effect method. We find that α of these W/CoFeB/MgO PMA \nfilms decreases with increasing Tann, reaches a minimum of α = 0.016 at Tann = 350 °C, and then \nincreases to 0.024 after post -annealing at 400 °C. The minimum α observed at 350 °C is \nrationalized by two competing effects  as Tann becomes higher : the enhanced crystallization of \nCoFeB and dead -layer growth occurring at the two interfaces of the CoFeB layer. We further \ndemonstrate that α of the 400 °C -annealed W/CoFeB/MgO film is comparable to that of a reference \nTa/CoFeB/MgO PMA fil m annealed at 300 °C , justif ying the enhanced thermal stability of the W -\nseeded CoFeB films.  2 \n I. INTRODUCTION  \nSince the first demonstration of perpendicular magnetic tunnel junctions with \nperpendicular magnetic anisotropy (PMA) Ta/CoFeB/MgO stacks  [1], interfacial PMA materials \nhave been extensively studied as promising candidates for ultra -high-density and low -power \nconsumption spintronic devices, including spin -transfer -torque magnetic rando m access memory \n(STT -MRAM)  [2,3] , electrical -field induced magnetization switching  [4-6], and spin -orbit torque \n(SOT) devices  [7-9]. An interfacial PMA stack typically consists of a thin ferromagnetic layer \n(e.g., CoFeB) sandwiched between a heavy metal layer ( e.g., Ta) and an oxide layer ( e.g., MgO). \nThe heavy metal layer interface  with the ferromagnetic layer  is responsible for the spin Hall effect , \nwhich is  favorable  for SOT and skyrmion devices  [10,11] . The critical switching current ( Jc0) \nshould be minimized to decrease the power consumption  of perpendicular STT -MRAM and SOT \ndevices . Reducing  Jc0 requires  the exploration of  new materials with low Gilbert damping  constant \n(α), large spin Hall angle ( θSHE), and large effective anisotropy ( Keff) [12,13] .  \nIn addition, spintronic devices need to sustain operation reliability for processing \ntemperatures as high as 400 °C  for their integration with existing CMOS fabrication technologies,  \nproviding the standard  back -end-of-line process compatibility  [14]. Based on this requirement, the \nmagnetic properties of a PMA material shou ld be thermally stable at annealing temperature s (Tann) \nup to  400 °C. Unfortunately, Ta/CoFeB/MgO  PMA films commonly used in spintronic devices \ncannot survive with Tann higher than 350 °C, due to Ta diffusion or CoFeB oxidation at the \ninterfaces  [15,16] . The diffusion of Ta atoms can act as scattering  sites to increase the spin -flip \nprobability  [17] and lead to a higher Gilbert Damping constant ( α), a measure of the energy \ndissipation from the magnetic precession into phonons or magnons  [18].  3 \n Modifying the composition of thin-film stack s can prevent heavy metal diffusion , which  \nis beneficial to both lowering  α and improving thermal stability  [19]. Along this line , new  \ninterfacial PMA stacks have been developed, such as  Mo/CoFeB/MgO, to circumvent the \nlimitation on device processing temperatures  [20,21] . While Mo/CoFeB/MgO films can indeed \nexhibit PMA at temperatures higher than 400 °C , they cannot be used for SOT devices due to the \nweak spin Hall effect of the Mo layer  [20,21] . Recently, W/CoFeB/MgO PMA thin films have \nbeen proposed because of their PMA property at high post -annealing temperature  [22], and  the \nlarge spin Hall angle of the W laye r (θSHE  ≈ 0.30)  [23], which is twice that of a Ta layer \n(θSHE  ≈ 0.12 ~ 0.15)  [9]. While  there have been a few scattered studies demonstrating the promise \nof fabricating SOT devices using the W/CoFeB/MgO stacks, special attention  has been given to \ntheir PMA properties and functionalities as SOT devic es [24]. A systematic investigation is lacking \non the effect of Tann on α of W/CoFeB/MgO PMA thin films.  \n \nII. SAMPLE PREPERATION AND MAGNETIC CHARACTERIZATION  \nIn this work, we grow a series of W(7)/Co 20Fe60B20(1.2)/MgO(2)/Ta(3) thin films on \nSi/SiO 2(300) substrates (thickness in nanometers) with a magnetron sputtering system \n(<5×1 08 Torr). These films are then post -annealed at varying temperatures (Tann = 250 ~ 400  °C) \nwithin a high -vacuum furnace (<1×106 Torr)  and their magnetic properties and damping constants \nas a function of Tann are systematically investigated . For comparison, a reference sample of \nTa(7)/Co 20Fe60B20(1.2)/MgO(2)/Ta(3) is also prepared to examine the effect of seeding layer to \nthe damping constant of these PMA films. The saturation magnetization ( Ms) and anisotropy  of \nthese films are measured with the Vibrating Sample Magnetometer (VSM)  module of a Physical 4 \n Property Measurement System . Figure  1 plots the magnetic hysteresis  loops and  associated  \nmagnetic properties extracted from VSM measurements.  \n \n \nFigure 1. Room temperature magnetic hysteresis loops of W/CoFeB/MgO PMA thin films post -\nannealed at ( a) 250 °C, (b) 300 °C, (c) 350 °C, and ( d) 400 °C. Black and red curves  denote external \nmagnetic field ( Hext) applied along and perpendicular to the film plane, res pectively . (e-g) Plots of \nthe saturation magnetization ( Ms), effective interfacial anisotropy ( Keff × t), and interfacial \nanisotropy ( Ki) as functions of Tann. \n \nWith the increase of Tann, Ms for the W/ CoFeB /MgO films decreases from ~780  to \n~630  emu/cm3 [Fig. 1 (e)]. The effective interfacial anisotropy  [(Keff × t) depicted in Fig. 1 (f)] \nshows a n increasing trend  with Tann (from ~0.18  to ~0.34 erg/cm2 when Tann increases from 250 to \n350 °C) and saturates at Tann = 350 °C. The positive values of Keff × t suggest that these \nW/CoFeB /MgO films maintain high PMA properties at elevated temperatures including 400  °C, \ndemonstrating their enhanced thermal stability compared to Ta/ CoFeB /MgO films that can only \nsustain PMA up to 350 ° C. Removing the influence  of the demagnetization energy from Keff × t 5 \n results in the  interfacial anisotropy ( Ki), which changes from 0.6 to 0.7 erg/cm2 with the increase \nof Tann up to 350  °C and then decreases to ~0.6 erg/cm2 at Tann = 400  °C [Fig. 1(f)]. Details about \nthe determination of Ki and Keff are provided in Section S1 of the Supp lemental  Material  (SM).  \n \n \nFigure 2 . The dead -layer extraction results. ( a), (b), (c), and ( d) represent the series of samples \nannealed at 250, 300, 350, and 400 C respectively. The tdead value is the extrapolated x-axis \nintercept from the linear fitting of the thickness -dependent saturation magnetization area product \n(Ms×t). \n \nWe attribute the decrease of Ki at high Tann to the growth of a dead layer at the CoFeB  \ninterfaces, which becomes prominent at higher Tann. To quantitatively determine the thickness of \nthe dead  layer as Tann increases, we prepare four sets of PMA stacks of \nW(7)/CoFeB( t)/MgO(2)/Ta( 3). One set contains five stacks with varying thicknesses for the \nCoFeB layer ( t = 1.2, 1.5, 1.8, 2.2, and 2.5 nm) and is post -annealed at a fixed Tann. Four Tann of 6 \n 250, 300, 350, and 400 °C  are used for four sets of the PMA stacks, respectively. The anne aling \nconditions are the same as those for the W(7)/CoFeB(1.2)/MgO(2)/Ta(3) samples for discussed \npreviously. We measure the magnetic hysteresis loops  of these samples using VSM and plot their \nsaturation magnetization area product (\nsMt ) as a function of film thickness ( t) in Fig. 2. Linear \nextrapolati on of the \nsMt  data provides the dead -layer thickness, at which the magnetization \nreduces to zero  as illustrate d by the x-axis intercept in Fig. 2.  \n \nIII. TR-MOKE MEASUREM ENTS  \nThe magnetization dynamics of these PMA thin films are determined using the all-optical  \nTime-Resolved Magneto -Optical Kerr Effect (TR -MOKE) method  [25-29]. This  pump -probe  \nmethod utilizes ultra -short laser pulses to thermally demagnetize the sample  and probe  the \nresulting Kerr rotation angle ( θK). In the polar -MOKE configuration,  θK is proportional to the \nchange of the out-of-plane  component of magnetization [Mz in Fig. 3(a)] [30]. Details  of the TR -\nMOKE setup are provided  in Section S2 of the SM.  \nThe TR-MOKE signal is fitted to the equation \n//\nK sin 2t C tA Be D ft e      , \nwhere A, B, and C are the offset, amplitude, and exponential decaying constant of the thermal \nbackground , respectively . D denotes the amplitude  of oscillations , f is the resonance  frequency , φ \nis a phase shift (related to the demagnetization process), and  is the relaxation time of \nmagnetization precession.  Directly from TR -MOKE measurements, an effective damping constant \n(αeff) can be extracted based on the relationship αeff = 1/(2πf). However, αeff is not an intrinsic \nmaterial property; rather, it depends on measurement conditions, such as the applied field direction \n[θH in Fig. 3(a)], the magnitude of the applied field ( Hext), and inhomogeneities of the sample ( e.g. \nlocal variation in the magnetic properties of the sample)  [31,32] .  7 \n  \n \nFigure  3. (a) Definition of the parameters and angles used in TR -MOKE experiments. The red \ncircle indicates the magnetization precession. θ is the equilibrium direction of the magnetization . \nθK is measured by the probe beam at a given time delay (Δ t). (b) The TR -MOKE data (open \nsymbols) and model fitting of  θK (black curves) for the 400 °C sample at 76° , for varying Hext from \n2.0 to 20 kOe.  \n \n \nTo obtain the Gilbert damping constant, the inhomogene ous contribution needs to be \nremoved from αeff, such that the remaining value of damping is a n intrinsic  material property  and \nindependent of the measurement conditions. To determine the inhomogeneous broadening in the 8 \n sample, the effective anisotropy field (\nk,eff eff s 2/ H K M ) needs to be pre -determined from either \n(1) the magnetic hysteresis loops; or (2) the fitting results of  f vs. Hext obtained from TR -MOKE.  \nThe resonance frequency, f, can be related to Hext through the Smit -Suhl approach  by identifying \nthe second derivatives of the total magnetic free energy, which combines a Zeeman energy, an \nanisotropy energy, and a demagnetization energy [33-35]. For a perpen dicularly magnetized thin \nfilm,  f is defined by Eqs. ( 1-4) [35]. \n12 f H H\n\n,      (1) \n 2\n1 ext H k,eff cos cos H H H      \n,     (2) \n  2 ext H k,eff cos cos 2 H H H      \n,     (3) \n  ext H k,eff2 sin sin 2HH  \n.             (4) \nThis set of equations permits calculat ion of  f with the material gyromagnetic ratio ( ), Hext, \nθH, Hk,eff, and the angle between the equilibrium magnetization di rection and the surface normal \n[θ, determined by Eq. ( 4)]. The measured values of f as a function of Hext can be fitted to Eq. (1) \nby treating and Hk,eff as fitting parameters. To minimize the fitting errors resulting from the \ninhomogeneous broadening effect that is pronounced at the low fields, we use measured \nfrequencies  at high fields ( Hext > 10 kOe) to determine Hk,eff. \nWith a known value of Hk,eff , the Gilbert damping constant of the sample can be determined \nthrough a fitting of the inverse relaxation time (1/ ) to Eq. (5). The two terms of Eq. (5) take into \naccount , respectively,  contributions from the intrinsic Gilbert damping of the materials (first term) \nand inhomogeneous broadening (second term)  [31]:  \n 1 2 k,eff\nk,eff1 1 1\n22dH H HdH   \n,     (5) 9 \n where H1 and H2 are related to the curvature of the magnetic free energy surface as defined by \nEqs. (2) and (3) [35,36] . The second term on the right side of Eq. (5) capture s the inhomogeneous \neffect by attributing it to a spatial variation in the magnetic properties (Δ Hk,eff), analogous  to the \nlinewidth broadening effect in F erromagnetic Resonance  measurements  [37]. The magnitude of \nk,eff/d dH\n can be calculated once the relationship of ω vs. Hext is determined with a numerical \nmethod. Both α and Δ Hk,eff (the inhomogeneous term related to the amount of spatial variation in \nHk,eff) are determined via the fitting of the measured 1/  based on Eq. ( 5). In this way, we can \nuniquely extract the field -independent  α, as an intrinsic material property, from the ef fective \ndamping ( αeff), which is directly obtained from TR -MOKE and dependent on Hext. \nIt should be noted here that the inhomogeneous broadening of the magnetization precession \nis presumably due to the multi -domain structure of the materials, which becomes  negligible in the \nhigh-field regime ( Hext >> Hk,eff) as the magnetization direction  of multiple magnetic domains  \nbecomes uniform. This is also reflected by the fact that the derivative in the second term of Eq.  (5) \napproaches zero for the high -field regim e [38]. \n \nIV. RESULTS AND DISCUSSION  \nThe measurement method is validated  by measuring  the Tann = 400 C at multiple angles \n(θH) of the external magnetic field direction.  By repeating this meas urement at varying  θH, we can \nshow that α is an intrinsic material property , independent of θH. Figure 4(a) plots the resonance \nfrequenc ies derived from TR -MOKE and model fittings for the 400 °C sample at two field \ndirections ( θH = 76° and 89° ).  For the data acquired at θH = 89°, a minimum f occurs at Hext ≈ Hk,eff. \nThis corresponds to the smallest amplitude of magnetization precession, when the equilibrium \ndirection of the magnetization is aligned with the applied field direction at the magnitud e of Hk,eff 10 \n [35]. The dip at this local minimum  diminishes when θH decreases, as reflected by the comparison \nbetween the red ( θH = 89°) and blue ( θH = 76°) lines in Fig. 4(a). With the Hk,eff extracted from the \nfitting of frequency data with θH = 89°, we generate the plot of theoretically predicted f vs. Hext \n[θH = 76° theory, blue line in Fig. 4(a)], which agrees well with experimental data [open square s \nin Fig. 4(a)].  \n \n \nFigure 4. (a) Measured f vs. Hext results for the 400 C sample at θH = 89° (open circles) and \nθH = 76° (open squares) and corresponding modeling at θH = 89° (red line) and θH = 76° (blue line) . \n(b) The measured inverse of relaxation time  (1/) at θH = 89° (open symbols) and the fitting of 1/  \nbased on Eq. (5) (dotted line). For reference, the first term of 1/  in Eq. (5) is also plotted (solid \nline), which accounts for the contribution from the Gilbert damping  only. (c) αeff as a function of \nHext for θH = 89° (red circles). Black circles are the extracted Gilb ert damping, which is \nindependent of Hext. The black dotted line shows the average of this extracted damping; ( d) and ( e) \ndepict similar plots of 1/  and damping constants for θH = 76°. Error bars in ( b) through ( e) come \nfrom the uncertainty in the mathematical fitting.  11 \n  \nThe inverse relaxation time (1/ should also have a minimum  value near Hk,eff for θH = 89° \nif the damping was purely from  Gilbert damping [as shown by the solid lines in Figs.  4(b) and \n4(d)]; however, the measured data do  not follow this trend. Adding the inhomogeneous term \n[dotted lines in Figs. 4(b) and 4(d)] more accurately describes the field  dependen ce of the measured \n1/[open symbols  in Figs. 4(b) and 4(d)] It should be noted that the dip of the predicted 1/ occurs \nwhen the frequency derivative term in Eq.  (5) approaches  zero; however,  this is not captured by \nthe measurement due to the finite interval over which we vary  Hext. Figures 4(c) and 4(e) depict  \nthe field -dependent effective damping ( αeff) and the Gilbert damping ( α) as the intrinsic material’s \nproperty obtained from fitting the measured 1/.  \nWith the knowledge that the value of α extracted with this method is the intrinsic material \nproperty, we repeat this data reduction technique for the annealed W/CoFeB /MgO samples \ndiscussed  in Fig. 1. The symbols in Fig. 5 represent  the resonance frequency and damping \nconstants (both effective damping and Gilbert damping) for all samples measured at θH ≈ 90°. The \nfittings for the resonance frequency [red lines , from Eq.  (1)] are also shown to demonstrate  the \ngood agreement between our TR -MOKE measurement and theoretical prediction. The \nuncertainties of f, , and Hk,eff are calculated from the least-square s fitting uncertainty and the \nuncertainty of measuring Hext with the Hall sensor.  12 \n \n \nFigure 5 . Results for f (a-d) and αeff (e-h, on a log scale) for individual samples. For comparison, \nthe Gilbert damping constant α is also plotted by subtracting the inhomogeneous terms from αeff. \nThe dashed line in (e -h) indicat es the average α. All samples are measured at θH = 90° except for \nthe 400 C sample ( θH = 89°).  \n \nThe summary of the anisotropy and damping measured via TR -MOKE is shown in Fig. 6. \nFigure 6(a) plots Hk,eff obtained from VSM (black open circle s) and TR -MOKE  (blue open \nsquares) , both of which exhibit a monotonic increasing trend as Tann becomes higher.  \nDiscrepancies in Hk,eff from these two methods can be attributed to the difference in the size of the \nprobing region , which is highly localized in TR -MOKE but sample -averaged in VSM. Since Hk,eff \ndetermined  from  TR-MOKE is obtained from fitting the measured frequency for a localized region, \nwe expect these values  more consistently  describ e the magnetization precession th an those \nobtained from VSM. The increase in Hk,eff  with Tann can be partially attributed to the crystallization 13 \n of the CoFeB layer  [32]. For temperatures higher than 350 °C, this increasing trend  of Hk,eff begins \nto lessen, presumably  due to the diffusion of W atoms into the CoFeB layer , which is more \npronounced at higher Tann. The W diffusion process  is also responsible for the decrease in Ms of \nthe CoFeB layer as Tann increases  [Fig. 1 (e)]. Subsequently, the  decrease in Ms leads to a further -\nreduced demagnetizing energy  and thus a  larger Hk,eff.  \nSimilar observation of Ms has been reported in literature for Ta/CoFeB/MgO PMA \nstructures and attribute d to the growth of a dead layer  at the heavy metal/CoFeB inter face [1]. \nFigure 6(b) summarizes tdead as a function of  Tann with tdead increas ing from 0.17 to 0.53 nm as Tann \nchanges fr om 250 to 400  C, as discussed in Section II.  \n \n \nFigure 6. Summary of the magnetic properties of W -seeded CoFeB as a function of Tann. (a) The \ndependence of Hk,eff on Tann obtained from both the VSM (black open circles) and TR-MOKE \nfitting (blue  open squares ). (b) The dependence of dead -layer thickness on Tann. (c) Damping \nconstants as a function of Tann. The minim um damping constant of α = 0.016 occurs at 350°C. The \nvalues for the all samples are obtained from measurements at θH = ~90°. For comp arison, α of the  \nreference Ta/CoFeB/MgO PMA sample annealed at 300 °C is also shown as a red triangle in ( c). 14 \n  \nFigure  6(c) depicts the dependence of α on Tann, which first decreases with Tann, reaches a \nminimum of 0.016 at 350 °C, and then increases as Tann rises to 400 °C. Similar trends have been \nobserved for Ta/CoFeB/MgO previously (minimum α at Tann = 300 °C) [32]. We speculate that \nthis dependence of damping on Tann is due to two competing effects: (1) the inc rease in \ncrystallization in the CoFeB layer with Tann which reduces the damping, and (2) the growth of a \ndead layer, which results from the diffusion of W and B atoms and is prominent at higher Tann. At \nTann = 400 °C, the dead -layer formation leads to a la rger damping presumably due to an increase \nin scattering sites (diffused atoms) that contribute to spin -flip events, as described by the Elliot -\nYafet relaxation mechanisms [17]. The observation that our W -seeded samples still sustain \nexcellent PMA properties at Tann = 400 °C confirms  their enhanced thermal stability, compared \nwith Ta/CoFeB/MgO stacks which fail at Tann = 350 °C or higher.  \nThe damping constants are comparable for the W/CoFeB/MgO and Ta/CoFeB/MgO films \nannealed at 300 °C, both of which are higher than that of the W/ CoFeB /MgO PMA with the \noptimal Tann of 350 °C.  Nevertheless,  our work focuses on the enhanced thermal stability of W -\nseeded CoFeB PMA films while still maintaining a relatively low damping constant. Such an \nadvantage enables W -seeded CoFeB layers to be viable and promising alternatives to \nTa/CoFeB/MgO , which is currently widely used in spintronic devices.  \n \nV. CONCLUSION  \nIn summary, we deposit a series of W -seeded CoFeB  PMA films with varying annealing \ntemperatures up to 400  °C and conduct ultrafast all -optical TR -MOKE measurements to study \ntheir magnetization precession dynamics. The Gilbert damping, as a n intrinsic  material property, \nis proven to be independent of meas urement conditions, such as the amplitudes and directions of 15 \n the applied field. The damping constant varies with Tann, first decreasing and then increasing, \nleading to a minimum of α = 0.016 for the sample anneale d at 350  °C. Due to the  dead -layer \ngrowth , the damping constant slightly increases to α = 0.024 at Tann = 400  °C, comparable to the \nreference Ta/ CoFeB /MgO PMA film annealed at 300°C, which demonstrates the improved \nenhanced thermal stability of W/ CoFeB /MgO over the Ta/ CoFeB /MgO structures. This strongly \nsuggests the great potential of W/ CoFeB /MgO PMA material systems for future spintronic device \nintegration that requires materials to sustain a processing temperature as high as 400 °C.  \n  16 \n Acknowledgements  \nThis work is supported by C -SPIN  (award #: 2013 -MA-2381) , one of six centers of STARnet, a \nSemiconductor Research Corporation program, sponsored by MARCO and DARPA. The authors \nwould like to thank Prof.  Paul Crowell and Dr. Changjiang Liu for valuable discussions.  \n \nSupplemental  Materials  Available:  Complete description of the TR -MOKE mea surement \nmethod, the angular dependent result summary , and the  description for determining the  interface \nanisotropy.  \n \n  17 \n References  \n[1] S. Ikeda, K. Miura, H. Yamamoto, K. Mizunuma, H. D. Gan, M. Endo, S. Kanai, J. \nHayakawa, F. Matsukura, and H. Ohno, A perpendicular -anisotropy CoFeB -MgO magnetic tunnel \njunction, Nat. Mater. 9, 721 (2010).  \n[2] H. Sato, E. C. I. Enobio, M. Yamanouchi, S. Ik eda, S. Fukami, S. Kanai, F. Matsukura, \nand H. Ohno, Properties of magnetic tunnel junctions with a MgO/CoFeB/Ta/CoFeB/MgO \nrecording structure down to junction diameter of 11nm, Appl. Phys. Lett. 105, 062403 (2014).  \n[3] J. Z. Sun, S. L. Brown, W. Chen, E.  A. Delenia, M. C. Gaidis, J. Harms, G. Hu, X. Jiang, \nR. Kilaru, W. Kula, G. Lauer, L. Q. Liu, S. Murthy, J. Nowak, E. J. O’Sullivan, S. S. P. Parkin, R. \nP. Robertazzi, P. M. Rice, G. Sandhu, T. Topuria, and D. C. Worledge, Spin -torque switching \nefficiency  in CoFeB -MgO based tunnel junctions, Phys. Rev. B 88, 104426 (2013).  \n[4] J. Zhu, J. A. Katine, G. E. Rowlands, Y. -J. Chen, Z. Duan, J. G. Alzate, P. Upadhyaya, J. \nLanger, P. K. Amiri, K. L. Wang, and I. N. Krivorotov, Voltage -Induced Ferromagnetic Resonan ce \nin Magnetic Tunnel Junctions, Phys. Rev. Lett. 108, 197203 (2012).  \n[5] W. G. Wang, M. G. Li, S. Hageman, and C. L. Chien, Electric -field-assisted switching in \nmagnetic tunnel junctions, Nat. Mater. 11, 64 (2012).  \n[6] S. Kanai, M. Yamanouchi, S. Ikeda, Y . Nakatani, F. Matsukura, and H. Ohno, Electric \nfield-induced magnetization reversal in a perpendicular -anisotropy CoFeB -MgO magnetic tunnel \njunction, Appl. Phys. Lett. 101, 122403 (2012).  \n[7] G. Q. Yu, P. Upadhyaya, Y. B. Fan, J. G. Alzate, W. J. Jiang, K . L. Wong, S. Takei, S. A. \nBender, L. T. Chang, Y. Jiang, M. R. Lang, J. S. Tang, Y. Wang, Y. Tserkovnyak, P. K. Amiri, \nand K. L. Wang, Switching of perpendicular magnetization by spin -orbit torques in the absence of \nexternal magnetic fields, Nat. Nanotech nol. 9, 548 (2014).  \n[8] D. Bhowmik, L. You, and S. Salahuddin, Spin Hall effect clocking of nanomagnetic logic \nwithout a magnetic field, Nat. Nanotechnol. 9, 59 (2014).  \n[9] L. Q. Liu, C. F. Pai, Y. Li, H. W. Tseng, D. C. Ralph, and R. A. Buhrman, Spin -Torq ue \nSwitching with the Giant Spin Hall Effect of Tantalum, Science 336, 555 (2012).  \n[10] W. J. Jiang, P. Upadhyaya, W. Zhang, G. Q. Yu, M. B. Jungfleisch, F. Y. Fradin, J. E. \nPearson, Y. Tserkovnyak, K. L. Wang, O. Heinonen, S. G. E. te Velthuis, and A. Hof fmann, \nBlowing magnetic skyrmion bubbles, Science 349, 283 (2015).  \n[11] G. Yu, P. Upadhyaya, Q. Shao, H. Wu, G. Yin, X. Li, C. He, W. Jiang, X. Han, P. K. Amiri, \nand K. L. Wang, Room -Temperature Skyrmion Shift Device for Memory Application, Nano \nLetters 17, 261 (2017).  \n[12] Z. T. Diao, Z. J. Li, S. Y. Wang, Y. F. Ding, A. Panchula, E. Chen, L. C. Wang, and Y. M. \nHuai, Spin -transfer torque switching in magnetic tunnel junctions and spin -transfer torque random \naccess memory, J. Phys.: Condens. Matter 19, 1652 09 (2007).  \n[13] S. Manipatruni, D. E. Nikonov, and I. A. Young, Energy -delay performance of giant spin \nHall effect switching for dense magnetic memory, Appl. Phys. Express 7, 103001 (2014).  \n[14] A. J. Annunziata, P. L. Trouilloud, S. Bandiera, S. L. Brown, E. Gapihan, E. J. O'Sullivan, \nand D. C. Worledge, Materials investigation for thermally -assisted magnetic random access \nmemory robust against 400°C temperatures, J. Appl. Phys. 117, 17B739 (201 5). \n[15] N. Miyakawa, D. C. Worledge, and K. Kita, Impact of Ta Diffusion on the Perpendicular \nMagnetic Anisotropy of Ta/CoFeB/MgO, IEEE Magn. Lett. 4, 1000104 (2013).  18 \n [16] S. Yang, J. Lee, G. An, J. Kim, W. Chung, and J. Hong, Thermally stable perpendicul ar \nmagnetic anisotropy features of Ta/TaO x/Ta/CoFeB/MgO/W stacks via TaO x underlayer insertion, \nJ. Appl. Phys. 116, 113902 (2014).  \n[17] R. J. Elliott, Theory of the Effect of Spin -Orbit Coupling on Magnetic Resonance in Some \nSemiconductors, Phys. Rev. 96, 266 (1954).  \n[18] H. Suhl, Theory of the magnetic damping constant, I EEE Trans. Magn. 34, 1834 (1998).  \n[19] S. Ikeda, J. Hayakawa, Y. Ashizawa, Y. M. Lee, K. Miura, H. Hasegawa, M. Tsunoda, F. \nMatsukura, and H. Ohno, Tunnel magnetoresistance of 604% at 300K  by suppression of Ta \ndiffusion in CoFeB∕MgO∕CoFeB pseudo -spin-valves annealed at high temperature, Appl. Phys. \nLett. 93, 082508 (2008).  \n[20] T. Liu, Y. Zhang, J. W. Cai, and H. Y. Pan, Thermally robust Mo/CoFeB/MgO trilayers \nwith strong perpendicular magn etic anisotropy, Sci. Rep. 4, 5895 (2014).  \n[21] H. Almasi, D. R. Hickey, T. Newhouse -Illige, M. Xu, M. R. Rosales, S. Nahar, J. T. Held, \nK. A. Mkhoyan, and W. G. Wang, Enhanced tunneling magnetoresistance and perpendicular \nmagnetic anisotropy in Mo/CoFeB/M gO magnetic tunnel junctions, Appl. Phys. Lett. 106, 182406 \n(2015).  \n[22] S. Cho, S. -h. C. Baek, K. -D. Lee, Y. Jo, and B. -G. Park, Large spin Hall magnetoresistance \nand its correlation to the spin -orbit torque in W/CoFeB/MgO structures, Sci. Rep. 5, 14668 (2015).  \n[23] C. F. Pai, L. Q. Liu, Y. Li, H. W. Tseng, D. C. Ralph, and R. A. Buhrman, Spin transfer \ntorque devices utilizing the giant spin Hall effect of tungsten, Appl. Phys. Lett. 101, 122404 \n(2012).  \n[24] C. He, A. Navabi, Q. Shao, G. Yu, D. Wu, W. Zhu , C. Zheng, X. Li, Q. L. He, S. A. Razavi, \nK. L. Wong, Z. Zhang, P. K. Amiri, and K. L. Wang, Spin -torque ferromagnetic resonance \nmeasurements utilizing spin Hall magnetoresistance in W/Co 40Fe40B20/MgO structures, Appl. \nPhys. Lett. 109, 202404 (2016).  \n[25] M. van Kampen, C. Jozsa, J. T. Kohlhepp, P. LeClair, L. Lagae, W. J. M. de Jonge, and B. \nKoopmans, All -Optical Probe of Coherent Spin Waves, Phys. Rev. Lett. 88, 227201 (2002).  \n[26] S. Mizukami, D. Watanabe, T. Kubota, X. Zhang, H. Naganuma, M. Oogane, Y.  Ando, \nand T. Miyazaki, Laser -Induced Fast Magnetization Precession and Gilbert Damping for CoCrPt \nAlloy Thin Films with Perpendicular Magnetic Anisotropy, Appl. Phys. Express 3, 123001 (2010).  \n[27] S. Mizukami, F. Wu, A. Sakuma, J. Walowski, D. Watanabe, T. Kubota, X. Zhang, H. \nNaganuma, M. Oogane, Y. Ando, and T. Miyazaki, Long -Lived Ultrafast Spin Precession in \nManganese Alloys Films with a Large Perpendicular Magnetic Anisotropy, Phys. Rev. Lett. 106, \n117201 (2011).  \n[28] S. Iihama, S. Mizukami, N. Inami , T. Hiratsuka, G. Kim, H. Naganuma, M. Oogane, T. \nMiyazaki, and A. Yasuo, Observation of Precessional Magnetization Dynamics in L1 0-FePt Thin \nFilms with Different L1 0 Order Parameter Values, Jpn. J. Appl. Phys. 52, 073002 (2013).  \n[29] J. Zhu, X. Wu, D. M.  Lattery, W. Zheng, and X. Wang, The Ultrafast Laser Pump -probe \nTechnique for Thermal Characterization of Materials With Micro/nanostructures, Nanoscale \nMicroscale Thermophys. Eng. , (2017) . [published online]  \n[30] C. Y. You and S. C. Shin, Generalized anal ytic formulae for magneto -optical Kerr effects, \nJ. Appl. Phys. 84, 541 (1998).  \n[31] P. Krivosik, N. Mo, S. Kalarickal, and C. E. Patton, Hamiltonian formalism for two \nmagnon scattering microwave relaxation: Theory and applications, J . Appl. Phys. 101, 0839 01 \n(2007).  19 \n [32] S. Iihama, S. Mizukami, H. Naganuma, M. Oogane, Y. Ando, and T. Miyazaki, Gilbert \ndamping constants of Ta/CoFeB/MgO(Ta) thin films measured by optical detection of precessional \nmagnetization dynamics, Phys. Rev. B 89, 174416 (2014).  \n[33] C. Kittel, On the Theory of Ferromagnetic Resonance Absorption, Phys. Rev. 73, 155 \n(1948).  \n[34] L. Ma, S. F. Li, P. He, W. J. Fan, X. G. Xu, Y. Jiang, T. S. Lai, F. L. Chen, and S. M. Zhou, \nTunable magnetization dynamics in disordered FePdPt ternary alloys: Effects of spin orbit \ncoupling, J . Appl.  Phys. 116, 113908 (2014).  \n[35] S. Mizukami, Fast Magnetization Precession and Damping for Magnetic Films with High \nPerpendicular Magnetic Anisotropy, J. Magn. Soc. Jpn. 39, 1 (2015).  \n[36] H. Suhl, Ferromagnetic Reso nance in Nickel Ferrite Between One and Two \nKilomegacycles, Phys. Rev. 97, 555 (1955).  \n[37] D. S. Wang, S. Y. Lai, T. Y. Lin, C. W. Chien, D. Ellsworth, L. W. Wang, J. W. Liao, L. \nLu, Y. H. Wang, M. Z. Wu, and C. H. Lai, High thermal stability and low Gilb ert damping constant \nof CoFeB/MgO bilayer with perpendicular magnetic anisotropy by Al capping and rapid thermal \nannealing, Appl. Phys. Lett. 104, 142402 (2014).  \n[38] A. Capua, S. H. Yang, T. Phung, and S. S. P. Parkin, Determination of intrinsic damping \nof perpendicularly magnetized ultrathin films from time -resolved precessional magnetization \nmeasurements, Phys. Rev. B 92, 224402, 224402 (2015).  \n "    },    {        "title": "1709.08194v1.Suppression_of_Recurrence_in_the_Hermite_Spectral_Method_for_Transport_Equations.pdf",        "content": "arXiv:1709.08194v1  [math.NA]  24 Sep 2017SUPPRESSION OF RECURRENCE IN THE HERMITE-SPECTRAL\nMETHOD FOR TRANSPORT EQUATIONS\nZHENNING CAI AND YANLI WANG\nAbstract. We study the unphysical recurrence phenomenon arising in th e numerical simu-\nlation of the transport equations using Hermite-spectral m ethod. From a mathematical point\nof view, the suppression of this numerical artifact with filt ers is theoretically analyzed for\ntwo types of transport equations. It is rigorously proven th at all the non-constant modes are\ndamped exponentially by the filters in both models, and forma lly shown that the filter does\nnot affect the damping rate of the electric energy in the linea r Landau damping problem.\nNumerical tests are performed to show the effect of the filters .\nKeywords: Hermite spectral method; filter; recurrence\n1.Introduction\nWe consider a system with a large number of microscopic particles, an d the motion of these\nparticles is governed by a force field. Instead of the state of ever y individual particle, we are\ninterested in the collective behavior of these particles, such as the local density and the mean\nvelocity. To obtain such information, the system needs to be prope rly modeled before carrying\nout the simulation. Compared with tracking the positions and velocitie s of all the particles as\nin the method of molecular dynamics, a more efficient method is to use t he kinetic theory to\ndescribe the system in a statistical way. The basic idea of the kinetic theory is to introduce a\nvelocity distribution function f(x,ξ,t), which denotes the number density of particles in the\nposition-velocity space, and the governing equation for fis\n(1.1)∂f\n∂t+ξ·∇xf+E·∇ξf= 0, t∈R+,x∈RN,ξ∈RN,\nwheretdenotes the time, xdenotes the position, and ξstands for the velocity of the particles.\nThe force field is given by E. In this paper, we consider the one-dimensional case with periodic\nboundary condition in space, and thus the equation for f(x,ξ,t) can be rewritten as\n∂f\n∂t+ξ∂f\n∂x+E∂f\n∂ξ= 0, t∈R+, x∈R, ξ∈R, (1.2a)\nf(x,ξ,t) =f(x+D,ξ,t),∀(x,ξ,t)∈R×R×R+, (1.2b)\nwhereDis the period, and we assume that Eis also periodic and independent of the velocity ξ,\nbut may be a function of tandx. A typical example of this model is the Vlasov-Poisson (VP)\nequation arising from the astrophysics and plasma physics, which mo dels the system formed\nby a large number of charged particles, and the force is generated by a self-consistent electric\nfield. Moreover, Landau damping is one of the fundamental problem s in the applications of the\nVP equation. However, in the numerical simulations of Landau dampin g, it is observed that\nan unphysical phenomenon called “recurrence” occurs for most g rid-based solvers [8].\nThis work is supported by National University of Singapore S tartup Fund under Grant No. R-146-000-241-\n133. Yanli Wang is also supported by the National Natural Sci entific Foundation of China (Grant No. 11501042)\nand China Scholarship Council.\n12 ZHENNING CAI AND YANLI WANG\nThe recurrence is an unphysical periodic behavior in the numerical s olutions of the VP\nequation. It can be demonstrated by the simple advection equation (E= 0 in (1.2)) whose\nexactsolutionis f(x,ξ,t) =f(x−ξt,ξ,0). It showsthat anyspatialwaveinthe initial condition\nwill cause a wave in the velocity domain in the evolution of the solution, a nd the frequency\nof the wave gets higher when tincreases. If the velocity domain is discretized by a fixed grid,\nthe exact solution cannot be well resolved when tis large. Particularly, the numerical solution\nmay look smoother than the exact solution and therefore appears similar to the solution at\na previous time. Such phenomenon has been reported in a number of research works with\ndifferent grid-based numerical methods [10, 26, 9, 28, 22]. The app earance of the recurrence\nmay be postponed by using a larger number of velocity grids [30, 15], which also introduces\nlarger computational cost. To avoid the recurrence, the particle -in-cell method [2, 18, 6] can\nbe adopted and it is reported in [4] that the numerical result does no t present recurrence.\nHowever, since the particle-in-cell method is a stochastic method, only half-order convergence\ncan be achieved. An idea of combining the two types of methods is intr oduced in [1], where the\nauthors suppress the recurrence by introducing some randomne ss into the grid-based methods.\nIn this paper, we consider another type of methods called transfo rm methods [5, 13], where\nthe distribution function is mapped to the frequency space and the Fourier modes are solved\ninstead of the values on the grid points. Especially, we adopt the Her mite-spectral method\nintroduced by [19] as the asymmetric Hermite method. The similar idea is adopted in [3] to\nget a slightly nonlinear discretization. For transform methods, one can suppress recurrence by\nintroducing filters to the numerical methods [23, 7], or adding artific ial collisions to the model\n[4, 24]. The suppression of the recurrence is numerically analyzed in [1 6], where it is shown\nthat the collision has a damping effect for the high-frequency modes , so that the distribution\nfunction is smoothed out and the filamentation is weakened. Howeve r, a theoretical study of\nits underlying mathematical mechanism is still missing in the literature.\nTo fill the vacancy, we are going to conduct a theoretical analysis o n the relation between\nthe filters and the recurrence. The analysis is performed on two ty pes of transport equations\nincluding the advection equation and Vlasov-Poisson equation. For b oth types of equations, it\nis shown by eigenvalue analysis that all the non-constant modes in th e discrete system converge\nto zero exponentially as the time goes to infinity, and therefore the damping effect is rigorously\nproven. Moreover, it is formally shown that the filter does not chan ge the damping rate of the\nelectric energy in the case of linear Laudau damping. Our numerical r esults are consistent with\nthe analysis. In the tests for linear Landau damping, numerical res ults with high quality are\nobserved with the filter introduced in [20].\nThe rest of this paper is organized as follows. In Section 2, we briefly introduce the Hermite-\nspectral method and the filters. In Section 3 and 4, two types of e quations are analyzed\nrespectively. Some numerical experiments are performed in Sectio n 5, and the concluding\nremarks are given in Section 6.\n2.Hermite-spectral method and filtering\nIn this section, we focus on the velocity discretization of the Vlasov equation. Following [3],\nwe consider the following approximation of the distribution function1:\n(2.1) f(x,ξ,t)≈1√\n2πM/summationdisplay\ni=0fi(x,t)Hei(ξ)exp/parenleftbigg\n−ξ2\n2/parenrightbigg\n,\n1In [3], the basis functions are translated and scaled in orde r to adapt the functions, while in this paper,\nsuch adaption is removed for easier analysis, and the result ing equations can be regarded as a linearized version\nof the model in [3].SUPPRESSION OF RECURRENCE FOR TRANSPORT EQUATIONS 3\nwhereHei(ξ) is the normalized Hermite polynomials defined by\n(2.2) Hen(ξ) =(−1)n\n√\nn!exp/parenleftbiggξ2\n2/parenrightbiggdn\ndxnexp/parenleftbigg\n−ξ2\n2/parenrightbigg\n,\nand they have the following properties:\n(1) Orthogonality:\n(2.3)1√\n2π/integraldisplay\nRHem(ξ)Hen(ξ)exp/parenleftbigg\n−ξ2\n2/parenrightbigg\ndξ=δmn,∀m,n∈N;\n(2) Recursion relation:\n(2.4)√\nn+1Hen+1(ξ) =ξHen(ξ)−√nHen−1(ξ),∀n∈N;\n(3) Differential relation:\n(2.5)He′\nn+1(ξ) =√\nn+1Hen(ξ),\nd\ndξ/parenleftbigg\nHen(ξ)exp/parenleftbigg\n−ξ2\n2/parenrightbigg/parenrightbigg\n=−√\nn+1Hen+1(ξ)exp/parenleftbigg\n−ξ2\n2/parenrightbigg\n,∀n∈N.\nUsing these properties, the equations for the coefficients fi(x,t) can be derived by inserting\n(2.1) into (1.2) and integrating the result against Hek(ξ) withk= 0,···,M. By defining\nf= (f0,f1,···,fM)T, we have the following evolution equations:\n(2.6)∂f\n∂t+A∂f\n∂x−EBf= 0,\nwhereAandBare (M+1)×(M+1) matrices defined by\n(2.7) A=\n0 1 0 0 ...0\n1 0√\n2 0 ...0\n0√\n2 0√\n3...0\n0...............\n......0√\nM−1 0√\nM\n0...0 0√\nM0\n,B=\n0 0 0 ...0\n1 0 0 ...0\n0√\n2 0 ...0\n...............\n0...0√\nM0\n.\nThe system (2.6) is the semi-discrete transport equation after sp ectral discretization of the\nvelocity variable. As will be seen below, such discretization suffers fr om a deficiency called\n“recurrence phenomenon” [3, 23], which causes the non-physical echo of the electric energy\nwhen simulating the plasma. In [23], the authors proposed the filtere d spectral method, and in\nthe numerical results, the recurrence was clearly suppressed. H ere we adopt the similar method\nand apply the filter after every time step. In detail, let fn= (fn\n0,···,fn\nM)Tbe the numerical\nsolution of (2.6) at the nth time step, and suppose a numerical scheme for (2.6) is\n(2.8) fn+1=Q(fn).\nWhen a filter is applied, the above scheme is altered as\n(2.9)fn,∗=Q(fn),\nfn+1\ni=σM(i)fn,∗\ni, i= 0,···,M,\nwhere the filter σM(i) satisfies\n(2.10) σM(0) = 1,lim\nM→∞σM(i) = 1,∀i∈N.4 ZHENNING CAI AND YANLI WANG\nThe filter is often interpreted as an operator with the effect of diffu sion [12]. Especially, when\nwe take the exponential filter\n(2.11) σM(i) = exp( −α(i/M)p),\nthe method (2.9) is actually computing the solution to the modified pro blem\n(2.12)∂f\n∂t+ξ∂f\n∂x+E∂f\n∂ξ=−α(−1)p\n∆tMpDpf,\nwhereDis a linear operator defined by\n(2.13) Df(x,ξ,t) =∂\n∂ξ/bracketleftbigg\nexp/parenleftbigg\n−ξ2\n2/parenrightbigg∂\n∂ξ/parenleftbigg\nexp/parenleftbiggξ2\n2/parenrightbigg\nf(x,ξ,t)/parenrightbigg/bracketrightbigg\n.\nThe time step ∆ tneeds to be chosen to ensure stability. Since /ba∇dblB/ba∇dbl2=√\nMand the maximum\neigenvalue of Ais the maximum zero of HeM+1(ξ), which grows asymptotically as O(√\nM), we\nchoose the time step as ∆ t∼O(M−1/2) in this work. Thus one sees that when Mgets larger,\nthe equation formally converges to the transport equation (1.2) if p >1/2.\nWhen the original transport equation (1.2) is replaced by (2.12), th e semi-discrete system\n(2.6) changes to\n(2.14)∂f\n∂t+A∂f\n∂x−EBf=Hf,\nwhereH=−∆t−1αdiag{0,(1/M)p,···,[(M−1)/M]p,1}. In general, we assume that\n(2.15) H= diag{h0,h1,···,hM}\nis an (M+1) by ( M+1) diagonal matrix with non-positive diagonal entries, and in order to\nkeep the conservation of total number of particles, we require th at the first entry h0= 0.\nBelowwearegoingtoremovethe spatialderivativeby Fourierseries expansion. The periodic\nboundary condition (1.2b) shows that fis also periodic with respect to x. Thus we have the\nfollowing series expansions:\n(2.16) f=/summationdisplay\nm∈Zˆf(m)exp(imkx), E=/summationdisplay\nm∈ZˆE(m)exp(imkx), k= 2π/D,\nand Parseval’s equality shows that\n(2.17) /ba∇dblf/ba∇dbl2\n2=D/summationdisplay\nm∈Z/vextenddouble/vextenddouble/vextenddoubleˆf(m)/vextenddouble/vextenddouble/vextenddouble2\n2,/ba∇dblE/ba∇dbl2\n2=D/summationdisplay\nm∈Z|ˆE(m)|2.\nSubstituting (2.16) into (2.14), we get the equations for the Fourie r coefficients ˆf(m):\n(2.18)∂ˆf(m)\n∂t+imkAˆf(m)−/summationdisplay\nl∈ZˆE(l)Bˆf(m−l)=Hˆf(m), m∈Z.\nBased on the form (2.18), we will show in the following sections that th e filterHcan suppress\nthe recurrence phenomenon, especially in the simulation of Landau d amping.\n3.Advection Equation\n3.1.Recurrence without filter. We will begin our discussion with a simple case E= 0, and\nthus the transport equation (1.2) becomes\n(3.1)∂f\n∂t+ξ∂f\n∂x= 0, t∈R+, x∈D, ξ∈R,SUPPRESSION OF RECURRENCE FOR TRANSPORT EQUATIONS 5\nwhereD= [0,D] and the periodic boundary condition is imposed. It is known that the\nrecurrence can be observed in such a simple advection equation with initial value\n(3.2) f(x,ξ,0) =1√\n2π(1+ǫcos(kx))exp/parenleftbigg\n−ξ2\n2/parenrightbigg\n, k=2π\nD,\nand the recurrence time can be exactly given if the velocity is discret ized with a uniform grid\n[25, 1]. For Hermite-spectral method, the system (2.18) is corres pondingly reduced to\n(3.3)∂ˆf(m)\n∂t+imkAˆf(m)=Hˆf(m), m∈Z.\nIf no filter is applied, H= 0, and then the solution to the above system is\n(3.4) ˆf(m)(t) = exp( −imktA)ˆf(m)(0) =RTexp(−imktΛ)Rˆf(m)(0),\nwhereRis an (M+ 1)×(M+ 1) orthogonal matrix satisfying RΛRT=A, andΛis an\n(M+1)×(M+1) real diagonal matrix due to the symmetry of the real matrix A. The equality\n(3.4) indicates that\n(3.5)/vextenddouble/vextenddouble/vextenddoubleˆf(m)(t)/vextenddouble/vextenddouble/vextenddouble2\n2=/vextenddouble/vextenddouble/vextenddoubleˆf(m)(0)/vextenddouble/vextenddouble/vextenddouble2\n2,∀m∈Z, t∈R+.\nTo observe the recurrence, we suppose that the exact solution t o (3.1) and (3.2) can be\nwritten as\n(3.6) f(x,ξ,t) =1√\n2πM/summationdisplay\ni=0/summationdisplay\nm∈Zˆf(m)\nex,i(t)exp(imkx)Hei(ξ)exp/parenleftbigg\n−ξ2\n2/parenrightbigg\n.\nBy straightforward calculation, we have\n(3.7) ˆf(m)\nex,i(t) =\n\nδi0, ifm= 0,\nǫ\n2(imkt)i\n√\ni!exp/parenleftbigg\n−k2t2\n2/parenrightbigg\n,ifm=±1,\n0, otherwise .\nTherefore\n(3.8)M/summationdisplay\ni=0|ˆf(±1)\nex,i(t)|2=ǫ2\n4M/summationdisplay\ni=0(kt)2i\ni!exp(−k2t2),\nwhich decays to zero as t→ ∞. The relation (3.5) shows that this property is not maintained\nafter discretization. In fact, if (2 π)−1ktΛis close to an integer matrix for some t, thenˆf(m)(t)\nis close to ˆf(m)(0) for all m∈Z, which turns out to be a “recurrence”. Some illustration will\nbe given in Section 5.\n3.2.Suppression of recurrence with filter. When a filter is applied, the solution to (3.3)\nis\n(3.9) ˆf(m)(t) = exp/parenleftig\n(−imkA+H)t/parenrightig\nˆf(m)(0).\nIn this subsection, we assume that His a filter whose first diagonal entry is zero and last\ndiagonal entry is nonzero. Actually, almost all filters have such a fo rm so that the high-\nfrequency modes can be damped while the low-frequency modes are not disturbed. Based on\nthis assumption, we have the following theorem:\nTheorem 1. LetAm=−imkA+H. Then for all m∈Z\\{0}, all the eigenvalues of Amhave\nnegative real parts.6 ZHENNING CAI AND YANLI WANG\nThe above theorem shows that for all m∈Z\\{0},ˆf(m)(t)→0 ast→+∞, which fixes the\nundesired property (3.5), and agrees with the decaying behavior o f the exact solution to the\nadvection equation. Thus the recurrence is suppressed. The pro of of the theorem requires the\nfollowing lemma:\nLemma 1. LetM= (aij)N×Nbe a symmetric tridiagonal matrix with nonzero subdiagonal\nentries. Define pn(λ),n= 1,···,Nas the characteristic polynomial of the nth order leading\nprinciple submatrix of M. Then the following statements hold:\n•The roots of pnandpn+1are interlacing;\n•Ifλis an eigenvalue of M, then the associated eigenvector is r= (r1,···,rN)Twith\nr1= 1andrn= (−1)n−1pn−1(λ)/(a12a23···an−1,n)forn >1;\n•Ifr= (r1,···,rN)Tis an eigenvector of M, thenrN/\\e}atio\\slash= 0.\nIn the above lemma, the first statement is a well-known result for th e interlacing system.\nWe refer the readers to [14] for the proof. The second result can be found in [29] and it can\nalso be checked directly using the definition of eigenvectors. The th ird statement is obviously\na result of the first two statements.\nNow we give the proof of Theorem 1 as below:\nProof of Theorem 1. Letλbe an eigenvalue of Amfor some m∈Z\\{0}. We first show that\nReλ/lessorequalslant0. Suppose ris the associated eigenvector. Using Amr=λr, we have that\n(3.10) 0 /greaterorequalslant2r∗Hr=r∗(A∗\nm+Am)r= (λ+¯λ)r∗r= 2/ba∇dblr/ba∇dbl2Reλ.\nThus Re λ/lessorequalslant0.\nIt remains only to show that Re λ/\\e}atio\\slash= 0. If there exist λI∈Randr∈CM+1, such that\nAmr= iλIr, then\n(3.11) i λI/ba∇dblr/ba∇dbl2=r∗Amr=−imkr∗Ar+r∗Hr.\nThe symmetry of both AandHyieldsr∗Hr= 0, which is equivalent to Hr= 0 since H\nis diagonal. Recalling that the last diagonal entry of His assumed to be strictly negative, we\nknow that the last component of ris zero. Furthermore, we have\n(3.12) Ar=i\nmk(Am−H)r=−λI\nmkr,\nwhich indicates that ris an eigenvector of A. According to Lemma 1, the last component of\nrmust be nonzero, which is a contradiction. Therefore Amdoes not have purely imaginary\neigenvalues, which concludes the proof. /square\nThefollowingtheoremshowsthattheconvergenceratehasalower boundforallnon-constant\nFourier modes:\nTheorem 2. For allm∈Z\\{0}, suppose\n(3.13) Am:=−imkA+H=RmJmR−1\nm,\nwhereJmis the Jordan normal form of Am, and every column of Rmis a unit vector. Then\nthere exists a constant C(0)>0, such that\n(3.14) /ba∇dblRm/ba∇dbl2/lessorequalslantC(0),/vextenddouble/vextenddoubleR−1\nm/vextenddouble/vextenddouble\n2/lessorequalslantC(0).\nAnd there exists a constant λ(0)>0, such that for any m∈Z\\{0}, all the eigenvalues of Am\nhave real parts less than −λ(0).SUPPRESSION OF RECURRENCE FOR TRANSPORT EQUATIONS 7\nProof.Consider the matrix\n(3.15) Bm:=i\nmkAm=A+i\nmkH, m∈Z\\{0}.\nApparently the characteristic polynomial of Bmconverges to the characteristic polynomial of\nAasm→ ∞. Thus all the eigenvalues of Bmalso converge to the eigenvalues of Aasm→ ∞.\nLemma 1 implies that all the eigenvalues of Aare distinct. Therefore there exists an m0>0\nsuch that all the eigenvalues of Bmare distinct if |m|> m0. The relation between Amand\nBmindicates that all the eigenvalues of Amare also distinct when |m|> m0. In this case, the\nsimilarity transformation (3.13) becomes a diagonalization of Am, and every column of Rmis\na unit eigenvector of Am(orBm).\nTo show the bound (3.14), it is sufficient to show that /ba∇dblRm/ba∇dbl2and/ba∇dblR−1\nm/ba∇dbl2have a uniform\nupper bound for all |m|> m0. Letr(m)\nibe theith column of Rm, and suppose Bmr(m)\ni=\nµ(m)\nir(m)\ni. Sinceexchangingtwocolumnsof Rmdoesnotchangethenorms /ba∇dblRm/ba∇dbl2and/ba∇dblR−1\nm/ba∇dbl2,\nwe can assume\n(3.16) lim\nm→∞µ(m)\ni=λi,\nwhereλiis theith eigenvalue of A. Using the fact that r(m)\niis a unit vector, we have that\n0 = lim\nm→∞/parenleftig\nBm−µ(m)\niI/parenrightig\nr(m)\ni= lim\nm→∞(A−λiI)r(m)\ni.\nHence,\n(3.17) lim\nm→∞r(m)\ni=ri,\nwhereriis the unit eigenvector of Aassociated with the eigenvalue λi. ThusRmhas a limit R\nasm→ ∞, and the limit diagonalizes AasA=RΛR−1, which naturally leads to the bound\n(3.14).\nTo show that the bound λ(0)exists, we use the unity of vectors r(m)\nito get\n(3.18) lim\nm→∞Reλ(m)\ni= lim\nm→∞Re/parenleftig/parenleftig\nr(m)\ni/parenrightig∗\nAmr(m)\ni/parenrightig\n= lim\nm→∞/parenleftig\nr(m)\ni/parenrightig∗\nHr(m)\ni=r∗\niHri<0.\nThe last inequality comes from the proof of Theorem 1. The existenc e of the negative limit\nshows the existence of the negative upper bound. /square\nThe above theorem gives an upper bound for the real parts of the eigenvalues. A direct\ncorollary is\nCorollary 1. ∀m∈Zandt >0,/ba∇dblexp(tAm)/ba∇dbl2is uniformly bounded, and if m/\\e}atio\\slash= 0, it holds\nthat\n(3.19) /ba∇dblexp(tAm)/ba∇dbl2/lessorequalslantC(1)/parenleftbig\ntM+1/parenrightbig\nexp/parenleftig\n−λ(0)t/parenrightig\n,\nwhereC(1)= (M+1)/parenleftbig\nC(0)/parenrightbig2, and the constants λ(0)andC(0)are introduced in Theorem 2.\nThe estimate (3.19) is a result of Theorem 2 and the following lemma:\nLemma 2. For any matrix M, suppose Jis its Jordan normal form and M=XJX−1. The\nfollowing estimate holds for the norm of exp(tM):\n(3.20) /ba∇dblexp(tM)/ba∇dbl2/lessorequalslantβ/ba∇dblX/ba∇dbl2/ba∇dblX−1/ba∇dbl2max\n0/lessorequalslanti/lessorequalslantβ−1ti\ni!e−αt,\nwhereβis the maximum dimension of the Jordan blocks, and αis the maximum real part of\nthe eigenvalues of M.8 ZHENNING CAI AND YANLI WANG\nThe lemma can be found in [21]. The uniform boundedness in Corollary 1 is an immediate\nresult of (3.19) and /ba∇dblexp(tA0)/ba∇dbl2=/ba∇dblexp(tH)/ba∇dbl2= 1.\nThe estimate (3.19) shows the linearly exponential decay of the non -constant Fourier modes.\nCompared with (3.8), the decay rate is still not fast enough, which in dicates that the recurrence\nis not fully removed. However, when Mis sufficiently large, the filtered spectral method can\ngive accurate approximation of the decay rate up to some time T, and after time T, the values\nof both the exact solution and the numerical solution are already sm all enough, and therefore\nthe numerical result can still be considered as accurate, although the decay rate may not be\nexact. Examples will be given in Section 5 to show the aforementioned behavior.\n3.3.Advection Equation with an Exponentially Decaying Force. The above result can\nbe extended to the case with a given decaying force field. Here we as sume that\n(3.21) E(x,t) = exp(−α(t))w(x,t), x∈D,\nwhereα(t)/greaterorequalslantαEt >0 andw(x,t)∈L∞(D×[0,+∞)). Such a force field mimics the electric\nforce field in the Vlasov-Poissonequations, where the self-consist ent force decays exponentially.\nTherefore, it can be expected that the behavior of this equation is similar to the linear Landau\ndamping. Again, we study the equations of Fourier coefficients (2.18 ) instead of the original\nsystem (2.14). It will be shown that the system (2.14) has a steady state solution in which only\nthe constant modes are nonzero. Here we only consider the case w ith filter, which means that\nthe last diagonal entry of His strictly negative.\nFor the purely advective equation, each ˆf(m)can be considered independently, while in this\ncase, the Fourier coefficients ˆf(m)are fully coupled for all m∈Z. Therefore, we define a Hilbert\nspaceHwhose elements have the form\n(3.22) ˆg=/parenleftig\n···,ˆg(−m),···,ˆg(−1),ˆg(0),ˆg(1),···,ˆg(m),···/parenrightig\n,\nwhere each ˆg(m)is a vector in CM+1. For any vector in H, below we always use the superscript\n“(m)” to denote its mth component as in (3.22). The inner product of His defined as\n(3.23) /a\\}b∇acketle{tˆg1,ˆg2/a\\}b∇acket∇i}ht=/summationdisplay\nm∈Z/parenleftbigˆg(m)\n1/parenrightbig∗ˆg(m)\n2,∀ˆg1,ˆg2∈H.\nThereforebyParseval’sinequality (2.17), Hcorrespondstothe space[ L2(D)]M+1in the original\nspace. For the purpose of uniformity, the norm in the Hilbert space His denoted as /ba∇dbl·/ba∇dbl2below.\nTo represent the system (2.14), we introduce two operators ˆAandˆB(t), which are transfor-\nmations on H, defined by\n(3.24)˜g=ˆAg⇐⇒˜g(m)=Amg(m),∀m∈Z,\n˜g=ˆB(t)g⇐⇒˜g(m)=/summationdisplay\nl∈Zˆw(l)(t)Bg(m−l),∀m∈Z.\nwhere the matrix Amis defined in Theorem 1 as Am=−imkA+H, and ˆw(l)(t) arethe Fourier\ncoefficients for w(x,t):\n(3.25) w(x,t) =/summationdisplay\nm∈Zˆw(m)(t)exp(ikx), m∈Z.\nThus (2.14) can be written as\n(3.26)∂ˆf(t)\n∂t=ˆAˆf(t)+exp(−α(t))ˆB(t)ˆf(t).SUPPRESSION OF RECURRENCE FOR TRANSPORT EQUATIONS 9\nHereˆf(·) is a map from R+toH. Applying Duhamel’s principle, we can obtain its integral\nform as\n(3.27) ˆf(t) = exp/parenleftig\ntˆA/parenrightig\nˆf(0)+/integraldisplayt\n0exp(−α(s))exp/parenleftig\n(t−s)ˆA/parenrightig\nˆB(s)ˆf(s)ds.\nFor the operators exp( tˆA) andˆB(t), we claim that\nLemma 3. For allt/greaterorequalslant0, the operators exp(tˆA)andˆB(t)are uniformly bounded, i.e. there\nexist constants CAandCBsuch that\n(3.28)/vextenddouble/vextenddouble/vextenddoubleexp(tˆA)/vextenddouble/vextenddouble/vextenddouble\n2/lessorequalslantCA,/vextenddouble/vextenddouble/vextenddoubleˆB(t)/vextenddouble/vextenddouble/vextenddouble\n2/lessorequalslantCB,∀t >0.\nThis lemma can be easily obtained by Corollary 1 and the boundedness o fw(x,t). The\ndetailed proof is left for the readers. Below we state the main result of this subsection:\nTheorem 3. Letˆf(t)be the solution to (3.27). There exists ˆf∞∈Hsuch that\n(3.29) lim\nt→+∞/ba∇dblˆf(t)−ˆf∞/ba∇dbl2= 0,\nandˆf∞satisfies\n(3.30) ˆf(m)\n∞= 0,∀m∈Z\\{0}.\nThis theorem shows that the steady solution of (2.14) exists and is a constant. It is known\nfrom Section 3.1 that when recurrence appears, some non-const ant modes will never decay,\nwhich causes the phenomenon that similar solutions appear again and again when all these\nmodes are close to the peaks of the waves. Consequently, no stea dy state solution exists in such\na case. In this sense, the theorem implies the suppression of recur rence by the filter. Before\nproving this theorem, we first show the boundedness of the solutio n:\nLemma 4. Letˆfbe the solution to (3.27). Then there exists a constant Cfsuch that\n(3.31)/vextenddouble/vextenddouble/vextenddoubleˆf(t)/vextenddouble/vextenddouble/vextenddouble\n2/lessorequalslantCf/vextenddouble/vextenddouble/vextenddoubleˆf(0)/vextenddouble/vextenddouble/vextenddouble\n2.\nProof.Taking the norm on both sides of (3.27) and plugging in the inequalities ( 3.28), we\nobtain\n/vextenddouble/vextenddouble/vextenddoubleˆf(t)/vextenddouble/vextenddouble/vextenddouble\n2/lessorequalslantCA/vextenddouble/vextenddouble/vextenddoubleˆf(0)/vextenddouble/vextenddouble/vextenddouble\n2+CA/integraldisplayt\n0exp(−α(s))/vextenddouble/vextenddouble/vextenddoubleˆBˆf(s)/vextenddouble/vextenddouble/vextenddouble\n2ds\n/lessorequalslantCA/vextenddouble/vextenddouble/vextenddoubleˆf(0)/vextenddouble/vextenddouble/vextenddouble\n2+CACB/integraldisplayt\n0exp(−αEs)/vextenddouble/vextenddouble/vextenddoubleˆf(s)/vextenddouble/vextenddouble/vextenddouble\n2ds(3.32)\nBy Gronwall’s inequality, we immediately have the estimation (3.31) with t he constant Cf=\nCAexp(/ba∇dblw/ba∇dbl∞CACB/αE). /square\nNow we present the proof of Theorem 3:\nProof of Theorem 3. Letˆb(t) =ˆB(t)ˆf(t). The boundedness of ˆB(t) and Lemma 4 yield that\n(3.33) /ba∇dblˆb(t)/ba∇dbl2/lessorequalslantCBCf/ba∇dblˆf(0)/ba∇dbl2.\nWriting the integral system (3.27) as\n(3.34) ˆf(m)(t) = exp(tAm)ˆf(m)(0)+/integraldisplayt\n0exp(−α(s))exp((t−s)Am)ˆb(m)(s)ds,∀m∈Z,10 ZHENNING CAI AND YANLI WANG\nwe can bound/vextenddouble/vextenddouble/vextenddoubleˆf(m)(t)/vextenddouble/vextenddouble/vextenddouble\n2withm∈Z\\{0}by\n\n/summationdisplay\nm∈Z\\{0}/vextenddouble/vextenddouble/vextenddoubleˆf(m)(t)/vextenddouble/vextenddouble/vextenddouble2\n2\n1/2\n/lessorequalslantC(1)(tM+1)exp( −λ(0)t)\n/summationdisplay\nm∈Z\\{0}/vextenddouble/vextenddouble/vextenddoubleˆf(m)(0)/vextenddouble/vextenddouble/vextenddouble2\n2\n1/2\n+/integraldisplayt\n0C(1)[(t−s)M+1]exp/parenleftBig\n−λ(0)(t−s)/parenrightBig\nexp(−αEs)\n/summationdisplay\nm∈Z\\{0}/vextenddouble/vextenddouble/vextenddoubleˆb(m)(s)/vextenddouble/vextenddouble/vextenddouble2\n2\n1/2\nds\n/lessorequalslantC(1)(tM+1)exp( −λ(0)t)/parenleftbigg\n/bardblˆf(0)/bardbl2+/integraldisplayt\n0exp/parenleftBig\n(λ(0)−αE)s/parenrightBig\n/bardblˆb(s)/bardbl2ds/parenrightbigg\n/lessorequalslantC(1)(tM+1)/parenleftbigg\nexp(−λ(0)t)+CBCf\n|λ(0)−αE|/bracketleftBig\nexp(−λ(0)t)+exp(−αEt)/bracketrightBig/parenrightbigg\n.(3.35)\nThe right hand side goes to zero as tgoes to infinity. Thus it remains only to prove ˆf(0)(t)\nhas a limit.\nWhenm= 0, the equation (3.34) becomes\n(3.36) ˆf(0)(t) = exp(tH)ˆf(0)(0)+/integraldisplayt\n0exp(−α(s))exp((t−s)H)ˆb(0)(s)ds.\nFor any i= 0,···,M, ifhi<0, we can still get ˆf(0)\ni(+∞) = 0 using the same method as in\n(3.35). If hi= 0,\n(3.37) lim\nt→+∞ˆf(0)\ni(t) =ˆf(0)\ni(0)+/integraldisplay+∞\n0exp(−α(s))ˆb(0)\ni(s)ds.\nThe uniform boundedness of ˆb(0)\niandα(t)/greaterorequalslantαEt >0 shows the convergence of the integral on\nthe right hand side, which concludes the proof. /square\nRemark 1.In Theorem 3, one can also observe an exponential convergence r ate to the steady\nstate solution. However, the convergence rate depends on both the eigenvalue bound λ(0)and\nthe decay rate of the force field αE, whereas in Section 3.2, the convergence rates depend only\non the eigenvalue bounds. In fact, if the force field does not decay , the steady state solution\nmay not exist. A simple example is to let Ebe a constant. Then the distribution function will\nkeep moving in the velocity space, and there is no mechanism which bala nces such a force to\nachieve the steady state. It can also be expected that current v elocity discretization cannot\nwell describe the system when tis large.\n4.Linear Landau Damping in the Vlasov-Poisson Equation\nIn this section, we will generalize the result to the linear Landau damp ing problem, for which\nthe recurrence in the electric energy has been widely observed [24 , 10]. To study the linear\nLandau damping, we consider the dimensionless Vlasov-Poisson (VP) equation which models\nthe motion of a collection of charged particles in the self-consistent electric field. For a given\ndistribution function f(x,ξ,t), the self-consistent electric field Esc(x,t) is given by\n(4.1)∂Esc(x,t)\n∂x=/parenleftbigg/integraldisplay\nRf(x,ξ,t)dξ−1\nD/integraldisplay\nD×Rf(x,ξ,t)dξdx/parenrightbigg\n.\nwith constraint\n(4.2)/integraldisplay\nDEsc(x,t)dx= 0.SUPPRESSION OF RECURRENCE FOR TRANSPORT EQUATIONS 11\nThus the Vlasov-Poisson equation can be written as (1.2) with E(x,t) =Esc(x,t). The initial\ncondition is a uniform Gaussian distribution with perturbed electric ch arge density in the x-\nspace:\n(4.3) f(x,ξ,0) =1√\n2π[1+ǫexp(ikx)]exp/parenleftbigg\n−ξ2\n2/parenrightbigg\n.\nWhenǫis small, it is known that the total electric energy\n(4.4) E(t) =/radicaligg/integraldisplay\nD|Esc(x,t)|2dx\ndecays exponentially. For this example, we say that a recurrence e xists in some numerical\nmethods if the electric energy does not decay with time.\n4.1.Asymptotic expansion and recurrence. Following the classical analysis [11], we ex-\npand the coefficients f= (f0,f1,···,fM)Tin terms of ǫ:\n(4.5) f(x,t) =ˆf(0)(t)+ǫˆf(1)(t)exp(ikx)+ǫ2ˆf(2)(t)exp(2ikx)+···,\nwhereˆf(m)= (ˆf(m)\n0,ˆf(m)\n1,···,ˆf(m)\nM)T. Correspondingly, the electric field E(x,t) is expanded\nas\n(4.6) E(x,t) =ǫˆE(1)(t)exp(ikx)+ǫ2ˆE(2)(t)exp(2ikx)+ǫ3ˆE(3)(t)exp(3ikx)+···.\nNote that the definitions of ˆf(m)andˆE(m)are slightly different from those defined in (2.16).\nIn this section, these symbols denote the Fourier coefficients scale d byǫm. Inserting (4.5)(4.6)\nand (2.1) into (4.1), we get that\n(4.7) ˆE(m)=−i\nmkˆf(m)\n0,∀m= 1,2,···.\nThe equations for the coefficients of all orders can be obtained by s ubstituting (4.5) and\n(4.6) into (2.14) and matching the terms with the same orders of ǫ:\nO(1) :∂ˆf(0)\n∂t=Hˆf(0),ˆf(0)(0) = (1,0,···,0)T.\nO(ǫ) :∂ˆf(1)\n∂t+ikAˆf(1)+i\nkˆf(1)\n0Bˆf(0)=Hˆf(1),ˆf(1)(0) = (1,0,···,0)T.\nO(ǫm) :∂ˆf(m)\n∂t+imkAˆf(m)+i\nmkˆf(m)\n0Bˆf(0)+i\nkm−1/summationdisplay\nl=11\nlˆf(l)\n0Bˆf(m−l)=Hˆf(m),ˆf(m)(0) = 0.\nThe last equation holds for all m/greaterorequalslant2. Recalling that His a diagonal matrix with its first entry\nbeing zero, we can easily obtain from the O(1) equation that ˆf(0)(t)≡ˆf(0)(0) = (1,0,···,0)T.\nTherefore the other two equations can be rewritten as\n∂ˆf(1)\n∂t+ik/parenleftbigg\nA+1\nk2G/parenrightbigg\nˆf(1)=Hˆf(1),ˆf(1)(0) = (1,0,···,0)T; (4.8)\n∂ˆf(m)\n∂t+imk/parenleftbigg\nA+1\n(mk)2G/parenrightbigg\nˆf(m)+i\nkm−1/summationdisplay\nl=1ˆf(l)\n0Bˆf(m−l)=Hˆf(m),ˆf(m)(0) = 0. (4.9)12 ZHENNING CAI AND YANLI WANG\nHere we have used the symbol Gto denote the ( M+1)×(M+1) matrix with only one nonzero\nentry locating at the second row and first column:\n(4.10) G=\n0 0···0\n1 0···0\n0 0···0\n............\n0 0···0\n.\nWhenH= 0, the recurrence phenomenon already exists in the first-order equation (4.8).\nTo observe this, we introduce the following lemma:\nLemma 5. For any m >0, there exists a diagonal matrix Dmsuch that such that the matrix\nDm/parenleftbigg\nA+1\n(mk)2G/parenrightbigg\nD−1\nm\nis symmetric and tridiagonal, and therefore the matrix A+(mk)−2Gis real diagonalizable.\nProof.The matrix Dmcan be explicitly written as\n(4.11) Dm= diag/braceleftigg\n1,mk/radicalbig\n(mk)2+1,mk/radicalbig\n(mk)2+1,···,mk/radicalbig\n(mk)2+1/bracerightigg\n.\nIt can then be easily verified that\nDm/parenleftbigg\nA+1\n(km)2G/parenrightbigg\nD−1\nm=\n0/radicalbig\n1+(mk)−20 0 ...0/radicalbig\n1+(mk)−2 0√\n2 0 ...0\n0√\n2 0√\n3...0\n0...............\n...... 0√\nM−1 0√\nM\n0 ... 0 0√\nM0\n.\n/square\nThe above lemma shows that when H= 0, all the eigenvalues of i k(A+k−2G) are purely\nimaginary, which indicates that ˆf(1)does not decay as time increases. Consequently, the\nrelation (4.7) shows that the electric energy will not decay, resultin g in the recurrence in the\nsimple discretization of velocity without using filters.\n4.2.Suppression of recurrence. This section focuses on the effect ofthe filter. As in Section\n3.2, we assume again that the filter matrix His a negative semidefinite diagonal matrix with\nits first diagonal entry being zero and last diagonal entry being non zero. In this section, we are\ngoing to show an exponential decay of the solution, which suppress es the recurrence. In detail,\nwe have the following theorem:\nTheorem 4. Letf(x,t)be the series (4.5)whereˆf(0)(t)≡(1,0,···,0)Tandˆf(m)withm/greaterorequalslant1\nare solved from the equations (4.8)(4.9) . There exist constants C(3)andλ(1)such that\n(4.12)/vextenddouble/vextenddouble/vextenddoublef(·,t)−ˆf(0)/vextenddouble/vextenddouble/vextenddouble\n2/lessorequalslantC(3)/parenleftbig\ntM+1/parenrightbig\nexp/parenleftig\n−λ(1)t/parenrightig\n,\nifǫis sufficiently small.\nSimilar to the case ofthe advectionequation, the decay rate λ(1)is associatedwith the eigen-\nvalues of matrices appearing in the equations. The following lemma give s a precise definition\nofλ(1):SUPPRESSION OF RECURRENCE FOR TRANSPORT EQUATIONS 13\nLemma 6. For allm∈Z\\{0}, define\n(4.13) Gm:=−imk/parenleftbigg\nA+1\n(mk)2G/parenrightbigg\n+H.\nThen there exists a uniform constant λ(1)>0, such that all the eigenvalues of Gmhave real\nparts less than −λ(1). Furthermore, there exists a constant C(4)such that\n(4.14) /ba∇dblexp(tGm)/ba∇dbl2/lessorequalslantC(4)(tM+1)exp/parenleftig\n−λ(1)t/parenrightig\n,∀m∈Z\\{0}.\nProof.Consider the matrix\n(4.15) DmGmD−1\nm=−imk/parenleftbigg\nDm/parenleftbigg\nA+1\n(mk)2G/parenrightbigg\nD−1\nm/parenrightbigg\n+H,\nwhich has the same eigenvalues as Gm. Due to Lemma 5, we see that the above matrix is\nsymmetric and tridiagonal, and all its subdiagonal entries are nonze ro. Thus, we can use the\nsame strategy as in the proof of Theorem 1 to show that all the eige nvalues of (4.15) have\nnegative real parts. Similarly, since\nlim\nm→∞Dm/parenleftbigg\nA+1\n(mk)2G/parenrightbigg\nD−1\nm=A,\nshowing the existence of a uniform bound λ(1)is an analog of the proof of Theorem 2. The\ndetails are left to the readers.\nTo show the inequality (4.14), we first follow the proof of Corollary 1 t o get\n(4.16)/vextenddouble/vextenddoubleexp(tDmGmD−1\nm)/vextenddouble/vextenddouble\n2/lessorequalslantC(2)/parenleftbig\ntM+1/parenrightbig\nexp/parenleftig\n−λ(1)t/parenrightig\n,∀m∈Z\\{0}.\nwhereC(2)is determined in the same way as C(1)in Corollary 1 and Theorem 2. Hence,\n/ba∇dblexp(tGm)/ba∇dbl2/lessorequalslant/ba∇dblD−1\nm/ba∇dbl2/ba∇dblexp(tDmGmD−1\nm)/ba∇dbl2/ba∇dblDm/ba∇dbl2\n=/radicaligg\n1+1\n(mk)2/ba∇dblexp(tDmGmD−1\nm)/ba∇dbl2/lessorequalslant/radicalbigg\n1+1\nk2C(2)(tM+1)exp/parenleftig\n−λ(1)t/parenrightig\n.\nThus (4.14) holds for C(4)=√\n1+k−2C(2). /square\nIn the above lemma, the estimate (4.14) gives the basic form of the d ecay rate. To turn\n(4.14) into an estimate of the solution (4.12), we follow the steps belo w:\n(1) Show that each non-constant term in the series (4.5) decays e xponentially.\n(2) Show the convergence of the series for small ǫ.\nThe first step is established by proving the following theorem:\nTheorem 5. Letˆf(m),m/greaterorequalslant1be the solutions to the equations (4.8)(4.9) . For each positive\nintegerm, there exists a constant K(m)such that\n(4.17)/vextenddouble/vextenddouble/vextenddoubleˆf(m)(t)/vextenddouble/vextenddouble/vextenddouble\n2/lessorequalslantK(m)(tM+1)exp( −λ(1)t).\nProof.Note that the equation (4.8) is linear. Therefore we can take m= 1 in (4.14) to get\n(4.18)/vextenddouble/vextenddouble/vextenddoubleˆf(1)(t)/vextenddouble/vextenddouble/vextenddouble\n2=/vextenddouble/vextenddouble/vextenddoubleexp(G1t)ˆf(1)(0)/vextenddouble/vextenddouble/vextenddouble\n2/lessorequalslantC(4)(tM+1)exp/parenleftig\n−λ(1)t/parenrightig\n.\nThus (4.17) is proven for m= 1 by taking K(1)=C(4).\nForm/greaterorequalslant2, we use induction and suppose that the inequality\n(4.19)/vextenddouble/vextenddouble/vextenddoubleˆf(j)(t)/vextenddouble/vextenddouble/vextenddouble\n2/lessorequalslantK(j)/parenleftbig\ntM+1/parenrightbig\nexp/parenleftig\n−λ(1)t/parenrightig\n,14 ZHENNING CAI AND YANLI WANG\nholds for all j < m. To estimate ˆf(m)(t), we write the explicit solution of (4.9) as\n(4.20) ˆf(m)(t) =/integraldisplayt\n0exp((t−s)Gm)pm(s)ds,\nwhere we have used the definition\n(4.21) pm(t) :=−i\nkm−1/summationdisplay\nl=1ˆf(l)\n0(t)Bˆf(m−l)(t)\nfor conciseness. Since /ba∇dblB/ba∇dbl2=√\nM, the vector pmcan be bounded by\n/ba∇dblpm(t)/ba∇dbl2=/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble−i\nkm−1/summationdisplay\nl=11\nlˆf(l)\n0(t)Bˆf(m−l)(t)/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble\n2/lessorequalslant√\nM\nkm−1/summationdisplay\nl=11\nl/vextenddouble/vextenddouble/vextenddoubleˆf(l)(t)/vextenddouble/vextenddouble/vextenddouble\n2/vextenddouble/vextenddouble/vextenddoubleˆf(m−l)(t)/vextenddouble/vextenddouble/vextenddouble\n2\n/lessorequalslant√\nM\nkm−1/summationdisplay\nl=11\nlK(l)K(m−l)/parenleftbig\ntM+1/parenrightbig2exp/parenleftig\n−2λ(1)t/parenrightig\n,(4.22)\nwhere the last inequality is an application of the inductive hypothesis. Now we can take the\nnorm on both sides of (4.20) and plug in the inequalities (4.14) and (4.22 ) to get\n/vextenddouble/vextenddouble/vextenddoubleˆf(m)(t)/vextenddouble/vextenddouble/vextenddouble\n2/lessorequalslant/integraldisplayt\n0/ba∇dblexp((t−s)Gm)/ba∇dbl2/ba∇dblpm(s)/ba∇dbl2ds\n/lessorequalslantC(5)m−1/summationdisplay\nl=11\nlK(l)K(m−l)/parenleftbig\ntM+1/parenrightbig\nexp/parenleftig\n−λ(1)t/parenrightig\n,(4.23)\nwhere\n(4.24) C(5)=√\nM\nkC(4)/integraldisplay∞\n0/parenleftbig\nsM+1/parenrightbig2exp/parenleftig\n−λ(1)s/parenrightig\nds.\nThe inequality (4.23) shows that (4.17) holds for\n(4.25) K(m)=C(5)m−1/summationdisplay\nl=11\nlK(l)K(m−l)=C(5)m−2/summationdisplay\nl=01\nl+1K(l+1)K(m−l−1).\nTherefore for all positive integer m, the estimate (4.17) holds according to the principle of the\nmathematical induction. /square\nFrom the above theorem, we see that if the distribution function (4 .5) is approximated by\na truncation of the series, then such a finite series converges to a constant as t→ ∞. To\nprove such a property for the infinite series (4.5), we still need to s tudy the magnitude of each\ncoefficient K(m). The recursion relation (4.25) reminds us of Jonah’s theorem introd uced in\n[17]:\nLemma 7 (Jonah).Ifm1/greaterorequalslant2m2, then\n(4.26)/parenleftbiggm1+1\nm2/parenrightbigg\n=m2/summationdisplay\nl=01\nl+1/parenleftbigg2l\nl/parenrightbigg/parenleftbiggm1−2l\nm2−l/parenrightbigg\n.\nBy this lemma, a general formula of K(m)can be explicitly written, and then it can be\nproperly bounded. The details are listed in the following proof:SUPPRESSION OF RECURRENCE FOR TRANSPORT EQUATIONS 15\nProof of Theorem 4. We first claim that\n(4.27) K(m+1)=1\n2m/parenleftbigg2m\nm/parenrightbigg\nK(1)/parenleftig\nC(5)K(1)/parenrightigm\n.\nApparently the above equality holds for m= 0. Ifm >0, we just need to verify that (4.27)\nfulfills (4.25). By inserting (4.27) into (4.25) and cancelling out some co nstants on both sides,\nwe obtain\n(4.28)1\n2/parenleftbigg2m\nm/parenrightbigg\n=m−1/summationdisplay\nl=01\nl+1/parenleftbigg2l\nl/parenrightbigg/parenleftbigg2(m−l−1)\nm−l−1/parenrightbigg\n.\nTo verify this equality, we apply Jonah’s theorem (4.26) and let m1= 2(m−1),m2=m−1.\nThus the right hand side of (4.28) matches the right hand side of (4.2 6). The left hand sides\nare also equal since\n(4.29)1\n2/parenleftbigg\n2m\nm/parenrightbigg\n=1\n2(2m)!\nm!m!=1\n22m\nm(2m−1)!\nm!(m−1)!=/parenleftbigg\n2m−1\nm−1/parenrightbigg\n=/parenleftbigg\nm1+1\nm2/parenrightbigg\n.\nBased on (4.27), it is easy to bound K(m)by\n(4.30) K(m+1)= 2m(2m−1)!!\n(2m)!!K(1)/parenleftig\nC(5)K(1)/parenrightigm\n/lessorequalslantK(1)/parenleftig\n2C(5)K(1)/parenrightigm\n.\nNow we can use (4.5) to get\n/vextenddouble/vextenddouble/vextenddoublef(·,t)−ˆf(0)/vextenddouble/vextenddouble/vextenddouble\n2/lessorequalslant√\nD+∞/summationdisplay\nm=0ǫm+1/vextenddouble/vextenddouble/vextenddoubleˆf(m+1)/vextenddouble/vextenddouble/vextenddouble\n2/lessorequalslant√\nD+∞/summationdisplay\nm=0ǫm+1K(m+1)(tM+1)exp( −λ(1)t)\n/lessorequalslantǫDK(1)+∞/summationdisplay\nm=0/parenleftig\n2ǫC(5)K(1)/parenrightigm\n(tM+1)exp( −λ(1)t).(4.31)\nNow it is clear that when ǫ </bracketleftbig\n2C(5)K(1)/bracketrightbig−1, the inequality (4.12) holds for\n/square (4.32) C(3)=ǫDK(1)\n1−2ǫC(5)K(1).\n4.3.Analysis on the damping rate of the electric energy. In this section, we will analyze\nhow the filter affects the damping rate and the oscillation frequency of the electric energy in\nthe problem of Landau damping. Below we will first review the analysis o n the original VP\nequation, and then the filtered equation will be discussed.\n4.3.1.Analysis on the VP equation. In the linear Landau damping problem, the plasma wave\nhas very small amplitude. Thus we can assume that\n(4.33) f(x,ξ,t) =f(0)(ξ)+ǫf(1)(x,ξ,t),\nwheref(0)(ξ) is the background distribution, which is uniform in x, andf(1)(x,ξ) has the\norder of magnitude O(1). Inserting this equation into the VP equation and ignoring the te rms\nquadratic in ǫ, we get the following linearized VP equation:\n(4.34)∂f(1)\n∂t+ξ∂f(1)\n∂x+E(1)∂f(0)\n∂ξ= 0,∂E(1)\n∂x=/integraldisplay\nRf(1)dξ.\nIf we further assume that the plasma wave being considered takes the form of a plane wave\ntraveling in the x-direction:\n(4.35) f(1)(x,t,ξ) =ˆf(1)(ξ)exp(−iωt+ikx), E(1)(x,t) =ˆE(1)exp(−iωt+ikx),16 ZHENNING CAI AND YANLI WANG\nwe obtain the dispersion relation\n(4.36)1\nk2/integraldisplay\nR1\nξ−ω/k∂f(0)\n∂ξdξ= 1.\nLetω=ωp+iγbe the solution to (4.36). Then γandωpare respectively the damping rate\nand the oscillation frequency of the electric energy. We refer the r eaders to [11] for the more\ndetails on the dispersion relation.\n4.3.2.Analysis on the filtered VP equation. In order to take into account the filters, we assume\nthat the filter (2.11) is used and consider the “modified equation” (2 .12), so that a similar\nanalysis can be carried out. Here we also assume that the distributio n function is a small\nperturbation of the background distribution (4.33), and the linear ized VP equation with filter\ncan be written as\n(4.37)∂f(1)\n∂t+ξ∂f(1)\n∂x+E(1)∂f(0)\n∂ξ=Hf(1),∂E(1)\n∂x=/integraldisplay\nRf(1)dξ,\nwhere the operator His defined by H= (−1)p+1α(∆tMp)−1Dp, and we have assumed Hf(0)=\n0, which holds for the initial condition (4.3). Let\n(4.38) g(x,ξ,t) = exp(−tH)f(1)(x,ξ,t).\nThen the function gformally satisfies\n(4.39) exp( tH)∂g\n∂t=−/parenleftbigg\nξexp(tH)∂g\n∂x+E(1)∂f(0)\n∂ξ/parenrightbigg\n,∂E(1)\n∂x=/integraldisplay\nRexp(tH)gdξ.\nNow we assume that the equation (4.39) has a plane-wave solution:\n(4.40) g(x,ξ,t) = ˆg(ξ)exp(−iωt+ikx), E(1)(x,t) =ˆE(1)exp(−iωt+ikx),\nwhich changes (4.39) into\n(4.41) i( ξk−ω)exp(tH)ˆg=−ˆE(1)∂f(0)\n∂ξ,ikˆE(1)=/integraldisplay\nRexp(tH)ˆgdξ.\nBy dividing the first equation by ξk−ωand integrating with respect to ξ, the amplitude of the\nelectric field ˆE(1)can be cancelled out and we get the result\n(4.42) k=/integraldisplay\nR1\nξk−ω∂f(0)\n∂ξdξ,\nwhich turns out to be exactly the same as the result of the original V P equation (4.36). Such\nanalysis shows that the modified equation and the originalVP equatio n havethe same damping\nrate and oscillation frequency for the electric field. Therefore add ing filters does not ruin the\nmajor behavior of the linear Laudau damping.\nRemark2.The aboveanalysisis onlyformallycorrect, since in (4.38), the opera torexp(−tH) is\ngenerallyan unbounded operator. Thereforeit still remainsto pro vethatf(1)liesin the domain\nof this operator for all xandt, which is not yet done. Currently we are not too concerned\nabout this point since the main idea of this section is to provide a clue fo r the reliability of\nfilters.SUPPRESSION OF RECURRENCE FOR TRANSPORT EQUATIONS 17\n5.Numerical Results\nIn order to verify the theoretical results and show how the recur rence is suppressed by the\nfilter, the numerical experiments for both the advection equation and the Vlasov equation\nare carried out. In the literature, the recurrence for the Landa u damping problem is usually\nobservedfrom the evolution ofthe self-consistentelectric energ yE(t) defined in (4.4). Therefore\nin all the numerical tests, we are going to use the same quantity (4.4 ) to show the effect of\nthe filter. Note that for the advection equations considered in Sec tion 3, we do not have\nE(x,t) =Esc(x,t) as in the Vlasov equation, and thus the quantity E(t) is merely a functional\nof the distribution function f(x,ξ,t) defined by (4.1) (4.4), and does not appear explicitly in\nthe transport equation.\nThe filter we adopt in our numerical tests is the one proposed in [20], w hich is identical to\nthe filter defined in (2.11) with α= 36 and p= 36. The corresponding filter matrix Hcan be\nwritten as (2.15) with\n(5.1) hi=−36∆t−1(i/M)36, i= 0,···,M.\nAs mentioned in Section 2, the time step ∆ tis set as ∆ t=C/√\nM, and the constant Cis\nchosen as 0 .5 in all the tests. Since it is much easier to obtain E(t) from the Fourier coefficients\nˆf(m), we solve (2.18) instead of the original transport equation (1.2) in a ll our experiments.\n5.1.Advection equation. In this section, we focus on the advection equation (3.1) with\ninitial condition (3.2). This initial condition gives the following Fourier co efficients:\nˆf(0)(0) = (1,0,···,0)T,ˆf(−1)(0) =ˆf(1)(0) = (ǫ/2,0,···,0)T,ˆf(m)(t) = 0,|m|/greaterorequalslant2.\nIn this case, the equations (3.3) shows that each ˆf(m)can be solved independently, and when\n|m|/greaterorequalslant2, the initial condition shows that ˆf(m)(t)≡0. Consequently, the function E(t) has a\nsimple form\n(5.2) E(t) =/radicaligg\nD\nk2/parenleftbigg/vextendsingle/vextendsingle/vextendsingleˆf(−1)\n0(t)/vextendsingle/vextendsingle/vextendsingle2\n+/vextendsingle/vextendsingle/vextendsingleˆf(1)\n0(t)/vextendsingle/vextendsingle/vextendsingle2/parenrightbigg\n.\nThe exact solution of E(t) can be directly obtained from (3.7) as\n(5.3) E(t) =ǫ\nk/radicalbigg\nD\n2exp(−k2t2/2).\nTo obtain E(t) with the spectral method, wejust need to apply (3.9) for m=±1, which requires\ndiagonalization of matrices A±1.\nIn this example, we set D= 4πandk= 2π/D= 1/2. We first verify that all the eigenvalues\nofA±1have negative real parts. Since A1=A∗\n−1, the eigenvalues of A1are the complex\nconjugates of those of A−1. Therefore it is sufficient to just look at A1. The distribution of\nthe eigenvalues of A1on the complex plane is plotted in Figure 1, and the eigenvalues of i kA\n(the case without filter) are also given as a reference. As predicte d, all the eigenvalues of i kA\nlocate exactly on the imaginary axis, while all the eigenvalues of A1have strictly negative real\nparts.\nThetimeevolutionsof EanditslogarithmareplottedrespectivelyinFigure2and3. Initially,\nbothnumericalsolutionswithandwithoutthefilterdecayinthesame wayastheexactsolution.\nWhen the evolution reaches the first “critical” point, almost simultan eously, both solutions\nleave the exact solution and exhibit a fast growth. For the case with out filter, the value of\nE(t) bounces almost back to its initial value before the next decay, and such a process repeats\nperiodically, which implies a constantly appearing recurrence. In gen eral, larger number of18 ZHENNING CAI AND YANLI WANG\n-40 -30 -20 -10 0 10 20-6-4-20246\nw/o filter\nw/ filter\n0-6-4-20246\n(a)M= 30-40 -30 -20 -10 0 10 20-8-6-4-202468\nw/o filter\nw/ filter\n0-8-6-4-202468\n(b)M= 60\nFigure 1. The eigenvalues of i kAandA1withM= 30 and 60. The circles\nare the eigenvalues of i kAand the plus signs are the eigenvalues of A1.\nmoments leads to a longer recurrence time, which can also be observ ed by comparing the\nresults with M= 30 and M= 60. When the filter is applied, the solutions behave similarly\nexcept that the peak values get smaller as tgets larger, due to the damping effect of the filter.\nIt can also be observedthat the damping rate for M= 60 is higher than that for M= 30, which\nindicates that for the same filter, systems with more moments may h ave better suppression of\nrecurrence. We comment here that the recurrence is not complet ely eliminated since E(t) is\nstill not monotonically decreasing. Nevertheless, from Figure 2, on e sees that the difference\nbetween the numerical solution and the analytical solution is negligible whentis large.\n0 20 40 60 80 100012345 w/o filter\nw/ filter\nexact\n(a)M= 300 20 40 60 80 100012345 w/o filter\nw/ filter\nexact\n(b)M= 60\nFigure 2. The time evolution of E(t)/ǫwithM= 30 and 60.\n5.2.Advection equation with given force field. In this section, some numerical experi-\nments for the transport equation with a given force field (3.21) are performed. Here we choose\nα(t) andw(t) respectively as\n(5.4) α(t) =γt, w (t) =ǫcos(ωt)exp(ikx),SUPPRESSION OF RECURRENCE FOR TRANSPORT EQUATIONS 19\n0 20 40 60 80 100-30-25-20-15-10-505\nw/o filter\nw/ filter\nexact\n(a)M= 300 20 40 60 80 100-50-45-40-35-30-25-20-15-10-505\nw/o filter\nw/ filter\nexact\n(b)M= 60\nFigure 3. The time evolution of log( E(t)/ǫ) withM= 30 and 60.\n0 10 20 30 40 50 60-16-14-12-10-8-6-4-202\nPSfrag replacements\nw/o filter mc= 3\nw/o filter mc= 5\nw/ filter mc= 3\nw/ filter mc= 5\n(a)M= 300 10 20 30 40 50 60-18-16-14-12-10-8-6-4-202\nPSfrag replacements\nw/o filter mc= 3\nw/o filter mc= 5\nw/ filter mc= 3\nw/ filter mc= 5\n(b)M= 60\nFigure 4. The evolution of log E(t) for the example in Section 5.2.\nso that it mimics the Landau damping process. The period and the wav e number are again\nchosen as D= 4π,k= 0.5. For Laudau damping, these parameters give the decay rate\nγ= 0.15336 and the plasma oscillation frequency ω= 1.416 (see Section 5.3 for details), which\nare also adopted in definition of w(t). The initial condition is given by (4.3). To see the\ncontribution from waves with higher frequencies, we set ǫto be 0.9. In order to carry out the\nnumerical simulation, only Fourier modes with |m|/lessorequalslantmcare taken into account.\nTwo numbers of moments M= 30 and M= 60 are simulated with mc= 3 and mc= 5.\nThe time evolution of the logarithm of Eis plotted in Figure 4, which also shows clearly how\nthe filter improves the solution. Figure 5 also shows such an effect fr om another point of view,\nwhere all the non-constant modes are taken into account. For th e case with filters, the decay\nof each mode is given in Figure 6. A reference decay rate given by the largest real part of the\neigenvalues of Amis also provided as a reference. It is seen that the decay rates of a ll the\nFourier modes are controlled by this reference line.\n5.3.Linear Landau damping. The numerical effect of the filter in the simulation of linear\nLandau damping has been studied in a number of previous works [23, 2 4]. Here some simple20 ZHENNING CAI AND YANLI WANG\n0 10 20 30 40 50 60-8-6-4-2024\nPSfrag replacements\nw/o filter mc= 3\nw/o filter mc= 5\nw/ filter mc= 3\nw/ filter mc= 5\ny=−λ(1)t\n(a)M= 300 10 20 30 40 50 60-8-6-4-2024\nPSfrag replacements\nw/o filter mc= 3\nw/o filter mc= 5\nw/ filter mc= 3\nw/ filter mc= 5\ny=−λ(1)t\n(b)M= 60\nFigure 5. The evolution of log( /ba∇dblˆf−ˆf(0)/ba∇dbl2) for the example in Section 5.2.\n0 10 20 30 40 50 60-35-30-25-20-15-10-505\nPSfrag replacements\nlog/bardblˆf(1)/bardbl2\nlog/bardblˆf(2)/bardbl2\nlog/bardblˆf(3)/bardbl2\nlog/bardblˆf(4)/bardbl2\nlog/bardblˆf(5)/bardbl2\ny=−λ(1)t\n(a)M= 30,mc= 50 10 20 30 40 50 60-35-30-25-20-15-10-505\nPSfrag replacements\nlog/bardblˆf(1)/bardbl2\nlog/bardblˆf(2)/bardbl2\nlog/bardblˆf(3)/bardbl2\nlog/bardblˆf(4)/bardbl2\nlog/bardblˆf(5)/bardbl2\ny=−λ(1)t\n(b)M= 60,mc= 5\nFigure 6. The evolution of log( /ba∇dblˆf(m)/ba∇dbl2) for the example in Section 5.2.\nresults are presented for our particular discretization (2.1). The parameters are chosen as\nǫ= 0.001,D= 4πandk= 1/2. Since ǫis quite small, it is sufficient to consider the leading\norder term ǫˆE(1)(t)exp(ikx) in the expansion (4.6).\nTheevolutionoflog E(t)withM= 90and M= 120isplottedinFigure7,wherethesolutions\nwith and without filters are both given. We can find that the two solut ions are almost the same\nbefore the recurrence occurs. Similar to in the last subsection, af ter the recurrence time, the\nresults without filters become unreliable, while the results with filters still decay exponentially.\nTo be precise, the numerical decay rate is obtained by least-squar e fitting of peak value points\nbefore time tF. Two different values of tF(respectively before and after recurrence) are used to\nfit the slope, and the results are compared with the theoretical de cay rate γ= 0.15336 (see e.g.\n[27]). Particularly, the second tFis chosen around twice the recurrence time. Table 1 shows\nthat before recurrence, the decay rates of both methods matc h very well with the theoretical\nresult, and after the recurrence time, an accurate decay rate c an still be kept for a long timeSUPPRESSION OF RECURRENCE FOR TRANSPORT EQUATIONS 21\nby filtering. In this example, the filter improves the numerical result dramatically, and such a\nmethod looks very promising in the numerical simulation of plasmas.\n0 20 40 60 80 100-24-22-20-18-16-14-12-10-8-6-4\nw/o filter\nw/ filter\n(a)M= 900 20 40 60 80 100-24-22-20-18-16-14-12-10-8-6-4\nw/o filter\nw/ filter\n(b)M= 120\nFigure 7. The evolution of log E(t) for the Laudau damping problem.\nTable 1. The numerical decay rates of the electric energy\nMw/o filter w/ filter w/ filter Mw/o filter w/ filter w/ filter\n30tF= 12 tF= 12tF= 2460tF= 22 tF= 22tF= 44\n0.155038 0.1550545 0.200223 0.1540681 0.1540679 0.166939\n90tF= 26 tF= 26tF= 52120tF= 30 tF= 30tF= 60\n0.154173 0.154173 0.152892 0.153780 0.153780 0.153629\n6.Conclusion\nIn this paper, we have systematically analyzed the effect of the filte r on the numerical\nsolutions to the transport equations using Hermite-spectral met hod. The theoretical analysis\non two types of transport equations is proposed respectively. It is both rigorously proven and\nnumerically validated that the filter makes all the non-constant mod es damp exponentially, and\ntherefore suppresses the recurrence. The example of Landau d amping shows that the filter has\nonly negligible effect on the damping rates of the electric energy, whic h illustrates that adding\nfilter is a promising method to relieve the recurrence phenomenon. A nalysis on the case with\nmagnetic field will be studied in the future.\nReferences\n[1] H. Abbasi, M. H. Jenab, and H. H. Pajouh. Preventing the re currence effect in the Vasov simulation by\nrandomizing phase-point velocities in phase space. Phys. Rev. E. , 84(3):036702, 2011.\n[2] C. K. Birdsall and A. B. Langdon. Plasma physics via computer simulation . Institute of Physics Publishing,\nBristol/Philadelphia, 1991.\n[3] Z. Cai, R. Li, and Y. Wang. Solving Vlasov equation using N Rxxmethod. SIAM J. Sci. Comput. ,\n35(6):A2807–A2831, 2013.\n[4] E. Camporeale, G. L. Delzanno, B. K. Bergen, and J. D. Moul ton. On the velocity space discretization for\nthe Vlasov-Poisson system: Comparison between implicit He rmite spectral and Particle-in-Cell methods.\nComput. Phys. Commun. , 198:47–58, 2016.22 ZHENNING CAI AND YANLI WANG\n[5] C. Z. Cheng and G. Knorr. The integration of the Vlasov equ ation in configuration space. J. Comput.\nPhys., 22:330–351, 1976.\n[6] S. Dawson and M. F. Wheeler. Particle simulation of plasm a.Reviews of Modern Physics , 55:403–447,\n1983.\n[7] L. Einkemmer and A. Ostermann. A strategy to suppress rec urrence in grid-based Vlasov solvers. Eur.\nPhys. J. D. , 68(7):197, 2014.\n[8] L. Einkemmer and A. Ostermann. Convergence analysis of a discontinuous Galerkin/Strang splitting ap-\nproximation for the Vlasov–Poisson equations. SIAM J. Numer. Anal. , 52(2):757–778, 2014.\n[9] F.Filbetand E.Sonnendr¨ ucker. ComparisonofEulerian Vlasovsolvers. Comput. Phys. Comm. , 150(3):247–\n266, 2003.\n[10] F. Filbet, E. Sonnendr¨ ucker, and P. Bertrand. Conserv ative numerical schemes for the Vlasov equation. J.\nComput. Phys. , 172:166–187, 2001.\n[11] R. J. Goldston and P. H. Rutherford. Introduction to plasma physics . Institute of Physics Pub, 2016.\n[12] D. Gottlieb and J. S. Hesthaven. Spectral methods for hy perbolic problems. J. Comput. Appl. Math. ,\n128:83–131, 2001.\n[13] F. C. Grant and M. R. Feix. Fourier-Hermite solutions of the Vlasov equations in the linearized limit. Phy.\nFluids, 10(4):696–702, 1967.\n[14] R. W. Hamming. Introduction to applied numerical analysis . Taylor & Francis/Hemisphere, 1989.\n[15] R. E. Heath, I. M. Gamba, P. J. Morrison, and C. Michler. A discontinuous Galerkin method for the\nVlasov-Poisson system. J. Comput. Phys. , 231(4):1140–1174, 2012.\n[16] P. P. Hilscher, K. Imadera, J. Q. Li, and Y. Kishimoto. Th e effect of weak collisionality on damped modes\nand its contribution to linear mode coupling in gyrokinetic simulation. Phys. Plasmas , 20:082127, 2013.\n[17] P. Hilton and J. Pedersen. The Ballot problem and Catala n numbers. Nieuw Archief voor Wiskunde ,\n8(2):209–216, 1990.\n[18] R. W. Hockney and J. W. Eastwood. Computer simulation using particles . McGraw-Hill, NewYork, 1981.\n[19] J. P. Holloway. Spectral velocity discretizations for the Vlasov-Maxwell equations. Transport. Theor. Stat. ,\n25(1):1–32, 1996.\n[20] T. Hou and R. Li. Computing nearly singular solutions us ing pseudo-spectral methods. J. Comput. Phys. ,\n226(1):379–397, 2007.\n[21] B.K˚ agstr¨ om. Bounds and perturbation bounds forthe m atrix exponential. BIT Numer. Math. , 17(1):39–57,\n1977.\n[22] T. Nakamura and T. Yabe. Cubic interpolated propagatio n scheme for solving the hyper-dimensional\nVlasov-Poisson equation in phase space. Comput. Phys. Commun , 120:122–154, 1999.\n[23] J. T. Parker and P. J. Dellar. Fourier-Hermite spectral representation for the Vlasov-Poisson system in the\nweakly collisional limit. J. Plasma Phys. , 81(02):305810203, 2015.\n[24] O. Pezzi, E. Camproeale, and F. Valentini. Collisional effects on the numerical recurrence in Vlasov-Poisson\nsimulation. Phys. Plasmas. , 23:022103, 2016.\n[25] E. Pohn, M. Shoucri, and G. Kamelander. Eulerian Vlasov codes.Comput. Phys. Comm. , 166(2):81–93,\n2005.\n[26] J. Qiu and C. Shu. Positivity preserving semi-Lagrangi an discontinuous Galerkin formulation: Theoretical\nanalysis and application to the Vlasov-Poisson system. J. Comput. Phys. , 230(23):8386 – 8409, 2011.\n[27] E. Sonnendr¨ ucker. Approximation num´ erique des ´ equ ations de Vlasov-Maxwell. Notes du cours de M2 ,\n2010.\n[28] E. Sonnendr¨ ucker, J. Roche, P. Betrand, and A. Ghizzo. The semi-Lagrangian method for the numerical\nresolution of Vlasov equations. J.Comput. Phys , 149(2):201–220, 1998.\n[29] J. H. Wilkinson. The algebraic eigenvalue problem . Clarendon, 1965.\n[30] T. Zhou, Y. Guo, and C. W. Shu. Numerical study on Landau d amping. Physica D , 157(4):322–333, 2001.\n(Zhenning Cai) Department of Mathematics, National University of Singapore , Level 4, Block\nS17, 10 Lower Kent Ridge Road, Singapore 119076\nE-mail address :matcz@nus.edu.sg\n(Yanli Wang) School of Mathematics Science, Peking University, Beijing, China, 100871\nE-mail address :wylmath@pku.edu.cn"    },    {        "title": "1709.10365v1.Non_local_Gilbert_damping_tensor_within_the_torque_torque_correlation_model.pdf",        "content": "Non-local Gilbert damping tensor within the torque-torque correlation model\nDanny Thonig,1,\u0003Yaroslav Kvashnin,1Olle Eriksson,1, 2and Manuel Pereiro1\n1Department of Physics and Astronomy, Material Theory, Uppsala University, SE-75120 Uppsala, Sweden\n2School of Science and Technology, Orebro University, SE-701 82 Orebro, Sweden\n(Dated: July 19, 2018)\nAn essential property of magnetic devices is the relaxation rate in magnetic switching which\ndepends strongly on the damping in the magnetisation dynamics. It was recently measured that\ndamping depends on the magnetic texture and, consequently, is a non-local quantity. The damping\nenters the Landau-Lifshitz-Gilbert equation as the phenomenological Gilbert damping parameter\n\u000b, that does not, in a straight forward formulation, account for non-locality. E\u000borts were spent\nrecently to obtain Gilbert damping from \frst principles for magnons of wave vector q. However,\nto the best of our knowledge, there is no report about real space non-local Gilbert damping \u000bij.\nHere, a torque-torque correlation model based on a tight binding approach is applied to the bulk\nelemental itinerant magnets and it predicts signi\fcant o\u000b-site Gilbert damping contributions, that\ncould be also negative. Supported by atomistic magnetisation dynamics simulations we reveal the\nimportance of the non-local Gilbert damping in atomistic magnetisation dynamics. This study gives\na deeper understanding of the dynamics of the magnetic moments and dissipation processes in real\nmagnetic materials. Ways of manipulating non-local damping are explored, either by temperature,\nmaterials doping or strain.\nPACS numbers: 75.10.Hk,75.40.Mg,75.78.-n\nE\u000ecient spintronics applications call for magnetic ma-\nterials with low energy dissipation when moving magnetic\ntextures, e.g. in race track memories1, skyrmion logics2,3,\nspin logics4, spin-torque nano-oscillator for neural net-\nwork applications5or, more recently, soliton devices6. In\nparticular, the dynamics of such magnetic textures |\nmagnetic domain walls, magnetic Skyrmions, or magnetic\nsolitons | is well described in terms of precession and\ndamping of the magnetic moment mias it is formulated\nin the atomistic Landau-Lifshitz-Gilbert (LLG) equation\nfor sitei\n@mi\n@t=mi\u0002\u0012\n\u0000\rBeff\ni+\u000b\nms@mi\n@t\u0013\n; (1)\nwhere\randmsare the gyromagnetic ratio and the\nmagnetic moment length, respectively. The precession\n\feldBeff\niis of quantum mechanical origin and is ob-\ntained either from e\u000bective spin-Hamilton models7or\nfrom \frst-principles8. In turn, energy dissipation is\ndominated by the ad-hoc motivated viscous damping in\nthe equation of motion scaled by the Gilbert damping\ntensor\u000b. Commonly, the Gilbert damping is used as\na scalar parameter in magnetization dynamics simula-\ntions based on the LLG equation. Strong e\u000borts were\nspend in the last decade to put the Gilbert damping\nto a \frst-principles ground derived for collinear mag-\nnetization con\fgurations. Di\u000berent methods were pro-\nposed: e.g. the breathing Fermi surface9{11, the torque-\ntorque correlation12, spin-pumping13or a linear response\nmodel14,15. Within a certain accuracy, the theoretical\nmodels allow to interpret16and reproduce experimental\ntrends17{20.\nDepending on the model, deep insight into the fun-\ndamental electronic-structure mechanism of the Gilbertdamping\u000bis provided: Damping is a Fermi-surface ef-\nfect and depending on e.g. scattering rate, damping\noccurs due to spin-\rip but also spin-conservative tran-\nsition within a degenerated (intraband, but also inter-\nband transitions) and between non-degenerated (inter-\nband transitions) electron bands. As a consequence of\nthese considerations, the Gilbert damping is proportional\nto the density of states, but it also scales with spin-orbit\ncoupling21,22. The scattering rate \u0000 for the spin-\rip tran-\nsitions is allocated to thermal, but also correlation ef-\nfects, making the Gilbert damping strongly temperature\ndependent which must be a consideration when applying\na three-temperature model for the thermal baths, say\nphonon14, electron, and spin temperature23. In particu-\nlar, damping is often related to the dynamics of a collec-\ntive precession mode (macrospin approach) driven from\nan external perturbation \feld, as it is used in ferromag-\nnetic resonance experiments (FMR)24. It is also estab-\nlished that the Gilbert damping depends on the orien-\ntation of the macrospin25and is, in addition, frequency\ndependent26.\nMore recently, the role of non-collective modes to the\nGilbert damping has been debated. F ahnle et al.27\nsuggested to consider damping in a tensorial and non-\nisotropic form via \u000bithat di\u000bers for di\u000berent sites i\nand depends on the whole magnetic con\fguration of the\nsystem. As a result, the experimentally and theoret-\nically assumed local Gilbert equation is replaced by a\nnon-local equation via non-local Gilbert damping \u000bijac-\ncounting for the most general form of Rayleigh's dissi-\npation function28. The proof of principles was given for\nmagnetic domain walls29,30, linking explicitly the Gilbert\ndamping to the gradients in the magnetic spin texture\nrm. Such spatial non-locality, in particular, for discrete\natomistic models, allows further to motivate energy dis-arXiv:1709.10365v1  [cond-mat.mtrl-sci]  29 Sep 20172\nij\nαij\nq\nFIG. 1: Schematic illustration of non-local energy dissipation\n\u000bijbetween site iandj(red balls) represented by a power\ncord in a system with spin wave (gray arrows) propagation q.\nsipation between two magnetic moments at sites iand\nj, and is represented by \u000bij, as schematically illustrated\nin Fig. 1. An analytical expression for \u000bijwas already\nproposed by various authors14,31,32, however, not much\nwork has been done on a material speci\fc, \frst-principle\ndescription of the atomistic non-local Gilbert damping\n\u000bij. An exception is the work by Gilmore et al.32who\nstudied\u000b(q) in the reciprocal space as a function of the\nmagnon wave vector qand concluded that the non-local\ndamping is negligible. Yan et al.29and Hals et al.33, on\nthe other hand, applied scattering theory according to\nBrataas et al.34to simulate non-collinearity in Gilbert\ndamping, only in reciprocal space or continuous meso-\nscopic scale. Here we come up with a technical descrip-\ntion of non-locality of the damping parameter \u000bij, in\nreal space, and provide numerical examples for elemental,\nitinerant magnets, which might be of high importance in\nthe context of ultrafast demagnetization35.\nThe paper is organized as follows: In Section I, we\nintroduce our \frst-principles model formalism based on\nthe torque-torque correlation model to study non-local\ndamping. This is applied to bulk itinerant magnets bcc\nFe, fcc Co, and fcc Ni in both reciprocal and real space\nand it is analysed in details in Section II. Here, we will\nalso apply atomistic magnetisation dynamics to outline\nthe importance in the evolution of magnetic systems. Fi-\nnally, in the last section, we conclude the paper by giving\nan outlook of our work.\nI. METHODS\nWe consider the torque-torque correlation model in-\ntroduced by Kambersk\u0013 y10and further elaborated on by\nGilmore et al.12. Here, \fnite magnetic moment rotations\ncouple to the Bloch eigenenergies \"n;kand eigenstates\njnki, characterised by the band index nat wave vec-tork, due to spin-orbit coupling. This generates a non-\nequilibrium population state (a particle-hole pair), where\nthe excited states relax towards the equilibrium distribu-\ntion (Fermi-Dirac statistics) within the time \u001cn;k=1=\u0000,\nwhich we assume is independent of nandk. In the adi-\nabatic limit, this perturbation is described by the Kubo-\nGreenwood perturbation theory and reads12,36in a non-\nlocal formulation\n\u000b\u0016\u0017(q) =g\u0019\nmsZ\n\nX\nnmT\u0016\nnk;mk+q\u0000\nT\u0017\nnk;mk+q\u0001\u0003Wnk;mk+qdk:\n(2)\nHere the integral runs over the whole Brillouin zone\nvolume \n. A frozen magnon of wave vector qis consid-\nered that is ascribed to the non-locality of \u000b. The scat-\ntering events depend on the spectral overlap Wnk;mk+q=R\n\u0011(\")Ank(\";\u0000)Amk+q(\";\u0000) d\"between two bands \"n;k\nand\"m;k+q, where the spectral width of the electronic\nbandsAnkis approximated by a Lorentzian of width \u0000.\nNote that \u0000 is a parameter in our model and can be spin-\ndependent as proposed in Ref. [37]. In other studies, this\nparameter is allocated to the self-energy of the system\nand is obtained by introducing disorder, e.g., in an al-\nloy or alloy analogy model using the coherent potential\napproximation14(CPA) or via the inclusion of electron\ncorrelation38. Thus, a principle study of the non-local\ndamping versus \u0000 can be also seen as e.g. a temperature\ndependent study of the non-local damping. \u0011=@f=@\"is\nthe derivative of the Fermi-Dirac distribution fwith re-\nspect to the energy. T\u0016\nnk;mk+q=hnkj^T\u0016jmk+qi, where\n\u0016=x;y;z , are the matrix elements of the torque oper-\nator ^T= [\u001b;Hso] obtained from variation of the mag-\nnetic moment around certain rotation axis e.\u001band\nHsoare the Pauli matrices and the spin-orbit hamilto-\nnian, respectively. In the collinear ferromagnetic limit,\ne=ezand variations occur in xandy, only, which al-\nlows to consider just one component of the torque, i.e.\n^T\u0000=^Tx\u0000i^Ty. Using Lehmann representation39, we\nrewrite the Bloch eigenstates by Green's function G, and\nde\fne the spectral function ^A= i\u0000\nGR\u0000GA\u0001\nwith the\nretarded (R) and advanced (A) Green's function,\n\u000b\u0016\u0017(q) =g\nm\u0019Z Z\n\n\u0011(\")^T\u0016^Ak\u0010\n^T\u0017\u0011y^Ak+qdkd\":(3)\nThe Fourier transformation of the Green's function G\n\fnally is used to obtain the non-local Gilbert damping\ntensor23between site iat positionriand sitejat position\nrj,\n\u000b\u0016\u0017\nij=g\nm\u0019Z\n\u0011(\")^T\u0016\ni^Aij\u0010\n^T\u0017\nj\u0011y^Ajid\": (4)\nNote that ^Aij= i\u0000\nGR\nij\u0000GA\nji\u0001\n. This result is consis-\ntent with the formulation given in Ref. [31] and Ref. [14].\nHence, the de\fnition of non-local damping in real space3\nand reciprocal space translate into each other by a\nFourier transformation,\n\u000bij=Z\n\u000b(q) e\u0000i(rj\u0000ri)\u0001qdq: (5)\nNote the obvious advantage of using Eq. (4), since it\nallows for a direct calculation of \u000bij, as opposed to tak-\ning the inverse Fourier transform of Eq. (5). For \frst-\nprinciples studies, the Green's function is obtained from\na tight binding (TB) model based on the Slater-Koster\nparameterization40. The Hamiltonian consists of on-site\npotentials, hopping terms, Zeeman energy, and spin-orbit\ncoupling (See Appendix A). The TB parameters, includ-\ning the spin-orbit coupling strength, are obtained by \ft-\nting the TB band structures to ab initio band structures\nas reported elsewhere23.\nBeyond our model study, we simulate material spe-\nci\fc non-local damping with the help of the full-potential\nlinear mu\u000en-tin orbitals (FP-LMTO) code \\RSPt\"41,42.\nFurther numerical details are provided in Appendix A.\nWith the aim to emphasize the importance of non-\nlocal Gilbert damping in the evolution of atomistic\nmagnetic moments, we performed atomistic magnetiza-\ntion dynamics by numerical solving the Landau-Lifshitz\nGilbert (LLG) equation, explicitly incorporating non-\nlocal damping23,34,43\n@mi\n@t=mi\u00020\n@\u0000\rBeff\ni+X\nj\u000bij\nmj\ns@mj\n@t1\nA:(6)\nHere, the e\u000bective \feld Beff\ni =\u0000@^H=@miis allo-\ncated to the spin Hamiltonian entails Heisenberg-like ex-\nchange coupling\u0000P\nijJijmi\u0001mjand uniaxial magneto-\ncrystalline anisotropyP\niKi(mi\u0001ei)2with the easy axis\nalongei.JijandKiare the Heisenberg exchange cou-\npling and the magneto-crystalline anisotropy constant,\nrespectively, and were obtained from \frst principles44,45.\nFurther details are provided in Appendix A.\nII. RESULTS AND DISCUSSION\nThis section is divided in three parts. In the \frst part,\nwe discuss non-local damping in reciprocal space q. The\nsecond part deals with the real space de\fnition of the\nGilbert damping \u000bij. Atomistic magnetization dynam-\nics including non-local Gilbert damping is studied in the\nthird part.\nA. Non-local damping in reciprocal space\nThe formalism derived by Kambersk\u0013 y10and Gilmore12\nin Eq. (2) represents the non-local contributions to the\nenergy dissipation in the LLG equation by the magnonwave vector q. In particular, Gilmore et al.32con-\ncluded that for transition metals at room temperature\nthe single-mode damping rate is essentially independent\nof the magnon wave vector for qbetween 0 and 1% of\nthe Brillouin zone edge. However, for very small scat-\ntering rates \u0000, Gilmore and Stiles12observed for bcc Fe,\nhcp Co and fcc Ni a strong decay of \u000bwithq, caused by\nthe weighting function Wnm(k;k+q) without any sig-\nni\fcant changes of the torque matrix elements. Within\nour model systems, we observed the same trend for bcc\nFe, fcc Co and fcc Ni. To understand the decay of the\nGilbert damping with magnon-wave vector qin more de-\ntail, we study selected paths of both the magnon qand\nelectron momentum kin the Brillouin zone at the Fermi\nenergy\"Ffor bcc Fe (q;k2\u0000!Handq;k2H!N),\nfcc Co and fcc Ni ( q;k2\u0000!Xandq;k2X!L) (see\nFig. 2, where the integrand of Eq. (2) is plotted). For\nexample, in Fe, a usually two-fold degenerated dband\n(approximately in the middle of \u0000H, marked by ( i)) gives\na signi\fcant contribution to the intraband damping for\nsmall scattering rates. There are two other contributions\nto the damping (marked by ( ii)), that are caused purely\nby interband transitions. With increasing, but small q\nthe intensities of the peaks decrease and interband tran-\nsitions become more likely. With larger q, however, more\nand more interband transitions appear which leads to an\nincrease of the peak intensity, signi\fcantly in the peaks\nmarked with ( ii). This increase could be the same or-\nder of magnitude as the pure intraband transition peak.\nSimilar trends also occur in Co as well as Ni and are\nalso observed for Fe along the path HN. Larger spectral\nwidth \u0000 increases the interband spin-\rip transitions even\nfurther (data not shown). Note that the torque-torque\ncorrelation model might fail for large values of q, since\nthe magnetic moments change so rapidly in space that\nthe adiababtic limit is violated46and electrons are not\nstationary equilibrated. The electrons do not align ac-\ncording the magnetic moment and the non-equilibrium\nelectron distribution in Eq. (2) will not fully relax. In\nparticular, the magnetic force theorem used to derive\nEq. (3) may not be valid.\nThe integration of the contributions in electron mo-\nmentum space kover the whole Brillouin zone is pre-\nsented in Fig. 3, where both `Loretzian' method given\nby Eq. (2) and Green's function method represented\nby Eq. (3) are applied. Both methods give the same\ntrend, however, di\u000ber slightly in the intraband region,\nwhich was already observed previously by the authors\nof Ref. [23]. In the `Lorentzian' approach, Eq. (2), the\nelectronic structure itself is una\u000bected by the scattering\nrate \u0000, only the width of the Lorentian used to approx-\nimateAnkis a\u000bected. In the Green function approach,\nhowever, \u0000 enters as the imaginary part of the energy\nat which the Green functions is evaluated and, conse-\nquently, broadens and shifts maxima in the spectral func-\ntion. This o\u000bset from the real energy axis provides a more\naccurate description with respect to the ab initio results\nthan the Lorentzian approach.4\nΓHq(a−1\n0)\nΓ H\nk(a−1\n0)\nFe\nΓX\nΓ X\nk(a−1\n0)\n Co\nΓX\nΓ X\nk(a−1\n0)\n Ni\n(i) (ii) (ii)\nFIG. 2: Electronic state resolved non-local Gilbert damping obtained from the integrand of Eq. (3) along selected paths in the\nBrillouin zone for bcc Fe, fcc Co and fcc Ni. The scattering rate used is \u0000 = 0 :01 eV. The abscissa (both top and bottom in\neach panels) shows the momentum path of the electron k, where the ordinate (left and right in each panel) shows the magnon\npropagation vector q. The two `triangle' in each panel should be viewed separately where the magnon momentum changes\naccordingly (along the same path) to the electron momentum.\nWithin the limits of our simpli\fed electronic structure\ntight binding method, we obtained qualitatively similar\ntrends as observed by Gilmore et al.32: a dramatic de-\ncrease in the damping at low scattering rates \u0000 (intra-\nband region). This trend is common for all here ob-\nserved itinerant magnets typically in a narrow region\n0<jqj<0:02a\u00001\n0, but also for di\u000berent magnon propa-\ngation directions. For larger jqj>0:02a\u00001\n0the damping\ncould again increase (not shown here). The decay of \u000b\nis only observable below a certain threshold scattering\nrate \u0000, typically where intra- and interband contribu-\ntion equally contributing to the Gilbert damping. As\nalready found by Gilmore et al.32and Thonig et al.23,\nthis point is materials speci\fc. In the interband regime,\nhowever, damping is independent of the magnon propa-\ngator, caused by already allowed transition between the\nelectron bands due to band broadening. Marginal vari-\nations in the decay with respect to the direction of q\n(Inset of Fig. 3) are revealed, which was not reported be-\nfore. Such behaviour is caused by the break of the space\ngroup symmetry due to spin-orbit coupling and a selected\nglobal spin-quantization axis along z-direction, but also\ndue to the non-cubic symmetry of Gkfork6= 0. As a re-\nsult, e.g., in Ni the non-local damping decays faster along\n\u0000Kthan in \u0000X. This will be discussed more in detail in\nthe next section.\nWe also investigated the scaling of the non-local\nGilbert damping with respect to the spin-orbit coupling\nstrength\u0018dof the d-states (see Appendix B). We observe\nan e\u000bect that previously has not been discussed, namely\nthat the non-local damping has a di\u000berent exponential\nscaling with respect to the spin-orbit coupling constant\nfor di\u000berentjqj. In the case where qis close to the Bril-\nlouin zone center (in particular q= 0),\u000b/\u00183\ndwhereas\nfor wave vectors jqj>0:02a\u00001\n0,\u000b/\u00182\nd. For largeq,\ntypically interband transitions dominate the scatteringmechanism, as we show above and which is known to\nscale proportional to \u00182. Here in particular, the \u00182will\nbe caused only by the torque operator in Eq. (2). On the\nother hand, this indicates that spin-mixing transitions\nbecome less important because there is not contribution\nin\u0018from the spectral function entering to the damping\n\u000b(q).\nThe validity of the Kambserk\u0013 y model becomes ar-\nguable for\u00183scaling, as it was already proved by Costa\net al.47and Edwards48, since it causes the unphysical\nand strong diverging intraband contribution at very low\ntemperature (small \u0000). Note that there is no experi-\nmental evidence of such a trend, most likely due to that\nsample impurities also in\ruence \u0000. Furthermore, various\nother methods postulate that the Gilbert damping for\nq= 0 scales like \u00182 9,15,22. Hence, the current applied\ntheory, Eq. (3), seems to be valid only in the long-wave\nlimit, where we found \u00182-scaling. On the other hand,\nEdwards48proved that the long-wave length limit ( \u00182-\nscaling) hold also in the short-range limit if one account\nonly for transition that conserve the spin (`pure' spin\nstates), as we show for Co in Fig. 11 of Appendix C. The\ntrends\u000bversusjqjas described above changes drastically\nfor the `corrected' Kambersk\u0013 y formula: the interband re-\ngion is not a\u000bected by these corrections. In the intraband\nregion, however, the divergent behaviour of \u000bdisappears\nand the Gilbert damping monotonically increases with\nlarger magnon wave vector and over the whole Brillouin\nzone. This trend is in good agreement with Ref. [29].\nFor the case, where q= 0, we even reproduced the re-\nsults reported in Ref. [21]; in the limit of small scattering\nrates the damping is constant, which was also reported\nbefore in experiment49,50. Furthermore, the anisotropy\nof\u000b(q) with respect to the direction of q(as discussed\nfor the insets of Fig. 3) increases by accounting only for\npure-spin states (not shown here). Both agreement with5\n510−22Fe\n0.000\n0.025\n0.050\n0.075\n0.100\nq: Γ→H\n2510−2α(q)Co\nq: Γ→X\n510−225\n10−310−210−110+0\nΓ (eV)Ni\nq: Γ→X\nFIG. 3: (Color online) Non-local Gilbert damping as a func-\ntion of the spectral width \u0000 for di\u000berent reciprocal wave vector\nq(indicated by di\u000berent colors and in units a\u00001\n0). Note that q\nprovided here are in direct coordinates and only the direction\ndi\u000bers between the di\u000berent elementals, itinerant magnets.\nThe non-local damping is shown for bcc Fe (top panel) along\n\u0000!H, for fcc Co (middle panel) along \u0000 !X, and for fcc Ni\n(bottom panel) along \u0000 !X. It is obtained from `Lorentzian'\n(Eq. (2), circles) and Green's function (Eq. (3), triangles)\nmethod. The directional dependence of \u000bfor \u0000 = 0:01 eV is\nshown in the inset.\nexperiment and previous theory motivate to consider \u00182-\nscaling for all \u0000.\nB. Non-local damping in real space\nAtomistic spin-dynamics, as stated in Section I (see\nEq. (6)), that includes non-local damping requires\nGilbert damping in real-space, e.g. in the form \u000bij. This\npoint is addressed in this section. Such non-local con-\ntributions are not excluded in the Rayleigh dissipation\nfunctional, applied by Gilbert to derive the dissipation\ncontribution in the equation of motion51(see Fig. 4).\nDissipation is dominated by the on-site contribution\n-101 Fe\nαii= 3.552·10−3\n˜αii= 3.559·10−3\n-101αij·10−4Co\nαii= 3.593·10−3\n˜αii= 3.662·10−3\n-10\n1 2 3 4 5 6\nrij/a0Ni\nαii= 2.164·10−2\n˜αii= 2.319·10−2FIG. 4: (Color online) Real-space Gilbert damping \u000bijas\na function of the distance rijbetween two sites iandjfor\nbcc Fe, fcc Co, and fcc Ni. Both the `corrected' Kambersk\u0013 y\n(red circles) and the Kambersk\u0013 y (blue squares) approach is\nconsidered. The distance is normalised to the lattice constant\na0. The on-site damping \u000biiis shown in the \fgure label. The\ngrey dotted line indicates the zero line. The spectral width is\n\u0000 = 0:005 eV.\n\u000biiin the itinerant magnets investigated here. For both\nFe (\u000bii= 3:55\u000110\u00003) and Co ( \u000bii= 3:59\u000110\u00003) the\non-site damping contribution is similar, whereas for Ni\n\u000biiis one order of magnitude higher. O\u000b-site contri-\nbutionsi6=jare one-order of magnitude smaller than\nthe on-site part and can be even negative. Such neg-\native damping is discernible also in Ref. [52], however,\nit was not further addressed by the authors. Due to\nthe presence of the spin-orbit coupling and a preferred\nglobal spin-quantization axis (in z-direction), the cubic\nsymmetry of the considered itinerant magnets is broken\nand, thus, the Gilbert damping is anisotropic with re-\nspect to the sites j(see also Fig. 5 left panel). For ex-\nample, in Co, four of the in-plane nearest neighbours\n(NN) are\u000bNN\u0019\u00004:3\u000110\u00005, while the other eight are\n\u000bNN\u0019\u00002:5\u000110\u00005. However, in Ni the trend is opposite:\nthe out-of-plane damping ( \u000bNN\u0019\u00001:6\u000110\u00003) is smaller\nthan the in-plane damping ( \u000bNN\u0019 \u00001:2\u000110\u00003). In-\nvolving more neighbours, the magnitude of the non-local6\ndamping is found to decay as 1=r2and, consequently, it\nis di\u000berent than the Heisenberg exchange parameter that\nasymptotically decays in RKKY-fashion as Jij/1=r353.\nFor the Heisenberg exchange, the two Green's functions\nas well as the energy integration in the Lichtenstein-\nKatsnelson-Antropov-Gubanov formula54scales liker\u00001\nij,\nG\u001b\nij/ei(k\u001b\u0001rij+\b\u001b)\njrijj(7)\nwhereas for simplicity we consider here a single-band\nmodel but the results can be generalized also to the multi-\nband case and where \b\u001bdenotes a phase factor for spin\n\u001b=\";#. For the non-local damping the energy integra-\ntion is omitted due to the properties of \u0011in Eq. (4) and,\nthus,\n\u000bij/sin\u0002\nk\"\u0001rij+ \b\"\u0003\nsin\u0002\nk#\u0001rij+ \b#\u0003\njrijj2:(8)\nThis spatial dependency of \u000bijsuperimposed with\nRuderman-Kittel-Kasuya-Yosida (RKKY) oscillations\nwas also found in Ref. [52] for a model system.\nFor Ni, dissipation is very much short range, whereas in\nFe and Co `damping peaks' also occur at larger distances\n(e.g. for Fe at rij= 5:1a0and for Co at rij= 3:4a0).\nThe `long-rangeness' depends strongly on the parameter\n\u0000 (not shown here). As it was already observed for the\nHeisenberg exchange interaction Jij44, stronger thermal\ne\u000bects represented by \u0000 will reduce the correlation length\nbetween two magnetic moments at site iandj. The same\ntrend is observed for damping: larger \u0000 causes smaller\ndissipation correlation length and, thus, a faster decay\nof non-local damping in space rij. Di\u000berent from the\nHeisenberg exchange, the absolute value of the non-local\ndamping typically decreases with \u0000 as it is demonstrated\nin Fig. 5.\nNote that the change of the magnetic moment length\nis not considered in the results discussed so far. The\nanisotropy with respect to the sites iandjof the non-\nlocal Gilbert damping continues in the whole range of the\nscattering rate \u0000 and is controlled by it. For instance, the\nsecond nearest neighbours damping in Co and Ni become\ndegenerated at \u0000 = 0 :5 eV, where the anisotropy between\n\frst-nearest neighbour sites increase. Our results show\nalso that the sign of \u000bijis a\u000bected by \u0000 (as shown in\nFig. 5 left panel). Controlling the broadening of Bloch\nspectral functions \u0000 is in principal possible to evaluate\nfrom theory, but more importantly it is accessible from\nexperimental probes such as angular resolved photoelec-\ntron spectroscopy and two-photon electron spectroscopy.\nThe importance of non-locality in the Gilbert damping\ndepend strongly on the material (as shown in Fig. 5 right\npanel). It is important to note that the total | de\fned as\n\u000btot=P\nj\u000bijfor arbitrary i|, but also the local ( i=j)\nand the non-local ( i6=j) part of the Gilbert damping do\nnot violate the thermodynamic principles by gaining an-\ngular momentum (negative total damping). For Fe, the\n-101\n1. NN.\n2. NN.Fe\n34567αii\nαtot=/summationtext\njαijαq=0.1a−1\n0αq=0\n-10αij·10−4Co\n123456\nαij·10−3\n-15-10-50\n10−210−1\nΓ (eV)Ni\n5101520\n10−210−1\nΓ (eV)FIG. 5: (Color online) First (circles) and second nearest\nneighbour (triangles) Gilbert damping (left panel) as well as\non-site (circles) and total Gilbert (right panel) as a function of\nthe spectral width \u0000 for the itinerant magnets Fe, Co, and Ni.\nIn particular for Co, the results obtained from tight binding\nare compared with \frst-principles density functional theory\nresults (gray open circles). Solid lines (right panel) shows the\nGilbert damping obtained for the magnon wave vectors q= 0\n(blue line) and q= 0:1a\u00001\n0(red line). Dotted lines are added\nto guide the eye. Note that since cubic symmetry is broken\n(see text), there are two sets of nearest neighbor parameters\nand two sets of next nearest neighbor parameters (left panel)\nfor any choice of \u0000.\nlocal and total damping are of the same order for all\n\u0000, where in Co and Ni the local and non-local damp-\ning are equally important. The trends coming from our\ntight binding electron structure were also reproduced by\nour all-electron \frst-principles simulation, for both de-\npendency on the spectral broadening \u0000 (Fig. 5 gray open\ncircles) but also site resolved non-local damping in the\nintraband region (see Appendix A), in particular for fcc\nCo.\nWe compare also the non-local damping obtain from\nthe real and reciprocal space. For this, we used Eq. (3)\nby simulating Nq= 15\u000215\u000215 points in the \frst magnon\nBrillouin zone qand Fourier-transformed it (Fig. 6). For7\n-1.0-0.50.00.51.0αij·10−4\n5 10 15 20 25 30\nrij/a0FFT(α(q));αii= 0.003481\nFFT(G(k));αii= 0.003855\nFIG. 6: (Color online) Comparing non-local Gilbert damping\nobtained by Eq. (5) (red symbols) and Eq. (4) (blue symbols)\nin fcc Co for \u0000 = 0 :005 eV. The dotted line indicates zero\nvalue.\nboth approaches, we obtain good agreement, corroborat-\ning our methodology and possible applications in both\nspaces. The non-local damping for the \frst three nearest\nneighbour shells turn out to converge rapidly with Nq,\nwhile it does not converge so quickly for larger distances\nrij. The critical region around the \u0000-point in the Bril-\nlouin zone is suppressed in the integration over q. On\nthe other hand, the relation \u000btot=P\nj\u000bij=\u000b(q= 0)\nfor arbitrary ishould be valid, which is however violated\nin the intraband region as shown in Fig. 5 (compare tri-\nangles and blue line in Fig. 5): The real space damping\nis constant for small \u0000 and follows the long-wavelength\nlimit (compare triangles and red line in Fig. 5) rather\nthan the divergent ferromagnetic mode ( q= 0). Two\nexplanations are possible: i)convergence with respect to\nthe real space summation and ii)a di\u000berent scaling in\nboth models with respect to the spin-orbit coupling. For\ni), we carefully checked the convergence with the summa-\ntion cut-o\u000b (see Appendix D) and found even a lowering\nof the total damping for larger cut-o\u000b. However, the non-\nlocal damping is very long-range and, consequently, con-\nvergence will be achieved only at a cut-o\u000b radius >>9a0.\nForii), we checked the scaling of the real space Gilbert\ndamping with the spin-orbit coupling of the d-states\n(see Appendix B). Opposite to the `non-corrected' Kam-\nbersk\u0013 y formula in reciprocal space, which scales like\n\u00183\nd, we \fnd\u00182\ndfor the real space damping. This indi-\ncates that the spin-\rip scattering hosted in the real-space\nGreen's function is suppressed. To corroborate this state-\nment further, we applied the corrections proposed by\nEdwards48to our real space formula Eq. (4), which by\ndefault assumes \u00182(Fig. 4, red dots). Both methods, cor-\nrected and non-corrected Eq. (4), agree quite well. The\nsmall discrepancies are due to increased hybridisations\nand band inversion between p and d- states due to spin-\norbit coupling in the `non-corrected' case.\nFinally, we address other ways than temperature (here\nrepresented by \u0000), to manipulate the non-local damping.\nIt is well established in literature already for Heisenberg\nexchange and the magneto crystalline anisotropy that\n-0.40.00.40.81.2αij·10−4\n1 2 3 4 5 6 7\nrij/a0αii= 3.49·10−3αii= 3.43·10−3FIG. 7: (Color online) Non-local Gilbert damping as a func-\ntion of the normalized distancerij=a0for a tetragonal dis-\ntorted bcc Fe crystal structure. Here,c=a= 1:025 (red circles)\nandc=a= 1:05 (blue circles) is considered. \u0000 is put to 0 :01 eV.\nThe zero value is indicated by dotted lines.\ncompressive or tensial strain can be used to tune the mag-\nnetic phase stability and to design multiferroic materials.\nIn an analogous way, also non-local damping depends on\ndistortions in the crystal (see Fig. 7).\nHere, we applied non-volume conserved tetragonal\nstrain along the caxis. The local damping \u000biiis marginal\nbiased. Relative to the values of the undistorted case,\na stronger e\u000bect is observed for the non-local part, in\nparticular for the \frst few neighbours. Since we do a\nnon-volume conserved distortion, the in-plane second NN\ncomponent of the non-local damping is constant. The\ndamping is in general decreasing with increasing distor-\ntion, however, a change in the sign of the damping can\nalso occur (e.g. for the third NN). The rate of change\nin damping is not linear. In particular, the nearest-\nneighbour rate is about \u000e\u000b\u00190:4\u000110\u00005for 2:5% dis-\ntortion, and 2 :9\u000110\u00005for 5% from the undistorted case.\nFor the second nearest neighbour, the rate is even big-\nger (3:0\u000110\u00005for 2:5%, 6:9\u000110\u00005for 5%). For neigh-\nbours larger than rij= 3a0, the change is less signi\fcant\n(\u00000:6\u000110\u00005for 2:5%,\u00000:7\u000110\u00005for 5%). The strongly\nstrain dependent damping motivates even higher-order\ncoupled damping contributions obtained from Taylor ex-\npanding the damping contribution around the equilib-\nrium position \u000b0\nij:\u000bij=\u000b0\nij+@\u000bij=@uk\u0001uk+:::. Note that\nthis is in analogy to the magnetic exchange interaction55\n(exchange striction) and a natural name for it would\nbe `dissipation striction'. This opens new ways to dis-\nsipatively couple spin and lattice reservoir in combined\ndynamics55, to the best of our knowledge not considered\nin todays ab-initio modelling of atomistic magnetisation\ndynamics.\nC. Atomistic magnetisation dynamics\nThe question about the importance of non-local damp-\ning in atomistic magnetization dynamics (ASD) remains.8\n0.40.50.60.70.80.91.0M\n0.0 0.5 1.0 1.5 2.0 2.5 3.0\nt(ps)0.5\n0.1\n0.05\n0.01αtot\nαij\n0.5 1.0 1.5 2.0 2.5 3.0\nt(ps)Fe\nCo\nFIG. 8: (Color online) Evolution of the average magnetic mo-\nmentMduring remagnetization in bcc Fe (left panel) and\nfcc Co (right panel) for di\u000berent damping strength according\nto the spectral width \u0000 (di\u000berent colors) and both, full non-\nlocal\u000bij(solid line) and total, purely local \u000btot(dashed line)\nGilbert damping.\nFor this purpose, we performed zero-temperature ASD\nfor bcc Fe and fcc Co bulk and analysed changes in the\naverage magnetization during relaxation from a totally\nrandom magnetic con\fguration, for which the total mo-\nment was zero (Fig. 8)\nRelated to the spectral width, the velocity for remag-\nnetisation changes and is higher, the bigger the e\u000bective\nGilbert damping is. For comparison, we performed also\nASD simulations based on Eq. (2) with a scalar, purely\nlocal damping \u000btot(dotted lines). For Fe, it turned out\nthat accounting for the non-local damping causes a slight\ndecrease in the remagnetization time, however, is overall\nnot important for relaxation processes. This is under-\nstandable by comparing the particular damping values\nin Fig. 5, right panel, in which the non-local part ap-\npear negligible. On the other hand, for Co the e\u000bect\non the relaxation process is much more signi\fcant, since\nthe non-local Gilbert damping reduces the local contribu-\ntion drastically (see Fig. 5, right panel). This `negative'\nnon-local part ( i6=j) in\u000bijdecelerates the relaxation\nprocess and the relaxation time is drastically increased\nby a factor of 10. Note that a `positive' non-local part\nwill accelerate the relaxation, which is of high interest for\nultrafast switching processes.\nIII. CONCLUDING REMARKS\nIn conclusion, we have evaluated the non-locality of\nthe Gilbert damping parameter in both reciprocal and\nreal space for elemental, itinerant magnets bcc Fe, fcc\nCo and fcc Ni. In particular in the reciprocal space,\nour results are in good agreement with values given in\nthe literature32. The here studied real space damping\nwas considered on an atomistic level and it motivates\nto account for the full, non-local Gilbert damping in\nmagnetization dynamic, e.g. at surfaces56or for nano-\nstructures57. We revealed that non-local damping canbe negative, has a spatial anisotropy, quadratically scales\nwith spin-orbit coupling, and decays in space as r\u00002\nij.\nDetailed comparison between real and reciprocal states\nidenti\fed the importance of the corrections proposed by\nEdwards48and, consequently, overcome the limits of the\nKambersk\u0013 y formula showing an unphysical and experi-\nmental not proved divergent behaviour at low tempera-\nture. We further promote ways of manipulating non-local\nGilbert damping, either by temperature, materials dop-\ning or strain, and motivating `dissipation striction' terms,\nthat opens a fundamental new root in the coupling be-\ntween spin and lattice reservoirs.\nOur studies are the starting point for even further in-\nvestigations: Although we mimic temperature by the\nspectral broadening \u0000, a precise mapping of \u0000 to spin\nand phonon temperature is still missing, according to\nRefs. [14,23]. Even at zero temperature, we revealed a\nsigni\fcant e\u000bect of the non-local Gilbert damping to the\nmagnetization dynamics, but the in\ruence of non-local\ndamping to \fnite temperature analysis or even to low-\ndimensional structures has to be demonstrated.\nIV. ACKNOWLEDGEMENTS\nThe authors thank Lars Bergqvist, Lars Nordstr om,\nJustin Shaw, and Jonas Fransson for fruitful discus-\nsions. O.E. acknowledges the support from Swedish Re-\nsearch Council (VR), eSSENCE, and the KAW Founda-\ntion (Grants No. 2012.0031 and No. 2013.0020).\nAppendix A: Numerical details\nWe performkintegration with up to 1 :25\u0001106mesh\npoints (500\u0002500\u0002500) in the \frst Brillouin zone for bulk.\nThe energy integration is evaluated at the Fermi level\nonly. For our principles studies, we performed a Slater-\nKoster parameterised40tight binding (TB) calculations58\nof the torque-torque correlation model as well as for the\nGreen's function model. Here, the TB parameters have\nbeen obtained by \ftting the electronic structures to those\nof a \frst-principles fully relativistic multiple scattering\nKorringa-Kohn-Rostoker (KKR) method using a genetic\nalgorithm. The details of the \ftting and the tight binding\nparameters are listed elsewhere23,59. This puts our model\non a \frm, \frst-principles ground.\nThe tight binding Hamiltonian60H=H0+Hmag+\nHsoccontains on-site energies and hopping elements H0,\nthe spin-orbit coupling Hsoc=\u0010S\u0001Land the Zeeman\ntermHmag=1=2B\u0001\u001b. The Green's function is obtained\nbyG= (\"+ i\u0000\u0000H)\u00001, allows in principle to consider\ndisorder in terms of spin and phonon as well as alloys23.\nThe bulk Greenian Gijin real space between site iandj\nis obtained by Fourier transformation. Despite the fact\nthat the tight binding approach is limited in accuracy, it\nproduces good agreement with \frst principle band struc-\nture calculations for energies smaller than \"F+ 5 eV.9\n-1.5-1.0-0.50.00.51.01.5\n5 10 15 20 25 30\nrij(Bohr radii)Γ≈0.01eVTB\nTBe\nDFT\nαDFT\nii= 3.9846·10−3\nαTB\nii= 3.6018·10−3-1.5-1.0-0.50.00.51.01.5\nΓ≈0.005eV\nαDFT\nii= 3.965·10−3\nαTB\nii= 3.5469·10−3αij·10−4\nFIG. 9: (Colour online) Comparison of non-local damping ob-\ntained from the Tight Binding method (TB) (red \flled sym-\nbols), Tight Binding with Edwards correction (TBe) (blue\n\flled symbols) and the linear mu\u000en tin orbital method (DFT)\n(open symbols) for fcc Co. Two di\u000berent spectral broadenings\nare chosen.\nEquation (4) was also evaluated within the DFT and\nlinear mu\u000en-tin orbital method (LMTO) based code\nRSPt. The calculations were done for a k-point mesh\nof 1283k-points. We used three types of basis func-\ntions, characterised by di\u000berent kinetic energies with\n\u00142= 0:1;\u00000:8;\u00001:7 Ry to describe 4 s, 4pand 3dstates.\nThe damping constants were calculated between the 3 d\norbitals, obtained using using mu\u000en-tin head projection\nscheme61. Both the \frst principles and tight binding im-\nplementation of the non-local Gilbert damping agree well\n(see Fig. 9).\nNote that due to numerical reasons, the values of\n\u0000 used for the comparisons are slightly di\u000berent in\nboth electronic structure methods. Furthermore, in the\nLMTO method the orbitals are projected to d-orbitals\nonly, which lead to small discrepancies in the damping.\nThe atomistic magnetization dynamics is also per-\nformed within the Cahmd simulation package58. To\nreproduce bulk properties, periodic boundary condi-\ntions and a su\u000eciently large cluster (10 \u000210\u000210)\nare employed. The numerical time step is \u0001 t=\n0:1 fs. The exchange coupling constants Jijare\nobtained from the Liechtenstein-Kastnelson-Antropov-\nGubanovski (LKAG) formula implemented in the \frst-\nprinciples fully relativistic multiple scattering Korringa-\nKohn-Rostoker (KKR) method39. On the other hand,\nthe magneto-crystalline anisotropy is used as a \fxed pa-\nrameter with K= 50\u0016eV.\n012345678α·10−3\n0.0 0.02 0.04 0.06 0.08 0.1\nξd(eV)2.02.22.42.62.83.03.2γ\n0.0 0.1 0.2 0.3 0.4\nq(a−1\n0)-12-10-8-6-4-20αnn·10−5\n01234567\nαos·10−3 1.945\n1.797\n1.848\n1.950\n1.848\n1.797\n1.950FIG. 10: (Color online) Gilbert damping \u000bas a function of\nthe spin-orbit coupling for the d-states in fcc Co. Lower panel\nshows the Gilbert damping in reciprocal space for di\u000berent\nq=jqjvalues (di\u000berent gray colours) along the \u0000 !Xpath.\nThe upper panel exhibits the on-site \u000bos(red dotes and lines)\nand nearest-neighbour \u000bnn(gray dots and lines) damping.\nThe solid line is the exponential \ft of the data point. The\ninset shows the \ftted exponents \rwith respect wave vector\nq. The colour of the dots is adjusted to the particular branch\nin the main \fgure. The spectral width is \u0000 = 0 :005 eV.\nAppendix B: Spin-orbit coupling scaling in real and\nreciprocal space\nKambersk\u0013 y's formula is valid only for quadratic spin-\norbit coupling scaling21,47, which implies only scattering\nbetween states that preserve the spin. This mechanism\nwas explicitly accounted by Edwards48by neglecting the\nspin-orbit coupling contribution in the `host' Green's\nfunction. It is predicted for the coherent mode ( q= 0)21\nthat this overcomes the unphysical and not experimen-\ntally veri\fed divergent Gilbert damping for low tem-\nperature. Thus, the methodology requires to prove the\nfunctional dependency of the (non-local) Gilbert damp-\ning with respect to the spin-orbit coupling constant \u0018\n(Fig. 10). Since damping is a Fermi-surface e\u000bects, it\nis su\u000ecient to consider only the spin-orbit coupling of\nthe d-states. The real space Gilbert damping \u000bij/\u0018\r\nscales for both on-site and nearest-neighbour sites with\n\r\u00192. For the reciprocal space, however, the scaling is\nmore complex and \rdepends on the magnon wave vec-\ntorq(inset in Fig. 10). In the long-wavelength limit,\nthe Kambersk\u0013 y formula is valid, where for the ferromag-\nnetic magnon mode with \r\u00193 the Kambersk\u0013 y formula\nis inde\fnite according to Edwards48.10\n10−32510−2α(q)\n10−310−210−110+0\nΓ (eV)0.000\n0.025\n0.050\n0.075\n0.100\nq: Γ→XCo\nFIG. 11: (Colour online) Comparison of reciprocal non-local\ndamping with (squares) or without (circles) corrections pro-\nposed by Costa et al.47and Edwards48for Co and di\u000berent\nspectral broadening \u0000. Di\u000berent colours represent di\u000berent\nmagnon propagation vectors q.\nAppendix C: Intraband corrections\nFrom the same reason as discussed in Section B, the\nrole of the correction proposed by Edwards48for magnon\npropagations di\u000berent than zero is unclear and need to\nbe studied. Hence, we included the correction of Ed-\nward also to Eq. (3) (Fig. 11). The exclusion of the spin-\norbit coupling (SOC) in the `host' clearly makes a major\nqualitative and quantitative change: Although the in-\nterband transitions are una\u000bected, interband transitions\nare mainly suppressed, as it was already discussed by\nBarati et al.21. However, the intraband contributions are\nnot totally removed for small \u0000. For very small scat-\ntering rates, the damping is constant. Opposite to the\n`non-corrected' Kambersk\u0013 y formula, the increase of the\nmagnon wave number qgives an increase in the non-\nlocal damping which is in agreement to the observation\nmade by Yuan et al.29, but also with the analytical modelproposed in Ref. [52] for small q. This behaviour was ob-\nserved for all itinerant magnets studied here.\nAppendix D: Comparison real and reciprocal\nGilbert damping\nThe non-local damping scales like r\u00002\nijwith the dis-\ntance between the sites iandj, and is, thus, very long\nrange. In order to compare \u000btot=P\nj2Rcut\u000bijfor arbi-\ntraryiwith\u000b(q= 0), we have to specify the cut-o\u000b ra-\ndius of the summation in real space (Fig. 12). The inter-\nband transitions (\u0000 >0:05 eV) are already converged for\nsmall cut-o\u000b radii Rcut= 3a0. Intraband transitions, on\nthe other hand, converge weakly with Rcutto the recipro-\ncal space value \u000b(q= 0). Note that \u000b(q= 0) is obtained\nfrom the corrected formalism. Even with Rcut= 9a0\nwhich is proportional to \u001930000 atoms, we have not\n0.81.21.62.0αtot·10−3\n4 5 6 7 8 9\nRcut/a00.005\n0.1\nFIG. 12: Total Gilbert damping \u000btotfor fcc Co as a function\nof summation cut-o\u000b radius for two spectral width \u0000, one in\nintraband (\u0000 = 0 :005 eV, red dottes and lines) and one in the\ninterband (\u0000 = 0 :1 eV, blue dottes and lines) region. The\ndotted and solid lines indicates the reciprocal value \u000b(q= 0)\nwith and without SOC corrections, respectively.\nobtain convergence.\n\u0003Electronic address: danny.thonig@physics.uu.se\n1S. S. P. Parkin, M. Hayashi, and L. Thomas, Science 320,\n190 (2008), URL http://www.sciencemag.org/cgi/doi/\n10.1126/science.1145799 .\n2J. Iwasaki, M. Mochizuki, and N. Nagaosa, Nature\nNanotech 8, 742 (2013), URL http://www.nature.com/\ndoifinder/10.1038/nnano.2013.176 .\n3A. Fert, V. Cros, and J. Sampaio, Nature Nanotech 8,\n152 (2013), URL http://www.nature.com/doifinder/10.\n1038/nnano.2013.29 .\n4B. Behin-Aein, D. Datta, S. Salahuddin, and S. Datta,\nNature Nanotech 5, 266 (2010), URL http://www.nature.\ncom/doifinder/10.1038/nnano.2010.31 .\n5N. Locatelli, A. F. Vincent, A. Mizrahi, J. S. Fried-\nman, D. Vodenicarevic, J.-V. Kim, J.-O. Klein, W. Zhao,\nJ. Grollier, and D. Querlioz, in DATE 2015 (IEEE Confer-ence Publications, New Jersey, ????), pp. 994{999, ISBN\n9783981537048, URL http://ieeexplore.ieee.org/xpl/\narticleDetails.jsp?arnumber=7092535 .\n6K. Koumpouras, D. Yudin, C. Adelmann, A. Bergman,\nO. Eriksson, and M. Pereiro (2017), 1702.00579, URL\nhttps://arxiv.org/abs/1702.00579 .\n7O. Eriksson, A. Bergman, L. Bergqvist, and\nJ. Hellsvik, Atomistic Spin Dynamics , Foundations\nand Applications (Oxford University Press, 2016),\nURL https://global.oup.com/academic/product/\natomistic-spin-dynamics-9780198788669 .\n8V. P. Antropov, M. I. Katsnelson, B. N. Harmon,\nM. van Schilfgaarde, and D. Kusnezov, Phys. Rev. B 54,\n1019 (1996), URL http://link.aps.org/doi/10.1103/\nPhysRevB.54.1019 .\n9V. Kambersk\u0013 y, Phys. Rev. B 76, 134416 (2007),11\nURL http://link.aps.org/doi/10.1103/PhysRevB.76.\n134416 .\n10V. Kambersk\u0013 y, Czech J Phys 34, 1111 (1984), URL https:\n//link.springer.com/article/10.1007/BF01590106 .\n11V. Kambersk\u0013 y, Czech J Phys 26, 1366 (1976), URL http:\n//link.springer.com/10.1007/BF01587621 .\n12K. Gilmore, Y. U. Idzerda, and M. D. Stiles, Phys. Rev.\nLett. 99, 027204 (2007), URL http://link.aps.org/doi/\n10.1103/PhysRevLett.99.027204 .\n13A. A. Starikov, P. J. Kelly, A. Brataas, Y. Tserkovnyak,\nand G. E. W. Bauer, Phys. Rev. Lett. 105, 236601 (2010),\nURL http://link.aps.org/doi/10.1103/PhysRevLett.\n105.236601 .\n14H. Ebert, S. Mankovsky, D. K odderitzsch, and P. J. Kelly,\nPhys. Rev. Lett. 107, 066603 (2011), URL http://link.\naps.org/doi/10.1103/PhysRevLett.107.066603 .\n15S. Mankovsky, D. K odderitzsch, G. Woltersdorf, and\nH. Ebert, Phys. Rev. B 87, 014430 (2013), URL http:\n//link.aps.org/doi/10.1103/PhysRevB.87.014430 .\n16M. A. W. Schoen, D. Thonig, M. L. Schneider, T. J. Silva,\nH. T. Nembach, O. Eriksson, O. Karis, and J. M. Shaw,\nNat Phys 12, 839 (2016), URL http://www.nature.com/\ndoifinder/10.1038/nphys3770 .\n17J. Chico, S. Keshavarz, Y. Kvashnin, M. Pereiro,\nI. Di Marco, C. Etz, O. Eriksson, A. Bergman, and\nL. Bergqvist, Phys. Rev. B 93, 214439 (2016), URL http:\n//link.aps.org/doi/10.1103/PhysRevB.93.214439 .\n18P. D urrenfeld, F. Gerhard, J. Chico, R. K. Dumas, M. Ran-\njbar, A. Bergman, L. Bergqvist, A. Delin, C. Gould,\nL. W. Molenkamp, et al. (2015), 1510.01894, URL http:\n//arxiv.org/abs/1510.01894 .\n19M. A. W. Schoen, J. Lucassen, H. T. Nembach, T. J. Silva,\nB. Koopmans, C. H. Back, and J. M. Shaw, Phys. Rev. B\n95, 134410 (2017), URL http://link.aps.org/doi/10.\n1103/PhysRevB.95.134410 .\n20M. A. W. Schoen, J. Lucassen, H. T. Nembach, B. Koop-\nmans, T. J. Silva, C. H. Back, and J. M. Shaw, Phys. Rev.\nB95, 134411 (2017), URL http://link.aps.org/doi/10.\n1103/PhysRevB.95.134411 .\n21E. Barati, M. Cinal, D. M. Edwards, and A. Umerski,\nPhys. Rev. B 90, 014420 (2014), URL http://link.aps.\norg/doi/10.1103/PhysRevB.90.014420 .\n22F. Pan, J. Chico, J. Hellsvik, A. Delin, A. Bergman, and\nL. Bergqvist, Phys. Rev. B 94, 214410 (2016), URL https:\n//link.aps.org/doi/10.1103/PhysRevB.94.214410 .\n23D. Thonig and J. Henk, New J. Phys. 16, 013032\n(2014), URL http://iopscience.iop.org/article/10.\n1088/1367-2630/16/1/013032 .\n24Z. Ma and D. G. Seiler, Metrology and Diagnostic\nTechniques for Nanoelectronics (Pan Stanford, Taylor\n& Francis Group, 6000 Broken Sound Parkway NW,\nSuite 300, Boca Raton, FL 33487-2742, 2017), ISBN\n9789814745086, URL http://www.crcnetbase.com/doi/\nbook/10.1201/9781315185385 .\n25D. Steiauf and M. F ahnle, Phys. Rev. B 72, 064450 (2005),\nURL https://link.aps.org/doi/10.1103/PhysRevB.\n72.064450 .\n26D. Thonig, J. Henk, and O. Eriksson, Phys. Rev. B 92,\n104403 (2015), URL http://link.aps.org/doi/10.1103/\nPhysRevB.92.104403 .\n27M. F ahnle and D. Steiauf, Phys. Rev. B 73, 184427 (2006),\nURL https://link.aps.org/doi/10.1103/PhysRevB.\n73.184427 .\n28E. Minguzzi, European Journal of Physics 36, 035014(2015), URL http://adsabs.harvard.edu/cgi-bin/\nnph-data_query?bibcode=2015EJPh...36c5014M&link_\ntype=EJOURNAL .\n29Z. Yuan, K. M. D. Hals, Y. Liu, A. A. Starikov, A. Brataas,\nand P. J. Kelly, Phys. Rev. Lett. 113, 266603 (2014),\nURL http://link.aps.org/doi/10.1103/PhysRevLett.\n113.266603 .\n30H. Nembach, J. Shaw, C. Boone, and T. Silva, Phys. Rev.\nLett. 110, 117201 (2013), URL http://link.aps.org/\ndoi/10.1103/PhysRevLett.110.117201 .\n31S. Bhattacharjee, L. Nordstr om, and J. Fransson, Phys.\nRev. Lett. 108, 057204 (2012), URL http://link.aps.\norg/doi/10.1103/PhysRevLett.108.057204 .\n32K. Gilmore and M. D. Stiles, Phys. Rev. B 79,\n132407 (2009), URL http://link.aps.org/doi/10.1103/\nPhysRevB.79.132407 .\n33K. M. D. Hals, A. K. Nguyen, and A. Brataas, Phys. Rev.\nLett. 102, 256601 (2009), URL https://link.aps.org/\ndoi/10.1103/PhysRevLett.102.256601 .\n34A. Brataas, Y. Tserkovnyak, and G. E. W. Bauer, Phys.\nRev. B 84, 054416 (2011), URL https://link.aps.org/\ndoi/10.1103/PhysRevB.84.054416 .\n35R. Chimata, E. K. Delczeg-Czirjak, A. Szilva, R. Car-\ndias, Y. O. Kvashnin, M. Pereiro, S. Mankovsky,\nH. Ebert, D. Thonig, B. Sanyal, et al., Phys. Rev. B 95,\n214417 (2017), URL http://link.aps.org/doi/10.1103/\nPhysRevB.95.214417 .\n36K. Gilmore, Ph.D. thesis, scholarworks.montana.edu\n(2008), URL http://scholarworks.montana.edu/\nxmlui/bitstream/handle/1/1336/GilmoreK1207.pdf?\nsequence=1 .\n37K. Gilmore, I. Garate, A. H. MacDonald, and M. D. Stiles,\nPhys. Rev. B 84, 224412 (2011), URL http://link.aps.\norg/doi/10.1103/PhysRevB.84.224412 .\n38M. Sayad, R. Rausch, and M. Pottho\u000b, Phys. Rev. Lett.\n117, 127201 (2016), URL http://link.aps.org/doi/10.\n1103/PhysRevLett.117.127201 .\n39J. Zabloudil, L. Szunyogh, R. Hammerling, and P. Wein-\nberger, Electron Scattering in Solid Matter , A Theoretical\nand Computational Treatise (2006), URL http://bookzz.\norg/md5/190C5DE184B30E2B6898DE499DFB7D78 .\n40J. C. Slater and G. F. Koster, Phys. Rev. 94,\n1498 (1954), URL http://link.aps.org/doi/10.1103/\nPhysRev.94.1498 .\n41J. M. Wills and B. R. Cooper, Phys. Rev. B 36,\n3809 (1987), URL https://link.aps.org/doi/10.1103/\nPhysRevB.36.3809 .\n42H. Dreyss\u0013 e, Electronic Structure and Physical Properties of\nSolids (Springer, 2000), URL http://bookzz.org/md5/\n66B5CD9A4859B7F039CB877263C4C9DD .\n43C. Vittoria, S. D. Yoon, and A. Widom, Phys. Rev. B\n81, 014412 (2010), URL https://link.aps.org/doi/10.\n1103/PhysRevB.81.014412 .\n44D. B ottcher, A. Ernst, and J. Henk, Journal of\nMagnetism and Magnetic Materials 324, 610 (2012),\nURL http://linkinghub.elsevier.com/retrieve/pii/\nS0304885311006299 .\n45A. Szilva, M. Costa, A. Bergman, L. Szunyogh,\nL. Nordstr om, and O. Eriksson, Phys. Rev. Lett. 111,\n127204 (2013), URL http://link.aps.org/doi/10.1103/\nPhysRevLett.111.127204 .\n46I. Garate, K. Gilmore, M. D. Stiles, and A. H. MacDonald,\nPhys. Rev. B 79, 104416 (2009), URL http://link.aps.\norg/doi/10.1103/PhysRevB.79.104416 .12\n47A. T. Costa and R. B. Muniz, Phys. Rev. B 92,\n014419 (2015), URL http://link.aps.org/doi/10.1103/\nPhysRevB.92.014419 .\n48D. M. Edwards, J. Phys.: Condens. Matter 28, 086004\n(2016), URL http://iopscience.iop.org/article/10.\n1088/0953-8984/28/8/086004 .\n49M. Oogane, T. Wakitani, S. Yakata, R. Yilgin, Y. Ando,\nA. Sakuma, and T. Miyazaki, Jpn. J. Appl. Phys. 45,\n3889 (2006), URL http://iopscience.iop.org/article/\n10.1143/JJAP.45.3889 .\n50S. M. Bhagat and P. Lubitz, Phys. Rev. B 10,\n179 (1974), URL https://link.aps.org/doi/10.1103/\nPhysRevB.10.179 .\n51T. L. Gilbert, IEEE Trans. Magn. 40, 3443 (2004), URL\nhttp://adsabs.harvard.edu/cgi-bin/nph-data_query?\nbibcode=2004ITM....40.3443G&link_type=EJOURNAL .\n52N. Umetsu, D. Miura, and A. Sakuma, J. Phys. Soc. Jpn.\n81, 114716 (2012), URL http://journals.jps.jp/doi/\n10.1143/JPSJ.81.114716 .\n53M. Pajda, J. Kudrnovsk\u0013 y, I. Turek, V. Drchal, and\nP. Bruno, Phys. Rev. B 64, 174402 (2001), URL https:\n//link.aps.org/doi/10.1103/PhysRevB.64.174402 .\n54A. I. Liechtenstein, M. I. Katsnelson, V. P. Antropov, and\nV. A. Gubanov, Journal of Magnetism and Magnetic Ma-\nterials 67, 65 (1987), URL http://linkinghub.elsevier.com/retrieve/pii/0304885387907219 .\n55P.-W. Ma, S. L. Dudarev, and C. H. Woo, Phys. Rev. B\n85, 184301 (2012), URL http://link.aps.org/doi/10.\n1103/PhysRevB.85.184301 .\n56L. Bergqvist, A. Taroni, A. Bergman, C. Etz, and O. Eriks-\nson, Phys. Rev. B 87, 144401 (2013), URL http://link.\naps.org/doi/10.1103/PhysRevB.87.144401 .\n57D. B ottcher, A. Ernst, and J. Henk, J. Phys.: Condens.\nMatter 23, 296003 (2011), URL http://iopscience.iop.\norg/article/10.1088/0953-8984/23/29/296003 .\n58D. Thonig, CaHmd - A computer program package for\natomistic magnetisation dynamics simulations. , Uppsala\nUniversity, Uppsala, 2nd ed. (2013).\n59D. Thonig, O. Eriksson, and M. Pereiro, Sci. Rep.\n7, 931 (2017), URL http://www.nature.com/articles/\ns41598-017-01081-z .\n60T. Schena, Ph.D. thesis, fz-juelich.de (2010), URL http://\nwww.fz-juelich.de/SharedDocs/Downloads/PGI/PGI-1/\nEN/Schena_diploma_pdf.pdf?__blob=publicationFile .\n61A. Grechnev, I. Di Marco, M. I. Katsnelson, A. I.\nLichtenstein, J. Wills, and O. Eriksson, Phys. Rev. B\n76, 035107 (2007), URL https://link.aps.org/doi/10.\n1103/PhysRevB.76.035107 ."    },    {        "title": "1710.10833v2.Probe_of_Spin_Dynamics_in_Superconducting_NbN_Thin_Films_via_Spin_Pumping.pdf",        "content": "1\n \n \nProbe of Spin Dynamics in Superconducting NbN Thin\n \nFilms via \n \nSpin Pumping\n \n \nYunyan\n \nYao\n1,2\n,\n \nQi\n \nSong\n1,2\n,\n \nYota\n \nTakamura\n3,4\n,\n \nJuan\n \nPedro\n \nCascales\n3\n,\n \nWei\n \nYuan\n1,2\n,\n \nYang\n \nMa\n1,2\n,\n \nYu\n \nYun\n1,2\n,\n \nX.\n \nC.\n \nXie\n1,2\n,\n \nJagadeesh\n \nS.\n \nMoodera\n3,5\n,\n \nand\n \nWei\n \nHan\n1,2*\n \n1\nInternational Center for Quantum Materials, School of Physics, Peking University, Beijing \n100871, China\n.\n \n2\nCollaborative Innovation Center of Quantum Matter, Beijing 100871, China\n.\n \n3\nPlasma Science and Fusion\n \nCenter and Francis Bitter Magnet Laboratory, Massachusetts \nInstitute of Technology, Cambridge, MA 02139, USA\n.\n \n4\nSchool of Engineering, Tokyo Institute of Technology, Tokyo 152\n-\n8550, Japan.\n \n5\nDepartment of Physics, Massachusetts Institute of Technology, Camb\nridge, MA 02139, USA\n \n*Correspondence to: \nweihan@pku.edu.cn (W.H.)\n \n \n \nAbstract\n:\n \nThe emerging field of superconductor (SC) spintronics has attracted \nintensive attentions recently\n. \nMany fantastic spin dependent properties in SC\ns\n \nhave been discovered, including large \nmagnetoresistance, long spin lifetimes and the giant spin Hall effect\n, \netc\n. Regarding the spin \ndynamic\ns\n \nin \ns\nuperconducting \nthin \nfilms, \nfew studies has been reported yet. \nHere, we report the \ninvestig\nation of the spin d\nynamics in a\n \ns\n-\nwave superconducting NbN film via spin pumping from \nan adjacent insulating ferromagnet GdN\n \nfilm\n. \nA\n \nprofound coherence peak of the Gilbert damping \nof GdN \nis observed slightly below the superconducting critical temperature of the NbN\n, \nwhich \nag\nrees well\n \nwith recent\n \ntheoretical \nprediction\n \nfor \ns\n-\nwave SCs \nin the presence of impurity spin\n-\norbit \nscattering\n. \nThis observation is also \na manifestation of the dynamic\n \nspin injection\n \ninto \nsuperconducting NbN thin film.\n \nOur results \ndemonstrate that spin pumping could be used \nt\no probe \nthe dynamic spin susceptibility\n \nof \nsuperconducting thin films, thus pave the \nway for future \ninvestigation of spin dynamics \nof\n \ninterfacial and \ntwo \ndimensional\n \ncrystalline \nSC\ns\n.\n \n 2\n \n \nI.\n \nINTRODUCTION\n \nThe \ninterplay between s\nuperconductivity and \nspintronics \nhas\n \nbeen intensively investigated \nin \nthe last decade\n \n[\n1\n-\n3\n]\n.\n \nIn the content of superconductivity, the \nferromagnetic \nmagnetization \nhas\n \nbeen found to \nplay\n \nan important role in the superconducting critical temperature (T\nC\n) in the \nferromagnet\n \n(FM)\n/\nsuperconductor (\nSC\n)\n \njunctions\n \n[\n4\n-\n9\n]\n.\n \nBesides, \nunusual\n \nspin\n-\npolarized\n \nsupercurrent \nhas been observed \nin \nferromagnetic \nJosephson junctions\n \n[\n2\n,\n3\n,\n10\n-\n13\n]\n. \nFurthermore, \nnon\n-\nabelian Majorana fermions have been proposed for \np\nx\n \n+ ip\ny\n \nsuperconductor surfaces driven \nby magnetic proximity effect, whic\nh has the potential for topological quantum computation \n[\n14\n]\n. \nIn the content of spintronics\n \n[\n15\n,\n16\n]\n, \nthe co\noper pairs \nin \nSCs\n \nhave exhibited many fa\nntas\ntic spin\n-\ndependent properties\n \nthat \nare promising \nfor future\n \ninformation technologies.\n \nFor instance, l\narge \nmagnetoresistance has been \nachi\ne\nved\n \nin the spin valve with \nSCs \nbetween two ferromagnetic films\n \n[\n7\n,\n17\n]\n, \nextremely \nl\nong spin lifetimes \nand large spin Hall \neffect have \nbeen discovered for the \nquasiparticles\n \nof\n \nSCs\n, which exceeds several orders compared to the normal states\n \nabove T\nC\n \n[\n18\n,\n19\n]\n. \nDespite these inte\nnsive studies of the spin\n-\ndependent \npr\noperties \nin the FM/SC junctions\n, \nthe spin dynamics of \nthe \nsupercond\nu\nc\nting thin films\n \nhas not been \nreported\n \nyet.\n \n \nVery \nintriguingly\n, \nit has been\n \nproposed \nrecently \nt\no investigate\n \nthe\n \nspin dynamics in \nsuperconducting \nfilms via spin pumping\n \n[\n20\n]\n, a well\n-\nestablished technique to \nperform d\nynamic\n \nspin injection and to probe the dynamic spin susceptibility in various materials, including metal, \nse\nmiconductors, Rashba 2DEGs, and topological insulators, etc\n \n[\n21\n-\n30\n]\n.\n \n \nIn this letter\n, we report the \nexperimental investigation \nof \nthe spin dynamics in \ns\n-\nwav\ne \nsuperconducting NbN thin film\ns\n \nvia spin pumping. \nA profou\nnd coherence peak of the Gilbert \ndamping\n \nof GdN\n \nis observed slightly below the superconducting critical temperature\n \n(T\nC\n)\n \nof the \nNbN \nin the \nNbN/GdN/NbN\n \ntrilayer samples\n, which indicates \ndynamic spin injection into the \nN\nbN \nthin film\n. Besides, \nthe \ninterface\n-\nenhanced Gilbert damping\n \nprobes \ndynamic spin susceptibility\n \nin \nthe \nNbN thin films \nin the presence of impurity spin\n-\norbit\n \nscattering\n, \nwhich is consistent w\nith \nthe\n \nrecent theoretical study\n \n[\n20\n]\n. \nOur experimental results further \ndemonstrat\ne that\n \nspin pumping \ncould \nbe\n \na \npowerful tool to study the spin dynamics\n \nin \nthe emerging two dimensional SCs\n \n[\n31\n-\n33\n]\n. \n \nII. EXPERIMENTAL DETAILS\n \nThe NbN (t)/GdN (d)/NbN (t) trilayer samples \nare\n \ngrown on Al\n2\nO\n3\n \n(~5 nm)\n-\nbuffered thermally \noxidized Si substrates by d.c. reactive magnetron sputtering at 300 \n°\nC in an ultrahigh vacuum 3\n \n \nchamber. The \nNbN layers \nare\n \ndeposited from a pure Nb target (99.95%) in Ar and N\n2\n \ngas mixture \nat a pressure of 2.3 mTorr (20% N\n2\n), and \nGdN films \nare\n \ndeposited from a pure Gd target (99.9%) \nin Ar and N\n2\n \ngas mixture at a pressure of 2.8 mTorr (6% N\n2\n).\n \nThe NbN and GdN layers \nare \nof \ntextured crystalline quality with a preferred direction along (111)\n-\norientation, as evidenced by X\n-\nray diffraction results (\nF\nig. S1)\n \n[\n34\n]\n. After the growth, a thin Al\n2\nO\n3\n \nlayer (~ 10 nm) was deposited \nin situ\n \nas a capping layer to avoid sample degradation with air exposure. The Curie temperature of \nthe GdN film was determined via the offset\n-\nmagnetization as a function of the temperature us\ning \na Magnetic Properties Measurement System (MPMS; Quantum Design).\n \nThe T\nC\n \nof the SC NbN \nthin films was measured by four\n-\nprobe resistance technique as a function of the temperature in a \nPhysical Properties Measurement System (PPMS; Quantum Design) using s\ntandard ac lock\n-\nin \ntechnique at low frequency of 7 Hz.\n \nThe FMR spectra of the multilayer samples were measured \nusing the coplanar wave guide technique with a vector network analyzer (VNA, Agilent E5071C) \nin the variable temperature insert of PPMS. The samp\nles were attached to the coplanar wave guide \nusing insulating silicon paste. During the measurement, the amplitudes of forward complex \ntransmission coefficients (S\n21\n) were recorded as a functio\nn of the\n \nin\n-\nplane\n \nmagnetic field from ~ \n4\n000 to 0 Oe at various\n \ntemperatures under different microwave frequencies\n \nand microwave power \nof 5 dBm\n.  \n \nIII. RESULTS and DISCUSSION\n \nFig\n.\n \n1\n(a)\n \nillustrates \nspin pumping and \nthe interfacial \ns\n-\nd\n \nexchange coupling between \nspins\n \nin \nthe NbN layer and \nm\nagnetic moment\ns\n \nin the GdN laye\nr\n. \nDue to the interfacial \ns\n-\nd\n \nexchange \ninteraction (\nJ\nsd\n), the time\n-\ndependent magnetization in the GdN layer pumps a quasiparticles\n-\nmediated spin current into the NbN layer, and the spin\n-\nflip scattering of quasiparticles \naccompanies a quantum process of mag\nnon annihilation in the GdN layer, giving rise to enhanced \nGilbert damping\n \n[\n20\n]\n. Hence, the interfacial s\n-\nd exchange coupling provides the route to detect the \nspin dynami\ncs of the NbN layer by measuring the magnetization dynamics of GdN\n.\n \nThe spin \npumping is performed by measuring Gilbert damping of the GdN \nfilm \nin the NbN (t)/GdN (d)/NbN \n(t) trilayer heterostructures\n \nvia ferromagnetic resonance (FMR) technique \n[\n35\n]\n. \nNbN is a\n \ns\n-\nwave \nSC\n \nwith short coherence length of ~ 5 nm\n \nand \nspin diffusion length of ~ 7 nm\n \n[\n19\n,\n36\n]\n. \nGdN is an \ninsulating FM\n. \nA 10 nm \nNbN film is used f\nor the NbN (t)/GdN (d)/NbN (t) \nsamples to justify the \nassumption that the spin backflow from SC is small in the theoretical study \n[\n20\n]\n.\n \nIn a typical sample \nof N\nbN (10)/GdN (5)/NbN (10)\n \n(with thickness in nm)\n, the Curie temperature (T\nCurie\n) of GdN is 4\n \n \ndetermined to be ~ 38 K \nfrom \nthe temperature dependence of \nmagnetic moments\n \n(Fig. 1\n(b)\n), and \nT\nC\n \nof NbN is \nobtained to be \n~ 10.8 K\n \nvia\n \nresistivity vs. temperature \nmeasurement \n(Fig. 1\n(c)\n)\n. \nThe \nT\nC\n \nis slightly affected by the \nexternal in\n-\nplane magnetic\n \nfield of 4000 Oe\n \ndue to the large critical \nfield.\n \n \nFig. 1\n(d)\n \nshows a typical FMR signal (S\n21\n) vs.\n \nthe magnetic field measured \nat \nT\n \n= \n10 K, with \na microwave excitation \nfrequency (\nf\n) of 15 GHz. \nThe half linewidth (\nH\n\n) could be obtained by \nthe Lorentz fitting of the \nFMR \nsignal\n \nfollowing the relationship\n \n[\n37\n]\n:\n \n2\n21 0\n2 2\n( )\n( ) ( )\nres\nH\nS S\nH H H\n\n\n  \n \n \n \n \n \n(1)\n \nwhere S\n0\n \nis the coefficient for the transm\nitted microwave power, \nH\n \nis the external magnetic field, \nand \nres\nH\nis the resonance magnetic field. Gilbert damping (\n\n) is determined using numerical \nfitting of\n \nH\n\nvs. \nf\n \n(Fig. 1\n(e)\n) \nbased \non the spin\n-\nrelaxation\n \nmechanism \n(\nFig. S2\n)\n \n[\n34\n,\n38\n]\n:\n \n0\n2\n4 2\n1 (2 )\nf f\nH H A\nf\n  \n  \n   \n\n \n \n \n \n(2)\n \nwhere \n0\nH\n\n \nis related to the inhomogeneous properties,  \n\n \nis the gyromagnetic ratio, \nA\n \nis t\nhe \nspin\n-\nrelaxation coefficient, and \n\n \nis the \nspin\n-\nrelaxation time constant\n. \n \nThe Gilbert damping \nof GdN \nis studied\n \nas a function of the temperature for two trilayer samples\n \nwith the same interface\n:\n \nNbN (2)/GdN (5)/NbN (2) and NbN (10)/GdN (5)/NbN (10)\n. \nThe 2 nm \nnm Nb\nN layer\n \nis not superconducting\n \ndown to 2 K \nwhile the \nNbN (10)/GdN (5)/NbN (10)\n \nexhibits \na T\nC\n \nof ~ 10.8 K \n(Fig. 2(a)\n). \nThe non\n-\nsuperconducting \nfeature of 2 nm NbN\n \nsample \nand lower \nT\nC\n \nof t\nhe 10 nm NbN \nsample \ncompared to over 16 K\n \nfor \nsingly crystalline \nNbN\n \ncould be \nattributed\n \nto \nreduced thickness, \npolycrystalline property, and magnetic proximity effect\n \n[\n1\n,\n36\n,\n39\n]\n. \nInterestingly, \na profound\n \ncoherence peak of the Gilbert damping \nis observed\n \nin the NbN (10)/GdN (5)/NbN (10)\n, \nbut not on \nNbN (2)/GdN (5)/NbN (2), \nas shown in Fig. 2\n(b)\n. \nThis feature is also evident \nfrom the \ntemperature dependence of the half\n \nlinewidth\n \n(Fig. S\n3\n)\n \n[\n34\n]\n.\n \nThe peak of the Gilbert damping in \nthe NbN (10)/GdN (5)/NbN (10) is observed at ~ 8.5 K, which is slightly below the T\nC\n \n(~ 10.8\n \nK) \nof the 10 nm NbN \nlayer\ns\n. These results indicate the successful dynamic spin injection into the 10 \nnm NbN \nsuperconducting \nlayer, which authenticates a charge\n-\nfree method to injec\nt spin\n-\npolarized \ncarriers into\n \nSC\ns\n \nbeyond previous repor\nts of electrical spin injection\n \n[\n18\n,\n40\n-\n42\n]\n. Furthermore, the 5\n \n \nobservation of the profound coherence peak \nat \nT\n \n= \n~\n \n0.8\n \nT\nC\n \nis expe\ncted \nbased on \nrecent theoretical \nstudies \nof the spin dynamic\ns for \ns\n-\nwave \ns\nupercond\nu\nc\nting thin films\n \nvia spin pumping\n \n[\n20\n]\n. \nSince \nthe thickness of NbN layer (d = 10 nm) i\ns longer than \nits\n \nspin diffusion length of ~ 7 nm, the spin \nbackflow effect, which \nis expected to\n \nreduce\n \nthe damping peak,\n \nis negligible\n \nhere\n \n[\n20\n,\n21\n]\n. \nAccording to the theory, \nGilbert damping is related to the interfacial \ns\n-\nd\n \nexchange interaction and \nthe imaginary part of the dynamic spin susceptibility of the SC\n \n[\n20\n]\n.\n \n \n \n \n \n \n2\nIm ( )\nR\nsd q\nq\nJ x\n \n\n\n \n \n \n \n \n(3)\n \nFor the \ns\n-\nwave superconducting NbN thin films, the superconducting gap \n∆\n \nforms\n \nbelow T\nC\n. At \nthe temperature slightly below T\nC\n, two coherence peaks of the density states exist around the edge \nof the superconducting gap following the BCS theory\n \n[\n43\n]\n, and these peaks in turn give rise to the \nenhancement of t\nhe dynamic spin susceptibility\n \nin the pre\nse\nnce of \nimpurity spin\n-\norbit scattering\n.\n \nQuantitatively\n,\n \nthe ratio of the peak value of Gil\nb\nert damping over \nthat \nat \nT\n \n~ \nT\nC\n \nis\n \n~ 1.8\n, which \nalso\n \nagree\ns\n \nwell wit\nh the theoretical calculation\n \nusing \nprevious\n \nexperimental values of spin \ndiffusion length (~ 7 nm), phase\n \ncoherence length (~ 5 nm), and mean free path (~ 0.3 nm)\n \nfor \nNbN\n \nthin films \n[\n19\n,\n20\n,\n36\n]\n. \nAs the temperature further decreases, the number of quasiparticles \ndecreases \nrapidly \nas the \n∆\n \ngrows, \ngiving rise to the fast decrease of Gilbert damping\n \nbelow \nT\n \n=\n \n~ \n7\n \nK\n. \n \nNext, the thickness of th\ne GdN laye\nr is varied to further study spin pumping\n \nand the spin \ndynamics in NbN layer. For\n \nboth \ns\namples of NbN (10)/GdN (d)/NbN (10) with d \n= \n10\n \nnm and d \n=\n \n30 nm, a profound coherence peak of Gilbert damping is observed slightly below the \nsuperconducting \ntemperat\nure of NbN, as shown in Figs. 3\n(a)\n \nand 3\n(b)\n \n(red circles). While for all \nthe samples of NbN (2)/GdN (d)/NbN (2), no such coherence peak of the Gilbert damping is \nnoticeable (green circles\n \nin Figs. 3(a) and 3(b)\n). The role of the effective \nmagnetization\n \nin the \nobserved \ncoherence peak\n \nhas been ruled out since it exhibits similar temperature dependence for \nNbN (2\n)/GdN (d\n)/NbN (2) and NbN (10)/G\ndN (d\n)/NbN (10) sampl\nes (\nF\nig. S\n4\n)\n \n[\n34\n]\n. \nClearly, \na \nmore profound damping peak for \nthe d = 10 nm sample\n \nis \nobserved\n \ncompared to \nd = 5 nm and \nd = \n30 nm \nsamples. \nFig. 3\n(c)\n \nshows t\nhe ratio of the peak Gilbert damping over the value at \nT\n \n= \n~ \nT\nC\n \nas a function of the GdN thickness\n, which\n \nis in the range from \n1.8 to 2.8\n. \nThe T\nC\n \nexhibit little \nvariation as a function of the GdN thickness\n \n(Fig. S5)\n \n[\n34\n]\n. \nFor a deeper understanding of the \nunderlying mechanism\n \nto account for the thickness dependence of the \nratio,\n \nfurther\n \ntheoretical and 6\n \n \nexperim\ne\nn\nt\nal\n \nstudies \nw\nould be essential\n. One possible cause might be related to the interface \nproximity exchange effect and/or \nthe \npresence of magnetic loose spins leading to scattering at the \ninterface, which could a\nffect the spin diffusion length, \ncoherence length\n, and mean free path\n \nof the \nN\nbN layer. \n \nTo further confirm \nthat \nthe observed coherence peak of the Gilbert damping aris\nes\n \nfrom the\n \ns\n-\nd\n \nexchange interaction\n \nat the SC/FM interface\n, \nthe interface\n-\ninduced Gilbert damping\n \n(\nS\n\n)\n \nis \nstudied as a function of temperatur\ne. \nS\n\n \nis obtained from the thickness dependence of \ntotal \nGilbert \ndamping \nat each temperature\n \n[\n34\n]\n, as shown in \nFig. 4\n(a)\n. \nT\nhe \ncontribution \nfrom GdN itself is \nsignificantly small\n \ncompared to\n \nS\n\n. \nThe unpe\nrturbed peak at ~ 8.5 K (Fig. 4\n(b)\n) unambiguously \ndemonstrates that the origin of the coherence peak in the Gilbert damping is indeed due to the \ninterfacial \ns\n-\nd\n \nexchange interaction between the magnetization of GdN and the spins \nof the \nquasiparticles\n \nin superconducting NbN thin films\n. \nThe \nslightly upturn of the interface\n-\ninduced \nGilbert damping starting at \nT\n \n= 11 K might be associated with the fluctuation superconductivity \nand/or the higher superconducting transition temperature for some regions of the NbN films than \nthe zero resistance tem\nperature. \nFor comparison, the interface\n-\ninduced Gilbert damping in the \nsamples of NbN (2)/GdN (d)/NbN (2) does not \nexhibit\n \nany \nsignature\n \nof coherence peak \nbetween \n4 and 15 K\n \n(\nF\nig. S6\n)\n \n[\n34\n]\n.\n \n \nNoteworthy is that our results are essentially different from previous reports of spin pumping \ninto \nsuperconducting\n \nNb films using the ferromagnetic metal permalloy\n \n(Py)\n, where a monotonic \ndecrease of\n \nthe Gilbert damping is reported when the temperature decreases\n \n[\n44\n,\n45\n]\n. \nThis feature \nis also \nconfirmed\n \nin our studies \nby measuring the temperature dependence of\n \nGilbert damping of \nPy in \nNb\n(100)/Py(20\n)/Nb(100)\n \n(F\nig. S7)\n \n[\n34\n]\n. \nSince \nP\ny\n \nis a \nFM\n \nmetal, the interface \nexchange \ninteraction\n \nwould be \nsignificant\n \nto \nstrongly\n \nor even \ncompletely\n \nsuppress\n \nthe\n \nsuperconducting gap\n \nof Nb\n \nat the \nNb/Py interface \n[\n17\n,\n20\n,\n43\n]\n. \nHence, no coherence peak of Gilbert damping is expected\n \nbased on the \ntheoretical\n \nstudy \nthat assumes the completely suppression of the gap at the interface\n \n[\n45\n]\n. \nBesid\nes, the strong suppression of the superconducting gap at the Nb/Py interface is in good \nagreement of previous tunnel spectroscopy measurements of vanishing superconducting gap at the \ninterface between Nb and Ni\n \n[\n4\n6\n]\n. \nHowever, \nfor \nFM insulating GdN, charge carriers from NbN do \nnot penetrate into GdN, thus not weakening the \nsuperconductivity\n \nat the interface\n, resulting the \nsurvival of the\n \nsuperconducting gap \n[\n47\n,\n48\n]\n. \n 7\n \n \nThe experimental \ndemonstration of the dynamic spin \nsusceptibility\n \nin \nsuperconducting\n \nthin \nfilms\n \nvia\n \nspin pumping \ncould be \nessential\n \nfor the field of superconductors.\n \nFor bulk SCs,\n \ndynamic \nspin susceptibility has been studied from the \ntemperature \ndependent \nspin \nrel\na\nxation rate via\n \nthe\n \nnuclear magnetic resonance\n \n[\n43\n,\n49\n,\n50\n]\n. \nIt also provides an avenue for  identifying unconventional \nsuperconducting paring mechanisms\n \n[\n51\n]\n, while limited to mostly bulk SCs due to low signal\n-\nto\n-\nnoise ratio\n \n[\n43\n,\n49\n-\n51\n]\n. S\npin pumping \nmethod \nhas much better \nsignal\n-\nto\n-\nnoise ratio\n \nso it could be \nvery \nsensitive\n \nto \nprobe the dynamic spin susceptibility\n \nof \nsuperconducting \nthin films\n.\n \nFurthermore, \nthe \nspin pumping offers a special \ntechnique \nto probe the \npair breaking strength, \nthe impurity s\npin\n-\norbit scattering, and the \nmagnetic proximity effect in the SC/FM junctions, which is \nalso \nof \nconsiderable interest\n \nfor the field of SC \nspintronics\n \n[\n1\n-\n3\n]\n. \n \nIV. CON\nCLUSION\n \nIn conclusion, \nthe \nspin \ndynamic\ns\n \nof \nsuperconducting \nNbN \nfilm\ns\n \nare\n \ninvestigated\n \nvia spin \npumping from \nan adjacent FM insulating layer\n. \nA profound coherence peak of the Gilbert damping \nis observed below T\nC\n,\n \nwhich indicates the dynamic spin injection \ninto the superconducting NbN \nfilms.\n \nOur results \npaves the way for\n \nfuture investigation of interface impurity spin\n-\norbit scattering, \nand pair breaking strength in FM/SC junctions, as w\nell as the spin dynamics in the \ninterfacial and \ntwo dimensional \ncrystalline SCs.\n \nIt may also be useful for the search for Majorana fermions in the \nSC/ferromagnetic insulator heterostructures.\n \n \n \nAcknowledgments\n \nWe acknowledge the fruitful discussion with Yuan Li and Tao Wu. Y.Y., Q.S., W.Y., Y.M., \nY.Y., X.C.X., and W.H.\n \nacknowledge the financial support from National Basic Research \nPrograms of China (973 program Grant Nos. 2015CB921104 and 2014CB920902) and National \nNatural Science Foundation of China (NSFC Grant No. 11574006). Y.T., J.P.C, and J.S.M. \nacknowledge the gra\nnts NSF DMR\n-\n1700137 and ONR N00014\n-\n16\n-\n1\n-\n2657. Y.T. also \nacknowledges the JSPS Overseas Research Fellowships and the Fundacion Seneca (Region de \nMurcia) posdoctoral fellowship (19791/PD/15). W.H. also acknowledges the support by the 1000 \nTalents Program for\n \nYoung Scientists of China.\n \n 8\n \n \nReferences\n:\n \n[1]\n \nA. I. Buzdin, Proximity effects in superconductor\n-\nferromagnet heterostructures.\n \nRev. Mod. \nPhys.\n \n77\n, 935 (2005).\n \n[2]\n \nJ. Linder and J. W. A. Robinson, Superconducting spintronics.\n \nNat. Phys.\n \n11\n, \n307 (2015).\n \n[3]\n \nM. Eschrig, Spin\n-\npolarized supercurrents for spintronics.\n \nPhys. Today\n \n64\n, 43 (2011).\n \n[4]\n \nJ. Y. Gu, C. Y. You, J. S. Jiang, J. Pearson, Y. B. Bazaliy, and S. D. Bader, Magnetization\n-\nOrientation Dependence of the Superconducting Transition Te\nmperature in the Ferromagnet\n-\nSuperconductor\n-\nFerromagnet System: CuNi/Nb/CuNi.\n \nPhys. Rev. Lett.\n \n89\n, 267001 (2002).\n \n[5]\n \nZ. Yang, M. Lange, A. Volodin, R. Szymczak, and V. V. Moshchalkov, Domain\n-\nwall \nsuperconductivity in superconductor\n-\nferromagnet hybrids.\n \nNa\nt. Mater.\n \n3\n, 793 (2004).\n \n[6]\n \nI. C. Moraru, W. P. Pratt, and N. O. Birge, Magnetization\n-\nDependent T\nC\n \nShift in \nFerromagnet/Superconductor/Ferromagnet Trilayers with a Strong Ferromagnet.\n \nPhys. Rev. Lett.\n \n96\n, 037004 (2006).\n \n[7]\n \nG.\n-\nX. Miao, K. Yoon, T. S. Sant\nos, and J. S. Moodera, Influence of Spin\n-\nPolarized Current \non Superconductivity and the Realization of Large Magnetoresistance.\n \nPhys. Rev. Lett.\n \n98\n, 267001 \n(2007).\n \n[8]\n \nH. Jeffrey Gardner, A. Kumar, L. Yu et al., Enhancement of superconductivity by a \nparallel \nmagnetic field in two\n-\ndimensional superconductors.\n \nNat. Phys.\n \n7\n, 895 (2011).\n \n[9]\n \nJ. Q. Xiao and C. L. Chien, Proximity Effects in Superconductor/Insulating\n-\nFerromagnet \nNbN/GdN Multilayers.\n \nPhys. Rev. Lett.\n \n76\n, 1727 (1996).\n \n[10]\n \nT. Kontos, M. April\ni, J. Lesueur, F. Genêt, B. Stephanidis, and R. Boursier, Josephson \nJunction through a Thin Ferromagnetic Layer: Negative Coupling.\n \nPhys. Rev. Lett.\n \n89\n, 137007 \n(2002).\n \n[11]\n \nR. S. Keizer, S. T. B. Goennenwein, T. M. Klapwijk, G. Miao, G. Xiao, and A. Gupta,\n \nA \nspin triplet supercurrent through the half\n-\nmetallic ferromagnet CrO\n2\n.\n \nNature\n \n439\n, 825 (2006).\n \n[12]\n \nJ. W. A. Robinson, J. D. S. Witt, and M. G. Blamire, Controlled Injection of Spin\n-\nTriplet \nSupercurrents into a Strong Ferromagnet.\n \nScience\n \n329\n, 59 (2010).\n \n[13]\n \nK. Senapati, M. G. Blamire, and Z. H. Barber, Spin\n-\nfilter Josephson junctions.\n \nNat. Mater.\n \n10\n, 849 (2011).\n \n[14]\n \nJ. D. Sau, R. M. Lutchyn, S. Tewari, and S. Das Sarma, Generic New Platform for \nTopological Quantum Computation Using Semiconductor Hetero\nstructures.\n \nPhys. Rev. Lett.\n \n104\n, \n040502 (2010).\n \n[15]\n \nS. A. Wolf, D. D. Awschalom, R. A. Buhrman et al., Spintronics: A Spin\n-\nBased Electronics \nVision for the Future.\n \nScience\n \n294\n, 1488 (2001).\n \n[16]\n \nI. Zutic, J. Fabian, and S. Das Sarma, Spintronics: Fundame\nntals and applications.\n \nRev. \nMod. Phys.\n \n76\n, 323 (2004).\n \n[17]\n \nG.\n-\nX. Miao, A. V. Ramos, and J. S. Moodera, Infinite Magnetoresistance from the Spin \nDependent Proximity Effect in Symmetry Driven bcc\n-\nFe/V/Fe Heteroepitaxial Superconducting \nSpin Valves.\n \nPhys. Rev. Lett.\n \n101\n, 137001 (2008).\n \n[18]\n \nH. Yang, S.\n-\nH. Yang, S. Takahashi, S. Maekawa, and S. S. P. Parkin, Extremely long \nquasiparticle spin lifetimes in superconducting aluminium using MgO tunnel spin injectors.\n \nNat. \nMater.\n \n9\n, 586 (2010).\n \n[19]\n \nT. Waka\nmura, H. Akaike, Y. Omori et al., Quasiparticle\n-\nmediated spin Hall effect in a \nsuperconductor.\n \nNat. Mater.\n \n14\n, 675 (2015).\n \n[20]\n \nM. Inoue, M. Ichioka, and H. Adachi, Spin pumping into superconductors: A new probe \nof spin dynamics in a superconducting thin f\nilm.\n \nPhys. Rev. B\n \n96\n, 024414 (2017).\n 9\n \n \n[21]\n \nY. Tserkovnyak, A. Brataas, G. E. W. Bauer, and B. I. Halperin, Nonlocal magnetization \ndynamics in ferromagnetic heterostructures.\n \nRev. Mod. Phys.\n \n77\n, 1375 (2005).\n \n[22]\n \nY. Ohnuma, H. Adachi, E. Saitoh, and S. Maeka\nwa, Enhanced dc spin pumping into a \nfluctuating ferromagnet near T\nC\n.\n \nPhys. Rev. B\n \n89\n, 174417 (2014).\n \n[23]\n \nK. Ando, S. Takahashi, J. Ieda et al., Inverse spin\n-\nHall effect induced by spin pumping in \nmetallic system.\n \nJ. Appl. Phys.\n \n109\n, 103913 (2011).\n \n[24]\n \nK.\n \nAndo, S. Takahashi, J. Ieda et al., Electrically tunable spin injector free from the \nimpedance mismatch problem.\n \nNat. Mater.\n \n10\n, 655 (2011).\n \n[25]\n \nY. Shiomi, K. Nomura, Y. Kajiwara et al., Spin\n-\nElectricity Conversion Induced by Spin \nInjection into Topologi\ncal Insulators.\n \nPhys. Rev. Lett.\n \n113\n, 196601 (2014).\n \n[26]\n \nJ. C. R. Sánchez, L. Vila, G. Desfonds et al., Spin\n-\nto\n-\ncharge conversion using Rashba \ncoupling at the interface between non\n-\nmagnetic materials.\n \nNat. Commun.\n \n4\n, 3944 (2013).\n \n[27]\n \nQ. Song, J. Mi, D. \nZhao et al., Spin injection and inverse Edelstein effect in the surface \nstates of topological Kondo insulator SmB\n6\n.\n \nNat. Commun.\n \n7\n, 13485 (2016).\n \n[28]\n \nE. Lesne, Y. Fu, S. Oyarzun et al., Highly efficient and tunable spin\n-\nto\n-\ncharge conversion \nthrough Rashba\n \ncoupling at oxide interfaces.\n \nNat. Mater.\n \n15\n, 1261 (2016).\n \n[29]\n \nQ. Song, H. Zhang, T. Su et al., Observation of inverse Edelstein effect in Rashba\n-\nsplit \n2DEG between SrTiO\n3\n \nand LaAlO\n3\n \nat room temperature.\n \nSci. Adv.\n \n3\n, e1602312 (2017).\n \n[30]\n \nR. Ohshima, Y. \nAndo, K. Matsuzaki et al., Realization of d\n-\nelectron spin transport at room \ntemperature at a LaAlO\n3\n/SrTiO\n3\n \ninterface.\n \nNat. Mater.\n \n16\n, 609 (2017).\n \n[31]\n \nN. Reyren, S. Thiel, A. D. Caviglia et al., Superconducting Interfaces Between Insulating \nOxides.\n \nScience\n \n317\n, 1196 (2007).\n \n[32]\n \nS. He, J. He, W. Zhang et al., Phase diagram and electronic indication of high\n-\ntemperature \nsuperconductivity at 65K in single\n-\nlayer FeSe films.\n \nNat. Mater.\n \n12\n, 605 (2013).\n \n[33]\n \nY. Saito, T. Nojima, and Y. Iwasa, Highly crystalline \n2D superconductors.\n \nNat. Rev. Mater.\n \n2\n, 16094 (2016).\n \n[34]\n \nSee Supplemental Material at [URL will be inserted by publisher] for the crystalline \nstructure characterization, the analysis of Gilbert damping, the measurement of effective \nmagnetization, the det\nermination of interface\n-\ninduced Gilbert damping, Temperature dependence \nof Gilbert damping of Nb/Py/Nb trilayer samples, and the impact of spin pumping exepriments on \nsuperconducting temperature of the NbN layers.\n \n[35]\n \nY. Zhao, Q. Song, S.\n-\nH. Yang et al., \nExperimental Investigation of Temperature\n-\nDependent Gilbert Damping in Permalloy Thin Films.\n \nScientific Reports\n \n6\n, 22890 (2016).\n \n[36]\n \nS. P. Chockalingam, M. Chand, J. Jesudasan, V. Tripathi, and P. Raychaudhuri, \nSuperconducting properties and Hall effect o\nf epitaxial NbN thin films.\n \nPhys. Rev. B\n \n77\n, 214503 \n(2008).\n \n[37]\n \nZ. Celinski, K. B. Urquhart, and B. Heinrich, Using ferromagnetic resonance to measure \nthe magnetic moments of ultrathin films.\n \nJ. Magn. Magn. Mater.\n \n166\n, 6 (1997).\n \n[38]\n \nC. L. Jermain, S. V. \nAradhya, N. D. Reynolds et al., Increased low\n-\ntemperature damping \nin yttrium iron garnet thin films.\n \nPhys. Rev. B\n \n95\n, 174411 (2017).\n \n[39]\n \nA. M. Goldman, Superconductor\n-\nInsulator Transitions.\n \nInt. J. Mod. Phys. B\n \n24\n, 4081 \n(2010).\n \n[40]\n \nC. H. L. Quay, D. Chev\nallier, C. Bena, and M. Aprili, Spin imbalance and spin\n-\ncharge \nseparation in a mesoscopic superconductor.\n \nNat. Phys.\n \n9\n, 84 (2013).\n 10\n \n \n[41]\n \nF. Hübler, M. J. Wolf, D. Beckmann, and H. v. Löhneysen, Long\n-\nRange Spin\n-\nPolarized \nQuasiparticle Transport in Mesoscopic\n \nAl Superconductors with a Zeeman Splitting.\n \nPhys. Rev. \nLett.\n \n109\n, 207001 (2012).\n \n[42]\n \nT. Wakamura, N. Hasegawa, K. Ohnishi, Y. Niimi, and Y. Otani, Spin Injection into a \nSuperconductor with Strong Spin\n-\nOrbit Coupling.\n \nPhys. Rev. Lett.\n \n112\n, 036602 (2014).\n \n[43]\n \nM. Tinkham, \nIntroduction to superconductivity \n(Dover Publications, 2004).\n \n[44]\n \nC. Bell, S. Milikisyants, M. Huber, and J. Aarts, Spin Dynamics in a Superconductor\n-\nFerromagnet Proximity System.\n \nPhys. Rev. Lett.\n \n100\n, 047002 (2008).\n \n[45]\n \nJ. P. Morten, A.\n \nBrataas, G. E. W. Bauer, W. Belzig, and Y. Tserkovnyak, Proximity\n-\neffect\n–\nassisted decay of spin currents in superconductors.\n \nEurophys. Lett.\n \n84\n, 57008 (2008).\n \n[46]\n \nM. A. Sillanpää, T. T. Heikkilä, R. K. Lindell, and P. J. Hakonen, Inverse proximity effect\n \nin superconductors near ferromagnetic material.\n \nEur. Phys. Lett.\n \n56\n, 590 (2001).\n \n[47]\n \nP. G. De Gennes, Coupling between ferromagnets through a superconducting layer.\n \nPhys. \nLett.\n \n23\n, 10 (1966).\n \n[48]\n \nB. Li, N. Roschewsky, B. A. Assaf et al., Superconducting\n \nSpin Switch with Infinite \nMagnetoresistance Induced by an Internal Exchange Field.\n \nPhys. Rev. Lett.\n \n110\n, 097001 (2013).\n \n[49]\n \nY. Masuda and A. G. Redfield, Nuclear Spin\n-\nLattice Relaxation in Superconducting \nAluminum.\n \nPhys. Rev.\n \n125\n, 159 (1962).\n \n[50]\n \nL. C. \nHebel and C. P. Slichter, Nuclear Spin Relaxation in Normal and Superconducting \nAluminum.\n \nPhys. Rev.\n \n113\n, 1504 (1959).\n \n[51]\n \nN. J. Curro, T. Caldwell, E. D. Bauer et al., Unconventional superconductivity in PuCoGa\n5\n.\n \nNature\n \n434\n, 622 (2005).\n \n[52]\n \nP. E. Seiden\n, Ferrimagnetic Resonance Relaxation in Rare\n-\nEarth Iron Garnets.\n \nPhys. Rev.\n \n133\n, A728 (1964).\n \n[53]\n \nM. Sparks, \nFerromagnetic Relaxation Theory \n(McGraw\n-\nHill, New York, 1964).\n \n[54]\n \nR. Arias and D. L. Mills, Extrinsic contributions to the ferromagnetic resonan\nce response \nof ultrathin films.\n \nPhys. Rev. B\n \n60\n, 7395 (1999).\n \n[55]\n \nK. Lenz, H. Wende, W. Kuch, K. Baberschke, K. Nagy, and A. Jánossy, Two\n-\nmagnon \nscattering and viscous Gilbert damping in ultrathin ferromagnets.\n \nPhys. Rev. B\n \n73\n, 144424 (2006).\n \n[56]\n \nO. d'Al\nlivy Kelly, A. Anane, R. Bernard et al., Inverse spin Hall effect in nanometer\n-\nthick \nyttrium iron garnet/Pt system.\n \nAppl. Phys. Lett.\n \n103\n, 082408 (2013).\n \n[57]\n \nX. Liu, W. Zhang, M. J. Carter, and G. Xiao, Ferromagnetic resonance and damping \nproperties of Co\nFeB thin films as free layers in MgO\n-\nbased magnetic tunnel junctions.\n \nJ. Appl. \nPhys.\n \n110\n, 033910 (2011).\n \n[58]\n \nJ. F. Sierra, V. V. Pryadun, S. E. Russek et al., Interface and Temperature Dependent \nMagnetic Properties in Permalloy Thin Films and Tunnel Junct\nion Structures.\n \nJ. Nanosci. \nNanotechnol.\n \n11\n, 7653 (2011).\n \n[59]\n \nC. Kittel, On the Theory of Ferromagnetic Resonance Absorption.\n \nPhys. Rev.\n \n73\n, 155 \n(1948).\n \n[60]\n \nG. Counil, J.\n-\nV. Kim, T. Devolder, C. Chappert, K. Shigeto, and Y. Otani, Spin wave \ncontributions\n \nto the high\n-\nfrequency magnetic response of thin films obtained with inductive \nmethods.\n \nJ. Appl. Phys.\n \n95\n, 5646 (2004).\n \n[61]\n \nM. L. Schneider, T. Gerrits, A. B. Kos, and T. J. Silva, Gyromagnetic damping and the role \nof spin\n-\nwave generation in pulsed induct\nive microwave magnetometry.\n \nAppl. Phys. Lett.\n \n87\n, \n072509 (2005).\n 11\n \n \n[62]\n \nP. Krivosik, N. Mo, S. Kalarickal, and C. E. Patton, Hamiltonian formalism for two \nmagnon scattering microwave relaxation: Theory and applications.\n \nJ. Appl. Phys.\n \n101\n, 083901 \n(2007).\n \n \n \n \n \n \n 12\n \n \nFigure 1\n \n \nFig. 1. \nSpin pumping into the \nsuperconducting\n \nNbN thin films. \n(a)\n \nSchematic of the\n \ninterfacial\n \ns\n-\nd\n \nexchange interaction (\nJ\nsd\n) between the spins in NbN layer and the rotating magnetization of GdN \n13\n \n \nlayer under the ferromagnetic resonance\n \n(FMR)\n \nconditions. \n(b) \nThe magnetic moment as a \nfunction of the temperature. Inset: The magnetic hysteresis loop at \nT\n \n= \n5 K. \n(c) \nThe four\n-\nprobe \nresistance as a function of the temperature\n \nwith \nin\n-\nplane \nmagnetic field at 0 and 4000 Oe\n. \n(d) \nThe \ntypical \nFMR \nsignal\n \nmeasured at \nT\n \n= \n10 K and \nf\n \n= \n15 GHz. The red line indicates the Lorentz fitting \ncurve to obtain the half linewidth based on equation (1). \n(e)\n \nThe half linewidth\n \nas a function of the \nmicrowave\n \nfrequency at \nT\n \n= \n10K. The red solid line is \nthe fitting c\nurve based on spin\n-\nrelaxation \nmodel. The results in Fig. 1\n(b\n-\nd)\n \nare obtained on the NbN (10)/GdN (5)/NbN (10) sample.\n \n \n \n 14\n \n \nFigure 2\n \n \n \n \n \nFig. 2. \nSpin dynamics \nof \nthe \nsuperconducting\n \nNbN thin films probed via \nspin pumping\n. The \nnormalized four\n-\nprobe resistance\n \n(\na\n)\n \nand Gilbert damping (\nb\n) \nas a function of the temperature for \nthe samples of NbN (2)/GdN (5)/NbN (2) and NbN (10)/GdN (5)/NbN (10), respectively. \n \n \n15\n \n \n \n \nFigure 3\n \n \nFig. 3. \nThe GdN thickness effect on spin dynamics \nof \nthe superconducting \nNbN thin films\n. \n(a\n-\nb)\n \nThe Gilbert damping as a function of the temperature for the samples of NbN (\nt\n)/GdN (10)/NbN \n(\nt), \nand NbN (\nt\n)/GdN (30)/NbN (\nt\n). Insets\n: The normalized four\n-\nprobe resistance as a function of \nthe temperature for the samples of NbN (10)/GdN (10)/NbN (10) and NbN (10)/GdN (30)/NbN \n(10). \n(c)\n \nThe \nratio of \nthe \npeak Gilbert damping \nover \nthe value\n \nat \nT\n \n= \n~ T\nC\n \n(\npeak C\n/ (~ T )\n \n) \nas a \nfun\nction of the GdN layer thickness.\n \n16\n \n \nFigure 4\n \n \n \nFig. 4. \nThe interface\n-\ninduced Gilbert damping\n \nat the \nNbN/GdN interface\n. \n(a)\n \nThe Gilbert damping \nas a function of the GdN thickness for the samples of NbN (10)/GdN (d)/NbN (10) \nat \nT\n \n= \n11\n,\n \n8.5\n, \nand 6 K\n, respect\nively\n. \n(b)\n \nThe interface\n-\ninduced Gilbert damping as a function of temperature\n \nfor \nNbN (10)/GdN (d)/NbN (10)\n. \n \n \n \n1\n \nSupplementary Materials for:\n \n \nProbe of Spin Dynamics in Superconducting NbN Thin Films via \n \nSpin Pumping\n \n \nYunyan\n \nYao\n1,2\n,\n \nQi\n \nSong\n1,2\n,\n \nYota\n \nTakamura\n3,4\n,\n \nJuan\n \nPedro\n \nCascales\n3\n,\n \nWei\n \nYuan\n1,2\n,\n \nYang\n \nMa\n1,2\n,\n \nYu\n \nYun\n1,2\n,\n \nX.\n \nC.\n \nXie\n1,2\n,\n \nJagadeesh\n \nS.\n \nMoodera\n3,5\n,\n \nand\n \nWei\n \nHan\n1,2*\n \n1\nInternational Center for Quantum Materials, School of Physics, Peking University, Beijing \n100871, China.\n \n2\nCollaborative Innovation Center of Quantum Matter, Beijing 100871, Chi\nna.\n \n3\nPlasma Science and Fusion Center and Francis Bitter Magnet Laboratory, Massachusetts \nInstitute of Technology, Cambridge, MA 02139, USA.\n \n4\nSchool of Engineering, Tokyo Institute of Technology, Tokyo 152\n-\n8550, Japan.\n \n5\nDepartment of Physics, Massachusetts\n \nInstitute of Technology, Cambridge, MA 02139, USA\n \n*Correspondence to: \nweihan@pku.edu.cn (W.H.)\n \n 2\n \nS1.\n \nDetermination\n \nof\n \nGilbert\n \ndamping\n \nA\n \nnonlinear\n \nbehavior\n \nof\n \nhalf\n \nlinewidth\n \n(\nH\n\n)\n \nvs.\n \nmicrowave\n \nfrequency\n \n(\nf\n)\n \nis\n \nobserved\n \non\n \nthe\n \nNbN\n \n(t)/GdN\n \n(d)/NbN\n \n(t)\n \nsamples,\n \nas\n \nshown\n \nin\n \nFig.\n \nS2(a\n-\nc).\n \nTo\n \nour\n \nbest\n \nknowledge,\n \nt\nhis\n \nnonlinear\n \nbehavior\n \ncould\n \nbe\n \nattributed\n \nto\n \ntwo\n \nmechanisms,\n \nnamely\n \nspin\n-\nrelaxation\n \n[\n38\n,\n52\n]\n,\n \nand\n \ntwo\n-\nmagnon\n \nscattering\n \n[\n53\n,\n54\n]\n.\n \nFor\n \nthe\n \nspin\n-\nrelaxation\n \nmechanism,\n \nH\n\n \nis\n \nrelated\n \nto\n \na\n \ntemperature\n-\ndependent\n \nspin\n-\nrelaxation\n \ntime\n \nconstant\n \n(\n\n),\n \nand\n \ncan\n \nbe\n \nexpressed\n \nby\n \n[\n38\n]\n:\n \n \n \n0\n2\n4 2\n1 (2 )\nf f\nH H A\nf\n  \n  \n    \n\n \n \n(S1)\n \nwhere\n \n0\nH\n\n \nis\n \nassociated\n \nwith\n \nthe\n \ninhomogeneous\n \nproperties,\n \n\n \nis\n \nthe\n \ngyromagnetic\n \nratio,\n \nand\n \nA\n \nis\n \nthe\n \nspin\n-\nrelaxation\n \ncoefficient.\n \nWhile,\n \nthe\n \nrelationship\n \nof\n \nH\n\n \nand\n \nf\n \nbased\n \non\n \ntwo\n-\nmagnon\n \nscattering\n \nmechanism\n \ncan\n \nbe\n \nexpressed\n \nby\n \n[\n54\n,\n55\n]\n:\n \n \n\n\n\n\n2\n2\n1\n2\n2\n(2 ) 2 2\nsin\n(2 ) 2 2\neff eff\neff eff\nf M M\nH\nf M M\n    \n    \n\n \n \n \n \n(S2)\n \nwhere\n \n\n \nis\n \nthe\n \ntwo\n-\nmagnon\n \nscattering\n \ncoefficient,\n \nand\n \neff\nM\nis\n \nthe\n \neffective\n \nmagnetization.\n \nClearly,\n \nas\n \nshown\n \nin\n \nFig.\n \nS2(a\n-\nc),\n \ntwo\n-\nmagnon\n \nscattering\n \nmechanism\n \nfails\n \nto\n \ngive\n \nreasonable\n \nfittings\n \n(blue\n \nlines)\n \nto\n \nour\n \nexperimental\n \nresults\n \n(black\n \nsquares).\n \nWhileas,\n \nour\n \nexperimental\n \nresults\n \nagree\n \nwell\n \nwith\n \nthe\n \nfitting\n \ncurves\n \nbased\n \non\n \nthe\n \nspin\n-\nrelaxation\n \nmechanism\n \n(red\n \nlines).\n \nBesides,\n \nit\n \nis\n \nobserved\n \nthat\n \nthe\n \nnonlinearity\n \nof\n \nH\n\n \nvs.\n \nf\n \nincreases\n \nas\n \nthe\n \nGdN\n \nlayer\n \nthickness\n \nincreases,\n \nwhich\n \nis\n \nalso\n \nopposite\n \nto\n \nthe\n \ntwo\n-\nmagnon\n \nscat\ntering\n \nmechanism\n.\n \nSince\n \ntwo\n-\nmagnon\n \nscattering\n \narises\n \nfrom\n \nthe\n \ndefects\n \nat\n \nthe\n \ninterface,\n \na\n \nmore\n \nconspicuously\n \nnonlinear\n \nbehavior\n \nis\n \nexpected\n \nas\n \nthe\n \nFM\n \nfilms\n \nbecome\n \nthinner\n \n[\n55\n-\n57\n]\n.\n \nAnother\n \nfeature\n \nthat\nH\n\n \ndecreases\n \nas\n \nf\n \nincreases\n \n(Fig.\n \nS2(c))\n \ncannot\n \nbe\n \nexplained\n \nby\n \nthe\n \ntwo\n-\nmagnon\n \nscattering\n \nmechanism\n \neither.\n \nHence,\n \nthe\n \nspin\n-\nrelaxation\n \nmechanism\n \nis\n \nused\n \nto\n \nanalyze\n \nthe\n \nexperimental\n \nresults\n \nof\n \nH\n\n \nvs.\n \nf\n,\n \nand\n \nGilbert\n \ndamping\n \ncould\n \nbe\n \nobtained\n \nsubsequently.\n \n \nS2.\n \nRole\n \nof\n \nthe\n \neffective\n \nmagnetization\n \n 3\n \nIn\n \nprevious\n \nstudies,\n \nit\n \nha\ns\n \nbeen\n \nsuggested\n \nthat\n \nthe\n \nenhancement\n \nof\n \nthe\n \nGilbert\n \ndamping\n \ncould\n \nbe\n \nalso\n \nassociated\n \nwith\n \nthe\n \nchange\n \nof\n \neff\nM\n \n[\n35\n,\n58\n]\n.\n \nTo\n \nrule\n \nout\n \nthe\n \neffect\n \nof\n \neff\nM\n,\n \nwe\n \nsystemically\n \nstudy\n \nthe\n \ntemperature\n \ndependence\n \nof\n \neff\nM\nbetween\n \nthe\n \nNbN\n \n(2)/GdN\n \n(d)/NbN\n \n(2)\n \nand\n \nNbN\n \n(10)/GdN\n \n(d)/NbN\n \n(10)\n \nsamples.\n \nThe\n \n4\neff\nM\n\n \nis\n \nbased\n \non\n \nthe\n \nKittel\n \nformula\n \n[\n59\n]\n:\n \n \n \n \n \n \n \n1\n2\n( )[ ( 4 )]\n2\nres res eff\nf H H M\n\n\n\n \n \n \n \n               \n(S3)\n \nwhere\n \nf\n \nis\n \nthe\n \nmicrowave\n \nfrequency,\n \nand\n \nres\nH\nis\n \nthe\n \nresonance\n \nmagnetic\n \nfield.\n \nAs\n \nshown\n \nin\n \nfig.\n \nS4(a),\n \nthe\n \nfitting\n \ncurve\n \n(red\n \nline)\n \nagrees\n \nwell\n \nwith\n \nthe\n \nexperimental\n \nresults.\n \nSimilar\n \ntemperature\n \ndependences\n \nof\n \n4\neff\nM\n\nfor\n \nthe\n \nNbN\n \n(2)/GdN\n \n(d)/NbN\n \n(2)\n \nand\n \nNbN\n \n(10)/GdN\n \n(d)/NbN\n \n(10)\n \nsamples\n \nare\n \nobserved\n \n(Figs.\n \nS4(c\n-\nd)).\n \nThis\n \nobservation\n \nconfirms\n \nthat\n \nthe\n \nobserved\n \ncoherence\n \npeak\n \nof\n \nthe\n \nGilbert\n \ndamping\n \nat\n \nT\n \n=\n \n~\n \n8.5\n \nK\n \nis\n \nnot\n \nrelated\n \nto\n \nthe\n \neffective\n \nmagnetization.\n \n \nS3.\n \nDetermination\n \nof\n \ninterface\n-\ninduced\n \nGilbert\n \ndamping\n \nPrevious\n \nstudies\n \nhave\n \nidentified\n \nthe\n \nmajor\n \nsources\n \nthat\n \ncontribute\n \nto\n \nthe\n \nGilbert\n \ndamping\n \nof\n \nthe\n \nFM\n \nfilms,\n \nincluding\n \nbulk\n \ndamping\n \n(\nB\n\n)\n \nand\n \nthe\n \ninterface\n-\ninduced\n \nGilbert\n \ndamping\n \n(\nS\n\n)\n \nthat\n \nis\n \ndue\n \nto\n \nspin\n \npumping\n \nand\n \nother\n \ninterface\n \neffects\n \n[\n20\n-\n22\n]\n.\n \nAs\n \nshown\n \nin\n \nFig.\n \n4(a)\n \nand\n \nFig.\n \nS6(a),\n \na\n \nslightly\n \nincrease\n \nof\n \nthe\n \nGilbert\n \ndamping\n \nis\n \nobserved\n \nas\n \nthe\n \nthickness\n \nof\n \nthe\n \nGdN\n \nlayer\n \nincreases\n \nfrom\n \n20\n \nto\n \n30\n \nnm.\n \nThis\n \nfeature\n \ncould\n \nbe\n \nassociated\n \nwith\n \nnon\n-\nhomogeneous\n \nfield\n \nexcitation\n \nwhen\n \nthe\n \nw\nidth\n \nof\n \ncoplanar\n \nwave\n \nguide\n \nis\n \nvery\n \nsmall\n \ncompared\n \nto\n \nthe\n \nsamples\n \n[\n60\n,\n61\n]\n,\n \nand\n \nthe\n \neddy\n \ncurrent\n \nloss\n \neffect\n \n[\n62\n]\n.\n \nBoth\n \nof\n \nthese\n \neffects\n \ncontribute\n \nto\n \nthe\n \nGilbert\n \ndamping\n \nthat\n \nis\n \nproportional\n \nto\n \nd\n2\n.\n \nHence,\n \nthe\n \ntotal\n \nGilbert\n \ndamping\n \ncan\n \nbe\n \nexpressed\n \nby:\n \n2\n1\n( ) '\nB S\nd\nd\n   \n  \n \n \n \n \n(S4)\n \nAnd\n \nthe\n \ninterface\n-\ninduced\n \nGilbert\n \nda\nmping\n \nand\n \nbulk\n \ndamping\n \ncould\n \nbe\n \nobtained\n \nsubsequently.\n \nBased\n \non\n \nthe\n \nfitted\n \nresults,\n \nB\n\n \nis\n \nsignificantly\n \nsmall\n \ncompared\n \nto\n \nS\n\n.\n \nThe\n \n~10%\n \nerror\n \nbar\n \nof\n \nS\n\n \nat\n \neach\n \ntemperature\n \nmakes\n \nit\n \nhard\n \nto\n \nobtain\n \nthe\n \naccurate\n \nvalues\n \nof\n \nB\n\n,\n \nwhich\n \nrequire\n \nfuture\n \nstudies.\n \nComparing\n \nthe\n \ninterface\n-\ninduced\n \nGilbert\n \ndamping\n \nfor\n \nNbN\n \n(10)/GdN\n \n(d)/NbN\n \n(10)\n \nand\n \nNbN\n 4\n \n(2)/GdN\n \n(d)/NbN\n \n(2)\n \nsamples\n \n(Fig.\n \n4(b)\n \nand\n \nFi\ng.\n \nS6(b)),\n \nthe\n \nprofound\n \ncoherence\n \npeak\n \nis\n \nonly\n \nobserved\n \non\n \nNbN\n \n(10)/GdN\n \n(d)/NbN\n \n(10)\n \nsamples\n \nat\n \n~\n \n8.5\n \nK.\n \nThis\n \nobservation\n \nunambiguously\n \ndemonstrates\n \nthat\n \nthe\n \norigin\n \nof\n \nthe\n \ncoherence\n \npeak\n \nin\n \nthe\n \nGilbert\n \ndamping\n \narises\n \nfrom\n \nthe\n \ninterfacial\n \ns\n-\nd\n \nexchange\n \ninter\naction\n \nbetween\n \nthe\n \nmagnetic\n \nmoments\n \nof\n \nGdN\n \nand\n \nthe\n \nspins\n \nof\n \nthe\n \nquasiparticles\n \nof\n \nNbN\n \nin\n \nits\n \nsuperconducting\n \nstate.\n \nS4.\n \nThe\n \nimpact\n \nof\n \nspin\n \npumping\n \nexperiments\n \non\n \nthe\n \nT\nC\n \nBeyond\n \nthe\n \nmagnetic\n \nfield\n \neffect\n \non\n \nthe\n \nsuperconducting\n \nfilms\n \n(Fig.\n \n1(c)),\n \nthe\n \nspin\n \npumping\n \nexperiments\n \nwith\n \nvarious\n \nmicrowave\n \nexcitations\n \ndo\n \nnot\n \naffect\n \nthe\n \nsuperconducting\n \nfilms\n \neither.\n \nAs\n \nshown\n \nin\n \nFigs.\n \nS8(a)\n \nand\n \nS8(b),\n \nthe\n \nT\nC\n \nexhibit\n \na\n \nsmall\n \nvariation\n \np\nrobed\n \non\n \nthe\n \ntypical\n \nsample\n \nNbN\n \n(10)/GdN\n \n(5)/NbN\n \n(10).\n \n \n \n \n 5\n \nFigure\n \nS\n1\n \n \n \n \n \nFigure\n \nS\n1.\n \nC\nrystalline\n \nstructure\n \nof\n \nthe\n \n(111)\n-\ntextured\n \nGdN\n \nand\n \nNbN\n \nfilms.\n \n(a\n-\nb)\n,\n \nXRD\n \nresults\n \nmeasured\n \non\n \nthe\n \nNbN\n \n(10)/GdN\n \n(50)/NbN\n \n(10)\n \nsample.\n \nTwo\n \nmain\n \nobservable\n \npeaks\n \ncorrespond\n \nto\n \nGdN\n \n(111)\n \nand\n \nNbN\n \n(111).\n \n \n \n \n6\n \nFigure\n \nS\n2\n \n \n \n \nFig\nure\n \nS\n2\n.\n \nComparison\n \nof\n \nthe\n \nspin\n-\nrelaxation\n \nand\n \ntwo\n-\nmagnon\n \nscattering\n \nmechanisms\n.\n \n(a\n-\nc)\n \nThe\n \nexperimental\n \nresults\n \nof\n \n\nH\n \nvs.\n \nf\n \nat\n \nT\n \n=\n \n10\n \nK\n \nand\n \nthe\n \nfitting\n \ncurves\n \n(red\n/green\n \nlines\n \nfor\n \nthe\n \nspin\n \nrelaxation/\ntwo\n-\nmagnon\n \nscattering)\n \non\n \nNbN\n \n(10)/GdN\n \n(d)/NbN\n \n(10)\n \nsamples\n \nwith\n \nd\n \n=\n \n5,\n \n10,\n \nand\n \n30\n \nnm\n \nrespectively.\n \n \n7\n \n \nFigure\n \nS\n3\n \n \n \n \nFigure\n \nS\n3.\n \nHalf\n \nlinewidth\n \nas\n \na\n \nfunction\n \nof\n \ntemperature\n \nfor\n \nNbN\n \n(10\n)/GdN\n \n(\n5)/NbN\n \n(10\n)\n \n(a)\n \nand\n \nNbN\n \n(10)/GdN\n \n(10)/NbN\n \n(10)\n \n(b)\n \nsamples.\n \n \n \n \n8\n \nFigure\n \nS\n4\n \n \n \n \nFigure\n \nS\n4.\n \nC\nharacterization\n \nof\n \neffective\n \nmagnetization.\n \n(a)\n \nThe\n \ndetermination\n \nof\n \neff\nM\n \nvia\n \nKittel\n \nf\normula\n \n(red\n \nline)\n \nfrom\n \nthe\n \nexperimental\n \nresults\n \nof\n \nf\n \nvs.\n \nres\nH\n.\n \n(b\n-\nd)\n,\n \n4\neff\nM\n\n \nas\n \na\n \nfunction\n \nof\n \ntemperature\n \nfor\n \nthe\n \nNbN\n \n(\n2\n)/GdN\n \n(d)/N\nbN\n \n(2\n)\n \n(\ngreen\n \ncircles)\n \nand\n \nNbN\n \n(\n10\n)/GdN\n \n(d)/N\nbN\n \n(10\n)\n \nsamples\n \n(red\n \ncircles)\n \nwith\n \nd\n \n=\n \n5\n \nnm\n \n(\nb\n)\n,\n \n10\n \nnm\n \n(\nc\n),\n \nand\n \n30\n \nnm\n \n(\nd\n),\n \nrespectively.\n \n \n \n9\n \n \nFigure\n \nS5\n \n \n \n \n \n \n \n \nFigure\n \nS\n5.\n \nT\nC\n \nof\n \nNbN\n \nas\n \na\n \nfunction\n \nof\n \nthe\n \nGdN\n \nthickness\n.\n \nThese\n \nresults\n \nare\n \nobtained\n \non\n \nthe\n \nNbN\n \n(10)/GdN\n \n(d)/NbN\n \n(10)\n \nsamples\n.\n \n \n \n \n10\n \nFigure\n \nS\n6\n \n \n \nFigure\n \nS6.\n \nI\nnterface\n-\ninduced\n \nGilbert\n \ndamping\n \nin\n \nNbN\n \n(2)/GdN\n \n(d)/NbN\n \n(2)\n.\n \n(a)\n \nThe\n \nGilbert\n \ndamping\n \nconstant\n \nas\n \na\n \nfunction\n \nof\n \nthe\n \nGdN\n \nthickness\n \nat\n \nT\n \n=\n \n11,\n \n8,\n \nand\n \n6\n \nK\n,\n \nrespectively\n.\n \nSolid\n \nlines\n \nare\n \nthe\n \nfitting\n \ncurves\n \nbased\n \non\n \nthe\n \nequation\n \n(S4)\n.\n \n(b)\n \nThe\n \ninterface\n-\ninduced\n \nGilbert\n \ndamping\n \nas\n \na\n \nfunction\n \nof\n \ntemperature.\n \n \n \n11\n \nFigure\n \nS7\n \n \n \n \n \n \nFigure\n \nS\n7.\n \nTemperature\n \ndependence\n \nof\n \nGilbert\n \ndamping\n \nof\n \nPy\n \nin\n \nthe\n \nNb\n \n(100)/Py\n \n(20)/Nb\n \n(100)\n \nsample.\n \n \n \n12\n \nFigure\n \nS8\n \n \n \n \n \n \nFigure\n \nS\n8\n.\n \nThe\n \nimpact\n \nof\n \nspin\n \npumping\n \nexperiments\n \non\n \nthe\n \nT\nC\n \non\n \nthe\n \ntypical\n \nNbN\n \n(10\n)/GdN\n \n(5\n)/NbN\n \n(10)\n \nsample\n.\n \n(a)\n \nThe\n \nfour\n-\nprobe\n \nresistance\n \nvs.\n \ntemperature\n \nunder\n \n4000\n \nOe\n \nwith\n \nvarious\n \nmicrowave\n \nexcitation\n \nfrequencies.\n \n(b)\n \nThe\n \nfour\n-\nprobe\n \nresistance\n \nvs.\n \ntemperature\n \naround\n \nthe\n \nFMR\n \nresonance\n \nconditions\n \nof\n \nGdN\n.\n \n \n \n \n"    },    {        "title": "1711.06759v2.Shot_noise_of_charge_and_spin_transport_in_a_junction_with_a_precessing_molecular_spin.pdf",        "content": "Shot noise of charge and spin transport in a junction with a precessing molecular spin\nMilena Filipović and Wolfgang Belzig\nFachbereich Physik, Universität Konstanz, D-78457 Konstanz, Germany\n(Dated: April 6, 2018)\nMagnetic molecules and nanomagnets can be used to influence the electronic transport in meso-\nscopic junction. In a magnetic field the precessional motion leads to resonances in the dc- and\nac-transport properties of a nanocontact, in which the electrons are coupled to the precession.\nQuantities such as the dc conductance or the ac response provide valuable information, such as\nthe level structure and the coupling parameters. Here, we address the current-noise properties\nof such contacts. This encompasses the charge current and spin-torque shot noise, which both\nshow a steplike behavior as functions of bias voltage and magnetic field. The charge-current noise\nshows pronounced dips around the steps, which we trace back to interference effects of electrons\nin quasienergy levels coupled by the molecular spin precession. We show that some components\nof the noise of the spin-torque currents are directly related to the Gilbert damping and hence are\nexperimentally accessible. Our results show that the noise characteristics allow us to investigate in\nmore detail the coherence of spin transport in contacts containing magnetic molecules.\nI. INTRODUCTION\nShot noise of charge current has become an active re-\nsearchtopicinrecentdecades, sinceitenablestheinvesti-\ngation of microscopic transport properties, which cannot\nbe obtained from the charge current or conductance.1\nIt has been demonstrated that spin-flip induced fluctua-\ntions in diffusive conductors connected to ferromagnetic\nleads enhance the noise power, approaching the Pois-\nsonian value.2,3Accordingly, the Fano factor defined as\nF=S(0)=ejIj, which describes the deviation of the shot\nnoise from the average charge current, equals 1 in this\ncase. On the other hand, it has been shown that shot\nnoise in a ferromagnet-quantum-dot-ferromagnet system\nwith antiparallel magnetization alignments can be sup-\npressed due to spin flip, with F < 1=2.4\nThe quantum-interference phenomenon, which is a\nmanifestation of the wave nature of electrons, has at-\ntracted a lot of attention. The quantum-interference ef-\nfects occur between coherent electron waves in nanoscale\njunctions.5Quantum interference in molecular junc-\ntions influences their electronic properties.6–10The Fano\neffect11due to the interference between a discrete state\nand the continuum has an important role in investigation\nof the interference effects in nanojunctions, which behave\nin an analogous way, and are manifested in the conduc-\ntance or noise spectra.5,12,13Particularly interesting ex-\namples involve spin-flip processes, such as in the presence\nof Rashba spin-orbit interaction,14,15a rotating magnetic\nfield,16or in the case of the magnetotransport.17–19\nIn the domain of spin transport it is interesting to\ninvestigate the noise properties, as the discrete na-\nture of electron spin leads to the correlations between\nspin-carrying particles. The spin current is usually\na nonconserved quantity that is difficult to measure,\nand its shot noise depends on spin-flip processes lead-\ning to spin-current correlations with opposite spins.20–22\nThe investigation of the spin-dependent scattering, spin\naccumulation,23and attractive or repulsive interactions\nin mesoscopic systems can be obtained using the shotnoise of spin current,24as well as measuring the spin re-\nlaxation time.20,24Even in the absence of charge cur-\nrent, a nonzero spin current and its noise can still\nemerge.22,25,26Several works have studied the shot noise\nof a spin current using, e.g., the nonequilibrium Green’s\nfunctions method and scattering matrix theory.22,27–29\nIt was demonstrated that the magnetization noise\noriginates from transferred spin current noise via\na fluctuating spin-transfer torque in ferromagnetic-\nnormal-ferromagnetic systems,30and magnetic tunnel\njunctions.31Experimentally, Spin Hall noise measur-\nments have been demonstrated,32and in a similar fash-\nion the spin-current shot noise due to magnon currents\ncan be related to the nonquantized spin of interact-\ning magnons in ferri-, ferro-, and antiferromagnets.33,34\nQuantum noise generated from the scatterings between\nthe magnetization of a nanomagnet and spin-polarized\nelectrons has been studied theoretically as well.35,36The\nshot noise of spin-transfer torque was studied recently\nusing a magnetic quantum dot connected to two non-\ncollinear magnetic contacts.29According to the defini-\ntion of the spin-transfer torque,37,38both autocorrela-\ntions and cross-correlations of the spin-current compo-\nnents contribute to the spin-torque noise.\nIn this article, we study theoretically the noise of\ncharge and spin currents and spin-transfer torque in a\njunction connected to two normal metallic leads. The\ntransport occurs via a single electron energy level inter-\nacting with a molecular magnet in a constant magnetic\nfield. The spin of the molecular magnet precesses around\nthe magnetic field with the Larmor frequency, which is\nkept undamped, e.g., due to external driving. The elec-\ntronic level may belong to a neighboring quantum dot or\nit may be an orbital of the molecular magnet itself. The\nelectroniclevelandthemolecularspinarecoupledviaex-\nchange interaction. We derive expressions for the noise\ncomponents using the Keldysh nonequilibrium Green’s\nfunctions formalism.39–41The noise of charge current is\ncontributed by both elastic processes driven by the bias\nvoltage, and inelastic tunneling processes driven by thearXiv:1711.06759v2  [cond-mat.mes-hall]  5 Apr 20182\nµLΓLΓRBS(t)µRJgµBevacRcos(Ωt+φR)∼eVµR0s(t)\nFIG. 1. (Color online) Tunneling through a single molecular\nlevel with energy \u000f0in the presence of a precessing molecular\nspin~S(t)in a constant magnetic field ~B, connected to two\nmetallic leads with chemical potentials \u0016\u0018,\u0018=L;R. The\nmolecular level is coupled to the spin of the molecule via ex-\nchange interaction with the coupling constant J. The applied\ndc-bias voltage eV=\u0016L\u0000\u0016R, and the tunnel rates are \u0000\u0018.\nmolecular spin precession. We observe diplike features\nin the shot noise due to inelastic tunneling processes and\ndestructive quantum interference between electron trans-\nport channels involved in the spin-flip processes. The\ndriving mechanism of the correlations of the spin-torque\ncomponents in the same spatial direction involves both\nprecession of the molecular spin and the bias voltage.\nHence, they are contributed by elastic and inelastic pro-\ncesses, with the change of energy equal to one or two Lar-\nmor frequencies. The nonzero correlations of the perpen-\ndicular spin-torque components are driven by the molec-\nular spin precession, with contributions of spin-flip tun-\nneling processes only. These components are related to\nthe previously obtained Gilbert damping coefficient,42,43\nwhich characterize the Gilbert damping term of the spin-\ntransfer torque,44–46at arbitrary temperature.\nThe article is organized as follows. The model and\ntheoretical framework based on the Keldysh nonequi-\nlibrium Green’s functions formalism39–41are given in\nSec. II. Here we derive expressions for the noise of spin\nand charge currents. In Sec. III we investigate and an-\nalyze the properties of the charge-current shot noise. In\nSec. IV we derive and analyze the noise of spin-transfer\ntorque. The conclusions are given in Sec. V.\nII. MODEL AND THEORETICAL\nFRAMEWORK\nThe junction under consideration consists of a nonin-\nteracting single-level quantum dot in the presence of a\nprecessing molecular spin in a magnetic field along the\nz-axis,~B=B~ ez, coupled to two noninteracting leads\n(Fig. 1). The junction is described by the Hamiltonian\n^H(t) =X\n\u00182fL;Rg^H\u0018+^HT+^HD(t) +^HS;(1)where\n^H\u0018=X\nk;\u001b\u000fk\u0018^cy\nk\u001b\u0018^ck\u001b\u0018 (2)\nis the Hamiltonian of contact \u0018=L;R. The spin-(up or\ndown) state of the electrons is denoted by the subscript\n\u001b=\";#= 1;2 =\u00061. The tunnel coupling between the\nquantum dot and the leads reads\n^HT=X\nk;\u001b;\u0018[Vk\u0018^cy\nk\u001b\u0018^d\u001b+V\u0003\nk\u0018^dy\n\u001b^ck\u001b\u0018];(3)\nwith spin-independent matrix element Vk\u0018. The creation\n(annihilation) operators of the electrons in the leads and\nthequantumdotaregivenby ^cy\nk\u001b\u0018(^ck\u001b\u0018)and^dy\n\u001b(^d\u001b). The\nHamiltonian of the electronic level equals\n^HD(t) =X\n\u001b\u000f0^dy\n\u001b^d\u001b+g\u0016B^~ s~B+J^~ s~S(t):(4)\nThe first term in Eq. (4) is the Hamiltonian of the non-\ninteracting single-level quantum dot with energy \u000f0. The\nsecond term describes the electronic spin in the dot,\n^~ s= (~=2)P\n\u001b\u001b0(~ \u001b)\u001b\u001b0^dy\n\u001b^d\u001b0, in the presence of a constant\nmagnetic field ~B, and the third term represents the ex-\nchange interaction between the electronic spin and the\nmolecular spin ~S(t). The vector of the Pauli matrices is\ngiven by ^~ \u001b= (^\u001bx;^\u001by;^\u001bz)T. The g-factor of the electron\nand the Bohr magneton are gand\u0016B, whereasJis the\nexchange coupling constant between the electronic and\nmolecular spins.\nThe last term of Eq. (1) can be written as\n^HS=g\u0016B~S~B; (5)\nand represents the energy of the molecular spin ~Sin\nthe magnetic field ~B. We assume that j~Sj\u001d~and ne-\nglecting quantum fluctuations treat ~Sas a classical vari-\nable. The magnetic field ~Bgenerates a torque on the\nspin~Sthat causes the spin to precess around the field\naxis with Larmor frequency !L=g\u0016BB=~. The dy-\nnamics of the molecular spin is kept constant, which\ncan be realized, e.g., by external rf fields47to cancel\nthe loss of magnetic energy due to the interaction with\nthe itinerant electrons. Thus, the precessing spin ~S(t)\npumps spin currents into the leads, but its dynamics\nremains unaffected by the spin currents, i.e., the spin-\ntransfer torque exerted on the molecular spin is compen-\nsated by the above mentioned external means. The un-\ndamped precessional motion of the molecular spin, sup-\nported by the external sources, is then given by ~S(t) =\nS?cos(!Lt)~ ex+S?sin(!Lt)~ ey+Sz~ ez, with\u0012the tilt an-\ngle between ~Band~S, andS?=Ssin(\u0012)the magnitude\nof the instantaneous projection of ~S(t)onto thexyplane.\nThe component of the molecular spin along the field axis\nequalsSz=Scos(\u0012).3\nThe charge- and spin-current operators of the lead \u0018\nare given by the Heisenberg equation39,40\n^I\u0018\u0017(t) =q\u0017d^N\u0018\u0017\ndt=q\u0017i\n~[^H;^N\u0018\u0017]; (6)\nwhere [;]denotes the commutator, while ^NL\u0017=P\nk;\u001b;\u001b0^cy\nk\u001bL(\u001b\u0017)\u001b\u001b0^ck\u001b0Lis the charge ( \u0017= 0andq0=\n\u0000e) and spin ( \u0017=x;y;zandq\u00176=0=~=2) occupation\nnumber operator of the contact \u0018. Here ^\u001b0=^1is the\nidentity matrix. Taking into account that only the tun-\nneling Hamiltonian ^HTgenerates a nonzero commutator\nin Eq. (6), the current operator ^I\u0018\u0017(t)can be expressed\nas\n^I\u0018\u0017(t) =\u0000q\u0017i\n~X\n\u001b;\u001b0(\u001b\u0017)\u001b\u001b0^I\u0018;\u001b\u001b0(t); (7)\nwhere the operator component ^I\u0018;\u001b\u001b0(t)equals\n^I\u0018;\u001b\u001b0(t) =X\nk[Vk\u0018^cy\nk\u001b\u0018(t)^d\u001b0(t)\u0000V\u0003\nk\u0018^dy\n\u001b(t)^ck\u001b0\u0018(t)]:(8)\nThe nonsymmetrized noise of charge and spin current\nis defined as the correlation between fluctuations of cur-\nrentsI\u0018\u0017andI\u0010\u0016,1,40\nS\u0017\u0016\n\u0018\u0010(t;t0) =h\u000e^I\u0018\u0017(t)\u000e^I\u0010\u0016(t0)i; (9)\nwith\u0017=\u0016= 0for the charge-current noise. The fluctu-\nation operator of the charge and spin current in lead \u0018is\ngiven by\n\u000e^I\u0018\u0017(t) =^I\u0018\u0017(t)\u0000h^I\u0018\u0017(t)i: (10)\nUsing Eqs. (7) and (10), the noise becomes\nS\u0017\u0016\n\u0018\u0010(t;t0) =\u0000q\u0017q\u0016\n~2X\n\u001b\u001b0X\n\u0015\u0011(\u001b\u0017)\u001b\u001b0(\u001b\u0016)\u0015\u0011S\u001b\u001b0;\u0015\u0011\n\u0018\u0010(t;t0);\n(11)\nwhereS\u001b\u001b0;\u0015\u0011\n\u0018\u0010(t;t0) =h\u000e^I\u0018;\u001b\u001b0(t)\u000e^I\u0010;\u0015\u0011(t0)i. The for-\nmal expression for S\u0017\u0016\n\u0018\u0010(t;t0)is given by Eq. (A10) in\nthe Appendix, where it is obtained using Eq. (11) and\nEqs. (A1)–(A9).\nUsing Fourier transformations of the central-region\nGreen’s functions given by Eqs. (A6)–(A8) and self-\nenergies in the wide-band limit, the correlations given\nby Eq. (A9) can be further simplified. Some correla-\ntion functions are not just functions of time difference\nt\u0000t0. Thus, as in Ref. 48, we used Wigner representa-\ntion assuming that in experiments fluctuations are mea-\nsured on timescales much larger than the driving pe-\nriodT= 2\u0019=!L, which is the period of one molecular\nspin precession. The Wigner coordinates are given by\nT0= (t+t0)=2and\u001c=t\u0000t0, while the correlation func-\ntions are defined as\nS\u001b\u001b0;\u0015\u0011\n\u0018\u0010(\u001c) =1\nTZT\n0dth\u000e^I\u0018;\u001b\u001b0(t+\u001c)\u000e^I\u0010;\u0015\u0011(t)i:(12)The Fourier transform of S\u001b\u001b0;\u0015\u0011\n\u0018\u0010(\u001c)is given by\nS\u001b\u001b0;\u0015\u0011\n\u0018\u0010(\n;\n0) = 2\u0019\u000e(\n\u0000\n0)S\u001b\u001b0;\u0015\u0011\n\u0018\u0010(\n);(13)\nwhere\nS\u001b\u001b0;\u0015\u0011\n\u0018\u0010(\n) =Z\nd\u001cei\n\u001cS\u001b\u001b0;\u0015\u0011\n\u0018\u0010(\u001c):(14)\nFor the correlations which depend only on t\u0000t0, the\nWigner representation is identical to the standard repre-\nsentation.\nThe symmetrized noise of charge and spin currents\nreads1,40\nS\u0017\u0016\n\u0018\u0010S(t;t0) =1\n2hf\u000e^I\u0018\u0017(t);\u000e^I\u0010\u0016(t0)gi;(15)\nwheref;gdenotes the anticommutator. According to\nEqs. (11), (12), (14), and (15), in the Wigner represen-\ntation the nonsymmetrized noise spectrum reads\nS\u0017\u0016\n\u0018\u0010(\n) =Z\nd\u001cei\n\u001cS\u0017\u0016\n\u0018\u0010(\u001c)\n=Z\nd\u001cei\n\u001c1\nTZT\n0dth\u000e^I\u0018\u0017(t+\u001c)\u000e^I\u0010\u0016(t)i\n=\u0000q\u0017q\u0016\n~2X\n\u001b\u001b0X\n\u0015\u0011(\u001b\u0017)\u001b\u001b0(\u001b\u0016)\u0015\u0011S\u001b\u001b0;\u0015\u0011\n\u0018\u0010(\n);(16)\nwhile the symmetrized noise spectrum equals\nS\u0017\u0016\n\u0018\u0010S(\n) =1\n2[S\u0017\u0016\n\u0018\u0010(\n) +S\u0016\u0017\n\u0010\u0018(\u0000\n)]\n=\u0000q\u0017q\u0016\n2~2X\n\u001b\u001b0X\n\u0015\u0011(\u001b\u0017)\u001b\u001b0(\u001b\u0016)\u0015\u0011S\u001b\u001b0;\u0015\u0011\n\u0018\u0010S(\n);\n(17)\nwhereS\u001b\u001b0;\u0015\u0011\n\u0018\u0010S(\n) = [S\u001b\u001b0;\u0015\u0011\n\u0018\u0010(\n) +S\u0015\u0011;\u001b\u001b0\n\u0010\u0018(\u0000\n)]=2. The\nexperimentallymosteasilyaccessiblequantityisthezero-\nfrequency noise power.\nIII. SHOT NOISE OF CHARGE CURRENT\nFor the charge-current noise, it is convenient to drop\nthe superscripts \u0017=\u0016= 0. The charge-current noise\nspectrum can be obtained as24\nS\u0018\u0010(\n) =\u0000e2\n~2[S11;11\n\u0018\u0010+S11;22\n\u0018\u0010+S22;11\n\u0018\u0010+S22;22\n\u0018\u0010](\n):(18)\nInthissection, weanalyzethezero-frequencynoisepower\nof the charge current S\u0018\u0010=S\u0018\u0010(0)at zero temperature.\nTakingintoaccountthatthermalnoisedisappearsatzero\ntemperature, the only contribution to the charge-current\nnoise comes from the shot noise. The tunnel couplings\nbetween the molecular orbital and the leads, \u0000\u0018(\u000f) =\n2\u0019P\nkjVk\u0018j2\u000e(\u000f\u0000\u000fk\u0018), are considered symmetric and in\nthe wide-band limit \u0000L= \u0000R= \u0000=2.4\n-4-2024-2-1012\neV@e0DIL@10-2e0DHaL\nmL,R=e1±eV2mL,R=e0±eV2mL,R=±eV2mL,R=eV,0\n4202401\neVΕ0102Ε0b\nΜL,RΕ1eV2ΜL,RΕ0eV2ΜL,ReV2ΜL,ReV, 0SLL\nFIG. 2. (Color online) (a) Charge current ILand (b) auto-correlation shot noise SLLas functions of bias-voltage eV. All\nplots are obtained at zero temperature, with ~B=B~ ez. The other parameters are \u0000L= \u0000R= \u0000=2,\u0000 = 0:05\u000f0,!L= 0:5\u000f0,\nJ= 0:01\u000f0,S= 100, and\u0012=\u0019=2. The molecular quasienergy levels are located at \u000f1= 0:25\u000f0,\u000f2= 0:75\u000f0,\u000f3= 1:25\u000f0, and\n\u000f4= 1:75\u000f0.\nThe average charge current from lead \u0018can be ex-\npressed as\nI\u0018=e\u0000\u0018\u0000\u0010\n~Zd\u000f\n2\u0019[f\u0018(\u000f)\u0000f\u0010(\u000f)]\n\u0002X\n\u001b\u001b0\n\u001b6=\u001b0jG0r\n\u001b\u001b(\u000f)j2[1 +\r2jG0r\n\u001b0\u001b0(\u000f+\u001b0!L)j2]\nj1\u0000\r2G0r\u001b\u001b(\u000f)G0r\n\u001b0\u001b0(\u000f+\u001b0!L)j2;(19)\nwhere\u00186=\u0010, whileG0r\n\u001b\u001b(\u000f)are matrix elements of\n^G0r(\u000f) = [\u000f\u0000\u000f0+iP\n\u0018\u0000\u0018=2\u0000^\u001bz(g\u0016BB+JSz)=2]\u00001.49,50\nIn the above expression, f\u0018(\u000f) = [e(\u000f\u0000\u0016\u0018)=kBT+ 1]\u00001is\nthe Fermi-Dirac distribution of the electrons in lead \u0018,\nwithkBthe Boltzmann constant and Tthe tempera-\nture. The conservation of the charge current implies that\nSLL(0) +SLR(0) = 0. Thus, it is sufficient to study only\none correlation function.\nTunning the parameters in the system such as the bias\nvoltageeV=\u0016L\u0000\u0016R(where\u0016Land\u0016Rare the chemical\npotentials of the leads), ~B, and the tilt angle \u0012, the shot\nnoise can be controlled and minimized. The shot noise\nin the small precession frequency limit !L\u001ckBTis in\nagreement with Ref. 22 for eV= 0.\nIn Fig. 2(a) we present the average charge current as\na staircase function of bias voltage, where the bias is\nvaried in four different ways. In the presence of the ex-\nternal magnetic field and the precessing molecular spin,\nthe initially degenerate electronic level with energy \u000f0\nresults in four nondegenerate transport channels, which\nhas an important influence on the noise. Each step corre-\nsponds to a new available transport channel. The trans-\nport channels are located at the Floquet quasienergies43\n\u000f1=\u000f0\u0000(!L=2)\u0000(JS=2),\u000f2=\u000f0+ (!L=2)\u0000(JS=2),\n\u000f3=\u000f0\u0000(!L=2)+(JS=2), and\u000f4=\u000f0+(!L=2)+(JS=2),\nwhich are calculated using the Floquet theorem.16,51–54\nThe correlated current fluctuations give nonzero noise\npower, which is presented in Fig. 2(b). The noise power\nshowsthemolecularquasienergyspectrum, andeachstepor diplike feature in the noise denotes the energy of a new\navailable transport channel. The noise has two steps and\ntwo diplike features that correspond to these resonances.\nCharge current and noise power are saturated for large\nbias voltages. If the Fermi levels of the leads lie below the\nresonances, the shot noise approaches zero for eV!0\n[red and dashed pink lines in Fig. 2(b)]. This is due to\nthe fact that a small number of electron states can par-\nticipate in transport inside this small bias window and\nboth current and noise are close to 0. If the bias voltage\nis varied with respect to the resonant energy \u000f1such that\n\u0016L;R=\u000f1\u0006eV=2[dot-dashed blue line in Fig. 2(b)], or\nwith respect to \u000f0such that\u0016L;R=\u000f0\u0006eV=2[green line\nin Fig. 2(b)], we observe a valley at zero bias eV= 0,\nwhich corresponds to \u0016L=\u0016R=\u000f1in the first case, and\nnonzero noise in the second case. For eV= 0, the charge\ncurrent is zero, but the precession-assisted inelastic pro-\ncesses involving the absorption of an energy quantum !L\ngive rise to the noise here.\nAt small bias voltage, the Fano factor F=SLL=ejILj\nis inversely proportional to eVand hence diverges as\neV!0, indicating that the noise is super-Poissonian,\nas depicted in Fig. 3. Due to absorption (emission)\nprocesses16and quantum interference effects, the Fano\nfactor is a deformed steplike function, where each step\ncorresponds to a resonance. As the bias voltage is in-\ncreased, the noise is enhanced since the number of the\ncorrelated electron pairs increases with the increase of\nthe Fermi level. For larger bias, due to the absorption\nand emission of an energy quantum !L, electrons can\njump to a level with higher energy or lower level during\nthe transport, and the Fano factor F < 1indicates the\nsub-Poissonian noise. Around the resonances \u0016L;R=\u000fi,\ni= 1;2;3;4, the probability of transmission is very high,\nresulting in the small Fano factor. Elastic tunneling con-\ntributes to the sub-Poissonian Fano factor around the\nresonances and competes with the spin-flip events caused\nby the molecular spin precession. However, if the reso-5\n4 2 0 2 40.40.60.81.01.21.4\neV Ε0ΜL,RΕ1eV2ΜL,RΕ0eV2ΜL,ReV2ΜL,ReV,0F\nFIG. 3. (Color online) Fano factor Fas a function of bias-\nvoltageeV. All plots are obtained at zero temperature, with\n~B=B~ ez. The other parameters are set to \u0000 = 0:05\u000f0,\u0000L=\n\u0000R= \u0000=2,!L= 0:5\u000f0,J= 0:01\u000f0,S= 100, and\u0012=\u0019=2.\nThe positions of the molecular quasienergy levels are \u000f1=\n0:25\u000f0,\u000f2= 0:75\u000f0,\u000f3= 1:25\u000f0, and\u000f4= 1:75\u000f0.\n0.0 0.5 1.0 1.5 2.0 2.5 3.01\nΩLΕ0102Ε0\nΘ 2Θ 4Θ 6Θ 15SLL\n0\nFIG. 4. (Color online) Shot noise of charge current SLLas a\nfunction of the Larmor frequency !Lfor different tilt angles\n\u0012, with~B=B~ ez, at zero temperature. The other parameters\nare\u0000 = 0:05\u000f0,\u0000L= \u0000R= \u0000=2,\u0016L= 0:75\u000f0,\u0016R= 0:25\u000f0,\nJ= 0:01\u000f0, andS= 100. For!L=\u0016L\u0000\u0016R, we observe a\ndip due to destructive quantum interference.\nnant quasienergy levels are much higher than the Fermi\nenergyoftheleads, theprobabilityoftransmissionisvery\nlow and the Fano factor is close to 1, as shown in Fig. 3\n(red line). This means that the stochastic processes are\nuncorrelated. If the two levels connected with the inelas-\ntic photon emission (absorption) tunnel processes, or all\nfour levels, lie between the Fermi levels of the leads, the\nFano factor approaches 1/2, which is in agreement with\nRef. 55. For eV=\u000f3[see Fig. 3 (red line)] a spin-down\nelectron can tunnel elastically, or inelastically in a spin-\nflip process, leading to the increase of the Fano factor.\nSpin-flip processes increase the electron traveling time,\nleading to sub-Poissonian noise. Similarly, the Pauli ex-\nclusionprincipleisknowntoleadtosub-Poissoniannoise,\nsince it prevents the double occupancy of a level.\n0.0 0.5 1.0 1.5 2.012\nΜΕ0102Ε0ΜΜ LΜR\n 3Ε0 0.25Ε0 0.05Ε0SLL\n0FIG. 5. (Color online) Shot noise of charge current SLLas\na function of the chemical potential of the leads \u0016=\u0016L=\n\u0016R, with~B=B~ ez, for three different couplings \u0000, where\n\u0000L= \u0000R= \u0000=2, at zero temperature. The other parameters\nare!L= 0:5\u000f0,J= 0:01\u000f0,S= 100, and\u0012=\u0019=2. The\nmolecular quasienergy levels are positioned at \u000f1= 0:25\u000f0,\n\u000f2= 0:75\u000f0,\u000f3= 1:25\u000f0, and\u000f4= 1:75\u000f0.\nThe precessing molecular spin induces quantum inter-\nference between the transport channels connected with\nspin-flip events and the change of energy by one energy\nquantum!L, i.e., between levels with energies \u000f1and\n\u000f2=\u000f1+!L, or\u000f3and\u000f4=\u000f3+!L. The destruc-\ntive quantum-interference effects manifest themselves in\nthe form of diplike features in Fig. 2(b). When one or\nboth pairs of the levels connected with spin-flip events\nenter the bias-voltage window, then an electron from the\nleft lead can tunnel through both levels via elastic or in-\nelastic spin-flip processes. Different tunneling pathways\nending in the final state with the same energy destruc-\ntively interfere, similarly as in the Fano effect.11Namely,\nthe state with lower energy \u000f1(or\u000f3) mimics the discrete\nstate in the Fano effect. An electron tunnels into the\nstate\u000f1(or\u000f3), undergoes a spin flip and absorbs an en-\nergy quantum !L. The other state with energy \u000f2(or\u000f4)\nis an analog of the continuum in the Fano effect, and the\nelectron tunnels elastically through this level. These two\ntunneling processes (one elastic and the other inelastic)\ninterfere, leading to a diplike feature in the noise power.\nIf we vary, for instance, the bias-voltage as eV=\u0016L,\nwhere\u0016R= 0[Fig. 2(b), red line], we observe diplike\nfeatures for eV=\u000f2andeV=\u000f4.\nThe destructive interference effect is also presented in\nFig. 4, where noise power SLLis depicted as a function\nof!L. Here, we observe a dip due to the quantum-\ninterference effect around !L= 0:5\u000f0, which corresponds\nto\u0016L=\u000f2and\u0016R=\u000f1. The other two steps in Fig. 4\noccur when the Fermi energy of the right or left lead\nis in resonance with one of the quasienergy levels. The\nmagnitude of the precessing component of the molecular\nspin, which induces spin-flip processes between molecu-\nlar quasienergy levels, equals JSsin(\u0012)=2. Therefore, the\ndip increases with the increase of the tilt angle \u0012, and it6\nis maximal and distinct for \u0012=\u0019=2.\nFinally, in Fig. 5 we plotted the noise power of charge\ncurrentSLLas a function of \u0016=\u0016L=\u0016Rat zero tem-\nperature. It shows a nonmonotonic dependence on the\ntunneling rates \u0000. For small \u0000(Fig. 5, red line) the noise\nis increased if \u0016is positioned between levels connected\nwith spin-flip events, and is contributed only by absorp-\ntion processes of an energy quantum !Las we vary the\nchemical potentials. For larger \u0000(Fig. 5, green line), the\ncharge-current noise is increased since levels broaden and\noverlap, and more electrons can tunnel. With further in-\ncrease of \u0000(Fig. 5, dotted blue line) the noise starts to\ndecrease, and it is finally suppressed for \u0000\u001d!Lsince\na current-carrying electron sees the molecular spin as\nnearly static in this case, leading to a reduction of the\ninelastic spin-flip processes.\nIV. SHOT NOISE OF SPIN CURRENT AND\nSPIN-TRANSFER TORQUE\nIn this section we present the spin-current noise spec-\ntrum components and relations between them. Later we\nintroduce the noise of spin-transfer torque and we inves-\ntigate the zero-frequency spin-torque shot noise at zero\ntemperature. The components of the nonsymmetrized\nspin-current noise spectrum read\nSxx\n\u0018\u0010(\n) =\u00001\n4[S12;21\n\u0018\u0010+S21;12\n\u0018\u0010](\n); (20)\nSxy\n\u0018\u0010(\n) =\u0000i\n4[S12;21\n\u0018\u0010\u0000S21;12\n\u0018\u0010](\n); (21)\nSzz\n\u0018\u0010(\n) =\u00001\n4[S11;11\n\u0018\u0010\u0000S11;22\n\u0018\u0010\u0000S22;11\n\u0018\u0010+S22;22\n\u0018\u0010](\n);\n(22)\nwhere Eq. (22) denotes the noise of the zcomponent of\nthe spin current.22,24Since the polarization of the spin\ncurrent precesses in the xyplane, the remaining com-\nponents of the spin-current noise spectrum satisfy the\nfollowing relations:\nSyy\n\u0018\u0010(\n) =Sxx\n\u0018\u0010(\n); (23)\nSyx\n\u0018\u0010(\n) =\u0000Sxy\n\u0018\u0010(\n); (24)\nSxz\n\u0018\u0010(\n) =Szx\n\u0018\u0010(\n) =Syz\n\u0018\u0010(\n) =Szy\n\u0018\u0010(\n) = 0:(25)\nTaking into account that the spin current is not a\nconserved quantity, it is important to notice that the\ncomplete information from the noise spectrum can be\nobtained by studying both the autocorrelation noise\nspectrumSjk\n\u0018\u0018(\n)and cross-correlation noise spectrum\nSjk\n\u0018\u0010(\n),\u00106=\u0018. Therefore, it is more convenient to in-\nvestigate the spin-torque noise spectrum, where both au-\ntocorrelation and cross-correlation noise components of\nspin currents are included. The spin-transfer torque op-\nerator can be defined as\n^Tj=\u0000(^ILj+^IRj); j =x;y;z ;(26)while its fluctuation reads\n\u000e^Tj(t) =\u0000[\u000e^ILj(t) +\u000e^IRj(t)]: (27)\nAccordingly, the nonsymmetrized and symmetrized spin-\ntorque noise can be obtained using the spin-current noise\ncomponents as\nSjk\nT(t;t0) =h\u000e^Tj(t)\u000e^Tk(t0)i\n=X\n\u0018\u0010Sjk\n\u0018\u0010(t;t0); j;k =x;y;z ;(28)\nSjk\nTS(t;t0) =1\n2[Sjk\nT(t;t0) +Skj\nT(t0;t)]; (29)\nwith the corresponding noise spectrums given by\nSjk\nT(\n) =X\n\u0018\u0010Sjk\n\u0018\u0010(\n); (30)\nSjk\nTS(\n) =X\n\u0018\u0010Sjk\n\u0018\u0010S(\n): (31)\nAccording to Eqs.(23),(24), and (30), Sxx\nT(\n) =Syy\nT(\n)\nandSyx\nT(\n) =\u0000Sxy\nT(\n).\nIn the remainder of the section we investigate the zero-\nfrequency spin-torque shot noise Sjk\nT=Sjk\nT(0)at zero\ntemperature, where Sxx\nT(0) =Sxx\nTS(0),Syy\nT(0) =Syy\nTS(0),\nSzz\nT(0) =Szz\nTS(0), whileSxy\nT(0)is a complex imaginary\nfunction, and Sxy\nTS(0) = 0 according to Eqs. (24) and\n(31). Since Sxx\nT(0) =Syy\nT(0), all results and discussions\nrelated toSxx\nT(0)also refer to Syy\nT(0).\nSpincurrents I\u0018xandI\u0018yareperiodicfunctionsoftime,\nwith periodT= 2\u0019=!L, whileI\u0018zis time-independent.\nIt has already been demonstrated that spin-flip pro-\ncesses contribute to the noise of spin current.22The pres-\nence of the precessing molecular spin affects the spin-\ncurrent noise. Since the number of particles with differ-\nent spins changes due to spin-flip processes, additional\nspin-current fluctuations are generated. Currents with\nthe same and with different spin orientations are corre-\nlated during transport. Due to the precessional motion\nof the molecular spin, inelastic spin currents with spin-\nflip events induce noise of spin currents and spin-torque\nnoise, which can be nonzero even for eV= 0. The noise\ncomponent Sxy\nTis induced by the molecular spin preces-\nsion and vanishes for a static molecular spin. The noises\nof spin currents and spin-transfer torque are driven by\nthe bias voltage and by the molecular spin precession.\nHence, in the case when both the molecular spin is static\n(absence of inelastic spin-flip processes) and eV= 0(no\ncontribution of elastic tunneling processes), they are all\nequal to zero. The noise of spin-transfer torque can be\nmodified by adjusting system parameters such as the bias\nvoltageeV, the magnetic field ~B, or the tilt angle \u0012.\nIn Fig. 6 we present the zero-frequency spin-torque\nnoise components Sxx\nT=Syy\nT,ImfSxy\nTg, andSzz\nTas func-\ntions of the bias voltage eV=\u0016L\u0000\u0016R, for\u0016R= 0\nand different tilt angles \u0012between~Band~Sat zero tem-\nperature. They give information on available transport7\n0.0 0.5 1.0 1.51\neV Ε0ST102Ε0\nST,Θ0ST,Θ2ST,Θ ΜLeV\nΜR0\nImST,Θ2ST,Θ2zz xx\nxyxx\nxxjk\n0\nFIG.6. (Coloronline)Spin-torqueshot-noisecomponents Sjk\nT\nas functions of the bias voltage eVfor\u0016R= 0,\u0016L=eV. All\nplots are obtained at zero temperature, with ~B=B~ ez, and\n\u0000L= \u0000R= \u0000=2, for \u0000 = 0:05\u000f0. The other parameters are\nset to!L= 0:5\u000f0,J= 0:01\u000f0, andS= 100. The molecular\nquasienergylevelslieat \u000f1= 0:25\u000f0,\u000f2= 0:75\u000f0,\u000f3= 1:25\u000f0,\nand\u000f4= 1:75\u000f0.\nchannelsandinelasticspin-flipprocesses. Themagnitude\nof the torque noise at resonance energies \u000fi,i= 1;2;3;4,\nis determined by \u0012. In cases \u0012= 0and\u0012=\u0019, there\nare only two transport channels of opposite spins deter-\nmined by the resulting Zeeman field B\u0006JS=g\u0016B. The\ncomponent Sxx\nTshows two steps with equal heights lo-\ncated at these resonances, where the only contribution to\nthe spin-torque noise comes from elastic tunneling events\n(dotted purple and red lines in Fig. 6). For \u0012=\u0019=2, the\nelastic tunneling contributes with four steps with equal\nheights located at resonances \u000fi, but due to the contri-\nbutions of the inelastic precession-assisted processes be-\ntween quasienergy levels \u000f1(\u000f3) and\u000f2(\u000f4), the heights of\nthe steps in Sxx\nTare not equal anymore (dot-dashed pink\nline in Fig. 6). Here, we observed that the contribution\nof the inelastic tunneling processes to Sxx\nT, involving ab-\nsorption of an energy quantum !Land a spin-flip, shows\nsteps at spin-down quasienergy levels \u000f1and\u000f3, while it\nis constant between and after the bias has passed these\nlevels. The component Szz\nTshows similar behavior (green\nline in Fig. 6). As in the case of the inelastic tunneling\ninvolving the absorption of one energy quantum !L, in\nSxx\nT=Syy\nTwe observed inelastic spin-flip processes in-\nvolving the absorption of two energy quanta 2!Lin the\nform of steps at spin-down levels \u000f1,\u000f3,\u000f2\u00002!L, and\n\u000f4\u00002!L, which have negligible contribution compared\nto the other terms. These processes are a result of cor-\nrelations of two oscillating spin-currents. For large bias\nvoltage, the spin-torque noise components Sxx\nTandSzz\nT\nsaturate.\nThe behavior of the component ImfSxy\nTgis completely\ndifferent in nature. It is contributed only by one energy\nquantum!Labsorption (emission) spin-flip processes.\nInterestingly, we obtained the following relation between\nthe Gilbert damping parameter \u000b,42,43andImfSxy\nTgat\n3210123101\nΩLΕ0102Ε0STΘ2ImSTSTxxzzxySTjkFIG.7. (Coloronline)Spin-torqueshot-noisecomponents Sjk\nT\nas functions of the Larmor frequency !Lfor\u0012=\u0019=2,\u0016R= 0,\nand\u0016L= 1:5\u000f0. All plots are obtained for ~B=B~ ezat zero\ntemperature. The other parameters are \u0000L= \u0000R= \u0000=2,\n\u0000 = 0:05\u000f0,J= 0:01\u000f0, andS= 100.\narbitrary temperature\nImfSxy\nTg=!LSsin2(\u0012)\n2\u000b: (32)\nHence, the component ImfSxy\nTgis increased for Fermi\nlevelsoftheleadspositionedintheregionswhereinelastic\ntunneling processes occur (blue line in Fig. 6).\nThe spin-torque noise is influenced by the magnetic\nfield~Bsince it determines the spin-up and spin-down\nmolecular quasienergy levels. The dependence of Sxx\nT,\nImfSxy\nTg, andSzz\nTon the Larmor frequency !Lis de-\npicted in Fig. 7. The steps, dips, or peaks in the\nplots are located at resonant tunneling frequencies !L=\n\u0006j2\u0016L;R\u00002\u000f0\u0006JSj. For!L= 0there are only two\ntransport channels, one at energy \u000f0+JS=2, which is\nequal to the Fermi energy of the left lead, and the other\nat\u000f0\u0000JS=2located between \u0016Land\u0016R. The contribu-\ntions of the elastic spin transport processes through these\nlevelsresultindipsinthecomponents Sxx\nTandSzz\nT, while\nImfSxy\nTg= 0. For!=\u000f0corresponding to \u0016R=\u000f1and\n\u0016R=\u000f4\u00002!L, both the elastic and spin-flip tunneling\nevents involving the absorption of energy of one quantum\n!Lcontribute with a dip, while the spin-flip processes\ninvolving the absorption of an energy equal to 2!Lcon-\ntribute with a peak to the component Sxx\nT. For!L= 2\u000f0\nand!L= 3\u000f0corresponding to \u0016L=\u000f2and\u0016R=\u000f3,\nboth elastic and spin-flip processes with the absorption\nof an energy equal to !Lcontribute with a step, while the\ninelastic processes involving the absorption of an energy\n2!Lgive negligible contribution to Sxx\nT. The component\nSzz\nTshows dips at these two points, since here the domi-\nnantcontributioncomesfrominelastictunnelingspin-flip\nevents. The component Szz\nTis an even function of !L,\nwhile ImfSxy\nTgis an odd function of !L. The spin-torque\nnoiseSxx\nTis an even function of !Lfor\u0012=\u0019=2.\nThe spin-torque noise components as functions of \u0012for\n\u0016L=\u000f3and\u0016R= 0at zero temperature are shown in8\n0 1 2 3 4 5 62468\nΘ103Ε0b\nST\nΜLΕ3\nΜR0\nImSTSTxx\nxyzzSTjk\n0\nFIG. 8. (Color online) Spin-torque shot-noise components as\nfunctions of the tilt angle \u0012for\u0016L=\u000f3,\u0016R= 0. All plots\nare obtained at zero temperature, with ~B=B~ ez,\u0000 = 0:05\u000f0,\nand\u0000L= \u0000R= \u0000=2. The other parameters are !L= 0:5\u000f0,\nJ= 0:01\u000f0, andS= 100.\nFig. 8. The magnitudes and the appearance of the spin-\ntorque noise components at resonance energies \u000fican be\ncontrolled by \u0012, since it influences the polarization of the\nspin current. Here we see that both Szz\nTandImfSxy\nTg\nare zero for \u0012= 0and\u0012=\u0019, as the molecular spin is\nstatic and its magnitude is constant along zdirection in\nboth cases. These torque-noise components take their\nmaximum values for \u0012=\u0019=2, where both elastic and\ninelastic tunneling contributions are maximal. The com-\nponentSxx\nTtakes its minimum value for \u0012= 0and its\nmaximum value for \u0012=\u0019, with only elastic tunneling\ncontributions in both cases. For \u0012=\u0019=2, the inelastic\ntunneling events make a maximal contribution while en-\nergy conserving processes give minimal contribution to\nSxx\nT.\nV. CONCLUSIONS\nIn this article, we studied theoretically the noise of\ncharge and spin transport through a small junction, con-sisting of a single molecular orbital in the presence of a\nmolecular spin precessing with Larmor frequency !Lin\na constant magnetic field. The orbital is connected to\ntwo Fermi leads. We used the Keldysh nonequilibrium\nGreen’s functions method to derive the noise components\nof charge and spin currents and spin-transfer torque.\nThen, we analyzed the shot noise of charge current\nand observed characteristics that differ from the ones in\nthe current. In the noise power, we observed diplike fea-\ntures which we attribute to inelastic processes, due to\nthe molecular spin precession, leading to the quantum-\ninterference effect between correlated transport channels.\nSince the inelastic tunneling processes lead to a spin-\ntransfer torque acting on the molecular spin, we have\nalso investigated the spin-torque noise components con-\ntributed by these processes, involving the change of en-\nergy by an energy quantum !L. The spin-torque noise\ncomponents are driven by both the bias voltage and the\nmolecular spin precession. The in-plane noise compo-\nnentsSxx\nTandSyy\nTare also contributed by the processes\ninvolving the absorption of an energy equal to 2!L. We\nobtained the relation between ImfSxy\nTgand the Gilbert\ndamping coefficient \u000bat arbitrary temperature.\nTaking into account that the noise of charge and spin\ntransport can be controlled by the parameters such as\nbias voltage and external magnetic field, our results\nmight be useful in molecular electronics and spintron-\nics. The experimental observation of the predicted noise\npropertiesmightbeachallengingtaskduetocomplicated\ntunnelling processes through molecular magnets. Find-\ning a way to control the spin states of single-molecule\nmagnets in tunnel junctions could be one of the future\ntasks.\nACKNOWLEDGMENTS\nWe would like to thank Fei Xu for useful discussions.\nWe gratefully acknowledge the financial support from the\nDeutsche Forschungsgemeinschaft through the SFB 767\nControlled Nanosystems , the Center of Applied Photon-\nics, the DAAD through a STIBET scholarship, and an\nERCAdvancedGrant UltraPhase ofAlfredLeitenstorfer.9\nAPPENDIX: FORMAL EXPRESSION FOR THE NONSYMMETRIZED NOISE\nHere, we present the derivation of the formal expression for the nonsymmetrized noise S\u0017\u0016\n\u0018\u0010(t;t0). The correlation\nfunctionsS\u001b\u001b0;\u0015\u0011\n\u0018\u0010(t;t0), introduced in Eq. (11), can be expressed by means of the Wick’s theorem56as\nS\u001b\u001b0;\u0015\u0011\n\u0018\u0010(t;t0) =X\nkk0[Vk\u0018Vk0\u0010G>\n\u001b0;k0\u0015\u0010(t;t0)G<\n\u0011;k\u001b\u0018(t0;t)\n\u0000Vk\u0018V\u0003\nk0\u0010G>\n\u001b0\u0015(t;t0)G<\nk0\u0011\u0010;k\u001b\u0018 (t0;t)\n\u0000V\u0003\nk\u0018Vk0\u0010G>\nk\u001b0\u0018;k0\u0015\u0010(t;t0)G<\n\u0011\u001b(t0;t)\n+V\u0003\nk\u0018V\u0003\nk0\u0010G>\nk\u001b0\u0018;\u0015(t;t0)G<\nk0\u0011\u0010;\u001b(t0;t)]; (A1)\nwith the mixed Green’s functions defined, using units in which ~=e= 1, as\nG<\n\u0011;k\u001b\u0018(t;t0) =ih^cy\nk\u001b\u0018(t0)^d\u0011(t)i; (A2)\nG>\n\u001b0;k0\u0015\u0010(t;t0) =\u0000ih^d\u001b0(t)^cy\nk0\u0015\u0010(t0)i; (A3)\nwhile Green’s functions G<\nk\u001b\u0018;\u0011(t;t0) =\u0000[G<\n\u0011;k\u001b\u0018(t0;t)]\u0003andG>\nk0\u0015\u0010;\u001b0(t;t0) =\u0000[G>\n\u001b0;k0\u0015\u0010(t0;t)]\u0003. The Green’s functions\nof the leads and the central region are defined as\nG<\nk\u001b\u0018;k0\u001b0\u0010(t;t0) =ih^cy\nk0\u001b0\u0010(t0)^ck\u001b\u0018(t)i; (A4)\nG>\nk\u001b\u0018;k0\u001b0\u0010(t;t0) =\u0000ih^ck\u001b\u0018(t)^cy\nk0\u001b0\u0010(t0)i; (A5)\nG<\n\u001b\u001b0(t;t0) =ih^dy\n\u001b0(t0)^d\u001b(t)i; (A6)\nG>\n\u001b\u001b0(t;t0) =\u0000ih^d\u001b(t)^dy\n\u001b0(t0)i; (A7)\nGr;a\n\u001b\u001b0(t;t0) =\u0007i\u0012(\u0006t\u0007t0)hf^d\u001b(t);^dy\n\u001b0(t0)gi: (A8)\nSince the self-energies originating from the coupling between the electronic level and the lead \u0018are diagonal in the\nelectron spin space, their entries can be written as \u0006<;>;r;a\n\u0018(t;t0) =P\nkVk\u0018g<;>;r;a\nk\u0018(t;t0)V\u0003\nk\u0018, whereg<;>;r;a(t;t0)are\nthe Green’s functions of the free electrons in lead \u0018. Applying Langreth analytical continuation rules,57Eq. (A1)\ntransforms into\nS\u001b\u001b0;\u0015\u0011\n\u0018\u0010(t;t0) =Z\ndt1Z\ndt2\n\u0002\b\n[Gr\n\u001b0\u0015(t;t1)\u0006>\n\u0010(t1;t0) +G>\n\u001b0\u0015(t;t1)\u0006a\n\u0010(t1;t0)][Gr\n\u0011\u001b(t0;t2)\u0006<\n\u0018(t2;t) +G<\n\u0011\u001b(t0;t2)\u0006a\n\u0018(t2;t)]\n+[\u0006>\n\u0018(t;t1)Ga\n\u001b0\u0015(t1;t0) + \u0006r\n\u0018(t;t1)G>\n\u001b0\u0015(t1;t0)][\u0006<\n\u0010(t0;t2)Ga\n\u0011\u001b(t2;t) + \u0006r\n\u0010(t0;t2)G<\n\u0011\u001b(t2;t)]\n\u0000G>\n\u001b0\u0015(t;t0)[\u0006r\n\u0010(t0;t1)Gr\n\u0011\u001b(t1;t2)\u0006<\n\u0018(t2;t) + \u0006<\n\u0010(t0;t1)Ga\n\u0011\u001b(t1;t2)\u0006a\n\u0018(t2;t)\n+\u0006r\n\u0010(t0;t1)G<\n\u0011\u001b(t1;t2)\u0006a\n\u0018(t2;t)]\u0000[\u0006r\n\u0018(t;t1)Gr\n\u001b0\u0015(t1;t2)\u0006>\n\u0010(t2;t0)\n+\u0006>\n\u0018(t;t1)Ga\n\u001b0\u0015(t1;t2)\u0006a\n\u0010(t2;t0) + \u0006r\n\u0018(t;t1)G>\n\u001b0\u0015(t1;t2)\u0006a\n\u0010(t2;t0)]G<\n\u0011\u001b(t0;t)\t\n\u0000\u000e\u0018\u0010[\u000e\u0011\u001bG>\n\u001b0\u0015(t;t0)\u0006<\n\u0018(t0;t) +\u000e\u001b0\u0015\u0006>\n\u0018(t;t0)G<\n\u0011\u001b(t0;t)]: (A9)\nFinally, using Eqs. (11) and (A9), the obtained formal expression for the nonsymmetrized noise of charge current40,58\nand spin currents in standard coordinates tandt0can be written as\nS\u0017\u0016\n\u0018\u0010(t;t0) =\u0000q\u0017q\u0016\n~2TrnZ\ndt1Z\ndt2\n\u0002\b\n^\u001b\u0017[^Gr(t;t1)^\u0006>\n\u0010(t1;t0) +^G>(t;t1)^\u0006a\n\u0010(t1;t0)]^\u001b\u0016[^Gr(t0;t2)^\u0006<\n\u0018(t2;t) +^G<(t0;t2)^\u0006a\n\u0018(t2;t)]\n+ ^\u001b\u0017[^\u0006>\n\u0018(t;t1)^Ga(t1;t0) +^\u0006r\n\u0018(t;t1)^G>(t1;t0)]^\u001b\u0016[^\u0006<\n\u0010(t0;t2)^Ga(t2;t) +^\u0006r\n\u0010(t0;t2)^G<(t2;t)]\n\u0000^\u001b\u0017^G>(t;t0)^\u001b\u0016[^\u0006r\n\u0010(t0;t1)^Gr(t1;t2)^\u0006<\n\u0018(t2;t) +^\u0006<\n\u0010(t0;t1)^Ga(t1;t2)^\u0006a\n\u0018(t2;t) +^\u0006r\n\u0010(t0;t1)^G<(t1;t2)^\u0006a\n\u0018(t2;t)]\n\u0000^\u001b\u0017[^\u0006r\n\u0018(t;t1)^Gr(t1;t2)^\u0006>\n\u0010(t2;t0) +^\u0006>\n\u0018(t;t1)^Ga(t1;t2)^\u0006a\n\u0010(t2;t0) +^\u0006r\n\u0018(t;t1)^G>(t1;t2)^\u0006a\n\u0010(t2;t0)]^\u001b\u0016^G<(t0;t)\t\n\u0000\u000e\u0018\u0010^\u001b\u0017[^G>(t;t0)^\u001b\u0016^\u0006<\n\u0018(t0;t) +^\u0006>\n\u0018(t;t0)^\u001b\u0016^G<(t0;t)]o\n; (A10)10\nwhere Trdenotes the trace in the electronic spin space.\n1Y. M. Blanter and M. Büttiker, Physics Reports, 336, 2\n(2000).\n2E. G. Mishchenko, Phys. Rev. B 68, 100409(R) (2003).\n3W. Belzig and M. Zareyan, Phys. Rev. B 69, 140407\n(2004).\n4F. M. Souza, A. P. Jauho, and J. C. Egues, Phys. Rev. B\n78, 155303 (2008).\n5A. E. Miroschnichenko, S. Flach, and Y. S. Kivshar, Rev.\nMod. Phys. 82, 2257 (2010).\n6C. M. Guédon, H. Valkenier, T. Markussen, K. S. Thyge-\nsen, J. C. Hummelen, and S. J. van der Molen, Nature\nNanotech. 7, 305 (2012).\n7H. Vázquez, R. Skouta, S. Schneebeli, M. Kamenetska, R.\nBeslov, L. Venkataraman, and M. S. Hybertsen, Nature\nNanotech. 7, 663 (2012).\n8R. Stadler, Phys. Rev. B 80, 125401 (2009).\n9M. A. Ratner, J. Phys. Chem. 94, 4877 (1990).\n10T. Hansen, G. C. Solomon, D. Q. Andrews, and M. A.\nRatner, J. Chem. Phys. 131, 194704 (2009).\n11U. Fano, Phys. Rev. 124, 1866 (1961).\n12J. Faist, F. Capasso, C. Sirtori, K. W. West, and L. N.\nPfeiffer, Nature 390, 589 (1997).\n13O. Entin-Wohlman, Y. Imry, S. A. Gurvitz, and A.\nAharony, Phys. Rev. B 75, 193308 (2007).\n14D. Sánchez and L. Serra, Phys. Rev. B 74, 153313 (2006).\n15Piotr Stefański, J. Phys.: Cond. Matter 22, 505303 (2010).\n16B. H. Wu and C. Timm, Phys. Rev. B 81, 075309 (2010).\n17S. A. Gurvitz, D. Mozyrsky, and G. P. Berman, Phys. Rev.\nB72, 205341 (2005).\n18S. A. Gurvitz, IEEE Trans. Nanotechnology 4, 45 (2005).\n19D. Mozyrsky, L. Fedichkin, S. A. Gurvitz, and G. P.\nBerman, Phys. Rev. B 66, 161313 (2002).\n20A. Lamacraft, Phys. Rev. B 69, 081301(R) (2004).\n21M.ZareyanandW.Belzig, Europhys.Lett. 70, 817(2005).\n22B. Wang, J. Wang, and H. Guo, Phys. Rev. B 69, 153301\n(2004).\n23S. H. Ouyang, C. H. Lam, and J. Q. You, Euro. Phys. J.\nB64, 67 (2008).\n24O. Sauret and D. Feinberg, Phys. Rev. Lett. 92, 106601\n(2004).\n25M. J. Stevens, A. L. Smirl, R. D. R. Bhat, A. Najmale, J.\nE. Sipe, and H. M. van Driel, Phys. Rev. Lett. 90, 136603\n(2003).\n26A. Brataas, Y. Tserkovnyak, G. E. W. Bauer, and B. I.\nHalperin, Phys. Rev. B 66, 060404(R) (2002).\n27R. L. Dragomirova and B. K. Nikolić, Phys. Rev. B 75,\n085328 (2007).\n28Y. He, D. Hou, and R. Han, J. Appl. Phys. 101, 023710\n(2007).\n29Y. Yu, H. Zhan, L. Wan, B. Wang, Y. Wei, Q. Sun and J.\nWang, Nanotechnology 24, 155202 (2013).\n30J. Foros, A. Brataas, Y. Tserkovnyak, and G. E. W. Bauer,Phys. Rev. Lett. 95, 016601 (2005).\n31A. L. Chudnovskiy, J. Swiebodzinski, and A. Kamenev,\nPhys. Rev. Lett. 101, 066601 (2008).\n32A. Kamra, F. P. Witek, S. Meyer, H. Huebl, S. Geprägs, R.\nGross, G. E. W. Bauer, and S. T. B. Goennenwein, Phys.\nRev. B 90, 214419 (2014).\n33A. Kamra and W. Belzig, Phys. Rev. Lett. 116, 146601\n(2016); Phys. Rev. B 94, 014419 (2016).\n34A. Kamra and W. Belzig, Phys. Rev. Lett. 119, 197201\n(2017).\n35Y. Wang and L. J. Sham, Phys. Rev. B 85, 092403 (2012).\n36Y. Wang and L. J. Sham, Phys. Rev. B 87, 174433 (2013).\n37J.C. Slonczewski, J. Magn. Magn. Mater. 159, L1 (1996).\n38L. Berger, Phys. Rev. B 54, 9353 (1996).\n39A.-P.Jauho, N. S. Wingreen, and Y. Meir, Phys. Rev. B\n50, 5528 (1994).\n40A.-P. Jauho and H. Haug, Quantum Kinetics in Transport\nand Optics of Semiconductors (Springer, Berlin, 2008).\n41N. S. Wingreen, A.-P.Jauho, and Y. Meir, Phys. Rev. B\n48, 8487 (1993).\n42M.Filipović,C.Holmqvist,F.Haupt,andW.Belzig,Phys.\nRev. B 87, 045426 (2013); 88, 119901(E) (2013).\n43M. Filipović and W. Belzig, Phys. Rev. B 93, 075402\n(2016).\n44T. L. Gilbert, Phys. Rev. 100, 1243 (1955); T. Gilbert,\nIEEE Trans. Magn. 40, 3443 (2004).\n45Y. Tserkovnyak, A. Brataas, G. E. W. Bauer, and B. I.\nHalperin, Rev. Mod. Phys. 77, 1375 (2005).\n46D. C. Ralph and M. D. Stiles, J. Magn and Magn. Mater.\n320, 1190 (2008).\n47C. Kittel, Phys. Rev. 73, 155 (1948).\n48M. H. Pedersen and M. Büttiker, Phys. Rev. B 58, 12993\n(1998).\n49B. Wang, J. Wang, and H. Guo, Phys. Rev. B 67, 092408\n(2003).\n50N. Bode, L. Arrachea, G. S. Lozano, T. S. Nunner, and F.\nvon Oppen, Phys. Rev. B 85, 115440 (2012).\n51G. Floquet, Ann. Sci. Ecole Normale Supérieure 12, 47\n(1883).\n52Jon H. Shirley, PhD Thesis, California Institute of Tech-\nnology, (1963).\n53M. Grifoni and P. Hänggi, Phys. Rep. 304, 229 (1998).\n54S-I. Chu and D. A. Telnov, Phys. Rep. 390, 1 (2004).\n55A. Thielmann, M. H. Hettler, J. König, and G. Schön,\nPhys. Rev. B 68, 115105 (2003).\n56A. Fetter and J. D. Walecka, Quantum Theory of Many-\nParticle Systems (Dover Publications, Inc., Mineola, N.\nY., 2003).\n57D. C. Langreth, in Linear and Nonlinear Electron Trans-\nport in Solids , edited by J. T. Devreese and E. Van Doren\n(Plenum, New York, 1976).\n58Z. Feng, J. Maciejko, J. Wang, and H. Guo, Phys. Rev. B\n77, 075302 (2008)."    },    {        "title": "1711.07455v1.Spin_Pumping_in_Ion_beam_Sputtered_Co__2_FeAl_Mo_Bilayers_Interfacial_Gilbert_Damping.pdf",        "content": "Spin Pumping in Ion-beam Sputtered Co2FeAl/Mo Bilayer s: \nInterfacial Gilbert Damping  \nSajid Husain, Vineet Barwal, and Sujeet Chaudhary*  \nThin Film Laboratory, Department of Physics, Indian Institute of Technology Delhi, New Delhi 110016 (INDIA)  \nAnkit Kumar, Nilamani Behera, Serkan Akansel, and Peter Svedlindh  \nÅngström Laboratory, Department of Engineering Sciences, Box 534, SE -751 21 Uppsala, Sweden  \nAbstract  \nThe spin pumping mechanism and associated interfacial Gilbert damping are demonstrated  in \nion-beam sputtered Co2FeAl  (CFA)/Mo bilayer thin films employing ferromagnetic resonance  \nspectroscopy . The d ependence  of the net spin current transportation on Mo layer thickness, 0 to \n10 nm, and the enhancement of the net effective  Gilbert damping are reported . The experimental \ndata has been analyzed using spin pumping  theory  in terms of  spin current pumped through the \nferromagnet/nonmagnetic metal interface  to deduce the  effective  spin mixing conductance  and \nthe spin -diffusion length , which  are estimated to be 1.16(±0.19 )×1019 m−2 and 3.50±0.35nm, \nrespectively. The damping constant is found to be 8 .4(±0.3)×10-3 in the Mo(3.5nm) capped \nCFA(8nm) sample corresponding to a  ~42% enhancement of the original Gilbert damping \n(6.0(± 0.3)×10-3) in the uncapped CFA layer. This is  further confirm ed by insertin g a Cu dusting \nlayer which reduce s the spin transport across the CFA/Mo interface. The Mo layer thickness \ndependent net spin current density is found to lie in the ra nge of 1-3 MAm-2, which  also provides \nadditional quantitative evidence of spin pumping in this bilayer thin film system .  \n*Author for correspondence: sujeetc@physics.iitd.ac.in  \n \n  I. INTRODUCTION  \nMagnetic damping is an exceedingly importan t property for spintronic devices  due to its \ninfluence on  power  consumption  and information  writing in the spin-transfer torque random  \naccess memor ies ( STT-MRAMs)  [1][2]. It is therefore of high importance to study the  \ngeneration, manipulation , and detection of the flow of spin angular momentum to enable the \ndesign of efficient  spin-based magneti c memories and logic devices  [3]. The transfer of spin \nangular momentum known as  spin pumping in ferromag netic (FM)/ nonmagnetic (NM) bilayer s \nprovide s information of how the precession of the magnetization transfer s spin angular \nmomentum into the adjacent  nonmagnetic  metallic  layer  [4]. This transfer ( pumping ) of spin \nangular momentum slows down the precession and leads  to an enhance ment of  the effective \nGilbert damping constant  in FM/NM bilayers . This enhancement  has been  an area of intensive \nresearch since the novel mechanism (theory) of spin pumping was proposed by Arne Brataas et \nal.  [5] [6]. The amount of spin pumping  is quantified by the magnitude of the spin current \ndensity  at the FM/NM  interface  and theoretically  [7] described as \n4eff\nS effdgdt mJm\n  where  \nm is the magnetization unit vector, \neff\nSJ is the effective spin current density pumped into the NM \nlayer from the FM layer  (portrayed in Fig. 1), and \neffg  is the spin mixing conductance  which is \ndetermined  by the reflection  coefficient s of conductance channels at FM/NM interface  [5]. \nTo date, a number of  NM metals , such as  Pt, Au, [5], Pd  [8][9],-Ta [10] and Ru  [11], \netc. have been e xtensively investigated with regards to their performance as spin sink material  \nwhen in contact with a FM . It is to be noted here that none of the Pt, Pd, Ru, and Au is an earth \nabundant  material  [12]. Thus , there is a natural need  to search for new non-magnetic material s \nwhich could generate  large spin current at the FM/NM in terface . In this study , we have explored the potential of the transition metal molybdenum (Mo)  as a new candidate  material  for spin \npumping  owing to the fact that  Mo possesses a large spin-orbit coupling  [13]. To the best of our \nknowledge, Mo has not been used till date for the study  of spin pumping effect in a FM/NM  \nbilayer system .  \nIn a FM/NM  bilayer  and/or multilayer  system s, there are several mechanisms for \ndissipation of the spin angular momentum  which are categorized as intrinsic and extrinsic . In the \nintrinsic  category , the magnon -electron coupling , i.e., spin -orbit coupling (SOC)  contributes \nsignificant ly [14]. Among the extrinsic  category , the two-magnon scattering  (TMS ) mechanism \nis linked to the inhomogeneity  and interface/surface roughness  of the heterostructure , \netc.  [15] [16] [17]. For large SOC , interfacial d-d hybridization between the NM and FM layers \nis highly desirable  [16]. Thus, the FM -NM interfacial hybridization is expected to result in \nenhancement of the transfer of spin angular momentum from the FM to the NM layer , and hence \nthe NM layer can act as a spin reservoir  (sink)  [18]. But, the NM metallic layer  does not always \nact as a perfect spin reservoir  due to the spin accumulation effect which prevents transfer of \nangular momentum  to some extent and a s a result , a backflow of  spin-current towards the  \nFM  [6] is estabished . While the flow of spin angular momentum through the FM/NM  interface \nis determine d by the effective spin-mixing conductance  \n()effg  at the interface , the spin backflow  \nis governed  by the spin diffusion length \n()d . It is emphasized  here that t hese parameters  (\neffg  \nand \nd ) are primarily  tuned  by appropriate selection  of a suitable NM layer.  \nIn this work,  we have performed ferromagnetic resonance  (FMR) measurements to \nexplore the spin pumping phenomenon and associated interfacial Gilbert damping enhancement  \nin the Co2FeAl(8nm) /Mo(\nMot) bilayer system , \nMot is the thickness of Mo , which is varie d from 0 to 10  nm. The \nMot dependent net spin current transfer across the interface  and spin diffusion \nlength of Mo are estimated . The choice of employing the Heusler alloy CoFe 2Al (CFA) as a thin \nFM layer lies in its half metallic character anticipated at room temperature  [19] [20], a trait \nwhich is highly desirable in any spintronic device operating at room temperature.  \nII. EXPERIMENTAL DETAILS  \nThe CFA  thin films with fixed thickness of 8 nm were  grown  on naturally oxidized Si(100) \nsubstrate at 573K temperature using an ion-beam  sputtering deposition system ( NORDIKO -\n3450). The substrate temperature (573K) has been selected  following the growth optimization \nreported in our previous reports  [21] [20] [22]. On the top of the CFA layer , a Mo film with \nthickness \nMot  (\nMot=0, 0.5, 1.0, 1.5, 2.0, 3.0, 4, 5, 7, 8 and 10 nm) was deposited  in situ  at room \ntemperature . In addition , a trilayer structure of CFA(8)/Cu(1)/Mo(5) was also prepared to \nunderstand  and confirm  the effect of an additional interface on the Gilbert damping  (spin \npumping ). Numbers in parenthesis are film thicknesses in nm. All the samples were prepared  at a \nconstant working pressure  of ~8.5×10-5 Torr (base vacuum  ~ 1.010-7 Torr); Ar gas was directly \nfed at 4 sccm  into the rf-ion source  operated at 75W  with the deposition rate s of 0.03nm/s  and \n0.02nm/s for CFA and Mo, respectively . The deposition rate for Cu was 0.07nm/s at 80 W. The \nsamples were then cut to 1×4 mm2 to record the FMR spectra employing  a homebuilt  FMR set-\nup [21] [23]. The data was collected  in DC-magnetic field sweep mode by keeping the \nmicrowave frequency fixed . The saturation  magnetization was measured  using  the Quantum \nDesign make Physical Property Measurement System (Model PPMS Evercool -II) with the \nvibrating sample  magnetometer option (QD PPMS -VSM). The film density, thickness  and \ninterface /surface  roughness were estimated  by simulating the  specular X -ray reflectivity (XRR) \nspectra using the PANalytical X’Pert reflectivity software (Ver. 1.2 with segmented fit). To determine surface morphology /microstru cture (e.g., roughness) , topographical imaging  was \nperformed using the ‘Bruker dimension ICON  scan assist’  atomic fo rce microscope (AFM).  All \nmeasurements were performed at  room temperature.  \nIII. RESULTS AND DISCUSSIONS  \nA. X-ray Reflectivity and A tomic Force Microscopy : Interface/surface analysis  \nFigure  2 shows the specular XRR  spectra  recorded on all the CFA(8)/Mo(\nMot ) bilayer thin films . \nThe fitting parameters were accurately determined  by simu lating (red lines) the experimental  \ncurves  (filled circles)  and are presented  in Table -I. It is evident that for the smallest NM layer \nthickness , Mo(0.5nm) , the estimated value s of the roughness from XRR and AFM are slightly \nlarger in comparison to the thickness of the Mo layer which indicates that the surface coverage of \nMo layer is not enough to cover all of the  CFA  surface  in the CFA(8)/Mo(0.5) bilayer sample  \n(modeled in Fig. 3(a)). For \nMot  ≥ 1nm, the film roughness is smaller than the thickness \n(indicating that t he Mo layer coverage is uniform as modeled in Fig s. 3(b)-(c)). For the  thicker \nlayer s of Mo (\nMot≥ 5nm) the estimated values of the surface roughness as estimated from  both \nXRR and AFM are found to be similar ~0.6nm (c.f. the lowest right panel in Fig. 2).  \nB. Ferromagnetic Resonance  Study  \nThe FMR  spectra were  recorded  on al l sample s in 5 to 11 GHz range of microwave frequenc ies. \nFigure. 4(a) shows the FMR spectra  recorded on the CFA(8)/Mo( 5) bilayer thin film . The  FMR \nspectra \n()FMRI  were fitted with the derivative of symmetric and anti -symmetric Lorentzian \nfunction s to extract the line -shape parameters, i.e., resonant field \nrH  and linewidth\nH , given \nby [24] [21]: \n22\n2\n2222\n22()\n() ()22 2\n( ) ( )22FMR\ndc rdc r\ndcS ext A ext\next ext\nr dc rUIH\nHH HHH HH\nSA\nHF H F H\nHHSA\nHHHH\nH\n        \n                 \n          , (1) \nwhere \nS extFH  and \nA extFH  are the symmetric and anti -symmetric Lorentzian functions, \nrespectively, with S and A being  the corresponding coefficients.  Symbol ‘U’ refers to the raw \nsignal voltage from the VNA . The linewidth  \nH  is the full width at half maxim um (FWHM) , \nand \ndcH  is the applied DC-magnetic  field.  \nThe f vs. \n0 rH  plots are  shown  in Fig. 4(b). These are  fitted using t he Kittel’s formul a [25]: \n 0()2r K r K eff f H H H H M\n   \n ,  (2) \nwhere 𝛾 is the gyromagnetic ratio ; \n/Bg\n  (1.76×1011s-1T-1) with \ng being the Lande’s  \nsplitting factor ; taken as 2,  \n0 effM  is the effective saturation magnetization, and  \n0 KH  is the \nuniaxial anisotropy field. The value s of \n0 effM  are comparable to the values of \n0 SM  (obtained \nfrom VSM measurements) as is shown in Fig . 4(c). Figure 4(e) shows  the variation  of \n0 KH  \nwith \nMot from which the decrease in \n0 KH  with increas e in \nMot  is clearly evident . This observed \nreduction  in \n0 KH  could possibly stem from the spin a ccumulation increasing with increasing  \nMot\n [26]. The FMR spectra was also recorded on  CFA (8)/Cu(1)/Mo(5) trilayer thin film for the  \ncomparison with the results of CFA (8)/Mo(5) bilayer. The magnitudes of  \n0 effM  (\n0 kH ) for \nCFA (8)/Mo(5) and CFA (8)/Cu(1)/Mo(5) are found to be 1.33±0.08 T (0.55±0.15mT) and  1.30±0.04 T (3.21±0.13 mT) , respectivel y. Further, Fig. 4(d) shows the \n0 rH  vs. \nMot behavior at \ndifferent constant frequenc ies ranging between  5 to 11 GHz. The observed values of \n0 rH  are \nconstant for all the Mo capped layers which  clearly  indicate s that the dominant contribution to \nthe observed resonance spectra arises from the intrinsic effect, i.e., magnon -electron \nscattering  [27]. \nC. Mo t hickness -dependen t spin pumping  \nFigure 5(a) shows t he linewidth  \n0µH  vs. f (for clarity, the results are shown  only for a \nfew sel ected film samples ). The frequency dependent linewidth can mainly  have  two \ncontributions ; the intrinsic  magnon -electron scattering  contribution,  and the extrinsic  two-\nmagnon scattering  (TMS ) contribution. The extrinsic  TMS  contribution in linewidth has been \nanalysed  (not presented here) using the methods given by Arias and Mills  [28]. A similar \nanalysis was reported  in one of our previous studies on the CFA/Ta  system  [21]. For the present  \ncase, t he linewidth analysis shows that inclusion of the TMS part does not affect the Gilbert \ndamping , which means  the TMS  contribution  is negligible in our case.  Now , the effective  Gilbert \ndamping constant \neff can be estimated using,  \n0\n04efffHH\n  \n. (2) \nHere, \n0H  is the frequency independent contribution  from  sample  inhomogeneity , while the \nsecond term corresponds to the frequency dependent contribution  associated with the intrinsic \nGilbert relaxation . Here , \neff , defined as  \neff SP CFA   , is the effective  Gilbert damping which includes the intrinsic  value of CFA  \n()eff  and a spin pumping  contribution  (\nSP ) from the \nCFA/Mo  bilayer . \nThe extracted effective Gilbert damping constant values for different \nMot  are shown  in \nFig 5(b). An enhancement of the  Gilbert damping constant with the increase of the Mo layer \nthickness is clearly observed , which is anticipated  owing to the transfer  of spin angular momenta \nby spin pumping from CFA to the Mo layer at the CFA /Mo interface . The value  of \neff  is found \nto increase  up to  8.4(± 0.3)×10-3 with the increase in  \nMot (≥ 3.5nm) , which corresponds to ~42% \nenhancement of the damping constant due to spin pumpin g. It is remarkable that such a large \nchange in Gilbert damping is observed for the CFA/Mo  bilayer ; the change is comparabl e to \nthose reported when a high SOC NM such as  Pt [8], Pd [29] [9], Ru [11], and Ta [30] is \nemployed in FM/NM bilayer s. Here , we would like to mention that the enhancement of the \nGilbert dampin g can , in principle , also be explained  by extrinsic two-magnon scattering  (TMS ) \ncontribution s in CFA/Mo(\nMot ) bilaye rs by considering the variation  of \nrH  with NM \nthickness  [27]. In our case, the  \n0 rH  is constant  for all \nMot  (c.f. Fig . 4(d)).  Thus the extrinsic \ncontribution induced increase  in \neff  is negligibly small  and hence  the enhancement of the \ndamping is dominated by the  spin pumping  mechanism . The estimated values of  \n0 0µH  are \nfound to vary  from 0.6 to 2.5 mT in the CFA/Mo(\nMot ) thin films . The variation in \n0 0µH  is \nassigned  to the finite, but small, statistical variations in sputtering conditions between  samples \nwith different  \nMot. \nFurther, to affirm the spin pumping in the CFA/Mo bilayer system, a copper  (Cu) dusting \nlayer was inserted at the CFA/Mo interface. Fig ure 5(c) compares the linewidth  vs. f plot of the CFA (8)/Mo(5) and CFA (8)/Cu(1)/Mo(5) heterostructures. The Gilbert damping was found to \ndecrease from 8.4(± 0.3)×10-3 to 6.4(± 0.3)×10-3 after inserting the Cu (1) thin layer, which is \ncomparable to the value of the uncapped CFA (3.5) sample. It may be noted that Cu has a very \nlarge  spin diffusion length  (\nd~300nm)  but weak SOC  strength  [32]. Due to the weak SOC,  the \nasymmetry in the band structure at the FM/Cu interface would thus lead to a non -equilibrium \nspin accumulation  at the CFA/ Cu interface  [33]. This spin accumulation  opposes the transfer of \nangular momentum into the Mo layer  and hence the Gilbert damping value , after insertion of the \ndusting layer , is found very similar to that of the single layer  CFA film. It  is also known  that \nenhancement of damping in the FM layer (when coupled to the NM layer ) can occur due to  the \nmagnetic proximity effect  [34]. However, we did not find any evidence in favor of the proximity \neffect as the effective saturation magnetization  did not show any increase on the inserti on of the  \nultrathin  Cu dusting layer at CFA/Mo bilayer interface , which support s our claim of absence of  \nspin pumping in the CFA/Cu/Mo trilayer sample . \nThe flow of angular momentum across the FM/NM bilayer interface is determined  by the \neffective  complex spin-mixing conductance \ng Re(g ) Im(g )eff eff eff i   , defined as the flow of \nangular momentum per unit area through the FM/NM metal interface  created by the  precessing  \nmoment s in the FM layer . The term effective  spin-mixing conductance is being used  because it \ncontain s the forward and  backflow  of spin momentum at the FM/NM interface. The imaginary \npart of the spin-mixing conductance  is usually  assumed to be negligibly small   \nRe(g ) Im(g )eff eff \n as compared  to the real part  [35] [36], and therefore, to determine the real \npart of the spin-mixing conductance , the obtained  \nMot dependent G ilbert damping  is fit ted with \nthe relation  [29],  21Re(g ) 14Mo\ndt\nB\neff CFA eff\nS CFAgeMt \n   , (3) \nwhere \nCFAis the damping for a single layer CFA without Mo capping  layer, \nRe(g )eff  is given  \nin unit s of m-2, \nB  is the Bohr magneton,  and \nCFAt  is a CFA  layer thickness . The exponential \nterm describes the reflection of spin -current from Mo/air  interface . Figure 5(b) shows the  \nvariation of  the effective Gilbert damping constant with  \nMot and the fit using  Eqn. (3) (red line) . \nThe values of \nRe(g )eff  and \nd are found to be 1. 16(±0.19 )×1019 m-2 and 3.5±0.35 nm, \nrespectively. The  value of the spin-mixing conductance is comparable to those recent ly reported \nin FM/Pt (Pd) thin films  such as  Co/Pt ( 1-4 ×1019 m-2) [8] [33], YIG/Pt  (9.7 ×1018 m-2)  [37], \nFe/Pd (1×1020 m-2)  [9], and Py/Pd(Pt) (1.4(3.2) ×1018 m-2) [34].  \nWe now calculate the net intrinsic interfacial spin mixing conductance \nG  which \ndepends on the thickness and the nature of the NM layer  as per the relation   [9]  [38], \n11\n4( ) Re(g ) 1 tanh3Mo\nMo eff\ndtGt\n\n \n, (4) \nwhere \n24( / )Z e c\n  is a material dependent  param eter (Z is the atomic number of Mo i.e., 42 \nand c is the speed of light) whose value for Mo is 0.0088 . Using Eq.  (4), \n()Mo Gt values have \nbeen compu ted for variou s \nMot ; the results  are shown in Fig. 6(a). The \nMot  dependence of \nG  \nclearly suggest s that the spin mixing conductance  critically depend s on the NM layer properties . \nFor bilayers with \nMot  6 nm, \nG  attains its saturation value, which is quite  comparable with \nthose reported for Pd and Pt  [34]  [37]. Understandably, such a large value of the spin mixing  conductance  will yield a large spin current  into the adjacent NM layer  [6] [7] [37] [33]. In the \nnext section, we have estimated the spin current from the experimental FMR data and discuss the \nsame with regards to spin pumping in further detail . \nD. Spin current generation in Mo due to spin pumping  \nThe enhancement of the Gilbert damping  observed in the CFA (8)/Mo(\nMot) bilayers  (Fig. 5(b)) is \ngenerally  interpreted in terms of  the spin -current generated in Mo layer  by the spin pumping \nmechanism at the bilayer interface  (Fig. 1). The associated net effective  spin current density  in \nMo is described by  the relation  [38] [39]: \n \n00 02 22 2\n2\n0224 2( ) G ( )8 4eff eff rf eff\nS Mo Mo\neffeffMM h eJ t t\nM   \n\n\n, (5) \nwhere\n2f and \nrfhis the rf-field (26 A/m) in the strip -line of our co-planar waveguide.  \nG ( )Mot\n is the net intrinsic inte rfacial spin mixing conductance discussed  in the previous \nsection  (Fig.  6). The estimated values of  \n()eff\nS MoJt  for differen t microwave frequencies are shown  \nin Fig. 7. It is clearly  observed  that the spin current density increase s with the increase  in \nMot, the \nincrease becomes relatively less at higher  \nMot , which indicate s the progressive spin current \ngeneration in Mo . Such an appreciable change in current density  directly provide s evidence of \nthe interfacial enhancement of the Gilbert damping  in these CFA/Mo bilayers . \nFurther,  it would be interesting  to investigate the effect on the  spin current generation in \nMo layer if  an ultra thin dusting layer  of Cu is inserted at the CFA/Mo interface . In princip le, on \ninsertion of a thin Cu layer , the spin pumping should cease  because of the unmatched band \nstructure between the CFA /Cu and Cu/M o interfaces owing to the insignificant SOC in Cu. This is in consonance with the observed decrease in Gilbert damping back to  the value for the \nuncapped CFA layer  (c.f. Fig. 5(c) and associated discussion ). The spi n-mixing conductance of \nthe trilayer heterostructure can be evaluated by \n0 g/eff B eff S CFAg M t   [29], where \nsp eff CFA    \n is the spin-pump ing induced Gilbert damping contribution which for the \nCFA/Cu/Mo trilayer is quite small , i.e., 4.0(±0.3) ×10-4 after Cu insertion.  For the trilayer, \ngeff  is \nfound to be 1.49 (±0.12) ×1017 m-2 which is two order s of magnitude small er compared  to that of \nthe CFA/Mo bilayers. Furthermore, u sing the values of  \ngeff , \n0 effM  and \neff  for the \nCFA/Cu/Mo trilayer hetero structure in Eqn. (5) and for f = 9GHz, the spin current density is \nfound be 0.0278 (±0.001 3) MA/m2, which is two order  of magnitude smaller than that in the \nCFA/Mo bilayers. Thus, t he reduction in \neff  and \neff\nSJ  subsequent to Cu dusting is quite \ncomparable  to previously reported results  [33] [40]. \nIV. CONCLUSIONS  \nWe have systematically investigated the  changes in the spin dynamics in the ion-beam sputtered \nCo2FeAl ( CFA )/Mo(\nMot) bilayer s for various \nMot  at constant CFA thickness  of 8nm . Increasing \nthe Mo layer thickness to its spin diffusion length; CFA (8)/Mo(\nMot =\nd), the effective  Gilbert \ndamping constant increases to  8.4(± 0.3)×10-3 which  corresponds to about  ~42% enhancement  \nwith respect to  the \neff  value of 6.0(± 0.3)×10-3 for the uncapped CFA layer  (i.e., without the top \nMo layer ). We interpret our results based on the spin -pumping effect s where in the effective spin-\nmixing conductance , and spin -diffusion length are found to be 1.16(±0.19 )×1019 m−2 and \n3.50±0.35nm, respectively.  The spin pumping is further confirmed  by inserting  an ultrathin  Cu \nlayer at the CFA/Mo interface. The overall effect of the damping constant  enhancement observed when Mo is deposited  over CFA is remarkably  comparable to the far less -abundant  non-\nmagnetic  metals that are currently  being used  for spin pumping  applications . From this view  \npoint , the demonstration of the new material , i.e., Mo, as a suitable spin pumping medium is \nindispensable  for the development  of novel STT spintronic  devices . \nACKNOWLEDGMENT S \nOne of the authors SH acknowledge s the Department of Science and Technology , Govt. of  India \nfor providing the INSPIRE Fellowship. Authors thank the NRF  facilit ies of IIT Delhi for AFM \nimaging . This work was in part supported by Knut and Alice Wallenberg (KAW)  Foundation \nGrant No. KAW 2012.0031 . We also acknowledge the Ministry  of Information  Technology,  \nGovernment  of India  for providing the  financial  grant . \n  REFERENCES:  \n[1] W. Kang, Z. Wang, H. Zhang, S. Li, Y. Zhang, and W. Zhao, Glsvlsi 299 (2017).  \n[2] E. Eken, I. Bayram, Y. Zhang, B. Yan, W. Wu, and Y. Chen, Integr. VLSI J. 58, 253 \n(2017).  \n[3] S. Bhatti, R. Sbiaa, A. Hirohata, H. Ohno, S. Fukami, and S. N. Piramanayaga m, Mater. \nToday http://dx.doi.org/10.1016/j.mattod.2017.07.007 1 (2017).  \n[4] Y. Tserkovnyak, A. Brataas, and G. E. W. Bauer, Phys. Rev. Lett. 88, 117601 (2002).  \n[5] A. Brataas, Y. Tserkovnyak, G. E. W. Bauer, and B. I. Halperin, Phys. Rev. B 66, \n060404(R) (2002).  \n[6] Y. Tserkovnyak, A. Brataas, and G. E. W. Bauer, Phys. Rev. B 66, 224403 (2002).  \n[7] Y. Tserkovnyak,  a Brataas, G. E. W. Bauer, and B. I. Halperin, Rev. Mod. Phys. 77, 1375 \n(2005).  \n[8] S. Azzawi, A. Ganguly, M. Tokaç, R. M. Rowan -Robinson, J. S inha, A. T. Hindmarch, A. \nBarman, and D. Atkinson, Phys. Rev. B 93, 54402 (2016).  \n[9] A. Kumar, S. Akansel, H. Stopfel, M. Fazlali, J. Åkerman, R. Brucas, and P. Svedlindh, \nPhys. Rev. B 95, 64406 (2017).  \n[10] E. Barati and M. Cinal, Phys. Rev. B 95, 134440  (2017).  \n[11] H. Yang, Y. Li, and W. E. Bailey, Appl. Phys. Lett. 108, 242404 (2016).  \n[12] G. B. Haxel, J. B. Hedrick, and G. J. Orris, United States Geol. Surv. Fact Sheet 87, 4 \n(2002).  \n[13] J. R. Clevelandj and J. L. Stanford, Phys. Rev. Lett. 24, 1482 ( 1970).  [14] M. C. Hickey and J. S. Moodera, Phys. Rev. Lett. 102, 137601 (2009).  \n[15] M. D. Stiles and A. Zangwill, Phys. Rev. B 66, 14407 (2002).  \n[16] J. O. Rantschler, R. D. McMichael, A. Castillo, A. J. Shapiro, W. F. Egelhoff, B. B. \nMaranville, D. Pulugurtha, A. P. Chen, and L. M. Connors, J. Appl. Phys. 101, 33911 \n(2007).  \n[17] K. Lenz, H. Wende, W. Kuch, K. Baberschke, K. Nagy, and A. Jánossy, Phys. Rev. B 73, \n144424 (2006).  \n[18] A. Brataas, Y. V. Nazarov, and G. E. W. Bauer, Phys. Rev. Lett. 84, 2481 (2000).  \n[19] I. Galanakis, P. H. Dederichs, and N. Papanikolaou, Phys. Rev. B 66, 174429 (2002).  \n[20] A. Kumar, F. Pan, S. Husain, S. Akansel, R. Brucas , L. Bergqvist, S. Chaudhary, and P. \nSvedlindh, arXiv 1706.04670 (2017).  \n[21] S. Husain, S. Akansel, A. Kumar, P. Svedlindh, and S. Chaudhary, Sci. Rep. 6, 28692 \n(2016).  \n[22] S. Husain, A. Kumar, S. Akansel, P. Svedlindh, and S. Chaudhary, J. Magn. Magn. M ater. \n442, 288 (2017).  \n[23] N. Behera, S. Chaudhary, and D. K. Pandya, Sci. Rep. 6, 19488 (2016).  \n[24] L. H. Bai, Y. S. Gui, A. Wirthmann, E. Recksiedler, N. Mecking, C. -M. Hu, Z. H. Chen, \nand S. C. Shen, Appl. Phys. Lett. 92, 32504 (2008).  \n[25] C. Kittel, Phys. Rev. 73, 155 (1948).  \n[26] S. P. Chen and C. R. Chang, Phys. Status Solidi 244, 4398 (2007).  \n[27] N. Behera, A. Kumar, S. Chaudhary, and D. K. Pandya, RSC Adv. 7, 8106 (2017).  [28] R. Arias and D. Mills, Phys. Rev. B 60, 7395 (1999).  \n[29] J. M. Shaw, H. T. Nembach, and T. J. Silva, Phys. Rev. B 85, 54412 (2012).  \n[30] G. Allen, S. Manipatruni, D. E. Nikonov, M. Doczy, and I. A. Young, Phys. Rev. B 91, \n144412 (2015).  \n[31] S. Yakata, Y. Ando, T. Miyazaki, and S. Mizukami, Jpn. J. Appl. Phys. 45, 3892 (2006).  \n[32] S. Mizukami, Y. Ando, and T. Miyazaki, Phys. Rev. B 66, 104413 (2002).  \n[33] J.-C. Rojas -Sanchez, N. Reyren, P. Laczkowski, W. Savero, J. -P. Attane, C. Deranlot, M. \nJamet, J. -M. George, L. Vila, and H. Jaffres, Phys. Rev. Lett. 112, 1066 02 (2014).  \n[34] M. Caminale, A. Ghosh, S. Auffret, U. Ebels, K. Ollefs, F. Wilhelm, A. Rogalev, and W. \nE. Bailey, Phys. Rev. B 94, 14414 (2016).  \n[35] K. Xia, P. J. Kelly, G. E. W. Bauer, A. Brataas, and I. Turek, Phys. Rev. B 65, 2204011 \n(2002).  \n[36] W. Zhang, W. Han, X. Jiang, S. -H. Yang, and S. S. P. Parkin, Nat. Phys. 11, 496 (2015).  \n[37] M. Haertinger, C. H. Back, J. Lotze, M. Weiler, S. Geprägs, H. Huebl, S. T. B. \nGoennenwein, and G. Woltersdorf, Phys. Rev. B 92, 54437 (2015).  \n[38] H. Nakayama, K.  Ando, K. Harii, T. Yoshino, R. Takahashi, Y. Kajiwara, K. Uchida, Y. \nFujikawa, and E. Saitoh, Phys. Rev. B 85, 144408 (2012).  \n[39] K. Ando, S. Takahashi, J. Ieda, Y. Kajiwara, H. Nakayama, T. Yoshino, K. Harii, Y. \nFujikawa, M. Matsuo, S. Maekawa, and E. S aitoh, J. Appl. Phys. 109, 103913 (2011).  \n[40] T. Nan, S. Emori, C. T. Boone, X. Wang, T. M. Oxholm, J. G. Jones, B. M. Howe, G. J. \nBrown, N. X. Sun, and A. Samples, Phys. Rev. B 91, 214416 (2015).  Table: 1 Summary of  XRR simulated parameters , i.e., , \nFMt , \nMot , and σ for the bilayer thin \nfilms [Si/CFA( 8)/Mo(\nMot)]. Here , \nFMt , \nMot, and σ refer to the density, thickness, and \ninterface width of the individual layers , respectively.  \n CFA  (Nominal thickness = 8 nm) Mo MoOx  \nS.No.  \n1 \n2 \n3 \n4 \n5 \n6 \n7 \n8 \n9 \n10 (gm/cc)±0.06  \n7.35 \n7.31 \n7.50 \n7.50 \n7.00 \n7.00 \n7.29 \n7.22 \n7.00 \n7.64 tFM(nm)±0.01  \n7.00 \n8.17 \n7.22 \n8.18 \n7.00 \n8.28 \n8.00 \n7.79 \n8.12 \n8.00 σ(nm) ±0.03  \n0.20 \n0.35 \n0.80 \n0.37 \n1.00 \n0.56 \n0.44 \n0.98 \n0.15 \n0.17 (gm/cc)±0.0 5 \n6.05 \n8.58 \n10.50  \n9.94 \n9.50 \n10.45  \n9.43 \n10.50  \n9.29 \n9.23 tMo(nm)±0.01  \n0.58 \n1.00 \n1.50 \n2.00 \n3.00 \n3.46 \n4.86 \n6.47 \n8.21 \n10.26  σ(nm) ±0.03  \n0.94 \n0.54 \n0.52 \n0.64 \n0.60 \n0.78 \n0.26 \n0.67 \n0.64 \n0.67 (gm/cc)±0.06  \n4.07 \n4.04 \n5.00 \n4.38 \n5.17 \n4.38 \n6.50 \n4.81 \n4.00 \n5.00 t(nm)±0.01  \n0.97 \n0.82 \n1.08 \n0.85 \n1.00 \n1.01 \n0.98 \n1.03 \n0.96 \n1.17 σ(nm) ±0.03  \n0.59 \n0.35 \n0.5 \n0.45 \n0.37 \n0.4 \n0.56 \n0.62 \n0.8 \n0.73 \n  Figure captions  \nFIG. 1. (color online) Schematic of the CFA/Mo bilayer structure  used in our work portrayed \nfor an example of spin current density \neff\nSJ  generated at the CFA/Mo interface  by spin pumping . \nFIG. 2 XRR spectra and the AFM topographical images of Si/CFA( 8)/Mo(\nMot ). In the \nrespective XRR spectra, circles represent the recorded experimental data points, and lines \nrepresent the simulated profiles. The estimated values of the surface roughness  in the entire \nsample series as obtained from XRR and AFM topographical measurements are compared in the \nlowest  right panel. The simulated parameters are presented  in the Table -I. All AFM images were \nrecorded  on a scan area of 10×10 m2. \nFIG.3 : The atomic representation  (model)  of the growth  of th e Mo layer (yellow sphere) on \ntop of the CFA (blue spheres) layer. The film changes from discontinuous to continuous as the \nthickness of the Mo layer is increased. Shown are the 3 different growth stages of the films: (a) \nleast coverage (b) partial coverage and (c) full coverage . \nFIG. 4: (a) Typical FMR spectra recorded  at various frequencies  (numbers in graph  are the \nmicrowave frequencies in GHz)  for the Si/SiO 2/CFA(8)/Mo(5)  bilayer sample (symbols \ncorrespond to experimental data and red  lines are fits  to the Eqn. (1)) Inset: FMR spectra of CFA \nsingle layer (filled circles) and CFA(8)/Mo(2) bilayer (open circles) samples measured at 5GHz \nshowing the increase in linewidth due to spin pumping. (b) The resonance field  \n0 rH  vs. f for all \nthe samples ( red lines are the fits to the Eqn. (2).  (c) Effective magnetization (scale on left) and \nsaturation magnetization (scale on right) vs.\nMot . The solid line represents  the bulk value of the \nsaturation magnetization  of Co 2FeAl . (d) The resonance field \n0 rH  vs. \nMot at different constant frequencies  for CFA(8)/Mo(\nMot ) bilayer thin films.  (e) Anisotropy field \n0 KH  vs. \nMot . (f) \nComparison  of \n0 rH  vs. f for the CFA(8)/Mo(5) and CFA(8)/Cu(1)/Mo(5) samples.  \nFIG. 5: (a) Linew idth vs. frequency for Si/SiO 2/CFA(8)/Mo(\nMot ) bilayer thin films. (b) \nEffective  Gilbert damping constant vs. Mo layer thicknesses. (c) \n0H  vs. f for CFA(8)/ Mo(5) \nand CFA(8)/Cu(1)/Mo(5) films.  \nFIG. 6 : Intrinsic s pin-mixing conductance vs. \nMot of the CFA (8)/Mo(\nMot) bilayers . \nFIG. 7. The effective spin current density (generated in Mo) vs. \nMot at different microwave \nfrequencies calculated using Eqn. (5)  \n   \nFIG. 1  \n  \n \nFIG. 2 \n  \n \nFIG. 3 \n  \n \nFIG. 4  \n  \n \n \nFIG. 5 \n  \n \nFIG. 6  \n  \n \n \nFIG. 7  \n \n"    },    {        "title": "1712.03550v1.Magnetic_field_gradient_driven_dynamics_of_isolated_skyrmions_and_antiskyrmions_in_frustrated_magnets.pdf",        "content": " \nMagnetic  field gradient  driven dynamics of isolated skyrmions and antiskyrmions  in \nfrustrated magnets  \n \nJ. J. Liang1, J. H. Yu1, J. C hen1, M. H. Qin1,*, M. Zeng1, X. B. Lu1, X. S. Gao1,  \nand J. –M. Liu2,† \n1Institute for Advanced Materials , South China Academy of Advanced Optoelectronics  and \nGuangdong Provincial Key Laboratory of Quantum Engineering and Quantum Materials, \nSouth China Normal University, Guangzhou 510006, China  \n2Laboratory of Solid State Microstr uctures  and Innovative Center for Advanced \nMicrostructures , Nanjing University, Nanjing 210093, China  \n \n[Abstract]  The study of skyrmion/antiskyrmion motion in magnetic materials is very \nimportant in particular for the spintronics applications.  In this work , we stud y the dynamics of \nisolated skyrmions and antiskyrmions  in frustrated magnets  driven by magnetic field gradient , \nusing the Landau -Lifshitz -Gilbert simulations on the frustrated classical Heisenberg  model  on \nthe triangular lattice . A Hall-like motio n induced  by the gradient is revealed in bulk system, \nsimilar to that in the well -studied chiral magnets.  More interestingly, our work suggest s that \nthe lateral confinement in nano -stripes of  the frustrated system  can completely suppress the \nHall motion an d significantly speed up the motion along the gradient direction. The simulated \nresults  are well explained by the Thiele theory . It is demonstrated that t he acceleration of the \nmotion is mainly determined by the Gilbert damping constant , which provides use ful \ninformation  for finding potential materials for skyrmion -based spintronics .  \n \nKeywords: skyrmion dynamics, field gradient, frustrated magnets   \nPACS numbers: 12.39.Dc, 66.30.Lw .Kw, 75.10.Jm \n \n                                                        \nEmail: *qinmh@scnu.edu.cn , † liujm@nju.edu.cn  I. INTRODUCTION  \nMagnetic skyrmions  which are topological defects with vortex -like spin structures  have  \nattracted extensive  attention  since their discovery in chiral magnets  due to their interesting \nphysics and potential applications in spintronic devices.1-4 Specifically, the interesting  \ncharacters of skyrmions such  as the topological protection5, the ultralow critical currents \nrequired to drive skyrmions  (~105 Am-2, several orders of smaller than that for domain -wall \nmanipulation )3,6, and th eir nanoscale size  make s them proposed to be promising candidates \nfor low po wer consumption magnetic memories  and high -density data processing devices. \nTheoretically, the cooperation of the energy competition among the ferromagnetic, \nDzyaloshinskii -Moriya ( DM), and the Zeeman couplings and the thermal fluctuations is \nsuggested to stabilize the skyrmions .7,8 Moreover, the significant effects of the uniaxial stress \non the stabilization of the skyrmion lattice have been revealed in earlier works.9-12 On the \nskyrmion  dynamics , it has been suggested that the skyrmions in chiral magnets can be \neffectively modulated by spin-polarized current,13-16 microwave fields,17 magnetic field \ngradients,18,19 electric field gradients,20,21 temperature gradient s22 etc. So far, some of these \nmanipulations have been realized  in experiments.23  \nDefinitely , finding new magnetic systems with skyrmions is essential both in application \npotential and in basic physical research.24 More recently,  frustrated magnets  have been \nsuggested theoretically to host skyrmion lattice  phase . For example, skyrmion crystals an d \nisolated skyrmions have been reported in the frustrated Heisenberg model on the triangular \nlattice.25,26 In this system, i t is suggested that the skyrmion crystals  are stabilized by the \ncompeting ferromagnetic nearest -neighbor (NN) and antifer romagnetic next-nearest -neighbo r \n(NNN) interaction s and thermal fluctuations at finite  temperatures (T) under applied magnetic \nfield h. Furthermore, the uniaxial anisotropy strongly affect s the spin orders in triangular \nantiferromagnets  and stabilize s the isolated sk yrmions even at zero T.26,27  \nCompared with the skyrmions in chiral magnets, those in frustrated magnets  hold two \nadditional merits.  On the one hand, the skyrmion lattice constant is typically an order of \nmagnitude smaller than that of chiral magnets, and higher -density data processing devices  are \nexpected.  On the other hand, the skyrmions are with  two additional degrees -of-freedomvorticity and helicity ) due to the fact that the exchange interactions are \ninsensitive to the direction of spin rotation . As a  result, both skyrmion and anti skyrmion \nlattices are possible in frustrated magnets  which keep the Z2 mirror symmetry in the xy spin \ncomponent.  Furthermore , the dynamics of skyrmions /antiskyrmions  is probably  different from \nthat of chiral magnets , as revea led in earlier work which  studied the current -induced dynamics \nin nanostripes of frustrated magnets.28 It has been  demonstrated that the spin states formed at \nthe edges  create multiple edge channels and guide the skyrmion /antiskyrmion  motion.  \nIt is noted that spin-polarized  current  may not drive the skyrmion well for insulating \nmaterials , and other control parameters such as field gradient are preferred . In chiral magnets, \nfor example, the gradient  can induce a Hall -like motion of skyrmions, i. e., the mai n velocity \nv (perpendicular to the gradient direction ) is induced by the gradient, and a low velocity v|| \n(parallel to the gradient direction ) is induced by  the damping effect . Thus, the gradient -driven \nmotion of skyrmions and antiskyrmions in frustrated systems is also expected . Furthermore , it \nhas been suggested  that the confined geometry suppress es the current -induced Hall motion of \nskyrmions and speed s up the motion along the current  direction , which is instructive  for \nfuture application s.29 In some ex tent, the gradient -driven motion could also be strongly \naffected by confining potential in narrow constricted geometries.  Thus, as a first step, the \nfield-gradient -induced dynamics of skyrmions and antiskyrmions in bulk frustrated magnets \nas well as in constricted geometries urgently deserves to be revealed  theoretically . However, \nfew works on this subject have been reported, as far as we know.  \nIn this work , we stud y the skyrmion /antiskyrmion dynamics in frustrated magnets \ninduced by  magnetic field gradien ts using  Landau -Lifshitz -Gilbert  (LLG)  simulations and \nThiele approach  based on the frustrated classical Heisenberg model on two -dimensional \ntriangular lattice . A Hall-like motion is revealed in bulk system, similar to that in chiral \nmagnets. More interest ingly, our work demonstrates  that the edge  confinement in nanostripes \nof frustrated magnets c an completely suppress the Hall motion and significantly accelerate  the \nmotion along the gradient direction.  \nThe remainder of this manuscript is organized as foll ows: in Sec. II the model and the \ncalculation method will be described. Sec. III is attributed to the results and discussion, and \nthe conclusion is presented in Sec. IV .      \nII. MODEL AND METHODS   \nFollowing the earlier  work,28 we consider  the Hamiltonian  \n22'\n12\n, ,z z z\ni j i j i i i i\ni j i i i ijH J J h S D S D S          S  S S  S\n,     (1) \nwhere Si is the classical Heisenberg spin with unit length on site i. The first term is the \nferromagnetic NN interaction with J1 = 1 (we use J1 as the energy unit, for simplicity) , and t he \nsecond term is the antiferromagnetic NNN  interacti on with J2 = 0.5 , and the third term is the \nZeeman coupling with a linear gradient field h = h0 + g·r (h0 = 0.4, r is the coordinate , and g \nis the gradient vector with a strength  g) applied along the [001] direction ,28 and the fourth  \nterm is the bulk uniaxial anisotropy  energy with D = 0.2 , and the last term  is the easy plane \nanisotropy energy of  the edges  with D' = 2. D' is only consider ed at the edges for the  \nnanostripes  system , which may give  rise to several  types of  edge states, as uncovered in \nearlier work .28 However, it has been confirmed that the skyrmion s/antiskyrmions  in \nnanostripes move with the same speed when they are  captured by one of these  edge state s. In \nthis work, we mainly concern the gradient -driven moti on of isolated skyrmions /antiskyrmion.  \nWe study  the spin dynamics at zero T by numerically solving the LLG  equation: \nii\ni i idd\ndt dt   SSS f S\n,                                           (2) \nwith the local effective field fi = (∂H/∂ Si). Here, γ = 6 is the gyromagnetic ratio , α is the \nGilbert damping coefficient. We use the fourth -order Runge -Kutta method to solve the LLG \nequation.  The initial spin configurations are obtained by solving the LLG equation at  g = 0. \nSubsequently, t he spin dynamics  are investigated under gradient fields. Furthermore, the \nsimulated results are further explained using the approach proposed by Thiele.29 The \ndisplacement of the skyrmion/antiskyrmion  is characteri zed by the position of its  center  (X, \nY): (1 )d d (1 )d d\n,.\n(1 )d d (1 )d dzz\nzzx S x y y S x y\nXY\nS x y S x y\n\n\n                           (3) \nThen, the velocity v = (vx, vy) is numerically calculated by  \nd d , d d .xyv X t v Y t\n                                            (4) \nAt last, v and v|| are obtaine d through a s imple coordinate transformation .  \n \nIII. RESULTS AND DISCUSSION  \nFirst, we investigate  the spin configurations of possible isolated skyrmions  and \nantiskyrmions  with various vorticities and helicities obtained by LLG simulations of bulk \nsystem ( D' = 0) at zero g. Specifically, four typical isolated skyrmions with the topological  \ncharge Q = 1 have been observed in our simulations, as depicted  in Fig 1(a). The first two \nskyrmions are N éel-type ones with different helicities, and the remaining two sk yrmions are \nBloch -type ones. Furthermore, isolated antiskyrmoins are also possible in this system, and \ntheir spin configurations with Q = 1 are shown  in Fig. 1(b).  \nAfter the relaxation of the spin configurations  at g = 0, the magnetic field gradient is \napplied along the direction of θ = /6 (θ is the angle between the gradient vector  and the \npositive x axis, as shown in Fig. 2(a)) to study the dynamics of isolated skyrmions and \nantiskyrmions  in bulk system . The LLG simulation is performed on a 28 × 28 triangular \nlattice  with the periodic boundary condition  applied along  the y' direction  perpendicular  to the \ngradient . Furthermore , we constrain the spin directions  at the edge s along the x direction by Sz \n= 1 (red circles  in Fig. 2(a) ) to reduce  the finite lat tice size effect . Similar to that in chiral \nmagnets, the skyrmion /antiskyrmion motion can be also driven  by the magnetic field \ngradients  in frustrated magnets. Fig. 2 (b) and Fig. 2( c) give  respectively  the calculated v|| and \nv as functions of g at α = 0.04 . v|| of the skyrmion equals to that of the antiskyrmion , and \nboth v|| and v increase linearl y with g. For a fixed g, the value of v is nearly an order of \nmagnitude larger than that of v||, clearly exhibiting  a Hall -like motion of the skyrmions/a ntiskyrmions.  It is noted that  v is caused by the gyromagnetic force which \ndepends on the sign of the topological charge. Thus, along the y' direction, the skyrmion and \nantiskyrmion move oppositely  under the field gradient , the same as earlier report.18 Moreover, \nv|| is resulted  from the dissipative force which is  associated with the Gilbert damping. For \nexample , the linear dependence of v|| on the Gilbert damping constant α has been revealed in \nchiral magnets,18 which still hold s true for the frustrated magnets. The dependence of velocity \non α at g = 103 is depicted in Fig. 3 , which clearly demonstrates  that v|| increases linearly and \nv is almost invariant  with the increa se of α.  \nSubsequently , the simulated results are qualitatively  explained by Thiele equations:  \n|| || '', and ,HHv Gv Gv vXY        \n                        (5) \nwith the skyrmion/antiskyrmoin center ( X', Y') in the x'y' coordinate system.  Here,  \n2 2 2 S S S S S Sd S 4 , and d d . G r Q r rx y x x y y                   \n     (6) \nFor the  frustrated  bulk magnets with the magnetic field gradient applied along the x' direction, \nthere are   \n''1 , and  0.z\ni\niHHg S gqXY   \n                                (7)        \nFor α << 1, q is almost invariant and the velocities can be estimated from  \n|| 2,. v gq v gqGG \n \n                                            (8) \nThus, a proportional relation between the velocity and field gradient is clearly  demonstrated . \nFurthermore, v is inversely propor tional to G and/or the topological charge  Q, resulting in  \nthe fact that the skyrmion  and antiskyrmion  move along the y' direction oppositely, as \nrevealed in our simulations.  For current -induced motion of skyrmions, the lateral confine ment can suppress the Hall \nmotion and accelerate the motion  along the current direction.29-31 The confinement effects  on \nthe h-gradient driven  skyrmion/antiskyrmion motion  are also investigated in the nanostripes \nof frustrated magnets. For this case, the L LG simulation is performed on an 84 × 30 \ntriangular -lattice with an open boundary condition along the y direction. For convenience, the \nfield gradient is applied along the x direction.  The easy plane anisotropy  with D' = 2 is \nconsidered at the lateral edges , which gives rise to the edge state and in turn confines the \nskyrmions/antiskyrmions . Fig. 4(a) gives  the time dependence of the y coordinate of the \nskyrmion center for  α = 0.04 and g = 103. It is clearly shown that t he isolated skyrmion  \njumps into the channel at Y = 17 and then moves with a constant  speed  along the gradient  \n(negative x, for this case)  direction . Furthermore, the position of the channel changes only a \nlittle due to the small range of the gradient consi dered in this work , which never affects our \nmain conclusions.   \nMore interestingly, the skyrmion/antiskyrmion motion along the gradient direction can be \nsignificant accelerated  by the lateral confinement, as shown  in Fig. 4(b) which gives v|| (vx) as \na fun ction of g at α = 0.04. For a fixed g, v|| of the  nanostripes is almost two orders of \nmagnitude larger than that of bulk system. When the skyrmion/antiskyrmion is captured by \nthe edge state ( under which v = 0), the equation (5) gives  v|| = gqγ/αΓ. It is s hown that v|| is \ninversely proportional to α in this confined geometry, and small α result s in a high speed of \nmotion  of the skyrmion/antiskyrmion . The inversely proportional relation between v|| and α \nhas also been confirmed in our LLG simulations, as cle arly shown  in Fig. 4(c) which gives the \nsimulated v|| as a function  of 1/α at g = 103.  \nAt last, we study the effect of the reversed gradient g on the skyrmion/antiskyrmion \nmotion, and its trail is recorded  in Fig. 4(d). It is clearly shown that the rever sed gradient  \nmoves the skyrmion/antiskyrmion out of the former channel near one lateral edge and drives \nit to the new channel near the other lateral edge. Subsequently, the skyrmion/antiskyrmion is \ncaptured by the n ew channel and moves reversely, resulting  in the loop -like trail. As a result, \nour work suggests that one may modulate the moving channel by reversing the field gradient, \nwhich is meaningful for future applications such as  in data eras ing/restoring .  \nAnyway, it is suggested theoretically that the  confined geometry in nanostripes of frustrated magnets could  significantly speed up the field-driven motion of the isolated \nskyrmions/antiskyrmions, especially for system with small Gilbert damping constant. \nFurthermore , we would also like to point out th at th is acceleration  is probably available in \nother confined materials such as  chiral magnets, which deserves to be checked in future \nexperiments.  \n \nIV. CONCLUSION  \nIn conclusion, we have studied the magnetic -field-gradient -drive n motion of the isolated \nskyrmions and antiskyrmions in the frustrated triangular -lattice spin model using \nLandau -Lifshitz -Gilbert simulations and Thiele  theory . The Hall -like motion is revealed in \nbulk system, similar to that in chiral magnets. More interestingly, it is suggested  that the \nlateral confinement in the nanostripes of the frustrated system  can suppress the Hall motion \nand significantly speed up the motion along the gradient direction. The acceleration of the \nmotion is mainly determined by the Gilbert damping constant , whic h is helpful for finding \npotential materials for skyrmion -based spintronics .  \n \n \nAcknowledgement s: \nThe work is supported by the National Key Projects for Basic Research of China  (Grant \nNo. 2015CB921202 ), and the National Key Research Programme of China (Gra nt No. \n2016YFA0300101), and the Natural Science Foundation of China ( Grant No. 51332007 ), and \nthe Science and Technology Planning Project of Guangdong Province (Grant No. \n2015B090927006) . X. Lu also thanks for the support from the project for Guangdong \nProvince Universities and Colleges Pearl River Scholar Funded Scheme (2016).  \n \n References:  \n \n1. A. Neubauer, C. Pfleiderer, B. Binz, A. Rosch, R. Ritz, P. G. N iklowitz, and P. Bö ni 2009 \nPhys. Rev. Lett. 102 186602  \n2. H. Wilhelm, M. Baenitz, M. Schmidt, U.K. Rö ß ler,  A. A. Leonov, and A. N. Bogdanov \n2011 Phys. Rev. Lett. 107 127203  \n3. F. Jonietz et al 2010  Science 330 1648  \n4. S. Mü hlbauer et al 2009 Science 323 915 \n5. J. Hagemeister, N. Romming, K. von Bergmann, E.Y . Vedmedenko , and R. Wiesendanger \n2015 Nat. Commun. 6 8455  \n6. X.Z. Yu, N. Kanazawa, W.Z. Zhang, T. Nagai, T. Hara, K. Kimoto, Y . Matsui, Y . Onose , \nand Y . Tokura 2012 Nat. Commun. 3 988 \n7. S. D. Yi, S . Onoda, N . Nagaosa, and J . H. Han 2009  Phy. Rev. B  80 054416  \n8. S. Buhrandt and L . Fritz 2013 Phy. Rev. B  88 195137  \n9. K. Shibata et al 2015 Nat. Nanotech.  10 589 \n10. Y . Nii et al 2015 Nat. Commun.  6 8539   \n11. A. Chacon et al 2015 Phys. Rev. Lett.  115 267202  \n12. J. Chen, W. P. Cai, M. H. Q in, S. Dong, X. B. Lu, X. S. Gao , and J. M. Liu 2017 Sci. Rep. \n7 7392  \n13. J. Zang, M . Mostovoy, J . H. Han, and  N. Nagaosa 2011 Phys. Rev. Lett. 107 136804  \n14. J. Sampaio, V . Cros, S. Rohart, A. Thiaville , and A. Fert 2013 Nat. Nanotech. 8 839 \n15. J. Iwasaki, M. Mochizuki, and N. Nagaosa, 2013 Nat. Commun. 4 1463  \n16. S. Huang, C. Zhou, G. Chen, H. Shen, A. K. Schmid, K. Liu, and Y . Wu 2017 Phy. Rev. B \n96 144412  \n17. W. Wang, M. Beg, B. Zhang, W. Kuch, and H. Fangohr 2015 Phy. Rev. B 92 020403  \n18. C. Wang, D. Xiao , X. Chen, Y . Zhou, and Y . Liu 2017 New J. Phys 19 083008  \n19. S. Komineas and N. Papanicolaou, 2015 Phy. Rev. B 92 064412  \n20. P. Up adhyaya, G. Yu, P. K. Amiri, and K. L. Wang 2015 Phy. Rev. B 92 134411  \n21. Y.-H. Liu, Y . -Q. Li, and J. H. Han 2013 Phys . Rev. B 87 100402  \n22. L. Kong and J.  Zang 2013 Phys. Rev. Lett. 111 067203  23. W. Legrand et al 2017 Nano Lett. 17 2703 -2712  \n24. Joseph Barker, Oleg A . Tretiakov  2016 Phys. Rev. Lett. 116 147203  \n25. T. Okubo, S. Chung, and H. Kawamura 2012 Phys. Rev . Lett. 108 017206  \n26. A. O. Leonov and M. Mostovoy, 2015 Nat. Commun. 6 8275  \n27. S. Hayami, S. -Z. Lin, and C. D. Batista 2016 Phy. Rev. B 93 184413  \n28. A. O. Leonov and M.  Mostovoy 2017 Nat. Commun. 8 14394  \n29. J. Iwasaki, M. Mochizuki, and N. Nagaosa 2013 Nat. Nanotech. 8 742 \n30. I. Purnama, W. L. Gan, D. W. Wong,  and W. S. Lew 2015 Sci. Rep. 5 10620  \n31. X. Zhang, G. P. Zhao, H. Fangohr, J. P. Liu, W. X. Xia, J. Xia, and F. J. Morvan  2015 Sci. \nRep. 5 7643  \n \n \n \n \n \n FIGURE CAPTIONS  \n \nFig.1. Typical LLG snapshot  of the spin configuration s of skyrmion s and antiskyrmion s. (a) \nskyrmion  structure s and (b) antiskyrmion structure s with different helicit ies. \n \nFig.2. (a) Effective model on the triangular  lattice.  (b) v|| and ( c) v as functions of g at α = \n0.04 in bulk system . \n \nFig.3. (a) v|| and ( b) v as functions of  α at g = 103.  \n \nFig.4. (a) Time dependence of Y coordinate of the skyrmion center at  α = 0.01 and g = 103. v|| \nas a function of  (b) g at α = 0.04 , and (c) α at g = 103 in the nanostripes of frustrated magnets . \n(d) The trail of skyrmion /antiskyrmion . The red line records  the motion with gradient g along \nthe positive x direction , while  the blu e line records  the motion with the revers ed g.      \n \n  \n \n \nFig.1. Typical LLG snapshot  of the spin configuration s of skyrmion s and antiskyrmion s. (a) \nskyrmion  structure s and (b) antiskyrmion structures with different helicit ies.  \nFig.2. (a) Effective model on the triangular  lattice.  (b) v|| and (c) v as functions of g at α = \n0.04 in bulk system .  \nFig.3. (a) v|| and ( b) v as functions of α at g = 103.  \nFig.4. (a) Time dependence of Y coordinate of the skyrmion center at α = 0.01 and g = 103. v|| \nas a function of  (b) g at α = 0.04, and (c) α at g = 103 in the nanostr ipes of frustrated magnets . \n(d) The trail of skyrmion /antiskyrmion . The red line records  the motion with gradient g along \nthe positive x direction , while  the blu e line records  the motion with the reversed g. \n "    }]